Integral Equations and Operator Theory - Volume 70

Integr. Equ. Oper. Theory 70 (2011), 1–15 DOI 10.1007/s00020-011-1874-3 Published online April 2, 2011 c Springer Basel ...

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Integr. Equ. Oper. Theory 70 (2011), 1–15 DOI 10.1007/s00020-011-1874-3 Published online April 2, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Reducing Subspaces on the Annulus Ronald G. Douglas and Yun-Su Kim Abstract. We study reducing subspaces for an analytic multiplication operator Mzn on the Bergman space L2a (Ar ) of the annulus Ar , and we prove that Mzn has exactly 2n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry. Mathematics Subject Classification (2010). 47A15, 47B37, 47B38, 51D25. Keywords. Bergman spaces, bilateral weighted shifts, Hardy spaces, reducing subspaces.

1. Introduction Important themes in operator theory are determining invariant subspaces and reducing subspaces for concretely deﬁned operators. Our goal in this note is to determine the reducing subspaces for a power of certain multiplication operators on natural Hilbert spaces of holomorphic functions on an annulus. We begin with the Bergman space and Hardy space. Next, we consider a generalization to certain bilateral weighted shifts. Finally, we interpret our results in the context of complex geometry describing another approach to these questions. The motivation for these questions arises from some earlier results of Zhu [12], Stessin and Zhu [10], and other researchers [1,8]. In these studies, the annulus is replaced by the open unit disk, and one considers Mzn on the Hardy space H 2 or the Bergman space L2a . In particular, the lattice of reducing subspaces of the nth power of the multiplication operator, Mzn on L2a , was shown to be discrete and have precisely 2n elements. This contrasted with the case of the classic Toeplitz operator Tzn on the Hardy space H 2 for which this lattice is inﬁnite and isomorphic to the lattice of all subspaces Research was partially supported by a grant from the National Science Foundation.

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of Cn . Thus, as is true for many other questions, the situations on the unit disk and annulus are different. For 0 < r < 1, let Ar denote the annulus {z ∈ C : r < |z| < 1} in the complex plane C. Let L2 (Ar ) denote the usual L2 -space for planar Lebesgue measure on Ar and L2a (Ar ) be the closure of R(Ar ) in L2 (Ar ), where R(Ar ) is the space of all rational functions with poles outside the closure of Ar . We let PL2a (Ar ) be the orthogonal projection of L2 (Ar ) onto the Bergman space L2a (Ar ). For ϕ in H ∞ (Ar ), the space of bounded holomorphic functions on Ar , deﬁne the operator Mϕ on L2a (Ar ) so that Mϕ (f ) = ϕf for f in L2a (Ar ). We are concerned of Mzn = Mzn for n ≥ 2.

with determining the reducing subspaces

2. Reducing Subspaces for Mzn (n ≥ 2) We let Sk denote the subspaces of L2a (Ar ) generated by {z m ∈ L2a (Ar ) : m = k( mod n)} for 0 ≤ k < n. To study reducing subspaces for the multiplication operator Mzn on L2a (Ar ), we will use these n reducing subspaces Sk (0 ≤ k < n) for Mzn . Note that L2a (Ar ) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 , and so for any f ∈ L2a (Ar ), we have a unique orthogonal decomposition f = f0 + f1 + · · · + fn−1 ,

(2.1)

where fk ∈ Sk (0 ≤ k < n). In this section, we will need the following well known fact [4]. For completeness, we provide a proof. Lemma 1. If MF : Sk → L2a (Ar ) is a (bounded) multiplication operator by a function F on Ar , then F ∈ H ∞ (Ar ) and F ∞ ≤ MF . Proof. First, since F is the quotient of two analytic functions (F = / Ar , it is analytic on Ar . For a ﬁxed z ∈ Ar , let λz (MF z k )/z k ) and 0 ∈ denote the point-evaluation functional on L2a (Ar ) deﬁned by λz (f ) = f (z) for f ∈

L2a (Ar ).

Clearly, λz is bounded, and for fk ∈ Sk ,

|F (z)λz (fk )| = |F (z)fk (z)| = |λz (MF (fk ))| ≤ λz MF fk . It follows that |F (z)| λz ≤ λz MF for any z ∈ Ar . Therefore, |F (z)| ≤ MF for any z ∈ Ar , and F is analytic.

Another familiar result classiﬁes bilateral shifts up to unitarily equivalence.

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Proposition 2. ([10]) If S, T are two bilateral weighted shifts with weight sequences {vm }, {wm }, and if there exists an integer k such that |vm | = |wm+k |

for all m,

then S and T are unitarily equivalent. Moreover, the converse is true. Lemma 3. For i, j such that 0 ≤ i = j < n, if Mi = Mzn |Si and Mj = Mzn |Sj , then Mi and Mj are not unitarily equivalent. kn+i

Proof. Let eik = zzkn+i where k is an integer. Then, {eik : k ∈ Z} is an orthonormal basis for Si . First, we calculate the weights of the operator Mi . Since (k+1)n+i z i z (k+1)n+i = ei , Mi ek = kn+i kn+i z z k+1 the weights of Mi are

(k+1)n+i z λk = z kn+i

(2.2)

for k ∈ Z. Similarly, the weights of Mj are (k+1)n+j z μk = kn+j z

(2.3)

for k ∈ Z. 2(n+1) 2 1 Since z n = n+1 − r n+1 , by Proposition 2 we conclude that Mi and Mj are not unitarily equivalent. Recall that determining the reducing subspaces of Mzn is equivalent to ﬁnding the projections in the commutant of Mzn [5]. Thus, in the following proposition, we characterize every bounded linear operator T on L2a (Ar ) commuting with Mzn . Proposition 4. A bounded linear operator T on L2a (Ar ) commutes with Mzn if and only if there are functions Fi (0 ≤ i < n) in H ∞ (Ar ) such that Tf =

n−1

Fi fi ,

i=0

where fi (0 ≤ i < n) denotes the functions in Eq. (2.1). Proof. (⇐) Let MFi : L2a (Ar ) → L2a (Ar ) be the multiplication operator deﬁned by MFi (g) = Fi g for g ∈ L2a (Ar ). Then, n−1 n−1 sup T f ≤ Fi fi ≤ MFi sup fi f =1

i=0

≤

n−1

i=0

MFi f .

i=0

It follows that T < ∞. Clearly, T Mzn = Mzn T .

i=0,...,n−1

(2.4)

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(⇒) Assume that T is a (bounded) operator on L2a (Ar ) such that T Mzn = Mzn T . Then, T ∗ commutes with Mλ∗n −zn for any λ ∈ Ar . Clearly, for λ ∈ Ar , ker Mλ∗n −zn is generated by {kλωk : ωk = exp(2πik/n)(0 ≤ k < n)}, where kλωk is the Bergman kernel function at λωk . Since T ∗ kλ ∈ ker Mλ∗n −zn , we have T ∗ kλ =

n−1

ak (λ)kλωk ,

(2.5)

k=0

for uniquely determined complex numbers {ak (λ)}n−1 k=0 . If f ∈ L2a (Ar ) and z ∈ Ar , then, by Eq. (2.5), T f (z) = (T f, kz ) = (f, T ∗ kz ) =

n−1

ak (z)f (zωk ).

(2.6)

k=0

Since ωkn = 1 for any 0 ≤ k < n, fi (zωk ) = ωki fi (z) (0 ≤ i, k < n),

(2.7)

where fi (0 ≤ i < n) is the function deﬁned in Eq. (2.1).

n−1

n−1

n−1 Since k=0 ak (z)f (zωk ) = k=0 ak (z)f0 (zωk ) + k=0 ak (z)f1 (zωk ) +

n−1 · · · + k=0 ak (z)fn−1 (zωk ), (2.6) and (2.7) imply that T f (z) =

n−1 k=0

ak (z)f0 (z) +

n−1

ak (z)ωk f1 (z) + · · · +

k=0

n−1 k=0

ak (z)ωkn−1 fn−1 (z). (2.8)

For 0 ≤ k < n, a function Fk on Ar is deﬁned by Fk (z) =

n−1

ai (z)ωik .

(2.9)

i=0

Then, Eq. (2.8) implies that Tf =

n−1

Fi fi .

i=0

To ﬁnish this proof, we have to show that Fi (0 ≤ i < n) is in H ∞ (Ar ). k ) Since Fk (z) = T (z for 0 ≤ k < n, Fk is analytic on Ar . zk By Lemma 1, Fk ∞ ≤ MFk , i.e., Fk ∞ < ∞ for any 0 ≤ k < n. An analogous result is known for Toeplitz operators on the open unit disk [2]. Because Mzn and Mzn |S0 ⊕ Mzn |S1 ⊕ · · · ⊕ Mzn |Sn−1 are unitarily equivalent, in the following proposition, we determine the projections in the commutant of Mzn |S0 ⊕ Mzn |S1 ⊕ · · · ⊕ Mzn |Sn−1 .

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Proposition 5. For 0 ≤ k < n, let Mk = Mzn |Sk . If B = (Bij )(n×n) is a projection such that ⎞ ⎛ ⎛ 0 ··· 0 0 0 M0 M0 ⎟ ⎜ 0 M1 ⎜ 0 M1 0 ·· 0 ⎟B = B ⎜ ⎜ ⎝ · ⎝ · · · · · ⎠ · 0 0 · · · 0 Mn−1 0 0

··· 0 · ···

5

0 ·· · 0

0 0 ·

⎞ ⎟ ⎟, ⎠

Mn−1 (2.10)

∞

then there are holomorphic functions ϕij (0 ≤ i, j < n) in H (Ar ) such that Bij = Mϕij . Moreover, ϕii is a real-valued constant function on Ar for 0 ≤ i < n, and ϕij ≡ 0 for i = j. Proof. Since the operator B commutes with Mzn , by Proposition 4, we have Bf =

n−1

ϕi fi ,

i=0

where each ϕi ∈ H ∞ (Ar ). By Eq. (2.9), ϕi (z) =

n−1

ak (z)ωki .

k=0

We can now solve this set of n equations for ak (z) to conclude that the ak ’s are analytic and bounded

n−1 in Ar . Let ak (z) = i=0 aki (z) where aki ∈ Si (i = 0, 1, . . . , n − 1). Then, ⎛ n−1

n−1

n−1 n−1 ⎞ ··· k=0 ak0 k=0 ak(n−1) ωk k=0 ak1 ωk ⎟ ⎜ n−1

n−1

n−1 n−1 ⎟ ⎜ ··· k=0 ak1 k=0 ak0 ωk k=0 ak2 ωk ⎟ ⎜ B=⎜ ⎟. · · ··· · ⎟ ⎜ ⎠ ⎝ · · ··· ·

n−1

n−1

n−1 n−1 a a ω · · · a ω k=0 k(n−1) k=0 k(n−2) k k=0 k0 k It follows that B = (Mϕij )n−1 i,j=0 , where Mϕij : Sj → Si is the multiplication operator deﬁned by Mϕij (fj ) = ϕij fj for fj ∈ Sj . By Lemma 1, ϕij (0 ≤ i.j < n) is in H ∞ (Ar ). Since Mϕii : Si → Si and B is a projection, Mϕ∗ii = Mϕii . Thus, ϕii is a real-valued holomorphic function and hence ϕii is a constant function. We now prove that ϕij = 0 if i = j. Suppose that there are l and k in {0, 1, . . . , n − 1} such that l = k and ϕlk = 0. By Eq. (2.10), Ml Mϕlk = Mϕlk Mk and Mk Mϕkl = Mϕkl Ml .

(2.11)

Thus, Eq. (2.11) implies that Ml Mϕlk Mϕkl = Mϕlk Mk Mϕkl = Mϕlk Mϕkl Ml .

(2.12)

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Since Mϕkl = Mϕ∗lk , Mϕlk Mϕkl : Sl → Sl is a self-adjoint operator commuting with Ml . Then, in the same way as for ϕii , we conclude that Mϕlk Mϕkl = Mϕlk ϕkl is a constant multiple of the identity operator, i.e., Mϕlk Mϕkl = cl ISl for 0 ≤ l < n,

(2.13)

where ISl is the identity operator on Sl . Note that cl > 0, since Mϕlk Mϕkl is positive and ϕlk = 0. Equations (2.11) and (2.13) imply that Mk and Ml are unitarily equivalent which is a contradiction by Lemma 3. Finally, it is time to determine the reducing subspaces of the multiplication operators Mzn (n ≥ 2). Theorem 6. For a given n ≥ 2, the multiplication operator Mzn : L2a (Ar ) → L2a (Ar ) has 2n reducing subspaces with minimal reducing subspaces S0 , . . . , Sn−1 . Proof. Since Mzn and Mzn |S0 ⊕Mzn |S1 ⊕· · ·⊕Mzn |Sn−1 are unitarily equivalent, it is enough to consider the reducing subspaces of Mzn |S0 ⊕ Mzn |S1 ⊕ · · · ⊕ Mzn |Sn−1 . By Proposition 5, if B = (Bij )(n×n) is a projection satisfying Eq. (2.10), then ⎞ ⎛ c0 0 · · · 0 ⎜ 0 c1 · · · 0 ⎟ ⎟ ⎜ · · · · · · ⎟ B=⎜ ⎟, ⎜ ⎝ · · ··· · ⎠ 0 0 · · · cn−1 where ci (0 ≤ i < n) are real numbers. Since B 2 = B, it follows that ci = 0, 1 for 0 ≤ i < n. Therefore, the reducing subspaces of Mzn |S0 ⊕ Mzn |S1 ⊕ · · · ⊕ Mzn |Sn−1 are c0 S0 ⊕ c1 S1 ⊕ · · · ⊕ cn−1 Sn−1 , with ci = 0, 1. Thus, this theorem is proven.

(2.14)

3. Reducing Subspaces for Tzn Ball [1] and Nordgren [8] studied the problem of determining reducing subspaces for an analytic Toeplitz operator on the Hardy space H 2 (D) of the open unit disk. In this section, for n ≥ 2, we determine the reducing subspaces for the analytic Toeplitz operator Tzn on the Hardy space H 2 (Ar ) of the annulus Ar . Note that, for Tzn on H 2 (D), the problem has an easy but sufﬁcient answer, since Tzn and Tz ⊗ ICn are unitarily equivalent. Recall that the Hardy space H 2 (Ar ) is the closure of R(Ar ) in L2 (m), where m is linear Lebesgue measure on ∂Ar . Let Sk denote the subspaces of H 2 (Ar ) generated by {z m ∈ H 2 (Ar ) : m = k( mod n)} for 0 ≤ k < n. In the same way as in Sect. 2, we will use these

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n reducing subspaces Sk for the Toeplitz operator Tzn : H 2 (Ar ) → H 2 (Ar ) deﬁned by Tzn (f ) = z n f, for n ≥ 2. Note that H 2 (Ar ) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 , and so for any f ∈ H 2 (Ar ), we have a unique orthogonal decomposition f = f0 + f1 + · · · + fn−1 ,

(3.1)

where fk ∈ Sk (0 ≤ k < n). Proposition 7. For i, j such that 0 ≤ i = j < n, if Ti = Tzn |Si and Tj = Tzn |Sj , then Ti and Tj are not unitarily equivalent. Proof. Note that 2

z n H 2 (Ar ) =

1 2π

2π

|einθ |2 dθ +

0

1 2π

2π

r2n |einθ |2 dθ = 1 + r2n .

0

Then, in the same way as in Lemma 3, the result is proven.

Determining the reducing subspaces of Tzn is equivalent to ﬁnding projections in the commutant of Tzn . Since Tzn and Tzn |S0 ⊕ Tzn |S1 ⊕ · · · ⊕ Tzn |Sn−1 are unitarily equivalent, we consider the commutant of Tzn |S0 ⊕ Tzn |S1 ⊕ · · · ⊕ Tzn |Sn−1 in the following proposition. Since we also have a kernel function in this case, a description similar to that of the commutant of Mzn on the Bergman space L2a (Ar ) is obtained. Proposition 8. A bounded linear operator T on H 2 (Ar ) commutes with Tzn if and only if there are functions Gi (0 ≤ i < n) in H ∞ (Ar ) such that Tf =

n−1

Gi fi ,

i=0

where fi denotes the functions in Eq. (3.1). By Proposition 7, Tzn |Si and Tzn |Sj are not unitarily equivalent where 0 ≤ i = j < n. Thus, in the same way as in Proposition 5, we characterize a projection which is in the commutant of Tzn |S0 ⊕ Tzn |S1 ⊕ · · · ⊕ Tzn |Sn−1 . Proposition 9. For 0 ≤ k < n, let Tk = Tzn |Sk . If F = (Fij )(n×n) is a projection such that ⎛ ⎛ ⎞ T0 0 T0 0 · · · 0 0 ⎜ 0 T1 ⎜ 0 T1 0 ·· ⎟ 0 ⎜ ⎟F = F ⎜ ⎝ · ⎝ · · · · · ⎠ · 0 0 · · · 0 Tn−1 0 0

··· 0 · ···

0 ·· · 0

⎞ 0 0 ⎟ ⎟ , (3.2) · ⎠ Tn−1

then there are holomorphic functions ϕij (0 ≤ i, j < n) in H ∞ (Ar ) such that Fij = Tϕij . Moreover, ϕii is a real-valued constant function on Ar for 0 ≤ i < n, and ϕij ≡ 0 if i = j.

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Since Tzn and Tzn |S0 ⊕ Tzn |S1 ⊕ · · · ⊕ Tzn |Sn−1 are unitarily equivalent, we have the following result. Theorem 10. For a given n ≥ 2, the Toeplitz operator Tzn : H 2 (Ar ) → H 2 (Ar ) has 2n reducing subspaces with minimal reducing subspaces S0 , . . . , Sn−1 .

4. Reducing Subspaces for Bilateral Weighted Shifts Note that the multiplication operator Mz on the Bergman space L2a (Ar ) and the Toeplitz operator Tz on the Hardy space H 2 (Ar ) are both bilateral weighted shifts. Moreover, in Sects. 2 and 3, we showed that the lattice of reducing subspaces for the operators (Mz )n (= Mzn ) on the Bergman space L2a (Ar ) and (Tz )n (= Tzn ) on the Hardy space H 2 (Ar ), both have 2n elements for n ≥ 2. Thus, it is natural to ask the following question. Question: Let H be a separable Hilbert space, and S : H → H be a bilateral weighted shift. Then, for a given n ≥ 2, does the operator S n have a discrete lattice of 2n reducing subspaces? In [10], Stessin and Zhu answered this question for powers of unilateral weighted shifts generalizing the earlier results for Tz on H 2 and Mz on L2a . That some condition is necessary is shown by considering the weighted shift with weights (. . . , 12 , 2, 12 , 2, . . .). In this case T 2 = I and hence the lattice of reducing subspaces consists of all subspaces. In this section, we generalize their results ﬁnding hypotheses to answer this question in the aﬃrmative for certain bilateral weighted shifts with spectrum Ar . Let {β(m)} be a two-sided sequence of positive numbers such that sup λm = sup β(m + 1)/β(m) < ∞. m

m

(4.1)

We consider the space of two-sided sequences f = {fˆ(m)} such that 2 2 f = f β = |fˆ(m)|2 (β(m))2 < ∞. We shall use the notation f (z) =

fˆ(m)z m ,

whether or not the series converges for any (complex) value of z. We shall denote this space as L2 (β) for the Laurent series case. Recall that these spaces are Hilbert spaces with the inner product (f, g) = fˆ(m)ˆ g (m)(β(m))2 . (4.2) Let Mz : L2 (β) → L2 (β) be the linear transformation deﬁned by (Mz f )(z) = (4.3) fˆ(m)z m+1 . By (4.1), Mz is bounded [9] (Note that {λm } are the weights.). If gk (z) = z k , then {gk } is an orthogonal basis for L2 (β).

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We let Sk denote the subspace of L2 (β) generated by {gm ∈ L2 (β) : m = k(modn)} for 0 ≤ k < n. To study the reducing subspaces for the operator Mzn : L2 (β) → L2 (β) deﬁned by fˆ(m)z m+n (f ∈ L2 (β) , (4.4) (Mzn f )(z) = we will use the n reducing subspaces Sk (0 ≤ k < n) for Mzn . Note that L2 (β) = S0 ⊕ S1 ⊕ · · · ⊕ Sn−1 , and so for any f ∈ L2 (β), we have a unique orthogonal decomposition f = f0 + f1 + · · · + fn−1 , where fk ∈ Sk (0 ≤ k < n). Consider the multiplication of formal Laurent series, f g = h: m ˆ gˆ(m)z m = , h(m)z fˆ(m)z m where, for all m, ˆ h(m) =

fˆ(k)ˆ g (m − k).

(4.5)

(4.6)

(4.7)

k

In general, we will assume that the product (4.6) is deﬁned only if all the series (4.7) are absolutely convergent. L∞ (β) denotes the set of formal

ˆ Laurent series φ(z) = φ(m)z m (−∞ < m < ∞) such that φL2 (β) ⊂ L2 (β). ∞ If φ ∈ L (β), then the linear transformation of multiplication by φ on L2 (β) will be denoted by Mφ . Proposition 11 ([9]). If A is a bounded operator on L2 (β) that commutes with Mz , then A = Mφ for some φ ∈ L∞ (β). Proposition 12 ([7]). For φ ∈ L∞ (β), Mφ is a bounded linear transformation, and the matrix (amk ) of Mφ , with respect to the orthogonal basis {gk }, is given by ˆ − k). amk = φ(m

(4.8)

Proposition 13 ([9]). The operator Mz on L2 (β) is unitarily equivalent to the ˜ if and only if there is an integer k such that z on L2 (β) operator M ˜ β(n + k + 1) ˜ n = β(n + 1) = λn+k = λ ˜ β(n + k) β(n) ˜ = z k L2 (β), and for all n. Equivalently, L2 (β) ˜ , f 1 = z k f f ∈ L2 (β)

(4.9)

˜ where f 1 denotes the norm of f in L2 (β). Lemma 14. If β0 (k) = β(nk), then M0 = Mzn |S0 is unitarily equivalent to Mz on L2 (β0 ).

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Proof. Let T : S0 → L2 (β0 ) be the linear transformation deﬁned by T (z nm ) = z m ,

(4.10)

where m ∈ Z.

If f ∈ S0 , then f (z) = m fˆ(nm)z nm and, by Eq. (4.2), 2 2 |fˆ(nm)|2 (β(nm))2 = |fˆ(nm)|2 (β0 (m))2 = T f L2 (β0 ) . f S0 = m

m

(4.11)

Therefore, T is an isometry. Clearly, for a given g = m gˆ(m)z m ∈ L2 (β0 ),

ˆ nm ∈ S0 such that T (f ) = g, where there is an element f = m f (nm)z ˆ gˆ(m) = f (nm), i.e., T is onto. It follows that T is unitary. Clearly, Mz T = T M 0 .

(4.12)

Therefore, M0 = Mzn |S0 is unitarily equivalent to Mz on L2 (β0 ).

We focus on the bilateral shift operator Mz on L2 (β) with monotonically increasing weights {λn }. If the weights {λn } of Mz on L2 (β) satisfy |λn | ≤ crn for some c > 0 and lim λn = 1, n→∞

then σ(Mz ) is the annulus Ar [9]. We will call such operators Mz a monotonic-Ar weighted shift. First, we obtain the analogue of Propositions 4 and 5 in the case of monotonic-Ar weighted shifts. Proposition 15. For 0 ≤ i < n, let Mi = Mzn |Si , and assume that Mz is a monotonic-Ar weighted shift. If P = (Pij )(n×n) is a projection such that ⎛ ⎛ ⎞ ⎞ M0 0 ··· 0 0 0 ··· 0 0 M0 ⎜ ⎜ 0 M1 0 ·· 0 ⎟ 0 ⎟ ⎜ ⎟ P = P ⎜ 0 M1 0 ·· ⎟, ⎝ ⎝ · ⎠ · · · · · · · · · ⎠ 0 0 · · · 0 Mn−1 0 0 · · · 0 Mn−1 (4.13) then there are elements ϕij (0 ≤ i, j < n) in L∞ (β) such that Pij = Mϕij . Moreover, ϕii is a positive constant function for 0 ≤ i < n, and ϕij ≡ 0 for i = j. Proof. For a given 0 ≤ i < n, deﬁne a sequence of positive numbers {βi (k)} by βi (k) = β(nk + i) for k ∈ Z, and let Ti : Si → L2 (βi ) be the linear transformation deﬁned by Ti (z nm+i ) = z m , where m ∈ Z.

(4.14)

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If f ∈ Si , then f (z) = m fˆ(nm + i)z nm+i and, by Eq. (4.2), 2 Ti f L2 (βi ) = |fˆ(nm + i)|2 (βi (m))2 m

=

m

|fˆ(nm + i)|2 (β(nm + i))2 = f Si .

Thus, Ti is isometric, and for a given g = ˆ(m)z m ∈ L2 (βi ), there mg

ˆ nm+i ∈ Si such that Ti (f ) = g, where is an element f = m f (nm + i)z ˆ gˆ(m) = f (nm + i), i.e., Ti is unitary. Since Mz Ti = Ti Mi , we have Mi = Ti −1 Mz Ti . Hence, Mi Pii = Pii Mi implies that Ti −1 Mz Ti Pii = Pii Ti −1 Mz Ti and so Mz (Ti Pii Ti −1 ) = (Ti Pii Ti −1 )Mz . By Proposition 11, Ti Pii Ti −1 = Mϕii ,

(4.15)

∞

for some ϕii ∈ L (βi ). Thus, Pii is unitarily equivalent to the linear transformation Mϕii for some ϕii ∈ L∞ (βi ). By Proposition 12, since Mϕii (0 ≤ i < n) is self-adjoint, for any integers m and p, ϕˆii (m − p) = ϕˆii (p − m),

(4.16)

(Mϕii (z p ), z m ) = (z p , Mϕii (z m )).

(4.17)

and Equations (4.16) and (4.17) imply that ϕˆii (m − p)β(m)2 = ϕˆii (p − m)β(p)2 = ϕˆii (m − p)β(p)2 .

(4.18)

In Eq. (4.18), if m = p, without loss of generality, we assume that m < p. Then, by Eq. (4.18), β(p)2 β(m)2 β(m + 1)2 β(m + 2)2 β(p)2 = ϕˆii (m − p) · · · . β(m)2 β(m + 1)2 β(p − 1)2

ϕˆii (m − p) = ϕˆii (m − p)

Thus, since λk =

β(k+1) β(k)

for any k,

ϕˆii (m − p) = ϕˆii (m − p)λ2m λ2m+1 · · · λ2p−1 .

(4.19)

Since Mz is a monotonic-Ar weighted shift, by Eq. (4.19), we conclude that ϕˆii (m − p) = 0 if p = m. Clearly, ϕˆii (0) is a real number by Eq. (4.16). Thus, ϕii is a real-valued constant function, i.e., Mϕii = ci IH

(4.20)

for some ci ∈ R. By Eqs. (4.15) and (4.20), Pii = Mϕii . Finally, if Plk = 0 for some 0 ≤ l = k < n, in the same way as the proof of Proposition 5, we have that Mk and Ml are unitarily equivalent which is a contradiction, since the weights for Mk and Ml are completely different. Hence, Mk and Ml cannot be unitarily equivalent for any 0 ≤ k = l < n by Proposition 13. Therefore, Pij = 0 if i = j.

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In the next Theorem, we discuss the reducing subspaces of the bilateral weighted shift operator Mzn on L2 (β) for a monotonic-Ar weighted shift Mz . Since Mzn and Mzn |S0 ⊕ Mzn |S1 ⊕ · · · ⊕ Mzn |Sn−1 are unitarily equivalent, we have the following result. Theorem 16. If Mz is a monotonic-Ar weighted shift, then the operator Mzn : L2 (β) → L2 (β) has 2n reducing subspaces for n ≥ 2. Although we could state hypothesis for a version of Theorem 16 in terms of the weights as Stessin and Zhu [10] do for the case of unilateral weighted shifts, we state one concrete result which generalizes Theorems 6 and 10. An operator T is said to be hyponormal if [T ∗ , T ] = T ∗ T − T T ∗ ≥ 0 and strictly hyponormal if ker[T ∗ , T ] = {0}. One concrete application of Theorem 16 is the Corollary 17. Corollary 17. If Mz on L2 (β) is a strictly hyponormal operator such that σ(Mz ) = Ar , then Mzn has 2n reducing subspaces for n ≥ 2. Proof. The operator Mz is strictly hyponormal if and only if λn < λn+1 for all n. Since all subnormal weighted shift operators which are not isometric are strictly hyponormal, our earlier Theorems (Theorems 6 and 10) follow from Theorem 16.

5. Kernel Function Point of View In this section, we also assume that the shift operator Mz on L2 (β) is invertible. Then, σ(Mz ) is the annulus A = {z ∈ C : [r(Mz−1 )]−1 ≤ |z| ≤ r(Mz )}, where r(Mz )(r(Mz−1 )) denotes the spectral radius of Mz (Mz−1 , respectively) [9]. In this section, we focus on the shift operator Mz on L2 (β) with monotonic weights {λn }. In this section, we assume that the weights {λn } of Mz on L2 (β) are monotonic satisfying λn = 1 and lim λn = 1. n→∞ rn By a Laurent polynomial we mean a ﬁnite linear combination of the vectors {gn }(−∞ < n < ∞). Recall that for a complex number ω, λω denotes the functional of evaluation at ω, deﬁned on Laurent polynomials by λω (p) = p(ω). lim

n→−∞

Definition 18. ω is said to be a bounded point evaluation on L2 (β) if the functional λω extends to a bounded linear functional on L2 (β). In this section, the hypotheses on the weights imply that every point ω in Ar is a bounded point evaluation. Thus, we have the reproducing kernel kω for L2 (β) associated with the point ω ∈ Ar . Lemma 19. If MF : Sk → L2 (β) is a (bounded) multiplication operator by a function F on Ar , then F ∈ H ∞ (Ar ) and F ∞ ≤ MF .

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Proof. Since every point ω in Ar is a bounded point evaluation, it is proven in the same way as in Lemma 1. Proposition 20. A bounded linear operator T on L2 (β) commutes with Mzn if and only if there are functions φi (0 ≤ i < n) in H ∞ (Ar ) such that Tf =

n−1

φi fi ,

(5.1)

i=0

where fi (0 ≤ i < n) denotes the functions in Eq. (4.5). Proof. In the same way as in Proposition 4, we have analytic functions φi (0 ≤ i < n) on Ar satisfying Eq. (5.1). i ) for 0 ≤ i < n, by Lemma 19, φi ∈ H ∞ (Ar ). Since φi (z) = T (z zi Proposition 21. For 0 ≤ k < n, let Mk = Mzn |Sk . If B = (Bij )(n×n) is a projection such that ⎛ ⎞ ⎛ M0 0 ··· 0 0 0 M0 ⎜ 0 M1 ⎟ ⎜ 0 M1 0 ·· 0 ⎟B = B⎜ ⎜ ⎝ · ⎝ · · · · · ⎠ · 0 0 · · · 0 Mn−1 0 0

··· 0 · ···

0 ·· · 0

0 0 ·

⎞ ⎟ ⎟, ⎠

Mn−1 (5.2)

then there are holomorphic functions ϕij (0 ≤ i, j < n) in H ∞ (Ar ) such that Bij = Mϕij . Moreover, ϕii is a real-valued constant function on Ar for 0 ≤ i < n, and ϕij ≡ 0 for i = j. Proof. Since the weights {λn } of Mz on L2 (β) is monotonic, by Proposition 13, Mk and Ml are not unitarily equivalent for any 0 ≤ k = l < n. Thus, by the same way as in Proposition 5, it is proven. Theorem 22. For 0 ≤ i < n, let Mi = Mzn |Si . Then the bilateral weighted shift operator Mzn : L2 (β) → L2 (β) has 2n reducing subspaces for n ≥ 2. Proof. In the same way as in Theorem 6, it is proven.

6. A Complex Geometric Point of View The adjoint of a hyponormal weighted shift with spectrum equal to the closure of Ar for 0 < r < 1 and essential spectrum equal to ∂Ar belongs to a very special class of operators, B1 (Ar ). Recall that, for a bounded domain Ω in C and a positive integer n, the Bn (Ω)-class was introduced by M. Cowen and the ﬁrst author in [3] and consists of those bounded operators on a Hilbert space H that satisfy: (1) ran (T − ω) is closed for ω ∈ Ω, (2) dim ker (T − ω) = n for ω ∈ Ω, and (3) ω∈Ω ker(T − ω) = H.

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The operators Mz∗ and Tz∗ as well as the adjoints of the bilateral weighted shifts Mz∗ deﬁned in the previous sections with σ(Mz ) = Ar and σe (Mz ) = ∂Ar belong to B1 (Ar ), while their nth powers, Mz∗n , Tz∗n and Mz∗n , belong to Bn (Arn ). All operators T in B1 (Ar ) have a kernel function, kz , and ker (T n − ω) is the span of Γω = {kλωk : ωk = exp(2πik/n)(0 ≤ k < n)}, where n λ = ω. Thus, a holomorphic frame for the hermitian holomorphic bundle ET canonically deﬁned by T n is given by the sums of the appropriate functions in λ analogous to the subspace decomposition into powers of z, z k , where k ≡ i(mod n), and 0 ≤ i < n. In the general case, these sections do not correspond to reducing subspaces since these sections being pairwise orthogonal can be shown to be equivalent to T being a weighted shift. Operators in the commutant of T n correspond to anti-holomorphic bundle maps, which have a matrix representation once an antiholomorphic frame is chosen for ET . That is what was accomplished as a ﬁrst step in the earlier sections. Reducing subspaces correspond to projection-valued anti-holomorphic bundle maps and are determined by the value at a single point. Again, that is the result proved in each of the three cases in which the bundle ET is presented as the orthogonal direct sum of n antiholomorphic line bundles. The question of whether there are other reducing subspaces is equivalent to the issue of representing this bundle as a different orthogonal direct sum. These bundles all have canonical Chern connections and hence a corresponding curvature. The fact that no two of the operators obtained by restricting T n to one of these reducing subspaces are unitarily equivalent corresponds to the fact that the curvature has distinct eigenvalues at some point in Ar . This is a straightforward calculation in the case of the disk but much less so for the annulus. If we take a general T in B1 (Ar ), it seems that the lattice of reducing subspaces has 2k elements for some 0 < k ≤ n. That is the case for Toeplitz operators on the Hardy space Hω2 (Ar ), where ω ∈ Ar and the measure used to deﬁne Hω2 (Ar ) is harmonic measure on Ar for the point ω. It is not clear just how to settle the general case, however, since calculating the curvature is probably not feasible. (Note in this case Tz is not a bilateral weighted shift.) Thus, we need to develope other techniques to settle this question. A more general question concerns operators T in Bn (Ω) for more general Ω. For Tz ⊗ ICn on H 2 (D) ⊗ Cn , the lattice of reducing subspaces is continuous and inﬁnite with no discrete part. Does this happen for any other examples besides Tϕ ⊗ ICn where ϕ is in H ∞ (Ω)?

References [1] Ball, J.A.: Hardy space expectation operators and reducing subspaces. Proc. Am. Math. Soc. 47, 351–357 (1975) [2] Cowen, C.C.: Iteration and the solution of functional equations for functions analytic in the unit disc. TAMS 265, 69–95 (1971)

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[3] Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141, 188–261 (1978) [4] Davidson, K.R., Douglas, R.G.: The generalized Berezin transform and commutator ideal. Pac. J. Math. 222(1), 29–56 (2005) [5] Douglas, R.G.: Banach algebra techniques in operator theory. Springer, New York (1998) [6] Duren, P., Schuster, A.: Bergman Spaces. American Mathematical Society, Providence, RI (2004) [7] Jewell, N.P.: Multiplication by the coordinate functions on the hardy space of the unit sphere in Cn . Duke Math. J. 44, 839–851 (1977) [8] Nordgren, E.: Reducing subspaces of analytic Toeplitz operators. Duke Math. J. 34, 175–181 (1967) [9] Shields, A.L.: Weighted Shift Operators and Analytic Function Theory. Mathematical Survey Series 13 (1974) [10] Stessin, M., Zhu, K.: Reducing subspaces of weighted shift operators. Proc. Am. Math. Soc. 130, 2631–2639 (2002) [11] Thomson, J.: The commutant of certain analytic Toeplitz operators. Proc. Am. Math. Soc. 54, 165–169 (1976) [12] Zhu, K.: Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. 62, 553–568 (2000) Ronald G. Douglas Department of Mathematics Texas A&M University College Station, TX 77843-3368, USA e-mail: [email protected] Yun-Su Kim (B) Department of Mathematics University of Toledo Toledo, OH 43606-3390, USA e-mail: [email protected] Received: February 13, 2009. Revised: March 10, 2011.

Integr. Equ. Oper. Theory 70 (2011), 17–62 DOI 10.1007/s00020-011-1873-4 Published online March 25, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism Joseph A. Ball and Alexander Kheifets Abstract. It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identiﬁes solutions of the lifting problem with minimal unitary extensions of a partially deﬁned isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefﬁcient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefﬁcient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89–138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287–317, 2002) Mathematics Subject Classification (2010). Primary 47A20; Secondary 47A57. Keywords. Feedback connection, unitary coupling, unitary extension, wave operator.

1. Introduction One of the seminal results in the development of operator theory and its applications over the past half century is the Commutant Lifting Theorem: given contraction operators T , T on respective Hilbert spaces H , H with respective isometric dilations V and V on respective Hilbert spaces K ⊃ H The work of A. Kheifets was partially supported by the University of Massachusetts Lowell Research and Scholarship Grant, project number: H50090000000010.

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and K ⊃ H and given a contractive operator X : H → H such that XT = T X, then there exists a operator Y : K → K also with Y ≤ 1 such that Y V = V Y and XPH = PH Y (where PH and PH are the orthogonal projections of K to H and K to H respectively). It is well is an known that the general case can be reduced to the case where T = U+ and isometry on a Hilbert space K+ with unitary extension U on K ⊃ K+ with unitary lift U on K ⊃ K− . Then this where T is a coisometry on K− normalized commutant lifting problem can be formalized as follows: Problem 1.1. (Lifting Problem) Given two unitary operators U and U on ⊂ K and Hilbert spaces K and K , respectively, along with subspaces K+ K− ⊂ K that are assumed to be ∗-cyclic for U and U respectively (i.e., the is the whole space K and smallest reducing subspace for U containing K+ similarly for U and K− ) and such that U K+ ⊂ K+ ,

U ∗ K− ⊂ K− ,

(1.1)

→ K− which satisfies the intertwinand given a contractive operator X : K+ ing condition = PK U X, XU |K+ −

(1.2)

characterize all contractive intertwiners Y of (U , K ) and (U , K ) which lift X in the (Halmos) sense that Y |K = X. PK− +

(1.3)

An important special case of this theorem was ﬁrst proved by Sarason [50]; there he also explains the connections with classical Nevanlinna–Pick and Carath´eodory–Fej´er interpolation. Since the result was ﬁrst formulated and proved in its full generality by Sz.-Nagy and Foias [55] (see also [56]), applications have been made to a variety of other contexts, including Nevanlinna–Pick interpolation for operator-valued functions and best approximation by analytic functions to a given L∞ -function in L∞ -norm (the Nehari problem)—we refer to the books [18,19] for an overview of all these developments. Moreover, the theorem has been generalized to still other contexts, e.g., to representations of nest algebras/time-varying systems [17,19] as well as representations of more exotic Hardy algebras [10,13,21,39,41,46,54] with applications to more exotic Nevanlinna–Pick interpolation theorems [42,47– 49]. There has also appeared a weighted version [14,59] as well as a relaxed version [20,22,24,38] of the theorem leading to still other types of applications. There are now also results on linear-fractional parametrizations for the set of all solutions (see [3,33] for the Nehari problem—see also [45, Chapter 5] for an overview, see [18, Chapter XIV] and [19, Theorem VI.5.1] and the references there for the standard formulation Problem 1.1 of the Lifting Problem, see [23,24,57] for the relaxed version of the lifting theorem); in the context of classical Nevanlinna–Pick interpolation, such parametrization results go back to the papers of Nevanlinna [43,44]. The associated inverse problem asks for a characterization of which Redheffer linear-fractional coefﬁcient-matrices arise in this way for some Lifting Problem. The inverse problem has been studied much less than the direct

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problem; there are only a few publications in this direction ([6,30,31]. We refer also to [24,25,52,53]) for some special cases of the inverse Lifting Problem (Nehari problem and Nevanlinna–Pick/Carath´eodory–Schur interpolation problem), and the quite recent work [58] on the inverse version of the relaxed commutant lifting problem. Our contribution here is to further develop the ideas in [33,34] to obtain new results on the inverse problem (Theorem 8.4). The formulas here are in an abstract operator-theoretic formalism which we call “coordinate-free”; in part 2 of this paper we show how the results here can be embedded into Hellinger functional-model spaces and in this way can be used to recover various more concrete special cases which have already appeared in the literature. The starting point for our approach is the coupling method ﬁrst introduced by Adamjan et al. [2] and developed further in [4,5,16,22,26–29,32– 35,38,40] (some of these in several-variable or relaxed contexts—see also [51] for a nice exposition). In this approach one identiﬁes solutions of the Lifting Problem with minimal unitary extensions of an isometry constructed in a natural way from the problem data. We use here the term isometry (sometimes also called semiunitary operator) in the following technical sense: we are given a Hilbert space H0 and subspaces D and D∗ of H0 together with a linear operator V which maps D isometrically onto D∗ ; we then say that V is an isometry on H0 with domain D and codomain D∗ . By a minimal unitary extension of V we mean a unitary operator U on a Hilbert space K containing H0 as a subspace such that the restriction of U ∗ to D agrees with V and the smallest U-reducing subspace containing H0 is all of K. From the work of Arov and Grossman [7,8] and Katsnelson et al. [26], it is known that there is a special unitary colligation U0 (called the universal unitary colligation) so that any such unitary extension U ∗ of V arises as the lower feedback connection U ∗ = F (U0 , U1 ) of U0 with a free-parameter unitary colligation U1 (see Theorem 6.1). A special unitary extension of V is obtained as the unitary dilation U0∗ of the universal unitary colligation U0 (or, in the language of [9], U0 is the unitary evolution operator for the Lax–Phillips scattering system in which U0 is embedded). This special unitary extension U0∗ of V is called the universal unitary extension. Unlike other contexts where the “lurking isometry” approach has been used (see in particular [12,26,32,35,36]), the connection between unitary extensions U ∗ of V and (U , U )-intertwiners Y solving the Lifting Problem, as in [33,34], involves an extra step: computation of the lift Y from the unitary extension U is not immediately explicit but rather involves a wave-operator construction demanding computation of powers of U. The lift Y is uniquely determined from its moments wY (n) = i∗∗ Y n i where i∗ and i are certain isometric embedding operators (or scale operators in the sense of [15]). Calculation of such moments (the collection of which we call the symbol of the lift Y ) requires the computation of powers of U ∗ = F (U0 , U1 ) in terms of the coefﬁcients of the universal unitary colligation U0 determined by the problem data and the coefﬁcients of the free-parameter unitary colligation U1 (or in terms of its characteristic function ω(ζ)). In Sect. 2 we identify a general principle of independent interest for the explicit computation of the

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powers of an operator U ∗ given as a feedback connection U ∗ = F (U0 , U1 ) of two unitary colligations U0 and U1 . With the application of this general principle, we arrive at an explicit Redheffer-type linear-fractional parametrization of the set of symbols {wY (n)}n∈Z associated with the set of solutions Y of a Lifting Problem (see Theorem 7.1). The symbol for the Redheffer coefﬁcient matrix is a simple explicit formula in terms of the universal unitary extension U0 [see formula (8.40) in Theorem 8.6 below]. This general principle (already implicitly present in [33]) can be summarized as follows. Suppose that the operator U is given as the lower feedback connection U ∗ = F (U0 , U1 ) of two colligation matrices A0 B0 A1 B1 X0 X1 X0 X1 U0 = : : → . , U1 = → D D∗ D∗ D C0 D0 C1 D1 (In our context always have D0 = 0). Associated with any colligation we A B X X matrix U = : → is the discrete-time linear system E E∗ C D x(n + 1) x(n) ΣU : (1.4) =U , x(0) = x0 e∗ (n) e(n) which recursively deﬁnes what we call the augmented input–output map (extending the usual input–output map in the sense that it takes into account an initial condition x(0) = x0 not necessarily equal to zero as well as the internal state trajectory {x(n)}n∈Z ): W (U )+ W (U )+ {x(n)}n∈Z+ x0 0 2 + W (U ) := : → {e(n)}n∈Z+ {e∗ (n)}n∈Z+ W (U )+ W (U0 )+ 1 if {e(n), x(n), e∗ (n)}n∈Z+ solves the system equations (1.4). Then the general principle asserts: powers U n of U = F (U0 , U1 ) can be computed via performing a feedback connection at the system-trajectory level: x0 x = F W (U0 )+ , W (U1 )+ Un 0 . x1 n∈Z x1 +

The general structure for the universal unitary extension U0 with embedded subspaces related to the original problem data (U , K ), (U , K ) and

Δ

∗ for the free-parameter characteristic function can coefﬁcient spaces Δ, be viewed as a fourfold Adamjan and Arov (AA) unitary coupling in the general sense of [1]. In this setting one can identify the special geometry corresponding to the case where the fourfold AA-unitary coupling arises from a Lifting Problem. In this way we arrive at the inverse theorem (Theorem 8.4), specifically, a characterization of which Redheffer coefﬁcient matrices arise as the coefﬁcient matrix for the linear-fractional parametrization of the set of all solutions of some Lifting Problem (with given operators U , U and subspaces ⊂ K , K− ⊂ K ) generalizing results of [6,30,31] obtained in the context K+ of the Nehari Problem and the bitangential Nevanlinna–Pick problem. The solution of the inverse problem for the Lifting Problem as presented here appears to be quite different from the inverse problem considered

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in [58]. We discuss the connections between the results of this paper and those of [58] in detail in Remark 7.3 (for the direct problem). The connections for the inverse problem will be discussed in the second part of our paper where the Hellinger functional-model version of the results is available. The paper is organized as follows. After the present Introduction, in Sect. 2 we present the general principle for computation of powers of U = F (U0 , U1 ) via the trajectory-level feedback connection of the augmented input–output operator of U0 with that of U1 . Section 3 reviews results from [15] concerning general unitary scattering systems which will be needed in the sequel. Section 4 reviews basic ideas from [1] concerning the correspondence between contractive intertwiners Y of two unitary operators U and U on the one hand and unitary couplings U of U and U on the other. Section 5 adds the constraint that the intertwiner Y should be a lift of a given , U− of U , U contractive intertwiner X of restricted/compressed versions U+ and identiﬁes the correspondence between solutions Y of the lifting problem and unitary extensions U ∗ of the isometry V constructed directly from the data for the Lifting Problem. Section 6 recalls the result from [7,8] that such unitary extensions arise as the lower feedback connection of the universal unitary colligation U0 with a free-parameter unitary colligation U1 . Section 7 uses the general principle from Sect. 2 to obtain a parametrization for the set of symbols {wY (n)}n∈Z associated with solutions Y of the Lifting Problem. Section 8 introduces the universal unitary extension. Here the universal unitary extension is identiﬁed as the fourfold AA-unitary coupling of the two unitary operators U , U appearing in the Lifting-Problem data together with the bilateral shift operators associated with the input and output spaces for the free-parameter unitary colligation. Here the special geometric structure is identiﬁed which leads to the coordinate-free version of our inverse theorem (Theorem 8.4) characterizing which fourfold AA-unitary couplings arise in this way from a Lifting Problem. Here also is established the formula for the Redheffer-coefﬁcient matrix in terms of the universal unitary extension.

2. Calculus of Feedback Connection of Unitary Colligations Suppose that we are given linear spaces X0 , X 0 , X1 , X 1 , F, F∗ and linear operators presented in block matrix form A0 B0 A1 B1 X0 X1 X 0 X 1 U0 = : → , U1 = : → . F C D F F F C0 D0 1 1 ∗ ∗ We deﬁne the feedback connection U := F (U0 , U1 ) : it exists) by F (U0 , U1 )

x0 x1

=

x

0 x

1

X0 X1

→

X 0 X 1

(2.1)

(when

if there exist f ∈ F and f∗ ∈ F∗ so that

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A0 C0

B0 D0

x0 f

x

0 = f∗

and

A1 C1

B1 D1

x1 f∗

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x

1 . = f

(2.2)

We also deﬁne the elimination operator Γ (U0 , U1 ) (when it exists) by x0 f

1 so that (2.2) holds.(2.3) Γ (U0 , U1 ) : → if there exist x

0 , x x1 f∗ As explained in the following result, the feedback connection and elimination operator exist and are well-deﬁned as long as the operator I − D1 D0 is invertible as an operator on F. Theorem 2.1. Suppose that we are given block-operator matrices U0 and U1 as in (2.1). Assume (I − D1 D0 )−1 and hence also (I − D0 D1 )−1 exist as Then the feedback connection (2.2) is operators on F and F∗ respectively. x0 well-posed, i.e., for each ∈ X0 ⊕ X1 there exists a unique f ∈ F and x1 x

0 f∗ ∈ F∗ so that the equations (2.2) determine a unique ∈ X 0 ⊕ X 1 x

1 x0 which we then define to be F (U0 , U1 ) . More explicitly, the feedback x 1 x0 x

0 connection operator F (U0 , U1 ) : → is given by x1 x

1 A0 + B0 (I − D1 D0 )−1 D1 C0 B0 (I − D1 D0 )−1 C1 . F (U0 , U1 ) = B1 (I − D0 D1 )−1 C0 A1 + B1 (I − D0 D1 )−1 D0 C1 (2.4) The operator (2.3) which assigns instead the uniquely determined elimination x0 f to is then given explicitly by x1 f∗ (I − D1 D0 )−1 D1 C0 (I − D1 D0 )−1 C1 x0 f Γ (U0 , U1 ) = : → . x1 f∗ (I − D0 D1 )−1 C0 (I − D0 D1 )−1 D0 C1

(2.5)

x0 x

0 = means that there is f ∈ F x1 x

1 and f∗ ∈ F∗ so that (2.2) holds. From the second equation of the ﬁrst system in (2.2) we have Proof. The definition F (U0 , U1 )

f∗ = C0 x0 + D0 f. Plug this into the second equation of the second system to get f = C1 x1 + D1 f∗ = C1 x1 + D1 (C0 x0 + D0 f ) = D1 C0 x0 + C1 x1 + D1 D0 f. Under the assumption that I − D1 D0 is invertible we then can solve for f to get f = (I − D1 D0 )−1 D1 C0 x0 + (I − D1 D0 )−1 C1 x1 .

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Plug this into the second equation of the ﬁrst system in (2.2) to then get f∗ = C0 x0 + D0 (I − D1 D0 )−1 D1 C0 x0 + D0 (I − D1 D0 )−1 C0 x1 = (I − D0 D1 )−1 C0 x0 + (I − D0 D1 )−1 D0 C1 x1 . In this way we get the formula (2.5) for the elimination operator Γ (U0 , U1 ). It is now a simple matter to plug in these values for f, f∗ in terms of x0 , x1 into the ﬁrst equations in the two systems (2.2) to arrive at the formula (2.4) for the feedback connection operator F (U0 , U1 ). While the formula (2.4) exhibits F (U0 , U1 ) explicitly in terms of U0 and U1 , direct computation of powers U n (n = 2, 3, . . .) of U = F (U0 , U1 ) appears to be rather laborious. We next show how eﬃcient computation of powers F (U0 , U1 ) can be achieved by use of a feedback connection at the level of system trajectories. Toward this end, we ﬁrst introduce some useful notation. For G any linear space, we let G (Z) (alternatively often written as G Z in the literature) denote the space of all G-valued functions on the integers Z. Similarly we let G (Z+ ) be the space of all G-valued functions on the nonnegative integers Z+ ; we often identify G (Z+ ) with the subspace of G (Z) consisting of all G-valued functions on Z which vanish on the negative integers. Similarly, G (Z− ) is the space of all G-valued functions on Z− and is frequently identiﬁed with the subspace of G (Z) consisting of all G-valued functions on Z vanishing on Z+ . By P + and P − we denote the natural projections of G (Z) onto G (Z+ ) and G (Z− ), respectively. Sometimes we will use notations g + = P +g ,

g − = P −g .

(2.6)

We also consider the bilateral shift operator J : g → g , where g (n) = g (n − 1). Given a colligation matrix U of the form A B X X : → U= (2.7) E E∗ C D we may consider the associated discrete-time input/state/output linear system Ax(n) + Be(n) x(n) x(n + 1) =U = . (2.8) e(n) e∗ (n) Cx(n) + De(n) Given an initial state x(0) = x0 and an input string e ∈ E (Z+ ), the system equations (2.8) recursively uniquely determine the state trajectory x ∈ X (Z+ ) and the output string e∗ ∈ E∗ (Z+ ); explicitly we have n

x(n) = A x0 +

n−1

An−1−k Be(k),

k=0

e∗ (n) = CAn x0 +

n−1

k=0

CAn−1−k Be(k) + De(n)

for n = 0, 1, 2, . . . . (2.9)

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If we view elements of X (Z+ ) (and of E (Z+ ) and E∗ (Z+ )) as column vectors, then operators between these various spaces can be represented as block matrices. We may then write the content of (2.9) in matrix form as + W0 W2+ x(0) x = e e∗ W1+ W + where the block-operator matrix W0+ W2+ X X (Z+ ) + W := : → E (Z+ ) E∗ (Z+ ) W1+ W + is given explicitly by

⎡

⎡

W0+

⎤ IX ⎢A ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎥ . =⎢ ⎢ n−1 ⎥, ⎢A ⎥ ⎣ ⎦ .. . ⎡

W1+

W2+

0 ⎢B ⎢ ⎢ AB ⎢ . =⎢ ⎢ .. ⎢ n−1 ⎢A B ⎣

⎡

⎤

C ⎢ CA ⎥ ⎢ ⎥ ⎢ CA2 ⎥ ⎢ ⎥ . ⎥, =⎢ ⎢ .. ⎥ ⎢ ⎥ n−1 ⎢ CA ⎥ ⎣ ⎦ .. .

0 0 B .. . An−2 B .. .

W+

D ⎢ CB ⎢ ⎢ CAB ⎢ . =⎢ ⎢ .. ⎢ ⎢ CAn−1 B ⎣

⎤

... ... ...

0 0 0 .. . An−3 B .. .

0 D CB .. . CAn−2 B .. .

(2.10)

... .. .

B

0 ..

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

... .. .

.

(2.11) ⎤

0 0 D

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

..

n−3

CA .. .

B

. ... .. .

D ..

.

W0+

The operator is the (forward-time) initial-state/state-trajectory map, the operator W2+ is the input/state-trajectory map, the operator W1+ is the observation operator and the operator W + is what is traditionally known as the input–output map in the control literature. We note that the multiplication operator associated with W + after applying the Z-transform

f → f(n)z n n∈Z+

to the input and output strings e and e∗ respectively has multiplier + (z) = D + zC(I − zA)−1 B W equal to the characteristic function of the colligation U [also known as the transfer function of the linear system (2.8)]. We shall refer to the whole 2 × 2-block operator matrix W+ simply as the (forward-time) augmented input/output map associated with the colligation U . If the colligation matrix U (2.7) is invertible, then we can also run the system in backwards time: αx(n + 1) + βe∗ (n) x(n) −1 x(n + 1) = (2.12) =U γx(n + 1) + δe∗ (n) e∗ (n) e(n)

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αβ : X ⊕ E∗ → X ⊕ E. In this case, speciﬁcation of where we set U = γ δ an initial state x(0) and of the output string over negative time e∗ ∈ E∗ (Z− ) determines recursively via the backward-time system equations (2.12) the state-trajectory over negative time x− ∈ X (Z− ) and the input string over negative time e− ∈ E (Z− ). Explicitly we have −1

x(n) = αn x(0) +

n

αn−k βe∗ (−k),

(2.13)

k=1

e(−n) = γαn−1 x(0) +

n−1

γαn−1 βe∗ (−k) + δe∗ (−n) for n = 1, 2, . . . .

k=1

If we write elements x = {x(n)}n∈Z− of X (Z− ) as inﬁnite column matrices ⎤ ⎡ .. . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x = ⎢ x(−3) ⎥ ⎥ ⎢ ⎣ x(−2) ⎦ x(−1) then linear operators between spaces of the type X (Z− ) can be written as matrices with inﬁnitely many rows as one ascends to the top. Then the relations (2.13) can be expressed in 2 × 2-block operator matrix form as W0− W1− x− x(0) = , (2.14) e∗− e− W2− W − where the 2 × 2-block operator matrix W0− W1− X (Z− ) X − : → W := E∗ (Z− ) E (Z− ) W2− W − is given explicitly by ⎤ .. . ⎢ n⎥ ⎢α ⎥ ⎢ ⎥ ⎥ =⎢ ⎢ ... ⎥, ⎢ ⎥ ⎣ α2 ⎦ α ⎡

W0−

⎤ .. ⎢ .n⎥ ⎢ γα ⎥ ⎥ ⎢ ⎥ =⎢ ⎢ ... ⎥, ⎥ ⎢ ⎣ γα2 ⎦ γα

W1−

⎡

⎢ . ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

..

. ... .. .

β

⎡

⎡

W2−

..

W−

.. ⎢ . ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

..

δ

. ...

..

⎤

..

. αn−3 β .. .

. αn−2 β .. .

β

αβ β

..

..

⎥ αn−1 β ⎥ ⎥ ⎥ .. ⎥, (2.15) . ⎥ α2 β ⎥ ⎥ ⎦ αβ β ⎤

. γαn−3 β .. .

. γαn−2 β .. .

δ

γβ δ

γαn−1 β .. . γαβ γβ δ

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

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Here the operators W0− , W1− , W2− and W − are the backward-time versions of the initial-state/state-trajectory, input/state-trajectory, observation and input/output operators, respectively, and we refer to the aggregate operator W− simply as the backward-time augmented input–output map. Let us now suppose that U0 and U1 are two colligation matrices A0 B0 X0 X0 U0 = : → , (2.16) D D∗ C0 D0 A1 B1 X1 X1 : (2.17) U1 = → D∗ D C1 D1 such that I − D1 D0 and hence also I − D0 D1 are invertible on D and on D∗ respectively. Then the feedback connection U = F (U0 , U1 ) is well-deﬁned as an operator on X0 ⊕ X1 as explained in Theorem 2.1. Then we also have associated augmented input–output maps for U0 and U1 given by W (U0 )+ X0 (Z+ ) W (U0 )+ X0 0 2 + W(U0 ) = : → D (Z+ ) D∗ (Z+ ) W (U0 )+ W (U0 )+ 1 ⎡ W(U1 )+ = ⎣

W (U1 )+ 0

W (U1 )+ 2

W (U1 )+ 1

W (U1 )+

⎤ ⎦:

X1 → D∗ (Z+ )

(2.18)

X1 (Z+ ) . D (Z+ )

Under the assumption that ID (Z+ ) − W (U1 )+ W (U0 )+ is invertible on D (Z+ ),

(2.19)

it makes sense to form the feedback connection F (W(U0 )+ , W(U1 )+ ). The following lemma guarantees that this connection is well-posed whenever the connection F (U0 , U1 ) is well-posed. Lemma 2.2. Let U0 and U1 be as in (2.16) and (2.17) and assume that I − D1 D0 is invertible on D. Then also I − W (U1 )+ W (U0 )+ is invertible on D (Z+ ). Proof. From the formula for W+ in (2.10) and (2.11), we see that W (U1 )+ and W (U0 )+ are given by lower triangular Toeplitz matrices with diagonal entries equal to D1 and D0 respectively. Hence I − W (U1 )+ W (U0 )+ is also lower triangular Toeplitz with diagonal entry equal to I − D1 D0 . A general fact is that an operator on D (Z+ ) given by a lower triangular Toeplitz matrix with invertible diagonal entry is invertible on D (Z+ ). It follows that I − W (U1 )+ W (U0 )+ is invertible on D (Z+ ) as asserted. We now come to the main result of this section, namely: the computation of powers of F (U0 , U1 ) via the feedback connection F (W(U0 ), W(U1 )). For this purpose it is convenient to introduce the following general notation. For U an operator on a linear space K and G a subspace of K with i∗G : K → G the adjoint of the inclusion map iG : G → K, we deﬁne an operator ΛG,+ (U ) : K → G (Z+ ) (called the Fourier representation operator) by ΛG,+ (U ) : k → {i∗G U n k}n∈Z+ .

(2.20)

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Note that in case we take G = K we have simply ΛK,+ (U ) : k → {U n k}n∈Z+ . Theorem 2.3. Suppose that we are given two colligation matrices (2.16), (2.17) such that I − D1 D0 is invertible on D and we set U = F (U0 , U1 ) ∈ L(X0 ⊕ X1 ). Then the trajectory-level feedback connection operator F (W(U0 )+ , W(U1 )+ ) computes the powers of U = F (U0 , U1 ): ΛX0 ⊕X1 ,+ (U ) = F (W(U0 )+ , W(U1 )+ ) : X0 ⊕ X1 → X0 ⊕X1 (Z+ ). (2.21) Hence, after application of the natural identification between the spaces X0 ⊕X1 (Z+ ) and X0 (Z+ ) ⊕ X1 (Z+ ), we have the explicit formulas ΛX0 ⊕X1 ,+ (U )11 ΛX0 ⊕X1 ,+ (U ) = ΛX0 ⊕X1 ,+ (U )21

ΛX0 ⊕X1 ,+ (U )12 ΛX0 ⊕X1 ,+ (U )22

:

X0 X1

(Z ) → X0 + X1 (Z+ )

where the matrix entries ΛX0 ⊕X1 ,+ (U )ij (i, j = 1, 2) are given explicitly by + + + −1 ΛX0 ⊕X1 ,+ (U )11 = W (U0 )+ W (U1 )+ W (U0 )+ 0 + W (U0 )2 (I − W (U1 ) W (U0 ) ) 1 , + + + ΛX0 ⊕X1 ,+ (U )12 = W (U0 )+ 2 (I − W (U1 ) W (U0 ) )W (U1 )1 , + + −1 W (U0 )+ ΛX0 ⊕X1 ,+ (U )21 = W (U1 )+ 2 (I − W (U0 ) W (U1 ) ) 1 ,

(2.22)

+ + + −1 W (U0 )+ W (U1 )+ ΛX0 ⊕X1 ,+ (U )22 = W (U1 )+ 0 + W (U1 )2 (I − W (U0 ) W (U1 ) ) 1 .

Proof. Note that Lemma 2.2 guarantees that the trajectory-level feedback connection F (W(U0 )+ , W(U1 )+ ) is well-posed. By definition, we see that x {x0 (n)}n∈Z+ F (W(U0 )+ , W(U1 )+ ) 0 = (2.23) x1 {x1 (n)}n∈Z+ means that

x0 (n + 1) d∗ (n) x1 (n + 1) d(n)

=

=

A0 C0 A1 C1

x0 (n) , d(n) D0 B1 x1 (n) d∗ (n) D1 B0

(2.24)

for uniquely determined strings {d(n)}n∈Z+ ∈ D (Z+ ) and {d∗ (n)}n∈Z+ ∈ D∗ (Z+ ). As U = F (U0 , U1 ), the particular case n = 0 of the equations (2.24) is just the assertion that x0 (0) x0 (1) =U . x1 (1) x1 (0) Inductively assume that

x0 (n) x1 (n)

=U

n

x0 (0) . x1 (0)

(2.25)

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The nth equation in (2.24) amounts to the assertion that x0 (n) x0 (n + 1) =U . x1 (n + 1) x1 (n) Combining with the inductive assumption (2.25) then gives us that (2.25) holds with n + 1 in place of n and hence (2.25) holds for all n = 0, 1, 2, . . .. Note next that (2.25) combined with (2.23) amounts to the identity (2.21). The explicit formulas (2.22) then follow from formula (2.4) with W(U0 )+ , W(U1 )+ as in (2.18) in place of U0 , U1 . If U0 and U1 are invertible with α0 β0 α1 β1 X1 X0 X0 X1 −1 −1 , U1 = → U0 = : → : γ0 δ0 D∗ D γ1 δ1 D D∗ (2.26) with ID∗ − γ1 γ0 invertible, a similar analysis can be brought to bear to compute negative powers of U = F(U0 , U1 ). Let us introduce an operator ΛG,− : K → G (Z− ) [also called a Fourier representation operator along with (2.20)] by ΛG,− (U ) : k → {i∗G U n k}n∈Z− where in particular ΛK,− (U )k : → {U n k}n∈Z− . Then we have the following backward-time result parallel to Theorem 2.3. As the proof is completely analogous, we omit the details of the proof. Theorem 2.4. Suppose that we are given two colligation matrices U0 and U1 as in (2.16), (2.17) with inverses as in (2.26) such that I − δ1 δ0 is invertible on D∗ and we set U = F (U0 , U1 ) ∈ L(X0 ⊕ X1 ). Then the trajectory-level feedback connection operator F (W(U0 )− , W(U1 )− ) computes the negative powers of U = F (U0 , U1 ): ΛX0 ⊕X1 ,− (U ) = F (W(U0 )− , W(U1 )− ) : X0 ⊕ X1 → X0 ⊕X1 (Z− ). (2.27) Hence, application of the natural identification between X0 ⊕X1 (Z− ) and X0 (Z− ) ⊕ X1 (Z− ) leads to explicit formulas ΛX0 ⊕X1 ,− (U )11 ΛX0 ⊕X1 ,− (U ) = ΛX0 ⊕X1 ,− (U )21

ΛX0 ⊕X1 ,− (U )12 ΛX0 ⊕X1 ,− (U )22

:

X0 X1

(Z ) → X0 − X1 (Z− )

where the matrix entries ΛX0 ⊕X1 ,+ (U )ij (i, j = 1, 2) are given explicitly by − − − −1 ΛX0 ⊕X1 ,− (U )11 = W (U0 )− W (U1 )− W (U0 )− 0 + W (U0 )1 (I − W (U1 ) W (U0 ) ) 2 , − − − ΛX0 ⊕X1 ,− (U )12 = W (U0 )− 1 (I − W (U1 ) W (U0 ) )W (U1 )2 , − − −1 ΛX0 ⊕X1 ,− (U )21 = W (U0 )− W (U0 )− 1 (I − W (U0 ) W (U1 ) ) 2 ,

(2.28)

− − − −1 ΛX0 ⊕X1 ,+ (U )22 = W (U1 )− W (U0 )− W (U1 )− 0 + W (U1 )1 (I − W (U0 ) W (U1 ) ) 2 .

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3. Unitary Scattering Systems Following [15] we deﬁne a unitary scattering system to be a collection S of the form S = (U, Ψ; K, E)

(3.1)

where K (the ambient space) and E (the coefficient space) are Hilbert spaces, U is a unitary operator on K (called the evolution operator), and Ψ (called the scale operator) is an operator from E into K. A fundamental object associated with any unitary scattering system S (3.1) is its so-called characteristic function wS (ζ) deﬁned by wS (ζ) =

∞

n

Ψ∗ U n Ψζ +

n=1 ∗

=Ψ

∞

Ψ∗ U ∗n Ψζ n

n=0

−1

(I − ζU)

+ (I − ζU ∗ )−1 − I Ψ

= (1 − |ζ|2 )Ψ∗ (I − ζU)−1 (I − ζU ∗ )−1 Ψ.

(3.2)

From (3.2) we see that wS (ζ) is a positive harmonic operator-function (values are operators on E) wS (ζ) ≥ 0 for ζ ∈ D

∂2 wS (ζ) = 0 ∂ζ∂ζ

and

If we introduce the convention ζ [n] =

ζn ζ

for all ζ ∈ D. (3.3)

if n ≥ 0

−n

if n < 0

then the ﬁrst formula in (3.2) can be written more succinctly as wS (ζ) =

∞

(Ψ∗ U ∗n Ψ)ζ [n] .

n=−∞

We shall refer to the string of coefﬁcients {wS,n }n∈Z given by wS,n := Ψ∗ U ∗n Ψ

(3.4)

as the characteristic moment sequence. Let us now introduce the spectral measure EU (·) (see e.g. [37]) for U; we then deﬁne the characteristic measure σS for the unitary scattering system S to be the spectral measure for U compressed by the action of Ψ given by σS (·) = Ψ∗ EU (·)Ψ.

(3.5)

Thus the spectral measure EU is a strong Borel measure on the unit circle T with values equal to orthogonal projection operators in L(K) while the characteristic measure σS is a strong Borel measure on T with values equal to positive-semidefinite operators in L(E). Note that the characteristic function wS (ζ) can be expressed in terms of the characteristic measure σS via the

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Poisson integral:

⎡ ⎤ wS (ζ) = Ψ∗ ⎣ ((1 − ζt)−1 + (1 − ζt)−1 − 1) EU (dt)⎦ Ψ T

P(t, ζ) σS (dt)

=

(3.6)

T

where P(t, ζ) =

1 − |ζ|2 |1 − ζt|2

for t ∈ T

and

ζ∈D

(3.7)

is the classical Poisson kernel and where the Lebesgue integral converges in the strong operator topology. We say that two unitary scattering systems (U, Ψ; K, E)

and

(U , Ψ ; K , E)

(with the same coefﬁcient space E) are unitarily equivalent if there is a unitary map τ : K → K so that τ U = U τ,

τ Ψ = Ψ .

(3.8)

We say that a unitary scattering system S = (U, Ψ; K, E) is minimal in case the linear manifold ΨE ⊂ K is ∗-cyclic for U, i.e., the smallest subspace K0 containing ΨE and invariant for both U and U ∗ is the whole space K. The following elementary result makes precise the idea that the characteristic function is a complete unitary invariant for minimal unitary scattering systems. Proposition 3.1. Two unitarily equivalent unitary scattering systems S = (U, Ψ; K, E)

and

S = (U , Ψ ; K , E)

have the same characteristic functions (wS (ζ) = wS (ζ) for all ζ ∈ D). Conversely, if S = (U, Ψ; K, E) and S = (U , Ψ ; K , E) are two minimal unitary scattering systems with the same characteristic function, then S and S are unitarily equivalent. Proof. This is essentially Theorem 4.1 in [15]. For the reader’s convenience, we recall the proof here. If τ : K → K satisﬁes the intertwining conditions (3.8), then wS (ζ) = Ψ∗ (I − ζU )−1 + (I − ζU ∗ )−1 − I Ψ = Ψ∗ τ ∗ (I − ζU )−1 + (I − ζU ∗ )−1 − I τ Ψ = Ψ∗ (I − ζU)−1 + (I − ζU ∗ )−1 − I Ψ = wS (ζ). Conversely, suppose that S and S are minimal unitary scattering systems with the same coefﬁcient space E and with identical characteristic functions wS (ζ) = wS (ζ) for all ζ ∈ D. The identity wS (ζ) = wS (ζ) between

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harmonic functions implies that coefﬁcients of powers of ζ and of ζ match up: Ψ∗ U n Ψ = Ψ∗ U n Ψ

for

n = 0, ±1, ±2, . . . .

Note that U n Ψe, U m Ψ

e = Ψ∗ U n−m Ψe, e

= Ψ∗ U n−m Ψ e, e

= U n Ψ e, U m Ψ e

for all n, m = 0, ±1, ±2, . . . and e, e ∈ E, and hence the formula τ : U n Ψe → U n Ψ e

(3.9)

extends by linearity and continuity to deﬁne a well-deﬁned isometry from D = span{U n Ψe : n ∈ Z, e ∈ E} onto R := span{U n Ψ e : n ∈ Z, e ∈ E}. Under the assumption that both S and S are minimal, we see that D = K and R = K , and hence τ is unitary from K onto K . From the formula (3.9) for τ specialized to the case n = 0, we see that τ Ψ = Ψ . From the general case of the formula we see that τ Uk = U τ k in case k ∈ K has the form k = U n Ψ∗ e for some n ∈ Z and e ∈ E. By the minimality assumption on S the span of such elements is dense in K, and hence the validity of the intertwining τ Uk = U τ k extends to the case of a general element k of K. This concludes the proof of Proposition 3.1.

4. Intertwiners and Unitary Couplings of Unitary Operators In this section nwe present some preliminary material on unitary couplings due originally to Adamjan and Arov [1] which is needed for a reformulation of the Lifting Problem to be presented in the next section. Suppose that we are given unitary operators (U , K ) and (U , K ). We say that the collection (U, iK , iK ; K) is an Adamjan-Arov unitary coupling (or, more brieﬂy, AA-unitary coupling) of (U , K ) and (U , K ) if U is a unitary operator on the Hilbert space K and iK : K → K and iK : K → K are isometric embeddings of K and K respectively into K such that iK U = UiK ,

iK U = UiK .

(4.1)

In this case it is clear that Y = i∗K iK : K → K is contractive (Y ≤ 1) and that Y intertwines U with U since Y U = i∗K iK U = i∗K UiK = U i∗K iK = U Y. The following theorem provides a converse to this observation. To formalize the ideas, let us say that the AA-unitary coupling (U, iK , iK ; K) is minimal in case im iK + im iK is dense in K

(4.2)

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iK , iK ; K)

of and that two AA-unitary couplings (U, iK , iK ; K) and (U, (U , K ) and (U , K ) are unitarily equivalent if there is a unitary operator

such that τ: K →K

τ U = Uτ,

τ iK = iK ,

τ iK = iK .

(4.3)

Then we have the following fundamental connection between AA-unitary couplings and contractive intertwiners of two given unitary operators (see e.g. [1,16]). Theorem 4.1. Suppose that we are given two unitary operators U and U on Hilbert spaces K and K , respectively. Then there is a one-to-one correspondence between unitary equivalence classes of minimal AA-unitary couplings (U, iK , iK ; K) and contractive intertwiners Y : K → K of (U , K ) and (U , K ). More precisely: 1.

Suppose that A := (U, iK , iK ; K) is an AA-unitary coupling of (U , K ) and (U , K ). Define an operator Y = Y (A) : K → K via Y = Y (A) = i∗K iK .

2.

(4.4)

Then Y is a contractive intertwiner of (U , K ) and (U , K ). Equivalent AA-unitary couplings produce the same intertwiner Y via (4.4). Suppose that Y : K → K is a contractive intertwiner of (U , K ) and a Hilbert space K := K ∗Y K as the completion of the (U , K ). Define K space in the inner product K IK Y h k h k = (4.5) , ∗ , k h I k h Y K K ∗Y K K ⊕K k for k , h ∈ K and k , h ∈ K (with pairs with zero self inner k product identified with 0). Define an operator U = U ∗Y U densely on K by U k k U: (4.6) → k U k together with inclusion maps iK : K → K and iK : K → K given by 0 k iK : k → : k → . (4.7) , i K k 0 Then the resulting collection A = A(Y ) := (U ∗Y U , iK , iK ; K ∗Y K )

(4.8)

is a minimal AA-unitary coupling of (U , K ) and (U , K ) such that we recover Y as Y = i∗K iK . Any minimal AA-unitary coupling (U, iK , iK ; K) of (U , K ) and (U , K ) with intertwiner Y as in (4.4) is unitarily equivalent to the one defined by (4.5), (4.6), (4.7).

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Moreover, the maps A → Y (A) in (1) and Y → A(Y ) in (2) are inverse to each other and set up a one-to-one correspondence between contractive intertwiners Y and unitary equivalence classes of minimal AA-unitary couplings of (U , K ) and (U , K ). Proof. The proof of part (1) was already observed in the discussion preceding the statement of the theorem. Conversely, suppose that Y : K → K is any contractive intertwiner of (U , K ) and (U , K ) and let (U ∗Y U , iK , iK ; K ∗Y K ) be the AA-unitary coupling of (U , K ) and (U , K ) given by (4.8). From the form of the inner by product (4.5), we see that the maps iK : K → K and iK : K → Kgiven K (4.7) are isometric. By the definition of K as the completion of in the K K ∗Y K -inner product, we see that the span of the images im iK + im iK is dense in K := K ∗Y K by construction. By using the intertwining condition Y U = U Y together with the unitary property of U and U , we see that IK Y U k U h , ∗ U k U h Y IK K ⊕K U IK Y U 0 0 k h = , k h 0 U Y ∗ IK 0 U K ⊕K IK Y k h = , k h Y ∗ IK K ⊕K and hence the operator

U:

k k

U k → U k

(4.9)

extends to deﬁne a unitary operator on K = K ∗Y K . We have thus veriﬁed that (U ∗Y U , iK , iK ; K ∗Y K ) deﬁned as in (4.8) is a minimal AA-unitary coupling of (U , K ) and (U , K ), and statement (2) of the theorem follows. From the form of the inner product, we see that we recover Y as Y = i∗K iK . Suppose now that (U, iK , iK ; K) is any AA-unitary coupling of (U , K ) and (U , K ) and we set Y = i∗K iK : K → K . For k , ∈ K and k , ∈ K , we compute I Y k = k + i∗K iK k , K + i∗K iK k + k , K , ∗ k I Y = k , K + iK k , iK K + iK k , iK K + k , K = iK k + iK k , iK + iK K . We conclude that the map

τ:

k k

→ iK k + iK k

K of K ∗Y K onto im iK + im iK . Hence K τ extends to an isometric mapping of all of K ∗Y K onto the closure of

maps the dense subspace

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im iK + im iK and hence τ is unitary exactly when (U, iK , iK ; K) is a minimal AA-unitary coupling. Moreover, from the form of τ it is easily veriﬁed that τ (U ∗Y U ) = Uτ . By definition of τ , it transforms the embeddings of K and K into K ∗Y K to the embeddings of K and K into K. In this way we see that the above correspondence between contractive intertwiners and minimal AA-unitary couplings is bijective. bijective. Given an AA-unitary coupling (U, iK , iK ; K) of the unitary operators (U , K ) and (U , K ) and two subspaces G ⊂ K and G ⊂ K which are ∗-cyclic for U and U respectively, let iG : G → K and iG : G → K be the compositions of the inclusion of G into K with the inclusion of K into K and of the inclusion of G into K with the inclusion of K into K, respectively: iG := iG →K = iK →K iG →K , iG := iG →K = iK →K iG →K . Then we may view

SAA := U, iG iG ; K, G ⊕ G

as a unitary scattering system with characteristic measure i∗G EU (·) iG iG , σSAA (·) = ∗ iG

(4.10)

characteristic function i∗G [(I − ζU)−1 + (I − ζU ∗ )−1 − I] iG iG w SAA (ζ) = ∗ iG and characteristic moment-sequence {wSAA (n)}n∈Z with wSAA (n) =

i∗G i∗G

U ∗n iG iG .

We note that the (1,2)-entry in the nth characteristic moment wSAA ,n is closely associated with the intertwiner Y = i∗K iK associated with the AAunitary coupling (U, iK , iK ; K): [wSAA (n)]12 g , g = i∗G U ∗n iG g , g = Y U ∗n g , g = Y g , U ∗n g . Given any contractive intertwiner Y : K → K of (U , K ) and (U , K ), we refer to the bilateral sequence of operators {wY (n)}n∈Z given by wY (n) = [wSAA (n)]12 = i∗G →K U ∗n Y iG →K = i∗G →K Y U ∗n iG →K (4.11) as the symbol (associated with given subspaces G ⊂ K and G ⊂ K ) of the intertwiner Y . If G is ∗-cyclic for U and G is ∗-cyclic for U , then the subspaces K0 = span{U n g : g ∈ G and n ∈ Z}, K0 = span{U n g : g ∈ G and n ∈ Z}

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are equal to all of K and K respectively and the observation Y U n g , U m g = wY (m − n)g , g G shows that there is a one-to-one correspondence between symbols wY (with respect to the two ∗-cyclic subspaces G and G ) of Y and the associated contractive intertwiners Y . Moreover, we have the following characterization of which bilateral L(G , G )-valued sequences w = {w(n)}n∈Z arise as the symbol w = wY for some contractive intertwiner Y . Theorem 4.2. Suppose that {w(n)} is the symbol (4.11) for a contractive intertwiner Y of the unitary operators (U , K ) and (U , K ) associated with ∗-cyclic subspaces G ⊂ K and G ⊂ K . Then {w(n)}n∈Z is the sequence of trigonometric moments w(n) = t−n w(dt) T

(equal to the Fourier transassociated with an L(G , G )-valued measure w form of {w(n)}n∈Z ) such that σ w σ := (4.12) is a positive L(G , G ) − valued measure w ∗ σ where we have set σ = i∗G EU (·)iG ,

σ = i∗G EU (·)iG .

(4.13)

Conversely, the inverse Fourier transform w(n) := t−n w(dt) T

on T which in addition satisfies (4.12) is of any L(G , G )-valued measure w the symbol (associated with the subspaces G and G ) for a uniquely determined contractive intertwiner Y : K → K . Proof. The forward direction of the theorem is an immediate consequence of the results preceding the theorem. For the converse we use the Hellinger model. Suppose that w is a vector measure such that (4.12) holds, where σ and σ are the positive measures given by (4.13). Consider the Hellinger space Lσ (see [15]) with operator U σ being multiplication by the independent variable t. We deﬁne the embeddings iK : K → Lσ

and

iK : K → Lσ

as follows: ﬁrst we map K onto Lσ and K onto Lσ by Fourier representations k → i∗G E (dt)k ,

k → i∗G E (dt)k .

Then we embed Lσ and Lσ into Lσ by w σ , σ p → σ p → p p σ w ∗

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for arbitrary vector trigonometric polynomials p , p. As it was shown in [15] im iK + im iK is dense in Lσ . Thus we get a minimal AA-unitary coupling of U and U . The symbol of the contractive intertwiner associated with this AA-unitary coupling is just the trigonometric-moment sequence of the originally given measure w. Since the definition of symbol (4.11) can be rephrased as wY (n − m) g , g = Y U

∗n

g , U ∗m g ,

(4.14)

and the sets {U ∗n g } and {U ∗m g } (n running over Z, g over G and g over G ) have dense span in K and K , respectively, we see that the correspondence between intertwiners Y and symbols {w(n)}n∈Z is one-to-one. Moreover, the correspondence between symbols {w(n)}n∈Z and their Fourier transforms w is also one-to-one. Remark 4.3. The proof of Theorem 4.2 in fact shows that, given an AA-unitary coupling (U, iK , iK ; K) of two unitary operators (U , K ) and (U , K ) together with a choice of ∗-cyclic subspaces G ⊂ K and G ⊂ K , then the AA-unitary coupling (U, iK , iK ; K) is unitarily equivalent to the Hellingermodel AA-unitary coupling (U σ , iK →Lσ , iK →Lσ ; Lσ ) where σ = σSAA is given by (4.10).

5. Liftings and Unitary Extensions of an Isometry Defined by the Problem Data In this section we discuss, following [2,4,5,16,26–29,32–34], how solutions of the lifting problem can be identiﬁed with unitary extensions of a certain (partially deﬁned) isometry V which is constructed directly from the problem data. Introduce a Hilbert space H0 by K− H0 = clos (5.1) K+ with inner product given by I k− − = , X∗ k+ + H 0

X I

k− . , − k+ + K ⊕K −

+

Special subspaces of H0 are of interest: ∗ K− U K− D := clos D∗ := clos ⊂ H0 , ⊂ H0 . U K+ K+ Deﬁne an operator V : D → D∗ densely by ∗ ∗ U k− U k− 0 V = → . ∗ : 0 U U k+ k+

(5.2)

(5.3)

(5.4)

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∈ K− and k+ ∈ K+ . By the same computation as in (4.9), we see for k− that V extends to deﬁne an isometry from D onto D∗ . Notice also that V is completely determined by the problem data. Let us say that the operator U ∗ on K is a minimal unitary extension of V if U ∗ is unitary on K and there is an isometric embedding iH0 : H0 → K of H0 into K such that

iH0 V = U ∗ iH0 |D .

(5.5)

spann∈Z U n im iH0 = K.

(5.6)

and

In this situation note that we then also have iH0 V ∗ = UiH0 |D∗ .

(5.7)

are said Two such minimal unitary extensions (U ∗ , iH0 ; K) and (U ∗ , iH0 ; K)

to be unitarily equivalent if there is a unitary operator τ : K → K such that τ U ∗ = U ∗ τ,

τ iH0 = iH0 .

Then the connection between minimal unitary extensions of V and lifts of X is given by the following. Theorem 5.1. Suppose that we are given data for a Lifting Problem 1.1 as above. Assume that the subspaces K+ and K− are ∗-cyclic. Let V : D → D∗ be the isometry given by (5.4). Then there exists a canonical one-to-one correspondence between equivalence classes of minimal AA-unitary couplings (U, iK , iK ; K) of (U , K ) and (U , K ) such that the contractive intertwiner Y = i∗K iK lifts X on the one hand and equivalence classes of minimal unitary extensions (U ∗ , iH0 ; K) of V on the other. Specifically, if (U, iK , iK ; K) is a minimal AA-unitary coupling of (U , K ) and (U , K ) with associated contractive intertwiner Y = i∗K iK lifting X, then the mapping K− iH0 := iK iK K+ extends to an isometric embedding of H0 into K and (U ∗ , iH0 ; K) is a minimal unitary extension of V . Conversely, if (U ∗ , iH0 ; K) is a minimal unitary extension of V and if we define isometric embedding operators iK : K → K and iK : K → K via the wave operator construction ∗n k 0 U ∗n n iK k = s-limn→∞ U iH0 , iK k = s-limn→∞ U iH0 0 U n k defined initially only for ! k ∈ U ∗m K+ , m≥0

k ∈

! m≥0

U m K− ,

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and then extended uniquely to all of K and K respectively by continuity, then the collection (U, iK , iK ; K) is a minimal AA-unitary coupling of (U , K ) and (U , K ) with associated contractive intertwiner Y = i∗K iK lifting X. Proof. Suppose that (U, iK , iK ; K) is a minimal AA-unitary coupling of (U , K ) and (U , K ). Deﬁne the map k− + iK k+ . (5.8) iH0 : → iK k− k+ Since iK and iK are isometric then iH0 is isometric if and only if iK k− , iK k+

K = k− , k+

H0 := k− , Xk+

K .

This in turn means that the intertwiner Y = i∗K iK lifts X. Now, ∗ k− U k− + iK k+ = i = iK U ∗ k− iH0 V H0 U k+ k+ k ∗ ∗ = U (iK k− + iK U k+ ) = U iH0 − . U k+ This in turn means that (5.5) holds. Thus, U ∗ on K with embedding iH0 is a is ∗-cyclic for U on K , K− unitary extension of V . Since, by assumption, K+ is ∗-cyclic for U on K , and since the AA-unitary coupling (U, iK , iK ; K) is minimal, then spann∈Z U n im iH0 = K. Thus, (U ∗ , iH0 ; K) is a minimal unitary extension of V . Conversely, suppose that (U ∗ , iH0 ; K) is a minimal unitary extension of V . We now apply the construction of the wave operator from [34] (Section and 4), which simpliﬁes significantly in our situation due to the fact that K+ ∗n isometrically into H . For k ∈ U K and m ≥ n note K− are embedded 0 + 0 0 ∗m m that U iH0 is well-deﬁned (since U k ∈ K+ so ∈ H0 ) U m k U m k ∗ and independent of m [since U is an extension of V , see (5.7)]. Thus, the formula 0 ∗m (5.9) iK : k → lim U iH0 U m k m→∞ !∞ U ∗n K+ into K. By assumption, is a well-deﬁned isometry from n=0 !∞ ∗n U K+ is dense in K , and hence iK extends uniquely by continuity n=0 to an isometry (still denoted by iK ) from K into K. Similarly, the formula ∗m k U iK : k → lim U m iH0 (5.10) 0 m→∞

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gives rise to a well-deﬁned isometry from K into K. Definitions (5.9) and (5.10) imply that UiK = iK U

and

UiK = iK U .

We have thus arrived at an AA-unitary coupling (U, iK , iK ; K) of (U , K ) and (U , K ). To check the minimality of the AA-unitary coupling note that it follows from (5.9) and (5.10) that 0 n im iK = spann∈Z− U iH0 K+ and

K− . 0 0 K− is invariant for U ∗ , we Since iH0 is invariant for U and iH0 0 K+ conclude that K− n im iK + im iK ⊇ spann∈Z U iH0 . K+ im iK = spann∈Z+ U n iH0

Therefore, im iK + im iK ⊇ spann∈Z U n iH0 = K. The last equality is due to minimality of the extension. Thus, im iK + im iK = K and it follows that the AA-unitary coupling is minimal. Moreover, Y = i∗K iK lifts X since 0 k Y k+ , k−

K = iK k+ , iK k−

K = = Xk+ , k−

K− . , − k+ 0 H 0

The correspondences between AA-unitary couplings and unitary extensions deﬁned above are mutually inverse. Moreover, it is straightforward from the definitions of the equivalences that under these correspondences equivalent AA-unitary couplings go to equivalent unitary extensions and equivalent unitary extensions go to equivalent AA-unitary couplings. This completes the proof of Theorem 5.1.

6. Structure of Unitary Extensions In the previous section we obtained a correspondence between contractive intertwining lifts Y of X and minimal unitary extensions U ∗ of a isometry V on a Hilbert space H0 with domain D and codomain D∗ . In this section we indicate how one can parametrize all such minimal unitary extensions. We therefore suppose that we are given a Hilbert space H0 , two subspaces D and D∗ of H0 and an operator V which maps D isometrically onto D∗ : V : D → D∗ .

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In this situation we say that V is an isometry on H0 with domain D and codomain D∗ . We let Δ and Δ∗ be the respective orthogonal complements Δ := H0 D,

Δ∗ := H0 D∗ .

∗

Let U be a minimal unitary extension of V to a Hilbert space K, i.e., U is unitary on the Hilbert space K, K contains the space H0 as a subspace, the smallest subspace of K containing H0 and reducing for U is the whole space K and U ∗ when restricted to D ⊂ H0 ⊂ K agrees with V : U|D = V . We set H1 equal to K H0 and write K = H0 ⊕ H1 . We associate two unitary colligations U1 and U0 to the extension U ∗ as follows. Since U ∗ |D = V maps D onto D∗ and since U ∗ is unitary, necessarily U ∗ must map K D = Δ ⊕ H1 onto K D∗ = Δ∗ ⊕ H1 . To deﬁne the unitary colligation U1 , we introduce

of Δ and a second copy Δ

∗ of Δ∗ together with unitary a second copy Δ identiﬁcation maps

→ Δ ⊂ H0 ⊂ K, i : Δ

∗ → Δ∗ ⊂ H0 ⊂ K. (6.1) i : Δ Δ

Δ∗

We then deﬁne the colligation A1 U1 := C1 by

U1 =

B1 D1

:

i∗H1 i∗Δ

H1

Δ

→

H1

∗ Δ

(6.2)

U ∗ iH1 iΔ

(6.3)

∗

where iH1 : H1 → K = H0 ⊕ H1 is the natural inclusion map. We deﬁne a second colligation H0 H0 A0 B0 → : U0 =

∗

C0 D0 Δ Δ as follows. Note that the space H0 has two orthogonal decompositions H0 = D ⊕ Δ = D∗ ⊕ Δ∗ . If we use the ﬁrst orthogonal decomposition of H0 on the domain side and the second orthogonal decomposition of H0 on the range side, then we may

∗ → H0 ⊕ Δ

via the 3 × 3-block matrix deﬁne an operator U0 : H0 ⊕ Δ ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ D∗ D V 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ iΔ U0 = ⎣ 0 0 (6.4)

∗ ⎦ : ⎣ Δ ⎦ → ⎣ Δ∗ ⎦,

0 i∗ 0 Δ Δ∗ Δ

or, in colligation form,

A0 U0 = C0

B0 0

:

H0

∗ Δ

→

H0

Δ

(6.5)

where A0 |D = V,

A0 |Δ = 0,

C0 |D = 0,

B0 = iΔ

∗ with im B0 = Δ∗ ⊂ H0 .

C0 |Δ = i∗Δ

, (6.6)

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We note that the colligation U0 is deﬁned by the problem data (i.e., the isometry V with given domain D and codomain D∗ in the space H0 ) and is independent of the choice of unitary extension U ∗ . As one sweeps all possible unitary extensions U ∗ of V , the associated colligation U1 can be an arbitrary

and output space colligation of the form (6.2), i.e., one with input space Δ

∗ . Moreover, from the fact the colligation matrix U0 = A0 B0 has a zero Δ C0 0 for its (2, 2)-entry, we see from Theorem 2.1 that the feedback connection F (U0 , U1 ) is well-deﬁned for any colligation (in particular, for any unitary colligation) of the form (6.2). Also, from the very definitions, we see that if U1 is constructed from the unitary extension U ∗ as indicated in (6.3), then we recover U ∗ from U0 and U1 as the feedback connection U ∗ = F (U0 , U1 ) given by (2.2). The following result gives the converse. Theorem 6.1. The operator U ∗ on K is a unitary extension of V to a Hilbert space K if and only if, upon decomposing K as K = H0 ⊕ H1 , U ∗ can be written in the form U ∗ = F (U0 , U1 ) where U0 is the universal unitary colligation determined completely by the problem data as in (6.6) and U1 is a free-parameter unitary colligation of the form (6.2). Moreover, U ∗ is a minimal unitary extension of V , i.e., the smallest reducing subspace for U ∗ containing H0 is the whole space K := H0 ⊕ H1 , if and only if U1 is a simple unitary colligation, i.e., the smallest reducing subspace for A1 containing im B1 + im C1∗ is the whole space H1 . Proof. We already showed that every unitary extension U ∗ of V has the form U ∗ = F (U0 , U1 ) where U ∗ determines U1 according to (6.3). Conversely we now show that every lower feedback connection F (U0 , U1 ) (with arbitrary unitary colligation U1 of the form (6.2) produces a unitary extension U ∗ of V . From the formula (2.4) for the lower feedback connection applied to the case where D0 = 0, we see that A0 + B0 D1 C0 B0 C1 h0 h0 F (U0 , U1 ) = . (6.7) h1 h1 B1 C0 A1 Specializing to the case where h0 = d ∈ D ⊂ H0 and h1 = 0 and using the formulas (6.6) for A0 , B0 , C0 , we see that d A0 d Vd = F (U0 , U1 ) = 0 0 0 and it follows that F (U0 , U1 ) is an extension of V . Moreover, by plugging in the explicit formulas (6.6) for A0 , B0 , C0 into (6.7), it is straightforward to verify that we recover U1 from U ∗ := F (U0 , U1 ) via the formula (6.3) and that U ∗ is unitary exactly when U1 is unitary. It remains to check: U ∗ is a minimal extension of V if and only if U1 is a simple unitary colligation. Consider the minimal reducing subspace for U ∗ that contains H0 , then its orthogonal complement (which is a subspace

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of H1 ) also reduces U ∗ . From the definitions one sees that the latter is the zero subspace exactly when U1 is simple. Since unitary extensions U ∗ of V are given via the lower feedback connection F (U0 , U1 ), we may use the results of Theorems 2.3 and 2.4 to compute positive and negative powers of U ∗ . To simplify notation, we let S0+ S2+ H0 (Z+ ) H0 + S = , : → Δ Δ S1+ S +

∗ (Z+ )

(Z+ ) − H0 (Z− ) H0 S0 S1− − S = : (6.8) → S2− S − Δ Δ

∗ (Z− )

(Z− ) be the forward and backward augmented input–output operators for the universal colligation U0 and we let + H1 (Z+ ) Ω0 Ω+ H1 2 + Ω = : , → Δ Δ Ω+ Ω+

∗ (Z+ )

(Z+ ) 1 (6.9) − H1 Ω0 Ω− H1 (Z− ) 1 − Ω = : → Δ Δ Ω− Ω−

∗ (Z− )

(Z− ) 2 be the forward and backward augmented input–output operators for the freeparameter unitary colligation U1 . From the ﬁrst rows in the formulas (2.22) and (2.28), we read oﬀ that ΛH0 ,− (U ) H0 H0 (Z− ) ΛH0 (U ) = : → H1 ΛH0 ,+ (U ) H0 (Z+ ) is given by

ΛH0 (U ) =

S0− + S1− (I − Ω− S − )−1 Ω− S2− S0+ + S2+ (I − Ω+ S + )−1 Ω+ S1+

S1− (I − Ω− S − )−1 Ω− 2 . (6.10) S2+ (I − Ω+ S + )−1 Ω+ 1

From the second rows in the formulas (2.22) and (2.28) we read oﬀ that ΛH1 ,− (U ) H0 (Z ) : ΛH1 (U ) = → H1 − H1 ΛH1 ,+ (U ) H1 (Z+ ) is given by

− − −1 − Ω− S2 1 (I − S Ω ) ΛH1 (U ) = + Ω2 (I − S + Ω+ )−1 S1+

− − − −1 − − Ω− S Ω2 0 + Ω1 (I − S Ω ) + + + −1 + + . (6.11) Ω+ + Ω (I − S Ω ) S Ω1 0 2

7. Parametrization of Symbols of Intertwiners Assume that we are given the data set (X,

(U , K ),

(U , K ),

K+ ⊂ K ,

K− ⊂ K )

(7.1)

as in the Lifting Problem 1.1. If we are given ∗-cyclic subspaces G and G for U and U respectively, then the sets {U n g : n ∈ Z, g ∈ G },

{U n g : n ∈ Z, g ∈ G }

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have dense span in K and K respectively. If Y ∈ L(K , K ) is any operator satisfying the intertwining condition Y U = U Y , then the computation Y U n g , U m g K = U (n−m) Y g , g K = i∗G U (n−m) Y iG , g , g G shows that Y is uniquely determined by its symbol, i.e., the sequence wY = {Yn }n∈Z of operators Yn : G → G given by Yn = i∗G U ∗n Y iG = i∗G Y U ∗n iG . Therefore, in principle, to describe all contractive intertwining lifts Y of a → K− , it sufﬁces to describe all the symbols wY of contractive given X : K+ intertwining lifts Y . Such a description is given in the next result. Theorem 7.1. Suppose given data set (7.1) for a Lifting Probthat we are H0 H0 be the universal unitary colligation conlem 1.1. Let U0 :

→

Δ∗ Δ structed from the problem data as in (6.4) or (6.5) and (6.6) with associated augmented input–output maps S+ and S− as in (6.8). For U1 equal to a free-parameter unitary colligation of the form (6.2), let Ω+ and Ω− be the associated augmented input–output maps as in (6.9). Finally let G and G be a fixed pair of U − ∗-cyclic and U − ∗-cyclic subspaces of K and K and assume that , G ⊂ K+

G ⊂ K− .

Let iG : G → H0 be the inclusion map of G into H0 obtained as the inclusion followed by the inclusion of K+ into H0 , and, similarly, let iG of G in K+ be the inclusion of G in H0 obtained as the inclusion of G in K− followed ∗ ∗ by the inclusion of K− in H0 . Let IG = diagn∈Z {iG } be the coordinatewise projection of H0 (Z) onto G (Z). Then the L(G , G )-valued bilateral sequence w = {w(n)}n∈Z is the symbol w = wY (with respect to G and G ) for a contractive intertwining lift Y of X if and only if there exists a freeA1 B1

→ H1 ⊕ Δ

∗ so that w parameter unitary colligation U1 = : H1 ⊕ Δ C1 D1 (as an infinite column vector) has the form ∗ − IG S0 iG wY = . (7.2) IG∗ [S0+ + S2+ (I − Ω+ S + )−1 Ω+ S1+ ]iG Remark 7.2. There are various other formulations of the formula (7.2) for the parametrization of lifting symbols. If we deﬁne + ∗ s+ 0 = IG S0 iG ,

+ ∗ s+ 2 = IG S2 ,

+ s+ 1 = S1 iG ,

s+ = S + ,

(7.3)

then the formula (7.2) assumes the form − s (7.4) wY = 0+ + + −1 + + s0 + s+ Ω s1 2 (I − Ω s ) + + s0 s2 together with s− where the coefﬁcient matrix + 0 is completely deters+ 1 s mined from the problem data while Ω+ is the input–output map for the free-parameter unitary colligation U1 .

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If we consider G (Z+ ) as embedded in G (Z) in the natural way, we may rewrite in turn the formula (7.4) in the still more compact form wY = s0 + s2 (I − ωs)−1 ωs1 where we deﬁne

(s0 g )(m) =

(s− 0 g )(m) (s+ 0 g )(m)

(7.5)

for m < 0 for m ≥ 0,

s2 = ιs+ 2 where ι : G (Z+ ) → G (Z) is the natural inclusion, s = s+ ,

ω = Ω+ ,

s1 = s+ 1.

(7.6)

Proof. Theorem 5.1 gives an identiﬁcation between contractive intertwining lifts U ∗ and unitary extensions of the isometry V : D → D∗ on H0 constructed from the Lifting Problem data while Theorem 6.1 in turn gives a Redheffertype parametrization of all such unitary extensions. Moreover formula (6.10) tells us how to compute the powers of U ∗ = F (U0 , U1 ) followed by the projection to the subspace H0 . By definition the symbol wY is given by wY (n) = i∗G U ∗n iG . The parametrization result (7.2) now follows by plugging into (6.10) once we verify: i∗G S1− (I − Ω− S − )−1 S2− iG = 0.

(7.7)

We assert that in fact S2− iG = 0.

(7.8)

Indeed, by definition S2− h0 = {δ ∗ (n)}n∈Z− means that δ ∗ (n) is generated by the recursion h0 (n) = A∗0 h0 (n + 1), h0 (0) = h0 , δ ∗ (n) = B0∗ h0 (n) for n = −1, −2, . . . . If we set m = −n, this means simply that δ ∗ (−m) = B0∗ A∗m 0 h0 . K For the case where h0 = iG g ∈ iK+ + ⊂ R, we then have U g ∈ D∗ A∗0 h0 = V ∗ iG g = iK+ k K and, inductively, given that A∗m + ∈ iK+ + ⊂ D∗ , we have 0 h0 = iK+ ∗ k U k K A∗m+1 h0 = A∗0 iK+ k+ = iK+ + = V iK+ + ∈ iK+ + ⊂ D∗ . 0

As D∗ is orthogonal to the ﬁnal space Δ∗ for the isometry iΔ

∗ , it follows that, for m = 1, 2, . . . , ∗ ∗m B0∗ A∗m 0 iG g = iΔ

A0 iG g = 0 for m = 1, 2, . . . ∗

from which (7.8) and (7.7) follow.

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Remark 7.3. We note that the value of the symbol wY (n) is independent of the choice of lift Y for n ≤ 0. Indeed, for n ≤ 0, g ∈ G and g ∈ G (where and G ⊂ K− ), we have as always we are assuming that G ⊂ K+ n = Y g , U g K− wY (n)g , g G = U ∗n Y g , g K− n = Xg , U = Y g , U n g K− g K− since U n g ∈ K− for n ≤ 0 whenever g ∈ G ⊂ K− . Let us consider the special case where we take G := K+

and

G = E := K− U ∗ K− .

Then E is wandering for U and we may represent K as the direct-sum decomposition K = K− ⊕

∞ "

U n (U E ).

n=0

Then the Fourier representation operator Φ : k → {i∗E U ∗n+1 k }n∈Z+ is a coisometry mapping K onto 2E (Z+ ) with initial space equal to K #∞ n K− = n=0 U (U E ). If Y is any lift, then Y is uniquely determined by its restriction Y |K+ to K+ by the wave-operator construction; thus, to solve the Lifting Problem : K it sufﬁces to describe all Y |K+ + → K rather than all lifts Y : K → K . Moreover, if we use the Fourier representation operator Φ to identify K K− K− with 2E (Z+ ), then we have an identiﬁcation of K with . 2E (Z+ ) has a matrix Then, with this identiﬁcation in place, the restricted lift Y |K+ representation of the form K− X = Y |K+ : K+ → 2 . Y+ E (Z+ ) With this representation we lose no information concerning the lift Y despite may not be ∗-cyclic for U . the fact that in general G := E ⊂ K− If we use the parametrization from (7.2), the operator Y+ in turn has an inﬁnite column-matrix representation given by ⎡ ⎤ wY (1) ⎢ wY (2) ⎥ ⎢ ⎥ ⎢ ⎥ .. ∗ ∗ ⎢ ⎥ = J+ . IE (S0+ + S2+ (I − Ω+ S + )−1 Ω+ S1+ )iK+ Y+ = ⎢ ⎥ ⎢ wY (n + 1) ⎥ ⎣ ⎦ .. . where J+ is the shift operator on 2E (Z+ ) and where Ω+ is the input–output map associated with the free-parameter unitary colligation U1 . Finally, if we

46

J. A. Ball and A. Kheifets

IEOT

apply the Z-transform {e (n)}n∈Z+ →

∞

e (n)ζ n

n=0 to transform to the Hardy space HE2 , then the operator Y+ : K+ → → 2E (Z+ ) is given by multiplication by the HE2 induced by Y+ : K+ , E )-valued function L(K+

2E (Z+ )

−1 + Y+ (ζ) = ζ −1 [ s+ + s2 (ζ)(I − ω(ζ) s+ (ζ))−1 ω(ζ) s+ 0 (ζ) − s 0 (0)] + ζ 1 (ζ)]iK+

(7.9) where ∗ + s+ 0 (ζ) = iE S0 (ζ)iG ,

and where S+ (ζ) 0 S1+ (ζ)

s2 (ζ) = i∗E S2+ (ζ),

S2+ (ζ) S+ (ζ)

s1 (ζ) = S1+ (ζ)iG ,

(I − ζA0 )−1 = C0 (I − ζA0 )−1

s(ζ) = S+ (ζ)

ζ(I − ζA0 )−1 B0 ζC0 (I − ζA)−1 B0

is the frequency-domain version of the augmented input–output map associated with the unitary colligation U0 (and hence is completely determined from the problem data) and where ω(ζ) = D1 + ζC1 (I − ζA1 )−1 B1 is the characteristic function of the free-parameter unitary colligation U1 . Let us use the notation DX for the defect operator DX := (I − X ∗ X)1/2 of X. Further analysis shows that Y+ (ζ) has a factorization Y+ (ζ) = Y0+ (ζ)DX where the operator Γ : DX k+ → Y0+ (ζ)DX k+ deﬁnes a contraction operator from DX := RanDX (viewed as a space of constant functions) into HE2 . Then we have the following form for the parametrization of the lifts: Xk+ Y : k+ → , (7.10) Y0+ (ζ)DX k+ where Y+ (ζ) = Y0+ (ζ)DX is given by (7.9). In Sections 6 and 7 of Chapter XIV in [18] or Theorem VI.5.1 in [19], there are derived formulas for a Redheffer coefﬁcient matrix Ψ11 (ζ) Ψ12 (ζ) (7.11) Ψ(ζ) = Ψ21 (ζ) Ψ22 (ζ) so that the function Y0+ (ζ) is expressed by the formula Y0+ (ζ) = Ψ11 (ζ) + Ψ12 (ζ)(I − ω(ζ)Ψ22 (ζ))−1 ω(ζ)Ψ21 (ζ).

(7.12)

S. ter Horst (private communication) has veriﬁed that, after some changes of variable, the formula (7.12) agrees with (7.10). In this formulation of the Lifting Problem, the intertwining property (1.3) is encoded directly in terms of Y0+ (ζ) in the form Y0+ (ζ)DX U+ = i∗E X + ζY0+ (ζ)DX .

(7.13)

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47

Here the range of i∗E is the space E and E is identiﬁed as the subspace of constant functions in HE2 . Associated with the data of a Lifting Problem is an underlying isometry ρ : F → E ⊕ DX where F = RanDX U+

and deﬁned densely by ρDX U+ k+

ρ1 DX U k+ = ρ2 DX U k+

i∗E Xk+ := . DX k+

(7.14)

Then the form (7.13) of the intertwining condition can be expressed directly in terms of the isometry ρ in the form ρ1 + ζ · Y0+ (ζ)ρ2 = Y0+ (ζ)|F .

(7.15)

It is this formulation which has been extended to the context of the Relaxed Commutant Lifting problem in [22,23] and in addition to a Redheffer parametrization for the set of all solutions in [24,57]. For the relaxed problem, the underlying isometry ρ given by (7.14) is in general only a contraction rather than an isometry. The Redheffer coefﬁcient matrix (7.11) is a coisometry 2 2 to HE2 ⊕ HD (Ψ11 and Ψ21 are multiplication operators). from DX ⊕ HD ρ∗ T

8. The Universal Extension Theorem 7.1 obtained a parametrization of all symbols of solutions of the lifting problem (and therefore also of all lifts under the assumption that G and G are ∗-cyclic) via a Redheffer linear-fractional map acting on a freeparameter input–output map, or equivalently, a free-parameter Schur-class

and Δ

∗ . As has been observed function, acting between coefﬁcient spaces Δ before in a variety of contexts (see e.g. [18,19]), a special role is played by the lift associated with the free-parameter taken to be equal to 0 (the central lift). In this section we develop the special properties of the universal lift from the point of view of the ideas developed here. The ﬁrst step is to construct the simple unitary colligation having characteristic function equal to the zero function. Theorem 8.1. The essentially unique simple unitary colligation H10 H10 A10 B10 → : U10 =

∗ C10 D10 Δ Δ

(8.1)

having characteristic function equal to the zero function

→Δ

∗ ω10 (λ) = D0 + λC0 (I − λA0 )−1 B0 ≡ 0 : Δ is constructed as follows: take 2 ∗ Δ J− 0

∗ (Z− ) , H10 = 2 , A10 = ∗ Δ 0 J+

(Z+ ) (−1) $ % i

(0)∗ B10 = Δ , D10 = 0, , C10 = 0 iΔ

∗ 0

(8.2)

48

J. A. Ball and A. Kheifets

IEOT

where in general J− : (. . . , x(−2), x(−1)) → (. . . , x(−3), x(−2)) is the compressed forward shift on 2X (Z− ), J+ : (x(0), x(1), x(2), . . . ) → (0, x(0), x(1), . . . ) is the forward shift on 2X (Z+ ) (with coefficient space X clear from the con(−1)

into the subtext), where iΔ : x → (. . . , 0, x) is the natural injection of Δ

(0)

space of elements of 2Δ

(Z− ) supported on the singleton {−1}, and iΔ

∗ : x →

(x, 0, 0, . . . ) is the natural injection of Δ∗ into the subspace of elements of 2Δ

(Z+ ) supported on the singleton {0}. ∗

Proof. This is a straightforward veriﬁcation which we leave to the reader. We have seen in Theorem 6.1 that the operator U ∗ on K extends the isometry V on H0 having domain D ⊂ H0 and range D∗ ⊂ H0 if and only if U ∗ has a representation of the form U ∗ = F (U0 , U1 ), where U0 is the universalcolligation given by (6.4) or equivalently by (6.5) H1 H1 →

and (6.6) and where U1 :

is a free-parameter unitary colligaΔ Δ∗ tion, and, moreover, U ∗ is a minimal unitary extension of V if and only if U1 is a simple unitary colligation. We now consider the particular case where we take U1 equal to the simple unitary colligation with zero characteristic function U10 given as in Theorem 8.1. We refer to the resulting minimal unitary extension U0∗ := F (U0 , U10 ) as the central unitary extension. An application of the general formula (6.7) then gives A0 + B0 D10 C0 B0 C10 U0∗ = F (U0 , U10 ) = B10 C0 A10 ⎡ ⎤ ⎤ ⎡ (0)∗ H0 A0 0 iΔ

∗ iΔ

∗ ⎢ ⎥ ⎢ 2 (Z ) ⎥ ∗ = ⎣ i(−1) i∗ J− (8.3) − ⎦. ⎦ on K0 := ⎣ Δ

0

Δ Δ 2 ∗ (Z ) +

0 0 J Δ +

∗

with adjoint given by ⎡

A∗0 ⎢0 U0 = ⎣ (0) ∗ iΔ

iΔ ∗

(−1)∗

iΔ

iΔ

J− ∗

0

⎤ 0 0 ⎥ ⎦. J+

(8.4)

To analyze the ﬁner structure of the universal extension (U0 , K0 ) given by (8.4), let us deﬁne embedding operators

iΔ,0

: Δ → K0 ,

iΔ

∗ ,0 : Δ∗ → K0 ,

,0 : K iK− − → K0 ,

,0 : K iK+ + → K0

Vol. 70 (2011) by

⎡

0

Inverse Lifting Problem

⎤

⎤

⎡

⎢ (−1) ⎥ iΔ,0 ⎦,

=⎣ i

Δ 0

iΔ

∗ ,0

0 ⎢0 =⎣

(0) iΔ

∗

49

⎤ ⎤ ⎡ →H →H iK− iK+ 0 0 ⎥ ⎥ ⎢ ⎢ ,0 = ⎣ 0 ,0 = ⎣ 0 iK− ⎦, iK+ ⎦. 0 0 ⎡

⎥ ⎦,

Then the collection ,0 iK ,0 ; iΔ iΔ,0 S0 = (U0 ,

∗ ,0 iK− +

K0 ,

⊕Δ

∗ ⊕ K− Δ ⊕ K+ ) (8.5)

is a scattering system in the sense of Sect. 3 [see (3.1)]. Moreover, the oper ,0 , iK ,0 have unique respective isometric extensions ators iΔ,0

, iΔ

∗ ,0 , iK− + i : 2 (Z) → K0 , i : 2 (Z) → K0 , iK ,0 : K → K0 , iK ,0 : K → K0 Δ,0 Δ∗ ,0 Δ Δ ∗

which satisfy the respective intertwining conditions i J = U0i , Δ,0 Δ,0

iK ,0 U = U0 iK ,0 ,

i J = U0i , Δ∗ ,0 Δ∗ ,0

iK ,0 U = U0 iK ,0

where here we set J equal to the bilateral shift operator on any space of the form 2X (Z) (the coefﬁcient space X determined by the context). Then the collection SAA,0 = (U0 ,

i , Δ,0

i , Δ∗ ,0

iK ,0 ,

iK ,0 ;

K0 )

(8.6)

can be viewed as a fourfold AA-unitary coupling of the four unitary operators (J, 2Δ

(Z)),

(J, 2Δ

(Z)),

(U , K ),

∗

(U , K )

which has certain additional properties. The next theorem identiﬁes some of these additional properties. Theorem 8.2. The scattering system (8.5) and its extension to the fourfold AA-unitary coupling (8.6) associated with the universal extension (8.4) U0 for a Lifting Problem have the following properties: 1.

The density conditions im iK ,0 + im iK ,0 is dense in K0 ,

(8.7)

2 ,0 } = K0 span{im iK ,0 , im iK− iΔ

∗ ,0 (Δ

(Z+ )), ∗

,0 , im iK ,0 } = span{im iK+

2 K0 iΔ,0

(Δ

(Z− )),

(8.8)

and ,0 } ∩ span{im iK ,0 , im iK ,0 } span{im iK ,0 , im iK− + ,0 , im iK ,0 } = span{im iK+ −

2.

(8.9)

hold. The orthogonality conditions i (2 (Z+ )) ⊥ im iK ,0 , Δ∗ ,0 Δ − ∗

i (2 (Z− )) ⊥ im iK ,0 , Δ,0 Δ +

i (2 (Z− )) ⊥ i (2 (Z+ )) Δ,0 Δ Δ∗ ,0 Δ ∗

and

(8.10)

50

J. A. Ball and A. Kheifets i (2 (Z− )) ⊥ im iK ,0 , Δ,0 Δ −

3.

IEOT

i (2 (Z+ )) ⊥ im iK ,0 Δ∗ ,0 Δ +

(8.11)

∗

hold. The subspace identities U0∗ im iΔ

∗ ,0 = iH0 ,0 Δ∗ ,

U0 im iΔ,0 = iH0 ,0 Δ

(8.12)

hold Proof. For simplicity let us use the bold notation H0 = im iH0 ,0 to indicate the subspace H0 when viewed as a subspace of K0 . Property (8.7) is a consequence of the fact that the universal freeparameter unitary colligation U10 given by (8.1) and (8.2) is simple and hence (by Theorem 6.1) the unitary operator U ∗ = F (U0 , U10 ) is a minimal unitary extension of V . To check conditions (8.8), we use the orthogonal decomposition of K0 [see (8.3)] 2 (2 (Z+ )). K0 = iΔ,0

(Δ

(Z− )) ⊕ im iH0 ,0 ⊕ iΔ Δ ∗,0

(8.13)

∗

From the formula for U0∗ in (8.3), it is easily checked that the smallest U0 -invariant subspace H0+ containing H0 is 2 (2 (Z− )). H0+ = H0 ⊕ iΔ

∗ ,0 (Δ

(Z+ )) = K0 im iΔ,0 Δ ∗

On the other hand, by the construction this smallest U0 -invariant sub ,0 }. Combining space can also be identiﬁed as H0+ = span{im iK ,0 , im iK+ these observations gives the ﬁrst part of (8.8). The second part follows similarly by identifying the smallest U0∗ -invariant subspace of H0− con2 taining H0 as H0− = K0 iΔ

∗ ,0 (Δ

∗ (Z+ )) on the one hand and also as ,0 , im iK ,0 } on the other. To prove (8.9), note from the H0− = span{im iK− above discussion that 2 H0+ = H0 ⊕ iΔ

∗ ,0 (Δ

(Z+ )), ∗

2 H0− = H0 ⊕ iΔ,0

(Δ

(Z− )). 2 (2 (Z− )) are orthogonal to each other, it follows As iΔ

∗ ,0 (Δ

∗ (Z+ )) and iΔ,0

Δ that H0+ ∩ H0− = H0 , i.e., (8.9) holds. The orthogonality conditions (8.10) and (8.11) are clear from (8.13). In fact, the orthogonality conditions (8.11) hold in the stronger form

imiΔ

∗ ,0 ⊥ im iK ,0 ,

imiΔ,0 ⊥ im iK ,0 .

(8.14)

,0 U = U0 iK ,0 and is invariant under U and iK+ To see this, note that K+ + hence im iK+ ,0 is invariant under U0 and the ﬁrst of the orthogonality conditions (8.11) implies that U0∗ni (2 (Z+ )) is orthogonal to im iK ,0 . As the Δ∗ ,0

subspace

∗ Δ

+

∗n 2 ∞ ∗n 2 ∪∞

∗ ,0 (Δ

(J Δ n=0 U0 iΔ

(Z+ )) = ∪n=0 iΔ,0

(Z+ )) ∗

∗

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Inverse Lifting Problem

51

,0 . As is orthogonal to im iK+ is dense in imiΔ

∗ ,0 , we conclude that im iΔ ∗ ,0 im iΔ

∗ ,0 is reducing for U0 , we conclude that in fact im iΔ

∗ ,0 is orthogonal ,0 , i.e., to im iK ,0 , to the smallest U0 -reducing subspace containing im iK+ and the ﬁrst of conditions (8.14) follows. The second orthogonality condi is invariant tion in (8.14) follows similarly from the observation that im iK− under U ∗ . The subspace identities (8.12) can be read oﬀ from the definitions, in particular, the definition of U0 (6.4). Remark 8.3. One can easily verify that the orthogonality conditions (8.10) and (8.11) can be expressed in more succinct fashion as ∗n i∗K− ,0 U0 i

Δ∗ ,0 = 0

∗n i∗Δ,0 ,0

U0 iK+ ∗ ∗n iΔ,0

∗ ,0

U0 iΔ

for n ≤ 0,

(8.15)

=0

for n < 0,

(8.16)

=0

for n < 0,

(8.17)

for n ≥ 0,

(8.18)

and ∗n i∗Δ,0 ,0 = 0

U0 iK− ∗n i∗Δ ,0

∗ ,0 U0 iK+

=0

for n < 0.

(8.19)

Since actually the stronger relations (8.14) hold, the conditions (8.18) and (8.19) actually hold for all n ∈ Z: ∗n i∗Δ,0 ,0 = 0

U0 iK− ∗n i∗Δ ,0

∗ ,0 U0 iK+

=0

for all n ∈ Z, for all n ∈ Z,

(8.20) (8.21)

respectively. It is of interest that conversely the properties (8.7), (8.8), (8.10)–(8.12) can be used to characterize the universal extension U0 associated with a Lifting Problem. We present two versions of such a result. Theorem 8.4. Suppose that (U , K ) and (U , K ) are unitary operators and ⊂ K and K+ ⊂ K are ∗-cyclic subspaces with K− and K+ invariant that K− ∗

under U and U respectively. Suppose also that Δ and Δ∗ are two coefficient Hilbert spaces and that we are given a scattering system of the form

⊕Δ

∗ ⊕ K− ,0 iK ,0 ; iΔ iΔ,0 K0 , Δ ⊕ K+ ) S0 = (U0 ,

∗ ,0 iK− + ,0 and iK ,0 are isometric embedding operators of the where iΔ,0

, iΔ

∗ ,0 , iK− +

∗ , K− and K+ into K0 . We assume also that there is a respective spaces Δ, Δ fourfold AA-unitary coupling & ' i , iK ,0 , iK ,0 ; K0 SAA,0 = U0 , iΔ,0

, Δ∗ ,0

of the four unitary operators (J, 2Δ

(Z)),

(J, 2Δ

(Z)), ∗

(U , K ),

(U , K )

52

J. A. Ball and A. Kheifets

IEOT

which extends S0 in the sense that (−1)

iΔ,0 = iΔ,0

◦ iΔ

,

(0)

◦ i , iΔ

∗ ,0 = iΔ

∗ ,0 Δ ∗

= iK ,0 , iK ,0 |K− −

= iK . iK ,0 |K+ +

(8.22) Define subspaces H0 , D, D∗ , Δ and Δ∗ of K0 according to & ' ,0 ,0 , ,0 + iK ,0 (U K ) iK+ D = clos im iK− , H0 = clos im iK− + + (8.23) ' & ∗ ,0 (U ,0 , Δ = H0 D, Δ∗ = H0 D∗ K− ) + im iK+ D∗ = clos iK− and let iH0 ,0 : H0 → K0 be the isometric inclusion map. Suppose also that either one of the the following additional conditions holds: 1. 2.

Conditions (8.7), (8.10)–(8.12) all hold, or Conditions (8.8)–(8.11), and the following weaker form of (8.12) im iΔ

∗ ,0 ⊥ im UiΔ,0

(8.24)

hold. Then S0 and SAA,0 are equal to the scattering system and the fourfold AAunitary coupling associated with the universal extension U0∗ from some Lifting Problem. Remark 8.5. From the ﬁrst version of Theorem 8.4, we see that if (8.7), (8.10)–(8.12) hold, then also (8.14), (8.8) and (8.9) hold. From the second version, we see that if (8.8)–(8.11) and (8.24) hold, then also (8.14), (8.7) and (8.12) hold. In the proofs below it is convenient to use the bold notation H0 = im iH0 , D = iH0 ,0 (D), D ∗ = iH0 ,0 (D∗ ), Δ = iH0 (Δ), Δ∗ = iH0 ,0 (Δ∗ ) for the subspaces introduced in (8.23) when viewed as subspaces of K0 rather than of H0 , as well as the additional simpliﬁcations 2 G − = iΔ,0

(Δ

(Z− )) ⊂ K0 ,

2 G ∗+ = iΔ

∗ ,0 (Δ

(Z+ )) ⊂ K0 . ∗

Proof of version 1: The combined eﬀect of the hypotheses (8.10) and (8.11) is that the three subspaces H0 , G − , G ∗+ are pairwise orthogonal. Therefore the span of these subspaces K00 has an orthogonal decomposition K00 = H0 ⊕ G − ⊕ G ∗+ .

(8.25)

From the definitions we see that H0 has a twofold orthogonal decomposition as H0 = D ⊕ Δ = D ∗ ⊕ Δ ∗ .

(8.26)

Due to the intertwinings U0 iK ,0 = iK ,0 U ,

U0 iK = iK U

one can see that U0∗ (D) = D ∗

(8.27)

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Inverse Lifting Problem

53

and in fact we have the alternate characterizations of D and D ∗ : D = {h ∈ H0 : U0∗ h ∈ H0 },

D ∗ = {h∗ ∈ H0 : U0 h8 ∈ H0 }.

From the hypothesis (8.12) we know that & ' U0∗ im iΔ U0∗ Δ = im iΔ,0

∗ ,0 = Δ∗ ,

(8.28)

(8.29)

J ∗ and U0∗i = i J ∗ , we know and, from the intertwinings U0∗iΔ

∗ ,0 = iΔ Δ,0 Δ,0 ∗ ,0 that G − = U0∗ G − ⊕ imiΔ,0

,

G ∗+ = imiΔ

∗ ,0 ⊕ U0 G ∗+ .

(8.30)

From the orthogonal decompositions (8.25) and (8.26) for K0 and H0 combined with (8.27), (8.29) and (8.30) we see that K00 is reducing for U0 . From hypothesis (8.7) we conclude that in fact K00 = K0 and the decomposition (8.25) applies with K0 in place of K00 , i.e., we have K0 = G − ⊕ H0 ⊕ G ∗+ .

(8.31)

From (8.27) we see that we may deﬁne an isometry V on H0 with domain D and range D∗ by V d = d∗ if U0∗ iH0 ,0 d = iH0 ,0 d∗

for d ∈ D, d∗ ∈ D∗ .

It is now straightforward to check that necessarily U0∗ is a universal extension of the isometry V . Furthermore, one can check that V is the isometry constructed from the Lifting Problem data X = i∗K− ,0 iK ,0 , +

(U , K ),

(U , K ),

K+ ⊂ K ,

K− ⊂ K .

This completes the proof of the ﬁrst version of Theorem 8.4.

Proof of version 2: Using hypotheses (8.10) and (8.11) as in the proof of ver 0 by sion 1, we form the subspace K00 as in (8.25). If we deﬁne H

0 := K0 [G − ⊕ G ∗+ ], H then by definition we have

0 ⊕ G − ⊕ G ∗+ . K0 = H

(8.32)

For convenience let us introduce the temporary notation ,0 }, H0− = span{im iK ,0 , im iK−

(8.33)

,0 , im iK ,0 }. H0+ = span{im iK+

(8.34)

Note that H0− is the smallest U0∗ -invariant subspace of K0 containing H0 and that H0+ is the smallest U0 -invariant subspace of K0 containing H0 . Hypothesis (8.8) now takes the form K0 = H0− ⊕ G ∗+ ,

(8.35)

K0 = H0+ ⊕ G − .

(8.36)

Combining (8.35) and (8.36) with (8.32) gives

0 ⊕ G−, H0− = H

0 ⊕ G ∗+ . H0+ = H

54

J. A. Ball and A. Kheifets

IEOT

Since G − is orthogonal to G ∗+ in K0 , we then get

0. H0− ∩ H0+ = H

0 and hence also K00 = K0 We may now invoke (8.9) to conclude that H0 = H and K0 has the orthogonal decomposition (8.25) (with K0 in place of K00 ), i.e. (8.31) holds. From the third condition in (8.10) combined with (8.24), we see that in fact U0∗ im iΔ

∗ ,0 ⊥ G − ,

U im iΔ,0 ⊥ G ∗+ .

U ∗ im iΔ

∗ ,0 ⊥ G ∗+ ,

U im iΔ,0 ⊥ G−.

But also Hence we have U ∗ im iΔ

∗ ,0 ⊥ G − ⊕ G ∗+ ,

U im iΔ,0 ⊥ G − ⊕ G ∗+ .

From the orthogonal decomposition for K0 (8.13), we conclude that U0∗ im iΔ

∗ ,0 ⊂ H0 ,

U0 im iΔ,0 ⊂ H0 .

(8.37)

As in the proof of version (1), we see that D and D ∗ have the characterizations (8.28) and H0 has the two orthogonal decompositions (8.26). By combining these observations with the decomposition (8.25) for K0 and the fact that U0 is unitary on K00 , we see that the containments (8.37) actually force U0∗ im iΔ

∗ ,0 = Δ∗ ,

U0 im iΔ,0 = Δ,

i.e., (8.12) holds. From (8.12) combined with the already proved decomposition (8.31) for K0 we see that (8.7) holds as well. It now follows from the already proved version (1) of Theorem 8.4 that (U0 , K0 ) is the central lift with associated central scattering system S0 and fourfold AA-unitary coupling SAA,0 coming from a Lifting Problem as asserted. Theorem 7.1 gives a parametrization of the set of all symbols wY (with and G ⊂ K+ ) via a respect to a choice of two scale subspaces G ⊂ K− Redheffer-type linear-fractional-transformation (7.4) wY = s0 + s1 (I − ωs)−1 ωs2 where ω : Δ

(Z+ ) → Δ

∗ (Z+ ) is the input–output map for a free-parameter unitary colligation, and where the Redheffer coefﬁcient matrix [see (7.6)] G s0 s2 G (Z) → : Δ Δ s1 s

∗ (Z+ )

(Z) is completely determined from the Lifting-Problem data. Note that, if elements of the space G ⊕Δ

(Z) are expressed as inﬁnite column vectors, the ﬁrst column of the Redheffer coefﬁcient matrix s0 G (Z) :G → =: G ⊕Δ

(Z) Δ s1

(Z)

Vol. 70 (2011)

Inverse Lifting Problem

55

can be expressed naturally as a column matrix s0 s0 (n) . = coln∈Z s1 s1 (n) If we also view elements of Δ

∗ (Z+ ) as inﬁnite column vectors, then the second column of the Redheffer coefﬁcient matrix s2 : Δ

∗ (Z+ ) → G ⊕Δ

(Z) s can be expressed as an inﬁnite matrix which has Toeplitz structure: s s (n − m) s2 = 2 =: 2 for n ∈ Z, m ∈ Z+ . s n,m s n−m,0 s(n − m)

(8.38)

Let us deﬁne the Redheffer coefficient-matrix symbol to be simply the operator sequence s0 (n) s2 (n) . (8.39) s1 (n) s(n) n∈Z The following result shows how the Redheffer coefﬁcient-matrix symbol can be expressed directly in terms of the universal extension U0 . To this end we introduce the notation iG ,0 : G → K0 ,

iG ,0 : G → K0

for the inclusion of G in K0 obtained as the composition iG ,0 = iH0 ,0 iG of the inclusion of G in H0 followed by the inclusion of H0 in K0 , and similarly iG ,0 = iH0 ,0 iG . Theorem 8.6. The Redheffer coefficient-matrix symbol (8.39) for a Lifting Problem can be recovered directly from the central extension U0 [see (6.4)– (6.6)] according to the formula ∗ iG ,0 s0 (n) s2 (n) ∗n i i (8.40) = ∗ G ,0 Δ

∗ ,0 . ∗ U0 iΔ,0 U s1 (n) s(n) 0

Moreover, s2 (m) = 0 for m ≤ 0,

s1 (m) = 0

and

s(m) = 0

for m < 0, (8.41)

and also s(0) = 0.

(8.42)

Proof. We ﬁrst check that the formula (8.40) is correct for n < 0. By using the definitions (7.3) to unravel formula (7.6), we read oﬀ that, for n < 0, s0 (n) = i∗G ,0 U ∗n iG ,0 ,

s1 (n) = 0,

s2 (n) = 0,

s(n) = 0.

The ﬁrst formula matches with the upper left corner of (8.40) for n < 0. As and G ⊂ K+ by assumption, the other three blocks match up for G ⊂ K− n < 0 as a consequence of the identities (8.15)–(8.17).

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We next verify (8.40) for n ≥ 0. From the Toeplitz structure (8.38) we see that the validity of (8.40) for n ≥ 0 is equivalent to showing that ( + g i∗G ,0 g s0 s+ 2 ∗ i (8.43) . → U0 G ,0 iΔ :

∗ ,0 i∗ U0∗n i δ ∗ i δ ∗ s+ s+ 1

Δ,0

Δ∗ ,0

Δ∗ ,0

From the definition (7.3) we have + + s0 s+ IG 0 S0 2 + = 0 I s+ s S1+ 1

S2+ S+

iG 0

n∈Z+

0 . I

Combining this with (8.43), we see that it sufﬁces to show that ( i∗H0 ,0 h0 h0 S0+ S2+ iΔ

(0) ∗n i ∗ iΔ : → G ,0

∗ ,0 ∗ U0 i∗Δ,0 S1+ S + iΔ δ∗ δ ∗

(0)

U0 ∗

(8.44) . n∈Z+

S0+ S2+ From the definition of as the forward-time augmented input– S1+ S + output map for the unitary colligation U0 , we know that ( h0 h0 (n) S0+ S2+ iΔ

(0) ∗ =

S1+ S + i (0) δ ∗ δ(n) Δ∗

means that h0 (0) = h0 ,

h0 (1)

δ(0)

= U0

n∈Z+

h0 , δ ∗

h0 (n + 1)

δ(n)

= U0

h0 (n) 0

for n > 0. (8.45)

∗ , let us deﬁne h0 (n) ∈ H0 and δ(n) ∈ Δ

by Given h0 ∈ H0 and δ ∗ ∈ Δ i∗H ,0 h0 (n)

(8.46) = ∗ 0 ∗ U0∗n k where k = iH0 ,0 h0 + iΔ

∗ ,0 δ∗ . iΔ,0 δ(n)

U0

Then (8.40) follows if we can show that {h0 (n), δ(n)} n∈Z+ so deﬁned satisﬁes (8.45). The ﬁrst equality in (8.45) is immediate from the fact that im iH0,0 is orthogonal to im iΔ

∗ ,0 in K0 . The second equality in (8.45) is an easy consequence of the general identity i∗H0 ,0 U0 = ∗ (8.47) U0∗ iH0 ,0 iΔ

∗ ,0 , iΔ,0

connecting U0 and U0 . This identity in turn is an easy consequence of the formula (8.3) for U0∗ and is an analogue of the formula (6.3) connecting U1 and U in a more general context.

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The third equality in (8.45) can also be seen as a consequence of (8.47) as follows. For n > 0 we compute ∗ h0 (n) iH0 ,0 U0∗n k = U0 U0 0 0 ∗ iH0 ,0 = ∗ U0∗ iH0 ,0 i∗H0 ,0 U0∗n k (by (8.47)) iΔ,0

i∗H0 ,0 = ∗ (8.48) U0∗ PH0 U0∗n k iΔ,0

where k = iH0 ,0 h0 +iΔ

∗ ,0 δ∗ and where PH0 is the orthogonal projection of K0 onto im iH0 ,0 . To verify the third equality in (8.45) it remains only to show that the projection PH0 is removable in the last expression in (8.48). To this end, use the orthogonal decomposition 2 (2

K0 = iΔ,0

(Δ

(Z− )) ⊕ im iH0 ,0 ⊕ iΔ Δ ∗ ,0

∗ ,0

(Z+ )).

2 Note that U0∗n k ⊥ iΔ

∗ ,0 (Δ

(Z+ )), i.e., ∗

U0∗n k

2 ∈ iΔ,0

(Δ

(Z− )) ⊕ im iH0 ,0

(8.49)

for n > 0. Moreover it is easily checked 2 U0∗iΔ,0

(Δ

⊕ im iH0 ,0 .

(Z− )) ⊥ im iΔ,0

(8.50)

From conditions (8.49) and (8.50) we see that indeed the projection PH0 is removable in (8.48) and the third equation in (8.45) follows as required. Now that the validity of (8.40) is established, we see that s2 (0) = 0 as a consequence of (8.15) for the case n = 0. It remains to verify that s(0) = i∗Δ,0

∗ ,0 = 0, or equivalently,

U0 iΔ im iΔ,0 ⊥ U0∗ im iΔ

∗ ,0 . This can be seen as a direct consequence of the definition of U0∗ in (8.3).

Remark 8.7. In the proof of Theorem 8.6 it is shown that, given that U0 and U0 are related as in (8.47), then (8.46) implies (8.45). This observation can be seen as a special case of the following general result. Given a unitary operator U on K, a unitary colligation U of the form A B H H . U= : → E E∗ C D such that

i∗H U = ∗ U ∗ iH iE∗

iE .

where iH : H → K,

iE : E → K,

iE∗ : E∗ → K

(8.51)

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are isometric embedding operators with H → K is isometric, im iH ⊥ im iE so iH iE : E H im iH ⊥ im iE∗ so iH iE∗ : → K is isometric, E∗ then, for any k ∈ K, if (e, h, e∗ ) of E (Z) × H (Z) × E∗ (Z) is given by e(n) = i∗E U ∗n k, h(n) = i∗H U ∗n k,

(8.52)

e∗ (n) = i∗E∗ U ∗n+1 k, then (e, h, e∗ ) is a U -system trajectory, i.e., the system equations h(n + 1) h(n) =U e∗ (n) e(n)

(8.53)

hold for all n ∈ Z. Under these assumptions there is no a priori way to characterize which system trajectories (e, h, e∗ ) arise from a k ∈ K via formula (8.52). If we impose the additional structure: im iE and im iE∗ are wandering subspaces for U, so there exist uniquely determined isometric embedding operators iE : 2E (Z) → K,

iE∗ : 2E (Z) → K ∗

which extend iE and iE∗ in the sense that (0)

iE = iE ◦ iE ,

(−1)

iE∗ = iE∗ ◦ iE∗ ,

and K has the orthogonal decomposition K = im iH ⊕ iE∗ (2E∗ (Z− )) ⊕ iE (2E (Z+ )),

(8.54)

then one can characterize the system trajectories of the form (8.52) as exactly those of ﬁnite-energy in the sense that e ∈ 2E (Z)

and e∗ ∈ 2E∗ (Z),

(8.55)

and e∗ |Z− ∈ 2E∗ (Z− ).

(8.56)

or, equivalently, in the sense that e|Z+ ∈ 2E (Z+ )

With the additional wandering-subspace assumption and orthogonal-decomposition assumption (8.54) given above in place, then the map k → (e(n), h(n), e∗ (n)) deﬁned by (8.52) gives a one-to-one correspondence between elements k of K and ﬁnite-energy U -system trajectories (e, h, e∗ ). This last statement is essentially Lemma 2.3 in [9] and is the main ingredient in the coordinate-free approach in embedding a unitary colligation into a (discrete-time) Lax–Phillips scattering system. The reader can check that the situation in

in place of E∗ and Δ

∗ Theorem 8.6 meets all these assumptions (with Δ in place of E); the computation in the proof of Theorem 8.6 exhibits the

k = iH0 h0 + iΔ

∗ ,0 δ∗ corresponding to the ﬁnite-energy system trajectory supported on Z+ with initial condition h0 and impulse input supported at time

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n = 0 equal to δ ∗ . We invite the reader to consult [11] for an extension of these ideas to a several-variable context.

References [1] Adamjan, V.M., Arov, D.Z.: On unitary coupling of semiunitary operators. Dokl. Akad. Nauk. Arm. SSR XLIII(5), 257–263 (1966); English translation: Am. Math. Soc. Transl. 95, 75–129 (1970) [2] Adamjan, V.M., Arov, D.Z., Kre˘ın, M.G.: Inﬁnite Hankel matrices and generalized problems of Carath´eodory–Fej´er and I. Schur. Funkcional. Anal. Prilozhen. 2(4), 1–17 (1968); English translation: Funct. Anal. Appl. 2(2), 269–281 (1968) [3] Adamjan, V.M., Arov, D.Z., Kre˘ın, M.G.: Inﬁnite Hankel block matrices and related extension problems. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6, 87–112 (1971); English translation: Am. Math. Soc. Transl. 111(2), 133–156 (1978) [4] Arocena, R.: Unitary colligations and parametrization formulas. Ukrainian Math Zh. 46(3), 147–154 (1994); English translation: Ukrainian Math. J. 46, 151–158 (1994) [5] Arocena, R. : Unitary extensions of isometries and contractive intertwining dilations. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds.) The Gohberg Anniversary Collection: Volume II Topics in Analysis and Operator Theory, OT 41, pp. 13–23. Birkh¨ auser, Basel (1989) [6] Arov, D.Z.: Regular J-inner matrix-functions and related continuation problems. In: Helson, H., Sz.-Nagy, B., Vasilescu, F.-H. (eds.) Linear Operators in Functions Spaces, OT43, pp. 63–87. Birkh¨ auser, Basel (1990) [7] Arov, D.Z., Grossman, L.Z.: Scattering matrices in the theory of unitary extensions of isometric operators. Soviet Math. Dokl. 270, 17–20 (1983) [8] Arov, D.Z., Grossman, L.Z.: Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachrichten 157, 105–123 (1992) [9] Ball, J.A., Carroll, P.T., Uetake, Y.: Lax–Phillips scattering theory and well-posed linear systems: a coordinate-free approach. Math. Control Signals Syst. 20, 37–79 (2008) [10] Ball, J.A., Li, W.S., Timotin, D., Trent, T.T.: A commutant lifting theorem on the polydisc. Indiana University Math. J. 48(2), 653–675 (1999) [11] Ball, J.A., Sadosky, C., Vinnikov, V.: Scattering systems with several evolutions and multidimensional input/state/output systems. Integr. Equ. Oper. Theory 52, 323–393 (2005) [12] Ball, J.A., Trent, T.T. The abstract interpolation problem and commutant lifting: a coordinate-free approach. In: Bercovici, H., Foia¸s C. (eds.) Operator Theory and Interpolation: International Workshop on Operator Theory and Applications, IWOTA96, OT115, pp. 51–83. Birkh¨ auser, Basel-Boston (2000) [13] Ball, J.A., Trent, T.T., Vinnikov, V. : Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Bart, H., Gohberg, I., Ran, A.C.M. (eds.) Operator Theory and Analysis, OT122, pp. 83–138. Birkh¨ auser, Basel (2001) [14] Biswas, A., Foias, C., Frazho, A.E.: Weighted commutant lifting. Acta Sci. Math. (Szeged) 65, 657–686 (1999)

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[15] Boiko, S.S., Dubovoy, V.K., Kheifets, A.Ya.: Measure Schur complements and spectral functions of unitary operators with respect to different scales. In: (Beer-Sheva/Rehovot 1997) Operator Theory, System Theory and Related Topics, OT123. Birkh¨ auser, Basel, pp. 89–138 (2001) [16] Cotlar, M., Sadosky, C.: Integral representations of bounded Hankel forms deﬁned in scattering systems with a multiparametric evolution group. In: Gohberg, I., Helton, J.W., Rodman, L. Contributions to Operator Theory and its Applications, OT35, Birkh¨ auser, Basel (1988) [17] Davidson, K.: Nest Algebras: Triangular Forms for Operator Algebras on Hilbert Space. Longman Scientiﬁc & Technical, Essex (1988) [18] Foias, C., Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems, OT44. Birkh¨ auser, Basel (1990) [19] Foias, C., Frazho, A.E., Gohberg, I., Kaashoek, M.A.: Metric Constrained Interpolation, Commutant Lifting and Systems, OT100. Birkh¨ auser, Basel (1998) [20] Foias, C., Frazho, A.E., Kaashoek, M.A.: Relaxation of metric constrained interpolation and a new lifting theorem. Integr. Equ. Oper. Theory 42, 253–310 (2002) [21] Frazho, A.E.: Complements to models for noncommuting operators. J. Funct. Anal. 59(3), 445–461 (1984) [22] Frazho, A.E., ter Horst, S., Kaashoek, M.A.: Coupling and relaxed commutant lifting problem. Integr. Equ. Oper. Theory 54, 33–67 (2006) [23] Frazho, A.E., ter Horst, S., Kaashoek, M.A.: All solutions to the relaxed commutant lifting problem. Acta Sci. Math. (Szeged) 72, 299–318 (2006) [24] Katsnelson, V.: Left and right Blaschke–Potapov products and Arov-singular matrix valued functions. Integr. Equ. Oper Theory 13, 236–248 (1990) [25] Katsnelson, V.: Weighted spaces of pseudocontinuable functions and approximation by rational functions with prescribed poles. Z. Anal. Anwendungen 12, 27–47 (1993) [26] Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and the theory of extensions of isometric operators (Russian). In: Operators in function spaces and problems in function theory, vol. 146. “Naukova Dumka”, Kiev, 83–96 (1987); English translation in Topics in interpolation theory, OT95, pp. 283–298 Birkh¨ auser, Basel (1997) [27] Kheifets, A.: The Parseval equality in an abstract problem of interpolation, and the union of open systems I., Teor. Funktsii Funktsional. Anal. i Prilozhen. (Russian) 49, 112–120 (1988); English translation: J. Soviet Math. 49(4), 1114–1120 (1990) [28] Kheifets, A.: The Parseval equality in an abstract problem of interpolation, and the union of open systems II. Teor. Funktsii Funktsional. Anal. i Prilozhen (Russian) 50, 98–103 (1988); English translation: J. Soviet Math. 49(6), 1307–1310 (1990) [29] Kheifets, A.: Scattering matrices and Parseval equality in Abstract Interpolation Problem. PhD thesis, Kharkov State University (1990) [30] Kheifets, A.: On regularization of γ-generating pairs. J. Funct. Anal. 130(2), 310–333 (1995)

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[31] Kheifets, A.: Nehari’s interpolation problem and exposed points of the unit ball in the Hardy space H 1 . In: Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), vol. 11, pp. 145-151, Israel Math. Conf. Proc., Bar-Ilan Univ., Ramat Gan (1997) [32] Kheifets, A.: The abstract interpolation problem and applications. In: Holomorphic Spaces (Berkeley, CA, 1995), pp. 351–379, Math. Sci. Res. Inst. Publ., vol. 33. Cambridge Univ. Press, Cambridge (1998) [33] Kheifets, A.: Parametrization of solutions of the Nehari problem and nonorthogonal dynamics. In: Bercovici H., Foia¸s C. (eds.) Operator Theory and Interpolation, OT115. Birkh¨ auser, Basel, pp. 213–233 (2000) [34] Kheifets, A.: Abstract interpolation scheme for harmonic functions. In: Alpay, D., Gohberg, I., Vinnikov, V. (eds.) Interpolation Theory, Systems Theory and Related Topics. OT134, Birkh¨ auser, Basel, pp. 287–317 (2002) [35] Kheifets, A., Yuditskii, P.: An analysis and extension of V. P. Potapov’s approach to interpolation problems with applications to the generalized bi-tangential Schur-Nevanlinna–Pick problem and J-inner–outer factorization. In: Matrix and operator valued functions, OT72. Birkh¨ auser, Basel, pp. 133– 161 (1994) [36] Kupin, S.: Lifting theorem as a special case of abstract interpolation problem. Zeitschrift f¨ ur Analysis Und Ihre Anwendungen 15, 789–798 (1996) [37] Lax, P.D.: Functional Analysis. Wiley, New York (2002) [38] Li, W.S., Timotin, D.: The relaxed intertwining lifting in the coupling approach. Integr. Equ. Oper Theory 54, 97–111 (2006) [39] McCullough, S.A., Sultanic, S.: Ersatz commutant lifting with test functions. Complex Anal. Oper. Theory 1(4), 581–620 (2007) [40] Moran, M.D.: On intertwining dilations. J. Math. Anal. Appl. 141, 219–234 (1989) [41] Muhly, P.S., Solel, B.: Tensor algebras over C ∗ -correspondences: representations, dilations and C ∗ -envelopes. J. Funct. Anal. 158, 389–457 (1998) [42] Muhly, P.S., Solel, B.: Hardy algebras, W ∗ -correspondences and interpolation theory. Math. Annalen 330, 353–415 (2004) ¨ [43] Nevanlinna, R.: Uber beschr¨ ankte Funktionen, die in gegebene Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. Ser. A 13(1) (1919) ¨ [44] Nevanlinna, R.: Uber beschr¨ ankte Funktionen. Ann. Acad. Sci. Fenn. Ser. A 32(7) (1929) [45] Peller, V.: Hankel Operators and Their Applications, Springer Monographs in Mathematics. Springer, New York (2003) [46] Popescu, G.: Isometric dilations for inﬁnite sequences of noncommuting operators. Trans. Am. Math. Soc. 316, 523–536 (1989) [47] Popescu, G.: Interpolation problems in several variables. J. Math. Anal. Appl. 227, 227–250 (1998) [48] Popescu, G.: Multivariable Nehari problem and interpolation. J. Funct. Anal. 200, 536–581 (2003) [49] Popescu, G.: Entropy and Multivariable Interpolation, Memoirs of the American Mathematical Society Number. American Mathematical Society, Providence 868 (2006)

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[50] Sarason, D.: Generalized interpolation in H ∞ . Trans. Am. Math. Soc. 127, 179–203 (1967) [51] Sarason, D. : New Hilbert spaces from old. In: Ewing, J.H., Gehring, F.W. (eds.) Paul Halmos: Celebrating 50 Years of Mathematics, pp. 195–204. Springer, Berlin (1985) [52] Sarason, D.: Exposed points in H 1 , I. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds.) The Gohberg Anniversary Collection: Volume II: Topics in Analysis and Operator Theory, OT41, pp. 485–496. Birkh¨ auser, Basel (1989) [53] Sarason, D.: Exposed points in H 1 , II. In: Branges, L.de , Gohberg, I., Rovnyak, J. (eds.) Topics in Operator Theory: Ernst. D. Hellinger Memorial Volume, OT48, pp. 333–347. Birkh¨ auser, Basel (1990) [54] Sultanic, S.: Commutant lifting theorem for the Bergman space. Integr. Equ. Oper. Theory 35(1), 639–649 (2001) [55] Sz.-Nagy, B., Foias, C.: Dilatation des commutants d’op´erateurs. C.R. Acad. Sci. Paris, Serie A 266, 493–495 (1968) [56] Sz.-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland/American Elsevier, Amsterdam (1970) [57] ter Horst, S.: Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas. Mathematische Nachrichten (to appear, 2011) [58] ter Horst S.: Redheffer representations and relaxed commutant lifting theory (preprint, 2011) [59] Treil, S., Volberg, A. : A ﬁxed point approach to Nehari’s problem and its applications. In: Basor, E., Gohberg, I. (eds.) Toeplitz Operators and Related Topics: The Harold Widom Anniversary volume, OT71, pp. 165–186. Birkh¨ auser, Basel (1994) Joseph A. Ball Department of Mathematics Virginia Tech Blacksburg, VA 24061, USA e-mail: [email protected] Alexander Kheifets (B) Department of Mathematics University of Massachusetts Lowell Lowell, MA 01854, USA e-mail: alexander [email protected] Received: November 21, 2009. Revised: February 26, 2011.

Integr. Equ. Oper. Theory 70 (2011), 63–99 DOI 10.1007/s00020-011-1871-6 Published online March 15, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

On the Solvability of Singular Integral Equations with Reflection on the Unit Circle L. P. Castro and E. M. Rojas Dedicated to the memory of Professor Di´ omedes B´ arcenas Abstract. The solvability of a class of singular integral equations with reﬂection in weighted Lebesgue spaces is analyzed, and the corresponding solutions are obtained. The main techniques are based on the consideration of certain complementary projections and operator identities. Therefore, the equations under study are associated with systems of pure singular integral equations. These systems will be then analyzed by means of a corresponding Riemann boundary value problem. As a consequence of such a procedure, the solutions of the initial equations are constructed from the solutions of Riemann boundary value problems. In the ﬁnal part of the paper, the method is also applied to singular integral equations with the so-called commutative and anti-commutative weighted Carleman shifts. Mathematics Subject Classification (2010). Primary 45E05; Secondary 30E20, 30E25, 45E10, 47A68, 47G10. Keywords. Singular integral equation, reﬂection, shift, solvability, Riemann boundary value problem.

1. Introduction The formulation of linear boundary values problems (BVP’s) for analytic functions has a very long history which is usually considered to start with B. Riemann’s work [26,27]. It is also clear that D. Hilbert [14], C. Haseman (1907), T. Carleman [4], N.I. Muskhelishvili, F.D. Gakhov, I.N. Vekua (and their students) carried out significant research on problems of this type. Historically, the paper by Haseman [13] was the ﬁrst work in which the boundary This work was supported in part by Center for R&D in Mathematics and Applications, University of Aveiro, Portugal, through FCT—Portuguese Foundation for Science and Technology. .

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value problem with a shift (BVPS) was considered for analytic functions. Singular integral equations with shift (SIES’s) are connected with such BVP’s in a natural way. For instance, the study of singular integral operators with conjugation began in the years 40–50 of last century precisely with the investigation of boundary value problems for analytic functions with conjugation, namely by Markouchevitch [24], Vekua [30] and Boiarskii [3]. Vekua’s paper [29] is considered to be the ﬁrst one in which SIES’s were considered. Up to the present, many publications were devoted to these problems. An important part of these investigations was carried out in the following two main directions: (i) the study of the Fredholm theory of SIES’s and, in a more detailed way, (ii) the solvability theory of singular integral equations and BVPS. The following monographs synthesize these developments in a significant way: Gakhov [12], Karapetiants and Samko [16], Kravchenko and Litvinchuk [21,22], Mikhlin and Pr¨ ossdorf [25] (and cf., also the references therein). As a result of all knowledge on this topic, several applications and theories have been developed. Among these, we may point out the theory of the cavity currents in an ideal liquid, the theory of inﬁnitesimal bounds of surfaces with positive curvature, the contact theory of elasticity and physics of plasma. In this paper, we will reduce a class of SIES’s on a weighted Lebesgue space Lp (T, w) (1 < p < ∞) to a system of SIE’s by using some operator identities and projections, which will allow us to study the solutions of the initial equation throughout a Riemann boundary value problem. The projection method which we have in mind was also used by Le Huy Chuan et al. [8,9,28] for the case of linear fractional Carleman shift on the space of H¨ older continuous functions H μ (0 < μ < 1). We would like to remark that the classical method for the reduction of a singular integral operator with shift (SIOS) into singular integral equations without a shift is based on a procedure which requires the “duplication of the size of the space” in which the operators are deﬁned. As a consequence, it is obtained a pure vector singular integral operator which has the same Fredholm properties as the initial one but with a “double” symbol matrix. In much of the cases, the so-called Gohberg-Krupnik-Litvinchuk identity (see e.g., [16,17,20]) and other explicit operator equivalence relations (c.f., e.g., [5,17]) are main ingredients for such analysis. In this way, the solvability of a (scalar) SIES associated with the SIOS is equivalently reformulated as a matrix factorization problem for corresponding matrices (which are built based on the new matrix coefﬁcients); for these and other methods see, for instance, [6,10,19,20,23]. The techniques of the present paper avoid the use of the just mentioned (independent) matrix singular integral operators by relating solutions of the SIES to solutions of a pure system of two SIE which presents some dependencies between both equations. This allows a direct construction of the corresponding solutions by using an appropriate substituting ansatz which revels here to be a fundamental piece in the full process of ﬁnding solutions to the initial problem.

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2. The SIES’s Under Study In the ﬁrst part of the paper we will consider a SIES deﬁned on weighted Lebesgue spaces Lp (T, ρ), p ∈ (1, ∞), consisting of the measurable complexvalued functions over the unit circle T with the norm ⎛ f p,ρ = ⎝

⎞1/p |f (τ )|p ρ(τ )p dτ ⎠

,

T

for certain measurable functions ρ : T → [0, ∞] for which the preimage ρ−1 ({0, ∞}) has measure zero. Among those weights ρ, if choosing the ones which belong to the Muckenhoupt class then the Cauchy singular integral operator: (ST f )(t) =

f (τ ) 1 p.v. dτ πi τ −t

(2.1)

T

is a well-deﬁned, linear and bounded operator on Lp (T, ρ), p ∈ (1, ∞) (where the integral is understood in the Cauchy principal value). In the sequel, we will be also using the Banach algebra L∞ (T) of all essentially bounded and Lebesgue measurable complex-valued functions deﬁned on T (endowed with the essential supremum norm). As usual, we shall say that a function φ analytic in the unit disk D := {z ∈ C : |z| < 1} is an element of the Smirnov class of domains E p (D), 1 ≤ p < ∞, if it possible to ﬁnd an expanding sequence boundaries T such that: (i) D ∪ T ⊂ D; (ii) Dk with rectiﬁable k k k k Dk = D; (iii) supk Tk |f (t)|p dt < ∞. We will also consider the weighted Smirnov class

deﬁned by E p (D, ρ) := φ ∈ E 1 (D) : φ|T ∈ Lp (T, ρ) , where φ|T denotes the non-tangential limit of φ a.e. on T. Other remarkable sets associated with E p (D, ρ) are the set of analytic functions on D, the set of analytic functions on C\D and the set of analytic p p functions on C\D vanishing at infinity—here denoted by E+ (D, ρ), E− (D, ρ) •

p (D, ρ), respectively. The following subspaces of Lp (T, ρ), 1 ≤ p < and E− p ∞ will be useful in order to attain our goals: H+ (T, ρ) := PT (Lp (T, ρ)), p p p (T, ρ) := QT (L (T, ρ)) ⊕ C and H (T, ρ) := QT (Lp (T, ρ)), where PT and H − − QT are the Riesz projections deﬁned by the rule

PT =

1 (IT + ST ), 2

QT =

1 (IT − ST ), 2

(2.2)

with IT being the identity operator on Lp (T, ρ). It is clear that for the case of ρ(t) = 1, these spaces are the so-called Hardy spaces. Another definition of those spaces is the following: A function f belongs to the class p p (T, ρ), H p (T, ρ)), 1 ≤ p ≤ ∞, if there exists a function of class H+ (T, ρ) (H − − •

p p p (D, ρ)(E− (D, ρ), E− (D, ρ)) for which their boundary values coincide with E+ f at almost all t ∈ T.

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Among all the Muckenhoupt weights, from now on we will be only considering those of the form β(t ) 1 0 β(t0 ) 1 (t), (t) := |t − t0 | t − t0 where t0 ∈ T+ := {a + bi ∈ T : b > 0} and is a continuous function at t0 and such that (t0 ) = 0 (and the exponents β(t0 ) are then subjected to the fact that belongs to the Muckenhoupt class). Our main purpose is the solvability of the following singular integral equation, which cannot be reduced to a two-term boundary value problem (see [22]), deﬁned on the space Lp (T, ), p ∈ (1, ∞): a(t)ϕ(t) +

1 m

b(t) 1 ϕ(τ ) aj (t) (−1)2−k bj (τ )ϕ(τ )dτ = f (t) dτ + 2 πi τ − αk (t) πi k=0

T

j=1

T

(2.3) ∞

where a, b, a1 , . . . , am ∈ L (T) (and later on are required to satisfy some extra conditions; see (5.5)–(5.9)), b1 , . . . , bm are given functions satisfying b (τ )ϕ(τ )dτ < ∞, and αk (t) = α(αk−1 (t)) with α0 (t) = t, α(t) = 1/t. T j Note that due to H¨ older’s inequality the condition T bj (τ )ϕ(τ )dτ < ∞ is automatically satisﬁed if we assume that bj ∈ Lq (T, −1 ), where q = p/(p−1). In the second part of the paper, we will consider a more general equation than (2.3) deﬁned on the classic Lebesgue space Lp (T), p ∈ (1, ∞): 1 ϕ(τ ) b(t) 2−k k 1 dτ (−1) (v(t)) a(t)ϕ(t) + 2 πi τ − θk (t) k=0

+

m

j=1

aj (t) πi

T

bj (τ )ϕ(τ )dτ = f (t)

(2.4)

T

with elements a, b, a1 , . . . , am , b1 , . . . , bm as above. Here, v(t) is a complex valued function on T and θ is a Carleman shift function on T, which is a homeomorphism θ(t) : T −→ T preserving the orientation on T or changing the orientation on T into the opposite one, and satisfying θ(θ(t)) ≡ t. The function v(t) is chosen such that the weighted shift operator induced by v(t) and θ(t), and deﬁned by (W ϕ)(t) = v(t)ϕ(θ(t)), is bounded on Lp (T), 1 < p < ∞ (see e.g. [1,2,7,11,15,17,18,20,21] for different concrete examples of weighted shift operators of this form). A (weighted) shift operator is called a Carleman shift operator if W 2 = IT . It is well known that, for non-weighted shift operators, the commutator W ST −ST W is a compact operator. However, we are going to consider here weighted Carleman shift operators of commutative, or anti-commutative type. This means that we will be considering Carleman shift operators W satisfying the property W ST = ST W (“commutative case”) or W ST = −ST W (“anti-commutative case”). A classiﬁcation and properties of shift functions and corresponding shift operators can be found in [21].

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The paper is divided into seven sections, in the next four sections we will consider the equation (2.3) with the speciﬁc shift function α(t) = 1/t. Afterwards, we will rewrite the results on these sections for the case of equation (2.4) for commutative and anti-commutative Carleman shifts. In detail, Sect. 3 is devoted to the study of some complementary projections and their behavior (commuting properties) with the shift operator and the Cauchy singular integral operator. In Sect. 4 we will derive the reduction of equation (2.3) (by using the mentioned projections) into a pure system of singular integral equations. In Sect. 5, by using the reduction method of Sect. 4, we associate a BVP to the equation (2.3) and also give the explicit solution of such BVP. The representation of the solutions of equation (2.3) are characterized through the solutions of the Riemann boundary value problem in Sect. 6. Finally, in Sect. 7 we rewrite the results obtained in the previous sections to the cases of “commutative” and “anti-commutative” Carleman shifts within the framework of equation (2.4).

3. Projections and Singular Integral Operators with Reflection Let us consider the following complementary projections P1 :=

1 (IT − J) 2

and

P2 :=

1 (IT + J) 2

(3.1)

where J is the shift operator (Jϕ)(t) = ϕ (1/t) , t ∈ T. 2 Note that J k = j=1 (−1)kj Pj , k = 1, 2, and 2

Pk =

1 (−1)k(1−j) J j+1 , 2 j=1

k = 1, 2.

(3.2)

In that follows, we will denote the multiplication operator of a function ϕ ∈ Lp (T, ), by a function a ∈ L∞ (T), as (Ka ϕ)(t) = a(t)ϕ(t) –which is a linear and bounded operator on Lp (T, ). The next proposition presents some of the dependencies between the projections Pk (k = 1, 2) and the multiplication operator Ka . Proposition 3.1. Let a ∈ L∞ (T) be fixed. Then, for every (j, k), with k, j = {1, 2}, there exists an element b ∈ L∞ (T) such that Kb Xj ⊂ Xk and P k K a P j = Kb P j , where Xk = Pk (Lp (T, )). The function b will be denoted by akj and determined as follows akj (t) :=

2 1 (−1)(j−k)(m+1) a(αm+1 (t)), 2 m=1

t ∈ T.

(3.3)

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Proof. Based on the properties of the projections Pk given in (3.2), we directly obtain: 2 1 Pk Ka P j = (−1)k(1−m) J m+1 Ka Pj 2 m=1 =

2 1 (−1)k(1−m) a(αm+1 (·))J m+1 Pj 2 m=1

=

2 2

1 (−1)k(1−m) a(αm+1 (·)) (−1)s(m+1) Ps Pj 2 m=1 s=1

=

2 1 (−1)k(1−m) a(αm+1 (·))(−1)j(m+1) Pj 2 m=1

= akj (·)Pj = Kb P j . A more direct relation between the projections Pk and the multiplication operator with symbol b is now exhibited in the next result. Proposition 3.2. Let a ∈ L∞ (T) be fixed. Then, for any k, j ∈ {1, 2}, we have Pk Kakj = Kakj Pj where akj is determined by (3.3). Proof. For any ϕ ∈ Lp (T, ), we have 2 1 (−1)k(1−m) J m+1 (akj (t)ϕ(t)) 2 m=1 2 2

1 k(1−m) m+1 (j−k)(n+1) (−1) J (−1) a(αn+1 (t)) ϕ(t) 2 n=1 m=1 2 2

1 (−1)(j−k)(n+1) a(αn+1+m+1 (t)) (−1)k(1−m) ϕ(αm+1 (t)). 2 n=1 m=1

(Pk Kakj ϕ)(t) = Pk (akj (t)ϕ(t)) = =

1 2

=

1 2

Notice that for m = 1 we get a(αn+1+m+1 (t)) = a(αn+3 (t)) = a(αn+1 (t)) and, for m = 2, a(αn+1+m+1 (t)) = a(αn+4 (t)) = a(αn (t)). Thus 2 1 1 (Pk Kakj ϕ)(t) = (−1)(n+1)(j−k) a(αn+1 (t))ϕ(t) 2 2 n=1 2 1 k (n+1)(j−k) (−1) a(αn (t)) ϕ(α(t)) + (−1) 2 n=1 2 1 1 (−1)(n+1)(j−k) a(αn+1 (t))ϕ(t) = 2 2 n=1 2 1 j n(j−k) (−1) a(αn (t)) ϕ(α(t)) + (−1) 2 n=1

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= akj (t)(Pj ϕ)(t) = (Kakj Pj ϕ)(t). Therefore, Pk Kakj ≡ Kakj Pj .

Proposition 3.3. Let ϕ ∈ Lp (T, ). Then, for z ∈ C\{0}, we have (1) (ST Jϕ)(z) = (ST ϕ)(0) − (JST ϕ)(z). k (2) (Pk ST ϕ)(z) = (ST Pj ϕ)(z) + (−1) k, j = 1, 2, k = j. 2 (ST ϕ)(0), Proof. We start by recalling that in our case ϕ ∈ Lp (T, ) ⊂ L1 (T) and therefore, based on the Cauchy integral of ϕ, we may consider corresponding analytic functions in the unitary disk or its exterior. We will have this in mind in the following calculations. (i) Let f ( τ1 ) 1 dτ. (ST Jf )(z) = πi τ −z T

Putting τ =

1 x

− x12 dx,

and having dτ = we get 1 f (x) f (x) 1 1 1 (ST Jf )(z) = − − 2 dx = − dx 1 πi x πi zx − 1 x x −z T T 1 1 1 =− − + f (x)dx πi x x − z1 T 1 f (x) f (x) 1 dx − = dx πi x πi x − z1 T

T

= (ST f )(0) − (JST f )(z). Therefore, the proposition (1) is obtained. (ii) To carry out the second part, we perform the following computations: 2

1 (Pk ST f )(z) = (−1)k(1−j) (J j+1 ST f )(z) 2 j=1

1 (ST f )(z) + (−1)k (JST f )(z)) 2

1 (ST f )(z) + (−1)k [−(ST Jf )(z) + (ST f )(0)] . = 2 From here we conclude that (−1)k (ST f )(0), k, j = 1, 2, k = j. (Pk ST f )(z) = (ST Pj f )(z) + 2 =

4. The Reduction of Equation (2.3) to a System of Pure Singular Integral Equations In this section we will relate the solutions of the SIES (2.3) with the solutions of a pure system of SIE. First, with the help of projection P1 (given in the

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previous section), we rewrite equation (2.3) as follows m

1 aj (t) bj (τ )ϕ(τ )dτ = f (t). a(t)ϕ(t) + b(t)(P1 ST ϕ)(t) + πi j=1

(4.1)

T

Additionally, suppose that a(t) is a non-vanishing function on T. Denoting by Mbj , j = 1, . . . , m, the linear functional on Lp (T, ) deﬁned as 1 Mbj (ϕ) := bj (τ )ϕ(τ )dτ, πi T

and putting Mbj (ϕ) = λj ,

j = 1, . . . , m,

(4.2)

then (4.1) can be rewritten in the form a(t)ϕ(t) + b(t)(P1 ST ϕ)(t) = f (t) −

m

λj aj (t).

(4.3)

j=1

Lemma 4.1. Let ϕ ∈ Lp (T, ). Then, ϕ is a solution of (4.3) if and only if {ϕk = Pk ϕ, k = 1, 2} is a solution of the following system aα (t)ϕk (t) + [ab]k (t)[(ST ϕ2 )(t) − (ST ϕ2 )(1)] = [af ]k (t),

k = 1, 2, (4.4)

where aα (t) = a(t)a(α(t)) 2

1 (−1)(j+1)(1−k) a(αj (t))b(αj+1 (t)) (4.5) 2 j=1 2 m

1 k(1−j) [af ]k (t) = (−1) λv av (αj+1 (t)) a(αj (t)). f (αj+1 (t)) − 2 j=1 v=1 [ab]k (t) =

Proof. Suppose that ϕ ∈ Lp (T, ) is a solution of (4.3). Then, multiplying by a(α(t)), applying the projections Pk (k = 1, 2) to both sides of such equation and using Propositions 3.1 and 3.2, we have 2

1 (−1)(j+1)(1−k) a(αj+2 (t))b(αj+1 (t))(P1 ST ϕ)(t) 2 j=1 2 m

1 k(1−j) = (−1) λv av (αj+1 (t)) a(αj+2 (t)). f (αj+1 (t)) − 2 j=1 v=1

Pk (a(t)a(α(t))ϕ(t)) +

Since a(αj+2 (t)) = a(αj (t)), it follows that the equation above is equivalent to the following system aα (t)(Pk ϕ)(t) + [ab]k (t)(P1 ST ϕ)(t) = [af ]k (t),

k = 1, 2.

(4.6)

Using Proposition 3.3, we are able to rewrite the system (4.6) in the form 1 aα (t)(Pk ϕ)(t) + [ab]k (t) (ST P2 ϕ)(t) − (ST ϕ)(0) = [af ]k (t), k = 1, 2. 2

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Evaluating z = 1 in the equality (P1 ST ϕ)(z) = (ST P2 ϕ)(z) − 12 (ST ϕ)(0), we obtain 1 (P1 ST ϕ)(1) = (ST P2 ϕ)(1) − (ST ϕ)(0) 2 1 1 [(ST ϕ)(1) − (JST ϕ)(1)] = (ST P2 ϕ)(1) − (ST ϕ)(0) 2 2 1 0 = (ST P2 ϕ)(1) − (ST ϕ)(0). 2 Thus, (P1 ϕ, P2 ϕ) is a solution of (4.4). Conversely, suppose that there exists ϕ ∈ Lp (T, ) such that (P1 ϕ, P2 ϕ) is a solution of (4.4). Summing from 1 to 2, we obtain 2

[aα (t)(Pk ϕ)(t) + [ab]k (t) ((ST P2 ϕ)(t) − (ST ϕ2 )(1))] =

k=1

2

[af ]k (t).

k=1

(4.7) Now, note that 2

[ab]k (t) =

k=1

1 [a(α(t))b(t) + a(t)b(α(t)) + a(α(t))b(t) − a(t)b(α(t))] 2

= a(α(t))b(t).

(4.8)

Similarly,

2 2 m

1 k(1−j) [af ]k (t) = (−1) λv av (αj+1 (t)) a(αj (t)) f (αj+1 (t)) − 2 j=1 v=1 k=1 k=1 2 m

1 λv av (t) a(α(t)) = f (t) − 2 v=1 k=1 m

k λv av (α(t)) a(t) + (−1) f (α(t)) − 2

v=1

=

f (t) −

m

λv av (t) a(α(t)).

(4.9)

v=1

Thus, (4.7) is equivalent to the following equality aα (t)ϕ(t) + b(t)a(α(t))[(ST P2 ϕ)(t) − (ST ϕ2 )(1)]= f (t) −

m

λv av (t) a(α(t)).

v=1

By Proposition 3.3, this implies the desired form (4.3).

Lemma 4.2. If (ψ1 , ψ2 ) is a solution of system (4.4), then (P1 ψ1 , P2 ψ2 ) is also a solution of (4.4). Proof. Suppose that (ψ1 , ψ2 ) is a solution of the system (4.4). Applying the projections Pk to both sides of the kth equation of (4.4), we get aα (t)(Pk ψk )(t) + Pk [[ab]k (t)((ST ψ2 )(t) − (ST ψ2 )(1))] = Pk ([af ]k (t)). (4.10)

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Now, we claim that Pk ([ab]k )I = [ab]k P1

and Pk ([af ]k )(t) = [af ]k (t).

(4.11)

In fact, 1 a(α(t))b(t) + (−1)3(1−k) a(t)b(α(t)) f (t) 2 2 1 1 (−1)k(1−j) J j+1 a(α(t))b(t) + (−1)3(1−k) a(t)b(α(t)) f (t) 2 j=1 2 1 a(α(t))b(t) + (−1)3(1−k) a(t)b(α(t)) f (t) 4 + (−1)k a(t)b(α(t)) + (−1)3(1−k) a(α(t))b(t) f (α(t)) 1 a(α(t))b(t) + (−1)3(1−k) a(t)b(α(t)) f (t) 4 + (−1)k a(t)b(α(t)) − a(α(t))b(t) f (α(t)) 1 a(α(t))b(t)(f (t) − f (α(t))) 4 +a(t)b(α(t)) (−1)3(1−k) f (t) + (−1)k f (α(t)) 1 a(α(t))b(t)(f (t)−f (α(t))) + (−1)3(1−k) a(t)b(α(t))[f (t) − f (α(t))] 4 [ab]k (t)(P1 f )(t).

Pk ([ab]k (t)f )(t) = Pk = =

=

=

= =

On the other hand, Pk ([af ]k (t)) =

2 1 k(1−j) j+1 (−1) J ([af ]k (t)) 2 j=1

2 1 k(1−j) j+1 (−1) J 2 j=1 ⎛ ⎞ 2 m

1 k(1−j) f (αj+1 (t)) − ×⎝ (−1) λv av (αj+1 (t)) a(αj (t))⎠ 2 j=1 v=1 2 m

1 k(1−j) j+1 = f (t) − (−1) J λv av (t) a(α(t)) 4 j=1 v=1 m

k + (−1) f (α(t)) − λv av (α(t)) a(t)

=

1 = 4

v=1

f (t) −

+ (−1)

k

m

v=1

f (α(t)) −

1 = 4

k

f (t) −

m

f (t) − k

m

m

f (α(t)) −

λv av (α(t)) a(t)

v=1

λv av (α(t)) a(t)

λv av (t) a(α(t)) + (−1)

v=1

f (α(t)) −

k

λv av (t) a(α(t))

v=1

+ (−1)

m

v=1

+(−1)

λv av (t) a(α(t)) + (−1)

m

v=1

k

f (α(t)) −

λv av (α(t)) a(t) + f (t) −

m

v=1

m

v=1

λv av (α(t)) a(t)

λv av (t) a(α(t)) .

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Singular Integral Equations

Therefore, we have 1 Pk ([af ]k (t)) = 4

−

2 f (t) −

m

v=1

m

73

λv av (t) a(α(t)) + 2(−1)k f (α(t))

λv av (α(t)) a(t)

v=1

= [af ]k (t). Now, substituting (4.11) into (4.10), we obtain aα (t)(Pk ψk )(t) + [ab]k (t)P1 [(ST ψ2 )(t) − (ST ψ2 )(1)] = [af ]k (t). (4.12) Using Proposition 3.3, we have that (4.12) is equivalent to the following equation: 1 aα (t)(Pk ψk )(t) + [ab]k (t) (ST P2 ψ2 )(t) − (ST ψ2 )(0) − P1 (ST ψ2 )(1) 2 = [af ]k (t), for k = 1, 2. Also, from Proposition 3.3 we have that 1 (ST ψ2 )(0) + P1 (ST ψ2 )(1) = (ST P2 ψ2 )(1). 2 Thus, we obtain that (P1 ψ1 , P2 ψ2 ) is a solution of (4.4).

Theorem 4.3. The equation (4.3) has solutions in Lp (T, ) if and only if the following equation aα (t)ϕ2 (t) + [ab]2 (t)(ST ϕ2 )(t) − [ab]2 (t)(ST ϕ2 )(1) = [af ]2 (t)

(4.13)

has solutions. Moreover, if ϕ2 is a solution of equation (4.13), then equation (4.3) has a solution given by the formula m f (t) − j=1 λj aj (t) − b(t)(P1 ST ϕ2 )(t) . (4.14) ϕ(t) = a(t) Proof. Suppose that ϕ ∈ Lp (T, ) is a solution of equation (4.3). By Lemma 4.1 we know that (P1 ϕ, P2 ϕ) is a solution of system (4.4). Hence, P2 ϕ is a solution of (4.13). Conversely, suppose that ϕ2 is a solution of (4.13). In this case (4.4) has a solution determined by the formula ϕ1 (t) =

[af ]1 (t) − [ab]1 (t)[(ST ϕ2 )(t) − (ST ϕ2 )(1)] . aα (t)

(4.15)

By Lemma 4.2, we know that (P1 ϕ1 , P2 ϕ2 ) is also a solution of (4.4). Put ϕ=

2

P k ϕk .

(4.16)

k=1

It is clear that Pk ϕ = Pk ϕk . This means that (P1 ϕ, P2 ϕ) is a solution of (4.4). From Lemma 4.1 it follows that ϕ is a solution of (4.3). Moreover, from

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(4.11), (4.15) and (4.16), we get 2 2

[af ]k (t) − [ab]k (t)[(ST ϕ2 )(t) − (ST ϕ2 )(1)] Pk ϕk (t) = Pk ϕ(t) = aα (t) k=1

=

k=1

1 aα (t)

2

{[af ]k (t) − [ab]k (t)[(P1 ST ϕ2 )(t) − P1 (ST ϕ2 )(1)]} . (4.17)

k=1

Substituting (4.8) and (4.9) into (4.17), we obtain (4.14). Thus the proof is completed.

5. The Solutions of Equation (4.13) by Means of the Associated BVP In this section we are going to obtain the explicit solutions of equation (4.13). In view of this goal, we will reduce that equation to a Riemann boundary value problem. In order to establish the solutions of this problem we shall use the following factorization notion (cf. [23,25]): A factorization of an element ψ ∈ L∞ (T) in the space Lp (T, ρ)(1 < p < ∞) is a representation of the form ψ(t) = ψ+ (t)tℵ ψ− (t),

t ∈ T,

where p q −1 ψ+ ∈ H+ (T, ), ψ+ ∈ H+ (T, −1 ), q (T, −1 ), ψ −1 ∈ H p (T, ), ψ− ∈ H −

−

−

ℵ is an integer which is called the (p, ρ)-index of ψ, and q := p/(p − 1) is the conjugate exponent of p ∈ (1, ∞). Let us consider the equation (4.13), and deﬁne ϕ2 (τ ) 1 dτ, z ∈ C\T. (5.1) Φ2 (z) := 2πi τ −z T

According to Sokhotski–Plemelj formula, we have: − ϕ2 (t) = Φ+ 2 (t) − Φ2 (t)

− (ST ϕ2 )(t) = Φ+ 2 (t) + Φ2 (t),

(5.2) (5.3)

where Φ± 2 (t) denote the usual nontangential limits of Φ2 (z) for elements z ∈ D and z ∈ C\D, respectively. These instruments allow us to equivalently reduce equation (4.13) to the following boundary problem: Find a function Φ2 (z) sectionally analytic in the corresponding domains (Φ2 (z) = Φ+ 2 (z) for (z) for z ∈ C\D), vanishing at inﬁnity and z ∈ D and Φ2 (z) = Φ− 2 − Φ+ 2 (t) + ψ(t)Φ2 (t) = g(t)

(5.4)

imposed on their boundary values on T, where ψ(t) :=

[ab]2 (t) − aα (t) , [ab]2 (t) + aα (t)

and g(t) :=

[af ]2 (t) + 2λ0 [ab]2 (t) [ab]2 (t) + aα (t)

(5.5)

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with ψ(t) away from zero on T and λ0 := 12 (ST ϕ2 )(1). The problem is considered on Lp (T, ), 1 < p < ∞. This means that the function g belongs to •

p p Lp (T, ) and Φ± 2 must belong to the classes E+ (T, ) and E− (T, ), respectively. As about additional conditions, we will assume that the bounded and measurable function ψ, deﬁned in (5.5), admits a factorization

ψ(t) = ψ+ (t) tℵ ψ− (t)

(5.6)

in Lp (T, ). Moreover, we are also assuming that: −1 1 1 ψ+ g ∈ L1 (T) := H+ (T) ⊕ H− (T),

ϕ+ 0 ϕ− 0

:= :=

p −1 ψ+ PT ψ+ g ∈ H+ (T, ), p −1 −ℵ −1 ψ− t QT ψ+ g ∈ H− (T, ).

(5.7) (5.8) (5.9)

Adapting the methods of [23, §3.1] to the present situation of spaces with weights, it follows that the general solution of problem (5.4) is of the form + Φ+ 2 = ϕ0 + ψ+ pℵ−1 ,

− −1 −ℵ Φ− pℵ−1 , 2 = ϕ0 + ψ− t

(5.10)

where pℵ−1 (z) = p1 + p2 z + · · · + pℵ z ℵ−1 ,

if ℵ ≥ 1,

(5.11)

is a polynomial of degree not greater than ℵ − 1 in case that ℵ > 0, and equal to zero if ℵ ≤ 0. The solutions can be then written in the following form: Φ+ 2 (z) = ψ+ (z)[A(z) + λ0 B(z) + pℵ−1 (z)]

Φ− 2 (z)

=

−1 ψ− (z)z −ℵ [C(z)

+ λ0 D(z) + pℵ−1 (z)].

(5.12) (5.13)

Here, [af ]2 (·) −1 A(z) = PT ψ+ (·) (z), [ab]2 (·) + aα (·) [ab]2 (·) −1 (·) (z), B(z) = PT 2ψ+ [ab]2 (·) + aα (·) [af ]2 (·) −1 (·) (z), C(z) = QT ψ+ [ab]2 (·) + aα (·) [ab]2 (·) −1 (·) (z). D(z) = QT 2ψ+ [ab]2 (·) + aα (·)

(5.14) (5.15) (5.16) (5.17)

In view of not increasing the notation, notice that in the right-hand side of the last four identities we are using the same notation which is used for the Cauchy projections (on T) although these right-hand sides should be •

p p (D, ) or E− (D, ) of the corresponding read as the existing extensions to E± p p functions on H± (T, ) or H− (T, ). The same choice of notation is consistent in the remaining corresponding parts.

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− The function Φ2 = (Φ+ 2 + Φ2 )/2 is a solution of (5.4) if Φ2 (1) = − + Φ2 (1))/2 = λ0 holds. I.e., ℵ + − ψ (1) A(1) + λ B(1) + p + 0 j j=1 Φ2 (1) + Φ2 (1) = 2 2 ℵ −1 (1) C(1) + λ0 D(1) + j=1 pj ψ− = λ0 . + 2 Since the representation of the solutions depends on the (p, )-index ℵ, we divide the analysis in the following different cases. Case ℵ ≥ 0. We start by recalling that pℵ−1 (1) = 0 in case ℵ = 0. Moreover, ⎤ ⎤ ⎡ ⎡ ℵ ℵ

−1 pj ⎦ + ψ − (1) ⎣C(1) + λ0 D(1) + pj ⎦ = 2λ0 ψ+ (1) ⎣A(1) + λ0 B(1) +

(Φ+ 2 (1)

j=1

implies that

⎡ ψ+ (1) ⎣A(1) +

j=1

ℵ

⎤

⎡

−1 pj ⎦ + ψ − (1) ⎣C(1) +

j=1

ℵ

⎤ pj ⎦

j=1

−1 (1)D(1)]. = λ0 [2 − ψ+ (1)B(1) − ψ−

(5.18)

From here we need to consider the following two sub-cases: −1 (i) 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0. From equation (5.18), we get ℵ −1 −1 ψ+ (1)A(1) + ψ− (1)C(1) + (ψ+ (1) + ψ− (1)) j=1 pj λ0 = . (5.19) −1 2 − ψ+ (1)B(1) − ψ− (1)D(1) In this case the general solutions of problem (5.4) are given by the formulas Φ+ 2 (z) = ψ+ (z) [A(z) +

$ % ℵ −1 −1 (1)C(1)+ ψ+ (1)+ψ− (1) ψ+ (1)A(1)+ψ− j=1 pj −1 2−ψ+ (1)B(1)−ψ− (1)D(1)

+pℵ−1 (z)] , Φ− 2 (z)

=

−1 ψ− (z)z −ℵ

+

B(z)

(5.20)

[C(z)

−1 −1 ψ+ (1)A(1)+ψ− (1)C(1)+(ψ+ (1)+ψ− (1))

2−

−1 ψ+ (1)B(1)−ψ− (1)D(1)

ℵ

j=1

+pℵ−1 (z)] ,

pj

D(z) (5.21)

±1 ψ±

(ii)

where are the outer factors in the factorization of the function ψ given in (5.5), A, B, C and D are the functions deﬁned in (5.14)–(5.17) and pℵ−1 is a polynomial of degree less than or equal to ℵ − 1. −1 2−ψ+ (1)B(1)−ψ− (1)D(1) = 0. In this case we have from (5.18) that: −1 −1 ψ+ (1)A(1) + ψ− (1)C(1) + (ψ+ (1) + ψ− (1))

ℵ

j=1

pj = 0.

(5.22)

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Then, the general solutions of problem (5.4) are given by the formulas: Φ+ 2 (z) = ψ+ (z)[A(z) + λ0 B(z) + pℵ−1 (z)]

Φ− 2 (z)

=

−1 ψ− (z)z −ℵ [C(z)

+ λ0 D(z) + pℵ−1 (z)],

(5.23) (5.24)

±1 where ψ± are the outer factors in the factorization of the function ψ given in (5.5), A, B, C and D are the functions deﬁned in (5.14)–(5.17), λ0 is arbitrary and pℵ−1 (z) is a polynomial of degree less than or equal to ℵ − 1 with complex coefﬁcients satisfying condition (5.22).

Case ℵ < 0. The necessary condition for the problem (5.4) to be solvable is that (see [23]) −1 ψ+ (τ )g(τ )τ k dτ = 0, k = 0, . . . , −(ℵ − 1). T

This condition can be written as follows: −1 −1 ψ+ (τ )[af ]2 (τ )τ k ψ+ (τ )[ab]2 (τ )τ k dτ = −2λ0 dτ. [ab]2 (τ ) + aα (τ ) [ab]2 (τ ) + aα (τ ) T

(5.25)

T

In this case pℵ−1 (z) ≡ 0. So, we receive: (i)

−1 (1)D(1) = 0. In this case, by means of equa2 − ψ+ (1)B(1) − ψ− tion (5.18), we get

λ0 =

−1 (1)C(1) ψ+ (1)A(1) + ψ− . −1 2 − ψ+ (1)B(1) − ψ− (1)D(1)

Hence, (5.25) becomes into the following condition −1 −1 ψ+ (1)A(1) + ψ− ψ+ (τ )[af ]2 (τ )τ k (1)C(1) dτ = −2 −1 [ab]2 (τ ) + aα (τ ) 2 − ψ+ (1)B(1) − ψ− (1)D(1) T

× T

−1 ψ+ (τ )[ab]2 (τ )τ k dτ. [ab]2 (τ ) + aα (τ )

(5.26)

If condition (5.26) is satisﬁed, then the solution of the problem (5.4) is given by the following formulas −1 ψ+ (1)A(1) + ψ− (1)C(1) + Φ2 (z) = ψ+ (z) A(z) + B(z) −1 2 − ψ+ (1)B(1) − ψ− (1)D(1) −1 (1)A(1) + ψ (1)C(1) ψ + − − −1 Φ2 (z) = ψ− (z)z −ℵ C(z) + D(z) . −1 2 − ψ+ (1)B(1) − ψ− (1)D(1)

(ii)

±1 The elements ψ± are the outer factors in the factorization of the function ψ given in (5.5), A, B, C and D are the functions deﬁned in (5.14)– (5.17). −1 (1)D(1) = 0. From (5.22), we obtain 2 − ψ+ (1)B(1) − ψ− −1 ψ+ (1)A(1) + ψ− (1)C(1) = 0.

(5.27)

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L. P. Castro and E. M. Rojas

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If condition (5.25) and (5.27) are satisﬁed, then the solution of the problem (5.4) is given by Φ+ 2 (z) = ψ+ (z)[A(z) + λ0 B(z)]

−1 −ℵ [C(z) + λ0 D(z)], Φ− 2 (z) = ψ− (z)z ±1 where, as before, ψ± are the outer factors in the factorization of the function ψ given in (5.5), A, B, C and D are the functions deﬁned in (5.14)–(5.17), λ0 is determined from condition (5.25). In the next theorem we give the explicit representation of the solutions of equation (4.13).

Theorem 5.1. Let us suppose that the functions [ab]2 (t) ± aα (t) do not vanish 2 −aα p on T and that the function ψ = [ab] [ab]2 +aα admits a factorization in L (T, ), say ψ(t) = ψ+ (t)tℵ ψ− (t). −1 (1) If 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0 and ℵ ≥ 0, then equation (4.13) has solutions ϕ2 satisfying the following formula

(ST ϕ2 )(t) = ψ+ (t)[A(t) +

$ % ℵ −1 −1 ψ+ (1)A(1) + ψ− (1)C(1) + ψ+ (1) + ψ− (1) j=1 pj −1 2 − ψ+ (1)B(1) − ψ− (1)D(1)

−1 + pℵ−1 (t)] + ψ− (t)t−ℵ [C(t)

+

−1 −1 ψ+ (1)A(1) + ψ− (1)C(1) + (ψ+ (1) + ψ− (1))

2 − ψ+ (1)B(1) −

−1 ψ− (1)D(1)

ℵ

j=1

+ pℵ−1 (t)].

pj

B(t)

D(t) (5.28)

±1 ψ±

Here are the outer factors in the factorization of the function ψ, A, B, C and D are the functions defined in (5.14)–(5.17), and pℵ−1 is a polynomial of degree less than or equal to ℵ − 1. −1 (1)D(1) = 0 and ℵ < 0, then the equation (4.13) (2) If 2 − ψ+ (1)B(1) − ψ− is solvable if the condition (5.26) is satisfied. In this case, equation (4.13) has a unique solution which satisfies the formula (5.28) where pℵ−1 (t) ≡ 0. −1 (1)D(1) = 0 and ℵ ≥ 0, then the equation (4.13) (3) If 2 − ψ+ (1)B(1) − ψ− has solutions ϕ2 satisfying the following formula: (ST ϕ2 )(t) = ψ+ (t)[A(z) + λ0 B(t) + pℵ−1 (t)] −1 −ℵ +ψ− t [C(t) + λ0 D(t) + pℵ−1 (t)]

(5.29)

±1 where ψ± are the outer factors in the factorization of the function ψ, A, B, C and D are the functions defined in (5.14)–(5.17), λ0 is arbitrary and pℵ−1 is a polynomial of degree less than or equal to ℵ − 1 with complex coefficients satisfying condition (5.22). −1 (1)D(1) = 0 and ℵ < 0, then the equation (4.13) is (4) If 2−ψ+ (1)B(1)−ψ− solvable if the conditions (5.25) and (5.27) are satisfied. In this case, the equation (4.13) has a unique solution which satisfies the formula (5.29), where pℵ−1 (t) ≡ 0 and λ0 is determined from the condition (5.25).

Vol. 70 (2011)

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Proof. It is known that under conditions (5.7)–(5.9) the boundary value problem (5.4) deﬁned on Lp (T, ) has solutions given by (5.10) (see [23]). On the other hand, from the Sokhotski–Plemelj formulas (5.2) and (5.3), we have that equation (4.13) has a solution ϕ2 determined by − ϕ2 (t) = Φ+ 2 (t) − Φ2 (t). − The conclusions are obtained from the equality (ST ϕ2 )(t) = Φ+ 2 (t) + Φ2 (t), applying (for each case) the required conditions. In this way, equation (5.28) is obtained adding equations (5.20) and (5.21), and equation (5.29) from the sum of equations (5.23) and (5.24).

6. The Solutions of Equation (4.1) Satisfying Condition (4.2) As it was shown in previous sections, Theorems 4.3 and 5.1 prove that if [ab]2 (t) ± aα (t) = 0 on T and ψ deﬁned in (5.5) admits a factorization in Lp (T, ) (5.6), then equation (4.3) is solvable in closed form. In this section, we are going to study the solutions of (4.3) (or (4.1) considering (4.2)). As distinguished above, we consider the following cases: −1 (1) 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0, ℵ ≥ 0. From Theorems 4.3 and 5.1 we have that the solutions of (4.3) are given by the following formula: m f (t) − j=1 λj aj (t) − b(t)(P1 ST ϕ2 )(t) , (6.1) ϕ(t) = a(t)

where (ST ϕ2 )(t) is determined by (5.28). From (4.5), we rewrite (5.14)– (5.17) as 2 m 1 j=1 [f (αj+1 (·)) − v=1 λv av (αj+1 (·))] 2 −1 A(z) = PT ψ+ (·) [ab]2 (·) + aα (·) ×a(αj (·)) (z), C(z) = QT

1 −1 (·) 2 ψ+

2

j=1

[f (αj+1 (·)) −

m

v=1

λv av (αj+1 (·))]

[ab]2 (·) + aα (·)

×a(αj (·)) (z). Or, equivalently A(z) = Θ1 (z) − C(z) = Θ2 (z) −

m

v=1 m

v=1

λv Ξ1v (z),

(6.2)

λv Ξ2v (z),

(6.3)

80

L. P. Castro and E. M. Rojas where

−1 ψ+ (·) 12

2

j=1

f (αj+1 (·))a(αj (·))

IEOT

(z), [ab]2 (·) + aα (·) −1 1 2 ψ+ (·) 2 j=1 av (αj+1 (·))a(αj (·)) Ξ1v (z) = PT (z), [ab]2 (·) + aα (·) −1 1 2 ψ+ (·) 2 j=1 f (αj+1 (·))a(αj (·)) Θ2 (z) = QT (z), [ab]2 (·) + aα (·) −1 1 2 ψ+ (·) 2 j=1 av (αj+1 (·))a(αj (·)) Ξ2v (z) = QT (z). [ab]2 (·) + aα (·) Θ1 (z) = PT

(6.4)

(6.5)

(6.6)

(6.7)

Substituting (5.11), (6.2) and (6.3) into (5.28), we have −1 (t)t−ℵ Θ2 (t) (ST ϕ2 )(t) = ψ+ (t)Θ1 (t) + ψ−

+

−1 −1 ψ+ (1)A(1) + ψ− (1)C(1) + (ψ+ (1) + ψ− (1))

2 − ψ+ (1)B(1) −

−1 ψ− (1)D(1)

ℵ

j=1

pj

−1 × (ψ+ (t)B(t) + ψ− (t)t−ℵ D(t)) m

−1 − λv [Ξ1v (t)ψ+ (t) + ψ− (t)t−ℵ Ξ2v (t)] v=1 −1 + (ψ+ (t) + ψ− (t)t−ℵ )

ℵ

pj tj−1 .

j=1

Then, (6.1) can be rewritten in the following form: ϕ(t) =

−1 (t)t−ℵ Θ2 (t)] f (t) − b(t)P1 [ψ+ (t)Θ1 (t) + ψ− a(t) m −1 ψ+ (1)(Θ1 (1)− j=1 λj Ξ1j (1))+ψ− (1)(Θ2 (1) − m j=1 λj Ξ2j (1)) − −1 2 − ψ+ (1)B(1) − ψ− (1)D(1) −1 −1 (ψ+ (1)+ψ− (1)) ℵ (t)t−ℵ D(t)] b(t)P1 [ψ+ (t)B(t)+ψ− j=1 pj + −1 a(t) 2 − ψ+ (1)B(1)−ψ− (1)D(1)

−

m

λj

−1 aj (t) − b(t)P1 [Ξ1j (t)ψ+ (t) + Ξ2j (t)ψ− (t)t−ℵ ] a(t)

pj

−1 b(t)P1 [(ψ+ (t) + ψ− (t)t−ℵ )tj−1 ] , a(t)

j=1

−

ℵ

j=1

(6.8)

±1 with ψ± , B, D, Θ1 , Ξ1j , Θ2 , Ξ2j (j = 1, . . . , m) determined by (5.6), (5.15)–(5.17), (6.4), (6.5), (6.6) and (6.7), respectively, and p1 , . . . , pℵ are arbitrary. The function ϕ is a solution of the equation (4.1) if it satisﬁes condition (4.2), that is:

Mbj (ϕ) = λj ,

j = 1, . . . , m.

Vol. 70 (2011)

Singular Integral Equations

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Substituting (6.8) into the last condition, we obtain ⎡ m

−1 (1)Θ2 (1) − λj [ψ+ (1)Ξ1j (1) λk = dk − ⎣ψ+ (1)Θ1 (1) + ψ− j=1 −1 +ψ− (1)Ξ2j (1)]+

ℵ

⎤

−1 pj (ψ+ (1)+ψ− (1))⎦fk −

j=1

m

ekj λj −

j=1

ℵ

pj gkj

j=1

−1 = [dk − (ψ+ (1)Θ1 (1) + ψ− (1)Θ2 (1))fk ] m

−1 − λj [ekj − (ψ+ (1)Ξ1j (1) + ψ− (1)Ξ2j (1))fk ] j=1

−

ℵ

−1 pj [gkj + (ψ+ (1) + ψ− (1))fk ],

k = 1, . . . , m

(6.9)

j=1

where

dk (t) := Mbk

−1 (t)t−ℵ Θ2 (t)] f (t) − b(t)P1 [ψ+ (t)Θ1 (t) + ψ− a(t)

−1 (t)t−ℵ ] aj (t) − b(t)P1 [Ξ1j (t)ψ+ (t) + Ξ2j (t)ψ− ekj (t) := Mbk a(t) −1 (t)t−ℵ D(t)] b(t)P1 [ψ+ (t)B(t) + ψ− fk (t) := Mbk , −1 a(t)(2 − ψ+ (1)B(1) − ψ− (1)D(1)) $ % −1 b(t)P1 tj−1 (ψ+ (t) + ψ− (t)t−ℵ ) gkj (t) := Mbk . a(t)

, , (6.10)

Putting ⎛

⎞ λ1 %m $ ⎟ ⎜ −1 , E = eij − (ψ+ (1)Ξ1j + ψ− (1)Ξ2j )fi i,j=1 λ = ⎝ ... ⎠ λm m×1 ⎞ ⎛ ⎞ ⎛ −1 (1)Θ2 (1))f1 p1 d1 −(ψ+ (1)Θ1 (1)+ψ− ⎟ ⎜ ⎟ ⎜ .. P = ⎝ ... ⎠ , D=⎝ ⎠ .

−1 dm − (ψ+ (1)Θ1 (1) + ψ− (1)Θ2 (1))fm m×1 ⎞ ⎛ −1 −1 )f1 . . . g1ℵ + (ψ+ (1) + ψ− )f1 g11 + (ψ+ (1) + ψ− ⎟ ⎜ .. .. .. G =⎝ ⎠ . . . −1 −1 gm1 + (ψ+ (1) + ψ− )fm . . . gmℵ + (ψ+ (1) + ψ− )fm m×ℵ

pℵ

ℵ×1

(6.11) we write (6.9) in matricial form (Im×m + E)λ = D − GP.

(6.12)

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Here, Im×m is the (m × m)-identity matrix. So we can formulate that the function determined by (6.8) is a solution of (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the condition (6.12). −1 (2) 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0, ℵ < 0. From Theorems 4.3 and 5.1 it follows that equation (4.3) has solutions if and only if the condition (5.26) is satisﬁed. If this is the case, then pℵ−1 ≡ 0 and the solutions of (4.3) are given as follows: −1 (t)t−ℵ Θ2 (t)] f (t) − b(t)P1 [ψ+ (t)Θ1 (t) + ψ− a(t) m −1 (1) Θ2 (1)− m ψ+ (1) Θ1 (1) − j=1 λj Ξ1j (1) +ψ− j=1 λj Ξ2j (1)

ϕ(t) = −

−1 2 − ψ+ (1)B(1)−ψ− (1)D(1)

−1 b(t)P1 [ψ+ (t)B(t) + ψ− (t)t−ℵ D(t)] a(t) m −1

aj (t) − b(t)P1 [Ξ1j (t)ψ+ (t) + Ξ2j (t)ψ− (t)t−ℵ ] . λj − a(t)

×

(6.13)

j=1

Therefore, the function ϕ determined by (6.13) is a solution of the equation (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the matricial condition (Im×m + E)λ = D,

(6.14)

where E and D are determined by (6.11). On the other hand, substituting (4.5), (6.2) and (6.3) into (5.26), we obtain dk

−

m

ekv λv

−1 = − ψ+ (1)Θ1 (1) + ψ− (1)Θ2 (1) −

v=1

λm [ψ+ (1)Ξ1v (1)

v=1

−1 (1)Ξ2v (1)] +ψ−

m

fk ,

k = 1, . . . , −ℵ,

(6.15)

where dk

:=

−1 ψ+ (τ ) 12

ekv

:= T

fk

j=1

f (αj+1 (τ ))a(αj (τ ))

[ab]2 (τ ) + aα (τ )

T

2

−1 ψ+ (τ ) 12

2

j=1

τ k dτ,

av (αj+1 (τ ))a(αj (τ ))

[ab]2 (τ ) + aα (τ )

τ k dτ,

2

−1 j+1 a(αj (τ ))b(αj+1 (τ ))ψ+ (τ ) k j=1 (−1)

:= T

×

[ab]2 (τ ) + aα (τ ) 1 . −1 2 − ψ+ (1)B(1) − ψ− (1)D(1)

τ dτ

(6.16)

Vol. 70 (2011) Deﬁning

Singular Integral Equations

⎞ −1 (1)Θ2 (1))f1 d1 + (ψ+ (1)Θ1 (1) + ψ− ⎟ ⎜ .. D := ⎝ , ⎠ . −1 d−ℵ + (ψ+ (1)Θ1 (1) + ψ− (1)Θ2 (1))f−ℵ −ℵ×1 %j=1,...,m $ −1 (1)Ξ2j )fi i=1,...,−ℵ , E := eij + (ψ+ (1)Ξ1j + ψ−

83

⎛

(6.17)

we rewrite (6.15) in the matricial form: E λ = D .

(6.18)

Combining (6.14) and (6.18) we conclude that the function ϕ determined by (6.13) is a solution of (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the following matricial identity D Im×m + E λ = . (6.19) D (m−ℵ)×1 E (m−ℵ)×m −1 (1)D(1) = 0, ℵ ≥ 0. In such a case, the solution of (3) 2 − ψ+ (1)B(1) − ψ− the equation (4.3) is given by the formula

ϕ(t) =

−1 f (t) − b(t)P1 [ψ+ (t)Θ1 (t) + ψ− (t)t−ℵ Θ2 (t)] a(t) −1 b(t)P1 [ψ+ (t)B(t) + ψ− (t)t−ℵ D(t)] a(t) m −1

aj (t) − b(t)P1 [Ξ1j (t)ψ+ (t) + Ξ2j (t)ψ− (t)t−ℵ ] − λj a(t) j=1

−λ0

−

ℵ

j=1

pj

−1 b(t)P1 [(ψ+ (t) + ψ− (t)t−ℵ )tj−1 ] , a(t)

(6.20)

±1 where ψ± , B, D, Θ1 , Ξ1j , Θ2 , Ξ2j (j = 1, . . . , m) are determined by (5.6), (5.15)–(5.17), (6.4), (6.5), (6.6) and (6.7) respectively, λ0 is an arbitrary complex number, and p1 , . . . , pℵ satisfy the condition (5.22). Substituting (6.2) and (6.3) into (5.22), we obtain

ψ+ (1)Θ1 (1) +

−1 ψ− (1)Θ2 (1)

−

m

$ % −1 λv ψ+ (1)Ξ1v (1) + ψ− (1)Ξ2v (1)

v=1 −1 + (ψ+ (1) + ψ− (1))

ℵ

pj = 0.

(6.21)

j=1

The function ϕ is a solution of the equation (4.1) if it satisﬁes the condition (4.2). Substituting (6.20) into (4.2), we have λ k = dk − λ 0 hk −

ℵ

j=1

pj gkj −

m

j=1

λj ekj ,

k = 1, 2, . . . , m,

(6.22)

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where dk , ekj and gkj are determined by (6.10) and ( ) −1 b(t)P1 ψ+ (t)B(t) − ψ− (t)t−ℵ D(t) . hk (t) := Mbk a(t) Let

⎛

⎞ λ1 ⎜ ⎟ λ = ⎝ ... ⎠ , λm m×1 ⎞ d1 ⎟ ⎜ D = ⎝ ... ⎠ , dm m×1 ⎛

⎛

g11 ⎜ G = ⎝ ... gm1

... .. . ...

⎛

... .. . ...

e11 ⎜ .. E =⎝ . em1

⎞ e1m .. ⎟ . ⎠ emm m×m

⎞ p1 ⎜ ⎟ P = ⎝ ... ⎠ ⎛

pℵ ⎞ g1ℵ .. ⎟ , . ⎠ gmℵ m×ℵ

(6.23) ℵ×1

⎞ h1 ⎟ ⎜ H = ⎝ ... ⎠ . hm m×1 ⎛

Then, we rewrite (6.22) in the form (Im×m + E)λ = D − λ0 H − GP.

(6.24)

Combining (6.21) and (6.24), we conclude that the function ϕ determined by (6.20) is a solution of (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the following matricial condition: Im×m + Eλ = D − λ0 H − GP.

(6.25)

Here, ⎛ Im×m + E (m+1)×m = ⎝

Im×m + E

⎞

⎠, −1 −1 (1)Ξ21 (1), . . . , ψ+ (1)Ξ1m (1) + ψ− (1)Ξ2m (1) ψ+ (1)Ξ11 (1) + ψ− D H H = , D= , −1 0 (m+1)×1 (1)Θ2 (1) (m+1)×1 ψ+ (1)Θ1 (1) + ψ− G G = . (6.26) −1 −1 )p1 , . . . , −(ψ+ (1) + ψ− )pℵ −(ψ+ (1) + ψ− (m+1)×ℵ

−1 (4) 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0, ℵ < 0. Again, Theorems 4.3 and 5.1 give us that equation (4.3) has solutions if the conditions (5.25) and (5.27) are satisﬁed. Since pℵ−1 ≡ 0, then the solutions of (4.3) are expressed by:

ϕ(t) =

−1 f (t) − b(t)P1 [ψ+ (t)Θ1 (t) + ψ− (t)t−ℵ Θ2 (t)] a(t) −1 b(t)P1 [ψ+ (t)B(t) + ψ− (t)t−ℵ D(t)] a(t) m −1

aj (t) − b(t)P1 [Ξ1j (t)ψ+ (t) + Ξ2j (t)ψ− (t)t−ℵ ] . (6.27) − λj a(t) j=1

−λ0

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The function ϕ determined by (6.27) is a solution of the equation (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the identity Im×m + Eλ = D − λ0 H,

(6.28)

where Im×m + E, D and H are given by (6.26). On the other hand, (5.25) is equivalent to the condition m

dk − ekj λj = λ0 hk , k = 1, · · · − ℵ, (6.29) j=1

with

dk , ekj

hk = −

determined by (6.16) and 2 −1 j+1 a(αj (τ ))b(αj+1 (τ ))ψ+ (τ ) j=1 (−1) [ab]2 (τ ) + aα (τ )

T

Putting

⎞ d1 ⎟ ⎜ D = ⎝ ... ⎠ , d−ℵ −ℵ×1 ⎛ e11 . . . ⎜ .. .. E =⎝ . . ⎛

e−ℵ1

...

τ k dτ.

⎞ h1 ⎟ ⎜ H = ⎝ ... ⎠ , h−ℵ −ℵ×1 ⎞ e1m .. ⎟ , . ⎠ ⎛

e−ℵm

(6.30)

−ℵ×m

we have that (6.29) can be rewritten in the matricial form E λ = D − λ0 H .

(6.31)

Combining (6.28) and (6.31) we can say that the function ϕ determined by (6.27) is a solution of (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the following matricial identity: Im×m + E D H λ = − λ . 0 E D (m+1−ℵ)×1 H (m+1−ℵ)×1 (m+1−ℵ)×m (6.32) Theorem 6.1. Let us suppose that the functions [ab]2 (t) ± aα (t) do not vanish 2 −aα p on T, and that the function ψ = [ab] [ab]2 +aα admits a factorization in L (T, ), say ψ(t) = ψ+ (t)tℵ ψ− (t). −1 (1) If 2 − ψ+ (1)B(1) − ψ− (1)D(1) = 0, ℵ ≥ 0, set r = rank((Im×m + E)

G)m×(m+ℵ) ,

where E, G are determined by (6.11). Then, the equation (4.3) is solvable if and only if the matrix D determined by (6.11) satisfies the condition rank((Im×m + E)

G

D)m×(m+ℵ+1) = r.

If this is the case, then the solutions of the equation (4.3) are given by the formula (6.8), where (λ1 , . . . , λm , p1 , . . . , pℵ ) satisfies (6.12). Moreover, we can choose m + ℵ − r coefficients in {λ1 , . . . , λm , p1 , . . . , pℵ }

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which are arbitrary up to the circumstance of ϕ being uniquely determined by these coefficients. In particular, if r = m then the equation (4.3) is solvable for any function f . −1 (1)D(1) = 0, ℵ < 0. Put (2) 2 − ψ+ (1)B(1) − ψ− Im×m + E , r = rank E (m−ℵ)×m where E and E are determined by (6.11) and (6.17), respectively. The equation (4.3) is solvable if and only if the function f determines D and D by the formulas (6.11) and (6.17) which satisfy the following matricial condition Im×m + E D = r. (6.33) rank E D (m−ℵ)×(m+1) If this is the case, then the solutions of the equation (4.3) are given by the formula (6.13), where (λ1 . . . , λm ) satisfies (6.19). In particular, if r = m and the condition (6.33) is satisfied, then the equation (4.3) has a unique solution. −1 (1)D(1) = 0, ℵ ≥ 0. Choose (3) 2 − ψ+ (1)B(1) − ψ− r = rank(Im×m + E

H

G)(m+1)×(m+1+ℵ) ,

with Im×m + E, H, G determined by (6.26). The equation (4.3) is solvable if and only if the matrix D determined by (6.26) satisfies the condition $ % rank Im×m + E H G D (m+1)×(m+2+ℵ) = r. If the above condition is satisfied, then the solutions of equation (4.3) are given by the formula (6.20), where (λ0 , . . . , λm , p1 , . . . , pℵ ) satisfies (6.25). Moreover, we can choose m + 1 + ℵ − r coefficients in {λ0 , . . . , λm , p1 , . . . , pℵ } which are arbitrary so that ϕ is uniquely determined by these coefficients. In particular, if r = m + 1 then the equation (4.3) is solvable for any function f . −1 (1)D(1) = 0, ℵ < 0. Put (4) 2 − ψ+ (1)B(1) − ψ− Im×m + E H r = rank , E H where Im×m + E, H, E , H are determined by (6.26) and (6.30). Then, the equation (4.3) is solvable if and only if the function f determines D and D by the formulas (6.26) and (6.30) which satisfy the condition ⎞ ⎛ Im×m + E H D ⎠ rank ⎝ = r. (6.34) E H D (m+1−ℵ)×(m+2)

If the condition (6.34) is satisfied, then the solutions of the equation (4.3) are given by the formula (6.27), where (λ0 , . . . , λm ) satisfies (6.32). In

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particular, if r = m + 1 and the condition (6.34) is satisfied, then the equation (4.3) has a unique solution. Proof. (1.) From the assumption it follows that the equation (4.3) has solutions if and only if there exist (λ1 , . . . , λm ) and (p1 , . . . , pℵ ) which satisfy (6.12). We can rewrite (6.12) in the form λ = D. ((Im×m + E) G)m×(m+ℵ) P (m+ℵ)×1 λ Therefore, is a solution of the following equation P ((Im×m + E)

G)X = D.

(6.35)

It follows that the necessary and sufﬁcient condition for which the equation (4.3) has solutions, is that the equation (6.35) has solutions in Cm+ℵ . Since rank((Im×m + E) G D) = rank((Im×m + E) G) = r, then using (6.35) we can express r coefﬁcients in {λ1 , . . . , λm , p1 , . . . , pℵ } by m + ℵ − r remaining ones. In particular, if r = m then the equation (6.35) has solutions with any D. Therefore the equation (4.3) is solvable with any f . The cases (2.), (3.) and (4.) are proved in a similar way.

7. The Solvability of Equation (2.4) for Commutative or Anti-Commutative Carleman Shift In this section we are going to study the solvability of the equation (2.4) in the cases when the Carleman shift function θ is of commutative or anticommutative type. We will rewrite the corresponding results of the previous sections for each one of these cases. 7.1. Properties of the Solutions of Equation (2.4) Let us introduce the weighted Carleman shift operator on Lp (T) as in Sect. 2: (W ϕ)(t) = v(t)ϕ(θ(t)),

t ∈ T.

We are going to assume henceforth that W is of commutative or anti-commutative type. The operator W allow us to deﬁne the complementary projections P1 := satisfying W k =

2

1 (IT − W ) and 2

kj j=1 (−1) Pj ,

P2 :=

1 (IT + W ) 2

(7.1)

k = 1, 2.

(7.2)

k = 1, 2, and

2

Pk =

1 (−1)k(1−j) W j+1 , 2 j=1

Notice that Propositions 3.1 and 3.2 hold for these projections Pk , k = 1, 2. Now, the corresponding result to Proposition 3.3 has the following form:

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Proposition 7.1. Let ψ ∈ Lp (T). Then, for z ∈ C, we have (ST Pk ψ)(z), if W ST = ST W (Pk ST ψ)(z) = (ST P3−k ψ)(z), if W ST = −ST W .

IEOT

(7.3)

Proof. We directly have

1 (ST ϕ)(z) + (−1)k W (ST ϕ)(z) 2

1 (ST ϕ)(z) ± (−1)k (ST W ϕ)(z) = 2 1 = ST (ϕ ± (−1)k W ϕ)(z) 2

(Pk ST ψ)(z) =

in ST W = W ST and ST W = −W ST cases, respectively. From here, equality (7.3) follows. Now, using projection P1 , equation (2.4) is rewritten as m

1 a(t)ϕ(t) + b(t)(P1 ST ϕ)(t) + aj (t) πi j=1

bj (τ )ϕ(τ )dτ = f (t).

(7.4)

T

As in the previous sections we assume that a(t) is a non-vanishing function on T. We denote by Mbj , j = 1, . . . , m, the linear functional on Lp (T) deﬁned as 1 bj (τ )ϕ(τ )dτ. Mbj (ϕ) := πi T

Putting Mbj (ϕ) = λj ,

j = 1, . . . m,

(7.5)

then (7.4) appears with the form a(t)ϕ(t) + b(t)(P1 ST ϕ)(t) = f (t) −

m

λj aj (t).

(7.6)

j=1

A corresponding result to the former Lemma 4.1 appears now in the present case as follows: Proposition 7.2. Let ϕ ∈ Lp (T). Then ϕ is a solution of (7.6) if and only if {ϕk := Pk ϕ, k = 1, 2} is a solution of the following system or

aθ (t)ϕk (t) + [ab]∗3−k (t)(ST ϕ1 )(t) = [af ]k (t),

if ST W = W ST

aθ (t)ϕk (t) + [ab]∗3−k (t)(ST ϕ2 )(t) = [af ]k (t),

if ST W = −W ST

(7.7)

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where, for k = 1, 2, aθ (t) = a(θ(t))a(t) 2

1 (−1)(3−k)(1−j) [a(θj (t))b(θj+1 (t))] 2 j=1 ⎛⎡ ⎤ ⎞ m

[af ]k (t) = Pk ⎝⎣f (t) − λj aj (t)⎦ a(θ(t))⎠ . ∗

[ab]3−k (t) =

(7.8)

j=1

p

Proof. Suppose that ϕ ∈ L (T) is a solution of (7.6). Multiplying by a(θ(t)) and applying the projections Pk (k = 1, 2), we have

Pk (a(θ(t))a(t)ϕ(t) + a(θ(t))b(t)(P1 ST ϕ)(t)) = Pk

f (t) −

m

λj aj (t) a(θ(t)) .

j=1

(7.9)

By using (7.2), we can verify that Pk [a(θ(t))a(t)ϕ(t)](t) = a(θ(t))a(t)(Pk ϕ)(t) Pk [a(θ(t))b(t)(P1 ST ϕ)](t) = [ab]∗3−k (t)(P1 ST ϕ)(t). Therefore, we are able to rewrite (7.9) as

⎛⎡

a(θ(t))a(t)(Pk ϕ(t)) + [ab]∗3−k (t)(P1 ST ϕ)(t) = Pk ⎝⎣f (t) −

m

⎤

⎞

λj aj (t)⎦ a(θ(t))⎠ .

j=1

Now, by Proposition 7.1, we have that P1 ST = ST P1 for the ST W = W ST case and P1 ST = ST P2 for the case of ST W = −W ST . Thus (P1 ϕ, P2 ϕ) is a solution of (7.7). Conversely, suppose that there exists ϕ such that (P1 ϕ, P2 ϕ) is a solution of (7.7). Summing k from 1 to 2, we directly obtain that 2

(

) a(θ(t))a(t)ϕk (t) + [ab]∗3−k (t)(ST ϕi )(t)

k=1

=

2

⎛⎡ Pk ⎝⎣f (t) −

k=1

is equivalent to

m

⎤

⎞

λj aj (t)⎦ a(θ(t))⎠,

i = 1, 2,

j=1

⎡

a(θ(t))a(t)ϕ(t) + a(θ(t))b(t)(ST ϕi )(t) = ⎣f (t) −

m

⎤ λj aj (t)⎦ a(θ(t)),

j=1

i = 1, 2 and this implies that a(t)ϕ(t) + b(t)P1 (ST ϕ)(t) = f (t) −

m

λj aj (t).

j=1

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Proposition 7.3. If (φ1 , φ2 ) is a solution of the system (7.7), then (P1 φ1 , P2 φ2 ) is also a solution of (7.7). Proof. Let (φ1 , φ2 ) be a solution of the system (7.7). Applying the projections Pk to both sides of (7.7), we have $ % Pk aθ (t)φk (t) + [ab]∗3−k (t)(ST φi )(t) = Pk ([af ]k (t)), k, i = 1, 2. (7.10) Notice that Pk [aθ (t)φk ](t) = aθ (t)Pk φk (t) and 1 Pk ([ab]∗3−k (t)(ST φi ))(t) = [ab]∗3−k (t)(ST φi )(t) 2

+(−1)k [ab]∗3−k (θ(t))W (ST φi )(t) 1 = [ab]∗3−k (t) {(ST φi )(t) − W (ST φi )(t)}. (7.11) 2 Equality (7.11) holds because [ab]∗3−k (t) = (−1)3−k [ab]∗3−k (θ(t)). Thus the right-hand side of equality (7.11) can be rewritten as [ab]∗3−k (t)P1 (ST φi )(t). From (7.7), the value of the index i depends on the commuting property of the shift operator with ST . Therefore, Pk ([ab]∗3−k (t)(ST φi ))(t) = [ab]∗3−k (t)(ST Pi φi )(t). Finally, note that Pk [af ]k (t) = [af ]k (t). Therefore, (P1 φ1 , P2 φ2 ) is a solution of (7.7). A corresponding result to the previous Theorem 4.3 also holds in the present case, and takes the following form. Theorem 7.4. The equation (7.6) has solutions in Lp (T) if and only if the following equation aθ (t)ϕ1 (t) + [ab]∗2 (t)(ST ϕ1 )(t) = [af ]1 (t), or aθ (t)ϕ2 (t) + [ab]∗1 (t)(ST ϕ2 )(t) = [af ]2 (t),

if ST W = W ST (7.12) if ST W = −W ST

has solutions. Moreover, if ϕk (t)(k = 1, 2) is a solution of equation (7.12), then equation (7.6) has a solution given by the formula ⎧ m ⎨ f (t)− j=1 λj aj (t)−b(t)(P1 ST ϕ1 )(t) , if S W = W S T T (7.13) ϕ(t) = f (t)−m λj aj a(t) (t)−b(t)(P1 ST ϕ2 )(t) j=1 ⎩ , if ST W = −W ST . a(t) Proof. Suppose that ϕ ∈ Lp (T) is a solution of equation (7.6). By Proposition 7.2 we know that (P1 ϕ, P2 ϕ) is a solution of system (7.7). Hence, for the ST W = W ST case, P1 ϕ is a solution of (7.12) and P2 ϕ is the corresponding solution for the ST W = −W ST case. Conversely, suppose that ϕ1 is a solution of (7.12). Without loss of generality, we assume now that ST W = W ST (since the situation of ST W = −W ST is dealt similarly). In this case, the system (7.7) has a solution (ϕ1 , ϕ2 ) determined by [af ]2 (t) − [ab]∗1 (t)(ST ϕ1 )(t) . (7.14) ϕ2 (t) = aθ (t)

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By Proposition 7.3 we have that (P1 ϕ1 , P2 ϕ2 ) is also a solution of (7.7). Set ϕ = P1 ϕ1 + P2 ϕ2 . It is clear that Pk ϕ = Pk ϕk . This means that (P1 ϕ, P2 ϕ) is a solution of (7.9). From Proposition 7.2 it follows that ϕ is a solution of (7.7). Moreover, from (7.14), we obtain 2

[af ]k (t) − [ab]∗3−k (ST ϕ1 )(t) Pk ϕ(t) = . (7.15) aθ (t) k=1

As before, we can see that 2

⎡

Pk [af ]k (t) = ⎣f (t) −

⎤ λj aj (t)⎦ a(θ(t)),

j=1

k=1 2

m

Pk ([ab]∗3−k (ST ϕ1 )(t)) = a(θ(t))b(t)(P1 ST ϕ1 )(t).

k=1

Thus, substituting these in (7.15), we have m f (t) − j=1 λj aj (t) − b(t)(P1 ST ϕ1 )(t) . ϕ(t) = a(t) 7.2. The BVP Associated to Equation (7.12) Equation (7.12) can be solved as in Sects. 5 and 6. I.e., by means of an associated Riemann boundary value problem. Letting ⎧ [ab]∗2 (t) − aθ (t) ⎪ ⎪ , if ST W = W ST ⎪ ⎪ ⎨ [ab]∗2 (t) + aθ (t)

(t) = ⎪ ∗ ⎪ ⎪ [ab]1 (t) − aθ (t) ⎪ , if ST W = −W ST , ⎩ [ab]∗1 (t) + aθ (t) ⎧ [af ]1 (t) ⎪ ⎪ , if ST W = W ST ⎪ ∗ ⎪ ⎨ [ab]2 (t) + aθ (t) h(t) = ⎪ ⎪ [af ]2 (t) ⎪ ⎪ , if ST W = −W ST , ⎩ ∗ [ab]1 (t) + aθ (t) 1 ϕk (t) = (ϕ(t) + (−1)k v(t)ϕ(θ(t))), k = 1, 2 2 and ϕk (τ ) 1 dτ, z ∈ C\T. Πk (t) = 2πi τ −z T

We get that equation (7.12) is reduced to the problem − Π+ k (t) + (t)Πk (t) = h(t)

(7.16)

imposed on their boundary values on T, where k = 1 in case of W being a commutative weighted Carleman shift operator, and k = 2 in case W being a weighted Carleman shift operator of anti-commutative type.

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In order to solve this problem, we assume that the functions admit a factorization in Lp (T) (as the function ψ in Sect. 5). I.e.,

(t) = + (t)tℵ − (t).

(7.17)

Furthermore, we assume that: −1 1 1

+ h ∈ L1 (T) := (H+ (T) ⊕ H− (T)),

υ0+ υ0−

(7.18)

p −1

+ PT + h ∈ H+ (T), p −1 −ℵ −1

− t QT + h ∈ H− (T).

:= :=

(7.19) (7.20)

The general solutions of problem (7.16) (with k = 1 if ST W = W ST and k = 2 in case ST W = −W ST ) have the form + Π+ k = υ0 + + pℵ−1 ,

− −1 −ℵ Π− pℵ−1 k = υ0 + − t

where pℵ−1 (z) = p1 + p2 z + · · · + pℵ z ℵ−1 ,

if ℵ ≥ 1

is a polynomial of degree less than or equal to ℵ − 1 if ℵ > 0, and equal to zero if ℵ ≤ 0. The representation of the solutions can be rewritten in the following form: Π+ k (z) = + (z)[A(z) + pℵ−1 (z)]

Π− k (z)

=

−1

− (z)z −ℵ [C(z)

+ pℵ−1 (z)],

where the functions A and C are given by ⎧ [af ]1 (·) −1 ⎪ P

(·) ⎪ ∗ + [ab] (·)+aθ (·) (z), ⎨ T A(z) =

C(z) =

2

(7.21) (7.22)

if W ST = ST W

⎪ ⎪ ]2 (·) −1 ⎩ PT + (z), if W ST = −ST W , (·) [ab][af ∗ (·)+a (·) θ 1 ⎧ [af ]1 (·) −1 ⎪ ⎪ ⎨ QT + (·) [ab]∗ (·)+aθ (·) (z), if W ST = ST W

(7.23)

⎪ ⎪ −1 ⎩ QT + (·)

(7.24)

2

[af ]2 (·) [ab]∗ 1 (·)+aθ (·)

(z),

if W ST = −ST W .

In addition, for the case of ℵ < 0, the problem (7.16) is solvable if the following conditions hold true: −1

+ (τ )h(τ )τ k dτ = 0, k = 0, . . . , −(ℵ − 1). T

This condition can be rewritten as follows: ⎧ −1

+ (τ )[af ]1 (τ ) k ⎪ ⎪ τ dτ = 0, if W ST = ST W ⎪ ∗ ⎪ [ab]2 (τ ) + aθ (τ ) ⎪ ⎪ ⎨T ⎪ −1 ⎪

+ (τ )[af ]2 (τ ) k ⎪ ⎪ ⎪ ⎪ ∗ ⎩ [ab]1 (τ ) + aθ (τ ) τ dτ = 0,

(7.25) if W ST = −ST W ,

T

where k = 0, . . . , −(ℵ − 1). The following result summarize all the above mentioned.

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Theorem 7.5. Suppose that the functions [ab]∗k (t) ± aθ (t)(k = 1, 2) do not [ab]∗ −a vanish on T and that the functions = [ab]k∗ +aθθ (k = 1, 2) admit a factork

ization in Lp (T), say (t) = + (t)tℵ − (t). If the p-index ℵ is greater or equal to zero, then equation (7.12) has solutions ϕk (k = 1, 2) which satisfy the following formula −1 −ℵ t [C(t) + pℵ−1 (t)], (ST ϕk )(t) = + (t)[A(z) + pℵ−1 (t)] + −

(7.26)

±1 where ± are the outer factors (and their inverses) in the factorization of the functions . A and C are the functions defined in (7.23) and (7.24), and pℵ−1 is a polynomial of degree less than or equals to ℵ − 1 which is identically equal to zero if ℵ = 0. In the case that ℵ < 0, the equation (7.12) is solvable if the condition (7.25) is satisfied. In this case equation (7.12) has a unique solution which satisfies the formula (7.26), where pℵ−1 (t) ≡ 0.

Proof. We know that under conditions (7.18)–(7.20) the boundary value problem (7.16) deﬁned on Lp (T) has a solution given by (7.21) and (7.22). On the other hand, from the Sokhotski–Plemelj formulas we have that equation (7.12) has solutions ϕk (for k = 1, 2 depending on the commutative property of the shift operator W ) determined by −

(t) = Π+ k (t) − Πk (t). − The conclusions are obtained from (ST ϕk )(t) = Π+ k (t) + Πk (t), applying the required conditions.

7.3. The Explicit Solutions of Equation (2.4) Conditioned to (7.5) In this part, we will exhibit the explicit representation of the solutions of equation (2.4) satisfying condition (7.5). We are going to use the solutions given in Theorem 7.5 for such a goal. Since the representation of the solutions depend on the sign of the p-index ℵ of the factorization (7.17), we will consider the next two different cases: Case ℵ ≥ 0 From Theorem 7.4 we know that the solutions of equation (7.6) are given by ⎧ f (t)−m λ a (t)−b(t)(P S ϕ )(t) 1 T 1 j=1 j j ⎨ , if ST W = W ST a(t) (7.27) ϕ(t) = f (t)−m λ a (t)−b(t)(P S ϕ )(t) 1 T 2 ⎩ j=1 j j , if S W = −W S . T T a(t) Moreover, by Theorem 7.5 we know that −1 −ℵ (ST ϕk )(t) = + (t)[A(z) + pℵ−1 (t)] + − t [C(t) + pℵ−1 (t)]

(7.28)

where, similarly to (6.2) and (6.3), A and C have the form A(z) = Θ1 (z) − C(z) = Θ2 (z) −

m

ς=1 m

ς=1

λς Ξ1ς (z)

(7.29)

λς Ξ2ς (z),

(7.30)

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with Θ1 , Ξ1ς , Θ2 and Ξ2ς (v = 1, . . . , m) in this case deﬁned by the following identities ⎧ −1 + (·) 12 2j=1 (−1)j+1 f (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ PT (z), ⎪ [ab]∗ ⎪ 2 (·)+aθ (·) ⎪ ⎪ ⎪ ⎪ ⎨ if ST W = W ST Θ1 (z) = (7.31) −1 ⎪ ⎪ + (·) 12 2j=1 f (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ PT (z), ⎪ [ab]∗ ⎪ 1 (·)+aθ (·) ⎪ ⎪ ⎩ if ST W = −W ST , ⎧ −1 + (·) 12 2j=1 (−1)j+1 aς (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ (z), ⎪ [ab]∗ ⎪ PT 2 (·)+aθ (·) ⎪ ⎪ ⎪ ⎪ if ST W = W ST ⎨ Ξ1ς (z) = (7.32) −1 ⎪ ⎪ + (·) 12 2j=1 aς (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ PT (z), ⎪ [ab]∗ ⎪ 1 (·)+aθ (·) ⎪ ⎪ ⎩ if ST W = −W ST , ⎧ −1 2 j+1 1 + (·) 2 j=1 (−1) f (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ (z), ⎪ [ab]∗ ⎪ QT 2 (·)+aθ (·) ⎪ ⎪ ⎪ ⎪ if ST W = W ST ⎨ Θ2 (z) = (7.33) −1 ⎪ ⎪ + (·) 12 2j=1 (f (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ QT (z), ⎪ [ab]∗ ⎪ 1 (·)+aθ (·) ⎪ ⎪ ⎩ if ST W = −W ST , ⎧ −1 2 j+1 1 + (·) 2 j=1 (−1) aς (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ (z), ⎪ QT [ab]∗ ⎪ 2 (·)+aθ (·) ⎪ ⎪ ⎪ ⎪ if ST W = W ST ⎨ Ξ2ς (z) = (7.34) −1 ⎪ ⎪ + (·) 12 2j=1 aς (θj+1 (·))a(θj (·))(v(·))j−1 ⎪ ⎪ QT (z), ⎪ [ab]∗ ⎪ 1 (·)+aθ (·) ⎪ ⎪ ⎩ if ST W = −W ST . Substituting (7.29) and (7.30) into (7.28), we obtain (ST ϕk )(t) = + (t)Θ1 (t) +

−1 −ℵ

− t Θ2 (t)

−

m

λς ( + (t)Ξ1ς (t)

ς=1 −1 −1 + − (t)Ξ2ς (t)t−ℵ ) + ( + (t) + − (t)t−ℵ )

ℵ

pj tj−1 .

j=1

Then, we can rewrite (7.27) in the form ϕ(t) =

−1 f (t) − b(t)P1 [ + (t)Θ1 (t) + − (t)t−ℵ Θ2 (t)] a(t) m −1

aj (t) − b(t)P1 [Ξ1j (t) + (t) + Ξ2j (t) − (t)t−ℵ ] − λj a(t) j=1

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ℵ

j=1

pj

−1 b(t)P1 [( + (t) + − (t)t−ℵ )tj−1 ] a(t)

95

(7.35)

±1 with ± , Θ1 , Ξ1j , Θ2 , Ξ2j (j = 1, . . . , m) determined by (7.17), (7.31), (7.32), (7.33) and (7.34), respectively (where we recall that the form of these functions depend on the commutative nature of the weighted Carleman shift operator W , and p1 , . . . , pℵ are arbitrary). The function ϕ is a solution of the equation (2.4) if it satisﬁes the condition (7.5) that is:

Mbj (ϕ) = λj ,

j = 1, . . . , m.

Substituting (7.35) into the last condition, we obtain λ ι = dι −

m

j=1

λj eιj −

ℵ

pj gιj ,

ι = 1, . . . , m

(7.36)

j=1

where dι , eιj and gιj are given by −1 (t)t−ℵ Θ2 (t)] f (t) − b(t)P1 [ + (t)Θ1 (t) + − , dι (t) := Mbι a(t) −1 (t)t−ℵ ] aj (t) − b(t)P1 [Ξ1j (t) + (t) + Ξ2j (t) − eιj (t) := Mbι , a(t) $ % −1 b(t)P1 tj−1 ( + (t) + − (t)t−ℵ ) gιj (t) := Mbι . (7.37) a(t) So, we can rewrite equation (7.36) in the form of the following matricial identity: (Im×m + E)λ = D − GP

(7.38)

with E, D, G, P and λ as in (6.23) but with the entries in (7.37). Thus, we can formulate that the function determined by (7.35) is a solution of (4.1) if and only if (λ1 , . . . , λm ) satisﬁes the condition (7.38). Case ℵ < 0 In this case, from Theorems 7.4 and 7.5 we have that equation (7.6) has solutions if the condition (7.25) is satisﬁed. Since pℵ−1 ≡ 0, then the solutions of (7.6) are given by ϕ(t) =

−1 f (t) − b(t)P1 [ + (t)Θ1 (t) + − (t)t−ℵ Θ2 (t)] a(t) m −1

aj (t) − b(t)P1 [Ξ1j (t) + (t) + Ξ2j (t) − (t)t−ℵ ] . − λj a(t) j=1

(7.39) The equation (7.39) is a solution of equation (7.4) if and only if the element (λ1 , . . . , λm ) satisﬁes the following matricial condition (Im×m + E)λ = D.

(7.40)

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On the other hand, since condition (7.25) is necessary for the solvability of the problem (7.16), then we rewrite it (using (7.8)) as dη =

m

eης λς

(7.41)

ς=1

with dη and eης given by ⎧ 2 −1

+ (τ ) 12 j=1 (−1)j+1 f (θj+1 (t))a(θj (τ ))(v(τ ))j−1 η ⎪ ⎪ ⎪ τ dτ, ⎪ ⎪ [ab]∗2 (τ ) + aθ (τ ) ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ if ST W = W ST ⎨ (7.42) dη = 2 ⎪ −1 1 j−1 ⎪ ⎪

(τ ) f (θ (t))a(θ (τ ))(v(τ )) j+1 j ⎪ + j=1 2 ⎪ τ η dτ, ⎪ ⎪ ⎪ [ab]∗1 (τ ) + aθ (τ ) ⎪ ⎪ ⎪ ⎩T if ST W = −W ST

nd

eης

⎧ 2 −1

+ (τ ) 12 j=1 (−1)j+1 aς (θj+1 (t))a(θj (τ ))(v(τ ))j−1 η ⎪ ⎪ ⎪ τ dτ, ⎪ ⎪ [ab]∗2 (τ ) + aθ (τ ) ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ if ST W = W ST ⎨ = (7.43) 2 ⎪ −1 1 j−1 ⎪ ⎪ + (τ ) 2 j=1 aς (θj+1 (t))a(θj (τ ))(v(τ )) ⎪ ⎪ τ η dτ, ⎪ ⎪ ⎪ [ab]∗1 (τ ) + aθ (τ ) ⎪ ⎪ ⎪ ⎩T if ST W = −W ST .

As in (6.18), equality (7.41) can be written in the following matricial form D = E λ.

(7.44)

The matrices D and E are deﬁned as in (6.30) with entries given in (7.42) and (7.43). Combining (7.40) and (7.44), we conclude that ϕ determined by (7.39) is a solution of (7.4) if and only (λ1 , . . . λm ) satisﬁes the following matricial condition D Im×m + E λ= . (7.45) D (m−ℵ)×1 E (m−ℵ)×m Theorem 7.6. Suppose that the functions [ab]∗k (t)±aθ (t)(k = 1, 2) do not van[ab]∗ −a ish on T, and that the functions = [ab]k∗ +aθθ (k = 1, 2) admit a factorization in Lp (T), say (t) = + (t)tℵ − (t). (1) If ℵ ≥ 0, consider

k

r = rank((Im×m + E)

G)m×(m+ℵ) ,

where E and G are defined as in (6.23) but with entries given by (7.37). Then, the equation (7.6) is solvable if and only if the matrix D, determined as in (6.23) and having entries defined by (7.37), satisfies the

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condition rank((Im×m + E)

G

D)m×(m+ℵ+1) = r.

If this is the case, the solutions of the equation (7.6) are given by the formula (7.35), where (λ1 , . . . , λm , p1 , . . . , pℵ ) satisfies (7.38). Moreover, we can choose m + ℵ − r coefficients in {λ1 , . . . , λm , p1 , . . . , pℵ } which are arbitrary so that ϕ is uniquely determined by these coefficients. In particular, if r = m then the equation (7.6) is solvable for any function f. (2) If ℵ < 0, let Im×m + E , r = rank E (m−ℵ)×m with E and E as in (6.23) and (6.30) whose entries are given in (7.37) and (7.42), respectively. The equation (7.6) is solvable if and only if the function f determines D and D as in the formulas (6.23) and (6.30) (with entries on (7.37) and (7.42), respectively) which satisfy the following matricial condition Im×m + E D rank = r. (7.46) E D (m−ℵ)×(m+1) If this is the case, the solutions of the equation (7.6) are given by the formula (7.39), where (λ1 . . . , λm ) satisfies (7.45). In particular, if r = m and the condition (7.46) is satisfied, then the equation (7.6) has a unique solution. Proof. The proof runs analogously to the proof of Theorem 6.1.

Acknowledgements The authors are thankful to the referee for very constructive comments and suggestions.

References [1] Bastos, M.A., Fernandes, C.A., Karlovich, Y.I.: Spectral measures in C ∗ -algebras of singular integral operators with shifts. J. Funct. Anal. 242, 86–126 (2007) [2] Baturev, A.A., Kravchenko, V.G., Litvinchuk, G.: Approximate methods for singular integral equations with a non-Carleman shift. J. Integral Equations Appl. 8, 1–17 (1996) [3] Boiarskii, B.V.: On generalized Hilbert boundary value problems, Soobsh. Akad. Nauk Gruz. SSR, 25(4), 385–390 (1960) [4] Carleman, T.: Application de la th´eorie des ´equations int´egrales lin´eaires aux syst´emes d’ ´equations diﬀ´erentielles non lin´eaires. Acta Math. 59, 63–87 (1932) [5] Castro, L.P., Rojas, E.M.: Reduction of singular integral operators with ﬂip and their Fredholm property. Lobachevskii J. Math. 29, 119–129 (2008)

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[6] Castro, L.P., Rojas, E.M.: Explicit solutions of Cauchy singular integral equations with weighted Carleman shift. J. Math. Anal. Appl. 371, 128–133 (2010) [7] Castro, L.P., Rojas, E.M.: Invertibility of singular integral operators with ﬂip through explicit operator relations. In: Integral Methods in Science and Engineering, vol. 1, pp. 105–114. Birkh¨ auser, Boston, (2010) [8] Chaun, L.H., Mau, N.V., Tuan, N.M.: On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel. Complex Var. Elliptic Equ. 53, 117–137 (2008) [9] Chaun, L.H., Tuan, N.M.: On singular integral equations with the Carleman shifts in the case of the vanishing coefﬁcient. Acta Math. Vietnam. 28, 319– 333 (2003) [10] Clancey K.F., Gohberg I.: Factorization Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol. 3. Birkh¨ auser, Basel-Boston-Stuttgart (1981) [11] Ferreira, J., Litvinchuk, G.S., Reis, M.D.L.: Calculation of the defect numbers of the generalized Hilbert and Carleman boundary value problems with linear fractional Carleman shift. Integr. Equ. Oper. Theory 57, 185–207 (2007) [12] Gakhov, F.D.: Boundary Values Problems. Dover, New York (1990) [13] Haseman, C.: Anwendung der Theorie der Integralgleichungen auf einige Randwertaufgaben. Gottingen (1911) ¨ [14] Hilbert, D.: Uber eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie. Verh. d. 3. intern. Math. Kongr. Heidelb. pp 233–240 (1905, in German) [15] Karapetiants, N., Samko, S.: Singular integral equations on the real line with a fractional-linear Carleman shift. Proc. A. Razmadze Math. Inst. 124, 73– 106 (2000) [16] Karapetiants, N., Samko, S.: Equations with Involutive Operators. Birkh¨ auser, Boston (2001) [17] Karelin, A.A.: Applications of operator equalities to singular integral operators and to Riemann boundary value problems. Math. Nachr. 280, 1108–1117 (2007) [18] Kravchenko, V.G., Lebre, A.B., Litvinchuk, G.S., Teixeira, F.S.: A normalization problem for a class of singular integral operators with Carleman shift and unbounded coefﬁcients. Integr. Equ. Oper. Theory 21, 342–354 (1995) [19] Kravchenko, V.G., Lebre, A.B., Rodr´ıguez, J.S.: Factorization of singular integral operators with a Carleman shift and spectral problems. J. Integr. Equ. Appl. 13, 339–383 (2001) [20] Kravchenko, V.G., Lebre, A.B., Rodr´ıguez, J.S.: Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case. Math. Nachr. 280, 1157–1175 (2007) [21] Kravchenko, V.G., Litvinchuk, G.S.: Introduction to the Theory of Singular Integral Operators with Shift. Kluwer, Amsterdam (1994) [22] Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and its Applications, vol. 523. Kluwer, Dordrecht (2000) [23] Litvinchuk, G.S., Spitkovsky, I.M.: Factorization of Measurable Matrix Functions. Operator Theory: Advances and Applications, vol. 25. Birkh¨ auser, Basel-Boston (1987)

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[24] Markouchevitch, A.: Quelques remarques sur les int´egrales du type de Cauchy. Uˇcenye Zapiski Moskov. Gos. Univ. 100. Matematika 1, 31–33 (1946, in Russian, French summary) [25] Mikhlin, S.G., Pr¨ ossdorf, S.: Singular Integral Operators. Springer, Berlin (1980) [26] Riemann, B.: Lehrs¨ atze aus der analysis situs f¨ ur die Theorie der Integrale von zweigliedrigen vollst¨ andigen Diﬀerentialen. J. Reine Angew. Math. 54, 105–110 (1857, in German) [27] Riemann, B.: Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachtr¨ age [Collected mathematical works, scientiﬁc Nachlass and addenda] Based on the edition by Heinrich Weber and Richard Dedekind. Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], Suppl. 1. BSB B. G. Teubner Verlagsgesellschaft, Leipzig; Springer, Berlin (1990, in German) [28] Tuan, N.M.: On a class of singular integral equations with rotation. Acta Math. Vietnam. 21, 201–211 (1996) [29] Vekua, N.P.: On a generalized system of singular integral equations. Soobˇsˇceniya Akad. Nauk Gruzin. SSR. 9, 153–160 (1948, in Russian) [30] Vekua, N. P.: The Carleman boundary problem for several unknown functions. Soobˇsˇceniya Akad. Nauk Gruzin. SSR. 13, 9–14 (1952, in Russian) L. P. Castro (B) Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal e-mail: [email protected] E. M. Rojas Departamento de Matem´ aticas Pontiﬁcia Universidad Javeriana Bogot´ a Colombia e-mail: [email protected] Received: July 22, 2010. Revised: February 19, 2011.

Integr. Equ. Oper. Theory 70 (2011), 101–123 DOI 10.1007/s00020-010-1860-1 Published online February 11, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Contractivity of Projections Commuting with Inner Derivations on JBW∗-triples Remo V. H¨ ugli Abstract. It is shown that if P is a weak∗ -continuous projection on a JBW∗ -triple A with predual A∗ , such that the range P A of P is an atomic subtriple with ﬁnite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of P A, then P is contractive if and only if it commutes with all inner derivations of P A. This provides characterizations of 1-complemented elements in a large class of subspaces of A∗ in terms of commutation relations. Mathematics Subject Classification (2010). Primary 17C65; Secondary 47D27. Keywords. JBW∗ -triple, inner automorphism, inner derivation, contractive projection.

1. Introduction The central idea put forward in this article is that commutation of a projection P with inner derivations in certain Banach-Jordan structures is, under extensive conditions, equivalent with P being contractive. This underpins the principle of commutation as a simple algebraic characterization of the geometric property of contractivity. The said principle was ﬁrst observed in the context of Hilbert spaces equipped with the Jordan-triple product of type-I Cartan-factors [21]. Here we consider atomic subtriples with ﬁnite-dimensional Cartan-factors of any of the six types in a general JBW∗ -triple A. We extend those main results of [21] concerning the relationships of contractive projections on A and on the pre-dual A∗ with the group Inn(A) of inner automorphisms and its Banach-Lie algebra inn(A) of inner derivations of A. In general, the explicit characterization of the 1-complemented subspaces P E of a Banach space E and of the corresponding contractive projections P on E is a highly non-trivial problem. If E carries also algebraic structure related to the norm, then the problem might be solved in terms of algebraic conditions. The categories consisting of JBW∗ -triples A or of Partly supported by the Science Foundation of Ireland (grant no. 05/IN1/I853).

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the associated “pre-symmetric” spaces A∗ are particularly well suited for investigating such problems; these categories are stable under contractive projections, as seen from the conditional expectations by Friedman and Russo [11], Kaup [23] and Stach´ o [31], and they contain various operator algebras such as B(H, H ) (for complex Hilbert spaces H, H ), W ∗ -algebras and JBW∗ -algebras. The conditional expectations provide necessary conditions for subspaces of A and A∗ to be 1-complemented, but typically, little information is known about sufﬁcient conditions. As shown in [21] and further worked out here, commutation relations make available both, necessary and sufﬁcient conditions for contractivity of projections and 1-complementation of subspaces. The outlines of this paper are as follows: In Sect. 2 we present preliminary facts about contractive projections on normed spaces E and the geometric relations of Lp -orthogonality. Section 3 contains some basics on the theory of JB(W)∗ -triples and their inner derivations and inner automorphisms. It follows from results by Loos [26] that in the case of a ﬁnite dimensional Cartan-factor C, the group Inn(C) of inner automorphisms acts transitively on the rank-classes of tripotents of C (see e.g. [26, Theorem 5.3]). Besides commutation, this transitivity property, which has recently been shown to hold for all Cartan factors of arbitrary dimension by Mackey and this author [22], is another key-ingredient of the present article. Theorem 4.1 shows that, under some technical but quite mild assumptions, a weak∗ -continuous projection P on a JBW∗ -triple A onto a subtriple B = P A is contractive if and only if it commutes with all inner derivations of B. In the ﬁnal section, we focus on properties of the pre-adjoint P∗ : A → A∗ of P . In Lemma 4.3, the condition that P is the sum of coordinate projections with respect to a standard grid of P A is shown to be generic, and hence it does not restrict the generality of the arguments. Theorem 4.10 focuses on the subspaces of A∗ and gives criteria for these to be 1-complemented. For general information on Jordan-algebras and -triples we refer to the monographs [1,14–17,26,29,30,32,33]. Developments in the description of contractive projections on operator-structures which involve Jordan tripleproducts can be found in [9,27,28].

2. Contractive Projections We brieﬂy review some speciﬁc definitions and concepts that will be important in this text. The results presented in this section are either well known, or they follow from standard arguments in functional analysis. All vector spaces used here are over the ﬁeld C of complex numbers, though some results also hold for real spaces. When E is a vector space with a norm ., the closed unit ball and the unit sphere of E are denoted by E1 and S1 (E) respectively, and (a · x) denotes the dual paring of elements x in E and a in the Banach dual space E ∗ of E. For an arbitrary set K, let K f in denote the set of all ﬁnite subsets of K. Clearly, K f in has partial order with respect to set inclusion. There are three particular relations of metric orthogonality to

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be deﬁned on E. We use the same notation as that in our earlier articles. Two elements x and y of E are said to be L1 -orthogonal, denoted x ♦ y, if x ± y = x + y, and are said to be L∞ -orthogonal, denoted x y, if x ± y = max{x, y}. These relations can be generalized in an obvious way to Lp -orthogonality, for p ≤ 1 ≤ ∞, but we are mainly interested in the cases p = 1 and p = ∞, in which the Lp -complements G♦ and G of a non-empty subset G ⊆ E are deﬁned, respectively, by G♦ := {y ∈ E : x ♦ y ∀x ∈ G}, G := {y ∈ E : x y ∀x ∈ G}. Observe that in general, these complements of a subset G in E need not be subspaces of E. Let P1 (E) and Pb (E) denote the sets of contractive projections and of bounded projections on E respectively. An element P of P1 (E) is said to be a GL-projection if (P E)♦ ⊆ kerP (i.e. P ((P E)♦ ) = {0}). The set of all GL-projections on E is denoted by PGL (E). The above metric orthogonality relations are said to hold for two projections P and Q, if they hold for all elements x of P E and y of QE. It is immediate from the definition of GL-projections that, for P, Q ∈ PGL (E), the relation P ♦ Q implies that P and Q are orthogonal as projections in that P Q = QP = 0. For p > 1, this algebraic orthogonality holds for Lp -orthogonal contractive projections. In particular, for p = ∞, we state Proposition 2.1. For P, Q ∈ P1 (E) the relation P Q implies that P Q = QP = 0. Proof. For all x ∈ E, with x = 1, the contractivity and L∞ -orthogonality of P and Q imply that, P Qx2 + P Qx = (P Qx + 1)P Qx ≤ P QxQx + P Qx = max{P QxQx, P Qx} = P Qx. Therefore, P Qx is zero. Exchanging P and Q shows that also QP x is zero.

Remarkably, when E is the predual A∗ of a JBW∗ -triple, and P and Q are elements of PGL (A∗ ), then P A∗ = QA∗ implies that P = Q. This shows that PGL (A∗ ) has a partial order with respect to set inclusion of the ranges of its elements [4,6,19]. Consider a locally convex Hausdorﬀ topology τ on E. A family (Pk )k∈K of elements in B(E) is said to be τ -operator summable with sum P ∈ P B(E) if, for each x ∈ E, the net J → k∈J k x from the partial order (K f in , ⊆) (i.e. J ⊆ K and |J| < ∞) to E is τ -convergent and normbounded. If so, the uniform boundedness principle implies that the mapping x → (τ ) limJ∈K f in k∈J PG x is an element of B(E). Notice that normoperator summability in the above sense is the usual SOT-summability of the occurring sum. For a ﬁnite subset J ∈ K f in we deﬁne PJ := j∈J Pj . The following result, similar to [29, IV. 3.10], is used to obtain examples of weak∗ -summable families. Proposition 2.2. Let E be a Banach space, let (Pk )k∈K be a family of bounded projections on E having the properties that, for all j, k ∈ K with j = k,

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(Pk E) is a subspace of E which contains Pj (E), and there is a constant M with supk∈K Pk = M < ∞. Then, the following statements hold: . (1) Set S := lin k∈K Pk E , and suppose that, for all k ∈ K, the restriction Pk |S of Pk to S is contractive. Then, (Pk )k∈K is norm-operator summable with sum P of norm M . (2) Suppose that E is the dual of some Banach space E∗ , and each Pk is σ(E,E∗ ) . If, for all k ∈ K, σ(E, E∗ )-continuous. Set S := lin k∈K Pk E the restriction Pk |S of P to S is contractive, then (Pk )k∈K is σ(E, E∗ )operator summable with sum P of norm M . Proof. We give a proof of (2). A proof of (1) is obtained from similar, or partly more straightforward arguments. Consider the dual pair (E, E∗ ), and denote by S0 the annihilator of S in E∗ . Standard Banach space theory shows that S = (E∗ /S0 )∗ , i.e. that S∗ = E ∗ /S0 is a predual of S, and that the σ(S, S∗ )-continuous functionals are precisely the restrictions x|S for x ∈ E∗ . As shown in Proposition 2.1, L∞ -orthogonal contractive projections annihilate each other. It follows that PJ := j∈J Pj is a projection, and that if F, J ∈ K f in are disjoint sets, then PJ PF = PF PJ = 0. Therefore PF ∪J = PF + PJ . Since Pk (E) is a subspace, an induction argument implies that, for each a ∈ E ∗ , PJ (a) = max{Pj a, j ∈ J} ≤ M a.

(2.1)

It follows that PJ is a σ(E, E∗ )-continuous projection of norm at most M on E. Hence, PJ has a preadjoint, denoted QJ , on E∗ . To be precise, QJ is an element of Pb (E∗ ) such that (QJ )∗ = PJ , and QJ = PJ . Moreover QJ |S is the preadjoint of PJ |S on S∗ . Given disjoint sets J and K of K f in , consider elements x of QJ (E∗ ) and y of QK (E∗ ). From the relation PJ PF , we conclude that the unit-ball of (PJ + PK )(E) is given by PJ (E)1 ⊕ PK (E)1 = PJ∪K (E)1 , and hence that x + y = sup{|(x · a1 )| + |(y · a2 )| : a1 ∈ PJ (E)1 , a2 ∈ PK (E)1 } = sup{|((x ± y) · a)| : a ∈ PJ∪K (E)1 } ≤ sup{|((x ± y) · a)| : a ∈ E1 } = x ± y. This conﬁrms that the relation QF ♦ QJ holds. It follows that, for all elements x of E∗ and F ∈ K f in , Q{k} (x) = QF (x) = PF (x) ≤ M x. k∈F

For x ∈ E, deﬁne Mx := supF ∈K f in PF (x). Then, for all ε > 0, there exists Fε ∈ K f in such that Mx − ε ≤ QFε (x) ≤ Mx . For any J ∈ K f in with J ∩ Fε = ∅, we have that Mx − ε ≤ QFε ∪J (x) = QFε (x) + QJ (x) ≤ Mx , QJ (x) ≤ ε.

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This shows that the assignment F → QF (x) : K f in → E is a Cauchy-net in E∗ . The norm-limit Q(x) := limF →∞ QF (x) deﬁnes a projection Q on the predual E∗ of E. The adjoint P := Q∗ of Q is the σ(E ∗ , E)-operator sum of (Pk )k∈K . Clearly P is the sought after projection. Remark 2.3. 1. It is sufﬁcient to assume that Pk (E) be a subspace of lin k∈K Pk E in both statements above. 2. The equality in (2.1) does not follow simply because the images Pk (E) are mutually L∞ -orthogonal. The subspace property is needed here, as can be seen from (counter-)Example 3 below. 3. As a well known consequence of the classical Hahn-Banach theorem shows, each one-dimensional subspace Cx of E is the range of at least one contractive projection P on E, and that P is deﬁned, for y ∈ E, by setting P (y) = (a·y)x, when x and a ∈ E ∗ are such that a = x = (a·x) = 1. In the case in which E is the predual of a JBW∗ -triple, each one-dimensional subspace is the range of precisely one GL-projection on E, (see [6,19]). The concept of projective systems, given by Definition 2.4, combines Hahn-Banach extensions of coordinate functionals with the notion of summability of the corresponding projections. A projective system can be regarded as a generalized metrically orthogonal set. This is illustrated by the examples 1 and 2 below. Definition 2.4. Let τ be a locally convex Hausdorﬀ topology on E, compatible with the norm-topology in that the τ -continuous linear forms are also norm-continuous. A τ -projective system in E is a pair (S, x → Px ) where S is a subset S of the unit sphere S1 (E) of E, and the mapping x → Px is from S to P1 (E), with the properties that (1) for x ∈ S, the projection Px is a τ -continuous Hahn-Banach extension of the identity on Cx to E; (2) when x and y are distinct elements of S, then Px Py = 0; (3) the family (Px )x∈S is τ -operator summable. We will also call the set S by itself a τ -projective system to mean that there all the stated conditions. If these exists a map x → Px : S → P1 (E) satisfying hold, then the corresponding sum PS = x∈S Px is necessarily a projection, and will be referred to as the projection associated with the τ -projective system S. It is also easily seen that, for all x ∈ S, Px P S = P S P x = P x .

(2.2)

An alternative definition may be given as follows: Denote by E τ the τ ∗ τ τ -dual of E. The compatibility condition means that E ⊆ E . Let E⊗E be the τ -operator closure of the set of sums j∈J xj ⊗ fj ∈ B(E) where xj ∈ E, fj ∈ E τ and |J| < ∞. A τ -projective system is an element P of E⊗E τ of the form P = j∈J xj ⊗ fj , where xj ∈ S1 (E), fj ∈ S1 (E τ ) and (xi · fj ) = δ(i, j), for i, j ∈ J. Example 1. Let H be a Hilbert space with unit sphere S1 (H). For x ∈ H, let Px denote the orthoprojection onto Cx. Given elements x and y of S1 (H), we see that Px Py = 0 is equivalent with x, y = 0. Hence, if x and y are

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Hilbert-orthogonal, then ({x, y}, x → Px , y → Py ) is a norm-projective system with sum equal to the orthoprojection onto Cx ⊕ Cy. Definition 2.4 can be applied analogously to arbitrary orthonormal subsets of S1 (H). Example 2. Let (ak )k∈K be a family of norm-one elements in the dual E ∗ of a Banach space E. Suppose that for each j ∈ K the L∞ -complement a j of aj is a subspace of E ∗ containing (ak∈K\{j} ), and that for each j ∈ K, there exists a σ(E ∗ , E)-continuous Hahn-Banach extension Paj of the identity on Caj (or equivalently, that the functionals aj are norm-attaining at elements xj of E1 ). Then, by Proposition 2.2 ((ak )k∈K , k → Pak ) is a σ(E ∗ , E)-projective system. It is clear that the properties of being a projection or of being contractive does not in general extend from a set of operators to their sum, nor from the sum to the summands. It is instructive to consider perhaps less evident “counterexamples” concerning the contractivity of sums of projections. 4 Example 3. Consider the space R4 = k=1 Rek . For j = 1, 2, 3, 4, let Pk be the coordinate projection on R4 onto Rek . Deﬁne B1,2 , B1,3 , B2,3 and B0,4 to be the sets 4 αl el ∈ R4 : αl = 0 if l ∈ / {j, k}, max{|αj |, |αk |} = 1 Bj,k := l=1

((j, k) := (1, 2), (1, 3), (2, 3)), 4 4 e4 , −e4 , ek , − ek .

B0,4 :=

k=1

k=1

The convex hull B of the union of these sets is balanced and absorbing. The corresponding Minkowsky functional is a norm .B on R4 . Since the coordinate projections Pl map each Bj,k into B, they are contractive with respect to this norm. Moreover, the subspaces Re1 , Re2 and Re3 are mutually L∞ -orthogonal. 4 Clearly, the sum P = P1 + P2 + P3 is a projection. The element x := k=1 ek is such that xB = 1, but P xB = 3/2. Hence P is not contractive. Observe also that, for j = 1, 2, 3, 4, (Rej ) is not a subspace. The next result is a straightforward observation, but will help to clarify some subtleties in later calculations involving GL-projections on the predual of a JBW∗ -triple. We consider surjective linear mappings S : B → C between subspaces B and C of a normed space E, i.e. S is a “partial isometry” in the context of normed vector spaces. Such a mapping is necessarily invertible. Note also that S has a “partial” adjoint S : C ∗ → B ∗ , where the symbol , rather than ∗, is used to indicate an adjoint operator conﬁned to these subspaces. Similarly, S : C∗ → B∗ is deﬁned on the possible preduals if such exists; Since this is the case for JBW∗ -triples considered later, Proposition 2.5 is phrased for mapping to the predual level, but it obviously holds verbatim for the corresponding dual version (with subscript ∗ and as superscript etc.). For T ⊆ E∗ , let T 0 := {a ∈ E : (a · x) = 0, ∀x ∈ T } denote the annihilator of T in E.

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Proposition 2.5. Let E be a Banach space which is the dual of a normed vector space E∗ . Let P and R be contractive projections on E∗ having the same range P (E∗ ) = R(E∗ ) =: T . Set B := R∗ E, C := P ∗ E. By contractivity of R and P , it is unambiguous to identify the preduals B∗ and C∗ with T . Let S : B → C be a σ(B, B∗ ) − σ(C, C∗ )-continuous surjective linear isometry, with pre-adjont S : C∗ → B∗ . Then, the following conditions are equivalent: S = idT , (S − id|B )(B) ⊆ T 0 , ∗

∗

SR = P ,

S−1 = idT ,

(2.3)

(S −1 − id|C )(C) ⊆ T 0 ,

(2.4)

S −1 P ∗ = R∗ .

(2.5)

Proof. Obviously it is sufﬁcient to perform the proof for the left column: (2.3) ⇒ (2.4): The assumptions about continuity of S imply that for each x ∈ B∗ , the mapping b → (Sb, x) is an element of C∗ and is equal to S x by definition, that is (Sb · x) = (b · S x). Hence, if B∗ = C∗ and S = idT , then (Sb − b · x) = 0, for all b ∈ B and x ∈ T , proving (2.4). (2.4) ⇒ (2.5): The assumptions imply that, for all x ∈ E∗ and a ∈ E, (SR∗ (a) · x) = (P ∗ SR∗ (a) · x) = (a · RP (x)) = (a · P (x)) = (P ∗ (a) · x). Condition (2.4) is implicit in the second equality. This proves (2.5). (2.5) ⇒ (2.3): Condition (2.5) means that (SR∗ (a) · x) = (P ∗ (a) · x), for all a ∈ E and x ∈ E∗ . Choose a in B and x in T to obtain (2.3).

3. JB(W)∗ -triples, Inner Derivations and Grids The notation and terminology used here is largely the same as in [10,14, 26,29,30,32], to which we refer the reader for detailed accounts, including the complete axiomatic settings of the theory of Jordan-triple structures. In what follows, A denotes a JB∗ -triple or a JBW∗ -triple with triple product { . . . } : A3 → A. The predual of a JBW∗ -triple, unique up to isometric isomorphisms, is denoted by A∗ . Recall that the group Aut(A) of linear triple-automorphisms is a real Banach-Lie group with Banach-Lie algebra aut(A) consisting of the triple derivations, i.e. linear maps δ : A → A with the property that δ{a, b, c} = {δ(a), b, c} + {a, δ(b), c} + {a, b, δ(c)} [32], and that Aut(A) coincides with the group of bijective linear isometries of A [24]. The real vector space inn(A) of inner derivations of A is the Lie-subalgebra (and ideal) inn(A) of aut(A) spanned by the linear operators iD(a, a), where D(a, b) : A → A is deﬁned by D(a, b)(c) := {a, b, c} (for a, b, c ∈ A). The exponentials exp(itD(a, a)) (for t ∈ R) generate the group Inn(A) of inner automorphisms of A, which is a normal subgroup of Aut(A). The set of tripotents (the elements u ∈ A with {u, u, u} = u) is denoted by U(A). A non-zero element a ∈ A is said to be minimal if {a, A, a} = Ca. An atomic JBW∗ -triple A is the weak∗ -closed span of its minimal tripotents. The basic algebraic relations between elements u and v of U(A) are that

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of orthogonality u⊥v which holds if D(u v) = 0, collinearity (termed coorthogonality in [14]) uv if 2D(u u)v = v and 2D(v v)u = u, as well as the relation u v (read “u governs v”) if D(u u)v = v and 2D(v v)u = u. A maximal orthogonal set of minimal tripotents is said to be a frame, a key-concept in our subsequent considerations. Denote by F(A) the collection of all frames in A. By a well known completeness property of atomic ∗ JBW∗ -triples, any element a of A can be written as the weak -convergent sum a = u∈F αu u, for some frame F = F (a) ∈ F and non-negative coefﬁcients αu [12,14]. If a ∈ U(A), then αu ∈ {0, 1}, for all u ∈ F (a). The cardinality |{u ∈ F (a) : αu = 1}| is referred to as the rank of a. The rank r of A is the maximal rank of any of its elements, and we have that r = |F | for all F ∈ F [14,26,29]. For any cardinality r, let Ur (A) denote the set of tripotents having rank r. The set of minimal tripotents is U1 (A), and, using the conjugate-linear operator Q(a) : A → A deﬁned as Q(a)(b) := {a, b, a}, it can be seen that U1 (A) = {u ∈ U(A) : Q(u)(A) = Cu} [29]. Let u and v be tripotents of A, which are such that u ⊥ (v − u). Then, u is said to be less than or equal to v, denoted u ≤ v. This relation provides a partial order on the set U(A) of all tripotents in A, and the minimality in the above sense is equivalent to minimality of u in (U(A)\{0}, ≤). Classiﬁcation theory identiﬁes the Cartan-factors as the irreducible components of atomic JBW∗ -triples, and hence as important examples thereof. We present some fundamental facts about the Cartan-factors. For more details, including complete definitions see the sources cited in these paragraphs. Our focus is on the role played by grids and frames within the ﬁnite-dimensional cases. The standard references used for this paragraph are [3,14,29]. Let G and H be complex Hilbert spaces. A Cartan factor of type I, also referred to as rectangular type, is (isometrically) equal to the space B(G, H) of bounded operators from G to H is a JBW∗ -triple, and equipped with the triple product {a, b, c} = 12 (ab∗ c + cb∗ a). A rectangular grid is given by the (ﬁnite or inﬁnite) set of matrix units; Let (gi )i∈I and (hj )j∈J be orthonormal bases of G and H, respectively. Then, for (i, j) ∈ I × J, the matrix unit uij ∈ B(G, H) is a minimal tripotent of C, and GI := (uij )(i,j)∈I×J is a rectangular grid. Let a† denote conjugation and transposition of a ∈ B(H) := B(H, H), with respect to (hi )i∈I . The type II and III factors, referred to as symplectic and hermitian, are the subspaces of B(H) consisting of elements with the properties that a = −a† and a = a† , respectively. The corresponding standard grids are given by GII := (uij − uji )ij∈I,i<j and GIII := (uij + uji )i,j∈I,i<j ∪ (uii )i∈I , respectively. A hermitian grid GIII is non-orthocollinear since the relation holds for certain elements of GIII . It is worth highlighting the explicit algebraic relations in the non-orthocollinear cases (there is one more such case, the odd-dimensional spin grids). The tripotents uij = uji ∈ GIII satisfy ujk uij uii ,

for distinct i, j, k ∈ I,

(3.1)

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if {i, j} ∩ {k, l} = ∅.

109 (3.2)

For arbitrary indices i, j, k, l ∈ I, the triple products that are not of the form {uij , ujk , ukl } vanish. These products, if involving at least two different tripotents are 1 {uij , ujk , ukl } = uil , if i = l, (3.3) 2 (3.4) {uij , ujk , uki } = uii . For the case when I = {1, . . . , n} and J = {1, . . . , m}, we set r := min{m, n}. Then, F := (uii )i∈{1,...,r} constitutes a frame of type I as well as type III and F := (uij − uji )i<j, ∈I constitutes a frame of type III. It is seen that r, is the rank of C. A spin factor S (type IV) is obtained by setting S := H + ⊕ H − or S := Cu0 ⊕ H + ⊕ H − (the “even” and “odd” subtype), where H + and H − + are isomorphic Hilbert spaces. A basis (u+ is conjugate to a basis i )i∈I of H − − (ui )i∈I of H . We set u0 := (1, 0, 0) in the case of an odd spin factor. This − establishes a grid GIV := (u+ i )i∈I ∪ (ui )i∈I (∪u0 ) for S. The odd-dimensional spin grids are non-orthocollinear. The algebraic relations within the grids GIV are as follows: For j, k ∈ I and ε, μ ∈ {+, −}, u0 uεj ⊥ u−ε j , uεj

uμk ,

if j = k.

(3.5) (3.6)

The determining triple products are, {u0 , uεj , u0 } = u−ε j , 1 u0 , {uεj , u0 , u−ε j } = 2 μ −μ {uεj , uk , u−ε j } = −uk ,

(3.7) (3.8) (3.9)

and all other products involving three different tripotents of GIV vanish. A − (non-trivial) spin factor has rank 2, and (u+ i , ui ) is a frame, for any i ∈ I. 5 The type V or bi-Cayley factor C consists of the space of 1×2 matrices with entries in the (complex) split-octonions Os . Let {cεi : ε ∈ {−1, 1}, j ∈ {1, . . . , 4}} be the basis of Os as explicitly given in [29, III. Sect. 3.1], and ε∈{−1,1} set uεj := (cεj , 0) and uεj+4 := (0, cεj ). Then GV := (uεj )j∈{1,...,8} constitutes a grid for C 5 . The bi-Cayley factor has rank two, and for ﬁxed j, ε, the pair 5 (uεj , u−ε j ) is a frame in C . The type VI factor C 6 , or Albert factor, is represented as the space H3 (Os ) of hermitian 3 × 3 matrices over the split-octonions Os . Let : ε ε Os → Os be the linear mapping determined by cε1 = c−ε 1 , and cj = −cj (j ∈ {2, 3, 4}). A matrix (mij )3i,j=1 ∈ Mat3×3 (Os ) lies in C 6 if and only if 5 kε ε ε mij = mji . The elements ukε ij ∈ C deﬁned for i < j by uij := ck eij + ck eji k∈{1,...,4},ε∈{+,−}

and ujj := 1Os ejj (j = 1, 2, 3) form a grid GVI := (ukε ∪ ij )i<j∈{1,2,3} (ujj )j∈{1,2,3} of C 6 . It is known that C 6 is also a Jordan-algebra with product ◦ : (a, b) → 12 (ab + ba), and the triple product is obtained from the Jordan product by {a b c} = a ◦ (b ◦ c) + c ◦ (a ◦ b ) − b ◦ (a ◦ c), where is the

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standard conjugate involution on Os combined with transposition (cf. [29, III. (3.8)]). The factor C 5 is contained in C 6 as a subtriple via the embedding (a, b) → ae21 + a e12 + be31 + b e13 . The factor C 6 has rank 3 with a frame F = (ujj )j=1,2,3 . The following observation will be useful later on. The statement is obvious from the above remarks. Proposition 3.1. Each standard grid of a Cartan-factor C contains at least one frame F of C as a subset. The grids of type I, III, V and VI are said to be ortho-collinear, because their elements u, v say, satisfy either u⊥v or uv, but the relation u v is absent in these structures. As for u v, we make a further observation from the description of the standard grids. Proposition 3.2. Let G be a standard grid as described above. Then for u, v ∈ G, the relation u v implies that (1) for v † := Q(u)(v), the elements v, w+ := 12 (u + v + v † ) and w− := 1 † 2 (u − v − v ) are tripotents of rank 1, i.e. are minimal within the corresponding Cartan-factor C, (2) the rank of u in C is 2. On the other hand, if u does not govern any v ∈ G, i.e. if u v, ∀v ∈ G, then the rank of u in C is 1. Hence, for any grid G as described and arbitrary u ∈ G, we have that 1 ⇔ ∀v ∈ G : u v . (3.10) rankC (u) = 2 ⇔ ∃v ∈ G : u v Proof. (1): The relation u v occurs only in the grids of type II or IV. Explicitly, we ﬁnd that for type II, u = eij + eji and either v = eii , or v = ejj . For type IV, we have u = h0 , v = hi and v † = hi . It is easily checked by calculation that in each case, v is such that {v z v} = δ(v, z)v, for z ∈ G. Hence v is minimal. An easy way to see that w± := 12 (u ± (v + v † )) are minimal is to use the fact that the JB∗ -triple T (u, v) generated by u and v is isometrically isomorphic to the three-dimensional factor of type II, which is represented as a subtriple of B(C2 ), when we identify u idC2 , v e12 and v † e21 (this factor is also isomorphic to C 4 , with grid {h0 , h1 , h1 } as given above). It is thus straightforward to check that w± and v are tripotents of rank 1 in T (u, v). Therefore, these are mapped onto one-another by some elements φ of the inner automorphism-group Inn(T (u, v)) [22,26]. Since Inn(T (u, v)) is a subgroup of Inn(C), the rank r of v and w± also agrees in C. As shown, r = 1. (2): This follows from (1) and the fact that w+ ⊥w− and u = w+ + w− . To verify that u v, ∀v ∈ G implies rankC (u) = 1 is a matter of checking the possible cases within the hermitian grids and odd-dimensional spin grids. All other grids are ortho-collinear systems of minimal tripotents [14,29]. A standard grid of a ﬁnite dimensional Cartan factor C, contained in A a subtriple, gives rise to a projective system. This requires some calculations involving the fundamental algebraic relations (⊥, , ). We also need

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the extension properties that were found in [2] concerning weak∗ -continuous linear functionals on JBW∗ -triples. Lemma 3.3. Let C be a finite-dimensional Cartan factor which is a subtriple of a JB∗ -triple A, and let G be a standard grid of C. Then, G is a norm-projective system. If A is a JBW∗ -triple, then G is also a weak∗ -projective system. In particular, for each element u ∈ G, there exists a (σ(A, A∗ )-continuous) Hahn-Banach extension P˜u ∈ P1 (A) of the identity id|Cu on Cu to A, such that (1) the sum P˜ := u∈G P˜u is a (σ(A, A∗ )-continuous) projection, (2) the projection P˜u is an extension from C to A of the coordinate projection (denoted Pu ) with respect to G. Proof. It is clear that, G ⊆ S1 (A) and that for all u ∈ G, the coordinate projection Pu , w.r.t. the basis G is bounded on C with one dimensional range Cu. When v is another element of G then Pu Pv = Pv Pu = 0.

(3.11)

The elements u and v obey one of the relations u ⊥ v, u v, v u or u v, all of which imply that the restriction P |Cu⊕Cv is contractive on the two-dimensional subspace Cu ⊕ Cv. One way to see this is to use the fact that the subspace Cu ⊕ Cv can be isometrically embedded B(C2 ). An embedding is given for the case u ⊥ v by u → e11 , v → e22 ; for u v by y → e11 , v → e12 ; for u v by setting u → id, and v → e12 , where eij are the standard matrix units [29]. That Pu |Cu⊕Cv is contractive is veriﬁed with simple calculations in all of these cases. Since Pu can be represented as Pu = u ⊗ fu , where fu is the linear functional on C deﬁned for u, v ∈ G by fu (v) = δ(u, v)u, a Hahn-Banach extension f˜u of fu gives rise to the soughtafter element P˜u := u ⊗ f˜u . The relation (3.11) remains valid when we write over these projections the tilde indicating Hahn-Banach extensions. Since C is ﬁnite dimensional, P˜C := u∈G P˜u is a bounded projection on A. Suppose that A is a JBW∗ -triple. We use the main result of [2] which ascertains that each σ(C, C∗ )-continuous linear functional has a σ(A, A∗ )continuous Hahn-Banach extension. Since in our situation dimCk < ∞, every linear functional f on Ck is σ(C, C∗ )-continuous. Therefore, the Hahn-Banach extension P˜u of Pu and the resulting projection P˜ = u∈G P˜u can be taken to be σ(A, A∗ )-continuous on A. The significance of contractive projections to the theory of JB∗ - and JBW -triples is revealed, for instance, by the conditional expectations that are valid within these categories. By the results of [23] and [31], the range P A of a contractive projection P on a JB∗ -triple A is itself a JB∗ -triple when equipped with the restricted triple product {. . .}P A , deﬁned for elements a, b and c in P A by ∗

{a, b, c}P A := P {a, b, c}.

(3.12)

It follows immediately that when P is the adjoint P = R∗ of a contractive projection R on A∗ , then P A is a JBW∗ -triple with respect to the restricted

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triple-product. The conditional expectation formulas found in [11,13,23] are as follows: For all a, b, c ∈ A, P {P a, P b, P c} = P {P a, b, P c}, P {P a, P b, P c} = P {a, P b, P c}.

(3.13) (3.14)

The next result, is easily shown to be a consequence of the identity (3.14) above. The identity (3.13) implies commutation of P with Q(b) (for b ∈ P A), but we will not need this case here. For a proof see [21]. Lemma 3.4. ([21, Theorem 4.1]) Let A be a JB∗ -triple, and let P be a contractive projection on A. Then, the following conditions are equivalent. {P A, P A, P A} ⊆ P A, [D(b, b), P ] = 0, ∀b ∈ P A,

(3.15) (3.16)

[D(a, b), P ] = 0, ∀a, b ∈ P A.

(3.17)

Condition (3.15) means that P A is a subtriple. Since the operators iD(b, b) are inner derivations of A, the above forms of conditional expectations indicate that the sets Der(A) and P1 (A) of operators on A are interconnected via the commutation relations. In particular, for P ∈ Pb (A), a necessary condition for contractivity of P (i.e. P ∈ P1 (A)) is that P commutes with D(b, b) for all b ∈ P (A). We are going to investigate the possibilities of the converse implications; We make general assumptions under which the condition that [P, D(b, b)] = 0, ∀b ∈ P (A) and P ∈ Pb (A) is sufﬁcient for P to be contractive. We need some further information about the relationships of inner automorphisms and L∞ -orthogonality, which is equivalent with algebraic orthogonality for tripotent elements [20]. A crucial step, providing the transitivity of Inn(A) on the set of frames F(A), is given by the following lemma, stated for ﬁnite-rank triples, which obviously include the ﬁnite-dimensional cases. Lemma 3.5. Let A be a JBW∗ -triple of finite rank. Then, for any two frames F1 and F2 of A there exists an inner automorphism ϕ of A such that ϕ(F1 ) = F2 . Proof. A ﬁnite-rank JBW∗ -triple is necessarily atomic. Since every atomic JBW∗ -triple is the direct orthogonal sum of Cartan-factors, we may ﬁrst assume that A is a Cartan-factor. It was shown in [22], 4.3 that any two orthogonal sets U := {u1 , . . . , un } and V = {v1 , . . . , vn } of n < ∞ minimal tripotents in A are conjugate by an element ϕ of Inn(A). Since, in the present case A has ﬁnite rank r = |F1 | = |F2 |, setting F1 = U and F2 = V gives the desired statement. If A is not itself a Cartan-factor, it is equal w∗ to A = for an orthogonal family (Ck )k∈K of Cartan-factors. k∈K Ck Since A has ﬁnite rank, there can be at most ﬁnitely many such factors; A = C 1 ⊕ · · · ⊕ C m . Then, the frames Fj ∈ F(A) (with j = 1, 2) are of the form F = Fj1 × · · · × Fjm , with Fik ∈ F(C k ). Moreover, for n, j ∈ {1, . . . , m}, the group Inn(Cn ) acts as the identity on Cj when j = n. The argument used for A being a Cartan-factor can be applied componentwise, which completes the proof.

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The crucial condition that the manifold F(A) of all frames in A is Inn(A)-isotropic can also be derived from the results by Loos [26], for ﬁnite dimensional Jordan pairs (V, V − ). This applies to JB∗ -triples A with dim(A) < ∞.

4. Normal Contractive Projections and GL-projections A tripotent u in a JBW∗ -triple A is σ-finite, if any set of pairwise orthogonal tripotents all of which are less than or equal to u is of countable cardinality. The set of all σ-ﬁnite tripotents of A is denoted by Uσ (A). By the results of [18], a JBW∗ -triple A is the norm-closure of the linear span of the set U(A). As shown in [12], for each element x in the predual A∗ of A there exists a tripotent ex of A that is the smallest of all elements u of U(A) such that (u · x) equals x. The tripotent ex is said to be the support tripotent of x. Furthermore, by [7], a tripotent u of A is σ-ﬁnite if and only if it is equal to ex for some element x of A∗ . We now proceed to the ﬁrst main result. It provides a ﬁrst set of cases in which commutation of a projection with inner derivations is sufﬁcient for its contractivity. This can be seen as a converse version of Lemma 3.4. We point out that a similar result may hold within the category of JB∗ triples, with the norm-topology replacing the weak∗ -topology. But in the absence of weak∗ compactness of the unit-ball, that case would require further technical clariﬁcations which we will not pursue here. Instead we focus on JBW∗ -triples. Theorem 4.1. Let A be a JBW∗ -triple. Let B be an atomic subtriple such that w∗ B = k∈K Ck , for a family (Ck )k∈k of pairwise orthogonal, finite-dimensional Cartan factors with standard grids Gk , and corresponding projections P˜k = u∈Gk P˜u , where P˜u is a normal extension of the coordinate projection Pu , as in Lemma 3.3. Then the following conditions are equivalent: (1) The projection P = k∈K P˜k exists as a σ(A, A∗ )-operator limit and is contractive. (2) For all k ∈ K, the projection P˜k is contractive. (3) For all k ∈ K and δ ∈ inn(Ck ) ⊆ inn(A), we have that [P˜k , δ] = 0. Proof. (1) ⇒ (2): By Lemma 3.3, for u ∈ Gk the projection P˜u is contractive, and hence, by Proposition 2.2, annihilates the subset u . By [5], u coincides with the σ(A, A∗ )-closed subspace u⊥ . Hence, for j ∈ K\{k}, u contains Cj = P˜j (A). Clearly Pu also annihilates Pv for v ∈ Gk \{u}. This implies that P˜k P = P P˜k = P˜k , whenever P is well deﬁned. The restriction P˜k |B of P˜k to B is contractive because Ck Cj , for all j ∈ K\{k}. Hence if P contractive, so is P˜k . (2) ⇒ (1): For each k ∈ K, the range Pk (A) = Ck of Pk is a σ(A, A∗ )closed subtriple of A. By [5, Theorem 4.4], (Pk A) is equal to the algebraic orthocomplement (Pk A)⊥ of Pk A. Hence, (Pk A) is a subspace of A which contains Pj (A), for all j ∈ K\{k}. The desired statement follows from Proposition 2.2.

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(2) ⇒ (3): Notice that under the given assumptions, inn(Ck ) = lin{itD(b, b) : b ∈ Ck , t ∈ R}. Since Ck ⊆ A is a subtriple, the inclusion δ ∈ inn(Ck ) ⊆ inn(A) is canonical. The sought implication is a straightforward consequence of Lemma 3.4. (3) ⇒ (2): For each index k ∈ K, the grid Gk contains the frame F0 ∈ F(Ck ) as given by Proposition 3.1. Recall that P˜k = u∈Gk P˜u , and that P˜u extends the coordinate projection w.r.t. Gk onto Cu. Deﬁne P˜0 := u∈F0 P˜u . As seen earlier by Example 2 or Proposition 2.2, P˜0 is a contractive projection, and by Lemma 3.3, P˜0 P˜k = P˜k P˜0 = P˜0 . Suppose that P˜k commutes with D(b b), for b ∈ Ck . The weak∗ -continuity of D(b b) and P˜k implies that P˜k commutes also with exp itD(b b), hence with all elements ϕ of Inn(Ck ). Since Ck is a subtriple of A, we have that Inn(Ck ) ⊆ Inn(A). In particular, each such ϕ is a surjective isometry and a triple automorphism of A. Let a ∈ A be arbitrary. There exists a frame F1 in Ck such that P˜k (a) = u∈F1 αu u and αu ∈ R+ . Moreover, by Lemma 3.5 there exists an inner automorphism ϕ ∈ Inn(Ck ) ⊆ Inn(A) such that ϕF1 = F0 . Thus, the map P˜1 deﬁned by P˜1 := ϕ−1 P˜0 ϕ is a contractive projection on A with range P˜1 A = linF1 . It follows that P˜1 P˜k = ϕ−1 P˜0 ϕP˜k = ϕ−1 P˜0 P˜k ϕ = ϕ−1 P˜0 ϕ = P˜1 . Since the element P˜k (a) lies in P˜1 A, we see that P˜1 P˜k (a) = P˜k (a) and we conclude that P˜k (a) = P˜1 P˜k (a) = P˜1 (a) ≤ a. This gives the required statement.

Remark 4.2. Alternative formulations of Condition (3) might be convenient in technical arguments. For example (3) is equivalent to the condition that [P˜k , D(u, u)] = 0, for all k ∈ K, Fk ∈ F(Ck ), and u ∈ Fk . To see this, recall that each b ∈ Ck is such that b = u∈F αu u, for some frame F of Ck and ¯ u D(u u). This αu ∈ R+ . Orthogonality of F implies that D(b b) = u∈F αu α shows the stated equivalence. The results of the previous section are formulated in terms of a chosen standard grid for the range P A of the projection P . In that situation, P is a sum of contractive projections with one-dimensional ranges. Moreover, the range P A of P was supposed to be a subtriple of A. In this section we will show that these assumptions actually represent the generic situation. In particular, if R is any normal contractive projection on A such that its range RA is isometrically isomorphic to some atomic JBW∗ -triple whose Cartanfactors are ﬁnite-dimensional, then R is conjugate by a partial isometry S on A to a normal contractive projection P˜ = SR which is as in Theorem 4.1. Moreover, P˜ is uniquely determined by R. Recall that a GL-projection on E is deﬁned to be a contractive projection P such that (P E)♦ ⊆ kerP . As shown in [4, Theorem 3.5], for each

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contractive projection R on A∗ there exists a unique GL-projection, which we denote by Rγ , such that RA∗ = Rγ A∗ . Therefore, γ : R → Rγ is a mapping from the set of contractive projections P1 (A∗ ) to that of GL-projections PGL (A∗ ) of A∗ , and Rγγ = Rγ . The stated uniqueness of Rγ also implies that the restriction to PGL (A∗ ) of the natural pre-order relation ≤ on P1 (A∗ ) is a partial order, and that R → RA∗ is an order isomorphism from PGL (A∗ ) to the set of contractively complemented subspaces of A∗ ordered by set inclusion. It is conjectured that, but as of yet not known whether these partial orders are lattices. In [19, Thm. 6.5], the existence of the supremum was conﬁrmed for the case of an L1 -orthogonal family (Pk )k∈K , in GL(A∗ ), i.e. Pk (A∗ )♦Pl (A∗ ) for distinct indices k, l ∈ K, namely, Pk = Pk ∈ GL(A∗ ), (4.1) k∈K

k∈K

the sum being SOT-convergent. The conjecture is also supported by Lemma 4.3, somewhat weaker versions of which have been stated earlier (for an overview on GL-projections and related topics in JBW∗ -triples see [19]). Lemma 4.3. Let {Pk }k∈K be a family of GL-projections on A∗ . If (Pk )k∈K is norm-operator summable with

sum P a contractive projection on A∗ , then P is equal to the supremum k∈K Pk of {Pk }k∈K in PGL (A∗ ). Proof. Notice that taking the L-complement reverses set-inclusion. By assumption, the inclusion (Pk )♦ ⊆ kerPk holds for all k ∈ K. Hence, ♦ ♦

n (P A∗ )♦ = lin Pk A∗ ⊆ Pk A∗ = (Pk A∗ )♦ k∈K k∈K k∈K

⊆ kerPk ⊆ kerP. k∈K

This shows that P is a GL-projection on A∗ . If Q ∈ PGL (A∗ ) is such that, for all k ∈ K, Pk ≤ Q, then n lin Pk A∗ ⊆ QA∗ . k∈K

Theorem 3.5 [4] shows that P is the unique element of PGL (A∗ ) with range n lin k∈K Pk A∗ . It follows that P ≤ Q, and that P is the least upper bound of (Pk )k∈K in PGL (A∗ ). By Hahn-Banach, each one-dimensional subspace is the range of some contractive projection. Hence, for each non-zero element x of A∗ , there is a unique GL-projection onto Cx which will be denoted by Px , and which is explicitly given in terms of the support tripotent ex of x, for z ∈ A∗ by Px (z) :=

1 x (ex ⊗ x)(y) = (ex · z) . x x

(4.2)

For x = 0, P0 is deﬁned to be the zero-projection. The support space s(X) of σ(A,A∗ )

a non-empty subset X ⊆ A∗ is deﬁned by s(X) = lin{ex : x ∈ X} . By Theorem 4.6 in [6], for P ∈ P1 (A∗ ), the following conditions are equivalent

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P ∈ PGL (A∗ ),

s(P A∗ ) = P ∗ A,

s(P A∗ ) ⊇ P ∗ A,

s(P A∗ ) ⊆ P ∗ A.

(4.3)

That GL-projections on A∗ are highly compatible with taking support tripotents is also seen from the following fact from [6, Cor 4.8]: For any JBW∗ -triple Z with predual Z∗ , and x ∈ Z∗ , denote by eZ (x) the support tripotent of x in Z. Consider a contractive projection P on A∗ , and P ∗ A equipped with the restricted triple-product {...}P ∗ . Then, P lies in GL(A∗ ), if and only if eP

∗

A

(x) = eA (x),

∀x ∈ P A∗ .

(4.4)

The results of [6,8,11,12] show that, for each R ∈ P1 (A∗ ), the range R∗ A of the adjoint R∗ ∈ R1 (A) is a subspace of (Rγ A) ⊕ (Rγ A)⊥ , and that the restriction S := Rγ∗ |R∗ (A) is a weak∗ -continuous surjective isometry with inverse S −1 = R∗ |Rγ∗ (A) . Part (1) of the following proposition has been proved in [4], whereas part (2) seems not to have been stated explicitly earlier. Proposition 4.4. Let R be a projection defined on the predual A∗ of a JBW∗ triple A. (1) If R is such that s(RA∗ ) ⊆ R∗ A. Then R is a GL-projection on A∗ , in particular R is contractive. (2) If R is contractive, with corresponding GL-projection Rγ and partial isometry S := Rγ∗ |R∗ (A) , then S ∗ R∗ = Rγ∗ R∗ = Rγ∗ . Proof. (1): For any x ∈ A∗ , the assumption means that R∗ e(Rx) = e(Rx), and the fundamental properties of support-tripotents imply that Rx = (e(Rx) · Rx) = (R∗ e(x) · x) = (e(Rx) · x) ≤ x.

(4.5)

∗

(2): For any x, y ∈ R(A∗ ), we have that R e(x) − e(x) ⊥ e(y) ([11, Lemma 2.1]). By (4.3), and the fact that s(X)⊥ ⊆ X 0 , we see that (R∗ − idRγ∗ (A) )(Rγ∗ (A)) ⊆ R(A∗ )0 . Hence, (2) follows since R∗ |Rγ∗ (A) = S −1 is an isometry and from Proposition 2.5. With the previous results, we can generalize Corollary 5.1 in [21]. Corollary 4.5. Let X = (xk )k∈K be a family of elements in A∗ with corresponding support tripotents (ek )k∈K in A, and GL-projections (Pk )k∈K as .

defined by (4.2). Define the subspace T of A∗ by T := linX . Suppose that, (i) there exists a contractive projection Q on A∗ with range QA∗ = T ; (ii) for distinct indices j, k ∈ K, (ek · xj ) = 0. Then, the following results hold. (1) For the support-spaces s(X) and s(T ) of X and T as defined above, we have that s(X) = s(T ) = Qγ∗ (A), and in particular Qγ∗ (A) is a subtriple of A. (2) If |K| < ∞, then Qγ = k∈K Pk .

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Proof. (1): By [4, Theorem 3.5], it can be assumed that Q = Qγ . Moreover, we exclude the trivial case that for all k ∈ K, xk = 0. It follows from (4.3) that the subspaces s(T ) and Q∗ A of A coincide. It remains to be shown that this space also coincides with s(X). The inclusion s(X) ⊆ s(T ) is trivial from the definition of the support spaces. Since Q is contractive, Q∗ A and QA∗ form a dual pair. Denote by M◦ and N ◦ the annihilators within this dual pair of non-empty subsets M ⊆ Q∗ A and N ⊆ QA∗ . The assumption (ii) implies that (ek · xj ) = δ(k, j)xj . Hence, s(X)◦ ⊆ ({ek }k∈K )◦ = {ek }◦ = {0}. k∈K ∗

Since s(X) is a weak -closed subspace of s(T ) = Q∗ A, it follows that s(X) = (s(X)◦ )◦ = {0}◦ = Q∗ A.

(4.6)

(2): Since K is ﬁnite we set K = {1, . . . , n}. Since, by (ii.) Pk Pj = n δ(j, k)Pk , the continuous linear map P = k=1 Pk is a projection with ∗ range T . The adjoint P of P has range lin{e1 , . . . , en }. Moreover, (ii) implies also linear independence of the sets {x1 , . . . , xn } and {e1 , . . . , en } in A∗ and A respectively. It follows that dimQ∗ A = dimQA∗ = n, hence that s(T ) = lin{e1 , ..., en }. Consequently, s(P A∗ ) = s(T ) = lin{e1 , ..., en } = P ∗ A. By Lemma 4.4 P is a GL-projection, as required.

Proposition 4.6. Let x and y be elements in A∗ with support tripotents ex and ey , and projections Px , Py as defined by (4.2). Suppose that ex ey . Then, (ey · x) = 0 holds if and only if Px + Py is a projection, Proof. From the definition (4.2) of Px and Py it is clear that (ey · x) = 0 is equivalent to Py Px = 0. Recall that y = D(ey , ey )∗ y. The assumption that ex ey implies that 2D(ey , ey )ex = ex . Hence 2(ex · y) = (ex · y) = 0, and (Px + Py )2 = Px + Py + Py Px . The statement is now obvious. Lemma 4.7. Let C be a finite-dimensional Cartan-factor, G a corresponding standard grid. For u ∈ G, let xu ∈ C∗ be the coordinate functional of u with respect to G. Then u = eC (xu ), for all u ∈ G. Proof. Recall that, for any JBW∗ -triple Z with predual Z∗ and y ∈ Z∗ , there exists the least tripotent eZ (y) in Z at which y attains its norm [12]. We distinguish between two cases, in the ﬁrst of which, namely when rankC (u) = 1, minimality of u implies that u = eC (x). The second case, rankC (u) = 2, requires more analysis. Clearly, we have that e(xu ) ≤ u. Hence, we are to show that the implication v < u ⇒ |(v · xu )| < 1

(4.7)

holds for all v ∈ U(C). In fact, we will see that the inequality holds for all minimal v in U(C). Given a tripotent v in C with v < u, deﬁne w := u − v. Then we have w ∈ U(Ck ) ∩ u⊥ . Moreover, rankC (v) = rankC (w) = 1. Next, we use the well known Riesz-representation bx of xu ∈ Ck∗ which is obtained as follows: On a

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ﬁnite-dimensional Jordan∗ -triple Ck , the sesquilinear form (a, b) → trD(a, b) is positive definite, tr denoting the trace [26,30]. Hence, there exists a unique bx ∈ Ck such that (xu · a) = trD(a, bx ), for all a ∈ Ck . It turns out that, in the present case, bx = νu, for some positive normalizing constant ν. The values of some of the traces are to be calculated explicitly from the tabled relations (3.1)–(3.9). If C is hermitian (type III), then we have u = uij , for a ﬁxed (i, j) ∈ I × I (and uij = uji , I labeling GIII ). Then, for i, j, k distinct, (4.8) trD(uij , uij ) = |I| + 1, trD(uik , uij ) = trD(uii , uij ) = trD(ukl , uij ) = 0, (4.9) 1 1 (4.10) trD(uii , uii ) = |I| + . 2 2 By (4.8), we have ν = 1/(|I| + 1), for the hermitian factors. Also, (4.8) and (4.9) conﬁrm the representation of xu by bx = uij . For C an odd-dimensional spin factor (type IV), we have u = u0 , and for some i in the index set I of GIV , we ﬁnd that trD(u0 , u0 ) = 2|I| + 1 = dim(C), trD(uεi , u0 )

=

(4.11)

trD(u−ε i , u0 )

= 0, (4.12) 1 (4.13) = |I| + . 2 Hence, by (4.11), ν = 1/dimC. This and (4.12) conﬁrm that xu is represented by bx = u0 . Notice that, we always have |I| ≥ 2 in the non-trivial situations, and that traceD(z, z) is proportional to the rank of z, as one would expect. For an arbitrary v ∈ U1 (C) (of any of these types) the Cauchy-Schwartz inequality implies that trD(uεi , uεi )

|(xu · v)|2 = ν 2 |trD(v, u)|2 ≤ ν 2 trD(v, v)trD(u, u) = ν trD(v, v). (4.14) Hence, it remains to be shown that ν trD(v, v) is strictly less than one. To this end we recall that for all z ∈ U1 (C), there exists ϕ ∈ Aut(C) such that ϕz = v. Since, trD(v, v) = trD(ϕz, ϕz) = tr ϕD(z, z)ϕ−1 = trD(z, z),

(4.15)

it is sufﬁcient to check the desired inequality for one selected z ∈ U1 (C). This is easily done with z := uεi for type IV, [cf. (4.11), (4.13)], and with z := uii for type III [cf. (4.8), (4.10)]. We obtain the next theorem by combining the result on GL-projections with those of the previous section on commuting operators. Theorem 4.8. Let R be a bounded projection on the predual A∗ of a JBW∗ triple A such that range R∗ A of R∗ on A is isometrically isomorphic to some atomic JBW∗ -triple B (not necessarily a subtriple of A) consisting of finitedimensional Cartan-factors. Then R is contractive if and only if there exists

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k , (k, j) → Pkj ) in A∗ with sum P and a norm-projective system ((xjk )j=1,...,n k∈K with the properties that for all k ∈ K and j = 1, ..., nk ,

(1) The projection Pkj is the unique GL-projection with range Cxjk , i.e. Pkj z = (ejk · z)xjk , where ejk is the support tripotent of xjk in A. (2) For each k ∈ K, the family (ejk )j=1,...,nk is a standard grid of a Cartan A. factor Ck , with respect to the triple product nk in (3) For each k ∈ K, the projection Pk := j=1 Pkj commutes with D(erk esk ), for 1 ≤ r, s ≤ nk . (4) There exists a weak∗ -continuous surjective isometry S : R∗ A → P ∗ A such that P ∗ = SR∗ . nk j j If this is the case, then the projections P = k,j Pk and Pk = j=1 Pk γ −1 are GL-projections, P = R , and the isometry S and its inverse S are the restrictions P ∗ |R∗ A and R∗ |P ∗ A , respectively. The family (Pk )k∈K is L1 -orthogonal, i.e. Pk (A∗ )♦Pl (A∗ ), for k = l ∈ K. Proof. If R is contractive, then its image RA∗ is isometrically the predual of the JBW∗ -triple R∗ A with the restricted triple product {. . .}P ∗ . By assumption, there are mutually L∞ -orthogonal subspaces (Bk )k∈ of A, each Bk isometrically isomorphic to a ﬁnite-dimensional Cartan-factor, such that w∗ R∗ A = k∈K Bk , and the projection ρk from R∗ A to Bk is σ(R∗ A, RA∗ )continuous and contractive. Hence, ρk has a pre-adjoint ρk ∈ P1 (P A∗ ). The product ρk R is deﬁned on the whole of A∗ , i.e. is an element of P1 (A∗ ). Obviously, we have, for arbitrary a ∈ A, x ∈ A∗ , (ρk R∗ a · x) = (R∗ ρk R∗ a · x) = (a · Rρk R x) = (a · ρk R x). (4.16) Hence, (ρk R)∗ = ρk R∗ . The further strategy is to pass to the “regularized” version of ρk R, i.e. we use its GL-projection Pk := (ρk R)γ and we set Ck := Pk∗ (A). By (4.3) we conclude that Ck = Pk∗ A = s(Pk (A∗ )) = s(ρk P (A∗ )).

(4.17)

By [12, Proposition 2.2] and [6, Lemma 3.2] it follows that Ck is a subtriple of A, and that the restrictions Pk∗ |Bk and ρk R∗ |Ck are isometric isomorphisms between the JBW∗ -triples (ρk R∗ A, {. . .}Pk∗ ) and (Ck , {. . .}) and inverse of each other. In particular dim(Ck ) = dim(Bk ) < ∞. Since Ck has a grid Gk = (ujk )j∈{1,...,mk } as a basis, we conclude from Lemma 3.3 and Proposition 4.6 that there exist elements (xjk )j∈{1,...,mk } in S1 (Pk (A)) = S1 (A)∩Pk A, such that (uik · xjk ) = δ(i, j). By (4.4 ) and Lemma 4.7, we see that eCk (xik ) = eA (xik ) = uik .

(4.18)

So far we have veriﬁed properties (1) and (2). Clearly, (3) follows from Theorem 4.1. Property (4) is easily obtained by applying the GL-regularization to R, i.e. we set P := Rγ . Observe that the families (Ck )k∈K and (Pk )k∈K as Ck Cl , and Pk ♦Pl . constructed are orthogonal in that Ck ⊥Cl , equivalently By Proposition 2.2 or by (4.1), we see that P = k∈K Pk ∈ GL(A∗ ). Again, [12] Proposition 2.2 and [6] Lemma 3.2. show that P ∗ A is a subtriple of A, and

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that the restrictions P ∗ |R∗ A and R∗ |P ∗ A are isometric isomorphisms between the JBW∗ -triples (R∗ A, {. . .}) and (P ∗ A, {. . .}P ∗ ) and inverse of each other. Hence, we can set R∗ |P ∗ A = S −1 and P ∗ |R∗ A = S, and Proposition 4.4 conﬁrms that P ∗ = SR∗ . k , (k, j) → Pkj ) is a norm-projective Conversely, suppose that((xjk )j=1,...,n k∈K system in A∗ with sum P and with the properties (1)–(4). The condition (1) in combination with Definition 2.4 and (4.2) implies that (ejk ·xil ) = δ(j, i)δ(k, l). nk Hence, for each k ∈ K, the sum Pk := j=1 Pkj is a projection. This and condition (2) ensure that the assumptions of Theorem 4.1 are satisﬁed. Hence condition (3) and Theorem 4.1 imply that for each k ∈ K, the projection Pk is contractive. By Corollary 4.5, Pk is a GL-projection. By Proposition 2.2, or by (4.1), (Pk )k∈K is a summable family, and its sum P := k∈K P k lies in GL(A∗ ). In particular, P is contractive. When S is a weak∗ -continuous surjective isometry as in (4), then the projection R∗ = S −1 P ∗ is necessarily contractive. Theorem 4.8 characterizes normal contractive projections on A in terms of commutation. The following corollary states an important special case of the previous theorem, concerning ﬁnite-dimensional subspaces. Corollary 4.9. A finite-dimensional subspace T of A∗ is 1-complemented in A∗ if and only if the following conditions hold: (1) There exists a basis X = {x1 , . . . , xn } of T with corresponding support m tripotents with an L1 -orthogonal partitioning X = k=1 Xk , i.e. Xk ♦Xl for k = l. k (2) For each k ∈ {1, ....m, }, the support-tripotents (ekj )m j=1 of the elements k xj ∈ Xk form a standard-grid Gk in A and are such that (eki · xkj ) = m k 1 k δ(i, j), hence Pk := j=1 e ⊗ xkj is a projection. xk j j

(3) For all k ∈ {1, . . . , m} and i, j ∈ {1, . . . , mk } as above, Pk commutes with D(eki , ekj ).

Proof. Suppose that Q ∈ P1 (A∗ ) is such that Q(A∗ ) = T . We set R := Qγ∗ , and we apply Theorem 4.8 to obtain the properties (1)–(3). Conversely, these conditions and Theorem m 4.1 imply that each of the projections Pk is contractive. We set P := k=1 Pkγ , and R := P ∗ . Theorem 4.8 shows that P is the unique GL-projection with range T . The next theorem generalizes Corollary 4.9. It provides criteria on subspaces of the predual A∗ to be contractively complemented, and summarizes some of the earlier statements. Theorem 4.10. Let T be a subspace of the predual A∗ of a JBW∗ -triple A, . which is the norm-closure of the L1 -orthogonal sum T = ⊕k∈K Tk of subspaces Tk , in that Tj ♦Tk for j ∈ k, and dim Tk < ∞ for all k ∈ K. Then T is 1-complemented in A∗ if and only if each Tk satisfies the conditions (1), (2), (3) of Corollary 4.9.

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Proof. The assumption of L1 -orthogonality Tj Ti implies that the restricted, local projection πk from T onto Tk is contractive. Hence, any global contractive projection Q ∈ P1 (A∗ ) with range T provides the contractive projections πk Q on A∗ with range Tk , which have the properties (1), (2), (3). The converse is established by the same arguments as in Corollary 4.9, by which the conditions (1), (2) and (3) imply that each Tk is 1-complemented by some Qk ∈ P1 (A∗ ), hence also by the associated GL-projection Qγk . As shown, the duals Qγ∗ (of ﬁnite dimensions) k have mutually L∞ -orthogonal subtriples γ∗ as their ranges. Apply Theorem 4.8 to R := k∈K Qk to complete the proof. The key arguments in the main theorems are the completeness property of the set F(B) of frames of a subtriple B of A, the isotropy of F(B) with respect to Inn(A), as well as the discussed commutation properties. In this way, we have also shown that the commutation relations are complementary with conditional expectations in that the former are sufﬁcient, the latter necessary for contractivity of these projections. The global assumption made throughout article, that the subtriples in question have ﬁnitedimensional Cartan-factors, is of somewhat technical nature. We conjecture that this restriction can be eased, and that our main results hold with little manipulation for more general types of subtriples. Moreover, the commutation criteria and most of the precepts we used in this paper to obtain sufﬁcient criteria for contractivity of the projections can be applied to Banach spaces. The crucial algebraic condition of commutation may be stated for projections and subgroups of isometries. This generalizes the classical concept known as uniformly isotropic space [25]. We conclude by stating a future target in the context of JBW∗ -triples: Conjecture: A weak∗ -continuous projection P on a JBW∗ -triple A, the range P A of which is a subtriple, is contractive if and only if P commutes with all inner derivations of P A. Acknowledgments The author is grateful for the hospitality and support he received from the Department of Mathematics of Nanjing University, Nanjing PRC, where parts of this work were completed.

References [1] Braun, H., Koecher, M.: Jordan-Algebren. Springer, Berlin (1966) [2] Bunce, L.J.: Norm preserving extensions in JBW∗ -triple preduals. Quart. J. Math. 52, 133–136 (2000) [3] Dang, T., Friedman, Y.: Classiﬁaction of JBW∗ -triple factors and applications. Math. Scand. 61, 292–330 (1987) [4] Edwards, C.M., H¨ ugli, R.V.: Order structure of GL-projections on the predual of a JBW∗ -triple. Atti. Sem. Math. Fis. Univ. Modena LIII, Fasc 2, 271– 287 (2005)

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[5] Edwards, C.M., H¨ ugli, R.V.: M-orthogonality and holomorphic rigidity in complex Banach spaces. Acta Sci. Math. (Szeged) 70, 237–264 (2004) [6] Edwards, C.M., H¨ ugli, R.V., R¨ uttimann, G.T.: A geometric characterization of structural projections on a JBW∗ -triple. J. Funct. Anal. 202, 174–194 (2003) [7] Edwards, C.M., R¨ uttimann, G.T.: Orthogonal faces of the unit ball in a Banach space. Atti Sem. Math. Fis. Univ. Modena 49, 473–493 (2001) [8] Edwards, C.M., R¨ uttimann, G.T.: Structural projections on JBW∗ -triples. J. Lond. Math. Soc. 53, 354–368 (1996) [9] Eﬀros, E.G., Størmer, E.: Positive projections and Jordan structure in operator algebras. Math. Scand. 45, 127–138 (1979) [10] Faraut, J., Kaneyuki, S., Kor´ anyi, A., Lu, Q.-K., Roos, G.: Analysis and Geometry on Complex Homogeneous Domains, Progress in Mathematics. Birkh¨ auser, Boston, Basel, Berlin (2000) [11] Friedman, Y., Russo, B.: Conditional expectation and bicontractive projections on Jordan C ∗ -algebras and their generalisations. Math. Z. 194, 227– 236 (1987) [12] Friedman, Y., Russo, B.: Structure of the predual of a JBW∗ -triple. J. Reine Angew. Math. 356, 67–89 (1985) [13] FriedmanRusso, B.: Conditional expectation without order. Pacif. J. Math. 115, 351–360 (1984) [14] Friedman, Y., Scarr, T.: Physical Applications of Homogeneous Balls. Progress in Mathematical Physics, vol. 40, Birkh¨ auser, Boston (2005) [15] Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitman Advanced, Boston/London/Melbourne (1984) [16] Harris, L.A.: Bounded symmetric domains in inﬁnite dimensional spaces. In: Hayden.Suﬀridge.inf.hol Hayden T.L., Suﬀridge T.J. (eds.) Proceedings on Inﬁnite Dimensional Holomorphy, Lecture Notes in Mathematics, vol. 364(13– 40), Springer Berlin/Heidelberg/New York (1974) [17] Harris, L.A.: A generalisation of C∗ -algebras. Proc. Lond. Math. Soc. 42, 331– 361 (1981) [18] Horn, G.: Characterisation of the predual and ideal structure of a JBW∗ triple. Math. Scand. 61, 117–133 (1983) [19] H¨ ugli, R.V.: Structural projections on a JBW∗ -triple and GL-projections on its predual. J. Korean Math. Soc. 41(1), 107–130 (2004) [20] H¨ ugli, R.V.: Characterizations of tripotents in JB∗ -triples. Math. Scand. 99, 147–160 (2006) [21] H¨ ugli, R.V.: Collinear systems and normal contractive projection in JBW∗ triples. Int. Equ. Oper. Theory 58, 315–339 (2007) [22] H¨ ugli, R.V., Mackey, M.: Transitivity of inner automorphisms on manifolds of tripotents. Math. Z 262, 125–141 (2009) [23] Kaup, W.: Contractive projections on Jordan C∗ -algebras and generalisations. Math. Scand. 54, 95–100 (1984) [24] Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983) [25] Lewicki, G., Odyniec, W.: Minimal projections in Banach spaces. In: Springer Lecture Notes in Mathematics, vol. 1449. Springer, Berlin/Heidelberg (1987)

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[26] Loos, O.: Bounded Symmetric Domains and Jordan Pairs, Mathematical Lectures. University of California, Irvine (1977) ´ Russo, B.: Classiﬁcation of contractively complemented [27] Neal, M., Ricard, E, Hilbertian operator spaces. J. Funct. Anal. 237, 589–616 (2006) [28] Neal, M., Russo, B.: Contractive projections and operator spaces. Trans. Am. Math. Soc. 355(6), 2223–2262 (2003) [29] Neher, E.: Jordan triple systems by the grid approach. In: Lecture Notes in Mathematics, vol. 1280. Springer, Berlin/Heidelberg/New York (1987) [30] Satake, I.: Algebraic Structures of Symmetric Domains. Publ. Math. Soc. Japan, Kanˆ o Memorial Lectures. Iwanami Shoten and Princeton Univ. Press (1980) [31] Stach` o, L.L.: A projection principle concerning biholomorphic automorphisms. Acta Sci. Math. 44, 99–124 (1982) [32] Upmeier, H.: Symmetric Banach manifolds and Jordan∗ -algebras. In: NorthHolland Math. Studies, vol. 104, p. 444 (1985) [33] Upmeier, H.: Jordan algebras in analysis, operator theory and quantum mechanics. Providence Am. Math. Soc. 81–85 (1986) Remo V. H¨ ugli UCD School of Physics, Condensed Matter Theory group University College Dublin Belﬁeld D4 Dublin Ireland e-mail: [email protected] Received: August 2, 2010. Revised: December 22, 2010.

Integr. Equ. Oper. Theory 70 (2011), 125–149 DOI 10.1007/s00020-011-1862-7 Published online January 27, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals Kenneth R. Davidson and Ryan Hamilton Abstract. If A is a unital weak-∗ closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property A1 (1), then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has A1 (1), and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-∗ closed subalgebras of H ∞ acting on Hardy space or on Bergman space. Mathematics Subject Classification (2010) . Primary 47A57; Secondary 30E05, 46E22. Keywords. Nevanlinna–Pick interpolation, reproducing kernel.

1. Introduction The classical interpolation theorem for analytic functions is due to Pick in 1916. Suppose z1 , . . . , zn are distinct points in the complex open disk D and w1 , . . . , wn are complex numbers. The Nevanlinna–Pick (NP) interpolation theorem asserts that the positivity of the matrix 1 − wi wj 1 − zi zj is a necessary and sufﬁcient condition to ensure the existence of an analytic function f on D satisfying f (zi ) = wi for i = 1, . . . , n and f ∞ := sup{|f (z)| : z ∈ D} ≤ 1. Nevanlinna gave a new proof and provided a parameterization of all solutions a few years later. K.R. Davidson was partially supported by an NSERC grant. R. Hamilton was partially supported by an NSERC fellowship.

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The seminal work of Sarason [30] reformulated the Nevanlinna–Pick theorem in operator theoretic language. Let JE be the ideal of functions in H ∞ (D) that vanish on E = {z1 , . . . , zn }, and let M (E) = H 2 JE H 2 . Sarason showed that the NP theorem is equivalent to the representation f + JE → PM (E) Mf PM (E) being isometric, where Mf denotes the multiplication operator by f on Hardy space H 2 (D). He established this using a prototypical version of the commutant lifting theorem. The Szeg¨ o kernel kzS (w) = (1 − zw)−1 is the reproducing kernel for H 2 (D), and one easily sees that M (E) = span{kzS1 , . . . , kzSn }. The operator PM (E) Mf∗ PM (E) = Mf∗ |M (E) is diagonal with respect to the (nonorthogonal) basis kλS1 , . . . , kλSn since Mf∗ kzSi = f (zi )kzSi . We can summarize the classic NP theorem as the distance formula dist(f, JE ) = Mf∗ |M (E) . Abrahamse [1] proved a Nevanlinna–Pick type theorem on a multiply connected domain A. Here, a Pick matrix associated to a single kernel was not sufﬁcient to guarantee the existence of a solution. Instead, an entire family of kernels indexed by copies of the complex torus was required. These spaces arose as subspaces of L2 (∂A) which are rationally invariant. In the case of the annulus, these subspaces were classiﬁed by Sarason [29]. Analogous invariant subspaces exist for all nice ﬁnitely connected domains. Abrahamse establishes a factorization theorem which shows that certain linear functionals can be represented as rank one functionals on one of these subspaces. This was the ﬁrst appearance of a Nevanlinna–Pick family of kernels in the literature, and motivated the search for other Nevanlinna–Pick families associated to different algebras of functions. There is considerable literature concerned with interpolation in the multiplier algebra M(H) of a reproducing kernel Hilbert space H over a set X. Again, given E = {λ1 , . . . , λn } in X and scalars w1 , . . . , wn , we are interested in ﬁnding a function f ∈ M(H) such that f (λi ) = wi for 1 ≤ i ≤ n and Mf ≤ 1. We again deﬁne JE = {f ∈ M(H) : f |E = 0} and M (E) = span{kλi : λi ∈ E}. Such a Hilbert space is called a Nevanlinna–Pick kernel if the analogous distance formula holds: dist(Mf , JE ) = Mf∗ |M (E) . The kernel functions kλi form a basis for M (E) consisting of eigenvectors for Mf∗ , and so the right hand side is at most 1 if and only if the following matrix is positive (1 − wi wj )kλj , kλi . One can also consider matrix interpolation, and the classical theorem works just as well for matrix algebras over H ∞ . A kernel for which the classical formula holds for all matrix valued functions is called a complete NP kernel. Results of McCullough [21,22] and Quiggin [25,26], building on (unpublished) work by Agler, provide a classiﬁcation of complete NP kernels. Davidson and Pitts [17] show that the Drury–Arveson space or symmetric

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Fock space on the unit ball Bd of Cd is a complete NP kernel. Agler and McCarthy [2] showed that every complete NP kernel is equivalent to the restriction of the Drury–Arveson space to some subset of the ball. There is no known characterization of ordinary NP kernels. In [14], a Nevanlinna–Pick problem on the Hardy space was studied for the subalgebra H1∞ = {f ∈ H ∞ (D) : f (0) = 0}. Beurling’s theorem for the shift was used to characterize the invariant subspace lattice for this algebra. The main result in [14] furnishes a necessary and sufﬁcient condition for the existence of an interpolant in H1∞ . This requires the positivity of an entire family of Pick matrices (1 − wi wj )PL kzSj , kzSi for (a family of) invariant subspaces L of H1∞ . Again this is equivalent to a distance formula for the ideal JE = {f ∈ H1∞ : f |E = 0}: dist(Mf , JE ) =

sup

L∈Lat(H1∞ )

PLJE L Mf∗ |LJE L .

In this paper, we are interested in general interpolation problems of this type. Consider a reproducing kernel Hilbert space H with kernel k on some set X, and a unital weak-∗-closed subalgebra A of multipliers on H. Let E = {λ1 , . . . , λn } be a ﬁnite subset of points in X which are separated by A, and let w1 , . . . , wn be complex numbers. We seek a necessary and sufﬁcient condition to ensure the existence of a contractive multiplier in A which interpolates the given data. Set JE = {f ∈ A : f |E = 0}. It is elementary (see Lemma 2.8) to verify that dist(Mf , JE ) ≥

sup PLJE L Mf∗ |LJE L .

L∈Lat A

Our primary goal here is to ﬁnd a sufﬁcient condition to ensure that equality holds. When such a formula is satisﬁed, the algebra A is said to have an NP family of kernels. In this case, there is a corresponding family of Pick matrices for which simultaneous positivity of its members is a necessary and sufﬁcient condition for interpolation. Our main result, Theorem 3.3, states that if H is an reproducing kernel Hilbert space, and A is any unital weak-∗-closed algebra of multipliers that has the strong predual factorization property called property A1 (1), then A admits a Nevanlinna–Pick family. A weak-∗ closed subspace of operators A has property A1 (1) if every weak-∗ continuous functional ϕ on A with ϕ < 1 can be represented as a vector functional ϕ(A) = Ax, y for all A ∈ A with x y < 1. This property was introduced in [4,18]. While the property A1 (1) is very strong, many relevant examples of multiplier algebras have it. In particular, if the multiplier algebra has A1 (1), then the results apply to all unital weak-∗ closed subalgebras. A result of Arias and Popescu [5] shows that quotients of the non-commutative Toeplitz algebra have property A1 (1). We use this to show that the multiplier algebras of all complete NP spaces have A1 (1). Thus, these results apply in many well

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known contexts including all subalgebras of the multipliers on Drury–Arveson space. Theorem 3.3 provides an NP theorem for any weak-∗-closed subalgebra of H ∞ acting on Hardy space. This provides a generalization of the results in [14] and [27]. Raghupathi [28] shows that Abrahamse’s interpolation theorem is equivalent to a constrained interpolation problem on certain weak-∗-closed subalgebras of H ∞ . Consequently our distance formula implies Abrahamse’s result as well. Additionally, the above distance formula is still achieved by restricting to the cyclic invariant subspaces of A. In certain cases of interest, it sufﬁces to use cyclic vectors which do not vanish on E. In the case of Hardy space and Drury–Arveson space, this is accomplished by showing that cyclic subspaces generated by outer functions sufﬁce. This yields a simpliﬁcation in the statement of the theorems. The distance formula also formulates the classic Nevanlinna–Pick problem in terms of the Bergman kernel, whose multiplier algebra has a property much stronger than A1 (1). Indeed, Bergman space satisﬁes even a complete distance formula; see Sect. 7. While only of theoretical interest, since interpolation problems for H ∞ are better carried out on Hardy space, this result is surprising since Bergman space is not an NP kernel, and this failure persists even for 2-point interpolation. In Sect. 5, algebras of multipliers on complete Nevanlinna–Pick spaces will be studied. The multipliers of any such space are complete quotients of the noncommutative analytic Toeplitz algebra Ld [17], which has the property A1 (1) [16] and even the much stronger property X0,1 [12]. In the proof of Theorem 5.2, this is used to show that all complete NP kernel multiplier algebras inherit A1 (1). Consequently, any weak-∗closed algebra of multipliers on a complete NP space admits an NP family of kernels. Finally, in Sect. 7, matrix-valued Nevanlinna–Pick problems are studied. In order to retrieve a suitable NP theorem for matrices of arbitrary size, something much stronger than A1 (1) is required. However, by working with a suitable ampliation of the algebra, it is possible to retrieve an NP-type theorem with milder assumptions. Both of these results are summarized in Theorem 7.2. From this, a complete NP family of kernels is presented for the Bergman space as well as the classic matrix-valued NP theorem. A recent result of Ball et al. [7] regarding matrix-valued interpolation on the algebra H1∞ is generalized to arbitrary unital weak-∗ closed subalgebras of H ∞ .

2. Nevanlinna–Pick Families Let H denote a reproducing kernel Hilbert space of complex-valued functions on a set X, with kλ as the reproducing kernel at λ ∈ X. Let k(λ, μ) := kμ , kλ denote the associated positive kernel on X ×X. We deﬁne the multiplier algebra of H by M(H) = {f : X → C : f h ∈ H for all h ∈ H},

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where the product f g is deﬁned pointwise. Each multiplier f on H deﬁnes the corresponding multiplication operator Mf , which is bounded by an application of the closed graph theorem. We may then regard M(H) as an abelian subalgebra of B(H). It is well known that an operator T ∈ B(H) deﬁnes a multiplication operator on H if and only if each kernel function kλ is an eigenvector for T ∗ . From this it easily follows that M(H) is closed in the weak operator topology. We say that a unital subalgebra A of B(H) is a dual algebra if it is closed in the weak-∗ topology on B(H). If A is contained in the multiplier algebra of H, we say that A is a dual algebra of multipliers of H. Suppose that E is a ﬁnite subset of X. Let JE denote the weak-∗ closed ideal of multipliers in A that vanish on E. When E = {λ}, we write Jλ . If the context is clear, we may write J := JE . Suppose L is an invariant subspace of A and let PL denote the orthogonal projection of H onto L. Then L is also a reproducing kernel Hilbert space on the points in X which are not annihilated by L. The reproducing kernel on this set is given by PL kλ . Every f ∈ A deﬁnes a multiplier on L since (Mf |L )∗ PL kλ = PL Mf∗ PL kλ = PL Mf∗ kλ = f (λ)PL kλ . The following lemma shows that, in certain cases, it is possible to extend this kernel to all of X. Lemma 2.1. Suppose A is a dual algebra of multipliers on H. If L = A[h] is a cyclic invariant subspace of A, it is possible to extend the kernel function PL kλ (which is non-zero on {λ : h(λ) = 0}) to a kernel kλL defined on all of X so that kλL = 0 only when Jλ [h] = A[h]. For each f ∈ M(H), kλL satisfies the fundamental relation PL Mf∗ kλL = f (λ)kλL , and thus Mf kλL , kλL = f (λ)kλL 2

for all

λ ∈ X.

Proof. Since L = A[h] is a cyclic subspace and dim A/Jλ = 1, it follows that dim A[h]/Jλ [h] ≤ 1. If PL kλ = 0, then this is an eigenvector for PL A∗ as shown above. For f ∈ Jλ , kλL , f h = PL Mf∗ kλL , h = f (λ)kλL , h = 0. So PL kλ belongs to A[h] Jλ [h] and we set kλL = PL kλ . When PL kλ = 0 but dim A[h]/Jλ [h] = 1, we pick a unit vector kλL in dim A[h] Jλ [h]. Then for f ∈ M(H), f − f (λ)1 lies in Jλ . Hence Mf kλL , kλL = f (λ)kλL , kλL − (f − f (λ)1)kλL , kλL = f (λ). Also, Jλ [h] ∈ Lat(A) and thus PL Mf∗ kλL lies in LJλ [h] = CkλL . The previous computation shows that PL Mf∗ kλL = f (λ)kλL . This extends the kernel k L to all of X.

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This kernel allows us to evaluate the multipliers at a generally much larger subset of X (see Example 2.3 below) than just using PL kλ . However some continuity is lost for evaluation of functions in L. Since we are primarily interested in interpolation questions about the multiplier algebra, evaluation of the multipliers is more important. Definition 2.2. For any dual algebra A of multipliers on H and any cyclic invariant subspace L ∈ CycLat(A), let kλL denote the extended reproducing kernel on L constructed in Lemma 2.1 at the point λ. Example 2.3. In spaces of analytic functions, it is often possible to fully describe the kernel structure on L ∈ CycLat(A). Indeed, suppose that H = A2 (D) is the Bergman space, and A = H ∞ . Let L = H ∞ [h] for some non-zero function h ∈ A2 (D). Then XL = {λ ∈ D : PL kλ = 0} = {λ ∈ D : h(λ) = 0}. However, since the Bergman space consists of analytic functions, h vanishes only to some ﬁnite order on each of its zeros. It is routine to verify that for each n ≥ 0 and λ ∈ D, there is a function kλ,n ∈ A2 (D) such that h, kλ,n = h(n) (λ) for

h ∈ A2 (D).

Suppose that h vanishes at λ with multiplicity r ≥ 0. We claim that PL kλ,r = 0 and PL kλ,n = 0 for 0 ≤ n < r. Indeed, for any f ∈ H ∞ and n ≤ r, f h, kλ,n = (f h)(r) (λi ) r

r (j) = f (λ)h(r−j) (λ) j j=0 0 if 0 ≤ n < r = f (λ)h(r) (λ) if n = r. So PL kλ,n = 0 for 0 ≤ n < r. Set kλL = PL kλ,r /PL kλ,r . This calculation shows that if f ∈ Jλ , then f h, kλL = 0. So kλL belongs to A[h] Jλ [h]. Now for f, g ∈ H ∞ , f h, Mg∗ kλL = gf h, kλL = g(λ)f (λ)h(r) (λi ) = f h, g(λ)kλL . It follows that (Mg |L )∗ kλL = PL Mg∗ kλL = g(λ)kλL . Thus, g is a multiplier for this reproducing kernel. An identical construction is possible for any space of analytic functions on the unit disk in which the Taylor coefﬁcients of the power series expansion about 0 are continuous and composition by the M¨ obius automorphisms of the disk are bounded maps. Remark 2.4. The Bergman space is also a good place to illustrate why we require cyclic invariant subspaces. The Bergman shift B is a universal dilator for strict contractions [4]. For example, ﬁx a point λ ∈ D. Then B has an

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invariant subspace L such that N = L Jλ L is inﬁnite dimensional. Since (Mf |L )∗ |N = f (λ)IN , there is no canonical choice for kλL . Similarly, for any 0 < r < 1, we can obtain the inﬁnite ampliation rB (∞) as the compression of B to a semi-invariant subspace M , and this has the same issue for every |λ| < r. On the other hand, we can always identify a kernel structure on any invariant subspace L ∈ Lat A if we allow multiplicity. The subspaces Nλ = L Jλ L satisfy PNλ Mf∗ |Nλ = f (λ)PNλ . So if k is any unit vector in NL , we obtain Mf k, k = f (λ)

for all f ∈ A.

The spaces {Nλ : λ ∈ X} are linearly independent and together they span L. See the continued discussion later in Remark 2.9. Our primary interest will be Nevanlinna–Pick interpolation on some ﬁnite subset E = {λ1 , . . ., λn ) of X by functions in the algebra A. It could be the case that A fails to separate certain points in X, and so we impose the natural constraint that E contains at most one representative from any set of points that A identiﬁes. It follows that the kernels kλi form a linearly independent set. Indeed, since A separates these points, we can ﬁnd elements n p1 , . . ., pn ∈ A such that pi (λj ) = δij . Hence if i=1 αi kλi = 0, we ﬁnd that

n ∗ αi kλi = αi kλi for 1 ≤ i ≤ n. 0 = Mpi i=1

The quotient algebra A/JE is n-dimensional, and is spanned by the idempotents {pi + J : 1 ≤ i ≤ n}. We seek to establish useful formulae for the norm on A/J. These so-called operator algebras of idempotents have been studied by Paulsen in [23]. Definition 2.5. Given a ﬁnite subset E of X, set M = M (E) = span{kλ : λ ∈ E}. For L ∈ Lat(A), deﬁne ML = PL M

and NL = L JE L.

Observe that if L ∈ Lat A, then JE L is also invariant and is contained in L. Thus, NL is a semi-invariant subspace. In particular, PNL Mf PL = PNL Mf PNL and compression to NL is a contractive homomorphism. Likewise, M (E) is co-invariant, and so M (E) + L⊥ is co-invariant. Observe that ML = (M (E) + L⊥ ) ∩ L = L (M (E) + L⊥ )⊥ , and so it is also semi-invariant. Lemma 2.6. Given a finite subset E ⊂ X on which A separates points, and L = A[h] in CycLat(A), the space NL is a reproducing kernel Hilbert space over E with kernel {kλL : λ ∈ E}; and the non-zero elements of this set form a basis for NL .

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Also ML is a subspace of NL spanned by {kλL = PL kλ : λ ∈ E, h(λ) = 0}, and it is a reproducing kernel Hilbert space over {λ ∈ E : h(λ) = 0}. Proof. For each λ ∈ E, kλL ∈ L Jλ [h] ⊂ L JE [h] = NL . Let EL = {λ ∈ E : kλL = 0}. Then for λ ∈ E \ EL , L = Jλ L. Since JEL = λ∈E\EL Jλ , we see that JEL L = L. Hence we may factor JE = JEL JE\EL and note that JE L = JEL JE\EL L = JEL L. Now dim A/JEL = |EL |, so dim NL ≤ |EL |. But NL contains the non-zero vectors kλL for λ ∈ EL . For f ∈ A and λ ∈ EL , PNL Mf∗ kλL = PL PJ⊥E [h] Mf∗ kλL = PL Mf∗ kλL = f (λ)kλL . Because A separates the points of EL , it follows that these vectors are eigenfunctions for distinct characters of A, and thus are linearly independent. This set has the same cardinality as dim NL , and therefore it forms a basis. Now ML is spanned by {PL k λ : λ ∈ E}, and it sufﬁces to use the nonzero elements. These coincide with kλL on EL0 := {λ ∈ E : h(λ) = 0}. This is a subset of the basis for NL , and hence ML is a subspace of NL . It is also a reproducing kernel space on EL0 . The equality ML = NL holds in the following important case. The proof is immediate from the lemma. Corollary 2.7. Suppose that E is a finite subset of X on which A separates points, and L = A[h] in CycLat(A). If h does not vanish on E, then ML = NL . We will be searching for appropriate distance formulae from elements of A to an ideal JE in order to obtain Nevanlinna–Pick type results. We note certain easy estimates which always hold. Lemma 2.8. Suppose that A is a dual algebra on H, and let J be a wotclosed ideal. Set NL = L JL and ML = PL M (E) for L ∈ Lat A. Then the following distance estimates hold: dist(f, J) ≥

sup L∈Lat(A)

≥

PNL Mf PNL =

sup L∈CycLat(A)

sup L∈CycLat(A)

PML Mf PML =

PNL Mf PNL

sup L∈Lat(A)

PML Mf PML .

Proof. Suppose L is an invariant subspace of A. For Mf ∈ A and Mg ∈ J, compute Mf − Mg ≥ PNL (Mf − Mg )PNL = PNL Mf PNL ≥ PML Mf PML . Taking an infimum over g ∈ J and a supremum over Lat(A), we obtain dist(f, J) ≥

sup L∈Lat(A)

PNL Mf PNL ≥

sup L∈CycLat(A)

PNL Mf PNL .

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Since ML is contained in NL , we have sup L∈CycLat(A)

PNL Mf PNL ≥

sup L∈CycLat(A)

PML Mf PML .

Now consider an arbitrary element L ∈ Lat(A). Then PNL Mf PNL = PNL Mf PL = =

sup h=1, h∈L

≤ =

sup h=1, h∈L

sup h=1, h∈L

≤

sup h=1, h∈L

PNL Mf PL h

(PL − PJE L )Mf PA[h] h (PL − PJE A[h] )PNA[h] Mf PA[h] h PNA[h] Mf PA[h] h

sup L∈CycLat(A)

PNL Mf PL =

sup L∈CycLat(A)

PNL Mf PNL .

Similarly, PML Mf PML = PML Mf PL = =

sup h=1, h∈L

=

sup h=1, h∈L

=

sup h=1, h∈L

≤

sup h=1, h∈L

PML Mf PL h

PML Mf PA[h] h PML PNA[h] Mf PA[h] h PMA[h] Mf PA[h] h

sup L∈CycLat(A)

PML Mf PL =

sup L∈CycLat(A)

PML Mf PML .

So we may take the supremum over all invariant subspaces without changing these two distance estimates. When A is a dual algebra of multipliers, it is convenient to work with the adjoints due to their rich collection of eigenvectors. Lemma 2.8 gives us dist(f, J) ≥

sup L∈Lat(A)

PL Mf∗ |NL =

sup L∈CycLat(A)

PNL Mf∗ |NL .

Remark 2.9. This proposition shows that it sufﬁces to look at cyclic subspaces. In view of Lemma 2.1, this is of particular importance when dealing with algebras of multipliers. But in fact, there is little additional complication when the subspaces Nλ = L Jλ L have dimension greater than one. These subspaces are at a positive angle to each other even when they are inﬁnite dimensional because the restriction of PL Mf∗ to Nλ is just f (λ)PNλ . When A separates points λ and μ, the boundedness of Mf∗ yields a positive angle between eigenspaces. Moreover the spaces Nλ for λ ∈ X span L. We are interested in the norm PNL Mf∗ |NL . This is approximately achieved at a vector h = xλ where this is a ﬁnite sum of vectors xλ ∈ Nλ . Since f (λ)xλ , PL Mf∗ h =

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it follows that K = span{xλ : λ ∈ X} is invariant for PL Mf∗ for all Mf ∈ A. In particular, we obtain that PNL Mf∗ |NL ≤ 1 if and only if PK Mf∗ |K ≤ 1 for each subspace K of the form just described. This is equivalent to saying PK − PK Mf Mf∗ |K ≥ 0 because of semi-invariance. Because the xλ span K, this occurs if and only if 0 ≤ (I − Mf Mf∗ )xλj , xλi = 1 − f (λi )f (λj ) xλj , xλi . Thus, the norm condition is equivalent to the simultaneous positivity of a family of Pick matrices. Moreover, in the case of J = JE for a ﬁnite set E = {λ1 , . . . , λn } which is separated by A, this family of Pick matrices is positive if and only if we have positivity of the operator matrix . 1 − f (λi )f (λj ) PNλi PNλj n×n

For an arbitrary ideal J, we can take the supremum over all ﬁnite subsets E of X. Definition 2.10. The collection {k L : L ∈ CycLat(A)} is said to be a Nevanlinna–Pick family of kernels for A if for every ﬁnite subset E of X and every f ∈ A, (and NL = L JE L) dist(f, JE ) =

sup L∈CycLat(A)

PNL Mf∗ |NL .

The following routine theorem reveals why Nevanlinna–Pick families are given their name. Theorem 2.11. Let A be a dual algebra of multipliers on a Hilbert space H. The family {k L : L ∈ CycLat(A)} is a Nevanlinna–Pick family of kernels for A if and only if the following statement holds: Given E = {λ1 , . . . , λn } distinct points in X which are separated by A, and complex scalars w1 , . . . , wn , there is a multiplier f in the unit ball of A such that f (λi ) = wi for 1 ≤ i ≤ n if and only if the Pick matrices (1 − wi wj )k L (λi , λj ) n×n are positive definite for every L ∈ CycLat(A). Proof. Suppose that {k L : L ∈ CycLat(A)} is a Nevalinna–Pick family. If such a multiplier f exists, then the positivity of the matrices follows from standard results in reproducing kernel Hilbert spaces (see [3], for instance). On the other hand, suppose that all such matrices are positive definite. Since A separates the points in E, ﬁnd an arbitrary interpolant p ∈ A so that p(λi ) = wi (for example, consider the functions pi as deﬁned earlier, and let n p = i=1 wi pi ). It is a routine argument in this theory that PNL Mp∗ |NL ≤ 1 if and only if I − PNL Mp PNL Mp∗ PNL ≥ 0.

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Since kλLi spans NL , this holds if and only if 0 ≤ (I − PNL Mp PNL Mp∗ PNL )kλLj , kλLi = kλLj , kλLi − Mp∗ kλLj , Mp∗ kλLi = kλLj , kλLi − wj kλLj , wi kλLi = (1 − wi wj )k L (λi , λj ) . It follows that dist(p, JE ) =

sup L∈CycLat(A)

PL Mp∗ |NL ≤ 1.

The ideal JE is weak-∗-closed, and so the distance is attained at some g ∈ J. The multiplier f := p − g is in the unit ball of A and interpolates the given data. Conversely, suppose that the interpolation property holds. Fix f ∈ A such that supL∈CycLat(A) PL Mf∗ |NL = 1. By assumption, there is a multiplier g in the unit ball of A such that g(λi ) = f (λi ) for each 1 ≤ i ≤ n. Hence dist(f, JE ) ≤ f − (f − g) = g ≤ 1. By Lemma 2.8, dist(Mf , A) ≥ supL∈Lat(A) PL Mf∗ |NL = 1. Therefore, the family {k L : L ∈ CycLat(A)} is a Nevanlinna–Pick family of kernels for A. We are also interested when the subspaces ML sufﬁce to compute the distance. The same argument shows that these spaces form a Nevanlinna– Pick family for interpolation on the set E. This will occur if we can show that cyclic subspaces generated by elements h that do not vanish on E sufﬁce in the distance estimate. This is not always the case, but it does happen in important special cases.

3. Algebras with property A1 (1) Let A be a dual algebra in B(H). Given vectors x and y in H, let [xy ∗ ] denote the vector functional A → Ax, y for A ∈ A. Every weak-∗ continuous functional on A is a (generally inﬁnite) linear combination of these vector functionals. One says that A has property A1 (r) if, for each weak-∗ continuous functional ϕ on A with ϕ < 1 , there are vectors x and y so that ϕ = [xy ∗ ] and x y < r. Property A1 (r) implies that the weak-∗ and weak operator topologies of A coincide, since the weak operator continuous linearly functionals are precisely the ﬁnite linear combinations of vector functionals. It is well known that for any reproducing kernel Hilbert space, the multiplier algebra M(H) is reﬂexive. This is because Ckλ is invariant for M(H)∗ for each kernel function kλ . Hence, if T ∈ Alg(Lat M(H)), then kλ is an eigenvector for T ∗ . It follows that T is a multiplier on H by standard results.

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Suppose A is a dual subalgebra of M(H) that is relatively reﬂexive; that is, the algebra satisﬁes A = Alg(Lat A) ∩ M(H). The reﬂexivity of the whole multiplier algebra implies that A is actually reﬂexive. We saw that a multiplier f in A deﬁnes the multiplication operator Mf |L on every invariant subspace L of A. On the other hand, if f is simultaneously a multiplier on each L in Lat A, then it is clearly in M(H) and leaves every L invariant. Consequently, A is the largest algebra of multipliers for the family of subspaces Lat(A). It is clear that it sufﬁces to consider cyclic invariant subspaces. So we have M(L) = M(L). A= L∈Lat(A)

L∈CycLat(A)

Following the notation of Agler and McCarthy [3], CycLat A is called a realizable collection of reproducing kernel Hilbert spaces. The ﬁrst part of the following theorem is contained in Hadwin and Nordgren’s paper [18, Prop. 2.5(1)] on A1 (r) (which is called Dσ (r) there). The second part is from Kraus and Larson [20, Theorem 3.3]. We say that a subalgebra A of B is relatively hyper-reflexive if there is a constant C so that dist(B, A) ≤ C sup PL⊥ BPL for all B ∈ B. L∈Lat A

The optimal value of C is the relative hyper-reﬂexivity constant. Again, it is clear that cyclic subspaces sufﬁce. Theorem 3.1 (Hadwin–Nordgren, Kraus–Larson). Suppose B is a dual subalgebra of M(H) and has property A1 (r). Then every wot-closed unital subalgebra A of B is reflexive. Moreover, A is relatively hyper-reflexive in B with distance constant at most r. If B has property A1 (1), we obtain an exact distance formula. Corollary 3.2. Suppose that B has property A1 (1), and let A be a wot-closed unital subalgebra. Then dist(B, A) =

PL⊥ BPL

sup

for all

B ∈ B.

L∈CycLat(A)

We can use the same methods to obtain a distance formula to any weak-∗ closed ideal. We do not have a reference, so we supply a proof. The ideas go back to the seminal work of Sarason [30]. An argument similar to this one is contained in the proof of [17, Theorem 2.1]. Theorem 3.3. Suppose that A is a dual algebra on H with property A1 (1), and let J be any wot-closed ideal of A. Then we obtain dist(A, J) =

sup L∈CycLat(A)

PLJL Mf∗ |LJL .

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Proof. By Lemma 2.8, dist(A, J) dominates the right hand side. Conversely, given A ∈ A and ε > 0, pick ϕ ∈ A∗ such that ϕ|J = 0, ϕ < 1 + ε and |ϕ(A)| = dist(A, J). Using property A1 (1), we obtain vectors x and y with x = 1 and y < 1 + ε so that ϕ = [xy ∗ ]. Set L = A[x] in CycLat A. Since L is invariant, we can and do replace y by PL y without changing [xy ∗ ]. Since ϕ|J = 0, y is orthogonal to J[x] = JL. Hence y belongs to L JL. Therefore, dist(A, J) = |Ax, y | = |APL x, PNL y | = |PLJL APL x, y | = |PLJL APLJL x, y | ≤ PLJL APLJL x y < (1 + ε)PLJL A|LJL . It follows that the right hand side dominates dist(A, J).

We apply this to the context of interpolation in reproducing kernel Hilbert spaces. Theorem 3.4. Suppose A is a dual algebra of multipliers on H with property A1 (1). Let E be a finite subset of X which is separated by A. Then the following distance formula holds: dist(f, JE ) = sup{PNL Mf∗ |NL : L ∈ CycLat(A)}. That is, {k L : L ∈ CycLat(A)} is a Nevanlinna–Pick family of kernels for A. Corollary 3.5. Suppose A is a dual algebra of multipliers on a reproducing kernel Hilbert space H on X that has property A1 (1). Let {λ1 , . . ., λn } be distinct points in X separated by A and let {w1 , . . ., wn } be complex numbers. There is a multiplier f in the unit ball of A such that f (λi ) = wi for each i if and only if (1 − wi wj )k L (λi , λj ) ≥ 0 for all L ∈ CycLat(A). By replacing the property A1 (1) with A1 (r) for r > 1, one may repeat the proof of the previous theorem and obtain an analogous result where the norm of the interpolant is bounded by r. In applications, is it convenient to use the spaces ML instead of NL . In light of Corollary 2.7, this is possible for cyclic subspaces L = A[h] provided that h does not vanish on E. A reﬁnement of the A1 (1) property can make this possible. The distinction made in this theorem is that the kernel is formed as simply the compressions PL kλ . Theorem 3.6. Let A be a dual algebra of multipliers on H, and let E = {λ1 , . . ., λn } be a finite subset of X which is separated by A. Suppose that A has property A1 (1) with the additional stipulation that each weak-∗ continuous functional ϕ on A with ϕ < 1 can be factored as ϕ = [xy ∗ ] with x y < 1 such that x does not vanish on E. Then there is a multiplier in the unit ball of A with f (λi ) = wi for λi ∈ E if and only if (1 − wi wj )PL kλj , kλi ≥ 0 for all cyclic subspaces L = A[h] where h does not vanish on E.

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4. Subalgebras of H ∞ In this section, we discuss the algebra H ∞ and its subalgebras. In particular, we show how to recover results of Davidson et al. [14] and Raghupathi [27]. We also show how one can formulate a Nevanlinna–Pick theorem for H ∞ on Bergman space. While this is only of theoretical interest, since interpolation is better done on the Hardy space, this has been an open question for some time. Hardy Space Let H = H 2 be Hardy space, so that M(H) = H ∞ . It is well known that this algebra has property A1 (1) [8, Theorem 3.7]. Moreover, any weak-∗ continuous functional ϕ on H ∞ can be factored exactly as ϕ = [xy ∗ ] with x y = ϕ. Now if the inner-outer factorization of x is x = ωh, with h outer and ω inner, then ϕ = [hk ∗ ] where k = Mω∗ y. Thus, the factorization may be chosen so that h is outer, and in particular, does not vanish on D. Consider the classical Nevanlinna–Pick problem for a ﬁnite subset E of D. Following Sarason’s approach [30], let BE denoting the ﬁnite Blaschke product with simple zeros at E. Then JE = BE H ∞ . Beurling’s invariant subspace theorem for the unilateral shift shows that if L ∈ Lat(H ∞ ), then there is some inner function ω so that L = ωH 2 . In particular, every invariant subspace is cyclic. Observe that Nω := ωH 2 ωBE H 2 = ω(H 2 BE H 2 ) = ωM (E). Here M (E) = span{kλ : λ ∈ E} = H 2 BE H 2 . However Mω is an isometry in M(H 2 ), and hence commutes with M(H 2 ). So it provides a unitary equivalence between the compression of M(H 2 ) to Nω and to M (E). Therefore, for f ∈ H ∞ , Theorem 3.4 gives dist(f, BE H ∞ ) = sup{PNω Mf∗ |Nω : ω inner} = Mf∗ |M (E) . The classical Nevanlinna–Pick interpolation theorem now follows as in Corollary 3.5, as Sarason showed. Perhaps more importantly, Theorem 3.4 provides an interpolation theorem for any weak-∗-closed subalgebra of H ∞ . These constrained Nevanlinna– Pick theorems fashion suitable generalizations of the results seen in [14] and [27]. In fact, since any weak-∗ linear functional ϕ on A extends to a functional on H ∞ with arbitrarily small increase in norm, it can be factored as ϕ = [hk ∗ ] where h is outer and h k < ϕ+ε. Therefore, the more reﬁned Theorem 3.6 applies. We obtain: Theorem 4.1. Let A be a weak-∗-closed unital subalgebra of H ∞ . Let E = {λ1 , . . . , λn } be a finite subset of D which is separated by A. Then there is a multiplier in the unit ball of A with f (λi ) = wi for 1 ≤ i ≤ n if and only if (1 − wi wj )PL kλi , kλj ≥ 0 for all cyclic subspaces L = A[h] where h is outer.

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In [14], the algebra H1∞ = {f ∈ H ∞ : f (0) = 0} is studied. Beurling’s Theorem was used to show that there is a simple parameterization of the cyclic invariant subspaces H1∞ [h] for h outer, namely 2 Hα,β := span{α + βz, z 2 H 2 }

for all

(α, β) with |α|2 + |β|2 = 1.

It was shown that these provide a Nevanlinna–Pick family for H1∞ , and the consequent interpolation theorem. Raghupathi [27] carries out this program ∞ = C1 + BH ∞ where B is a ﬁnite Blaschke for the class of algebras HB product. Raghupathi [28] shows that Abrahamse’s interpolation result for multiply connected domains [1] is equivalent to the interpolation problem for certain weak-∗ subalgebras of H ∞ . Consequently, our distance formula includes Abrahamse’s theorem as a special case. Singly-generated Multiplier Algebras Suppose that T is an absolutely continuous contraction on H, and let AT be the unital, weak-∗-closed algebra generated by T . The Sz.Nagy dilation theory provides a weak-∗ continuous, contractive homomorphism Φ : H ∞ → AT given by Φ(f ) = PH f (U )|H , where U is the minimal unitary dilation of T . If Φ is isometric (i.e. Φ(f ) = f ∞ for every f ∈ H ∞ ), then Φ is also a weak-∗ homeomorphism. In addition, Φ being isometric ensures that the preduals (AT )∗ and L1 /H01 are isometrically isomorphic. In this case, we say that T has an isometric functional calculus. See [9] for relevant details. The following deep result of Bercovici [11] will be used. Theorem 4.2 (Bercovici). Suppose T is an absolutely continuous contraction on H and that T has an isometric functional calculus. Then AT has property A1 (1). We will use Theorem 4.2 to show that a wide class of reproducing kernel Hilbert spaces admit Nevanlinna–Pick families of kernels for arbitrary dual subalgebras of H ∞ . Example 4.3. Let μ be a ﬁnite Borel measure on D, and let P 2 (μ) denote the closure of the polynomials in L2 (μ). A bounded point evaluation for P 2 (μ) is a point λ for which there exists a constant M > 0 with |p(λ)| ≤ M pP 2 (μ) for every polynomial p. A point λ is said to be an analytic bounded point evaluation for P 2 (μ) if λ is in the interior of the set of bounded point evaluatons, and that the map z → f (z) is analytic on a neighborhood of λ for every f ∈ P 2 (μ). It follows that if λ is a bounded point evaluation, then there is a kernel function kλ in P 2 (μ) so that p(λ) = p, kλ . For an arbitrary f ∈ P 2 (μ), if we set f (λ) = f, kλ , then these values will agree with f a.e. with respect to μ. For both the Hardy space and Bergman space, the set of analytic bounded point evaluation is all of D. A theorem of Thomson shows that either P 2 (μ) = L2 (μ) or P 2 (μ) has analytic bounded point evaluations [31]. Let m be Lebesque area measure on the disk. For s > 0, deﬁne a weighted area measure on D by dμs (z) = (1 − |z|)s−1 dm(z). The monomials z n form an orthogonal basis for P 2 (μ). This includes the Bergman space for s = 1. For these spaces, every point in D is an analytic point evaluation.

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The following result appears as Theorem 4.6 in [3]. Theorem 4.4. Let μ be a measure on D such that the set of analytic bounded point evaluations of P 2 (μ) contains all of D. Then M(P 2 (μ)) is isometrically isomorphic and weak-∗ homeomorphic to H ∞ . Corollary 3.5 yields the following interpolation result for these spaces. Theorem 4.5. Let μ be a measure on D such that the set of analytic bounded point evaluations of P 2 (μ) contains all of D. Suppose A is a dual subalgebra of M(P 2 (μ)). Then A has a Nevanlinna–Pick family of kernels. Example 4.6. In particular, Theoorem 4.5 provides a Nevanlinna–Pick condition for Bergman space A2 := A2 (D), whose reproducing kernel kernel kλB = (1 − λz)−2 is not an NP kernel. In fact, the Bergman kernel fails to even have the two-point Pick property. See [3, Example 5.17] for details. The multiplier algebra of Bergman space has property A1 (1) as a consequence of much stronger properties (see Sect. 7), but the subspace lattice of the Bergman shift is immense.

5. Complete NP Kernels A reproducing kernel is a complete Nevanlinna–Pick kernel if matrix interpolation is determined by the positivity of the Pick matrix for the data. That is, if H is a Hilbert space with reproducing kernel k on a set X, E = {λ1 , . . . , λn } is a ﬁnite subset of X, and W1 , . . . , Wn are r × r matrices, then a necessary condition for there to be an element F ∈ Mr (M(H)) with F (λi ) = Wi and F ≤ 1 is the positivity of the matrix (Ir − Wi Wj∗ )k(λi , λj ) . We say that k is a complete NP kernel if this is also sufﬁcient. The Davidson and Pitts [17] showed that symmetric Fock space, now called the Drury–Arveson space Hd2 , on the complex ball Bd of Cd (including d = ∞) is a complete NP space with kernel 1 . k(w, z) = 1 − w, z The complete NP kernels were classiﬁed by McCullough [21,22] and Quiggin [25,26] building on work by Agler (unpublished). Another proof was provided by Agler and McCarthy [2], who noticed the universality of the Drury–Arveson kernel. Theorem 5.1. (McCullough, Quiggin, Agler-McCarthy) Let k be an irreducible kernel on X. Then k is a complete Nevanlinna–Pick kernel if and only if for some cardinal d, there is an injection b : X → Bd and a nowherevanishing function δ : X → C such that k(x, y) =

δ(x)δ(y) . 1 − b(x), b(y)

δ(x) In this case, the map kx → 1− z,b(x)

extends to an isometry of H into Hd2 .

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Since the span of kernel functions is always a co-invariant subspace for the space of multipliers, the complete NP kernel spaces are seen to correspond to certain co-invariant subspaces of Drury–Arveson space, i.e. span{kz : z ∈ b(X)}. The Davidson and Pitts [17] show that the multiplier algebras of these spaces are all complete quotients of the non-commutative analytic Toeplitz algebra, Ld , generated by the left regular representation of the free semi2 + group F+ d on the full Fock space (Fd ). It follows immediately that they are 2 complete quotients of M(Hd ). See also Arias and Popescu [6]. We will show that all such quotients of M(Hd2 ) have property A1 (1). The algebra Ld actually has property Aℵ0 [15] and even property X0,1 [12]. But these stronger properties do not extend to M(Hd2 ). More specifically, if J is a wot-closed ideal of Ld with range M = J 2 (F+ d ), then [17] shows that Ld /J is completely isometrically isomorphic to the compression of Ld to M ⊥ . Conversely, if M is an invariant subspace of both Ld and its commutant, the right regular representation algebras Rd , then J = {A ∈ Ld : Ran A ⊂ M } is a wot-closed ideal with range M . In particular, if C is the commutator ideal, it is shown that M(Hd2 ) Ld /C. Moreover the compression of both Ld and Rd to Hd2 agree with M(Hd2 ). So if N is a coinvariant subspace of Hd2 , then M = 2 (F+ d ) N is invariant for both Ld and Rd and the theory applies. The following result is due to Arias and Popescu [6, Prop.1.2]. We provide a proof for completeness. Theorem 5.2 (Arias–Popescu). Let J be any wot-closed ideal of Ld and let ⊥ M = J 2 (F+ d ). Then A = PM Ld |M ⊥ has property A1 (1). Proof. Let q : Ld → Ld /J A denote the canonical quotient map. Suppose ϕ is a weak-∗ functional on A with ϕ < 1. Then ϕ ◦ q is a weak-∗ continuous functional on (Ld )∗ of norm ¡1. Hence there are vectors x and y in 2 (F+ d) with ϕ ◦ q = [xy ∗ ] and xy < 1. Form the cyclic subspace L = Ld [x]. By [5,15] there is an isometry R ∈ Rd such that Ld [x] = R 2 (F+ d ). Let u be the vector such that x = Ru and set v = R∗ y. A direct calculation shows that [xy ∗ ] = [uv ∗ ]. Observe that Ld [u] = 2 (F+ d ). ∗ Obviously [uv ] annihilates J, and it follows that v is orthogonal to Ju = JLd u = J 2 (F+ d ) = M. ⊥ B|⊥ Thus, v ∈ M ⊥ . Now, for A ∈ A, pick B ∈ Ln with A = q(B) = PM M. ⊥ Then since M is coinvariant for Ld ,

ϕ(A) = ϕ ◦ q(B) = Bu, v ⊥ ⊥ ⊥ v = PM BPM u, v = Bu, PM ⊥ u), v . = A(PM ⊥ u)v ∗ ]. Therefore, A has property A1 (1). Hence ϕ = [(PM

The remarks preceding this theorem show that the multiplier algebra of every complete NP kernel arise in this way. We further reﬁne this to observe that the vector x in the factorization may be chosen so that it does not vanish.

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Corollary 5.3. The multiplier algebra M(H) of every complete NP kernel has property A1 (1). In particular, M(Hd2 ) has property A1 (1). Moreover, each ϕ ∈ M(H)∗ with ϕ < 1 can be represented as ϕ = [xy ∗ ] such that x does not vanish on X and x y < 1. Proof. We may assume that M(H) is the compression of Ld to the subspace M = span{kλ : λ ∈ X ⊂ Bn }. It remains only to verify that in the proof of ⊥ u does not vanish. Since u is a cyclic vector for Theorem 5.2, the function PM Ld , it is not orthogonal to any kλ because Ckλ is coinvariant. Therefore, for any λ ∈ X, ⊥ ⊥ ⊥ kλ = PM u, kλ = δ(λ)−1 (PM u)(λ), 0 = u, kλ = u, PM

where δ is the scaling function of the embedding in Theorem 5.1.

The proof actually shows that it sufﬁces to use vectors h which are cyclic for M(H). In the case of Drury–Arveson space, like for Hardy space, these vectors are called outer. Remark 5.4. The algebra M(Hd2 ) does not have property Aℵ0 . The reason is that algebras with this property have non-trivial invariant subspaces which are orthogonal [10]. But any two non-trivial invariant subspaces of M(Hd2 ) have non-trivial intersection. Indeed, if M ∈ Lat M(Hd2 ), then N = M + (Hd2 )⊥ is invariant for Ld . By [5,15], this space is the direct sum of cyclic invariant subspaces Ni = Ri 2 (F+ d ) for isometries Ri ∈ Rd . The compression PHd2 Ri |Hd2 is a multiplier Mfi in M(Hd2 ). Thus, M = PHd2 N = PHd2 Ri 2 (F+ d) =

i

i

PHd2 Ri PHd2 2 (F+ d)=

Mfi Hd2 .

i

In particular, every invariant subspace M contains the range of a non-zero multiplier Mf . Hence given two invariant subspaces M and N in Hd2 , we can ﬁnd non-zero multipliers f and g with Ran Mf ⊂ M and Ran Mg ⊂ N . So M ∩ N contains Ran Mf g . Based on heuristic calculations, we expect that M(Hd2 ) does not have property A2 (r) for any r ≥ 1, and likely not A2 . We are now in a position to apply Theorem 3.6. Theorem 5.5. Suppose k is an irreducible, complete Nevanlinna–Pick kernel on X. Then any dual subalgebra A of multipliers of H admits a Nevanlinna–Pick family of kernels. More specifically, if E = {λ1 , . . ., λn } is a finite subset of X which is separated by A and w1 , . . . , wn are scalars, then there is a multiplier f in the unit ball of A with f (λi ) = wi for 1 ≤ i ≤ n if and only if (1 − wi wj )PL kλj , kλi ≥ 0 for all cyclic invariant subspaces L = A[h] of A (and it suffices to use h which do not vanish on E).

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6. Finite Kernels In this section, we present numerical evidence that there is a ﬁnite dimensional multiplier algebra A with the property that CycLat(A) is not an NP family for A. In particular, this algebra does not have A1 (1). It does, however, have property A1 (r) for some r > 1. Suppose X = {λ1 , . . . , λN } is a ﬁnite set and k : X × X → C is an irreducible kernel. Let y1 , . . . , yN be vectors in CN such that k(λi , λj ) = yj , yi , and let {x1 , . . . , xN } be a dual basis for the yi . The space H = CN may be regarded as a reproducing kernel Hilbert space over X, with reproducing kernel at λi given by yi . The multiplier algebra M(H) is an N -idempotent operator algebra spanned by the rank one idempotents pi = xi yi∗ . If {ei } is the canonical orthonormal basis for CN , then one readily sees that M(H) is similar to the diagonal algebra DN via the similarity S deﬁned by Sei = xi . Since DN evidently has property A1 (1), it follows from elementary results on dual algebras that M(H) has A1 (r) for some r ≥ 1. If k is irreducible and a complete NP kernel, then Corollary 5.3 shows that M(H) has A1 (1). However, there are many kernels k that cannot be embedded in Drury–Arveson space in this way. We expect that many of these algebras fail to have A1 (1) and that the distance formula fails in such cases. Since A is similar to the diagonal algebra DN , the invariant subspaces are spanned by some subset of {x1 , . . . , xN }. Denote them by Lσ = span{xi : i ∈ σ}. For E ⊂ {1, . . . , N }, the ideal J = JE = span{pi : i ∈ E}. Then JLσ = Lσ\E , and Nσ := NLσ = Lσ Lσ\E . The distance formula is obtained as the maximum of compressions to these subspaces—so we need only consider the maximal ones. These arise from σ ⊃ E. For trivial reasons, the distance formula is always satisﬁed when N = 2 and N = 3. There is strong numerical evidence to suggest that the formula does hold for N = 4, though we do not have a proof. In the following 5-dimensional example, Wolfram Mathematica 7 was used to ﬁnd a similarity S such that the distance formula fails. Example 6.1. Deﬁne the similarity ⎡

3 ⎢0 ⎢ S=⎢ ⎢ −1 ⎣ −1 1

1 1 0 1 1

1 −2 −1 2 3

0 −1 1 1 1

⎤ −1 0 ⎥ ⎥ −1 ⎥ ⎥ −1 ⎦ −2

Let pi = xi yi∗ for 1 ≤ i ≤ 5 be the idempotents which span the algebra A := M(H). Let E = {1, 2, 3}, and form J = JE = span{p4 , p5 }. Consider the element A = −2p1 − 3p2 + 7p3 . We are interested in comparing maxσ PNσ APNσ with dist(A, J). As noted above, it sufﬁces to use

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maximal Nσ ’s formed by the cyclic subspaces that do not vanish on E, namely N{123} = span{x1 , x2 , x3 }, N{1234} = span{x1 , x2 , x3 , x4 } Cx4 , N{1235} = span{x1 , x2 , x3 , x5 } Cx5 , and N{12345} = span{x4 , x5 }⊥ = span{y1 , y2 , y3 }. For notational convenience, set Pσ := PNσ . The values of Pσ APσ were computed and rounded to four decimal places: P123 AP123 = 9.0096, P1234 AP1234 = 10.1306, P1235 AP1235 = 7.4595, P12345 AP12345 = 10.6632. By minimizing a function of two variables, the following distance estimate was obtained dist(A, J) ≈ 11.9346. Similar results appeared for many different elements of A, which indicate that CycLat(A) is not an NP family for A. Consequently, it must also fail to have A1 (1). We currently have no example of a dual algebra of multipliers on any H that fails to have A1 (r) for every r ≥ 1, or even fails to have A1 .

7. Matrix-valued Interpolation In this section, we will discuss matrix-valued Nevanlinna–Pick interpolation problems. The classical theorem for matrices says that given z1 , . . ., zn in the disk, and r × r matrices W1 , . . ., Wn , there is a function F in the the unit ball of Mr (H ∞ ) such that F (zi ) = Wi if and only if the Pick matrix Ir − Wi Wj∗ 1 − zi zj r×r is positive semidefinite. One can deﬁne a linear map R on M (E) ⊗ Cr by setting R(kλs i ⊗ u) = kλs i ⊗ Wi∗ u

for

1 ≤ i ≤ n and

u ∈ Cr .

Note that if F is an arbitrary interpolant, then R = MF∗ |M (E)⊗Cr . Now R ≤ 1 is equivalent to I − R∗ R ≥ 0, which is equivalent to the positivity condition above. Hence, this provides the complete distance formula: dist(F, Mr (J)) = MF∗ |M (E)⊗Cr . The same holds (by definition) for all complete NP kernels when the factor 1 1−zi zj is replaced by k(λi , λj ). The multiplier algebra of all complete NP kernels therefore satisfy the analogous distance formula. Our goal is to generalize the results of Sect. 2 to a matrix-valued setting by imposing stronger conditions on our algebras of multipliers. Let (H, k)

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be a reproducing kernel Hilbert space over X, and let r ≥ 1. We will consider the algebra Mr (M(H)) of r × r matrices of multipliers acting on the vector-valued space H (r) = H ⊗ Cr . For any non-zero vector u ∈ Cr , the functions kλ ⊗ u act as vector-valued kernel functions. We can therefore consider Mr (M(H)) as functions on X with values in Mr . They act as multipliers of H (r) , and inherit a norm as elements of Mr (B(H)) B(H (r) ). It is readily veriﬁed that for any multiplier F and any λ ∈ X, we have MF∗ (kλ ⊗ u) = kλ ⊗ F (λ)∗ u for

λ∈X

and

u ∈ Cr .

Conversely, any bounded operator on H (r) that satisﬁes these relations is a multiplier of H (r) . The algebra Mr (M(H)) is a unital, weak-∗-closed subalgebra of Mr (B(H)), and thus is a dual algebra. Consequently, we may apply the same heuristic as Sect. 2 when trying to compute distances. Any dual subalgebra A of M(H) determines the dual subalgebra Mr (A) of Mr (M(H)). Suppose that E = {λi : 1 ≤ i ≤ n} is a ﬁnite subset of X separated by A. Let JE be the ideal of functions in A vanishing on E. For F ∈ Mr (A), any subspace of the form L(r) for L ∈ Lat(A) is invariant for Mr (A). Conversely, any invariant subspace of Mr (A) takes this form. The subspace L(r) is cyclic if and only if L is r-cyclic because if x1 , . . . , xr is a cyclic set, then x = (x1 , . . . , xr ) is a cyclic vector for L(r) and vice versa. So in general we cannot deal only with cyclic invariant subspaces of the algebra A. We will have to deal with some multiplicity of the kernels on these spaces. This can be handled as in the discussion in Remark 2.9. We can apply Lemma 2.8 to Mr (A) and the ideal Mr (JE ). For any F ∈ Mr (A), we have dist(F, Mr (JE )) ≥ supL∈Lat(A) (PNL ⊗ Ir )MF (PNL ⊗ Ir ) ≥ supL∈Lat(A) (PML ⊗ Ir )Mf (PML ⊗ Ir ). Definition 7.1. If equality holds for this distance formula for every ﬁnite subset E which is separated by A, i.e. for any F ∈ Mr (A), and NL = L JE L for L ∈ Lat A, dist(F, Mr (JE )) =

sup L∈Lat(A)

(PNL ⊗ Ir )MF (PNL ⊗ Ir )

then we say that Lat A is an r × r Nevanlinna–Pick family for A. If this holds for all r ≥ 1, then we say that Lat A is a complete Nevanlinna–Pick family for A. Generally, property A1 (1) is not inherited by matrix algebras. Conway and Ptak [13] show that any absolutely continuous contraction in class C00 with an isometric functional calculus has X0,1 . This includes the Bergman shift B, and consequently the multiplier algebra of Bergman space. The property X0,1 implies that Mr (A) has A1 (1) for every r ≥ 1. On the other hand, it can be the case that some ﬁnite ampliation of the algebra will have A1 (1). Given a dual algebra A on a Hilbert space H, the k-th ampliation A(k) is an isometric representation of A on H (k) , the direct sum of k copies of H, with elements A(k) = A ⊕ · · · ⊕ A, the direct sum of k

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(k)

copies of A. The preduals A∗ and A∗ are isometrically isomorphic. However, a rank k functional on A converts to a rank one functional on A(k) since k Axi , yi = T (k) x, y i=1

where x = (x1 , . . . , xk ) and y = (y1 , . . . , yk ) in H (k) . If A has A1 , so does 2 Mr (A(r ) ) [9, Proposition 2.6], but the constants are not always good enough. The inﬁnite ampliation of any operator algebra has A1 (1). This is because weak-∗ continuous functional on B(H) can be represented by a trace class operator. Using the polar decomposition, this can be realized as ∞ ϕ = i=1 [xi yi∗ ] where i xi 2 = i yi 2 = ϕ. So ϕ(T ) =

∞

T xi , yi = T (∞) x, y

i=1

where x = (x1 , x2 , . . . ) and y = (y1 , y2 , . . . ) in H (∞) . In fact, this inﬁnite ampliation is easily seen to have property X0,1 . As in Theorem 3.4, if Mr (A) has property A1 (1), then we obtain an exact distance formula which yields a Nevanlinna–Pick type theorem for these algebras. The proof is the same. Theorem 7.2. Suppose A is a dual algebra of multipliers on H. If Mr (A) has property A1 (1), then Lat A is an r × r Nevanlinna–Pick family for A. More generally, if the ampliation Mr (A(s) ) has A1 (1), then Lat(A(s) ) is an r×r Nevanlinna–Pick family for A. In particular, Lat(A(∞) ) is a complete Nevanlinna–Pick family for any algebra of multipliers A. While it appears that ampliations of matrix algebras over some well known multiplier algebras have A1 (1), we are unaware of any general results of this kind. Such a result would be interesting. We will illustrate Theorem 7.2 with some examples. Bergman Space The Bergman shift B on A2 (D) has property X0,1 [13]. This is inherited by any dual subalgebra A of M(A2 (D)). Therefore, Mr (A) has property A1 (1) for all r ≥ 1. We obtain a formulation of the complete Nevanlinna–Pick interpolation for subalgebras of H ∞ in this context. Theorem 7.3. Let A be a dual subalgebra of M(A2 (D)) H ∞ . Let E = {z1 , . . . , zn } be points in D which are separated by A, and let W1 , . . . , Wn be r×r matrices. There is an element F ∈ Mr (A) with F (zi ) = Wi and F ≤ 1 if and only if the following holds: for each L ∈ Lat A, (setting Nzi = L Jzi L for 1 ≤ i ≤ n), we have (Ir − Wi Wj∗ ) ⊗ PNzi PNzj ≥ 0 for all L ∈ Lat A. n×n

Proof. Let A be any element of Mr (A) such that A(zi ) = Wi . Since Mr (A) has A1 (1), Theorem 3.3 implies that dist(A, Mr (JE )) =

sup L∈Lat(A)

(PNL ⊗ Ir )A(PNL ⊗ Ir ).

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As before, a necessary and sufﬁcient condition for interpolation with an element F ∈ Mr (A) of norm at most one is that dist(A, Mr (JE )) ≤ 1. Also arguing in a standard manner, using the semi-invariance of NL , (PNL ⊗ Ir )A(PNL ⊗ Ir )2 = (PNL ⊗ Ir )AA∗ (PNL ⊗ Ir ). This has norm at most 1 if and only if (PNL ⊗ Ir )(I − AA∗ )(PNL ⊗ Ir ) ≥ 0. As we observed in Remark 2.9, NL is spanned by the spaces Nzi for 1 ≤ i ≤ n. These subspaces are eigenspaces for (PL A|L )∗ , and thus they are independent, and at a positive angle to each other. So positivity of the operator above is equivalent to the positivity of (PNzi ⊗ Ir )(I − AA∗ )(PNzj ⊗ Ir ) = (Ir − Wi Wj∗ ) ⊗ PNλi PNλj because the restriction of (PL ⊗ Ir )A∗ to Nzj ⊗ Cr is just PNzj ⊗ Wj∗ .

Hardy Space We return to the case of subalgebras of H ∞ acting on Hardy space. In [14], for A = H1∞ := {f ∈ H ∞ : f (0) = 0}, it was shown that the distance formula for matrix interpolation fails for A. In our terminology, Lat H1∞ is not a complete Nevanlinna–Pick family. So we cannot drop the assumption that Mr (A) has A1 (1). Indeed, the unilateral shift fails to have even property A2 [8, Theorem 3.7]. We will show that with ampliations, a general result can be obtained. The following result should be well known, but we do not have a reference. A version of it appears as Theorem 4 in [30]. Lemma 7.4. Mr (H ∞ ) acting on (H 2 ⊗ Cr )(r) as r × r matrices over M(H 2 ) ampliated r times has property A1 (1). Proof. Form the inﬁnite ampliation Mr (M(H 2 )(∞) ). Then any weak-∗ continuous functional ϕ on Mr (H ∞ ) with ϕ < 1 can be represented as a rank one functional [xy ∗ ] on (H 2 ⊗ Cr )(∞) H 2(∞) ⊗ Cr with x y < 1. Write x = (x1 , . . . , xr ) and y = (y1 , . . . , yr ) with xi and yi in H 2(∞) so that if F = [fij ] ∈ Mr (H ∞ ), then ϕ(F ) =

r

Mfij xj , yi .

i,j=1

Let M = H ∞ [x1 , . . . , xr ]. By the Beurling-Lax-Halmos theory for shifts (∞) (s) of inﬁnite multiplicity [19], Mz |M is unitarily equivalent to Mz for some 2(r) s ≤ r. Thus, we may assume that xi and yj live in H . So this means that 2 r (r) x and y are then identiﬁed with vectors in (H ⊗ C ) as desired. We immediately obtain: Theorem 7.5. Suppose A is a dual subalgebra of H ∞ acting on H 2 . Then Lat(A(r) ) is an r × r Nevanlinna–Pick family for A.

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An additional application of the Beurling–Lax–Halmos result shows that Theorem 7.5 reduces to the matrix-valued Nevanlinna–Pick theorem when A = H ∞. In the case of H1∞ , this yields the result of Ball et al. [7]. They express their models as invariant subspaces of Mr (H 2 ) (in the Hilbert Schmidt norm) instead of H 2(r) ⊗ Cr , but this is evidently the same space. It sufﬁces to use subspaces which are cyclic for H ∞ . In much the same manner as [14], they obtain an explicit parameterization of these subspaces.

References [1] Abrahamse, M.B.: The Pick interpolation theorem for finitely connected domains. Mich. Math. J. 26, 195–203 (1979) [2] Agler, J., McCarthy, J.E.: Complete Nevanlinna–Pick kernels. J. Funct. Anal. 175, 111–124 (2000) [3] Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. In: Graduate Studies in Mathematics 44, Am. Math. Soc., Providence, RI (2002) [4] Apostol, C., Bercovici, H., Foia¸s, C., Pearcy, C.: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra I. J. Funct. Anal. 63, 369–404 (1985) [5] Arias, A., Popescu, G.: Factorization and reflexivity on Fock spaces. Integ. Equ. Oper. Theory 23, 268–286 (1995) [6] Arias, A., Popescu, G.: Noncommutative interpolation and Poisson transforms. Israel J. Math. 115, 205–234 (2000) [7] Ball, J., Bolotnikov, V., ter Horst, S.: A constrained Nevanlinna–Pick interpolation problem for matrix-valued functions, Indiana Univ. Math. J. (to appear). arXiv:0809.2345v1 [8] Bercovici, H., Foias, C., Pearcy, C.: Dilation theory and systems of simultaneous equations in the predual of an operator algebra. Mich. Math. J. 30, 335– 354 (1983) [9] Bercovici, H., Foias, C., Pearcy, C.L.: Dual algebras with applications to invariant subspaces and dilation theory. IN: CBMS Notes 56, Am. Math. Soc., Providence, RI (1985) [10] Bercovici, H.: A note on disjoint invariant subspaces. Mich. Math. J. 34, 435– 439 (1987) [11] Bercovici, H.: Factorization theorems and the structure of operators on Hilbert space. Ann. Math. 128(2), 399–413 (1988) [12] Bercovici, H.: Hyper-reflexivity and the factorization of linear functionals. J. Funct. Anal. 158, 242–252 (1998) [13] Conway, J.B., Ptak, M.: The harmonic functional calculus and hyperreflexivity. Pacif. J. Math. 204, No. 1 (2002) [14] Davidson, K.R., Paulsen, V., Raghupathi, M., Singh, D.: A constrained Nevanlinna–Pick interpolation problem. Indiana Univ. Math. J. 58, 709–732 (2009) [15] Davidson, K.R., Pitts, D.R.: Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. Lond. Math. Soc. 78, 401–430 (1999) [16] Davidson, K.R., Pitts, D.R.: The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311, 275–303 (1998)

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[17] Davidson, K.R., Pitts, D.R.: Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras. Integ. Equ. Oper. Theory 31, 321–337 (1998) [18] Hadwin, D.W., Nordgren, E.A.: Subalgebras of reflexive algebras. J. Oper. Theory 7, 3–23 (1982) [19] Halmos, P.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961) [20] Kraus, J., Larson, D.: Reflexivity and distance formulae. Proc. Lond. Math. Soc. 53, 340–356 (1986) [21] McCullough, S.: Carathodory interpolation kernels. Integ. Equ. Oper. Theory 15, 43–71 (1992) [22] McCullough, S.: The local de Branges-Rovnyak construction and complete Nevanlinna–Pick kernels. In: Algebraic Methods in Operator Theory, pp. 1524. Birkhauser, Basel (1994) [23] Paulsen, V.: Operator algebras of idempotents. J. Funct. Anal. 181, 209–226 (2001) [24] Paulsen, V.: An introduction to the theory of reproducing kernel Hilbert spaces. Course notes. http://www.math.uh.edu/vern [25] Quiggin, P.: For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integ. Equ. Oper. Theory 16, 244–266 (1993) [26] Quiggin, P.: Generalisations of Pick’s theorem to reproducing kernel Hilbert spaces. Ph.D. thesis, Lancaster University (1994) [27] Raghupathi, M.: Nevanlinna–Pick interpolation for C + BH ∞ . Integ. Equ. Oper. Theory 63(1), 103–125 (2009) [28] Raghupathi, M.: Abrahamse’s interpolation theorem and Fuchsian groups. J. Math. Anal. Appl. 355(1), 258–276 (2009) [29] Sarason, D.: The H p spaces of an annulus. Mem. Am. Math. Soc. 56, 1–78 (1965) [30] Sarason, D.: Generalized interpolation in H ∞ . Trans. Am. Math. Soc. 127, 179– 203 (1967) [31] Thomson, J.: Approximation in the mean by polynomials. Ann. Math. 133, 477–507 (1991) Kenneth R. Davidson and Ryan Hamilton (B) Department of Pure Mathematics University of Waterloo Waterloo, ON N2L-3G1, Canada e-mail: [email protected]; [email protected] Received: August 4, 2010. Revised: January 6, 2011.

Integr. Equ. Oper. Theory 70 (2011), 151–203 DOI 10.1007/s00020-011-1869-0 Published online February 22, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Representations of Hardy Algebras: Absolute Continuity, Intertwiners, and Superharmonic Operators Paul S. Muhly and Baruch Solel Abstract. Suppose T+ (E) is the tensor algebra of a W ∗ -correspondence E and H ∞ (E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T+ (E) on a Hilbert space to ultra-weakly continuous completely contractive representations of H ∞ (E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foia¸s functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra. Keywords. Tensor algebras, Hardy algebras, tree analysis, free holomorphic, absolute continuity.

1. Introduction Suppose ρ is a contractive representation of the disc algebra A(D) on a Hilbert space H, i.e., suppose ρ(f )B(H) ≤ f ∞ , where · B(H) is the operator norm on the space of bounded operators on H, B(H), and where · ∞ is the sup norm on A(D), taken over D. (By the maximum modulus principle, the supremum needs only to be evaluated over the circle, T.) Then ρ is completely determined by its value at the identity function z in A(D), T := ρ(z). Of course T is a contraction operator in B(H). On the other hand, given a contraction operator T in B(H), then von Neumann’s inequality guarantees that there is a unique contractive representation ρ of A(D) in B(H) such that T = ρ(z). A natural question arises: When does ρ extend to a representation of H ∞ (T) in B(H) that is continuous with respect to the weak-∗ topology on H ∞ (T) and the weak-∗ topology on B(H)? (We follow the convention of calling the weak-∗ topology on B(H) the ultra-weak topology.) Thanks to the Sz.-Nagy–Foia¸s functional calculus [26], a neat succinct answer may be given in terms of T , viz., ρ admits such an extension if and only if the The authors gratefully acknowledge support from the U.S.-Israel Binational Science Foundation. The second author gratefully acknowledges additional support from the Lowengart Research Fund.

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unitary part of T is absolutely continuous. In a bit more detail, recall that an arbitrary contraction operator T on a Hilbert space H decomposes uniquely into the direct sum T = Tcnu ⊕ U , where Tcnu is completely non unitary, meaning that there are no invariant subspaces for Tcnu on which Tcnu acts as a unitary operator, and where U is unitary. Thus the answer to the question is: ρ extends if and only if the spectral measure for U is absolutely continuous with respect to Lebesgue measure on T. The assertion that extension is possible when U is absolutely continuous is [26, Theorem III.2.1]. The assertion that if the extension is possible, then U is absolutely continuous is essentially [26, Theorem III.2.3]. Note in particular that since the eigenspace of any eigenvalue for T of modulus one must reduce T , it follows that when H is ﬁnite dimensional ρ extends to H ∞ (T) as a weak-∗ continuous representation if and only if the spectral radius of T is strictly less than one. And when dim(H) = 1, we recover the well-known fact that a character of A(D) extends to a weak-∗ continuous character of H ∞ (T) if and only if it comes from a point in D. We were drawn to thinking about this perspective on the Sz.-Nagy– Foia¸s functional calculus by recent work we have done in the theory of tensor and Hardy algebras. Suppose M is a W ∗ -algebra and that E is a W ∗ -correspondence over M in the sense of [15]. Then, in a fashion that will be discussed more thoroughly in the next section, one can form both the tensor algebra of E, T+ (E), and its ultra-weak closure, the Hardy algebra of E, H ∞ (E). If M = C = E, then T+ (E) = A(D) and H ∞ (E) = H ∞ (T). Every completely contractive representation ρ : T+ (E) → B(H) of T+ (E) on a Hilbert space H with the property that ρ restricted to (the copy of) M in T+ (E) is a normal representation of M on H, that we denote by σ, is determined uniquely by a contraction operator T : E ⊗σ H → H satisfying the intertwining equation Tσ E ◦ ϕ(·) = σ(·)T,

(1)

where ϕ gives the left action of M on E and where σ E is the induced representation of L(E) on E ⊗σ H deﬁned by the formula σ E (a) = a ⊗ IH , a ∈ L(E). And conversely, once σ is ﬁxed, each contraction T satisfying this equation determines a completely contractive representation of T+ (E). We write ρ = T × σ. The question we wanted to address, and which we will discuss here, is: What conditions must T satisfy so that T × σ extends from T+ (E) to an ultra-weakly continuous representation of H ∞ (E)? It is easy to see that if T < 1, then T × σ extends from T+ (E) to an ultra-weakly continuous representation of H ∞ (E) [15, Corollary 2.14]. Thus, the question is really about operators T that have norm equal to one. With quite a bit more work we showed in [15, Theorem 7.3] that if T is completely non-coisometric, meaning that there is no subspace of H that is invariant under T × σ(T+ (E))∗ to which T∗ restricts yielding an isometry mapping to E ⊗σ H, then T × σ extends to an ultra-weakly continuous representation of

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H ∞ (E). Thus it looks like we are well on the way to generalizing the theorems of Sz.-Nagy and Foia¸s that we cited above. “All we need is a good generalization of the notion of a completely non-unitary contraction and a good generalization of an absolutely continuous unitary operator.” It turns out, however, that things are not this simple. While a natural generalization of a unitary operator is a representation T × σ, where T is a Hilbert space isomorphism, it is not quite so clear what it means for T to be absolutely continuous. There is no evident notion of spectral measure for T in this case. Further, in the Sz.-Nagy–Foia¸s theory, it is important to know about the minimal unitary extension of the minimal isometric dilation of the contraction T , i.e., it is important to know about the minimal unitary dilation of T . However, it turns out in the theory we are describing, while there is always a unique (up to unitary equivalence) minimal isometric dilation of T there may be many “unitary” extensions of the isometric dilation. Straightforward definitions and results do not appear to exist. We are not the ﬁrst to ponder our basic question. We have received a lot of inspiration from two important papers [2,3]. In [2], Davidson et al. weren’t directly involved with this question, but they clearly were inﬂuenced by it. They considered the situation where M = C and E = Cd for a suitable d. (When d = ∞, we view Cd as 2 (N).) The tensor algebra, T+ (Cd ), in this case is the norm-closed algebra generated by the creation operators on the full Fock space F(Cd ). We ﬁx an orthonormal basis {ei }di=1 for Cd and let Li be the creation operator of tensoring with ei . Thus Li η = ei ⊗ η for all η ∈ F(Cd ). Then T+ (Cd ) is generated by the Li and coincides with Popescu’s noncommutative disc algebra, denoted Ad . The weakly closed algebra generated by the Li is denoted Ld and is called the noncommutative analytic Toeplitz algebra. It turns out that Ld is also the ultra-weak closure of T+ (Cd ), and so Ld coincides with H ∞ (Cd ). In their setting T+ (Cd ) and H ∞ (Cd ) are concretely deﬁned operator algebras since F(Cd ) is a Hilbert space. We single out this special representation of T+ (Cd ) and H ∞ (Cd ) with the notation λ—for left regular representation, which it is, if F(Cd ) is identiﬁed with the 2 -space of the free semigroup on d generators through a choice of basis. If ρ = T × σ is completely contractive representation of T+ (Cd ) on H, then σ must be a multiple of the identity representation of C, namely the Hilbert space dimension of H. Also, T is simply the row d-tuple of operators, (T1 , T2 , . . . , Td ), where Ti = ρ(Li ). As an operator from Cd ⊗ H to H, T has norm at most 1; that is (T1 , T2 , . . . , Td ) a row contraction. The Eq. (1) is automatic in this case. Davidson, Katsoulis and Pitts assume in their work that (T1 , T2 , . . . , Td ) is a row isometry, i.e., that T is an isometry. They are interested in how (T1 , T2 , . . . , Td ) relates to (L1 , L2 , . . . , Ld ). For this purpose they let S be the weakly closed subalgebra of B(H) that is generated by {T1 , T2 , . . . , Td } and the identity, and they let S0 be the weakly closed ideal in S generated by {T1 , T2 , . . . , Td }. Their principal result is [2, Theorem 2.6], which they call The Structure Theorem. It asserts that if N denotes the von Neumann algebra generated by S, and if p is the largest projection in N such that pSp is self-adjoint, then

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N p = k≥1 S0k ; Sp = N p, so in particular, pSp = pN p; p⊥ H is invariant under S and S = N p + p⊥ Sp⊥ ; and assuming p = I, p⊥ Sp⊥ is completely isometrically isomorphic and ultraweakly homeomorphic to Ld . Since Ld = H ∞ (Cd ), it follows that if p = 0, then the representation of T+ (Cd ) determined by the tuple (T1 , T2 , . . . , Td ) extends to H ∞ (Cd ) as an ultra-weakly continuous representation of H ∞ (Cd ). If p = 0, then the representation may still extend to H ∞ (Cd ), but the matter becomes more subtle. As the authors of [2] observe, this decomposition, is suggestive of certain aspects of absolute continuity in the setting of a single isometry. This point is taken up in [3], where Davidson, Li and Pitts say that a vector x in the Hilbert space of (T1 , T2 , . . . , Td ), H, is absolutely continuous if the vector functional on T+ (Cd ) it determines can be represented by a vector functional on Ld , i.e., if there are vectors ξ, η ∈ F(Cd ) such that (ρ(a)x, x) = (λ(a)ξ, η) for all a ∈ T+ (Cd ). The collection of all such vectors x is denoted Vac (ρ). This set is, in fact, a closed subspace of H, and the representation ρ extends to H ∞ (Cd ) as an ultraweakly continuous representation if and only if Vac (ρ) = H. One of their main results is [3, Theorem 3.4], which implies that Vac (ρ) = H if and only if the structure projection for the representation determined by (L1 ⊕ T1 , L2 ⊕ T2 , . . . , Ld ⊕ Td ) acting on F(Cd ) ⊕ H is zero. A central role is played in [3] by the operators that intertwine λ and ρ, i.e., operators X : F(Cd ) → H that satisfy the equation ρ(a)X = Xλ(a), for all a ∈ T+ (Cd ). Theorem 2.7 of [3] shows that Vac (ρ) is the union of the ranges of the X’s that intertwine λ and ρ. In an aside [3, Remark 2.12], the authors note that Popescu [20, Theorem 3.8] has shown that if X is an intertwiner then XX ∗ is a nonnegative operator on H that satisﬁes the two conditions:

1. 2. 3. 4.

Φ(XX ∗ ) ≤ XX ∗

(2)

and Φk (XX ∗ ) → 0

(3) ∗

in the strong operator topology, where Φ(Q) := TQT , and conversely every nonnegative operator Q on H that satisﬁes (2) and (3) can be factored as Q = XX ∗ , where X is an intertwiner. After contemplating this connection between [3] and [20], we realized that there is a very tight connection among all the various constructs we have discussed and that they all can be generalized to our setting of tensor and Hardy algebras associated to a W ∗ -correspondence. This is what we do here. In the next section, we draw together a number of facts that we will use in the sequel. Most are known from the literature. In Sect. 3 we develop the notion of absolute continuity ﬁrst for isometric representations of T+ (E), i.e. for representations ρ = T × σ where T is an isometry. In Sect. 4, we study absolute continuity in the context of an arbitrary completely contractive representation ρ. Here we show that ρ extends from T+ (E) to an ultraweakly continuous completely contractive representation of H ∞ (E) if and only if ρ

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is absolutely continuous, i.e., if and only if Vac (ρ) = H. It turns out that the absolutely continuous subspace Vac (ρ) is really an artifact of the completely positive map attached to T. This fact, coupled to our work in [16], which shows that every completely positive map on a von Neumann algebra gives rise to a W ∗ -correspondence and a representation of it, enables us to formulate a notion of absolute continuity for an arbitrary completely positive map. This formulation is made in Sect. 5, where other corollaries of Sects. 3 and 4 are drawn. Section 6 is something of an interlude, where we deal with an issue that does not arise in [2,3]. The analogue of the representation λ in our theory is a representation of T+ (E) that is induced from a representation of M in the sense of Rieffel [21,22]. Owing to the possibility that the center of M is non-trivial, an induced representation need not be faithful. This fact creates a number of technical problems for us with which we deal in Sect. 6. The ﬁnal section (Sect. 7), is devoted to our generalization of the Structure Theorem of Davidson et al. [2, Theorem 2.6] and to its connection with the notion of absolute continuity.

2. Background and Preliminaries It will be helpful to have at our disposal a number of facts developed in the literature. Our presentation is only a survey, and a little discontinuous. Certainly, it is not comprehensive, but we have given labels to paragraphs for easy reference in the body of the paper. 2.1 Throughout this paper, M will denote a ﬁxed W ∗ -algebra. We do not preclude the possibility that M may be ﬁnite dimensional. Indeed, as we have indicated, the situation when M = C can be very interesting. However, we want to think of M abstractly, as a C ∗ -algebra that is a dual space, without regard to any Hilbert space on which M might be represented. We will reserve the term “von Neumann algebra” for a concretely represented W ∗ -algebra. The weak-∗ topology on a W ∗ -algebra or on any of its weak-∗ closed subspaces will be referred to as the ultra-weak topology. If S is a subset u−w for its ultra-weak closure. of a W ∗ -algebra, we shall write S To eliminate unnecessary technicalities we shall always assume M is σ-ﬁnite in the sense that every family of mutually orthogonal projections in M is countable. Alternatively, to say M is σ-ﬁnite is to say that M has a faithful normal representation on a separable Hilbert space. So, unless explicitly indicated otherwise, every Hilbert space we consider will be assumed to be separable. 2.2 In addition, E will denote a W ∗ -correspondence over M in the sense of [15]. This means ﬁrst that E is a (right) Hilbert C ∗ -module over M that is self-dual in the sense that each (right) module map Φ from E into M is induced by a vector in E, i.e., there is an η ∈ E such that Φ(ξ) = η, ξ , for all ξ ∈ E. Our basic reference for Hilbert C ∗ - and W ∗ -modules is [13]. It is shown in [13, Proposition 3.3.4] that when E is a self-dual Hilbert module over a W ∗ -algebra M , then E must be a dual space. In fact, it may be viewed

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as an ultra-weakly closed subspace of a W ∗ -algebra. Further, every continuous module map on E is adjointable [13, Corollary 3.3.2] and the algebra L(E) consisting of all continuous module maps on E is a W ∗ -algebra [13, Proposition 3.3.4]. To say that E is a W ∗ -correspondence over M is to say, then, that E is a self-dual Hilbert module over M and that there is an ultraweakly continuous ∗-representation ϕ : M → L(E) such that E becomes a bimodule over M where the left action of M is determined by ϕ, a·ξ = ϕ(a)ξ. We shall assume that E is essential or non-degenerate as a left M -module. This is the same as assuming that ϕ is unital. We also shall assume that our W ∗ -correspondences are countably generated as self-dual Hilbert modules over their coefﬁcient algebras. This is equivalent to assuming that L(E) is σ-ﬁnite. 2.3 In this paper, we will be studying objects of various kinds, algebras, ∗-algebras, modules, etc. and we will be considering various types of linear maps of such objects to spaces of bounded operators on Hilbert spaces. We will be especially interested in spaces of intertwining operators between such maps. For this purpose, we introduce the following notation. Suppose X is an object of one of the various kinds we are considering in this paper, e.g., an algebra, a bimodule, etc., and suppose that for i = 1, 2, ρi : X → B(Hi ) is a map of X to bounded linear operators on the Hilbert space Hi . Then we write I(ρ1 , ρ2 ) for the space of all bounded linear operators X : H1 → H2 such that Xρ1 (ξ) = ρ2 (ξ)X for all ξ ∈ X . That is, I(ρ1 , ρ2 ) is the intertwining space or the space of all intertwiners of ρ1 and ρ2 . If X is a C ∗ -algebra and ρ1 and ρ2 are C ∗ -representations, then I(ρ1 , ρ2 ) has the structure of a Hilbert W ∗ -module over the commutant of ρ1 (X ), ρ1 (X ) . The ρ1 (X ) -valued inner product on I(ρ1 , ρ2 ) is given by the formula X, Y := X ∗ Y and the right action of ρ1 (X ) on I(ρ1 , ρ2 ) is given by the formula X · a := Xa, X ∈ I(ρ1 , ρ2 ), a ∈ ρ1 (X ) . Of course, the product, Xa, is just the composition of operators. It is quite clear that I(ρ1 , ρ2 ) with these operations is a Hilbert C ∗ -module over ρ1 (X ) , but what makes it a self-dual Hilbert module is the fact that it is ultra-weakly closed in B(H1 , H2 ). See [13, Theorem 3.5.1]. If X is a C ∗ -algebra and ρ1 and ρ2 are two C ∗ -representations, then it is clear that if I(ρ1 , ρ2 )∗ denotes the space {X ∗ | X ∈ I(ρ1 , ρ2 )}, then I(ρ1 , ρ2 )∗ = I(ρ2 , ρ1 ). Consequently, either I(ρ1 , ρ2 ) and I(ρ2 , ρ1 ) are both nonzero or both are zero (in the latter case, ρ1 and ρ2 are called disjoint.) However, there are situations that we shall encounter (see Remark 4.9.) where X is an operator algebra and the ρi are completely contractive representations of X with the property I(ρ1 , ρ2 ) is nonzero, but I(ρ2 , ρ1 ) = {0}. Thus, in the non-self-adjoint setting, one has to take extra care when manipulating intertwining spaces. 2.4 The concept of a W ∗ -correspondence over a W ∗ -algebra is really a special case of the very useful more general notion of a W ∗ -correspondence from one W ∗ -algebra to another. Specifically, if M1 and M2 are W ∗ -algebras, a W ∗ -correspondence from M1 to M2 is a self-dual Hilbert module F over M2 endowed with a normal representation ϕ : M1 → L(F ). We always assume

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that ϕ is unital. In particular, every normal representation of M on a Hilbert space H makes H a W ∗ -correspondence from M to C, and conversely, a W ∗ -correspondence from M to C is just a normal representation of M on a Hilbert space. Also, observe that if ρ1 and ρ2 are two representations of a C ∗ -algebra A on Hilbert spaces H1 and H2 , respectively, then I(ρ1 , ρ2 ) is a W ∗ -correspondence from ρ2 (A) to ρ1 (A) . We know already that I(ρ1 , ρ2 ) is a self-dual Hilbert C ∗ -module over ρ1 (A) . The left action of ρ2 (A) on I(ρ1 , ρ2 ) is given by composition of maps: b · X := bX for all b ∈ ρ2 (A) and all X ∈ I(ρ1 , ρ2 ). Correspondences can be “composed” through the process of balanced tensor product, but a little care must be taken. That is, if F is a W ∗ -correspondence from M1 to M2 and if G is a W ∗ -correspondence from M2 to M3 then one can form their balanced C ∗ -tensor product, balanced over M2 , but in general it won’t be a W ∗ -correspondence from M1 to M3 . So for us, the tensor product F ⊗M2 G, balanced over M2 , will be the unique self-dual completion of the balanced C ∗ -tensor product of F and G. (See [13, Theorem 3.2.1].) It will be a W ∗ -correspondence from M1 to M3 . As an example, one can easily check that I(ρ2 , ρ3 ) ⊗ρ2 (A) I(ρ1 , ρ2 ) is naturally isomorphic to I(ρ1 , ρ3 ), where the ρi are all C ∗ -representations of a C ∗ -algebra A; the isomorphism sends X ⊗ Y to XY , where XY is just the ordinary product of X and Y . 2.5 As a special case of the composition of W ∗ -correspondences, we ﬁnd the notion of induced representation in the sense of Rieffel [21,22]. If F is a self-dual Hilbert C ∗ -module over the W ∗ -algebra M , then inter alia F is a W ∗ -correspondence from L(F ) to M . So if σ is a normal representation of M on a Hilbert space H, then σ makes H a W ∗ -correspondence from M to C. Thus we get a correspondence F ⊗M H from L(F ) to C. If we want to think of F ⊗M H in terms of representations, then the normal representation of L(F ) associated with F ⊗M H is denoted σ F and is called the representation of L(F ) induced by σ. Evidently, σ F is given by the formula σ F (T )(ξ ⊗ h) = (T ξ) ⊗ h. We will usually write F ⊗M H as F ⊗σ H. It is a consequence of [22, Theorem 6.23] that the commutant of σ F (L(F )) is IF ⊗ σ(M ) . 2.6 Putting together the structures we have discussed so far, we come to a central concept to our theory. Returning to our correspondence E over M , let σ be a normal representation of M on a Hilbert space H. Form the induced representation σ E of L(E) and form the normal representation of M, σ E ◦ ϕ, which acts on E ⊗σ H. (Recall that the left action of M on E is given by the normal representation ϕ : M → L(E).) We deﬁne E σ to be I(σ, σ E ◦ ϕ) and we call E σ the σ-dual of E. This is a W ∗ -correspondence over σ(M ) . The bimodule actions are given by the formula a · X · b := (IE ⊗ a)Xb,

a, b ∈ σ(M ) ,

X ∈ I(σ, σ ◦ ϕ).

2.7 Along with E, we may form the (W ∗ -)tensor powers of E, E ⊗n . They will be understood to be the self-dual completions of the C ∗ -tensor powers of E. Likewise, the Fock space over E, F(E), will be the self-dual completion

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of the Hilbert C ∗ -module direct sum of the E ⊗n : F(E) = M ⊕ E ⊕ E ⊗2 ⊕ E ⊗3 ⊕ · · · We view F(E) as a W ∗ -correspondence over M , where the left and right actions of M are the obvious ones, i.e., the diagonal actions, and we shall write ϕ∞ for the left diagonal action of M . For ξ ∈ E, we shall write Tξ for the so-called creation operator on F(E) deﬁned by the formula Tξ η = ξ ⊗ η, η ∈ F(E). It is easy to see that Tξ is in L(F(E)) with norm ξ, and that Tξ∗ annihilates M , as a summand of F(E), while on elements of the form ξ ⊗ η, ξ ∈ E, η ∈ F(E), it is given by the formula Tξ∗ (ζ ⊗ η) := ϕ∞ ( ξ, ζ )η. Definition 2.1. If E is a W ∗ -correspondence over a W ∗ -algebra M , then the tensor algebra of E, denoted T+ (E), is deﬁned to be the norm-closed subalgebra of L(F(E)) generated by ϕ∞ (M ) and {Tξ | ξ ∈ E}. The Hardy algebra of E, denoted H ∞ (E), is deﬁned to be the ultra-weak closure in L(F(E)) of T+ (E). 2.8 Plenty of examples are given in [15] and discussed in detail there. More will be given below, but we now want to describe some properties of the representation theory of T+ (E) and H ∞ (E) that we shall use. Details for what we describe are presented in Section 2 of [15]. If ρ is a completely contractive representation of T+ (E) on a Hilbert space H, then σ := ρ ◦ ϕ∞ is a C ∗ -representation of M on H. We shall consider only those completely contractive representations of T+ (E) with the property that ρ◦ϕ∞ is an ultraweakly continuous representation of M . This is not a significant restriction. In particular, it is not a restriction at all, if H is assumed to be separable, since every C ∗ -representation of a σ-ﬁnite W ∗ -algebra without a ﬁnite type I summand on a separable Hilbert space is automatically ultra-weakly continuous [27, Theorem V.5.1]. In addition to the representation σ of M, ρ deﬁnes a bimodule map T from E to B(H) by the formula T (ξ) := ρ(Tξ ). To say that T (·) is a bimodule map means that T (ϕ(a)ξb) = σ(a)T (ξ)σ(b) for all a, b ∈ M and for all ξ ∈ E. The assumption that ρ is completely contractive guarantees that T is completely contractive with respect to the unique operator space structure on E that arises from viewing E as a corner of its linking algebra. On the other hand, the complete contractivity of T is equivalent to the assertion that the linear map T deﬁned initially on the algebraic tensor product E ⊗ H to H by the formula T(ξ ⊗ h) = T (ξ)h

(4)

has norm at most one and extends to a contraction, mapping E ⊗σ H to H, that satisﬁes the Eq. (1) by [16, Lemma 2.16]. That lemma, coupled with [15, Theorem 2.8], also guarantees, conversely, that if T is a contraction from

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E ⊗σ H to H satisfying Eq. (1), then the Eq. (4) deﬁnes a completely contractive bimodule map that together with σ can be extended to a completely contractive representation of T+ (E) on H. For these reasons we call the pair (T, σ) (or the pair (T, σ)) a completely contractive covariant representation of (E, M ) and we call the representation ρ the integrated form of (T, σ) and write ρ = T ×σ. From Eq. (1) we see that T∗ lies in the space we have denoted E σ . So, if we write D(E σ ) for the open unit ball in E σ and D(E σ ) for its norm closure, then all the completely contractive representations ρ of T+ (E) such ∗ that ρ◦ϕ∞ = σ are parametrized bijectively by D(E σ∗ ) = D(E σ )∗ = D(E σ ) . 2.8.1 In the special case when (E, M ) is (Cd , C), a representation σ of C on a Hilbert space H is quite simple; it does the only thing it can: σ(c)h = ch, h ∈ H, and c ∈ C. In this setting, E ⊗σ H is just the direct sum of d copies of H and T is simply a d-tuple of operators (T1 , T2 , . . . , Td ) such that T is a row contraction. The map T , then, is given by i Ti Ti∗ ≤ 1, i.e. the formula T (ξ) = ξi Ti , where ξ = (ξ1 , ξ2 , . . . , ξd ) ∈ Cd . The space E σ is column space over B(H), Cd (B(H)) and D(E σ ) is simply the unit ball in Cd (B(H)). 2.8.2 If (T, σ) is a completely contractive covariant representation of (E, M ) on a Hilbert space H, then while it is not possible to form the powers of T, which maps E ⊗σ H to H, we can form the generalized powers of T, which map E ⊗n ⊗ H to H, inductively as follows: Since M ⊗σ H is isomorphic to H 0 is just the identity map. Of course, T 1 = T. via the map a ⊗ h → σ(a)h, T ⊗n+1 For n > 0, Tn+1 := T1 (IE ⊗ Tn ), mapping E ⊗ H to H. This sequence of maps satisﬁes a semigroup-like property T n+m = Tm (IE ⊗m ⊗ Tn ) = Tn (IE ⊗n ⊗ Tm ),

(5)

where we identify E ⊗m ⊗(E ⊗n ⊗σ H) and E ⊗n ⊗(E ⊗m ⊗σ H) with E ⊗(n+m) ⊗σ n are all contractions, they may be used to H [17, Section 2]. Since the maps T promote (T, σ) to a completely contractive covariant representation (Tn , σ) of E ⊗n on H, simply by setting n (ξ1 ⊗ ξ2 ⊗ · · · ⊗ h). Tn (ξ1 , ξ2 , . . . , ξn )h := T (ξ1 )T (ξ2 ) · · · T (ξn )h = T 2.8.3 If (T, σ) is a completely contractive covariant representation of (E, M ) on a Hilbert space H, then (T, σ) induces a completely positive map ΦT on σ(M ) deﬁned by the formula ΦT (a) := T(IE ⊗ a)T∗

a ∈ σ(M ) .

(6)

Indeed, ΦT is clearly a completely positive map from σ(M ) into B(H), since a → IE ⊗ a is faithful normal representation of σ(M ) (onto the commutant of σ E (L(E))), as we have noted earlier, and T is a bounded linear map from E ⊗σ H to H. To see that its range is contained in σ(M ) , simply note that Tσ E ◦ ϕ = σ T. So, if a ∈ σ(M ) and if b ∈ M , then σ(b)ΦT (a) = σ(b)T(IE ⊗ a)T∗ = Tσ E ◦ ϕ(b)(IE ⊗ a)T∗ = T(IE ⊗ a)σ E ◦ ϕ(b)T∗ =

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T(IE ⊗ a)T∗ σ(b) = ΦT (a)σ(b), which shows that the range of ΦT is contained in σ(M ) . Furthermore, we see that n (IE ⊗n ⊗ a)T n ΦnT (a) = T

∗

(7)

for all n. This is an immediate application of paragraph 2.8.2 (see [16, Theorem 3.9] for details.) 2.9 An important tool used in the analysis of T+ (E) and H ∞ (E) is the “spectral theory of the gauge automorphism group”. What we need is developed in detail in [15, Section 2]. We merely recall the essentials that we will use. Let Pn denote the projection of F(E) onto E ⊗n . Then Pn ∈ L(F(E)) and the series ∞ eint Pn Wt := n=0

converges in the ultra-weak topology on L(F(E)). The family {Wt }t∈R is an ultra-weakly continuous, 2π-periodic unitary representation of R in L(F(E)). Further, if {γt }t∈R is deﬁned by the formula γt = Ad(Wt ), then {γt }t∈R is an ultra-weakly continuous group of ∗-automorphisms of L(F(E)) that leaves invariant T+ (E) and H ∞ (E). Indeed, the subalgebra of H ∞ (E) ﬁxed by {γt }t∈R is ϕ∞ (M ) and γt (Tξ ) = e−it Tξ , ξ ∈ E. Associated with {γt }t∈R we have the “Fourier coefﬁcient operators” {Φj }j∈Z on L(F(E)), which are deﬁned by the formula 1 Φj (a) := 2π

2π

e−ijt γt (a) dt,

a ∈ L(F(E)),

(8)

0

where the integral converges in the ultra-weak topology. An alternate formula for Φj is Pk+j aPk . Φj (a) = k∈Z ∞

Each Φj leaves H (E) invariant and, in particular, Φj (Tξ1 Tξ2 · · · Tξn ) = Tξ1 Tξ2 · · · Tξn if and only if n = j and zero otherwise. Associated with the Φj are the “arithmetic mean operators” {Σk }k≥1 that are deﬁned by the formula |j| (1 − )Φj (a), Σk (a) := k |j|
a ∈ L(F(E)). For a ∈ L(F(E)), limk→∞ Σk (a) = a, where the limit is taken in the ultra-weak topology. 2.10 We follow the terminology advanced in [18]. A completely contractive covariant representation (T, σ) of (E, M ) on a Hilbert space H is called isometric (resp. fully coisometric) in case T : E ⊗ H → H is an isometry (resp. a coisometry). It is not difﬁcult to see that a completely contractive covariant representation (T, σ) is isometric if and only if its integrated form T × σ is a completely isometric representation. Further, this happens if and only if T × σ is the restriction to T+ (E) of a C ∗ -representation of the C ∗ -subalgebra

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T (E) of L(F(E)) generated by T+ (E). This C ∗ -algebra is called the Toeplitz algebra of E. (See [18, Section 2].) A special kind of isometric covariant representations are constructed as follows. Let π0 : M → B(H0 ) be a normal representation of M on the Hilbert F (E) space H0 , and let H = F(E) ⊗π0 H0 . Set σ = π0 ◦ ϕ∞ ⊗ IH0 , and deﬁne S : E → B(H) by the formula S(ξ) = Tξ ⊗ IH0 , ξ ∈ E. Then it is immediate that (S, σ) is an isometric covariant representation and we say that it is induced by π0 . We also will say S × σ is induced by π0 . In a sense that will become clear, the representation π0 should be viewed as a generalization of the multiplicity of a shift. n∗ → 0 n S If (S, σ) is an induced isometric covariant representation then S ⊗k ∗ strongly as n → ∞ because Sn Sn is the projection onto k≥n E ⊗π0 H0 . In general, an isometric covariant representation (S, σ) and its integrated form n∗ → 0 strongly as n → ∞. n S are called pure if S It is clear that induced covariant representations are analogues of shifts. Corollary 2.10 of [17] shows that every pure isometric covariant representation of (E, M ) is unitarily equivalent to an isometric covariant representation that is induced by a normal representation of M . We therefore will usually say simply that a pure isometric covariant representation is induced. In Theorem 2.9 of [17] we proved a generalization of the Wold decomposition theorem that asserts that every isometric covariant representation of (E, M ) decomposes as the direct sum of an induced isometric covariant representation of (E, M ) and an isometric representation if (E, M ) that is both isometric and fully coisometric. 2.11 We will need an analogue of a unilateral shift of inﬁnite multiplicity. Lemma 2.2. For i = 1, 2, let πi : M → B(Hi ) be a normal representation of M on the Hilbert space Hi , and let (Si , σi ) be the isometric covariant representation of (E, M ) induced by πi . 1. If X : H1 → H2 intertwines π1 and π2 , then IF (E) ⊗X intertwines (S1 , σ1 ) and (S2 , σ2 ). In particular, if π1 and π2 are unitarily equivalent, then so are (S1 , σ1 ) and (S2 , σ2 ). 2. If (S1 , σ1 ) and (S2 , σ2 ) are unitarily equivalent, then so are π1 and π2 . Proof. The ﬁrst assertion is a straightforward calculation. The key point is that IF (E) ⊗ X represents a bounded operator from F(E) ⊗π1 H1 to F(E) ⊗π2 H2 precisely because X intertwines π1 and π2 . The second assertion may be seen as follows. Let U be a Hilbert space isomorphism from F(E) ⊗π1 H1 to F(E) ⊗π2 H2 such that U (S1 × σ1 ) = (S2 × σ2 )U and let H0∞ (E) = {a ∈ H ∞ (E) | Φ0 (a) = 0}, where Φ0 is the zeroth Fourier coefﬁ u−w = k≥1 E ⊗k . It follows that for cient operator (8). Then H0∞ (E)F(E) i = 1, 2, (Si × σi )(H0∞ (E))(F(E) ⊗πi Hi ) = ( k≥1 E ⊗k ) ⊗πi Hi and, since U (S1 × σ1 ) = (S2 × σ2 )U , it follows also that U carries ( k≥1 E ⊗k ) ⊗π1 H1 onto ( k≥1 E ⊗k ) ⊗π2 H2 . Consequently, U restricts to a Hilbert space isomorphism from M ⊗π1 H1 H1 onto M ⊗π2 H2 H2 which, because U (S1 × σ1 ) = (S2 × σ2 )U , must satisfy U π1 = π2 U .

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We shall ﬁx, once and for all, a representation (S0 , σ0 ) that is induced by a faithful normal representation π of M that has infinite multiplicity. That is, (S0 , σ0 ) acts on a Hilbert space of the form F(E) ⊗π K0 , where π : M → B(K0 ) is an inﬁnite ampliation of a faithful normal representation of M . Then σ0 := π F (E) ◦ ϕ∞ , while S0 (ξ) := Tξ ⊗ IK0 , ξ ∈ E. The following proposition will be used in a number of arguments below. Proposition 2.3. The representation (S0 , σ0 ) is unique up to unitary equivalence and every induced isometric covariant representation of (E, M ) is unitarily equivalent (in a natural way) to a restriction of (S0 , σ0 ) to a subspace of the form F(E) ⊗π K, where K is a subspace of K0 that reduces π. Proof. The uniqueness assertion is an immediate consequence of Lemma 2.2 and the structure of isomorphisms between von Neumann algebras [4, Theorem I.4.4.3]. If (R, ρ) is an induced representation acting, say, on the Hilbert space H, then there is a normal representation ρ0 of M on a Hilbert space H0 and a Hilbert space isomorphism W : H → F(E) ⊗ρ0 H0 such that W R(ξ)W −1 = Tξ ⊗ IH0 and such that W ρ(a)W −1 = ϕ∞ (a) ⊗ IH0 , ξ ∈ E F (E) and a ∈ M . That is, W (R × ρ)W −1 is the induced representation ρ0 of L(F(E)) restricted to T+ (E) acting on F(E) ⊗ρ0 H0 . (The proofs of these assertions are easy; details are contained in the proof of [17, Theorem 2.9].) Since π is a faithful normal representation of M on K0 with inﬁnite multiplicity, one knows that ρ0 is unitarily equivalent to π0 restricted to a reducing F (E) F (E) subspace K of K0 . It follows that ρ0 is unitarily equivalent π0 , which, in turn, is unitarily equivalent to the restriction of π F (E) to F(E) ⊗π K. Stringing the unitary equivalences together completes the proof. Definition 2.4. We shall refer to (S0 , σ0 ) as the universal induced covariant representation of (E, M ). By Proposition 2.3, (S0 , σ0 ) does not really depend on the choice of representation π used to deﬁne it. It will serve the purpose in our theory that the unilateral shift of inﬁnite multiplicity serves in the structure theory of single operators on Hilbert space. 2.12 A key tool in our theory is the following result that we proved as [15, Theorem 2.8]. Theorem 2.5. Let (T, σ) be a completely contractive covariant representation of (E, M ) on a Hilbert space H. Then there is an isometric covariant representation (V, ρ) of (E, M ) acting on a Hilbert space K containing H such that if P denotes the projection of K onto H, then 1. P commutes with ρ(M ) and ρ(a)P = σ(a)P, a ∈ M , and 2. for all η ∈ E, V (η)∗ leaves H invariant and P V (η)P = T (η)P . The representation (V, ρ) may be chosen so that the smallest subspace of K that contains H is all of K. When this is done, (V, ρ) is unique up to unitary equivalence and is called the minimal isometric dilation of (T, ρ). There is an explicit matrix form for (V, ρ) similar to the classical Sch¨aﬀer matrix for the unitary dilation of a contraction operator. We will need parts

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of it, so we present the essentials here. For more details, see Theorem 3.3 and Corollary 5.21 in [18] and the proof of Theorem 2.18 in [16], in particular. Let Δ = (I − T˜∗ T˜)1/2 and let D be its range. Then Δ is an operator on E ⊗σ H and commutes with the representation σ E ◦ ϕ of M , by Eq. (3.6). Write σ1 for the restriction of σ E ◦ ϕ to D. Form K = H ⊕ F(E) ⊗σ1 D, where the tensor product F(E) ⊗σ1 D is balanced over σ1 . The representation ρ is F (E) just σ ⊕ σ1 ◦ ϕ∞ . For V , we form E ⊗ρ K and deﬁne V : E ⊗ρ K → K matricially as ⎤ ⎡ ˜ T 0 0 ··· ⎥ ⎢ .. ⎥ ⎢Δ 0 0 . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ 0 I 0 .. ⎥ ⎢ (9) V˜ := ⎢ ⎥, . .. ⎥ ⎢0 0 I 0 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. 0 0 I . . . ⎦ ⎣ .. .. . . where the identity operators in this matrix must be interpreted as the operators that identify E⊗σn+1 (E ⊗n ⊗σ1 D) with E ⊗(n+1) ⊗σ1 D, and where, in turn, ⊗n σn+1 = σ1E ◦ ϕ(n) . (Here ϕ(n) denotes the representation of M in L(E ⊗n ) given by the formula ϕ(n) (a)(ξ1 ⊗ ξ2 · · · ⊗ ξn ) = (ϕ(a)ξ1 ) ⊗ ξ2 ⊗ · · · ξn ).) Then it is easily checked that V˜ is an isometry and that the associated covariant representation (V, ρ) is the minimal isometric dilation (T, σ). 2.13 Another important tool in our analysis is the commutant lifting theorem [18, Theorem 4.4]. The way we stated this result in [18], it may be difﬁcult to recognize relations with the classical theorem. So we begin with a more “down to earth” statement. It is exactly what is proved in [18]. Theorem 2.6. Let (T, σ) be a completely contractive representation of (E, M ) on a Hilbert space H and let (V, ρ) be the minimal isometric dilation of (T, σ) acting on the space K containing H. If X is an operator on H that commutes with (T ×σ)(T+ (E)) then there is an operator Y on K such that the following statements are satisfied: 1. 2. 3. 4.

Y = X. Y commutes with (V × ρ)(T+ (E)). Y leaves K H invariant. P Y |H = X, where P is the projection of K onto H.

Suppose, now, that for i = 1, 2 (Ti , σi ) is a completely contractive covariant representation of (E, M ) on a Hilbert space Hi . Then we shall write I((T1 , σ1 ), (T2 , σ2 )) for the set of operators X : H1 → H2 such that XT1 (ξ) = T2 (ξ)X for all ξ ∈ E and Xσ1 (a) = σ2 (a)X for all a ∈ M . That is, I((T1 , σ1 ), (T2 , σ2 )) = I(T1 , T2 ) ∩ I(σ1 , σ2 ) = I(T1 × σ1 , T2 × σ2 ). Theorem 2.7. For i = 1, 2, let (Ti , σi ) is a completely contractive covariant representation of (E, M ) on a Hilbert space Hi , let (Vi , ρi ) be the minimal

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isometric dilation of (Ti , σi ) acting on the space Ki , and let Pi be the orthogonal projection of Ki onto Hi . Further, let I((V1 , ρ1 ), (V2 , ρ2 ); H1 , H2 ) = {X ∈ I((V1 , ρ1 ), (V2 , ρ2 )) | XH1⊥ ⊆ H2⊥ }. Then the map Ψ from B(K1 , K2 ) to B(H1 , H2 ) defined by Ψ(X) = P2 XP1 maps I((V1 , ρ1 ), (V2 , ρ2 ); H1 , H2 ) onto I((T1 , σ1 ), (T2 , σ2 )) and for every Y ∈ I((T1 , σ1 ), (T2 , σ2 )), there is an X ∈ I((V1 , ρ1 ), (V2 , ρ2 ); H1 , H2 ) with X = Y such that Ψ(X) = Y . Otherwise stated, the restriction of Ψ to I((V1 , ρ1 ), (V2 , ρ2 ); H1 , H2 ) is a complete quotient map that carries I((V1 , ρ1 ), (V2 , ρ2 ); H1 , H2 ) onto I((T1 , σ1 ), (T2 , σ2 )). Theorem 2.7 is not the statement of Theorem 4.4 of [18], but it is a consequence of [18, Theorem 4.4] and the following standard matrix trick: Write the sum (T1 , σ1 ) ⊕ (T2 , σ2 ) acting on H1 ⊕ H2 (T1 , σ1 ) 0 matricially as . Then the minimal isometric dilation of 0 (T2 , σ2 ) (T1 , σ1 )⊕(T2 , σ2 ) is (V1 , ρ1 )⊕(V2 , ρ2 ). Further, from the matricial perspective (T1 , σ1 ) 0 it is clear that the commutant of is 0 (T2 , σ2 ) I((T2 , σ2 ), (T1 , σ1 )) (T1 , σ1 ) I((T1 , σ1 ), (T2 , σ2 ))

(T2 , σ2 )

(V1 , ρ1 ) 0 . With these obserand similarly for the commutant of 0 (V2 , ρ2 ) vations, it is easy to see how Theorem 2.7 follows from [18, Theorem 4.4].

2.14 The following lemma will be used at several points in the text. Lemma 2.8. Suppose A and B are ultra-weakly closed linear subspaces of B(H) and ψ : A → B is a bounded linear map whose restriction to the unit ball of A is continuous with respect to the weak operator topologies on A and B. Then ψ is continuous with respect to the ultra-weak topologies on A and B. Proof. Let f be in B∗ and consider the linear functional f ◦ψ (on A). Since the weak operator topology and the ultra-weak topology agree on any bounded ball of B, they agree on ψ(A1 ), where A1 is the unit ball of A. Consequently, f ◦ ψ|A1 is ultra-weakly continuous. It now follows from [4, Theorem I.3.1, (ii4) ⇒ (ii1)] that f ◦ ψ is ultra-weakly continuous. Since f was an arbitrary ultra-weakly continuous functional, ψ is continuous with respect to the ultra-weak topologies.

3. Absolute Continuity and Isometric Representations Throughout this paper we shall use the following standard notation. If ξ and η are vectors in a Hilbert space H, then ωξ,η will denote the functional on B(H) deﬁned by the formula ωξ,η (X) := Xξ, η , X ∈ B(H). If ξ = η, then we shall simply write ωξ for ωξ,η . Definition 3.1. Let (T, σ) be a completely contractive covariant representation of (E, M ) on a Hilbert space H.

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(1) A vector x ∈ H is said to be absolutely continuous in case the functional ωx ◦(T ×σ) on T+ (E) can be written as ωξ,η ◦(S0 ×σ0 ) for suitable vectors ξ and η in the Hilbert space F(E) ⊗π K0 of the universal representation (S0 , σ0 ). That is, (T × σ)(a)x, x = (S0 × σ0 )(a)ξ, η = (a ⊗ I)ξ, η ,

a ∈ T+ (E).

(2) We write Vac (or Vac (T, σ)) for the set of all absolutely continuous vectors in H. (3) We say that (T, σ) and T × σ are absolutely continuous in case Vac = H. We will show eventually that Vac (T, σ) is a closed linear subspace of H, as one might expect. However, this will take a certain amount of preparation and development. At the moment, all we can say is that Vac (T, σ) is closed under scalar multiplication. One of our principal goals is to show that a completely contractive representation T × σ of T+ (E) extends to an ultra-weakly continuous completely contractive representation of H ∞ (E) if and only if (T, σ) is absolutely continuous. The following remark suggests that one should think of being absolutely continuous as being a “local” phenomenon. Remark 3.2. A vector x ∈ H is absolutely continuous if and only if the functional ωx ◦ (T × σ) extends to an ultra-weakly continuous linear functional on H ∞ (E). To see this, ﬁrst observe that any faithful normal representation of M induces a faithful representation of L(F(E)) that is also a homeomorphism with respect the ultra-weak topologies. Consequently, S0 × σ0 extends to a faithful representation of H ∞ (E) that is a homeomorphism between the ultra-weak topology on H ∞ (E) and the ultra-weak topology on B(F(E) ⊗π K0 ) restricted to the range of S0 × σ0 . Further, since π is assumed to have inﬁnite multiplicity, every ultra-weakly continuous linear functional on (S0 × σ0 )(H ∞ (E)) is a vector functional, that is, it is of the form ωξ,η for vectors ξ and η in F(E) ⊗π K0 . Consequently, if ωx ◦ (T × σ) extends to an ultra-weakly continuous linear functional on H ∞ (E), it must be representable in the form ωξ,η ◦ (S0 × σ0 ), for suitable vectors ξ and η in F(E) ⊗π K0 . Thus, x is an absolutely continuous vector. On the other hand, if x is an absolutely continuous vector, then ωx ◦ (T × σ) can be written as ωξ,η ◦ (S0 × σ0 ), by definition, and so extends to an ultra-weakly continuous linear functional on H ∞ (E). To show that a completely contractive representation of T+ (E) is absolutely continuous if and only if it extends to an ultra-weakly continuous representation of H ∞ (E), we begin by proving this for completely isometric representations. This fact and the technology used to prove it will be employed in the next subsection to obtain the general result. Definition 3.3. Let (S, σ) be an isometric representation of (E, M ) on H. A vector x ∈ H is said to be a wandering vector for (S, σ), or for S × σ, if for every k and every ξ ∈ E ⊗k , the spaces σ(M )x and Sk (ξ)σ(M )x are orthogonal spaces.

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Lemma 3.4. Let (S, σ) be an isometric representation of (E, M ) on the Hilbert space H. If x ∈ H is a wandering vector for (S, σ), then the representation S × σ, restricted to the closed invariant subspace of H generated by x is pure and, therefore, an induced representation. Proof. Suppose x is a wandering vector. Then the closed invariant subspace ∞ M generated by x is the orthogonal sum k=0 ⊕[S(E ⊗k )σ(M )x]. If we write H0 for [σ(M )x] and if π0 denotes the restriction of σ to H0 , then it is easy to check that S ×σ, restricted to M, is unitarily equivalent to the representation induced by π0 . The proof of the next proposition was inspired by the proof of [2, Theorem 1.6]. Proposition 3.5. Let (S, σ) be an isometric representation on the Hilbert space H and suppose x ∈ Vac (S, σ). Then (i) x lies in the closure of the subspace of H ⊕ (F(E) ⊗π K0 ) generated by the wandering vectors for the representation (S × σ) ⊕ (S0 × σ0 ). (ii) There is an X ∈ I((S0 , σ0 ), (S, σ)) such that x lies in the range of X. Proof. Since x ∈ H is absolutely continuous for (S, σ), we can ﬁnd vectors ζ, η in F(E) ⊗π K0 such that (S × σ)(a)x, x = (S0 × σ0 )(a)ζ, η for a ∈ T+ (E). Write τ for the representation (S × σ) ⊕ (S0 × σ0 ), and then note that τ (a)(x ⊕ ζ), (x ⊕ (−η)) = (S × σ)(a)x, x − (S0 × σ0 )(a)ζ, η = 0 (10) for all a ∈ T+ (E). Let N be the closed, τ (T+ (E))-invariant subspace generated by x ⊕ ζ, [τ (T+ (E))(x ⊕ ζ)], and write (R, ρ) for the isometric representation associated with the restriction of τ to N . Thus R(ξ) = (S(ξ) ⊕ S0 (ξ))|N = (S(ξ) ⊕ (Tξ ⊗ IK0 ))|N and ρ = (σ ⊕ σ0 )|N = (σ ⊕ (ϕ∞ ⊗ IK0 ))|N . We shall use Corollary 2.10 of [17] to show that (R, ρ) is an induced representation. For that we must show that ∞ R(E ⊗k )(N ) = {0}. k=0

But, since R(ξk ) = S(ξk ) ⊕ (Tξk ⊗ IK ), it is immediate that ∞

R(E ⊗k )(N ) ⊆ H ⊕ {0}.

(11)

k=0

Since (R, ρ) is an isometric representation, we can apply the Wold decomposition theorem ([17, Theorem 2.9]) and write (R, ρ) = (R1 , ρ1 ) ⊕ (R2 , ρ2 ) where (R1 , ρ1 ) is an induced isometric representation and where (R, ρ2 ) is fully coisometric. This decomposition enables us, then, to write x ⊕ ζ = λ + μ, where λ and μ are the projections of x ⊕ ζ into the induced and the coisometric subspaces, respectively, for (R, ρ). (We want to emphasize that we are not claiming λ lies in H and μ lies in F(E) ⊗π K0 ; i.e., the Wold direct sum decomposition may be different from the direct sum decomposition H ⊕ (F(E) ⊗π K0 ).) It follows from Eq. (11), however, that μ is of the form h ⊕ 0 for some h ∈ H and, thus, λ = (x − h) ⊕ ζ. Since μ is orthogonal to λ, h is orthogonal to x − h. Also, it follows from Eq. (10) that h ⊕ 0 is

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orthogonal to x ⊕ (−η). Thus h is orthogonal to x. Since it is also orthogonal to x − h, h = 0, implying that μ = 0 and, thus, that the representation (R, ρ) is induced. But that implies that every vector in N is in the closure of the subspace spanned by the wandering vectors of τ . In particular, x ⊕ ζ lies there. Now note that we could have replaced ζ and η by tζ and t−1 η for any t > 0. We would then ﬁnd that x⊕tζ lies in the span of the wandering vectors for every t > 0. Letting t → 0, we ﬁnd that x lies in that span, completing the proof of (i). To prove (ii), ﬁrst let X0 be the projection of H ⊕ (F(E) ⊗π K0 ) onto H, restricted to N . Then x = X0 (x ⊕ ζ). By construction, X0 intertwines R × ρ and S × σ. However, by Lemma 2.3 and the fact that (R, ρ) is induced, we ﬁnd that R × ρ is unitarily equivalent to a summand of S0 × σ0 . Taking the equivalence into account, X0 can be exchanged for an X that intertwines S0 × σ0 and S × σ, and has x in its range. Lemma 3.6. Suppose (S, σ) is an isometric representation of (E, M ) on a Hilbert space H and suppose that X : F(E) ⊗π K0 → H is an element of the intertwining space I((S0 × σ0 ), (S × σ)). If X denotes the closure of the range of X, then X is invariant under (S, σ) and the restriction (R, ρ) of (S, σ) to X is an isometric representation. Also, R × ρ admits a unique extension to a representation of H ∞ (E) on X that is ultra-weakly continuous and completely isometric. Consequently, X = Vac (R, ρ) ⊆ Vac (S, σ). Proof. The fact that R × ρ admits such an extension is proved in [15, Lemma 7.12]. It then follows that for x ∈ X , the functional ωx ◦ (R × ρ) = ωx ◦ (S × σ) extends to an ultra-weakly continuous functional on H ∞ (E). By Remark 3.2, this proves the last statement of the lemma. Theorem 3.7. If (S, σ) is an isometric covariant representation of (E, M ), then Vac (S, σ) = {Ran(X) | X ∈ I((S0 , σ0 ), (S, σ))}, and so in particular Vac (S, σ) is a closed, σ(M )-invariant subspace of H. Proof. We already noted that Vac (S, σ) is closed under scalar multiplication. To see that it is closed under addition, ﬁx x, y ∈ Vac (S, σ). Then, by Proposition 3.5(ii), there are operators X, Y ∈ I((S0 , σ0 ), (S, σ)) such that x = X(ξ) and y = Y (η) for suitable vectors ξ and η in F(E) ⊗π K0 . Since π has inﬁnite multiplicity, we may assume that the initial spaces of X and Y are orthogonal and, in particular, that ξ and η are orthogonal. It follows, then, that if we set Z := X ⊕ Y , then Z ∈ I((S0 , σ0 ), (S, σ)), and Z(ξ + η) = x + y. Lemma 3.6 implies that x + y ∈ Vac (S, σ). But also, Ran(X) ⊆ Vac (S, σ) for every X ∈ I((S0 , σ0 ), (S, σ)), by Lemma 3.6. Thus, it remains to show that Vac (S, σ) is closed. To this end, suppose {xn }n∈N ⊆ Vac (S, σ) is a sequence that converges to x in H. Then the ultra-weakly continuous linear functionals ωxn ◦ (S × σ) converge in norm to ωx ◦ (S × σ), since, in general ωx ◦ (S × σ) − ωy ◦ (S × σ) ≤ x − y. But the ultra-weakly continuous

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linear functionals on H ∞ (E) form a norm closed subset of the dual space of H ∞ (E). Thus ωx ◦(S ×σ) is extends to an ultra-weakly continuous functional on H ∞ (E). By Remark 3.2, x ∈ Vac (S, σ). Corollary 3.8. If (S, σ) is an isometric representation of (E, M ) on H, then Vac (S, σ) = H ∩ span{the wandering vectors of ρ := (S × σ) ⊕ (S0 × σ0 )} ⊇ span{the wandering vectors of S × σ}. Proof. If x is a wandering vector for S × σ, then the restriction of (S, σ) to the smallest S × σ- invariant subspace N spanned by x is an induced isometric representation, by Lemma 3.4. Let X be the inclusion of N into H. Since the restriction of S × σ to N is pure, we may view X as an element of I((S0 , σ0 ), (S, σ)) by Proposition 2.3. Then it follows from Lemma 3.6 that x ∈ Vac (S, σ). Now suppose x ⊕ ζ ∈ H ⊕ (F(E) ⊗π K0 ) is a wandering vector for ρ. Then the same argument shows that the functional ωx⊕ζ ◦ ρ on T+ (E) extends to an ultra-weakly continuous linear functional on H ∞ (E). The same applies to the functional ωζ ◦ (S × σ). Thus ωx ◦ (S × σ) extends to an ultra-weakly continuous functional on H ∞ (E). By Proposition 3.5 and Theorem 3.7 x ∈ Vac (S, σ). Remark 3.9. In general, the closed linear span of the wandering vectors of S×σ is a proper subspace of Vac (S, σ). Indeed, it can be zero and yet Vac (S, σ) is the whole space. Let S be the unitary operator on L2 of the upper half of the unit circle with Lebesgue measure that is given by multiplication by the independent variable. Then S is an absolutely continuous unitary operator, but it has no wandering vectors. Theorem 3.10. For an isometric representation (S, σ) of (E, M ) on a Hilbert space H the following assertions are equivalent: (1) S × σ admits an ultra-weakly continuous extension to a completely isometric representation of H ∞ (E) on H. (2) (S, σ) is absolutely continuous (i.e., Vac (S, σ) = H). (3) H is contained in the closed linear span of the wandering vectors of (S, σ) ⊕ (S0 , σ0 ). Proof. It is clear that (1) implies (2). The equivalence of (2) and (3) follows from Corollary 3.8. It is left to show that (2) implies (1). So assume (2) holds and for every X ∈ I((S0 , σ0 )(S, σ)), let Ran(X) be the closure of the range of X. It follows from the assumption that Vac (S, σ) = H and Proposition 3.5 that H is spanned by the family of subspaces {Ran(X) | X ∈ I((S0 , σ0 )(S, σ))} and, furthermore, the restriction of S × σ to each subspace Ran(X) in this family extends to an ultra-weakly continuous, completely isometric, representation of H ∞ (E) that we shall denote by (S × σ)X . We need to show that these “restriction representations” can be glued together to form an ultra-weakly continuous completely contractive extension of S ×σ. To this end, ﬁx an operator a ∈ H ∞ (E) and recall that Σk (a) denotes the k th -arithmetic mean of the Taylor series of a. The Σk (a) all lie in T+ (E), satisfy the inequality Σk (a) ≤ a, and converge to a in the

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ultra-weak topology on H ∞ (E). Since the sequence (S × σ)(Σk (a)) is uniformly bounded in B(H), it has an ultra-weak limit point in B(H). Any two limit points must agree on each of the spaces Ran(X) since the restrictions (S × σ)(Σk (a))|Ran(X) must converge to (S × σ)X . Since the spaces Ran(X) span H, we see that there is only one limit point Θ(a) of the sequence (S × σ)(Σk (a)). Thus the sequence {(S × σ)(Σk (a))}k∈N converges ultra-weakly to Θ(a). Moreover, for x ∈ Ran(X), Θ(a)x = (S × σ)X (a)x. The same sort of reasoning shows that Θ, so deﬁned, is a completely isometric representation of H ∞ (E) on H that extends S × σ. It remains to show that Θ is ultra-weakly continuous. For this it sufﬁces to show that if {aα }α∈A is a bounded net H ∞ (E) converging ultra-weakly in H ∞ (E) to an element a ∈ H ∞ (E), then {Θ(aα )}α∈A converges ultra-weakly to Θ(a). Since Θ is continuous, {Θ(aα )}α∈A is a bounded net and so we need only show that it converges weakly to Θ(a). But for any x ∈ H, we can ﬁnd an X ∈ I((S0 , σ0 )(S, σ)) so that x ∈ Ran(X) by Lemma 3.5. We conclude, then, that ωx ◦ Θ(aα ) = ωx ◦ (S × σ)X (aα ) → ωx ◦ (S × σ)X (a) = ωx ◦ Θ(a). Thus Θ(aα ) → Θ(a) weakly. It follows (Lemma 2.8) that Θ is σ-weakly continuous on H ∞ (E). This proves that (2) implies (1).

4. Completely Contractive Representations and Completely Positive Maps As we saw in paragraph 2.8.3, if (T, σ) is a completely contractive covariant representation of (E, M ) on a Hilbert space H, then (T, σ) induces a completely positive map ΦT on σ(M ) deﬁned by the formula ΦT (a) := T(IE ⊗ a)T∗

a ∈ σ(M ) .

(12)

One of our goals is to show that the absolutely continuous subspace Vac (T, σ) can be described completely in terms of ΦT . We therefore want to begin by showing that given a contractive, normal, completely positive map Φ on a W ∗ -algebra M and a normal representation ρ of M on a Hilbert space H, then there is a canonical way to view ρ ◦ Φ as a ΦT for a certain T attached to a completely contractive covariant representation (T, σ) of a natural correspondence over ρ(M ) . We will then prove that Vac (T, σ) is an artifact of Φ. For this ﬁrst step, it will be convenient for later use to omit the assumption that our completely positive maps are contractions in the following theorem. Theorem 4.1. Given a normal completely positive map Φ on a W ∗ -algebra M and a normal ∗-representation ρ of M on a Hilbert space H, there is a canonical triple (E, η, σ), where E is a W ∗ -correspondence over the commutant of ρ(M ), ρ(M ) , σ is a normal ∗-representation of ρ(M ) , and where η is an element of E σ , such that ρ(Φ(a)) = η ∗ (IE ⊗ ρ(a))η

(13)

for all a ∈ M . The triple (E, η, σ) is essentially unique in the following sense: If (E1 , η1 , σ1 ) is another triple consisting of a W ∗ -correspondence E1 over ρ(M ) , a normal ∗-representation σ1 of ρ(M ) and an element η1 of E1 σ1

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such that ρ(Φ(a)) = η1 ∗ (IE1 ⊗ ρ(a))η1 for all a ∈ M , then the kernel of η1∗ is of the form σ E1 (q1 )E1 ⊗H for a projection q1 ∈ L(E1 ) and there is an adjointable, surjective, bi-module map W : E1 → E such that η1∗ = η ∗ (W ⊗ I) and such that W ∗ W = IE1 − q1 . Further, σ1 differs from σ by an automorphism of ρ(M ) , i.e., σ1 = σ ◦ α for a suitable automorphism α of ρ(M ) . Proof. We present an outline of the existence of (E, η, σ) since parts of the argument will be useful later. The details may be found in [16]. The uniqueness is proved in [24, Theorem 2.6] and we omit those details here. First, recall Stinespring’s dilation theorem [25] and Arveson’s proof of it [1]. Form the Stinespring space M ⊗ρ◦Φ H, which is the completion of the algebraic tensor product M H in the inner product derived from the formula a ⊗ h, b ⊗ k := h, ρ ◦ Φ(a∗ b)k , and view M as acting on M ⊗ρ◦Φ H through the Stinespring representation, π: π(a)(b ⊗ h) = ab ⊗ h. Let V be the map from H to M ⊗ρ◦Φ H deﬁned by the formula V h = I ⊗ h. Then the equation π(a)V h, V k = (a ⊗ h), 1 ⊗ k = h, ρ ◦ Φ(a∗ )k = ρ ◦ Φ(a)h, k ,

(14)

which is valid for all a ∈ M and h, k ∈ M , shows that V is bounded, with 1 norm Φ(I) 2 , and that V ∗ π(a)V = ρ(Φ(a))

(15)

for all a ∈ M . Then let E be the intertwining space I(ρ, π), i.e., I(ρ, π) = {X ∈ B(H, M ⊗ρ◦Φ H) | Xρ(a) = π(a)X, for all a ∈ M }. As we noted in paragraph 2.3, this space is a W ∗ -correspondence from the commutant of π(M ), π(M ) , to the commutant of ρ(M ), ρ(M ) . However, the map a → IM ⊗ a is normal representation of ρ(M ) into the commutant of π(M ), and so by restriction, E = I(ρ, π) becomes a W ∗ -correspondence over ρ(M ) . The bimodule structure is given by the formula a · X · b = (IM ⊗ a)Xb, a, b ∈ ρ(M ) . We let σ be the identity representation of ρ(M ) on H. To deﬁne η ∈ E σ , we ﬁrst observe that there is a Hilbert space isomorphism U : E ⊗σ H → M ⊗ρ◦Φ H deﬁned by the formula U (X ⊗h) := Xh. The fact that U is isometric is immediate from the way the ρ(M ) -valued inner product on E is deﬁned. The fact that U is surjective is Lemma 2.10 of [16]. Further, a straightforward computation shows that U (IE ⊗ ρ(a))U −1 = π(a) for all a ∈ M . Indeed, if X ∈ E = I(ρ, π) and if h ∈ H, then for a ∈ M , U (IE ⊗ ρ(a))(X ⊗ h) = Xρ(a)h = π(a)Xh = π(a)U (X ⊗ h). That is, U (IE ⊗ ρ(·)) = π(·)U . Second, we note that since IM ⊗ σ(a) lies in π(M ) for all a ∈ ρ(M ) , a similar calculation shows that U (ϕ(·) ⊗ IH ) = (IM ⊗ σ(·))U . Finally, observe from the definition of V that V σ(·) = IM ⊗

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σ(·)V . Now set η = U ∗ V . Then η ∈ E σ since for all a ∈ ρ(M ) , ησ(a) = U ∗ V σ(a) = U ∗ (IM ⊗ σ(a))V = (ϕ(a) ⊗ IH )U ∗ V = (ϕ(a) ⊗ IH )η. Also, η ∗ (IE ⊗ ρ(a))η = V ∗ U (IE ⊗ ρ(a))U ∗ V = V ∗ π(a)V = ρ ◦ Φ(a). Remark 4.2. The cb-norm of any completely positive map is the norm of its value at the identity. So Φη cb = η ∗ ησ(M ) = η2E σ . Consequently, Φη is contractive and completely positive if and only if η ∈ D(E σ ). We thus see that every contractive completely positive map Φ on a W ∗ -algebra can be realized in terms of a completely contractive covariant representation of the natural W ∗ -correspondence E we just constructed from it. We call E the Arveson–Stinespring correspondence associated to Φ (see [16]). It depends on a choice of a representation of M , but that will only be emphasized when necessary. The ultra-weakly continuous, completely contractive covariant representation (T, σ) of (E, ρ(M ) ) such that ρ ◦ Φ = Φη , where T = η ∗ is called the identity representation. The advantage of the identity representation of a completely positive map through Eq. (13) in Theorem 4.1 over the Stinespring representation, Eq. (15), is that one can express the powers of Φ in terms of it as we discussed in paragraph 2.8.3. In this setting, Eq. 7 becomes ∗

n (IE ⊗n ⊗ ρ(a))T n . ρ(Φn (a)) = T Example 4.3. To illustrate these constructs in a concrete example, let M = ∞ ({1, 2, . . . , n}) and let σ represent M on the Hilbert space Cn as diagonal matrices. Thus σ(ϕ) = diag(ϕ1 , ϕ2 , . . . , ϕn ). Of course, σ(M ) is the masa Dn consisting of all diagonal and so σ(M ) = Dn , too. Also, let A = (aij ) be an n × n sub-Markov matrix. This means that the aij are all non-negative, and that for each i, j aij ≤ 1. Such a matrix determines a completely positive, contractive map Φ on Dn through the formula ⎞ ⎛ Φ(d) := diag ⎝ a1j dj , a2j dj , . . . , anj dj ⎠, j

j

j

where d = (d1 , d2 , . . . , dn ). We let εi be the diagonal matrix with zeros everywhere but in the ith row and column, where it is a one, and we let {ei }ni=1 be the standard basis for Cn . Then the vectors εi ⊗ ej , i, j = 1, 2, . . . , n, span σ(M ) ⊗Φ Cn , and an easy calculation shows that εi ⊗ ej , εk ⊗ el = aji if and only (i, j) = (k, l) and is zero otherwise. It follows that in σ(M ) ⊗Φ Cn , εi ⊗ ej is nonzero if and only if (j, i) lies in the support of A, which we denote by G1 . (The reason for the super script is that we are going to view G1 as the edge set of a graph. The vertex set, G0 , is {1, 2, . . . , n}.) The calculation just completed shows that −1

{aji 2 εi ⊗ ej | (j, i) ∈ G1 } is an orthonormal basis for σ(M ) ⊗Φ Cn = Dn ⊗Φ Cn . We let λ be the representation of σ(M ) = Dn on σ(M ) ⊗Φ Cn is given by the formula

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λ(d)(εi ⊗ ej ) = di εi ⊗ ej . We also let ι be the identity representation of σ(M ) = Dn on Cn . Then the Arveson–Stinespring correspondence in this case is E = I(ι, λ). An operator X from Cn to σ(M ) ⊗Φ Cn is given by a matrix that we shall write [X((i, j), k)](i,j)∈G1 k∈{1,2,...,n} . The formula for X((i, j), k) is, of course, −1

X((i, j), k) = Xek , aji 2 εi ⊗ ej . Since an X in E intertwines the identity representation of σ(M ) = Dn on Cn and λ, it follows from this equation that X((i, j), k) is zero unless i = k, when X ∈ E. Thus E may be viewed as a space of functions supported on ∗ G1 . Now for X and Y in E, X Y (i, j) = (k,l)∈G1 X((k, l), i)Y ((k, l), j). Since X((k, l), i) = 0, unless k = i and since Y ((k, l), j) = 0, unless k = j, we ∗ see that X ∗ Y (i, j) = 0, unless i = j, in which case we ﬁnd that X Y (i, i) = n 1 l=1 X((i, l), i)Y ((i, l), i). So, if X(i,j) , (i, j) ∈ G , is deﬁned by the formula −1

X(i,j) ((k, l), m) = aji 2 , when k = m = i and l = j, and zero otherwise, then {X(i,j) }(i,j)∈G1 is an orthonormal basis for E. It follows, then, that {X(i,j) ⊗ei }(i,j)∈G1 is an orthonormal basis for E ⊗σ(M ) Cn (owing to the fact that X ⊗ dh = X · d ⊗ h for all X ⊗ h ∈ E ⊗σ(M ) Cn and for all d ∈ Dn .) The map U : E ⊗σ(M ) Cn → σ(M ) ⊗Φ Cn = Dn ⊗Φ Cn is given by the formula U (X ⊗ h) = Xh and so, at the level of coordinates, we ﬁnd that U (X ⊗ h)(i, j) = X((i, j), i)h(i), where (j, i) lies in G1 . In particular, we see that U (X(i,j) ⊗ ei )(i, j) = X((i, j), i) = −1

−1

aji 2 so that U (X(i,j) ⊗ei ) = aji 2 (εi ⊗ej ). The map V : Cn → σ(M ) ⊗Φ Cn = Dn ⊗Φ Cn is deﬁned by the formula V h = 1 ⊗ h, where in this case, 1 denotes the identity matrix. Recapitulating an earlier calculation we see that V ∗ (εi ⊗ ej ), ek = εi ⊗ ej , V ek = εi ⊗ ej , 1 ⊗ ek = ej , Φ(ε∗i )ek = ej , aki ek . With all the pieces calculated, we see that the map T : E ⊗σ(M ) Cn → Cn is deﬁned on basis vectors for E ⊗σ(M ) Cn by the equation T(X(i,j) ⊗ ei ) = V ∗ U (X(i,j) ⊗ ei ) −1

= V ∗ (aji 2 εi ⊗ ej ) −1

= aji aji 2 ej 1

2 = aji ej .

(16)

We will use these calculations in later examples. Definition 4.4. Let Φ be a completely positive operator on a W ∗ -algebra M . An element Q ∈ M is called a superharmonic operator in case Q ≥ 0 and Φ(Q) ≤ Q.

(17)

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If, in addition, the sequence {Φn (Q)}n∈N converges to zero strongly, then we say that Q is a pure superharmonic operator. A superharmonic operator Q such that Φ(Q) = Q is called harmonic. If M is L∞ (X, μ) for some probability space (X, μ), then a superharmonic operator is a superharmonic function in the sense of Markov processes. (See [23, Deﬁnition 2.1.1].) Remark 4.5. There is an analogue of the Riesz decomposition theorem for superharmonic functions, viz: If Q is a superharmonic operator for Φ, then Q decomposes uniquely as Q = Qp + Qh , where Qp is a pure superharmonic operator for Φ and Qh is a harmonic operator for Φ. Indeed, simply set Qh := Q − lim Φn (Q) and Qp := Q − Qh . Our next goal is to describe all the pure superharmonic operators for a given completely positive map, Φ, say. We will assume that we are given some W ∗ -correspondence E over M , a normal representation σ : M → B(H) and an element η ∈ E σ so that Φ is realized as Φη acting on σ(M ) through the formula Φη (a) := η ∗ (IE ⊗ a)η,

a ∈ σ(M ) ,

(18)

as in Theorem 4.1. The data (E, η, σ) need not be the data constructed in that result; it can be quite arbitrary. However, since we are not assuming that Φη and η have norm at most one, some additional preparation is necessary. Since η ∗ is a bounded linear map from E ⊗σ H to H that satisﬁes the equation η ∗ σ E ◦ ϕ = ση ∗ , [18, Lemma 3.5] implies that if we deﬁne η∗ by the formula η∗ (ξ)h = η ∗ (ξ ⊗ h),

ξ ⊗ h ∈ E ⊗σ H,

η∗

then is a completely bounded bimodule map with cb-norm η ∗ = η. We shall refer to the pair (η∗ , σ) as a completely bounded covariant representation of (E, M ). Although η∗ need not extend to a completely bounded representation of T+ (E), as would be the case if η ≤ 1, we still can promote ⊗n ∗ η∗ to a map η , via the formula n on each of the tensor powers of E, E ∗ ∗ ∗ ∗ η n (ξ1 ⊗ ξ2 ⊗ · · · ⊗ ξn ) := η (ξ1 )η (ξ2 ) · · · η (ξn ). ⊗n ∗ When this is done, the map η to B(H) whose n is a bimodule map from E ⊗n ∗ ⊗σ H to H is given by the formula associated linear map η n from E

∗ ∗ ∗ ∗ η n (ξ1 ⊗ ξ2 ⊗ · · · ⊗ ξn ⊗ h) = η (ξ1 ⊗ η (ξ2 ⊗ (· · · ⊗ η (ξn ⊗ h) · · · ) = η ∗ (IE ⊗ η ∗ )(IE ⊗2 ⊗ η ∗ ) · · · (IE ⊗(n−1) ⊗ η ∗ )(ξ1 ⊗ ξ2 ⊗ · · · ⊗ ξn ⊗ h), ∗ ∗ ∗ ∗ ∗ i.e., η n = η (IE ⊗ η )(IE ⊗2 ⊗ η ) · · · (IE ⊗(n−1) ⊗ η ). To lighten the notation, we drop the “hat” and “tilde”, and simply write ηn∗ = η ∗ (IE ⊗ η ∗ )(IE ⊗2 ⊗ n in η ∗ ) · · · (IE ⊗(n−1) ⊗ η ∗ ). This is entirely consistent with what we used for T ∗ ∗ paragraph 2.8.3. Further, we may then also deﬁne ηn := (ηn ) , which yields ηn = (IE ⊗(n−1) ⊗ η)(IE ⊗(n−2) ⊗ η) · · · (IE ⊗ η)η

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as expected. We let η0 be the map from H to M ⊗σ H that identiﬁes H with M ⊗σ H in the customary fashion. With this notation, we ﬁnd that Φnη (a) = ηn∗ (IE ⊗n ⊗ a)ηn

(19)

for all a ∈ σ(M ) and all n ≥ 0. Theorem 4.6. Let (η∗ , σ) be a completely bounded covariant representation of (E, M ) on the Hilbert space H and let Φη be the completely positive map on σ(M ) defined by (18). An operator Q in σ(M ) is a pure superharmonic operator for Φη if and only if Q = CC ∗ for an operator C ∈ I((S0 , σ0 ), (η∗ , σ)). 1 In this event, if r = (Q − Φ(Q)) 2 , then (IF (E) ⊗ r)C(η) is a bounded linear operator defined on all of H, mapping H to F(E) ⊗σ H, and C ∗ may be written as C ∗ = (IF (E) ⊗ v)(IF (E) ⊗ r)C(η), where v is any partial isometry in I(σ, π) whose initial projection contains the range projection of r. Proof. Suppose Q ∈ σ(M ) has the form Q = CC ∗ , C ∈ I((S0 , σ0 ), (η∗ , σ)). Then by Eq. (19) we may write Φnη (Q) = ηn∗ (IE ⊗n ⊗ Q)ηn

= ηn∗ (IE ⊗n ⊗ C)(IE ⊗n ⊗ C ∗ )ηn 0 )n ((S 0 )n )∗ C ∗ = C(S = CPn C ∗ ,

where here we use Pn to denote the projection onto k≥n E ⊗k ⊗π K0 . Since the Pn decrease strongly to zero, the operators Φnη (Q) decrease strongly to zero, as n → ∞. Thus Q is pure superharmonic for Φη . For the converse, suppose Q ∈ σ(M ) is a given pure superharmonic write r2 := Q − Φη (Q). The “purity” of Q guarantees operator for Φη and n 2 that Q = n≥0 Φη (r ), where the series converges in the strong operator topology. Indeed, the nth partial sum of the series is Q − Φn+1 (Q). Let R η be the closure of the range of r. Since r ∈ σ(M ) , R reduces σ and so we get a new normal representation, σR , of M by restricting σ(·) to R. Choose an isometry v from R into K0 that is in I(σR , π). (Such a choice is possible by the definition of π.) Deﬁne C ∗ : H → F(E) ⊗π K0 by the formula C ∗ x : = (IF (E) ⊗ v) (IE ⊗n ⊗ r)ηn x n≥0

= (IF (E) ⊗ vr)C(η)x. A straightforward calculation shows that this series converges and that the sum deﬁnes a bounded operator C ∗ that satisﬁes the equation CC ∗ x = ηn∗ (IE ⊗n ⊗ r2 )ηn x n≥0

=

Φnη (r2 )x

n≥0

= Qx. It is also clear that C ∈ I((S0 , σ0 ), (η∗ , σ)).

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Theorem 4.6 has its roots in work of Kato [11]. Indeed, he might have called the operator r a smooth operator with respect to η ∗ (See [11, p. 545].) The proof of the theorem that we presented is a minor modiﬁcation of Douglas’s proof of Theorem 5 in [5]. Popescu proved Theorem 4.6 in the setting of free semigroup algebras as [20, Theorem 3.7] and developed a number of other important features of Φη in that setting. Many of them extend to our context, but we will not pursue all of them here. Our primary objective is to prove the following theorem that identiﬁes Vac (T, σ) for a completely contractive representation (T, σ) of (E, M ). Theorem 4.7. Let (T, σ) be a completely contractive representation of (E, M ) on the Hilbert space H, let (V, ρ) be the minimal isometric dilation of (T, σ) acting on a Hilbert space K containing H, and let P denote the projection of K onto H. Then K H is contained in Vac (V, ρ) and the following sets are equal. (1) (2) (3) (4) (5)

Vac (T, σ). H ∩ Vac (V, ρ). P Vac (V, ρ). {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))}. span{Ran(Q) | Q is a pure superharmonic operator for ΦT }.

In particular, (T, σ) is absolutely continuous if and only if (V, ρ) is absolutely continuous. Proof. First, observe that the orthogonal complement of H in K, H ⊥ , is 1 F(E) ⊗σ1 D, where D is the closure of the range of Δ = (IE⊗H − T∗ T) 2 and where σ1 is the restriction of σ E ◦ ϕ(·) to D. (See paragraph 2.12.) The restriction of (V, ρ) to H ⊥ = F(E)⊗σ1 D is just the representation induced by σ1 and, therefore, is absolutely continuous. Thus K H ⊆ Vac (V, ρ). To see the equality of the indicated subspaces, begin by noting that the coincidence of the two spaces Vac (T, σ) and H ∩ Vac (V, ρ) is an immediate consequence of the fact that for every vector x ∈ H the two functionals ωx ◦ (T × σ) and ωx ◦ (V × ρ) are equal. This, in turn, is clear because for such an x, P x = x, where P is the projection from K onto H. Consequently, ωx ◦ (V × ρ)(·) = V × ρ(·)x, x = P V × ρ(·)P x, x = T × σ(·)x, x = ωx ◦ (T × σ)(·). Note in particular, by Theorem 3.7, the fact that Vac (T, σ) = H ∩ Vac (V, ρ) shows that Vac (T, σ) is a closed subspace of H. Clearly, H ∩ Vac (V, ρ) is contained in P Vac (V, ρ). On the other hand, if x = P y, with y ∈ Vac (V, ρ), then by Proposition 3.5, there is an X ∈ I((S0 , σ0 ), (V, ρ)) and a z ∈ F(E) ⊗π K0 such that y = Xz. Since H ⊥ is invariant under V ×ρ, we see that (T ×σ)P X = P (V ×ρ)P X = P (V ×ρ)X = P X(S0 × ρ). Thus x = P Xz lies in {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))}. On the other hand, the commutant lifting theorem, [18, Theorem 4.4], implies that every operator C ∈ I((S0 , σ0 ),(T, σ)) has the form P X for an operator X ∈ I((S0 , σ0 ), (V, ρ)). Thus, {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))} ⊆ P {Ran(X) | X ∈ I((S0 , σ0 ), (V, ρ))} = P Vac (V, ρ), where the last equation is justiﬁed by Theorem 3.7. Thus {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))} = P Vac (V, ρ). To see that P Vac (V, ρ) = H ∩ Vac (V, ρ), note that we showed that

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Vac (V, ρ) contains H ⊥ and so the projection onto Vac (V, ρ) commutes with P . Consequently, H ∩ Vac (V, ρ) = P Vac (V, ρ), and so the ﬁrst four sets (1)–(4) are equal. From Theorem 4.6, we know that if C ∈ I((S0 , σ0 ), (T, σ)), then CC ∗ is pure superharmonic for ΦT . Although the range of CC ∗ may be properly contained in the range of C it is dense in the range of C, and so we see that {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))} ⊆ span{Ran(Q) | Q is a pure superharmonic operator for ΦT }.

(20)

The opposite inclusion is an immediate consequence of the opposite implication in Theorem 4.6, which shows that every pure superharmonic Q ∗ for ΦT has the form Q = CC for a suitable C ∈ I((S0 , σ0 ), (T, σ)), and the fact, already proved, that {Ran(C) | C ∈ I((S0 , σ0 ), (T, σ))} is the closed linear space Vac (T, σ). Corollary 4.8. Suppose (T, σ) is a completely contractive covariant representation of (E, M ) on H. Then: 1. Vac (T, σ) = 0 if and only if I((S0 , σ0 ), (T, σ)) = {0}. 2. If T < 1, then (T, σ) is absolutely continuous. Proof. The ﬁrst assertion is immediate from part (4) of Theorem 4.7. The second assertion is immediate from part (5) since when T < 1, the identity n → 0 is a pure superharmonic for operator ΦT by Eq. 7 and the fact that T as n → ∞. Remark 4.9. With Corollary 4.8 in hand, it is easy to pick up on a point raised at the end of paragraph 2.3. Let (T, σ) be completely contractive covariant representation of (E, M ) on a Hilbert space H and assume that T < 1. Then (T, σ) is absolutely continuous by Corollary 4.8, which means that I((S0 , σ0 ), (T, σ)) is quite large. On the other hand, it is easy to see that if T < 1, then I((T, σ), (S0 , σ0 )) = 0. Indeed, if C ∈ I((T, σ), (S0 , σ0 )), then 0 (IE ⊗ C). From this it follows that C Tn = (S0 )n (IE ⊗n ⊗ C) for all C T = S is isometric, this equation implies that n. Since each (S) n (S0 )∗n C Tn = IE ⊗n ⊗ C. We conclude that C = 0, since the left hand side of this equation goes to zero in norm, while the right hand side has norm C for all n. We note in passing that when Theorem 4.7 is specialized to the setting when M = E = C, it yields an improvement of [5, Corollary 5.5] in the following sense: If W is a unitary operator on a Hilbert space H, then its absolutely continuous subspace is the closed span of the ranges of all the pure superharmonic operators with respect to the automorphism of B(H) induced by W ; it is also the union of the ranges of all the operators that intertwine the unilateral shift of inﬁnite multiplicity and W . Recall that if A is an algebra of operators on a Hilbert space H, then a subspace M of H is called hyperinvariant for A if and only if M is invariant under every operator in A and every operator in the commutant of A.

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One important feature of this notion is that when A is generated by single normal operator T , say, then the hyperinvariant subspaces of A are precisely the spectral subspaces of T . Thus in a sense, hyperinvariant subspaces for an algebra should be viewed as analogues of spectral subspaces for an operator. One needs to take this extended perspective with a grain of salt, however, since spectral subspaces need not be central, i.e., the projection P onto a hyperinvariant subspace need not lie in the center of A. Nevertheless, knowing that a subspace is hyperinvariant for an algebra is useful information. Evidently, if (T, σ) is a completely contractive covariant representation of (E, M ) then Vac (T, σ) is invariant under T × σ(T+ (E)) by part (4) of Theorem 4.7. Indeed, if h ∈ Vac (T, σ), then there is a vector x ∈ F(E) ⊗π K0 and an operator C ∈ I((S0 , σ0 ), (T, σ))} such that h = Cx. Then for ξ ∈ E and a ∈ M, T (ξ)h = T (ξ)Cx = CS0 (ξ)x and σ(a)h = σ(a)Cx = Cπ F (E) (a)x are in Vac (T, σ). The next result, a consequence of Theorems 4.7 and 4.6, shows that Vac (T, σ) is hyperinvariant for T × σ(T+ (E)). Theorem 4.10. For i = 1, 2, let (Ti , σi ) be a completely contractive covariant representation of (E, M ) on a Hilbert space Hi and suppose that R : H1 → H2 intertwines (T1 , σ1 ) and (T2 , σ2 ). Then RVac (T1 , σ1 ) ⊆ Vac (T2 , σ2 ). In particular, the absolutely continuous subspace of a completely contractive covariant representation of (E, M ) is hyperinvariant for its image. Proof. By Theorem 4.7, Vac (Ti , σi ) = {Ran(C) | C ∈ I((S0 , σ0 ), (Ti , σi ))}, i = 1, 2. If C ∈ I((S0 , σ0 ), (T1 , σ1 )), then RC(S0 ×σ0 ) = R(T1 ×σ1 )C = (T2 × σ2 )RC, which shows that RI((S0 , σ0 ), (T1 , σ1 )) ⊆ I((S0 , σ0 ), (T2 , σ2 )). Since R(Ran(C)) = Ran(RC), we conclude that RVac (T1 , σ1 ) ⊆ Vac (T2 , σ2 ). We come now to the main result of this section, which provides criteria for deciding when a completely contractive covariant representation of T+ (E) extends to an ultra-weakly continuous representation of H ∞ (E). Theorem 4.11. Let (T, σ) be a completely contractive covariant representation of T+ (E) on a Hilbert space H and let (V, ρ) be its minimal isometric dilation acting on a Hilbert space K containing H. Then the following assertions are equivalent. 1. T × σ extends to an ultra-weakly continuous, completely contractive representation of H ∞ (E). 2. (T, σ) is absolutely continuous. 3. span{Ran(Q) | Q is a pure superharmonic operator for ΦT } = H. 4. V × ρ extends to an ultra-weakly continuous, completely contractive representation of H ∞ (E). 5. (V, ρ) is absolutely continuous. 6. H is contained in Vac (V, ρ). Of course, we could add a number of other conditions to this list. However, these are the principal ones and more important, none refers to the “external construct” (S0 , σ0 ). That is to say, all the conditions listed refer to intrinsic features of the representation (T, σ).

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Proof. Of course, much of the proof amounts to assembling pieces already proved. Thus, (2) and (3) are equivalent by virtue of Theorem 4.7. Likewise, (4) and (5) are equivalent by Theorem 3.10. The equivalence of (5) and (6) follows from the observation that (V, ρ) is absolutely continuous if and only if H ⊆ Vac (V, ρ). (This is because H ⊥ ⊆ Vac (V, ρ), as we noted in the proof of Theorem 4.7.) Thus (5) and (6) are equivalent. Conditions (2) and (6) are equivalent by virtue of the equation Vac (T, σ) = H ∩ Vac (V, ρ) proved in Theorem 4.7. Thus conditions (2)–(6) are equivalent. But (1) certainly implies (2). On the other hand, if (2) holds, so does (4). If V × ρ denotes the ultra-weakly continuous extension of V × ρ to H ∞ (E), then it is clear that P (V × ρ)|H is an ultra-weakly continuous extension of T × σ to H ∞ (E).

5. Further Corollaries and Examples 5.1. Invariant States In this subsection we collect a number of results that show how the notion of absolute continuity relates to the notion of an invariant state for a completely positive map. Definition 5.1. Let Φ be a normal completely positive map on a W ∗ -algebra M . We let Pac = Pac (Φ) denote the smallest projection in M that dominates the range projection of each pure superharmonic element of Φ and we call Pac the absolutely continuous projection for Φ. If M is represented faithfully on a Hilbert space H by a normal representation ρ, and if Φ is contractive, then the range of ρ(Pac ) is the absolutely continuous subspace Vac (T, σ) for the identity representation (T, σ) of (E, ρ(M ) ), where E is the Arveson–Stinespring correspondence determined by Φ (and ρ), by Theorem 4.1 and Theorem 4.7. So the terminology is consistent with the developments in Sects. 4 and 5, and it makes sense without having to assume Φ is contractive. Definition 5.2. If Φ be a normal completely positive map on a W ∗ -algebra M , then a normal state ω on M is called periodic of period k with respect to Φ, if k is the least positive integer such that ω ◦ Φk = ω. We denote the collection of all normal period states of period k for Φ by Pk or by Pk (Φ, M ). Recall that if ω is a normal state on a W ∗ -algebra N then there is a largest projection e ∈ N such that ω(e) = 0. The projection e⊥ := 1 − e is called the support projection for ω, which we shall denote by supp(ω). As is customary, we often identify a projection with its range and we shall think of supp(ω) as a subspace of whatever Hilbert space on which N may be found to be acting. Our aim now is to prove Theorem 5.3. Let Φ be a normal completely positive map on the W ∗ -algebra M . If ω is a normal state on M that is periodic for Φ, then its support projection is orthogonal to Pac (Φ).

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The proof is based on the following lemma. Lemma 5.4. For each k ≥ 2, Pac (Φk ) = Pac (Φ). Proof. It is clear that Pac (Φ) ≤ Pac (Φk ) since every superharmonic operator for Φ is superharmonic for Φk . The problem we face with trying to prove the reverse inclusion is that in general a pure superharmonic operator for Φk is not evidently a pure superharmonic operator for Φ. So what we prove is that if Q is a pure superharmonic operator for Φk , then there is a pure superharmonic operator R for Φ such that Q ≤ R. This will show that Pac (Φk ) ⊆ Pac (Φ). Our choice for R is Q + Φ(Q) + · · · + Φk−1 (Q). Evidently, this operator dominates Q. So it sufﬁces to show that it is a pure superharmonic operator for Φ. Since Φ(Q + Φ(Q) + · · · + Φk−1 (Q)) = Φ(Q) + Φ2 (Q) + · · · + Φk (Q) and Φk (Q) ≤ Q by hypothesis, we see that Φ(Q + Φ(Q) + · · · + Φk−1 (Q)) = Φ(Q) + Φ2 (Q) + · · · + Φk (Q) ≤ Φ(Q) + Φ2 (Q) + · · · + Φk−1 (Q) + Q = Q + Φ(Q) + · · · + Φk−1 (Q). Thus Q + Φ(Q) + · · · + Φk−1 (Q) is superharmonic for Φ. This means that the sequence {Φn (Q + Φ(Q) + · · · + Φk−1 (Q))}n≥0 is decreasing. Thus to show it tends strongly to zero, it sufﬁces to show that a subsequence tends to zero strongly. But the sequence {Φnk (Q + Φ(Q) + · · · + Φk−1 (Q))}n≥0 has this property since Φnk (Q + Φ(Q) + · · · + Φk−1 (Q)) = Φnk (Q) + Φ(Φnk (Q)) + · · · + Φk−1 (Φnk (Q)) and each term on the right hand side tends to zero monotonically as n → ∞. Proof of Theorem 5.3. Since Pac (Φk ) = Pac (Φ) by Lemma 5.4, it sufﬁces to show that if ω is an invariant state for Φ then supp(ω) ⊥ Pac (Φ). But if Q is a pure superharmonic operator for Φ, then the equation ω ◦ Φ = ω implies that ω(Q) = 0. Thus the range projection of Q is orthogonal to the support of ω, and so supp(ω) ⊥ Pac (Φ). Recall from [15, p. 404 ﬀ.] that if (T, σ) is an ultra-weakly continuous completely contractive representation of (E, M ) acting on the Hilbert space H, then there is a largest subspace H1 of H that is invariant under (T × σ)(T+ (E))∗ on which T∗ acts isometrically. It is given by the equation ∗

n h = h, for all n ≥ 0}. H1 = {h ∈ H | T The representations (T, σ) and T × σ are called completely non-coisometric (abbreviated c.n.c.) in case H1 = {0}. We record for reference the following theorem that is proved as part of Theorem 7.3 in [15]. Theorem 5.5. If (T, σ) is a completely non coisometric, ultra-weakly continuous, completely contractive representation of (E, M ) on a Hilbert space H, then T × σ extends to an ultra-weakly continuous, completely contractive representation of H ∞ (E), and so (T, σ) is absolutely continuous, by Theorem 4.11.

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Of course, if T < 1, then (T, σ) is completely non coisometric. Thus, Theorem 5.5 improves upon Corollary 4.8. Theorem 5.6. Let (T, σ) be an ultra-weakly continuous, completely contractive covariant representation of (E, M ) on a Hilbert space H. If σ(M ) is finite dimensional, then T × σ extends to an ultra-weakly continuous representation of H ∞ (E) on H if and only if (T, σ) is completely non coisometric. Proof. We know that T × σ extends if (T, σ) is completely non coisometric regardless of the dimension of σ(M ) . So we attend to the reverse implication. If (T, σ) has a nonzero coisometric part H1 then we can compress (T, σ) to H1 to get a new representation (T1 , σ1 ) that is fully coisometric, ∗ 1 T 1 = IH . Of course σ1 (M ) will be ﬁnite dimensional, too. So, it i.e., T 1 sufﬁces to assume from the outset that (T, σ) is fully coisometric, which is tantamount to assuming ΦT is unital. But a unital completely positive map on a ﬁnite dimensional W ∗ -algebra admits an invariant normal state. The support of such a state, as we have seen in Theorem 5.3, must be orthogonal to Vac (T, σ) and so (T, σ) cannot be absolutely continuous. By Theorem 4.11, we conclude that T × σ does not admit an extension to an ultra-weakly continuous representation of H ∞ (E). Corollary 5.7. If (T1 , T2 , . . . , Td ) is a row contraction where the Ti act on a finite dimensional Hilbert space, then the map which takes the ith generator Si of H ∞ (Cd ) to Ti extends to an ultra-weakly continuous representation of H ∞ (Cd ) if and only if (T1 , T2 , . . . , Td ) is completely non coisometric. Example 5.8. As an extremely simple, yet somewhat surprising concrete 0 1 0 0 example, consider the case when d = 2 and T1 = and T2 = 0 0 1 0 are acting on H = C2 . Then (T1 , T2 ) is a row coisometry and so is not absolutely continuous. In fact, since ΦT preserves the trace, as is easy to see, the absolutely continuous subspace of C2 reduces to zero. Thus these matrices do not come from an ultra-weakly continuous representation of H ∞ (C2 ) even though T1 and T2 are both nilpotent. Example 5.9. Suppose the W ∗ -algebra M is given and that E comes from a unital endomorphism α, i.e., suppose E =α M , which is M as a (right) Hilbert module over M , with the left action of M given by α. If (T, σ) is an ultra-weakly continuous, completely contractive covariant representation of (E, M ) on a Hilbert space H, then the map x ⊗ h → σ(x)h extends to an isomorphism from E ⊗σ H to H that allows us to view T as an operator T0 on H that has the property T0 σ(α(a)) = σ(a)T0 for all a ∈ M . The map ΦT , then, is given by the formula ΦT (x) = T0 xT0∗ . If T0 is a unitary operator, then [5, Corollary 5.5] tells us that Vac (T, σ) is contained in the absolutely continuous subspace for T0 . It is, however, quite possible for the two spaces to be distinct. Indeed, let M be L∞ (T) with σ the

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multiplication representation on L2 (T). Let α be implemented by an irrational rotation on T, say z → eiθ z. Finally, let T0 be the unitary operator on L2 (T) deﬁned by the equation T0 ξ(z) = zξ(eiθ z). Then σ(M ) = L∞ (T) and because ΦT (ϕ)(z) = ϕ(eiθ z), as may be easily calculated, we see that Vac (T, σ) = {0} because integration against Lebesgue measure gives an invariant faithful normal state on M . On the other hand, it is easy to see that T0 is unitarily equivalent to the bilateral shift of multiplicity one. Indeed, T0 leaves H 2 (T) invariant and H 2 (T) T0 H 2 (T) = H 2 (T) zH 2 (T), where we have identiﬁed z with multiplication by z. It is an easy matter to check that D := H 2 (T) T0 H 2 (T) is a complete wandering ⊕ subspace for T0 , i.e., L2 (T) = k∈Z T0k D. Thus, we see that T0 is an absolutely continuous unitary operator. 5.2. Markov Chains Our next example connects the theory we have been developing with the theory of Markov chains. Recall from Example 4.3 that a sub-Markov matrix is an n × n matrix A with non-negative entries aij with the property that the row-sums j aij ≤ 1. We think of A as deﬁning a completely positive map Φ on the W ∗ -algebra of all diagonal n ×n matrices, D n . If d = diag{d1 , d2 , . . . , dn }, then Φ(d) := diag{ j a1j dj , j a2j dj , . . . , j anj dj }. We want to describe the absolutely continuous projection Pac (Φ). Recall that the norm of A as an operator on ∞ ({1, 2, . . . , n}) is at most 1 and if the norm of A is 1 then 1 is an eigenvalue for A. In this event, we can ﬁnd an element z = (z1 , z2 , . . . , zn ) such that zi ≥ 0 for all i and such that Az = z. This is a consequence of the Perron-Frobenius theory (see [9, pp. 64,65].) It applies as well to the transpose of A. We will call such an eigenvector a non-negative left eigenvector for A. We will call the support of z the set of i ∈ {1, 2, . . . , n} such that zi = 0, and we will say that z has full support if zi = 0 for all i. The analysis in Section III.4 of [9] shows that after conjugating A by a permutation matrix, A has the form described in the following lemma: Lemma 5.10. If A is a sub-Markov matrix with spectral radius 1, then there is a permutation matrix S so that SAS −1 has the (block) lower triangular form ⎡ ⎤ A11 ⎢ 0 ⎥ A22 0 ⎢ ⎥ ⎢ ⎥ .. (21) ⎢ 0 ⎥, . ⎢ ⎥ ⎣ 0 ⎦ 0 0 Ak−1k−1 U Akk where the Aii ’s are square matrices (and U is a rectangular matrix of the appropriate size) such that: 1. For i = 1, 2, . . . , k − 1, each Aii has 1 as an eigenvalue, and the corresponding left eigenvector is nonnegative, with full support.

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2. Akk has spectral radius less than 1. Definition 5.11. We call the matrix (21) the canonical form of the sub-Markov matrix A. The set of indices Ei such that the matrix entries of Aii are indexed by Ei × Ei will be called the support of Aii as well as the support of the left Perron-Frobenius eigenvector z (i) for Aii , corresponding to 1, i = 1, 2, . . . , k − 1. Lemma 5.12. Let Φ be a normal completely positive map on a W ∗ -algebra N and let p be a central projection of N that is invariant under Φ in the sense that Φ(p) ≤ p. Then pPac (Φ) = Pac (Φ|pN p ). Proof. Suppose Q is a pure superharmonic element of N for Φ. Then the sequence {Φn (Q)}n∈N decreases to zero. So Φ(pQp) = pΦ(pQp)p ≤ pΦ(pQp+ p⊥ Qp⊥ )p = pΦ(Q)p ≤ pQp. Therefore, since Φ(Q) is also pure superharmonic for Φ, Φ2 (pQp) = Φ(Φ(pQp)) = Φ(pΦ(pQp)p) ≤ Φ(pΦ(Q)p) ≤ pΦ(Φ(Q))p ≤ Φ2 (Q). Continuing in this fashion, we see that Φn (pQp) ≤ Φn (Q) for all n, and so the range projection of pQp is less than or equal to Pac (Φ|pN p ). That is, pPac (Φ) ≤ Pac (Φ|pN p ). The reverse inequality is clear, since if pQp is pure superharmonic for Φ|pN p , then it certainly is pure superharmonic for Φ. Theorem 5.13. Let Φ be the completely positive map on ∞ ({1, 2, . . . , n}) induced by a sub-Markov matrix A and assume A is written in its canonical form (21). Then Pac (Φ) is the support projection of Akk . Proof. If E is the union of the support projections for Ai , i = 1, 2, . . . , k − 1, then the projection 1E in ∞ ({1, 2, . . . , n}) is coinvariant for A and the sum k−1 of the vectors z := i=1 z (i) determines an invariant state ωz for Φ via fork−1 (i) mula ωz (d) = i=1 z · di , where di is the restriction of d to the support of z (i) , Ei , and z (i) · di denotes the dot product of the two tuples. The state ωz is faithful on 1E ∞ ({1, 2, . . . , n}), and so Pac (Φ) ≤ 1Ek = 1E c , by Theorem 5.3. On the other hand, since 1Ek ∞ ({1, 2, . . . , n}) is invariant for Φ, and since the spectral radius of Ak , which is the matrix of the restriction of Φ to 1Ek ∞ ({1, 2, . . . , n}), is less than 1, it is clear that Φn (1Ek ) → 0 in norm. This implies that 1Ek is pure superharmonic for Φ and, therefore, that 1Ek ≤ Pac (Φ). 5.3. Similarity of Representations Many of the results in this subsection are analogues of theorems in Popescu’s paper [20] and many of his proofs work here, as well. We focus on those features and proofs that take advantage of our perspective that focuses on the connection between completely positive maps and intertwiners. So a number of our arguments are different from the ones given in [20]. Suppose σ : M → B(H) is a normal representation of M on the Hilbert space H and suppose η ∈ E σ . Then we may form the completely bounded bimodule map η∗ : E → B(H) discussed in the paragraph before Theorem 4.6. Let T+0 (E) be the linear span of ϕ∞ (M ) and the operators {Tξ | ξ ∈ E ⊗n , n ≥ 1}. Then T+0 (E) is the algebra generated by ϕ∞ (M ) and {Tξ | ξ ∈ E}, and given σ

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and η∗ , we can deﬁne a representation of T+0 (E) on H, denoted η∗ × σ by the formulae η∗ × σ(ϕ∞ (a)) = σ(a),

a ∈ M,

and η∗ × σ(Tξ1 Tξ2 · · · Tξn ) = η∗ (ξ1 )η∗ (ξ2 ) · · · η∗ (ξn ),

ξi ∈ E, i = 1, 2, . . . , n.

We are interested in understanding when η∗ × σ extends to a completely bounded representation of T+ (E) in B(H). Thanks to the famous theorem of Paulsen [19], this will happen if and only if η∗ × σ is similar to a completely contractive representation of T+ (E). By [18, Theorem 3.10] (see also paragraph (2.8), this will happen if and only if η∗ × σ is similar to ζ∗ × σ1 , for a ζ ∈ D(E σ1 )∗ . Thus, we are led to investigating the similarity properties of the completely bounded maps η∗ × σ. For this purpose, observe that if η1∗ × σ1 is similar to η2∗ × σ1 then there is an invertible operator S on H such that η1∗ × σ1 (a)S = S η2∗ × σ2 (a) for all a ∈ T+ (E). In particular, when a ∈ ϕ∞ (M ), σ1 (a)S = Sσ2 (a), i.e., σ1 and σ2 are similar. Since they are ∗-maps, σ1 and σ2 must be unitarily equivalent. But observe that if U is a Hilbert space isomorphism from the space of σ1 , H1 , to the space of σ2 , then the map η → (IE ⊗ U )ηU −1 is a complete isometric isomorphism between E σ1 and E σ2 that is also a homeomorphism for the ultra-weak topologies. Since we will be interested here in the norm properties of η’s and related constructs the difference between the duals of unitarily equivalent representations may safely be ignored. Thus, in particular, when given a similarity S between η1∗ × σ1 and η2∗ × σ2 we may identify σ1 and σ2 with one σ and assume that S lies in σ(M ) . In this case, η1∗ (I ⊗ S) = Sη2∗ , i.e. S −1 η1∗ (I ⊗ S) = η2∗ . Conversely, any invertible S in the commutant of σ(M ) that satisﬁes the equation S −1 η1∗ (I ⊗ S) = η2∗ implements a similarity between η1∗ × σ(·) and η2∗ × σ(·). Definition 5.14. We introduce several terms we will use in the sequel. 1. If η and ζ are two elements of E σ , then we shall say they are similar if there is an invertible operator S in σ(M ) such that S −1 ·η ·S = (IE ⊗S)−1 ηS = ζ. 2. Let Φ1 and Φ2 be two completely positive maps deﬁned on a W ∗ -algebra N . We say they are similar if and only if there is an invertible operator R ∈ N such that −1 ◦ Φ1 ◦ ψR = Φ2 , ψR

where ψR is the complete positive map on M deﬁned by the formula ψR (a) = RaR∗ . 3. We say that an η ∈ E σ belongs to class C·0 in case the identity is a pure superharmonic operator for Φη . Remark 5.15. Several points may be helpful. 1. We have taken the definition of similarity for completely positive maps from Popescu [20].

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2. It is easy to see that if elements η, ζ ∈ E σ are similar, then Φη and Φζ are similar. Indeed, if R is an invertible element of σ(M ) , then −1 ψR ◦ Φη ◦ ψR (a) = R−1 (η ∗ (IE ⊗ RaR∗ )η)R∗−1

= R−1 η ∗ (IE ⊗ R)(IE ⊗ a)(IE ⊗ R∗ )ηR∗−1 = Φζ ,

where ζ = (IE ⊗R∗ )ηR∗−1 . The converse assertion is not true owing to the nonuniqueness of representing a completely positive map Φ in the form Φη . 3. In fact, it is convenient to use bimodule notation: η and ζ are similar if and only if there is an invertible r such that r · η · r−1 = ζ. Observe that in this case, r · ηn · r−1 = ζn for all n. 4. The notion of a C·0 element of E σ is borrowed from [15, Deﬁnition 7.14]. There the norm of the element was assumed to be at most 1. It turns out that when M = C = E, so that η is really an operator on Hilbert space, then the terminology we have adopted agrees with that of Sz.-Nagy and Foia¸s in [26]. 5. In the terminology of [20], a completely positive map Φ is called pure if and only if I is a pure superharmonic operator for Φ. Our ﬁrst result gives a necessary and sufﬁcient condition for η∗ × σ to extend to a completely bounded representation on T+ (E). It was inspired by [20, Theorem 5.13]. However, our proof is somewhat different. Theorem 5.16. Let σ : M → B(H) be a normal representation and let η ∈ E σ . Then the following conditions are equivalent. 1. The representation η∗ × σ extends to a completely bounded representation of T+ (E). 2. η is similar to a ζ ∈ D(E σ ). 3. Φη admits an invertible superharmonic operator. Proof. If η∗ × σ extends to a completely bounded representation of T+ (E), then η∗ × σ must be similar to a completely contractive representation ρ of T+ (E) by Paulsen’s famous theorem [19]. Since ρ must be of the form ζ∗ × σ1 for some ζ of norm at most 1 in E σ1 , σ and σ1 are similar, and therefore unitarily equivalent. As we noted above, we may identify σ with σ1 and conclude that η is similar to a point in D(E σ )∗ . Thus 1. implies 2. The converse is immediate, since as we noted above, a similarity between two points η1 and η2 in E σ implements a similarity between η1∗ × σ and η2∗ × σ. Suppose η is similar to a ζ in D(E σ )∗ , say r·η·r−1 = ζ for some r ∈ σ(M ) . Then r·η = ζ ·r and so Φη (r∗ r) = Φr·η (I) = Φζ·r (I) = r∗ Φζ (I)r ≤ r∗ r, since ζ ≤ 1. Thus r∗ r is superharmonic for Φη and since r is invertible by assumption, 3. is proved. Finally, suppose Φη admits an invertible super harmonic operator, 1 say R. If r = R 2 and if we set ζ = r · η · r−1 , then ζ · r = r · η, Φζ (I) = r−1 Φζ·r (I)r−1 = r−1 Φr·η (I)r−1 = r−1 Φη (r2 )r−1 ≤ r−1 r2 r−1 = I, since r2 is superharmonic for Φη . Thus ζ ≤ 1.

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The following theorem identiﬁes when an η ∈ E σ is similar to a C·0 element in D(E σ ). It is was inspired by [20, Theorem 5.11]. Again, the proof is somewhat different. Theorem 5.17. Let σ be a normal representation of M on a Hilbert space H and let η be an element of E σ . Then the following assertions about η are equivalent. 1. η is similar to a C·0 element of D(E σ ). 2. There is a positive element r ∈ σ(M ) and positive numbers a and b so that ∞ aIH ≤ Φnη (r) ≤ bIH . (22) n=0

3. There is an invertible pure superharmonic operator for Φη . Proof. The equivalence of 1. and 3. is an easy calculation of the sort that we performed above. If r · η · r−1 = ζ, then r · ηn = ζn · r for all n ≥ 0 and we further have Φnη (r∗ ar) = Φr·ηn (a) = Φζn ·r (a) = r∗ Φζn (a)r = r∗ Φnζ (a)r,

(23)

for all a ∈ σ(M ) . Now suppose that 1. holds, then with a = 1, we see that r∗ r is an invertible superharmonic operator for Φη because Φη (r∗ r) = r∗ Φζ (I)r ≤ r∗ r, since ζ ≤ 1. On the other hand, because ζ is a C·0 element of E σ , r∗ r is a pure superharmonic operator: Φnη (r∗ r) = r∗ Φnζ (I)r → 0 weakly and, therefore, strongly. Thus, 3. is satisﬁed. The argument is essentially reversible: Suppose 3. holds and let a be a positive invertible superharmonic operator for Φη . If we let r be the positive square root of a, then r is invertible and we may let ζ = r · η · r−1 . Since a is superharmonic, we conclude that r2 ≥ Φη (r2 ) = rΦζ (I)r, which shows that Φζ (I) ≤ I, because r is positive and invertible, and this implies that ζ ∈ D(E σ ). On the other hand, we conclude from these calculations and the assumption that a = r2 is a pure superharmonic operator for Φη , that Φnζ (I) = r−1 Φnη (a)r → 0 weakly as n → ∞. Thus ζ is a C·0 element of D(E σ ), as was required. Suppose assertion 2. is satisﬁed and let r, a and b be as in equation ∞ (22). Then the series n=0 Φnη (r) converges strongly to an operator R that ∞ is invertible in σ(M ) . Now Φη (R) = n=1 Φnη (r) = R − r ≤ R. Thus R is ∞ superharmonic for Φη . But also Φnη (R) = k=n Φkη (r) and this sequence of ∞ operators converges strongly to zero, since the series n=0 Φnη (r) converges strongly. Thus condition 3. is satisﬁed. Suppose condition 3. is satisﬁed, let R be an invertible pure superharmonic operator ∞ for Φη and set r := R−Φη (R). Then r is positive semidefinite and R = n=0 Φnη (r). Since R is assumed invertible, the inequality (22) is satisﬁed for suitable a and b. The next theorem uses intertwiners to describe when an η ∈ E σ is similar to a ζ in the open unit disc D(E σ )∗ . It is similar in spirit to [20, Theorem 5.9], but arguments use different technology. Recall that (S0 , σ0 ) denotes the

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universal isometric induced representation. Also, we let P0 be the orthogonal projection of F(E) ⊗π K0 onto the zeroth summand, M ⊗π K0 K0 . Theorem 5.18. Let σ be a normal representation of M on a Hilbert space H and let η be a point in E σ . Then η is similar to a point in the open unit disc D(E σ ) if and only if there is a C ∈ I((S0 , σ0 ), (η ∗ , σ)) such that CP0 C ∗ is invertible. Proof. Suppose there is an invertible r ∈ σ(M ) such that r · η · r−1 = ζ, with ζ < 1. Then r · ηn = ζn · r for all n and we see that Φnη (r∗ r) = r∗ Φnζ (I)r ≤ r∗ rζ ∗ ζn , which shows both that r∗ r is a pure superharmonic operator for Φη . So by Theorem 4.6 there is a C ∈ I((S0 , σ0 ), (η ∗ , σ)) such that r∗ r = CC ∗ . But also, ∗ ∗ ∗ ∗ ∗ 0 S CP0 C ∗ = C(I − S 0 )C = CC − η CC η = r∗ r − Φη (r∗ r) = r∗ r − r∗ Φζ (I)r ≥ (1 − ζ2 )r∗ r.

Since r is invertible, so is CP0 C ∗ . Conversely, suppose that there is a C ∈ I((S0 , σ0 ), (η ∗ , σ)) such that CP0 C ∗ is invertible and let b ∈ R satisfy the inequality CP0 C ∗ ≥ bI > 0. b Also let t be a positive number less than CC ∗ (< 1). Our objective is to 1

show that if r = (CC ∗ ) 2 and if ζ = r · η · r−1 , then ζ2 ≤ 1 − t. Recall that CP0 C ∗ = CC ∗ − Φη (CC ∗ ). By definition of t, CC ∗ ≤ CC ∗ I ≤ bt I. Therefore CC ∗ − bI ≤ CC ∗ − tCC ∗ . But then (1 − t)CC ∗ − Φη (CC ∗ ) ≥ [CC ∗ − bI] − Φη (CC ∗ ) > 0. Consequently, Φη (CC ∗ ) ≤ (1 − t)CC ∗ . Now CC ∗ = r2 and ζ is deﬁned to be r · η · r−1 . We have ζ2 = Φζ (I) = r−1 Φη (r2 )r−1 ≤ r−1 ((1 − t)r2 )r−1 = 1 − t. Thus, 2. implies 1.

Our ﬁnal theorem in this vein has no analogue in [20], but it is in the spirit of that paper. The proof rests on the main results proved to this point. Theorem 5.19. Let σ : M → B(H) be a normal representation of M on the Hilbert space H, and let η ∈ E σ . Then the following assertions are equivalent. 1. η is similar to an absolutely continuous ζ ∈ D(E σ ). 2. Φη admits an invertible superharmonic operator and H = {Ran(C) | C ∈ I((S0 , σ0 ), (η ∗ , σ))}. 3. Φη admits an invertible superharmonic operator and H = {Ran(Q) | Q ∈ σ(M ) , Q − pure superharmonic for Φη }. Proof. Because of the hypotheses in 2. and 3. that Φη admits an invertible superharmonic function, we know from Theorem 5.16 that η is similar to a contraction in each of the situations. The point of 2. is that if η ∗ is similar to a point ζ ∗ ∈ D(E σ )∗ then there is an invertible r ∈ σ(M ) such that r(η ∗ × σ)r−1 = (ζ ∗ × σ), and so a C satisﬁes C(S0 × σ0 ) = (η ∗ × σ)C if and only if rC satisﬁes the equation rC(S0 × σ0 ) = r(η ∗ × σ)r−1 (rC) = ∗ (ζ ∗ × σ)(rC), i.e., if and if η and ζ only rC lies in I((S0 , σ0 ), (ζ∗ , σ)). Thus are similar, the spaces {Ran(C) | C ∈ I((S0 , σ0 ), (η , σ))} and {Ran(C) | C ∈ I((S0 , σ0 ), (ζ ∗ , σ))} are identical. Similarly, if η and ζ are similar, then

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the spaces {Ran(Q) | Q ∈ σ(M ) , Q − pure superharmonic for Φη } and {Ran(Q) | Q ∈ σ(M ) , Q−pure superharmonic for Φζ } are identical. Thus the theorem is an immediate consequence of Theorem 4.7.

6. Induced Representations and their Ranges In a sense, this section is an interlude that develops some ideas that will be used in the next section on the structure theorem. However, we believe the results in it are of sufﬁcient interest in themselves that we want to develop them separately. Throughout this section, τ will be a normal representation of our W ∗ -algebra M on a Hilbert space H and τ F (E) will be the induced representation of L(F(E)) acting on the Hilbert space F(E) ⊗τ H. The support projection of τ will be denoted e. This is a central projection in M and e⊥ is the projection onto the kernel of τ, ker(τ ). The problem we want to address is this. Problem 6.1. Determine when the image of H ∞ (E) under τ F (E) is ultraweakly closed. Of course, τ F (E) is a normal representation of L(F(E)) and so the image of L(F(E)) in B(F(E) ⊗τ H) is ultra-weakly closed, since L(F(E)) is a W ∗ -algebra. Also, of course, if τ is injective, then so is τ F (E) and, consequently, τ F (E) is isometric and an ultra-weak homeomorphism. In this event, τ F (E) (H ∞ (E)) is an ultra-weakly closed subalgebra of B(F(E) ⊗τ H). In particular, if M is a factor, then τ F (E) (H ∞ (E)) is ultra-weakly closed. The problem, then, is to determine what happens when the kernel of τ, e⊥ M , is non-trivial. In this case, the projection onto the kernel of τ F (E) is IF (E) ⊗ e⊥ and the problem is to see how it interacts with τ F (E) (H ∞ (E)). We have no examples of representations τ where the image τ F (E) (H ∞ (E)) fails to be ultra-weakly closed, but we are able to provide useful, very general conditions on e that guarantee that τ F (E) (H ∞ (E)) is ultra-weakly closed. We adopt the following terminology, which is suggested by [7]. Definition 6.2. A projection e in the center of M, Z(M ), that satisﬁes ξe = ϕ(e)ξe for all ξ ∈ E will be called an E-saturated projection. If e also satisﬁes ξe = ϕ(e)ξ for all ξ ∈ E, e will be called an E-reducing projection. Example 6.3. If α is an endomorphism of M and if E is the correspondence then a central projection e ∈ M is E-saturated if and only if α(e)ae = ae for all a ∈ M . That is, e is E-saturated if and only if e ≤ α(e). Moreover, e will be E-reducing if and only if e is ﬁxed by α, e = α(e).

αM ,

The meaning of the E-saturation condition for the present discussion may be further clariﬁed by the following two lemmas. Lemma 6.4. A projection e in the center of M, Z(M ), is an E-saturated projection in M if and only if ϕ∞ (e) is an invariant projection for H ∞ (E) in the sense that H ∞ (E)ϕ∞ (e) = ϕ∞ (e)H ∞ (E)ϕ∞ (e).

(24)

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Thus, if e is E-saturated, then in any completely contractive representation ρ of H ∞ (E), the range of ρ(ϕ∞ (e)) is an invariant invariant subspace ρ(H ∞ (E)). Proof. First, recall that for a, b ∈ M and ξ ∈ E, Tϕ(a)ξb = ϕ∞ (a)Tξ ϕ∞ (b). Consequently, if e is E saturated, so that by definition ξe = ϕ(e)ξe for all ξ ∈ E, it follows that for all ξ ∈ E, Tξ ϕ∞ (e) = Tξe ϕ∞ (e) = Tϕ(e)ξe ϕ∞ (e) = ϕ∞ (e)Tξ ϕ∞ (e). Since ϕ∞ (e) obviously commutes with ϕ∞ (M ), Eq. 24 is veriﬁed. For the converse assertion, simply write out the matrices for Tξ , ξ ∈ E, and ϕ∞ (e) with respect to the direct sum decomposition of F(E) = ⊗n and compute what it means for the equation Tξ ϕ∞ (e) = ϕ∞ (e) n≥0 E Tξ ϕ∞ (e) to hold. Note that the same argument shows e is E-reducing if and only if ϕ∞ (e) commutes with H ∞ (E). In fact, as we shall see in a moment, if e is E-reducing, then ϕ∞ (e) lies in the center of L(F(E)). Lemma 6.5. If e is E-saturated, then the space Ee becomes a W ∗ -correspondence over M e. Moreover, the left action of M on E restricts to a unital left action of M e on Ee, and if we define π : M e → L(F(E)) and V : Ee → L(F(E)) by the formulae π(me) := ϕ∞ (me) and V (ξe) := Tξe , then the pair (V, π) is an ultra-weakly continuous, isometric, covariant representation of (Ee, M e) in L(F(E)), whose image is contained in H ∞ (E). Proof. The calculation, V (ξe)∗ V (ηe) = (Tξ ϕ∞ (e))∗ Tη ϕ∞ (e) = ϕ∞ (e)Tξ∗ Tη ϕ∞ (e) = ϕ∞ (e)ϕ∞ ( ξ, η )ϕ∞ (e) = π( ξe, ηe M e ), shows that V is isometric. The bimodule property is immediate. The ultraweak continuity is an immediate consequence of [15, Lemma 2.5, Remark 2.6]. Remark 6.6. Strictly speaking, of course, (V, π) is an isometric representation of (Ee, M e) into the abstract W ∗ -algebra, L(F(E)), so to apply the theory from [15] here and elsewhere, one should compose (V, π) with a faithful normal representation of L(F(E)) on Hilbert space. The details are easy and may safely be omitted. Later, however, it will prove useful to use that device. Anticipating results to be proved shortly (Lemma 6.10), we call (V, π) or V × π the canonical embedding of T (Ee) in T (E). We will see that V × π is faithful on the Toeplitz algebra, T (Ee), and extends to a completely isometric, ultra-weakly continuous representation of H ∞ (Ee), mapping it into H ∞ (E). The following lemma may be known, but we do not have a reference. It will be helpful to have the details in hand.

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Lemma 6.7. Let F be a C ∗ -Hilbert module over a C ∗ -algebra N . For a ∈ N , define Ra : F → F by the formula Ra ξ = ξa, ξ ∈ F . Then Ra is a bounded C-linear operator on F with norm at most a. If a lies in the center of N, Z(N ), then Ra is a bounded adjointable operator on F that lies in the center of L(F ). Proof. For ξ ∈ F and a ∈ N , we have Ra ξ, Ra ξ = ξa, ξa = a∗ ξ, ξ a ≤ a∗ ξ2 a ≤ a2 ξ2 , which shows that Ra is a continuous C-linear operator with norm bounded by a. To see that Ra ∈ L(F ) when a ∈ Z(N ), simply observe that for ξ and η in F , Ra ξ, η = ξa, η = a∗ ξ, η = ξ, η a∗ = ξ, ηa∗ = ξ, Ra∗ η . This shows that Ra is adjointable, with adjoint Ra∗ and this, in turn, shows that Ra is N -linear. Thus, Ra ∈ L(F ). (Of course, the fact that a lies in Z(N ) also implies directly that Ra is N -linear.) However, since elements of L(F ) are N -module maps, i.e., T (ξb) = (T ξ)b, it is immediate that Ra lies in the center of L(F ). Among other things, the following lemma solves Problem 6.1 under the hypothesis that the support projection of the representation τ is E-reducing. Lemma 6.8. Let τ be a normal representation of the W ∗ -algebra M on a Hilbert space K and let e be its support projection. Let q be the smallest projection in M such that ϕ∞ (q)Re = Re . Then the following assertions hold: (1) ker(τ F (E) ) = {R ∈ L(F(E)) | RRe = 0}. (2) The ultra-weakly closed ideal H ∞ (E) ∩ ker(τ F (E) ) = {R ∈ H ∞ (E) : Rϕ∞ (q) = 0} = H ∞ (E)ϕ∞ (q ⊥ )H ∞ (E)

u−w

,

in H ∞ (E) is generated by ϕ∞ (q ⊥ ). (3) The subspace ϕ∞ (q)F(E) of F(E) is invariant for τ F (E) (H ∞ (E)) and the map X → τ F (E) (X)|ϕ∞ (q)F(E) is an injective completely contractive representation of H ∞ (E). (4) If e is E-saturated then e = q, which implies that e is F(E)-saturated, i.e., ϕ∞ (e)F(E)e = F(E)e. (5) If e is E-reducing, the three projections Re , ϕ∞ (e), and ϕ∞ (q) coincide and lie in the center of L(F(E)). Consequently, τ F (E) (H ∞ (E)) is ultraweakly closed and the restriction of τ F (E) to H ∞ (E)ϕ∞ (e) is completely isometric. Proof. For R ∈ L(F(E)), R ⊗τ IK = 0 if and only if for every k ∈ K and η ∈ F(E), 0 = Rη ⊗ k, Rη ⊗ k = k, τ ( Rη, Rη )k , that is, if and only if Rη, Rη ∈ M e⊥ . This happens if and only if R = RRe⊥ . It follows that ker(τ F (E) ) = {R ∈ L(F(E)) | RRe = 0}

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and H ∞ (E) ∩ ker(τ F (E) ) = {R ∈ H ∞ (E) | RRe = 0}.

(25)

Now choose a faithful normal representation, σ, of M on a Hilbert space H. Then σ F (E) is a ∗ -isomorphism of L(F(E)) onto L(F(E)) ⊗ IH (and is therefore also a homeomorphism with respect to the ultra-weak topologies.) Also note that IF (E) ⊗ σ(e) = Re ⊗ IH = σ F (E) (Re ) by Lemma 6.7. Set g = {u(σ F (E) (Re ))u∗ | u ∈ σ F (E) (ϕ∞ (M )) , uisunitary }. The range of the projection u(σ F (E) (Re ))u∗ is u(Re F(E) ⊗σ H) and, thus, ϕ∞ (q ⊥ ) ⊗ IH = σ F (E) (ϕ∞ (q ⊥ )) vanishes on it. Hence g ≤ σ F (E) (ϕ∞ (q)). By construction g is a projection in the center of σ F (E) (ϕ∞ (M )), so we can write g = σ F (E) (ϕ∞ (z)) for some projection z ∈ Z(M ). But then ϕ∞ (q − z) vanishes on Re F(E), which implies that q = z, since q is the smallest projection in M with this property. Consequently, σ F (E) (ϕ∞ (q)) = {u(σ F (E) (Re ))u∗ | u ∈ σ F (E) (ϕ∞ (M )) , u is unitary}, (26) and q ∈ Z(M ). Next we want to show that if R ∈ H ∞ (E) and if RRe = 0, then Rϕ∞ (q) = 0. To this end, we use the gauge automorphism group and the notation developed in paragraph 2.9. Observe that Wt commutes with Re for all t ∈ T, since Wt ∈ L(F(E)) and Re ∈ Z(L(F(E))). Consequently, γt (R)Re = Wt RRe Wt∗ = 0, and so Φk (R)Re = 0 for all k, by Eq. (8). Since Φk (R)∗ Φk (R) ∈ ϕ∞ (M ), Eq. (26) implies that Φk (R)ϕ∞ (q) = 0 for every k. Consequently each of the Cesaro sums Σn (R) := Σ0≤j
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representation to ϕ∞ (q)H ∞ (E)ϕ∞ (q) is a composition of two completely contractive injective maps and is, therefore, completely contractive and injective. For assertion (5), note that, if e is E-saturated then, for every n ≤ 1 and every ξ ∈ E ⊗n , ξe = ϕn (e)ξe. Thus F(E)e = ϕ∞ (e)F(E)e and so q = e. For assertion (6), assume that e is E-reducing. Then it is easy to verify that Re = ϕ∞ (e) and, in particular, Re lies in the center of H ∞ (E). The map τ F (E) , restricted to the von Neumann algebra L(F(E))Re , is an injective ultra-weakly continuous representation and, thus, is a homeomorphism with respect to the ultra-weak topologies onto its image. Consequently, its restriction to H ∞ (E)ϕ∞ (e) is completely isometric and has a ultra-weakly closed image which is τ F (E) (H ∞ (E)). This proves (6). The following Theorem solves Problem 6.1 under the hypothesis that the support projection is saturated. Theorem 6.9. If the support projection e of a normal representation τ of M is E-saturated, then τ F (E) (H ∞ (E)) is ultra-weakly closed and the restriction of τ F (E) to H ∞ (E)ϕ∞ (e) is completely isometric. We require two lemmas. Lemma 6.10. If e is an E-saturated central projection in M , then the canonical embedding V × π of T (Ee) into T (E), defined in Lemma 6.5, is faithful. Proof. To show V × π is faithful, it sufﬁces to prove that if σ0 is a faithful F (E) normal representation of M on the Hilbert space H and if π0 = σ0 ◦π F (E) ◦ V , then V0 × π0 is a faithful representation of T (Ee) and if V0 = σ0 F (E) is a faithful normal representation of on F(E) ⊗σ0 H. This is because σ0 L(F(E)). We use [8, Theorem 2.1], which asserts in our setting, that V0 × π0 will be faithful if the representation of M e obtained by restricting π0 (M e) to (F(E) ⊗σ0 H) (V0 (Ee)(F(E) ⊗σ0 H)) is faithful. For this, observe that because e is E-saturated F (E)

(V (Ee))(F(E) ⊗σ0 H) V0 (Ee)(F(E) ⊗σ0 H) = σ0 = Ee ⊗ F(E) ⊗σ0 H = E ⊗ ϕ∞ (e)F(E) ⊗σ0 H = E ⊗ F(E)e ⊗σ0 H = E ⊗ F(E) ⊗ σ0 (e)H, ⊗n , we see that by point (4) of Lemma 6.8. Since E ⊗ F(E) = n≥1 E (F(E) ⊗σ0 H) (V0 (Ee)(F(E) ⊗σ0 H)) contains M ⊗σ0 σ0 (e)H as a summand, to which π0 (M e) restricts. But that restriction is obviously faithful on M e since it is just left multiplication on M ⊗σ0 σ0 (e)H = M e ⊗σ0 H by elements of the form me ⊗ IH , m ∈ M , and σ0 is faithful. Lemma 6.11. Suppose τ is a normal representation of M , suppose its support projection e is E-saturated, and let τe be the restriction of τ to M e. Then the map that sends ξe ⊗τe k in F(Ee) ⊗τe K to ξ ⊗τ k in F(E) ⊗τ K extends to a well-defined Hilbert space isomorphism U mapping F(Ee) ⊗τe K onto F(E) ⊗τ K such that, for every X ∈ T+ (Ee), τeF (Ee) (X) = U ∗ τ F (E) ((V × π)(X))U,

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where (V, π) is the canonical embedding of (Ee, M e) in T (E). The image of T+ (Ee) under V × π is T+ (E)ϕ∞ (e) = ϕ∞ (e)T+ (E)ϕ∞ (e). Proof. The fact that U is well-deﬁned and isometric is an easy calculation. F (Ee) The fact that U is surjective is immediate. Observe that τe (Tθe )(ξe ⊗τe k) = θe ⊗ ξe ⊗τe k. By definition of U , this last expression is U ∗ (θe ⊗ (ξ ⊗τ k)) = U ∗ (θ ⊗ ϕ∞ (e)ξ ⊗τ k) = U ∗ τ F (E) (Tθ )(ϕ∞ (e)ξ ⊗τ k) = U ∗ τ F (E) (Tθ ϕ∞ (e))U (ξe ⊗τe k) = U ∗ τ F (E) ((V × π)(Tθe ))U (ξe ⊗τe k). F (Ee)

Similarly, τe (ϕe∞ (me)) = U ∗ τ F (E) ((V × π)(ϕ∞ (me))U for all me ∈ M e, where ϕe∞ denotes the action of M e on F(Ee). F (E)

Proof of Theorem 6.9. Since τe is a faithful representation of M e, τe is F (E) a completely isometric map of H ∞ (Ee) and its image, τe (H ∞ (Ee)), is F (E) ultra-weakly closed. Since both τ F (E) and τe are ultra-weakly continuous, we have τ F (E) (T+ (E))

u−w

= τ F (E) (H ∞ (E))

u−w

and F (Ee)

τe

u−w

(T+ (Ee))

F (Ee)

= τe

(H ∞ (Ee))

u−w

u−w

= τeF (Ee) (H ∞ (Ee)). u−w

= τ F (E) (H ∞ (E)ϕ∞ (e)) since ϕ∞ (e⊥ ) is the Now τ F (E) (H ∞ (E)) F (E) projection onto the kernel of τ . Also, Lemma 6.11 implies that (V × π) u−w = H ∞ (E)ϕ∞ (e). Con(T+ (Ee)) = T+ (E)ϕ∞ (e). So (V × π)(T+ (Ee)) sequently, from Lemma 6.11, we conclude that τ F (E) (H ∞ (E)ϕ∞ (e) F (Ee) U τe (H ∞ (Ee)) U ∗ .

u−w

=

u−w

, there is a net {Xn } of elements of Thus, given Z ∈ τ F (E) (H ∞ (E)) T+ (Ee) that converges ultra-weakly to X ∈ H ∞ (Ee) such that Xn ≤ X F (Ee) for all n and Z = U τe (X)U ∗ . The net {(V × π)(Xn )} is bounded in T+ (E)ϕ∞ (e) and so has a subnet {(V ×π)(Xnα )} that converges ultra-weakly to some Y ∈ H ∞ (E)ϕ∞ (e), with Y ≤ X. However, from the ultra-weak F (Ee) continuity of the maps τ F (E) and τe , we conclude that Z = U τeF (Ee) (X)U ∗ = lim U τeF (Ee) (Xnα )U ∗ = lim τ F (E) (Xnα ) = τ F (E) (Y ) belongs to τ F (E) (H ∞ (E)ϕ∞ (e)). This shows that τ F (E) (H ∞ (E)) is ultraweakly closed. We also have Y ≤ X = Z ≤ τ F (E) (Y ) ≤ Y . Thus Y = τ F (E) (Y ). If we started with a given Y ∈ H ∞ (E)ϕ∞ (e) and Z = τ F (E) (Y ), we would be able to conclude that Y = Z so that the map τ F (E) , restricted to H ∞ (E)ϕ∞ (e), is isometric. One can argue similarly to show that it is completely isometric.

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7. The Structure Theorem We have two goals in this section. The ﬁrst is Theorem 7.8, which is a generalization of [2, Theorem 2.6], that Davidson, Katsoulis and Pitts call the Structure Theorem. This theorem provides a very special matricial decomposition for the ultra-weak closure of (S × σ)(T+ (E)), where (S, σ) is an isometric representation of (E, M ). Our second goal is Theorem 7.10, which uses Theorem 7.8 to complement Theorem 3.1, showing that (S, σ) is absolutely continuous if and only if it S × σ is completely isometrically and ultra-weakly homeomorphically equivalent to a subrepresentation of an induced representation. Suppose, then, that (S, σ) is an isometric representation of (E, M ) acting on a Hilbert space H. We shall write S for the ultra-weak closure of (S × σ)(T+ (E)). It is thus the ultra-weakly closed algebra generated by the operators {σ(a), S(ξ) | ξ ∈ E, a ∈ M }. Also, we shall write S0 for the ultra-weakly closed algebra generated by {S(ξ) | ξ ∈ E}. Evidently, S0 is an ultra-weakly closed, two-sided ideal in S. The decomposition that Davidson, Katsoulis and Pitts advanced centers on understanding the position of S0 in S. In particular, it is important to know when S0 is a proper ideal of S. Our analysis follows a similar route, but it is made more complicated by the presence of M . Observe that since S0 is an ultra-weakly closed 2-sided ideal in S, σ −1 (S0 ) is an ultra-weakly closed two sided ideal in M and hence is of the form pM for a suitable projection in the center of M . Definition 7.1. The projection e in M with the property that σ −1 (S0 ) = e⊥ M is called the model projection for S (or for (S, σ)). The reason for terminology will become clear through a brief outline for our analysis that we hope will be helpful when following our arguments. As in [2], we let N be thevon Neumann algebra generated by S. We will see in ∞ Proposition 7.5 that k=1 S0k is a left ∞ideal in N . Consequently, there is a projection P ∈ S ∩ σ(M ) such that k=1 S0k = N P = SP . Following [2] we call this projection the structure projection for S. We will show in Theorem 7.8 that (I − P )H = P ⊥ H is invariant under S and that S = N P + P ⊥ SP ⊥ . That is, in matrix form, PNP 0 S= . P ⊥ N P P ⊥ SP ⊥ We will see in Lemma 7.2 that e is E-saturated in the sense of Definition 6.2 and that it is the support projection for an induced representation τ F (E) that we will soon describe. We will see in Lemma 7.6 that σ(e) is the central support of P ⊥ in σ(M ). In part (3) of Theorem 7.8 we will see that P ⊥ SP ⊥ is completely isometrically isomorphic and ultra-weakly homeomorphic to τ F (E) (H ∞ (E)). Thus τ F (E) (H ∞ (E)) serves as a model for P ⊥ SP ⊥ . Further, it will follow from Theorem 7.9 that if σ(e⊥ ) = P = 0, then the representation (S, σ) is absolutely continuous.

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Concerning this last statement, one might be inclined at ﬁrst to believe that once one knows that S is completely isometrically isomorphic and ultraweakly homeomorphic to the ultra-weak closure of the range of an induced representation, then (S, σ), itself, must be an induced representation, but examples constructed by Davidson et al. in [2, Section 3] show this is not the case. Thus our Theorem 7.9 leads to a much more inclusive result and one may wonder about a converse: Does every absolutely continuous, isometric representation (S, σ) have a vanishing structure projection? There answer, in general, is “no”. Indeed, even in the case when A = C = E, the answer is “no” for classical reasons. In this situation, it follows from Szeg¨ o’s theorem that if S(1) is an absolutely continuous unitary operator whose spectrum does not cover the circle, then P = I. It is of interest to determine more precisely when this may happen in the general setting. To proceed with the details, we ﬁx the isometric representation (S, σ) and we begin by analyzing the model projection e. For this purpose we observe that since we are working with ultra-weakly closed algebras, ultra-weakly closed ideals and related constructs involving the ultra-weak topology, it will be convenient to employ an inﬁnite multiple (S (∞) , σ (∞) ) of (S, σ), which acts on the countably inﬁnite direct sum of copies of H, H (∞) . In general, for x ∈ B(H), we write x(∞) for its inﬁnite ampliation in B(H (∞) ). Likewise for a space of operators X ⊆ B(H), we write X (∞) for {x(∞) | x ∈ X}. Note that for x ∈ H ∞ (E), we have S (∞) × σ (∞) (x) = (S × σ(x))∞ . In particular, S (∞) (ξ) = S(ξ)(∞) , for all ξ ∈ E. We are especially interested in the set M that consists of all subspaces M of H (∞) that reduce σ (∞) (M ) and are wandering subspaces for S (∞) in the sense that M, [S (∞) (E)M], [S (∞) (E)S (∞) (E)M], etc. are all orthogonal, and their direct sum is a subspace of H (∞) that is invariant for the algebra S (∞) × σ (∞) (H ∞ (E)). This subspace must be invariant for the ultraweak closure of S (∞) × σ (∞) (H ∞ (E)), which is the same as S (∞) . It is easily seen, then, that M ⊕ [S (∞) (E)M] ⊕ [S (∞) (E)S (∞) (E)M] ⊕ · · · = [S ∞ (M)]. For M in M, we denote the restriction of (S (∞) , σ (∞) ) to [S (∞) (M)] by (∞) (∞) (∞) (∞) (SM , σM ). Evidently, (SM , σM ) is unitarily equivalent to the induced F (E) where τM is the restriction of σ (∞) to M. In more representation τM (∞) (E)M] ⊕ [S (∞) (E)S (∞) (E)M] ⊕ · · · = [S ∞ (M)], the detail, since M ⊕ [S map which takes h0 ⊕ S (∞) (ξ1 )h1 ⊕ S (∞) (ξ21 )S (∞) (ξ22 )h2 ⊕ · · · ⊕ S (∞) (ξn1 )S (∞) (ξn2 ) · · · S (∞) (ξnn )hn in [S ∞ (M)] to h0 ⊕ ξ1 ⊗ h1 ⊕ ξ21 ⊗ ξ22 ⊗ h2 ⊕ · · · ⊕ ξn1 ⊗ ξn2 ⊗ · · · ξnn ⊗ hn in F(E) ⊗τM M is well deﬁned and extends to a Hilbert space isomorphism UM : [S (∞) (M)] → F(E) ⊗τM M such that ∗ (ϕ∞ (a) ⊗ IM )UM = σ (∞) (a)|[S (∞) M] , a ∈ M, UM

and ∗ UM (Tξ ⊗ IM )UM = S (∞) (ξ)|[S (∞) M] , ξ ∈ E.

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(See [15, Remark 7.7]). Although it leads to non-separable spaces, in general, and introduces a lot of redundancy into our analysis, it is convenient to let E be the external direct sum of all the spaces M in M, writing E :=

⊕

M,

M∈M

and setting τ :=

⊕

τM .

(27)

M∈M

Then τ F (E) gives representations of T+ (E) and H ∞ (E) on F(E) ⊗τ E. Fur ther, if we write U := ⊕UM , then U implements a unitary equivalence be (∞) (∞) tween τ F (E) and M∈M ⊕(SM ×σM ) = M∈M ⊕(S (∞) ×σ (∞) )|[S (∞) M]. The map Φ : B(H) → B(F(E) ⊗τ E) deﬁned by the formula Φ(x) = U

! ⊕P[S (∞) M] x(∞) |[S (∞) M] U ∗ ,

(28)

where P[S (∞) M] is the projection of H (∞) onto [S (∞) M], is a completely positive map that is continuous with respect to the ultra-weak topologies. When restricted to S, ! (29) Φ(x) = U ⊕x(∞) |[S (∞) M] U ∗ , x ∈ S, so this restriction, also denoted Φ, is a homomorphism that satisﬁes Φ(σ(a)) = ϕ∞ (a) for a ∈ M and Φ(S(ξ)) = Tξ ⊗IE for ξ ∈ E. Thus Φ(S) is ultra-weakly dense in the ultra-weak closure of τ F (E) (T+ (E)). (Note that each element in this ultra-weak closure can be approximated by Cesaro means calculated with respect to the gauge automorphism group, and each such mean lies in Φ(S)). Lemma 7.2. The model projection e for S is the support projection of τ and is E-saturated. Proof. Recall that by definition, e⊥ is the central cover of {a ∈ M | σ(a) ∈ S0 }, which is an ultra-weakly closed ideal in M . Since ker(τ ) is also an ultraweakly closed ideal in M , it sufﬁces to show that each contains the same projections p. If σ(p) ∈ S0 , then for M ∈ M, τM (p)M = σ (∞) (p)M ⊆ ⊥ S0 (M) ⊆ M . But τM (p)M ⊆ M. Thus τM (p) = 0. Since τ is deﬁned to be M∈M τM , p ∈ ker(τ ). On the other hand, if σ(p) is not in S0 , then there is an ultra-weakly continuous linear functional f such that f (σ(p)) = 0 and f (a) = 0 for a ∈ S0 . But then we can write f (x) = x(∞) ξ, η for suitable vectors ξ, η ∈ H (∞) , and ﬁnd that η is orthogonal to the σ (∞) (M )-invariant subspace [(S0 )(∞) ξ], on the one hand, but is not orthogonal to σ (∞) (p)ξ, on (∞) the other. Consequently, the space M := [σ (∞) (p)S (∞) ξ] [σ (∞) (p)S0 ξ]

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is non zero, and by construction, M is a σ (∞) (M )-invariant wandering subspace. That is, M lies in M, and evidently, τM (p) = IM = 0. Thus ker(τ ) = {a ∈ M | σ(a) ∈ S0 }. To see that e is E-saturated, i.e., to see that ϕ(e⊥ )ξe = 0 for all ξ ∈ E, note that σ(e⊥ ) lies in S0 . Thus, for every ξ, η ∈ E, S(η)∗ σ(e⊥ )S(ξ) ∈ S0 . But S(η)∗ σ(e⊥ )S(ξ) = σ( η, ϕ(e⊥ )ξ ) and so it follows from the definition of e that η, ϕ(e⊥ )ξe = η, ϕ(e⊥ )ξ e = 0 for all ξ, η ∈ E. Thus ϕ(e⊥ )ξe = 0 for all ξ ∈ E. The representation τ acts on a non-separable space, so it may be comforting to know that for all intents and purposes we have in mind, it is possible to replace τ with a representation on a separable space. This an immediate consequence of the following proposition. Proposition 7.3. Suppose τ1 and τ2 are two normal representations of M on Hilbert spaces H1 and H2 , respectively, and suppose that ker(τ1 ) = ker(τ2 ). F (E) F (E) Then the map Ψ that sends τ1 (R) to τ2 (R), R ∈ L(F(E)), is a normal ∗-isomorphism from the the von Neumann algebra L(F(E)) ⊗τ1 IH1 onto the von Neumann algebra L(F(E)) ⊗τ2 IH2 . Further, the restriction of Ψ F (E) to the ultra-weak closure of τ1 (H ∞ (E)) is a completely isometric, ultraF (E) (H ∞ (E)) onto the ultra-weak closure of weak homeomorphism from τ1 F (E) ∞ (H (E)). τ2 Proof. For R ∈ L(F(E)), ξ ∈ F(E) and h ∈ Hi , i = 1, 2, Rξ ⊗ h, Rξ ⊗ h = h, τi ( Rξ, Rξ )h . Thus R ⊗τi IHi = 0 if and only if Rξ, Rξ ⊆ ker(τi ) for all ξ ∈ E. Since we assume that ker(τ1 ) = ker(τ2 ), the map Ψ is a well deﬁned, injective map. It is clearly a ∗ -homomorphism and normality is also easy to check. We turn next to the problem of identifying the structure projection P for S. Proposition 7.4. If Φ is the map defined by Eq. (28), then S ∩ ker(Φ) =

∞

S0k .

k=1

Proof. For x∈ S0k and M ∈ M, x(∞) [S (∞) M] ⊆ [(S0k )(∞) M]. Since the ∞ k (∞) intersection, k [(S0 ) M], is {0}, we ﬁnd that k=1 S0k ⊆ ker(Φ). On the ∞ k other hand, if x ∈ S is not in k=1 S0 , then there is a k ≥ 1 such that x is not in S0k . Consequently, there is an ultra-weakly continuous linear functional f such that f vanishes on S0k but f (x) = 0. Since f is ultra-weakly continuous, we may ﬁnd vectors ξ, η in H (∞) such that f (y) = y (∞) ξ, η for all y ∈ B(H). It follows that x(∞) ξ is not in [(S0k )(∞) ξ], proving that N := [S (∞) ξ] [(S0k )(∞) ξ] = {0}. In fact, we ﬁnd that x(∞) N [(S0k )(∞) ξ].

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We can write N as a direct sum of wandering spaces in M: N = ([S (∞) ξ] [(S0 )(∞) ξ]) ⊕ ([(S0 )(∞) ξ] [(S02 )(∞) ξ]) ⊕ · · · ⊕ ([(S0k−1 )(∞) ξ] [(S0k )(∞) ξ]), which shows that x is not in the kernel of Φ. ∞ Proposition 7.5. The space k=1 S0k is an ultra-weakly closed left ideal in the von Neumann algebra N generated by S. Thus ∞

S0k = N P = SP

(30)

k=1

for some projection P ∈ S ∩ σ(M ) . Recall that P is called the structure projection for S (and for (S, σ).) Proof. We ﬁrst claim that the map Φ deﬁned in Eq. (28) satisﬁes the equation Φ(S(ξ)∗ R) = (Tξ∗ ⊗ IE )Φ(R) , ξ ∈ E, R ∈ S. Since Φ is an ultra-weakly continuous linear map, it sufﬁces to prove the claim for R = σ(a), a ∈ M , and for R = S(η1 )S(η2 ) · · · S(ηk ) for η1 , η2 , . . . ηk ∈ E. For R = σ(a), Φ(σ(a)) = ϕ∞ (a) ⊗ IE and, thus both sides of the equation are equal to 0. For R = S(η1 )S(η2 ) · · · S(ηk ), we have Φ(S(ξ)∗ R) = Φ(σ( ξ, η1 )S(η2 ) · · · S(ηk )) = (ϕ∞ ( ξ, η1 ) ⊗ IE )Φ(S(η2 ) · · · S(ηk )) = (Tξ∗ ⊗ IE )(Tη1 ⊗ IE )Φ(S(η2 ) · · · S(ηk )) = (Tξ∗ ⊗ IE )Φ(R),

∞ Φ ∩ S), which proves the claim. Consequently,for R ∈ k=1 S0k (= ker ∞ ∞ Φ(S(ξ)∗ R) = 0 and, thus, S(ξ)∗ R ∈ k=1 S0k . Clearly S(ξ)R ∈ k=1 S0k ∞ k for such R and, therefore, ∩k=1 S0 is an ideal in N . Since ker(Φ) ∩ S is an ideal in S, it follows that P ∈ σ(M ) . Lemma 7.6. The structure projection P and the model projection e for the isometric representation (S, σ) are related: e is the smallest projection in M satisfying σ(e) = {v(I − P )v ∗ : v is a unitary in σ(M ) }. In particular, σ(e) is the central support in σ(M ) of P ⊥ . So, if P = 0, then e is the support projection of σ; thus if P = 0 and the restriction of S × σ to M is faithful, then e = I. Proof. If z is a projection in M with σ(z) ∈ S0 then, for k ≥ 1, σ(z) = σ(z)k ∈ S0k and, thus, σ(z) ∈ N P and σ(z) ≤ P . The converse also holds since if σ(z) = σ(z)P , then σ(z) ∈ S0 . Thus it follows from Lemma 7.2 that e⊥ is the largest projection z in M such σ(z) ≤ P . Equivalently, e is the smallest projection p in M such that σ(p) ≥ I − P . The statement of the lemma is now immediate.

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The next result was inspired by [2, Theorem 1.1]. Our arguments are adaptations of those in [2]. Theorem 7.7. Let (S, σ) be an isometric representation of (E, M ), write S for the ultra-weak closure of (S × σ)(T+ (E)) and let e be the model projection for S. Suppose τ is a normal representation of M on a Hilbert space K and that the support projection of τ is e. Suppose, also, that there is given an ultra-weakly continuous, completely contractive homomorphism ψ from S to τ F (E) (H ∞ (E)) such that ψ ◦ (S × σ) = τ F (E) on T+ (E). Then ψ is surjective and the map ψ : S/ ker(ψ) → τ F (E) (H ∞ (E)) obtained by passing to the quotient is a complete isometry and a homeomorphism with respect to the ultra-weak topologies on S/ ker(ψ) and τ F (E) (H ∞ (E)). Proof. Recall the unitary group {Wt } in L(F(E)), and the gauge automorphism group {γt }, γt = AdWt , discussed in paragraph 2.9. Recall in particular that ⎛ ⎞ 2π 1 |j| ⎝ 1− Σk (a) := e−int ⎠ γt (a)dt 2π k 0

|j|
(n)

(n)

(n)

for a ∈ L(F(E)). We write Wt for diag(Wt , . . . , Wt ), γt for AdWt ⎛ ⎞ 2π 1 |j| (n) (n) ⎝ Σk (A) := 1− e−ijt ⎠ γt (A)dt, 2π k 0

and

|j|
(n)

(n)

for A ∈ Mn (L(F(E))). We ﬁnd that Σk is a contractive map and Σk (A) → A in the ultra-weak topology as k → ∞. (n) Fix A ∈ Mn (ϕ∞ (e)H ∞ (E)ϕ∞ (e)) and write Xk := (S ×σ)(n) (Σk (A)). (n) Note that Σk (A) ∈ Mn (T+ (E)) so that the last expression is well deﬁned (n) (n) and ψ (n) (Xk ) = (τ F (E) )(n) (Σk (A)). Thus, Xk = (S × σ)(n) (Σk (A)) ≤ (n) Σk (A) ≤ A (here we used the fact that (S × σ)(n) is contractive on (n) ϕ∞ (e)T+ (E)ϕ∞ (e) and Σk (A) lies in Mn (ϕ∞ (e)T+ (E)ϕ∞ (e)) ). Now, the sequence Xk has a subnet Xkα that is ultra-weakly convergent to some X ∈ Mn (S) with X ≤ A. Since ψ is ultra-weakly continuous, it follows that (n)

ψ (n) (X) = lim ψ (n) (Xkα ) = lim(τ F (E) )(n) (Σkα (A)) = (τ F (E) )(n) (A), where the limits are all taken in the ultra-weak topology. Thus ψ (n) is surjective for every n ≥ 1. Since e is E-saturated, Theorem 6.9 implies that τ F (E) , restricted to ∞ H (E)ϕ∞ (e), is a complete isometry and, thus, A ≥ X ≥ ψ (n) (X) = (τ F (E) )(n) (A). Therefore, in this case we get X = (τ F (E) )(n) (A). This shows that the map ψ from S/ ker(ψ) onto τ F (E) (H ∞ (E)), induced by ψ, is a complete isometry.

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The fact that this map is an ultra-weak homeomorphism follows from the ultra-weak continuity of ψ as in the proof of [2, Theorem 1.1]. To be more precise, the map ψ is the dual of a map ψ∗ from the predual of τ F (E) (H ∞ (E)) to the predual of S/ ker(ψ). This map is injective (as ψ is surjective ) and contractive and, thus, has a bounded inverse (ψ∗ )−1 . The dual of this inverse Thus ψ is an ultra-weak homeomorphism. is the inverse of ψ. The following theorem is our analogue of [2, Theorem 2.6]. Theorem 7.8. Let (S, σ) be an isometric representation of (E, M ) on H, let S be the ultra-weak closure of (S × σ)(T+ (E)), let N be the von Neumann algebra generated by S, and let P be the structure projection for S. Then in addition to Eq. (30), the following assertions hold: (1) (2) (3) (4)

P ⊥ H is invariant for S. SP ⊥ = P ⊥ SP ⊥ . S = N P + P ⊥ SP ⊥ . If τ is any normal representation of M whose support projection is the model projection for S, then the map Ψ from (S × σ)(T+ (E))P ⊥ to τ F (E) (H ∞ (E)) defined by the equation Ψ((S × σ)(a)P ⊥ ) = τ F (E) (a),

a ∈ T+ (E),

is a completely isometric isomorphism from SP ⊥ onto τ F (E) (H ∞ (E)) that is a homeomorphism with respect to the ultra-weak topologies on SP ⊥ and τ F (E) (H ∞ (E)). Proof. Suppose ﬁrst that τ is the representation deﬁned in (27) and that Φ is the map deﬁned in (29). Since ker(Φ) is an ideal in S and the structure projection, P , belongs to it, P SP ⊥ ⊆ ker(Φ) = N P . But then P SP ⊥ = 0. Thus P ⊥ H is invariant for S and SP ⊥ = P ⊥ SP ⊥ . Now, Φ is a completely contractive, ultra-weakly continuous, homomorphism mapping S into the σ-weak closure of τ F (E) (T+ (E)). So we may apply Theorem 7.7 to Φ (which plays the role of ψ in the statement of Theorem 7.7) to conclude that the map induced by Φ on SP ⊥ S/ ker(Φ) is completely isometric and an ultra-weak homeomorphism onto the ultra-weak closure of τ F (E) (T+ (E))—which is τ F (E) (H ∞ (E))) by Theorem 6.9 and the fact that the model projection for S is E-saturated, Lemma 7.2. But once we have proven the result with the special representation τ from (27), we can replace it with any representation with the same support, viz. the model projection of S, thanks to Proposition 7.3. To connect the structure projection with the absolutely continuous subspace of an isometric representation, we employ the universal isometric representation (S0 , σ0 ) from paragraph 2.11 and use results developed in Sect. 3. Recall that (S0 , σ0 ) is the representation induced by a representation π on a Hilbert space K0 , and so (S0 , σ0 ) acts on the Hilbert space F(E) ⊗π K0 .

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Theorem 7.9. Let (S, σ) be an isometric representation of (E, M ) on a Hilbert space H, let P be its structure projection, and let Q be the structure projection for the representation (S × σ) ⊕ (S0 × σ0 ), acting on H ⊕ F(E) ⊗π K0 . Then (1) Vac (S, σ) = H Q(H ⊕ (F(E) ⊗π K0 )), (2) P ⊥ H ⊆ Vac (S, σ), and (3) the range of Q is contained in the range of P , viewed as a subspace of H ⊕ (F(E) ⊗π K0 ). Proof. We shall write S for the ultra-weak closure of S × σ(T+ (E)). To simplify the notation, we shall write ρ for (S × σ) ⊕ (S0 × σ0 ) and we shall write Sρ for the ultra-weak closure of ρ(T+ (E)). We ﬁrst want to prove that the range of Q is contained in H. For this purpose and for later use, we shall write PH for the projection of H ⊕ (F(E) ⊗π K0 ) onto H. Since ρ(a) commutes with PH , for every a ∈ T+ (E), the von Neumann algebra generated by Sρ , Nρ , commutes with PH . In particular, Q commutes with PH . So it sufﬁces to show that if ξ is a vector in H ⊕ (F(E) ⊗π K0 ) such that ξ = Qξ lies in H ⊥ , then ξ = 0. So ﬁx such a vector ξ. Then ξ ∈ F(E) ⊗π K0 and, for every η ∈ E ⊗m , ρ(Tη )ξ ∈ (E ⊗m ⊕ E ⊗(m+1) ⊕ · · · ) ⊗π K0 . Thus, for ⊗m ⊕ E ⊗(m+1) ⊕ · · · ) ⊗π K0 . It follows that, if everyX ∈ (Sρ )m 0 , Xξ ∈ (E m X ∈ m (Sρ )0 = Nρ Q, then Xξ = 0. In particular, Qξ = 0. This shows that the range of Q is contained in H. Next observe that the wandering vectors of any isometric representation are orthogonal to the range of its structure projection. To prove this, we will work with (S, σ), but the argument is quite general. So suppose ξ ∈ H is a wandering vector for (S, σ). Then M0 := [σ(M )ξ] is a σ(M )-invariant wandering subspace for (S, σ), M := {η (∞) : η ∈ M0 } is in M, and ξ (∞) ∈ [S (∞) M]. Since the structure projection P of (S, σ) lies in the kernel of Φ by Proposition 7.4, P (∞) |[S (∞) M] = 0 and, thus, P ξ = 0. Thus the span of the wandering vectors for ρ is orthogonal to the range of Q. But by Corollary 3.8, the span of the wandering vectors for ρ is Vac (S, σ) ∩ H. Thus Vac (S, σ) ⊆ PH Q⊥ (H ⊕ (F(E) ⊗ K)). To show the reverse inclusion, suppose ξ = PH Q⊥ ξ. Then ξ = Q⊥ ξ, as Q ≤ PH . Let f be the linear functional on T+ (E) deﬁned by the equation f (a) = ρ(a)ξ, ξ = ρ(a)Q⊥ ξ, Q⊥ ξ , a ∈ T+ (E). We want to use Theorem 7.8 to show that ξ lies in Vac (S, σ) by showing that f extends to an ultra-weakly continuous linear functional on H ∞ (E). In the notation of that theorem, replace S × σ with ρ and let eρ be the model projection for ρ. Also, let τ be any normal representation of M with support equal eρ . Then by part (4) of Theorem 7.8 there is a completely isometric isomorphism Ψ from Sρ Q⊥ onto τ F (E) (H ∞ (E)) that is an ultra-weak homeomorphism such that Ψ(ρ(a)Q⊥ ) = τ F (E) (a) for all a ∈ H ∞ (E). We conclude, then, that f (a) = ρ(a)Q⊥ ξ, Q⊥ ξ = Ψ−1 ◦ τ F (E) (a)ξ, ξ , a ∈ T+ (E), extends to an ultra-weakly continuous linear functional on H ∞ (E). By Remark 3.2, ξ ∈ Vac (S, σ). Thus we conclude that Vac (S, σ) = H Q(H ⊕ (F(E) ⊗π K0 )). This proves point (1).

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The argument needed to prove that P ⊥ H ⊆ Vac (S, σ) is similar. Let ξ ∈ P ⊥ H and consider the functional g(a) := (S × σ)(a)ξ, ξ = (S × σ)(a)P ⊥ ξ, P ⊥ ξ for a ∈ T+ (E). Using part (4) of Theorem 7.8 (and the F (E) notation there), we may write g(a) = Ψ−1 (τe (a))ξ, ξ for any normal representation τe of M whose support projection is the model projection for S × σ. This shows that g is ultra-weakly continuous and so ξ ∈ Vac (S, σ). Thus P ⊥ H is contained in Vac (S, σ), proving point (2). Since P ⊥ H ⊆ Vac (S, σ) = H Q(H ⊕ (F(E) ⊗π K0 )), we see that the range of PH Q is contained in the range of P . Since, however, the range of Q is contained in H, as we proved at the outset, we conclude that the range of Q is contained in the range of P , proving point (3). As a corollary we conclude with following theorem, which complements Theorem 3.10 in the sense that if (S, σ) is an ultra-weakly continuous, isometric representation of (E, M ), then (S, σ) is absolutely continuous if and only if S × σ ⊕ S0 × σ0 is completely isometrically equivalent to an induced representation of (E, M ). Thus, (S, σ) is absolutely continuous if and only if it is completely isometrically equivalent to a subrepresentation of an induced representation. Theorem 7.10. Let (S, σ) be an ultra-weakly continuous, isometric covariant representation of (E, M ) on a Hilbert space and denote by ρ the representation S × σ ⊕ S0 × σ0 . Then the following assertions are equivalent: (1) (S, σ) is absolutely continuous. (2) The structure projection for ρ is zero. (3) If τ is any representation of M whose support projection is the model projection of S × σ, then there is an ultra-weakly homeomorphic, completely isometric isomorphism Ψ from τ F (E) (H ∞ (E)) onto the ultra-weak closure Sρ of ρ(T+ (E)) such that Ψ ◦ τ F (E) (a) = ρ(a), for all a ∈ T+ (E). Proof. Let P be the structure projection for S × σ. If (S, σ) is absolutely continuous then Vac (S, σ) = H and it follows from Theorem 7.9 (1) that the range of Q is orthogonal to H. Since, however, the range of Q is contained in the range of P by part (3) of that lemma, we conclude that Q = 0. This proves that (1) implies (2). The implication, (2) ⇒ (3), follows from Theorem 7.8 (4). Now assume that (3) holds. Then Ψ−1 (Sρ ) = τ F (E) (H ∞ (E)), and for F (E) ∞ of the space Hm (E), which conm ≥ 0, Ψ−1 ((Sρ )m 0 ) is the image under τ ∞ sists of all operators X ∈ H (E) with the property that Φk (X) = 0, k ≤ m, where Φk is the k th Fourier operator (8). Then " # −1 m Ψ (Sρ )0 Ψ−1 ((Sρ )m = 0 ) m

m

=

m

" τ

F (E)

∞ (Hm (E))

=τ

F (E)

m

# ∞ Hm (E)

.

∞ Since m Hm (E) = {0}, m (Sρ )m 0 = {0} and so Q = 0. This proves that (3) implies (2). Since Theorem 7.9 (1) shows that (2) implies (1), the proof is complete.

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Remark 7.11. In [12], M. Kennedy deftly uses technology from the theory of dual operator algebras to show that in the setting when M = C and E = Cd , the ultra-weak closure of the image of an isometric representation of T+ (Cd ) is ultraweakly homeomorphic and completely isometrically isomorphic to H ∞ (Cd ) if and only if the representation is absolutely continuous. Whether his result can be generalized to the setting of this paper remains to be seen.

References [1] Arveson, W.: Subalgebras of C ∗ -algebras. Acta Math. 123, 141–224 (1969) [2] Davidson, K., Katsoulis, E., Pitts, D.: The structeure of free semigroup algebras. J. Reine Angew. Math. 533, 99–125 (2001) [3] Davidson, K., Li, J., Pitts, D.: Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras. J. Funct. Anal. 224, 160–191 (2005) [4] Dixmier, J.: Von Neumann Algebras. In: North-Holland Mathematical Library, vol. 27. North Holland, Amsterdam (1981) [5] Douglas, R.G.: On the operator equation S ∗ XT = X and related topics. Acta Sci. Math. (Szeged) 30, 19–32 (1969) [6] Douglas, R.G.: On majorization, factorization, and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–415 (1966) [7] Fowler, N., Muhly, P.S., Raeburn, I.: Representations of Cuntz-Pimsner algebras. Indiana Univ. Math. J. 52, 569–605 (2003) [8] Fowler, N., Raeburn, I.: The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48, 155–181 (1999) [9] Gantmacher, F.: Applications of the Theory of Matrices. Wiley, New York (1959) [10] Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. 1. Academic Press, New York (1983) [11] Kato, T.: Smooth operators and commutators. Studia Math. 31, 535–546 (1968) [12] Kennedy, M.: Absolutely continuous presentations of the non-commutative disk algebra. Preprint. arXiv: 1001.3182v1 (2010) [13] Manuilov, V., Troitsky, E.: Hilbert C ∗ -Modules. In: Translations of Mathematical Monographs, vol. 226. Amer. Math. Soc., Providence (2005) [14] Muhly, P.S., Solel, B.: The Poisson kernel for Hardy algebras. Complex Anal. Oper. Theory 3, 221–242 (2009) [15] Muhly, P.S., Solel, B.: Hardy algebras, W ∗ -correspondences and interpolation theory. Math. Ann. 330, 353–415 (2004) [16] Muhly, P.S., Solel, B.: Quantum Markov processes (correspondences and dilations). Int. J. Math. 13, 863–906 (2002) [17] Muhly, P.S., Solel, B.: Tensor algebras, induced representations, and the Wold decomposition. Can. J. Math. 51, 850–880 (1999) [18] Muhly, P.S., Solel, B.: Tensor algebras over C ∗ - correspondences (Representations, dilations, and C ∗ - envelopes). J. Funct. Anal. 158, 389–457 (1998)

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[19] Paulsen, V.: Every completely polynomially bounded operator is similar to a contraction. J. Funct. Anal. 55, 1–17 (1984) [20] Popescu, G.: Similarity and ergodic theory of positive linear maps. J. Reine Angew. Math. 561, 87–129 (2003) [21] Rieffel, M.: Morita equivalence for C ∗ -algebras and W ∗ -algebras. J. Pure Appl. Alg. 5, 51–96 (1974) [22] Rieffel, M.: Induced representations of C ∗ -algebras. Adv. Math. 13, 176– 257 (1974) [23] Revuz, D.: Markov Chains. North Holland/Elsevier, New York (1984) [24] Shalit, O., Solel, B.: Subproduct Systems. Doc. Math. 14, 801–868 (2009) [25] Stinespring, F.: Positive functions on C ∗ -algebras. Proc. Am. Math. Soc. 6, 211–216 (1955) [26] Sz.-Nagy, B., Foia¸s, C.: Harmonic Analysis of Operators on Hilbert Space. Elsevier, New York (1970) [27] Takesaki, M.: Theory of Operator Algebras I. Springer-Verlag, New York (1979) Paul S. Muhly (B) Department of Mathematics University of Iowa Iowa City, IA 52242 USA e-mail: [email protected]; [email protected] Baruch Solel Department of Mathematics Technion 32000 Haifa Israel e-mail: [email protected] Received: June 9, 2010. Revised: January 17, 2011.

Integr. Equ. Oper. Theory 70 (2011), 205–226 DOI 10.1007/s00020-011-1878-z Published online April 13, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Semigroups of Partial Isometries and Symmetric Operators R. T. W. Martin Abstract. Let {V (t)| t ∈ [0, ∞)} be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let V (−t) := V ∗ (t) for t ∈ [0, ∞). It is shown that if V (t) is a partial isometry for all t ∈ [−t0 , t0 ], t0 > 0, then the pointwise two-sided derivative of V (t) exists on a dense domain of vectors. This derivative B is necessarily a densely deﬁned symmetric operator. This result can be viewed as a generalization of Stone’s theorem for one-parameter strongly continuous unitary groups, and is used to establish sufﬁcient conditions for a self-adjoint operator on a Hilbert space K to have a symmetric restriction to a dense linear manifold of a closed subspace H ⊂ K. A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the 2 independent variable in K := ⊕n i=1 L (R) to certain model subspaces of the Hardy space of n−compenent vector valued functions which are analytic in the upper half plane is presented. Mathematics Subject Classification (2010). 47B25 (symmetric and selfadjoint operators (unbounded)), 47D06 (one-parameter semigroups and linear evolution equations), 30H10 (Hardy spaces). Keywords. Symmetric operators, Hardy space, model subspaces, strongly continuous one parameter operator semigroups, partial isometries.

1. Introduction Let S := {V (t)| t ∈ [0, ∞)} be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space. That is, the map t → V (t) is strongly continuous for t ∈ [0, ∞), V (·) is contraction valued and ({V (t)}, ·) is a representation of the semigroup ([0, ∞), +): V (t)V (s) = V (t + s) for all t, s ≥ 0 and V (0) = I. For brevity, we will sometimes call such semigroups contraction semigroups. A complete classiﬁcation of semigroups S which consist solely of partial isometries has been provided in [4]. The simplest examples of such semigroups S are the semigroups of truncated shifts acting on L2 (R). Given an interval I ⊂ R, each V (t) ∈ SI , the semigroup of truncated shifts on the interval I,

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acts by ﬁrst projecting L2 (R) onto L2 (I), translating the resulting function to the right by t and then projecting back onto L2 (I). It is not difﬁcult to verify that SI is a semigroup of partial isometries. This example will be discussed in greater detail below in Sect. 1.1. The results of [4] show that all semigroups of partial isometries are built up from this example. Indeed, the main theorem of [4] asserts that if S is a semigroup of partial isometries, then it can be written as the direct sum of a purely unitary semigroup, a purely isometric semigroup, a purely co-isometric semigroup and a semigroup which can be expressed as a direct integral of truncated shifts. Here by a purely unitary semigroup we mean a contraction semigroup consisting solely of unitary operators. As shown in Sect. 1.1 below, the semigroup of truncated shifts has the property that its two-sided derivatives of all powers exist on a dense domain of vectors, and that its two-sided derivative is a symmetric operator. It follows from the decomposition result of [4], described in the above paragraph, that any semigroup of partial isometries has this property. The goal of this paper is to show that one can weaken the assumption that V (t) ∈ S is a partial isometry for every t ∈ [0, ∞) to obtain a strictly larger class of contraction semigroups which still retain the property that their two-sided derivatives of all powers are densely deﬁned. Deﬁne V (−t) := V ∗ (t). Given a contraction semigroup S := {V (t)| t ∈ [0, ∞)}, suppose V (t) is a partial isometry for all t ∈ [0, t0 ), t0 > 0. We will say such a semigroup is partially isometric in a neighbourhood of zero, or partially isometric for small time. Under this weaker assumption it will be shown that there is a dense domain D ⊂ H such that for all φ ∈ D, the two-sided φ exists, and in fact that BD ⊂ D so that all derivative Bφ := limt→0 (V (t)−I) t powers of the symmetric operator B exist on D. This result can be seen as a generalization of Stone’s theorem for strongly continuous one-parameter unitary groups of operators, and will be proven by suitably adapting the classical proof of Stone’s theorem as presented in [12, VIII.4] and using some elementary semigroup theory. Sufﬁcient conditions for a self-adjoint operator D on a Hilbert space K to have a symmetric restriction to a dense linear manifold (a not necessarily closed linear subspace) of a closed proper subspace H ⊂ K will be obtained as a consequence of these investigations (see Theorem 3). This paper will proceed as follows. The example of the semigroup of truncated shifts is presented below in Sect. 1.1, Sect. 2 reviews some basic facts about contraction semigroups and symmetric operators, Sect. 3 develops some basic theory on the two-sided derivative of a semigroup of contractions, Sect. 4 shows that contraction semigroups which consist of partial isometries for small time have densely deﬁned two-sided derivatives of all powers, and in the ﬁnal section, Sect. 5 we present a large class of examples of these semigroups from the Nagy-Foias model theory for contraction semigroups on Hilbert space. These examples show that there are many contraction semigroups S which are partially isometric for small time but not for all time. A complete characterization of semigroups which are partially isometric for small time in the case where the co-generator of the semigroup has an inner Nagy-Foias characteristic function is provided by Corollary 3.

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1.1. A Motivating Example: The Semigroup of Truncated Shifts Let H := L2 (R) and let D and M be the essentially self-adjoint derivative and multiplication operators on the domain C0∞ (R) of inﬁnitely differentiable functions with compact support deﬁned by D φ = iφ and M φ(x) = xφ(x) for all φ ∈ C0∞ (R). Let D and M be the self-adjoint closures of D and M , respectively. It is not hard to check that the unitary group U (t) := eitD generates right translations, i.e. for any f ∈ L2 (R), U (t)f (x) = f (x − t)a.e. Now let P := χ[a,b] (M ) be the projector onto L2 [a, b]. Here χΩ denotes the characteristic function of the Borel set Ω. Let V : R → B(H) be the strongly continuous operator-valued map deﬁned by V (t) := P U (t)P . It is straightforward to visualize the action of V (t) for t ≥ 0 as translation to the right truncated to the interval [a, b]. It is further easy to verify that S[a,b] := {V (t)| t ∈ [0, ∞)} is a contraction semigroup and that each V (t) is a partial isometry. The semigroup S[a,b] is nilpotent, for each t > 0, V (t) is a nilpotent operator since V (t) = 0 for all t ≥ b − a. The semigroup S[a,b] is called a semigroup of truncated shifts on the interval I = [a, b]. Further observe that if f is an arbitrary element of C0∞ (a, b), then 1 V (t) − I V (t) − I 1 lim f= lim f = if = Df. i t→0+ t i t→0− t

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This shows that the strong, two-sided derivative of V (t)f exists for any f ∈ C0∞ (a, b) which is a dense linear manifold in L2 [a, b]. This also implies that the self-adjoint derivative operator D on L2 (R) has a restriction to a dense domain in L2 [a, b] for any interval [a, b]. It is clear that this restriction is a symmetric operator. A natural question raised by the above example is the following. If {V (t)| t ∈ [0, ∞)} is a strongly continuous one-parameter semigroup of contractions on a separable Hilbert space H such that each V (t) is a partial isometry for t ∈ [0, t0 ], t0 > 0, does this imply that (with the definition V (−t) := V ∗ (t)) there is a dense linear manifold D ⊂ H on which the two-sided derivative of V (t) exists? Moreover, let D and {U (t) = eitD | t ∈ R} denote a closed self-adjoint operator and the unitary group it generates. If S ⊂ H is a closed subspace with self-adjoint projector P , and {V (t) := P U (t)P | t ∈ [0, ∞)} is a semi-group consisting of partial isometries for t ∈ [0, t0 ], t0 > 0, does this imply that D has a symmetric restriction to a dense domain in S? The results of this paper provide aﬃrmative answers to both of these questions. Remark 1.1.1. Since M and D are unitarily equivalent by Fourier transform, and L2 [0, t0 ] = L2 [0, ∞) L2 [t0 , ∞) = L2 [0, ∞) eit0 D L2 [0, ∞), it follows that the compression of the unitary group {eitM |t ≥ 0} to the model subspace Ke2it0 z = H 2 (U) eit0 M H 2 (U) is unitarily equivalent by Fourier transform to the semigroup of truncated shifts on L2 [0, t0 ]. Here H 2 (U) denotes the Hardy space of scalar valued functions on the upper half plane U. In Sect. 5 we will construct a large class of semigroups which are partially isometric for small time by considering the compression of the unitary group of M to more general model subspaces.

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2. Notation and Preliminaries Before proceeding further, it will be convenient to establish some notation and recall some basic facts about semigroups of contractions, dilation theory and symmetric operators. Standard references include [10,12,14] for contraction semigroups and dilation theory and [1,12] for symmetric operators and their self-adjoint extensions. Let T be a contraction on H. If 1 ∈ / σp (T ) where the point spectrum σp (T ) is the set of eigenvalues of T , then the Hardy functional calculus can 1+z be used to deﬁne T (t) := exp(itμ−1 (T )), μ−1 (z) = i 1−z for each t ≥ 0. The functional calculus further implies {T (t)| t ∈ [0, ∞)} is a strongly continuous one-parameter contraction semigroup. Conversely, given any such one-parameter contraction semigroup, S = {T (t)| t ≥ 0}, the limit T := z−1+s always exists in the strong operator lims→0+ ϕs (T (s)) where ϕs (z) := z−1−s topology. This limit, T , is a contraction on H such that 1 ∈ / σp (T ), and such that T (t) = exp(itμ−1 (T )). The contraction semigroup S and T , which is called the co-generator of S, determine each other uniquely by these formulas. The closed and densely deﬁned operator A := μ−1 (T ) will be called the generator of S, so that V (t) = eitA and iA is equal to the two-sided derivative of V (t) at zero acting on a dense set of vectors. Given a contraction T on H, recall that a unitary operator U on K ⊃ H is called a unitary dilation of T if T k = PH U k |H for all k ∈ N ∪ {0}. The dilation U is called minimal if K is the closure of the linear span of U k H; k ∈ Z. Any contraction T has a minimal unitary dilation, and this minimal unitary dilation is unique up to a certain natural unitary equivalence [10, Theorem 4.3]. Similarly, any one-parameter contraction semigroup S = {T (t)| t ∈ [0, ∞)} has a unique minimal unitary dilation G = {U (t)| t ∈ R} where G is a strongly continuous one-parameter unitary group on some Hilbert space K ⊃ H obeying PH U (t)|H = V (t) for t ≥ 0. It is not difﬁcult to show that a unitary group G is a (minimal) unitary dilation of a contraction semigroup S if and only if the co-generator U of G is a (minimal) unitary dilation of T , the co-generator of S. A subspace S is called semi-invariant for a semigroup of operators S on H ⊃ S if the compression of S to S is again a semigroup. A subspace S is semi-invariant if and only if it can be written as S = S1 S2 where the Si are invariant for S and S1 ⊃ S2 [13]. In general, if G is a unitary group, then its compression SS := PS G|S to a subspace S ⊂ K is a semigroup (so that G is a unitary dilation of SS ) if and only if S is semi-invariant for G. Let H be a separable Hilbert space. Recall that a linear transformation B is called symmetric if Bφ, ψ = φ, Bψ for all φ, ψ ∈ Dom(B). Even if a symmetric linear transformation B is densely deﬁned, it is not necessarily selfadjoint. The deﬁciency subspaces of a closed symmetric linear transformation B are deﬁned as D± := Ran(B ± i)⊥ , and the deﬁciency indices (n+ , n− ) of B are deﬁned by n± := dim (D± ). We will generally call a symmetric linear transformation a linear operator if it is densely deﬁned. A closed symmetric linear operator B is self-adjoint if and only if its deﬁciency indices are both zero.

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Given a closed symmetric linear transformation B, the isometric linear transformation μ(B) = (B − i)(B + i)−1 , μ(z) = z−i z+i is called the Cayley transform of B and is an isometry from Ran(B + i) onto Ran(B − i). 1+z , the inverse of μ with respect to composiIf we deﬁne μ−1 (z) := i 1−z −1 tion, then B = μ (μ(B)). Note here that if V := μ(B), then μ−1 (V ) = z+i and its inverse with respect to (μ† )−1 (V ∗ ), where μ† (z) := μ(z) = z−i z+1 † −1 composition is (μ ) (z) = i z−1 . This can be computed from the fact that B = μ−1 (V ) is symmetric so that V = μ(B), V ∗ = μ† (B) and (μ† )−1 (V ∗ ) = (μ† )−1 (μ† (B)) = B. If A, B are two linear transformations we will use the notation A ⊃ B or B ⊂ A to denote the fact that Dom(B) ⊂ Dom(A) and A|Dom(B) = B. In this case B will be called a restriction of A, and A an extension of B.

3. A Symmetric Restriction Suppose {U (t) = eitD | t ∈ R} is the unitary group generated by a self-adjoint operator D acting on a dense domain of a separable Hilbert space K, that P is a projection, P K = H, and that {V (t) := P U (t)P | t ∈ [0, ∞)} is a contraction semi-group. Hence H is semi-invariant for the unitary group generated by φ D. Recall that the generator A of V (t) is deﬁned by iAφ = lims→0+ V (s)−I s on the domain of all φ for which this limit exists. The generator of the adjoint semigroup {V ∗ (t)| t ∈ [0, ∞)} is A = −A∗ and is calculated as iA φ = lims→0+ V (−s)−I φ = lims→0− I−Vs (s) φ = −iA∗ φ s on the set of all φ for which this limit exists. Note that these derivatives are one-sided. One can use V (t) to deﬁne a closed symmetric linear transformation as follows. Let V (s) − I φ ∈ H ⊂ Dom(A) ∩ Dom(A∗ ), (2) Dom(B) := φ ∈ H lim s→0 s φ for all φ ∈ Dom(B). That is, B is deﬁned and deﬁne Bφ := 1i lims→0 V (s)−I s as −i times the strong two-sided derivative of V (t) at 0. Then B is symmetric since Dom(B) is, by definition, the set of all φ ∈ Dom(A) ∩ Dom(A∗ ) for which Aφ = A∗ φ. Explicitly, if φ ∈ Dom(B) this means that V (s) − I φ = Bφ s V (s) − I V (−t) − I = −i lim φ = −i lim φ − + s −t s→0 t→0 V ∗ (t) − I = i lim φ = i(iA)∗ φ = A∗ φ. t t→0+ Hence for all φ, ψ ∈ Dom(B), Aφ = −i lim+ s→0

Bφ, ψ = Aφ, ψ = φ, A∗ ψ = φ, Bψ .

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In general, Dom(B) will not be dense and may even be {0}. The main result of Sect. 4 will be the proof of the fact that if {V (t)| t ∈ [0, ∞)} is

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a semi-group consisting of partial isometries for small time then B, the two sided derivative of V (t) at 0, is a densely deﬁned symmetric operator. Theorem 1. The linear transformation B is closed and is the restriction of the self-adjoint operator D to Dom(B) ⊂ H, B = D|Dom(B) . Proof. The fact that B is closed follows easily from the fact that A and A∗ are closed. By the definition of B, its graph is equal to Γ(B) = Γ(A) ∩ Γ(A∗ ) which is closed since the graphs Γ(A) and Γ(A∗ ) of A and A∗ are closed. Hence B is closed. Let T denote the co-generator of the semi-group {V (t) = P eitD P | t ≥ 0}. By definition, T = (A − i)(A + i)−1 = μ(A), where μ(z) = z−i z+i . Since B ⊂ A is symmetric, its Cayley transform μ(B) is an isometry from Ran(B + i) to Ran(B − i). Since {U (t)| t ∈ R} is a unitary dilation of {V (t)| t ≥ 0}, it further follows that μ(D) is a unitary dilation of the contraction μ(A). This can be deduced, for example, from the fact that a co-generator T acting on H and the semigroup V (t) it generates determine each other by the formulas −1 z−1+t and V (t) = eitμ (T ) and T = SOT − lims→0+ ϕt (V (t)) where ϕt (z) := z−1−t SOT−lim denotes the limit in the strong operator topology [14, Section III.8]. If G := {U (t)| t ∈ R} acting on K ⊃ H is the minimal unitary dilation of S := {V (t)| t ≥ 0}, if the co-generator of G is U , and if P is the projection of K onto H, then T 2 = SOT − lims→0+ SOT − limt→0+ ϕs (P U (s)P )ϕt (P U (t)P ). Here we are using the fact that multiplication on bounded subsets of L(H), the Banach space of bounded linear operators on H, is continuous in the strong operator topology. Since ϕs is analytic on D for any s > 0, the semigroup property of V (t) implies that this can be written as SOT − lims→0+ SOT − limt→0+ P ϕs (U (s))ϕt (U (t))P = P U 2 P . In conclusion T 2 = P U 2 P , and similarly one can show T k = P U k P so that U is a unitary dilation of T . Hence, P (D − i)(D + i)−1 |H = (A − i)(A + i)−1 and, in particular, P (D − i)(D + i)−1 |Ran(B+i) = (B − i)(B + i)−1 |Ran(B+i) . Here, P is the projector onto H ⊂ K. Since μ(D) = (D − i)(D + i)−1 is unitary and μ(B) is an isometry from Ran(B + i) to Ran(B − i), it follows that for any ψ ∈ Ran(B + i) of unit norm, that 1 = μ(B)ψ2 = P μ(D)ψ2 , while 1 = μ(D)ψ2 = P μ(D)ψ2 + (I − P )μ(D)ψ2 . This proves that P μ(D)|Ran(B+i) = μ(D)|Ran(B+i) = μ(B). Hence, Dom(B) = (I − μ(B))Ran(B + i) = (I − μ(D))Ran(B + i) = 2i(D + i)−1 Ran(B + i) ⊂ Dom(D). Furthermore, since D = i(I + μ(D))(I − μ(D))−1 , and B = i(I + μ(B))(I − μ(B))−1 , it follows that given any φ ∈ Dom(B), that φ = (I − μ(B))ψ = (I − μ(D))ψ where ψ ∈ Ran(B + i) and, Dφ = i(I + μ(D))ψ = i(I + μ(B))ψ = i(I + μ(B))(I − μ(B))−1 (I − μ(B))ψ = Bφ.

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Remark 3.0.2. Similarly one can show that if Dom(B k ) ⊂ Dom(B), k ∈ N is deﬁned as the set of all φ ∈ H on which the kth two-sided derivative of V (t)

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at zero exists, then B k with domain Dom(B k ) is a closed symmetric linear transformation and B k = Dk |Dom(B k ) . Proposition 1. The deficiency indices n± of B are equal to the dimensions of Dom(A)/Dom(B) and Dom(A∗ )/Dom(B), respectively. Proof. Since σ(A) ⊂ U it follows that Ran(A + i) = H. By definition n+ = dim(Ran(B + i)⊥ ). Since B is closed and B +i is bounded below, Ran(B + i) is a closed subspace. Since (A + i)−1 is a bijection from H onto Dom(A) and B ⊂ A it follows that (A + i)−1 Ran(B + i)⊥ is the linear manifold of vectors in Dom(A) which are linearly independent modulo Dom(B). This proves the claim for n+. The same logic using A∗ ⊃ B proves the analogous statement about n− . Recall that if T is a contraction on H, then one deﬁnes the defect operators DT := (1 − T ∗ T )1/2 , and DT ∗ = (1 − T T ∗ )1/2 , the defect spaces DT := Ran(DT ) and DT ∗ := Ran(DT ∗ ) and the defect indices of T as dT := dim (DT ) and dT ∗ := dim (DT ∗ ). The next theorem shows that if T is the co-generator of a contraction semigroup S, then the defect spaces of T are the deﬁciency subspaces of B. ⊥ is an isometry. Claim 1. T D⊥ T = DT ∗ and T |D⊥ T

Proof. By [14, I.3.1], T DT = DT ∗ T , so if φ ∈ D⊥ T = Ker(DT ) it follows ⊥ that DT ∗ T φ = T DT φ = 0. This shows that T D⊥ T ⊂ DT ∗ , and similarly ∗ ⊥ ⊥ ⊥ T DT ∗ ⊂ DT . Conversely if φ ∈ DT ∗ = Ker(DT ∗ ) = Ker((1 − T T ∗ )1/2 ) = ⊥ Ker(1 − T T ∗ ) then φ = T T ∗ φ, and since T ∗ φ ∈ D⊥ T it follows that DT ∗ = ⊥ is an isometry is obvious. T DT . That T |D⊥ T The two-sided derivative, B is the maximal symmetric linear transformation which is also a restriction of D. To see this note that if B were another symmetric restriction of D to a linear subspace Dom(B ) ⊂ H, then for any φ ∈ Dom(B ), the two-sided derivative of V (t)φ = P U (t)P φ at zero exists and is equal to B φ = Dφ. Hence by the definition of B, φ ∈ Dom(B) and Bφ = B φ so that B ⊂ B. Theorem 2. If T = μ(A) is the co-generator of a semigroup S = {V (t)| t ≥ 0} then the deficiency subspaces of B are D+ = DT and D− = DT ∗ . The sym which metric linear transformation B has no proper symmetric extension B obeys B = D|Dom(B) . Proof. Since μ(B) = T |Dom(μ(B)) and μ(B) is an isometry from its domain ⊥ onto its range, it is clear that D⊥ + = Dom(μ(B)) ⊂ DT and that Ran(μ(B)) = ∗ ⊥ ⊥ ⊥ and Dom(μ(B) ) ⊂ DT ∗ . To prove that DT = D+ , observe that V := T |D⊥ T are isometries from their domains onto their ranges, and that (V )∗ = T ∗ |D⊥ T∗ z+i . It follows that T = μ(A) and that T ∗ = μ† (A∗ ) where μ† (z) = μ(z) = z−i −1 B := μ (V ) is a symmetric restriction of A, and that B = (μ† )−1 ((V )∗ ) is also a symmetric restriction of A∗ . Note here that (μ† )−1 ((V )∗ ) = μ−1 (V ) as discussed in Sect. 2.

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Hence Dom(B ) ⊂ Dom(A) ∩ Dom(A∗ ) and B φ = Aφ = A∗ φ for all φ ∈ Dom(B ). By the definition of B, Dom(B ) ⊂ Dom(B) and B ⊂ B so ⊥ ⊥ ⊥ that D⊥ T = Dom(μ(B )) ⊂ Dom(μ(B)) = D+ and similarly DT ∗ ⊂ D− . As observed at the beginning of the proof the opposite inclusions hold and it can be concluded that D+ = DT and D− = DT ∗ . which agreed with D on its domain If B had a proper closed extension B Dom(B) ⊂ H, then μ(B) would be a proper closed isometric extension of Since T = μ(A) = PH μ(D)|H μ(B) which agrees with μ(D) on Dom(μ(B)). However, it is it follows easily that T |Dom(μ(B)) = μ(D)|Dom(μ(B)) = μ(B). ⊂ D⊥ which equals Dom(μ(B)) by the previous then clear that Dom(μ(B)) T = B. part of the proof. Hence B Remark 3.0.3. Although we shall not have further use for this fact, it is not (t) φ = V (t)Bφ difﬁcult to check that for any φ ∈ Dom(B), lim→0 V (t+)−V and that this is equal to AV (t)φ if t > 0 and to A∗ V (t)φ if t < 0.

4. One-Parameter Operator Semigroups Which Are Partially Isometric for Small Time In this section {V (t)| t ∈ [0, ∞)} denotes a one-parameter strongly continuous contraction semigroup which consists of partial isometries for t ∈ [0, t0 ], t0 > 0. As in the previous section, let V (−t) := V ∗ (t), and deﬁne Dom(B) as φ exists. The main result of the set of all vectors φ for which lims→0 V (s)−I s this section is the following: Proposition 2. If {V (t)| t ∈ [0, ∞)} is a one-parameter strongly continuous contraction semigroup on H and V (t) is a partial isometry for all t ∈ [0, t0 ], t0 > 0, then Dom(B), the domain of the two-sided symmetric derivative B of V (t) at 0, is dense in H. This will be proven using some basic semigroup theory and some wellknown and easily established facts about semigroups of partial isometries. Combined with Theorem 1 of the previous section, this will imply: Theorem 3. Let H ⊂ K be a closed subspace with projector P . Let D be the closed self-adjoint generator of a one-parameter strongly continuous unitary group {U (t)| t ∈ R} on K. If {V (t) = P U (t)P | t ∈ [0, ∞)} is a strongly continuous one-parameter semigroup which consists of partial isometries in some neighbourhood of 0, then D has a symmetric restriction B to a dense domain Dom(B) ⊂ H. 4.1. Basic Properties This section establishes some basic properties which will be needed in the proof of Proposition 2. Although the material of this section seems to be well-known and overlaps with [2, Section 2], we include it here for the sake of completeness.

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Fix t0 > 0 so that V (t) is a partial isometry for all t ∈ [0, t0 ]. Let P (t) := V (t)V ∗ (t) and Q(t) = V ∗ (t)V (t) for all t ∈ [0, t0 ] so that P (t) is the projection onto Ran(V (t)) = Ker(V ∗ (t))⊥ and Q(t) is the projection onto Ker(V (t))⊥ = Ran(V ∗ (t)). Lemma 1. If 0 ≤ s ≤ t ≤ t0 then Q(s) ≥ Q(t) and P (s) ≥ P (t). Proof. If t = s then the claim holds trivially. If s < t then Q(s)Q(t) = Q(s)V ∗ (t)V (t) = Q(s)V ∗ (s)V ∗ (t − s)V (t). ∗

∗

(6)

∗

Since Q(s) is the projector onto Ran(V (s)), Q(s)V (s) = V (s). It follows that Q(s)Q(t) = V ∗ (s)V ∗ (t − s)V (t) = Q(t).

(7)

Furthermore, since Q(s) projects onto Ran(V ∗ (s)) = Ker(V (s))⊥ , V (s)Q(s) = V (s), and, Q(t)Q(s) = V ∗ (t)V (t − s)V (s)Q(s) = V ∗ (t)V (t − s)V (s) = V ∗ (t)V (t) = Q(t). (8) This proves that Q(t) ≤ Q(s). The proof of the analogous result for P (t) is similar and omitted.

Lemma 2. For any 0 ≤ s ≤ t ≤ t0 , Q(t − s)V (s)Q(t) = V (s)Q(t) and P (t − s)V ∗ (s)P (t) = V ∗ (s)P (t). In other words, this claim says that if φ ∈ Ker(V (t))⊥ then V (s)φ ∈ Ker(V (t − s))⊥ and if φ ∈ Ker(V ∗ (t))⊥ then V ∗ (s)φ ∈ Ker(V ∗ (t − s))⊥ , whenever 0 ≤ s ≤ t ≤ t0 . Proof. V (t) is an isometry from Ker(V (t))⊥ onto Ran(V (t)). Choose any / Ker(V (t − s))⊥ then φ ∈ Ker(V (t))⊥ , then φ = V (t)φ. Now if V (s)φ ∈ φ > V (t − s)(V (s)φ) = V (t)φ, which is a contradiction. The proof of the other half of the lemma is directly analogous.

(9)

The projection valued functions P (t) and Q(t) are clearly continuous in the strong operator topology since multiplication on bounded subsets of L(H) (the Banach space of bounded linear operators on H) is continuous in the strong operator topology. Lemma 3. The projector R(t) onto Ran(Q(t)) ∩ Ran(P (t)) is strongly continuous for t ∈ [0, t20 ] and R(0) = I. Proof. Since V (0) = V ∗ (0) = I, it follows that P (0) = I = Q(0) and that R(0) = I. Now by von Neumann’s alternating projection theorem R(t) = SOT − lim (Q(t)P (t))n . n→∞

(10)

Here SOT − lim denotes the limit in the strong operator topology. If n = 2 and t ∈ [0, t0 /2] then

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(Q(t)P (t))2 = Q(t)P (t)Q(t)P (t) = V ∗ (t)V (2t)V ∗ (2t)V (2t)V ∗ (t) = V ∗ (t)P (2t)V (2t)V ∗ (t) = V ∗ (t)V (2t)V ∗ (t) = Q(t)P (t).

(11)

It follows that (Q(t)P (t)) = Q(t)P (t) for all n ∈ N and t ∈ [0, t0 /2]. The alternating projection theorem then implies that R(t) = Q(t)P (t) for all t ∈ [0, t0 /2]. The projection R(t) is clearly continuous in the strong operator topology for t in this range. n

Remark 4.1.1. It follows easily from Lemmas 1 and 3 that R(t) ≤ R(s) for 0 ≤ s ≤ t ≤ t20 . 4.2. A Densely Defined Two-Sided Derivative Consider the following set in H: D := {φ ∈ H | ∃φ ∈ (0, t0 /2] s.t. R(s)φ = φ ∀ 0 ≤ s ≤ φ }.

(12)

Given any φ ∈ H and > 0, since R(t) is strongly continuous on [0, t0 /2] and R(0) = I, there is a δ ∈ [0, t0 /2] such that 0 ≤ s ≤ δ implies that R(s)φ − φ < . Let ψ = R(δ)φ so that, by Remark 4.1.1, R(y)ψ = ψ for all 0 ≤ y ≤ δ. This shows that ψ ∈ D and that D is dense in H. Lemma 4. If φ ∈ D , 0 ≤ t ≤ φ /2, and 0 ≤ s ≤ φ /2, then Q(s)V (t)φ = V (t)φ and P (s)V ∗ (t)φ = V ∗ (t)φ. Proof. This follows from earlier results. First, by definition, φ = R(s)φ for all 0 ≤ s ≤ φ . By Lemma 2, it follows that Q(φ − t)V (t)φ = V (t)φ for all t ∈ [0, φ /2]. Since Lemma 1 implies that Q(s) ≥ Q(φ −t) ∀ s, t ∈ [0, φ /2], it follows that Q(s)V (t)φ = Q(s)Q(φ − t)V (t)φ = Q(φ − t)V (t)φ = V (t)φ.The remaining half of the proof uses similar reasoning. Now given any φ ∈ D , consider the element φ

∞ φf :=

2 f (t)V (t)φdt =

f (t)V (t)φdt,

(13)

− 2φ

−∞

where f is any function in C0∞ (− 2φ , 2φ ), the space of inﬁnitely differentiable functions with compact support in (−φ /2, φ /2) ⊂ (−t0 /2, t0 /2). Let D denote the linear manifold of all ﬁnite linear combinations of such elements.

Lemma 5. D is dense in H. The proof of this lemma is standard and uses a simple resolution of the identity. It is very similar to an argument often used in the proof of Stone’s theorem [11, VIII.8], and is omitted. All of the tools needed for the proof of Proposition 2 have ﬁnally been assembled. In the following proof of this proposition it will be shown that the set D ⊂ Dom(B) so that Dom(B) is indeed dense. Proof. (of Proposition 2) Let D and D be the sets of vectors constructed above. Given any φf ∈ D, assume s > 0, and consider

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∞ ∞ V (s) − I V (s + t) − V (t) φdt + f (t) f (−t)V ∗ (t)φdt s s 0

∞

=

f (t) −∞

V (s + t) − V (t) s

s +

f (−t) 0

0

φdt

V (s − t)P (t) − V ∗ (t) s

(14)

0 φdt −

f (t) −s

V (s + t) − V (t) φdt s (15)

∞ −s P (s)V ∗ (t − s) − V ∗ (t) V (s + t) − V (t) + f (−t) φdt − φdt. f (t) s s s

−∞

(16)

Consider the lines (a) = (14), (b) = (15), and (c) =(16) separately. First ∞ (a) = −∞

V (s + t) − V (t) φdt = f (t) s

∞ −∞

f (t − s) − f (t) s→0+ V (t)φdt −→ φ−f . s (17)

Notice that since f has support only on [−φ /2, φ /2], so does −f , so that φ−f ∈ D. Next, consider s s V (s − t)P (t) − V ∗ (t) V (s − t) − V ∗ (t) φdt − f (−t) φdt (b) = f (−t) s s 0

=

1 s

0

s f (−t)V (s − t) (P (t) − I) φdt.

(18)

0

Since φ ∈ D , as soon as s ≤ φ , P (t)φ = φ for all t ∈ [0, s] so that (b) vanishes in the limit as s → 0+ . Finally, ∞ ∞ P (s)V ∗ (t − s) − V ∗ (t) V ∗ (t − s) − V ∗ (t) φdt − f (−t) φdt (c) = f (−t) s s s

=

P (s) − I s

∞

s

f (−t)V ∗ (t − s)φdt

s

=

P (s) − I s

P (s) − I = s

∞

f (−y − s)V ∗ (y)φdy

0 φ /2

f (−y − s)V ∗ (y)dy.

0

(19)

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The last line above follows from the fact that f (t) has support only on [−φ /2, φ /2] so that f (−y −s) is non-zero only if −φ /2−s ≤ y ≤ φ /2−s ≤ φ /2. By Lemma 4, it follows that as soon as s ≤ φ /2, P (s)V ∗ (t)φ = V ∗ (t)φ for all t ∈ [0, φ /2] so that (c) = 0 as soon as s ≤ φ /2. This allows one to conclude that (V (s) − I) φf = φ−f (20) lim s s→0+ for all φf ∈ D. To show that the strong two-sided limit of (V (s)−I) as s → 0 exists on s D, it remains to show that V (s) − I φf = φ−f . s Let y > 0 and s = −y. Equation (21) becomes lim

s→0−

lim+

y→0

V (−y) − I I − V ∗ (y) φf = lim+ φf . −y y y→0

(21)

(22)

The proof that the above is equal to φ−f is a straightforward calculation which is directly analogous to the ﬁrst part of the proof. For the sake of brevity we omit this calculation. In conclusion any φf belongs to Dom(B) and since D is the linear span of such elements D ⊂ Dom(B). Combining Proposition 2 with Theorem 1 now yields Theorem 3. Recall that the symmetric operator B on Dom(B) is deﬁned by Bφ := φ for φ ∈ Dom(B). The proof of Proposition 2 above actu−i lims→0 V (s)−I s ally shows that BD ⊂ D. Such a linear manifold D is called an analytic domain for the operator B, since any power of B is deﬁned on D. Combining this fact with Remark 3.0.2 yields the following corollary. Corollary 1. Let D be a self-adjoint operator on a separable Hilbert space H. If S ⊂ H is semi-invariant for the unitary group G := {U (t) := eitD | t ∈ R} generated by D, and if the compression S := PS G|S of G to S, is partially isometric for small time, then the kth two-sided derivative, B k , of V (t) at zero is a densely defined closed symmetric operator, B k ⊂ Dk and D ⊂ Dom(B k ) for all k ∈ N.

5. A Class of Examples of Contraction Semigroups Which Are Partially Isometric for Small Time In this section, a large class of examples of one-parameter contraction semigroups which are partially isometric in a neighbourhood of 0 is presented. Note that if V (t) is a partial isometry for all t ∈ [0, t0 ), then it is not hard to verify that it is a partial isometry for all t ∈ [0, t0 ] (and in fact for all t ∈ [−t0 , t0 ]). Let Hn := ⊕nj=1 L2 (R) where n ∈ N ∪ {∞}. If n = ∞ then this is ∞ 2 an l direct sum, i.e. if f = (fj )j∈N ∈ H then f 2 = j=1 fj 2L2 (R) . As before Hn2 (U) ⊂ Hn will denote the Hardy space of Cn -valued functions on

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U. Recall that Hn2 (U) is deﬁned as the set of all Cn valued analytic functions (C∞ := l2 (N)) on U whose L2 norms on lines parallel to R in the upper half plane are uniformly bounded. Every element of Hn2 (U) can be extended to the closed upper half-plane and this extension deﬁnes a unique element of Hn . Let Θ be a non-constant inner function. That is, Θ is a non-constant L(Cn )-valued function which is analytic and bounded in U and unitary almost everywhere on R. Here L(Cn ) denotes the space of bounded linear operators on Cn . Note that the standard reference [14] calls Θ inner provided it is isometric a.e. on R, which differs from our definition in the case n = ∞. Our definition of inner is called inner from both sides in [14]. This more restrictive definition will be sufﬁcient for our purposes. We also use Θ to denote the operator of multiplication by Θ on Hn2 (U). The space of all L(Cn ) valued functions which are analytic and bounded in U will be denoted Hn∞ (U), n ∈ N ∪ {∞}. Let M denote the operator of multiplication by the independent variable on Hn . The operator μ(M )|Hn2 (U) , where μ(z) := z−i z+i is the image of the shift (multiplication by the independent variable z) on Hn2 (D) under the canonical isometry of Hn2 (D) onto Hn2 (U) [5, Chapter 8]. For this reason this operator 2 . will be called the shift on Hn2 (U). Let μ(M )θ denote its compression to KΘ 2 2 Claim 2. If w ∈ U and f ∈ KΘ is such that f (w) = 0, then (M −w)−1 f ∈ KΘ .

Proof. Clearly g(z) :=

f (z) z−w

∈ Hn2 (U). If h ∈ Hn2 (U) then

g, Θh = f, Θ(M − w)−1 h = 0, −1

since (M − w)

h∈

Hn2 (U)

Dom(VΘ∗ )

for any h ∈

Hn2 (U).

It follows that g ∈

(23) 2 KΘ .

2 KΘ |

f (i) = 0}, and deﬁne VΘ∗ := z+i where μ† (z) := μ(z) = z−i and

Now deﬁne := {f ∈ ∗ ∗ μ(M ) |Dom(VΘ∗ ) . Here μ(M ) = μ† (M ) z+1 (μ† )−1 (z) = i z−1 is the inverse of μ† (z) with respect to composition. The 2 above claim shows that Ran(VΘ∗ ) ⊂ KΘ . The operator VΘ∗ is an isometry from its domain onto its range with adjoint VΘ = μ(M )|Ran(VΘ∗ ) , Dom(VΘ ) = Ran(VΘ∗ ). Finally deﬁne M Θ := M |Dom(M Θ ) = (μ† )−1 (VΘ∗ )|Dom(M Θ ) where Dom(M Θ ) := (VΘ∗ − 1)Dom(VΘ∗ ). Proposition 3. M Θ is a closed simple symmetric linear transformation with deficiency indices (n, n). Recall that a symmetric operator is called simple if it has no self-adjoint restriction to a proper subspace of the Hilbert space. It will be convenient to establish a few additional facts before proceeding with the proof of Proposition 3. Given f ∈ Hn2 (U), deﬁne †f := f † by f † (z) = f (z). Observe that † : Hn2 (U) → Hn2 (L), where L denotes the open lower half plane, is an anti-linear, idempotent and onto isometry. Here, Hn2 (L) is the usual Hardy space of Cn -valued functions on L. Given an inner function Θ ∈ Hn∞ (U), note that its transpose ΘT is also inner. Let CΘ := † ◦ Θ∗ . 2 Claim 3. A function f belongs to KΘ if and only if both f and CΘ f belong 2 ∗ 2 to Hn (U). The map CΘ = † ◦ Θ = ΘT ◦ † is an anti-linear isometry of KΘ 2 onto KΘT .

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2 . Then f ∈ Hn2 (U) and given any h ∈ Proof. First suppose that f ∈ KΘ 2 ∗ Hn (U), 0 = f, Θh = Θ f , h . Hence Θ∗ f ∈ L2n (R) Hn2 (U) = Hn2 (L), so that † ◦ Θ∗ f ∈ Hn2 (U). Conversely if f ∈ Hn2 (U) and (Θ∗ f )† ∈ Hn2 (U), given any h ∈ Hn2 (U), since † ◦ († ◦ Θ∗ )f = Θ∗ f ∈ Hn2 (L), 0 = Θ∗ f , h = f, Θh . 2 . This proves that f ∈ KΘ T That CΘ = Θ ◦ † is an easy computation. For example, suppose n = 2. Given any f ∈ H22 (U ), view f as an 2−component column vector with entries fi , and Θ as an 2 × 2 matrix. Then, for x ∈ R,

† f1 (x) Θ11 (x) Θ21 (x) CΘ f (x) = † ◦ Θ∗ f (x) = f2 (x) Θ12 (x) Θ22 (x) Θ11 (x) Θ21 (x) f1 (x) = = ΘT (x)f † (x). Θ12 (x) Θ22 (x) f2 (x)

(24) To see that CΘ is an isometry, choose an arbitrary f ∈ Hn2 (U) and compute CΘ f , CΘ f = ΘT f † , ΘT f † = f † , f † = f, f . Finally to show 2 2 2 2 onto KΘ that CΘ maps KΘ T , it sufﬁces to show that CΘ KΘ ⊂ KΘT and that 2 CΘT CΘ |KΘ2 = PΘ , the projector onto KΘ . The rest then follows by symmetry. 2 , CΘ f , ΘT h = ΘT f † , ΘT h = Given any h ∈ Hn2 (U), and f ∈ KΘ † † 2 2 2 ⊂ KΘ f , h = 0, since f ∈ Hn (L). This proves that CΘ KΘ T . Finally, 2 T ∗ T † † † † for any f ∈ Hn (U), CΘT CΘ f = ((Θ ) Θ f ) = (f ) = f , so that CΘT CΘ |KΘ2 = PΘ . Anti-linearity is clear. Claim 4. The anti-linear isometry CΘ maps Ran(VΘ∗ ) onto Dom(VΘ∗T ). 2 2 Since, by Claim 3, CΘ is an isometry of KΘ onto KΘ T , the above claim ∗ ⊥ ∗ ⊥ implies that CΘ (Ran(VΘ )) = Dom(VΘT ) . z+i g(z), where both f and g are Proof. Given any f ∈ Ran(VΘ∗ ), f (z) = z−i 2 2 elements of KΘ . By Claim 3, CΘ f ∈ KΘT , and CΘ f (z) = ΘT (z)f † (z) = z−i T † z+i Θ (z)g (z). This shows that CΘ f vanishes at z = i, and so belongs to ∗ Dom(VΘT ) by definition. Conversely suppose that f ∈ Dom(VΘ∗T ). We claim that CΘT f ∈ Ran(VΘ∗ ) so that by the proof of the previous claim, CΘ CΘT f = f ∈ 2 ∗ 2 CΘ Ran(VΘ∗ ) and CΘ is onto. Now VΘ∗T f ∈ KΘ T and g := CΘT VΘT f ∈ KΘ z−i † by the previous claim. Since g(z) = z−i z+i Θ(z)f (z) = z+i CΘT f (z), and 2 CΘT f ∈ KΘ , it follows that g(i) = 0 so that g ∈ Dom(VΘ∗ ) and VΘ∗ g = CΘT f ∈ Ran(VΘ∗ ).

We are now ready to prove Proposition 3: Proof. (of Proposition 3) Since Dom(VΘ∗ ) is closed, it follows that VΘ∗ and hence that M Θ = i(VΘ∗ + 1)(VΘ∗ − 1)−1 is closed. Here (VΘ∗ − 1)−1 is deﬁned on Dom(M Θ ) = Ran(VΘ∗ − 1). An easy calculation which we omit veriﬁes that M Θ is symmetric. Further note that D+ = Dom(VΘ )⊥ and D− = Ran(VΘ )⊥ . The operator M Θ must be simple. If not MΘ would have a self-adjoint restric2 so that its Cayley transform VΘ and hence VΘ∗ tion to some subspace S ⊂ KΘ

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2 . Then for any ﬁxed f in S and would have a unitary restriction to S ⊂ KΘ z+i k ∗ k k ∈ N it would follow that (VΘ ) f ∈ S. However (VΘ∗ )k f (z) = ( z−i ) f (z). Since f is analytic in the upper half plane, it would follow that f has a zero of inﬁnite order at z = i, and hence must vanish identically. (j) i 1−Θ(z)Θ(i) ej for 1 ≤ j ≤ n where ej is the jth standard Let ki (z) := 2π z−i n (j) 2 2 belongs to KΘ and given any f ∈ KΘ ,f = basis vector in C . Each k n (j) (j) (j) (j) f e , f, k = f (i), inner products with the k evaluate the jth j i i j=1 (j)

component at the point z = i. It is clear that each ki , for 1 ≤ j ≤ n (j) belongs to D− . Moreover, the ki , 1 ≤ j ≤ n are linearly independent. To see this note that if this were not the case then there would be a non-zero v = 0 for all z ∈ U. In particular it would vector v ∈ Cn such that 1−Θ(z)Θ(i) z−i ∗ follow that Θ(i)v = Θ(x) v a.e. x ∈ R with respect to Lebesgue measure. Since we assume that Θ is non-constant this is not possible. We conclude (j) 2 that the ki , 1 ≤ j ≤ n are linearly independent so that n− ≥ n. If f ∈ KΘ (j) is orthogonal to all the ki then f vanishes identically at z = i. Hence f ∈ Dom(VΘ∗ ) = D⊥ − . It follows that n− = n. Using the same arguments as above, and Claim 4 yields n+ = dim

(D+ ) = dim Ran(VΘ∗ )⊥ = dim (CΘT Dom(VΘ∗T ))⊥ = dim Dom(VΘ∗T )⊥ = n, so that n = n+ = n− . This completes the proof. This symmetric linear transformation M Θ will not be densely deﬁned in general. Necessary and sufﬁcient conditions on Θ for M Θ to be densely deﬁned when n < ∞ can be found in [8, Theorem 5.1.4], see also [7] for the scalar (n = 1) case. The fact that M Θ is densely deﬁned in the cases of interest here, when Θ satisﬁes the conditions of Proposition 4, follows from Theorem 3 of this paper. Remark 5.0.1. The operator M Θ is in fact the maximal symmetric restriction 2 . To see this suppose that M were another of M to a linear subspace of KΘ 2 . The Cayley transform symmetric restriction of M to a linear subspace of KΘ μ(M ) : Ran(M + i) → Ran(M − i) would then be an isometric restriction of the unitary operator μ(M ) so that μ(M )∗ would be an isometric restricz+i tion of μ(M )∗ . It follows that μ(M )∗ acts as multiplication by μ† (z) = z−i 2 2 on its domain in KΘ . Since elements of KΘ are analytic in U it follows that 2 | f (i) = 0} = Dom(μ(MΘ )∗ ). Hence μ(M )∗ is a Dom(μ(M )∗ ) ⊂ {f ∈ KΘ ∗ restriction of μ(MΘ ) and M = (μ† )−1 (μ(M )∗ ) is a restriction of MΘ . Now let VΘ (t) := PΘ eitM |KΘ2 , t ∈ R, where PΘ denotes the projection 2 onto KΘ . When there is no possibility of confusion, we will simply write V (t) for VΘ (t). For t ≥ 0, {V (t)| t ∈ [0, ∞)} is the semigroup co-generated 2 . This follows easily from the by μ(M )Θ , the compression of the shift to KΘ fact that μ(M )Θ = PΘ μ(M )|KΘ2 , the fact that μ(M ) is a unitary dilation of 2 μ(M )Θ , the fact that KΘ is invariant for μ(M )∗ and hence semi-invariant for μ(M ), and the Hardy functional calculus [14]. Let Θ ∈ Hn∞ (U) be an inner function. Decompose Cn into subspaces Sj with the property that each Sj reduces Θ(t) almost everywhere t ∈ R, (and

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each Sj contains no proper reducing subspace with this property) so that we can write Θ(z) = ⊕kj=1 Θj (z) on Cn = ⊕kj=1 Sj , with k ≤ n. Recall here that we deﬁned C∞ := l2 (N). This further yields the decomposition Hn2 (U) = ⊕kj=1 H 2 (Sj ) where H 2 (Sj ) is the Hardy space of Sj valued functions analytic 2 2 2 in U, and KΘ = ⊕kj=1 KΘ with each KΘ ⊂ H 2 (Sj ) a co-invariant subspace j j for the shift on H 2 (Sj ). Each Θj belongs to H ∞ (Sj ) and is inner. The remainder of this section is devoted to establishing the following proposition. Two inner functions Θ1 , Θ2 are said to be relatively prime if there is no non-constant inner function Θ such that both Θ−1 Θ1 and Θ−1 Θ2 are inner [14, pg. 111], [9, Chapter 1]. Proposition 4. Let Θ be a non-constant inner function. The semigroup defined by V (t) := PΘ eitM |KΘ2 , t ∈ [0, ∞) consists of partial isometries for t ∈ [0, t0 ] if and only if Θ(z) = ⊕kj=1 Θj (z) = ⊕kj=1 eitj z Φj (z), tj ≥ 0 up to multiplication by a constant inner function. Here Φj is relatively prime to all inner functions of the form eisz 1Sj ; s ≥ 0 acting on Sj , and either: 1.

2.

itM |K2 Φj ≡ 1Sj . In this case Vj (t) := PΘ je

Θj

is a partial isome-

j = try for all t ∈ [0, ∞) and Vj (t) = 0 for t ∈ (tj , ∞). Here Θ diag(1S1 , . . . , 1Sj−1 , Θj , 1Sj+1 , . . . ), and each Sj is one-dimensional so that Θj (z) = eitj z . Or, Φj = 1Sj in which case tj ≥ t0 and there is no tj > tj for which Vj (t) is a partial isometry for all t ∈ [0, tj ]. In particular there is no t > inf tj for which V (t) is a partial isometry for all t ∈ [0, t ].

Any Θ satisfying the conditions of the above written in block form as ⎛ it1 z e Φ1 (z) 0 0 it2 z ⎜ Φ (z) 0 0 e 2 ⎜ .. Θ(z) = ⎜ ⎜ . 0 0 ⎝ .. .. . .

proposition can thus be 0 0 0

⎞ ··· ···⎟ ⎟ ⎟, ···⎟ ⎠

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with respect to a suitable basis. Note that in Case 1 of this proposition the semigroup S = {V (t)| t ∈ [0, ∞)} consists of partial isometries for all time, and can be decomposed as the direct sum of semigroups Sj with elements Vj (t) := Peitj z eitM |K 2it z , e

j

where Peitj z is the orthogonal projector of H 2 (U) ⊂ L2 (R) onto Ke2itj z = H 2 (U)eitj z H 2 (U). As discussed in Remark 1.1.1, each Sj is unitarily equivalent to the semigroup of truncated shifts S[0,tj ] on the interval [0, tj ] so that S is unitarily equivalent to the direct sum of truncated shifts. This is in agreement with the more general result of [4] which shows that any semigroup which is partially isometric for all time and has no isometric, co-isometric or unitary part is unitarily equivalent to a direct integral of truncated shifts. This proposition provides a large class of semigroups which are partial isometries for small time but not for all time. For example, if n = 1, then any inner function can be canonically factorized into the product of a singular

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inner function with a Blaschke product, Θ(z) = χB(z)S(z), χ ∈ T. Here the Blaschke product is ∞ 1 − z/zn , (26) 1 − z/zn n=1 1 ⊂ U and Im zn < ∞, and the singular inner function

B(z) :=

where (zn )n∈N is given by

⎛ eiσz exp ⎝i

∞

−∞

⎞ 1 + tz dν(t)⎠, t−z

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where σ ≥ 0, and ν is positive and singular with respect to Lebesgue measure on R [3, Theorem 11.6], [5, pgs. 63–70]. Any product of the above form deﬁnes an inner function in H ∞ (U) and any such inner function is completely and uniquely (up to a constant χ ∈ T) determined by its zeroes (zn ) ⊂ U, the number σ ≥ 0 and the singular measure ν. It is not difﬁcult to see that two such inner functions Θ, Θ are relatively prime provided that their sets of zeroes (zn ) and (zn ) do not intersect, that the positive constants σ, σ appearing in their exponential parts are not both non-zero, and provided that their singular measures ν, ν have no non-zero common minorant [14, pg. 110]. In particular, any non-constant scalar inner function Φ which has zero exponential part (i.e. σ = 0) is relatively prime to all inner functions of the from eiσz , σ > 0. By part (2) of the above proposition, since Φ is non-constant, the inner function Θ(z) = eiσz Φ(z) where σ > 0 is such that the semigroup deﬁned by V (t) = PΘ eitM |KΘ2 consists of partial isometries for t ∈ [0, σ] but not for t > σ. 2 2 = ⊕kj=1 KΘ , and V (t) consists of partial isometries for Proof. Since KΘ j t ∈ [0, t0 ] if and only if each Vj (t) := PΘj eitM |KΘ2 consists of partial isomj

etries for t ∈ [0, t0 ], we can assume without loss of generality that there is no non-trivial subspace of Cn which reduces Θ(t) almost everywhere with respect to Lebesgue measure for t ∈ R. To prove the sufﬁciency part of the proposition, since we can assume without loss of generality that Θ is an irre ducible block, we can assume that k = 1, Sk = Cn and Θ = eit z Φ. In case (1) it follows that Θ(z) = eit z is a scalar inner function. Here t ≥ 0 since Θ is inner. In this case, as discussed in Remark 1.1.1, {V (t)| t ∈ [0, ∞)} is unitarily equivalent to a nilpotent semigroup of truncated shifts. This semigroup consists of partial isometries for all t ≥ 0, and V (t) = 0 for t ≥ t . Next suppose as in the second case of the above proposition that Θ(z) = eitz Φ(z), where Φ = 1n . Let Θs (z) := eisz Φ(z) so that Θt = Θ. Now 2 2 for any 0 ≤ s ≤ t, KΘ ⊃ KΘ . This yields t−s

2 2 V (s) KΘ = PΘ eisz (H 2 ΘH 2 ) (H 2 Θt−s H 2 ) KΘ t−s

(28) = PΘ eisz Θt−s H 2 Θt H 2 = PΘ Θ(H 2 eisz H 2 ) = {0}. 2 . Recall that PΘ is the projector onto KΘ

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Also, eisz (H 2 Θt−s H 2 ) = eisz H 2 ΘH 2 2 = KΘ Ke2isz ,

(29)

so that V (s)|KΘ2 = e |KΘ2 which is an isometry. These calculations t−s t−s prove the sufﬁciency. We now establish the necessity. The least common inner multiple, Θ of two inner functions Θ1 and Θ2 is deﬁned as the inner function with the fol−1 lowing two properties. First Θ−1 1 Θ and Θ2 Θ are both inner functions, and if Θ is any other inner function with this ﬁrst property then Θ−1 Θ is inner [9]. We will use the following elementary facts: isz

Lemma 6. Θ1 H 2 ⊃ Θ2 H 2 if and only if Θ−1 1 Θ2 is inner. If Θ1 Θ2 −Θ2 Θ1 = 0, then Θ1 H 2 ∩Θ2 H 2 = ΘH 2 . Here Θ is the least common inner multiple of Θ1 and Θ2 . The proof of this lemma is standard, and is omitted. We will say an inner function Θ1 is (right) divisible by an inner function Θ2 , or that Θ2 divides Θ1 , if Θ−1 2 Θ1 is inner. Write Θ as Θ(z) = E(z)Φ(z) where E(z) = diag(eis1 z , eis2 z , . . .), si ≥ 0 and Φ is an inner function such that there is no non-trivial diagonal inner function E1 (z) = diag(eit1 z , eit2 z , . . .) of the same form as E(z) for which E1−1 Φ is inner. As discussed at the beginning of the proof, we can assume that there is no subspace of Cn (n = rank(Θ)) which reduces Θ and hence Φ for almost all t ∈ R.

2 Claim 5. Let s := inf i si and F (z) := e−is z Θ. Then V (t)(KΘ KF2 ) = {0} for all t ∈ [s , ∞).

The proof of this claim follows from the same calculation used above in equation (28). Claim 6. Let F be as in Claim 5. For any t ∈ [s , ∞), V (t)f = 0 for any non-zero f ∈ KF2 . Proof. Suppose to the contrary that there is a f ∈ KF2 for which V (t)f = 0, for some t ∈ [s , ∞). Then 0 = eitz f , q for all q ∈ KF2 so that if s := s + t then h(z) := eisz f (z) = Θ(z)g(z) for some g ∈ H 2 . Hence h ∈ eisz H 2 ∩ ΘH 2 = Θ H 2 , where Θ is the least common inner multiple of Θ and eisz . It is not difﬁcult to show that Θ (z) = ei(s−s )z Θ(z) = eisz F (z). Indeed, clearly both eisz and Θ divide Θ , and if Θ1 is another inner function which is divisible by both eisz and Θ, then Θ1 = eisz G, where G is inner, and G is divisible by Φ, so that (Θ )−1 Θ1 is inner. It follows that h = eisz f belongs to Θ H 2 so that eisz f (z) = isz e F (z)g(z) for some g ∈ H 2 . Hence f ∈ KF2 ∩ F H 2 = {0}. We are now in position to complete the proof of the proposition. To complete the proof it sufﬁces to show that if s < t0 , then F = 1. By Claim 6, V (t)f = 0 for any t ∈ [s , ∞) and f ∈ KF2 , while by Claim 5, V (s)g = 0 2 KF2 and s ∈ [s , ∞). Since V (t) is a partial isometry for each for all g ∈ KΘ t ∈ [s , t0 ], V (t)|KΘ2 KF2 = 0 and Ker(V (t)) ∩ KF2 = {0} for all t in this range,

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it follows that V (t)|KF2 is an isometry for all t ∈ [s , t0 ]. By the semigroup property V (t)|KF2 is an isometry for all t ∈ [s , ∞). Actually V (t)|KF2 is an isometry for all t ≥ 0, since if g ∈ KF2 is such that g = 1 but V (t)g = 1, then V (t)g < 1 as V (t) is a contraction, and for all k ∈ N, V (kt)g = V (t)k g < 1 which contradicts the fact that V (kt)|KF2 is an isometry for large enough k ∈ N. It follows that V (s)|KF2 = PF V (s)|KF2 = PF eisM |KF2 . Moreover since V (t) is an isometry for each t ≥ 0, it follows that V (t) = eitM |KF2 , and that KF2 is both invariant and co-invariant for the semigroup {eitM | t ∈ [0, ∞)}, and hence reduces the shift μ(M )|Hn2 (U) . Standard arguments using the Wold decomposition for μ(M )|Hn2 (U) , and the fact that F is unitary almost everywhere x ∈ R shows that this can only happen if F = 1 (up to a constant inner function). In conclusion it has been shown that either Θ(z) = eitz for some t ≥ 0 or Θ(z) = eis z G(z) for some inner G = 1 which is relatively prime to all eitz 1 and s ≥ t0 . Moreover, the same arguments as above show that if G = 1 then V (t) cannot consist of partial isometries for all t ∈ [0, s] if s > s . This completes the proof of the proposition. 5.1. Further Consequences and Results Theorem 3 combined with the earlier results of this section, implies that given 2 for which Θ(z) = diag(eit1 z F1 (z), eit2 z F2 (z), . . .) any model subspace KΘ where each Fi is relatively prime to eitz for all t ≥ 0, and either ti > 0 or Fi = 1 is such that the symmetric multiplication operator M Θ is a densely 2 . In fact Corollary 1 shows that all powers of M Θ are deﬁned operator in KΘ 2 . densely deﬁned symmetric operators in KΘ A semigroup {V (t)| t ∈ [0, ∞)} is said to be completely non-unitary (c.n.u.) if there is no orthogonal decomposition of H as S ⊕ S ⊥ for which V (t) decomposes as V (t) = U (t) ⊕ V1 (t) with U (t) unitary for all t ≥ 0. A semigroup is c.n.u. if and only if its co-generator T is c.n.u. where a c.n.u. contraction is deﬁned analogously in the obvious way. In the particular case where the semigroup {V (t)| t ≥ 0} is c.n.u. and consists solely of isometries, the following assertion can be made. Corollary 2. If {V (t)| t ∈ [0, ∞)} is c.n.u and isometric [co-isometric] for all t ≥ 0 then B, the symmetric two-sided derivative of V (t) at 0 has deficiency indices (0, m) [(n, 0)], where m ∈ N ∪ {∞}. In this case B is unitarily d d m 2 n n 2 equivalent to ⊕m j=1 i dx on ⊕j=1 L [0, ∞) [⊕j=1 i dx on ⊕j=1 L (−∞, 0]]. This indicated unitary transformation takes {V (t)| t ∈ [0, ∞)} onto the direct sum of m copies of the semigroup of right translations on L2 [0, ∞) [n copies of the semigroup of right translations on L2 (−∞, 0]], and B = A, the generator of {V (t)| t ∈ [0, ∞)}. The bulk of the assertions made in the statement of the corollary above are established using different methods in [14, III.9]. The remaining assertions can be pieced together from [1, Sect. 82], and Sect. 3 of this paper. For this reason we omit the proof of this corollary. For a proof of the above

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corollary using the methods of this paper, we direct the interested reader to [6, Chapter 12.6]. The following corollary provides a complete characterization of oneparameter contraction semigroups which are partially isometric for small time, and whose co-generators are c.n.u. and have an inner Nagy-Foias characteristic function. Corollary 3. Let {V (t)| t ≥ 0} be a semigroup of contractions which is partially isometric for t ∈ [0, t0 ), t0 > 0, and let T be its co-generator. Suppose that T is c.n.u. and that the characteristic function ΘT coincides with an inner function. Then rank(ΘT ) = n where n = n+ = n− and n± are the deficiency indices of B. Furthermore, T is unitarily equivalent to the com2 ⊂ Hn2 (U) where Θ coincides with ΘT ◦ μ, pressed shift μ(M )Θ acting on KΘ and satisfies Proposition 4. Recall that the Nagy-Foias characteristic function of a contraction T is deﬁned as ΘT (z) = (−T + zDT ∗ (1 − zT ∗ )−1 DT )|DT ,

(30)

and is a contractive analytic function for z ∈ D, the unit disc, with domain DT and range DT ∗ . Any two contractive analytic functions Θ1 and Θ2 , both deﬁned on D or on U, and whose domains and ranges are separable Hilbert spaces are said to coincide provided that there are ﬁxed constant unitaries U1 , U2 such that U1 Θ1 U2 = Θ2 . As in the case of the upper half-plane, a L(Cn ) valued contractive analytic function Θ on D is called inner if it is unitary almost everywhere on the unit circle T, i.e. if and only if Θ ◦ μ is an inner function on U. By [14, Theorem VI.4.1], for a c.n.u. contraction T, ΘT coincides with an inner function if and only if σ(T ) ∩ T has zero Lebesgue measure. Proof. Let Θ be an inner function on D which coincides with ΘT . Since Θ is inner, it follows that dim (DT ) = dim (DT ∗ ). By Theorem 2, we have that n+ = dim (DT ) = dim (DT ∗ ) = n− . It follows from the model theory for contractions that T is unitarily equivalent to the compression of multiplication by z on Hn2 (D) to the coinvariant subspace Hn2 (D)Θ Hn2 (D) (see [14, Theorem VI.2.3]). Applying the canonical isometry from Hn2 (D) onto Hn2 (U) shows that T is unitarily equiv2 where Θ = Θ ◦ μ, μ(z) = alent to μ(M )Θ , the compression of the shift to KΘ z−i z+i . Since the semigroup {VΘ (t)| t ∈ [0, ∞)} co-generated by μ(M )Θ is partially isometric in a neighbourhood of 0, Θ must satisfy Proposition 4. The ﬁnal result below characterizes one-parameter semigroups of partial isometries which are quasi-nilpotent. Recall that an operator T is said to be nilpotent if T k = 0 for some k ∈ N, and quasi-nilpotent if σ(T ) = {0}. Corollary 4. If {V (t)| t ∈ [0, ∞)} is a quasi-nilpotent semigroup of partial isometries, then it is unitarily equivalent to the semigroup obtained as the 2 compression of eitM to KΘ where Θ(z) = diag(eit1 z , eit2 z , . . .), and ti ≥ 0. If supi ti = t∞ < ∞ then each V (t) is nilpotent and V (t) = 0 for all t > t∞ .

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In other words any such semigroup is a direct sum of one-dimensional truncated shifts. For an interesting decomposition of arbitrary one-parameter semigroups of partial isometries into a direct integral of truncated shifts see [4]. Proof. Since σ(V (t)) = {0} for all t > 0, it follows from the spectral mapping theorem that if T is the co-generator of {V (t)|t ≥ 0}, that σ(T ) = {1}. Hence σ(T ) ∩ T has zero Lebesgue measure. The assumptions of the previous corollary are satisﬁed and V (t) is unitarily equivalent to the compression of eitM to 2 where Θ ∈ Hn∞ (U) is inner, and rank(Θ) = n = n± . However, since some KΘ V (t) is a partial isometry for every t ≥ 0, it follows from Proposition 4, that Θ has the form Θ(z) = diag(eit1 z , eit2 z , . . .), and the remaining assertions also follow from the same proposition.

6. Outlook Corollary 3 has provided a large class of semigroups which are partially isometric for small time. Namely the semigroup generated by the compression of 2 where Θ(z) = diag(eit1 z F1 (z), eit2 z F2 (z), . . .) and each Fi (z) the shift to KΘ is relatively prime to every eitz 1; t ≥ 0 consists of partial isometries for all t ∈ [0, inf j tj ]. It seems possible that if the Fj in the above block-diagonal form of Θ were allowed to be more general contractive analytic functions than inner functions, that the compression of the shift to the de Branges-Rovnyak space 2 deﬁned using the contractive analytic function Θ may still generate a KΘ semigroup which will be partially isometric for small time. It would be interesting to investigate this and to see whether this could lead to a more complete characterization of this class of contraction semigroups.

References [1] Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space, Two Volumes Bound as One. Dover Publications, New York (1993) [2] Bracci, L., Picasso, L.E.: Representations of semigroups of partial isometries. Bull. Lond. Math. Soc. 30, 792–802 (2007) [3] Duren, W.: Theory of Hp Spaces. Academic Press, New York (1970) [4] Embry, M.R., Lambert, A.L., Wallen, L.J.: A simpliﬁed treatment of the structure of semigroups of partial isometries. Mich. Math. J. 22, 175–179 (1975) [5] Hoﬀman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliﬀs (1962) [6] Martin, R.T.W.: Bandlimited functions, curved manifolds and self-adjoint extensions of symmetric operators (Ph.D. thesis). University of Waterloo, Waterloo (2008). http://hdl.handle.net/10012/3698 [7] Martin, R.T.W.: Representation of symmetric operators with deﬁciency indices (1, 1) in de Branges space. Op. Th. Comp. Anal. (2009)

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[8] Martin, R.T.W. (2010) Unitary perturbations of compressed n-dimensional shifts. Op. Th. Comp. Anal. (submitted) [9] Nikolskii, N.K.: Treatise on the Shift Operator. Springer, New York (1986) [10] Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, New York (2002) [11] Reed, M., Simon, B.: Methods of Modern Mathematical Physics v.1: Functional Analysis. Academic Press, New York (1972) [12] Reed, M., Simon, B.: Methods of Modern Mathematical Physics v.2: Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1999) [13] Sarason, D.: On spectral sets having connnected complement. Acta Sci. Math. 26, 289–299 (1965) [14] Sz.-Nagy, B., Foia¸s, C.: hARMONIC Analysis of Operators on Hilbert Space. American Elsevier Publishing Company, New York (1970) R. T. W. Martin (B) Department of Mathematics and Applied Mathematics University of Cape Town Cape Town, South Africa e-mail: [email protected] Received: June 17, 2010. Revised: March 14, 2011.

Integr. Equ. Oper. Theory 70 (2011), 227–263 DOI 10.1007/s00020-010-1844-1 Published online November 25, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces Joseph A. Ball, Vladimir Bolotnikov and Sanne ter Horst Abstract. Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83–96, Naukova Dumka, Kiev, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges–Rovnyak space H(KS ) associated with an operator-valued Schur class function S. A description of all solutions is obtained in terms of functions from an associated de Branges–Rovnyak space satisfying only a bound on the de Branges–Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609–618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in H(KS ). In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator. Mathematics Subject Classification (2010). 46E22, 47A57, 30E05. Keywords. de Branges–Rovnyak space, abstract interpolation problem, boundary interpolation, operator-argument interpolation, Redheffer transformations, Toeplitz kernels.

1. Introduction De Branges–Rovnyak spaces play a prominent role in Hilbert space approaches to H ∞ -interpolation. However, very little work exists on interpolation for functions in de Branges–Rovnyak spaces themselves. In this paper we pursue our studies of interpolation problems for functions in de Branges–Rovnyak spaces, which started in [4]. We consider a norm constrained interpolation problem (denoted by AIPH(KS ) in what follows), which is sufﬁciently ﬁne so V. Bolotnikov was supported by National Science Foundation Grant DMS 0901124.

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as to include on the one hand interpolation problems with operator argument (considered for the nondegenerate and scalar-valued case in [4]) and, on the other hand, boundary interpolation problems; it is only recent work [8–10,20] which has led to a systematic understanding of boundary-point evaluation on de Branges–Rovnyak spaces from an operator-theoretic point of view. In order to state the interpolation problem we ﬁrst introduce some definitions and notations. As usual, for Hilbert spaces U and Y the symbol L(U, Y) stands for the space of bounded linear operators mapping U into Y, abbreviated to L(U) in case U = Y. Following the standard terminology, we deﬁne the operator-valued Schur class S(U, Y) to be the class of analytic functions S on the open unit disk D whose values S(z) are contraction operators in L(U, Y), with again the abbreviation S(U) in place of S(U, U). By HU2 we denote the standard Hardy space of analytic U-valued functions on D with square-summable sequence of Taylor coefﬁcients. We also make use of the notation HolU (D) for the space of all U-valued holomorphic functions on the unit disk D. Among several alternative characterizations of the Schur class there is one in terms of positive kernels and associated reproducing kernel Hilbert spaces: A function S : D → L(U, Y) is in the Schur class S(U, Y) if and only if the associated de Branges–Rovnyak kernel KS (z, ζ) =

IY − S(z)S(ζ)∗ 1 − z ζ¯

(1.1)

is positive (precise definitions are recalled at the end of this Introduction). This positive kernel gives rise to a reproducing kernel Hilbert space H(KS ), the de Branges–Rovnyak space deﬁned by S (see [13]). On the other hand, the kernel (1.1) being positive is equivalent to the operator MS : f → Sf of multiplication by S being a contraction in L(HU2 , HY2 ); then the general complementation theory applied to the contractive operator MS : HU2 → HY2 provides the characterization of H(KS ) as the operator range H(KS ) = 1 Ran(I − MS MS∗ ) 2 ⊂ HY2 with the lifted norm 1

(I − MS MS∗ ) 2 f H(KS ) = (I − p)f HY2 1

where p here is the orthogonal projection onto Ker(I − MS MS∗ ) 2 . Upon set1 ting f = (I − MS MS∗ ) 2 h in the last formula we get (I − MS MS∗ )hH(KS ) = (I − MS MS∗ )h, hHY2 .

(1.2)

The data set of the problem AIPH(KS ) is a tuple D = {S, T, E, N, x}

(1.3)

consisting of a Schur-class function S ∈ S(U, Y), Hilbert space operators T ∈ L(X ), E ∈ L(X , Y), N ∈ L(X , U), and a vector x ∈ X . With this data set we associate the observability operators OE,T : x → E(I − zT )−1 x and ON,T : x → N (I − zT )−1 x,

(1.4)

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which we assume map X into HolY (D) and HolU (D). We also associate with the data set D the L(X , Y)-valued function F S (z) = (E − S(z)N )(I − zT )−1

(1.5)

along with the multiplication operator MF S : x → F S x, mapping X into HolY (D). Using the notation (1.4), we can write MF S as MF S = OE,T − MS ON,T : X → HolY (D).

(1.6)

HY2 ,

Observe that, for an operator A : X → H(KS ) ⊂ the adjoint operator can be taken in the metric of HY2 as well as in the metric of H(KS ) which are not the same unless S is inner (i.e., the multiplication operator MS : HU2 → HY2 is an isometry). To avoid confusion, in what follows we use the notation A∗ for the adjoint of A in the metric of HY2 and A[∗] for the adjoint of A in the metric of H(KS ). Definition 1.1. We say that the data set (1.3) is AIPH(KS ) -admissible if: 1. The operators OE,T and ON,T map X into HolY (D) and HolU (D), respectively (in other words, (E, T ) and (N, T ) are analytic output pairs). 2. The operator MF S maps X into H(KS ). [∗] 3. The operator P := MF S MF S satisﬁes the Stein equation P − T ∗ P T = E ∗ E − N ∗ N.

(1.7)

We are now ready to formulate the problem AIPH(KS ) : Given an AIPH(KS ) -admissible data set (1.3), find all f ∈ H(KS ) such that [∗]

MF S f = x

and

f H(KS ) ≤ 1.

(1.8)

The AIPH(KS ) -problem as formulated here does not appear to be an interpolation problem, but in Sect. 6 we show that indeed the operatorargument Nevanlinna–Pick interpolation problem can be seen as a particular instance of the AIPH(KS ) -problem. This operator-argument problem was considered in [4] for scalar-valued functions and for the nondegenerate case where the solution P of the Stein equation (1.7) is positive definite (i.e., invertible). The eventual parametrization for the set of all solutions, which we obtain in Theorem 5.1 below, is connected with previously appearing representations for almost invariant subspaces and Toeplitz kernels in terms of an isometric multiplier between two de Branges–Rovnyak spaces. As another application of the AIPH(KS ) problem, we obtain an alternative characterization of Toeplitz kernels (in Corollary 7.5 below) in terms of an explicitly computable isometric multiplier on an appropriate de Branges–Rovnyak space; this is a reﬁnement of the characterization due to Dyakonov [17]. At one level the interpolation problem AIPH(KS ) is straightforward since de Branges–Rovnyak spaces are Hilbert spaces and consequently the set of all norm-constrained solutions splits as the orthogonal direct sum of the

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unique minimal-norm solution and the set of all functions satisfying the homogeneous interpolation condition and the complementary norm constraint. By viewing (1.8) as a special case of a basic linear operator equation discussed in Sect. 2, we get some general results on the AIPH(KS ) -problem in Sect. 3. These results make no use of condition (3) (i.e., the Stein equation (1.7)) in the definition of AIPH(KS ) -admissibility, and can be easily extended to a more general framework of contractive multipliers between any two reproducing kernel Hilbert spaces (not necessarily of de Branges–Rovnyak type). By using the full strength of AIPH(KS ) -admissibility, in Sect. 5 we obtain a more explicit formula (see Theorem 5.1 below) for the parametrization of the solution set by using the connection with an associated Schur-class Abstract Interpolation Problem and its known Redheffer transform solution as worked out in [24]. The latter problem and its solution through the associated Redheffer transform is recalled in Sect. 4. This section also includes an analysis of the conditions under which the Redheffer transform is injective, a property which does not happen in general. The paper concludes with three sections that discuss the various applications of the AIPH(KS ) -problem mentioned above. The notation is mostly standard. We just mention that an operator X ∈ L(Y), for some Hilbert space Y, is called positive semidefinite in case Xy, y ≥ 0 for all y ∈ Y and positive definite if X is positive semidefinite and invertible in L(X ). Also, in general, given a function K deﬁned on a Cartesian product set Ω × Ω with values in L(Y), we say that K is a positive kernel if any one of the following equivalent conditions hold: 1. K is a positive kernel in the sense of Aronszajn: given any ﬁnite collection of points ω1 , . . . , ωN in Ω and vectors y1 , . . . , yN in the Hilbert coefﬁcient space Y, it holds that N

K(ωi , ωj )yj , yi Y ≥ 0.

i,j=1

2. K is the reproducing kernel for a reproducing kernel Hilbert space H: there is a Hilbert space H(K) whose elements are Y-valued functions on Ω so that (i) for each ω ∈ Ω and y ∈ Y the Y-valued function kω y given by kω y(ω ) = K(ω , ω)y is an element of H(K), and (ii) the functions kω y have the reproducing property for H(K): f, kω yH(K) = f (ω), yY for all f ∈ H(K). 3. K has a Kolmogorov decomposition: there is an auxiliary Hilbert space K and a function H : Ω → L(K, Y) so that K can be expressed as K(ω , ω) = H(ω )H(ω)∗ . These equivalences are well-known straightforward extensions of the ideas of Aronszajn [2] to the case of operator-valued kernels in place of scalar-valued kernels. Next we mention that on occasion we view a vector x in a Hilbert space X as an operator from the scalars C into X : x maps the scalar c ∈ C to the

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vector cx ∈ X . Then x∗ denotes the adjoint operator mapping X back to C: x∗ (y) = y, x ∈ C. We will use the notation x∗ for this operator rather than the more cumbersome ·, x. Finally we note that a crucial tool for many of the results of this paper is the manipulation of 2× 2 block matrices centeringaroundthe so-called Schur A B with A invertible, complement. Given any 2 × 2 block matrix M = C D we deﬁne the Schur complement of D (with respect to M ) to be the matrix SM (D) := D − CA−1 B. In case D is invertible, we deﬁne the Schur complement of A (with respect to M ) to be the matrix SM (A) := A − BD−1 C. Our main application is to the case where M = M ∗ is self-adjoint (so A = A∗ , D = D∗ and C = B ∗ ). Assuming A is invertible, we may factor A as A = |A|1/2 J|A|1/2 where J := sign(A) and the factorization 0 J 0 A B |A|1/2 |A|1/2 J|A|−1/2 B = B∗ D 0 I B ∗ |A|−1/2 J I 0 D − B ∗ A−1 B shows that M ≥ 0 (i.e., M is positive-semidefinite) if and only if A ≥ 0 (so J = I) and the Schur complement D − B ∗ A−1 B of D is positive semidefinite. Similarly, in case D is invertible, we see that M ≥ 0 if and only if D ≥ 0 and the Schur complement of A, namely, A − BD−1 B ∗ , is positive semidefinite. In fact, these results go through without the invertibility assumption on A or D, using Moore–Penrose inverses instead.

2. Linear Operator Equations The problem AIPH(KS ) is a particular case of the following well-known norm constrained operator problem: Given A ∈ L(H2 , H3 ) and B ∈ L(H1 , H3 ), with H1 , H2 and H3 given Hilbert spaces, describe the operators X ∈ L (H1 , H2 ) that satisfy AX = B

and

X ≤ 1.

(2.1)

The solvability criterion is known as the Douglas factorization lemma [16]. Lemma 2.1. There exists an X ∈ L(H1 , H2 ) satisfying (2.1) if and only if AA∗ ≥ BB ∗ . In this case, there exists a unique X ∈ L(H1 , H2 ) satisfying (2.1) and the additional constraints RanX ⊂ RanA∗ and KerX = KerB. In case AA∗ ≥ BB ∗ , Lemma 2.1 guarantees the existence of (unique) contractions X1 ∈ L(H1 , RanA) and X2 ∈ L(H2 , RanA) so that 1

(AA∗ ) 2 X1 = B,

1

(AA∗ ) 2 X2 = A,

KerX1 = KerB,

KerX2 = KerA. (2.2)

By construction, X2 is a coisometry. The next lemma gives a description of the operators X ∈ L(H1 , H2 ) satisfying (2.1) in terms of the operators X1 and X2 .

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Lemma 2.2. Assume AA∗ ≥ BB ∗ and let X ∈ L(H1 , H2 ). Then the following statements are equivalent: 1. X satisfies conditions (2.1). 2. The operator ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ B∗ X∗ IH1 H1 H1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ AA A ⎦ : ⎣H3 ⎦ → ⎣H3 ⎦ ⎣B ∗ IH2 H2 H2 X A

(2.3)

is positive semidefinite. 3. X is of the form 1

1

X = X2∗ X1 + (I − X2∗ X2 ) 2 K(I − X1∗ X1 ) 2

(2.4)

where X1 and X2 are defined as in (2.2) and where the parameter K is an arbitrary contraction from Ran(I − X1∗ X1 ) into Ran(I − X2∗ X2 ). Moreover, if X satisfies (2.1), then X is unique if and only if X1 is isometric on H1 or X2 is isometric on H2 . Proof. Note that positivity of the block-matrix in (2.3) is equivalent to positivity of the Schur complement of IH2 in (2.3), namely I − X ∗ X B ∗ − X ∗ A∗ I B∗ X∗ ∗ X A = − ≥ 0. (2.5) B AA∗ A B − AX 0 Because of the zero in the (2, 2)-entry of the left hand side of (2.5), we ﬁnd that the inequality (2.5) holds precisely when B − AX = 0

and I − X ∗ X ≥ 0,

which is equivalent to (2.1). On the other hand, condition (2.3) is equivalent, by taking the Schur complement of AA∗ in (2.3) and making use of (2.2), to I − X1∗ X1 X ∗ − X1∗ X2 I X∗ X1∗ X1 X2 = ≥ 0. − X I X2∗ X − X2∗ X1 I − X2∗ X2 By Theorem XVI.1.1 from [18], the latter inequality is equivalent to the representation (2.4) for X with K some contraction in L(Ran(I − X1∗ X1 ), Ran(I − X2∗ X2 )). Moreover, X and K in (2.4) determine each other uniquely. The last statement in the lemma now follows from representation (2.4). 1

Note that since X2 is a coisometry, it follows that (I − X2∗ X2 ) 2 is the orthogonal projection onto H1 KerA = H1 KerX1 . This implies that for each K in (2.4) and each h ∈ H1 , we have 1

1

Xh2 = X2∗ X1 h2 + (I − X2∗ X2 ) 2 K(I − X1∗ X1 ) 2 h2 , so that X2∗ X1 is the minimal norm solution to the problem (2.1).

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3. The AIPH(KS ) -Problem as a Linear Operator Equation In this section we consider data sets D = {S, T, E, N, x} satisfying conditions (1) and (2) in the definition of an AIPH(KS ) -admissible data set but not necessarily condition (3); condition (3) of an AIPH(KS ) -admissible data set (i.e., the Stein equation (1.7)) comes to the fore for the derivation of the more explicit results to be presented in Sect. 5. We still speak of the AIPH(KS ) -problem for this looser notion of admissible data set. Deﬁne F S as in (1.5). If we apply Lemma 2.1 to the case where [∗]

A = MF S : H(KS ) → X ,

B=x∈X ∼ = L(C, X ),

(3.1)

then we see that solutions X : C → H(KS ) to problem (2.1) necessarily have the form of a multiplication operator Mf for some function f ∈ H(KS ). This observation leads to the following solvability criterion. Theorem 3.1. The problem AIPH(KS ) has a solution if and only if P ≥ xx∗ ,

where

[∗]

P := MF S MF S .

(3.2)

Remark 3.2. Observe that for the unconstrained version of the problem AIPH(KS ) , the existence criterion follows immediately from the definition [∗]

(3.2) of P : there is a function f ∈ H(KS ) such that MF S f = x if and only if 1 x ∈ RanP 2 . The next theorem characterizes solutions to the problem AIPH(KS ) in terms of a positive kernel. We emphasize that characterizations of this type go back to the Potapov’s method of fundamental matrix inequalities [29]. Here the notation x∗ associated with a vector x ∈ X follows the conventions explained at the end of the Introduction. Theorem 3.3. A function f : D → Y is a solution of the problem AIPH(KS ) with data set (1.3) if and only if the kernel ⎤ ⎡ 1 x∗ f (ζ)∗ ⎥ ⎢ P F S (ζ)∗ ⎦ K(z, ζ) = ⎣ x (z, ζ ∈ D), (3.3) f (z)

F S (z)

KS (z, ζ)

is positive on D × D. Here P , F S and KS are given by (3.2), (1.5) and (1.1), respectively. Proof. By Lemma 2.2 specialized to A and B as in (3.1) and X = Mf , we conclude that f is a solution to the problem AIPH(KS ) (that is, it meets conditions (1.8)) if and only if the following operator is positive semidefinite: ⎡ ⎤ ⎡ ⎤ [∗] [∗] 1 x∗ 1 x∗ Mf Mf ⎢ ⎥ ⎢ ⎥ [∗] [∗] [∗] ⎢ P := ⎢ MF S MF S MF S ⎥ P MF S ⎥ ⎣ x ⎦=⎣ x ⎦ ≥ 0. Mf MF S IH(KS ) Mf MF S IH(KS )

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We next observe that for every vector g ∈ C ⊕ X ⊕ H(KS ) of the form ⎡ ⎤ cj r ⎢ ⎥ xj g(z) = (cj ∈ C, yj ∈ Y, xj ∈ X , zj ∈ D) (3.4) ⎣ ⎦ j=1

KS (·, zj )yj

the identity Pg, gC⊕X ⊕H(KS ) =

r c cj K(zj , z ) xy , xyj

j,=1

j

C⊕X ⊕Y

(3.5)

holds. Since the set of vectors of the form (3.4) is dense in C ⊕ X ⊕ H(KS ), the identity (3.5) now implies that the operator P is positive semidefinite if and only if the quadratic form on the right hand side of (3.5) is nonnegative, i.e., if and only if the kernel (3.3) is positive on D × D. For the rest of this section we assume that the operator P in (3.2) is 1 positive definite. Then the operator MF S P − 2 is an isometry and the space 1

N = {F S (z)x : x ∈ X } with norm F S xH(S) = P 2 xX

(3.6)

is isometrically included in H(KS ). Moreover, the orthogonal complement of S ) with reproducing N in H(KS ) is the reproducing kernel Hilbert space H(K kernel S (z, ζ) = KS (z, ζ) − F S (z)P −1 F S (ζ)∗ . K (3.7) Theorem 3.4. Assume that condition (3.2) holds and that P is positive defi S (z, ζ) be the kernel defined in (3.7). Then: nite. Let K 1. All solutions f to the problem AIPH(KS ) are described by the formula f (z) = F S (z)P −1 x + h(z)

(3.8)

S ) subject to where h is a free parameter from H(K 1 1 − P − 2 x2 . hH(K S ) ≤ 2. The problem AIPH(KS ) has a unique solution if and only if 1

P − 2 x = 1

or

S (z, ζ) ≡ 0. K

(3.9)

Proof. It is readily seen that 1 X1 = P − 2 x ∈ X ∼ = L(C, X ),

1

[∗]

X2 = P − 2 MF S ∈ L(H(KS ), X )

are the operators X1 and X2 from (2.2) after specialization to the case (3.1). 1 The second statement now follows from Lemma 2.2, since P − 2 x ∈ 1 L(C, X ) being isometric means that P − 2 x = 1 and, on the other hand, the 1 isometric property for the operator MF S P − 2 means that the space N deﬁned S ) = H(KS )N = {0} in (3.6) is equal to the whole space H(KS ). Thus H(K or KS ≡ 0. In the present framework, the parametrization formula (2.4) takes the form

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235

1 [∗] 1 x + 1 − P − 2 x2 · (I − MF S P −1 MF S ) 2 K

(3.10)

S ) by where K is equal to the operator of multiplication Mk : C → H(K − 12 a function k ∈ H(KS ) with k ≤ 1. Since MF S P is an isometry, the second term on the right hand side of (3.10) is equal to the operator Mh of S ) such that h = hH(K ) ≤ multiplication by a function h ∈ H(K S H(KS ) 1 − 2 1 − P 2 x . Remark 3.5. The second term h on the right hand side of (3.8) represents in fact the general solution of the homogeneous interpolation problem (with [∗] S ), interpolation condition MF S f = 0). If h runs through the whole space H(K [∗]

then formula (3.8) produces all functions f ∈ H(KS ) such that MF S f = x. This unconstrained interpolation problem has a unique solution if and only S (z, ζ) ≡ 0. Thus, the second condition in (3.9) provides the uniqueness if K [∗]

of an f subject to MF S f = x in the whole space H(KS ), not just in the unit S (z, ζ) ≡ 0, then the unconstrained problem has inﬁnitely ball of H(KS ). If K many solutions and, as in the general framework, the function F S (z)P −1 x 1 has the minimal possible norm. Since MF S P − 2 is an isometry, it follows 1 1 from (3.6) that MF S P −1 xH(KS ) = P − 2 x. Thus, if P − 2 x = 1, then uniqueness occurs since the minimal norm solution already has unit norm.

4. Redheffer Transform Related to the AIP-Problem on S(U , Y) To obtain a more explicit parametrization of the solution set to the AIPH(KS ) -problem, we need some facts concerning the Abstract Interpolation Problem for functions in the Schur class S(U, Y) (denoted as the AIPS(U ,Y) -problem) from [24] (see also [26,28]) which we now recall. We consider the data set D = {P, T, E, N }

(4.1)

consisting of operators P, T ∈ L(X ), E ∈ L(X , Y), N ∈ L(X , U) such that the pairs (E, T ) and (N, T ) are output analytic and P is a positive semidefinite solution of the Stein equation (1.7). A set with these properties is called AIPS(U ,Y) -admissible. AIPS(U ,Y) : Given an AIPS(U ,Y) -admissible data set (4.1), find all functions S : D → L(U, Y) such that the kernel P F S (ζ)∗ (z, ζ) → (z, ζ ∈ D) (4.2) F S (z) KS (z, ζ) is positive on D × D, or equivalently, find all functions S ∈ S(U, Y) so that the operator MF S = OE,T − MS ON,T : x → F S (z)x maps X into H(KS ) [∗] and satisfies MF S MF S ≤ P . Here F S is the function defined in (1.5). The equivalence of the two above formulations follows from a general result on reproducing kernel Hilbert spaces; see [6].

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The parametrization of the solutions through an associated Redheffer transform is recalled in Theorem 4.1 below. To state the result we need to construct the Redheffer transform. Observe that (1.7) can equivalently be written as 1

1

P 2 x2 + N x2 = P 2 T x2 + Ex2

x ∈ X.

for all

1 2

Let X0 = Ran P . With some abuse of notation we will occasionally view P as an operator mapping X into X0 , or X0 into X , while still using P = P ∗ . The above identity shows that there exists a well deﬁned isometry V with domain DV and range RV equal to 1 1 X0 X0 P2 P 2T DV = Ran ⊆ ⊆ and RV = Ran , U Y N E respectively, which is uniquely determined by the identity 1 1 P 2x P 2Tx V = for all x ∈ X . Nx Ex We then deﬁne the defect spaces X0 Δ := DV and U

Δ∗ :=

X0

(4.3)

RV ,

Y

(4.4)

and Δ ∗ denote isomorphic copies of Δ and Δ∗ , respectively, with and let Δ unitary identiﬁcation maps i:Δ→Δ

∗. and i∗ : Δ∗ → Δ

With these identiﬁcation maps we deﬁne a unitary colligation matrix U from ∗ = X ⊕ U ⊕ Δ ∗ onto RV ⊕ Δ∗ ⊕ Δ =X ⊕Y ⊕Δ by DV ⊕ Δ ⊕ Δ ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ DV RV V 0 0 ⎢ ⎢Δ ⎥ ∗⎥ ⎢ Δ ⎥ U = ⎣ 0 0 i∗ ⎦ : ⎣ (4.5) ⎦ → ⎣ ∗ ⎦, 0 i 0 Δ∗ Δ which we also decompose as ⎡ A B1 ⎢ U = ⎣ C1 D11 C2

D21

B2

⎤ ⎡

X0

⎤

⎡

X0

⎤

⎥ ⎢ ⎥ ⎢ ⎥ D12 ⎦ : ⎣ U ⎦ → ⎣ Y ⎦ . ∗ 0 Δ Δ

(4.6)

Write Σ for the characteristic function associated with this colligation U, i.e., D11 D12 C1 Σ(z) = (z ∈ D), (4.7) (I − zA)−1 B1 B2 +z D21 0 C2 and decompose Σ as Σ(z) =

Σ11 (z)

Σ12 (z)

Σ21 (z)

Σ22 (z)

:

U ∗ Δ

→

Y . Δ

(4.8)

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A straightforward calculation based on the fact that U is coisometric gives C1 I − Σ(z)Σ(ζ)∗ (4.9) (I − zA)−1 (I − ζA∗ )−1 C1∗ C2∗ , = 1 − zζ C2 ∗ , Y ⊕Δ). which implies in particular that Σ belongs to the Schur class S(U ⊕Δ Moreover, it follows from the construction that Σ22 (0) = 0. These facts imply that the Redheffer linear fractional transform S = RΣ [E] := Σ11 + Σ12 (I − EΣ22 )−1 EΣ21

(4.10)

Δ ∗ ). The next theis well deﬁned for every Schur-class function E ∈ S(Δ, Δ ∗) orem (see [24] for the proof) shows that the image of the class S(Δ, under the Redheffer transform RΣ is precisely the solution set of the problem AIPS(U ,Y) . Theorem 4.1. Given an AIPS(U ,Y) -admissible data set (4.1), let RΣ be the Redheffer transform constructed as in (4.7), (4.10). A function S : D → L(U, Y) is a solution of the problem AIPS(U ,Y) if and only if S = RΣ [E] for Δ ∗ ). some E ∈ S(Δ, The function Σ11 appears as a solution upon taking E ≡ 0, and is called the central solution of the problem AIPS(U ,Y) . In case the problem has only one solution, this solution must be the central solution. Proposition 4.2. Let Σ be the characteristic function of the unitary colligation U in (4.6), decomposed as in (4.8), and let S = RΣ [E] ∈ S(U, Y) for a Δ ∗ ). Define the functions given E ∈ S(Δ, G(z) = Σ12 (z)(I − E(z)Σ22 (z))−1 , Γ(z) = (C1 + G(z)E(z)C2 ) (I − zA)−1 .

(z ∈ D)

(4.11)

Then G defines a contractive multiplier MG : H(KE ) → H(KS ), Γ a contractive multiplier MΓ : X0 → H(KS ), and the operator H(KE ) MG M Γ : (4.12) → H(KS ) X0 is coisometric. Furthermore, we have 1

F S (z) = Γ(z)P 2

(4.13)

for each z ∈ D, where F S is the function defined in (1.5). In particular, MΓ [∗] is an isometry and MG a partial isometry if and only if P = MF S MF S with 1 MF S : X0 → H(KS ) defined by MF S = MΓ P 2 . Proof. The identity I G(z)E(z) Σ(z) = S(z) G(z) is an immediate consequence of (4.10) and the definition of G(z) in (4.11). Using this identity one can easily compute that

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I − S(z)S(ζ)∗ = G(z)(I − E(z)E(ζ)∗ )G(ζ)∗ + I G(z)E(z)

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I E(ζ)∗ G(ζ)∗

S(ζ)∗ − S(z) G(z) G(ζ)∗

= G(z) (I − E(z)E(ζ)∗ ) G(ζ)∗

∗

+ I G(z)E(z) (I − Σ(z)Σ(ζ) )

I

;

E(ζ)∗ G(ζ)∗

(see also [7, Lemma 8.3]). Dividing both sides of the latter identity by 1 − zζ leads to I ∗ KS (z, ζ) = G(z)KE (z, ζ)G(ζ) + I G(z)E(z) KΣ (z, ζ) . E(ζ)∗ G(ζ)∗ By replacing KΣ (z, ζ) by the expression on the right hand side of (4.9), we get KS (z, ζ) = G(z)KE (z, ζ)G(ζ)∗ + Γ(z)Γ(ζ)∗ .

(4.14)

It is easy to verify that ∗ MG : KS (·, ζ)y → KS (·, ζ)G(ζ)∗ y,

MΓ∗ : KS (·, ζ)y → Γ(ζ)∗ y ∗ from which one can deduce that the context of (4.14) is that MG MΓ∗ is isometric on span{KS (·, ζ)y : ζ ∈ D, y ∈ Y} = H(KS ), i.e., MG MΓ is coisometric as asserted. In particular, we see that MG and MΓ are contractions. To verify (4.13), recall that the very construction of the colligation U implies that ⎡ ⎤ ⎡ 1 ⎤ A B1 1 P 2T ⎢ ⎥ P2 ⎢ ⎥ = ⎣ E ⎦, ⎣C1 D11 ⎦ N C2 D21 0 1

1

1

1

so that AP 2 − P 2 T = B1 N, C1 P 2 = E − D11 N, C2 P 2 = −D21 N . Making use of the latter equalities and of realization formulas for Σ11 and Σ21 in (4.7) we compute for z ∈ D, C1 C1 C1 1 1 1 −1 12 (I −zA) P (I − zT ) = P2 +z (I −zA)−1 (AP 2 −P 2 T ) C2 C2 C2 E − D11 N C1 = (I − zA)−1 B1 N −z C2 D21 N E Σ11 (z) = N. (4.15) − Σ21 (z) 0 Upon multiplying the left-hand side expression in (4.15) by I G(z)E(z) on 1 the left and by (I − zT )−1 on the right, we get Γ(z)P 2 by definition (4.11).

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Applying the same multiplications to the right hand side expression gives, on account of (4.11) and (4.10), E(I − zT )−1 − (Σ11 (z) + G(z)E(z)Σ21 (z))N (I − zT )−1 = (E − S(z)N )(I − zT )−1 1

which is F S (z). Thus Γ(z)P 2 = F S (z), and (4.13) follows. It is now straightforward to verify that MΓ is an isometry if and only [∗] if P = MF S MF S , while, since (4.12) is a coisometry, MΓ being an isometry implies that MG is a partial isometry. 4.1. Injectivity of RΣ and MG In this subsection we focus on two questions: (1) when is the above con Δ ∗ ), structed Redheffer transform RΣ injective, and, (2) for a given E ∈ S(Δ, when is the multiplication operator MG : H(KE ) → H(KS ) from Proposi[∗] tion 4.2 injective (and thus an isometry if P = MF S MF S )? The next lemma provides the basis for the results to follow. Lemma 4.3. Assume MΣ12 : HolΔ ∗ (D) → HolY (D) has a trivial kernel and MΣ21 : HolU (D) → HolΔ (D) has dense range. Then the Redheffer trans Δ ∗ ) the multiplication operator form RΣ is injective, and for any E ∈ S(Δ, MG : H(KE ) → H(KS ) has trivial kernel. Proof. The identity MS = MΣ11 + MΣ12 (I − ME MΣ22 )−1 ME MΣ21 may not hold if we consider the multiplication operators as acting between the appropriate H 2 -spaces, since I − ME MΣ11 may not be boundedly invertible, but the identity does hold when the multiplication operators are viewed as operators between the appropriate linear spaces of holomorphic functions on D, i.e., ME : HolΔ MS : HolΔ ∗ (D) → HolY (D), ∗ (D) → HolΔ (D), HolU (D) HolY (D) MΣ11 MΣ12 : → . HolΔ MΣ21 MΣ22 HolΔ ∗ (D) (D) Note that I − ME MΣ22 is invertible as a linear map on HolΔ ∗ (D) since Σ22 (0) = 0 and E(z) and Σ22 (z) are both contractive for z ∈ D. Now assume Δ ∗ ) so that RΣ [E] = RΣ [E ]. By the assumptions on MΣ and E, E ∈ S(Δ, 12 MΣ21 it follows that (I − ME MΣ22 )−1 ME = (I − ME MΣ22 )−1 ME = ME (I − MΣ22 ME )−1 , and thus ME − ME MΣ22 ME = ME (I − MΣ22 ME ) = (I − ME MΣ22 )ME = ME − ME MΣ22 ME . Hence E = E . Since MΣ12 : HolΔ ∗ (D) → HolY (D) has a trivial kernel, so does MΣ12 (I − ME MΣ22 )−1 when viewed as an operator acting on HolΔ ∗ (D),

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Δ ∗ ). In particular, G(z)h(z) ≡ 0 independently of the choice of E ∈ S(Δ, implies h = 0 for any h ∈ H(KE ). The proof of the above lemma does not take into account the particularities of the Redheffer transform associated with the problem AIPS(U ,Y) constructed in (4.3)–(4.6), besides the fact that Σ22 (0) = 0. As we shall see, for the coefﬁcients in the Redheffer transform we consider, MΣ21 always has dense range, while MΣ12 has a trivial kernel if the operator T ∗ is injective. As preparation for this result, we need the following lemma. Lemma 4.4. Let D21 and D12 be the operators in the unitary colligation (4.6). Then 1 ker T ∗ P 2 |X0 ∗ ker D21 = {0} and ker D12 = i∗ . (4.16) {0} be such that D∗ δ = 0U . By construction (4.5), the vector Proof. Let δ ∈ Δ 21 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∗ X0 C2 δ x0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ⎥ U δ=⎢ (4.17) ⎣D21 δ ⎦ = ⎣ 0 ⎦ ∈ ⎣ U ⎦ ∗ 0 Δ 0 belongs to Δ, which means (by definition (4.4) of Δ) that ⎡ ⎤ ⎡ 12 ⎤ x0 P x 1 = x0 , P 2 xX0 0 = ⎣ 0 ⎦ , ⎣ Nx ⎦ 0 0 X ⊕U ⊕Δ 0

∗

1 2

1

for every x ∈ X0 , which is equivalent to P x0 = 0. Since P 2 |X0 is injective, we get x0 = 0. Thus U∗ δ = 0 by (4.17) and consequently, δ = 0, since U is ∗ = {0}. unitary. So KerD21 To prove the second equality X in (4.16) we ﬁrst observe that a vector x0 X0 [ y ] ∈ Y belongs to Δ∗ := Y0 RV if and only if 1

T ∗ P 2 x0 + E ∗ y = 0.

(4.18)

∗ so that D12 δ∗ = 0Y . Then, by (4.5) and (4.6), Now let δ∗ ∈ Δ B2 x0 X0 ∗ Uδ∗ = i∗ δ∗ = δ∗ = ∈ Δ∗ ⊂ , with x0 = B2 δ∗ . D12 0 Y Since [ x00 ] is in Δ∗ , it follows from (4.18) that T ∗ P 2 x0 = 0, i.e., x0 ∈ 1 ker T ∗ P 2 . Thus, since U is unitary and [ x00 ] ∈ Δ∗ , we see that δ∗ = U∗ [ x00 ] = 1 i∗ [ x00 ] for some x0 ∈ ker T ∗ P 2 . 1 Conversely, for every x0 ∈ Ker(T ∗ P 2 |X0 ), the vector [ x00 ] belongs to ∗ and we have, on Δ∗ (by (4.18)) so that its image δ∗ = i∗ [ x00 ] belongs to Δ account of (4.5)–(4.6), B2 x B2 x x δ∗ = i∗ 0 = i∗∗ i∗ 0 = 0 . D12 D12 0 0 0 1

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Equating the bottom entries we get D12 δ∗ = 0 which completes the proof of the characterization of KerD12 . Theorem 4.5. Let RΣ be the Redheffer transform associated with the Schur class function Σ defined in (4.7) from the AIPH(KS ) -admissible data set Δ ∗ ) → S(U, Y) is injec(4.1). Assume that T ∗ is injective. Then RΣ : S(Δ, tive, and for E ∈ S(Δ, Δ∗ ) the multiplication operator MG : H(KE ) → [∗] H(KS ) has trivial kernel. If in addition P = MF S MF S , then MG is an isometry. Proof. It is easy to see that MΣ21 has dense range if and only if Σ21 (0) = D21 has dense range and that ker Σ12 (0) = ker D12 = {0} implies that MΣ12 has a trivial kernel. The converse of the latter statement is not true in general. Moreover, the last statement of the theorem is immediate from Proposition 4.2. Thus Theorem 4.5 follows immediately from Lemma 4.4. Remark 4.6. In case the operator (I − ωT )−1 is bounded for some ω ∈ D, we can deﬁne = 1 − |ω|2 · E(I − ωT )−1 , N = 1 − |ω|2 · N (I − ωT )−1 E z−ω T = (ωI − T )(I − ωT )−1 , S(z) =S . 1 − zω It is not hard to verify that if D = {S, T, E, N, x} is an AIPH(KS ) -admis = {S, T, E, N , x} is also AIPH(K ) -admissisible data set, then the set D S ble and moreover, a function f solves the problem AIPH(KS ) if and only z−ω Therefore, ) solves the problem AIPH(KS ) with data set D. f(z) := f ( 1−zω up to a suitable conformal change of variable, we get all the conclusions in Theorem 4.5 under the assumption that (I − ωT )−1 ∈ L(X) and (T ∗ − ωI) is injective for some ω ∈ D. In case T ∗ is not injective, and neither is (T ∗ − ωI) for some ω ∈ D so that (I − ωT ) is invertible in L(X ), it may still be possible to reach the conclusion of Theorem 4.5 under weaker assumptions on the operator T . We start with a preliminary result. Lemma 4.7. The operator MΣ12 : HolΔ ∗ (D) → HolY (D) is not injective if and only if there is a sequence {gn }n≥1 of non-zero vectors in X0 such that 1

g1 ∈ Ker(T ∗ P 2 ),

1

1

P 2 gn = T ∗ P 2 gn+1 (n ≥ 1),

1

lim sup B2∗ gn+1 n ≤ 1. n→∞

(4.19) ∞

1 k

Proof. Let h(z) = k=0 z k hk ∈ HolΔ ∗ (D), i.e., lim supk→∞ hk ≤ 1 and for n ≥ 0, X0 1 xn xn hn = i∗ with ∈ , subject to T ∗ P 2 xn + E ∗ yn = 0. yn yn Y (4.20)

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Note that xn and yn are retrieved from hn by the identity B2 xn ∗ hn . = i ∗ hn = yn D12

(4.21)

Deﬁne gn ∈ X0 by gn =

n−1

An−k−1 xk ∈ X0

for n ≥ 1,

(4.22)

k=0

or equivalently via the recursion g1 = x0 ,

gn+1 = xn + Agn

for n ≥ 1.

(4.23)

Since U in (4.5) and (4.6) is unitary, it follows that ∗ B2∗ A = −D12 C1 .

(4.24)

Moreover, since U is connected with V as in (4.5) and V is given by (4.3), we see that 1 1 x0 A P 2 x0 P 2T C1 x0 , x0 = U 0 , U N x0 E C2

0

0

=

0

1 x0 P 2 x0 1 0 = P 2 x0 , x0 , N x0 0

0

from which we conclude that 1

1

T ∗ P 2 A + E ∗ C1 = P 2 .

(4.25)

Hence for n ≥ 1, 1

T ∗ P 2 gn+1 =

n

1

1

n−1

1

n−1

T ∗ P 2 An−k xk = T ∗ P 2 xn +

k=0

1

T ∗ P 2 An−k xk

k=0

= T ∗ P 2 xn +

1

(P 2 − E ∗ C1 )An−k−1 xk

k=0 1

1

= T ∗ P 2 xn − E ∗ C1 gn + P 2 gn 1

= −E ∗ (yn + C1 gn ) + P 2 gn

(4.26)

where we used the relation between xn and yn in (4.20) for the last step. Moreover, using the identity in (4.24), we get xn ∗ ∗ ∗ ∗ B2 gn+1 = B2 (xn + Agn ) = B2 D12 −C1 gn xn ∗ ∗ = B2∗ D12 (yn + C1 gn ) − D12 yn x ∗ (yn + C1 gn ) = i∗ n − D12 yn ∗ = hn − D12 (yn + C1 gn ).

(4.27)

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Now assume that h = 0 and Σ12 (z)h(z) ≡ 0, i.e., (D12 + zC1 (I − zA)−1 B2 )h(z) ≡ 0.

(4.28)

Using the power series representations ∞ ∞ hn z n and Σ12 (z) = D12 + z k C1 Ak−1 B2 h(z) = n=0

k=1

for h and Σ12 and recalling (4.21), it follows that (4.28) is equivalent to the following system of equations: y0 = D12 h0 = 0,

yn + C1 gn = D12 hn +

n−1

C1 An−k−1 B2 hk = 0

for n ≥ 1.

k=0

(4.29) Without loss of generality we may, and will, assume that h0 = 0; otherwise replace h by h(z) = z − h(z) for ∈ Z+ sufﬁciently large. Then [ xy00 ] = i∗∗ h0 = 0 since h0 = 0 by assumption. But y0 = 0 by (4.29) and hence x0 = 0. From 1 the constraint in (4.20) we see that 0 = x0 ∈ Ker T ∗ P 2 . Moreover, the second identity in (4.29) combined with (4.26) and (4.27) yields 1

1

T ∗ P 2 gn+1 = P 2 gn ,

B2∗ gn+1 = hn .

(4.30)

The second of identities (4.30) then gives us lim sup B2∗ gn+1 1/n = lim sup hn 1/n ≤ 1. n→∞

(4.31)

n→∞

1

Finally, observe that, since g1 = x0 = 0 and Ker P 2 |X0 = {0}, the recursive 1 1 relation T ∗ P 2 gn+1 = P 2 gn [the ﬁrst of identities (4.30)] implies that gn = 0 for all n ≥ 1. We conclude that the sequence {gn }n≥1 has all the desired properties. Conversely, assume {gn }n≥1 is a sequence in X0 satisfying (4.19). Deﬁne x0 = g1 ,

xn = gn+1 − Agn ,

yn = −C1 gn

y0 = 0,

for n ≥ 1.

Applying (4.26) to gn for n ≥ 1 and using (4.25), we ﬁnd that 1

1

1

T ∗ P 2 gn+1 = P 2 gn = T ∗ P 2 Agn + E ∗ C1 gn 1

1

= T ∗ P 2 gn+1 − T ∗ P 2 xn + E ∗ C1 gn . 1

1

We conclude that T ∗ P 2 xn + E ∗ yn = T ∗ P 2 xn − E ∗ C1 gn = 0. For n = 0 1 the identity T ∗ P 2 xn + E ∗ yn = 0 follows from the ﬁrst of conditions (4.19). Hence we obtain that [ xynn ] ∈ Δ∗ . ∗ , and h(z) = ∞ z k hk . As before, xn Now deﬁne hn = i∗ [ xynn ] ∈ Δ k=0 and yn are retrieved from hn by (4.21), and from the definition of xn it follows that the sequence {gn }n≥1 is retrieved by (4.23). Moreover, the definition of yn shows that (4.29) holds and, in combination with the computation (4.27), 1 that B2∗ gn+1 = hn . The latter implies that lim supk→∞ hk k ≤ 1, via (4.19) and the identities in (4.31). In particular, h ∈ HolΔ ∗ (D). The fact that (4.29) holds now is equivalent to Σ12 (z)h(z) ≡ 0. Note that x0 = g1 = 0 and hence h0 = i∗ [ xy00 ] = 0. Thus h = 0 and it follows that MΣ12 : HolΔ ∗ (D) → HolY (D) is not injective.

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Based on the previous result, we obtain the following relaxation of the the condition on T in Theorem 4.5. Theorem 4.8. Let D = {P, T, E, N } be an AIPS(U ,Y) -admissible data set, let Σ be constructed as in (4.7) and let us assume that T meets the condition ⎛ ⎞ ⎝ Ran(T ∗ )k ⎠ KerT ∗ = {0}. (4.32) k≥1

Then the operator MΣ12 : HolΔ ∗ (D) → HolY (D) is injective. Proof. Assume MΣ12 : HolΔ ∗ (D) → HolY (D) is not injective. By Lemma 4.7, there exists a nonzero sequence {gn }n≥1 in X0 satisfying (4.19). By the ﬁrst 1 relation in (4.19), P 2 g1 ∈ ker T ∗ . On the other hand, iterating the second con1 1 1 dition in (4.19) gives P 2 g1 = (T ∗ )n P 2 gn+1 for each n ≥ 1. Since P 2 g1 = 0, ∗ n ∗ it follows that Ran(T ) ∩ ker T = {0} for each n ≥ 1. The latter is in con tradiction with (4.32). Thus MΣ12 : HolΔ ∗ (D) → HolY (D) is injective. It is easy to see that not only the injectivity of MΣ12 , but all the conclusions of Theorem 4.5 hold with the condition that T ∗ is injective replaced by the weaker condition (4.32). Although condition (4.32) is far from being necessary, it guarantees injectivity of MΣ12 for important particular cases: 1. T ∗ is injective (so Ker T ∗ = {0}), 2. T ∗ is nilpotent (so ∩k≥1 Ran(T ∗ )k = {0}), and 3. dim X < ∞, or, more generally e.g., T = λI + K with 0 = λ ∈ C and ˙ Ker(T ∗ )p once p is sufﬁciently large). K compact (so X = Ran(T ∗ )p + The question of ﬁnding a condition that is both necessary and sufﬁcient for injectivity of MΣ12 remains open.

5. Description of all Solutions of the Problem AIPH(KS ) We now present the parametrization of the solution set to the problem AIPH(KS ) . The proof relies on Theorem 3.3, Theorem 4.1 and Proposition 4.2. By Theorem 3.3, the solution set to the problem AIPH(KS ) coincides with the set of all functions f : D → Y such that the kernel K(z, ζ) deﬁned in (3.3) is positive on D × D. In particular, the function S must be such that the kernel (4.2) is positive meaning that S must be a solution to the associated problem AIPS(U ,Y) . By Theorem 4.1, there exists a Schur-class function E such that S = RΣ [E] where RΣ is the Redheffer transform constructed in be the unique vector in (4.3)–(4.10). Deﬁne G and Γ as in (4.11) and let x 1 . Making use of equalities (4.13) and (4.14) we can write X0 so that x = P 2 x K(z, ζ) in the form ⎡ ⎤ 1 ∗ P 2 1 x f (ζ)∗ ⎢ 1 ⎥ 1 ⎥. K(z, ζ) = ⎢ P P 2 Γ(ζ)∗ ⎣P 2x ⎦ 1 ∗ ∗ 2 f (z) Γ(z)P G(z)KE (z, ζ)G(ζ) + Γ(z)Γ(ζ)

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The positivity of the latter kernel is equivalent to positivity of the Schur complement of P with respect to K(z, ζ), that is, to the condition ∗ Γ(ζ)∗ f (ζ)∗ − x 1 − x2 0 (z, ζ ∈ D). (5.1) f (z) − Γ(z) x G(z)KE (z, ζ)G(ζ)∗ We arrive at the following result. Theorem 5.1. Let {S, T, E, N, x} be an AIPH(KS ) -admissible data set and [∗] let us assume that P := MF S MF S ≥ xx∗ with F S as in (1.5). Let Σ be constructed as in (4.3)–(4.8), let E be a Schur-class function such that S = be the unique vector in RΣ [E], let G and Γ be defined as in (4.11) and let x 1 . Then: X0 so that x = P 2 x 1. The set of solutions f of the problem AIPH(KS ) is given by the formula f (z) = Γ(z) x + G(z)h(z) with parameter h in H(KE ) subject to hH(KS ) ≤ 2. For f defined by (5.2)

(5.2) 1 − x2 .

2 + MG h2 = x2 + PH(KE )ker MG h2 f 2H(KS ) = MΓ x

(5.3)

x is the unique minimal-norm solution. and hence fmin (z) = Γ(z) x = 1 3. The problem AIPH(KS ) admits a unique solution if and only if or Ran MF S = H(KS ). Proof. As we have seen, a function f : D → Y solves the problem AIPH(KS ) if and only if (5.1) holds, that is, if and only if the function g := f − MΓ x with reproducing kerbelongs to the reproducing kernel Hilbert space H(K) ζ) = G(z)KE (z, ζ)G(ζ)∗ and satisﬁes g ≤ 1 − x2 . The nel K(z, H(K) tells us that range characterization of H(K) = {G(z)h(z) : h ∈ H(KE )} with norm MG h = (I − q)hH(K ) H(K) E H(K) where q is the orthogonal projection onto the subspace ker MG ⊂ H(KE ). is of the Therefore, the function g = f − MΓ x form g = MG h for some 1 − h ∈ H(KE ) such that hH(KE ) = gH(K) x2 . This proves the ≤ characterization of solutions through (5.2). [∗] Since P = MF S MF S , it follows from Proposition 4.2 that the operator (4.12) is a coisometry and MΓ is an isometry. From this combination the x and the orthogonality between the minimal-norm solution fmin (z) = Γ(z) remainder on the right hand side of (5.2), as well as the second identity in (5.3), is evident. Since MG is a partial isometry, it follows from (5.2) that the problem x = 1 (because AIPH(KS ) admits a unique solution if and only if either then h = 0 ∈ H(KE ) is the only admissible parameter), H(KE ) = {0} (i.e., if E is an unimodular constant) or MG = 0. Since the operator (4.12) is a coisometry and because MΓ is an isometry, the last two cases are covered by the condition that MΓ is unitary. Due to the relation between F S and Γ [see (4.13)], this is equivalent to MF S having dense range.

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Although the correspondence E → S = RΣ [E] established by formula (4.10) is not one-to-one in general, it follows from the proof of Theorem 5.1 that in order to ﬁnd all solutions f of the problem AIPH(KS ) it sufﬁces to take into account just one parameter E so that S = RΣ [E], rather than all. The further analysis in Sect. 4, i.e., Theorem 4.5 and Lemma 4.8, provide conditions under which the Schur class function E in Theorem 5.1 is unique. Theorem 5.2. Let (1.3) be an AIPH(KS ) -admissible data set satisfying condition (3.2) and assume that the operator T ∗ satisfies condition (4.32). Then: 1. There is a unique Schur-class function E such that S = RΣ [E], where RΣ is the Redheffer transform constructed from the data set (1.3) via (4.3)–(4.10). 2. The parametrization h → f in Theorem 5.1, via formula (5.2), of the solutions f to the problem AIPH(KS ) is injective. That is, the operator MG : H(KE ) → H(KS ) is isometric so that in addition f 2H(K

S)

= x2X0 + h2H(K ) .

(5.4)

S

Remark 5.3. Given an AIPH(KS ) -admissible data set (S, E, N, T, x), it is straightforward that (E, N, T, P ) with P = F S[∗] F S is an AIPS(U ,Y) admissible data set and that S is a solution for the associated problem AIPS(U ,Y) . Now consider another solution S ∈ S(U, Y) of the problem [∗] AIPS(U ,Y) . Unlike for S, this solution S satisﬁes M M S ≤ P and equalFS

F

ity may not hold. We may then still ask the question for which functions This f : D → Y the kernel in (3.3) is positive, with S is replaced by S. question turns out to to be equivalent to that of determining the f ∈ H(KS ) [∗] with f H(KS ) ≤ 1 and such that the vector M S f is close to x in the sense F that ! " 12 [∗] [∗] # M S f = x + 1 − f 2H(K ) P − M S MF S x (5.5) F

S

F

# ∈ X with # for some x x ≤ 1. The solutions to this problem can still be Δ ∗ ) so that S = RΣ [E] parameterized by formula (5.2), with now E ∈ S(Δ, in the definition of Γ and G, with the twist that in this case, because we may [∗] not have M S MF S = P , there is no guarantee that we have orthogonality as F is the solution with in (5.3), nor is it clear if the ‘central’ solution f = MΓ x minimal norm. To conclude this section we will brieﬂy discuss the interplay between the [∗] uniqueness of S as a solution of the problem AIPS(U ,Y) (with P = MF S MF S of the form (3.2)) and the determinacy of the related (unconstrained) problem AIPH(KS ) . We will assume that the operator T meets the condition (4.32), leaving the general case open. Under this assumption, there are only three uniqueness and semi-uniqueness cases. Recall that Δ and Δ∗ are the defect spaces of the isometry (4.3). Case 1: Let Δ∗ = {0}. Then S = Σ11 is the unique solution of the problem AIPS(U ,Y) . Furthermore, we conclude from (4.11) that Γ(z) = C1 (I − zA)−1 ,

G(z) ≡ 0

and H(KE ) = {0}.

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By Theorem 5.1, the unconstrained problem AIPH(KS ) has a unique solution 1 where x is the unique vector in X0 such that P 2 x = x. f (z) = C1 (I −zA)−1 x Case 2: Let Δ = {0}. In this case still S = Σ11 is the unique solution of the problem AIPS(U ,Y) . Also we have Γ(z) = C1 (I − zA)−1 . However, we 2 now have G = Σ12 and H(KE ) = HΔ ∗ . By Theorem 5.1, all solutions f to the unconstrained problem AIPH(KS ) are given by + Σ12 (z)h(z), f (z) = C1 (I − zA)−1 x

(5.6)

2 HΔ ∗.

is as above and where h varies in where x One can see that the same description holds if the spaces (4.4) are nontrivial and S = Σ11 is the central (but not unique) solution to the associated problem AIPS(U ,Y) . Case 3: Let Δ and Δ∗ be nontrivial and let us assume that S is an extremal solution to the problem AIPS(U ,Y) (in the sense that the unique E such that S = RΣ [E] is a coisometric constant). Then the unconstrained problem AIPH(KS ) has a unique solution since in this case H(KE ) = {0}.

6. Interpolation with Operator Argument In this section we show that the interpolation problem with operator argument in the space H(KS ) can be embedded into the general scheme of the problem AIPH(KS ) considered above. Recall that a pair (E, T ) with E ∈ L(Y, X ) and T ∈ L(X ) is called an analytic output pair if the observability operator OE,T maps X into HolY (D). The starting point for the operatorargument point-evaluation is a so-called output-stable pair (E, T ) which is an analytic output pair with the additional property that OE,T ∈ L(X , HY2 ): OE,T : x → E(I − zT )−1 x =

∞

z n ET n x ∈ HY2 .

(6.1)

n=0

Given such an output-stable pair (E, T ) and a function f ∈ HY2 , we deﬁne the left-tangential operator-argument point-evaluation (E ∗ f )∧L (T ∗ ) of f at (E, T ) by (E ∗ f )∧L (T ∗ ) =

∞

T ∗n E ∗ fn

n=0

The computation ∞ ∗n ∗ T E fn , x n=0

= X

if

f (z) =

∞

fn z n .

(6.2)

n=0

∞ n=0

fn , ET n xY = f, OE,T xHY2

shows that the output-stability of the pair (E, T ) is exactly what is needed for the inﬁnite series in the definition of (E ∗ f )∧L (T ∗ ) in (6.2) to converge in the weak topology on X . The same computation shows that tangential evaluation with operator argument amounts to the adjoint of OE,T : ∗ f (E ∗ f )∧L (T ∗ ) = OE,T

for f ∈ HY2 .

(6.3)

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Evaluation (6.2) applies to functions from de Branges–Rovnyak spaces H(KS ) as well, since H(KS ) ⊂ HY2 , and suggests the following interpolation problem. OAPH(KS ) : Given S ∈ S(U, Y), T ∈ L(X ), E ∈ L(Y, X ) and x ∈ X so that the pair (E, T ) is output stable, find all functions f ∈ H(KS ) such that f H(KS ) ≤ 1

and

∗ (E ∗ f )∧L (T ∗ ) = OE,T f = x.

(6.4)

In the scalar-valued case U = Y = C, the latter problem has been considered recently in [4], with the additional assumption that P > 0. Similarly to the situation in [4], the operator-valued version contains left-tangential Nevanlinna–Pick and Carath´eodory-Fej´er interpolation problems as particular cases corresponding to special choices of E and T . We now show that on the other hand, the problem OAPH(KS ) can be considered as a particular case of the problem AIPH(KS ) . Lemma 6.1. Let (E, T ) be an output stable pair with E ∈ L(Y, X ) and T ∈ L(X ), let S ∈ S(U, Y) be a Schur-class function and let N ∈ L(X , U) be defined by ∞ ∞ Sj∗ ET j , where S(z) = Sj z j (6.5) N := j=0

j=0

or equivalently, via its adjoint ∗ MS |U : U → X . N ∗ = OE,T

(6.6)

Then the data set D = {S, T, E, N, x} is AIPH(KS ) -admissible for every [∗] ∗ x ∈ X . Furthermore, MF S = OE,T |H(KS ) , so that the interpolation conditions (6.4) coincide with those in (1.8). Proof. For N deﬁned as in (6.5), the pair (N, T ) is output stable (cf. [3, Proposition 3.1]) and the observability operator ON,T : x → N (I − zT )−1 x equals ON,T = MS∗ OE,T : X → HU2 .

(6.7)

S

With N as above, we now deﬁne F by formula (1.5). For the multiplication operator (1.6) we have, on account of (6.7), MF S = OE,T − MS ON,T = (I − MS∗ MS )OE,T

(6.8)

which together with the range characterization of H(KS ) implies that MF S maps X into H(KS ). Furthermore, it follows from (1.5), (1.2) and (6.2) that F S x2H(KS ) = (I − MS MS∗ )OE,T x, OE,T xHY2

∗ ∗ = (OE,T OE,T − ON,T ON,T )x, xX

for every x ∈ X . The latter equality can be written in operator form as [∗]

∗ ∗ OE,T − ON,T ON,T . P := MF S MF S = OE,T

(6.9)

It follows from the series representation (6.1) and the definition of inner product in HY2 that ∗ OE,T OE,T =

∞ n=0

T ∗n E ∗ ET n

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with convergence in the strong operator topology. Using the latter series expansion one can easily verify the identity ∗ ∗ OE,T − T ∗ OE,T OE,T T = E ∗ E. OE,T

(6.10)

Since the pair (N, T ) is also output stable, we have similarly ∗ ∗ ON,T − T ∗ ON,T ON,T T = N ∗ N. ON,T

(6.11)

Subtracting (6.11) from (6.10) and taking into account (6.9) we conclude that P satisﬁes the Stein identity (1.7). Thus, the data set D is AIPH(KS ) admissible. In view of (6.8) and (1.2), the equalities [∗]

MF S f, xX = f, MF S xH(KS ) = f, (I − MS MS∗ )OE,T xH(KS ) ∗ = f, OE,T xHY2 = OE,T f, xX [∗]

∗ hold for all f ∈ H(KS ) and x ∈ X . Therefore, MF S = OE,T |H(KS ) .

As a consequence of Lemma 6.1, the solutions to the problem OAPH(KS ) are obtained from Theorem 5.1, after specialization to the case under consideration. We do not state this specialization of Theorem 5.1 here because the formulas do not significantly simplify. Instead we now discuss the operatorargument interpolation problem for functions in HY2 , that is, the problem OAPH(KS ) with S ≡ 0. As we shall see, in that case the problems AIPH(KS ) and OAPH(KS ) coincide. Consider an AIPH(KS ) -admissible data set {S, T, E, N, x} with S ≡ 0 ∈ S(U, Y). Then H(KS ) = HY2 and F S = OE,T . Thus condition (2) in Definition 1.1 just says that F S = OE,T is in L(X , HY2 ), and thus that (E, T ) ∗ OE,T is output-stable. The third condition states that P = F S[∗] F S = OE,T ∗ satisﬁes the Stein equation (1.7). This implies that necessarily N = 0 = (E ∗ S)∧L (T ∗ ), and it follows that the problem AIPH(KS ) reduces to the problem OAPH(KS ) with data T , E and x, and S ≡ 0. We now specify Theorem 5.1 to this case, with the additional assumption that P is positive definite. Theorem 6.2. Given an output stable pair (E, T ) with E ∈ L(Y, X ) and ∗ OE,T and that P is T ∈ L(X ), and x ∈ X . Assume that xx∗ ≤ P := OE,T 2 positive definite. Then the set of all f ∈ HY satisfying f HY2 ≤ 1

and

(E ∗ f )L (T ∗ ) = x

is given by the formula f (z) = E(I − zT )−1 P −1 x + B(z)h(z)

(6.12)

where h is a free parameter from the ball $ % h ∈ HY2 0 : h2H 2 ≤ 1 − x∗ P −1 x ⊂ HY2 Y0

for an auxiliary Hilbert space Y0 ; here B(z) is the inner function in the Schur class S(Y0 , Y) determined uniquely (up to a constant unitary factor on the right) by the identity KB (z, ζ) :=

IY − B(z)B(ζ)∗ = E(I − zT )−1 P −1 (I − ζT ∗ )−1 E ∗ . 1 − zζ

(6.13)

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Proof. As remarked above, we are considering the problem AIPH(KS ) with data set {S, T, E, N, x} where S ≡ 0 and N = 0. Then, for x ∈ X , we have N −1 ∗n ∗ n P x, x = lim T E ET x, x N →∞

= lim

n=0

N −1

N →∞

T

∗n

∗

n

(P − T P T )T x, x

n=0 1

= P x, x − lim P 2 T N x2 , N →∞

1 2

N

2

and we conclude that P T x → 0 as N → ∞. The assumption that 1 P > 0 implies that P 2 is invertible and we conclude that T N x2 → 0 as well, i.e., that T is strongly stable. The fact that N = 0 yields that in the construction of the unitary colligation U in (4.3)–(4.6), 1 DV = X , Δ = U and the isometry V is deﬁned by 1 the identity V P 2 = P E2 T . Moreover, in the unitary colligation U we have B1 = 0, D11 = 0 and C2 = 0, and A and C1 can be computed explicitly as 1

1

1

A = P 2 TP−2 ,

C1 = EP − 2 .

As T is strongly stable, we conclude that A is strongly stable as well. The unitary colligation U then collapses to ⎤ ⎡ A 0 B2 ⎥ ⎢ 0 D12 ⎦ U = ⎣C1 (6.14) 0

D21

and Σ(z) has the form Σ11 (z) Σ12 (z) 0 Σ(z) = = D21 Σ21 (z) Σ22 (z)

0

D12 + zC1 (I − zA)−1 B2 0

From the special form (6.14) of U, it follows that D21 and

A

. B2

are C1 D12 unitary. As A is strongly stable, it is then well known that Σ12 is inner, OC1 ,A maps X isometrically into the de Branges–Rovnyak space H(KΣ12 ) = 2 HY2 Σ12 HΔ ∗ , and hence the operator 2 HΔ ∗ MΣ12 OC1 ,A : (6.15) → HY2 X is unitary. Note that the Redheffer transform RΣ reduces to S(z) = RΣ [E](z) = Σ12 (z)E(z)D21

(z ∈ D).

Since S ≡ 0, we have S = RΣ [E] when E ≡ 0. In fact, because Σ12 is inner and D21 unitary, the Redheffer transform RΣ is one-to-one, and thus E ≡ 0 2 and the is the only E ∈ S(Δ, Δ∗ ) with RΣ [E] ≡ 0. Then H(KE ) = HΔ ∗

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function G in (4.11) is equal to Σ12 , and thus is inner. To complete the proof, 1 1 = P − 2 x, so that note that Γ = F S P − 2 and x 1

1

Γ(z) x = F S (z)P − 2 P − 2 x = OE,T P −1 x = E(I − zT )−1 P −1 x = C1 (I − zA)−1 x. Thus (5.2) coincides with (6.12) with B = Σ12 . The coisometric property of the unitary operator (6.15) expressed in reproducing kernel form gives us I − Σ12 (z)Σ12 (ζ)∗ = C1 (I − zA)−1 (I − ζA∗ )−1 C1∗ 1 − zζ = E(I − zT )−1 P −1 (I − ζT ∗ )−1 E ∗ and we see that B := Σ12 is determined from the data set as in (6.13) in Theorem 6.2.

7. Homogeneous Interpolation and Toeplitz Kernels Let S ∈ S(U, Y) be an inner function, i.e., MS ∈ L(HU2 , HY2 ) is an isometry. Then MS HU2 is a closed, invariant subspace of the shift operator Mz on HY2 . By the Beurling–Lax–Halmos theorem, this is the general form of a closed shift-invariant subspace of HY2 . Moreover, the de Branges–Rovnyak space KS := H(KS ) is the orthogonal complement of MS HU2 : KS = HY2 MS HU2 and provides a general form for closed backward shift-invariant subspaces of HY2 . Let, in addition, B ∈ S(W, Y) be inner, so that we have shift invariant 2 subspaces MS HU2 and MB HW and backward shift invariant subspaces KS 2 2 and KB of HY . Characterizations of the intersections MS HU2 ∩ MB HW and KS ∩ KB in terms of S and B are well-known (see e.g., [30]). In this section we characterize the space 2 MS,B := KS ∩ MB HW .

(7.1)

Let us introduce the operators T ∈ L(KB ), E ∈ L(KB , Y), and N ∈ L(KB , U) by h(z) − h(0) , E : h → h(0), z N : h(z) = hj z j → Sj∗ hj where S(z) = Sj z j .

T : h(z) →

j≥0

j≥0

(7.2) (7.3)

j≥0

The operator T is strongly stable (i.e., limn→∞ T n h = 0 for each h ∈ X = KB ) and the pair (E, T ) is output-stable. With N deﬁned in accordance with (6.5), the data set D = {S, E, N, T, x = 0} is AIPH(KS ) -admissible, by ∗ Lemma 6.1. Furthermore the adjoint OE,T : HY2 → KB of the observability operator O E,T amounts to the orthogonal projection PKB onto KB . Indeed, if h(z) = j≥0 hj z j ∈ KB , then ET j h = hj for j ≥ 0 and hence (OE,T h)(z) = (ET j h)z j = hj z j = h(z). j≥0

j≥0

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∗ f = 0 if and only if f ∈ HY2 KB = Therefore, for an f ∈ HY2 we have OE,T 2 MB HW . It is now easily checked that the space (7.1) is characterized as & ' ∗ f =0 , (7.4) MS,B = f ∈ KS : OE,T

i.e., as the solution set of the (unconstrained) homogeneous problem OAPH(KS ) with the data set {S, E, T, x = 0}. The operator P deﬁned by formulas (6.9) now amounts to the compression of the operator IHY2 − MS MS∗ to the subspace KB : P = IKB − PKB MS MS∗ |KB .

(7.5)

Theorem Given inner functions S ∈ S(U, Y) and B ∈ S(W, Y), let Σ11 7.1. Σ12 Σ = Σ21 Σ22 be the characteristic function of the unitary colligation U associated via formulas (4.3)–(4.6) to the tuple {P, T, E, N } given in (7.2), (7.3), (7.5). Then the space MS,B given by (7.1) is given explicitly as MS,B = G · H(KE )

(7.6)

∗ ) such that S = RΣ [E] and where E is the unique function in S(U ⊕ Δ −1 G(z) = Σ12 (z)(I − E(z)Σ22 (z)) . Furthermore MG is a unitary operator from H(KE ) onto MS,B . Proof. The parametrization formula (7.6) follows from (7.4) upon applying Theorem 5.1. The fact that in the present situation the Redheffer transform RΣ is one-to-one was established in [25] (see also [28, Theorem 5.8]). Thus the parameter E such that S = RΣ [E] is uniquely determined. It is also shown in [28, Proposition 5.9] that # 12 (z) Σ12 (z) = B(z)Σ

(7.7)

# 12 is a ∗-outer function in S(Δ ∗ , Y). From this identity and the defiwhere Σ 2 . Secondly, nition of MS,B we see directly that MS,B is contained in MB HW # we see from the ∗-outer property of Σ12 and the factorization (7.7) of Σ12 that the operator of multiplication by G(z) = Σ12 (z)(I − E(z)Σ22 (z))−1 is injective. Since we know that MG is a partial isometry, it now follows that MG : H(KE ) → MS,B is unitary. Remark 7.2. One gets the same parametrization of MS,B in case S ∈ S(U, Y) is not inner. As the following result indicates, spaces of the form MS,B come up in the description of kernels of Toeplitz operators. To formulate the result let us say that the triple (S, B, Γ) is an admissible triple if 1. 2.

S and B in S(Y) with Y ﬁnite-dimensional are inner (i.e., S and B assume unitary values almost everywhere on the unit circle T), and ∞ ∞ )±1 , i.e., both Γ and Γ−1 are in HL(Y) . Γ ∈ (HL(Y)

We also need the following result from [5]. Theorem 7.3. (See [5, Theorem 4.1].) Let > 0 and suppose that Y ∼ = Cn is a finite-dimensional coefficient Hilbert space. Suppose also that Φu ∈ L∞ L(Y) has

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unitary values almost everywhere on T. Then there exists almost everywhere ∞ with L−1 , K −1 in L∞ invertible functions L, K ∈ HL(Y) L(Y) such that Φu = L∗ K almost everywhere on T and such that L∞ , K∞ , L−1 ∞ , K −1 ∞ < 1 + . 2 For Φ a function in L∞ L(Y) , the associated Toeplitz operator TΦ on HL(Y) is deﬁned by

TΦ (f ) = PHY2 (Φ · f ). We consider such operators only for the case where Φ is invertible almost everywhere on the unit circle and in addition det Φ∗ Φ is log-integrable: ( det (Φ(ζ)∗ Φ(ζ)) |dζ| > −∞. T

We are now ready to state our result concerning Toeplitz kernels. Here we use the notation L∞ L(Y) to denote the space of essentially uniformly bounded measurable L(Y)-valued functions on the unit circle T. Theorem 7.4. Let the coefficient Hilbert space Y be finite-dimensional. A sub∗ space M ⊂ HY2 has the form M = Ker TΦ for some Φ ∈ L∞ L(Y) with det Φ Φ log-integrable on T if and only if there is an admissible triple (S, B, Γ) so that M has the form M = ΓB −1 · MS,B := ΓB −1 · (KS ∩ MB HY2 ). ∗ Proof. Suppose that Φ ∈ L∞ L(Y) with det Φ Φ log-integrable. Then there ∞ exists an outer function F ∈ HL(Y) solving the spectral factorization problem

Φ(ζ)Φ(ζ)∗ = F (ζ)∗ F (ζ) almost everywhere on T (see e.g. [31]). If we set Φu := F ∗−1 Φ, then Φu is unitary-valued on T and we have the factorization Φ = F ∗ Φu . By Theorem 7.3, we may factor Φu as Φu = L∗ K ∞ with L, K ∈ HL(Y) and L−1 , K −1 ∈ L∞ L(Y) . Let L = Li Lo and K = Ki Ko be the inner-outer factorizations of L and K (again we refer to [31] for details on matrix-valued Hardy space theory). Then Φ has the representation

Φ = F ∗ L∗o L∗i Ki Ko . Suppose now that f ∈ HY2 is in Ker TΦ . This condition can be equivalently written as F ∗ L∗o L∗i Ki Ko f ∈ HY2⊥ , or L∗o L∗i Ki Ko f ∈ F ∗−1 HY2⊥ ∩ L2Y .

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Since F −1 is an outer Nevanlinna-class function, it follows that F ∗−1 HY2⊥ ∩ L2Y = HY2⊥ and we are left with L∗o L∗i Ki Ko f ∈ HY2⊥ . By a similar argument (even easier since L−1 is bounded), we deduce that, o equivalently, L∗i Ki K0 f ∈ HY2⊥ , or Ki Ko f ∈ Li HY2⊥ . As Ki Ko f ∈ HY2 , we actually have Ki Ko f ∈ Li HY2⊥ ∩ HY2 = KLi . Clearly Ki Ko f ∈

Ki HY2

(7.8)

and hence (7.8) takes the sharper form

Ki Ko f ∈ KLi ∩ Ki HY2 =: MLi ,Ki . Solving for f gives f ∈ Ko−1 Ki−1 MLi ,Ki = ΓB −1 · MS,B where we set (S, B, Γ) equal to the admissible triple (Li , Ki , Ko−1 ). Conversely, all the steps are reversible: if f ∈ Ko−1 Ki−1 MLi ,Ki , then f ∈ Ker TΦ . Conversely, suppose that (S, B, Γ) is any admissible triple. Deﬁne Lo ∈ ∞ )±1 as any outer solution of the spectral factorization problem (HL(Y) Lo L∗o = S ∗ BΓ∗ ΓB ∗ S and set Φu = L∗o S ∗ BΓ−1 . Then one can check that Φu is even unitary-valued on T and that Ker TΦu = ΓB −1 · MS,B . Theorem 7.4 combined with Theorem 7.1 leads to the following Corollary, where the free-parameter space MS,B in Theorem 7.4 is replaced by the arguably easier free-parameter space H(KE ). Corollary 7.5. Assume that the coefficient Hilbert space Y ∼ = Cn has finite 2 dimension. A subspace M ⊂ HY is a Toeplitz kernel, i.e., M = Ker TΦ , ∗ for an L∞ L(Y) -function Φ pointwise-invertible on T with det Φ Φ log-integra∞ ±1 ble if and only if there is a function Γ ∈ (HL(Y) ) , inner functions S and B in S(Y), a function E in the Schur class S(W, V) for some auxiliary Hilbert spaces W and V, and a function G : D → L(W, C) such that MG : g(z) → G(z)g(z) maps the de Branges–Rovnyak space H(KE ) isometrically onto (H 2 S · H 2 ) ∩ B · H 2 , so that M = ΓB −1 G · H(KE ). Here the function G can be constructed explicitly from the pair (S, B) by applying the construction in Theorem 7.1. In particular, there exist auxiliary coefficient Hilbert spaces W andV of dimension at most equal to dim(H 2 Σ Σ 11 12 ∈ S(Y ⊕ W, Y ⊕ V) so that S · H 2 ) + 1 and a function Σ = Σ21

Σ22

G(z) = Σ12 (z)(I − E(z)Σ22 (z))−1 S(z) = Σ11 (z) + Σ12 (z)(I − E(z)Σ22 (z))−1 E(z)Σ21 (z).

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Proof. Simply plug in the representation of a space MS,B in Theorem 7.1 into the parametrization of Ker TΦ in Theorem 7.4. Remark 7.6 . A subspace M of H 2 is said to be nearly invariant for the backward shift Mz∗ if f (z)/z ∈ M whenever f ∈ M and f (0) = 0. In [23], Hitt obtained the following characterization of almost invariant subspaces: a subspace M ⊂ H 2 is nearly invariant if and only if there is an inner function u with u(0) = 0 and a holomorphic function g on the disk D so that M = Mg · (H 2 Mu H 2 )

(7.9)

where g is such that the multiplication operator Mg : h(z) → g(z)h(z) acts isometrically from H 2 Mu H 2 into H 2 . Theorem 0.3 from [19] characterizes which functions g are such that Mg acts contractively from H 2 Mu H 2 into H 2 for a given inner function u with u(0) = 0: such a g must have the form g(z) = a1 (z)(1 − u(z)b1 (z))−1 (7.10) a (z) 1 in the Schur class S(C, C2 ). It is not hard to for a function σ(z) = b1 (z)

see that Mg : H 2 Mu H 2 → H 2 is isometric exactly when in addition |a1 (ζ)|2 + |b1 (ζ)|2 = 1

for almost all ζ ∈ T

from which it follows also that g2 = 1. If one starts with g ∈ H 2 of unit norm for which Mg : H 2 Mu H 2 → H 2 is isometric, one can construct the representation (7.10) for g as follows. Let g have inner-outer factorization g = ω · f with ω inner and f outer with f (0) > 0. Let F denote the Herglotz integral of |f |2 , i.e., for z ∈ D we set ( ζ +z |dζ| |f (ζ)|2 . F (z) = ζ −z 2π T

g22

f 22

The fact that = = 1 implies that F (0) = 1; we also note that −1 F (z) has positive real part for z in D. If we then set b = F F +1 , then b is in ∞ the unit ball of H and satisﬁes b(0) = 0. The fact that Mg is isometric from H 2 Mu H 2 into H 2 forces b to be divisible by u, so we can factor b as b = ub1 with b1 in the unit ball of H ∞ . Let a be the unique outer function with |a(ζ)|2 = 1 − |b(ζ)|2 for almost all ζ ∈ T and with a(0) > 0. Set a1 (z) = ω(z)a(z). Then g has the representation (7.10) with this choice of a1 and b1 . The characterization of isometric multipliers from H 2 Mu H 2 into H 2 in this form together with the application to Hitt’s theorem is one of the main results of Sarason’s paper [32]. A direct proof for the special case where u(z) = z appears in [33, Lemma 2, p. 488] in connection with a different problem, namely, the characterization of Nehari pairs. Following the terminology of [27], we say that pair of H ∞ functions (a, b) is a γ-generating pair if (i) (ii) (iii) (iv)

a and b are functions in the unit ball of H ∞ , a is outer and a(0) > 0, b(0) = 0, and |a|2 + |b|2 = 1 almost everywhere on the unit circle T.

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Note that the pair of functions (a, b) appearing above in the representation (7.10) (with a1 = ωa and b = ub1 ) is γ-generating. It is not hard to see that the kernel of any bounded Toeplitz operator Ker Tφ ⊂ H 2 with φ ∈ L∞ is always nearly invariant; hence any Toeplitz kernel M = Ker Tφ is in particular of the form (7.9) as described above. The result of Hayashi in [22] is the following characterization of which nearly invariant subspaces are Toeplitz kernels: the subspace M ⊂ H 2 is the kernel of some bounded Toeplitz operator Tφ if and only if M has the form (7.9) with ω(z) = 1 for some γ-generating pair and inner function u with u(0) = 0 "2 ! a b is an exposed subject to the additional condition that the function 1−zu

point of the unit ball of H 1 . These results have now been extended to the matrix-valued case in [14] and [15]. The paper [17] of Dyakonov obtains the alternative characterization of Toeplitz kernels given in Theorem 7.4 for the scalar case; our proof is a simple adaptation of the proof in [17] to the matrix-valued case, with the matrix-valued factorization result from [5] (Theorem 7.3 above) replacing the special scalar-valued version of the result due to Bourgain [12]. The advantage of this characterization of Toeplitz kernels (as opposed to the earlier results of Hayashi [21] for the scalar case and of Chevrot [15] for the matrix-valued case) is the avoidance of mention of H 1 -exposed points (as there is no useable characterization of such objects). Moreover Dyakonov formulates his results for subspaces of H p rather than just H 2 ; we expect that our Theorem 7.4 extends in the same way to the H p setting, but we do not pursue this generalization here as Theorem 7.1 is at present formulated only for the H 2 setting. Note that our characterization of Toeplitz kernels (Corollary 7.5 above) brings us back to the formulations of Hayashi and Sarason for characterizations of nearly invariant subspaces/Toeplitz kernels in two respects: (1) the characterization involves a multiplication operator which is unitary from some model space of functions to the space to be characterized, and (2) there is an explicit parametrization of which such multipliers have this unitary property.

8. Boundary Interpolation In this section we consider a boundary interpolation problem in a de Branges– Rovnyak space H(KS ). For the sake of simplicity we focus on the scalar-valued case; it is a routine exercise to extend the results presented here to the matrix- or operator-valued case by using the notation and machinery from (j) [8–10]. In what follows, fj (z) = f j!(z) stands for the j-th Taylor coefﬁcient at z ∈ D of an analytic function f . By fj (t0 ) we denote the boundary limit fj (t0 ) := lim fj (z) z→t0

(8.1)

as z tends to a boundary point t0 ∈ T nontangentially, provided the limit exists and is ﬁnite.

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The next theorem collects from the existing literature several equivalent characterizations of the higher order Carath´eodory–Julia condition for a Schur-class function s ∈ S lim inf z→t0

∂ 2n 1 − |s(z)|2 < ∞, ∂z n ∂ z¯n 1 − |z|2

(8.2)

where now z tends to t0 ∈ T unrestrictedly in D. Theorem 8.1. Let s ∈ S, t0 ∈ T and n ∈ N. The following are equivalent: 1. s meets the Carath´eodory–Julia condition (8.2). n 2. The function ∂∂ζ¯n KS (·, ζ) stays bounded in the norm of H(KS ) as ζ tends radially to t0 . 3. It holds that k

1 − |ak |2 + |t0 − ak |2n+2

(2π 0

dμ(θ) <∞ |t0 − eiθ |2n+2

(8.3)

where the numbers ak come from the Blaschke product of the inner-outer factorization of s: ⎫ ⎧ 2π ( iθ ⎬ ⎨ )a ¯k z − ak e +z dμ(θ) . · · exp − s(z) = ⎭ ⎩ ak 1 − z¯ ak eiθ − z k

0

4. The boundary limits sj := sj (t0 ) exist for j = 0, . . . , n and the functions Kt0 ,j (z) :=

j z j− sj zj − s(z) · j+1 (1 − zt0 ) (1 − zt0 )j+1− =0

(j = 0, . . . , n) (8.4)

belong to H(Ks ). 5. The boundary limits sj := sj (t0 ) exist for j = 0, . . . , n and the function Kt0 ,n (z) defined via formula (8.4) belongs to H(Ks ). 6. The boundary limits sj := sj (t0 ) exist for j = 0, . . . , 2n + 1 and are such that |s0 | = 1 and the matrix ⎡ ⎡ ⎤ ⎤ s1 · · · sn+1 s0 . . . sn ⎢ . ⎢ . ⎥ .. ⎥ ⎢ ⎥ . . ... ⎥ (8.5) Psn (t0 ) := ⎢ . ⎦ Ψn (t0 ) ⎣ ⎣ .. ⎦ sn+1

···

s2n+1

0

s0

is Hermitian, where the first factor is a Hankel matrix, the third factor is an upper triangular Toeplitz matrix and where Ψn (t0 ) is the upper triangular matrix given by Ψn (t0 ) =

[Ψj ]n j,=0

,

Ψj

+j+1 = (−1) , t j 0

0 ≤ j ≤ ≤ n.

(8.6)

7. For every f ∈ H(Ks ), the boundary limits fj (t0 ) exist for j = 0, . . . , n. Moreover, if one of the conditions (1)–(7) is satisfied, and hence all, then:

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(a) The matrix (8.5) is positive semidefinite and equals n Psn (t0 ) = Kt0 ,i , Kt0 ,j H(Ks ) i,j=0 .

IEOT

(8.7)

(b) The functions (8.4) are boundary reproducing kernels in H(Ks ) in the sense that f, Kt0 ,j H(Ks ) = fj (t0 ) := lim

z→t0

f (j) (z) j!

for

j = 0, . . . , n.

(8.8)

Proof. Equivalences (1)⇐⇒(4)⇐⇒(5), implication (5)=⇒(6) and statements (a) and (b) were proved in [8]; implication (6)=⇒(1) and equivalence (1)=⇒(7) appear in [11] and [10], respectively. Equivalence (1)⇐⇒(7) was established in [1] for s inner and extended in [20] to general Schur class functions. Equivalence (2)⇐⇒(7) was shown in [34, Section VII]. Theorem 8.1 suggests a boundary interpolation problem for functions in H(Ks ) with data set Db = {s, t, k, {fij }},

(8.9)

consisting of two tuples t = (t1 , . . . , tk ) ∈ Tk and n = (n1 , . . . , nk ) ∈ Nk , a doubly indexed sequence {fij } (with 0 ≤ j ≤ ni and 1 ≤ i ≤ k) of complex numbers and of a Schur-class function s subject to the Carath´eodory–Julia conditions lim inf z→ti

∂ 2ni 1 − |s(z)|2 <∞ ∂z ni ∂ z¯ni 1 − |z|2

for i = 1, . . . , k,

(8.10)

or one of the equivalent conditions from Theorem 8.1. We consider the problem BPH(Ks ) : Given a date set (8.9) satisfying (8.10), find all f ∈ H(Ks ) such that f H(Ks ) ≤ 1 and fj (ti ) := lim

z→ti

f (j) (z) = fij j!

for

j = 0, . . . , ni

and

i = 1, . . . , k. (8.11)

According to Theorem 8.1, conditions (8.10) guarantee that all the boundary limits in (8.11) exist as well as the boundary limits sij := sj (ti ) for j = 0, . . . , 2ni + 1 k

and i = 1, . . . , k.

(8.12)

We let N = i=1 (ni + 1) denote the total number of interpolation conditions in (8.11) and we let X = CN . With the data set (8.9), we associate the matrices ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ 0 T1 E E1 E2 . . . Ek ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎥ and ⎢ T =⎢ ⎣ N ⎦ = ⎣ N1 N2 . . . Nk ⎦, (8.13) . ⎦ ⎣ x∗1 x∗2 . . . x∗k x∗ 0 Tk

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t¯i

⎢ ⎢ ⎢0 Ti = ⎢ ⎢. ⎢. ⎣. 0

1

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...

t¯i ..

.

..

...

0

.

⎤ 0 .. ⎥ ⎥ .⎥ ⎥ ⎥ ⎥ 1⎦ t¯i

⎡ and

Ei

⎤

⎡

1

⎢ ⎥ ⎢ ⎣ Ni ⎦ = ⎣ si,0 x∗i f i,0

259

⎤

0

...

0

si,1

...

⎥ si,ni ⎦.

f i,1

...

f i,ni

(8.14)

Now we deﬁne the function F s by formula (1.5) with E, T and N given by (8.13) and (8.14), and show that F s can be expressed in terms of boundary kernels as (8.15) F s (z) := (E − s(z)N )(I − zT )−1 = Kt1 ,n1 (z) . . . Ktk ,nk (z) where Kti ,ni (z) = Kti ,0 (z)

Kti ,1 (z)

...

Kti ,ni (z)

for

i = 1, . . . , k,

and where the functions Kti ,j are the boundary kernels deﬁned (8.4). Indeed, it follows from definitions (8.14) that ⎡ 1 z ni . . . ⎢ 1 − z t¯i (1 − z t¯i )ni +1 Ei ⎢ −1 ni (I − zTi ) = ⎢ si, z ni − Ni ⎣ si,0 ... 1 − z t¯i (1 − z t¯i )ni +1−

(8.16)

via formula ⎤ ⎥ ⎥ ⎥. ⎦

(8.17)

=0

Multiplying the latter equality by 1 −s(z) on the left and taking into account (8.16) and explicit formulas (8.4) for Kti ,j we obtain (Ei − s(z)Ni ) (I − zTi )

−1

= Kti ,ni (z),

(8.18)

and equality (8.15) now follows from the block structure (8.13) of T , E and N. Now we will show that the problem AIPH(KS ) with the {s, T, E, N, x} taken in the form (8.13), (8.14) is equivalent to the problem BPH(Ks ) . We ﬁrst check that the data is AIP-admissible. The ﬁrst requirement is self-evident since all the eigenvalues of T fall onto the unit circle and therefore (I − zT )−1 is a rational functions with no poles inside D. However, it is worth noting that the pair (E, T ) is not outputstable and so BPH(Ks ) cannot be embedded into the scheme of the problem OAPH(Ks ) of Sect. 6. To verify that the requirements (2) concerning F s are also fulﬁlled, we ﬁrst observe from (8.15) and (8.16) that for a generic vector x = Col1≤i≤k Col0≤j≤ni xij in X , F s (z)x =

ni k

Kti ,j (z)xij .

(8.19)

i=1 j=0

Now it follows from statement (3) in Theorem 8.1 that the operator MF s maps X into H(Ks ).

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[∗]

Furthermore, due to (8.19), the operator P = MF s MF s admits the following block matrix representation with respect to the standard basis of X = CN : r=0,...,nj k P = [Pij ]i,j=1 where Pij = Kti , , Ktj ,r H(Ks ) =0,...,n (8.20) i

and the explicit formulas for Pij in terms of boundary limits (8.12) are (see [9] for details): ⎡ ⎤ sj,0 . . . sj,nj ⎢ .. ⎥ .. ⎥ (8.21) Pij = Hij · Ψnj (tj ) · ⎢ . . ⎦ ⎣ 0

sj,0 n

i where Ψnj (tj ) is deﬁned via formula (8.6), where Hii = [si,+r+1 ],r=1 is a Hankel matrix and where the matrices Hij (for i = j) are deﬁned entry-wise by r si, m+r− [Hij ]r,m = (−1)r− m (ti − tj )m+r−+1 =0 m sj, m+r− − (−1)r (8.22) r (ti − tj )m+r−+1

=0

for r = 0, . . . , ni and m = 0, . . . , nj . It was shown in [9] that the matrix P of the above structure satisﬁes the Stein identity (1.7), with T , E and N given by (8.13), (8.14), whenever P is Hermitian. This works since in the present situation, P is positive semidefinite. Thus, the data set {s, T, E, N, x} is AIP-admissible. By the reproducing property (8.8), representation (8.18) implies that for every f ∈ H(Ks ), [∗] MF s f, xX

= f, M

Fs

xH(Ks ) =

ni k

fj (ti )xij .

i=1 j=0

On the other hand, for x deﬁned in (8.15) and (8.16), x, xX =

ni k

fi,j xij .

i=1 j=0

It follows from the two last equalities that interpolation conditions (8.11) are equivalent to the equality [∗]

MF s f, xX = x, xX [∗]

holding for every x ∈ X , i.e., the equality MF s f = x holds. We now conclude that the problem AIPH(KS ) with the data set {s, T, E, N, x} taken in the form (8.13), (8.14) is equivalent to the BPH(Ks ) . Thus, the problem BPH(Ks ) has a solution if and only if P ≥ xx∗ where P is defined in terms of boundary limits (8.12) for s as in (8.21) and where x is defined in (8.13), (8.14). If this is the case, the solution set for the problem BPH(Ks ) is parametrized as in Theorem 5.1.

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References [1] Ahern, P.R., Clark, D.N.: Radial limits and invariant subspaces. Am. J. Math. 92, 332–342 (1970) [2] Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337– 404 (1950) [3] Ball, J.A., Bolotnikov, V.: Interpolation problems for Schur multipliers on the Drury-Arveson space: from Nevanlinna-Pick to Abstract Interpolation Problem. Integr. Equ. Oper. Theory 62(3), 301–349 (2008) [4] Ball, J.A., Bolotnikov, V., ter Horst, S.: Interpolation in de Branges-Rovnyak spaces. Proc. Am. Math. Soc. 139(2), 609–618 (2011) [5] Barclay, S.: A solution to the Douglas-Rudin problem for matrix-valued functions. Proc. Lond. Math. Soc. 99(3), 757–786 (2009) [6] Beatrous, F., Burbea, J.: Positive-definiteness and its applications to interpolation problems for holomorphic functions. Trans. Am. Math. Soc. 284(1), 247– 270 (1984) [7] Bolotnikov, V., Dym, H.: On degenerate interpolation maximum entropy and extremal problems for matrix Schur functions. Integr. Equ. Oper. Theory 32(4), 367–435 (1998) [8] Bolotnikov, V., Kheifets, A.: A higher order analogue of the Carath´eodory– Julia theorem. J. Funct. Anal. 237(1), 350–371 (2006) [9] Bolotnikov, V., Kheifets, A.: The higher order Carath´eodory–Julia theorem and related boundary interpolation problems. In: Recent Advances in Matrix and Operator Theory, pp. 63–102. Oper. Theory Adv. Appl., vol. 179. Birkh¨ auser Verlag, Basel (2008) [10] Bolotnikov, V., Kheifets, A.: Carath´eodory–Julia type theorems for operator valued Schur functions. J. Anal. Math. 106, 237–270 (2008) [11] Bolotnikov, V., Kheifets, A.: Carath´eodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk. Math. Nachr. 282(11), 1513–1536 (2009) [12] Bourgain, J.: A problem of Douglas and Rudin on factorization. Pac. J. Math. 121, 47–50 (1986) [13] de Branges, L., Rovnyak, J.: Square Summable Power Series. Holt, Rinehart and Winston, New York (1966) [14] Chalendar, I., Chevrot, N., Partington, J.R.: Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces. J. Oper. Theory 63(2), 403– 415 (2010) [15] Chevrot, N.: Kernel of vector-valued Toeplitz operators. Integr. Equ. Oper. Theory 67, 57–78 (2010) [16] Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17, 413–415 (1966) [17] Dyakonov, K.: Kernels of Toeplitz operators via Bourgain’s factorization theorem. J. Funct. Anal. 170(1), 93–106 (2000) [18] Foias, C., Frazho, A.E.: The commutant lifting approach to interpolation problems. In: Oper. Theory Adv. Appl., vol. 44. Birkh¨ auser Verlag, Basel (1990) [19] Frazho, A.E., ter Horst, S., Kaashoek, M.A.: Relaxed commutant lifting: an equivalent version and a new application. In: Recent Advances in Operator

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Theory and Applications, pp. 157–168. Oper. Theory Adv. Appl., vol. 187. Birkh¨ auser Verlag, Basel (2009) Fricain, E., Mashreghi, J.: Boundary behavior of functions in the de BrangesRovnyak spaces. Complex Anal. Oper. Theory 2(1), 87–97 (2008) Hayashi, E.: The kernel of a Toeplitz operator. Integr. Equ. Oper. Theory 9, 588–591 (1986) Hayashi, E.: Classiﬁcation of nearly invariant subspaces of the backward shift. Proc. Am. Math. Soc. 110(2), 441–448 (1990) Hitt, D.: Invariant subspaces of H 2 of an annulus. Pac. J. Math. 134(1), 101– 120 (1988) Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and extension theory of isometric operators. In: Marchenko, V.A. (ed.) Operators in Spaces of Functions and Problems in Function Theory, vol. 146, pp. 83–96. Naukova Dumka, Kiev, 1987. English transl. In: Dym, H., Fritzsche, B., Katsnelson, V., Kirstein, B. (eds.) Topics in Interpolation Theory, pp. 283–298, Oper. Theory Adv. Appl., vol. 95. Birkh¨ auser, Basel (1997) Kheifets, A.: Generalized bi-tangential Schur-Nevanlinna-Pick problem and related Parceval equality. Manuscript No. 3108-889, deposited at VINITI (1989) Kheifets, A.: The abstract interpolation problem and applications. In: Sarason, D., Axler, S., McCarthy, J. (eds.) Holomorphic Spaces, pp. 351–379. Cambridge University Press, Cambridge (1998) Kheifets, A.: On a necessary but not sufﬁcient condition for a γ-generating pair to be a Nehari pair. Integr. Equ. Oper. Theory 21, 334–341 (1995) Kheifets, A., Yuditskii, P.: An analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bitangential Schur–Nevanlinna–Pick problem and J-inner–outer factorization. In: Gohberg, I., Sakhnovich, L.A. (eds.) Matrix and Operator Valued Functions, pp. 133–161, Oper. Theory Adv. Appl., vol. 72. Birkh¨ auser Verlag, Basel (1994) Kovalishina, I.V., Potapov, V.P.: Seven papers translated from the Russian. In: American Mathematical Society Translations, vol. 138. AMS, Providence (1988) Nikolskii, N.: Treatise on the shift operator. Spectral function theory. Grundlehren der Mathematischen Wissenschaften, 273. Springer-Verlag, Berlin (1986) Rosenblum, M., Rovnyak, J.: Hardy Classes and Operator Theory. Oxford University Press (1985) Sarason, D.: Nearly invariant subspaces of the backward shift. In: Gohberg, I., Helton, J.W., Rodman, L. (eds.) Contributions to Operator Theory and its Applications, pp. 481–493. Oper. Theory Adv. Appl., vol. 35. Birkh¨ auser Verlag, Basel (1988) Sarason, D.: Exposed points in H 1 , I, In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds.) The Gohberg Anniversary Collection: Volume II: Topics in Analysis and Operator Theory, pp. 485–496. Oper. Theory Adv. Appl., vol. 41. Birkh¨ auser Verlag, Basel (1989) Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. Wiley, New York (1994)

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¨ [35] Schur, I.: Uber Potenzreihen, die im Innern des Einheitskreises beschr¨ ankt sind. I. J. Reine Angew. Math. 147, 205–232 (1917) Joseph A. Ball Department of Mathematics Virginia Tech, Blacksburg VA 24061-0123, USA e-mail: [email protected] Vladimir Bolotnikov (B) Department of Mathematics The College of William and Mary Williamsburg VA 23187-8795, USA e-mail: [email protected] Sanne ter Horst Mathematical Institute Utrecht University PO Box 80 010 3508 TA, Utrecht The Netherlands e-mail: [email protected] Received: August 11, 2010. Revised: October 19, 2010.

Integr. Equ. Oper. Theory 70 (2011), 265–279 DOI 10.1007/s00020-010-1849-9 Published online December 4, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Numerical Ranges of C0(N ) Contractions Chafiq Benhida, Pamela Gorkin and Dan Timotin Abstract. A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C0 (N ) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu. Mathematics Subject Classification (2000). 47A12, 47A20. Keywords. Contraction, unitary dilation, numerical range.

1. Introduction Suppose H, H are separable Hilbert spaces; we will denote by L(H, H ) the space of bounded linear operators T : H → H and L(H) = L(H, H). The numerical range of an operator T ∈ L(H) is the set W (T ) := {T x, x : x = 1}. Much is known about this set; for example, it is convex, in the ﬁnitedimensional case it is compact, and if T is normal, the closure of W (T ) is the convex hull of the spectrum of T . In general, however, the numerical range is difﬁcult to compute. In this paper, we study new ways of obtaining the numerical range of a contraction T from the numerical ranges of certain unitary dilations of T . If there is a Hilbert space K containing H and an operator T˜ ∈ L(K) such that T = PH T˜|H, where PH denotes the orthogonal projection onto H, the operator T is said to dilate to the operator T˜. (We note that we are considering the so-called weak dilations here, and not power dilations treated in Sz.-Nagy dilation theory.) The operator T˜ is said to be a dilation of T ; more precisely, if dim(K H) = k, then T˜ is called a k-dilation. We will be interested in unitary dilations. A result of Halmos [14, Problem 222(a)] shows that every contraction T has unitary dilations. It is easy to see that W (T ) ⊆ ∩{W (U ) : U is a unitary dilation of T }.

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Choi and Li showed that, in fact, W (T ) = ∩{W (U ) : U ∈ L(H ⊕ H) is a unitary dilation of T }, answering a question raised by Halmos (see, for example, [13]). We note that in the case that H is n-dimensional, these unitary dilations are n-dilations; that is, the dilations are of size 2n × 2n. Before Choi and Li’s work was completed, Gau and Wu [9] studied the so-called compressions of the shift on ﬁnite-dimensional spaces and their numerical ranges. If Sn is the class of all completely non-unitary contractions T (that is, T ≤ 1 and T has no eigenvalue of modulus one) on an n-dimensional space with rank(I − T ∗ T ) = 1, Gau and Wu [9, Corollary 2.8] showed that, in fact, if T ∈ Sn , then W (T ) = {W (U ) : U is an (n + 1)-dimensional unitary dilation of T }. (There is no need to take the closure in the case of ﬁnite-dimensional spaces.) Thus, the unitary dilations may be chosen to be 1-dilations when rank(I − T ∗ T ) = 1. An extension of this result can be found in [8]: namely, if T is an n × n contraction with rank(I − T ∗ T ) = k, then W (T ) = {W (U ) : U ∈ Mn+k is a unitary k-dilation of T }. (1.1) It is easy to see that if rank(I − T ∗ T ) = k, then T has no unitary -dilations for < k, which explains why Gau, Li and Wu refer to (1.1) in [8] as the most “economical” solution to the Halmos problem. We also refer the reader to the papers [10–12], and [20] for work related to this discussion. These authors, as well as others, (in particular, [16,17], and [5]) have studied this problem from a geometric point of view. The analogue of Sn on a space of inﬁnite dimension is the class of contractions with rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1 for which T n and T ∗n tend strongly to 0. It is well known (see, for instance, [19]) that such a T is unitarily equivalent to some model operator Sθ deﬁned as follows: suppose S is the unilateral shift on H 2 . For θ an inner function on the unit disc D, deﬁne Kθ = H 2 θH 2 and Sθ = PKθ S|Kθ . (The operator Sθ is often called a compression of the shift.) Noting that when θ(0) = 0 all unitary 1-dilations of Sθ are equivalent to rank-1 perturbations of Szθ , the authors of [3] show that when θ = B is a Blaschke product we have W (SB ) = {W (U ) : U a rank-1 perturbation of SzB }. Our goal in this paper is to extend these results to operator-valued inner functions. After two preliminary sections, the main results appear in Sect. 4, where we show that the closure of the numerical range of SΘ , where Θ is an inner function in H 2 (CN ), is the intersection of the closures of the numerical ranges of an appropriate family of unitary dilations of SΘ (see Corollary 4.7). In Theorem 4.8 this result is extended to a larger class of contractions, called C0 (N ) (see Definition 2.3). In Section 5, we describe the spectrum of the unitary dilations, obtaining a generalization of the scalar case. We conclude the paper with a brief discussion of a conjecture about the numerical ranges of contractions with ﬁnite defect index.

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2. Preliminaries 2.1. Matrix-Valued Analytic Functions The basic reference that we will use for matrix-valued analytic functions (or, equivalently, functions with values in L(CN )) is [15]; our definitions are simpler since we will consider only bounded (in the operator norm) analytic functions F : D → MN (the set of N × N matrices). These share certain factorization properties similar to those of scalar analytic functions. A bounded analytic matrix-valued function F : D → MN is called outer if det F (z) is outer, and inner if the boundary values (which can be deﬁned as radial limits almost everywhere) are isometries for almost all eit ∈ T. It is known [15, Theorem 5.4] that any analytic bounded F can be factorized as F = ΘE

(2.1)

ˆ E, ˆ then Θ ˆ = ΘV , E ˆ = V ∗E where Θ is inner and E is outer, and, if F = Θ for some constant unitary V . The inner function appearing in (2.1) can be further factorized in two parts. Recall that a Blaschke–Potapov factor b(P, λ)(z) determined by a point λ ∈ D and an orthogonal projection P on CN is an inner function given by the formulas b(P, λ)(z) =

|λ| λ − z ¯ P + (I − P ) for λ = 0, λ 1 − λz

b(P, 0) = zP + (I − P ).

A ﬁnite Blaschke–Potapov product is a product Bn (z) = b(P1 , λ1 )(z) · · · b(Pn , λn )(z), for some λj , Pj , j = 1, . . . , n. If (λj ) is a Blaschke sequence in D (that is, N j (1 − |λj |) < ∞), while Pj is an arbitrary sequence of projections on C , then the sequence Bn (z) converges at each point z ∈ D to B(z), where B is b(Pj , λj ). A function that can be written an inner function denoted by j

as B(z)V , where V is a constant unitary, is called an (inﬁnite) Blaschke– Potapov product. The convergence is uniform on all compact subsets of D. (Note that such a function is sometimes called a left Blaschke–Potapov product; we will not have the occasion to use right Blaschke–Potapov products.) Finally, an inner function Θ is called singular if det Θ(z) = 0 for all z ∈ D. With these definitions, Theorem 4.1 in [15] states that any inner function Θ decomposes as Θ = BS, where B is a (ﬁnite or inﬁnite) Blaschke–Potapov product and S is singular. As in the case of inner–outer factorization, the decomposition is unique up to a unitary constant; more precisely, if we also ˆ Sˆ with B ˆ a Blaschke–Potapov product and Sˆ singular, then have Θ = B ∗ ˆ ˆ B = BV and S = V S for some constant unitary V . The next lemma is a Frostman-type theorem that follows from [15]. Lemma 2.1. Every inner function Θ in H 2 (CN ) is a uniform limit of infinite Blaschke–Potapov products.

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¯ −1 is inner and I − λΘ is outer; thus Proof. For λ ∈ D, (Θ − λI)(I − λΘ) ¯ ¯ −1 (I − λΘ) Θ − λI = (Θ − λI)(I − λΘ) is the inner–outer factorization of Θ − λI. But Corollary 6.1 from [15] says that for a dense set of λ ∈ D the inner factor of Θ − λI is a Blaschke–Potapov product. If we take a sequence λn → 0 with this property and we denote the corresponding Blaschke–Potapov product by B (n) , then ¯ n Θ)−1 , B (n) = (Θ − λn I)(I − λ whence ¯ n Θ) = lim B (n) . Θ = λn I + B (n) (I − λ n→∞

2.2. Model Spaces Let E, E∗ be Hilbert spaces. Suppose we are given an operator-valued inner function Θ(z) : E → E∗ . The model space associated to it is KΘ := H 2 (E∗ ) ΘH 2 (E), The operator TΘ : H 2 (E) → H 2 (E∗ ) deﬁned by TΘ f = Θf is an isometry, and we have PKΘ = I − TΘ T∗Θ .

(2.2)

In particular, Θ(z) = zIE∗ : E∗ → E∗ is inner; we will denote the corresponding TΘ simply by Tz . The model operator SΘ is the compression of Tz to KΘ ; that is, SΘ = PKΘ Tz PKΘ |KΘ . An inner function Θ is called pure if it has no constant unitary direct summand; this is equivalent to assuming Θ(0)x < x for all x = 0. A general inner function is the direct sum of a pure inner function and a unitary constant; from the point of view of model spaces and operators we may consider only pure inner functions. Thus, from now on, we assume that Θ is a pure inner function. Recall that the defect operators and spaces of a contraction T are deﬁned by DT = (I − T ∗ T )1/2 and DT = ran DT . The next lemma shows how one can identify the defect spaces of SΘ ; a good reference is [7, Section 1]. Lemma 2.2. Suppose Θ(z) : E → E∗ is a pure inner function; in particular, DΘ(0) and DΘ(0)∗ have dense ranges. Define the maps ι : E → H 2 (E∗ ), ι∗ : E∗ → H 2 (E∗ ) (on dense domains) by 1 (Θ(z) − Θ(0))ξ, ξ ∈ E; z ι∗ (DΘ(0)∗ ξ∗ ) = (I − Θ(z)Θ(0)∗ )ξ∗ , ξ∗ ∈ E∗ . ι(DΘ(0) ξ) =

(2.3)

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Then ι and ι∗ are isometries with ranges DSΘ and DSΘ∗ respectively, and the following diagram is commutative: ι

E −−−−→ DSΘ ⏐ ⏐ ⏐−Θ(0) ⏐S . Θ

(2.4)

ι∗

E∗ −−−−→ DSΘ∗ In particular, dim DSΘ = dim E and dim DSΘ∗ = dim E∗ . We will occasionally write ιΘ and ιΘ ∗ to indicate the dependence on Θ. From the Sz-Nagy–Foias theory it follows that any C.0 contraction T (that is, a contraction such that the powers of the adjoint tend strongly to 0) is unitarily equivalent to some SΘ , where we can take E = DT and E∗ = DT ∗ . We are actually interested in the particular case when dim DT = dim DT ∗ = N < ∞. The following definition appears in [19]. Definition 2.3. A contraction T ∈ L(H) is said to be of class C0 (N ) if dim DT = dim DT ∗ = N < ∞ and T n , T ∗n tend strongly to 0. If T ∈ C0 (N ), then T is unitarily equivalent to SΘ , with Θ(z) : CN → C . In this case Θ(0) is a strict contraction, and formulas (2.3) are deﬁned on all of CN . The next lemma collects a few facts that we shall use. N

Lemma 2.4. Suppose Θ(z) : CN → CN is an inner function. (i) KΘ is finite dimensional if and only if Θ is a finite Blaschke–Potapov product. (ii) If Θn → Θ in the uniform norm, then PKΘn → PKΘ uniformly. (iii) Let bj = b(Pj , λj ). If B = bk is an infinite Blaschke–Potapov product k

and Bn = b 1 · · · bn , then (a) KB = n KBn ; (b) Bn ξ → Bξ in H 2 (CN ), for any ξ ∈ CN . Proof. Statement (i) can be found, for instance, in [18, Ch.2, Lemma 5.1], while (ii) follows from (2.2). (iii), a standard normal family argument shows that BH 2 (CN ) =

As for 2 N n Bn H (C ), and therefore (a) follows by passing to orthogonal complements. ˜n , where B ˜n is also an inﬁnite Blaschke–Potapov For (b), write B = Bn B product. If B(0) is invertible, the pointwise convergence of Bn to B implies ˜n (0) → ICN , whence (taking norms and scalar products in H 2 (CN )) that B ˜n ξ Bn ξ − Bξ2 = 2ξ2 − 2Bn ξ, Bξ = 2ξ2 − 2ξ, B ˜n (0)ξ → 0. = 2ξ2 − 2ξ, B In the general case, write Bn = CDn , where C contains the Blaschke–Potapov factors b(P, λ) corresponding to λ = 0. We have then B = CD (with D an inﬁnite Blaschke–Potapov product), while the previous argument shows that Dn ξ → Dξ in H 2 (CN ). Multiplying with the inner function C yields the result.

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3. Unitary N -Dilations The next result is folklore; we give a short proof. Proposition 3.1. Suppose T ∈ L(H) is a contraction such that dim DT = dim DT ∗ = N < ∞. If U ∈ L(H ⊕ E) is a unitary N -dilation of T , then there exist unitary operators ω : E → DT , ω∗ : E → DT ∗ , such that T DT ∗ ω U= . (3.1) −ω∗∗ T ∗ ω ω∗∗ DT Conversely, any choice of ω : E → DT , ω∗ : E → DT ∗ yields, through formula (3.1), a unitary N -dilation of T . Proof. Theorem 1.3 of [1] says that if T T12 :H⊕E →H⊕E T21 T22 is a contraction, then there exist contractions Γ1 : E → DT ∗ , Γ2 : DT → E and Γ : DΓ1 → DΓ∗2 such that T12 = DT ∗ Γ1 , T21 = Γ2 DT and T22 = −Γ2 T ∗ Γ1 + DΓ∗2 ΓDΓ1 . We apply this result to U . Since U (x ⊕ 0)2 = T x2 + Γ2 DT x2 ≤ T x2 + DT x2 = x2 , and the ﬁrst column of U is an isometry, the last term is equal to the ﬁrst; so the middle inequality is an equality. This means that Γ2 acts isometrically on the image of DT ; but this is precisely DT , whence Γ2 has to be an isometry. In fact, Γ2 is unitary, since it acts between spaces of the same dimension N . Similarly, we obtain that Γ1 is unitary, which implies Γ acts between 0 spaces. The result follows if we let ω = Γ1 and ω∗ = Γ∗2 . The converse is immediate. We can write (3.1) as T I 0 U= DT 0 ω∗∗

DT ∗ −T ∗

I 0

0 . ω

The next corollary follows immediately from this formula. Corollary 3.2. Suppose T ∈ L(H) is a contraction with dim DT = dim DT ∗ = N < ∞ and T u12 u= ∈ L(H ⊕ E) u21 u22 is a unitary N -dilation of T . Then any unitary N -dilation U of T on H ⊕ E (for some Hilbert space E with dim E = N ) is given by the formula I 0 I 0 U= u (3.2) 0 Ω∗∗ 0 Ω where Ω, Ω∗ : E → E are unitaries. We are interested in consequences for SΘ . The notation below refers to that of Lemma 2.2.

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Lemma 3.3. All unitary N -dilations of SΘ to KΘ ⊕ CN can be indexed by unitaries Ω, Ω∗ : CN → CN , according to the formula

ι∗ DΘ(0)∗ Ω SΘ UΩ,Ω∗ = . (3.3) Ω∗∗ DΘ(0) ι∗ Ω∗∗ Θ(0)∗ Ω Proof. Let us apply Proposition 3.1 (the part stated as a converse) to the case T = SΘ , E = CN , ω = ι∗ , ω∗ = ι. We obtain the following N -dilation of T to H ⊕ CN :

SΘ DSΘ∗ ι∗ V = ∗ . ∗ ι DSΘ −ι∗ SΘ ι∗ The commutative diagram (2.4) yields the relations SΘ ι = −ι∗ Θ(0), ∗

∗ SΘ ι∗

∗ ι∗ SΘ = −Θ(0)∗ ι∗∗ .

(3.4)

∗

= Θ(0) . It follows immediately that −ι ∗ ∗ From (3.4) we have ι∗ SΘ SΘ ι = Θ(0)∗ Θ(0), whence ι∗ (IH − SΘ SΘ )ι = ∗ ∗ ICN −Θ(0) Θ(0) and thus ι DSΘ ι = DΘ(0) . Multiplying the last relation with ι∗ on the right, and taking into account that ιι∗ = PDSΘ and DSΘ PDSΘ = DSΘ , we obtain ι∗ DSΘ = DΘ(0) ι∗ . A similar computation yields DSΘ∗ ι∗ = ι∗ DΘ(0)∗ , and thus

V =

SΘ DΘ(0) ι∗

ι∗ DΘ(0)∗ . Θ(0)∗

Applying Corollary 3.2 to E = E = CN and u = V ﬁnishes the proof.

We have thus parametrized all unitary N -dilations of SΘ to KΘ ⊕CN by pairs of unitaries on CN . If we are interested only in classes of unitary equivalence, we may take a single unitary as parameter, since UΩ,Ω∗ , UΩΩ∗∗ ,I , and UI,Ω∗ Ω∗ are all unitarily equivalent, and therefore have the same numerical range. In the sequel, we let

ι∗ DΘ(0)∗ Ω SΘ Θ UΩ := . (3.5) DΘ(0) ι∗ Θ(0)∗ Ω

4. The Main Result Let K denote the complete metric space of all nonempty compact subsets of C, endowed with the Hausdorﬀ distance d. Suppose that A ∈ K and τ : X → K is a continuous mapping deﬁned on some compact space X. We will say that τ wraps A if for each open half-plane H in C that contains A there exists x ∈ X such that τ (x) ⊂ H.

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Lemma 4.1. Let An , A ∈ K with An → A. Let X be a compact space, and τn , τ : X → K be continuous mappings such that τn → τ uniformly on X. Suppose that for each n, τn wraps An . Then τ wraps A. Proof. If H is an open half-plane and A ⊂ H, let H be a slight translate of H towards A such that we still have A ⊂ H . For n sufﬁciently large An ⊂ H . It follows then from the assumption that for each n sufﬁciently large there exists xn ∈ X such that τn (xn ) ⊂ H . Letting x be a limit point of xn in X, a simple /2 argument shows that τ (x) ⊂ H. Remark 4.2. Suppose A ⊂ τ (x) for all x, and A and τ (x) are convex for all x. If τ wraps A, then A = ∩x∈X τ (x). The converse is not true, as can easily be seen by considering A to be the intersection of two line segments. However, the result that we quote below (in Theorem 4.6, Step 1) from [8] actually yields a wrapping property of A, not only intersection. The following simple lemma will be used in Sect. 6. Lemma 4.3. Suppose A ∈ K and τ : X → K wraps A. If B ∈ K, A˜ := co(A, B), ˜ τ˜(x) = co(τ (x), B), then τ˜ wraps A. ˜ Then it contains A and B. By Proof. Take a half-plane H that contains A. hypothesis, there exists x ∈ X such that τ (x) ⊂ H. Since H is convex, it follows that τ˜(x) ⊂ H, which proves the lemma. The elements of K that we will consider are closures of numerical ranges. The next lemma states some continuity properties for these sets. Lemma 4.4. (i) Let T, S ∈ L(H). Then d(W (T ), W (S)) ≤ T − S. (ii) If Hn ⊂ Hn+1 ⊂ · · · ⊂ H and n Hn = H, then for all T ∈ L(H), W (T ) = W (PHn T PHn |Hn ). n

In particular, d(W (PHn T PHn |Hn ), W (T )) → 0. Suppose T ∈ L(H), and P, Q are orthogonal projections on H, with P − Q < 1. Then 2 d(W (P T P |P H), W (QT Q|QH)) ≤ T · P − Q 1 + . (1 − P − Q)2 (iii)

In particular, if Pn , P are orthogonal projections and Pn → P uniformly, then d(W (Pn T Pn |Pn H), W (P T P |P H)) → 0. In the sequel, we will let X denote the space of unitary operators on CN and we deﬁne τ Θ (Ω) = W (UΩΘ ), where UΩΘ is given by (3.5). The next lemma singles out a technical argument that will be used twice in the proof of Theorem 4.6. Lemma 4.5. Suppose that Θ, Θn : D → L(CN ) are inner functions, such that (a) Θn ξ → Θξ in H 2 (CN ), for any ξ ∈ CN ; (b) d(W (SΘn ), W (SΘ )) → 0;

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if we define Vn,Ω =

SΘn DΘ(0) ι∗ PKΘn

273

PKΘn ι∗ DΘ(0)∗ Ω , Θ(0)∗ Ω

then d(W (Vn,Ω ), W (UΩΘ )) → 0 uniformly in Ω.

If τ Θn wraps W (SΘn ) for all n, then τ Θ wraps W (SΘ ). Θ n Proof. Condition (a) implies, by formulas (2.3), that ιΘn → ιΘ and ιΘ ∗ → ι∗ ; whence Vn,Ω −UΩΘn → 0 uniformly in Ω. Therefore d(W (Vn,Ω ), W (UΩΘn )) →

0 uniformly in Ω, which, together with (c), yields d(W (UΩΘ ), W (UΩΘn )) → 0 uniformly in Ω. We may then apply Lemma 4.1 with An = W (SΘn ), A = W (SΘ ), τn = τ Θn , and τ = τ Θ . With these notations, the assumption of Lemma 4.5 becomes that τn wraps An , and it follows that τ wraps A. Theorem 4.6. For any inner function Θ : D → L(CN ), the map τ Θ wraps W (SΘ ). Proof. The proof will be done in three steps. Step 1. In case Θ is a ﬁnite Blaschke–Potapov product, the space KΘ is ﬁnite dimensional and the statement is a consequence of [8, Theorem 1.2] (see Remark 4.2). Step 2. To pass to inﬁnite Blaschke–Potapov products, suppose that Θ = B and Θn = Bn (where the notation is as in Lemma 2.4 (iii)). We want to use Lemma 4.5. Condition (a) therein is satisﬁed by Lemma 2.4 (iii)(b). Applying Lemma 4.4 (ii) to T = SB , H = KB , Hn = KBn , we obtain d(W (SΘn ), W (SΘ )) → 0, and therefore (b) is also satisﬁed. Finally, to obtain (c), we apply Lemma 4.4 (ii) again, this time to T = UΩB , H = KB ⊕ CN , Hn = KBn ⊕ CN . By Step 1 we know that τ Bn wraps W (SBn ) for all n, and Lemma 4.5 implies that τ B wraps W (SB ). Step 3. According to Lemma 2.1, we take a sequence of Blaschke–Potapov products Θn that tend uniformly to an arbitrary inner function Θ. Condition (a) in Lemma 4.5 is obviously satisﬁed. By Lemma 2.4 (ii), we have PKΘn → PKΘ uniformly. Since SΘn = PKΘn Tz PKΘn |KΘn and SΘ = PKΘ Tz PKΘ |KΘ , Lemma 4.4 (iii), applied to H = H 2 (CN ), T = Tz , Pn = PKΘn , and P = PKΘ , yields condition (b) in Lemma 4.5. To obtain (c), apply Lemma 4.4 (iii) again, this time to H = H 2 (CN ) ⊕ C , Pn = PKΘn ⊕CN , P = PKΘ ⊕CN , and

ι∗ DΘ(0)∗ Ω Tz T = . DΘ(0) ι∗ Θ(0)∗ Ω N

Once again, we use the fact that PKΘn → PKΘ uniformly to conclude that (c) is also satisﬁed. By Step 2 we know that τ Θn wraps W (SΘn ) for all n, and Lemma 4.5 implies that τ Θ wraps W (SΘ ). The proof of the theorem is ﬁnished.

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The next corollary is a consequence of Remark 4.2. Corollary 4.7. Suppose Θ : D → L(CN ) is an inner function. Then W (SΘ ) = W (UΩΘ ),

(4.1)

Ω

where UΩΘ is defined by (3.5), while the intersection is taken with respect to all unitary operators Ω on CN . Since C0 (N ) contractions are unitarily equivalent to model operators SΘ , with Θ : D → L(CN ) inner, we may extend the result to this class. Theorem 4.8. Suppose T ∈ L(H) is a contraction of class C0 (N ), U is the set of unitary N -dilations of T to H ⊕ CN , and τ : U → K is defined by τ (U) = W (U). Then τ wraps W (T ). In particular, W (T ) = W (U). U∈U

5. Spectrum and Numerical Range of N -Dilations In the case that Θ is a ﬁnite (scalar) Blaschke product, the spectrum of the extensions UΩΘ can be identiﬁed precisely. Since UΩΘ is a unitary operator, the numerical range is the (closed) convex hull of the spectrum, and we obtain a complete description of W (UΩΘ ), (see, for example, [3,5], and [9]). The same can be done in the case of a general matrix-valued inner function, by relating these functions to perturbations of a “slightly larger” model operator. We need some preliminary material, for which the reference is [7]. If T ∈ L(H) is a contraction, then T (DT ) ⊂ DT ∗ , T (DT⊥ ) ⊂ DT⊥∗ , and T acts unitarily from DT⊥ onto DT⊥∗ . For A : DT → DT ∗ , deﬁne T [A] ∈ L(H) by the formula Ax if x ∈ DT , T [A]x = (5.1) T x if x ∈ DT⊥ . It is easy to see that T [A] is a contraction (respectively isometry, coisometry, unitary) if and only if A is a contraction (respectively isometry, coisometry, unitary). We will be interested in the particular situation when T = SΞ , with Ξ(z) = zΘ(z), with Θ : D → L(CN ) an inner function and A unitary. According to formulas (2.3), we have then DSΞ = ΘCN , DSΞ∗ = CN . Thus a unitary mapping A : DT → DT ∗ is given by AΘ(z)ξ = ωξ, with ω : CN → CN unitary. We will write A = Aω . Since KΞ = KΘ ⊕ ΘCN = zKΘ ⊕ CN , we have unitary operators J, J∗ : KΘ ⊕ CN → KΞ deﬁned by J(f ⊕ ξ) = f + Θξ,

J∗ (f ⊕ ξ) = zf + ξ.

We will write ZΞ (ω) := J ∗ SΞ [Aω ]J ∈ L(KΘ ⊕ CN ).

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With these notations, the lemma below follows from [7, Theorem 3.6]. For a more general result, see [2, Theorem 4.5]. Lemma 5.1. With the above assumptions, the spectrum of ZΞ (ω) is the union of the sets of points ζ ∈ T at which Ξ has no analytic continuation and the set of points ζ ∈ T at which Ξ has an analytic continuation but Ξ(ζ) − ω is not invertible. In particular, if Ξ is a finite Blaschke–Potapov product, then σ(ZΞ (ω)) = {ζ ∈ T : det(Ξ(ζ) − ω) = 0}. The relation with N -dilations is given by the next proposition. Proposition 5.2. Suppose Θ : D → L(CN ) is an inner function. Define Ξ(z) = zΘ(z). Then UΩΘ = ZΞ (Ω). Proof. We have ZΞ (Ω) = J ∗ SΞ [AΩ ]J = (J ∗ J∗ )(J∗∗ SΞ [AΩ ]J).

(5.2)

Since SΞ [AΩ ]J(f ⊕ ξ) = SΞ [AΩ ](f + Θξ) = zf + Ωξ = J∗ (f ⊕ Ωξ), it follows that (J∗∗ SΞ [AΩ ]J)(f ⊕ ξ) = f ⊕ Ωξ.

(5.3)

∗ N N To compute J J∗ : K Θ ⊕ C → KΘ ⊕ C , denote the correspondA11 A12 ing matrix by , and let Pi denote the projection on the i-th A21 A22 component in KΘ ⊕ CN . We have

A11 (f ) = P1 (J ∗ J∗ (f ⊕ 0)) = P1 (J ∗ zf ) = PKΘ zf = SΘ f. Further, J∗ (0⊕ξ) = ξ, viewed as a constant function in KΞ . This decomposes with respect to KΞ = KΘ ⊕ ΘCN as ξ = (1 − ΘΘ(0)∗ )ξ + ΘΘ(0)∗ ξ = ι∗ DΘ(0)∗ ξ + ΘΘ(0)∗ ξ. It follows that J ∗ J∗ (0 ⊕ ξ) = ι∗ DΘ(0)∗ ξ ⊕ Θ(0)∗ ξ, and thus A12 = ι∗ DΘ(0)∗ ,

A22 = Θ(0)∗ .

To obtain A21 , we work now with the adjoint map J∗∗ J. We have J(0 ⊕ ξ) = Θξ, and the last function decomposes with respect to KΞ = zKΘ ⊕ CN as Θ − Θ(0) Θξ = z ξ + Θ(0)ξ = zιDΘ(0) ξ + Θ(0)ξ, z whence J∗∗ J(0 ⊕ ξ) = ιDΘ(0) ξ ⊕ Θ(0)ξ. Therefore A∗21 = ιDΘ(0) ,

A21 = DΘ(0) ι∗ .

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Finally,

∗

J J=

SΘ

DΘ(0) ι∗

ι∗ DΘ(0)∗ Θ(0)∗

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(5.4)

Now the proof follows by comparing equations (5.2), (5.3), and (5.4) with (3.5). From Lemma 5.1 and Proposition 5.2, the ﬁnal result about spectrum and numerical range of N -dilations follows. Theorem 5.3. With the above notations, the spectrum σ(UΩΘ ) is the union of the sets of points ζ ∈ T at which Θ has no analytic continuation and the set of points ζ ∈ T at which Θ has an analytic continuation but ζΘ(ζ) − Ω is not invertible, while W (UΩΘ ) is the closed convex hull of σ(UΩΘ ). In particular, if Θ is a finite Blaschke–Potapov product, then σ(UΩΘ ) = {ζ ∈ T : det(ζΘ(ζ) − Ω) = 0} and W (UΩΘ ) is the closed convex hull of the zeros of the polynomial det(ζΘ(ζ) − Ω). The scalar case of Theorem 5.3 is contained in [11, Theorem 6.3].

6. Final Remarks It seems natural to formulate the following conjecture, which would complement Choi and Li’s answer to Halmos’ question. Conjecture 6.1. Suppose T ∈ L(H) is a contraction with dim DT = dim DT ∗ = N < ∞, U is the set of unitary N -dilations of T to H ⊕ CN , and τ : U → K is deﬁned by τ (U) = W (U). Then τ wraps W (T ). In particular, W (T ) = W (U) (6.1) U∈U

Note that the conjecture is open even for N = 1. The main points that have been settled are presented below. In the sequel T ∈ L(H) will be a contraction with dim DT = dim DT ∗ = N < ∞. 6.1. Theorem 4.8 shows that the conjecture is true for C0 (N ) contractions. 6.2. As we show below, if we add a unitary operator to one for which the conjecture holds, the conjecture will still hold. Lemma 6.2. If T1 = T ⊕ V , where V is unitary, and Conjecture 6.1 is true for T , then it is true for T1 .

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Proof. The unitary N -dilations of T ⊕ V , for V unitary, are exactly U ⊕ V , with U a unitary N -dilation of T . Since the numerical range of a direct sum is the convex hull of the numerical ranges of the components, the statement follows from Lemma 4.3. Again by [19], it is known that an arbitrary contraction is the direct sum of a completely nonunitary contraction and a unitary; it follows then from Lemma 6.2 that it is enough to prove Conjecture 6.1 for a completely nonunitary T . 6.3. We now specialize to the case N = 1. Suppose T is a completely nonunitary contraction with scalar characteristic function θ [19]; now θ is an arbitrary function in the unit ball of H ∞ . Then T is unitarily equivalent to the model operator Tθ ∈ L(Kθ ), where Kθ = (H 2 ⊕ L2 (Δ)) {θf ⊕ (1 − |θ|2 )1/2 f : f ∈ H 2 }, with Δ = {ζ ∈ T : |θ(ζ)| < 1}, while Tθ (f ⊕ g) = PKθ (zf ⊕ ζg). If θ is inner, then Δ = ∅ and we are back in the C0 (1) case discussed in 6.1. On the other hand, the spectrum of Tθ may be precisely identiﬁed in terms of the characteristic function: σ(Tθ ) is the union of the zeros of θ inside D and the complement of the open arcs of T on which |θ(ζ)| = 1 and through which θ has an analytic extension outside the unit disk (see again [19] for a general statement; in the scalar case it was known earlier and is usually called the Livsic–Moeller theorem). In particular, it follows that Conjecture 6.1 can be settled for a situation at the opposite extreme of the case in which θ is inner. Namely, if |θ(ζ)| < 1 almost everywhere on T, then σ(Tθ ) ⊃ T. In this case, Conjecture 6.1 is trivially true: W (T ) as well as every W (U ) must equal D. A ﬁnal remark: the case dim DT = dim DT ∗ = N < ∞ is the only one in which we can hope to obtain the numerical range of T by using “economical” unitary dilations. If dim DT = dim DT ∗ , or if both dimensions are inﬁnite, then it is easy to see that for any unitary dilation U ∈ L(K) of T ∈ L(H) one must have dim(K H) = ∞.

Acknowledgements We thank the referee for useful remarks and, in particular, for pointing out the need for a slight change to the proof of Theorem 4.6.

References [1] Arsene, Gr., Gheondea, A.: Completing matrix contractions. J. Oper. Theory 7, 179–189 (1982)

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[2] Ball, J.A., Lubin, A.: On a class of contractive perturbations of restricted shifts. Pac. J. Math. 63, 309–323 (1976) [3] Chalendar, I., Gorkin, P., Partington, J.R.: Numerical ranges of restricted shifts and unitary dilations. Oper. Matrices 3, 271–281 (2009) [4] Choi, M.-D., Li, C.-K.: Constrained unitary dilations and numerical ranges. J. Oper. Theory 46, 435–447 (2001) [5] Daepp, U., Gorkin, P., Voss, K.: Poncelet’s theorem, Sendov’s conjecture and Blaschke products. J. Math. Anal. App. 365, 93–102 (2010) [6] Durszt, E.: On the numerical range of normal operators. Acta Sci. Math. (Szeged) 25, 262–265 (1964) [7] Fuhrmann, P.A.: On a class of finite dimensional contractive perturbations of restricted shifts of finite multiplicity. Israel J. Math. 16, 162–175 (1973) [8] Gau, H.-L., Li, C.-K, Wu, P.Y.: Higher-rank numerical ranges and dilations. J. Oper. Theory 63, 181–189 (2010) [9] Gau, H.-L., Wu, P.Y.: Numerical range of S(φ). Linear Multilinear Algebra 45(1), 49–73 (1998) [10] Gau, H.-L., Wu, P.Y.: Numerical range circumscribed by two polygons. Linear Algebra Appl. 382, 155–170 (2004) [11] Gau, H.-L., Wu, P.Y.: Numerical range and Poncelet property. Taiwanese J. Math. 7(2), 173–193 (2003) [12] Gau, H.-L., Wu, P.Y.: Dilation to inflations of S(φ). Linear Multilinear Algebra 45(2–3), 109–123 (1998) [13] Halmos, P.R.: Numerical ranges and normal dilations. Acta Sci. Math. (Szeged) 25, 1–5 (1964) [14] Halmos, P.R.: A Hilbert space problem book, 2nd edn. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, 17. Springer, New York, Berlin (1982) [15] Katsnelson, V.E., Kirstein, B.: On the theory of matrix-valued functions belonging to the Smirnov class. Topics in interpolation theory 299–350, Oper. Theory Adv. Appl., 95 (1997) [16] Mirman, B., Borovikov, V., Ladyzhensky, L., Vinograd, R.: Numerical ranges, Poncelet curves, invariant measures. Linear Algebra Appl. 329(1–3), 61–75 (2001) [17] Mirman, B.: Numerical ranges and Poncelet curves. Linear Algebra Appl. 281(1–3), 59–85 (1998) [18] Peller, V.V.: Hankel Operators and their Applications. Springer, Berlin (2003) [19] Sz-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North-Holland Publishing, Amsterdam (1970) [20] Wu, P.Y.: Polygons and numerical ranges. Amer. Math. Mon. 107(6), 528–540 (2000)

Chafiq Benhida UFR de Math´ematiques, Universit´e des Sciences et Technologies de Lille, 59655 Villeneuve D’Ascq Cedex, France e-mail: [email protected]

Vol. 70 (2011)

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Pamela Gorkin Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA e-mail: [email protected] Dan Timotin (B) Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania e-mail: [email protected] [email protected] Received: August 27, 2010. Revised: November 2, 2010.

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Integr. Equ. Oper. Theory 70 (2011), 281–300 DOI 10.1007/s00020-010-1850-3 Published online December 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces Anahit Galstian Dedicated to Professor Kunihiko Kajitani on the occasion of his 70th birthday Abstract. The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given. Mathematics Subject Classification (2000). Primary 35L80, 35S10; Secondary 45D05. Keywords. Pseudo-differential operators, exponential function, Gevrey classes, hyperbolic operators, multiple characteristics.

1. Introduction The remarkable progress made in the theory of linear partial differential operators over the last three decades is essentially due to the extensive application of the microlocal analysis. In particular, the Gevrey pseudo-differential operators and related wave front sets were used beginning in the 1970s as a natural version of the corresponding C ∞ techniques. Nowadays they play a definite and important role, in connection with the study of the nonlinear partial differential operators and linear operators with multiple characteristics. Unlike in the Sobolev spaces, the spaces of the Gevrey functions allow one to appeal to the pseudodifferential operators of inﬁnite order, which in many cases is crucial for the solvability of the problem for the above mentioned equations. Kajitani and Wakabayashi in [6] introduced the pseudo-differential operators of inﬁnite order with a symbol in a Gevrey class. In particular, Kajitani in [7] treated pseudo-differential operators of inﬁnite order as Fourier integral operators

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with the complex-valued phase functions. He assumed that the order of an imaginary part of those phase functions does not exceed 1/θ, where θ is an exponent of a Gevrey class of functions. In particular, by utilizing such operators, Kajitani [5] proved a local solvability of the Cauchy problem for the nonlinear hyperbolic systems in Gevrey classes. In fact, operators of inﬁnite order appear in the construction of the fundamental solution for the Cauchy problem for operators with multiple characteristics whose lower-order terms satisfy the so-called Gevrey-type Levi conditions. These conditions for the second order equation Dt2 u − t2l Dx2 u + atk Dx u = f imply 1 ≤ θ < (2l − k)/(l − k − 1), where θ is the Gevrey exponent. They guarantee the well-posedness of the Cauchy problem in Gevrey class G(θ) (Ω) [3,4,14]. Meanwhile, according to the well-known H¨ ormander’s theorem, the wave front set of the solution is contained in the characteristic set of the principal symbol of the operator. This hints at the structure of the fundamental solution of the Cauchy problem. More precisely, it suggests that the fundamental solution can be written as a product of the Fourier integral operator with zero-order elliptic symbol and the pseudo-differential operator. If k ≥ l − 1, then the last pseudo-differential operator is of ﬁnite order [15], while for l, k, and θ satisfying the above mentioned inequalities, it is a pseudo-differential operator of inﬁnite order [11,16]. This operator is one of the main factors of the fundamental solution constructed in [16] as well as an important component of the proof of the uniqueness theorem for the degenerate elliptic operators [1]. It turns out that this operator is an exponential function of some other pseudo-differential operator appearing after the so-called perfect diagonalization and an application of the Egorov’s theorem. For details see [17]. Thus, the main ingredient of such construction is the exponential function of the pseudo-differential operator. The case of the operators with x-independent coefﬁcients, which is applicable to the Gevrey classes of functions, is given in Section 5 [16]. In that article the Cauchy problem by means of the Fourier transform is reduced to the ordinary differential equation with the large parameter and a single turning point. Then the representation of any solution of that ordinary differential equation, in the terms of the amplitude functions and phase functions, is given. The inverse Fourier transform, applied to the last representation, leads to the fundamental solution that is written as the sum of Fourier integral operators of inﬁnite order.

2. Preliminaries and Main Result First, we state some definitions and facts from [8–10,18]. Let Ω be an open set in Rn , K a compact subset of Ω and let θ > 1, A > 0. We denote by G(θ),A (K) the set of all complex valued functions ϕ ∈ C ∞ (Ω) such that ϕK,A = A−|α| α!−θ sup |Dxα ϕ(x)| < +∞. α∈Zn +

x∈K

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We set G(θ) (Ω) = lim ←−

lim G(θ),A (K), −→

K→Ω A→+∞

(θ),A

G0

(K) = G(θ),A (K) ∩ C0∞ (K),

and (θ)

G0 (Ω) = lim −→

lim G(θ),A (K) ∩ C0∞ (K). −→

K→Ω A→+∞

(θ)

The spaces of the ultradistributions of order θ on Ω, G(θ) (Ω), G0 (Ω), are (θ) deﬁned as the duals of G(θ) (Ω), G0 (Ω), respectively. G(θ) (Ω) can be identi (θ) ﬁed with the subspace of ultradistributions of G0 (Ω) with compact support. (θ) (θ) (θ) G0 (Ω), G0 (Ω), G(θ) (Ω) and G0 (Ω) are complete, Montel and Schwartz spaces. (θ) The Fourier transform of a function u ∈ G0 (Rn ) is the function u ˆ(ξ) = u(exp(−i ·, ξ)), ξ ∈ Rn . This function can be extended to an entire analytic function in Cn , which is called the Fourier–Laplace transform of u. A theorem (θ) analogous to the Paley–Wiener theorem holds for the elements of G0 (Rn ) (θ) and G0 (Rn ) [8,9]. Let Ω be an open set in Rn and let θ, ρ, δ be real numbers such that θ > 1, 0 ≤ δ < ρ ≤ 1, θρ ≥ 1. ∞,θ Definition 2.1. ([10,18]) We shall denote by Sρ,δ (Ω) the space of all func∞ n tions p ∈ C (Ω × R ) satisfying the following condition: for every compact subset K ⊂ Ω there exist constants C > 0 and B ≥ 0 and for every ε > 0 there exists a constant cε such that

sup Dξα Dxβ p(x, ξ) ≤ cε C |α+β| α!β!θ(ρ−δ) (1 + |ξ|)−ρ|α|+δ|β| exp(ε|ξ|1/θ )

x∈K

for every α, β ∈ Zn+ and for every ξ ∈ Rn such that |ξ| ≥ B|α|θ . (θ)

(θ)

∞,θ For every p ∈ Sρ,δ (Ω) and u ∈ G0 we can deﬁne on G0 (Ω) the operator: (θ) −n u(ξ)dξ, u ∈ G0 (Ω). (2.1) ei<x,ξ> p(x, ξ)ˆ P u(x) = (2π) ∞,θ (Ω) we denote the space of all operators P of the form (2.1) with By OP Sρ,δ

∞,θ p ∈ Sρ,δ (Ω). We shall write P = p(·, D). We have the following important

∞,θ estimate for such operators [10,18]. Let p ∈ Sρ,δ (Ω) and let K be a compact subset of Ω and A > 0. There exist positive constants ε0 , C, B such that: ≤ C exp(−2ε0 |η|1/θ + ε0 |ξ|1/θ )vK,A ei<x,η> p(x, ξ)ˆ v (x)dx (θ),A

for every ξ ∈ Rn , |η| > B, v ∈ G0 (Ω). The last estimate implies that these (θ) operators are continuous linear operators from G0 (Ω) to G(θ) (Ω), which can (θ) be extended to the operators from G(θ) (Ω) to G0 (Ω) [10,18]. We denote (θ) by V θ (Ω) the space of the last operators from G(θ) (Ω) to G0 (Ω). For

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the symbols of operators there is an asymptotic expansion similar to C ∞ case [12]. ∞,θ Definition 2.2. [10,18] We denote by F Sρ,δ (Ω) the space of all formal sums ∞ n p (x, ξ), where p ∈ C (Ω × R ) satisﬁes the following condition: for j j j≥0 every compact set K ⊂ Ω there exist constants C > 0 and B ≥ 0 and for every ε > 0 there exists a constant cε , such that for any α, β ∈ Zn+ and every ξ ∈ Rn with |ξ| ≥ B(|α| + j)θ :

sup |Dξα Dxβ pj (x, ξ)| ≤ cε C |α+β|+j α!(β!j!)θ(ρ−δ)

x∈K

×(1 + |ξ|)−ρ|α|+δ|β|−(ρ−δ)j exp(ε|ξ|1/θ ).

(2.2)

∞,θ pj and j≥0 qj from F Sρ,δ (Ω) are said to be equivalent j≥0 pj ∼ j≥0 qj if for any compact set K ⊂ Ω we can ﬁnd C > 0 and B ≥ 0 and for every ε > 0 we can ﬁnd cε such that sup |Dξα Dxβ (pj (x, ξ) − qj (x, ξ))| ≤ Cε C |α+β|+s α!β!s!θ(ρ−δ)

Then,

x∈K

j≥0

j<s

×(1+|ξ|)−ρ|α|+δ|β|−(ρ−δ)s exp(ε|ξ|1/θ ), for every α, β ∈ Zn+ and for |ξ| ≥ B(s + |α|)θ . ∞,θ Theorem 2.3. ([10,18]) Let j≥0 pj ∈ F Sρ,δ (Ω). For every open relatively

∞,θ compact subset Ω of Ω there exists p ∈ Sρ,δ (Ω) which is equivalent to j≥0 pj . Ω

∞,θ It can be proved (see, e.g. [10,18]) that if p ∼ 0 in F Sρ,δ (Ω), then for every compact set K ∈ Ω there exist positive constants c, h, B such that:

sup |Dxβ p(x, ξ)| ≤ c|β|+1 β!θ(ρ−δ) (1 + |ξ|)δ|β| exp(−h|ξ|1/θ )

x∈K

for every β ∈ Zn+ , |ξ| ≥ B. We denote by VRθ (Ω) the subspace of V θ (Ω) of all operators which can be extended as continuous linear maps from G(θ) into (θ) θ G . The operator P ∈ VR (Ω) is said to be θ-regularizing on Ω. If for every compact set K ⊂ Ω there exist positive constants c, h, B for which sup |Dxβ p(x, ξ)| ≤ c|β|+1 β!θ exp(−h|ξ|1/θ ),

x∈K

|ξ| ≥ B,

(2.3)

(θ)

then, the operator P deﬁned on G0 (Ω) by (2.1) is θ-regularizing. More∞,θ (Ω), then this is true even if (2.3) is satisﬁed only when over, if p ∈ Sρ,δ θ |ξ| ≥ c1 |ξ| , where the constant c1 depends on K. ∞,θ Hence, if p ∼ 0 in F Sρ,δ (Ω) then p(·, D) ∈ VRθ (Ω) (see, [10,18]). ∞,θ For p, q ∈ Sρ,δ (Ω) we deﬁne Leibniz product p ◦ q to be the formal sum r , where: j≥0 j (α!)−1 ∂ξα p(x, ξ)Dxα q(x, ξ). rj (x, ξ) =

|α|=j

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The properties of the Leibniz product in the Gevrey classes are entirely analogous to the ones for C ∞ functions. (See, e.g. [10,12,18].) # ∞,θ For p ∈ F S (Ω) we deﬁne p to be formal sum j j j≥0 j≥0 ρ,δ j≥0 qj , where qj (x, ξ) = (α!)−1 ∂ξα Dxα ph (x, −ξ). |α|+h=j

∞,θ It is proved in [10,18] that if j≥0 pj ∈F Sρ,δ (Ω), then also ( j≥0 pj )# ∈ ∞,θ F Sρ,δ (Ω) and (( j≥0 pj )# )# ∼ j≥0 pj . Now we enlarge our class of operators by considering operators of the form: (2.4) Au(x) = (2π)−n ei<x−y,ξ> u(y)a(x, y, ξ)dy dξ. ∞,θ In this definition we use the amplitude function a(x, y, ξ) ∈ Sρ,δ (Ω × Ω),

∞,θ where Sρ,δ (Ω × Ω) denotes the space of all functions a(x, y, ξ) ∈ C ∞ (Ω × Ω × n R ) satisfying the following condition: for every compact set W ⊂ Ω×Ω there exist constants C ≥ 0, B ≥ 0 and for every ε > 0 there exists a constant cε , such that:

sup Dξα Dxβ Dyγ a(x, y, ξ) ≤ Cε C |α+β+γ| α!(β!γ!)θ(ρ−δ)

x∈K

×(1 + |ξ|)−ρ|α|+δ|β+γ| exp(ε|ξ|1/θ ), for every α, β, γ ∈ Zn+ and for every ξ ∈ Rn such that |ξ| ≥ B|α|θ . (θ) It can be proved that A : G0 (Ω) −→ Gθ (Ω) is continuous. We denote ∞,θ (Ω × Ω) the space of all operators of the form (2.4). OP Sρ,δ ∞,θ Next we deﬁne the Schwartz kernel K of A ∈ OP Sρ,δ (Ω × Ω), more θ precisely, K is the θ-ultradistribution in G0 (Ω) such that

K, v ⊗ u = Au, v, One can write formally K(x, y) = (2π)−n

(θ)

u, v ∈ G0 (Ω).

ex−y,ξ a(x, y, ξ)dξ.

∞,θ Moreover, the kernel of A ∈ OP Sρ,δ (Ω × Ω) is in G(θ) (Ω × Ω \ Δ) [10,18]. Let

∞,θ A ∈ OP Sρ,δ (Ω × Ω). For every open relatively compact subset Ω of Ω there

∞,θ exists p(·, D) ∈ OP Sρ,δ (Ω ) such that A − p(·, D) ∈ VRθ (Ω ). Moreover, if a(x, y, ξ) is the amplitude of A, then p(x, ξ) ∼ α (α!)−1 Dξα Dyα a(x, y, ξ)|y=x ∞,θ in F Sρ,δ (Ω ). (θ)

(θ)

A linear continuous operator A from G0 (Ω) to G0 (Ω) is proper if its kernel K is a proper θ-ultradistribution on Ω × Ω, i.e. if supp K has compact intersection with H ×Ω and with Ω×H for every compact set H ⊂ Ω. Alterna(θ) (θ) tively, A is proper if A and t A are continuous maps from G0 (Ω) to G0 (Ω).

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∞,θ (Ω × Ω) with θ(ρ − δ) > 1 there exists a properly supFor A ∈ OP Sρ,δ

∞,θ ported operator A1 ∈ OP Sρ,δ (Ω × Ω) such that A − A1 ∈ VRθ (Ω). The prod-

∞,θ uct p(·, D)q(·, D) of the properly supported operator p(·, D) ∈ OP Sρ,δ (Ω)

∞,θ with operator q(·, D) ∈ OP Sρ,δ (Ω) obeys representation p(·, D)q(·, D) = ∞,θ t(·, D) + R where t(x, ξ) ∼ p(x, ξ) ◦ q(x, ξ) in F Sρ,δ (Ω) and R ∈ VRθ (Ω). Now we formulate our main result. Consider the Cauchy problem

Dt Q(t, s) + R(t, s)Q(t, s) + R0 (t, s) ∈ C([0, T1 ]; VRθ1 (Rn )), (2.5) Q(s, s) = 0 (0 ≤ s ≤ t ≤ T1 ),

where R(t, s), R0 (t, s) are matrix PDO of inﬁnite order with the symbols r(t, s, x, ξ), r0 (t, s, x, ξ), respectively. These symbols belong to C([0, T1 ] × ∞,θ [0, T1 ]; Sρ,δ (Rn )), where θ > 1, 0 ≤ δ < ρ ≤ 1, θ(ρ − δ) > 1, θ1 > 1, θ1 ≥ θ. Theorem 2.4. We assume that for every compact subset K ⊂ Rn there exist constants C > 0 and B ≥ 0 and for every ε > 0, there exists a constant cε , such that for any α, β, for all 0 ≤ s ≤ t ≤ T, ξ ∈ Rn , |ξ| ≥ B|α|θ , x ∈ Rn , the inequalities sup Dξα Dxβ r0 (t, s, x, ξ) ≤ cε C |α+β| α!β!θ(ρ−δ) ξ−ρ|α|+δ|β|

x∈K

× exp(ε ξ1/θ1 )gε (t, ξ), sup x∈K

Dξα Dxβ r(t, s, x, ξ) T1

≤C

|α+β|

α!β!

θ(ρ−δ)

−ρ|α|+δ|β|

ξ

(2.6) gε (t, ξ),

gε (τ, ξ)dτ ≤ ε ξ1/θ1+ C ln ξ,

(2.7)

(2.8)

0

T1

gε (τ, ξ)dτ ≤ C ξ1/θ1 ,

(2.9)

0

gε (t, ξ) ≤ cε exp(ε ξ1/θ1 ),

(2.10)

are fulfilled. Then there exists a solution Q(t, s) of the problem (2.5) with the symbol q(t, s, x, ξ) such that for all 0 ≤ s ≤ t ≤ T, ξ ∈ Rn , |ξ| ≥ B|α|θ , x ∈ Ω ⊂ Rn with some constants C1 , C2 and 0 ≤ δ1 < ρ1 ≤ 1, ρ1 − δ1 + θ11 ≤ ρ − δ, the inequalities |α+β|

sup Dξα Dxβ q(t, s, x, ξ) ≤ cε C1

x∈K

ξ−ρ1 |α|+δ1 |β|

×α!(β!)θ1 (ρ1 −δ1 ) exp(3ε ξ1/θ1 ) are fulfilled for all k = 0, 1, . . . , uniformly with respect to (t, s) ∈ [0, T1 ] × [0, T1 ]. Thus, (Ω)). q(t, s, x, ξ) ∈ C([0, T1 ] × [0, T1 ]; Sρ∞,θ 1 ,δ1 (θ1 )

The solution is unique modulo VR

(Rn ).

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3. The Proof of Theorem 2.4 Existence. We select proper representatives of the equivalence classes of operators R(t, s) and R0 (t, s) and we construct proper operator Q(t, s). We shall seek the solution in the form q ∼ q0 + q 1 + q 2 + · · · , where Dt qk + rqk + rk = 0, k−1

rk =

l=0 |α|=k−l

qk (s, s) = 0,

1 α α ∂ rDx ql , α! ξ

t qk = −i

rk (s1 )ds1 +

k = 1, 2, . . . , t l

(−i)

l=2

s sl−1

···

∞

k = 0, 1, 2, . . . ,

(3.1) (3.2)

s1 ds1

s

ds2 s

dsl r(s1 ) · · · r(sl−2 )rk (sl−1 ).

(3.3)

s

t We introduce the operator (Ir)(t) = s r(s1 )ds1 . If g is a scalar function, then IgIg · · · Ig = (Ig)l /l!. We rewrite (3.3) in the form l

qk = −iIrk +

∞ (−i)l Ir · · · IrIrk .

(3.4)

l=2

Lemma 3.1. There exist constants C1 , C2 and B ≥ 0 and for every ε > 0, there exists a constant cε , such that for any α, β, k (k = 1, 2, . . .), for all 0 ≤ s ≤ t ≤ T , ξ ∈ Rn , |ξ| ≥ B|α|θ , x ∈ Rn , the following estimates hold: sup Dξα Dxβ rk (t, s, x, ξ)

x∈K

(|α| + k)! ((|β| + k)!)θ(ρ−δ) k! ∞ (Ig)l , (3.5) × exp(ε ξ1/θ1 )gε (t, ξ)(1 + Ig)k−1 l! |α+β|

≤ cε C1

C2k ξ−ρ|α|+δ|β|−(ρ−δ)k

l=0

sup Dξα Dxβ qk (t, s, x, ξ)

x∈K

(|α| + k)! ((|β| + k)!)θ(ρ−δ) k! ∞ (Ig)l . (3.6) × exp(ε ξ1/θ1 )Ig(1 + Ig)k l! |α+β|

≤ cε C1

C2k ξ−ρ|α|+δ|β|−(ρ−δ)k

l=1

Proof. We prove the lemma by induction on k. From (3.4) we have q0 =

∞ l=1

(−i)l Ir · · · IrIr0 .

l

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Therefore, it follows that sup Dξα Dxβ q0

x∈K

≤

∞ l=1

×

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

α! α1 ! · · · αl−1 !(α − α1 − · · · − αl−1 )!

β! β1 ! · · · βl−1 !(β − β1 − · · · − βl−1 )!

≤

l−1

α−

α

×Dξα1 Dxβ1 Ir · · · Dξ l−1 Dxβl−1 Ir Dξ

j=1

αj

β− l−1 j=1 βj

Dx

Ir0

|α+β| cε C1

ξ−ρ|α|+δ|β| |α|!(|β|!)θ(ρ−δ) exp(ε ξ1/θ1 ) |α1 +···+αl−1 +β1 +···+βl−1 | ∞

×

l=1

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

|α+β|

≤ cε C1

C C1

ξ−ρ|α|+δ|β| |α|!|β|!θ(ρ−δ) exp(ε ξ1/θ1 )

∞ (Ig)l l=1

l!

(Ig)l l!

.

Since by means of the appropriate choice of the constant C1 we can get an inequality |α1 +···+αl−1 +β1 +···+βl−1 | C ≤ 1, C 1 α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

if |α + β| = 0. [For α = β = 0, the estimate is easily seen from (2.6), (2.7)]. Next, we consider the case of k = 1. According to (3.2) 1 γ (∂ r)(Dxγ q0 ), r1 = γ! ξ |γ|=1

therefore sup Dξα Dxβ r1 ≤

x∈K

|γ|=1 α1 ≤α β1 ≤β

β! 1 α! α1 !(α − α1 )! β1 !(β − β1 )! γ!

×Dξα1 +γ Dxβ1 r Dξα−α1 Dxβ−β1 +γ q0 |α+β|

C2 ξ−ρ|α|+δ|β|−(ρ−δ) gε (t, ξ) exp(ε ξ1/θ1 ) ∞ (Ig)l ×(|α| + 1)!((|β| + 1)!)θ(ρ−δ) l! l=1 |γ| |α +β | 1 1 C CC1 1 α! × C1 C2 α1 !(α − α1 )! γ!

≤ cε C1

|γ|=1 α1 ≤α β1 ≤β

×

θ(ρ−δ) (α1 + γ)!|α − α1 |! β1 ! (γ + β − β1 )! β! β1 !(β − β1 )! (|α| + 1)! (|β| + 1)!

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|α+β|

C2 ξ−ρ|α|+δ|β|−(ρ−δ) gε (t, ξ) exp(ε ξ1/θ1 ) ∞ (Ig)l θ(ρ−δ) . ×(|α| + 1)!((|β| + 1)!) l!

≤ cε C1

l=1

Here ﬁrst we chose constant C1 then constant C2 such that (|γ| > 0) Cn |α1 +β1 | nCC1 |γ| < 1. C1 C2

|γ|=1 α1 ≤α β1 ≤β

We consider now q1 . From (3.3) we have q1 =

∞ l=1

Ir . . . IrIr1 .

l

Therefore, we obtain |α+β|

sup Dξα Dxβ q1 ≤ cε C1

x∈K

C2 ξ−ρ|α|+δ|β|−(ρ−δ)

× exp(εξ1/θ1 )(|α| + 1)!((|β| + 1)!)θ(ρ−δ) |α1 +···αl−1 +β1 +···+βl−1 | ∞ C × C 1 α1 ,...,αl−1 l=1

×

β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

∞ |α|! (Ig)m+l (m + l)! |α 1 |! · · · |α − α1 − · · · − αl−1 |! m=1

|α1 |! · · · |αl−1 |!(|α − α1 − · · · − αl−1 | + 1)! (|α| + 1)! |β|! × |β1 |! · · · |β − β1 − · · · − βl−1 |! θ(ρ−δ) |β1 |! · · · |βl−1 |!(|β − β1 − · · · − βl−1 | + 1)! × (|β| + 1)! ×

|α+β|

C2 ξ−ρ|α|+δ|β|−(ρ−δ) exp(εξ1/θ1 ) ∞ (Ig)k+1 ×(|α| + 1)!((|β| + 1)!)θ(ρ−δ) (k + 1)! k=1 |α1 +···αl−1 +β1 +···+βl−1 | k C × C 1 α1 ,...,αl−1

≤ cε C1

l=1

β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

(|α − α1 − · · · − αl−1 | + 1) (|β − β1 − · · · − βl−1 | + 1) (|α| + 1) (|β| + 1) θ(ρ−δ)−1 |β1 |! · · · |βl−1 |!(|β − β1 − · · · − βl−1 | + 1)! × (|β| + 1)! ×

|α+β|

C2 ξ−ρ|α|+δ|β|−(ρ−δ) exp(εξ1/θ1 )(|α| + 1)! ∞ (Ig)k+1 ×((|β| + 1)!)θ(ρ−δ) (k + 1)!

≤ cε C1

k=1

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A. Galstian k ×

IEOT

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

l=1

C C1

|α1 +···αl−1 +β1 +···+βl−1 |

|α+β|

C2 ξ−ρ|α|+δ|β|−(ρ−δ) exp(εξ1/θ1 ) ∞ k(Ig)k+1 ×(|α| + 1)!((|β| + 1)!)θ(ρ−δ) . (k + 1)!

≤ cε C1

k=1

In the last inequality we employed the following estimate: |α1 +···αl−1 +β1 +···+βl−1 | k C ≤k C1 α1 ,...,αl−1 l=1

β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

with the appropriate choice of the constant C1 if |α + β| = 0. Consequently, |α+β|

sup Dξα Dxβ q1 ≤ cε C1

x∈K

C2 ξ−ρ|α|+δ|β|−(ρ−δ) exp(ε ξ1/θ1 )

×(|α| + 1)!((|β| + 1)!)θ(ρ−δ) ≤

∞ (k + 1)(Ig)k+1

(k + 1)!

k=1 |α+β| −ρ|α|+δ|β|−(ρ−δ) cε C1 C2 ξ exp(ε ξ1/θ1 ) ∞ (Ig)k θ(ρ−δ)

×(|α| + 1)!((|β| + 1)!)

Ig

k!

k=1

.

For α = β = 0, we have sup q1 ≤

x∈K

∞

Ir · · · IrIr1

l=1

≤ cε ξ−(ρ−δ) C2 exp(ε ξ1/θ1 )

∞ l=1

≤ cε ξ−(ρ−δ) C2 exp(ε ξ1/θ1 )Ig

Ig · · · Ig l

∞ (Ig)k k=1

k!

∞ (Ig)m m! m=1

.

Thus, (3.6) is proved when k = 1. We assume now that (3.5), (3.6) have been proved for k and next we prove them for k + 1. Indeed, we have rk+1 =

k

l=0 |γ|=k+1−l

1 γ (∂ r)(Dxγ ql ). γ! ξ

It follows |α+β|

sup Dξα Dxβ rk+1 ≤ cε C1

x∈K

C2k+1

(|α| + k + 1)! ((|β| + k + 1)!)θ(ρ−δ) (k + 1)!

× ξ−ρ|α|+δ|β|−(ρ−δ)(k+1) exp(ε ξ1/θ1 )gε (t, ξ) ⎧ ⎨ C |α1 +β1 | CC |γ| 1 (k + 1)! 1 × ⎩ C1 C2 γ! l! |γ|=k+1 α1 ≤α β1

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(γ + α1 )!(|α − α1 |)! β! α! α1 !(α − α1 )! (|α| + k + 1)! β1 !(β − β1 )! θ(ρ−δ) ∞ β1 !(|β − β1 + γ|)! (Ig)m × (|β| + k + 1)! m! m=1 k C |α1 +β1 | CC1 |γ| (k + 1)! + C1 C2 l! ×

l=1 |γ|=k+1−l α1 ≤α β1

1 α! (γ + α1 )!(|α − α1 | + l)! β! γ! α1 !(α − α1 )! (|α| + k + 1)! β1 !(β − β1 )! θ(ρ−δ) ∞ β1 !(|β − β1 + γ| + l)! (Ig)m l−1 . Ig(1 + Ig) × (|β| + k + 1)! m! m=1 ×

Due to the choice of C1 , C2 for all 0 ≤ l ≤ k we obtain C |α1 +β1 | CC1 |γ| 1 α! (k + 1)! C1 C2 γ! α1 !(α − α1 )! l! |γ|=k+1−l α1 ≤α β1 ≤β

β! (γ + α1 )!(|α − α1 | + l)! × (|α| + k + 1)! β1 !(β −β1 )!

β1 !(|β −β1 +γ|+l)! (|β|+k+1)!

θ(ρ−δ) ≤ 1.

Indeed, it is easy to see that θ(ρ−δ) β1 !(|β − β1 + γ| + l)! β! β1 !(β − β1 )! (|β| + k + 1)! θ(ρ−δ)−1 (|β − β1 | + 1) · · · (|β − β1 | + k + 1) |β1 |!(|β −β1 |+k+1)! ≤ ≤ 1. (|β| + 1) · · · (|β| + k + 1) (|β|+k+1)! Further we have 1 α! (γ + α1 )!(|α − α1 | + l)! (k + 1)! γ! α1 !(α − α1 )! (|α| + k + 1)! l! (|α − α | + |α |) · · · (|α − α | + 1)(|α − α1 | + l) · · · (l + 1) 1 1 1 ≤ n|α1 +γ| (|α1 | + |α − α1 | + k + 1) · · · (k + 1) ≤ n|α1 +γ| . Since the inequalities Cn 1 ≤ , C1 3

CC1 n 1 ≤ C2 3

imply

Cn |α| α

C2

≤ 2n ,

therefore, we obtain |α+β|

sup Dξα Dxβ rk+1 ≤ cε C1

x∈K

C2k+1

(|α| + k + 1)! ((|β| + k + 1)!)θ(ρ−δ) (k + 1)!

× ξ−ρ|α|+δ|β|−(ρ−δ)(k+1) exp(ε ξ1/θ1 )gε (t, ξ) ∞ k (Ig)m l−1 . Ig(1 + Ig) × 1+ m! m=1 l=1

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Further, we consider 1+

k

Ig(1+Ig)l−1 = (1+Ig)(1 + Ig+Ig(1 + Ig)+· · · + Ig(1 + Ig)k−2 )

l=1

= (1+Ig)2(1+ Ig+Ig(1 + Ig) + · · · + Ig(1 + Ig)k−3 ) = (1 + Ig)k . Finally, we obtain |α+β|

sup Dξα Dxβ rk+1 ≤ Cε C1

x∈K

C2k+1

(|α| + k + 1)! (|β| + k + 1)!θ(ρ−δ) (k + 1)!

× ξ−ρ|α|+δ|β|−(ρ−δ)(k+1) exp(ε ξ1/θ1 ) ∞ (Ig)m . ×gε (t, ξ)(1 + Ig)k m! m=1 Thus, (3.5) is proved. Consider now qk+1 deﬁned as follows: t qk+1 = −i

∞ rk+1 (s1 )ds1 + (−i)l ds1 . . . dsl r(s1 ) . . . r(sl−1 )rk+1 (sl ). sl−1

t

l=2

s

s

s

We have |α+β|

sup Dξα Dxβ qk+1 ≤ cε C1

x∈K

C2k+1 ξ−ρ|α|+δ|β|−(ρ−δ)(k+1) (|α| + k + 1)! ((|β| + k + 1)!)θ(ρ−δ) (k + 1)! ∞ ∞ (Ig)m+l

× exp(ε ξ1/θ1 ) ×(1 + Ig)k

l=1 m=1

×

(m + l)!

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

C C1

|α1 +···αl−1 +β1 +···+βl−1 |

α! (|α − α1 − · · · − αl−1 | + k + 1)! × αl−1 !(α − α1 − · · · − αl−1 )! (|α| + k + 1)! β! × β1 ! · · · βl−1 !(β − β1 − · · · − βl−1 )! θ(ρ−δ) β1 ! · · · βl−1 !(|β − β1 − · · · − βl−1 | + k + 1)! . × (|β| + k + 1)! We claim that by means of the appropriate choice of the constants C1 and C2 we can show that |α1 +···αl−1 +β1 +···+βl−1 | C S:= C1 α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

×

(|α − α1 − · · · − αl−1 | + k + 1)! α! αl−1 !(α − α1 − · · · − αl−1 )! (|α| + k + 1)!

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β! β1 ! · · · βl−1 !(β − β1 − · · · − βl−1 )! θ(ρ−δ) β1 ! · · · βl−1 !(|β − β1 − · · · − βl−1 | + k + 1)! ≤ 1. × (|β| + k + 1)!

×

Indeed,

S≤

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

C C1

|α1 +···αl−1 +β1 +···+βl−1 |

α1 αl−1 α! ··· α1 ! · · · αl−1 !(α − α1 − · · · − αl−1 )! |α1 |! |αl−1 |! |α1 |! · · · |αl−1|! (|α − α1 − · · · − αl−1 | + k + 1)! × (|α| + k + 1)! |β|! × |β1 |! · · · |β − β1 − · · · − βl−1 |! θ(ρ−δ) |β1 |! · · · |βl−1 |!(|β − β1 − · · · − βl−1 | + k + 1)! × (|β| + k + 1)! |α1 +···αl−1 +β1 +···+βl−1 | C ×

≤

α1 ,...,αl−1 β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

C1

(|α − α1 − · · · − αl−1 | + 1) · · · (|α − α1 − · · · − αl−1 | + k + 1) (|α| + 1) · · · (|α| + k + 1) (|β − β1 − · · · − βl−1 | + 1) · · · (|β − β1 − · · · − βl−1 | + k + 1) × (|β| + 1) · · · (|β| + k + 1) θ(ρ−δ)−1 |β1 |! · · · |βl−1 |!(|β − β1 − · · · − βl−1 | + k + 1)! × (|β| + k + 1)! |α1 +···αl−1 +β1 +···+βl−1 | C ≤ ≤1 C 1 α1 ,...,αl−1 ×

β1 ,...,βl−1 α1 ≤α,...,αl−1 ≤α β1 ≤β,...,βl−1 ≤β

if |α + β| = 0. [For α = β = 0, the estimate is easily seen from (2.6), (2.7)]. Thus, we obtain sup Dξα Dxβ qk+1

x∈K

(|α| + k + 1)! (k + 1)! ∞ ∞ (Ig)m+l θ(ρ−δ) 1/θ1 k exp(ε ξ )(1 + Ig) ×((|β| + k + 1)!) (m + l)! m=1 |α+β|

≤ cε C1

C2k+1 ξ−ρ|α|+δ|β|−(ρ−δ)(k+1)

l=1

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(|α| + k + 1)! (k + 1)! ∞ (Ig)k . ×((|β| + k + 1)!)θ(ρ−δ) exp(ε ξ1/θ1 )Ig(1 + Ig)k k! |α+β|

≤ cε C1

C2k+1 ξ−ρ|α|+δ|β|−(ρ−δ)(k+1)

k=1

The lemma is proved. The conclusion of the proof of the existence. From (3.6) it follows that sup Dξα Dxβ qk (t, s, x, ξ)

x∈K

|α+β|

≤ cε C1

(|α| + k)! (β!k!)θ(ρ−δ) k! + C ln ξ)

C2k ξ−ρ|α|+δ|β|−(ρ−δ)k

× exp(ε ξ1/θ1 )(ε ξ1/θ1

×(1 + C ξ1/θ1 )k exp(ε ξ1/θ1 + C ln ξ) (|α| + k)! |α+β| k (β!k!)θ(ρ−δ) C2 ξ−ρ|α|+δ|β|−(ρ−δ)k+C ≤ cε C1 k! × exp(2ε ξ1/θ1 )(ε ξ1/θ1 + C ln ξ)(1 + C ξ1/θ1 )k (|α| + k)! |α+β| k (β!k!)θ(ρ−δ) ≤ cε C1 C2 ξ−ρ|α|+δ|β|−(ρ−δ)k+C k! × exp(2ε ξ1/θ1 )(ε ξ1/θ1 + C ln ξ)2k C k ( ξ1/θ1 )k (|α| + k)! |α+β| ˜ k C2 ξ−ρ|α|+δ|β|+(1/θ1 −(ρ−δ))k ≤ cε C1 k! ×(β!k!)θ(ρ−δ) exp(3ε ξ1/θ1 )(ε ξ1/θ1 + C ln ξ) for all k = 0, 1, . . ., uniformly with respect to (t, s) ∈ [0, T1 ] × [0, T1 ]. Thus, 1 qk ∈ C([0, T1 ] × [0, T1 ]; Sρ∞,θ (Rn )), where 1/θ1 + ρ1 − δ1 < ρ − δ. The last 1 ,δ1 inequality is equivalent to (2.2). Therefore, from the Theorem 2.3, we get 1 (Ω)), q(t, s, x, ξ) ∈ C([0, T1 ] × [0, T1 ]; Sρ∞,θ 1 ,δ1 which is equivalent to j≥0 pj |Ω , where Ω ⊂ Rn . Thus, the existence of the solution Q(t, s) is proved. Proof of the uniqueness. We note that, according to the existence part of the theorem, the Cauchy problem

Dt Q(t, t ) + Q(t, t )R(t, s) + R0 (t, s) ∈ C([0, T1 ] ; VRθ (Rn )), 0 ≤ t ≤ t ≤ T1 , Q(t , t ) = 0,

has a solution for every t ∈ [s, T1 ]. Further, for every t ∈ [s, T1 ] the problem

Dt V (t, t ) + V (t, t )R# (t, s) ∈ C([0, T1 ] ; VRθ (Rn )), (3.7) 0 ≤ t ≤ t ≤ T 1 , V (t , t ) = I, has a solution as well. By applying a conjugation to (3.7), we obtain

Dt V # (t, t ) − R(t, s)V # (t, t ) ∈ C([0, T1 ] ; VRθ (Rn )), V # (t , t ) = I, 0 ≤ t ≤ t ≤ T 1 .

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Deﬁne an operator G as follows:

Gf (t ) :=

t

f (t)V # (t, t ) dt.

s

If we set f (t) := Dt u(t) + u(t)R(t, s), then we obtain

t

Gf (t ) = −iu(t ) + iu(s)V # (s, t ) − u(t) Dt V # (t, t ) − R(t, s)V # (t, t ) dt. s

Hence, u(t ) = V # (s, t )u(s) + iG (Dt u(t) + R(t, s)u(t))

t +i

u(t) Dt V # (t, t ) − R(t, s)V # (t, t ) dt.

(3.8)

s

Therefore, if Q1 (t, s) and Q2 (t, s) are two solutions to the problem (2.5). Then we plug in (3.8) the distribution-valued function u(t) = (Q1 (t, s) − Q2 (t, s)) u0 ,

u0 ∈ G(θ) (Rn ),

and arrive at Q1 (t, s) − Q2 (t, s) ∈ Ct1 ([0, T0 ]; VRθ1 (Rn )).

The theorem is proved.

4. Applications In this section we consider the Cauchy problem for the hyperbolic equation Dt2 u − λ2 (t)

n

θ

Dx2j u + a(t)λ2 (t)Λ(t)− θ−1

i=1

n

bj (t)Dxj u = 0,

(4.1)

i=1

u(s, x) = u0 (x), Dt u(s, x) = u1 (x), (θ)

(4.2)

where s, t ∈ [0, T ], u0 (x), u1 (x) ∈ G0 (R) . We describe the operator of (4.1) by means of the real-valued function λ ∈ C ∞ ([0, T ]) such that λ(0) = λ (0) = 0, λ (t) > 0 when t = 0. In the following λ means the same as dλ/dt. For λ(t) t we deﬁne Λ(t) = 0 λ(r) dr and assume that the function λ2 Λθ/(1−θ) belongs to C ∞ ([0, T ]) and that the following estimates cλ(t)/Λ(t) ≤ λ (t)/λ(t) ≤ c0 m(t)λ(t)Λθ/(1−θ) (t) for all t ∈ (0, T ], k−1 |λ(k) (t)| ≤ ck m(t)λ(t)Λθ/(1−θ) (t) |λ (t)|

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are satisﬁed for all k = 2, 3, . . . , t ∈ (0, T ], with positive constants c, c0 , ck , where c > θ/(2(θ − 1)). Here m(t)Λθ/(1−θ) (t) is a monotone function and m(t) tends to 0 as t → 0. Further we assume that θ

a(t)λ2 (t)Λ(t)− θ−1

n

bj (t) ∈ C ∞ ([0, t]),

(4.3)

i=1

k |Dtk a(t)| + |Dtk bj (t)| ≤ Ck m(t)λ(t)Λθ/(1−θ) (t) a1 (t) := max |a(t)| → 0 τ ≤t

as

for all

t ∈ (0, T ],

t → 0.

Our aim is the construction of real-valued phase functions Φk , k = 1, 2, and of amplitude functions Ajk such that for a given s ∈ J solution u(t, x) of (4.1) can be represented as 1 2 1 u(t, x) = ei(x·ξ−y·ξ+Φk (t,s,ξ)) Ajk (t, s, ξ)uj (y) dξ dy, (2π)n j=0 k=1Rn Rn

(4.4) where the phase functions Φkl (t, s, ξ) are deﬁned by t Φk (t, s, ξ) = (−1) |ξ| k

λ(τ ) dτ,

k = 1, 2.

(4.5)

s

The amplitude functions Ajkl (t, s, ξ) are such that for every given positive number ε there exists a positive constant K, such that, for every j, k, l, p, r, α, p + r ≤ 2, the inequality |Dtp Dsr Dξα Ajk (t, s, ξ)| ≤ Cp,r,α ξK+(p+r)/m−|α|(θ−2)/θ exp (p + r)ε ξ1/θ (4.6) holds for all s, t ∈ [0, T ], ξ ∈ R . We follow the method of zones, which is introduced in [15,17], and give only few steps of the construction, which are different from the case of C ∞ classes. For a positive constant N , N ≥ 1, we deﬁne tξ,θ as a root of the equation n

Λ(t)θ ξθ−1 = N θ−1

(4.7)

with respect to t. Here ξ = (1 + |ξ|2 )1/2 . For a positive M we denote by RnM the set {ξ ∈ Rn | ξ ≥ M }. Then, for every given positive numbers M and N we deﬁne the so-called pseudo-differential and hyperbolic zones Zpd (M, N, θ) = {(t, ξ) ∈ J × Rn

| Λ(t)θ ξθ−1 ≤ N θ−1 , ξ ≥ M },

Zhyp (M, N, θ) = {(t, ξ) ∈ J × Rn

| Λ(t)θ ξθ−1 ≥ N θ−1 , ξ ≥ M }.

Furthermore, the continuous roots of the equation n n θ ξj2 + a(t)λ2 (t)Λ(t)− θ−1 bj (t)ξj = 0, τ 2 − λ2 (t) j=1

(4.8)

j=1

(these are the zeros of the complete symbol) are denoted by τl (t, x), l = 1, 2.

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Proposition 4.1. There exist positive constants M , N , and δ1 such that the zeros τ1 (t, ξ), τ2 (t, ξ), are smooth functions on the set Zh (M, N, θ), τ1 , τ2 ∈ C ∞ (Zh (M, N, θ)), and for every k, α the inequalities k |Dtk Dξα τl (t, ξ)| ≤ Ck,α ξ1−|α| λ(t) m(t)λ(t)Λθ/(1−θ) (t) , |τ1 (t, ξ) − τ2 (t, ξ)| ≥ δ1 λ(t) ξ,

k θ |Dtk Dξα Im τl (t, ξ)| ≤ o(1)Ck,α ξ−|α| λ(t) m(t)λ(t)Λθ/(1−θ) (t) Λ(t) 1−θ ,

hold for all (t, ξ) ∈ Zh (M, N, γ) and all l = 1, 2. Here o(1) → 0 as t → 0. To construct fundamental solution to this problem one faces the problem (2.5) with the function

2θ(θ−1) ρt (t, ξ) θ θ−1 − θ−2 gε (t, ξ) = χ Λ(t) ξ ε /2 ρ(t, ξ) + ρ(t, ξ) 2θ(θ−1) θ θ θ−1 − θ−2 1−θ a(t)λ(t)Λ(t) + 1 − χ 2Λ(t) ξ ε θ θ a(t)ε 2−θ , (4.9) ×χ a(t)ε 2−θ + C 1 − χ where a(t) := max{a1 (t), m(t)} and the function ρ(t, ξ) is deﬁned as a positive root of the equation θ

ρ2 − 1 − ξλ2 (t)Λ(t)− θ−1 = 0, while χ ∈ C0∞ (R) is a cutoﬀ function such that χ(z) = 0 if |z| ≥ 1, and χ(z) = 1 if |z| ≤ 1/2. Lemma 4.2. Function gε (t, ξ) of (4.9) satisfies the conditions (2.8), (2.9) and (2.10) of Theorem 3.1. Proof. Let be given a positive number ε. Consider T gε (t, ξ) dt. 0

First, we evaluate the integral T 2θ(θ−1) ρt (t, ξ) χ Λ(t)θ ξθ−1 ε− θ−2 /2 ρ(t, ξ) + dt ρ(t, ξ) 0

tξ,θ ρt (t, ξ) ≤ ρ(t, ξ) + dt, ρ(t, ξ) 0

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where the point tξ,θ is determined by the equation Λ(tξ,θ )θ ξθ−1 = 2ε It follows from (4.7) that 2ε

2θ(θ−1) θ−2

2θ(θ−1) θ−2

.

= N θ−1 and we obtain

tξ,θ tξ,θ tξ,θ θ 1 ρt (t, ξ) dt + λ(t) ξ 2 Λ(t)− 2(θ−1) dt ρ(t, ξ) + dt ≤ ρ(t, ξ) 0

0

0

+ ln ρ(tξ,θ , ξ) ≤ 1 + ln ρ(tξ,θ , ξ) + ≤ C + C ln ξ +

2

θ−2 1 2(θ − 1) Λ(tξ,θ ) 2(θ−1) ξ 2 θ−2

2θ 2 −θ−2 2θ(θ−1)

1 (θ − 1) ε ξ θ . θ−2

Further, T

2θ(θ−1) θ θ 1 − χ 2Λ(t)θ ξθ−1 ε− θ−2 a(t)λ(t)Λ(t) 1−θ χ a(t)ε 2−θ dt

0

T ≤

θ θ a(t)λ(t)Λ(t) 1−θ χ a(t)ε 2−θ dt

tξ,θ

≤ε

T

θ θ−2

θ

λ(t)Λ(t) 1−θ dt, tξ,θ

where the point tξ,θ is determined by the equation 4Λ(tξ,θ )θ ξθ−1 = ε

2θ(θ−1) θ−2

.

Hence θ

T

θ

θ

λ(t)Λ(t) 1−θ dt = ε θ−2

ε θ−2 tξ,θ

θ

≤ ε θ−2 ≤ The lemma is proved.

1 1 1 Λ(T ) 1−θ − Λ(tξ,θ ) 1−θ 1−θ 1 1 Λ(tξ,θ ) 1−θ θ−1

2 1 1 2 θ(θ−1) ε ξ θ . θ−1

The necessity of the condition (4.3) for the representation (4.4) is proved in [2].

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Acknowledgments The author would like to express her sincere gratitude to the referee for his/her valuable comments and suggestions, which helped to improve the text.

References [1] Galstian, A., Yagdjian, K.: Uniqueness of the solution of the Cauchy problem for degenerating elliptic equation. Sov. J. Contemp. Math. Anal. 25(2), 85– 90 (1990) [2] Ishida, H., Yagdjian, K.: On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity. Publ. Res. Inst. Math. Sci. 38(2), 265–287 (2002) [3] Ivrii, V.: Conditions for well-posedness in Gevrey classes of the Cauchy problem for hyperbolic equations with characteristics of variable multiplicity (Russian). Sib. Mat. Zh. 17(6), 1256–1270 (1976) [4] Ivrii, V.: Well-posedness in Gevrey classes of the Cauchy problem for nonstrictly hyperbolic operators (Russian). Izv. Vyssh. Uch. Zaved. 2, 26–35 (1978) [5] Kajitani, K.: Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes. Hokkaido Math. J. 12(3), 434–460 (1983) [6] Kajitani, K., Wakabayashi, S.: Propagation of singularities for several classes of pseudo differential operators. Bull. Sci. Math. 115, 397–449 (1991) [7] Kajitani, K., Nishitani, T.: The hyperbolic Cauchy problem. In: Lecture Notes in Mathematics, vol. 1505. Springer-Verlag, Berlin (1991) [8] Komatsu, H.: Ultradistributions I. Structures theorems an a characterization. J. Fac. Sci. Univ. Tokyo 20, 25–105 (1973) [9] Komatsu, H.: Ultradistributions II. The kernel theorem and ultradistributions with support in a submanifold. J. Fac. Sci. Univ. Tokyo 24, 607–628 (1977) [10] Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, River Edge (1993) [11] Shinkai, K., Taniguchi, K.: Fundamental solution for a degenerate hyperbolic operator in Gevrey classes. Publ. RIMS 28, 169–205 (1992) [12] Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981) [13] Treves, F.: Introduction to the Theory of Pseudodifferential and Fourier Integral Operator, vols. 1 & 2. Plenum Press, New York (1980) [14] Yagdjian, K.: The Cauchy problem for weakly hyperbolic equation in the classes of Gevrey functions (Russian). Izvestiya Akademii Nauk Armyanskoi SSR, Matematika 13, 3–22 (1978) [15] Yagdjian, K.: Pseudodifferential operators with a parameter and the fundamental solution of the Cauchy problem for operators with multiple characteristics. Sov. J. Contemp. Math. Anal. 21(4), 1–29 (1986) [16] Yagdjian, K.: Gevrey asymptotic representation of the solutions of equations with one turning point. Math. Nachr. 183, 295–312 (1996) [17] Yagdjian, K.: The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach. Akademie Verlag, Berlin (1997)

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[18] Zanghirati, L.: Pseudodifferential operators of infinite order and Gevrey classes. Ann. Univ. Ferrara Sez. VII (N.S.) 31, 197–219 (1985) Anahit Galstian (B) Department of Mathematics University of Texas-Pan American 1201 West University Dr. Edinburg, TX USA e-mail: [email protected] Received: August 29, 2010. Revised: November 10, 2010.

Integr. Equ. Oper. Theory 70 (2011), 301–305 DOI 10.1007/s00020-011-1884-1 Published online May 12, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

A Generalization of Stenger’s Lemma to Maximal Dissipative Operators M. A. Nudelman Abstract. It is shown that for any maximal dissipative operator A in some Hilbert space H, which is the orthogonal sum H = F ⊕G of two Hilbert spaces F , G with dim G < ∞, the compression T := PF A|dom A∩F of A to F is again a maximal dissipative operator in F . Mathematics Subject Classification (2010). Primary 47B25; Secondary 47A48. Keywords. Stenger’s lemma, maximal dissipative operator, passive resistance system.

1. Introduction Let the Hilbert space H be the orthogonal sum of its subspaces F, G: H=F ⊕G

where dim G < ∞,

(1.1)

and let PF denote the orthogonal projection in H onto F. According to [1, Lemma 2.1], for every dense linear manifold D of H, the intersection D ∩ F is dense in F. Hence for every linear operator A in H with dense domain domA ⊂ H, the set dom A ∩ F is dense in F. Stenger’s lemma [2] states that, if A is a self-adjoint operator in H, then T := PF A|domA∩F is a self-adjoint operator in F, see also [3,4]. This implies that the main operator of a conservative resistance system [5] is correctly defined and of the form iT where T is a self-adjoint operator in the corresponding state space. In the present note we generalize Stenger’s lemma to the case of a maximal dissipative operator A; here dissipative means that (Ax, x) ≥ 0,

x ∈ dom A.

In the following, for a linear operator B, we denote by dom B, ran B, ker B the domain, range, and kernel of B, respectively.

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2. The Main Result Let A be a maximal dissipative operator which acts in the Hilbert space H of the form (1.1) with dim G < ∞. In view of [1, Lemma 2.1] (see above), the set dom A ∩ F is dense in F. We denote by H+ (A) the domain of the operator A, endowed with the graph norm h 2+ := h 2 + Ah 2 ,

h ∈ dom A.

(2.1)

The closedness of the operator A implies that H+ (A) is a Hilbert space, and also that M := F ∩ dom A,

(2.2)

equipped with the inner product (2.1), is a closed subspace of H+ (A). We write H+ (A) = M [+] U, where [+] denotes the orthogonal sum in H+ (A). Lemma 2.1. The subspace U of H+ (A) is finite-dimensional and dim U = dim G.

(2.3)

Proof. Since dom A is dense in H and dim G < ∞, the image of the orthogonal projection of dom A onto the subspace G of H is the whole space G. Therefore the image of the orthogonal projection of U onto the subspace G is also the whole space G. It remains to note that a system of vectors in U is linearly independent if and only if their projections onto the subspace G are linearly independent. Theorem 2.2. Let A be a maximal dissipative operator in the Hilbert space H. If H = F ⊕G and dim G < ∞, then dom A∩F is dense in F and the operator T := PF A|dom A∩F is maximal dissipative in F. For the proof of Theorem 2.2 we need two lemmas in which the operator Z := PF (A + iIH ), considered as a bounded operator from H+ (A) to F, plays a role; here IH is the identity operator on H. Thus, for any vector m + u ∈ H+ (A),

m ∈ M, u ∈ U,

the formula Z(m + u) = (T + iIF )m + Bu

(2.4)

defines an operator B acting from the space U to the space F. Denote K := {u ∈ U| Bu ∈ ran (T + iIF )},

κ := dim K.

Then u ∈ K if and only if there exists m ∈ M such that m+u ∈ ker Z. Indeed, and, by (2.4), if u ∈ K then there exists m ∈ M such that Bu = (T + iIF )m we can set m = −m. Conversely, if m ∈ M, u ∈ U and m + u ∈ ker Z, then

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the relation (T + iIF )m + Bu = 0 implies that Bu ∈ ran (T + iIF ) and hence u ∈ K. Lemma 2.3. dim ker Z = κ. Proof. As was shown above, k ∈ K if and only if there exists an m ∈ M such that m + k ∈ ker Z. Since the operator A is dissipative, the operator T is also dissipative, and hence ker(T + iIF ) = {0}. This implies that a system of vectors k1 , k2 , . . . , kn ∈ K is linearly independent if and only if the corresponding system of vectors k1 + m1 , k2 + m2 , . . . , kn + mn ∈ ker Z is linearly independent. Indeed, if n n n λj (kj + mj ) = λj kj + λ j mj , 0= j=1

n

j=1

n

n

j=1

then j=1 λj kj = 0 since j=1 λj kj ∈ U, j=1 λj mj ∈ M and M ∩ U = {0}. n Conversely, let j=1 λj kj = 0. In view of (2.4), n

λj (Bkj + (T + iIF )mj ) = 0,

j=1

and hence 0=

n

n λj T + iIF mj = (T + iIF ) λ j mj ,

j=1

which implies that

n

j=1

j=1

λj mj = 0.

Lemma 2.4. ran B ⊂ ran (T + iIF ). Proof. Consider the linear manifold W := (A + iIH ) ker Z. By Lemma 2.3, dim W ≤ dim ker Z = κ.

(2.5)

According to (2.4), W consists of those and only those elements of ran (A + iIH )

(2.6)

that are of the form 0 ⊕ u,

u ∈ U.

In view of the maximal dissipativity of the operator A, the set (2.6) coincides with H, hence W = U and dim W = dim U. Taking into account the relations (2.5), (2.3), we obtain κ = dim U.

(2.7)

Since K is the dimension of the subspace of U which consists of all elements u with Bu ∈ ran (T + iIF ), (2.7) implies that this subspace coincides with U. Now let x ∈ ran B. Then there exists u ∈ U such that B u = x. Since the set of elements u ∈ U with Bu ∈ ran (T + iIF ) coincides with U, we have B u = x ∈ ran (T + iIF ).

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Proof of Theorem 2.2. That dom T = dom A ∩ F is dense in F was mentioned at the beginning of this section. Since A is dissipative, the operator T is also dissipative. It remains to prove the relation ran (T + iIF ) = F.

(2.8)

To this end, consider f ∈ F. Since the operator A is maximal dissipative, there exists an element m + u ∈ M [+] U (= H+ (A)) such that (A + iIH ) (m + u) = f ⊕ 0 ∈ F ⊕ G. In accordance with (2.4), f = (T + iIF )m + Bu. By Lemma 2.4, Bu = (T + iIF ) r for some r ∈ M. Thus f = (T + iIF )(m + r), and (2.8) is proved.

Acknowledgments The author gratefully acknowledges the support of an anonymous referee who pointed out Stenger’s lemma to him and helped to improve the presentation of this note.

References [1] Gohberg, I.C., Krein, M.G.: Fundamental aspects of defect numbers, root numbers and indexes of linear operators. Uspekhi Mat. Nauk. 12 2(74), 43–118 (1957) (Russian) (Engl. translation: AMS Translations, ser. 2, vol. 13, 185–264 (1960)) [2] Stenger, W.: On the projection of a selfadjoint operator. Bull. Am. Math. Soc. 74, 369–372 (1968) [3] Dijksma, A., Langer, H., de Snoo, H.S.V.: Unitary colligations in Πκ -spaces, ˇ characteristic functions and Straus extensions. Pacif. J. Math. 125(2), 347–362 (1986) [4] Dijksma, A., Luger, A., Shondin, Yu.: Minimal models for Nk∞ -functions. Oper. Theory Adv. Appl. 163, 97–134 (2005) [5] Arov, D.Z.: Passive linear stationary dynamical systems. Sibirsk. Math. J. 20(2), 211–228 (Russian) (1979)

Vol. 70 (2011)

A Generalization of Stenger’s Lemma

M. A. Nudelman Integrated Banking Information Systems Troitskaya 13 Odessa 65125 Ukraine e-mail: [email protected] Received: January 2, 2010. Revised: April 6, 2011.

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Integr. Equ. Oper. Theory 70 (2011), 307–322 DOI 10.1007/s00020-010-1858-8 Published online February 6, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Oblique Projections and Sampling Problems Gustavo Corach and Juan I. Giribet Abstract. In this work, the consistent sampling requirement of signals is studied. We establish how this notion is related with certain set of projectors which are selfadjoint with respect to a semi-inner product. We extend previous results and present some new problems related with sampling theory. Mathematics Subject Classification (2010). Primary 94A20; Secondary 47A58. Keywords. Consistent sampling, oblique projections.

1. Introduction In signal processing language, sampling is an operation which converts a continuous signal (i.e., function) into a discrete one. This is a previous step which allows the analysis of a signal in the computer. The classical sampling scheme is based on the Whittaker–Kotelnikov-Shannon theorem. Recall that the Paley–Wiener space of band-limited functions is the space PW of π all f ∈ L2 (R) which can be written as f (t) = −π g(ω)eiωt dω, t ∈ R for some g ∈ L2 (R). The Whittaker–Kotelnikov-Shannon theorem establishes that it is possible to reconstruct any signal f ∈ PW from its values at the converges integers {f (n)}n∈Z . More precisely, the series n∈Z f (n) sin(π(t−n)) π(t−n) 2 uniformly to f (t). If a signal f ∈ L (R) does not belong to the Paley–Wiener space, a common strategy in signal processing applications is to apply a low ) to the signal pass ﬁlter (i.e., the operator mapping f → f, sin(π(.−n)) π(.−n) f , obtaining a new one g. Although the signal recovered from the samples {g(n)}n∈Z in general does not coincide with the original f , it turns out that it is always a good approximation of it. In fact, the recovered signal is the orthogonal projection of the original signal in PW. For a detailed exposition of these facts see [21]. An usual way of representing the samples of a signal f is the following: given vectors {vn }n∈N which spans a closed subspace This research was partially supported by CONICET PIP2010, ANPCYT PICT 209/06 and 008/08, PUNQ 0530/07, UBACyT I023.

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S (sampling subspace), these samples are given by {fn }n∈N = { f, vn }n∈N [21]. On the other hand, given samples {fn }n∈N , the reconstructed signal fˆ is given by fˆ = n∈N fn wn , where {wn }n∈N spans a closed subspace R (reconstruction subspace). In the classical sampling scheme the reconstruction and the sampling subspaces are assumed to be the same. However, in many applications, this not the case, and then it is not always possible to recover the best approximation of the original signal. Thus, different sampling techniques must be used. In [22], Unser and Aldroubi introduced the idea of consistent sampling. Here, the reconstructed signal fˆ is not generally the best approximation of the original signal, but f and fˆ are forced to have the same samples. Originally, this idea has been studied in shift invariant spaces. Later, in [10,11], the consistent sampling has been studied in abstract Hilbert spaces by Y. Eldar for ﬁnite dimensional spaces and by Y. Eldar and T. Werther for inﬁnite dimensional spaces, respectively. An underlying assumption in these works is that the Hilbert space H of signals can be decomposed as H = R S ⊥ (where is the direct sum). Under this assumption, it is shown that the consistent sampling requirement is related with the (unique) pro⊥ jection P with range R(P ) = R and nullspace N (P ) = S . More precisely, the unique signal fˆ ∈ R that satisﬁes fˆ, vn = f, vn , for every n ∈ N, is given by fˆ = P f . However, in some signal applications the hypothesis R ∩ S ⊥ = {0} is not satisﬁed. Hirabayashi and Unser in [13], proved that in this case there are inﬁnitely many projections P such that fˆ = P f satisﬁes the consistent sampling requirement. The main goal of this paper is to give an interpretation of the consistent sampling in terms of the notion of compatibility between a closed subspace S of a Hilbert space H and a positive semi-definite operator A acting on H. This notion, deﬁned in [5] and developed later in [2–4,6], has a completely different origin. In [19], Pasternak-Winiarski studied, for a ﬁxed subspace S, the analyticity of the map A → PA,S which associates to each positive invertible operator A the orthogonal projection onto S under the (equivalent) inner product f, g A = Af, g , for f, g ∈ H. These perturbations of the inner product occur quite frequently, the reader is referred to [16–18] for many applications. The notion of compatibility appears when A is allowed to be any positive semi-definite operator, not necessarily invertible (and even, a selfadjoint bounded linear operator). More precisely, A and S are said to be compatible if there exists a (bounded linear) projection Q with range S which satisﬁes AQ = Q∗ A (i.e., Q is Hermitian with respect to the semiinner product , A ). Unlike for invertible A’s, it may happen that there is no such Q or that there is an inﬁnite number of them. However, there exists an angle condition between S ⊥ and A(S) which determines the existence of such projections. As far as we know, with the exception of [11], the consistent sampling requirement has not been studied in the context of perturbed inner spaces. In [11], the assumption R ∩ S ⊥ = {0} forces the perturbations of the inner product to be deﬁned by positive invertible operators, i.e., the consistent

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sampling idea is studied in an equivalent Hilbert space. But, as we show in this work, the consistent sampling requirement in semi-inner product spaces allows a simpler and more general way for studying this problem. It is important to remark that, although some of the results given in [13] can be directly extended to inﬁnite dimensional spaces by means of Moore– Penrose pseudo inverses, for some others results this approach is not possible. For instance, it is easy to characterize, in inﬁnite dimensional spaces, the (possibly inﬁnite) set of oblique projections related with the consistent sampling idea. Here the notion of compatibility plays an important role because, as we show below, this notion is closely related with the notion of angle condition between subspaces. By this reason, if S ⊥ ∩ R = {0} there are inﬁnitely many oblique projections related with the consistent sampling requirement; in [13] criteria for selecting one among these projections have been considered. These criteria are motivated by signal processing applications. Let C be the set of all projections that satisﬁes the consistent sampling requirement. In [13], given a subspace M ⊆ R all projections Q ∈ C such that M ⊆ R(Q) are characterized. In this paper, we study a more general problem: given a subspace M ⊆ R, we give necessary and sufﬁcient conditions for the existence of a projection Q0 ∈ C such that f − Q0 f ≤ f − Qf for every f ∈ M and every Q ∈ P, and, if those conditions hold, we characterize the set of all these Q0 . As a corollary we study the case Q0 f = f , for every f ∈ M. Also motivated by signal processing applications, we characterize those projections in C which minimize the so called aliasing norm [10,14,15]. The paper is organized as follows: Sect. 2 contains the preliminaries. In Sect. 3 we present some variational problems in the set P(A, S). These problems are related with two sampling problems presented in Sect. 5. In Sect. 4, we present the notion of consistent sampling and the link with the compatibility. We characterize the set of operators satisfying a consistent sampling requirement and characterize some operators with particular properties. In Sect. 5, we study when it is possible to impose to the consistent sampling requirement the additional property of recovering the best approximation of a signal, for certain set of signals. Finally, in Sect. 6 a problem related with the sampling of perturbed signals is presented.

2. Oblique Projections and Compatibility In this section, we present a survey of useful results concerning the existence of projections which are orthogonal, in some sense, with respect to a ﬁxed positive semi-definite operator. We start with some notation. Along this work H denotes a (complex, separable) Hilbert space with inner product , . Given Hilbert spaces H and K, L(H, K) denotes the space of bounded linear operators from H into K and L(H) = L(H, H). If T ∈ L(H) then T ∗ denotes the adjoint operator of T, R(T ) stands for the range of T and N (T ) for its nullspace. If S is a closed subspace of H and T is a closed subspace of K, then L(S, T ) will be identiﬁed with the subspace of L(H, K) consisting of all T ∈ L(H, K) such that R(T ) ⊆ T and S ⊥ ⊆ N (T ).

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Let GL(H) denote the group of invertible operators of L(H), L(H)+ the cone of (semi-definite) positive operators of L(H), GL(H)+ = L(H)+ ∩GL(H) and Q the set of projections of L(H), i.e., Q = {Q ∈ L(H) : Q2 = Q}. If S and T are two (closed) subspaces of H, denote by S T the direct sum of S and T , S ⊕ T the (direct) orthogonal sum of them and S T = S ∩ (S ∩ T )⊥ . If H = S T , the (oblique) projection PS//T on to S along T is the projection uniquely determined by R(PS//T ) = S and N (PS//T ) = T . In particular, PS = PS//S ⊥ is the orthogonal projection on to S. Two notions of angles between subspaces are recalled now. The reader is referred to [1,7,8,12] for proofs. Given two subspaces S, T , the cosine of the Friedrichs angle θ(S, T ) ∈ [0, π/2] between them is deﬁned by c(S, T ) = sup{| f, g | : f ∈ S T , f < 1, g ∈ T S, g < 1}. Remark 2.1. From the fact that c(S, T ) = c(S ⊥ , T ⊥ ), the following conditions are equivalent: 1. c(S, T ) < 1; 2. S + T is closed; 3. S ⊥ + T ⊥ is closed; 4. c(S ⊥ , T ⊥ ) < 1. The Dixmier angle between S and T is the angle in [0, π/2] whose cosine is deﬁned by c0 (S, T ) = sup{| f, g | : f ∈ S, f < 1, g ∈ T , g < 1}. Observe that, in general c(S, T ) ≤ c0 (S, T ) and if S ∩ T = {0} then the equality holds. Notice that c0 (S, T ) < 1 ⇐⇒ S ∩ T = {0} and S + T is closed. Observe that, by its definition, c0 (S, T ) is monotone in each variable: if S ⊆ S and T ⊆ T then c0 (S, T ) ≤ c0 (S , T ). However, this is not true, in general, for Friedreich’s cosine. In [22], Unser and Aldroubi introduced a notion of largest angle or uniform cosine angle between two subspaces S and T : this is the angle in [0, π/2] whose cosine is R(S, T ) = inf { PT s , s ∈ S, s = 1}. This notion, widely used in signal processing literature, is related with the Dixmier angle. In fact, R(S, T ) = 1 − c0 2 (S, T ⊥ ), and all relevant properties of R can be easily deduced from those of c0 . Given A ∈ L(H)+ consider f, g A = Af, g , for every f, g ∈ H. Then , A deﬁnes a semi-inner product on H. There is also a seminorm associated 1/2 to , A , namely f A = Af, f for every f ∈ H. An operator T ∈ L(H) is A-selfadjoint if T f, g A = f, T g A , for every f, g ∈ H. The following lemma characterizes the A-selfadjoint projections. Lemma 2.2. (Krein, [5]) Let A ∈ L(H)+ and Q ∈ Q. Then the following conditions are equivalent: a. AQ = Q∗ A, i.e., Q is A-selfadjoint, b. N (Q) ⊆ A−1 (R(Q)⊥ ).

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Definition 1. If A ∈ L(H)+ and S is a closed subspace of H, the pair (A, S) is said to be compatible if there exists Q ∈ Q such that R(Q) = S and AQ = Q∗ A. The following result gives a list of equivalent conditions to the compatibility. Lemma 2.3. [5] Let A ∈ L(H)+ and S be a closed subspace of H. The following conditions are equivalent: 1. 2. 3.

The pair (A, S) is compatible, H = S + A−1 (S ⊥ ), c0 (S ⊥ , A(S)) < 1.

The next theorem, due to Douglas [9] plays a relevant role in what follows. Theorem 2.4. (Douglas, [9]) Let A, B ∈ L(H). Then the following conditions are equivalent: 1. 2. 3.

R(A) ⊆ R(B). There exists a positive number λ such that AA∗ ≤ λBB ∗ . There exists D ∈ L(H) such that AD = B.

Moreover, in this case there exists a unique solution D of the equation AX = B such that R(D) ⊆ N (A)⊥ . D is called the reduced solution of the equation AX = B. If A† denotes the Moore–Penrose inverse of A, then D = A† B. It also satisﬁes that D 2 = inf{λ : AA∗ ≤ λBB ∗ }. Corollary 2.5. [5] Let S be a closed subspace of H and write A ∈ L(H)+ , in terms of the decomposition H = S ⊕ S ⊥ , as a b A= ∗ (1) b c Then the pair (A, S) is compatible if and only if R(b) ⊆ R(a). Denote P(A, S) = {Q ∈ Q : R(Q) = S and AQ = Q∗ A}, i.e., P(A, S) is the set of A-selfadjoint projections with range S. From now on, denote N = S ∩ N (A). If N = {0} then P(A, S) is a singleton (see Theorem 2.7). Remark 2.6. Using the

decomposition given in Eq. (1), we can write I x P(A, S) = {Q = : x ∈ L(S ⊥ , S) and ax = b}. 0 0 The set P(A, S) can be empty, a singleton (for example, if A is positive definite) or an inﬁnite set. Indeed, this set is an aﬃne manifold. The next theorem provides a parametrization of P(A, S), if A and S are compatible.

I d , where d is the reduced solution of the equation Let PA,S = 0 0 ax = b. We recall the main properties of the projection PA,S .

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Theorem 2.7. [5,6] Let A ∈ L(H)+ and S be a closed subspace of H such that (A, S) is compatible. Then PA,S ∈ P(A, S) is the projection onto S with nullspace A−1 (S ⊥ ) N . The set P(A, S) is an aﬃne manifold and it can be parametrized as P(A, S) = PA,S + L(S ⊥ , N ), or, in terms of the matrix decomposition given above, I d+z P(A, S) = , 0 0 with R(z) ⊆ N (a). Moreover, PA,S has minimal norm in P(A, S), but it is not the unique with this property, in general. However, for every f ∈ H it holds that (I − PA,S )f is the unique minimal norm element in the set {(I − Q)f : Q ∈ P(A, S)}.

(2)

In the following, we recall basic definitions and results related to frames of closed subspaces. Definition 2. Let S be a closed subspace of H. The set V = {vn }n∈N ⊆ S is a frame for S if there exist numbers γ1 , γ2 > 0 such that γ1 f 2 ≤ | f, vn |2 ≤ γ2 f 2 , for every f ∈ S. (3) n∈N

If the set V = {vn }n∈N is also linearly independent then it is called a Riesz basis of S. Let S be a closed subspace of H and let V = {vn }n∈N be a frame for S. Let B = {en }n∈N be the canonical orthonormal basis of 2 . The unique operator F ∈ L(2 , H) such that F en = vn , for every n ∈ N is called the synthesis operator of V. The adjoint operator F ∗ ∈ L(H, 2 ) is called the analysis ∗ operator of V, and it is given by F f = n∈N f, vn en . Finally, the operator T = F F ∗ ∈ L(H) is called the frame operator of V and, from Eq. (3), it satisﬁes: γ1 PS ≤ PS T PS ≤ γ2 PS and then 1/γ1 PS ≤ PS T † PS ≤ 1/γ2 PS , where T † denotes the Moore-Penrose inverse of T .

3. Minimization Problems in the Set P(A, S) The purpose of this section is to study the following problem which, as will be shown below, is related with some signal processing applications. As it was stated in Sect. 2, the reduced solution of ax = b gives the element PA,S ∈ P(A, S). In [4], another interesting characterization of the element PA,S is given in terms of the solution of a variational problem. It comes from the following properties of PS ⊥ and PS : if S is a closed subspace, it is easy to see that the orthogonal projection PS is the unique solution of the problem min

Q∈Q,R(Q)=S

QQ∗ ,

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in a similar way, PS ⊥ is the unique solution of min

Q∈Q,N (Q)=S

Q∗ Q;

In [4] the following results are proven. Let A = {Q ∈ Q : R(Q) ⊆ M⊥ , N (Q) = S}. Proposition 3.1. Let S and M be two closed subspaces of H such that H = S + M⊥ . Then P0 = PM⊥ S//S is the unique solution of min Q∗ Q.

Q∈A

Corollary 3.2. Let A ∈ L(H)+ and S be a closed subspace of H such that the pair (A, S) is compatible. Then PA,S ∈ P(A, S) is the unique solution of min

(I − Q)∗ (I − Q)

Q∈P(A,S)

These results suggest the following problem. Problem 3.3. Let A ∈ L(H)+ and S be a closed subspace of H, such that the pair (A, S) is compatible. Characterize the closed subspaces M for which there exists Q0 ∈ P(A, S) such that

Q0 f ≤ Qf , for every Q ∈ P(A, S), f ∈ M

(4)

For each such M, determine the set of all Q0 . Observe that, if Q0 f ≤ Qf for every f ∈ M and Q ∈ P(A, S), then PM Q∗0 Q0 PM f, f ≤ PM Q∗ QPM f, f for every f ∈ H, i.e., Q0 ∈ P(A, S) is a solution of min

Q∈P(A,S)

PM Q∗ QPM .

The following Theorem gives necessary and sufﬁcient conditions for the existence of solutions of Problem 3.3. Furthermore, it is shown that the solutions of Problem 3.3 can be related with the solution of certain operator equations. Theorem 3.4. Let A ∈ L(H)+ and S be a closed subspace of H, such that the pair (A, S) is compatible. Given a closed subspace M of H, there exists Q0 ∈ P(A, S) satisfying (4) if and only if M⊥ + N ⊆ M⊥ + S ⊥ . Moreover, Q0 = PA,S + W0 PS ⊥ , where W0∗ ∈ L(H) is a solution of the equation, PM PS ⊥ X = PM PN .

(5)

Proof. By Theorem 2.7, any Q ∈ P(A, S) can be written as Q = PA,S + W PS ⊥ , with R(W ) ⊆ N = N (A) ∩ S = A−1 (S ⊥ ) ∩ S. Observe that, if Q0 ∈ P(A, S) is decomposed as Q0 = PA,S + W0 PS ⊥ , with R(W0 ) ⊆ N , then Q0 is a solution of (4) if and only if W0 ∈ L(H, N ) is such that

PA,S PM f − W0 PS ⊥ PM f ≤ PA,S PM f − W PS ⊥ PM f ,

(6)

for every f ∈ H and for every bounded linear operator W with R(W ) ⊆ N . Given f ∈ H, observe that if f ∈ N (PS ⊥ PM ), then (6) holds for every W ∈ L(H, N ). By the other hand, if f ∈ / N (PS ⊥ PM ), given η ∈ N there

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exists W ∈ L(H, N ) such that η = W PS ⊥ PM f . Then, W0 ∈ L(H, N ) satisﬁes (6) if and only if

PA,S PM f − W0 PS ⊥ PM f ≤ PA,S PM f − η ,

(7)

for every η ∈ N . Therefore, W0 PS ⊥ PM f ∈ N is the best approximation of PA,S PM f in the subspace N , i.e., W0 PS ⊥ PM f = PN PA,S PM f = PN PM f , since PN PA,S = PN . Thus, we look for those W0 ∈ L(H) such that, PM PS ⊥ W0∗ = PM PN . By Theorem 2.4, the above equation has a solution if and only if R(PM PN ) ⊆ R(PM PS ⊥ ) or, analogously, if and only if M⊥ + N ⊆ M⊥ + S ⊥ . Observe that the set of solutions of (5) is the aﬃne manifold: (PM PS ⊥ )† PM PN + L(H, N (PM PS ⊥ )) Examples. The following are sufﬁcient conditions for the existence of solutions of (4). 1. If N = {0}, it obviously follows that M⊥ + N ⊆ M⊥ + S ⊥ . Recall that, in this case, P(A, S) = {PA,S }, so that (4) has a unique solution for every M. 2. If S and M are closed subspaces such that c0 (S, M) < 1, then H = S ⊥ + M⊥ (see Remark 2.1). Thus M⊥ + N ⊆ M⊥ + S ⊥ , and there exists a solution of Eq. (4). In this case, if X is a solution of (5) then X ∈ (PM PS ⊥ )† PN + L(H, S + S ⊥ ∩ M⊥ ), since N ((PM PS ⊥ )† ) = M⊥ . Proposition 3.5. Let (A, S) be a compatible pair. Then there exists Q0 ∈ P(A, S) such that Q0 PM = 0 if and only if M ⊆ A−1 (S ⊥ ) and c0 (M, S) < 1. Proof. Suppose that M ⊆ A−1 (S ⊥ ) and c0 (M, S) < 1; from the second condition it follows that M ∩ S = {0} and M S is closed. Let T = A−1 (S ⊥ ) (S M); by Lemma 2.3 it follows that H = S T M. Deﬁne Q0 = PS//M+T ; by Lemma 2.2 it follows that Q0 ∈ P(A, S). Furthermore, for every f ∈ M, Q0 f = 0. Conversely, if there exists Q0 ∈ P(A, S) such that Q0 PM = 0, then M ⊆ N (Q0 ) ⊆ A−1 (S ⊥ ). Furthermore, since c(N (Q0 ), S) = c0 (N (Q0 ), S) < 1, it follows that c0 (M, S) ≤ c0 (N (Q0 ), S) < 1. The following result characterizes the set of projections Q ∈ P(A, S) that satisﬁes QPM = 0. Proposition 3.6. Let A ∈ L(H)+ and S be a closed subspace of H such that the pair (A, S) is compatible. Let M ⊆ A−1 (S ⊥ ) be a closed subspace of H such that c0 (M, S) < 1. Then, the set of projections Q ∈ P(A, S) satisfying QPM = 0 is the aﬃne manifold PS//M+T + L(S ⊥ ∩ M⊥ , N ), where the (closed) subspace T ⊆ A−1 (S ⊥ ) is any complement of N M in A−1 (S ⊥ ). Proof. Since M ⊆ A−1 (S ⊥ ) satisﬁes c0 (M, S) < 1, it follows that c0 (M, N ) < 1 and, then, M N ⊆ A−1 (S ⊥ ) is closed. Let T be a complementary subspace of M N in A−1 (S ⊥ ), and let Q0 = PS//MT . By Lemma 2.3, it follows that Q0 ∈ P(A, S) and for every f ∈ M, Q0 f = 0. Let

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T ∈ L(S ⊥ ∩ M⊥ , N ); since T ∈ L(S ⊥ , N ), it follows that Q = Q0 + T ∈ P(A, S). Furthermore, if f ∈ M, it is easy to see that Qf = 0. Then every Q ∈ Q0 + L(S ⊥ ∩ M⊥ ) ⊆ P(A, S), satisﬁes QPM = 0. Conversely, suppose that Q, P ∈ P(A, S) satisfy QPM = 0. Then, by Theorem 2.7, Q − P ∈ L(S ⊥ , N ). If f ∈ M, it follows that Qf = P f = (Q − P )f = 0, i.e., M + S ⊆ N (Q − P ). Then Q − P ∈ L(S ⊥ ∩ M⊥ , N ).

4. Consistent Sampling In this section we study a generalization of a sampling problem proposed by Unser and Aldroubi in [22] and generalized in [10,11,13]. More specifically, let f be a signal (a vector) of a suitable Hilbert space H, and let S and R be closed subspaces of H, called, respectively, the sampling and reconstruction subspaces. Given Vs = {vn }n∈N , a frame of S, with synthesis operator H, g = H ∗ f = N f, vn en is called the samples of f . As an inverse procedure, we have the reconstruction process. Given Vr = {wn }n∈N , a frame of R with synthesis operator F , given the samples g ∈ 2 , the reconstructed signal associated to them is f = F g. In some signal applications we deal with the problem: given the synthesis and analysis operators, ﬁnd a ﬁlter (a bounded linear operator) X ∈ L(2 ) such that the signal reconstructed from the ﬁltered samples, i.e., the signal reconstructed from the samples h = Xg, has good (in a sense to be established) approximation properties. In particular, the consistent sampling condition imposes that, given the samples g = H ∗ f , the reconstructed signal f˜ = F Xg = F XH ∗ f has the same samples as f , for every f ∈ H (i.e., H ∗ f = H ∗ f˜). From the point of view of its samples, the original signal and the reconstructed one are undistinguished. If the consistent sampling condition is satisﬁed, then, for every f ∈ H, it holds that H ∗ f = H ∗ f˜, so that (F XH ∗ )2 f = F XH ∗ (F XH ∗ f ) = F XH ∗ f˜ = F XH ∗ f, i.e., F XH ∗ is a projection. Moreover, if f ∈ N (F XH ∗ ), then F XH ∗ f = 0 = H ∗ F XH ∗ f = H ∗ f˜ = H ∗ f , thus f ∈ N (H ∗ ) = R(H)⊥ = S ⊥ . Furthermore, since S ⊥ ⊆ N (F XH ∗ ), it follows that F XH ∗ is a projection with N (F XH ∗ ) = S ⊥ . Based on this idea we give the following definition. Definition 3. Given H, F ∈ L(K, H), the operator X ∈ L(K) satisﬁes the consistent sampling requirement for H and F if X ∈ CS(F, H) := {X ∈ L(K) : F XH ∗ ∈ Q, N (F XH ∗ ) = N (H ∗ )}. Since R(F XH ∗ ) ⊆ R(F ), if there exists an operator X ∈ L(K) satisfying the consistent sampling requirement, then H = R(F ) + N (H ∗ ). The converse is also true because, if H = R(F ) + N (H ∗ ) it follows that X = F † PR(F )N (H ∗ )//N (H ∗ ) H ∗ † ∈ L(K) is well deﬁned and it is easy to see that satisﬁes the consistent sampling requirement. In [10,11,22] the consistent sampling requirement has been studied under the condition that H = N (H ∗ ) R(F ). Recently, in [13], this notion has been studied (for ﬁnite dimensional spaces) without the assumption R(F )∩N (H ∗ ) = {0}, and it was shown that the set of projections of the type

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F XH ∗ , with N (F XH ∗ ) = N (H ∗ ) can be inﬁnite. In this work we relate the consistent sampling condition in inﬁnite dimensional spaces with the notion of compatibility introduced in [5] and developed in [2,4,6]. We show that the results given in [13] can be easily obtained using some characterizations of the set P(A, N (H ∗ )), for an appropriate A ∈ L(H)+ . The following set will be useful to characterize, in terms of the compatibility, the projections associated with the consistent sampling requirement. Given a closed range operator H ∈ L(H, K), and a closed subspace W of K let AH (W) := {A ∈ L(H)+ : A−1 (R(H)) = W} Observe that, if R is a closed subspace of H such that H = N (H ∗ ) + −1 = R(PR⊥ PN (H ∗ ) )⊥ = R, then PR⊥ ∈ AH (R), because PR ⊥ (R(H)) ⊥ N (PN (H ∗ ) PR⊥ ) = R R ∩ R(H) = R. Also notice that, in this case, for any A ∈ AH (R) the pair (A, N (H ∗ )) is compatible (see Lemma 2.3). Lemma 4.1. Consider operators F, H ∈ L(K, H) with closed range such that H = R(F ) + N (H ∗ ) and A ∈ AH (R(F )). Then X ∈ CS(F, H) if and only if F XH ∗ = I − Q, for some Q ∈ P(A, N (H ∗ )). Proof. Suppose that X ∈ L(K) is such that I − F XH ∗ ∈ P(A, N (H ∗ )). Then, R(I − F XH ∗ ) = N (F XH ∗ ) = N (H ∗ ), and therefore X ∈ CS(F, H). Conversely, if X ∈ CS(F, H) then E = F XH ∗ is a projection with N (E) = N (H ∗ ). Furthermore N (I − E) = R(E) ⊆ R(F ) = A−1 (R(H)). Then, by Lemma 2.3, Q = I − E ∈ P(A, N (H ∗ )), since R(H) = R(H) = N (H ∗ )⊥ The above Lemma allows us to give a parametrization of the operators that satisﬁes a consistent sampling requirement. A similar result, for ﬁnite dimensional spaces, can be found in [13]. Theorem 4.2. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ), T = R(F ) N (H ∗ ) and R = R(F ∗ ) ∩ F −1 (N (H ∗ )). Then, CS(F, H) = F † PT //N (H ∗ ) H ∗ † + L(R(H ∗ ), R) + L(K, N (F )) + L(N (H), K) Moreover, for the operator X0 = F † PT //N (H ∗ ) H ∗ † ∈ CS(F, H) it holds that

F X0 H ∗ f ≤ F XH ∗ f for every X ∈ CS(F, H) and f ∈ H. Proof. Let A ∈ AH (R(F )), so that R(F ) = A−1 (R(H)). Since H = R(F ) + N (H ∗ ), it follows from Definition 1 that the pair (A, N (H ∗ )) is compatible. Suppose that X ∈ CS(F, H); by Lemma 4.1, I − F XH ∗ ∈ P(A, N (H ∗ )). Then, by Theorem 2.7, I − F XH ∗ = PA,N (H ∗ ) + W , for some W ∈ L(R(H), N (H ∗ )∩A−1 (R(H))) (i.e., for some W ∈ L(R(H), N (H ∗ )∩R(F ))). Recall that PA,N (H ∗ ) = PN (H ∗ )//T , where T = A−1 (R(H)) N (H ∗ ) = R(F ) N (H ∗ ); then PN (F )⊥ XPN (H)⊥ = F † F XH ∗ H ∗ † = F † (I − PN (H ∗ )//T )H ∗ † − F † W H ∗ † . Furthermore, since R(H ∗ † ) = N (H ∗ )⊥ = R(H) and N (H ∗ † )⊥ = R(H ∗ ), it follows that ˜, PN (F )⊥ XPN (H)⊥ = F † PT //N (H ∗ ) H ∗ † + F † W

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˜ ∈ L(R(H ∗ ), N (H ∗ ) ∩ R(F )). Then, with W X ∈ H † PT //N (H ∗ ) F ∗ † + L(K, N (F )) + L(N (H), K) + L(R(H ∗ ), R), (8) since R(F † ) = N (F )⊥ = R(F ∗ ). Conversely, if X ∈ L(K) satisﬁes condition (8), then F XH ∗ = PR(F ) PT //N (H ∗ ) PR(H) + F W H ∗ , for some W ∈ L(R(H ∗ ), R). Combining the inclusions R(PT //N (H ∗ ) ) ⊆ R(F ), R(H)⊥ = N (PT //N (H ∗ ) ) and R(F W ) ⊆ N (H ∗ ) ∩ R(F ), we get I − F XH ∗ = ˜ , for some W ˜ ∈ L(N (H ∗ )⊥ , N (H ∗ ) ∩ R(F )). This means that PN (H ∗ )//T + W I − F XH ∗ ∈ P(A, N (H ∗ )) for any A ∈ AH (R(F )). In virtue of Lemma 4.1, it follows that X ∈ CS(F, H). Now, let A ∈ AH (R(F )) and X ∈ F † PT //N (H ∗ ) H ∗ † + L(K, N (H)) + L(N (H), K) + L(R(H ∗ ), R). From the paragraph above, it follows that F XH ∗ = I − Q for some Q ∈ P(A, N (H ∗ )). Observe that F X0 H ∗ = I − PA,N (H ∗ ) and, by Theorem 2.7, it follows that F X0 H ∗ f ≤ F XH ∗ f , for every f ∈ H. Observe that, if N (F ) = {0} and N (H) = {0} then CS(F, H) = F † PR(F )N (H ∗ )//N (H ∗ ) H ∗ † + L(K, F −1 (N (H ∗ ))). Remark 4.3. to (1), A ∈ AH (R(F )) has a decompo Assume that, according

a b 0 d sition A = ∗ , where d is the Douglas solution of . Then X0 = 0 0 b c the equation ax = b. Remark 4.4. An important magnitude related with signal processing applications is the aliasing norm (see [10,14,15]): suppose that T ∈ L(H) is the operator that assign to every signal f ∈ H the reconstructed signal fˆ = T f . The aliasing norm is given by T PR(T )⊥ . Given X ∈ CS(F, H), let aX be the aliasing norm corresponding to F XH ∗ . Then, by Theorem 4.2, the operator X0 = F † PR(F )N (H ∗ )//N (H ∗ ) H ∗ † satisﬁes aX0 ≤ aX , for every X ∈ CS(F, H). Suppose that F, F ∈ L(K, H) and H, H ∈ L(K, H) satisfy R(F ) = R(F ) and R(H) = R(H ). Although, in general CS(F, H) = CS(F , H ), it is possible to establish a relation between these sets.

Lemma 4.5. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ) and let A ∈ AH (R(F )). Given Q ∈ P(A, N (H ∗ )) there is a unique X ∈ CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )) such that Q = I − F XH ∗ . †

Proof. Given Q ∈ P(A, N (H ∗ )), it is easy to see that F † (I − Q)H ∗ ∈ CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )). Suppose that Q = I − F XH ∗ = I − F X H ∗ , with X, X ∈ CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )). Then, 0 = F (X − X )H ∗ = F † F (X − X )H ∗ H ∗ † = PR(F ∗ ) (X − X )PR(H ∗ ) = X − X .

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Proposition 4.6. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ). Suppose that H ∈ L(K, H) and F ∈ L(K, H) satisfy R(F ) = R(F ) and R(H) = R(H ). Then there is a bijection between CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )) and CS(F , H ) ∩ L(R(H ∗ ), R(F ∗ )). Proof. Let S = R(H)⊥ = R(H )⊥ , T = R(F ) = R(F ) and A ∈ AH (T ). Fix X ∈ CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )). By Lemma 4.1 it holds Q = I − F XH ∗ ∈ P(A, S) and by Lemma 4.5, there exists a unique X ∈ CS(F , H )∩ L(R(H ∗ ), R(F ∗ )) such that Q = I −F X H ∗ . Let F(X ) = F † F X H ∗ H ∗ † . It follows that F is a bijection between CS(F, H) ∩ L(R(H ∗ ), R(F ∗ )) and † CS(F , H ) ∩ L(R(H ∗ ), R(F ∗ )), and F −1 (X) = F † F XH ∗ H ∗ .

5. Best Approximation of Signals with Consistency Requirement As it was mentioned in the introduction, if one can not recover a signal f ∈ H by its samples, a common strategy in signal processing is to apply a ﬁlter and thus recover the orthogonal projection of f in PW. Although in the consistent sampling requirement there is a projection Q involved, the situation is different. Given F, H ∈ L(K, H) such that H = R(F ) + N (H ∗ ), Lemma 4.1 establishes that, for a signal f ∈ H, the reconstructed signal f˜ ∈ R(F ) is given by f˜ = (I − Q)f , for some Q ∈ P(A, N (H ∗ )), with A ∈ AH (R(F )). Since, generally, Q is not selfadjoint, the reconstructed signal is not the best approximation of the signal f ∈ H in R(F ). If R(F ) ∩ N (H ∗ ) = {0}, there is a certain degree of freedom in choosing the projection Q ∈ P(A, N (H ∗ )). It is an interesting question if for certain signals belonging to a given (closed) subspace M, it is possible to ﬁnd a Q0 ∈ P(A, N (H ∗ )), such that the recovered signal (I −Q0 )f be the best approximation of the signal f , over all possible reconstructed signals that satisﬁes the consistent sampling requirement. In other words, we are interested in ﬁnding Q0 ∈ P(A, N (H ∗ )) such that f − (I − Q0 )f ≤ f − (I − Q)f , for every f ∈ M and for every Q ∈ P(A, N (H ∗ )). The solution of this problem is based on Theorem 3.4. Theorem 5.1. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ), and let M be a closed subspace of H. There exists X0 ∈ CS(F, H) such that f −F X0 H ∗ f ≤ f −F XH ∗ f for every X ∈ CS(F, H) and for every f ∈ M, if and only if M⊥ + N (H ∗ ) ∩ R(F ) ⊆ M⊥ + R(H). Moreover, F X0 H ∗ = PR(F )N (H ∗ )//N (H ∗ ) + W0 PR(H) , where W0∗ ∈ L(H) is a solution of the equation PM PR(H) X = PM PN (H ∗ )∩R(F ) . Proof. It is a consequence of Theorem 3.4 and Lemma 4.1.

In [13], the following problem is studied in ﬁnite dimensional spaces. Given a closed subspace M of H, ﬁnd an operator X ∈ L(K) satisfying the consistent sampling requirement and such that every signal f ∈ M be perfectly recovered. This problem can be restated as follows: ﬁnding Q ∈ P(A, N (F ∗ )) (with A ∈ AF (R(H))) such that f − (I − Q)f = Qf = 0,

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for every f ∈ M. Notice that, if A, B ∈ AF (R(H)) then P(A, N (F ∗ )) = P(B, N (F ∗ )). This problem is a particular case of Theorem 5.1 (see Proposition 3.5). Corollary 5.2. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ), and let M be a closed subspace of H. Then there exists X0 ∈ CS(F, H) such that every f ∈ M is perfectly recovered (i.e., f − F X0 H ∗ f = 0), if and only if M ⊆ R(F ) and c0 (M, N (H ∗ )) < 1. Proof. It is a consequence of Proposition 3.5 and Lemma 4.1.

Let M ⊆ R(F ) be a closed subspace such that c0 (M, N (H ∗ )) < 1, and suppose that T is a closed subspace such that M T N (H ∗ ) = H. Let A ∈ AH (T M); observe that, P(A, N (H ∗ )) is a singleton because N = N (H ∗ ) ∩ A−1 (R(H)) = {0} (see Theorem 2.7). Also notice that Q ∈ P(A, N (H ∗ )) satisﬁes that every f ∈ M is perfectly recovered, since N (Q) = A−1 (R(H)) = M T . Based on Proposition 3.6, given a closed subspace M of H, it is possible to characterize the operators X ∈ CS(F, H) that perfectly recover the signals f ∈ M. Proposition 5.3. Consider closed range operators F, H ∈ L(K, H) such that H = R(F ) + N (H ∗ ), and a closed subspace M ⊆ R(F ). Then X ∈ CS(F, H) satisﬁes f − F XH ∗ f = 0 for every f ∈ M if and only if, given a complementary subspace T of M + N (H ∗ ) ∩ R(F ) in R(F ) and T ∈ L(R(H) ∩ M⊥ , N (H ∗ ) ∩ R(F )), it holds PN (F )⊥ XPN (H)⊥ = F † PMT //N (H ∗ ) H ∗ † − F † T H ∗ † . Proof. Let A ∈ AH (R(F )), since H = R(F ) + N (H ∗ ) it follows that the pair (A, N (H ∗ )) is compatible. Suppose that X ∈ CS(F, H). By Lemma 4.1, it holds I − F XH ∗ ∈ P(A, N (H ∗ )), and, by Proposition 3.6, we get I − F XH ∗ = PN (H ∗ )//MT + T ; here T is a closed subspace such that R(F ) = T M N (H ∗ ) ∩ R(F ) and T ∈ L(R(H) ∩ M⊥ , R(H) ∩ R(F )), since R(H) is a closed subspace. Then, PN (F )⊥ XPN (H)⊥ = F † F XH ∗ H ∗ † = F † (I − PN (H ∗ )//MT − T )H ∗ † = F † PMT //N (H ∗ ) H ∗ † − F † T H ∗ † . Conversely, let X ∈ L(K) be such that it holds PN (F )⊥ XPN (H)⊥ = † F PMT //N (H ∗ ) H ∗ † − F † T H ∗ † , with T ∈ L(R(H) ∩ M⊥ , R(H) ∩ R(F )). It is easy to see that I − F XH ∗ = PMT //N (H ∗ ) + T , i.e., I − F XH ∗ ∈ P(A, N (H ∗ )), for any A ∈ AH (R(F )). Then, by Lemma 4.1, X ∈ CS(F, H). Furthermore, if f ∈ M, we get (I − F XH ∗ )f = PMT //N (H ∗ ) f = f , i.e., every signal f ∈ M is perfectly recovered.

6. Consistent Sampling of Perturbed Signals Suppose that a signal f ∈ L2 (R) is perturbed by a stochastic process δf (see below for a proper definition). In this section, we are interested in studying the existence of a consistent sampling requirement which is unbiased for a

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certain family of signals, and the inﬂuence of the perturbation δf on the reconstructed signal is minimal. In order to be more precise, we ﬁx some terminology. Let μ be a Lebesgue–Stieltjes measure on R and let H be the Hilbert space L2 (μ). Suppose that (Ω, F, P ) is a probability space; if z : Ω → R is P -measurable then the expectation of z is E(z) = Ω z(ω)dP (ω). Let δx : R × Ω → R be a μ × P -measurable function with the following properties: 1. 2. 3.

for almost every t ∈ R, E(δx(t, .)) = 0, for almost every ω ∈ Ω, δx(., ω) ∈ H, E( δx 2 ) = Ω R |δx(ω, t)|2 dμ(t)dP (ω) < ∞.

The variance operator A ∈ L(H)+ of δx is deﬁned by Ax = E( x, δx δx) = δx(ω, .) δx(ω, t)x(t)dμ(t)dP (ω), Ω

R

for every x ∈ H. As it is shown in [20, Lemma 2], the variance operator A is a positive trace class operator. Problem 6.1. Let H = L2 (R), F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ). Given a closed subspace M of H, let J = {X ∈ CS(F, H) : E(F XH ∗ (f + δf )) = f for every f ∈ M}. Find X0 ∈ J such that E( F X0 H ∗ δf 2 ) ≤ E( F X0 H ∗ δf 2 ) for every X ∈ J . Remark 6.2. Observe that, since E(F XH ∗ (f + δf )) = F XH ∗ f + E(F XH ∗ δf ) = F XH ∗ f , Problem 6.1 has a solution if every signal in M can be perfectly recovered. Then, by Corollary 5.2, a necessary condition is that M ⊆ R(F ) and c0 (M, N (H ∗ )) < 1. Problem 6.1 is related with the V-approximation processes studied by A. Sard in [20] and generalized in [2]: Definition 4. Given a closed subspace M of H and δf with the above assumptions and variance operator V ∈ L(H)+ , let U = {T ∈ L(H) : E(T (f + δf ) = f, for every f ∈ M}. Then T ∈ U is a V -approximation process over M if E T δf 2 ≤ E U δf 2 for every U ∈ U. In [2, Theorem 3.4], it is shown that T ∈ L(H) is a V -approximation process over M if and only if T can be decomposed as T = I − P ∗ + W , for some P ∈ P(V, M⊥ ) and W ∈ L(N (V ) ∩ M⊥ , M⊥ ). In the following we will assume that N (V ) = {0}, so that T is a V -approximation process over M if and only if T = I − P ∗ , for some P ∈ P(V, M⊥ ). Theorem 6.3. Let F, H ∈ L(K, H) be closed range operators such that H = R(F ) + N (H ∗ ) and M ⊆ R(F ) be a closed subspace such that c0 (M, N (H ∗ )) < 1. Suppose that δf is a stochastic process (with the above assumptions) and variance V ∈ L(H)+ , with N (V ) = {0}. Then Problem 6.1 has a solution if M N (H ∗ ) = H and R(H) ⊆ V −1 (M).

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Proof. Since M ⊆ R(F ) and c0 (M, N (H ∗ )) < 1, by Corollary 5.2 it follows that there exists X ∈ CS(F, H) such that M ⊆ R(F XH ∗ ), i.e., for every f ∈ M, F XH ∗ f = f . Then, by Remark 6.2, it follows that E(F XH ∗ (f + δf )) = f , for every f ∈ M. Furthermore, by Lemma 4.1, I − F XH ∗ ∈ P(A, N (H ∗ )) with A ∈ AH (R(F )). Then, it follows that F XH ∗ = PM//N (H ∗ ) , because M N (H ∗ ) = H. On the other hand, Q = I − (F XH ∗ )∗ = I − PR(H)//M⊥ = PM⊥ //R(H) . Since R(H) ⊆ V −1 (M), by Lemma 2.2 it follows that Q ∈ P(V, M⊥ ); then F XH ∗ = I − Q∗ with Q ∈ P(V, M⊥ ) and by [2, Theorem 3.4] the projection F XH ∗ is a V -approximation process over M⊥ . Therefore F XH ∗ is a solution of Problem 6.1.

References [1] B¨ ottcher, A., Spitkovsky, I.M.: A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432, 1412–1459 (2010) [2] Corach, G., Giribet, J.I., Maestripieri, A.: Sard’s approximation processes and oblique projections. Stud. Math. 194, 65–80 (2009) [3] Corach, G., Maestripieri, A.: Weighted generalized inverses, oblique projections and least squares problems. Numer. Funct. Anal. Optim. 26, 659–673 (2005) [4] Corach, G., Maestripieri, A.: Redundant decompositions, angles between subspaces and oblique projections. Publ. Math. 54, 461–484 (2010) [5] Corach, G., Maestripieri, A., Stojanoﬀ, D.: Oblique projections and Schur complements. Acta Sci. Math. (Szeged) 67, 337–356 (2001) [6] Corach, G., Maestripieri, A., Stojanoﬀ, D.: Oblique projections and abstract splines. J. Approx. Theory 117, 189–206 (2002) [7] Deutsch, F.: The angle between subspaces of a Hilbert space. Approximation Theory, Wavelets and Applications (Maratea, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, pp. 107–130. Kluwer, Dordrecht (1995) [8] Deutsch, F.: Best approximation in inner product spaces. CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, vol. 7, xvi+338 pp. ISBN: 0-387-95156-3. Springer, New York (2001) [9] Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert spaces. Proc. Am. Math. Soc. 17, 413–416 (1966) [10] Eldar, Y.: Sampling and reconstruction in arbitrary spaces and oblique dual frame vectors. J. Fourier Anal. Appl. 1, 77–96 (2003) [11] Eldar, Y., Werther, T.: General framework for consistent sampling in Hilbert spaces. Int. J. Wavelets Multiresolut. Inf. Process. 3, 497–509 (2005) [12] Gal´ antai, A.: Subspaces, angles and pairs of orthogonal projections. Linear Multilinear Algebra 56(3), 227–260 (2008) [13] Hirabayashi, A., Unser, M.: Consistent sampling and signal recovery. IEEE Trans. Signal Proc. 55(8), 4104–4115 (2007) [14] Hogan, J.A., Lakey, J.: Sampling and aliasing without translation-invariance. In: Proceedings of Conference on SampTA, pp. 61–66 (2001) [15] Jenssen, A.J.E.M.: The Zak transform and sampling theorems for wavelets subspaces. IEEE Trans. Signal Proc. 41, 3360–3364 (1993)

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[16] Knyazev, A.V., Argentati, M.E.: Principal angles between subspaces in an A-based scalar product: algoritheorems and perturbation estimates. SIAM J. Sci. Comput. 23, 2008–2040 (2002) [17] Odzijewicz, A.: On reproducing kernels and quantization of states. Commun. Math. Phys. 114, 577–597 (1988) [18] Odzijewicz, A.: Coherent states and geometric quantization. Commun. Math. Phys. 150, 385–413 (1992) [19] Pasternak-Winiarski, Z.: On the dependence of the orthogonal projector on deformations of the scalar product. Stud. Math. 128, 1–17 (1998) [20] Sard, A.: Approximation and variance. Trans. Am. Math. Soc. 73, 428–446 (1952) [21] Unser, M.: Sampling 50 years after shannon. Proc. IEEE 88, 569–587 (2000) [22] Unser, M., Aldroubi, A.: A general sampling theory for nonideal acquisition devices. IEEE Trans. Signal Proc. 42, 2915–2925 (1994) Gustavo Corach (B) and Juan I. Giribet Instituto Argentino de Matem´ atica - CONICET University of Buenos Aires Buenos Aires Argentina e-mail: [email protected]; [email protected] Received: August 23, 2010. Revised: December 26, 2010.

Integr. Equ. Oper. Theory 70 (2011), 323–331 DOI 10.1007/s00020-011-1879-y Published online April 19, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Refinement Equations and Feller Operators Janusz Morawiec and Rafal Kapica Abstract. Results on the existence and non-existence of nontrivial L1 -solutions of the reﬁnement equation f (x) = | det K(ω)|f (K(ω)x − L(ω))dP (ω) Ω

are obtained by considering ﬁxed points a Feller operator acting on measures associated with this equation. Mathematics Subject Classification (2010). Primary 37A30; Secondary 28A80 · 39B12. Keywords. Reﬁnement equations, Feller operators, L1 -solutions, iterated function systems, attractors.

1. Introduction The discrete reﬁnement equation f (x) =

cn f (kx − n)

n∈Z

is applicable in wavelet theory, spline theory, subdivision schemes in approximation theory, computer graphics, combinatorial number theory and many others (see e.g. [1,2,5,14,16]). An interesting generalization of the discrete reﬁnement equation is the two-direction reﬁnement equation cn,1 f (kx − n) + cn,−1 f (−kx − n) f (x) = n∈Z

n∈Z

of which applications still are being discovered (see e.g. [17,18]). Continuous counterpart of the discrete reﬁnement equation is the continuous reﬁnement equation f (x) = g(y)f (kx − y)dy R

This research was supported by Silesian University Mathematics Department (Refinement Equations and Markov Operators program).

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which also has many important applications (see e.g. [3,4,13]). A natural extension all of the above equations is the reﬁnement equation of the form f (x) = | det K(ω)|f (K(ω)x − L(ω))dP (ω), (1.1) Ω

where, from now on, we assume that (Ω, A, P ) is a complete probability space, m is a positive integer, K : Ω → Rm×m , L : Ω → Rm are measurable functions with det K(ω) = 0 for ω ∈ Ω. We say that f ∈ L1 (Rm ) is an L1 -solution of the reﬁnement equation if (1.1) holds for almost all x ∈ R with respect to the m-dimensional Lebesgue measure lm . First of all notice that if f, g : Rm → R are representatives of an L1 -function and (1.1) holds for almost all x ∈ R, then g(x) = Ω | det K(ω)|g(K(ω)x − L(ω))dP (ω) for almost all x ∈ R, by completeness of the probability space (Ω, A, P ); see [10] for details in case m = 1. This observation allows us to consider the problem of the existence of nontrivial L1 -solutions of Eq. (1.1) in which we are interested in this paper. Our purpose is to show a deep connection between the problem of the existence of nontrivial L1 -solutions of Eq. (1.1) and the problem of deciding whether the attractor of a suitable deﬁned Feller operator is absolutely continuous. We also prove that under appropriate assumptions the support of every L1 -solution of Eq. (1.1) is contained in the attractor of suitable chosen iterated function systems. Note that using the Fourier transform it is easy to prove that Eq. (1.1) has no nontrivial L1 -solution if P (det K = 0) > 0. Hence it is justiﬁed our assumption that det K(ω) = 0 for ω ∈ Ω.

2. Notations and Definitions Fix a metric space (X, d). Denote by K the family of all non-empty and compact subsets of X and by ρ the Hausdorﬀ metric on K. Denote by Mf in the family of all ﬁnite Borel measures on X and by M1 its subfamily of all probability measures. As usually denote by B(X) the space of all bounded Borel measurable functions g : X → R and by C(X) its subspace of all continuous functions; both equipped with the supremum norm. Following [6] deﬁne the Fortet–Mourier metric on Mf in by ⎫ ⎧ ⎬ ⎨ μ1 − μ2 F = sup g(x)dμ1 (x) − g(x)dμ2 (x) : g ∈ F , ⎭ ⎩ X

X

where F = {g ∈ C(X) : |g(x)| ≤ 1 and |g(x) − g(y)| ≤ d(x, y) for x, y ∈ X}. The space M1 equipped with the Fortet–Mourier metric is complete if (X, d) is complete. A linear transformation M : Mf in → Mf in is said to be a Markov operator, if M (M1 ) ⊂ M1 . A Markov operator M : Mf in → Mf in is called a Feller operator if there exists an operator U : B(X) → B(X), dual to M , such that U (C(X)) ⊂ C(X) and

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U g(x)dμ(x) =

X

g(x)dM μ(x)

for g ∈ B(X), μ ∈ Mf in .

X

We say that a Markov operator M is asymptotically stable if there exists μ∗ ∈ M1 such that M μ∗ = μ∗ and lim M n μ − μ∗ F = 0

n→∞

for μ ∈ M1 .

The measure μ∗ , called the attractor of M , is unique, if it exist. The condition limn→∞ M n μ − μ∗ F = 0 does not imply that M μ∗ = μ∗ , however this implication holds for Feller operators. Deﬁne the support of μ ∈ Mf in by suppμ = {x ∈ X : μ(B(x, r)) > 0 for r > 0}, where B(x, r) stands for the open ball centered at x and radius r. An iterated function system (shortly IFS) is a family {Si : i ∈ I} of continuous maps from X into X. Given IFS {Si : i ∈ I} deﬁne the Hutchinson operator F : X → X by F (A) = cl Si (A), i∈I

where the symbol cl denotes closure. We say that IFS {Si : i ∈ I} is asymptotically stable if there exists a set A∗ ∈ K such that F (A∗ ) = A∗ and lim ρ(F n (A), A∗ ) = 0

n→∞

for A ∈ K.

The set A∗ , called the attractor of {Si : i ∈ I}, is unique, if it exists. Given a matrix A ∈ Rm×m deﬁne, as usually, its operator norm by AL = sup{Ax : x ∈ Rm with x = 1}, where · denotes the Euclidean norm in Rm .

3. Iterated Function Systems and Refinement Equations For every ω ∈ Ω deﬁne a map Sω : Rm → Rm by Sω (x) = K(ω)−1 (x + L(ω)). Obviously, {Sω : ω ∈ Ω} forms IFS. To prove the main result of this section we need the following lemma. Lemma 3.1. Assume that F is the Hutchinson operator corresponding to the IFS {Sω : ω ∈ Ω}. If k = sup K(ω)−1 L : ω ∈ Ω < +∞, then F (K) ⊂ K if and only if s = sup {Sω (0) : ω ∈ Ω} < +∞.

(3.1)

Proof. (⇒) Since F (K) ⊂ K we can ﬁnd r > 0 such that F ({0}) ⊂ B(0, r). Hence s ≤ r. (⇐) It is clear that if a set A ⊂ Rm is non-empty and closed, then so is F (A). Fix a set A ∈ K and choose r > 0 such that A ⊂ B(0, r). Next

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Sω (A) and choose ω0 ∈ Ω and y ∈ A such that x = Sω0 (y).

x ≤ Sω0 (y) − Sω0 (0) + Sω0 (0) ≤ K(ω0 )−1 L y + s ≤ kr + s. Consequently, ω∈Ω Sω (A) ⊂ B(0, kr + s).

Theorem 3.2. Assume that F is the Hutchinson operator corresponding to the IFS {Sω : ω ∈ Ω}. If (3.1) holds and

k = sup K(ω)−1 L : ω ∈ Ω < 1, (3.2) then the IFS {Sω : ω ∈ Ω} is asymptotically stable. Proof. The proof is quite standard. Namely, the space K endowed with the Hausdorﬀ metric is complete and F (K) ⊂ K, by Lemma 3.1. Now it is enough to use the Banach ﬁxed point theorem observing ﬁrst that by basic properties of the Hausdorﬀ metric we have Sω (A), Sω (B) ρ(F (A), F (B)) = ρ ω∈Ω

ω∈Ω

≤ sup{ρ(Sω (A), Sω (B)) : ω ∈ Ω} ≤ kρ(A, B) for A, B ∈ K.

If the IFS {Sω : ω ∈ Ω} is asymptotically stable, then its attractor A∗ can be constructed as follows. Fix x ∈ Rm and take all partial limits of the sequence (Sωn ◦ · · · ◦ Sω1 (x) : n ∈ N), where (ωn : n ∈ N) runs over the set ΩN .

4. Markov Operators and Refinement Equations We ﬁrst introduce a Markov operator associated with Eq. (1.1). Lemma 4.1. The operator M : Mf in → Mf in given by M μ(A) = χA (Sω (x))dμ(x)dP (ω) for Borel sets A ⊂ Rm (4.1) Ω Rm

is a Feller operator with a dual operator U : B(Rm ) → B(Rm ) given by for x ∈ Rm . (4.2) U g(x) = g(Sω (x))dP (ω) Ω

Proof. It is easy to check that M is a Markov operator. To prove that it is a Feller operator observe ﬁrst that U (C(Rm )) ⊂ C(Rm ), by the Lebesgue dominated convergence theorem. Fix μ ∈ Mf in and a Borel measurable set A ⊂ Rm . By the Fubini theorem we have U χA (x)dμ(x) = M μ(A) = χA (x)dM μ(x), Rm

Rm

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and then by standard arguments we conclude that U g(x)dμ(x) = g(x)dM μ(x) for g ∈ B(Rm ). Rm

Rm

The proof is complete.

The next result is a counterpart of Theorem 3.2 for the Markov operator given by (4.1). Theorem 4.2. Assume k = K(ω)−1 L dP (ω) < 1 Ω

and

Sω (0)dP (ω) < +∞.

s= Ω

(4.3) Then the Markov operator given by (4.1) is asymptotically stable. Proof. For a direct proof see e.g. [12]. Our proof is an immediate application of an interesting consequence of an extension of the Banach contraction principle from [11]. Fix g ∈ C(Rm ) such that |g(x) − g(y)| ≤ x − y for x, y ∈ Rm . By Lemma 4.1 we have |U g(x) − U g(y)| ≤ Sω (x) − Sω (y)dP (ω) ≤ kx − y. Ω

The second part of (4.3) allow us to apply Corollary 3.3 from [11], which completes the proof. In the next result we state a characterization of the existence of nontrivial and non-negative L1 -solutions of Eq. (1.1). Theorem 4.3. Assume (4.3). Let μ∗ be the attractor of the Markov operator given by (4.1). Then: (i) μ∗ is either absolutely continuous with respect to lm or mutually singular with lm . (ii) μ∗ is absolutely continuous with respect to lm if and only if there exists a (unique) density f ∈ L1 (Rm ) satisfying (1.1). Proof. Assertion (i) is folklore but since a ready reference to it was hard to locate we decided to include a brief proof of it for the sake of completeness. By the Lebesgue decomposition theorem there exist a unique p ∈ [0, 1], a unique probability Borel measure μa absolutely continuous with respect to lm and a unique probability Borel measure μs mutually singular with lm such that μ∗ = pμa + (1 − p)μs .

(4.4) 1

By the Radon–Nikodym theorem there exists a unique density f ∈ L (Rm ) such that μa (A) = f (x)dx for Borel sets A ⊂ Rm . (4.5) A

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Deﬁne a function g : Rm → [0, +∞] by g(x) = | det K(ω)|f (Sω−1 (x))dP (ω). Ω

Simple calculation shows that M μa (A) = g(x)dx

for Borel sets A ⊂ Rm .

(4.6)

A

(i)

Assume that p = 0 in representation (4.4). Since pμa + (1 − p)μs = μ∗ = M μ∗ = pM μa + (1 − p)M μs ,

(ii)

we have M μa ≤ μa and, in consequence, M μa = μa . By the uniqueness of μ∗ we conclude that μ∗ = μa . If p = 1 in representation (4.4), then using (4.5) and (4.6) we obtain f (x)dx = μa (A) = M μa (A) = | det K(ω)|f (Sω−1 (x))dP (ω)dx A

A Ω m

for Borel sets A ⊂ R . Hence (1.1) holds. Conversely, if there exists a density f ∈ L1 (Rm ) satisfying (1.1), then formula (4.5) deﬁnes a probability Borel measure on Rm and | det K(ω)|f (Sω−1 (x))dP (ω)dx μa (A) = A Ω

χA (Sω (x))dμ(x)dP (ω) = M μa (A)

= Ω Rm

for Borel sets A ⊂ Rm . Hence μa = M μa , and by the uniqueness of μ∗ we get μ∗ = μa . In particular, μ∗ is absolutely continuous with respect to lm . Before we formulate next result observe that (4.3) holds if (3.1) and (3.2) are satisﬁed. Theorem 4.4. Assume (3.1) and (3.2). Let A∗ be the attractor of the IFS {Sω : ω ∈ Ω} and let μ∗ be the attractor of the Markov operator given by (4.1). Then suppμ∗ ⊂ A∗ .

(4.7)

Proof. Arguing as in the proof of Lemma 4.1 we obtain that the formula μ(A) = M χA (Sω (x))dμ(x)dP (ω) for Borel sets A ⊂ A∗ Ω A∗

deﬁnes a Feller operator acting on Mf in (A∗ ). Since A∗ is compact, so also is M1 (A∗ ). This allow us to repeat the same arguments as in the proof of is asymptotically stable. Let μ Theorem 4.2 to obtain that M ∗ be the attractor of M .

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For every Borel set A ⊂ Rm put ν(A) = μ ∗ (A ∩ A∗ ). Clearly, ν ∈ M1 . Moreover, μ ν(A) = M ∗ (A ∩ A∗ ) =

χA∩A∗ (Sω (x))d μ(x)dP (ω) Ω A∗

χA (Sω (x))dν(x)dP (ω) = M ν(A)

= Ω

Rm

for Borel sets A ⊂ Rm . Hence ν = M ν, and by the uniqueness of μ∗ we have μ∗ ⊂ A∗ . μ∗ = ν. Consequently, suppμ∗ = suppν = supp In general inclusion (4.7) may be proper; for example if P ({ω0 }) = 1 and x0 is a unique ﬁx point of the map Sω0 for some ω0 ∈ Ω, then suppμ∗ = {x0 }. On the other hand, the attractor A∗ of {Sω : ω ∈ Ω} contains all ﬁx points all of the maps Sω . However, in the case where Ω = {ω1 , . . . , ωN } and P ({ωn }) > 0 for n ∈ {1, . . . , N } we have A∗ = suppμ∗ (see e.g. [7]). We end this section by summarizing previous results. Corollary 4.5. Assume (4.3). If the attractor μ∗ of the Markov operator given by (4.1) is absolutely continuous with respect to lm , then Eq. (1.1) has a unique non-negative L1 -solution f ∈ L1 (Rm ) with f 1 = 1. Moreover, if (3.1) and (3.2) hold, then f is compactly supported with suppf ⊂ A∗ , where A∗ is the attractor of the IFS {Sω : ω ∈ Ω}.

5. Examples of Applications Now we can try to use some results on Markov operators to get information on the existence and non-existence of nontrivial L1 -solutions of Eq. (1.1). To formulate our ﬁrst result we need the definition of a non-singular measure. We say that a measure μ ∈ Mf in (Rm ) is non-singular if there exists an absolutely continuous measure μa ∈ Mf in (Rm ) such that μa (Rm ) > 0 and μa (A) ≤ μ(A) for Borel sets A ⊂ Rm . According to Theorem 12.7.2 from [12] we can formulate the following result which extends [8, Corollary 3.2], [4, Corollary 15] and [9, Corollary 3]. Theorem 5.1. Assume that Ω L(ω)dP (ω) < +∞ and K is constant with K −1 L < 1. If the law of L is non-singular, then Eq. (1.1) has a nontrivial and non-negative L1 -solution f : Rm → R. Moreover, if sup{L(ω) : ω ∈ Ω} < +∞, then suppf ⊂ A∗ , where A∗ is the attractor of the IFS {Sω : ω ∈ Ω}. Our second application concerns the following case of Eq. (1.1) f (x) =

N n=1

cn am n f (an Rn x − bn ),

(5.1)

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N where cn > 0, n=1 cn = 1, an > 1, bn ∈ Rm , Rn is an orthogonal matrix (i.e. Rn−1 = RnT ) for n ∈ {1, . . . , N }. The associated IFS consists of maps T {S1 , . . . , SN } deﬁned by Sn (x) = a−1 n Rn (x + bn ) for n ∈ {1, . . . , N }. Using the Fourier transform one can prove that the space of all L1 -solutions of Eq. (5.1) is at most one-dimensional and that each member of this space is of constant sign. Now, according to [15, Theorem 1.1] we have the following result. N cn > 1 then Eq. (5.1) has no nontrivial Theorem 5.2. (i) If n=1 (cn am n) 1 L -solution. N cn = 1 and there exists n ∈ {1, . . . , N } with cn am (ii) If n=1 (cn am n) n = 1, then Eq. (5.1) has no nontrivial L1 -solution. 1 (iii) If cn am n = 1 for n ∈ {1, . . . , N }, then Eq. (5.1) has a nontrivial L solution if and only if lm (A∗ ) > 0, where A∗ is the attractor of the IFS {S1 , . . . , SN }. Moreover, if f : Rm → R is an L1 -solution of Eq. (5.1), then there exists α ∈ R such that f = αχA∗ . Note that in assertion (iii) of Theorem 1.1 from [15] instead of lm (A∗ ) > 0 we have the open set condition, which reads as follows: There N exists an open set U ⊂ Rm such that n=1 Sn (U ) ⊂ U and Sn (U )∩Sk (U ) = ∅ for n = k.

References [1] Benedetto, J.J., Frazier, M.W. (eds.): Wavelets: Mathematics and Applications. CRC Press, Boca Raton (1994) [2] Cavaretta, D., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Amer. Math. Soc 93, 1–186 (1991) [3] Chui, C.K., Shi, X.: Continuous two-scale equations and dyadic wavelets. Adv. Comput. Math. 2, 185–213 (1994) [4] Derfel, G., Dyn, N., Levin, D.: Generalized reﬁnement equations and subdivision processes. J. Approx. Theory 80, 272–297 (1995) [5] Dubickas, A., Xu, Z.: Reﬁnement equations and spline functions. Adv. Comput. Math 32, 1–23 (2010) [6] Fortet, R., Mourier, B.: Convergence de la r´epartition empirique versa la r´epar´ tition th´eor´ethque. Ann. Sci. Ecole Norm. Sup. 70, 267–285 (1953) [7] Hutchinson, J.E.: Fractals and self-similarity. Indiana Math. J. 30, 713–743 (1981) [8] Jia, R.Q., Lee, S.L., Sharma, A.: Spectral properties of continuous reﬁnement operators. Proc. Amer. Math. Soc. 126, 729–737 (1998) [9] Kapica, R., Morawiec, J.: Probability distribution functions of the Grinceviˇcjus series. J. Math. Anal. Appl. 342, 1380–1387 (2008) [10] Kapica, R., Morawiec, J.: On a reﬁnement type equation. J. Appl. Anal. 14, 251–257 (2008) [11] Lasota, A.: From fractals to stochastic differential equations. In: Chaos–The Interplay Between Stochastic and Deterministic Behaviour (Karpacz, 1995). Lecture Notes in Phys., 457, Springer, Berlin, pp. 235–255 (1995)

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[12] Lasota, A., Mackey, M.C.: Chaos, Fractals and Noice. Stochastic Aspects of Dynamics. Springer, Berlin (1994) [13] Lee, D.M., Lee, J.G., Yoon, S.H.: A construction of multiresolution analysis by integral equations. Proc. Am. Math. Soc. 130, 3555–3563 (2002) [14] Micchelli, C.A.: Mathematical Aspects of Geometric Modeling, CBMS-NSF Series in Applied Mathematics 65. SIAM Publ, Philadelphia (1995) [15] Ngai, S.M., Wang, Y.: Self-similar measures associated to IFS with non-uniform contraction ratios. Asian J. Math. 9, 227–244 (2005) [16] Protasov, V.: On the asymptotics of the binary partition function. Mat. Zametki 76, 151–156 (2004) [17] Xie, C.: Construction of biorthogonal two-direction reﬁnable function and two-direction wavelet with dilation factor m. Comput. Math. Appl. 56, 1845– 1851 (2008) [18] Yang, S.Z., Li, Y.: Two-direction reﬁnable functions and two-direction wavelets with dilation factor m. Appl. Math. Comput. 188, 1908–1920 (2007) Janusz Morawiec (B) and Rafal Kapica Institute of Mathematics Silesian Univeristy Bankowa 14 40-007 Katowice Poland e-mail: [email protected]; [email protected] Received: August 26, 2010. Revised: March 31, 2011.

Integr. Equ. Oper. Theory 70 (2011), 333–361 DOI 10.1007/s00020-011-1864-5 Published online February 1, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Fractional Abstract Cauchy Problems Li Kexue and Peng Jigen Abstract. This paper is concerned with fractional abstract Cauchy problems with order α ∈ (1, 2). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (F ACP0 ) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (F ACPf ). Mathematics Subject Classification (2010). Primary 34A08, Secondary 47D06. Keywords. Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, Caputo fractional derivative, fractional solution operator, fractional abstract Cauchy problem.

1. Introduction In this paper we are concerned with the well-posedness of the homogeneous fractional Cauchy problem C α Dt u(t) = Au(t), t ∈ [0, T ], (F ACP0 ) u(0) = 0, u (0) = x, and the existence and uniqueness of the mild solutions and strong solutions of the inhomogeneous fractional abstract Cauchy problem C α Dt u(t) = Au(t) + Jt2−α f (t), t ∈ [0, T ], (F ACPf ) u(0) = 0, u (0) = x, where 1 < α < 2, A : D(A) ⊂ X → X is a densely deﬁned closed linear operator, X is a Banach space, C Dtα is the α-order Caputo fractional derivative operator, Jt2−α is the (2 − α)-order Riemann–Liouville fractional integral This work was supported by the Natural Science Foundation of China under the contact No. 60970149.

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operator, f : [0, T ] → X, x ∈ X. As a special case, when A is a negative number, (F ACPf ) is able to model processes intermediate between exponential decay (α = 1) and pure sinusoidal oscillation (α = 2), for more details about fractional ordinary differential equations, we refer to [1]. Fractional differential equations has attracted much attention in recent years, see the books of Podlubny [2], Hilfer [3], Kilbas et al. [4] and the papers of Delbosco and Rodino [5], Eidelman and Kochubei [6], Anh and Leonenko [7], Niu and Xie [8]. Fractional derivatives can describe the properties of memory and heredity of materials, which is the major advantage of fractional derivatives compared with integer-order derivatives. Practical problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions. Initial conditions for the Caputo fractional derivatives are expressed in terms of initials of integer order derivatives. Bazhlekova [9] studied the following fractional order abstract Cauchy problem α Dt u(t) = Au(t), t ≥ 0, (1.1) u(0) = x, u(k) (0) = 0, k = 1, . . . , m − 1, where m = α, the smallest integer greater than or equal to α, Dα t is the Caputo fractional derivative operator deﬁned by m−1 tk α u(k) (0) . u(t) − Dα t u(t) = Dt k! k=0

The notion of solution operator is introduced in [9] as follows: Definition 1.1. Let α > 0. A family {Sα (t)}t≥0 ⊂ B(X) of bounded linear operators on X is called a solution operator for (1.1) if the following three conditions are satisﬁed: (a) Sα (t) is strong continuous for t ≥ 0 and Sα (0) = I (the identity operator on X), (b) Sα (t)D(A) ⊂ D(A) and ASα (t)x = Sα (t)Ax for all x ∈ D(A), t ≥ 0; (c) For all x ∈ D(A), t ≥ 0, Tα (t)x is a solution of u(t) = x + Jtα Au(t). Following [10], problem (1.1) is well-posed if and only if it admits a solution operator, just like the ﬁrst order homogeneous abstract Cauchy problem is well-posed if and only if its coefﬁcient operator generates a strongly continuous semigroup. The notion of α-times resolvent families (or solution operators) is introduced in Li and Zheng [11]. Li et al. [14] showed that under general conditions, elliptic operators with zero boundary condition can generate fractional resolvent families. Li et al. [13] considered fractional order evolution equation Dα u(t) = Au(t); u(0) = u0 , u (0) = 0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π, Dα is the Caputo fractional derivative operator, 1 < α < 2. They showed that such

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equations are well-posed in the sense that there exists an α-times resolvent family for A. Chen and Li [12] presented a purely algebraic notion, α-resolvent operator function: Definition 1.2. Let α > 0. A function Sα : R+ → B(X) of bounded linear operators on X is called an α-resolvent operator function if the following conditions are satisﬁed: (a) Sα (·) is strongly continuous on R+ and Sα (0) = I (the identity operator on X), (b) Sα (t)Sα (s)=Sα (s)Sα (t) for all t, s ≥ 0, (c) the functional equation Sα (s)Jtα Sα (t) − Jsα Sα (s)Sα (t) = Jtα Sα (t) − Jsα Sα (s) holds for all t, s ≥ 0. It is proved that a family {Sα (t)}t≥0 is an α-resolvent operator function if and only if it is a solution operator for a certain fractional Cauchy problem. Moreover, Chen and Li [12] introduced the concept of integrated α-resolvent operator function: Definition 1.3. Let α > 0, β ≥ 0. A function Sα,β : R+ → B(X) of bounded linear operators on X is called a β-times integrated α-resolvent operator function if the following conditions are satisﬁed: (a) Sα,β (·) is strong continuous on R+ and Sα,β (0) = gβ+1 (0)I, (b) Sα,β (s)Sα (t)=Sα (t)Sα (s) for all s, t ≥ 0, (c) the functional equation Sα,β (s)Jtα Sα,β (t)−Jsα Sα,β (s)Sα,β (t) = gβ+1 (s)Jtα Sα,β (t)−gβ+1 (t)Jsα Sα,β (s) holds for all s, t ≥ 0. They gave basic properties and analyticity criteria of fractional resolvent operator functions and discussed the relations between integrated resolvent families and resolvent families. Li et al. [15] studied the fractional powers of generators of fractional resolvent family, they showed that if −A generates a bounded α-times resolvent family for some α ∈ (0, 2], then −Aβ generates an analytic γ-times 2π−πγ ), γ ∈ (0, 2]. Moreover, they discussed the resolvent family for β ∈ (0, 2π−πα relations of solutions of fractional Cauchy problems and Cauchy problems of ﬁrst order. Umarov [16] presented a fractional generalizations of Duhamel’s principle for the Cauchy problem for general inhomogeneous fractional distributed order differential-operator equations of the form ⎧

μ ⎨ ∧ L [u] ≡ f (α, A)D∗α u(t)d ∧ (α) = h(t), t > 0, 0 ⎩ u(0) = ϕk , k = 0, . . . , m − 1, where μ ∈ (m − 1, m], A : D → X is a closed linear operator with domain D ⊂ X, X is a reﬂexive Banach space, f (α, A) is a family of operators with

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the symbol f (α, z) continuous in the variable α ∈ [0, μ], analytic in the variable z ∈ G ⊂ C, ∧ is a ﬁnite measure deﬁned on [0, μ], and D∗α is the α-order Caputo fractional differential operator. Fractional Cauchy Problems are useful in physics to model anomalous diffusion. Zaslavsky [17] introduced the fractional kinetic equation t−β ∂ β g(x, t) = Lg(x, t) + p (x) 0 ∂tβ Γ(1 − β)

(1.2)

for Hamiltonian chaos, where 0 < β < 1, L is the generator of a Feller semigroup {T (t)}t≥0 and p0 ∈ C ∞ (R1 ) is an arbitrary initial condition. Here β t)/∂tβ is the inverse Laplace transform of sβ g˜(x, s), where g˜(x, s) = ∂

∞g(x, −st e g(x, t)dt is the usual Laplace transform. Stochastic solutions of frac0 tional Cauchy problems are subordinated processes. Baeumer and Meerschaert [18] considered the general case where L is the generator of a Feller semigroup {T (t)}t≥0 associated with some inﬁnitely divisible law on Rd . Assume gβ is the density of the stable subordinator with Laplace transform

that ∞ −st e gβ (t)dt = exp(−sβ ), they proved that if q(x, t) = T (t)f (x) solves 0 ∂q(x, t) = Lq(x, t); q(x, 0) = p0 (x) ∂t for all t > 0 and x ∈ Rd , then g(x, t) = S(t)f (x) solves the fractional Cauchy problem (1.2), where S(t) is deﬁned by ∞ T ((t/s)β )gβ (s)f ds.

S(t)f = 0

Meerschaert et al. [19] developed classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D ⊂ Rd with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. More precisely, they proved that u(t, x) = TD ((t/l)β )f (x)gβ (l)dl 0

is the solution of

⎧ β ∂ u(t,x) ⎪ ⎪ ⎨ ∂tβ = Δu(t, x), x ∈ D, t > 0. ⎪ ⎪ ⎩

u(t, x) = 0, x ∈ ∂D, t > 0, u(0, x) = f (x), x ∈ D,

where 0 < β < 1, ∂ β /∂tβ is the Caputo fractional derivative operator, D is a bounded domain with boundary ∂D ∈ C 1,β , C 1,β is the space consists of functions whose ﬁrst order partial derivatives are uniformly H¨ older continuous with exponent β in D, TD (t) is the killed semigroup of Brownian motion {Xt } in D, Et is the process inverse to a stable subordinator of index β ∈ (0, 1), gβ is the density of the stable subordinator with Laplace transform

∞ e−st gβ (t)dt = exp(−sβ ), and f satisﬁes some regularity. 0

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The aim of this paper is to develop an operator theory to study fractional Cauchy problems of order α ∈ (1, 2). In Sect. 2 we present some basic definitions and preliminary facts which will be used throughout the following sections. In Sect. 3, We introduce the notion of fractional solution operator, obtain its some properties, and give a generation theorem for exponentially bounded fractional solution operator. In Sect. 4, we study the homogeneous cauchy problem (F ACP0 ) and the inhomogeneous fractional Cauchy problem (F ACPf ), we prove that problem (F ACP0 ) is well-posed iﬀ its coefﬁcient operator A generates an α-order fractional solution operator, and obtain sufﬁcient conditions for the existence and uniqueness of problem (F ACPf ).

2. Preliminaries In this section, we introduce some definitions, notations, and preliminary facts which are used throughout this paper. Let α > 0, m = α denote the smallest integer greater than or equal to α. Let X be a Banach space, R+ = [0, ∞). By L1 ((0, T ); X) we denote the space of all Bochner integrable functions u : (0, T ) → X, it is a Banach space with the norm T u(t)dt

u1 = 0

By C([0, T ]; X), resp. C 1 ([0, T ]; X), we denote the space of functions u : [0, T ] → X, which are continuous, resp. continuously differentiable. C([0, T ]; X) and C 1 ([0, T ]; X) are Banach spaces endowed with the norms uC = sup u(t)X , uC 1 = sup (u(t)X + u (t)X ). t∈[0,T ]

t∈[0,T ]

If A is a linear operator in X, the resolvent set ρ(A) of A is the set of all complex numbers λ for which (λI − A)−1 is a bounded linear operator in X. R(λ, A) = (λI − A)−1 denotes the resolvent operator of A. Let N denote the sets of natural numbers, N0 = N ∪ {0}. We use the abbreviation t (f ∗ g)(t) =

f (t − τ )g(τ )dτ 0

for the convolution. Let I = (0, T ), or I = [0, T ], or I = R+ , m ∈ N, 1 ≤ p < ∞. The Sobolev spaces can be deﬁned as following (see [20, Appendix]): W m,p (I; X) = {u| ∃ϕ ∈ L1 (I; X) : u(t) =

m−1 k=0

Note that, ϕ(t) = u(m) (t), ck = u(k) (0).

ck

tm−1 tk + ∗ ϕ(t), t ∈ I}. k! (m − 1)!

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Definition 2.1. The α-order Riemann–Liouville fractional integral is deﬁned by t 1 α (t − τ )α−1 u(τ )dτ, u ∈ L1 ((0, T ); X), (2.1) Jt u(t) = Γ(α) 0

where Γ(α) is the Gamma function. The Riemann–Liouville integral can be written as Jtα u(t) = (gα ∗ u)(t), where

gα (t) =

Set

Jt0 u(t)

tα−1 Γ(α) ,

(2.2)

t > 0, t ≤ 0.

0,

= u(t), the integral operators Jtα satisfy the semigroup property, Jtα Jtβ = Jtα+β , α, β ≥ 0.

(2.3)

Definition 2.2. The α-order Riemann–Liouville fractional derivative is deﬁned by t dm 1 α Dt u(t) = (t − τ )m−α−1 u(τ )dτ, (2.4) Γ(m − α) dtm 0

where u ∈ L1 ((0, T ); X), gm−α ∗ u ∈ W m,1 ((0, T ); X). When α = m, m ∈ dm N, N denotes the sets of natural numbers, deﬁne Dtm = dt m. The Riemann–Liouville derivative operator Dtα is a left inverse of the integral operator Jtα but in general not a right inverse, that is, Dtα Jtα u = u, u ∈ L1 ((0, T ); X),

(2.5)

and (Jtα Dtα u)(t) = u(t) −

m−1

(gm−α ∗ u)(k) (0)gα+k+1−m (t),

(2.6)

k=0

where u ∈ L1 ((0, T ); X) gm−α ∗ u ∈ W m,1 ((0, T ); X). Definition 2.3. The α-order Caputo fractional derivative is deﬁned by

m−1 C α α (k) Dt u(t) = Dt u(t) − u (0)gk+1 (t) , (2.7) k=0 1

((0, T ); X), gm−α ∗ u ∈ W m,1 ((0, T ); X). where u ∈ L ((0, T ); X) ∩ C The Caputo derivative operator C Dtα is also a left inverse of the integral operator Jtα but in general not a right inverse, that is, m−1

C

Dtα Jtα u = u, u ∈ L1 ((0, T ); X),

C

Dtα u)(t) = u(t) −

(2.8)

and (Jtα

m−1

u(k) (0)gk+1 (t),

(2.9)

k=0

where u ∈ L1 ((0, T ); X) ∩ C m−1 ((0, T ); X), gm−α ∗ u ∈ W m,1 ((0, T ); X).

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The Laplace transform formula for the Riemann–Liouville fractional integral is deﬁned by 1 (λ). (2.10) L{Jtα u(t)} = α u λ where u (λ) is the Laplace of u deﬁned by ∞ (2.11) u (λ) = e−λt u(t)dt, Reλ > ω, 0

where Reλ represents the real part of the complex number λ. Definition 2.4. The Mittag-Lefﬂer functions are deﬁned by Eα,β (z) =

∞ k=0

zk , α, β > 0, z ∈ C, Γ(αk + β)

where C denotes the set of complex numbers. The Mittag-Lefﬂer function are related to the Laplace integral ∞ λα−β , Reλ > ω 1/α , ω > 0, e−λt tβ−1 Eα,β (ωtα )dt = α λ −ω 0

and to the following asymptotic formulas as z → ∞. If 0 < α < 2, β > 0, then 1 1 (2.12) Eα,β (z) = z (1−β)/α exp(z 1/α ) + εα,β (z), |argz| ≤ απ, α 2 1 Eα,β (z) = εα,β (z), |arg(−z)| < (1 − α)π, 2

(2.13)

where εα,β (z) = −

N −1 n=1

z −n + O(|z|−N ), z → ∞. Γ(α − βn)

3. Fractional Solution Operator In this section, we introduce the notion of fractional solution operator, obtain its some properties, give a generation theorem for exponentially bounded fractional solution operator and present two examples illustrating the abstract theory. Definition 3.1. Let 1 < α < 2, a family {Tα (t)}t≥0 ⊂ B(X) of all bounded linear operators on X is called an α-order fractional solution operator if it satisﬁes the following assumptions: (1) Tα (t) is strongly continuous on R+ and limt→0+ Tαt(t) x = x for all x ∈ X; (2) Tα (s)Tα (t) = Tα (t)Tα (s) for all t, s ≥ 0; (3) Tα (s)Jtα Tα (t) − Jsα Tα (s)Tα (t) = sJtα Tα (t) − tJsα Tα (s) for all t, s ≥ 0.

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The linear operator A deﬁned by D(A) = {x ∈ X : lim

t→0+

Tα (t)x − tx exists} tα+1

and Ax = Γ(α + 2) lim

t→0+

Tα (t)x − tx , for x ∈ D(A) tα+1

is the generator of the α-order fractional solution operator Tα (t), D(A) is the domain of A. Remark 3.2. Deﬁne Tα (0)x = dTαdt(t)x |t=0 . From (1) of Definition 3.1, it is evident that Tα (0) = 0, then we have Tα (0) = I (the identity operator on X). Remark 3.3. The fractional integrals in Definition 3.1 are understood strongly in the sense of Bochner. Remark 3.4. It is clear to see that the notion of fractional solution operator is just the special case of β-times integrated α-resolvent operator function for β = 1. However, it also should be pointed out that the equality limt→0+ Tαt(t) x = x (∀x ∈ X) in Definition 3.1 is essential in the proof of properties (b), (c), (d) of Propositon 3.7, which are necessary for the equivalency of the well-posedness of Cauchy problem (F ACP0 ) to the existence of an α-order fractional solution operator with coefﬁcient operator A as the generator. Definition 3.5. If an α-order fractional solution operator Tα (t) satisﬁes Tα (t) ≤ M eωt , t ≥ 0.

(3.1)

for some constants ω ≥ 0 and M ≥ 1, then it is said to be exponentially bounded. An operator A is said to belong C α (M, ω) if A generates a fractional solution operator Tα (t) satisfying (3.1). Denote C α (ω) = {C α (M, ω); M ≥ 1}. Proposition 3.6. Let A be the generator of an α-order fractional solution operator Tα (t) on X. Then supt∈[0,T ] Tα (t) < ∞ for every T > 0. Proof. Any given T > 0, deﬁned a mapping S : X → C([0, T ]; X) by (Sx)t = Tα (t)x, t ∈ [0, T ]. It is easy to show that S is linear and closed, hence by the closed graph theorem S is bounded, there exists a constant M > 0 such that supt∈[0,T ] Tα (t)x ≤ M x for all x ∈ X. Therefore, by the uniform boundedness theorem it follows that supt∈[0,T ] Tα (t) < ∞. Proposition 3.7. Let Tα (t) be an α-order fractional solution operator on X, and let A be its generator. Then (a) Tα (t) commutes with A, which means that Tα (t)(D(A)) ⊂ D(A) and ATα (t)x = Tα (t)Ax for all x ∈ D(A) and t ≥ 0. (b) for all x ∈ D(A) and t ≥ 0, Tα (t)x = tx + AJtα Tα (t)x.

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(c) for all x ∈ D(A) and t ≥ 0, Tα (t)x = tx + Jtα Tα (t)Ax, and Tα (·)x ∈ C 1 (R+ ; X). (d) A is closed and densely defined. Proof. (a) Let x ∈ D(A), for t ≥ 0, s ≥ 0, by (2) of Definition 3.1, (Tα (s) − s)Tα (t)x = Tα (t)(Tα (s)x − sx), hence Γ(α + 2) lim

s→0+

Tα (s)Tα (t)x − sTα (t)x = Tα (t)Ax. sα+1

That is, Tα (t)x ∈ D(A) and ATα (t)x = Tα (t)Ax for all x ∈ D(A) and t ≥ 0. (b) For all x ∈ X and s ≥ 0, Jsα Tα (s)x −x sα+1 s Γ(α + 2) (s − τ )α−1 Tα (τ )xdτ − x = Γ(α)sα+1

Γ(α + 2)

0

=

=

Γ(α + 2) Γ(α)s Γ(α + 2) Γ(α)s

1 (1 − τ )α−1 Tα (sτ )xdτ − x 0

1 (1 − τ )α−1 Tα (sτ )xdτ 0

Γ(α + 2) − Γ(α) =

Γ(α + 2) Γ(α)

1 (1 − τ )α−1 τ xdτ 0

1 (1 − τ )α−1 τ 0

Tα (sτ ) x − x dτ. sτ

(3.2)

By (1) of Definition 3.1, we have lim

s→0+

Tα (sτ ) x = x, sτ

Apply dominated convergence theorem to (3.2) to conclude that lim Γ(α + 2)

s→0+

Jsα Tα (s)x = x. sα+1

(3.3)

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Thus, using (3) of Definition 3.1 and 3.3, we get Tα (s)Jtα Tα (t)x − sJtα Tα (t)x s→0+ sα+1 Jsα Tα (s)(Tα (t)x − tx) = Γ(α + 2) lim s→0+ sα+1 = Tα (t)x − tx,

AJtα Tα (t)x = Γ(α + 2) lim

(3.4)

therefore (b) holds. (c) For x ∈ D(A), the limit Tα (s)x − sx s→0+ sα+1 lim

exists, then the function Tα (s)x − sx s→0+ sα+1 is bounded for sufﬁciently small s > 0. For t ≥ 0, by dominated convergence theorem, we get f (s) = lim

Tα (t)x − tx = AJtα Tα (t)x Γ(α + 2) Tα (s) − s = lim Γ(α) s→0+ sα+1 =

=

Γ(α + 2) lim Γ(α) s→0+ Γ(α + 2) Γ(α)

t (t − τ )α−1 Tα (τ )xdτ 0

t (t − τ )α−1 Tα (τ ) 0

t (t − τ )α−1 Tα (τ ) lim

s→0+

0

Tα (s)x − sx dτ sα+1 Tα (s)x − sx dτ sα+1

= Jtα Tα (t)Ax.

(3.5)

From (3.5) it follows that Tα (t)x is differentiable for t ≥ 0 and d d Tα (t)x = (tx + Jtα Tα (t)Ax) dt dt d = x + Jtα Tα (t)Ax dt = x + Jtα−1 Tα (t)Ax t 1 = x+ (t − τ )α−2 Tα (τ )Axdτ Γ(α − 1) 0

= x+

tα−1 Γ(α − 1)

1 (1 − τ )α−2 Tα (tτ )Axdτ. 0

Now use dominated convergence theorem to (3.6) to conclude that Tα (·)x ∈ C 1 (R+ ; X), x ∈ D(A).

(3.6)

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(d) Let xn ∈ D(A), xn → x and Axn → y as n → ∞. Using (3.5) and dominated convergence theorem, we have Tα (t)x − tx = lim Tα (t)xn − txn n→∞

1 = lim n→∞ Γ(α) =

1 Γ(α)

t (t − τ )α−1 Tα (τ )Axn dτ 0

t (t − τ )α−1 Tα (τ )ydτ 0

= Jtα Tα (t)y.

(3.7)

Using (3.7) and (3.3), we have Tα (t)x − tx tα+1 α J Tα (t)y = Γ(α + 2) lim t α+1 t→0+ t = y.

Ax = Γ(α + 2) lim

t→0+

The closeness of A is proved. For every x ∈ X, set xt = Jtα Tα (t)x, from the proof of (b) it follows that J α Tα (t)x → x as t → 0+. So A is densely deﬁned. xt ∈ D(A), and Γ(α+2) t tα+1 From Definition 3.1, it is clear to see that an α-order fractional solution operator possesses unique generator. The following proposition shows that the converse conclusion is also true. Proposition 3.8. Let A be the generator of an α-order fractional solution operator Tα (t) on X, then Tα (t) is unique. Proof. If Tα (t) and Sα (t) are both α-order fractional solution operator generated by A, then for x ∈ D(A), by property (c) of Proposition 3.7, we have t ∗ Tα (t)x = (Sα (t) − Jtα Sα (t)A) ∗ Tα (t)x = Sα (t) ∗ Tα (t)x − gα (t) ∗ Sα (t) ∗ ATα (t)x = Sα (t) ∗ Tα (t)x − Sα (t) ∗ gα (t) ∗ ATα (t)x = Sα (t) ∗ (Tα (t)x − gα (t) ∗ ATα (t)x) = t ∗ Sα (t)x, from Titchmarsh’s theorem (see [21, p. 166]), it follows that Tα (t)x = Sα (t)x for each x ∈ D(A), t ≥ 0. Hence Tα (t) = Sα (t) by the density of D(A). Lemma 3.9. ([22, Corollary 1.2]) Let G be a Banach space. Let a ≥ 0 and r : (a, ∞) → G be an infinitely differentiable function. For M ≥ 0, ω ∈ (−∞, a] the following assertions are equivalent. (i) (λ − ω)n+1 r(n) (λ)/n! ≤ M, λ > a, n ∈ N0 . (ii) There exists a function F : [0, ∞) → G satisfying F (0) = 0 and lim sup(1/h)F (t + h) − F (t) ≤ M eωt (t ≥ 0) h↓0

(3.8)

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such that ∞ r(λ) = λ

e−λt F (t)dt (λ > a).

(3.9)

0

Moreover, r has an analytic extension to {λ ∈ C : Reλ > ω} which is given by (3.9) if Reλ > 0. It is well-known that both strongly continuous semigroup and strongly continuous cosine function are necessarily exponentially bounded. However, whether an α-order fractional solution operator Tα (t) is exponentially bounded is unknown in general. In fact, if Tα (t) is exponentially bounded, we have the following generation theorem. Theorem 3.10. Let X be a Banach space. Let A be a closed linear operator with dense domain D(A) ⊂ X. Then A ∈ C α (M, ω) if and only if (ω α , ∞) ⊂ ρ(A) and

dn α−2 M n! (λ R(λα , A)) ≤ , λ > ω, n ∈ N0 . n dλ (λ − ω)n+1

(3.10)

Proof. (Necessary) Suppose A ∈ C α (M, ω) and Tα (t) is the fractional solution operator generated by A. For x ∈ X, λ > ω, we deﬁne ∞ R(λ)x = e−λt Tα (t)xdt. (3.11) 0

Since Tα (t) ≤ M e , R(λ) is well-deﬁned for every λ satisfying λ > ω. By properties (b), (c) of Proposition 3.7 and the identity (2.10), it follows that ωt

λα R(λ)x − λα−2 x = AR(λ)x, x ∈ X, λα R(λ)x − λα−2 x = R(λ)Ax, x ∈ D(A), Thus, λα I − A is invertible and R(λ) = λα−2 R(λα , A), that is {λα : λ > ω} ⊂ ρ(A)

(3.12)

and ∞ λ

α−2

α

R(λ , A)x =

e−λt Tα (t)xdt, λ > ω, x ∈ X.

0

From (3.13), we have d α−2 d (λ R (λα , A)) x = dλ dλ

∞

−λt

e

∞ Tα (t)xdt = −

0

te−λt Tα (t)xdt.

0

Proceeding by induction we get dn α−2 (λ R(λα , A))x = (−1)n dλn

∞ 0

tn e−λt Tα (t)xdt.

(3.13)

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Therefore, dn n (λα−2 R(λα , A))x = dλ

∞ 0

tn e−λt Tα (t)xdt

∞

≤M

tn e(ω−λ)t xdt

0

≤

M n! x. (λ − ω)n+1

(3.14)

(Sufﬁciency) By Lemma 3.9, for M ≥ 1, ω ≥ 0, there exists a function S : [0, ∞) → B(X) satisfying S(0) = 0, and lim sup(1/h)S(t + h) − S(t) ≤ M eωt (t ≥ 0)

(3.15)

h↓0

such that ∞ λ

α−3

α

R(λ , A) =

e−λt S(t)dt (λ > ω).

(3.16)

0

From (3.16), we see that S(t) commutes with A and ∞ ∞ 1 1 −λt e S(t)dt = 3 + α e−λt S(t)Adt, λ λ 0

(3.17)

0

Making inverse Laplace transform to (3.17) we obtain that S(t)x =

t2 x + Jtα S(t)Ax, 2

(3.18)

where x ∈ D(A), t ≥ 0. From (3.18) and note that α ∈ (1, 2), we get d d S(t)x = tx + dt dt t = tx + 0

t gα (t − s)S(s)Axds 0

d gα (t − s)S(s)Axds dt

1 = tx + Γ(α − 1) Set G(t)x =

t 0

(t − s)

α−2

t (t − s)α−2 S(s)Axds.

(3.19)

0

S(s)Axds, then

d S(t)x = tx + G(t)x (3.20) dt we shall prove that G(t)x is continuous on R+ for every x ∈ D(A). According to (3.15), there exists a sufﬁciently small δ > 0 such that for h ∈ (0, δ),

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S(t + h) − S(t) ≤ M heωt , t ≥ 0.

(3.21)

For Δt ∈ (0, δ) and x ∈ D(A), we have t+Δt

(t + Δt − s)α−2 S(s)Ax

G(t + Δt)x − G(t)x = 0

t −

(t − s)α−2 S(s)Ax 0

t

(t − s)α−2 − (t + Δt − s)α−2 S(s)Axds

≤ 0

t+Δt

(t + Δt − s)α−2 S(s)Axds.

+

(3.22)

t

By dominated convergence theorem,

t 0

(t − s)α−2 − (t + Δt − s)α−2 S(s)Axds → 0

(3.23)

as Δt → 0. By (3.21), for s ∈ [t, t + Δt] we have S(s) ≤ S(s) − S(t) + S(t) ≤ M Δteωt + S(t).

(3.24)

By (3.24), t+Δt

(t + Δt − s)α−2 S(s)Axds ≤

t

(Δt)α−1 (M Δteωt + S(t) ) Ax

α−1

→0 (3.25)

as Δt → 0. Put (3.23) and (3.25) into (3.22) to conclude that G(t + Δt)x − G(t)x → 0 as Δt → 0+. By the same way we can show that G(t + Δt)x − G(t)x → 0 as Δt → 0−. Hence G(t)x is continuous on R+ for every x ∈ D(A). Since D(A) is dense in X, from (3.20) it follows that S(t) is continuous differentiable on R+ . d Deﬁne Tα (t)x = dt S(t)x = S (t)x, t ≥ 0, x ∈ X. Then Tα (t) is continuous on d R+ . By (3.19), it is clear that dt S(t)|t=0 = 0 and for x ∈ D(A), we have tα−1 Tα (t) x=x+ t Γ(α − 1)

1 s(1 − s)α−2 0

S(ts) Axds. ts

(3.26)

Note that 1 < α < 2 and S(0) = S (0) = 0, applying dominated convergence theorem to (3.26), we obtain lim

t→0+

Tα (t) x = x, x ∈ X. t

(3.27)

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By (3.15), we see that Tα (t) ≤ M eωt .

(3.28)

From the fact that S(t) commutes with A and A is closed, it follows that d Tα (t) commutes with A. Since Tα (t)x = dt S(t)x and S(0) = 0, we have

t S(t)x = 0 Tα (τ )xdτ , this together with (3.19) yields Tα (t)x = tx + Jtα Tα (t)Ax.

(3.29)

The closedness of A and the density of D(A) imply that for all x ∈ X Tα (t)x = tx + AJtα Tα (t)x.

(3.30)

Since Tα (t) commutes with A and A is closed, from (3.30) it follows that Jsα Tα (s)Tα (t)x = tJsα Tα (s)x + Jsα Tα (s)AJtα Tα (t)x = tJsα Tα (s)x + AJsα Tα (s)Jtα Tα (t)x = tJsα Tα (s)x + Tα (s)Jtα Tα (t)x − sJtα Tα (t)x. (3.31) We next show that Tα (t) commutes with Tα (s). For all x ∈ D(A) and t, s ≥ 0, by (3.29) we have Tα (t)Tα (s)x = tTα (s)x + gα (t) ∗ Tα (t)ATα (s)x = tTα (s)x + gα (t) ∗ ATα (t)Tα (s)x,

(3.32)

and Tα (s)Tα (t)x = tTα (s)x + gα (t) ∗ Tα (s)ATα (t)x = tTα (s)x + gα (t) ∗ ATα (s)Tα (t)x,

(3.33)

From (3.32) and (3.33) we see that both w1 (t) = Tα (t)Tα (s)x and w2 (t) = Tα (s)Tα (t)x are solutions of u(t) = tTα (s)x + gα (t) ∗ Au(t)

(3.34)

Hence for x ∈ D(A) we have tTα (s)x ∗ w1 (t) = (w2 (t) − gα (t) ∗ Aw2 (t)) ∗ w1 (t) = w2 (t) ∗ w1 (t) − gα (t) ∗ Aw2 (t)) ∗ w1 (t) = w2 (t) ∗ (w1 (t) − gα (t) ∗ Aw1 (t)) = w2 (t) ∗ tTα (s)x.

(3.35)

From (3.35) it follows that tTα (s)x ∗ (w1 (t) − w2 (t)) = 0,

(3.36)

by Titchmarsh’s theorem (see [21, p. 166]) we get w1 (t) − w2 (t) = 0, that is Tα (t)Tα (s)x = Tα (t)Tα (s)x. By density of D(A), for all t, s ≥ 0, we obtain Tα (t)Tα (s) = Tα (t)Tα (s).

(3.37)

By (3.27), (3.28), (3.31), (3.37) and the fact that Tα (t) is strongly continuous on R+ , we conclude that A ∈ C α (M, ω). Therefore, the proof of Theorem 3.10 is completed.

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Theorem 3.11. Let 1 < α < 2. Then A ∈ C α (M, ω) if and only if (ω α , ∞) ⊂ ρ(A) and there is a strongly continuous vector-valued function T (t) satisfying T (t) ≤ M eωt , M ≥ 1, t ≥ 0, and such that ∞ λ

α−2

α

R(λ , A)x =

e−λt T (t)xdt, λ > ω, x ∈ X.

(3.38)

0

Proof. Suppose there exists a function T (t) and a linear operator A satisfy the conditions above. For λ > ω, R(λα , A) is differentiable any number of times for λ > ω. Then, by differentiating λα−2 R(λα , A)x n-times for λ, we obtain dn α−2 (λ R(λα , A))x = (−1)n dλn

∞

tn e−λt T (t)xdt.

0

Hence, we have

dn α−2 M n! (λ R(λα , A)) ≤ , λ > ω, n ∈ N0 . n dλ (λ − ω)n+1

From Theorem 3.10, it follows that A ∈ C α (M, ω). Let Tα (t) be the corresponding α-order fractional solution operator. Then Tα (t) and T (t) both satisfy (3.38), by the uniqueness theorem for Laplace transforms, we see that Tα (t) = T (t). The converse has already proven in Theorem 3.10. Remark 3.12. Note that for α = 2, Theorem 3.11 is consistent with the characterization by Laplace transform of strongly continuous 1-times integrated cosine function (see Arendt-Kellermann [23]). Proposition 3.13. Let 1 < α < 2. If A is the generator of an exponentially bounded α-times resolvent family Sα (t), then A is the generator of an exponentially bounded α-order fractional solution operator Tα (t). Proof. Assume there are constants M ≥ 1, ω ≥ 0 such that Sα (t) ≤ M eωt , then by relation (2.6) in [9] we have (ω α , ∞) ⊂ ρ(A) and ∞ λ

α−1

α

R(λ , A) =

e−λt Sα (t)xdt, λ > ω, x ∈ X.

0

Deﬁne t Sα (τ )xdτ, x ∈ X.

Tα (t)x = 0

(3.39)

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It is clear that Tα (t) is exponentially bounded. By the convolution property of Laplace transforms and (3.39), we have ∞ ∞ −λt e Tα (t)xdt = e−λt (1 ∗ Sα )(t)xdτ 0

0

1 = λ =λ

∞

0 α−2

e−λt Sα (t)xdt R(λα , A).

From Theorem 3.10, it follows that A is generator of the exponentially bounded α-order fractional solution operator Tα (t). Example 3.14. Consider the equation ⎧ C α ⎨ Dt u(t, x) = Au(t, x), t ≥ 0, u(t, 0) = u(t, 1) = 0, ⎩ u(0, x) = 0, ∂u ∂t u(t, x)|t=0 = f (x),

(3.40)

2

∂ where α ∈ (1, 2), A := eiθ ∂x 2 (θ ∈ [0, π)) 2 is deﬁned in X =L (0, 1) with domain D(A) = {ϕ ∈ W 2,2 (0, 1), ϕ(0) = ∞ ϕ(1) = 0}, f (x) = n=1 cn sin nπx, n ∈ N. It is easy to see that A has eigenvalues −eiθ n2 π 2 with eigenfunctions sin nπx, n ∈ N . The solution of (3.40) is

u(t, x) =

∞

tEα,2 (−eiθ n2 π 2 tα )cn sin nπx.

n=0

From the asymptotic formulas of the Mittag-Lefﬂer function (2.12) and (2.13), it follows that A ∈ C α (0) if and only if |θ| ≤ (1 − α2 )π. Example 3.15. Assume that A is a self-adjoint operator on a Hilbert space H and A is bounded above; i.e., (Ax, x) ≤ ωx2 for all x ∈ D(A) and some ω ∈ R, where (·) is the inner product in H. Then for α ∈ (1, 2), A generates an exponentially bounded α-order fractional solution operator. Proof. Since A satisﬁes the conditions mentioned above, we have from Example 3.14.16 in [24] that A generates a cosine function. Thus, by Theorem 3.1 in [9], A generates an exponentially bounded α-times resolvent family for α ∈ (1, 2). It follows from Proposition 3.13 that A generates an exponentially bounded α-order fractional solution operator.

4. Fractional Cauchy Problems In this section, we devoted to building the relationship between fractional solution operator and the Cauchy problem (F ACP0 ). Besides, we give sufﬁcient conditions to guarantee the existence and the uniqueness of mild solutions and strong solutions of the Cauchy problem (F ACPf ).

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Definition 4.1. A function u ∈ C([0, T ]; X) is called a mild solution of (F ACP0 ) if Jtα u(t) ∈ D(A) and u(t) = tx + AJtα u(t), t ∈ [0, T ]. Definition 4.2. A function u ∈ C([0, T ]; X) is called a strong solution of (F ACP0 ) if u ∈ C([0, T ]; D(A))∩C 1 ([0, T ]; X), g2−α ∗(u−tx) ∈ C 2 ([0, T ]; X) and (F ACP0 ) holds. Definition 4.3. A function u ∈ C([0, T ]; X) is called a mild solution of (F ACPf ) if Jtα u(t) ∈ D(A) and u(t) = tx + AJtα u(t) + t ∗ f (t), t ∈ [0, T ]. Definition 4.4. A function u ∈ C([0, T ]; X) is called a strong solution of (F ACPf ) if u ∈ C([0, T ]; D(A))∩C 1 ([0, T ]; X), g2−α ∗(u−tx) ∈ C 2 ([0, T ]; X) and (F ACPf ) holds. Lemma 4.5. Let A be the generator of an α-order fractional solution operator Tα (t) on X. Let x ∈ X and u ∈ C([0, T ]; X) be a mild solution of (F ACP0 ). Then u(t) = Tα (t)x is a unique mild solution of (F ACP0 ) on [0, T ]. Proof. By property (b) of Proposition 3.7 and Definition 4.1, it follows that t ∗ u(t) = (Tα (t) − Agα (t) ∗ Tα (t)) ∗ u(t) = Tα (t) ∗ u(t) − Tα (t) ∗ Agα (t) ∗ u(t) = Tα (t) ∗ (u(t) − Agα (t) ∗ u(t)) = t ∗ Tα (t)x, we have u(t) = Tα (t)x, t ∈ [0, T ]. Let u(t), v(t) be two mild solution of (F ACP0 ) and let w(t) = u(t) − v(t), then w(t) = AJtα w(t), hence w(t) = 0. Lemma 4.6. Let A be the generator of an α-order fractional solution operator Tα (t) on X. Let x ∈ D(A), then u(t) = Tα (t)x is a unique strong solution of (F ACP0 ) on [0, T ]. Proof. By property (c) of Proposition 3.7, for x ∈ D(A), Tα (·)x ∈ C 1 ([0, T ]; X).

(4.1)

From (1) of Definition 3.1, we have Tα (0) = 0. Then Tα (t)x = x. t→0+ t

Tα (0)x = 0, Tα (0)x = lim

(4.2)

Since Tα (t) is strongly continuous on R+ and A is closed, it follows that Tα (·)x ∈ C([0, T ]; D(A)), x ∈ D(A).

(4.3)

Using (2.5), (2.7) and properties (c), (a) of Proposition 3.7, we have C

Dtα Tα (t)x = Dtα (Tα (t)x − Tα (0)x − tTα (0)x) = Dtα (Tα (t)x − tx) = Dtα Jtα Tα (t)Ax = Dtα Jtα ATα (t)x = ATα (t)x.

(4.4)

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It follows from (2.3) and property (c) of Proposition 3.7 that Jt2−α (Tα (t)x − tx) = Jt2−α Jtα Tα (t)Ax = Jt2 Tα (t)Ax,

then d2 2−α J (Tα (t)x − tx) = Tα (t)Ax. (4.5) dt2 t From (4.1)–(4.5), it follows that u(t) = Tα (t)x is a strong solution of (F ACP0 ) for all x ∈ D(A) and t ∈ [0, T ]. Note that every strong solution of (F ACP0 ) is also a mild solution, hence strong solutions of (F ACP0 ) are unique. Definition 4.7. The problem (F ACP0 ) is said to be well-posed if for any x ∈ D(A) there exists a unique strong solution u(t; x) on [0, ∞), and {xn } ⊂ D(A), xn → 0 imply that u(t; xn ) → 0 as n → ∞ in X, uniformly on compact intervals. To study the relationship between well-posedness of (F ACP0 ) and existence of fractional solution operator for coefﬁcient operator A, we consider the Volterra equation t gα (t − s)Au(s)ds, t ∈ [0, T ].

u(t) = tx +

(4.6)

0

Definition 4.8. A function u ∈ C([0, T ]; X) is called (a) strong solution of (4.6) if u ∈ C([0, T ]; D(A)) and (4.6) holds on [0, T ]. (b) mild solution of (4.6) if Jtα u(t) ∈ D(A) and u(t) = tx + AJtα u(t) on [0, T ]. Definition 4.9. Equation (4.6) is called well-posed if for every x ∈ D(A) there is a unique strong solution u(t; x) on R+ of (4.6), and {xn } ⊂ D(A), xn → 0 imply u(t; xn ) → 0 in X uniformly on compact intervals. Remark 4.10. Applying (2.9), we see that the Cauchy problem (F ACP0 ) is well-posed in sense of Definition 4.7 if and only if (4.6) is well-posed in sense of Definition 4.9. Theorem 4.11. The fractional abstract Cauchy problem (F ACP0 ) is wellposed if and only if A generates an α-order fractional solution operator Tα (t). Proof. (Sufﬁciency). Suppose A generates an α-order fractional solution operator Tα (t). By Lemma 4.6, for every x ∈ D(A) the function u(t) = Tα (t)x is a unique strong solution of (F ACP0 ). Continuous dependence of the solutions on x follows from uniform boundedness of Tα (t) on compact intervals of R+ by Proposition 3.6. (Necessity). Assume that (F ACP0 ) is well-posed, then (4.6) is wellposed by Remark 4.10. For every x ∈ D(A), by u(t; x) we denote a unique strong solution of (4.6). We Deﬁne a mapping Tα (t) : D(A) → D(A) by Tα (t)x = u(t; x), x ∈ D(A), t ≥ 0. From the uniqueness of the solutions of (4.6) it follows that Tα (t) is well deﬁned. Obviously, Tα (t) is linear. By

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the definition of Tα (t), it follows that Tα (t)x ∈ C(R+ ; D(A)) and Tα (0)x = 0, Tα (0)x = x for every x ∈ D(A). By density of D(A), we have Tα (0)x = 0, Tα (0)x = lim

t→0

Tα (t) x = x, x ∈ X. t

(4.7)

We show that Tα (t) is uniformly bounded on compact intervals of R+ . If this is false then there is a sequence {tn } ⊂ [0, T ] and {yn } ⊂ D(A), yn = 1, such that T (tn )yn ≥ n for every n ∈ N . Let xn = ynn , then xn ∈ D(A), xn → 0, by the definition of Tα (t) we get the contradiction 1 ≤ Tα (tn )xn = u(tn ; xn ) → 0 as n → ∞. So Tα (t) is uniformly bounded on compact intervals of R+ . This implies that Tα (t) can be extended to all of X, Tα (t)x is continuous for every x ∈ X. By (a) of Definition 4.8, it follows that for every x ∈ D(A), Tα (t)x = tx + Jtα ATα (t)x = tx + AJtα Tα (t)x, t ≥ 0.

(4.8)

For every x ∈ X, since Tα (t) is bounded and A is a closed and densely deﬁned we have Jtα Tα (t)x ∈ D(A) and AJtα Tα (t) = Tα (t) − t is strongly continuous for t ≥ 0, hence u(t; x) = Tα (t)x is a mild solution of (4.6). Next, we prove the mild solutions of (4.6) are unique. In fact, let u1 , u2 ∈ C(R+ ; X) be two mild solutions of (4.6). Then u = u1 − u2 ∈ C([0, ∞); X) and u(t) = AJtα u(t) for all t ≥ 0. Let v(t) = Jtα u(t), then v(t) is a strong solution of (4.6) with v(0) = v (0) = 0. It is obvious that u = 0 is a strong solution of (4.6) with x = 0, by uniqueness of the strong solutions, we have v(t) = 0, then u(t) = Dtα v(t) = 0. For x ∈ D(A), both u(t; Ax) and Au(t; x) are mild solutions of (4.6) with tx replaced by tAx, therefore Tα (t)Ax = u(t; Ax) = Au(t; x) = ATα (t)x, t ≥ 0.

(4.9)

For all x ∈ D(A), t, s ≥ 0, by (4.8) and (4.9) it follows that Tα (t)Tα (s)x = tTα (s)x + gα (t) ∗ Tα (t)ATα (s)xdτ = tTα (s)x + gα (t) ∗ ATα (t)Tα (s)xdτ, and Tα (s)Tα (t)x = tTα (s)x + gα (t) ∗ Tα (s)Tα (t)Ax = tTα (s)x + gα (t) ∗ ATα (s)Tα (t)x.

By (a) of Definition 4.8, both Tα (·)Tα (s)x and Tα (s)Tα (·)x are strong solutions of (4.6) with initial conditions Tα (0)Tα (s)x = Tα (s)Tα (0)x = 0 and Tα (0)Tα (s)x = Tα (s)Tα (0) = Tα (s)x, hence Tα (t)Tα (s)x = Tα (s)Tα (t)x for every x ∈ D(A) by the well-posedness of (4.6). Since D(A) is dense in X, we have Tα (t)Tα (s) = Tα (s)Tα (t), t, s ≥ 0.

(4.10)

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Finally, we prove that Tα (t) satisﬁes (3) of Definition 3.1. For x ∈ D(A), it follows from (4.8) and (4.9) that Tα (s)Jtα Tα (t) − sJtα Tα (t) = AJsα Tα (s)Jtα Tα (t) = Jsα Tα (s)AJtα Tα (t) = Jsα Tα (s)Tα (t) − tJsα Tα (s).

(4.11)

Density of D(A) implies that (4.11) holds for all x ∈ X. By (4.7), (4.10) and (4.11), it follows that Tα (t) is an α-order fractional resolvent. The proof is therefore complete. Theorem 4.12. Let A be the generator of an α-order fractional solution operator Tα (t) on X. Then for every f ∈ L1 ([0, T ]; X) the problem (F ACPf ) has a unique mild solution u given by t Tα (t − τ )f (τ )dτ, t ∈ [0, T ].

u(t) = Tα (t)x +

(4.12)

0

Proof. Uniqueness: Let u1 , u2 ∈ C([0, T ]; X) be two mild solutions of (F ACPf ). Then w := u1 − u2 ∈ C([0, T ]; X) and AJtα w(t) = w(t) for all t ∈ [0, T ]. It follows from Lemma 4.5 that w ≡ 0. Existence: We have seen that Tα (·)x is a mild solution of the homogeneous fractional Cauchy problem (F ACP0 ). It remains to show that v(t) =

t T (t − s)f (s)ds is a mild solution of (F ACPf ). Since Tα (t) is strongly con0 α tinuous on R+ and f ∈ L1 ([0, T ]; X), by Proposition 1.3.4 in [22], we have v ∈ C([0, T ]; X). Using properties (d), (b) of Proposition 3.7 we obtain AJtα v(t) = A(gα ∗ Tα ∗ f ) = (Agα ∗ Tα ) ∗ f = (Tα ∗ f )(t) − t ∗ f (t) = v(t) − t ∗ f (t).

The proof is therefore completed.

Theorem 4.13. Let A be the generator of an α-order fractional solution operator Tα (t) and x ∈ D(A). Assume that one of the following two conditions is satisfied: (i) f ∈ D(A), f ∈ C([0, T ]; X) and Af ∈ C([0, T ]; X), or (ii) g2−α ∗ f ∈ W 2,1 ([0, T ]; X), f ∈ W 1,1 ([0, T ]; X). Then (F ACPf ) has a unique strong solution u defined by t Tα (t − s)f (s)ds, t ≥ 0.

u(t) = Tα (t)x +

(4.13)

0

Proof. For the uniqueness, let u1 , u2 ∈ C([0, T ]; X) be two strong solutions of (F ACPf ). Then w := u1 − u2 ∈ C([0, T ]; X), w(0) = w (0) = 0 and

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Dtα w(t) = Aw(t) for all t ∈ [0, T ]. It follows from Lemma 4.6 that w ≡ 0. For the existence, let t v(t) = Tα (t − s)f (s)ds.

C

0

Case (i): Since f ∈ D(A), then from property (c) of proposition 3.7 and Proposition 1.3.4 in [24] it follows that v ∈ C 1 ([0, T ]; X)

(4.14)

and t

v (t) =

Tα (t − s)f (s)ds.

(4.15)

0

It is clear that v(0) = v (0) = 0. We have seen that Tα (t)x is a strong solution of the homogeneous Cauchy problem (F ACP0 ) for x ∈ D(A). Then we only need to show that v(t) is a strong solution of (F ACPf ) with x = 0. Since v ∈ C([0, T ]; X), A is closed, then t Tα (t − s)Af (s)ds.

Av(t) =

(4.16)

0

Since Af ∈ C([0, T ]; X), Proposition 1.3.4 in [24] shows that Av ∈ C([0, T ]; X).

(4.17)

By Definition 2.3, we have C

Dtα v(t) = Dtα (v(t) − v(0) − v (0)t) t 1 d2 = (t − r)1−α v(r)dr. Γ(2 − α) dt2 0

d d 1 ( = Γ(2 − α) dt dt

t

(t − r)1−α v(r)dr).

0

Let S(t) =

1 d Γ(2 − α) dt

t

(t − r)1−α v(r)dr.

0

From Fubini’s theorem, we see that t r d 1 S(t) = (t − r)1−α Tα (r − s)f (s)dsdr Γ(2 − α) dt 0

=

d 1 Γ(2 − α) dt

0

t t 0

s

(t − r)1−α Tα (r − s)f (s)drds

(4.18)

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=

=

Fractional Abstract Cauchy Problems

d 1 Γ(2 − α) dt 1 Γ(2 − α)

t 0

t t−s (t − s − τ )1−α Tα (τ )f (s)dτ ds 0

0

⎛ t−s ⎞ d ⎝ (t − s − τ )1−α Tα (τ )f (s)dτ ⎠ ds dt 0

1 lim + Γ(2 − α) s→t−0

=

1 Γ(2 − α)

t 0

t−s (t − s − τ )1−α Tα (τ )f (s)dτ. 0

⎛ t−s ⎞ d ⎝ (t − s − τ )1−α Tα (τ )f (s)dτ ⎠ ds dt 0

1 + lim (t − s)2−α Γ(2 − α) s→t−0

=

1 Γ(2 − α)

t 0

355

1

(1 − τ )1−α Tα ((t − s)τ )f (s)dτ

0

⎛ t−s ⎞ d ⎝ (t − s − τ )1−α Tα (τ )f (s)dτ ⎠ ds dt

(4.19)

0

By (4.18) and (4.19), we have C

Dtα v(t) = =

d S(t) dt 1 Γ(2 − α)

t 0

⎛ t−s ⎞ d ⎝ (t − s − τ )1−α Tα (τ )f (s)dτ ⎠ ds dt2 2

0

d 1 lim + Γ(2 − α) s→t−0 dt

t−s (t − s − τ )1−α Tα (τ )f (s)dτ 0

= I1 + I2 .

(4.20)

Since f ∈ D(A), by property (c), (d) of Proposition 3.7, we get

I1 =

1 Γ(2 − α)

t 0

⎛ t−s ⎞ d ⎝ (t − s − τ )1−α Tα (τ )f (s)dτ ⎠ ds dt2 2

0

t (Drα Tα (r)f (s)|r=t−s ) ds

= 0

t

t (Drα (r)f (s)|r=t−s )ds

= 0

+

(Tα (r)Af (s)|r=t−s )ds 0

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=

1 Γ(2 − α)

=

Jt2−α f (t)

t

(t − s)1−α f (s)ds +

0

IEOT

t Tα (t − s)Af (s)ds 0

+ Av(t).

(4.21)

By dominated convergence theorem, we have ⎛ ⎞ 1 d ⎝ 1 lim (t − s)2−α (1 − τ )1−α Tα ((t − s)τ )f (s)dτ ⎠ I2 = Γ(2 − α) s→t−0 dt 0

2−α lim (t − s)2−α = Γ(2 − α) s→t−0

1

1−α

(1 − τ )

1 lim (t − s)2−α Γ(2 − α) s→t−0

τ

0

d 1 lim (t − s)2−α + Γ(2 − α) s→t−0 dt =

1

1

Tα ((t − s)τ )f (s) (t − s)τ

dτ

(1 − τ )1−α Tα ((t − s)τ )f (τ )dτ

0

(1 − τ )1−α

0

d Tα ((t − s)τ )f (τ )dτ dt

(4.22)

By property (c) of Proposition 3.7, for f ∈ D(A), Tα (·)f ∈ C 1 ([0, T ]; X), this together with (4.22) yield 1 lim (t − s)2−α I2 = Γ(2 − α) s→t−0

1

(1 − τ )1−α

0

1 lim (t − s)2−α lim = s→t−0 Γ(2 − α) s→t−0

1

d Tα ((t − s)τ )f (τ )dτ dt

(1 − τ )1−α

0

d Tα ((t − s)τ )f (τ )dτ dt

= 0.

(4.23) It follows from (4.20), (4.21) and (4.23) that C

Let h(t) =

Dtα v(t) = Av(t) + Jt2−α f (t).

Jt2−α f (t),

then h(t) is continuous on [0, T ]. In fact,

1 h(t) = Γ(2 − α) =

(4.24)

t2−α Γ(2 − α)

t

(t − τ )1−α f (τ )dτ

0

1

(1 − τ )1−α f (tτ )dτ.

0

By dominated convergence theorem, it is clear that g(t) is continuous on [0, T ]. Since Av ∈ C([0, T ]; X) and g2−α ∗ f = h ∈ C([0, T ]; X), by (4.24), we obtain g2−α ∗ v ∈ C 2 ([0, T ]; X). Considering that v ∈ C([0, T ]; D(A)) ∩ C 1 ([0, T ]; X), g2−α ∗ v ∈ C 2 ([0, T ]; X) and v(t) satisﬁes (F ACPf ) with x = 0, therefore v is a strong solution of (F ACPf ) with x = 0.

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Case (ii): Since T (t) is strongly continuous and f ∈ W 1,1 ([0, T ]; X), then v is differentiable and t d v (t) = Tα (s)f (t − s)ds dt 0

t =

Tα (s)f (t − s)ds + Tα (t)f (0).

(4.25)

0

From Proposition 1.3.4 in [24], it follows that Tα ∗ f ∈ C([0, T ]; X), this together with (4.25) yield v ∈ C 1 ([0, T ]; X).

(4.26)

By dominated convergence theorem, we have (g2−α ∗ f )(0) = lim (g2−α ∗ f )(s) s→0+

1 = lim s→0+ Γ(2 − α) s2−α s→0+ Γ(2 − α)

s

(s − τ )1−α f (τ )dτ

0

1

= lim

(1 − τ )1−α f (sτ )dτ

0

= 0.

(4.27)

Since g2−α ∗ f ∈ W 2,1 ([0, T ]; X), from (2.6), (4.27), it follows that f (s) = Jsα Dsα f (s) + (g2−α ∗ f )(0)gα−1 (s) + (g2−α ∗ f ) (0)gα (s) = Jsα Dsα f (s) + (g2−α ∗ f ) (0)gα (s).

Then t Tα (t − s)f (s)ds

v(t) = 0

t =

Tα (t − s) (Jsα Dsα f (s) + (g2−α ∗ f ) (0)gα (s)) ds

0

t

t Tα (t −

=

s)Jsα Dsα f (s)ds

0

=

=

1 Γ(α) 1 Γ(α)

+

gα (s)T (t − s)(g2−α ∗ f ) (0)ds

0

t t 0

(s − τ )α−1 Tα (t − s)Dτα f (τ )dsdτ + Jtα Tα (t)(g2−α ∗ f ) (0)

τ

t−τ t (t − τ − r)α−1 Tα (r)Dτα f (τ )drdτ + Jtα Tα (t)(g2−α ∗ f ) (0). 0

0

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Hence the closedness of A and property (b) of proposition 3.7 imply that v(t) ∈ D(A) and 1 A Av(t) = Γ(α)

t−τ t (t − τ − r)α−1 Tα (r)Dτα f (τ )drdτ 0

0

+ AJtα Tα (t)(g2−α ∗ f ) (0) t−τ t 1 A (t − τ − r)α−1 Tα (r)Dτα f (τ )drdτ = Γ(α) 0

0

+ Tα (t)(g2−α ∗ f ) (0) − t(g2−α ∗ f ) (0) t = (Tα (t − τ )Dτα f (τ ) − (t − τ )Dτα f (τ ))dτ 0

+ Tα (t)(g2−α ∗ f ) (0) − t(g2−α ∗ f ) (0) t = Tα (t − τ )Dτα f (τ )dτ − Jt2 Dtα f (t) 0

+ Tα (t)(g2−α ∗ f ) (0) − t(g2−α ∗ f ) (0) t = Tα (t − τ )Dτα f (τ )dτ − Jt2−α f (t) + t(g2−α ∗ f ) (0) 0

+ Tα (t)(g2−α ∗ f ) (0) − t(g2−α ∗ f ) (0) t = Tα (t − τ )Dτα f (τ )dτ − Jt2−α f (t) + Tα (t)(g2−α ∗ f ) (0).

(4.28)

0

By Definition 2.3, we have Dtα v(t) = Dtα (v(t) − v(0) − tv (0)) = Dtα v(t) t α = Dt Tα (τ )f (t − τ )dτ 0

t r d2 1 = (t − r)1−α Tα (s)f (r − s)dsdr Γ(2 − α) dt2 0 0 ⎛ ⎞ t r d ⎝d 1 = (t − r)1−α Tα (s)f (r − s)dsdr⎠. Γ(2 − α) dt dt 0

0

Let d 1 w(t) = Γ(2 − α) dt

t r 0

0

(t − r)1−α Tα (s)f (r − s)dsdr.

(4.29)

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Using Fubini’s theorem, we obtain d 1 w(t) = Γ(2 − α) dt

=

=

d 1 Γ(2 − α) dt 1 Γ(2 − α)

t t 0

(t − r)1−α Tα (s)f (r − s)drds

s

t Tα (s) 0

t−s (t − s − r)1−α f (r)drds 0

t Tα (s) 0

d dt

t−s

(t − s − r)1−α f (r)drds

0

1 Tα (s) lim + s→t−0 Γ(2 − α)

=

1 Γ(2 − α)

t Tα (s) 0

d dt

t−s (t − s − r)1−α f (r)dr 0

t−s

(t − s − r)1−α f (r)drds

0

1 Tα (s) lim (t − s)2−α + s→t−0 Γ(2 − α)

=

1 Γ(2 − α)

t Tα (s) 0

d dt

1

(1 − r)1−α f ((t − s)r)dr

0

t−s

(t − s − r)1−α f (r)drds.

0

From (4.29) and (4.30), we see that C

Dtα v(t) =

d w(t) dt

d 1 = Γ(2 − α) dt

=

1 Γ(2 − α)

t 0

d T (s) dt

t T (s) 0

d2 dt2

t−s (t − s − r)1−α f (r)drds 0

t−s (t − s − r)1−α f (r)drds 0

d 1 T (t) lim + s→t−0 dt Γ(2 − α) 1 = Γ(2 − α)

t 0

d2 T (t − τ ) 2 dτ

t−s (t − s − r)1−α f (r)dr 0

τ 0

(τ − r)1−α f (r)drdτ

(4.30)

360

L. Kexue and P. Jigen d 1 T (t) lim + s→0+ ds Γ(2 − α) t =

s

IEOT

(s − r)1−α f (r)dr

0

T (t − τ )Dτα f (τ )dτ + Tα (t)(g2−α ∗ f ) (0).

(4.31)

0

Therefore, by (4.28) and (4.31), we have C

Dtα v(t) = Av(t) + Jt2−α f (t).

Since Av ∈ C([0, T ]; X), g2−α ∗f ∈ C([0, T ]; X), then g2−α ∗v ∈ C 2 ([0, T ]; X). Considering that v ∈ C([0, T ]; D(A)) ∩ C 1 ([0, T ]; X), g2−α ∗ v ∈ C 2 ([0, T ]; X) and v satisﬁes (F ACPf ) with x = 0, therefore v is a strong solution of (F ACPf ) with x = 0. The proof is completed. Acknowledgments The authors would like to thank the reviewer for his/her insightful comments that helped to improve the paper.

References [1] Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, New York (1996) [2] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) [3] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) [4] Kilbas, A.A., Srivastava H.M., Trujillo J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006) [5] Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996) [6] Eidelman, S.D., Kochubei, A.N.: Cauchy problems for fractional diffusion equations. J. Differ. Eq. 199, 211–255 (2004) [7] Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 104, 1349–1387 (2001) [8] Niu, M., Xie, B.: A fractional partial differential equations driven by space-time white noise. Proc. Am. Math. Soc. 138, 1479–1489 (2010) [9] Bazhlekova, E.: Fractional Evolution Equations in Banach Spaces. Ph.D. Thesis, Eindhoven University of Technology (2001) [10] Pr¨ uss, J.: Evolutionary Integral Equations and Applications. Birkh¨ auser, Basel, Berlin (1993) [11] Li, M., Zh, Q.: On spectral inclusions and approximation of α-times resolvent families. Semigroup Forum 69, 356–368 (2004)

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[12] Chen, C., Li, M.: On fractional resolvent operator functions. Semigroup Forum 80, 121–142 (2010) [13] Li, F.-B., Li, M., Zheng, Q.: Fractional evolution equations governed by coercive differential operators. Abstr. Appl. Anal. 2009, Article ID 438690 (2009) [14] Li, M., Li, F.-B., Zheng, Q.: Elliptic operators with variable coefficients generating fractional resolvent families. Int. J. Evol. Eq. 2, 195–204 (2007) [15] Li, M., Chen, C., Fu-Bo, Li.: On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, 2702–2726 (2010) [16] Umarov, S.: On fractional Duhamel’s principle and its applications. ArXiv 1004: 2098 [17] Zaslavsky, G.M.: Fractional kinetic equation for Hamiltonian chaos. Phys. D 76, 110–122 (1994) [18] Baeumer, B., Meerschaert, M.M.: Stochastic solutions for fractional Cauchy problems. Frac. Calc. Appl. Anal. 4, 481–500 (2001) [19] Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy Problems on Bounded Domains. Ann. Probab. 37, 979–1007 (2009) [20] Brezis, H.: Op´erateurs Maximaux Monotones et Semi-groupes de Contrations dans les Espaces de Hilbert. Math. Studies 5, North-Holland, Amsterdam (1973) [21] Yosida, K.: Functional Analysis, 6th Edition. Springer, New York (1980) [22] Arendt, W.: Vector-value Laplace transforms and Cauchy problems. Israel J. Math. 59, 327–352 (1987) [23] Arendt, W., Kellerman, H.: Integrated solutions of Volterra integrodifferential equations and applications. In: Da Prato, G., Iannelli, M. (eds.) Volterra Intergral differential Equations in Banach Spaces and Applications, pp. 21– 51. Longman Sci. Tech, Harlow, Essex (1989) [24] Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems. Monogr. Math. vol. 96. Birkh¨ auser, Basel (2001) L. Kexue (B) and P. Jigen Department of Mathematics Xi’an Jiaotong University Xi’an 710049 China e-mail: [email protected]; [email protected] Received: August 31, 2010. Revised: January 9, 2011.

Integr. Equ. Oper. Theory 70 (2011), 363–378 DOI 10.1007/s00020-010-1851-2 Published online December 4, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

A Metric for Unbounded Linear Operators in a Hilbert Space Go Hirasawa Abstract. We introduce a metric in the set S(H) of all semiclosed operators in a Hilbert space H, and its topological structures are studied. Mathematics Subject Classification (2000). Primary 47A65; Secondary 47A05. Keywords. De Branges space, semiclosed subspace, semiclosed operator, q-metric.

1. Introduction Let H be an inﬁnite dimensional, complex Hilbert space and s be an operator from a domain dom(s) ⊆ H to H. Throughout this paper, an operator means a linear operator. The operator s is said to be closed if the graph {(u, su) ∈ H × H : u ∈ dom(s)} is a closed subspace in the product Hilbert space H × H. Closed operators in Hilbert spaces are enthusiastically studied. In fact, many important operators are closed operators. Nevertheless, the sum of two closed operators need not be closed, and therefore, we often need to be careful to consider the sum of closed operators. Though the sum of two closed operators are necessarily semiclosed operators deﬁned in the next section. In this sense, it seems quite natural to consider semiclosed operators. In this paper, we deﬁne a metric, which we call the “q-metric”, on S(H) the set of all semiclosed operators in H. Then the set of all closed and densely deﬁned operators in H, denoted by CD(H), is a subset of S(H). Among other things, we shall prove that CD(H) is an open subset of S(H) under the topology induced by the q-metric. That is, CD(H) is stable under the small perturbation of the q-metric. The q-metric is a natural metric in a following sense: On B(H) the set of all bounded operators on H, the q-metric coincides with the metric induced by the operator norm. G. Hirasawa was partially supported by the Grant-in-Aid for Scientific Research(C) No. 20540154, from the Japan Society for the Promotion of Science.

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2. Preliminaries The rest of this paper, H is an inﬁnite dimensional, complex Hilbert space. 1 We denote by (·, ·) the original inner product in H and put · := (·, ·) 2 . Let B(H) be the set of all bounded operators on H. Then T ∈ B(H) is said to be positive, in short T ≥ 0, if (T u, u) ≥ 0 for all u ∈ H. Let M be a subspace in H. Then M is said to be a semiclosed subspace in H if there exists an inner product (·, ·)M on M such that M is a complete inner product space with respect to (·, ·)M and that the inclusion mapping J : (M, · M ) → H is continuous with respect to the norm · M induced by (·, ·)M . That is, u ≤ cuM , u ∈ M for some c > 0. When the inclusion mapping J is continuous, we write J : (M, · M ) → H. In this paper, we call · M a Hilbert norm on a semiclosed subspace M . It is easy to see that a closed subspace is a semiclosed subspace. Semiclosed subspaces are characterized by operator ranges. For if M is a semiclosed subspace in H, then it is the operator 1 1 range of the positive bounded operator (JJ ∗ ) 2 , that is, M = (JJ ∗ ) 2 H for the inclusion mapping J : (M, · M ) → H. Conversely, if M is an operator range for some S ∈ B(H), i.e., M = SH := {Su : u ∈ H}, then the inner product (·, ·)M deﬁned by (Su, Sv)M := (u, v)

u, v ∈ (ker S)⊥

gives Hilbert space structures for M = SH so that (M, · M ) → H. Therefore M is semiclosed. Definition. For T ∈ B(H), we deﬁne the inner product (·, ·)T on the operator range T H by (T u, T v)T := (u, v)u, v ∈ (ker T )⊥ . Then (T H, (·, ·)T ) is a complete inner product space, that is, a Hilbert space and (T H, (·, ·)T ) → H. We call (T H, (·, ·)T ) de Branges space induced by T and denote by M(T ). The relationship between semiclosed subspaces and de Branges spaces is given by the following lemma. Lemma 2.1. ([1]) Let M be a semiclosed subspace in H and let · M be a Hilbert norm on M such that J : (M, ·M ) → H. Then there exists a unique positive bounded operator T ∈ B(H) such that (M, · M ) = M(T ) ∗

(isometrically isomorphic),

1 2

where T is given by (JJ ) . When we handle continuously embedded Hilbert spaces in H, it is sufﬁcient to consider de Branges spaces induced by positive bounded operators. De Branges spaces originated from the solutions of the Bieberbach conjecture. Since then, it has been applied to various area in mathematics (cf. [1,9]). Now we are ready to give the definition of semiclosed operators. Definition. Let s : dom(s) → H be an operator with a domain dom(s) ⊆ H. Then s is said to be a semiclosed operator if the graph {(u, su) : u ∈ dom(s)} is a semiclosed subspace in the product Hilbert space H × H. Since closed subspaces are semiclosed subspaces, closed operators are semiclosed operators by the definition. A semiclosed operator s : dom(s) → H

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is characterized in several ways. Instead of the original norm of H, we often consider another norm · dom(s) on dom(s). In this case, we deﬁne s˜ : (dom(s), · dom(s) ) → H by s˜u = su for u ∈ dom(s). Theorem 2.2. ([8]) Let s : dom(s) → H be an operator with a domain dom(s) ⊆ H. Then the following conditions are equivalent. (1) (2)

The operator s is a semiclosed operator in H. The domain dom(s) of s is a semiclosed subspace in H, so that s˜ : (dom(s), · dom(s) ) → H

(3)

is a bounded operator with respect to some (equivalently, any) Hilbert norm · dom(s) on dom(s). The operator s is represented by a quotient of bounded operators, namely there exist bounded operators A, B ∈ B(H) such that ker A ⊆ ker B, dom(s) = AH, sAH = BH and sAu = Bu for all u ∈ H; in this case we write s = B/A.

If s is a semiclosed operator, then the domain dom(s) is a semiclosed subspace in H by Theorem 2.2. According to Lemma 2.1, there exists a unique positive bounded operator A ≥ 0 such that (dom(s), · dom(s) ) is isometrically isomorphic to de Branges space M(A). If we set B := sA, then we see that B ∈ B(H) by the semiclosed graph theorem, and consequently s is uniquely represented by a quotient B/A of bounded operators A and B. A general theory of quotients are argued in [5,7]. Let S(H) be the set of all semiclosed operators in H. Then, it is known that the set S(H) is closed under additions and multiplications [8]. But this does not mean that the set S(H) forms a vector space. The uniqueness of the zero element in S(H) does not hold. We denote by CD(H) the set of all closed and densely deﬁned operators in H. By the definition, CD(H) ⊂ S(H). 1

Lemma 2.3. (cf. [7]) If s ∈ CD(H), then (I + s∗ s)− 2 is a positive bounded operator so that a Hilbert space (dom(s), ·graph ) is isometrically isomorphic 1 to de Branges space M((I + s∗ s)− 2 ), where · graph is the graph norm of s 1 defined by f graph := (f 2 + sf 2 ) 2 for f ∈ dom(s). Proof. In [7], it is shown that a closed and densely deﬁned operator s in H is 1 represented by s = B/(I − B ∗ B) 2 for a unique pure contraction B = s(I + 1 1 1 1 s∗ s)− 2 . Then it follows from (I + s∗ s)− 2 = (I − B ∗ B) 2 that (I + s∗ s)− 2 H = 1 1 (I − B ∗ B) 2 H = dom(s), and (f = (I − B ∗ B) 2 u, u ∈ H) 1

f 2graph = f 2 + sf 2 = (I − B ∗ B) 2 u2 + Bu2 1

= u2 = (I − B ∗ B) 2 u2

1

(I−B ∗ B) 2

for any f ∈ dom(s).

= f 2

1

(I+s∗ s)− 2

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3. The Metric Between Semiclosed Subspaces Let M be a semiclosed subspace in H, equivalently a bounded operator range. Then, by Lemma 2.1, there exists a bijective mapping · M → T from the set of Hilbert norms {·M : (M, ·M ) → H} to the set of positive bounded operators {T ≥ 0 : M = T H}. When M is closed, the norm · (restricted to M ) is corresponding to the orthogonal projection PM onto M [cf. (3.1)]. The following is basic in our arguments. For any semiclosed subspace M , we choose a Hilbert norm · M from the set of all Hilbert norms on M , and let α be its correspondence M → · M (equivalently, M → T ≥ 0 from the above arguments). That is, a correspondence α is a choice function to choose a Hilbert norm from each semiclosed subspace. From now, we shall deﬁne the metric for semiclosed subspaces by a correspondence α. Let M1 and M2 be any semiclosed subspaces in H. Then, by a correspondence α, there exist unique positive bounded operators T1 and T2 which correspond to M1 and M2 , respectively. In this case, we deﬁne the real valued function ρα by the following. ρα (M1 , M2 ) := T1 − T2 . Here · is the operator norm of B(H). Then, we have Theorem 3.1. For each correspondence α, the function ρα is a metric on the set of all semiclosed subspaces in a Hilbert space H. Especially, it coincides with a gap metric on the set of all closed subspaces. Proof. It easily follows from the definition of the function that ρα is a metric on the set of all semiclosed subspaces. When M is a closed subspace in H, (M, · ) is isometrically isomorphic to de Branges space M(PM ) induced by the orthogonal projection PM onto M . Because u2PM = PM u2PM = (PM u, PM u)PM = (u, u) = u2

(3.1)

for u ∈ M = (ker PM )⊥ . Therefore, the metric between closed subspaces M1 and M2 is given by ρα (M1 , M2 ) = PM1 − PM2 which stands for the gap metric. This completes the proof. In the Hilbert space L2 (R), Sobolev spaces are typical examples of continuously embedded Hilbert space in L2 (R), so that Sobolev spaces are expressed by de Branges spaces. Thus, we can consider an approach of operational methods to some structures for Sobolev spaces. Indeed, the ρ-metric for Sobolev spaces of Fourier type is shown to be characterized by their Bessel kernels [2] in the next section. Now on, we shall calculate a value of the ρ-metric between Sobolev spaces of standard type. For m ≥ 1, a subspace 1 d dom(D1m ) = {f ∈ L2 (R) : D1 f, . . . , D1m f ∈ L2 (R)} D1 = i dx is a Hilbert space with a standard Hilbert structure 1

f W m,2 := (f 2 + D1 f 2 + · · · + D1m f 2 ) 2 .

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Here, D1 : dom(D1 ) → L2 (R) is a differential operator in a weak derivative sense. Clearly (dom(D1m ), · W m,2 ) → L2 (R). Therefore we see that dom(D1m ) is a semiclosed subspace in L2 (R) and we call W m,2 (R) := (dom(D1m ), · W m,2 ) the standard Sobolev spaces with the order m. Since Jm : W m,2 (R) → L2 (R), the standard Sobolev space is de Branges space. It follows from Lemma 2.1 that W m,2 (R) = M(Am ) for ∗ 12 ) . A positive bounded operator Am in detail is given by the Am = (Jm Jm following lemma. Lemma 3.2. Let W m,2 (R) (m ≥ 1) be the standard Sobolev spaces. Then W m,2 (R) = M(Am ) (isometrically isomorphic), where Am ≥ 0 is given by 1

Am = (I + D12 + · · · + D12m )− 2 .

(3.2)

Proof. For any f ∈ W m,2 (R), we see the following equations from selfadjointness of D1 . f 2W m,2 = f 2 + D1 f 2 + · · · + D1m f 2 1

= f 2 + (D1∗ D1 + · · · + D1∗ m D1m ) 2 f 2 1

= f 2 + (D12 + · · · + D12m ) 2 f 2 = f 2 + sf 2 , = f 2graph

1

s := (D12 + · · · + D12m ) 2

(the graph norm of s).

We have that W m,2 (R) is isometrically isomorphic to (dom(s), · graph ). By Lemma 2.3, we further have that W m,2 (R) is isometrically isomorphic to 1 M((I + s∗ s)− 2 ). Hence, it follows from the uniqueness condition of positivity that we obtain 1

1

Am = (I + s∗ s)− 2 = (I + D12 + · · · + D12m )− 2 . Example 3.1. (One dimensional cases for the standard Sobolev spaces) Let α be the choice function that we choose the standard Sobolev norm · W m,2 from each semiclosed subspace dom(D1m ), and we suitably choose a Hilbert norm from each semiclosed subspace except for dom(D1m ). Under this correspondence α, we shall calculate the metric ρα (dom(D11 ), dom(D12 )) which is simply denoted by ρ(W 1,2 (R), W 2,2 (R)). In the followings, we use this notation so long as without confusions. By (3.2), ρ(W 1,2 (R), W 2,2 (R)) = A1 − A2 = =

sup g∈L2 ,g≤1

=

sup g ∈L2 , g ≤1

sup g∈L2 ,g≤1

A1 g − A2 g

1

1

(I + D12 )− 2 g − (I + D12 + D14 )− 2 g 1

1

(1 + ξ 2 )− 2 g − (1 + ξ 2 + ξ 4 )− 2 g 1

1

= (1 + ξ 2 )− 2 − (1 + ξ 2 + ξ 4 )− 2 ∞

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The value of the Last term is about 0.229365 . . .. In above equations, · means the Fourier transform (3.3). Next, we change the standard norms for the norms (3.4) used by the Fourier transform 1 f (ξ) := f (x)e−ix·ξ dx, f ∈ L2 (RN ) for N ≥ 1. (3.3) N (2π) 2 RN

Then, is the value in Example 3.1 still the same number ? To reply for this question, we start to consider from situations of more general settings, multidimensional cases for Sobolev spaces of Fourier type. For σ > 0, we deﬁne Sobolev space H σ (RN ) by σ f ∈ L2 (RN ) : (1 + |ξ|2 ) 2 f ∈ L2 (RN ) , f H σ . H σ (RN ) := Here σ f H σ := (1 + |ξ|2 ) 2 f

(3.4) N

for ξ = (ξ1 , . . . , ξN ) ∈ R . To calculate the metric and |ξ| := + · · · + between Sobolev spaces H σ (RN ), we let α be the choice function that we choose the Sobolev norm · H σ from each semiclosed subspace σ f ∈ L2 (RN ) : (1 + |ξ|2 ) 2 f ∈ L2 (RN ) , (σ > 0) (3.5) 2

ξ12

2 ξN

and we suitably choose a Hilbert norm from each semiclosed subspace except for semiclosed subspaces (3.5). The following Lemma 3.3 shows that H σ (RN ) is expressed by de Brang σ ) for a unique positive bounded operator A

σ which is nothing es space M(A but the Bessel potential of the order σ. Since the norm · H σ is not a form of the graph norm, the same argument in the proof of getting Am in Lemma 3.2 cannot be applied here.

σ ) Lemma 3.3. H σ (RN ) is isometrically isomorphic to de Branges space M(A for a unique positive bounded operator

σ = (I − Δ)− σ2 , A N where Δ = j=1 ∂ 2 /∂x2j .

(the Bessel potential of the order σ)

σ L2 (RN ) = (I − Δ)− σ2 L2 (RN ) as a set in L2 (RN ), Proof. Since H σ (RN ) = A it is sufﬁcient to prove the equality of norms · H σ = · A σ . For any

σ )⊥ (= L2 (RN )) which f ∈ H σ (RN ), there exists a unique element g ∈ (ker A

satisﬁes f = Aσ g. Then

σ g = g = f A σ = A g = (1 + |ξ|2 ) 2 (1 + |ξ|2 )− 2 g Aσ σ

σ

σ = (1 + |ξ|2 ) 2 f = f H σ .

The next example is an answer for a previous question.

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Example 3.2. (Multi dimensional cases for Sobolev norms of Fourier type) The metric for Sobolev spaces H σ (RN ) is determined by Lemma 3.3 as follows. (1) For the cases of σ = 1 and σ = 2,

1 − A

2 =

1 g − A

2 g ρ(H 1 (RN ), H 2 (RN )) = A sup A g∈L2 ,g≤1

=

sup g∈L2 ,g≤1

=

sup g ∈L2 , g ≤1

1

(I − Δ)− 2 g − (I − Δ)−1 g 1

(1 + |ξ|2 )− 2 g − (1 + |ξ|2 )−1 g 1

= (1 + |ξ|2 )− 2 − (1 + |ξ|2 )−1 ∞ = 0.25. (2)

For 0 < σ1 < σ2 ,

σ − A

σ ρ(H σ1 (RN ), H σ2 (RN )) = A 1 2

σ1

= (1 + |ξ|2 )− 2 − (1 + |ξ|2 )− 1 2 σ σ−σ σ σ−σ σ1 2 1 σ1 2 1 = − . σ2 σ2

σ2 2

∞

Since the value of (1) follows from the result of (2), it is sufﬁcient to show the result of (2) whose proof is stated in the Appendix. Proposition 3.4. A distance ρ(H m (RN ), H m+1 (RN )) is monotone decreasing and convergent to 0 as m → ∞. Proof. Let am := ρ(H m (RN ), H m+1 (RN )). Then m m+1 m m mm − = . am = m+1 m+1 (m + 1)m+1 Thus, we have am+1 (m + 1)m+1 (m + 1)m+1 = · am (m + 2)m+2 mm m + 1 (m + 1)m (m + 1)m+1 · = · m + 2 (m + 2)m+1 mm m+1 m 1 m ) (1 + m m+1 ( m ) m+1 · m+2 m+1 = · = . 1 m + 2 ( m+1 ) m + 2 (1 + m+1 )m+1 1 m ) } (m ≥ 1) is monotone increasing, then we Note that a sequence {(1 + m see am+1 /am < 1 for m ≥ 1, and clearly am → 0 as m → ∞. Hence we have the desired results.

4. A Result of N. Aronszajn–K. Smith In this section, we shall give a new view point (Remark 4.1) for a result [2] of N. Aronszajn and K. Smith from which we can clarify the relation between the ρ-metric and the Bessel kernel.

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Theorem 4.1. ([2]) For Sobolev space H σ (RN ) (σ > 0), there exists uniquely Gσ ∈ L1 (RN ) such that H σ (RN ) = {Gσ ∗ g : g ∈ L2 (RN )}

and

Gσ ∗ gH σ = g.

Here Gσ is given by Gσ (x) =

1 2

N +σ−2 2

π

N 2

|x|

Γ( σ2 ) −N 2

σ (ξ) = (2π) G

σ−N 2

K N −σ (|x|) 2

and

σ

(1 + |ξ|2 )− 2 .

In above, Kν (·) (ν ∈ R) is the modified Bessel function of the third kind, and Gσ (x − y)g(y)dy. convolutions are defined by (Gσ ∗ g)(x) = RN

Remark 4.1. It is easy to understand a structure of this theorem from a view point of de Branges space as below. For any f ∈ H σ (RN ), there is a unique

σ g = (I − Δ)− σ2 g from Lemma 3.3. Then, by g ∈ L2 (RN ) such that f = A the Fourier transform σ N N σ f(ξ) = (1 + |ξ|2 )− 2 g(ξ) = (2π) 2 (2π)− 2 (1 + |ξ|2 )− 2 g(ξ) N σ (ξ) = (2π) 2 G g (ξ) = G σ ∗ g.

σ g. From this equation, we see that H σ (RN ) Hence, we have f = Gσ ∗ g = A is an operator range of the convolution operator Gσ ∗ · for the Bessel kernel Gσ . Therefore, it follows from the definition of de Branges norm and Lemma

σ g = g. 3.3 that f H σ = Gσ ∗ gH σ = A Aσ Theorem 4.2. Let Gσ be the Bessel kernel for the space H σ (RN ). For σ1 > 0 and σ2 > 0,

ρ(H σ1 (RN ), H σ2 (RN )) = (2π) 2 G σ1 − Gσ2 ∞ . N

Proof. From the definition of the ρ-metric and Remark 4.1,

σ − A

σ ρ(H σ1 (RN ), H σ2 (RN )) = A 1

=

sup g∈L2 ,g≤1

=

sup g ∈L2 , g ≤1

2

Gσ1 ∗ g − Gσ2 ∗ g

(2π) 2 G − (2π) 2 G σ1 g σ2 g N

N

N

= (2π) 2 G σ1 − Gσ2 ∞ .

5. The Metric Between Semiclosed Operators We shall introduce the metric in the set of all semiclosed operators based on the ρ-metric. Let α be any correspondence as stated in Sect. 3. Let s be any semiclosed operator with a domain dom(s) in H. Then since dom(s) is semiclosed, there exists a Hilbert norm · dom(s) on it which is given by the correspondence α in advance. Hence there exists a unique positive bounded operator A such that (dom(s), ·dom(s) ) is isometrically isomorphic to M(A).

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Put B := sA, then B ∈ B(H). This means that s is uniquely represented up α to α by a quotient B/A, so that we denote s = B/A. For any semiclosed operators s, t ∈ S(H), α

s = B/A,

α

t = D/C

for A ≥ 0, B, C ≥ 0, D ∈ B(H). Then we deﬁne the real valued function by qα (s, t) := max(A − C, B − D). We remark that a term A − C stands for the distance between domains of s and t under the correspondence α, that is, ρα (dom(s), dom(t)). The function qα seems to be special, but it is so natural in the sense of the following theorem: Theorem 5.1. For each correspondence α, the function qα is the metric in the set S(H) of all semiclosed operators in H. If (M :=) dom(s) = dom(t) for s, t ∈ S(H), then qα (s, t) = s − tα , where · α is the operator norm of the mapping from Hilbert space M equipped with a Hilbert norm · M to H: sα :=

su , u∈dom(s) uM sup

(5.1)

where · M is determined by α. Especially, if S, T ∈ B(H), then qα (S, T ) = S − T . Proof. Since qα is clearly a metric in S(H), we prove the rest part. First, we α show the following equality for s = B/A with dom(s) = AH. sα = B.

(5.2)

From Lemma 2.1, we see sα = =

su su = sup u u M A u∈AH u∈dom(s) sup

sup u=Av,v∈(ker A)⊥

=

sup v∈(ker A)⊥

( · A : de Branges norm)

(B/A)Av AvA

Bv Bv = sup AvA v ⊥ v∈(ker A)

= B (by (ker A)⊥ ⊇ (ker B)⊥ ). α

α

Put t = D/A. Then, since s − t = B/A − D/A = (B − D)/A, we have α qα (s, t) = B −D = s−tα by (5.2). Finally, if S, T ∈ B(H), then S = S/I, α T = T /I. Hence we see qα (S, T ) = S − T . Example 5.1. Let α be the same ones in Example 3.1. We shall calculate the metric qα (D1 , D2 ) for differential operators 1 d d2 and D2 = − 2 i dx dx with domains dom(D1 ) and dom(D2 ), respectively. Domains mean subspaces of L2 (R). But we consider their domains as Hilbert spaces W 1,2 (R) and D1 =

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W 2,2 (R). Then, by Lemma 3.2, there exist unique positive bounded operators A1 and A2 such that W 1,2 (R) = M(A1 ) and W 2,2 (R) = M(A2 ). This means that the differential operators have unique forms of quotients: α α D1 = B/A1 and D2 = D/A2 . Since ρ(W 1,2 (R), W 2,2 (R)) = A1 − A2 is about 0.229365 . . ., the rest problem is to calculate the term B − D. B − D =

sup g∈L2 ,g≤1

=

sup g∈L2 ,g≤1

=

D1 A1 g − D2 A2 g 1

sup g∈L2 ,g≤1

=

Bg − Dg

1

D1 (I + D12 )− 2 g − D12 (I + D12 + D14 )− 2 g 1

sup g ∈L2 , g ≤1

1

ξ(1 + ξ 2 )− 2 g − ξ 2 (1 + ξ 2 + ξ 4 )− 2 g 1

1

= ξ(1 + ξ 2 )− 2 − ξ 2 (1 + ξ 2 + ξ 4 )− 2 ∞ = 2. Hence, we see that qα (D1 , D2 ) = max{A1 − A2 , B − D} = 2. Remark 5.1. If we change W m,2 (R) for H m (R) (m = 1, 2), then we also have the same value of the metric by Example 3.2. Example 5.2. Let α and other settings be the same ones in Example 5.1. We α α shall calculate the metric qα (D1 , D1 + D2 ). Let D1 = B/A1 and D1 + D2 = F/A2 . B − F = D1 A1 − (D1 + D2 )A2 = sup D1 A1 g − (D1 + D2 )A2 g g∈L2 ,g≤1

=

1

sup g∈L2 ,g≤1

1

D1 (I + D12 )− 2 g − D1 (I + D12 + D14 )− 2 g 1

−D12 (I + D12 + D14 )− 2 g =

1

sup g∈L2 ,g≤1

1

ξ(1 + ξ 2 )− 2 g − ξ(1 + ξ 2 + ξ 4 )− 2 g 1

−ξ 2 (1 + ξ 2 + ξ 4 )− 2 g 1

1

1

= ξ(1 + ξ 2 )− 2 − ξ(1 + ξ 2 + ξ 4 )− 2 − ξ 2 (1 + ξ 2 + ξ 4 )− 2 ∞ = 2. Hence, we see that qα (D1 , D1 + D2 ) = max{A1 − A2 , B − F } = 2. Example 5.3. Let α and other settings be the same ones in Example 5.1. We shall calculate the metric qα (D2 , D2 + D1 ). Since dom(D2 ) ⊂ dom(D1 ), we see that qα (D2 , D2 + D1 ) = D1 |dom(D2 ) α by Theorem 6.7. That is, D1 |dom(D2 ) α = =

sup f ∈W 2,2 ,f W 2,2 ≤1

sup g∈L2 ,g≤1

D1 f =

D1 A2 g =

sup f =A2 g,A2 gA2 ≤1

sup g ∈L2 , g ≤1

1 1 = ξ(1 + ξ 2 + ξ 4 )− 2 ∞ = √ . 3

D1 A2 g 1

ξ(1 + ξ 2 + ξ 4 )− 2 g

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6. Topological Structures In this section, we start to study the topological structures for the metric space (S(H), qα ). For a time, we explain the topology of the article [4]. We already have introduced the topology O in S(H) in that paper. Let α be any α correspondence. For ε > 0, a neighborhood of s = B/A is deﬁned by α

V (s; α, ε) := {t ∈ S(H) : t = D/A, B − D < ε}.

(6.1)

Although the above definition of a neighborhood depends on the correspondence α, the topology followed by (6.1) is independent of α. Hence we simply denote it by O. Roughly speaking, the topology O is for constant domain cases unlike the case of the topology induced from the metric qα . Following (2) of Theorem 2.2, a semiclosed operator s is a bounded operator s˜ ∈ B(M, H) from a Hilbert space M := dom(s) with some (equivalently, any) Hilbert norm to H. A Hilbert norm on M is determined by α in advance. It is proved that the set B(M, H) is a connected component in O for each semiclosed subspace M in H. The topology (S(H), O) is shown to be metrizable and one of their metrics is called the d-metric. A sequence sn ∈ S(H) converges to s ∈ S(H) in the d-metric if and only if the domain of sn coincides with the domain of s for sufﬁciently large number n and sn converges to s in the operator norm, that is, sn − sα → 0 (n → ∞) as deﬁned by (5.1). From here, we study some properties of a metric space (S(H), qα ). For any α, the topology O is stronger than the topology induced from the metric qα . Indeed, O is the weakest topology containing topologies induced from qα for all α. This reads as follows. Theorem 6.1. Let sn , s ∈ S(H) (n ≥ 1). Then the following conditions are equivalent. (1) A sequence sn → s in O as n → ∞. (2) For any α, qα (sn , s) → 0 as n → ∞. α

α

Proof. Suppose the condition (1). For any α, put sn = Bn /An and s = B/A. Then we see that Bn converges to B as n → ∞ and An = A for sufﬁciently large number n by the definition of a neighborhood (6.1). Hence clearly qα (sn , s) = Bn − B → 0 as n → ∞. Conversely, suppose the condition (2). Then we have dom(sn ) = dom(s) for sufﬁciently large number n. For, assume that it does not hold, then without loss of generality we have dom(sn ) = dom(s) for all n. Then we see that An = A for all n and An → A, α α Bn → B as n → ∞ for sn = Bn /An , s = B/A. However, if we exchange the Hilbert norm equipped with dom(s) which corresponds to A for another Hilbert norm which corresponds to C( = A), then we have An → C as n → ∞ by the assumption (2), and this is a contradiction. This means that sn → s in O as n → ∞. For each semiclosed subspace M in H, the set B(M, H) is not necessarily a connected component in (S(H), qα ). But we have Theorem 6.2. For any α, the set B(H) is a connected component in (S(H), qα ).

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Proof. First we note that any B ∈ B(H) is represented by a quotient B = α B/I for any α. Suppose that a sequence Bn = Bn /I ∈ B(H) converges to α s = B/A ∈ S(H) in the qα -metric, then clearly A = I, so that s ∈ B(H). α Hence B(H) is closed. For each B = B/I ∈ B(H), any semiclosed operator α s = D/C which is sufﬁciently close to B/I is a member of B(H). Because, C is invertible (C −1 ∈ B(H)) by I − C is sufﬁciently small, so that s = DC −1 ∈ B(H). Hence B(H) is open. Since B(H) is clearly connected, it is a connected component. In [4, Theorem 3.7], it is proved that the set C(H) of all closed operators in H is an open subset in (S(H), O). But, the set C(H) is not necessarily open in the topology induced by qα . Concerning this argument, we have Theorem 6.4. To prove it, we give a lemma. α

Lemma 6.3. For any α, let s ∈ S(H) and s = B/A. Then, s is a closed and 1 densely defined operator in H if and only if Rs := (A2 + B ∗ B) 2 is invertible, −1 that is, Rs ∈ B(H). Proof. In [7, Theorem 1], it is shown that a quotient D/C is a closed operator 1 if and only if C ∗ H + D∗ H = (C ∗ C + D∗ D) 2 H is closed in H. Now, suppose that s is a closed and densely deﬁned operator in H. Then it follows that Rs has closed range. From the kernel condition ker A ⊆ ker B of a quotient B/A, we have the condition ker Rs = ker A by the equation Rs u2 = Au2 + Bu2 . Since s is densely deﬁned, AH is dense in H, that is, ker A = {0}. Hence Rs H = H, or Rs is invertible. Conversely, suppose that Rs is invertible. Obviously Rs has closed range, so that s is closed, and a density of dom(s)(= AH) follows from the condition ker A = ker Rs = {0}. Theorem 6.4. For any α, the set CD(H) is an open subset in (S(H), qα ). Proof. We note that the set of all invertible element of B(H) is an open subset α with respect to the operator norm in B(H). For any s = B/A ∈ CD(H), Rs is α invertible by Lemma 6.3. Suppose that t = D/C ∈ S(H) is sufﬁciently close 1 to s, that is, A − C and B − D are very small. Then Rt = (C 2 + D∗ D) 2 is close to Rs . Hence, Rt is invertible, or t ∈ CD(H). Remark 6.1. Though the set C(H) is not necessarily open in (S(H), qα ), the situation is improved under the constant domain condition : If s ∈ C(H), then there exists ε > 0 such that t ∈ S(H) with dom(s) = dom(t) and qα (s, t) < ε implies t ∈ C(H). For, since C(H) is an open subset in (S(H), O), there exists ε > 0 such that V (s; α, ε) ⊂ C(H). This means that desired assertions hold by (6.1). Now we show that the qα -metric is stronger than the gap metric in CD(H) by using the following Lemma 6.5. Let s, t ∈ CD(H), then the gap metric (cf. [6]) between s and t is deﬁned by the operator norm Ps − Pt , where Ps and Pt are the orthogonal projection onto the graph of s and t respectively.

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α

Lemma 6.5. ([4, Lemma 3.9]) For any α, let s ∈ CD(H) and s = B/A, and 1 let Rs = (A2 + B ∗ B) 2 . Then the orthogonal projection onto the graph {(Au, Bu) ∈ H × H : u ∈ H} of the quotient B/A is given by the operator matrix A(Rs−1 )2 A A(Rs−1 )2 B ∗ B(Rs−1 )2 A

B(Rs−1 )2 B ∗

in the product Hilbert space H × H. Theorem 6.6. For any α, the topology induced by the qα -metric is stronger than that induced by the gap metric g in CD(H), that is, if qα (sn , s) → 0 implies g(sn , s) → 0 as n → ∞ for sn , s ∈ CD(H). α

Proof. Suppose that qα (sn , s) → 0 as n → ∞. We put sn = Bn /An and α s = B/A. Then An → A and Bn → B as n → ∞. Obviously, since 1 1 → Rs−1 in Rsn = (A2n + Bn∗ Bn ) 2 → (A2 + B ∗ B) 2 = Rs , we see that Rs−1 n −1 2 −1 2 −1 2 ∗ −1 2 ∗ B(H). Hence An (Rsn ) An → A(Rs ) A, An (Rsn ) Bn → A(Rs ) B and Bn (Rs−1 )2 Bn∗ → B(Rs−1 )2 B ∗ . From Lemma 6.5, this completes the proof. n Concerning the perturbation of semiclosed operators, we have the following. Theorem 6.7. (cf. [4, Theorem 4.1]) Let s and t be semiclosed operators such that dom(s) ⊆ dom(t). Then, for each α, the distance s and s + t is characterized by the operator norm · α of the perturbed term: qα (s, s + t) = t|dom(s) α , where t|dom(s) is the operator t restricted to the domain of s. Proof. Let s and t be any semiclosed operators such that dom(s) ⊆ dom(t). α α Put s = B/A and t = D/C. Since dom(s) ⊆ dom(t), we have AH ⊆ CH. Hence, by Douglas’s majorization theorem [3], there exists a bounded operator X such that A = CX. Then s + t = B/A + D/C = B/A + DX/CX α

= B/A + DX/A = (B + DX)/A, and since domains of s and s + t are the same, qα (s, s + t) = s − (s + t)α by Theorem 5.1. Then s − (s + t)α = B/A − ((B + DX)/A)α = DX/Aα = DX

by (5.2). α

On the other hand, since t|dom(s) = DX/CX = DX/A for the same X as above, consequently we have qα (s, s + t) = t|dom(s) α . Corollary 6.8. Let s be a closed operator and let t be a semiclosed operator with dom(s) ⊆ dom(t), and we put s(κ) = s + κt for a complex number κ. Then s(κ) is closed for sufficiently small of absolute value |κ|.

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Proof. From Theorem 6.7, we have qα (s, s(κ)) = |κ|t|dom(s) α . Topological properties of the set of all closed operators which are deﬁned on the same domain are stated in Remark 6.1. Since domains of s and s(κ) are the same, it follows that s(κ) is closed for sufﬁciently small |κ|. Finally, we are end this section to study connections between the relative bounds and the q-metric. Let s and t be semiclosed operators. t is said to be relative bounded with respect to s or simply s-bounded if there exist non-negative constants a and b such that tu ≤ au + bsu,

u ∈ dom(s) ⊆ dom(t).

(6.2)

The relative bound of t with respect to s or simply s-bound is deﬁned by the infimum b satisfying (6.2). If s is closed, then t is s-bounded if and only if dom(s) ⊆ dom(t). The following theorem and corollary are concerned with [6, p. 206 Theorem 2.24]. Though closed operators are treated on a Banach space in that theorem, we here handle a more general class S(H) in a Hilbert space. Theorem 6.9. Let s ∈ S(H), and let tn ∈ S(H) (n ≥ 1) be a s-bounded sequence, that is, there exist non-negative constants an and bn such that tn u ≤ an u + bn su,

u ∈ dom(s) ⊆ dom(tn ).

(6.3)

Then, if an → 0 and bn → 0 imply qα (s, s + tn ) → 0 as n → ∞ for any α. α

α

Proof. Let α be any correspondence and ﬁxed. Put s = B/A and tn = Bn /An . Since dom(s) ⊆ dom(tn ), there exists a bounded operator Xn such that A = An Xn for each n. Then the next condition follows from (6.3) by putting u = Av, v ∈ H. Bn Xn v ≤ an Av + bn Bv,

v ∈ H.

Thus we have Bn Xn ≤ an A + bn B. This implies that Bn Xn → 0 α as n → ∞ by hypothesis. Note that tn |dom(s) =Bn Xn /An Xn = Bn Xn /A. Then it follows from (5.2) and Theorem 6.7 that Bn Xn = tn |dom(s) α = qα (s, s + tn ) → 0. Corollary 6.10. Let s ∈ C(H) and tn ∈ S(H) (n ≥ 1) be a s-bounded sequence. If an → 0, bn → 0 as above, then s + tn is closed for sufficiently large number n and s + tn converges to s in the gap metric. Proof. By Theorem 6.9 and Remark 6.1, we see that s + tn is closed for sufﬁciently large number n. Hence, it follows from Theorem 3.8 [4] that s + tn converges to s in the gap metric. Acknowledgments I would like to thank the referee for suggesting the proof in Appendix, which is simpler than my original one. And I wish to express my deep gratitude to people who have encouraged me up to now.

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7. Appendix Proof of Example 3.2. Since the value of (1) of Example 3.2 immediately follows from the result of (2), we only prove (2) here. Put r := 1 + |ξ|2 ≥ 1. f (r) := r−

σ1 2

− r−

σ2 2

≥ 0.

Since f (r) takes the minimum value at r = 1, it is sufﬁcient to consider on (1, ∞). Then we have σ1 σ1 σ2 σ2 f (r) = − r− 2 −1 + r− 2 −1 2 2 σ2 −σ1 1 − σ1 −1 = r 2 (−σ1 + σ2 r− 2 ). 2 Let r0 := r > 0 be the positive number such that −σ1 + σ2 r− σ −σ σ1 − 2 1 . r0 2 = σ2

σ2 −σ1 2

= 0, or (7.1)

Clearly 1 < r < r0 or r0 < r implies f (r) > 0 or f (r) < 0, respectively. Thus f (r) takes the maximum value at r0 . σ σ σ σ −σ − 21 − 22 − 21 − 22 1 − r0 = r0 f (r0 ) = r0 1 − r0 σ σ σ2 − σ1 σ1 − 21 − 21 = r0 (1 − ) = r0 . σ2 σ2 1 σ σ−σ σ σ1 2 1 − 21 From (7.1), we see that r0 = . Hence we have σ2 σ1 1 σ σ−σ σ1 2 1 σ2 − σ1 σ1σ2 −σ1 f (r0 ) = = (σ2 − σ1 ) σ2 σ2 σ2 σ2 σ2 −σ1 1 2 σ σ−σ σ σ−σ σ1 2 1 σ1 2 1 = − . σ2 σ2 Therefore we conclude that 2 −

(1 + |ξ| )

σ1 2

2 −

− (1 + |ξ| )

σ2 2

∞ = f (r0 ) =

σ1 σ2

1 σ σ−σ 2

1

−

σ1 σ2

2 σ σ−σ 2

1

.

References [1] Ando, T.: De Branges Spaces and Analytic Operator Functions. Lecture Note. Hokkaido University, Sapporo, Japan (1990) [2] Aronszajn, N., Smith, K.: Theory of Bessel potentials (I). Ann. Inst. Fourier. Grenoble 11, 385–475 (1961) [3] Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966)

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[4] Hirasawa, G.: A topology for semiclosed operators in a Hilbert space. Acta Sci. Math. (Szeged) 73, 271–282 (2007) [5] Izumino, S.: Quotients of bounded operators. Proc. Am. Math. Soc. 106, 427–435 (1989) [6] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1980) [7] Kaufman, W.E.: Representing a closed operator as a quotient of continuous operators. Proc. Am. Math. Soc. 72, 531–534 (1978) [8] Kaufman, W.E.: Semiclosed operators in Hilbert space. Proc. Am. Math. Soc. 76, 67–73 (1979) [9] Sarason, D.: Sub-Hardy Hilbert spaces in the unit disk. In: University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10. Wiley, New York (1994) Go Hirasawa (B) Faculty of Engineering Ibaraki University 4-12-1 Nakanarusawa Hitachi, Japan e-mail: [email protected] Received: September 1, 2010. Revised: November 5, 2010.

Integr. Equ. Oper. Theory 70 (2011), 379–394 DOI 10.1007/s00020-011-1876-1 Published online April 5, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

A Perturbation Theory for Core Operators of Hilbert-Schmidt Submodules Michio Seto Abstract. We discuss a perturbation theory for core operators of HilbertSchmidt submodules in the Hardy space over the bidisk. It is shown that eigenvalues of core operators perturbed by automorphisms of the bidisk have a certain continuity, and also we show that the family of eigenspaces of simple eigenvalues has a local cross section. Mathematics Subject Classification (2010). Primary 47B32; Secondary 47B35. Keywords. Hilbert modules, Hardy spaces, Toeplitz operators, Perturbation theory.

1. Introduction Let D be the open unit disk in the complex plane C, and let H 2 (D) be the Hardy space over D. The Hardy space over the bidisk D2 will be denoted by H 2 for short. Then z = (z1 , z2 ) will denote the variable of functions in H 2 , and kλ will denote the reproducing kernel of H 2 at λ = (λ1 , λ2 ) in D2 . We note that H 2 can be deﬁned as the tensor product Hilbert space H 2 (D) ⊗ H 2 (D). Let A denote the bidisk algebra. Then, under pointwise multiplication, H 2 becomes a Hilbert module over A. A closed subspace M of H 2 is called a submodule if M is invariant under the module action, that is, a submodule is an invariant subspace of H 2 under multiplication of each function in A. [S] denotes the submodule generated by a set S, and we set Mλ = [(z1 − λ1 )M + (z2 − λ2 )M] for a submodule M and λ = (λ1 , λ2 ) in D2 . Rf denotes the compression of a Toeplitz operator Tf into M, that is, we set Rf = PM Tf |M where PM is the orthogonal projection of H 2 onto a submodule M. The rank of a submodule M is the least cardinality of a generating set of M as a Hilbert module, and This research was partially supported by Grant-in-Aid for Young Scientists (B) (21740099).

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which will be denoted by rank M. The common zero set of a submodule M is denoted by Z(M), that is, we set Z(M) = {λ ∈ D2 : f (λ) = 0 for any f ∈ M}. In the theory of Hardy space H 2 (D), it is known that every submodule corresponds to an inner function. This is the famous Beurling theorem, which is one of the fundamental results giving rich connection between Hilbert space operator theory and complex analysis. Recently, Drury-Arveson space has been considered as a proper multivariable generalization of H 2 (D) from this point of view (see [1,2,13]). On the other hand, submodules in other Hilbert spaces consisting of holomorphic functions have been studied from the perspective suggested by Douglas, Paulsen and their collaborators (see [4]). On H 2 (D2 ), although we have been able to deal with submodules generated by polynomials by Guo’s theory (see [2]), structure of general submodules would be still unknown. In this situation, some eﬀective approaches to them have been given by Yang in a series of his papers ([18–21]). Particularly, in [20], he deﬁned a new class of submodules said to be Hilbert-Schmidt as follows: Definition 1.1 (Yang [20]). A submodule M in H 2 is said to be HilbertSchmidt if ∗ ∗ ∗ ∗ ΔM 0 = I − Rz1 Rz1 − Rz2 Rz2 + Rz1 Rz2 Rz1 Rz2

is Hilbert-Schmidt, where I denotes the identity operator on M. The above ΔM 0 has been called the core operator or the defect operator of a submodule M, and which was studied by Guo [7], Guo–Yang [9] and Yang [19–21]. There are two advantages in dealing with Hilbert-Schmidt submodules. First, they are characterized by integral operators encoding data on reproducing kernels on submodules. Secondly, it is a quite large class. In fact, Yang showed that “almost” all submodules are Hilbert-Schmidt in [19]. For example, submodules generated by polynomials and Rudin’s famous inﬁnitely generated submodule are Hilbert-Schmidt. In this paper, we are going to study eigenvalues and eigenspaces of the following operator valued function which is a kind of generalization of a core operator: Definition 1.2. For a submodule M in H 2 and any λ = (λ1 , λ2 ) in D2 , we set ∗ ΔM λ = I − Rbλ1 (z1 ) Rbλ

1

(z1 )

− Rbλ2 (z2 ) Rb∗λ

+ Rbλ1 (z1 ) Rbλ2 (z2 ) Rb∗λ

where

(bλ1 (z1 ), bλ2 (z2 )) =

1

∗ (z1 ) Rbλ

2

2

(z2 )

(z2 ) ,

z1 − λ 1 z2 − λ 2 , 1 − λ 1 z1 1 − λ 2 z2

.

We will abbreviate ΔM λ to Δλ if no confusion occurs. Since (bλ1 (z1 ), bλ2 (z2 )) deﬁnes an automorphism of D2 (i.e., a biholomorphic map acting on D2 ), Δλ can be seen as a core operator perturbed by an automorphism. Although there exist other automorphisms of D2 , namely

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rotations and the permutation of variables, looking at the definition of Δλ , one can verify that it is enough to consider linear fractional transformations. This paper has been organized as follows. In Sect. 2, we shall give some observation on Hilbert functions and module rank, and a reason why we study not only Δ0 but Δλ as well is explained. In Sect. 3, basic properties of Δλ are given. Especially, we show that Δλ is continuous with respect to the Hilbert-Schmidt norm if M is Hilbert-Schmidt. Section 4 is the main part of the paper. We show that eigenvalues of the operator valued function Δλ have a certain continuity, and also we show that the family of eigenspaces of simple eigenvalues have a local cross section. In Sect. 5, we deal with two concrete examples. In Sect. 6, as concluding remarks, we give some comments toward future research.

2. Some Relation between Hilbert Function and Module Rank In this section, we will observe some relation between Δλ and module rank. First, we shall introduce Hilbert functions along the line suggested by Douglas–Yan [5], and calculate them in some special class of submodules. Definition 2.1. Let M be a submodule, and let Iλ be the maximal ideal of the bidisk algebra A corresponding to λ = (λ1 , λ2 ). Then the Hilbert function of M with respect to Iλ is deﬁned as follows: HIλ ,M (k) = dim M/[Iλk · M], where we set Iλk · M = (z1 − λ1 )k M + (z1 − λ1 )k−1 (z2 − λ2 )M + · · · + (z2 − λ2 )k M for any k in N. We note that the following inequality is well known: k(k + 1) rank M. dim M/[Iλk · M] ≤ 2 Let q be an inner function in H 2 (D). Then K(q) will denote the quotient module of qH 2 (D) in H 2 (D), that is, we set K(q) = H 2 (D)/qH 2 (D) = H 2 (D) qH 2 (D), and kλ1 = kλ1 (z1 ) will denote the reproducing kernel of H 2 (D) at λ1 in D. The following lemma would be well known. Lemma 2.2. Let q be an inner function in H 2 (D), and we set K = K(bnλ1 ) ∩ K(q). Then (i) if (ii) if (iii) if f

bjλ1 kλ1 is in K then bj−1 λ1 kλ1 is in K for any 1 ≤ j ≤ n − 1, there exists a non-zero function f in K then kλ1 is in K, kλ1 , bλ1 kλ1 , . . . , bjλ1 kλ1 are in K and there exists a non-zero function in K which is orthogonal to them, then bj+1 λ1 kλ1 is in K.

Definition 2.3. A sequence of one-variable inner functions {qj }∞ j=0 is called an inner sequence if qj /qj+1 is also inner for any j ≥ 0.

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It is easy to see that every inner sequence deﬁnes a submodule M in H 2 as follows: M=

∞

⊕qj H 2 (D) ⊗ z2j .

j=0

For the detail of these submodules, see [15] and [16]. We deﬁne k subsets of integers for each inner sequence {qj }∞ j=0 and λ1 in D as follows: qj−k qj−k+1 S1 (λ1 ) = j ≥ 1 : mλ1 ≥ k, mλ1 ≥ k − 1, . . . qj qj qj−2 qj−1 . . . , mλ1 ≥ 2, mλ1 ≥1 qj qj qj−k qj−k+1 S2 (λ1 ) = j ≥ 2 : mλ1 ≥ k − 1, mλ1 ≥ k − 2, . . . qj qj qj−3 qj−2 . . . , mλ1 ≥ 2, mλ1 ≥1 qj qj .. . qj−k Sk (λ1 ) = j ≥ k : mλ1 ≥1 , qj where mλ1 (f ) denotes the zero multiplicity of f at λ1 and we set q−j = 0 for j in N. Theorem 2.4. Let Iλ be the maximal ideal of A corresponding to λ = (λ1 , 0) in D2 , and let {qj }∞ j=0 be an inner sequence. Then the Hilbert function of M=

∞

⊕qj H 2 (D) ⊗ z2j

j=0

with respect to Iλ is as follows: k

HIλ ,M (k) =

k(k + 1) + |Si (λ1 )|. 2 i=1

Proof. For simplicity, we shall show this formula in the case where k = 3. First, we note that M/[Iλ3 · M]

= M [Iλ3 · M]

= M [(z1 − λ1 )3 M + (z1 − λ1 )2 z2 M + (z1 − λ1 )z22 M + z23 M]

= (M b3λ1 M) ∩ (M b2λ1 z2 M) ∩ (M bλ1 z22 M) ∩ (M z23 M) =: M1 ∩ M2 ∩ M3 ∩ M4 .

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Then, identifying qj H 2 (D) ⊗ z2j with H 2 (D) as Hilbert spaces, we have M1 ∼ = K(b3λ1 ) M2 ∼ = H 2 (D) M3 ∼ = H 2 (D) M4 ∼ = H 2 (D) 1st

3 ⊕ K(b λ1 )

⊕ K b2λ1 qq01

⊕ H (D)

3 ⊕ K(b λ1 )

⊕ K b2λ1 qq12

⊕ K bλ1 qq02

⊕ H 2 (D) 2nd

⊕ H 2 (D) 3rd

2

3 ⊕ K(b λ1 )

⊕ K b2λ1 qq23

⊕ K bλ1 qq13

⊕ K qq03 4th

⊕

···

⊕

···

⊕

···

⊕

··· ··· .

Let Kj denote the Hilbert space obtained by taking the intersection ∞ of the (j + 1)-st column in the above. Then M/[Iλ3 · M] is isomorphic to j=0 ⊕Kj as Hilbert spaces. Therefore, by Lemma 2.2, we have HIλ1 ,M (3) =

∞

dim Kj

j=0

qj−3 qj−2 qj−1 = 1 + j ≥ 1 : mλ1 ≥ 3, mλ1 ≥ 2, mλ1 ≥ 1 qj qj qj (2.1) q q j−3 j−2 + 2 + j ≥ 2 : mλ1 (2.2) ≥ 2, mλ1 ≥ 1 qj qj qj−3 + 3 + j ≥ 3 : mλ1 (2.3) ≥ 1 , qj where we note that (2.1), (2.2) and (2.3) are counting numbers of j s such that b2λ1 kλ1 , bλ1 kλ1 and kλ1 belong to Kj , respectively. Thus we obtain the formula. Further, in the case where k = 1, one can verify the following simple formula. Theorem 2.5. Let M be the same submodule as in Theorem 2.4, and let λ = (λ1 , λ2 ) be in D2 . Then HIλ ,M (1) = dim M/Mλ 1 + |{j ≥ 1 : (qj−1 /qj )(λ1 ) = 0}| (λ2 = 0) =

0). 1 (λ2 = Next, we shall introduce Rudin’s module. Set αn = 1 − n−3 (n ∈ N), and let bαn be the Blaschke factor whose zero is αn , and we set qj =

∞

n=j

bn−j αn .

Then {qj }∞ j=0 is an inner sequence and the corresponding submodule has been called Rudin’s module. Although the original construction given in [12] is different from the above, this deﬁnes the same submodule (see, [14,16], for the details of this construction). The striking fact on Rudin’s module is that

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the rank is inﬁnity. Indeed, using the formula in Theorem 2.4 in the case where λ = (αn , 0), we have 1 2 1 HI(αn ,0) ,M (k) = k + n + k 2 2 k(k + 1) rank M. ≤ 2 As n tends to inﬁnity in the above, we have rank M = ∞. Thus it is reasonable to conjecture that Hilbert function might know something on module rank. Especially, the values of HIλ ,M (1) on Z(M) will be enough interesting. Indeed, since Z(M) = {αn }∞ n=1 × {0}, we have

HIλ ,M (1) = dim M/Mλ =

n+1 1

(λ = (αn , 0) ∈ Z(M)) (λ ∈ Z(M))

by Theorem 2.5. Now, the author would like to mention that HI(αn ,0) ,M (1) is just the dimension of the eigenspace of the operator Δ(αn ,0) = I − Rbαn (z1 ) Rb∗αn (z1 ) − Rz2 Rz∗2 +Rbαn (z1 ) Rz2 Rb∗αn (z1 ) Rz∗2

with eigenvalue 1. In fact, one can verify that M/M(αn ,0) = ker(I − Δ(αn ,0) ) (see [8,9]). Therefore, it will be worth while investigating the following operator valued function: Δλ = I − Rbλ1 (z1 ) Rb∗λ

∗ (z1 ) − Rbλ2 (z2 ) Rbλ2 (z2 ) +Rbλ1 (z1 ) Rbλ2 (z2 ) Rb∗λ (z1 ) Rb∗λ (z2 ) . 1 2 1

We should mention that Guo and Wang have given more general framework in [8] from another point of view.

3. Basic Properties of Δλ Let kξM denote the reproducing kernel of M at ξ in D2 . Then, trivially, we have PM kξ = kξM . The following theorem is straightforward generalization of facts given in Guo–Yang [9]. Theorem 3.1 (Guo–Yang [9]). For any λ = (λ1 , λ2 ) and ξ = (ξ1 , ξ2 ) in D2 , we set Dλ (ξ, z) = (Δλ kξM )(z). Then we have the following:

(i) Dλ (ξ, z) = 1 − bλ1 (ξ1 )bλ (z1 ) 1 − bλ2 (ξ2 )bλ2 (z2 ) kξM (z).

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(ii) Let m be the normalized Lebesgue measure on the distinguished boundary T2 of D2 . Then (Δλ f )(ξ) = f (·), Dλ (ξ, ·) L2 = f (z)Dλ (ξ, z) dm(z). T2

(iii) For 0 ≤ r < 1, Dλ (rz, rz) converges to the Poisson kernel at λ as r tends to 1 for almost all z in T2 . (iv) Δλ = 1 for any λ in D2 . (v) If Δλ is a trace class operator, then tr Δλ = 1. (vi) Δλ 22 ≤ 3 dim(H 2 /M) + 1 for any λ in D2 . 1 is unitarily equivalent to (vii) If M1 is unitarily equivalent to M2 then ΔM λ M2 2 Δλ for every λ in D . Proof. Proofs of (i), (ii), (iv), (v), (vi) and (vii) are the same as the original. We shall show only (iii). Setting ξ = rz, by (i), we have

Dλ (ξ, ξ) = 1 − bλ1 (ξ1 )bλ (ξ1 ) 1 − bλ2 (ξ2 )bλ2 (ξ2 ) kξM (ξ) = (1 − |bλ1 (ξ1 )|2 )(1 − |bλ2 (ξ2 )|2 )kξM 2 =

(1 − |bλ1 (ξ1 )|2 )(1 − |bλ2 (ξ2 )|2 ) (1 − |ξ1 |2 )(1 − |ξ2 |2 )kξM 2 . (1 − |ξ1 |2 )(1 − |ξ2 |2 )

By the Julia-Carath´eodory theorem (or direct calculation), we have that (1 − |bλ1 (ξ1 )|2 )(1 − |bλ2 (ξ2 )|2 ) (1 − |ξ1 |2 )(1 − |ξ2 |2 ) converges to the Poisson kernel at λ as |ξ| tends to 1. On the other hand, by Theorem 2.1 in [9], we have (1 − |ξ1 |2 )(1 − |ξ2 |2 )kξM 2 → 1 (|ξ| → 1) almost everywhere. Hence we have (ii).

Remark 3.2. Note that Δλ 2 is not constant. We will see it in Sect. 5. By the theory of integral operators, if Δ0 is Hilbert-Schmidt, then there exists a function C0 in L2 (T4 ) such that (Δ0 f )(z) = f (·), C0 (z, ·) 2

for almost all z in T . Then, by (ii) of Theorem 3.1, D0 can be identiﬁed with C0 . Therefore, a submodule M is Hilbert-Schmidt if and only if D0 is in L2 (T4 ). In the following argument, we set

Fλ (ξ, z) = 1 − bλ1 (ξ1 )bλ1 (z1 ) 1 − bλ2 (ξ2 )bλ2 (z2 ) . Then we have Dλ (ξ, z) = Fλ (ξ, z)kξM (z).

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Lemma 3.3. For any fixed λ and μ in D2 , Fλ (ξ, z)/Fμ (ξ, z) can be continuously extended to the closure of D4 and Fλ (ξ, z) (1 + |λ1 |)(1 + |λ2 |)(1 + |μ1 |)(1 + |μ2 |) ≤ max (1 − |λ1 |)(1 − |λ2 |)(1 − |μ1 |)(1 − |μ2 |) . ξ,z∈D2 Fμ (ξ, z) Moreover Fλ /F0 is a C ∞ -function with four real variables λ = (x1 + iy1 , x2 + iy2 ) in D2 . Proof. For j = 1, 2, we have 1 − bλj (ξj )bλj (zj ) 1 − bμj (ξj )bμj (zj ) =

(1 − λj ξj )(1 − λj zj ) − (ξj − λj )(zj − λj ) (1 − μj ξj )(1 − μj zj ) · (1 − μξj )(1 − μj zj ) − (ξj − μj )(zj − μj ) (1 − λj ξj )(1 − λj zj )

=

1 − |λj |2 (1 − μj ξj )(1 − μj zj ) · . 1 − |μj |2 (1 − λj ξj )(1 − λj zj )

This concludes the proof.

The next theorem shows that the concept of Hilbert-Schmidt submodules is independent of choice of coordinates. Theorem 3.4. If Δμ is Hilbert-Schmidt for some μ, then Δλ is so for any λ. Proof. Since Dλ (ξ, z) = Fλ (ξ, z)kξM (z) Fλ (ξ, z) Fμ (ξ, z)kξM (z) Fμ (ξ, z) Fλ (ξ, z) Dμ (ξ, z), = Fμ (ξ, z)

=

we have that Dλ is in L2 (T4 ) if Dμ is so by Lemma 3.3. This concludes the proof. The following is one of the main results. Theorem 3.5. Let M be a Hilbert-Schmidt submodule. Then Δλ converges to Δμ with respect to the Hilbert-Schmidt norm as λ tends to μ. Moreover Δλ is an operator valued C ∞ -function with four real variables λ = (x1 + iy1 , x2 + iy2 ), and those derivatives are also Hilbert-Schmidt. Proof. First, we shall show the continuity of Δλ with respect to the HilbertSchmidt norm. Since 2 |Dλ (ξ, z) − Dμ (ξ, z)|2 dm(z)dm(ξ) Δλ − Δμ 2 = T2 T2

by the theory of integral operators on L2 , it sufﬁces to show that the integral on the right hand side converges to 0 as λ tends to μ. Let U be an open disk

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centered at μ such that its closure U is contained in D2 . By Lemma 3.3, we have that Fλ (ξ, z) D0 (ξ, z) |Dλ (ξ, z)| = F0 (ξ, z) (1 + |λ1 |)(1 + |λ2 |) ≤ |D0 (ξ, z)| (1 − |λ1 |)(1 − |λ2 |) ≤ CK |D0 (ξ, z)| for any λ in U and almost all ξ and z in T2 , where CK is some constant depending only on the compact set K = U . Therefore we have 2 |D0 (ξ, z)|2 |Dλ (ξ, z) − Dμ (ξ, z)|2 ≤ 4CK

for any λ in U , almost all ξ and z in T2 . Moreover we have Fλ (ξ, z) D0 (ξ, z) F0 (ξ, z) Fμ (ξ, z) D0 (ξ, z) → F0 (ξ, z) = Dμ (ξ, z)

Dλ (ξ, z) =

as λ tends to μ for almost all ξ and z in T2 . Therefore, we have |Dλ (ξ, z) − Dμ (ξ, z)|2 dm(z)dm(ξ) → 0 T2 T2

as λ tends to μ by Lebesgue dominated convergence theorem. This concludes the continuity of Δλ . The differentiability of Δλ is shown similarly. Indeed, by Lemma 3.3, Fλ /F0 is a C ∞ -function with four real variables λ = (x1 + iy1 , x2 + iy2 ), and it is easy to see that those partial derivatives are bounded on any compact set contained in D2 . Therefore partial derivatives of Δλ are Hilbert-Schmidt operators deﬁned by those of Dλ . For instance, letting ∂Δλ /∂x1 denote the Hilbert-Schmidt operator deﬁned by the kernel function ∂(Fλ /F0 ) ∂Dλ = D0 , ∂x1 ∂x1 and setting λ + h = ((x1 + h) + iy1 , x2 + iy2 ) for sufﬁciently small h, we have 2 Δλ+h − Δλ ∂Δλ − h ∂x1 2 2 Dλ+h − Dλ ∂Dλ − = dm(z)dm(ξ) h ∂x1 T2 T2

=

(Fλ+h /F0 ) − (Fλ /F0 ) ∂(Fλ /F0 ) 2 |D0 |2 dm(z)dm(ξ). − h ∂x1

T2 T2

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Therefore we have

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2 Δλ+h − Δλ ∂Δλ →0 − h ∂x1 2

as h tends to 0 by Lebesgue dominated convergence theorem.

4. Eigenvalues and Eigenspaces of Δλ Let σ(T ) denote the spectrum of a Hilbert-Schmidt operator T . Then the determinant of T can be deﬁned (see [17], for the details), which will be denoted by det2 (T ). This is the key ingredient of this section. Theorem 4.1 (Theorem 9.2 in [17]). Let S and T be Hilbert-Schmidt operators, and we set σ(T ) = {σk (T )}k counting multiplicity. Then det2 (I + T ) has the following properties: an entire function, (i) det2 (I + wT ) is (ii) det2 (I + wT ) = k (1 + σk (T )w) exp(−σk (T )w), (iii) | det2 (I + S) − det2 (I + T )| ≤ S − T 2 exp(C2 (S2 + T 2 + 1)2 ), where C2 is some constant independent of S and T . Lemma 4.2. Set dλ (w) = det2 (I + wΔλ ). Then dλ converges to dμ uniformly on any compact subset of C as λ tends to μ. Proof. By (iii) in Theorem 4.1, we have the following inequality: |dλ (w) − dμ (w)| = | det2 (I + wΔλ ) − det2 (I + wΔμ )| ≤ |w|Δλ − Δμ 2 exp(C2 (wΔλ 2 + wΔμ 2 + 1)2 ).

By Theorem 3.5, this concludes the proof.

For any a in C, Ua will denote an open disk centered at a, and Ua will denote its closure. Throughout this section, σ(λ) will denote a non-zero eigenvalue of Δλ . Lemma 4.3. Let M be a Hilbert-Schmidt submodule, and let σ(μ) be a nonzero eigenvalue of Δμ for a fixed μ. Then there exist open disks Uσ(μ) and Uμ such that Uσ(μ) ∩ σ(Δλ ) = ∅

and

∂Uσ(μ) ∩ σ(Δλ ) = ∅

for any λ in Uμ . Proof. Since σ(μ) is isolated, there exists Uσ(μ) such that Uσ(μ) ∩ σ(Δμ ) = {σ(μ)}

and

∂Uσ(μ) ∩ σ(Δμ ) = ∅.

Taking Uμ sufﬁciently small, we have that σ(Δλ ) ∩ Uσ(μ) = ∅ for any λ in Uμ by Theorem 1.14 in Chapter VIII of [11] and Theorem 3.5. We assume that there were λn for each n in N such that λn converged to μ and ∂Uσ(μ) ∩ σ(Δλn ) = ∅. Let ηn be an element in ∂Uσ(μ) ∩ σ(Δλn ). Then we note that dλn (−1/ηn ) = 0 by (ii) in Theorem 4.1. Taking a subsequence of {ηn }n∈N if

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necessary, we might assume that ηn converged to some η in ∂Uσ(μ) . Then we would have that dμ (−1/η) = 0, because |dμ (−1/η)| ≤ |dμ (−1/η) − dμ (−1/ηn )| + |dμ (−1/ηn ) − dλn (−1/ηn )| ≤ |dμ (−1/η) − dμ (−1/ηn )| + sup |dμ (−1/w) − dλn (−1/w)| w∈∂Uσ(μ)

→0 as n tends to inﬁnity by Lemma 4.2, that is, η would be in σ(Δμ ) by (ii) in Theorem 4.1. This is a contradiction. Theorem 4.4. Let M be a Hilbert-Schmidt submodule, and let σ(μ) be a nonzero eigenvalue of Δμ with multiplicity m for a fixed μ. Then there exist an open disk Uμ and a set valued continuous function Σ(λ) defined on Uμ such that its value consists of m eigenvalues of Δ(λ), counting multiplicity. Proof. Let Uσ(μ) and Uμ be the pair of open disks obtained in Lemma 4.3. Then, for any λ in Uμ , we set 1 P (λ) = (wI − Δλ )−1 dw. 2πi ∂Uσ(μ)

We note that P (λ) is the orthogonal projection of M onto ker(σ(λ)I − Δλ ) σ(λ)∈Uσ(μ)

because Δλ is self-adjoint. Shrinking Uμ if necessary, we may assume that P (λ) − P (μ) < 1 for any λ in Uμ . Then dim P (λ)M = dim P (μ)M = m for any λ in Uμ by Lemma 4.10 in Chapter I of [11]. Therefore, counting multiplicity, m eigenvalues σ1 (λ), . . . , σm (λ) of Δλ are determined in Uσ(μ) uniquely, and we set Σ(λ) = {σj (λ)}m j=1 . It follows that Σ(λ) is upper and lower semi-continuous at μ. Similarly we can show the continuity of Σ(λ) for any λ in Uμ . Thus we conclude the proof. Theorem 4.5. Let σ(μ) be a non-zero simple eigenvalue of Δμ for a fixed μ. Then there exist open disks Uσ(μ) and Uμ , and a vector valued C ∞ -function e(λ) with four real variables λ = (x1 + iy1 , x2 + iy2 ) defined on Uμ such that Uσ(μ) ∩ σ(Δλ ) = {σ(λ)} and Ce(λ) = ker(σ(λ)I − Δλ ) for any λ in Uμ . Proof. Let Uσ(μ) and Uμ be the pair of open disks obtained in Lemma 4.3. We set 1 (wI − Δλ )−1 dw P (λ) = 2πi ∂Uσ(μ)

for any λ in Uμ . By the same argument in the proof of Theorem 4.4, we may assume that P (λ) − P (μ) < 1 and dim P (λ)M = 1 for any λ in Uμ . Hence we have that Uσ(μ) ∩σ(Δλ ) = {σ(λ)}. Moreover P (λ) is an orthogonal projection valued C ∞ -function by Theorem 3.5, and the range is ker(σ(λ)I − Δλ ).

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Let e = e(μ) be a unit vector in ker(σ(μ)I − Δμ ) and we set e(λ) = P (λ)e for any λ in Uμ . Then, since e(λ) − e = P (λ)e − P (μ)e ≤ P (λ) − P (μ) < 1, e(λ) is non-zero. This concludes the proof.

The statement of Theorem 4.5 is translated into geometric language as follows: Corollary 4.6. Let M be a Hilbert-Schmidt submodule, and let σ(λ), e(λ) and Uμ be the same as those obtained in Theorem 4.5. Then the mapping π : ker(σ(λ)I − Δλ ) → {λ} defines a trivial line bundle over Uμ , and e(λ) is a cross section. Remark 4.7. By a fact in p. 52 in [6] or Example 3 (a) in [18], under the following condition: there exists a non-zero function ϕλ in M ∩ H ∞ such that ϕλ (λ) = 0 for any λ in D2 \ Z(M), it is easy to see that the mapping π : ker(I − Δλ ) → {λ} deﬁnes a line bundle over D2 \ Z(M). In fact, kλM is a cross section and CkλM = ker(I − Δλ ). The author does not know whether Corollary 4.6 gives a local coordinate of a line bundle over D2 \ S for some thin set S. Remark 4.8. All results obtained in this section hold in the Hardy space over the general polydisk. However, on the unit disk, since Δλ is an orthogonal projection, the corresponding result is quite trivial. Therefore, Theorem 4.5 will be peculiar to multivariable setting. Remark 4.9. The assumption that σ(μ) is simple can not be removed in Theorem 4.5. Roughly speaking, eigenvalues are continuous, however eigenvectors are not continuous. We will see it in the next section by giving examples. This phenomenon is well known in perturbation theory for linear operators (see pp. 110–111 in [11]).

5. Two Examples In this section, we exhibit eigenvalues and eigenvectors of Δλ on two HilbertSchmidt submodules. Example. Let {qj }j≥0 be an inner sequence. If M=

∞

⊕qj H 2 (D) ⊗ z2j ,

j=0

then, for any λ = (λ1 , 0), we have σ(Δλ ) = {±σj (λ1 )}j≥1 ∪ {0, 1},

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σj (λ1 ) =

391

2

1 − |(qj−1 /qj )(λ1 )| .

If σj (λ1 ) = 1 then the eigenfunction corresponding to σj (λ1 ) is 2 (qj−1 /qj )(λ1 )qj Kλ1 + −1 + 1 − |(qj−1 /qj )(λ1 )| qj−1 Kλ1 ⊗ z2 , where we set Kλ1 = 1 − |λ1 |2 kλ1 (z1 ). Let ej (λ1 ) denote the above eigenfunction. Note that ej (λ1 ) converges to 0 as σj (λ1 ) tends to 1. However, for any λ = (λ1 , λ2 ), by Theorem 1.2, we have 1 + |{j ≥ 1 : (qj−1 /qj )(λ1 ) = 0}| (λ2 = 0) dim ker(I − Δλ ) = 1 (λ2 = 0). Moreover, in the former case, by Theorem 1.1 in [15], ker(I − Δλ ) = Ckλ1 (z1 )q0 (z1 ) ⊕ ⊕Ckλ1 (z1 )qj (z1 )z2j , j where the sum j is taken only over j s such that (qj−1 /qj )(λ1 ) = 0, and we note that kλM = 0 if and only if the sum j is non-trivial. Example. Let q1 = q1 (z1 ) and q2 = q2 (z2 ) be one-variable inner functions, and let M be the submodule generated by q1 and q2 (see [10] and [19] for the details). Then M can be decomposed as follows: M = (q1 H 2 (D) ⊗ q2 H 2 (D)) ⊕ (q1 H 2 (D) ⊗ K(q2 )) ⊕ (K(q1 ) ⊗ q2 H 2 (D)). With respect to the above decomposition, Δλ has the following matrix representation: ⎞ ⎛ |q1 (λ1 )|2 + |q2 (λ2 )|2 − 1 q2 (λ2 ) 1 − |q2 (λ2 )|2 q1 (λ1 ) 1 − |q1 (λ1 )|2 ⎟ ⎜ 1 − |q2 (λ2 )|2 0 ⎠, ⎝ q2 (λ2 ) 1 − |q2 (λ2 )|2 q1 (λ1 ) 1 − |q1 (λ1 )|2 0 1 − |q1 (λ1 )|2 where we omit the zero matrix corresponding to the kernel of Δλ . Hence we have σ(Δλ ) = {0, 1, ±σ(λ)}, where we set σ(λ) =

(1 − |q1 (λ1 )|2 )(1 − |q2 (λ2 )|2 ).

This calculation has been done already in the case where (λ1 , λ2 ) = (0, 0) by Yang in [20]. If σ(λ) = 1 then the eigenfunction corresponding to σ(λ) is

q1 (z1 )q2 (z2 ) 1 − |q2 (λ2 )|2 − 1 − |q1 (λ1 )|2 e(λ) = (1 − λ1 z1 )(1 − λ2 z2 ) q1 (z1 )(1 − q2 (λ2 )q2 (z2 )) q2 (λ2 ) − 2 1 − |q2 (λ2 )| (1 − λ1 z1 )(1 − λ2 z2 ) q2 (z2 )(1 − q1 (λ1 )q1 (z1 )) q1 (λ1 ) + . 2 1 − |q1 (λ1 )| (1 − λ1 z1 )(1 − λ2 z2 )

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Note that e(λ) converges to 0 as σ(λ) tends to 1. However, 2 (if q1 (λ1 ) = q2 (λ2 ) = 0) dim ker(I − Δλ ) = 1 (otherwise). Moreover, in the former case, ker(I − Δλ ) = span{q1 (z1 )kλ , q2 (z2 )kλ } and which is equivalent to that kλM = 0.

6. Concluding Remarks We shall give some remarks toward future research. First, there is a famous theory for vector bundles deﬁned by a certain class of Hilbert space operators, namely Cowen–Douglas theory [3]. Every Cowen–Douglas class operator deﬁnes an Hermitian holomorphic vector bundle. Since every Hermitian holomorphic vector bundle has a unique canonical connection which preserves its structure, a curvature is determined uniquely. The main result of Cowen–Douglas theory is that the curvature is a complete unitary invariant for any Cowen–Douglas class operator. On the contrary, line bundles obtained in Sect. 5 are not holomorphic because Δλ is not holomorphic. Therefore connection is not determined uniquely. Moreover, they are not deﬁned on the whole spectrum D2 even on ker(I − Δλ ). According to the observation in Sect. 2, singular ﬁbers know something on module rank. To know the relation between singular ﬁbers and module rank explicitly is interesting. Next, it is not difﬁcult to generalize our result in the Cn -valued Hardy space setting. Let H 2 ⊗ Cn denote the Cn -valued Hardy space over D2 . Then the definition of Hilbert-Schmidt submodules, core operators and so on are the same as those in the cases where n = 1, and we have Theorem 6.1. Let M be a Hilbert-Schmidt submodule in H 2 ⊗Cn , and let σ(μ) be a non-zero eigenvalue of Δμ with multiplicity n for a fixed μ. Then there exist an open disk Uμ and vector valued C ∞ -functions e1 (λ), . . . , en (λ) with four real variables λ = (x1 + iy1 , x2 + iy2 ) defined on Uμ such that {e(λ)}nj=1 is a basis of ker(σ(λ)I − Δλ ) for any λ in Uμ . Corollary 6.2. Let M be a Hilbert-Schmidt submodule in H 2 ⊗ Cn , and let σ(λ), {ej (λ)}nj=1 and Uμ be the same as those obtained in Theorem 6.1. Then the mapping π : ker(σ(λ)I − Δλ ) → {λ} defines a trivial vector bundle over Uμ , and {ej }nj=1 is a local frame. Acknowledgements This research was started at when the author visited Saarland University. The author expresses gratitude to Professor E. Albrecht and Professor J. Eschmeier for their comments and suggestions, and also thanks to them and the members of Department Mathematics of Saarland University for their kind hospitality.

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References [1] Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, 44. American Mathematical Society, Providence (2002) [2] Chen, X., Guo, K.: Analytic Hilbert modules. Chapman & Hall/CRC Research Notes in Mathematics, 433. Chapman & Hall/CRC, Boca Raton (2003) [3] Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141(3-4), 187–261 (1978) [4] Douglas, R.G., Paulsen, V.: Hilbert modules over function algebras. Pitman Research Notes in Mathematics Series, vol. 217. Longman Scientific & Technical, Harlow; copublished in the United States with Wiley, Inc., New York (1989) [5] Douglas, R.G., Yan, K.: Hilbert-Samuel polynomials for Hilbert modules. Indiana Univ. Math. 42, 811–820 (1993) [6] Gleason, J., Richter, S., Sundberg, C.: On the index of invariant subspaces in spaces of analytic functions of several complex variables. J. Reine Angew. Math. 587, 49–76 (2005) [7] Guo, K.: Defect operators, defect functions and defect indices for analytic submodules. J. Funct. Anal. 213, 380–411 (2004) [8] Guo, K., Wang, P.: Defect operators and Fredholmness for Toeplitz pairs with inner symbols. J. Operator Theory 58, 251–268 (2007) [9] Guo, K., Yang, R.: The core function of submodules over the bidisk. Indiana Univ. Math. J. 53(1), 205–222 (2004) [10] Izuchi, K., Nakazi, T., Seto, M.: Backward shift invariant subspaces in the bidisc II. J. Operator Theory 51, 361–376 (2004) [11] Kato, T.: Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics. Springer, Berlin (1995) [12] Rudin, W.: Function theory in polydiscs. W. A. Benjamin, Inc., New York (1969) [13] Sawyer, E.T.: Function theory: interpolation and corona problems. Fields Institute Monographs, vol. 25, American Mathematical Society, Providence (2009) [14] Seto, M.: A new proof that Rudin’s module is not finitely generated. Oper. Theory Adv. Appl. 127, 195–197 (2008) [15] Seto, M.: Infinite sequences of inner functions and submodules in H 2 (D2 ). J. Oper. Theor. 61(1), 75–86 (2009) [16] Seto, M., Yang, R.: Inner sequence based invariant subspaces in H 2 (D2 ). Proc. Amer. Math. Soc. 135(8), 2519–2526 (2007) [17] Simon, B.: Trace ideals and their applications. 2nd edn. Mathematical Surveys and Monographs, vol. 120. American Mathematical Society, Providence (2005) [18] Yang, R.: Operator theory in the Hardy space over the bidisk III. J. Funct. Anal. 186(2), 521–545 (2001) [19] Yang, R.: Hilbert-Schmidt submodules and issues of unitary equivalence. J. Oper. Theor. 53(1), 169–184 (2005) [20] Yang, R.: The core operator and congruent submodules. J. Funct. Anal. 228(2), 469–489 (2005) [21] Yang, R.: A note on classification of submodules in H 2 (D2 ). Proc. Amer. Math. Soc. 137(8), 2655–2659 (2009)

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Michio Seto (B) Department of Mathematics Shimane University Matsue 690-8504 Japan e-mail: [email protected] Received: September 2, 2010. Revised: March 15, 2011.

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Integr. Equ. Oper. Theory 70 (2011), 395–418 DOI 10.1007/s00020-011-1881-4 Published online May 4, 2011 c The Author(s) This article is published with open access at Springerlink.com 2011

Integral Equations and Operator Theory

Right Invertible Multiplication Operators and Stable Rational Matrix Solutions to an Associate Bezout Equation, I: The Least Squares Solution A. E. Frazho, M. A. Kaashoek, A. C. M. Ran Abstract. In this paper a state space formula is derived for the least squares solution X of the corona type Bezout equation G(z)X(z) = Im . Here G is a (possibly non-square) stable rational matrix function. The formula for X is given in terms of the matrices appearing in a state space representation of G and involves the stabilizing solution of an associate discrete algebraic Riccati equation. Using these matrices, a necessary and sufﬁcient condition is given for right invertibility of the operator of multiplication by G. The formula for X is easy to use in Matlab computations and shows that X is a rational matrix function of which the McMillan degree is less than or equal to the McMillan degree of G. Mathematics Subject Classification (2010). Primary 47B35, 39B42; Secondary 47A68, 93B28. Keywords. Right invertible multiplication operator, Toeplitz operators, Bezout equation, Stable rational matrix functions, State space representation, Discrete algebraic Riccati equation, Stabilizing solution.

1. Introduction Throughout this paper G is a stable rational m × p matrix function. Here stable means that G has all its poles in |z| > 1, inﬁnity included. In particular, G is a rational matrix-valued H ∞ function. In general, p will be larger than m, and thus G will be a “fat” non-square matrix function. We shall be dealing with the corona type Bezout equation G(z)X(z) = Im ,

z ∈ D.

(1.1)

∞

Equation (1.1)—for arbitrary H functions—has a long and interesting history, starting with Carleson’s corona theorem [4] (for the case when m = 1) The research of the first author was partially supported by a visitors grant from NWO (Netherlands Organisation for Scientific Research).

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and Fuhrmann’s extension to the matrix-valued case [10]. The topic has beautiful connections with operator theory (see [1,22,24], the books [15,19–21], and the more recent papers [26–28]). Rational matrix equations of the form (1.1) also play an important role in solving systems and control theory problems, in particularly, in problems involving coprime factorization, see, e.g., [30, Section 4.1], [13, Section A.2], [31, Chapter 21]). For matrix polynomials (1.1) is closely related to the Sylvester resultant; see, e.g., Section 3 in [12] and the references in this paper. The operator version of the corona theorem tells us that (1.1) has a p×m matrix valued H ∞ solution X if and only if the operator MG of multiplication by G mapping the Hardy space H 2 (Cp ) into the Hardy space H 2 (Cm ) is right invertible. The necessity of this condition is trivial; sufﬁciency can be proved by using the commutant lifting theorem (see, e.g., [21, Theorem 3.6.1]). In our case, because G(z) is rational, a simple approximation argument (see the paragraph after Proposition 2.1 below) shows that the existence of a H ∞ solution implies the existence of a rational H ∞ - solution. Assuming that MG is right invertible, let X be the p×m matrix function deﬁned by ∗ ∗ −1 (MG MG ) Ey, X(·)y = MG

y ∈ Cm .

(1.2)

Here E is the canonical embedding of Cm into H 2 (Cm ), that is, (Ey)(z) = y for each z ∈ D and y ∈ Cm . We shall see (Theorem 1.1 or Proposition 2.1 below) that the function X determined by (1.2) is a stable rational matrix ∗ ∗ −1 (MG MG ) is the unique function satisfying (1.1). Note that the operator MG 2 m (Moore-Penrose) right inverse of MG mapping H (C ) onto the orthogonal complement of Ker MG in H 2 (Cp ). This implies that the solution X of (1.1) deﬁned by (1.2) has an additional minimality property, namely given a stable rational matrix solution V of (1.1) we have X(·)uH 2 (Cp ) ≤ V (·)uH 2 (Cp )

for each u in Cm ,

(1.3)

or equivalently 1 2π

2π 0

1 X(e ) X(e ) dt ≤ 2π it ∗

it

2π

V (eit )∗ V (eit ) dt.

(1.4)

0

Moreover, equality holds in (1.4) if and only if V = X. For this reason we refer to the matrix function X deﬁned by (1.2) as the least squares solution of ∗ ∗ −1 (MG MG ) (1.1). We note that the use of the Moore-Penrose right inverse MG is not uncommon in the analysis of the corona problem (see, e.g., Section 1 in [27]). Let us now describe the main result of the present paper. The starting point is a state space representation of G. As is well-known from mathematical systems theory, the fact that G is a stable rational matrix function, allows us (see, e.g., Chapter 1 of [5] or Chapter 4 in [2]) to write G in the following form: G(z) = D + zC(In − zA)−1 B.

(1.5)

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Here A, B, C, D are matrices of appropriate sizes, In is an identity matrix of order n, and the n × n matrix A is stable, that is, A has all its eigenvalues in the open unit disc D. In the sequel we shall refer to the right hand side of (1.5) as a stable state space representation. State space representations are not unique. By definition the smallest n for which G has a stable state space representation of the form (1.5) is called the McMillan degree of G, denoted by δ(G). From the stability of the matrix A in (1.5) it follows that the symmetric Stein equation P − AP A∗ = BB ∗

(1.6)

has a unique solution P . Given this n×n matrix P we introduce two auxiliary matrices: R0 = DD∗ + CP C ∗ ,

Γ = BD∗ + AP C ∗ .

(1.7)

The following theorem is our main result. Theorem 1.1. Let G be the m × p rational matrix function given by the stable state space representation (1.5). Let P be the unique solution of the Stein equation (1.6), and let the matrices R0 and Γ be given by (1.7). Then equation (1.1) has a stable rational matrix solution if and only if (i) the discrete algebraic Riccati equation Q = A∗ QA + (C − Γ∗ QA)∗ (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA)

(1.8)

has a (unique) stabilizing solution Q, that is, Q is an n × n matrix with the following properties: (a) R0 − Γ∗ QΓ is positive definite, (b) Q satisfies the Riccati equation (1.8), (c) the matrix A − Γ(R0 − Γ∗ QΓ)−1 (C − Γ∗ QA) is stable; (ii) the matrix In − P Q is non-singular. Moreover, (i) and (ii) are equivalent to MG being right invertible. Furthermore, if (i) and (ii) hold, then the p×m matrix-valued function X defined by (1.2) is a stable rational matrix solution of (1.1) and X admits the following the state space representation: (1.9) X(z) = Ip − zC1 (In − zA0 )−1 (In − P Q)−1 B D1 , where A0 = A − Γ(R0 − Γ∗ QΓ)−1 (C − Γ∗ QA), C1 = D∗ C0 + B ∗ QA0 , with C0 = (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA), D1 = (D∗ − B ∗ QΓ)(R0 − Γ∗ QΓ)−1 + C1 (In − P Q)−1 P C0∗ . Finally, X is the least squares solution of (1.1), the McMillan degree of X is less than or equal to the McMillan degree of G, and 1 2π

2π 0

X(eit )∗ X(eit ) dt = D1∗ Ip + B ∗ Q(In − P Q)−1 B D1 .

(1.10)

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The necessary and sufﬁcient state space conditions for the existence of a stable rational matrix solution and the formula for the least squares solution given in the above theorem are new. They resemble analogous conditions and formulas appearing in the state space theory of discrete H 2 and H ∞ optimal control; see [16,17,23], Chapter 21 in the book [31], see also [6] for the continuous time analogues. However, the algebraic Riccati equation in Theorem 1.1 is of the stochastic realization type with the solution Q being positive semidefinite, while the H ∞ or H 2 control Riccati equations in the mentioned references are of the LQR type again with the solutions being positive semidefinite (see, e.g., [14, Chapter 5] for the LQR type, and [14, Chapter 6] for the stochastic realization type). It is easy to rewrite the stochastic realization Riccati equation into the LQR type, but then the condition on the stabilizing solution being positive semidefinite changes into negative semidefinite. As far as we know there is no direct way to reduce the problem considered in the present paper to a standard H 2 control problem or to a coprime factorization problem. Concerning the latter, the discrete time analogue of the coprime method employed in [30, Section 4.1] could be used to obtain a parametrization of all stable rational solutions of (1.1). However, minimal H 2 -solutions are not considered in [30], and to the best of our knowledge coprime factorization does not provide a method to single out such a solution. Moreover, it is not clear whether or not the minimal H 2 -solution X considered in the present paper does appear among the solutions obtained by using the discrete time analogue of the state space formulas given in [30, Section 4.2]; see the ﬁnal part of Example 2 in Sect. 5 for a negative result in this direction. We remark that Theorem 1.1 provides a computationally feasible way to check whether or not for a given m × p stable rational matrix function G the multiplication operator MG is right invertible and to obtain the least squares solution in that case. Indeed, ﬁrst one constructs a realization (1.5) in the standard way. Next, one solves (1.6) for P , for instance by using the Matlab command dgram or dlyap. With P one constructs the matrices R0 and Γ as in (1.7). Then solve the algebraic Riccati equation (1.8) for Q, either using the Matlab command dare or an iterative method. Finally, one checks that one is not an eigenvalue of P Q. Continuing in this way one also computes the least squares solution X given by (1.9). In the subsequent paper [9], assuming MG is right invertible, we shall present a state space description of the set of all stable rational matrix solutions of equation (1.1) and a full description of the null space of MG . In that second paper we shall also discuss the connection with the related Tolokonnikov lemma [25] for the rational case. The paper consists of ﬁve sections, the ﬁrst being the present introduction. Sections 2 and 3 have a preparatory character. The basic operator theory results on which Theorem 1.1 is based are presented in Sect. 2. In Sect. 3 we explain the role of the stabilizing solution Q of the Riccati equation appearing in Theorem 1.1. Also a number of auxiliary state space formulas are presented in this third section. The proof of Theorem 1.1 is given in Sect. 4. In Sect. 5 we present two examples, and illustrate the comment on MatLab procedures made above.

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2. The Underlying Operator Theory Results We begin with some terminology and notation. Let F be any m×p matrix-valued function of which the entries are essentially bounded on the unit circle T. Recall (see, e.g., Chapter XXIII in [11]) that the block Toeplitz operator deﬁned by F is the operator TF given by ⎤ ⎡ F0 F−1 F−2 · · · ⎢F F F ···⎥ ⎥ 2 p ⎢ 1 0 −1 2 m ⎥ TF = ⎢ (2.1) ⎢ F2 F1 F0 · · · ⎥ : + (C ) → + (C ). ⎦ ⎣ .. .. .. . . . . . . Here . . . , F−1 , F0 , F1 , . . . are the block Fourier coefﬁcients of F . By HF we denote the block Hankel operator determined by the block Fourier coefﬁcients Fj with j = 1, 2, . . ., that is, ⎤ ⎡ F 1 F2 F3 · · · ⎢F F F ···⎥ ⎥ 2 p ⎢ 2 3 4 2 m ⎥ (2.2) HF = ⎢ ⎢ F3 F4 F5 · · · ⎥ : + (C ) → + (C ). ⎦ ⎣ .. .. .. . . . . . . ˜ for the canonical embedding of Cm onto the ﬁrst coorWe shall write E 2 ˜ is just equal to the operator from dinate space of + (Cm ). Note that TF∗ E m 2 p C into + (C ) deﬁned by the ﬁrst column of TF∗ . The identity operator on 2+ (Cm ) or 2+ (Cp ) will be denoted by I. The symbol In stands n × n identity matrix or the identity operator on Cn . Let G be a stable rational m × p matrix function. In this case HG is an operator of ﬁnite rank and its rank is equal to the McMillan degree δ(G). Furthermore, the multiplication operator MG used in the previous section is unitarily equivalent to the block Toeplitz operator TG . In fact, MG FCp = FCm TG , where for each positive integer k the operator FCk is the Fourier transform mapping 2+ (Ck ) onto the Hardy space H 2 (Ck ). In what follows it ˜ = E, will be more convenient to work with TG than with MG . Note that FCm E where E is the embedding operator appearing in (1.2). Furthermore, the ∗ −1 expression MG (MG MG ) , also appearing in (1.2), can be derived from ∗ ∗ −1 (MG MG ) FCm = FCp TG∗ (TG TG∗ )−1 . MG

(2.3)

The following result provides the operator theory background for the proof of Theorem 1.1. Proposition 2.1. Let G be a stable rational m × p matrix function, and let R be the rational m × m matrix function given by R(z) = G(z)G∗ (z). Then the following four statements are equivalent. (a) The equation GX = I has a stable rational matrix solution. (b) The Toeplitz operator TG is right invertible. (c) The Toeplitz operator TR is invertible and the same holds true for the ∗ −1 T R HG . operator I − HG

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Moreover, if one of these conditions is satisfied, then TG TG∗ is invertible, its inverse is given by ∗ −1 ∗ −1 (TG TG∗ )−1 = TR−1 + TR−1 HG (I − HG TR HG )−1 HG TR ,

and the function X = satisfying (1.1).

˜ FCp TG∗ (TG TG∗ )−1 E

(2.4)

is a stable rational matrix function

We note that the equivalence of (a) and (b) in the above proposition is known. In fact, (a) implies (b) is trivial, and (b) implies the existence of a H ∞ solution. But if (1.1) has an H ∞ solution, then it also has a stable rational matrix solution. The latter follows from a simple approximation argument. To see this, given an H ∞ function F and 0 < r < 1, let us write Fr for the function Fr (z) = F (rz). Now assume that X is an H ∞ solution of (1.1). Then Gr (z)Xr (z) = Im , and hence G(z)Xr (z) = Im − (G(z) − Gr (z)) Xr (z),

|z| < 1.

Since G is rational, Gr (z) → G(z) uniformly on |z| ≤ 1 for r → ∞. Furthermore, Xr ∞ → X∞ for r → ∞, and the sequence {Xr ∞ }r≥1 is uniformy bounded. Thus there exists r◦ such that (G − Gr◦ )Xr◦ ∞ < 1/2. Since Xr◦ (z) is continuous on |z| ≤ 1, there exists a stable rational matrix ˜ such that Xr − X ˜ ∞ is strictly less than (4 + 4G∞ )−1 . Now function X ◦ note that ˜ ˜ G(z)X(z) = Im + (G(z) − Gr (z)) Xr (z)−G(z) Xr (z) − X(z) , |z| < 1. ◦

◦

◦

˜ ∞ < 3/4. Hence GX ˜ is a staMoreover, (G − Gr◦ )Xr◦ ∞ + G(Xr◦ − X) ble rational matrix function which has a stable rational matrix inverse. This ˜ X) ˜ −1 is a stable rational matrix solution of (1.1).1 implies that X(G In order to prove Proposition 2.1 it will be convenient to prove the following lemma ﬁrst. Lemma 2.2. Let G be a stable rational m×p matrix function, and let R be the rational m × m matrix function given by R(z) = G(z)G∗ (z). Assume TR is ∗ −1 TR HG is contained invertible. Then TG has closed range, the spectrum of HG in the closed interval [0, 1], and ∗ −1 dim Ker TG∗ = dim Ker I − HG (2.5) TR HG < ∞. In particular, TG is semi-Fredholm. Proof. We shall need the identity ∗ TR = TGG∗ = TG TG∗ + HG HG .

(2.6)

This identity can be found, for example, in [11], see formula (4) in Section XXIII.4 of [11]. It was proved there for the case when the entries of TG and HG are square matrices, but the general case can be reduced to the square case by adding zero rows or columns to the entries. Since TR is assumed to be invertible, (2.6) yields ∗ ∗ TG TG∗ = TR − HG HG . (2.7) = TR I − TR−1 HG HG 1

We thank the referee for providing the above argument.

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Recall that HG has ﬁnite rank. Thus the ﬁrst equality in (2.7) shows that TG TG∗ is a ﬁnite rank perturbation of an invertible operator. Hence TG TG∗ is a Fredholm operator of index zero. As is well-known, the latter implies that TG has closed range (cf., Exercise 2 on page 283 of [11]). Next we use the fact that Ker TG is perpendicular to Im TG∗ . This implies that the operator TG is one-to-one on Im TG∗ , and therefore Ker TG∗ = Ker TG TG∗ . Since is dim Ker TG TG∗ is ﬁnite, the same holds true for dim Ker TG∗ . Furthermore, we can use the second identity in (2.7) to show that ∗ dim Ker TG∗ = dim Ker TG TG∗ = dim Ker TR I − TR−1 HG HG ∗ = dim Ker I − TR−1 HG HG ∗ −1 = dim Ker I − HG T R HG . This proves (2.5). ∗ −1 It remains to prove that the spectrum of HG TR HG is contained in the ∗ −1 closed interval [0, 1]. Since HG TR HG is selfadjoint, it sufﬁces to show that ∗ −1 TR HG is at most one. To do this we use the fact the spectral radius of HG that TR is strictly positive, which implies that TR factors as TR = Λ∗ Λ, with Λ being an invertible operator. For instance, for Λ we can take the square root of TR . Multiplying (2.6) from the left by Λ−1 and from the right by Λ−∗ yields the identity ∗ −1 Λ = Λ−∗ TG TG∗ Λ−1 . I − Λ−∗ HG HG

(2.8)

The right hand side of the latter identity is non-negative, and hence the ∗ −1 Λ is a contraction. In particular, its spectrum is in the operator Λ−∗ HG HG ∗ −1 Λ ) ≤ 1. Here rspec (K) stands for closed unit disc, that is, rspec (Λ−∗ HG HG the spectral radius of the operator K. But the spectral radius of a product of two operators is independent of the order of the operators. Thus ∗ −1 ∗ −1 TR HG ) = rspec (HG Λ )(Λ−∗ HG ) rspec (HG ∗ −1 Λ ). = rspec (Λ−∗ HG HG

∗ −1 We conclude rspec (HG TR HG ) ≤ 1, as desired.

(2.9)

Proof of Proposition 2.1. We split the proof into three parts. The equivalence (a) ⇒ (b) is trivial. The ﬁrst part of the proof deals with (b) ⇒ (c). In the second part, assuming (c) holds, we derive (2.4), and in the third part, again assuming (c), we prove the ﬁnal statement of the theorem and (c) ⇒ (a). On the way we give a new proof of (b) ⇒ (a) not using the corona theorem as was done in the paragraph directly after Proposition 2.1. Part 1. Assume that TG is right invertible. Then TG TG∗ is strictly positive. ∗ is non-negative, it follows from (2.6) that TR is strictly positive. As HG HG ∗ −1 TR HG is a ﬁnite rank operator, In particular, TR is invertible. Since HG ∗ −1 ∗ −1 TR HG is one-to-one. The I − HG TR HG is invertible if and only if I − HG ∗ fact that TG is right invertible implies that Ker TG consists of the zero element ∗ −1 TR HG is indeed one-to-one. only, and hence formula (2.5) shows that I − HG −1 ∗ Thus I − HG TR HG invertible, and (c) is proved.

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Part 2. In this part we assume (c) and derive (2.4). Assume that TR is invert∗ −1 TR HG . Hence we ible and that the same holds true for the operator I − HG ∗ can apply Lemma 2.2 to show that TG is a one-to-one operator with closed range. This implies that TG is surjective, and hence TG TG∗ is invertible. But then we can use (2.7) to show that ∗ −1 −1 TR (TG TG∗ )−1 = I − TR−1 HG HG −1 −1 ∗ −1 ∗ TR = I + TR HG (I − HG TR HG )−1 HG ∗ −1 ∗ −1 = TR−1 + TR−1 HG (I − HG TR HG )−1 HG TR .

Thus the inverse of TG TG∗ is given by (2.4). Note that the above also shows (c) ⇒ (b), and thus (b) and (c) are equivalent. Part 3. In this part we assume (c) holds and derive (a). To do this it remains to prove the ﬁnal statement of the theorem. For this purpose we need the following terminology. A vector x in 2+ (Cm ) is said to be a rational vector whenever FCm x is a stable rational m × 1 matrix function. If F is a rational m × p matrix function without poles on the unit circle T, then TF maps rational vectors in 2+ (Cp ) into rational vectors in 2+ (Cm ) and the range of HF consists of rational vectors only. These facts are well-known; for the statement about the range of HF see the remark made at the end of the second paragraph of Sect. 3. We ﬁrst show that (TG TG∗ )−1 maps rational vectors into rational vectors. To do this, let x be a rational vector in 2+ (Cm ). Put ∗ −1 ∗ −1 y = HG (I − HG TR HG )−1 HG TR x.

Thus (TG TG∗ )−1 x = TR−1 (x + y). Since G is a stable rational matrix function and y is in the range of HG , we know (see the previous paragraph) that y is a rational vector. Thus we have to show TR−1 (x + y) is a rational vector. Note that x + y is a rational vector. As R is positive definite on the unit circle, R admits a spectral factorization relative to the unit circle. It follows that TR−1 can be written as TR−1 = T T ∗ where T is a Toeplitz operator deﬁned by a stable rational matrix function (see Theorem 3.2 below for more details). Thus both T and T ∗ are Toeplitz operators deﬁned by a rational matrix function without poles on the unit circle. But such Toeplitz operators map rational vectors into rational vectors (see the previous paragraph). We conclude that TR−1 (x + y) is a rational vector, and thus (TG TG∗ )−1 x is a rational vector. Now put ˜ and X ˜ = TG∗ (TG TG∗ )−1 E. ˜ ˜ = (TG TG∗ )−1 E Ξ From the result of the previous paragraph we know that for each u in Cp the ˜ and recall ˜ is a rational vector in 2+ (Cp ). Note that Xu ˜ = T ∗ Ξu, vector Ξu G that a Toeplitz operator deﬁned by a rational matrix function maps rational ˜ is also a rational vector. This implies vectors into rational vectors. Hence Xu ˜ is a stable rational matrix function. From that X = FCp X ˜ = TG T ∗ (TG T ∗ )−1 E ˜ = E, ˜ TG X G

G

it follows that G(z)X(z) = Im . Thus (a) holds and the ﬁnal statements of the theorem are proved.

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Both Proposition 2.1 and Lemma 2.2 hold in greater generality. For instance, Lemma 2.2 remains true when G is an m × p matrix-valued H ∞ function continuous on the closed unit disk. Also the equivalence of (a), (b) and (c) in Proposition 2.1 as well as formula (2.4) remain true for such a function G, provided one allows in (a) for H ∞ solutions.

3. Preliminaries About the Riccati Equation In this section we clarify the role of the Riccati equation (1.8), and present some auxiliary state space formulas. Throughout this and the following sections we assume that G is given by the stable state space representation (1.5). With this representation we associate the operators ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ n 2 m ⎥ Wobs = ⎢ (3.1) ⎢ CA2 ⎥ : C → + (C ) ⎣ ⎦ .. .

Wcon = B AB A2 B A3 B · · · : 2+ (Cp ) → Cn . (3.2) The fact that the matrix A is stable implies that these operators are welldeﬁned and bounded. We call Wobs the observability operator and Wcon the controllability operator corresponding to the state space representation (1.5). Since for j = 1, 2, . . . the j-th Taylor coefﬁcient of G at zero is given by CAj−1 B it follows from (3.1) and (3.2) that ⎤ ⎡ G1 G2 G3 · · · ⎢G G G ···⎥ ⎥ ⎢ 2 3 4 ⎥ HG = ⎢ (3.3) ⎢ G3 G4 G5 · · · ⎥ = Wobs Wcon . ⎦ ⎣ .. .. .. . . . . . . From (3.3) it is clear that rank HG is ﬁnite and the range of HG consists of rational vectors. Recall that P is the unique solution of the Stein equation ∞

P − AP A∗ = BB ∗ .

(3.4)

∗ , where Wcon is deﬁned by (3.2). Thus P = ν=0 Aν BB ∗ (A∗ )ν = Wcon Wcon Recall that P is unique because A is stable.

Lemma 3.1. Let G be the m × p rational matrix function given by the stable state space representation (1.5), and let P be the unique solution of the Stein z −1 )∗ . Then R equation (3.4). Put R(z) = G(z)G∗ (z), where G∗ (z) = G(¯ admits the following representation R(z) = zC(In − zA)−1 Γ + R0 + Γ∗ (zIn − A∗ )−1 C ∗ ,

(3.5)

where R0 = DD∗ + CP C ∗ ,

Γ = BD∗ + AP C ∗ .

(3.6)

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Proof. From (1.5) we see that R(z) = G(z)G∗ (z) = G(z)D∗ + G(z)B ∗ (zIn − A∗ )−1 C ∗ .

(3.7)

We ﬁrst prove that G(z)B ∗ (zIn − A∗ )−1 = C(In − zA)−1 P + Γ∗ (zIn − A∗ )−1 .

(3.8)

To do this observe that G(z)B ∗ (zIn − A∗ )−1 = DB ∗ (zIn − A∗ )−1 + zC(In − zA)−1 BB ∗ (zIn − A∗ )−1 . From (3.4) we see that zBB ∗ = P (zIn − A∗ ) + (In − zA)P A∗ , and thus z(In − zA)−1 BB ∗ (zIn − A∗ )−1 = (In − zA)−1 P + P A∗ (zIn − A∗ )−1 . Inserting the latter identity in the formula for G(z)B ∗ (zIn −A∗ )−1 we obtain G(z)B ∗ (zIn − A∗ )−1 = DB ∗ (zIn − A∗ )−1 + C(In − zA)−1 P + CP A∗ (zIn − A∗ )−1 . From the second identity in (3.6) we know that Γ∗ = DB ∗ + CP A∗ . Thus (3.8) holds. Using the representation (1.5) and inserting (3.8) in (3.7) yields G(z)G∗ (z) = G(z)D∗ + C(In − zA)−1 P C ∗ + Γ∗ (zIn − A∗ )−1 C ∗ = G(z)D∗ + CP C ∗ + zC(In − zA)−1 AP C ∗ + Γ∗ (zIn − A∗ )−1 C ∗ = DD∗ + CP C ∗ + zC(In − zA)−1 (BD∗ + AP C ∗ ) + Γ∗ (zIn − A∗ )−1 C ∗ . But DD∗ + CP C ∗ = R0 and BD∗ + AP C ∗ = Γ by (3.6). Thus (3.5) is proved. Following [8] we associate with the representation (3.5) the discrete algebraic Riccati equation Q = A∗ QA + (C − Γ∗ QA)∗ (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA).

(3.9)

Note that this is precisely the Riccati equation appearing Theorem 1.1. Using the symmetric version of Theorem 1.1 in [8] (see Section 14.7 in [3] or Sections 10.2 and 10.2 in [7]) we know that R(z) = G(z)G∗ (z) is positive definite for each z on the unit circle T if and only if the Riccati equation (3.9) has a solution Q satisfying (a) R0 − Γ∗ QΓ is positive definite, (b) Q satisﬁes the Riccati equation (3.9), (c) the matrix A − Γ(R0 − Γ∗ QΓ)−1 (C − Γ∗ QA) is stable. Moreover, this solution is unique and hermitian. In fact, ∗ Q = Wobs TR−1 Wobs .

(3.10)

Here TR is the block Toeplitz operator on 2+ (Cp ) deﬁned by the matrix function R, and Wobs is deﬁned by (3.2). The solution Q satisfying (a), (b), (c) above will be called the stabilizing solution of (3.9), cf., Section 13.5 in [18].

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In sequel, given the stabilizing solution Q of (3.9), we write A0 = A − ΓC0 ,

where

C0 = (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA).

(3.11)

Note that (c) tells us that A0 is stable. When (3.9) has a stabilizing Q, then (and only then) the function R admits a right spectral factorization relative to the unit circle T. Moreover, in that case, a right spectral factorization R(z) = Φ∗ (z)Φ(z) is obtained (see, e.g., Section 14.7 in [3]) by taking Φ(z) = Δ + zΔC0 (In − zA)−1 Γ,

where

Δ = (R0 − Γ∗ QΓ)1/2 .

(3.12)

Note that Δ is invertible, because R0 − Γ∗ QΓ is invertible. The ﬁrst identity in (3.11) then implies (cf., Theorem 2.1 in [2]) that Φ(z)−1 = Δ−1 − zC0 (In − zA0 )−1 ΓΔ−1 .

(3.13)

Since A and A0 are both stable, (3.12) and (3.13) both present stable state space representations, and hence Φ is invertible outer. (We call a square matrix-valued H ∞ function F invertible outer whenever F (z)−1 exists and is again an H ∞ function. Thus a square stable rational matrix function F is invertible outer whenever F (z)−1 exists and is a stable rational matrix function, i.e., F is invertible in the algebra of stable square rational matrix functions.) Given the right spectral factorization R(z) = Φ∗ (z)Φ(z) with Φ given by (3.12), the block Toeplitz operator TR factors as TR = L∗ L, where L = TΦ . Note that both TΦ and TΦ−1 are block lower triangular. We summarize the above results in the following theorem. Theorem 3.2. Let G be given by (1.5) with A stable, and put R(z) = G(z)G∗ (z). Then TR is invertible if and only if the Riccati equation (3.9) has a stabilizing solution Q. In that case, Q is uniquely determined by (3.10) and the inverse of TR is given by TR−1 = TΨ TΨ∗ . Here TΨ is the block lower triangular Toeplitz operator on 2+ (Cm ) defined by the stable rational matrix function Ψ(z) = Im − zC0 (In − zA0 )−1 Γ Δ−1 , where Δ = (R0 − Γ∗ QΓ)1/2 . (3.14) The following result is an addition to Lemma 2.2. Lemma 3.3. Let G be given by (1.5) with A stable, and let P be the unique solution of the Stein equation (3.4). Put R(z) = G(z)G∗ (z), and assume that TR is invertible, or equivalently, assume that the Riccati equation (3.9) has a stabilizing solution Q. Then the n × n matrix P Q has all its eigenvalues in the closed interval [0, 1], and dim Ker TG∗ = dim Ker (In − P Q).

(3.15)

∗ Proof. Recall that HG = Wobs Wcon and Q = Wobs TR−1 Wobs ; see (3.3) and (3.10). Using these identities we see that ∗ −1 ∗ ∗ ∗ HG TR HG = Wcon Wobs TR−1 Wobs Wcon = Wcon QWcon .

(3.16)

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∗ and the identity (2.5). It follows that Next we use P = Wcon Wcon ∗ −1 ∗ dim Ker TG∗ = dim Ker (I − HG TR HG ) = dim Ker (In − Wcon QWcon ) ∗ = dim Ker (In − Wcon Wcon Q) = dim Ker (In − P Q). ∗ −1 This proves (3.15). By Lemma 2.2 the spectral radius of I − HG TR HG is at most one. Hence (3.16) yields ∗ 1 ≥ rspec (I − Wcon QWcon ) = rspec (In − P Q).

Finally, note that the non-zero eigenvalues of P Q are equal to the non-zero eigenvalues of P 1/2 QP 1/2 . But the latter matrix is nonnegative (because Q is nonnegative by (3.10)), and thus all the eigenvalues of P Q belong to [0, 1], as desired. The following lemma will be useful in the next sections. Lemma 3.4. Let G be given by (1.5) with A stable, and let P be the unique solution of the Stein equation (3.4). Assume that R(z) = G(z)G∗ (z) is positive definite for each z on T, and let Q be the stabilizing solution of the Riccati equation (3.9). Then the following identities hold: G∗ (z)C0 (In − zA0 )−1 = C1 (In − zA0 )−1 + B ∗ (zIn − A∗ )−1 Q,

(3.17)

G(z)C1 (In − zA0 )−1 = C(In − zA)−1 (In − P Q),

(3.18)

R(z)C0 (In − zA0 )−1 = C(In − zA)−1 + Γ∗ (zIn − A∗ )−1 Q.

(3.19)

Here A0 and C0 are given by (3.11), the matrix Γ is defined by the second identity in (3.6), and C1 is given by C1 = D∗ C0 + B ∗ QA0 .

(3.20)

Furthermore, we have BC1 = A(In − P Q) − (In − P Q)A0 , DC1 = C(In − P Q), C1∗ C1 = (Q − QP Q) − A∗0 (Q − QP Q)A0 .

(3.21) (3.22) (3.23)

Proof. We begin the proof with the last three identities and then we proceed with the ﬁrst three. Using the definition of A0 and C0 in (3.11) together with the fact that Q is a hermitian matrix satisfying (3.9) we see that Q = A∗ QA0 + C ∗ C0 .

(3.24)

The latter identity will play an important role in deriving (3.17) and (3.23).

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Proof of (3.21). Using the definition of C1 in (3.20) and the Stein equation (3.4), we have BC1 = BD∗ C0 + BB ∗ QA0 = BD∗ C0 + (P − AP A∗ )QA0 = BD∗ C0 + P QA0 − AP A∗ QA0 = BD∗ C0 + P QA0 − AP (Q − C ∗ C0 ) ∗

[by (3.24)]

∗

= (BD + AP C )C0 + P QA0 − AP Q = ΓC0 + P QA0 − AP Q = A − A0 + P QA0 − AP Q = A(In − P Q) − (In − P Q)A0 .

Proof of (3.22). Notice that DC1 = DD∗ C0 + DB ∗ QA0 = DD∗ C0 + (Γ∗ − CP A∗ )QA0 ∗

∗

[by the second identity in (3.6)]

∗

= DD C0 + Γ QA0 − CP A QA0 = DD∗ C0 + Γ∗ Q(A − ΓC0 ) − CP (Q − C ∗ C0 ) = (DD∗ + CP C ∗ )C0 + Γ∗ QA − Γ∗ QΓC0 − CP Q = (R0 − Γ∗ QΓ)C0 + Γ∗ QA − CP Q [by the ﬁrst identity in (3.6) = C − Γ∗ QA + Γ∗ QA − CP Q [by the second identity in (3.11)] = C(In − P Q).

Proof of (3.23). We use C1∗ = C0∗ D + A∗0 QB and the previous identities for BC1 and DC1 above. This yields C1∗ C1 = C0∗ DC1 + A∗0 QBC1 = C0∗ C(In − P Q) + A∗0 Q (A(In − P Q) − (In − P Q)A0 ) = (C0∗ C + A∗0 QA)(In − P Q) − A∗0 Q(In − P Q)A0 = Q(In − P Q) − A∗0 Q(In − P Q)A0 = Q − QP Q − A∗0 (Q − QP Q)A0 .

[by (3.24)]

Proof of (3.17). Using the representation of G(z) given by (1.5), we obtain G∗ (z)C0 (In − zA0 )−1 = D∗ C0 (In − zA0 )−1 + B ∗ (zIn − A∗ )−1 C ∗ C0 (In − zA0 )−1 . According to (3.24), we have C ∗ C0 = Q − A∗ QA0 . It follows that C ∗ C0 = (zIn − A∗ )QA0 + Q(In − zA0 ). This yields (zIn − A∗ )−1 C ∗ C0 (In − zA0 )−1 = QA0 (In − zA0 )−1 + (zIn − A∗ )−1 Q.

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By using the latter identity in the formula for G∗ (z)C0 (In − zA0 )−1 above we obtain G∗ (z)C0 (In − zA0 )−1 = D∗ C0 (In − zA0 )−1 + B ∗ QA0 (In − zA0 )−1 + B ∗ (zIn − A∗ )−1 Q. As C1 = D∗ C0 + B ∗ QA0 , we have proved (3.17).

Proof of (3.18). Note that G(z)C1 (In − zA0 )−1 = DC1 (In − zA0 )−1 + zC(In − zA)−1 BC1 (In − zA0 )−1 . Using (3.21) we have zBC1 = zA(In − P Q) − z(In − P Q)A0 = (In − P Q)(In − zA0 ) − (In − zA)(In − P Q). This yields z(In − zA)−1 BC1 (In − zA0 )−1 = (In − zA)−1 (In − P Q) − (In − P Q)(In − zA0 )−1 . Thus G(z)C1 (In − zA0 )−1 = DC1 (In − zA0 )−1 + C(In − zA)−1 (In − P Q) − C(In − P Q)(In − zA0 )−1 . Now (3.22) shows that DC1 (In − zA0 )−1 − C(In − P Q)(In − zA0 )−1 = 0. Thus (3.18) holds. Proof of (3.19). Using (3.17) and (3.18) we have R(z)C0 (In − zA0 )−1 = G(z)G∗ (z)C0 (In − zA0 )−1 = G(z)C1 (In − zA0 )−1 + G(z)B ∗ (zIn − A∗ )−1 Q = C(In − zA)−1 (In − P Q) + G(z)B ∗ (zIn − A∗ )−1 Q. (3.25) Inserting the identity for G(z)B ∗ (zIn − A∗ )−1 given by (3.8) into (3.25) we obtain (3.19).

4. Proof of Theorem 1.1 It will be convenient to prove the following result ﬁrst. Theorem 4.1. Let G be given by (1.5) with A stable, and let P be the unique solution of the Stein equation (3.4). Then the operator TG is right invertible if and only if (i) the Riccati equation (3.9) has a stabilizing solution Q and (ii) the matrix In − P Q is non-singular.

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In that case the operator TG TG∗ is invertible and its inverse is given by (TG TG∗ )−1 = TΨ TΨ∗ + K(In − P Q)−1 P K ∗ .

(4.1)

Here TΨ is the block lower triangular Toeplitz operator on 2+ (Cm ) defined by the stable rational matrix function (3.14), and K is the observability operator defined by ⎡ ⎤ C0 ⎢C A ⎥ ⎢ 0 0⎥ n 2 m ⎥ K = W0, obs = ⎢ (4.2) ⎢ C0 A20 ⎥ : C → + (C ). ⎣ ⎦ .. . ˜ is a stable rational m × m matrix function, In that case Ξ = FCm (TG TG∗ )−1 E and Ξ admits the following state space representation: Ξ(z) = D0 + zC0 (In − zA0 )−1 B0 ,

(4.3)

where A0 and C0 are given by (3.11), and B0 = A0 (In − P Q)−1 P C0∗ − Γ(R0 − Γ∗ QΓ)−1 ,

D0 = C0 (In − P Q)−1 P C0∗ + (R0 − Γ∗ QΓ)−1 .

(4.4) (4.5)

Finally, it is noted that D0 is strictly positive. Proof. By Proposition 2.1 and Lemma 2.2 the operator TG is right invertible if and only if TR is invertible and dim Ker TG∗ = 0. But TR being invertible is equivalent to the requirement that the Riccati equation (3.9) has a stabilizing solution Q, and in that case, Lemma 2.2 tells us that dim Ker TG∗ = 0 if and only if In − P Q is non-singular. This proves the necessity and sufﬁciency of the conditions (i) and (ii). Now, assume that these two conditions are fulﬁlled. Then we know that TG TG∗ is invertible and its inverse is given by (2.4). We have to transform (2.4) into (4.1). Note that (3.19) tells us that TR W0, obs = Wobs . It follows that TR−1 HG = TR−1 Wobs Wcon = W0, obs Wcon . ∗ −1 ∗ We already know that HG TR HG = Wcon QWcon ; see (3.16). Since P = ∗ Wcon Wcon , we obtain ∗ −1 ∗ −1 TR−1 HG (I − HG TR HG )−1 HG TR

∗ ∗ = W0, obs Wcon (I − Wcon QWcon )−1 Wcon W0,∗ obs ∗ ∗ = W0, obs (In − Wcon Wcon Q)−1 Wcon Wcon W0,∗ obs

= W0, obs (In − P Q)−1 P W0,∗ obs = K(In − P Q)−1 P K ∗ . This takes care of the second term in the right hand side of (4.1). The ﬁrst term in the right hand side of (4.1) follows by applying Theorem 3.2 to the ﬁrst term in the right hand side of (2.4).

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˜ To do this It remains to derive the formula for Ξ = FCm (TG TG∗ )−1 E. ∗ ˜ ∗ we use (4.1). From (4.2) is clear that K E = C0 . We conclude that ˜ (z) = FCm K(In − P Q)−1 P C0∗ (z) FCm K(In − P Q)−1 P K ∗ E = C0 (In − zA0 )−1 (In − P Q)−1 P C0∗

= C0 (In − P Q)−1 P C0∗ +

+ zC0 (In − zA0 )−1 A0 (In − P Q)−1 P C0∗ . (4.6)

˜ Since T ∗ is block upper triangular with the matrix Now consider FCm TΨ TΨ∗ E. Ψ ∗ −1/2 ˜ = E(R ˜ 0 − Γ∗ QΓ)−1/2 . Finally, (R0 − Γ QΓ) on the main diagonal, TΨ∗ E because TΨ is the block Toeplitz operator deﬁned by Ψ, we obtain ˜ = Ψ(z)(R0 − Γ∗ QΓ)−1/2 (FCm TΨ TΨ∗ E)(z)

= (R0 − Γ∗ QΓ)−1 −zC0 (In − zA0 )−1 Γ(R0 − Γ∗ QΓ)−1 . (4.7)

˜ has the desired By adding (4.6) and (4.7) we see that Ξ = F(TG TG∗ )−1 E state space representation. To complete the proof, it is noted that C0 (In − P Q)−1 P C0∗ = C0 P 1/2 (In − P 1/2 QP 1/2 )−1 P 1/2 C0∗ is positive. Since (R0 − Γ∗ QΓ)−1 is strictly positive, it follows that D0 is strictly positive. Corollary 4.2. Let G be given by (1.5) with A stable. Then MG is right invertible if and only if G can be written as G(z) = DV (z), where D = G(0) has full row rank and V is an invertible outer stable rational matrix function. Moreover, in that case one can take for V the function given by V (z) = Ip + zC1 (In − P Q)−1 (In − zA)−1 B.

(4.8)

Here P and Q are as in Theorem 4.1 and C1 is defined by (3.20). Proof. Assume G(z) = DV (z) for some invertible outer stable rational matrix function V , and let D+ be any right inverse of D. Put U (z) = V (z)−1 D+ . Then G(z)U (z) = DV (z)V (z)−1 D+ = Im for each |z| ≤ 1. Thus MG MU = I, and MG is right invertible. Conversely, assume MG is right invertible. Let P and Q be as in Theorem 4.1. Then In − P Q is invertible. Let V be deﬁned by (4.8). By consulting (3.22), we obtain C = DC1 (In − P Q)−1 . Thus G(z) = D + zC(In − zA)−1 B = D + zDC1 (In − P Q)−1 (In − zA)−1 B = DV (z). It remains to show that V is invertible outer. We have V (z)−1 = Ip − zC1 (In − P Q)−1 (In − zA× )−1 B,

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where A× = A − BC1 (In − P Q)−1 = A − (A(In − P Q) − (In − P Q)A0 ) (In − P Q)−1 −1

= (In − P Q)A0 (In − P Q)

[by (3.21)]

.

Therefore A× is similar to the stable matrix A0 , and hence A× is stable. It follows that both V (z) and V (z)−1 are stable rational matrix functions. Thus V is invertible outer. Proof of Theorem 1.1. In view of Theorem 4.1 we only have to derive the ∗ ∗ −1 (MG MG ) E and to prove the statements in the ﬁnal formula for X = MG paragraph of the theorem. ∗ ∗ −1 ˜ It folFrom (2.3) we see that MG (MG MG ) E = FCp TG∗ (TG TG∗ )−1 E. ∗ ∗ −1 ˜ ˜ ˜ ˜ lows that X = FCp TG Ξ, where Ξ = (TG TG ) E. Put Ξ = FCm Ξ. According to Theorem 4.1, the function Ξ is given by (4.3). Note that ˜ = FCp TG∗ Ξ E. ˜ X = FCp TG∗ Ξ Lets us compute G∗ (z)Ξ(z). Using the state space representation (1.5) for G and the identity (3.17) we have G∗ (z)Ξ(z) = G∗ (z)D0 + zG∗ (z)C0 (In − zA0 )−1 B0 = D∗ D0 + B ∗ (zIn − A∗0 )−1 C ∗ D0

+ zC1 (In − zA0 )−1 B0 + zB ∗ (zIn − A∗ )−1 QB0

= D∗ D0 + B ∗ QB0 + zC1 (In − zA0 )−1 B0 + B ∗ (zIn − A∗0 )−1 C ∗ D0 + B ∗ (zIn − A∗ )−1 A∗ QB0 .

(4.9)

It follows that ˜ = D∗ D0 + B ∗ QB0 + zC1 (In − zA0 )−1 B0 . X(z) = (FCp TG∗ Ξ E)(z)

(4.10)

Recall that the operators D0 and B0 are given by (4.5) and (4.4), respectively. Since C1 = D∗ C0 + B ∗ QA0 , it is clear that D∗ D0 + B ∗ QB0 = D1 , where D1 is deﬁned in Theorem 1.1. The next step is to show that B0 = −(In − P Q)−1 BD1 . To accomplish this we compute BD1 . Let us set Λ = (R0 − Γ∗ QΓ)−1 . Then BD1 = B(D∗ − B ∗ QΓ)Λ + BC1 (In − P Q)−1 P C0∗

= BD∗ Λ − BB ∗ QΓΛ + BC1 (In − P Q)−1 P C0∗

= BD∗ Λ − P QΓΛ + AP A∗ QΓΛ + BC1 (In − P Q)−1 P C0∗

= (In − P Q)ΓΛ + (BD∗ − Γ)Λ + AP A∗ QΓΛ + BC1 (In − P Q)−1 P C0∗ .

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We proceed with (BD∗ − Γ)Λ + AP A∗ QΓΛ + BC1 (In − P Q)−1 P C0∗

= −AP C ∗ Λ + AP A∗ QΓΛ + BC1 (In − P Q)−1 P C0∗ = −AP C0∗ + BC1 (In − P Q)−1 P C0∗ = BC1 (In − P Q)−1 − A P C0∗

= (BC1 − A(In − P Q)) (In − P Q)−1 P C0∗

[by (3.21)]

= (A(In − P Q) − A(In − P Q) − (In − P Q)A0 ) (In − P Q)−1 P C0∗

= −(In − P Q)A0 (In − P Q)−1 P C0∗ . Thus

BD1 = (In − P Q)ΓΛ − (In − P Q)A0 (In − P Q)−1 P C0∗ = −(In − P Q)(−ΓΛ + A0 (In − P Q)−1 P C0∗ ) = −(In − P Q)B0 .

We conclude with the statements in the ﬁnal paragraph of the theorem. First we prove the result about McMillan degrees. To do this assume that the number n in the state space representation (1.5) is chosen as small as possible. In that case, δ(G) = n. Since the matrix A0 in the state space representation of X has the same size as A, we conclude that δ(X) ≤ n. Thus δ(X) ≤ δ(G), as desired. Finally, we prove (1.10). The left hand side of (1.10) can be written as D1∗ N D1 , where ∗

−1

N = Ip + B (In − QP ) From (3.23) we know that

∞

(A∗0 )ν C1∗ C1 Aν0 ν=0

∞

∗ ν ∗ ν ν=0 (A0 ) C1 C1 A0

(In − P Q)−1 B.

= Q − QP Q. It follows that

N = Ip + B ∗ (In − QP )−1 (Q − QP Q)(In − P Q)−1 B = Ip + B ∗ Q(In − P Q)−1 B. Thus D1∗ N D1 is equal to the right side of (1.10).

A direct proof that X is a solution of (1.1). Let X be as in Theorem 1.1. From our construction of X we know that X is a solution of (1.1). This fact can also be checked directly by using (3.18) and (3.22). To see this, recall that X is given by (1.9). By using (3.18) we compute that G(z)X(z) = G(z)D1 − zG(z)C1 (In − zA0 )−1 (In − P Q)−1 BD1 = DD1 + zC(zIn − A)−1 BD1 − zC(zIn − A)−1 BD1 = DD1 .

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It remains to show DD1 = Im . For this purpose we use (3.22). As before put Λ = (R0 − Γ∗ QΓ)−1 . We compute DD1 = (DD∗ − DB ∗ QΓ)Λ + DC1 (I − P Q)−1 P C0∗

= (DD∗ − DB ∗ QΓ)Λ + CP C0∗ = (DD∗ − Γ∗ QΓ + CP A∗ QΓ)Λ + CP (C ∗ − A∗ QΓ)Λ = (DD∗ + CP C ∗ − Γ∗ QΓ)Λ = (R0 − Γ∗ QΓ)Λ = Im .

Hence DD1 = Im , and G(z)X(z) = Im .

5. Two Examples In this section we present two examples. The ﬁrst is a simple example for which all computations can be carried out by hand. For the second example we use MatLab procedures to obtain the desired formulas. Example 1. Consider the 1×2 matrix function G(z) = [1+z 1 = 1. [1 + z − z] 1

−z]. Obviously,

Hence the equation G(z)X(z) = 1 has a stable rational matrix solution. The solution [1 1]T in the above equation is not the least squares solution but it is the optimal corona solution (that is, the solution of minimal H ∞ norm); see [29]. We shall use Theorem 1.1 to compute the least squares solution. A minimal realization of G is given by A = 0,

B = [1

− 1],

C = 1,

D = [1

0].

Solving the symmetric Stein equation (3.4) for this case, we see that P = 2. Since G(z)G∗ (z) = 3 + z + z −1 , we have R0 = 3 and Γ = 1. The Riccati equation (3.9) now becomes Q=

1 , 3−Q

√ √ and the stabilizing solution is given by q = 12 (3− 5). We see that qP = 3− 5 is in the open unit disc, as expected. Inserting this data into the formulas for C0 and A0 in (3.11), we obtain C0 = q and A0 = −q. Computing C1 and D1 from Theorem 1.1, and using the fact that q = 1/(3 − q), we arrive at q 1 1−q 2 − q =q , (5.1) C1 = 0 −1 q 2 q 1 1 1−q 1−q q= − q q+q D1 = (5.2) 0 −1 q q 1 − 2q 1 − 2q It follows that −(1 − P q)−1 BD1 = −(1 − P q)−1 q = −q(1 − 2q)−1 . Using Theorem 1.1, we see that for this case the least squares solution X of (1.1)

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is given by z C1 (1 − P q)−1 BD1 1 + zq 2 q q z 1−q 1−q = − q q 1 − 2q 1 + zq 1 − 2q 2 1 q z 1−q = q− . q 1 − 2q 1 + zq

X(z) = D1 −

In other words, q X(z) = 1 − 2q

1−q (1 + zq)−1 , q

where q =

√ 1 (3 − 5). 2

Let us check directly that X is indeed a solution of (1.1):

q 1 + z −z X(z) = ((1 + z)(1 − q) − zq) (1 + zq)−1 1 − 2q q (1 + z − q − 2qz)(1 + zq)−1 = 1 − 2q q ((1 − 2q)z + (1 − q)) (1 + zq)−1 = 1 − 2q q − q2 (1 + zq)−1 = 1. = qz(1 + zq)−1 + 1 − 2q The last equality holds because (q − q 2 )/(1 − 2q) = 1. To obtain this identity recall that q satisﬁes q = 1/(3 − q) or q 2 − 3q + 1 = 0. Example 2. Consider the 2 × 3 matrix function G(z) given by 1 z + z2 z2 G(z) = . 0 1+z z We have

2

1 z+z z 0 1+z z

2

⎤ 1 −z ⎢ ⎥ ⎣ 0 1 ⎦ = I2 . 0 −1

(5.3)

⎡

(5.4)

Hence the equation G(z)X(z) = I2 has a stable rational matrix solution. Our aim is to compute the least squares solution. To do this we apply the method provided by Theorem 1.1. A minimal realization for G is given by 100 01 010 . (5.5) A= , B= , C = I2 , D = 010 00 011 For this case the solution of the symmetric Stein equation (3.4) is given by 3 1 P = . 1 2

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Furthermore, one computes that

10 3 + 01 1 01 1 Γ = BD∗ + AP C ∗ = + 01 0

R0 = DD∗ + CP C ∗ =

41 , 13 2 13 = . 0 01 1 2

415

=

Since in this case all matrices are real, the unique stabilizing solution Q of the corresponding Riccati equation is real symmetric. Hence (cf., Section 12.7 in [18]) we can assume that Q is of the form q 1 q2 Q= , q2 q3 and one computes that the Riccati equation (3.9) takes the form q1 q2 00 1 0 = + q2 q3 0 q1 −q1 1 − 3q1 − q2 −1 4 − q1 1 −q1 1 − 3q1 − q2 × . 1 − 3q1 − q2 3 − 9q1 − 6q2 − q3 0 1 − 3q1 − q2 To ﬁnd the stabilizing solution by hand is a problem. However we can use the standard MatLab command ’dare’ from the MatLab control toolbox to compute the stabilizing solution Q for the case considered here. This yields: 0.2764 −0.1056 Q= . −0.1056 0.4223 By using this Q in (3.11) we obtain 0.0403 −0.1613 0.2764 A0 = , C0 = 0.1056 −0.4223 −0.1056

−0.1056 . 0.4223

Inserting this data in the formulas of Theorem 1.1 and using MatLab to make the computations we arrive at ⎡ ⎤ ⎡ ⎤ 1 0 0.2764 −0.1056 ⎢ ⎥ 1 C1 = ⎣ −0.0652 0.2610 ⎦ , D1 = ⎣ 0 ⎦, 0.0403 −0.1613 0 −0.6180 0 −4.6180 . −(I2 − P Q)−1 BD1 = 0 −2.6180 This then shows that the least squares solution X(z) is given by ⎡ ⎤ 1 −z 1 ⎢ ⎥ X(z) = ⎣ 0 1+0.3820z ⎦ . 0

(5.6)

−0.618 1+0.3820z

Remark on coprime factorization. In this ﬁnal remark we use Example 2 above to show that the least squares solution (5.6) cannot be derived via

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the double coprime factorization approach in Chapter 4 of [30]. To see this, put 2 1 z + z2 0 z −1 , P (z) = G1 (z) G2 (z) = . G1 (z) = , G2 (z) = z z 0 1+z 1+z Note that P (z) = G1 (z)−1 G2 (z) is a left coprime factorization. Using the matrices in (5.5), we see that G1 (z) = I2 + zC(I2 − zA)−1 B1 , G2 (z) = zC(I2 − zA)−1 B2 , 01 0 where B1 = and B2 = . 01 1 Furthermore, −1

P (z) = zC (I2 − zA1 )

B2 ,

with A1 = A − B1 C =

00 . 0 −1

Now let us apply the discrete time analogue of the results of Section 4.2 in [30] to this realization of P (z). Choose K = [k1 k2 ] such that 0 0 A2 := A1 + B2 K = is stable. (5.7) k1 −1 + k2 Put H1 (z) = I2 − zC(I2 − zA2 )−1 B1 ,

H2 (z) = zK(I2 − zA2 )−1 B1 .

Then, according to the discrete time analogue of Theorem 4 in Section 4.2 of [30] (see also Section 21.5.2 in [31]), we have G1 (z)H1 (z) + G2 (z)H2 (z) = I2 . Hence for any choice of k1 and k2 in (5.7), ⎤ ⎡ ⎤ ⎡ −1 1 0 −1 0 H1 (z) 01 0 ⎥ 1 ⎢ ⎥ ⎢ H(z) := = ⎣ 0 1 ⎦ + z ⎣ 0 −1 ⎦ 01 H2 (z) zk1 1 + z − zk2 0 0 k1 k2 is a stable rational matrix function satisfying G(z)H(z) = I2 . Moreover, δ(H) ≤ δ(G). However, for any choice of k1 and k2 the value of H at zero is different for the value at zero of X given by (5.6). Thus there is no choice of k1 , k2 such that H = X, and hence we cannot obtain the least-squares solution via the above coprime factorization method. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References [1] Arveson, W.: Interpolation problems in nest algebras. J. Funct. Anal. 20, 208– 233 (1975)

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[2] Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A.C.M.: Factorization of Matrix and Operator Functions: The State Space Method, OT 178. Birkh¨ auser Verlag, Basel (2008) [3] Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A.C.M.: A State Space Approach to Canonical Factorization: Convolution Equations and Mathematical Systems, OT 200. Birkh¨ auser Verlag, Basel (2010) [4] Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. Math. 76, 547–559 (1962) [5] Corless, M.J., Frazho, A.E.: Linear Sytems and Control. Marcel Dekker, New York (2003) [6] Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.A.: State-space solutions to standard H 2 and H ∞ control problems. IEEE Trans. Autom. Control 34, 831–847 (1989) [7] Frazho, A.E., Bosri, W.: An Operator Perspective on Signals and Systems, OT 204. Birkh¨ auser Verlag, Basel (2010) [8] Frazho, A.E., Kaashoek, M.A., Ran, A.C.M.: The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle. Integral Equ. Oper. Theory 66, 215–229 (2010) [9] Frazho, A.E., Kaashoek, M.A., Ran, A.C.M.: Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, II: description of all solutions (in preparation, 2011) [10] Fuhrmann, P.: On the corona theorem and its applications to spectral problems in Hilbert space. Trans. Am. Math. Soc. 132, 55–66 (1968) [11] Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. II, OT 63. Birkh¨ auser Verlag, Basel (1993) [12] Gohberg, I., Kaashoek, M.A., Lerer, L.: The resultant for regular matrix polynomials and quasi commutativity. Indiana Univ. Math. J. 57, 2783–2813 (2008) [13] Green, M., Limebeer, D.J.N.: Linear Robust Control. Prentice Hall, Englewood Cliﬀs (1995) [14] Heij, Chr., Ran, A.C.M., Schagen, F.van : Introduction to Mathematical Systems Theory. Linear Systems, Identiﬁcation and Control. Birkh¨ auser Verlag, Basel (2007) [15] Helton, J.W.: Operator Theory, Analytic Functions, Matrices and Electrical Engineering, Regional Conference Series in Mathematics, vol. 68. American Mathematical Society, Providence (1987) [16] Iglesias, P.A., Glover, K.: State-space approach to discrete-time H ∞ control. Int. J. Control 54, 1031–1073 (1991) [17] Iglesias, P.A., Mustafa, D.: State-space solution of the discrete-time minimum entropy control problem via separation. IEEE Trans. Autom. Control 38, 1525– 1530 (1993) [18] Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Clarendon Press, Oxford (1995) [19] Nikol’skii, N.K.: Treatise on the Shift Operator, Grundlehren, vol. 273. Springer, Berlin (1986) [20] Nikol’skii, N.K.: Operators, functions and systems. Math. Surveys Monographs, vol. 92. American Mathematical Society, Providence (2002)

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[21] Peller, V.V.: Hankel Operators and their Applications, Springer Monographs in Mathematics. Springer, Berlin (2003) [22] Schubert, C.F.: The corona theorem as an operator problem. Proc. Am. Math. Soc. 69, 73–76 (1978) [23] Stoorvogel, A.A.: The H∞ control problem: a state space approach. Dissertation, Eindhoven University of Technology, Eindhoven (1990) [24] Sz-Nagy, B., Foias, C.: On contractions similar to isometries and Toeplitz operators. Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 553–564 (1976) [25] Tolokonnikov, V.A.: Estimates in Carleson’s corona theorem. Ideals of the algeokefalvi-Nagy. Zap. Nauˇcn. Sem. Leningrad. Otdel. bra H ∞ , the problem of Sz˝ Mat. Inst. Steklov. (LOMI) 113, 178–198 (Russian) (1981) [26] Treil, S.: Lower bounds in the matrix corona theorem and the codimension one conjecture. GAFA 14, 1118–1133 (2004) [27] Treil, S., Wick, B.D.: The matrix-valued H p corona problem in the disk and polydisk. J. Funct. Anal. 226, 138–172 (2005) [28] Trent, T.T., Zhang, X.: A matricial corona theorem. Proc. Am. Math. Soc. 134, 2549–2558 (2006) [29] Trent, T.T.: An algorithm for the corona solutions on H ∞ (D). Integral Equ. Oper. Theory 59, 421–435 (2007) [30] Vidyasagar, M.: Control System Synthesis: A Factorization Approach. The MIT Press, Cambridge (1985) [31] Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall, Englewood Cliﬀs (1996) A.E. Frazho Department of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907, USA e-mail: [email protected] M.A. Kaashoek (B) and A.C.M. Ran Afdeling Wiskunde Faculteit der Exacte Wetenschappen Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: [email protected]; [email protected] Received: September 7, 2010. Revised: April 6, 2011.

Integr. Equ. Oper. Theory 70 (2011), 419–427 DOI 10.1007/s00020-011-1865-4 Published online February 4, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Upper Bounds for R-Linear Resolvents Olavi Nevanlinna Abstract. We discuss upper bounds for the resolvent of an R-linear operator in Cd . Mathematics Subject Classification (2010). 47A10. Keywords. Real linear operator, resolvent operator, spectrum.

1. Introduction The resolvent λ → (λI − A)−1 of a d × d complex matrix A is a matrix valued function with rational elements. Thus, in particular, all singularities are poles of at most order d, and the following lower and upper bound hold λI − Ad−1 1 ≤ (λI − A)−1 ≤ , dist(λ, σ(A)) dist(λ, σ(A))d

(1.1)

where σ(A) denotes the set of eigenvalues of A. In order to represent conveniently R-linear operators in Cd let us denote by τ the complex conjugation. Given two d × d complex matrices A, B we put A = A + Bτ

(1.2)

so that A maps a vector x ∈ Cd as x → Ax = Ax + Bx.

(1.3)

We deﬁne the spectrum σ(A) of A by σ(A) = {λ ∈ C | Ax = λx for some x = 0}.

(1.4)

The spectrum consists of at most d curves on C, it is compact but it can be empty [1,5]. In what follows the norm is the induced operator norm A = sup Ax. x=1

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The resolvent of A is the real analytic function λ → R(λ, A) = (λ − A)−1 deﬁned outside the spectrum. For |λ| > A we have j ∞ 1 1 A R(λ, A) = λ λ j=0

(1.5)

(1.6)

and thus in particular R(λ, A) ≤

1 . |λ| − A

(1.7)

From this one gets easily the analogue of the leftmost inequality in (1.1). However, unlike in the C-linear case, the spectrum of an R-linear operator can be empty. If we set the distance to an empty set to be inﬁnite, then the lower bound trivially holds but simultaneously, it is clear that the analogue of the upper bound cannot hold. In [4] a lower bound was presented, based on a set δ(A) which is always nonempty, contains the spectrum and equals it in the C-linear case. In this note we consider upper bounds. It is in order to point out that while the real analytic resolvent satisﬁes λR(λ, A) − AR(λ, A) = I

(1.8)

one can associate with A a complex analytic function, the cosolvent C(λ, A) as a solution of the Sylvester equation λC(λ, A) − C(λ, A)A = I

(1.9)

[4]. For |λ| > A we have C(λ, A) =

∞

λ−j−1 Aj

j=0

and all singularities are poles. It is possible to give an analogue of the Cauchy integral in which the kernel is the cosolvent and not the resolvent. Solution methods for R-linear problems were discussed in [1] and eigenvalue problems in [5]. Additional material on real linear operators can be found e.g. in [3,5–7] and the references given there.

2. Additional Preliminaries Let A = A + Bτ be given. If Ax = b, that is, Ax + Bx = b, then also Bx + Ax = b

(2.1)

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Upper Bounds for R-Linear Resolvents

A B

B A

The matrix

ψ(A) =

b x = . x b A B

B A

∈ C2d×2d

421

(2.2)

(2.3)

is said to give the C-linear representation of A, see [4]. There holds A = ψ(A). Lemma 2.1. The equation (2.1) has a unique solution x ∈ Cd for every b ∈ Cd if and only if the matrix ψ(A) is nonsingular. Proof. If ψ(A) is nonsingular, then for every b there exists a unique y1 y= y2 such that

b ψ(A)y = . b

But then also

ψ(A)

y1 − y2 y1 − y2

=0

and so y2 = y1 . Reversely, if ψ(A) is singular, there exists a nontrivial y1 y= y2 such that Ay1 + By2 = 0 and Ay2 + By1 = 0 so that A(y1 + y2 ) + B(y1 + y2 ) = 0, or A(y1 + y2 ) = 0. Thus A has a nontrivial kernel, except if y1 + y2 = 0. However, in that case we can set x = iy1 to obtain Ax = 0. Definition 2.2. We set for given real linear operator A in Cd zI − A −B p(z, w) = det −B wI − A and denote Σ(A) = {(z, w) ∈ C2 | p(z, w) = 0}.

(2.4)

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Thus p is a polynomial of two complex variables of degree 2d of the form p(z, w) = z d wd + r(z, w)

(2.5)

where r is a polynomial of degree at most 2d − 1. Unless otherwise explicitly mentioned, we measure the distances in C2 using the max-norm. Proposition 2.3. The polynomial p does not depend on the coordinate system in Cd . Moreover, σ(A) = {λ ∈ C | (λ, λ) ∈ Σ(A)}.

(2.6)

For |λ| > A we have dist((λ, λ), Σ(A)) ≥ |λ| − A.

(2.7)

Proof. A coordinate change by a similarity matrix S leads to the C-linear representation −1 zI − A −B 0 S S 0 0 S S −1 0 −B wI − A and so its determinant is independent of S. The second claim is obvious. The third one follows from inverting 1 I 0 I 0 ψ(A) − z 0 I 0 w1 I by the Neumann series which converges as ψ preserves the norm of A and the norm of the diagonal matrix equals the largest absolute value of the elements. It is clear that ψ preserves the norm if the operator is applied to vectors of the form (x, x) ∈ C2d where x ∈ Cd . However, observe that the operator norm of ψ(B) is obtained as square root of the largest eigenvalue of ψ(B)∗ ψ(B) and by Lemma (2.1) it sufﬁces to work with vectors of the form (x, x). Example 2.4. If

A=

0 −β

β 0

then −1

(λ − A)

1 = 2 |λ| + |β|2

τ

λ −βτ

(2.8)

βτ λ

while p(z, w) = (zw + |β|2 )2 . Thus σ(A) = ∅ whereas −|β|2 Σ(A) = z, | z = 0 . z Finally, observe that the distance in the max-norm from the origin to Σ(A) is |β|.

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3. First Upper Bound The idea of the upper bounds for the resolvent (λ−A)−1 starts as follows. We consider (ψ(λ − A))−1 and apply the simple identity between the eigenvalues and singular values of matrices. Denoting by sj ≥ sj+1 the singular values of ψ(λ − A) we have 2d−1 1 j=1 sj . (3.1) = s2d | det ψ(λ − A)| We arrive at the following bound. Proposition 3.1. For λ ∈ / σ(A) (λ − A)−1 ≤

λ − A2d−1 . |p(λ, λ)|

(3.2)

Proof. We have for all j sj ≤ ψ(λ − A) = λ − A and likewise 1 = (λ − A)−1 . s2d What remains is to bound |p(λ, λ)| from below. Lemma 3.2. Let · be a norm in C2 and put γ = (1, 1). Then, in that norm, for all (z, w) ∈ C2 2d 1 |p(λ, λ)| ≥ (dist((z, w), Σ(A)))2d . (3.3) γ Proof. Recall that p(z, w) = (zw)d + r(z, w) where r is at most of degree 2d − 1. Following [2] we consider points along the complex line (z, w) + ζ(1, 1) and put q(ζ) = p((z, w) + ζ(1, 1)) so that q is a polynomial satisfying q(ζ) = ζ 2d + lower order terms. Thus, denoting by ζj the zeros of q, |p(z, w)| = |q(0)| =

2d

|ζj |.

j=1

Since (z, w) + ζj (1, 1) ∈ Σ(A) we have |ζj |(1, 1) ≥ dist((z, w), Σ(A))

and the claim follows. We formulate our bound using max-norm in C2 as then γ = 1.

/ σ(A) Theorem 3.3. Let A be a real linear operator in Cd . Then for all λ ∈ (λ − A)−1 ≤

(λ − A)2d−1 . dist((λ, λ), Σ(A))2d

(3.4)

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Example 3.4. In Example 2.4 the distance from the origin to Σ(A) is |β|. Thus we obtain |β|3 1 = A−1 ≤ 4 (|β|) |β| so the upper bound is at origin an equality.

4. Second Upper Bound It is tempting to think that if σ(A) is not empty, one could control the resolvent by the distance from λ to it in C, in place of distance from (λ, λ) to Σ(A) in C2 . Example 4.1. Let A be the direct sum of complex scalar α and the operator in Example 2.4: ⎛ ⎞ α 0 0 ⎜ ⎟ 0 βτ ⎠ . A=⎝0 0 −βτ 0 If |α| > |β|, then the norm of the resolvent bears no relation to the distance to σ(A) near the origin. Thus we may hope for a bound which holds near σ(A). Theorem 4.2. For every real linear operator A in Cd there exists an open U ⊂ C such that σ(A) ⊂ U and for every λ ∈ U \ σ(A) we have 2d−1 (λ − A)−1 ≤ λ − A . dist(λ, σ(A))2d

(4.1)

Proof. If the spectrum is empty U can be taken to be the empty set. Otherwise, for each λ ∈ / σ(A) there exists a closest λ0 ∈ σ(A). Likewise, λ0 is then a closest point to λ in the conjugate of the spectrum. Put E = {(z, z) | z ∈ C} and denote Γ = Σ(A) ∩ E. Then the line

λ0 , λ0 + ζ λ − λ0 , λ − λ0

is orthogonal to the tangent of the curve Γ at (λ0 , λ0 ), or the point (λ0 , λ0 ) is isolated, in which case there is nothing to be shown. The technical part of the proof consists of showing that the line is normal not only to the curve Γ but to the whole tangent plane of Σ(A). Assume this has been done. Since Γ = Σ(A) ∩ E is compact there exists a δ > 0 such that along each such line the point (λ0 , λ0 ) ∈ Γ is a closest point to every point for |ζ| < δ. Thus, the claim follows from the previous theorem.

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Let us prove the technical part. It follows from the definition of p that interchanging the roles of z and w causes its coefﬁcients to be conjugated. Thus p(z, w) = (zw)d + lower order terms =

d

(αjk z j wk + αjk z k wj ).

(4.2)

j,k=0

Let w(z) denote one of the d roots such that p(z, w(z)) = 0. Denoting by ∂j the partial derivative with respect to the j’th variable we have ∂1 p + ∂2 p w = 0. If ∂2 p = 0 we have at (λ0 , λ0 ) ∈ Γ w (λ0 ) = −

∂1 p . ∂2 p

However, for E we have from (4.2) that ∂2 p = ∂1 p.

(4.3)

Thus w (λ0 ) is of modulus 1 and all we need its argument. If λ − λ0 = |λ − λ0 |eiθ and ∂1 p(λ0 , λ0 ) = ρeiϕ then the orthogonality implies that either ϕ = −θ or ϕ = π − θ. In either case w (λ0 ) = −e2iθ . The calculation above assumed that the partial derivatives do not vanish. This however can happen. But w is even then regular and of modulus 1. In fact, let z(w) be such that p(z(w), w) = 0 and thus z(w(z)) = z which implies at (z, w) = (λ0 , λ0 ) z (λ0 )w (λ0 ) = 1. But interchanging their roles we have using (4.2) z (λ0 ) = w (λ0 ) and thus |w (λ0 )| = 1. Suppose we make a small perturbation Δz to λ0 and denote λ1 = λ0 + Δz. Then the ﬁrst component satisﬁes |λ − λ1 | = | |λ − λ0 | − e−iθ Δz| while the second component satisﬁes |λ − w(λ1 )| = | |λ − λ0 | + e−iθ Δz + O((Δz)2 )|. Thus the maximum increases if Δz = 0.

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Remark 4.3. For normal operators A we have (λ − A)−1 =

1 , dist(λ, σ(A))

(4.4)

[4]. Here A is normal if there is a unitary matrix U such that U AU ∗ is a diagonal real linear operator. By Theorem 2.3 in [4] A = A + Bτ is normal if and only if A is normal and B, HB, KB are symmetric, where H and K denote the Hermitian and skew-Hermitian parts of A. Here e.g. B is symmetric, means that it equals its transpose B = B T , while Hermitian satisﬁes B = BT . The equality (4.4) follows immediately by inverting the diagonal elements. In particular, if α, β ∈ C, then 1 −1 . (λ − α − βτ ) = |λ − α| − |β| Observe however, that our method of proof in the bounds above give for circlets (λ − α − βτ )−1 ≤

λ − α − βτ ) |λ − α| + |β| = . 2 dist(λ, σ(α + βτ )) (|λ − α| − |β|)2

Here the second power in the nominator appears inevitably due to the technique of the proof. Since p(λ, λ) = |λ − α|2 − |β|2 there actually is a common factor 1 λ − α − βτ |λ − α| + |β| = = 2 2 |λ − α| − |β| |λ − α| − |β| |p(λ, λ)| but we do not know how this eﬀect could be utilized in the general case.

References [1] Eirola, T., Huhtanen, M., von Pfaler, J.: Solution methods for R-linear problems in Cn . SIAM J. Matrix Anal. Appl. 25, 804–828 (2004) [2] Grayson, M.A.: On the distance to the zero-set of a polynomial. J. Complex. 7, 97 (1991) [3] Huhtanen, M., Nevanlinna, O.: Approximating real linear operators. Stud. Math. 179, 7–25 (2007) [4] Huhtanen, M., Nevanlinna, O.: The real linear resolvent and cosolvent operators. J. Operator Theory 58(2), 229–250 (2007) [5] Huhtanen, M., von Pfaler, J.: The real linear eigenvalue problem in Cn . Linear Algebra Appl. 394, 169–199 (2005) [6] Huhtanen, M., Ruotsalainen, S.: Real linear operator theory and its applications. Integr. Equ. Operator Theory 69, 113–132 (2011) [7] Nevanlinna, O.: Circlets and diagonalization of real linear operators. Helsinki University of Technology, Institute of Mathematics, Research Reports A523 (2008) http://math.tkk.fi/reports/

Vol. 70 (2011)

Upper Bounds for R-Linear Resolvents

Olavi Nevanlinna (B) Department of Mathematics and Systems Analysis Aalto University P.O. Box 11100, Otakaari 1M, Espoo 00076 Aalto, Finland e-mail: [email protected]

Received: September 8, 2010. Revised: December 31, 2010.

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Integr. Equ. Oper. Theory 70 (2011), 429–450 DOI 10.1007/s00020-011-1870-7 Published online February 22, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Composition Operators on Hardy–Orlicz Spaces on the Ball St´ephane Charpentier Abstract. We give embedding theorems for Hardy–Orlicz spaces on the ball and then apply our results to the study of the boundedness and compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy– Orlicz spaces, different from H ∞ , on which every composition operator is bounded. Mathematics Subject Classification (2010). Primary 47B33; Secondary 32C22, 46E15. Keywords. Carleson measure, composition operator, Hardy–Orlicz space.

1. Introduction and Preliminaries 1.1. Introduction The continuity and compactness of composition operators Cφ , deﬁned by Cφ (f ) = f ◦ φ, have been extensively studied on common Banach spaces of analytic functions. On the Hardy spaces H p (BN ) of the unit ball N 2 N BN = z = (z1 , . . . zN ) ∈ C , |zi | < 1 i=1

of CN , and on the Bergman spaces Ap (BN ), 1 ≤ p < ∞, they have been characterized in terms of Carleson measures ([3]). In dimension one, the Littlewood subordination principle is the main tool to show that every composition operator is bounded, hence the corresponding Carleson characterization is always satisﬁed ([15]). Whatever the dimension, it appears that both boundedness and compactness of Cφ on H p (BN ) (resp. Ap (BN )) are independent of p. On the other hand, it is not difﬁcult to check that Cφ is compact on H ∞ if and only if φ∞ < 1, so that there is a “break” between H ∞ and H p (D) (resp. Ap (D)), for the compactness of Cφ .

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This observation motivated P. Lef´evre, D. Li, H. Queﬀ´elec and L. Rodr´ıguez-Piazza to study composition operators on the Hardy–Orlicz spaces H ψ (D) (resp. the Bergman–Orlicz spaces Aψ (D)) of the disk, which both provide an intermediate scale of spaces between H ∞ and H p (D) (resp. Ap (D)) and generalize the latter. Since 2006, they have produced several papers on this subject, e.g. [6–9], in which they have given characterizations of boundedness and compactness of Cφ on these spaces. Moreover, they were interested in the question of whether there are some Hardy–Orlicz spaces on which the compactness of Cφ is equivalent to that on H ∞ . In fact, they answer this question in the negative, by proving ([9, Theorem 4.1]) that, for every Hardy–Orlicz space H ψ (D), one can construct a surjective map φ : D → D which induces a compact composition operator Cφ on H ψ (D). This result extends that obtained by MacCluer and Shapiro for H p (D) ([11, Example 3.12]). The same problem in the Bergman–Orlicz framework has not yet been completely solved. It turns out that the study of composition operators on Hardy– Orlicz spaces may be more important in several variables than in one variable. Indeed, it is well-known that there exist symbols φ such that Cφ is not bounded on the classical Hardy spaces H p (BN ) (resp. Bergman spaces Ap (BN )), while every Cφ is bounded on H ∞ . To be precise, one may ask whether there exist conditions on ψ which ensure that every composition operator is bounded on H ψ (BN ) (resp. Aψ (BN )). In the case of weighted Bergman–Orlicz spaces, we show that the answer to that question is yes, using characterizations of boundedness and compactness of Cφ ([2]). The present paper establishes similar results for Hardy–Orlicz spaces. We organise our paper as follows: after recalling the definition of Hardy– Orlicz spaces and introducing the materials involved, we conclude the ﬁrst section by giving some topological and duality results. Section 2 is devoted to general adapted Carleson embedding theorems, which are the main tools to obtain, in Sect. 3, characterizations of boundedness and compactness of composition operators. New difﬁculties arise when attempting to apply our Carleson embedding theorems, due to the fact that Hardy–Orlicz spaces are not separable in general. We end the section with some consequences of these characterizations, in particular we exhibit a class of Orlicz functions ψ deﬁning Hardy–Orlicz spaces H ψ (BN ), on which every composition operator is bounded. It turns out that this condition is the same than that given for Bergman–Orlicz spaces in [2]. Notation. Throughout this paper, we will denote by dσN the normalized invariant measure on the unit sphere SN = ∂BN . Given two points z, w ∈ CN , the euclidean inner product of z and w will N be denoted by z, w , that is z, w = i=1 zi wi ; the notation |·| will stand for the associated norm, as well as for the modulus of a complex number. 1.2. Orlicz Spaces: Notations 1.2.1. Definitions. A strictly convex function ψ : R+ → R+ is called an Orlicz function if ψ (0) = 0, ψ is continuous at 0 and ψ(x) −−−−→ +∞. If x − x→+∞

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(Ω, P) is a probability space, the Orlicz space Lψ (Ω) associated to the Orlicz function ψ on (Ω, P) is the set of all (equivalence classes of) measurable func tions f on Ω such that there exists some C > 0, such that Ω ψ |fC| dP

is ﬁnite. Lψ (Ω) is a vector space, which can be normed with the so-called Luxemburg norm deﬁned by

|f | f ψ = inf C > 0, ψ dP ≤ 1 . C Ω It is well-known that Lψ (Ω), ·ψ is a Banach space and that, for every

Orlicz function ψ, the inclusions L∞ ⊂ Lψ (Ω) ⊂ Lp (Ω) hold. Moreover, if ψ (x) = xp , for some 1 ≤ p < ∞ and for every x ≥ 0, then Lψ (Ω) coincides with the usual Lebesgue space Lp (Ω). We also introduce the Morse–Transue space M ψ (Ω) as the subspace of ψ L (Ω) generated by L∞ (Ω), and for every Orlicz function ψ, we can consider its complementary function Φ deﬁned by Φ(y) = supx∈R+ {xy − ψ(x)}, which can be shown to be an Orlicz functiontoo. These ∗ two notions allow us to identify (isomorphically) the dual space M ψ (Ω) of M ψ (Ω) and LΦ , whenever both of these two spaces are normed with the Luxemburg norm, with the natural integral duality bracket (see e.g. [13, IV, 4.1, Theorem 7]). 1.2.2. Four Classes of Orlicz Functions. In order to distinguish the Orlicz spaces and to get a significant scale of intermediate spaces between L∞ and Lp (Ω), we need to classify the Orlicz functions with respect to their growth or their regularity. We essentially introduce four classes of Orlicz spaces. • The ﬁrst class is that of Orlicz functions which satisfy the so-called Δ2 -Condition which is a condition of moderate growth: an Orlicz function ψ satisﬁes the Δ2 -Condition if there exist x0 > 0 and a constant K > 1, such that ψ (2x) ≤ Kψ (x) for any x ≥ x0 . For example, x −→ axp (1 + b log (x)), p > 1, a > 0 and b ≥ 0, satisﬁes the Δ2 -Condition. Corollary 5, Chapter II of [13] reads: Proposition 1.1. Let ψ be an Orlicz function satisfying the Δ2 -Condition. Then there are some p > 1 and C > 0 such that ψ (x) ≤ Cxp , for x large enough. Therefore, Lp ⊂ Lψ ⊂ L1 , for some p > 1. •

The ∇2 -class consists of those Orlicz functions ψ such that there exist some β > 1 and some x0 > 0, such that ψ (βx) ≥ 2βψ (x), for x ≥ x0 . This a condition of regularity for ψ which is equivalent to the fact that its complementary function Φ satisﬁes the Δ2 -condition. For instance, if both Lψ (Ω) and LΦ (Ω) are normed with the Luxemburg norm, then Lψ (Ω) is isomorphic to the dual of LΦ (Ω), as soon as ψ satisﬁes the ∇2 -Condition. Moreover, we have the following interpolation theorem, which is not general, but which will be sufﬁcient for our purpose. It is nothing but [8, Proposition 3.5]:

Proposition 1.2. Let ψ be an Orlicz function which satisfies the ∇2 -Condition. Then every linear, or sub-linear, operator which is of weak-type (1, 1) and strong type (∞, ∞) is bounded from Lψ into Lψ .

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The two following conditions are also regularity conditions which are satisﬁed by most of the Orlicz functions that we are interested in: ψ satisﬁes the ∇0 -Condition if there exist some x0 > 0 and some constant C ≥ 1, such that for every x0 ≤ x ≤ y we have ψ (2Cy) ψ (2x) ≤ . ψ (x) ψ (y) Proposition 4.6 of [8] ensures that this latter condition is still equivalent to the condition that, for every (or equivalently one) β > 1, there exists a constant Cβ ≥ 1 such that ψ (βx) ψ (βCβ y) ≤ ψ (x) ψ (y)

•

for every x0 ≤ x ≤ y. We shall consider the following subclass: ψ satisﬁes the uniform ∇0 -Condition if it satisﬁes the ∇0 -Condition for a constant Cβ ≥ 1 independent of β > 1. One deﬁnes a last class of Orlicz functions which grow fast: we say that ψ belongs to the Δ2 -class if it satisﬁes one of the following equivalent conditions: 2 i. there exist C > 0 and x0 > 0, such that ψ (x) ≤ ψ (Cx) for every x ≥ x0 ; b ii. there exist b > 1, C > 0 and x0 > 0 such that ψ (x) ≤ ψ (Cx), for every x ≥ x0 ; b iii. for every b > 1, there exist Cb > 0 and x0,b > 0 such that ψ (x) ≤ ψ (Cb x), for every x ≥ x0,b . (See [13, Chapter II, Paragraph 2.5, pages 40 and further] or [5, Chapter I, Section 6, Paragraph 5].) Now, [13, Chapter II, Paragraph 2, Proposition 6]) says that an Orlicz function which satisﬁes the Δ2 -Condition needs to have at least an exponential growth:

Proposition 1.3. Let ψ be an Orlicz function which satisfies the Δ2 -Condition. There exist a > 0 and x0 > 0 such that ψ (x) ≥ eax , for every x ≥ x0 . If ψ satisﬁes the Δ2 -Condition, we shall say that Lψ (Ω) is a “small” Orlicz space, i.e. “far” from any Lp (Ω) and “close” to L∞ . Finally, we mention that these conditions are not independent (see [8, Proposition 4.7]): Proposition 1.4. Let ψ be an Orlicz function. 1. If ψ satisfies the uniform ∇0 -Condition, then it satisfies the ∇2 -Condition; 2. If ψ satisfies the Δ2 -Condition, then it satisfies the uniform ∇0 -Condition. For any 1 0 and b ≥ 1, x −→

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b

eax − 1 belongs to the Δ2 -Class (and then to the uniform ∇0 -Class), yet the Orlicz functions which can be written not to the Δ2 -one. In addition, b x → exp a (ln (x + 1)) − 1 for a > 0 and b ≥ 1, satisfy the ∇2 and ∇0 -Conditions, but do not belong to the Δ2 -Class. For a complete study of Orlicz spaces, we refer to [5] and [13]. We can also ﬁnd precise and useful information in [8], such as other classes of Orlicz functions and their links with each other. 1.3. Hardy–Orlicz Spaces on BN The definition of Hardy–Orlicz spaces on the ball is quite similar to that of classical Hardy spaces. With the notations above, (Ω, P) stands for (SN , dσN ). Given an Orlicz function ψ, the Hardy–Orlicz space H ψ (BN ) on BN is the vector space of analytic functions f : BN → C such that sup0
∗

If we denote by f H ψ := f ψ , then H ψ (BN ) endowed with the norm ·H ψ is a Banach space. The proof of this result is the same on BN or on D, and we refer to [8, Proposition 3.1] for details. For simplicity, we will sometimes denote by · ψ the norm on H ψ (BN ), emphasizing that H ψ (BN ) can be seen as a subspace of Lψ (SN ). Similarly, if f ∈ H ψ (BN ), we will write f ∗ (or even just f ) for the function equal to f in BN and equal to the boundary limits of f almost everywhere on SN . In the sequel, we will denote by Pz the invariant Poisson kernel at z ∈ BN , N 2 1 − |z| , z ∈ BN , ζ ∈ SN . Pz (ζ) = P (z, ζ) = 2 |1 − z, ζ | We will also use the notation uζ,r , ζ ∈ SN and 0 < r < 1, for the function 2N 1−r uζ,r (z) = , z ∈ BN . 1 − r z, ζ It is easy to check that uζ,r ∈ H ∞ and that uζ,r ∞ = 1. If z ∈ SN , then N 1−r P (rz, ζ), |uζ,r (z)| = 1+r N so that uζ,r L1 (SN ) = 1−r . 1+r

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As usual, we shall verify that the point evaluation linear functional maps δz at z ∈ BN , deﬁned by δz (f ) = f (z) for f ∈ H ψ (BN ) are bounded. Proposition 1.6. If z ∈ BN then N N 1 + |z| 1 + |z| 1 −1 −1 ψ ≤ δz (H ψ (BN ))∗ ≤ ψ . 4N 1 − |z| 1 − |z| Proof. Since H ψ (BN ) ⊂ H 1 (BN ), for every f ∈ H ψ (BN ), we have

f (z) = Pz (ζ) f (ζ) dσ (ζ). SN

N 1+|z| , we get By Jensen inequality and because Pz ∞ ≤ 1−|z| N f (z) 1 + |z| |f (ζ)| ≤ ψ (ζ) dσ (ζ) ≤ , P ψ z f ψ f ψ 1 − |z| SN

hence |f (z)| ≤ f ψ ψ −1

1 + |z| 1 − |z|

N .

For the lower bound, it is sufﬁcient to compute δz (uζ,r ) and uζ,r ψ , thanks to [8, Lemma 3.9], with r = |z| and ζ ∈ SN such that z, ζ = r. Contrary to what happens for the Hardy spaces, the ball algebra A (BN ) is not always dense in H ψ (BN ), as the following result indicates (from now on, HM ψ (BN ) denotes H ψ (BN ) ∩ M ψ (BN ).) Theorem 1.7 (Chapter IX, Theorem 4 of [13] for Ω = BN ). Let ψ be an Orlicz function. A (BN ) is dense in H ψ (BN ) if and only if H ψ (BN ) is separable, which in turn is equivalent to the fact that ψ satisfies the Δ2 -Condition. However, HM ψ (BN ) is always separable. The non-separability of H ψ (BN ) will cause some problems when we try to deduce results on composition operators from embedding theorems. Yet, we have the following result: Theorem 1.8. Let ψ be an Orlicz function. 1. HM ψ (BN ) is the closure of H ∞ in Lψ (SN ); in particular the set of all polynomials is dense in HM ψ (BN ) for the topology defined by ·ψ . More precisely, for every f ∈ HM ψ (BN ), fr − f ψ −−−→ 0. r→1

2. On the unit ball of H ψ (BN ) (or equivalently on every ball), the induced weak-star topology coincides with that of uniform convergence on compacta of BN . 3. H ψ (BN ) is weak-star closed in Lψ (SN ). ∗∗ 4. H ψ (BN ) can be isometrically identified with HM ψ (BN ) , whenever ψ satisfies the ∇2 -Condition. In particular, if ψ satisfies the ∇2 and Δ2 Conditions, then H ψ (BN ) is reflexive.

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Proof. (1) It sufﬁces to show that fr − f ψ −−−→ 0 for every f ∈ HM ψ r→1

(BN ); this can be done as in D, see [8, Proposition 3.4]. (2) This is a classical use of Poisson kernel and Proposition 1.6. (3) It is sufﬁcient to show that the balls of H ψ (BN ) are weak-star closed, i.e. compact. Using point (2) and the fact that the topology of uniform convergence on compact sets is metrizable, we are led to show that, if (fn )n is a sequence in the unit ball of H ψ (BN ), then we can extract from it a subsequence which converges uniformly on compacta to some f in the unit ball of H ψ (BN ). Now this follows from Proposition 1.6, which ensures that (fn )n is a normal family, and from Fatou’s lemma. ∗ ∗ (4) Since M ψ (SN ) = Lφ (SN ) and since Lφ (SN ) = Lψ (SN ), because the ∇2 -Condition ([13, IV, 4.1, Corollary 9, p. 111]), then ∗∗ ψ satisﬁes = Lψ (SN ). Now, using [4, V, 5.6, Corollary 6], we get M ψ (SN ) ∗∗ is the weak-star closure of HM ψ (BN ) in Lψ (BN ). that HM ψ (BN ) ψ Since H (BN ) is itself weak-star closed, according to 3), the proof will be ﬁnished if we show that every function f in H ψ (BN ) belongs to the weak-star closure of HM ψ (BN ) in Lψ (SN ). Now, let f ∈ H ψ (BN ); thanks to (1), fr −−−→ f uniformly on every compact set, and therer→1

fore (point 2) the convergence occurs for the weak-star topology, ∗∗since and fr ψ ≤ f ψ for every 0 < r < 1. Therefore f ∈ HM ψ (BN ) the proof is complete.

2. Embedding Theorems for Hardy–Orlicz Spaces As usual, our embedding theorems will involve some geometric conditions. The non-isotropic distance on the sphere SN , which we denote by d, is given by d (ζ, ξ) = |1 − ζ, ξ |, for (ζ, ξ) ∈ SN . It can be extended to BN , where it still satisﬁes the triangle inequality (e.g. [14, Paragraph 5.1]). For ζ ∈ BN and h ∈ ]0, 1], we deﬁne the non-isotropic “ball” of BN by

2 S (ζ, h) = z ∈ BN , d (ζ, z) < h . and its analogue in BN by

2 S (ζ, h) = z ∈ BN , d (ζ, z) < h .

Let us also denote by Q = S (ζ, h) ∩ SN the “true” balls in SN . Next, for ζ ∈ SN and h ∈ ]0, 1], we deﬁne the so-called Carleson windows z W (ζ, h) = z ∈ BN , 1 − |z| < h, ∈ Q (ζ, h) |z| and W (ζ, h) =

z z ∈ BN , 1 − |z| < h, ∈ Q (ζ, h) , |z|

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anyi approach region D (η) its analogue in BN . Finally, we introduce the Kor´ for η ∈ SN by

2 2 D (η) = z ∈ BN , d (z, η) < 1 − |z| . Given f continuous on BN , the maximal function Nf of f associated to the Kor´ anyi approach regions is deﬁned as follows: Nf (ξ) = sup {|f (w)|}, w∈D(ξ)

for ξ ∈ SN . The Hardy–Littlewood maximal function Mf of f ∈ L1 (SN ) is given by

1 |f | dσN , Mf (ξ) = sup δ>0 σ (Q (ξ, δ)) Q(ξ,δ)

for ξ ∈ SN . It is well-known that the sub-linear maximal operator M : f −→ Mf is of weak type (1, 1) on SN (see [16], Lemma 4.8) and that it maps L∞ (SN ) into itself boundedly. Therefore, Proposition 1.2 yields the following result concerning Hardy–Littlewood maximal function on Hardy–Orlicz spaces, under the ∇2 -condition: Theorem 2.1. Let ψ be an Orlicz function satisfying the ∇2 -condition. Then, the Hardy–Littlewood maximal operator M maps Lψ (SN ) into itself boundedly. More precisely, there exists a constant Cψ > 0 such that Mf ψ ≤ Cψ f ψ , for every f ∈ Lψ (SN ). Moreover, Nf is dominated by Mf in the following “Hardy-Orlicz sense”: Theorem 2.2. Let ψ be an Orlicz function. Then there exists a constant C > 0 such that Nf (ξ) ≤ CMf ∗ (ξ) ψ

for every f ∈ H (BN ) and for every ξ ∈ SN , where f ∗ ∈ Lψ (SN ) is the radial limit of f . Proof. First, f ∗ ∈ L1 (SN ) since f ∈ H ψ (BN ), so f ∗ dσN is a ﬁnite com4.10] to get NP [f ∗ ] ≤ plex Borel measure on SN and we can use [16, Theorem CMf ∗ for some constant C > 0, where P [f ∗ ] (z) = SN P (z, ζ) f ∗ (ζ) dσN (ζ) for z ∈ BN . We ﬁnish the proof by noticing that P [f ∗ ] = f , because f ∈ H 1 (BN ). We get a result similar to Theorem 2.1 for the maximal operator N associated to Kor´anyi approach regions: Corollary 2.3. Let ψ be an Orlicz function satisfying the ∇2 -condition. Then, the maximal operator N associated to Kor´ anyi approach regions maps H ψ ψ (BN ) into L (SN ) boundedly. More precisely, there exists a constant Cψ > 0 such that Nf Lψ (SN ) ≤ Cψ f H ψ (BN ) , for every f ∈ H ψ (BN ).

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Proof. This is an immediate consequence of Theorem 2.1 and Theorem 2.2, using the fact that f ∗ Lψ (SN ) = f H ψ (BN ) for every f ∈ H ψ (BN ). For any positive ﬁnite Borel measure μ on BN , we introduce the two following functions ρμ and Kμ : ρμ (h) = sup μ (W (ξ, h)) ξ∈SN

μ (W (ξ, t)) , for h ∈ (0, 1). tN 0
and Kμ (h) = sup

We recall that μ is a Carleson measure if Kμ (h) is ﬁnite for some h ∈ (0, 1). In the sequel, we will assume that the restrictions to SN of all the measures μ on BN that we will consider are absolutely continuous with respect to the invariant measure dσN on the sphere. Theorem 2.4. There exist two constants C˜ > 0 and C > 1 such that, for every f ∈ H 1 (BN ) and every positive finite Borel measure μ on BN , we have ˜ μ (Ch) σN ({Nf > t}) μ z ∈ BN , |z| > 1 − h and |f (z)| > t ≤ CK for every h ∈ (0, 1/C) and every t > 0. For the proof of this theorem, we will need a covering lemma: Lemma 2.5. Let g be a continuous function on BN , a > 0 and h ∈ (0, 1). Then, either |g (w)| < a in BN \ (1 − h) BN or there exist w1 , w2 , . . . in BN \ (1 − h) BN such that: 1. |g (wi )| ≥ a for every i ≥ 1; 2. the following inclusion holds: 2 w ∈ BN , |g (w)| ≥ a ∩ (BN \ (1 − h) BN ) ⊂ S wi , 4 1 − |wi | ;

2

3. the sets Q wi , 1 − |wi |

i≥1

, i ≥ 1, are pairwise disjoints.

Proof. This lemma is stated in a slightly different form and for h = 1/2 in [12]. There is no difﬁculty to extend it to the above form and for any h ∈ (0, 1). Proof of Theorem 2.4. We ﬁx t > 0. We may suppose that there exists a ∈ BN \ (1 − h) BN such that |f (a)| > t, with |a| > 1 − h. Then the previous lemma ensures that there exists (wi )i≥1 ⊂ BN \ (1 − h) BN such that 2 μ z ∈ BN , |z| > 1−h and |f (z)| > t ≤ μ S wi , 4 1 − |wi | . i≥1

(2.1) Moreover, because of the definition of the Kor´ anyi approach region D (η), η ∈ , and that of Q, we may verify that N S N f (η) ≥ t whenever η ∈ Q 2

2

wi , 1 − |wi | . Therefore, since the sets Q wi , 1 − |wi | are pairwise disjoints, we have 2 ≤ σN ({Nf ≥ t}). σN Q wi , 1 − |wi | (2.2) i≥1

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2 Now, the triangle inequality ensures that if we set r = 9 1 − |wi | , then wi wi 2 ,r ≤μ W , C0 r μ S wi , 4 1 − |wi | ≤μ S |wi | |wi | for some C0 > 1. By definition of Kμ and as r ≤ 2h, we can ﬁnd some absolute constant C > 1 (in fact, we can take C = 2C0 ) such that wi μ W |w , C0 r | i 2 μ S wi , 4 1 − |wi | ≤ C0N rN Kμ (Ch). ≤ C0N rN C0N rN (2.3) Now, by using [16, Lemma 4.6] and by homogeneity of the invariant measure on SN , we get wi 2 N . (2.4) r σN Q ,r σN Q wi , 1 − |wi | |wi | Hence, inequalities (2.1), (2.2), (2.3) and (2.4) give the existence of two constants C > 1 and C˜ > 0 such that ˜ μ (Ch) σN ({Nf ≥ t}). μ z ∈ BN , |z| > 1 − h and |f (z)| > t ≤ CK The next technical lemma is a consequence of Theorem 2.4. Lemma 2.6. Let μ be a finite positive Borel measure on BN and let ψ1 and ψ2 be two Orlicz functions. Let C ≥ 1 be the constant appearing in Theorem 2.4. Assume that there exist A > 0, η > 0 and hA ∈ (0, 1/C) such that Kμ (h) ≤ η

ψ2

1/hN Aψ1−1 (1/hN )

for every h ∈ (0, hA ). Then, there exist three constants B > 0, xA > 0 and C1 > 0 (where the latter does not depend on A, η and hA ) such that, for every f ∈ H ψ1 (BN ) with f ψ1 ≤ 1 and every Borel subset E of BN ,

|f | ψ2 dμ ≤ μ (E) ψ2 (xA ) + C1 η ψ1 (Nf ) dσN . B E

SN

The proof of this lemma follows very closely that of [8, Lemma 4.14], so we will only give a scheme of it, without going into details of computations. Proof. For f ∈ H ψ1 (BN ), f ψ1 ≤ 1, using integration by parts, we have

∞

ψ2 (|f |) dμ = E

ψ2 (t) μ ({|f | > t} ∩ E) dt. 0

We shall pay attention to μ ({|f | > t}). Proposition 1.6 ensures 1/N 1 |z| > 1 − , ψ1 2Nt+1

(2.5)

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whenever |f (z)| > t, since f ψ1 ≤ 1; hence Theorem 2.4 yields ⎛ 1/N ⎞ 1 ⎠ σN ({Nf > t}). μ ({|f | > t}) = Kμ ⎝C ψ1 2Nt+1

439

(2.6)

Now, we consider A, η, hA and E as in the statement of the lemma, and we set N C A −1 ψ ; xA := (C + 1) C N −1 1 hA the assumption ensures that if s ≥ xA , then ⎛ ⎛ ⎞1/N ⎞ (C+1)C N −1 ψ s ˜ A 1 ηC 1 ⎜ ⎟ ⎠ C+1 Kμ ⎝C ⎝ . ⎠≤ N (C+1)C N −1 C ψ2 C s ψ1 s A Therefore, applying (2.5) to with t=

A|f | , 2N +1 (C+1)C N −1

(2.7)

we deduce from inequalities (2.6)

2N +1 (C + 1) C N −1 s A

and (2.7) that

xA A ψ2 |f | dμ ≤ ψ2 (s) μ (E) ds 2N +1 (C + 1) C N −1 0 E N −1 ∞ (C+1)C

ψ1 s A 2N +1 (C + 1) C N −1 η C˜ C+1 s ds. ψ2 (s) + N Nf > σN C A ψ2 C s

xA

(2.8) We then conclude the proof, using convexity of ψ, as in the proof of [8, Lemma 4.14]. 2.1. The Canonical Embedding H ψ1 (BN ) → Lψ2 (μ). The main theorem of this section is the following: Theorem 2.7. Let μ be a finite positive Borel measure on BN and let ψ1 and ψ2 be two Orlicz functions; we suppose that ψ1 satisfies the ∇2 -condition. Then: 1. If inclusion H ψ1 (BN ) ⊂ Lψ2 (μ) holds and is continuous, then there exists some A > 0 such that 1 . ρμ (h) = Oh→0 (2.9) ψ2 Aψ1−1 (1/hN )

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2. If there exists some A > 0 such that Kμ (h) = Oh→0

1/hN −1 ψ2 Aψ1 (1/hN )

(2.10)

then inclusion H ψ1 (BN ) ⊂ Lψ2 (μ) holds and is continuous. 3. If in addition ψ1 = ψ2 = ψ satisfies the uniform ∇0 -Condition, then Conditions (2.9) and (2.10) are equivalent. Proof. Let us denote by jμ the embedding H ψ1 (BN ) → Lψ2 (μ) and Cμ its norm (possibly inﬁnite). Note that Cμ is ﬁnite as soon as jμ is well-deﬁned, by the closed graph theorem. (1) We assume that jμ is well-deﬁned. For a ∈ SN and h ∈ (0, 1), we deﬁne 1 ua,1−h , fa,h = ψ −1 hN 2N h . As we saw in the previous secwhere ua,1−h (z) = 1−(1−h)z,a

tion, f lies in the unit ball of H ψ1 (BN ), so that jμ (fa,h )Lψ2 (μ) = fa,h Lψ2 (μ) should not be larger than Cμ . It follows that BN ψ2 (|f | /Cμ ) dμ ≤ 1. Now, we may easily check that |1 − (1 − h) z, a | ≤ 2h whenever z ∈ S (a, h), hence |ua,1−h (z)| ≥ 41N and |f (z)| ≥ 41N ψ1−1 h1N for any z ∈ S (a, h). Consequently, integrating on S (a, h), we get

1 −1 1 1 −1 1 1≥ ψ2 ψ ψ dμ ≥ ψ2 μ (S (a, h)), 4N 1 hN 4N 1 hN S (a,h)

which yields Condition (2.9) and the ﬁrst part of the theorem. (2) This part will use Lemma 2.6. Since ψ1 satisﬁes ∇2 -Condition, Corollary 2.3 ensures that there exists some constant CM ≥ 1 such that, for any f ∈ Lψ1 (SN ), Nf Lψ1 (SN ) ≤ CM f H ψ1 (BN ) . We ﬁx f in the unit ball of H ψ1 (BN ) and we intend to show that f Lψ2 (μ) ≤ C0 for some C0 > 0 independent of f . We also introduce a constant C˜ ≥ 1 whose value will be ﬁxed later. Now, as Condition (2.9) is assumed to be satisﬁed, we shall apply Lemma 2.6 to f /CM , with E = BN , η and hA , and we get the existence of B > 0, xA > 0 and C1 > 0, all independent of f such that

|f | |f | 1 dμ ≤ ψ2 ψ2 dμ BCM C˜ BCM C˜ BN

BN

⎞ ⎛

1 1 ⎝ μ BN ψ2 (xA ) + C1 η ψ1 Nf dσN ⎠ ≤ CM C˜ SN

1 μ BN ψ2 (xA ) + C1 η . ≤ ˜ C

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Of course, C1 may be chosen so that C1 η ≥ 1 and putting C˜ = μ BN ˜ hence the second ψ2 (xA ) + C1 η ≥ 1, we get f Lψ2 (μ) ≤ C0 := BCM C, part of Theorem 2.7. (3) First, it is clear that Condition (2.10) implies Condition (2.9). The converse is based on the following claim: Claim. If Condition (2.9) is satisﬁed, then there exist some A as large as we want and η > 0 such that 1 ρμ (h) ≤ η −1 (2.11) ψ2 Aψ1 (hA /hN ) for some hA , 0 < hA ≤ 1 and for any 0 < h < hA . Proof of the claim. We assume that Condition 1 ρμ (h) ≤ η −1 ˜ ψ2 Aψ (1/hN )

(2.12)

1

holds for some A˜ ≥ 0, h˜A , 0 < h˜A ≤ 1, η > 0 and any 0 < h < h˜A . We ﬁx ≤ 1 such that A > 1 and we search for some constant hA,A ˜ ψ2

1 1 ≤ N −1 N ˜ −1 Aψ1 (1/h ) hA,A ψ2 Aψ1 ˜ /h

(2.13)

for 0 < h < hA,A ˜ . Now it is easy to verify that Inequality (2.13) is equivalent to ψ1−1 1/hN A 1 ≤ N ≤ hN ˜ A ˜ A,A hA,A ψ1−1 ˜ /h by concavity of ψ1−1 . Then the claim follows by choosing hA,A small enough. ˜ We ﬁnish the proof of the third point of the theorem. Let us assume that ψ1 = ψ2 = ψ belongs to the uniform ∇0 -Class and that Condition (2.9) is satisﬁed for some constant A > 0. The previous claim asserts that there exist B ≥ 1 and 0 < K = KB,A ≤ 1 such that 1 ρμ (h) ≤ η N ψ Bψ −1 (K/h) for every 0 < h < K. Therefore, we have ρμ (t) N 0
Kμ (h) = sup

1/tN 0
≤ η sup = η

1 ψ (x) , sup N ψ (Bx) x≥ψ −1 ((K/h)N ) K

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for any 0 < h ≤ K. Let C > 0 be the constant induced by the uniform ∇0 -Condition satisﬁed by ψ and let β be such that B = βC. According to the claim, B can be chosen large enough so that β > 1. Then, using uniform ∇0 -Condition, we get N ψ βψ −1 (K/h) ψ (Bx) ≤ N ψ (x) (K/h) N for any x ≥ ψ −1 (K/h) . Hence, for every 0 < h ≤ K, Kμ (h) ≤ η

1/hN 1/hN ≤ η N ψ (βK N ψ −1 (1/hN )) ψ βψ −1 (K/h)

by concavity of ψ −1 , which is (2.10).

The previous theorem leads us to introduce the ψ-Carleson measures: Definition 2.8. Let μ be a finite positive Borel measure on BN and let ψ be an Orlicz function. We say that μ is a ψ-Carleson measure if there exists A > 0, such that 1 μ (S (ξ, h)) = Oh→0 (2.14) ψ (Aψ −1 (1/hN )) uniformly with respect to ξ ∈ SN . We remark that (2.14) is equivalent to (2.9), then we have the following corollary, by noticing that the uniform ∇0 -Condition implies the ∇2 -Condition (Proposition 1.4): Corollary 2.9. Let μ be a finite positive Borel measure on BN and let ψ be an Orlicz function. Then, if ψ satisfies the uniform ∇0 -Condition, then inclusion H ψ (BN ) ⊂ Lψ (μ) holds (and is continuous) if and only if μ is a ψ-Carleson measure. 2.2. Compactness of the Canonical Embedding H ψ1 (BN ) → Lψ2 (μ). For this purpose, we need a criterion of compactness for embedding operators from H ψ1 (BN ) into Lψ2 (μ). First of all, we shall state the following proposition, whose proof surprisingly relies on the quite deep result by Alexandrov, which asserts that there exist non-constant inner functions on BN ([1]): Proposition 2.10. Let μ be a finite positive Borel measure on BN (whose restriction to SN is absolutely continuous with respect to σN ); let ψ1 and ψ2 be two Orlicz functions. If the canonical embedding H ψ1 (BN ) → Lψ2 (μ) is compact, then μ (SN ) = 0. Proof. We assume that jμ : H ψ1 (BN ) → Lψ2 (μ) is compact. We consider a non-constant inner function f : BN → C, thanks to Alexandrov’s result ([1]); up to composition of f with an automorphism of BN , we may assume that |f (0)| < 1. The sequence (f n )n lies in the unit ball of H ψ1 (BN ) hence, by compactness of jμ and up to the extraction of a subsequence, we may suppose that (jμ (f n ))n = (f n )n is a Cauchy sequence in L1 (μ), for Lψ2 (μ) ⊂ L1 (μ).

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Now, since μ|SN is absolutely continuous with respect to the invariant measure σN , and since f is inner, we must have f n − f m L1 (μ|S ) = 1 − f m−n L1 (μ ) −−−−−→ 0. |SN n,m→∞ N By contradiction, we assume that μ|SN > 0. We can then extract a subsequence (f nk )nk which converges to 1μ|SN -a.e. and by Egoroﬀ’s theorem, the convergence is uniform on a set E ⊂ SN of measure μ|SN positive. Because μ|SN is absolutely continuous with respect to σN , we have σN (E) > 0. Now, by subharmonicity of log |1 − f nk |, it follows that

nk nk log |1 − f nk | dσN . log |1 − f (0)| ≤ log |1 − f | dσN + E

SN \E

The right hand side of this inequality tends to −∞ as k → ∞, because log |1 − f nk | is uniformly convergent to −∞ on E and log |1 − f nk | ≤ log 2 a.e. We get a contradiction, for f nk (0) tends to 0 as k → ∞, since |f (0)| < 1. We give a necessary and sufﬁcient condition for the canonical embedding H ψ1 (BN ) → Lψ2 (μ) to be compact. Proposition 2.11. Let μ be a finite positive measure on BN and let ψ1 and ψ2 be two Orlicz functions. We suppose that the canonical embedding jμ : H ψ1 (BN ) → Lψ2 (μ) holds (and is bounded). 1. The two following assertions are equivalent: (a) The canonical embedding H ψ1 (BN ) → Lψ2 (μ) is compact; (b) Every sequence in the unit ball of H ψ1 (BN ), which is convergent to 0 uniformly on every compact subset of BN , is convergent to 0 in Lψ2 (μ). 2. If H ψ1 (BN ) is continuously embedded in Lψ2 (μ) and if limr→1− Ir =0, where Ir (f ) = f.χBN \rBN , then the canonical embedding H ψ1 (BN ) → Lψ2 (μ) is compact. The proof of this proposition uses classical arguments and is a direct adaptation of that of [8, Proposition 4.9], so we omit it. We shall now state and prove our main theorem about compactness of the canonical embedding H ψ1 (BN ) → Lψ2 (μ). Theorem 2.12. Let μ be a finite positive Borel measure on BN , and let ψ1 and ψ2 be two Orlicz functions. We assume that ψ1 satisfies the ∇2 -condition. Then: 1. If inclusion H ψ1 (BN ) ⊂ Lψ2 (μ) holds and is compact, then for every A > 0 we have 1 . ρμ (h) = oh→0 (2.15) ψ2 Aψ1−1 (1/hN )

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2. If

Kμ (h) = oh→0

IEOT

1/hN −1 ψ2 Aψ1 (1/hN )

(2.16)

for every A > 0, then inclusion H ψ1 (BN ) ⊂ Lψ2 (μ) holds and is compact. 3. If in addition ψ1 = ψ2 = ψ satisfies both ∇0 and ∇2 -Conditions then Conditions (2.15) and (2.16) are equivalent. Proof. (1) By contradiction, we assume that H ψ1 (BN ) → Lψ2 (μ) is compact, while Condition (2.15) is not satisﬁed: there exist ε0 ∈ (0, 1), A > 0, a sequence (hn )n ⊂ (0, 1) decreasing to 0, and a sequence ξn ⊂ SN , such that ε0 , μ (S (ξn , hn )) ≥ ψ2 Aψ1−1 (1/hN ) for any n ∈ N. We consider the test functions introduced in the proof of Theorem 2.7: fn = ψ1−1 1/hN n uξn ,1−hn . Each fn lays in the unit ball of H ψ1 (BN ) and (fn )n tends to 0 uniformly on every compact set of BN so that Proposition 2.11 ensures that (fn )n tends to 0 in Lψ2 (μ). Now, we showed in the proof of Theorem 2.7 that 1 1 |f (z)| ≥ N ψ1−1 , 4 hN for any z ∈ S (ξn , hn ). Hence N

4 A A −1 1 ψ2 |fn | dμ ≥ ψ2 ψ μ (S (ξn , hn )) ε0 ε0 1 hN BN

≥ ψ2

A −1 ψ ε0 1

≥1

1 hN

ε0 −1 ψ2 Aψ1 (1/hN )

using convexity of ψ2 . Therefore, fn Lψ2 (μ) ≥ 4Nε0A for any n, which contradicts the fact that fn Lψ2 (μ) tends to 0. 2) We assume that Condition (2.16) is satisﬁed. Theorem 2.7 ensures that inclusion H ψ1 (BN ) ⊂ Lψ2 (μ) holds (and is bounded). Thanks to Proposition 2.11, it is sufﬁcient to show that, for every ε > 0, Ir < ε when- ever r is closed enough to 1, where Ir : H ψ1 (BN ) → Lψ2 BN \rBN , μ is as in the second point of Proposition 2.11. Let η ∈ (0, 1) and let N +1 N −1 us set A = 2 (C+1)C > 0, where C is the constant involved in ε Theorem 2.4; Condition (2.16) yields the existence of a constant hA ∈ (0, 1/C) such that Kμ (h) ≤ η

ψ2

1/hN , Aψ1−1 (1/hN )

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for any h ≤ hA . Now, if f is in the unit ball of H ψ1 (BN ) and if r ∈ (0, 1) is given, Lemma 2.6, applied to E = BN \rBN and to f , provides a conN +1 N −1 stant B > 0, given by B = 2 (C+1)C = ε, and xA > 0, C1 > 0 A independent of f , such that

|f | |f | ψ2 ψ2 dμ = dμ ε B BN \rBN

BN \rBN

≤ μ BN \rBN ψ2 (xA ) + C1 η

ψ1 (Nf ) dσN .

SN

(2.17)

Furthermore, η > 0 is chosen in order that C1 η SN ψ1 (Nf ) dσN ≤ 12 (which is possible thanks to Corollary 2.3, since ψ1 satisﬁes the ∇2 Condition). For the end of the proof, we need a lemma: Lemma 2.13. Under the assumptions of Theorem 2.12, if Condition (2.16) is satisfied, then μ (SN ) = 0. Proof. For h ∈ (0, 1), let C (h) denote the minimum number of nonisotropic balls Q (ζ, h) needed to cover SN . Thanks to [14, Lemma 5.2.3], there exists C > 0, independent of h, such that C (h) ≤ hCN . Therefore, if Condition (2.16) holds, then ρμ (h) μ BN \ (1 − h) BN ≤ C N ≤ CKμ (h) −−−→ 0. h→0 h

We ﬁnish the proof of Theorem 2.12 by considering some r0 ∈ (0, 1) such that 1 μ BN \rBN ψ2 (xA ) ≤ 2 for any r, r0 < r < 1, since μ (SN ) = 0. We deduce that Ir (f )Lψ2 (μ) ≤ ε in (2.17), for each r > r0 , which ends the proof. 3) This is quite similar to [8, Theorem 4.11, 3)]. The third point of the theorem leads us to deﬁne what one calls vanishing ψ-Carleson measures: Definition 2.14. Let μ be a finite positive Borel measure on BN and let ψ be an Orlicz function. We say that μ is a vanishing ψ-Carleson measure if, for every A > 0, 1 μ (S (ξ, h)) = oh→0 (2.18) ψ (Aψ −1 (1/hN )) uniformly with respect to ξ ∈ SN . We now state the following corollary:

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Corollary 2.15. Let ψ be an Orlicz function and let μ be a finite positive Borel measure on BN . If ψ satisfies the ∇0 ∩ ∇2 -Condition, then the embedding H ψ (BN ) → Lψ (μ) holds and is compact if and only if μ is a vanishing ψ-Carleson measure.

3. Application to Composition Operators on Hardy–Orlicz Spaces In this section, we denote by φ : BN → BN a holomorphic map and by φ∗ the map which is equal to φ on BN and which is deﬁned almost everywhere on SN as the boundary limit of φ; we deﬁne the pull-back measure μφ on BN as the image of the normalized invariant measure σN on SN by φ∗ : −1 μφ (E) = σN φ∗ (E) ∩ SN for every Borel subset E ⊂ BN . We need a classical criterion of compactness for composition operators on Hardy–Orlicz spaces. Its proof is an easy adaptation of that of [3, Proposition 3.11]. Proposition 3.1. Let ψ be an Orlicz function and let φ : BN → BN be holomorphic. Cφ is compact on H ψ (BN ) if and only if, for every sequence (fn )n in the unit ball of H ψ (BN ) converging to 0 uniformly on every compact subset of BN , fn ◦ φ converges to 0 in H ψ (BN ). Due to the non-separability of small Hardy–Orlicz spaces, the following general theorem will not be an easy consequence of embedding theorems, as it is the case for classical Hardy spaces. Theorem 3.2. Let ψ be an Orlicz function which satisfies the ∇2 -Condition and let φ : BN → BN be holomorphic. 1. If ψ satisfies the uniform ∇0 -Condition, then Cφ is bounded from H ψ (BN ) into itself if and only if μφ is a ψ-Carleson measure. 2. If ψ satisfies the ∇0 -Condition, then Cφ is compact from H ψ (BN ) into itself if and only if μφ is a vanishing ψ-Carleson measure. Proof. The difﬁcult part of this theorem is the sufﬁcient part: if μφ is a ψ-Carleson measure (resp. vanishing ψ-Carleson measure) then under the uniform ∇0 -Condition (resp. ∇0 -Condition), Cφ is bounded (resp. compact) on H ψ (BN ). The converse is quite similar to the proof of 1) of Theorem 2.7 (resp. Theorem 2.12), using test functions. We turn to the proof of sufﬁciency in ψ satisﬁes ∇2 -Condition, ∗∗1). Since = H ψ (BN ), and therefore that Theorem 1.8 ensures that HM ψ (BN ) the bi-adjoint of Cφ|HM ψ is equal to Cφ itself. Hence, if Cφ is bounded from HM ψ (BN ) into itself, then it is bounded from H ψ (BN ) into itself (note that the converse is trivially true, since Cφ (f ) ∈ HM ψ (BN ) if f ∈ HM ψ (BN ), whenever Cφ is bounded on H ψ (BN )). So, for ψ satisfying the uniform ∇0 Condition, it is sufﬁcient to show that, if μφ is a ψ-Carleson measure, then Cφ is bounded from HM ψ (BN ) into itself. Thanks to Theorem 2.7, it is still

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sufﬁcient to prove that, for any f ∈ HM ψ (BN ), jμφ (f )ψ = Cφ (f )ψ , where jμφ is the embedding H ψ (BN ) → Lψ (μφ ). Now, it is not difﬁcult to show that jμ (f ) = Cφ (f ) , (3.1) φ

ψ

ψ

for any f in the ball algebra A (BN ), and (3.1) can be extended to HM ψ (BN ), by density of A (BN ) into HM ψ (BN ), which concludes (1). The proof of sufﬁciency for compactness, in (2), follows exactly the same argument as above and uses the fact that, if μφ is a vanishing ψCarleson measure then, under ∇0 -Condition, jμφ is compact (Theorem 2.12) so that Cφ is compact from HM ψ (BN ) into itself, because of Proposition 2.11 and Proposition 3.1. Remark 3.3. If we do not assume that ψ satisfies the uniform ∇0 -Condition (resp. the ∇0 -Condition), then Theorem 2.7 (resp. Theorem 2.12) provides a priori non-equivalent necessary and sufficient conditions to the boundedness (resp. compactness) of Cφ on H ψ (BN ). The following corollary is a particular case of Theorem 3.2: Corollary 3.4. Let ψ be an Orlicz function which satisfies the Δ2 and ∇2 -Conditions (i.e. H ψ (BN ) is reflexive) and let φ : BN → BN be holomorphic. 1. Cφ is bounded from H ψ (BN ) into itself if and only if μφ is a Carleson measure. 2. Cφ is compact from H ψ (BN ) into itself if and only if μφ is a vanishing Carleson measure. Proof. It sufﬁces to observe that 1 ≈ hN ψ (Aψ −1 (1/h)) for every A > 0, whenever ψ is an Orlicz function which satisﬁes the Δ2 -Condition (see Remark 2 (a) following [8, Theorem 4.11]). In particular, with these assumptions, we do not need to assume that ψ satisﬁes the ∇0 Condition. A ﬁrst consequence of the previous results is the following corollary: Corollary 3.5. Let φ : BN → BN be holomorphic and let ψ, ν be two Orlicz functions which both satisfy ∇2 -Condition. We assume that ν also satisfies the Δ2 -Condition. Then 1. If Cφ is bounded on H ν (BN ) (e.g. on any H p (BN )), then it is bounded on H ψ (BN ); 2. If in addition ψ satisfies the ∇0 -Condition and if Cφ is compact on H ψ (BN ), then it is compact on H ν (BN ) (e.g. on any H p (BN )). Proof. The ﬁrst point follows from Remark 3.3 and from the fact that if μ is a Carleson measure, i.e. if Kμ ≤ C for some constant C ≥ 1, then μ satisﬁes Condition (2.10) for some 0 < A ≤ 1.

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For the second point, it sufﬁces to show that Condition (2.16) implies that μ is a vanishing Carleson measure, which is trivial if we apply this condition with A = 1. We now turn to the most surprizing result of this paper: the existence of Hardy–Orlicz spaces on which composition operators are always bounded. We ﬁrst need a result quoted in [10] without proof: Proposition 3.6. If φ : BN → BN is holomorphic, then there exists a constant B > 0 such that μφ (S (ξ, h)) ≤ B.h

(3.2)

uniformly with respect ot ξ ∈ SN and for every 0 < h < 1. Proof. We ﬁx ξ ∈ SN and 0 < h < 1. We denote by χ(φ∗ )−1 (S (ξ,h)) the char−1 acteristic function of (φ∗ ) (S (ξ, h)). The formula of integration by slices ([14, Proposition 1.4.7, (1)]) yields

χ(φ∗ )−1 (S (ξ,h)) (ζ) dσN (ζ) μφ (S (ξ, h)) = SN

χ(φ∗ )−1 (S (ξ,h)) (uζ) dλ (u) dσN (ζ),

= SN T

where λ is the Lebesgue measure on the torus T. Let us observe that χ(φ∗ )−1 (S (ξ,h)) (uζ) = 1 is equivalent to |1 − φ∗ (uζ), ξ | < h.

(3.3)

For every ζ ∈ SN , let ϕζ,ξ : D → D be the function deﬁned by ϕζ,ξ (z) = φ (zζ), ξ for any z ∈ D. ϕζ,ξ is holomorphic and it is not difﬁcult to verify that ϕ∗ζ,ξ (u) = φ∗ (uζ), ξ for λ-almost every u ∈ T, where ϕ∗ζ,ξ is the λ-almost everywhere radial limit of ϕζ,ξ . Inequality (3.3) is then equivalent to ϕ∗ζ,ξ (u) ∈ S1 (1, h), where S1 (1, h) is the one-dimensional disk of radius h, centered at 1, intersected with D. Now, by the Littlewood subordination principle together with the classical (automatic) characterization of boundedness of Cφ on H p (D), there exists a constant B > 0, independent of ζ and ξ, such that −1 (S1 (1, h)) ≤ B.h, λ ϕ∗ζ,ξ which concludes the proof.

The main result of that section is the following Theorem 3.7, which roughly says that, although composition operators are no longer bounded on usual Hardy spaces H p (BN ), p < ∞ when N ≥ 2, they become again bounded on Hardy–Orlicz spaces H ψ (BN ) when the latter space is in some sense close enough to H ∞ (BN ), space for which boundedness trivially holds. Theorem 3.7. Let ψ be an Orlicz function satisfying the Δ2 -Condition. Then every composition operator on H ψ (BN ) is bounded.

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b

In particular, this works when ψ (x) = eax − 1, for a > 0 and b ≥ 1. Proof of Theorem 3.7. Recall that the Δ2 -condition implies both the uniform ∇0 -condition and the ∇2 -condition. Thus we may apply Theorem 3.2. Now, if we compare its result with that of Proposition 3.6, we observe that every composition operator will be bounded on H ψ (BN ) as soon as, for every B > 0, there exist A > 0 and h0 > 0 such that 1 , (3.4) Bh ≤ ψ (Aψ −1 (1/hN )) for any 0 < h ≤ h0 . Now, putting y = ψ −1 (1/h), then x = Ay in (3.4), an easy computation shows that 1 1 N ⇐⇒ ψ (x) ≤ N ψ (x/A). Bh ≤ −1 N ψ (Aψ (1/h )) B Thus, every composition operator is bounded on H ψ (BN ) whenever for every B > 0, there exist A > 0 such that 1 N ψ (x) ≤ N ψ (x/A) B for any x large enough. Finally, using the convexity of ψ, we may easily check that the last condition is satisﬁed as soon as there exists C > 0 such that N ψ (x) ≤ ψ (Cx) for any x large enough. When N > 1, this is nothing but 2 Δ -Condition, while this is a trivial condition when N = 1. Acknowledgements I am grateful to Fr´ed´eric Bayart, Alexandre Borichev and Daniel Li for their helpful comments. I would also like to thank the referee for carefully reading this paper and making many valuable suggestions.

References [1] Alexandrov, A.B.: Existence of inner functions in the unit ball. Math. USSR Sbornik 46, 143–159 (1983) [2] Charpentier S.: Composition operators on weighted Bergman–Orlicz spaces. Complex Anal. Oper. Theory (to appear) [3] Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995) [4] Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory. Pure and Appl Math. Vol. 7, Interscience, New York (1958) [5] Krasnosel´skii, M.A., Rutickii, Ya.B.: Convex Functions and Orlicz Spaces. P. Noordhoﬀ Ltd, Groningen (1961) [6] Lef´evre, P., Li, D., Queﬀ´elec, H., Rodr´ıguez-Piazza, L.: Compact composition operators on H 2 (D) and Hardy-Orlicz spaces. J. Math. Anal. Appl. 354, 360–371 (2009) [7] Lef´evre, P., Li, D., Queﬀ´elec, H., Rodr´ıguez-Piazza, L.: Composition operators on Bergman-Orlicz spaces, hal-00426831 (2009)

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[8] Lef´evre P., Li D., Queﬀ´elec H., Rodr´ı guez-Piazza L.: Composition operators on Hardy–Orlicz spaces, Mem. Am. Math. Soc. 207, No. 974, (2010) [9] Lef´evre, P., Li, D., Queﬀ´elec, H., Rodr´ıguez-Piazza, L.: Some revisited results about composition operators on Hardy spaces, hal-00448623, (2010) [10] MacCluer, B.D., Mercer, P.R.: Composition operators between hardy and weighted Bergman spaces on convex domains in Cn . Proc. Am. Math. Soc. 123(7), 2093–2102 (1995) [11] MacCluer, B.D., Shapiro, J.H.: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Can. J. Math. 38(4), 878– 906 (1986) [12] Power, S.C.: H¨ ormander’s Carleson theorem for the ball. Glasgow Math. J. 26, 13–17 (1985) [13] Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Pure and Applied Mathematics 146. Marcel Dekker Inc, New York (1991) [14] Rudin, W.: Function Theory in the Unit Ball of Cn . Springer, New York (1980) [15] Shapiro, J.H.: Composition Operators and Classical Function Theory, Universitext, Tracts in Mathematics. Springer, New York (1993) [16] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Springer, New York (2005) St´ephane Charpentier D´epartement de Math´ematiques Universit´e Paris-Sud Bˆ atiment 425 91405, Orsay France e-mail: [email protected] Received: October 27, 2010. Revised: February 1, 2011.

Integr. Equ. Oper. Theory 70 (2011), 451–483 DOI 10.1007/s00020-010-1861-0 Published online January 28, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data Alexei Yu. Karlovich, Yuri I. Karlovich and Amarino B. Lebre Abstract. Suppose α is an orientation preserving diffeomorphism (shift) of R+ = (0, ∞) onto itself with the only ﬁxed points 0 and ∞. We establish sufﬁcient conditions for the Fredholmness of the singular integral operator with shift (aI − bWα )P+ + (cI − dWα )P− p

acting on L (R+ ) with 1 < p < ∞, where P± = (I ± S)/2, S is the Cauchy singular integral operator, and Wα f = f ◦ α is the shift operator, under the assumptions that the coefﬁcients a, b, c, d and the derivative α of the shift are bounded and continuous on R+ and may admit discontinuities of slowly oscillating type at 0 and ∞. Mathematics Subject Classification (2010). Primary 45E05; Secondary 47A53, 47B35, 47G10, 47G30. Keywords. Orientation-preserving non-Carleman shift, Cauchy singular integral operator, slowly oscillating function, Mellin pseudodifferential operator, Fredholmness.

1. Introduction Let B(X) denote the Banach algebra of all bounded linear operators acting on a Banach space X, let K(X) be the closed two-sided ideal of all compact operators in B(X), and let B π (X) := B(X)/K(X) be the Calkin algebra of the cosets Aπ := A + K(X) where A ∈ B(X). An operator A ∈ B(X) is said to be Fredholm if its image is closed and the spaces ker A and ker A∗ are ﬁnite-dimensional.

This work is partially supported by “Centro de An´ alise Funcional e Aplica¸c˜ oes” at Instituto Superior T´ecnico (Lisboa, Portugal), which is financed by FCT (Portugal). The second author is also supported by the SEP-CONACYT Project No. 25564 (M´exico) and by PROMEP (M´ exico) via “Proyecto de Redes”.

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Through this paper we will assume that 1 < p < ∞. Let Cb (R+ ) denote the C ∗ -algebra of all bounded continuous functions on R+ := (0, +∞), and let α be an orientation-preserving diffeomorphism of R+ onto itself, which has only two ﬁxed points 0 and ∞. The function α is referred to as an orientation-preserving non-Carleman shift on R+ . If log α ∈ Cb (R+ ), then the shift operator Wα , given by Wα f = f ◦α, is an isomorphism of the Lebesgue space Lp (R+ ) onto itself. As is well known, the Cauchy singular integral operator S given by 1 (Sf )(t) := lim ε→0 πi

R+ \(t−ε,t+ε)

f (τ ) dτ τ −t

(t ∈ R+ )

is bounded on the Lebesgue space Lp (R+ ). Then the operators P± := 12 (I ±S) also are in B(Lp (R+ )). The Fredholm theory of singular integral operators with discontinuous coefﬁcients and shifts on Lebesgue spaces has the long and rich history. We mention the monographs by Gohberg and Krupnik [12], Mikhlin and Pr¨ ossdorf [28], and B¨ ottcher and the second author [5] for the Fredholm theory of singular integral operators with jump discontinuities (and without shifts); the books by Litvinchuk [27], Roch and Silbermann [32], Kravchenko and Litvinchuk [26], Antonevich [1], Karapetiants and Samko [14], and the references therein for the Fredholm theory of singular integral operators with shifts. In all these sources coefﬁcients of singular integral operators and derivatives of shifts are supposed to be either continuous or piecewise continuous. Singular integral operators with coefﬁcients admitting discontinuities of slowly oscillating type were considered in [6,7,30,31]. We also mention the works [2–4], where C ∗ -algebras of singular integral operators with piecewise slowly oscillating coefﬁcients and various classes of shifts with continuous derivatives were considered in the setting of L2 -spaces. Singular integral operators with shifts and slowly oscillating data were studied in [20–22,24] by applying the theory of pseudodifferential operators with so-called compound (double) non-regular symbols. This approach requires at least three times continuous differentiability of slowly oscillating shifts. This paper is devoted to singular integral operators with slowly oscillating coefﬁcients and slowly oscillating non-Carleman shifts preserving the orientation in the Lp (R+ )-setting. Our results generalize and complement those of [23] (see also [26, Chap. 4, Section 2]), where a Fredholm criterion was obtained for a singular integral operator with continuous coefﬁcients and a shift being an orientation-preserving diffeomorphism of [0, 1] onto itself with the only ﬁxed points 0 and 1. In contrast to previous applications of pseudodifferential operators, our approach here is based on a simpler theory of pseudodifferential operators with non-compound non-regular symbols combined with the Allan–Douglas local principle (see [9]) and invertibility results for functional operators. This allows us to reduce the smoothness of shifts to the existence of slowly oscillating ﬁrst derivatives.

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To formulate our main results explicitly, we need several definitions. Following [33], a function f ∈ Cb (R+ ) is called slowly oscillating (at 0 and ∞) if for each (equivalently, for some) λ ∈ (0, 1), lim osc(f, [λr, r]) = 0

r→s

(s ∈ {0, ∞}),

where osc(f, [λr, r]) := sup {|f (t) − f (τ )| : t, τ ∈ [λr, r]} is the oscillation of f on the segment [λr, r] ⊂ R+ . Obviously, the set SO(R+ ) of all slowly oscillating (at 0 and ∞) functions in Cb (R+ ) is a unital commutative C ∗ -algebra. This algebra properly contains C(R+ ), the C ∗ -algebra of all continuous functions on R+ := [0, +∞]. We say that an orientation-preserving non-Carleman shift α is slowly oscillating (at 0 and ∞) if log α ∈ Cb (R+ ) and α ∈ SO(R+ ). We denote by SOS(R+ ) the set of such shifts. As we will show in Sect. 2.2, the shifts α ∈ SOS(R+ ) are represented in the form α(t) = teω(t) for t ∈ R+ where ω ∈ SO(R+ ). Our ﬁrst concern is the invertibility of binomial functional operators with slowly oscillating coefﬁcients and a slowly oscillating shift. The following theorem was obtained in the Lp (0, 1)-setting in [16], the present version is derived from that one in Sect. 3.3. Theorem 1.1. Suppose a, b ∈ SO(R+ ) and α ∈ SOS(R+ ). The functional operator aI − bWα is invertible on the Lebesgue space Lp (R+ ) if and only if either −1/p > 0 (s ∈ {0, ∞}); inf |a(t)| > 0, lim inf |a(t)| − |b(t)| (α (t)) t→s

t∈R+

(1.1) or

−1/p < 0 (s ∈ {0, ∞}). inf |b(t)| > 0, lim sup |a(t)| − |b(t)| (α (t))

t∈R+

t→s

(1.2) If (1.1) holds, then (aI − bWα )−1 =

∞

(a−1 bWα )n a−1 I.

(1.3)

n=0

If (1.2) holds, then (aI − bWα )−1 = −Wα−1

∞

(b−1 aWα−1 )n b−1 I.

(1.4)

n=0

By M (A) denote the maximal ideal space of a unital commutative Banach algebra A. Identifying the points t ∈ R+ with the evaluation functionals t(f ) = f (t) for f ∈ C(R+ ), we get M (C(R+ )) = R+ . Consider the ﬁbers Ms (SO(R+ )) := ξ ∈ M (SO(R+ )) : ξ|C(R+ ) = s

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of the maximal ideal space M (SO(R+ )) over the points s ∈ {0, ∞}. By [20, Proposition 2.1], the set Δ := M0 (SO(R+ )) ∪ M∞ (SO(R+ ))

(1.5)

coincides with closSO∗ R+ \R+ where closSO∗ R+ is the weak-star closure of R+ in the dual space of SO(R+ ). Then M (SO(R+ )) = Δ ∪ R+ . In what follows we write a(ξ) := ξ(a) for every a ∈ SO(R+ ) and every ξ ∈ Δ. We now formulate the main result of the paper. Theorem 1.2. Suppose a, b, c, d ∈ SO(R+ ) and α ∈ SOS(R+ ). The singular integral operator N := (aI − bWα )P+ + (cI − dWα )P−

(1.6)

p

with the shift α is Fredholm on the space L (R+ ) if the following two conditions are fulfilled: (i) the functional operators A+ := aI − bWα and A− := cI − dWα are invertible on the space Lp (R+ ); (ii) for every pair (ξ, x) ∈ Δ × R, 1 + coth[π(x + i/p)] nξ (x) := a(ξ) − b(ξ)eiω(ξ)(x+i/p) 2 1 − coth[π(x + i/p)] = 0, (1.7) + c(ξ) − d(ξ)eiω(ξ)(x+i/p) 2 where ω(t) := log[α(t)/t] ∈ SO(R+ ). The paper is organized as follows. In Sect. 2 we collect properties of slowly oscillating functions and shifts. In particular, we prove that each slowly oscillating shift can be represented in the form α(t) = teω(t) where ω is a real-valued slowly oscillating function. Section 3 is devoted to the proof of Theorem 1.1. In Sect. 4 we collect properties of Mellin convolution operators with piecewise continuous symbols. In particular, we mention the well-known structure of the commutative algebra A generated by the operator S and the identity operator. This algebra contains the operator with ﬁxed singularities f (τ ) 1 dτ (t ∈ R+ ). (Rf )(t) = πi τ +t R+

Section 5 contains necessary facts from the theory of Mellin pseudodifferential operators with slowly oscillating symbols. Section 6 is dedicated to the localization with the aid of the Allan– Douglas local principle. We introduce the algebra Z generated by the compact operators and the operators I, S, and cR, where c are slowly oscillating functions. Further we introduce the algebra Λ of operators commuting with the elements of Z modulo compact operators. The Fredholmness of an operator A ∈ Λ is equivalent to the invertibility of the coset Aπ in the quotient algebra Λπ = Λ/K. The maximal ideal space of its central subalgebra Z π = Z/K is homeomorphic to the set {−∞, +∞} ∪ (Δ × R). Since N ∈ Λ, by the Allan–Douglas local principle, the Fredholmness of N is equivalent to the inπ π π , N π + J+∞ , and N π + Jξ,x vertibility of the local representatives N π + J−∞

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π π π , Λπ /J+∞ , and Λπ /Jξ,x with (ξ, x) ∈ Δ × R, in the local algebras Λπ /J−∞ π π π respectively, where J±∞ and Jξ,x are ideals of Λ . In Sect. 7 we prove that certain functions, playing an important role in the proof of Theorem 1.2, belong to the algebra of symbols of Mellin pseudodifferential operators commuting modulo compact operators. Section 8 is dedicated to the proof of Theorem 1.2. The invertibility of the functional operπ ators aI − bWα and cI − dWα imply the invertibility of the cosets N π + J+∞ π π π π π π and N + J−∞ in the local algebras Λ /J+∞ and Λ /J−∞ , respectively. On the other hand, using the technique of Mellin pseudodifferential operators we π is invertible in the show that nξ (x) = 0 implies that the coset N π + Jξ,x π π local algebra Λ /Jξ,x for (ξ, x) ∈ Δ × R. The proof is based on the important property: the product Wα R of the shift operator Wα and the operator R with ﬁxed singularities at 0 and ∞ is similar to a Mellin pseudodifferential operator. To ﬁnish the proof of Theorem 1.2, it remains to apply the Allan–Douglas local principle. Finally, we note that conditions (i) and (ii) of Theorem 1.2 are also necessary for the Fredholmness of the operator N . This statement is proved in [17].

2. Slowly Oscillating Functions and Shifts 2.1. Fundamental Property of Slowly Oscillating Functions Lemma 2.1. ([20, Proposition 2.2]). Let {ak }∞ k=1 be a countable subset of SO(R+ ) and s ∈ {0, ∞}. For each ξ ∈ Ms (SO(R+ )) there exists a sequence {tn } ⊂ R+ such that tn → s as n → ∞ and ξ(ak ) = lim ak (tn ) n→∞

k ∈ N.

for all

(2.1)

Conversely, if {tn } ⊂ R+ is a sequence such that tn → s as n → ∞, then there exists a functional ξ ∈ Ms (SO(R+ )) such that (2.1) holds. 2.2. Exponential Representation of Slowly Oscillating Shifts Lemma 2.2. An orientation-preserving non-Carleman shift α : R+ → R+ belongs to SOS(R+ ) if and only if α(t) = teω(t) ,

t ∈ R+ ,

(2.2)

1

for some real-valued function ω ∈ SO(R+ ) ∩ C (R+ ) such that the function t → tω (t) also belongs to SO(R+ ) and inf (1 + tω (t)) > 0.

t∈R+

(2.3)

Proof. Necessity. Let α ∈ SOS(R+ ). Then log α ∈ Cb (R+ ) and hence 0 < mα := inf α (t) ≤ sup α (t) =: Mα < ∞, t∈R+

(2.4)

t∈R+

α ∈ SO(R+ ), and α(0) = 0, α(∞) = ∞. As α(0) = 0, we have the representation t 1 1 α(t) = α (x)dx = α (tx)dx, t t 0

0

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which implies due to (2.4) that 0 < mα ≤ inf

t∈R+

α(t) α(t) ≤ sup ≤ Mα < ∞ t t∈R+ t

(2.5)

and α(t) α(τ ) − = t τ

1

(α (tx) − α (τ x)) dx.

(2.6)

0

Using (2.5) and (2.6), we conclude that the function D(t) := α(t)/t belongs to Cb (R+ ). Furthermore, from (2.6) it follows that 1 osc(D, [r/2, r]) ≤

sup 0

1 =

t,τ ∈[r/2,r]

|α (tx) − α (τ x)| dx

osc(α , [rx/2, rx])dx.

(2.7)

0

Since α ∈ SO(R+ ), we conclude that for every ε > 0 there exist positive numbers δ0 < δ∞ such that osc(α , [r/2, r]) < ε for all r ∈ (0, δ0 ) ∪ (δ∞ , ∞). Hence, for r ∈ (0, δ0 ) and all x ∈ (0, 1], osc(α , [rx/2, rx]) < ε,

(2.8)

which implies due to (2.7) that lim osc(D, [r/2, r]) = 0.

r→0

(2.9)

On the other hand, for r > δ∞ and all x ∈ (δ∞ /r, 1], we also have (2.8). Therefore, 1

osc(α , [rx/2, rx])dx ≤ ε(1 − δ∞ /r),

δ∞ /r δ ∞ /r

osc(α , [rx/2, rx])dx ≤ 2 α L∞ (R+ ) δ∞ /r,

0

whence 1

osc(α , [rx/2, rx])dx ≤ ε(1 − δ∞ /r) + 2 α L∞ (R+ ) δ∞ /r.

0

This implies in view of (2.7) that lim osc(D, [r/2, r]) = 0.

r→∞

(2.10)

Thus, by (2.9) and (2.10), the function D(t) = α(t)/t actually belongs to SO(R+ ).

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Since SO(R+ ) is a C ∗ -algebra, we infer from (2.5) that the function ω(t) := log[α(t)/t] also is in SO(R+ ). Thus α(t) = teω(t) where ω ∈ C(R+ ). Finally, since α , eω ∈ SO(R+ ), it follows from the equality α (t) = (1 + tω (t)) eω(t)

(2.11)

and (2.4), (2.5) that the function t → tω (t) also belongs to SO(R+ ) and (2.3) holds. Sufficiency. Let α be an orientation-preserving diffeomorphism of R+ onto itself, with the ﬁxed points 0 and ∞ only; and let α(t) = teω(t) , where the functions ω and t → tω (t) are in SO(R+ ) and (2.3) holds. Since 0 < inf eω(t) ≤ sup eω(t) < ∞, t∈R+

t∈R+

we infer from (2.3) and (2.11) that log α ∈ Cb (R+ ). Furthermore, as the functions eω and t → tω (t) are in SO(R+ ), from (2.11) it follows that α belongs to SO(R+ ). Thus, α ∈ SOS(R+ ). The representation (2.2) will be called the exponential representation of the slowly oscillating shift α and the function ω will be referred to as the exponent function of α. 2.3. Properties of Slowly Oscillating Shifts Lemma 2.3. If c ∈ SO(R+ ) and α ∈ SOS(R+ ), then c◦α ∈ SO(R+ ), c−c◦α ∈ Cb (R+ ), and limt→s (c(t) − c(α(t))) = 0 for s ∈ {0, ∞}. Proof. Obviously, c − c ◦ α ∈ Cb (R+ ). Since α ∈ SOS(R+ ), we deduce from (2.5) that mα t ≤ α(t) ≤ Mα t for every t ∈ R+ , where the positive numbers mα ≤ Mα are deﬁned in (2.4). Hence, for every t ∈ R+ we obtain osc(c ◦ α, [t/2, t]) ≤ osc(c, [(λ/2)r, r]), |c(t) − c(α(t))| ≤ osc(c, [λr, r]), (2.12) where λr = t min{mα , 1} and r = t max{Mα , 1}. Therefore we conclude that λ = min{mα , 1}/ max{Mα , 1} ∈ (0, 1) except for the trivial case α(t) = t. Since c ∈ SO(R+ ), from (2.12) it follows that lim osc(c ◦ α, [t/2, t]) = lim osc(c, [(λ/2)r, r]) = 0,

t→s

r→s

lim |c(t) − c(α(t))| = lim osc(c, [λr, r]) = 0

t→s

r→s

for s ∈ {0, ∞}.

Let β := α−1 be the inverse function to α. Lemma 2.4. If α ∈ SOS(R+ ), then β ∈ SOS(R+ ). Proof. Since log α ∈ Cb (R+ ) and α ∈ SO(R+ ), we infer from the relations β (t) =

1 , α (β(t))

|β (t) − β (τ )| =

|α (β(t)) − α (β(τ ))| α (β(t))α (β(τ ))

(t, τ ∈ R+ )

that log β ∈ Cb (R+ ) and β ∈ SO(R+ ) too. Thus β belongs to SOS(R+ ).

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3. Invertibility of Binomial Functional Operators 3.1. The Case of the Unit Interval Let Cb (I) denote the set of all bounded continuous functions on I := (0, 1). According to [33], a function ϕ ∈ Cb (I) is called slowly oscillating at 0 if lim osc(ϕ, [λr, r]) = 0

r→0

for every (equivalently, some) λ ∈ I. A function ϕ ∈ Cb (I) is called slowly oscillating at 1 if the function y → ϕ(1 − y) slowly oscillates at 0. Let SO(I) denote the set of all functions in Cb (I) that slowly oscillate at 0 and 1. In this subsection we assume that α is an orientation-preserving diffeomorphism of I onto itself that has only two ﬁxed points 0 and 1. According to [16], we say that α is a slowly oscillating shift if log α ∈ Cb (I) and α ∈ SO(I). In the latter case we will write α ∈ SOS(I). The shift operator Wα on the space Lp (I) is deﬁned by Wα f = f ◦ α. It is easy to see that Wα , Wα−1 ∈ B(Lp (I)) whenever α ∈ SOS(I). From Theorem 1.2, Lemma 2.2, and Proposition 5.2 of [16] we extract the following. Theorem 3.1. Suppose a, b ∈ SO(I) and α ∈ SOS(I). The functional operator aI − bWα is invertible on the Lebesgue space Lp (I) if and only if either −1/p > 0 (s ∈ {0, 1}); inf |a(y)| > 0, lim inf |a(y)| − |b(y)| (α (y)) y∈I

y→s

(3.1) or

−1/p inf |b(y)| > 0, lim sup |a(y)| − |b(y)| (α (y)) < 0 (s ∈ {0, 1}).

y∈I

y→s

(3.2) −1

If (3.1) holds, then (aI − bWα ) (aI − bWα )−1 is given by (1.4).

is given by (1.3). If (3.2) is fulfilled, then

3.2. Transplantation from the Half-Line to the Unit Interval Let η : [0, 1] → R+ = [0, +∞] be deﬁned by η(y) = y/(1 − y). Then its inverse is given by η −1 (t) = t/(1 + t). Consider the isometric isomorphism G : Lp (R+ ) → Lp (I) deﬁned by (Gϕ)(y) := (1 − y)−2/p ϕ[η(y)]

(y ∈ I).

Its inverse is given by (G−1 f )(t) := (1 + t)−2/p f [η −1 (t)] Let II be the identity operator on Lp (I) and ϕ(y) 1 dy (SI ϕ)(x) := πi y−x

(t ∈ R+ ).

(x ∈ I).

I

It is well known that the operator SI is bounded on Lp (I).

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Lemma 3.2. Suppose 1 < p < ∞. (a) We have GSG−1 = wp−1 SI wp II , where wp (y) := (1 − y)2/p−1 for y ∈ I. (b) If a ∈ L∞ (R+ ), then G(aI)G−1 = (a ◦ η)II . (c) If α : R+ → R+ is a diffeomorphism such that log α ∈ L∞ (R+ ), then GWα G−1 = cα,p Wα , where 2/p 1−α

(y) −1 for y ∈ I. α

:= η ◦ α ◦ η, cα,p (y) := 1−y The proof of this lemma is straightforward and therefore it is omitted. Lemma 3.3. Let 1 < p < ∞. (a) If a ∈ SO(R+ ), then a ◦ η ∈ SO(I).

∈ SOS(I), cα,p ∈ SO(I), and (b) If α ∈ SOS(R+ ), then α 0 < inf cα,p (y) ≤ sup cα,p (y) < +∞. y∈I

(3.3)

y∈I

Proof. (a) Let ψ(y) := 1 − y for y ∈ I. If y ∈ (0, 1/2], then y ≤ η(y) ≤ 2y and 1/(2y) ≤ η(ψ(y)) ≤ 1/y. Hence for λ ∈ (0, 1) and r ∈ (0, 1/2], [η(λr), η(r)] ⊂ [λr, 2r] = [(2−1 λ)2r, 2r], [(η ◦ ψ)(r), (η ◦ ψ)(λr)] ⊂ [(2r)−1 , (λr)−1 ] = [2−1 λ(λr)−1 , (λr)−1 ]. Therefore, osc(a ◦ η, [λr, r]) = osc(a, [η(λr), η(r)]) ≤ osc(a, [(2−1 λ)2r, 2r]),

(3.4)

osc(a ◦ η ◦ ψ, [λr, r]) = osc(a, [(η ◦ ψ)(r), (η ◦ ψ)(λr)]) ≤ osc(a, [2−1 λ(λr)−1 , (λr)−1 ]).

(3.5)

Since a ∈ SO(R+ ), we get lim osc(a, [(2−1 λ)2r, 2r]) = lim osc(a, [2−1 λ(λr)−1 , (λr)−1 ]) = 0.

r→0

(b)

r→0

These equalities and inequalities (3.4), (3.5) imply that a◦η and a◦η◦ψ slowly oscillate at zero. Thus a ◦ η ∈ SO(I). Part (a) is proved. Let y ∈ I. Since 1 − α

(y) = 1/[1 + (α ◦ η)(y)], we have α

(y) =

(α ◦ η)(y) = (α ◦ η)(y)c2 (y), (1 − y)2 [1 + (α ◦ η)(y)]2

(3.6)

where c(y) :=

1−α

(y) > 0. 1−y

By Lemma 2.2, α(t) = teω(t) with ω ∈ SO(R+ ). Hence, for t ∈ R+ , 1 + t−1 1+t = e−ω(t) . ω(t) 1 + te 1 + t−1 e−ω(t) Since SO(R+ ) is a C ∗ -algebra, e−ω ∈ SO(R+ ) ⊂ Cb (R+ ). Therefore (c ◦ η −1 )(t) =

1 + t−1 = 1. t→+∞ 1 + t−1 e−ω(t) This implies that the function c ◦ η −1 slowly oscillates at +∞. On the other hand, eω ∈ SO(R+ ) ⊂ Cb (R+ ). Then lim

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1+t = 1. 1 + teω(t) In particular, this implies that c ◦ η −1 slowly oscillates at zero. Thus c ◦ η −1 belongs to SO(R+ ). By part (a) of this lemma, c ∈ SO(I). Similarly it can be shown that 1/c ∈ SO(I). Since SO(I) is a C ∗ -algebra, we conclude that cα,p = c2/p ∈ SO(I) and c2 ∈ SO(I). By definition of a slowly oscillating shift, α ∈ SO(R+ ). Then from part (a) we deduce that α ◦ η ∈ SO(I) ⊂ Cb (I). Combining this observation with c2 ∈ SO(I) and (3.6), we conclude that α

∈ SO(I). We have already known that c, 1/c ∈ SO(I) ⊂ Cb (I). Hence (3.3) holds and log(c2 ) ∈ Cb (I). On the other hand, α ∈ SOS(R+ ). Then log α ∈ Cb (R+ ) and thus log(α ◦η) ∈ Cb (I). Taking into account (3.6), we get lim (c ◦ η −1 )(t) = lim

t→0

t→0

log(

α ) = log(α ◦ η) + log(c2 ) ∈ Cb (I),

which concludes the proof of α

∈ SOS(I). 3.3. Proof of Theorem 1.1 From Lemma 3.2(b), (c) it follows that G(aI − bWα )G−1 = (a ◦ η)II − (b ◦ η)cα,p Wα .

(3.7)

From Lemma 3.3 we know that a ◦ η ∈ SO(I), (b ◦ η)cα,p ∈ SO(I), α

∈ SOS(I), and 0 < C1 := inf cα,p (y) ≤ sup cα,p (y) =: C2 < +∞. y∈I

y∈I

It is easy to see that inf |(a ◦ η)(y)| = inf |a(t)|,

y∈I

t∈R+

inf |(b ◦ η)(y)| = inf |b(t)|.

y∈I

t∈R+

(3.8)

Hence C1 inf |b(t)| ≤ inf |(b ◦ η)(y)cα,p (y)| ≤ C2 inf |b(t)|. t∈R+

y∈I

t∈R+

(3.9)

Further, it can be checked straightforwardly that for every y ∈ I, α (y)) |(a ◦ η)(y)| − |(b ◦ η)(y)cα,p (y)| (

−1/p

= |(a ◦ η)(y)| − |(b ◦ η)(y)| ((α ◦ η)(y)) Therefore

−1/p

.

−1/p α (y)) lim inf |(a ◦ η)(y)| − |(b ◦ η)(y)cα,p (y)| (

y→0 y→0

−1/p , (3.10) = lim sup lim inf |a(t)| − |b(t)| (α (t)) t→0 t→0

−1/p lim sup lim inf |(a ◦ η)(y)| − |(b ◦ η)(y)cα,p (y)| (

α (y)) y→1 y→1

−1/p , (3.11) = lim sup lim inf |a(t)| − |b(t)| (α (t)) lim sup

t→∞

t→∞

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respectively. Equality (3.7) says that aI − bWα is invertible on Lp (R+ ) if and only if (a ◦ η)II − (b ◦ η)cα,p Wα is invertible on Lp (I). On the other hand, equality (3.8), inequalities (3.9), and equalities (3.10), (3.11) imply that the conditions of Theorems 1.1 and 3.1 are equivalent. Further, if (1.1) holds, then (3.1) is fulﬁlled. Then, by Theorem 3.1, n ∞ b◦η 1 −1 ((a ◦ η)II − (b ◦ η)cα,p Wα ) = cα,p Wα

II . (3.12) a ◦ η a ◦ η n=0 Applying Lemma 3.2(b), (c), we obtain ∞ n ∞ b◦η 1 −1 n −1 cα,p Wα

II = G (a bWα ) a I G−1 . (3.13) a ◦ η a ◦ η n=0 n=0 Combining (3.7), (3.12), and (3.13), we get (1.3). Analogously it can be shown that if (1.2) is fulﬁlled, then (aI − bWα )−1 is calculated by (1.4).

4. Convolution Operators 4.1. Fourier Convolution Operators Let F : L2 (R) → L2 (R) denote the Fourier transform, (F f )(x) := f (y)e−ixy dy (x ∈ R), R −1

2

2

and let F : L (R) → L (R) be the inverse of F . A function a ∈ L∞ (R) is called a Fourier multiplier on Lp (R) if the mapping f → F −1 aF f maps L2 (R) ∩ Lp (R) onto itself and extends to a bounded operator on Lp (R). The latter operator is then denoted by W 0 (a). We let Mp (R) stand for the set of all Fourier multipliers on Lp (R). One can show that Mp (R) is a Banach algebra under the norm a Mp (R) := W 0 (a) B(Lp (R)) . We denote by P C the C ∗ -algebra of all bounded piecewise continuous functions on R˙ = R ∪ {∞}. By definition, a ∈ P C if and only if a ∈ L∞ (R) and the one-sided limits a(x0 − 0) :=

lim

x→x0 −0

a(x),

a(x0 + 0) :=

lim

x→x0 +0

a(x)

˙ If a function a is given everywhere on R, then its total exist for each x0 ∈ R. variation is deﬁned by n V (a) := sup |a(xk ) − a(xk−1 )|, k=1

where the supremum is taken over all n ∈ N and −∞ < x0 < x1 < · · · < xn < +∞. If a has a ﬁnite total variation, then it has ﬁnite one-sided limits a(x − 0) and ˙ that is, a ∈ P C (see, e.g., [29, Chap. VIII, Sections 3 a(x + 0) for all x ∈ R,

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and 9]). The following theorem gives an important subset of Mp (R). Its proof can be found, e.g., in [8, Theorem 17.1]. Theorem 4.1. (Stechkin’s inequality) If a ∈ P C has finite total variation V (a), then a ∈ Mp (R) and a Mp (R) ≤ SR B(Lp (R)) a L∞ (R) + V (a) , where SR is the Cauchy singular integral operator on R. According to [8, p. 325], let P Cp be the closure in Mp (R) of the set of all functions a ∈ P C with ﬁnite total variation on R. Following [8, p. 331], put Cp (R) := P Cp ∩ C(R), where R := [−∞, +∞]. 4.2. Mellin Convolution Operators Let dμ(t) = dt/t be the (normalized) invariant measure on R+ . Consider the Fourier transform on L2 (R+ , dμ), which is usually referred to as the Mellin transform and is deﬁned by dt 2 2 f (t)t−ix . M : L (R+ , dμ) → L (R), (M f )(x) = t R+

It is an invertible operator, with inverse given by M

−1

2

2

: L (R) → L (R+ , dμ),

(M

−1

1 g)(t) = 2π

g(x)tix dx. R

Let E be the isometric isomorphism E : Lp (R+ , dμ) → Lp (R),

(Ef )(x) := f (ex ) (x ∈ R).

(4.1)

−1

Then the map A → E AE transforms the Fourier convolution operator W 0 (a) = F −1 aF to the Mellin convolution operator Co(a) := M −1 aM with the same symbol a. Hence the class of Fourier multipliers on Lp (R) coincides with the class of Mellin multipliers on Lp (R+ , dμ). The following result was obtained in [11, Proposition 1.6], its proof can also be found in [32, Proposition 12.7]. Theorem 4.2. (Duduchava) If a ∈ Cb (R+ ) and b ∈ P Cp are such that lim a(t) = lim a(t) = lim b(x) = lim b(x) = 0, t→+∞

t→0+0

x→−∞

x→+∞

then a Co(b) ∈ K(L (R+ , dμ)). p

4.3. The Algebra A Let A be a Banach algebra and S be a subset of A. By closA S we denote the closure of S in the norm of A. Following [9, Section 3.45], we denote by algA S the smallest closed subalgebra of A containing S and by idA S the smallest closed two-sided ideal of A containing S. Let 1 < p < ∞. Put B := B(Lp (R+ )),

K := K(Lp (R+ )),

A := algB {I, S}.

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Obviously, the algebra A is commutative. For β ∈ C, let ∞ f (τ ) 1 (t ∈ R+ ) (Rβ f )(t) := πi τ − eiβ t 0

and write R for Rπ . Further, put sp (x) := coth[π(x + i/p)],

rp,β (x) :=

e(x+i/p)(π−β) sinh[π(x + i/p)]

(x ∈ R)

and write rp for rp,π . Consider the isometric isomorphism Φ : Lp (R+ ) → Lp (R+ , dμ),

(Φf )(t) := t1/p f (t)

(t ∈ R+ ).

(4.2)

The following facts are well known. Their proofs can be found, e.g., in [32, Propositions 2.1–2.5] (see also [11]). Theorem 4.3. Suppose 1 < p < ∞. (a) The algebra A is the smallest closed subalgebra of B that contains the operators Φ−1 Co(a)Φ with a ∈ Cp (R). (b) If β ∈ C and Re β ∈ (0, 2π), then sp , rp,β ∈ Cp (R) and ΦSΦ−1 = Co(sp ),

ΦRβ Φ−1 = Co(rp,β ).

(c) The maximal ideal space of the commutative Banach algebra A is homeomorphic to R. In particular, an operator Φ−1 Co(a)Φ with a ∈ Cp (R) is invertible if and only if a(x) = 0 for all x ∈ R. Thus A is an inverse closed subalgebra of B. (d) An operator Φ−1 Co(a)Φ with a ∈ Cp (R) belongs to idA {R} if and only if a(−∞) = a(+∞) = 0. From s2p − rp2 = 1 and Theorem 4.3(b) it follows that 4P+ P− = 4P− P+ = I − S 2 = −R2 .

(4.3)

Let us describe the quotient algebra A := (A + K)/K. Since a Mellin convolution operator is Fredholm if and only if it is invertible, from Theorem 4.3 we obtain the following. π

Corollary 4.4. (a) The algebra Aπ is commutative and its maximal ideal space is homeomorphic to R. (b) The Gelfand transform of a coset (Φ−1 Co(a)Φ)π ∈ Aπ for a ∈ Cp (R) is given by −1 (Φ Co(a)Φ)π (x) = a(x) for x ∈ R. In particular, (S π )(±∞) = ±1,

(S π )(x) = sp (x)

for

x ∈ R,

for

x ∈ R.

and if β ∈ C and Re β ∈ (0, 2π), then (Rβπ )(±∞) = 0,

(Rβπ )(x) = rp,β (x)

(c) An operator H ∈ A belongs to idA {R} if and only if (H π )(−∞) = (H π )(+∞) = 0.

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5. Mellin Pseudodifferential Operators 5.1. Boundedness If a is an absolutely continuous function of ﬁnite total variation on R, then a ∈ L1 (R) and V (a) = |a (x)|dx R

(see, e.g., [29, Chap. VIII, Sections 3 and 9; Chap. XI, Section 4]). The set V (R) of all absolutely continuous functions of ﬁnite total variation on R forms a Banach algebra when equipped with the norm a V := a L∞ (R) + V (a) = a L∞ (R) + |a (x)|dx. R

Following [18,19], let Cb (R+ , V (R)) denote the Banach algebra of all bounded continuous V (R)-valued functions on R+ with the norm a(·, ·) Cb (R+ ,V (R)) = sup a(t, ·) V . t∈R+

C0∞ (R+ )

As usual, let be the set of all inﬁnitely differentiable functions of compact support on R+ . The following boundedness result for Mellin pseudodifferential operators was obtained in [19, Theorem 3.1] (see also [18, Theorem 3.1]). Theorem 5.1. If a ∈ Cb (R+ , V (R)), then the Mellin pseudodifferential operator Op(a), defined for functions f ∈ C0∞ (R+ ) by the iterated integral ix t 1 dτ for t ∈ R+ , [Op(a)f ](t) = dx a(t, x) f (τ ) 2π τ τ R

R+

extends to a bounded linear operator on the space Lp (R+ , dμ) and there is a number Cp ∈ (0, ∞) depending only on p such that Op(a) B(Lp (R+ ,dμ)) ≤ Cp a Cb (R+ ,V (R)) . 5.2. Compactness of Commutators Let SO(R+ , V (R)) denote the Banach subalgebra of Cb (R+ , V (R)) consisting of all V (R)-valued functions a on R+ that slowly oscillate at 0 and ∞, that is, C lim cmC r (a) = lim cmr (a) = 0,

r→0

where

r→∞

cmC r (a) = max a(t, ·) − a(τ, ·) L∞ (R) : t, τ ∈ [r, 2r] .

Let E(R+ , V (R)) be the Banach algebra of all V (R)-valued functions a belonging to SO(R+ , V (R)) and such that lim sup a(t, ·) − ah (t, ·) = 0 (5.1) |h|→0 t∈R+

V

where ah (t, x) := a(t, x + h) for all (t, x) ∈ R+ × R.

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The following result on compactness of commutators of Mellin pseudodifferential operators was obtained in [20, Theorem 3.5] (see also [18, Corollary 8.4]). Theorem 5.2. If a, b ∈ E(R+ , V (R)), then the commutator Op(a) Op(b) − Op(b) Op(a) is compact on the space Lp (R+ , dμ).

6. Localization 6.1. The Allan–Douglas Local Principle The Allan–Douglas local principle is one of the main tools in studying singular integral operators in the last decades. The aim of this section is to apply this principle to operators in the algebra F := algB aI, S, Wα , Wα−1 : a ∈ SO(R+ ) . Here is the formulation of the local principle taken from [9, Theorem 1.35(a)]. Let A be a Banach algebra with identity. A subalgebra Z of A is said to be a central subalgebra if za = az for all z ∈ Z and all a ∈ A. Theorem 6.1. (Allan–Douglas) Let A be a Banach algebra with identity e and let Z be a closed central subalgebra of A containing e. Let M (Z) be the maximal ideal space of Z, and for ω ∈ M (Z), let Jω refer to the smallest closed two-sided ideal of A containing the ideal ω. Then an element a is invertible in A if and only if a + Jω is invertible in the quotient algebra A/Jω for all ω ∈ M (Z). The algebra A/Jω is referred to as the local algebra of A at ω ∈ M (Z). Now we are going to construct an algebra Λ that contains the algebra F and such that the quotient algebra Λπ := Λ/K has a center properly containing Aπ . To this end we need several compactness results. 6.2. Compactness Results The following compactness results were obtained in [15, Corollaries 5.2, 5.3]. Theorem 6.2. Let 1 < p < ∞. (a) If a ∈ SO(I), then aSI − SI aII ∈ K(Lp (I)). (b) If α ∈ SOS(I), then Wα SI − SI Wα ∈ K(Lp (I)). Now we prove their counterparts for the case of R+ . Theorem 6.3. Let 1 < p < ∞. (a) If a ∈ SO(R+ ), then aS − SaI ∈ K. (b) If α ∈ SOS(R+ ), then Wα S − SWα ∈ K. Proof. (a) If a ∈ SO(R+ ), then a ◦ η ∈ SO(I) by Lemma 3.3(a). Then in view of Theorem 6.2(a), the operator (a ◦ η)SI − SI (a ◦ η)II is compact on L2 (I). For p = 2 from Lemma 3.2(a),(b) it follows that aS − SaI = G−1 [(a ◦ η)SI − SI (a ◦ η)II ]G ∈ K(L2 (R+ )).

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Since the operator aS − SaI is bounded on all spaces Lp (R+ ) with p ∈ (1, ∞), from the Krasnosel’skii interpolation theorem (see [25, Theorem 3.10]) it follows that the operator aS − SaI is compact on all spaces Lp (R+ ), p ∈ (1, ∞).

∈ SOS(R+ ) and cα,2 ∈ SO(I) due to If α ∈ SOS(R+ ), then α Lemma 3.3(b). From Theorem 6.2 it follows that K1 := Wα SI − SI Wα ∈ K(L2 (I)), K2 := cα,2 SI − SI cα,2 II ∈ K(L2 (I)). Then cα,2 Wα SI − SI cα,2 Wα = cα,2 K1 + K2 Wα ∈ K(L2 (I)). From Lemma 3.2(a),(c) we get Wα S − SWα = G−1 [cα,2 Wα SI − SI cα,2 Wα ]G ∈ K(L2 (R+ )). Since the operator Wα S − SWα is bounded on all spaces Lp (R+ ) for 1 < p < ∞, we conclude that the operator Wα S − SWα is compact on all spaces Lp (R+ ), 1 < p < ∞, by analogy with part (a).

Corollary 6.4. If a ∈ SO(R+ ) and α ∈ SOS(R+ ), then for every A ∈ A, aA − AaI ∈ K,

Wα A − AWα ∈ K.

Proof. It is easy to see that if B ∈ B is such that BS − SB ∈ K, then for every A ∈ A, the commutator BA − AB is compact. It remains to apply Theorem 6.3. Theorem 6.5. If a ∈ Cb (R+ ) and limt→s a(t) = 0 for s ∈ {0, ∞}, then aR is compact. Proof. By Theorem 4.3(b), rp ∈ Cp (R) ⊂ P Cp and R = Φ−1 Co(rp )Φ. It is easy to see that lim rp (x) = lim rp (x) = 0.

x→−∞

x→+∞

Hence, by Theorem 4.2, a Co(rp ) ∈ K(Lp (R+ , dμ)). Therefore the operator aR = aΦ−1 Co(rp )Φ = Φ−1 a Co(rp )Φ is compact on Lp (R+ ). Corollary 6.6. If a ∈ Cb (R+ ), limt→s a(t) = 0 for s ∈ {0, ∞} and H ∈ idA {R}, then aH ∈ K. Proof. Since A is commutative, we see that idA {R} = closA {RA : A ∈ A}. From this equality and Theorem 6.5 we immediately get the statement. 6.3. Algebras Z, D, and Λ Let us consider Z := algB {I, S, cR, K : c ∈ SO(R+ ), K ∈ K} , D := algB {aI, S : a ∈ SO(R+ )} , Λ := {A ∈ B : AC − CA ∈ K for all C ∈ Z} . Lemma 6.7. (a) The set Λ is a closed unital subalgebra of B. (b) The set K is a closed two-sided ideal of the algebra Λ.

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(c) An operator A ∈ Λ is Fredholm if and only if the coset Aπ := A + K is invertible in the quotient algebra Λπ := Λ/K. The proof is straightforward and therefore it is omitted. Theorem 6.8. We have K ⊂ Z ⊂ D ⊂ F ⊂ Λ. Proof. The inclusion K ⊂ Z follows from the definition of the algebra Z. The inclusion D ⊂ F is obvious. To prove that Z ⊂ D, it is sufﬁcient to show that all the generators of Z belong to D. Obviously, I, S ∈ D. Further, cR ∈ D for c ∈ SO(R+ ) because R ∈ A ⊂ D. It is well known that K ⊂ algB {aI, S : a ∈ C(R+ )} (see, e.g., [5, Lemma 8.23] for Carleson Jordan curves, in the present case the proof is analogous). Since C(R+ ) ⊂ SO(R+ ), from the above inclusion it follows that K ⊂ D. Thus, Z ⊂ D. Let us show F ⊂ Λ. By Corollary 6.4, aI, S ∈ Λ for a ∈ SO(R+ ) and Wα S − SWα ∈ K,

Wα R − RWα ∈ K.

(6.1)

Since α ∈ SOS(R+ ), from Lemma 2.4 we infer that α−1 ∈ SOS(R+ ), too. From Lemma 2.3 and Theorem 6.5 it follows that (c − c ◦ α−1 )R ∈ K.

(6.2)

Combining the second relation in (6.1) and relation (6.2), we get Wα cR − cRWα = Wα (c − c ◦ α−1 )R + c(Wα R − RWα ) ∈ K.

(6.3)

From (6.1) and (6.3) it follows that Wα ∈ Λ. Taking into account that Wα−1 = Wα−1 and repeating the above argument with α−1 ∈ SOS(R+ ) in place of α we ﬁnally get Wα−1 ∈ Λ. We have proved that all the generators of F belong to Λ. Thus F ⊂ Λ. From the above theorem it follows that the quotient algebras Z π := Z/K, Dπ := D/K, and Λπ := Λ/K are well deﬁned. Clearly, Z π lies in the center of Λπ . Our next aim is to describe the maximal ideal space of Z π . We start with a description of the maximal ideal space of the bigger algebra Dπ , which is commutative in view of Theorem 6.3(a). 6.4. Maximal Ideal Space of D π Theorem 6.9. The maximal ideal space M (Dπ ) of the commutative Banach algebra Dπ is homeomorphic to the set M := (M (SO(R+ )) × {−∞, +∞}) ∪ (Δ × R),

(6.4)

where Δ is given by (1.5). Proof. The proof is similar to that of [2, Theorem 6.3]. It is clear that C π := {(cI)π : c ∈ SO(R+ )} and Aπ are commutative Banach subalgebras of the algebra Dπ and their maximal ideal spaces can be identiﬁed with M (SO(R+ )) and R, respectively. Fix (ξ, x) ∈ M (SO(R+ )) × R and consider the maximal ideals {(cI)π : c ∈ SO(R+ ), c(ξ) = 0} of C π and {Aπ ∈ Aπ : (Aπ )(x) = 0} π denote the closed two-sided (not necessarily maximal) ideal of Aπ . Let Nξ,x π of D generated by the above maximal ideals of C π and Aπ . Identifying the π and taking into account pair (ξ, x) ∈ M (SO(R+ )) × R with the ideal Nξ,x

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(6.4) and M (SO(R+ )) = R+ ∪ Δ, we see that M (Dπ ) is homeomorphic to a subset of M (SO(R+ )) × R = (R+ × R) ∪ M. Observe that due to (4.3) any coset Dπ ∈ Dπ is of the form Dπ = (c+ P+ )π + (c− P− )π + Y π ,

(6.5)

where c± ∈ SO(R+ ), P± = (I ± S)/2, and Y π = lim

n→∞

mn

(cn,k Hn,k )π

(6.6)

k=1

with cn,k ∈ SO(R+ ), Hn,k ∈ idA {R}, and mn ∈ N. Fix (ξ, x) ∈ R+ × R. Given a coset Dπ of the form (6.5), (6.6), we can cn,k with compact support on R+ such that choose continuous functions

c± ,

±, H

n,k ∈ idA {R} such that cn,k (ξ) = cn,k (ξ) and operators H

c± (ξ) = c± (ξ),

π π π π

(H± )(x) = (P± )(x), (Hn,k )(x) = (Hn,k )(x). Then π π

± )π ,

± ) + (

c± )P± ] +

c± (P± − H c± H (c± P± )π = [(c± −

respectively, where the ﬁrst two terms on the right-hand side belong to the π

± )π are zero by Corollary 6.6. Hence, ideal Nξ,x , whereas the terms (

c± H π π π (c± P± ) ∈ Nξ,x . Analogously, (cn,k Hn,k )π ∈ Nξ,x for all pairs (n, k). Thus, π π Nξ,x for (ξ, x) ∈ R+ × R contains every coset D ∈ Dπ and hence cannot be a maximal ideal. In fact, it is not even a proper ideal, since it contains the unit I π . Consequently, M (Dπ ) ⊂ M. π is a proper ideal of Dπ Consider now a point (ξ, x) ∈ M. Then Nξ,x π π because it does not contain the unit I . Let us show that the ideal Nξ,x is π maximal. Assume the contrary: the ideal Nξ,x is not maximal. Then there is

π of Dπ that contains properly N π . For any Dπ ∈ Dπ a maximal ideal N ξ,x ξ,x π π π of the form (6.5), we have Dπ = (Dπ )(ξ, x)I π + Oξ,x , where Oξ,x ∈ Nξ,x , 1 + sp (x) 1 − sp (x) + c− (ξ) + (Y π )(ξ, x), 2 2 mn π cn,k (ξ)(Hn,k )(x). (Y π )(ξ, x) = lim

(Dπ )(ξ, x) = c+ (ξ)

n→∞

(6.7) (6.8)

k=1

π \N π is of the form Dπ = (δI)π + Oπ with Hence every coset Dπ ∈ N ξ,x ξ,x ξ,x π π π δ ∈ C\{0} and Oξ,x ∈ Nξ,x . But then Dπ − Oξ,x = (δI)π is invertible in Dπ

π . Thus N π is a maximal ideal of and this contradicts the maximality of N ξ,x ξ,x π π D for (ξ, x) ∈ M, and therefore M (D ) = M. Theorem 6.9 immediately implies the following. Corollary 6.10. A coset Dπ ∈ Dπ given by (6.5), (6.6) is invertible in the Banach algebra Dπ if and only if the Gelfand transform (Dπ )(ξ, x) given by (6.7), (6.8) does not vanish for every (ξ, x) ∈ M.

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6.5. Maximal Ideal Space of Z π Now we are ready to describe the maximal ideal space of Z π . Theorem 6.11. The maximal ideal space M (Z π ) of the commutative Banach algebra Z π is homeomorphic to the set {−∞, +∞} ∪ (Δ × R). Proof. From (4.3) it easily follows that any coset Z π ∈ Z π is of the form (6.5) where now c± are complex constants. Hence, Corollaries 4.4(b) and 6.10 imply that Z π is invertible in Dπ if and only if (Z π )(ξ, x) = c+ (1 + sp (x))/2 + c− (1 − sp (x))/2 + (Y π )(ξ, x) = 0 π −1

(6.9)

for every (ξ, x) ∈ M. If (6.9) holds, then (Z ) = (d+ P+ ) + (d− P− ) + Gπ , where d± ∈ SO(R+ ) and Gπ is a coset of the form (6.6). π From Corollary 4.4(b) we know that (Hn,k )(±∞) = 0 for every operator Hn,k in the representation (6.6). Hence for every coset Y π of the form (6.6), we deduce from (6.8) that (Y π )(ξ, ±∞) = 0

π

for ξ ∈ M (SO(R+ )).

π

(6.10)

Taking the Gelfand transform of the coset I π = Z π (Z π )−1 = (c+ d+ P+ )π + (c− d− P− )π + Qπ , where Qπ is of the form (6.6), at the points (ξ, ±∞) for ξ ∈ M (SO(R+ )), from Corollary 4.4(b) and (6.10) we see that c± d± (ξ) = 1 for all ∈ C\{0}, whence (Z π )−1 = ξ ∈ M (SO(R+ )). Therefore d± = c−1 ± −1 π −1 π π π π π c+ P+ + c− P− + G ∈ Z . Thus Z ∈ Z is invertible in Z π if and only if (6.9) holds for all (ξ, x) ∈ M. From (6.7) and (6.10) it follows that only one element in M (Z π ) correπ . sponds to any pair in the set M (SO(R+ ))×{±∞}. We will denote it by I±∞ π π Let Nξ,x be the maximal ideal of D corresponding to the point (ξ, x) ∈ M. It is clear that for every ξ ∈ M (SO(R+ )) we have π π I±∞ = Nξ,±∞ ∩ Z π = {Z π ∈ Z π : (Z π )(ξ, ±∞) = 0} .

Finally, π π Iξ,x = Nξ,x ∩ Z π = {Z π ∈ Z π : (Z π )(ξ, x) = 0}

is a maximal ideal of the algebra Z π for every (ξ, x) ∈ Δ×R. Thus, M (Z π ) = {−∞, +∞} ∪ (Δ × R). 6.6. Fredholmness of Operators of Local Type According to the proof of Theorem 6.11, π I±∞ := idZ π P∓π , (gR)π : g ∈ SO(R+ ) , π Iξ,x = {Z π ∈ Z π : (Z π )(ξ, x) = 0} π J±∞

for (ξ, x) ∈ Δ × R.

π Jξ,x

Further, let and be the closed two-sided ideals of the Banach algeπ π bra Λπ generated by the ideals I±∞ and Iξ,x of the algebra Z π , respectively, and put π Λπ±∞ := Λπ /J±∞ ,

π Λπξ,x := Λπ /Jξ,x

for the corresponding quotient algebras.

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Theorem 6.12. An operator A ∈ Λ is Fredholm on the space Lp (R+ ) if and only if the following two conditions are fulfilled: π are invertible in the quotient algebras Λπ±∞ , respec(i) the cosets Aπ + J±∞ tively; π is invertible in the quotient (ii) for every (ξ, x) ∈ Δ × R, the coset Aπ + Jξ,x π algebra Λξ,x .

Proof. By Lemma 6.7(c), the Fredholmness of an operator A ∈ Λ is equivalent to the invertibility of the coset Aπ = A + K in the quotient algebra Λπ . By Theorem 6.11, Z π is its central subalgebra, whose maximal ideal space is homeomorphic to the set {−∞, +∞} ∪ (Δ × R). Therefore, by the Allan– Douglas local principle (Theorem 6.1), the invertibility of the coset Aπ in the quotient algebra Λπ is equivalent to the invertibility of the local representaπ π π , Aπ + J+∞ , and Aπ + Jξ,x in the local algebras Λπ−∞ , Λπ+∞ , tives Aπ + J−∞ and Λπξ,x for all (ξ, x) ∈ Δ × R, respectively.

7. Some Functions in Cb (R+ , V (R)) and E(R+ , V (R)) 7.1. Functions sp , rp , and Slowly Oscillating Functions In this section we will prove that certain functions, playing a major role in the proof of Theorem 1.2, belong to E(R+ , V (R)) or to Cb (R+ , V (R)). Lemma 7.1. Let g ∈ SO(R+ ). Then the functions g(t, x) := g(t),

sp (t, x) := sp (x),

rp (t, x) := rp (x),

(t, x) ∈ R+ × R,

belong to the algebra E(R+ , V (R)). Proof. The statement is obvious for g. Let us prove it for sp and rp . Since sp and rp are constant in the ﬁrst variable, the only nontrivial property is (5.1). Because sp , rp ∈ V (R), we conclude that sp and rp belong to L1 (R). Taking into account that for X = Cb (R), L1 (R) and f ∈ X, f (· + h) − f (·) X → 0

as

|h| → 0

(see, e.g., [10, Chap. 2, Section 6] and also [34, Chap. III, Section 2] for L1 (R)), we conclude that sup sp (t, ·) − shp (t, ·) V = sp (· + h) − sp (·) L∞ (R)

t∈R+

+ sp (· + h) − sp (·) L1 (R) → 0 as |h| → 0. The same property is true with sp , sp , and sp replaced by rp , rp , and rp , respectively. Thus (5.1) holds for sp and rp . 7.2. A Function in the Algebra Cb (R+ , V (R)) We start with the following obvious auxiliary statement.

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Lemma 7.2. For every j ∈ N ∪ {0}, we have Cj∞ := sup |x + i/p|j |rp (x)| < ∞, x∈R Cj1 := |x + i/p|j |rp (x)| < ∞,

(7.1) (7.2)

R

M0 := sup |πsp (x)| < ∞.

(7.3)

x∈R

Lemma 7.3. Suppose α is a slowly oscillating shift and ω is its exponent function. The function (t, x) ∈ R+ × R,

c(t, x) := eiω(t)(x+i/p) rp (x),

(7.4)

belongs to the algebra Cb (R+ , V (R)). Proof. Through the proof we will assume that (t, x), (τ, x) ∈ R+ × R. Since ω ∈ SO(R+ ) is real-valued, we have M1 := sup |ω(t)|, t∈R+

M2 := sup (−ω(t)) < ∞.

(7.5)

t∈R+

Then |eiω(t)(x+i/p) | = e−ω(t)/p ≤ eM2 /p ,

(7.6)

|c(t, x)| ≤ eM2 /p |rp (x)|.

(7.7)

whence due to (7.4), From this estimate and (7.1) it follows that c(t, ·) L∞ (R) ≤ eM2 /p C0∞ .

(7.8)

It is easy to see that ∂c (t, x) = (iω(t) − πsp (x)) c(t, x). ∂x Hence from (7.7), (7.2), (7.3) and (7.5) we obtain ∂c V (c(t, ·)) = (t, x) dx ≤ (M1 + M0 )eM2 /p C01 . ∂x

(7.9)

(7.10)

R

Further, for t, τ ∈ R+ , we get

⎛

⎜ c(t, x) − c(τ, x) = i(x + i/p) ⎝

ω(t)

⎞ ⎟ eiθ(x+i/p) dθ⎠ rp (x).

ω(τ )

For every θ in the segment with the endpoints ω(τ ) and ω(t), we have |eiθ(x+i/p) | = e−θ/p ≤ eM2 /p . Hence |c(t, x) − c(τ, x)| ≤ eM2 /p |ω(t) − ω(τ )| |x + i/p| |rp (x)|.

(7.11)

From (7.1) and (7.11) we get for t, τ ∈ R+ , c(t, ·) − c(τ, ·) L∞ (R) ≤ C1∞ eM2 /p |ω(t) − ω(τ )|.

(7.12)

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From (7.9) it follows that ∂c ∂c (t, x) − (τ, x) = i (ω(t) − ω(τ )) c(t, x) ∂x ∂x + (iω(τ ) − πsp (x)) (c(t, x) − c(τ, x)) .

(7.13)

Combining (7.3) and (7.5) with inequalities (7.7) and (7.11), we deduce from (7.13) that ∂c (t, x) − ∂c (τ, x) ≤ eM2 /p |ω(t) − ω(τ )| ∂x ∂x × (|rp (x)| + (M1 + M0 )|x + i/p| |rp (x)|) . Therefore, taking into account (7.2) we infer from the latter inequality that for t, τ ∈ R+ , ∂c ∂c (τ, x) dx V (c(t, ·) − c(τ, ·)) = (t, x) − ∂x ∂x R

≤ (C01 + (M1 + M0 )C11 )eM2 /p |ω(t) − ω(τ )|. (7.14) Combining (7.12) and (7.14), we arrive at the estimate c(t, ·) − c(τ, ·) V ≤ L|ω(t) − ω(τ )| (C1∞

C01

(t, τ ∈ R+ ),

M0 )C11 )eM2 /p .

+ + (M1 + This inequality implies that where L := c is a continuous V (R)-valued function. Moreover, it is bounded in view of (7.8) and (7.10). Thus c ∈ Cb (R+ , V (R)). 7.3. Key Lemma The main result of this section is the following. Lemma 7.4. Suppose α is a slowly oscillating shift and ω is its exponent function. Let ξ ∈ Δ and 2

a(t, x) := eiω(t)(x+i/p) (rp (x)) ,

(t, x) ∈ (R+ × R) ∪ (Δ × R).

Then there exists a function bξ ∈ E(R+ , V (R)) such that a(t, x) − a(ξ, x) = (ω(t) − ω(ξ)) rp (x)bξ (t, x),

(t, x) ∈ R+ × R. (7.15)

Proof. We divide the proof into ten steps. Through the proof we will assume that (t, x), (τ, x) ∈ R+ × R and all estimates are uniform in t, τ, x. 1. Definition of bξ . Consider ω(t) iω(t)(x+i/p)

e

−e

iω(ξ)(x+i/p)

eiθ(x+i/p) dθ

= i(x + i/p) ω(ξ)

1 = (ω(t) − ω(ξ)) i(x + i/p)

Eξ (x, ω(t), y)dy, 0

where Eξ (x, θ, y) := ei[ω(ξ)+y(θ−ω(ξ))](x+i/p) ,

(x, θ, y) ∈ R × R × [0, 1].

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Then we obtain (7.15) with bξ (t, x) := rp (x)eξ (t, x),

(7.16)

1 eξ (t, x) := i(x + i/p)

Eξ (x, ω(t), y)dy.

(7.17)

0

2. Uniform estimate for bξ (t, x). From Lemma 2.1 we obtain −ω(ξ) ≤ M2 , where M2 is deﬁned by (7.5). Therefore, for θ = ω(ξ) + y(ω(t) − ω(ξ)) and y ∈ [0, 1], |Eξ (x, ω(t), y)| = |eiθ(x+i/p) | = e−θ/p ≤ eM2 /p . This estimate and (7.16), (7.17) imply that |bξ (t, x)| ≤ eM2 /p |x + i/p| |rp (x)|. 3. Uniform estimate for ∂eξ (t, x) = i ∂x

∂bξ ∂x (t, x).

(7.18)

Differentiating (7.17) we get

1 Eξ (x, ω(t), y)dy 0

1 i [ω(ξ) + y (ω(t) − ω(ξ))] Eξ (x, ω(t), y)dy.

+ i(x + i/p) 0

Integrating by parts the ﬁrst integral and splitting the second integral into two integrals, we get ∂eξ (t, x) = ieiω(t)(x+i/p) + iω(ξ)eξ (t, x). ∂x

(7.19)

Taking into account (7.16), the definition of rp , (7.19) and (7.4), we obtain ∂bξ drp ∂eξ (t, x) = (x)eξ (t, x) + rp (x) (t, x) ∂x dx ∂x = (−πsp (x) + iω(ξ)) bξ (t, x) + ic(t, x). From (7.20), (7.3), (7.5), (7.7) and (7.18) it follows that ∂bξ ≤ M3 |x + i/p| |rp (x)| + eM2 /p |rp (x)|, (t, x) ∂x

(7.20)

(7.21)

where M3 := (M0 + M1 )eM2 /p . ∂ 2 bξ ∂x2 (t, x).

From (7.20) and (7.9) it follows that ∂ 2 bξ 2 2 bξ (t, x) (t, x) = (πr (x)) + (−πs (x) + iω(ξ)) p p ∂x2 (7.22) + (−2πsp (x) + iω(ξ) + iω(t)) ic(t, x).

4. Uniform estimate for

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Taking into account (7.1), (7.3) and (7.5), we infer from (7.22), (7.7), and (7.18) that 2 ∂ bξ (7.23) ∂x2 (t, x) ≤ M4 |x + i/p| |rp (x)| + 2M3 |rp (x)|, where M4 := ((πC0∞ )2 + (M0 + M1 )2 )eM2 /p . 5. Uniform estimate for bξ (t, x) − bξ (τ, x). For y ∈ [0, 1] we have ω(t)

Eξ (x, ω(t), y) − Eξ (x, ω(τ ), y) = i(x + i/p)y

Eξ (x, θ, y)dθ.

ω(τ )

Then, taking into account (7.17), we get eξ (t, x) − eξ (τ, x) = −(x + i/p)2

1

⎛ ⎜ ⎝y

0

ω(t)

⎞ ⎟ Eξ (x, θ, y)dθ⎠ dy.

ω(τ )

As in Step 2, we obtain |Eξ (x, θ, y)| = |eiψ(x+i/p) | = e−ψ/p ≤ eM2 /p for ψ = ω(ξ) + y(θ − ω(ξ)) with y ∈ [0, 1] and θ in the segment with the endpoints ω(τ ) and ω(t). Therefore 2

1

|eξ (t, x) − eξ (τ, x)| ≤ |x + i/p|

yeM2 /p |ω(t) − ω(τ )|dy 0

M2 /p

= (e

/2)|ω(t) − ω(τ )| |x + i/p|2 .

Thus, by (7.16), |bξ (t, x) − bξ (τ, x)| ≤ eM2 /p |ω(t) − ω(τ )| |x + i/p|2 |rp (x)|. 6. Uniform estimate for

∂bξ ∂x (t, x)

−

∂bξ ∂x (τ, x).

(7.24)

From (7.20) it follows that

∂bξ ∂bξ (t, x) − (τ, x) = (−πsp (x) + iω(ξ)) (bξ (t, x) − bξ (τ, x)) ∂x ∂x +i (c(t, x) − c(τ, x)) . (7.25) Taking into account (7.3), (7.5), from (7.25), (7.24) and (7.11) we obtain ∂bξ ∂bξ ≤ M3 |x + i/p|2 |rp (x)| + eM2 /p |x + i/p| |rp (x)| (t, x) − (τ, x) ∂x ∂x ×|ω(t) − ω(τ )|. (7.26)

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7. Uniform estimate for bξ (t, x+h)−bξ (t, x). From (7.21) and (7.1) we obtain for h ∈ R, x+h ∂b ξ (t, y)dy |bξ (t, x + h) − bξ (t, x)| = ∂y x x+h x+h ≤ M3 |y + i/p| |rp (y)|dy + eM2 /p |rp (y)|dy x

x

≤ M5 |h|, where M5 :=

M3 C1∞

M2 /p

+e

(7.27)

C0∞ .

8. Proof of bξ ∈ Cb (R+ , V (R)). From (7.18) and (7.1) it follows that bξ (t, ·) L∞ (R) ≤ eM2 /p C1∞ .

(7.28)

Further, from (7.21) and (7.2) we get ∂bξ (t, x) dx ≤ M3 C11 + eM2 /p C01 . V (bξ (t, ·)) = ∂x

(7.29)

R

Combining (7.28) and (7.29), we obtain bξ Cb (R+ ,V (R)) = sup bξ (t, ·) L∞ (R) + V (bξ (t, ·)) t∈R

≤ eM2 /p C1∞ + M3 C11 + eM2 /p C01 < ∞.

(7.30)

From (7.1) and (7.24) we see that bξ (t, ·) − bξ (τ, ·) L∞ (R) ≤ eM2 /p C2∞ |ω(t) − ω(τ )|.

(7.31)

Further, from (7.26) and (7.2) it follows that for all t, τ ∈ R+ , ∂bξ ∂bξ (t, x) − (τ, x) dx V (bξ (t, ·) − bξ (τ, ·)) = ∂x ∂x R

≤ (M3 C21 + eM2 /p C11 )|ω(t) − ω(τ )|.

(7.32)

Combining (7.31) and (7.32) we see that for all t, τ ∈ R+ , bξ (t, ·) − bξ (τ, ·) V ≤ M6 |ω(t) − ω(τ )|, where M6 := e C2∞ + M3 C21 + eM2 /p C11 . From this inequality it follows that bξ is a continuous V (R)-valued function. Moreover, it is bounded in view of (7.30). Thus, bξ ∈ Cb (R+ , V (R)). M2 /p

9. Proof of bξ ∈ SO(R+ , V (R)). Estimate (7.31) immediately implies that M2 /p ∞ cmC C2 osc(ω, [r, 2r]), r (bξ ) ≤ e

r ∈ R+ .

Since ω ∈ SO(R+ ), from this estimate we obtain lim cmC r (bξ ) = lim osc(ω, [r, 2r]) = 0 (s ∈ {0, ∞}).

r→s

r→s

Thus, taking into account the result of Step 8, we conclude that bξ belongs to the algebra SO(R+ , V (R)).

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10. Proof of bξ ∈ E(R+ , V (R)). From (7.27) it follows that for h ∈ R, sup bξ (t, ·) − bhξ (t, ·) L∞ (R) ≤ M5 |h|.

(7.33)

t∈R+

On the other hand, from (7.23) we obtain for t ∈ R+ and h > 0,

∂bξ ∂bξ dx (t, x + h) − (t, x) ∂x ∂x R x+h x+h 2 2 ∂ bξ ∂ b ξ dx ≤ dy dx = (t, y)dy (t, y) 2 2 ∂y ∂y

V (bξ (t, ·) − bhξ (t, ·)) =

R

x

R

x

x+h x+h ≤ M4 |y + i/p| |rp (y)|dy dx + 2M3 |rp (y)|dy dx. R

x

R

x

(7.34)

Changing the order of integration and taking into account (7.2), we get for h ∈ R and j ∈ {0, 1}, y x+h j |y + i/p| |rp (y)|dy dx = |y + i/p|j |rp (y)|dx dy R

R y−h

x

=h

|y + i/p|j |rp (y)|dy = Cj1 h. (7.35)

R

Combining (7.34) and (7.35), we see that V (bξ (t, ·) − bhξ (t, ·)) ≤ M7 h (h > 0),

(7.36)

where M7 := M4 C11 + 2M3 C01 . Analogously it can be shown that V (bξ (t, ·) − bhξ (t, ·)) ≤ M7 (−h) (h < 0). From (7.36) and (7.37) we get for h ∈ R, sup V bξ (t, ·) − bhξ (t, ·) ≤ M7 |h|.

(7.37)

(7.38)

t∈R+

Combining (7.33) with (7.38), we arrive at the equality lim sup bξ (t, ·) − bhξ (t, ·) V = 0.

|h|→0 t∈R+

(7.39)

From Step 9 and equality (7.39) it ﬁnally follows that bξ ∈ E(R+ , V (R)).

8. Sufficient Conditions for Fredholmness π 8.1. Invertibility in the Quotient Algebras Λπ +∞ and Λ−∞

Theorem 8.1. Suppose a, b, c, d ∈ SO(R+ ), α ∈ SOS(R+ ), and the operator N is given by (1.6). (a) If the operator A+ := aI − bWα is invertible on the space Lp (R+ ), then π is invertible in the quotient algebra Λπ+∞ . the coset N π + J+∞

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(b) If the operator A− := cI − dWα is invertible on the space Lp (R+ ), then π is invertible in the quotient algebra Λπ−∞ . the coset N π + J−∞ Proof. (a) If A+ is invertible in B, then from Theorem 1.1 it follows that A−1 + ∈ F. Then from Theorem 6.8 we see that A± ∈ Λ and A+ is invertible in the algebra Λ. Hence the coset Aπ+ = A+ + K is invertible in the quotient π in the algebra Λπ , which implies the invertibility of the coset Aπ+ + J+∞ π quotient algebra Λ+∞ . It remains to observe that π π π N π + J+∞ = (A+ P+ + A− P− )π + J+∞ = Aπ+ + J+∞ .

Part (a) is proved. The proof of part (b) is analogous. 8.2. Invertibility in the Quotient Algebras Λπ ξ,x with (ξ, x) ∈ Δ × R

Lemma 8.2. Suppose α is a slowly oscillating shift and ω is its exponent function. For ξ ∈ Δ, let the function bξ be defined by (7.16), (7.17). Then the operator Φ−1 Op(bξ )Φ belongs to the algebra Λ. Proof. Let g ∈ SO(R+ ) and gp (t, x) := g(t)rp (x),

sp (t, x) = sp (x),

(t, x) ∈ R+ × R.

Lemmas 7.1 and 7.4 imply that the functions sp , gp , and bξ belong to the algebra E(R+ , V (R)). From this observation and Theorem 5.2 it follows that Op(sp ) Op(bξ ) − Op(bξ ) Op(sp ) ∈ K(Lp (R+ , dμ)), Op(gp ) Op(bξ ) − Op(bξ ) Op(gp ) ∈ K(Lp (R+ , dμ)).

(8.1)

Since E(R+ , V (R)) ⊂ Cb (R+ , V (R)), we infer from Theorem 5.1 that Bξ := Φ−1 Op(bξ )Φ ∈ B. Then relations (8.1) and the equalities S = Φ−1 Co(sp )Φ = Φ−1 Op(sp )Φ,

gR = gΦ−1 Co(rp )Φ = Φ−1 Op(gp )Φ

[see Theorem 4.3(b)] imply that SBξ − Bξ S ∈ K and gRBξ − Bξ gR ∈ K. Hence Bξ ∈ Λ. Lemma 8.3. Suppose α is a slowly oscillating shift and ω is its exponent function. If (ξ, x) ∈ Δ × R, then π (Wα R2 )π − eiω(ξ)(x+i/p) (rp (x))2 I π ∈ Jξ,x .

Proof. From [13, formula 3.194.4] it follows that for k > 0 and y ∈ R, 1 1 t1/p −iy dt 1 t = 1/p−iy · = ei(y+i/p) log k rp (y). πi 1 + kt t i sin[π(1/p − iy)] k R+

Taking the inverse Mellin transform, we get for k, t ∈ R+ , 1 1 t1/p = ei(y+i/p) log k rp (y)tiy dy. πi 1 + kt 2π R

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Assume that f ∈ C0∞ (R+ ). Since α(τ ) = eω(τ ) τ , from the above identity it follows that for t ∈ R+ , 1 f (τ )(t/τ )1/p f (τ )(t/τ )1/p dτ 1 dτ = (ΦWα RΦ−1 f )(t) = πi τ + α(t) πi 1 + eω(t) (t/τ ) τ R+

1 = 2π =

1 2π

R+

⎛ ⎝

R+

R

dy R

⎞ iy t dτ eiω(t)(y+i/p) rp (y) dy ⎠ f (τ ) τ τ eiω(t)(y+i/p) rp (y)

R+

iy t dτ f (τ ) . τ τ

Hence, for f ∈ C0∞ (R+ ), we have ΦWα RΦ−1 f = Op(c)f , where c(t, y) := eiω(t)(y+i/p) rp (y),

(t, y) ∈ (R+ × R) ∪ (Δ × R).

By Lemma 7.3, this function belongs to the algebra Cb (R+ , V (R)). Then from Theorem 5.1 it follows that Op(c) extends to a bounded operator on Lp (R+ , dμ) and therefore ΦWα RΦ−1 = Op(c).

(8.2)

On the other hand, from Theorem 4.3(b) and Lemma 7.1 it follows that ΦRΦ−1 = Co(rp ) = Op(rp ),

(8.3)

where rp (t, y) = rp (y) for (t, y) ∈ R+ × R. From (8.2) and (8.3) we obtain Wα R2 = Φ−1 Op(a)Φ,

(8.4)

where a(t, y) := c(t, y)rp (t, y) = eiω(t)(y+i/p) (rp (y))2 , (t, y) ∈ (R+ × R) ∪ (Δ × R). Let us represent this function in the form a(t, y) = a(t, y) − a(ξ, y) + c(ξ, y)rp (y) = (a(t, y) − a(ξ, y)) + (c(ξ, y) − c(ξ, x)) rp (t, y) + c(ξ, x) (rp (t, y) − rp (t, x)) + a(ξ, x).

(8.5)

From Lemma 7.4 and Theorem 4.3(b) it follows that Φ−1 Op (a − a(ξ, ·)) Φ = (ω − ω(ξ)) RΦ−1 Op(bξ )Φ, where bξ ∈ E(R+ , V (R)). By Lemma 8.2, the operator Φ−1 Op(bξ )Φ belongs to the algebra Λ. It is easy to see that by Corollary 4.4(b), π

([(ω − ω(ξ)) R] ) (ξ, x) = (ω(ξ) − ω(ξ)) rp (x) = 0. π and thus Therefore [(ω − ω(ξ))R]π ∈ Iξ,x −1 π π π Φ Op (a − a(ξ, ·)) Φ = (ω − ω(ξ)) RΦ−1 Op(bξ )Φ ∈ Jξ,x . (8.6)

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Since ω(ξ) ∈ R, we see that Re(π−iω(ξ)) = π ∈ (0, 2π). From Theorem 4.3(b) we conclude that Φ−1 Op(c(ξ, ·))Φ = Φ−1 Co(rp,π−iω(ξ) )Φ = Rπ−iω(ξ) ∈ A. Hence, by Corollary 4.4(b), π Φ−1 Op (c(ξ, ·) − c(ξ, x)) Φ (ξ, x) = [Rπ−iω(ξ) − c(ξ, x)I]π (ξ, x) = rp,π−iω(ξ) (x) − c(ξ, x) = 0. Therefore −1 π π Φ Op (c(ξ, ·) − c(ξ, x)) Φ = [Rπ−iω(ξ) − c(ξ, x)I]π ∈ Iξ,x and thus −1 π π π Φ Op [(c(ξ, ·) − c(ξ, x)) rp ] Φ = Rπ−iω(ξ) − c(ξ, x)I R ∈ Jξ,x . (8.7) Finally, in view of Corollary 4.4(b), −1 [Φ Op(rp − rp (x))Φ]π (ξ, x) = ([R − rp (x)I]π ) (ξ, x) = rp (x) − rp (x) = 0. Hence π π c(ξ, x)[Φ−1 Op(rp − rp (x))Φ]π ∈ Iξ,x ⊂ Jξ,x .

(8.8)

Combining (8.4)–(8.8), we arrive at

π π (Wα R2 )π − eiω(ξ)(x+i/p) (rp (x))2 I π = Φ−1 Op(a)Φ − a(ξ, x)I ∈ Jξ,x ,

which ﬁnishes the proof.

Theorem 8.4. Suppose a, b, c, d ∈ SO(R+ ), α ∈ SOS(R+ ), and the operator N is given by (1.6). If nξ (x) = 0 for some (ξ, x) ∈ Δ × R, where the function π is invertible in the quotient nξ is defined by (1.7), then the coset N π + Jξ,x π algebra Λξ,x . Proof. Fix (ξ, x) ∈ Δ × R and consider the operators 2 1 1 ± sp (x) H± := R2 . 2 rp (x)

(8.9)

Then from Corollary 4.4(b) it follows that π (H± )(ξ, x) =

1 ± sp (x) . 2

π Therefore (P± − H± )π ∈ Iξ,x and π

π , [(aI − bWα )(P+ − H+ ) + (cI − dWα )(P− − H− )] ∈ Jξ,x

whence π

π π N π + Jξ,x = [(aI − bWα )H+ + (cI − dWα )H− ] + Jξ,x .

(8.10)

Since H± ∈ idA {R} ⊂ A, we infer from Corollary 6.4 that (Wα H+ )π = (H+ Wα )π ,

(Wα H− )π = (H− Wα )π .

(8.11)

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Taking into account Corollary 4.4(b), it is easy to see that [(bH+ )π − (b(ξ)H+ )π ] (ξ, x) = 0,

[(dH− )π − (d(ξ)H− )π ] (ξ, x) = 0 (8.12)

and [(aH+ )π − (a(ξ)H+ )π ] (ξ, x) = 0,

[(cH− )π − (c(ξ)H− )π ] (ξ, x) = 0.

Hence π π (aH+ )π − (a(ξ)H+ )π , (cH− )π − (c(ξ)H− )π ∈ Iξ,x ⊂ Jξ,x .

(8.13)

Taking into account (8.11)–(8.12), we also see that π , (bWα H+ )π − (b(ξ)Wα H+ )π = [(bH+ )π − (b(ξ)H+ )π ]Wαπ ∈ Jξ,x

(8.14)

π Jξ,x .

(8.15)

(dWα H− ) − (d(ξ)Wα H− ) = [(dH− ) − (d(ξ)H− ) π

π

π

π

]Wαπ

∈

From (8.10) and (8.13)–(8.15) it follows that π

π π = (a(ξ)H+ − b(ξ)Wα H+ + c(ξ)H− − d(ξ)Wα H− ) + Jξ,x . N π + Jξ,x

(8.16) It is easy to see that π (1/rp (x))2 R2 − I (ξ, x) = 0. Hence π H± −

1 ± sp (x) π π π I ∈ Iξ,x ⊂ Jξ,x . 2

(8.17)

By Lemma 8.3 and (8.9), (Wα H± )π − eiω(ξ)(x+i/p)

1 ± sp (x) π π I ∈ Jξ,x . 2

(8.18)

Combining (8.16)–(8.18), we arrive at the relation π π N π + Jξ,x = nξ (x)I π + Jξ,x ,

where nξ (x) is given by (1.7). If nξ (x) = 0, then one can check straightforπ π is the inverse of the coset N π + Jξ,x in the wardly that (1/nξ (x))I π + Jξ,x π quotient algebra Λξ,x . 8.3. Proof of Theorem 1.2 If condition (i) of Theorem 1.2 is fulﬁlled, then by Theorem 8.1 the cosets π are invertible in the quotient algebras Λπ±∞ , respectively. On the N π + J±∞ other hand, if condition (ii) of Theorem 1.2 holds, then in view of Theoπ is invertible in the quotient algebra Λπξ,x for rem 8.4, the coset N π + Jξ,x every pair (ξ, x) ∈ Δ × R. Then, by Theorem 6.12, the operator N ∈ Λ is Fredholm.

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References [1] Antonevich, A.B.: Linear Functional Equations. Operator Approach. Operator Theory: Advances and Applications, vol. 83. Birkh¨ auser, Basel (1995) [2] Bastos, M.A., Fernandes, C.A., Karlovich, Yu.I.: C ∗ -algebras of integral operators with piecewise slowly oscillating coefﬁcients and shifts acting freely. Integr. Equ. Oper. Theory 55, 19–67 (2006) [3] Bastos, M.A., Fernandes, C.A., Karlovich, Yu.I.: Spectral measures in C ∗ -algebras of singular integral operators with shifts. J. Funct. Anal. 242, 86– 126 (2007) [4] Bastos, M.A., Fernandes, C.A., Karlovich, Yu.I.: C ∗ -algebras of singular integral operators with shifts having the same nonempty set of ﬁxed points. Complex Anal. Oper. Theory 2, 241–272 (2008) [5] B¨ ottcher, A., Karlovich, Yu.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics, vol. 154. Birkh¨ auser, Basel (1997) [6] B¨ ottcher, A., Karlovich, Yu.I., Rabinovich, V.S.: Mellin pseudodifferential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights. Manuscr. Math. 95, 363–376 (1998) [7] B¨ ottcher, A., Karlovich, Yu.I., Rabinovich, V.S.: The method of limit operators for one-dimensional singular integrals with slowly oscillating data. J. Oper. Theory 43, 171–198 (2000) [8] B¨ ottcher, A., Karlovich, Yu.I., Spitkovsky, I.M.: Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications, vol. 131. Birkh¨ auser, Basel (2002) [9] B¨ ottcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006) [10] DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993) [11] Duduchava, R.: On algebras generated by convolutions and discontinuous functions. Integr. Equ. Oper. Theory 10, 505–530 (1987) [12] Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations. Vols. 1 and 2. Operator Theory: Advances and Applications, vols. 53– 54. Birkh¨ auser, Basel (1992) [13] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007) [14] Karapetiants, N., Samko, S.: Equations with Involutive Operators. Birkh¨ auser, Boston (2001) [15] Karlovich, A.Yu., Karlovich, Yu.I.: Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts. In: Factorization, Singular Operators and Related Problems, pp. 111–129. Kluwer Academic Publishers, Dordrecht (2003) [16] Karlovich, A.Yu., Karlovich, Yu.I., Lebre, A.B.: Invertibility of functional operators with slowly oscillating non-Carleman shifts. In: Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, vol. 142, pp. 147–174 (2003) [17] Karlovich, A.Yu., Karlovich, Yu.I., Lebre, A.B.: Necessary conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data. Integr. Equ. Oper. Theory. arXiv:1010.5336 (2010) (submitted)

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[18] Karlovich, Yu.I.: An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. Lond. Math. Soc. (3) 92, 713–761 (2006) [19] Karlovich, Yu.I.: Pseudodifferential operators with compound slowly oscillating symbols. In: The Extended Field of Operator Theory. Operator Theory: Advances and Applications, vol. 171, pp. 189–224 (2006) [20] Karlovich, Yu.I.: Nonlocal singular integral operators with slowly oscillating data. In: Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol. 181, pp. 229–261 (2008) [21] Karlovich, Yu.I.: The Haseman boundary value problem with slowly oscillating data. In: Analytic Methods of Analysis and Differential Equations: Amade2006, pp. 81–110. Cambridge Scientiﬁc Publishers, Cambridge (2008) [22] Karlovich, Yu.I.: An algebra of shift-invariant singular integral operators with slowly oscillating data and its application to operators with a Carleman shift. In: Analysis, Partial Differential Equations and Applications. The Vladimir Maz’ya Anniversary Volume. Operator Theory: Advances and Applications, vol. 193, pp. 81–95 (2009) [23] Karlovich, Yu.I., Kravchenko, V.G.: Singular integral equations with nonCarleman shift on an open contour. Differ. Equ. 17, 1408–1417 (1981) [24] Karlovich, Yu.I., Lebre, A.B.: Algebra of singular integral operators with a Carleman backward slowly oscillating shift. Integr. Equ. Oper. Theory 41, 288– 323 (2001) [25] Krasnosel’skii, M.A., Zabreiko, P.P., Pustyl’nik, E.I., Sobolevskii, P.E.: Integral Operators in Spaces of Summable Functions. Noordhoﬀ International Publishing, Leyden (1976) [26] Kravchenko, V.G., Litvinchuk, G.S.: Introduction to the Theory of Singular Integral Operators with Shift. Mathematics and its Applications, vol. 289. Kluwer Academic Publishers, Dordrecht (1994) [27] Litvinchuk, G.S.: Boundary Value Problems and Singular Integral Equations with Shift. Nauka, Moscow (1977, in Russian) [28] Mikhlin, S.G., Pr¨ ossdorf, S.: Singular Integral Operators. Springer, Berlin (1986) [29] Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar Publishing Co., New York (1955) [30] Rabinovich, V.S.: Algebras of singular integral operators on compound contours with nodes that are logarithmic whirl points. Izv. Math. 60, 1261– 1292 (1996) [31] Rabinovich, V.S.: Mellin pseudodifferential operators techniques in the theory of singular integral operators on some Carleson curves. In: Differential and Integral Operators (Regensburg, 1995). Operator Theory: Advances and Applications, vol. 102, pp. 201–218 (1998) [32] Roch, S., Silbermann, B.: Algebras of Convolution Operators and their Image in the Calkin Algebra. Report MATH, 90-05. Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut f¨ ur Mathematik, Berlin (1990) [33] Sarason, D.: Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26, 817–838 (1977) [34] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

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Alexei Yu. Karlovich (B) Departamento de Matem´ atica Faculdade de Ciˆencias e Tecnologia Universidade Nova de Lisboa Quinta da Torre 2829-516 Caparica Portugal e-mail: [email protected] Yuri I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa C.P. 62209 Cuernavaca, Morelos M´exico e-mail: [email protected] Amarino B. Lebre Departamento de Matem´ atica Instituto Superior T´ecnico Universidade T´ecnica de Lisboa Av. Rovisco Pais 1049-001 Lisboa Portugal e-mail: [email protected] Received: September 28, 2010. Revised: December 29, 2010.

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Integr. Equ. Oper. Theory 70 (2011), 485–510 DOI 10.1007/s00020-011-1872-5 Published online March 11, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Noncommutative Fig`a-Talamanca–Herz Algebras for Schur Multipliers C´edric Arhancet Abstract. In this work, we introduce a noncommutative analogue of the Fig` a-Talamanca–Herz algebra Ap (G) on the natural predual of the operator space Mp,cb of completely bounded Schur multipliers on the Schatten space Sp . We determine the isometric Schur multipliers and prove that the space Mp of bounded Schur multipliers on the Schatten space Sp is the closure in the weak operator topology of the span of isometric multipliers. Mathematics Subject Classification (2000). Primary 46L51; Secondary 46M35, 46L07. Keywords. Fig` a-Talamanca–Herz algebra, noncommutative Lp -spaces, complex interpolation, Schur multipliers, operator spaces.

1. Introduction The Fourier algebra A(G) of a locally compact group G was introduced by Eymard [9]. The algebra A(G) is the predual of the group von Neumann then the fourier transalgebra V N (G). If G is abelian with dual group G, form induces an isometric isomorphism of L1 (G) onto A(G). In [10], A. Fig`aTalamanca showed, if G is abelian, that the natural predual of the Banach space of the bounded Fourier multipliers on Lp (G) is isometrically isomorphic to a space Ap (G) of continuous functions on G. Moreover A2 (G) = A(G) isometrically. In [12] and [9], Herz proved that the space Ap (G) is a Banach algebra for the usual product of functions (see also [Pie]). Hence Ap (G) is an Lp -analogue of the Fourier algebra A(G). These algebras are called Fig` a-Talamanca–Herz algebras. In [24], V. Runde introduced an operator space analogue OAp (G) of the algebra Ap (G). The underlying Banach space of OAp (G) is different from the Banach space Ap (G). Moreover, it is possible to show (in using a suitable variant of [15, Theorem 5.6.1]) that OAp (G) is the natural predual of the operator space of the completely bounded Fourier This work is partially supported by ANR 06-BLAN-0015.

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multipliers. We refer to [5,6,14] and [25] for other operator space analogues of Ap (G). The purpose of this article is to introduce noncommutative analogues of these algebras in the context of completely bounded Schur multipliers on Schatten spaces Sp . Recall that a map T : Sp → Sp is completely bounded if IdSp ⊗ T is bounded on Sp (Sp ). If 1 p < ∞, the operator space CB(Sp ) of completely bounded maps from Sp into itself is naturally a dual operator space. Indeed, we have a completely isometric isomorphism CB(Sp ) = denote the operator space projective tensor product. p∗ ∗ where ⊗ Sp ⊗S Moreover, we will prove that the subspace Mp,cb of completely bounded Schur multipliers is a maximal commutative subset of CB(Sp ). Consequently, the subspace Mp,cb is w*-closed in CB(Sp ). Hence Mp,cb is naturally a dual oper p∗ /(Mp,cb )⊥ ∗ . If we denote by ψp : Sp ⊗S p∗ → ator space with Mp,cb = Sp ⊗S S1 the map (A, B) → A ∗ B, where ∗ is the Schur product, we will show that (Mp,cb )⊥ = Ker ψp . Now, we deﬁne the operator space Rp,cb as the space p∗ /Ker ψp . We Im ψp equipped with the operator space structure of Sp ⊗S ∗ have completely isometrically (Rp,cb ) = Mp,cb . Moreover, by definition, we have a completely contractive inclusion Rp,cb ⊂ S1 . Recall that elements of S1 can be regarded as inﬁnite matrices. Our principal result is the following theorem. Theorem 1.1. Suppose 1 p < ∞. The predual Rp,cb of the operator space Mp,cb equipped with the usual matricial product or the Schur product is a completely contractive Banach algebra. In [27] and [17], Strichartz and Parott showed that if 1 p ∞, p = 2 every isometric Fourier multiplier on Lp (G) is a scalar multiple of an operator induced by a translation. In [10], A. Fig` a-Talamanca showed that the space of bounded Fourier multipliers is the closure in the weak operator topology of the span of these operators. We give noncommutative analogues of these two results. Theorem 1.2. 1. Suppose 1 p ∞. If p = 2, an isometric Schur multiplier on Sp is defined by a matrix [ai bj ] with ai , bj ∈ T. 2. Suppose 1 p < ∞. The space Mp of bounded Schur multipliers on Sp is the closure of the span of isometric Schur multipliers in the weak operator topology. The paper is organized as follows. In Sect. 2, we ﬁx notations and we show that the natural preduals of Mp and Mp,cb admit concrete realizations as spaces of matrices. We give elementary properties of these spaces. In Sect. 3, we show that the operator space Mp,cb equipped with the matricial product is a completely contractive Banach algebra. In Sect. 4, we turn to the Schur product. We observe that the natural predual Rp of the Banach space Mp of bounded Schur multipliers is a Banach algebra for the Schur product. Moreover, we show that the space Rp,cb equipped with the Schur product is a completely contractive Banach algebra.

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In Sect. 5, we determine the isometric Schur multipliers on Sp and prove that the space Mp is the closure in the weak operator topology of the span of isometric multipliers.

2. Predual of Spaces of Schur Multipliers Let us recall some basic notations. Let T = {z ∈ C | |z| = 1} and δij the symbol of Kronecker. If E and F are Banach spaces, B(E, F ) is the space of bounded linear maps between E and F . We denote by ⊗γ the Banach projective tensor product. If E, F and G are Banach spaces we have (E ⊗γ F )∗ = B(E, F ∗ ) isometrically. In particular, if E is a dual Banach space, B(E) is also a dual Banach space. If (E0 , E1 ) is a compatible couple of Banach spaces we denote by (E0 , E1 )θ the intermediate space obtained by complex interpolation between E0 and E1 . The readers are refereed to [3,7,18] and [23] for the details on operator spaces and completely bounded maps. We let CB(E, F ) for the space of all completely bounded maps endowed with the norm

T E − →F,cb = sup IdMn ⊗ u Mn (E)− →Mn (F ) . n1

When E and F are two operator spaces, CB(E, F ) is an operator space for the structure corresponding to the isometric identiﬁcations Mn (CB(E, F )) = CB (E, Mn (F )). The dual operator space of E is E ∗ = CB(E, C). If E and F are operator spaces then the adjoint map T → T ∗ from CB(E, F ) into CB(F ∗ , E ∗ ) is a complete isometry. I by RI the If I is a set, we by CI the operator space B C, 2 and denote I I operator space B 2 , C . We have a complete isometry B 2 = CB (CI ) (see [3, (1.14)]). The complex interpolated space between two compatible operator spaces E0 and E1 is the usual Banach space Eθ with the matrix norms corresponding to the isometric identiﬁcations Mn (Eθ ) = (Mn (E0 ), Mn (E1 ))θ . Let F0 , F1 be two another compatible operator spaces. Let ϕ : E0 + E1 → F0 + F1 be a linear map. If ϕ is completely bounded as a map from E0 into F0 , and from E1 into F1 , then, for any 0 θ 1, ϕ is completely bounded from Eθ into Fθ with

ϕ cb,Eθ − →Fθ ( ϕ cb,E0 − →F0 )

1−θ

θ

( ϕ cb,E1 − →F1 ) .

If E0 ∩ E1 is dense in both E0 and E1 , we have a completely contractive inclusion (CB(E0 ), CB(E1 ))θ ⊂ CB(Eθ ) (see [11, Lemma 0.2]). the operator space projective tensor product, by ⊗min We denote by ⊗ the operator space minimal tensor product, by ⊗h the Haagerup tensor product, by ⊗σh the normal Haagerup tensor product, by ⊗ the normal spatial tensor product, by ⊗w∗ h the weak* Haagerup tensor product and by

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⊗eh the extended Haagerup tensor product (see [3,8] and [26]). Suppose that E, F, G and H are operator spaces. If ϕ : E → F and ψ : G → H are completely bounded maps then the maps ϕ ⊗ ψ : E ⊗h G → F ⊗h H and → F ⊗H are completely bounded and we have ϕ ⊗ ψ : E ⊗G

ϕ ⊗ ψ cb,E⊗h G− →F ⊗h H ϕ cb,E→F ψ cb,G→H and ϕ cb,E→F ψ cb,G→H .

ϕ ⊗ ψ cb,E ⊗G − →F ⊗H If E, F are operator spaces, we have E ⊗h F ⊂ E ⊗w∗ h F completely isometrically [see [3] page 43]. If E, F and G are operator spaces, we denote by CB(E ×F, G) the space of jointly completely bounded map. We have G = CB (E, CB(F, G)) CB(E × F, G) = CB E ⊗F, ∗ = CB(E, F ∗ ) comcompletely isometrically. Consequently, we have E ⊗F pletely isometrically. In particular, if E is a dual operator space, CB(E) is also a dual operator space. At several times, we will use the next easy lemma left to the reader. Lemma 2.1. Suppose E and F are operator spaces. Let V : E → F and W : F → E be any completely contractive maps. Then the map ΘV,W : CB(E) −→ CB(F ) T −→ V T W is completely contractive. Moreover, if E and F are reflexive then this map is also w*-continuous. A Banach algebra A equipped with an operator space structure is called completely contractive if the algebra product (a, b) − → ab from A × A to A is a jointly completely contractive bilinear map. We equip I∞ with its natural operator space structure coming from its structure as a C ∗ -algebra and the Banach space I1 with its natural operator space structure coming from its structure of predual of I∞ . If I is an index set and if E is a vector space, we write MI (E) for the space of the I × I matrices with entries in E. We denote by Mﬁn I (E) the subspace of matrices with a ﬁnite number of non null entries. For I = {1, . . . , n}, we simplify the notations, we let Mn (E) for M{1,...,n} (E). We write Mﬁn for Mﬁn N (C). We use the inclusion MI ⊗ MI ⊂ MI×I with the identiﬁcation [A ⊗ B](t,r),(u,s) = atu brs . For all i, j, k, l ∈ I, the tensor eij ⊗ ekl identiﬁes to the matrix [δit δju δkr δls ](t,r),(u,s)∈I×I (see [7] page 5 for more information on these identiﬁcations). Given a set I, the set Pf (I) of all ﬁnite subsets of I is directed with respect to set inclusion. For J ∈ Pf (I) and A ∈ MI , we write TJ (A) for the matrix obtained from A by setting each entry to zero if its row and column index are not both in J. We call (TJ (A))J∈Pf (I) the net of ﬁnite submatrices of A.

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The Schatten-von Neumann class SpI , 1 p < ∞, is the space of those p1 compact operators A from I2 into I2 such that A SpI = Tr (A∗ A) 2 p < ∞. I of compact operators from I2 into I2 is equipped with the The space S∞ operator norm. For I = N, we simplify the notations, we let Sp for SpN . K I I K S∞ of compact operators from I2 ⊗2 K The space S∞ 2 into 2 ⊗2 2 is I K equipped with the operator norm. If 1 p < ∞, the space Sp Sp is the I K space of those compact operators C from I2 ⊗2 K 2 into 2 ⊗2 2 such that p1 p ∗

C SpI (SpK ) = (Tr ⊗ Tr )(C C) 2 < ∞. I Elements of Sp are regarded as matrices A = [aij ]i,j∈I of MI . If A ∈ SpI we denote by AT the operator of SpI whose the matrix is the matrix transpose of A. If 1 p ∞, A ∈ SpI and B ∈ SpI∗ , the operator AB T belongs to S1I . We let A, B SpI ,SpI∗ = Tr AB T . We have A, B SpI ,SpI∗ = limJ i,j∈J aij bij . I We equip S∞ with its natural operator space structure coming from its structure as a C ∗ -algebra. We equip S1I with its natural operator space I . If 1 < p < ∞, we give structure coming from its structure as dual of S∞ I I I , S1I 1 completely on Sp the operator space structure deﬁned by Sp = S∞ p

isometrically (see [23] page 140 for interesting remarks on thisdefinition). By the same way, we deﬁne an operator space structure on SpI SpK . We have completely isometrically SpI SpK = SpK SpI = SpI×K . We will often silently use these identiﬁcations. By the same way, we deﬁne SpI SpK (SpL ) and similar operator space structures. G. Pisier showed that a map T : SpI → SpI is completely bounded if IdSp ⊗ T is bounded on Sp SpI (see [21, Lemma 1.7]). The readers are refereed to [21] for the details on operator space structures on the Schatten-von Neumann class. We denote by ∗ the Schur (Hadamard) product: if A = [aij ]i,j∈I and B = [bij ]i,j∈I are matrices of MI we have A ∗ B = [aij bij ]i,j∈I . We recall that a matrix A of MI deﬁnes a Schur multiplier MA on SpI if for any B ∈ SpI the matrix MA (B) = A ∗ B represents an element of SpI . In this case, by the closed graph theorem, the linear map B → MA (B) is bounded on SpI . The notation MIp stands for the algebra of all bounded Schur multipliers on the Schatten space SpI . We denote by MIp,cb the space of completely bounded Schur multipliers on SpI . We give the space MIp,cb the operator space structure induced by CB SpI . For I = N, we simplify the notations, we let Mp for N I I MN p and Mp,cb for Mp,cb . Recall that if A ∈ Sp , we have MA ∈ Mp (see [3] page 225). If MC ∈ MIp , we have MC ∈ MIp∗ . Moreover, if A ∈ SpI and B ∈ SpI∗ , we have MC (A), B S I ,S I∗ = A, MC (B) S I ,S I∗ . p

p

p

p

If 1 p ∞, the Banach spaces MIp and MIp∗ are isometric and the operator spaces MIp,cb and MIp∗ ,cb are completely isometric. We have MI∞ = MI∞,cb isometrically (see e.g. [16, Remark 2.2] and [13, Lemma 2]). Moreover, we

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have MI∞,cb = I∞ ⊗w∗ h I∞ completely isometrically (see e.g. [26, Theorem 3.1]) and MI2 = I×I ∞ isometrically. If MA ∈ MIp is a Schur multiplier, we have MTJ (A) B(S I ) MA B(SpI ) p

for any ﬁnite subset J of I. Moreover, if MA ∈ MIp,cb , we have for any ﬁnite subset J of I the inequality MTJ (A) CB(S I ) MA CB(SpI ) . p

It is well-known that the map (A, B) → A ∗ B from SpI × SpI∗ into S1I is contractive. In order to study the preduals of MIp and MIp,cb , we need to show that this map is jointly completely contractive. Proposition 2.2. Suppose 1 p ∞. The bilinear map SpI × SpI∗ −→ S1I (A, B) −→ A ∗ B is jointly completely contractive. Proof. We denote β : I2 → I∞ the canonical contractive map. We have

β cb,CI →I∞ = β I2 →I∞ 1

and β cb,RI →I∞ = β I2 →I∞ 1

(see [3, (1.10)]). Then by tensoring, the map CI ⊗h RI → I∞ ⊗h I∞ is completely contractive. Now recall that we have a completely isometric canonical I ) map I∞ ⊗h I∞ → MI∞ and a completely isometric map T → T ∗ from CB(S∞ I into CB(S1 ). Then the map I = CI ⊗h RI −→ I∞ ⊗h I∞ −→ MI∞ −→ CB(S1I ) S∞

eij −→ ei ⊗ ej −→ Meij −→ Meij I is completely contractive. This means that the map A → MA from S∞ into I × CB(S1I ) is completely contractive. Then the map (A, B) → A ∗ B from S∞ S1I into S1I is completely jointly contractive. By the commutativity of ∗ and I the map from S1I × S∞ into S1I is also completely jointly contractive. ⊗, Finally, we obtain the result by bilinear interpolation (see [23] page 57 and [2] page 96).

Then we can deﬁne the completely contractive map pI∗ −→ S1I ψpI : SpI ⊗S A ⊗ B −→ A ∗ B. pI∗ , the map ψpI induces a conAs SpI ⊗γ SpI∗ embeds contractively into SpI ⊗S I I I traction from Sp ⊗γ Sp∗ into S1 , which we denote by ϕIp . We let ψp = ψpN . The following theorem (and the comments which follow) is a noncommutative version of a theorem of Fig` a-Talamanca [10]. This latter theorem states that the natural predual of the space of bounded Fourier multipliers admits a concrete realization as a space Ap (G) of continuous functions on G. In the pI∗ and B(SpI ), SpI ⊗γ SpI∗ sequel, we consider the dual pairs CB(SpI ), SpI ⊗S where 1 p < ∞.

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Theorem 2.3. Suppose 1 p < ∞. 1. The pre-annihilator MIp,cb of the space MIp,cb of completely bounded ⊥

2.

Schur multipliers on SpI is equal to Ker ψpI . We have a complete isometry pI∗ /Ker ψpI ∗ . MIp,cb = SpI ⊗S I The pre-annihilator Mp ⊥ of the space MIp of bounded Schur multipli ers on SpI is equal to Ker ϕIp . We have an isometry MIp = SpI ⊗γ SpI∗ / ∗ Ker ϕIp .

Proof. We will only prove the part 1. The proof of part 2 is similar. Let C = l I I I k=1 Ak ⊗ Bk ∈ Sp ⊗ Sp∗ . Note that, for all integers k, we have MAk ∈ Mp . If i, j are elements of I we have

l

Meij , C CB(S I ),S I ⊗S Meij , Ak ⊗ Bk I = p

p∗

p

I∗ CB(SpI ),SpI ⊗S p

k=1

=

l

eij ∗ Ak , Bk S I ,S I∗ p

k=1

=

l

eij , Ak ∗ Bk S I ,S I∗ p

k=1

=

p

eij ,

l

p

Ak ∗ Bk

k=1

= ψpI (C) ij .

I pI∗ , we have Meij , C By continuity, if C ∈ SpI ⊗S I = ψp (C) ij . CB(S I ),S I ⊗S p

p∗

p

We deduce that, if C ∈ Ker ψpI and MD ∈ MIp,cb , we have for all J ∈ Pf (I)

MTJ (D) , C CB(S I ),S I ⊗S I = 0. p

p

p∗

so

Now, it is easy to see that we have MTJ (D) −→ MD in CB(SpI ) (i.e., for J

wo

→ MD (A)). Then MTJ (D) −−→ MD in all A ∈ SpI , we have MTJ (D) (A) − J J I CB(Sp ). Moreover, recall that, for all J ∈ Pf (I), we have MTJ (D) MI p,cb

w∗

MD MIp,cb . Thus MTJ (D) −−→ MD . Consequently, if C ∈ Ker ψpI and MD ∈ MIp,cb we have

J

MD , C CB(SpI ),SpI ⊗S I∗ = lim MTJ (D) , C CB(S I ),S I ⊗S I∗ = 0. p p p J p ⊥ ⊂ Thus we have Ker ψpI ⊂ MIp,cb . Now we will show that Ker ψpI ⊥ ⊥ MIp,cb . Suppose that T ∈ Ker ψpI . If i, j, k, l are elements of I such that (i, j) = (k, l), the tensor eij ⊗ ekl belongs to Ker ψpI . Therefore we have T (eij ), ekl S I ,S I∗ = T, eij ⊗ ekl CB(S I ),S I ⊗S I∗ p

p

p

= 0.

p

p

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⊥ ⊂ MIp,cb . Since Hence T is a Schur multiplier. We conclude that Ker ψpI pI∗ we deduce that Ker ψpI is norm-closed in SpI ⊗S I ⊥ Mp,cb ⊥ ⊂ Ker ψpI = Ker ψpI . ⊥

Then the ﬁrst claim of part 1 of the theorem is proved. Now, we will show that MIp,cb is a maximal commutative subset of CB(SpI ). Let T : SpI → SpI be a bounded map which commutes with all Schur multipliers Meij : SpI → SpI where i, j ∈ I. Then, for all i, j, k, l ∈ I such that (i, j) = (k, l) we have

T (eij ), ekl S I ,S I∗ = T Meij (eij ), ekl S I ,S I p p p p∗

= Meij T (eij ), ekl S I ,S I p p∗

= T (eij ), Meij (ekl ) S I ,S I p

p∗

= 0. Hence T is a Schur multiplier. This proves the claim. Then MIp,cb is weak* closed in CB(SpI ). We immediately deduce the second claim of part 1 of the theorem. If 1 p < ∞, we deﬁne the operator space RIp,cb as the space Im ψpI I∗ /Ker ψpI . We let Rp,cb = equipped with the operator space structure of SpI ⊗S ∗ p I RN = MIp,cb . By definition, we p,cb . We have completely isometrically Rp,cb have a completely contractive inclusion RIp,cb ⊂ S1I . We deﬁne the Banach space RIp as the space Im ϕIp equipped with the norm of SpI ⊗γ SpI∗ /Ker ϕIp . I ∗ = MIp . We let Rp = RN p . We have isometrically Rp I By duality, well-known results on Mp and MIp,cb translate immediately into results on RIp and RIp,cb . If 1 p < ∞, there is a contractive inclusion RIp ⊂ RIp,cb . If 1 < p < ∞, the Banach spaces RIp and RIp∗ are isometric and the operator spaces RIp,cb and RIp∗ ,cb are completely isometric. We have a completely isometric isomorphism I1 ⊗h I1 −→ RI1,cb ei ⊗ ej −→ eij

(2.1)

and isometric isomorphisms I1 ⊗h I1 −→ RI1 ei ⊗ ej −→ eij

and

I×I −→ RI2 = RI2,cb 1 eij −→ eij .

Suppose 1 p q 2, we have injective contractive maps MI1 ⊂ MIp ⊂ MIq ⊂ MI2

and

MI1,cb ⊂ MIp,cb ⊂ MIq,cb ⊂ MI2,cb

(see [11] page 219). One more time, by duality, we deduce that we have injective contractive inclusions RI2 ⊂ RIq ⊂ RIp ⊂ RI1

and

RI2,cb ⊂ RIq,cb ⊂ RIp,cb ⊂ RI1,cb .

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Actually, the last inclusions are completely contractive. It is a part of Proposition 2.7. Suppose 1 p < ∞. By a well-known property of the Banach projective tensor product, an element C in S1I belongs to RIp if and only if there exists two sequences (An )n1 ⊂ SpI and (Bn )n1 ⊂ SpI∗ such that the series +∞ +∞ I I n=1 An ⊗ Bn converge absolutely in Sp ⊗Sp∗ and C = n=1 An ∗ Bn in S1I . Moreover, we have +∞ +∞

An SpI Bn SpI∗ | C = An ∗ Bn (2.2)

C RIp = inf n=1

n=1

where the infimum is taken over all possible ways to represent C as before. I ﬁn We observe that we have an inclusion Mﬁn I ⊂ Rp . It is clear that MI is dense I I in Rp and Rp,cb . Remark 2.4. The Banach spaces MIp and MIp,cb contain the space I∞ . We deduce that, if I is inﬁnite, the Banach spaces MIp , MIp,cb , RIp and RIp,cb are not reﬂexive. Now we make precise the duality between the operator spaces MIp,cb and on the one hand and the Banach spaces MIp and RIp on the other hand. I I Moreover, the next lemma speciﬁes the density of Mﬁn I in Rp and Rp,cb . RIp,cb

Lemma 2.5. Suppose 1 p < ∞. 1. If J is a finite subset of I, the truncation map TJ : RIp,cb → RIp,cb is completely contractive. Moreover, if A ∈ RIp,cb , we have in RIp,cb TJ (A) − → A.

(2.3)

J

2.

For any completely bounded Schur multiplier MA ∈ MIp,cb and any B ∈ RIp,cb , we have aij bij . (2.4) MA , B MI ,RI = lim p,cb

p,cb

J

i,j∈J

3.

If J is a finite subset of I, the truncation map TJ : RIp → RIp is contractive. Moreover, if A ∈ RIp , we have TJ (A) − → A in RIp .

4.

For any bounded Schurmultiplier MA ∈ MIp and any B ∈ RIp , we have MA , B MIp ,RIp = limJ i,j∈J aij bij .

J

Proof. We only prove the assertions for the operator space RIp,cb . If i, j are elements of I and MA ∈ MIp,cb , we have MA , eij MI

I p,cb ,Rp,cb

= MA , eij ∗ eij MI

I p,cb ,Rp,cb

= MA (eij ), eij S I ,S I∗ p

p

= aij . Then we deduce that, for all MA ∈ MIp,cb and all B ∈ Mﬁn I , we have MA , B MI ,RI = a b . Now, it is not difﬁcult to see that, for ij ij i,j∈I p,cb

p,cb

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any ﬁnite subset J of I, the truncation map TJ : SpI → SpI is completely contractive. Then, it follows easily that the truncation map TJ : MIp,cb → MIp,cb is completely contractive. Hence, by duality and by using the density of Mﬁn I in RIp,cb , we deduce that the truncation map TJ : RIp,cb → RIp,cb is completely I contractive. Furthermore, by density of Mﬁn I in Rp,cb , it is not difﬁcult to prove the assertion (2.3). Finally, the equality (2.4) is now immediate.

Finally, we end the section by giving supplementary properties of these operator spaces. For that, we need the following proposition inspired by [16, Proposition 2.4]. If x, y ∈ R, we denote by Mx,y : SpI → SpI the Schur multi plier associated with the matrix eixr eiys r,s∈I of MI and by M x,y : SpI → SpI the Schur multiplier associated with the matrix e−ixr e−iys r,s∈I of MI . It is easy to see that, for all x, y ∈ R, the maps Mx,y : SpI → SpI and M x,y : SpI → SpI are completely contractive. We denote by dx the normalized measure on [0, 2π]. Proposition 2.6. Suppose 1 p ∞. The space MIp,cb of completely bounded Schur multipliers on SpI is 1-completely complemented in the space CB SpI . Proof. Let T : SpI → SpI be a completely bounded map. For any A ∈ Mﬁn I the map [0, 2π] × [0, 2π] −→ SpI (x, y) −→ Mx,y T M x,y (A) is continuous and we have 2π 2π Mx,y T M x,y (A)dxdy 0

0

2π2π

SpI

Mx,y T M x,y (A) I dxdy S p

0

0

Mx,y S I − →SpI M x,y S I − →S I T SpI − →S I A SpI dxdy p

0

2π2π

p

p

p

0

T SpI − →SpI A SpI . By the previous computation, we deduce that there exists a unique linear map P (T ) : SpI → SpI such that for all A ∈ SpI we have 2π2π (P (T )) (A) =

Mx,y T M x,y (A)dxdy. 0

0

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l Moreover, for all k=1 Ak ⊗ Bk ∈ Mﬁn ⊗ SpI we have l Ak ⊗ Bk IdSp ⊗ P (T ) k=1 Sp (SpI ) l 2π2π = Ak ⊗ Mx,y T M x,y (Bk )dxdy k=1 0 0 Sp (SpI ) 2π 2π l IdSp ⊗ Mx,y T M x,y = Ak ⊗ Bk dxdy k=1 0 0 Sp (SpI ) l T cb,SpI →SpI Ak ⊗ Bk . k=1

Sp (SpI )

Thus we see that the linear map P (T ) is actually completely bounded and that we have P (T ) cb,S I →S I T cb,SpI →SpI . Now, for all r, s, k, l ∈ I we p p have 2π2π P (T )ers , ekl SpI ,S I∗ =

p

0

Mx,y T M x,y ers , ekl

0

2π2π =

SpI ,SpI∗

dxdy

e−ιxr e−ιys Mx,y T ers , ekl S I ,S I∗ dxdy p

0

p

0

⎞ ⎛ 2π 2π e−ιxr e−ιys eιxk eιyl dxdy ⎠ T ers , ekl =⎝ 0

SpI ,SpI∗

0

⎛ 2π ⎞ ⎛ 2π ⎞ = ⎝ eιx(k−r) dx⎠ ⎝ eιy(l−s) dy ⎠ T ers , ekl 0

SpI ,SpI∗

0

= δrk δsl T (ers ), ekl SpI ,S I∗ . p

Then the linear map P (T ) : SpI → SpI is a Schur multiplier. Moreover, if T : SpI → SpI is a Schur multiplier, we have P (T ) = T . Now, if T ∈ Mn CB(SpI ) and [Akl ]1k,lm ∈ Mm SpI , with the notations of Lemma 2.1, we have ⎡ ⎤ 2π2π ⎣ ⎦ Mx,y Tij M x,y (Akl )dxdy 1i,jn 0 0 I 1k,lm Mmn (Sp )

2π2π Mx,y Tij M x,y 1i,jn 0

0

Mn (CB(SpI ))

[Akl ] dxdy

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2π2π IdMn ⊗ ΘMx,y ,M x,y (T ) 0

0

Mn (CB(SpI ))

T Mn (CB(SpI )) [Akl ]1k,lm Mm (S I )

[Akl ] dxdy

by Lemma 2.1.

p

Thus we obtain

(IdMn ⊗ P )(T ) Mn (CB(SpI )) = [P (Tij )]1i,jn

Mn (CB(SpI ))

T Mn (CB(SpI )) . We deduce that the map P : CB SpI → MIp,cb is completely contractive. The proof is complete.

Proposition 2.7. 1.

We have completely isometric isomorphisms

I1 −→ RI2,cb I1 ⊗ ei ⊗ ej −→ eij 2.

and

I I×I ∞ −→ M2,cb

A −→ MA .

Suppose 1 p q 2. We have injective completely contractive maps

MI1,cb

⊂ MIp,cb ⊂ MIq,cb ⊂ MI2,cb

and

RI2,cb ⊂ RIq,cb ⊂ RIp,cb ⊂ RI1,cb .

Proof. 1) By minimality, we have a completely contractive map MI2,cb → I×I ∞ . We will show that the inverse map is completely contractive. We have a complete isometry I I×I ∞ −→ B S2 = CB(CI×I ) A −→ MA . Now we know that (RI×I )∗ = CI×I . Then we deduce a complete isometry I×I ∞ −→ CB(CI×I ) −→ CB(RI×I ) A −→ MA −→ (MA )∗ = MA .

By interpolation, we deduce a complete contraction I×I ∞ → (CB(CI×I ), CB(RI×I )) 1 . 2

S2I

2)

Recall that we have (CI×I , RI×I ) 1 = completely isometrically (see 2 [21] pages 137 and 140). Then we have a complete contraction (CB(CI×I ), CB(RI×I )) 1 → CB S2I . 2 Finally, we obtain a complete contraction I×I → CB S2I . We obtain ∞ − the other isomorphism by duality. Let 1 p q 2. Recall that we have a contraction from MIp,cb into completely MI2,cb (see [11] page 219). Moreover we have MI2,cb = I×I ∞ I isometrically. Thus we have a complete contraction Mp,cb → MI2,cb . Now, there exists 0 θ 1 with SqI = SpI , S2I θ . Moreover, the identity mapping MIp,cb → MIp,cb is completely contractive. By interpolation, we

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obtain a complete contraction MIp,cb → MIp,cb , MI2,cb . On one hand, θ we know that we have a complete contraction CB SpI , CB S2I θ → CB SpI , S2I θ = CB SqI . On the other hand, the space MIp,cb of completely bounded Schur mul tipliers is 1-completely complemented in the space CB SpI . Then we have a complete contraction MIp,cb , MI2,cb → MIq,cb . By composition, θ

we deduce that we have a complete contraction MIp,cb ⊂ MIq,cb . We obtain the other completely contractive maps by duality.

3. Non Commutative Fig`a-Talamanca–Herz Algebras We begin with the cases p = 1 and p = 2. Recall that we have a completely isometric isomorphism RI1,cb = I1 ⊗h I1 [see (2.1)] and a completely contractive inclusion RI1,cb ⊂ S1I . Hence, the trace on S1I induces a completely contractive functional Tr : I1 ⊗h I1 −→ C ei ⊗ ej −→ δij . By tensoring, we deduce a completely contractive map IdI1 ⊗ Tr ⊗IdI1 : I1 ⊗h I1 ⊗h I1 ⊗h I1 → I1 ⊗h I1 . By composition with the canonical completely contractive map I I 1 ⊗h I1 − 1 ⊗h I1 ⊗ → I1 ⊗h I1 ⊗h I1 ⊗h I1 we obtain a completely contractive map I 1 ⊗h I1 → I1 ⊗h I1 . IdI1 ⊗ Tr ⊗IdI1 : I1 ⊗h I1 ⊗ With the identiﬁcation RI1,cb = I1 ⊗h I1 , we obtain the completely contractive map I1,cb −→ RI1,cb RI1,cb ⊗R A ⊗ B −→ AB. RI1,cb

equipped with the matricial product is a comThis means that the space I1 pletely contractive Banach algebra. Now, recall that we have RI2,cb = I1 ⊗ I completely isometrically. Then, by a similar argument, R2,cb equipped with the matricial product is also a completely contractive Banach algebra. For other values of p, the proof is more complicated since we do not have any explicit description of RIp,cb . In the following proposition, we give a link between RIp,cb and RI×I p,cb . Proposition 3.1. Suppose 1 p < ∞. Then there exists a canonical complete contraction Ip,cb −→ RI×I RIp,cb ⊗R p,cb A ⊗ B −→ A ⊗ B.

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Proof. The identity mapping on SpI ⊗ SpI extends to a complete contraction pI → SpI (SpI ). Hence by tensoring, we obtain a completely contractive SpI ⊗S map pI∗ (SpI∗ ). pI ⊗S pI∗ ⊗S pI∗ → SpI (SpI )⊗S β : SpI ⊗S pI∗ − → RIp,cb is a complete quotient map. By [7, Proposition The map ψpI : SpI ⊗S 7.1.7], we obtain a complete quotient map pI∗ ⊗S pI ⊗S pI∗ → RIp,cb ⊗R Ip,cb . ψpI ⊗ ψpI : SpI ⊗S the map Finally, by the commutativity of ⊗, I I I I pI ⊗S pI∗ ⊗S pI∗ α : Sp ⊗Sp∗ ⊗Sp ⊗Sp∗ −→ SpI ⊗S A ⊗ B ⊗ C ⊗ D −→ A ⊗ C ⊗ B ⊗ D is completely isometric. We will prove that there exists a unique linear map such that the following diagram is commutative and that this map is completely contractive. pI∗ ⊗S pI ⊗S pI∗ SpI ⊗S

/ SpI ⊗S pI ⊗S pI∗ ⊗S pI∗

α

β

/ SpI (SpI )⊗S pI∗ (SpI∗ )

ψpI ⊗ψpI

ψpI×I

/ RI×I

Ip,cb RIp,cb ⊗R

p,cb

Ip,cb = SpI ⊗S pI∗ ⊗S pI ⊗S pI∗ /Ker ψpI ⊗ ψpI completely isoWe have RIp,cb ⊗R I metrically. It sufﬁces to show that Ker ψp ⊗ ψpI ⊂ Ker ψpI×I βα . By [7, Proposition 7.1.7], we have the equality pI∗ + SpI ⊗S pI∗ ⊗ Ker ψpI . Ker ψpI ⊗ ψpI = closure Ker ψpI ⊗ SpI ⊗S pI∗ ⊗S pI ⊗S pI∗ , it sufﬁces to Since the space Ker ψpI×I βα is closed in SpI ⊗S show that pI∗ + SpI ⊗S p∗ ⊗ Ker ψpI ⊂ Ker ψpI×I βα . Ker ψpI ⊗ SpI ⊗S pI∗ . There exists integers ni , mj , matrices Ak,i , Cl,j ∈ Let E ∈ Ker ψpI ⊗SpI ⊗S I I Sp and Bk,i , Dl,j ∈ Sp∗ such that the sequences mj n i Ak,i ⊗ Bk,i and Cl,j ⊗ Dl,j

k=1

are convergent in E=

l=1

i1

pI∗ , SpI ⊗S lim

ni

i→+∞

Ak,i ⊗ Bk,i

k=1

and

ψpI

lim

i→+∞

ni k=1

⊗

lim

j→+∞

j1

mj

Cl,j ⊗ Dl,j

l=1

Ak,i ⊗ Bk,i

= 0.

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Then, in the space S1I , we have ni

Ak,i ∗ Bk,i −−−−→ 0.

(3.1)

i→+∞

k=1

pI∗ → S1I , the seMoreover, note that, by continuity of the map ψpI : SpI ⊗S mj quence l=1 Cl,j ∗ Dl,j j1 is convergent. Now, we have ψpI×I βα(E)

= ψpI×I βα

lim

ni

i→+∞

Ak,i ⊗ Bk,i

⊗

k=1 mj

= lim

ni

lim

i→+∞ j→+∞

= lim

lim

i→+∞ j→+∞

= lim

lim

i→+∞ j→+∞

= lim

lim

i→+∞ j→+∞

= =0

lim

i→+∞

ni

k=1 l=1 mj ni k=1 l=1 mj ni k=1 l=1 mj ni

Cl,j ⊗ Dl,j

l=1

ψpI×I βα (Ak,i ⊗ Bk,i ⊗ Cl,j ⊗ Dl,j ) ψpI×I (Ak,i ⊗ Cl,j ⊗ Bk,i ⊗ Dl,j ) (Ak,i ⊗ Cl,j ) ∗ (Bk,i ⊗ Dl,j ) (Ak,i ∗ Bk,i ) ⊗ (Cl,j ∗ Dl,j )

k=1 l=1

Ak,i ∗ Bk,i

k=1

lim

j→+∞

mj

⊗

lim

j→+∞

mj

Cl,j ∗ Dl,j

l=1

by (3.1).

pI∗ ⊗ Ker ψpI ⊂ Ker ψpI×I βα by a similar computaWe prove that SpI ⊗S tion. The proof is complete. ﬁn ﬁn → Mﬁn Now, we deﬁne the map V : Mﬁn I ⊗ MI − I ⊗ MI by V (eij ⊗ ekl ) = δkl eik ⊗ ekj . ﬁn Proposition 3.2. With respect to trace duality, the map W : Mﬁn I ⊗ MI → ﬁn ﬁn MI ⊗ MI defined by

W (eij ⊗ ekl ) = δjk eil ⊗ ejj is the dual map of V . Moreover, the map V induces a partial isometry V : S2I ⊗2 S2I → S2I ⊗2 S2I . Proof. For all i, j, k, l, r, s, t, u ∈ I, we have Tr V (eij ⊗ ekl )(ers ⊗ etu )T = δkl Tr (eik ⊗ ekj ) eTrs ⊗ eTtu = δkl Tr eik eTrs Tr ekj eTtu = δklst δir δju

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and

T = δst Tr (eij ⊗ ekl )(eru ⊗ ess )T Tr (eij ⊗ ekl ) (W (ers ⊗ etu )) = δst Tr eij eTru Tr ekl eTss = δklst δir δju .

We conclude that W is the dual map of V . The fact that V induces a partial isometry is clear. ﬁn ﬁn ﬁn Proposition 3.3. Suppose 1 p ∞. The maps V : Mﬁn I ⊗MI → MI ⊗MI ﬁn ﬁn ﬁn ﬁn and W : MI ⊗ MI → MI ⊗ MI admit completely contractive extensions V : SpI (SpI ) → SpI (SpI ) and W : SpI (SpI ) → SpI (SpI ).

Proof. We ﬁrst prove that V and W admit completely contractive extensions I I I I (S∞ ) into S∞ (S∞ ). Suppose that B = i,j,k,l∈J bijkl ⊗ eij ⊗ ekl ∈ from S∞ ﬁn ∈ Mﬁn for all i, j, k, l ∈ J. Note Mﬁn ⊗ Mﬁn I ⊗ MI with J ∈ Pf (I) and bijkl J J (S∞ ) is unitary. Then we have that the matrix U = r,s∈J ers ⊗ esr of S∞

(IdS∞ ⊗ V )(B) S∞ (S I (S I )) = b ⊗ e ⊗ e ijkk ik kj ∞ ∞ i,j,k∈J I (S I )) S∞ (S∞ ⎛ ⎞⎞ ⎛ ⎞ ∞ ⎛ ⎝ ⎝ = ers ⊗ esr ⎠⎠ ⎝ bijkk ⊗ eik ⊗ ekj ⎠ IS∞ ⊗ r,s∈J i,j,k∈J I (S I )) S∞ (S∞ ∞ = bijkk ⊗ ers eik ⊗ esr ekj r,s,i,j,k∈J I (S I )) S∞ (S∞ ∞ = bijkk ⊗ ekk ⊗ eij i,j,k∈J I (S I )) S∞ (S∞ ⎞ ∞ ⎛ ⎠ ⎝ = ekk ⊗ bijkk ⊗ eij I k∈J i,j∈I I )) S∞ (S∞ (S∞ = max b ⊗ e ijkk ij k∈J I I i,j∈I S∞ (S∞ )

B S∞ (S∞ (S∞ ))

(submatrices)

and

(IdS∞ ⊗ W )(B) S∞ (S∞ = b ⊗ e ⊗ e I (S I )) ijjl il jj ∞ i,j,l∈J I (S I )) S∞ (S∞ ∞ ⎞ ⎛ ⎝ = bijjl ⊗ eil ⊗ ejj ⎠ (IS∞ ⊗ U ) (IS∞ ⊗ U ) i,j,l∈J I I S∞ (S∞ (S∞ ))

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= b ⊗ e e e ⊗ e e e ijjl rs il tu sr jj ut r,s,i,j,l,t,u∈J I (S I )) S∞ (S∞ ∞ = bijjl ⊗ ejj ⊗ eil i,j,l∈J I (S I )) S∞ (S∞ ⎞ ∞ ⎛ ⎠ ⎝ = ejj ⊗ bijjl ⊗ eil I j∈J i,l∈J I )) S∞ (S∞ (S∞ = max bijjl ⊗ eil j∈J i,l∈J I ) S∞ (S∞ bijkl ⊗ ekj ⊗ eil (submatrices) i,j,k,l∈J I I S∞ (S∞ (S∞ )) ⎛ ⎞⎞ ⎛ ⎞ ⎛ ⎝IS∞ ⊗ ⎝ ⎠ ⎠ ⎝ ⎠ e ⊗ e b ⊗ e ⊗ e = rs sr ijkl kj il r,s∈J i,j,k,l∈J I (S I )) S∞ (S∞ ∞ = bijkl ⊗ ers ekj ⊗ esr eil r,s,i,j,k∈J I I S∞ (S∞ (S∞ ))

= B S∞ (S∞ (S∞ )) .

Then we deduce the claim. Hence, by duality, the maps V ∗ : S1I S1I → S1I S1I and W ∗ : S1I S1I → S1I S1I are completely contractive. Moreover, we know that W = V ∗ . By interpolation between p = 1 and p = ∞, we obtain that the maps V : SpI SpI → SpI SpI and W : SpI SpI → SpI SpI are completely contractive. Now, we deﬁne the linear map Δ : MI −→ MI×I A −→ [ats δur ](t,r),(u,s)∈I×I . Proposition 3.4. Let 1 p ∞. Suppose that MA : SpI → SpI is a completely bounded Schur multiplier on SpI associated with a matrix A of MI . Then the map V MA ⊗ IdSpI W is a bounded Schur multiplier on SpI (SpI ). Its associated matrix is Δ(A). Proof. If i, j, k, l ∈ I and MA ∈ MIp,cb , we have MΔ(A) (eij ⊗ ekl ) = [ats δur ](t,r),(u,s)∈I×I ∗ [δit δju δkr δls ](t,r),(u,s)∈I×I = δjk ail [δit δju δkr δls ](t,r),(u,s)∈I×I = δjk ail eik ⊗ ekl

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and

V MA ⊗ IdSpI W (eij ⊗ ekl ) = δjk V MA ⊗ IdSpI (eil ⊗ ejj ) = δjk ail V (eil ⊗ ekk ) = δjk ail eik ⊗ ekl .

Recall that, for all operator spaces E and F , the map R ⊗ T → R ⊗ T is completely contractive from CB(E) ⊗CB(F ) into CB (E ⊗min F ) and from CB(E)⊗CB(F ) into CB E ⊗F (see [4, Proposition 5.11]). Proposition 3.5. Suppose 1 p ∞. Let I, J be any sets. The map CB SpI −→ CB SpI (SpJ ) T −→ T ⊗ IdSpJ is a complete contraction.

I J J pI S p = S∞ ⊗min SpI and S1J SpI = S1J ⊗S Proof. By definition, we have S∞ completely isometrically. Then we obtain two complete contractions J J I I CB SpI −→ CB S∞ (Sp ) ⊗CB Sp −→ CB S∞ J ⊗ T J ⊗ T −→ IdS∞ T −→ IdS∞ and

I CB SpI −→ CB S1J ⊗CB Sp −→ CB S1J (SpI ) −→ IdS1J ⊗ T. T −→ IdS1J ⊗ T

By interpolation, we obtain a completely contractive map J I → CB S∞ (Sp ) , CB S1J (SpI ) 1 . CB SpI − p

We conclude by composing with the complete contraction J I CB S∞ (Sp ) , CB S1J (SpI ) 1 − → CB SpJ (SpI ) p

and by using Fubini’s theorem (see [21, Theorem 1.9]).

Remark 3.6. If the set J is not empty, it is easy to see that this map is completely isometric. The next theorem is the principal result of this paper. Theorem 3.7. Suppose 1 p < ∞. The space RIp,cb equipped with the usual matricial product is a completely contractive Banach algebra. More precisely, if A and B are matrices of RIp,cb and i, j ∈ I, the limit limJ k∈J aik bkj ex ists. Moreover, the matrix A.B of MI defined by [A.B]ij = limJ k∈J aik bkj belongs to RIp,cb . Finally, the map Ip,cb −→ RIp,cb RIp,cb ⊗R A⊗B −→ AB is completely contractive.

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Proof. We have already seen that it sufﬁces to prove the theorem with 1 < p < ∞. If MA ∈ MIp,cb , by Proposition 3.4, we have the following commutative diagram SpI (SpI )

MΔ(A)

/ SpI (SpI ) O

W

V

SpI (SpI )

/ SpI (SpI ).

MA ⊗IdS I

p

By Proposition 3.5, the map MA → MA ⊗ IdSpI is completely contractive

from MIp,cb into MI×I . Moreover it is easy to see that this map is w*-con p,cb I I tinuous. Since Sp Sp is reﬂexive, by Lemma 2.1 and by composition, the map MA → MΔ(A) from MIp,cb into MI×I p,cb is a complete contraction and I×I is w*-continuous. We denote by Δ∗ : Rp,cb − → RIp,cb its preadjoint. Now, by Lemma 2.5, we have for all i, j ∈ I and for all matrices A, B of Mﬁn I

[Δ∗ (A ⊗ B)]ij = Meij , Δ∗ (A ⊗ B) MI ,RI p,cb p,cb

= MΔ(eij ) , A ⊗ B MI×I ,RI×I p,cb p,cb = M[δit δjs δur ](t,r),(u,s)∈I×I , [atu brs ](t,r),(u,s)∈I×I I×I I×I = lim J

Mp,cb ,Rp,cb

air brj

r∈J

= [A.B]ij . Thus we conclude that, if A, B ∈ Mﬁn I , we have Δ∗ (A ⊗ B) = AB. By Propﬁn I I osition 3.1 and by density of Mﬁn I ⊗ MI in Rp,cb ⊗Rp,cb , we deduce that the map Δ

∗ I×I ﬁn −→ RIp,cb Mﬁn I ⊗ MI −→ Rp,cb − A ⊗ B −→ A ⊗ B −→ AB

Ip,cb into RIp,cb . Moreover, admits a unique bounded extension from RIp,cb ⊗R this map is completely contractive. Finally, we complete the proof by a straightforward approximation argument using Lemma 2.5. Remark 3.8. We do not know if the space RIp equipped with the usual matricial product is a Banach algebra. The Banach space analogue of Proposition 3.5 is false. It is the reason which explains that the method does not work for RIp . However, note that if MIp = MIp,cb isometrically we have RIp = RIp,cb isometrically. For 1 < p < ∞, p = 2 the equality MIp = MIp,cb is a classical open question.

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4. Schur Product In this section, we replace the matricial product by the Schur product. First, it is easy to show the following proposition. Proposition 4.1. Suppose 1 p < ∞. The Banach space RIp equipped with the Schur product is a commutative Banach algebra. Proof. It sufﬁces to use the equality (2.2) and the fact that SpI equipped with the Schur product is a Banach algebra (see [3] page 225). Now we will show the completely bounded analogue of this proposition. We deﬁne the pointwise product I1 −→ I1 P : I1 ⊗ ei ⊗ ej −→ δij ei . This map is well-deﬁned and is completely contractive (see [3] page 211). Then, by tensoring, we obtain a completely contractive map I1 ⊗h I1 ⊗ I1 → I1 ⊗h I1 . P ⊗ P : I1 ⊗ (4.1) By [8, Theorem 6.1], the map I ∞ ⊗I∞ ⊗σh I∞ ⊗I∞ −→ I∞ ⊗σh I∞ ⊗ I∞ ⊗σh I∞ a ⊗ b ⊗ c ⊗ d −→ a ⊗ c ⊗ b ⊗ d is completely contractive. Moreover, by [8, (5.23)], we have the following commutative diagram I / I∞ ⊗σh I∞ ⊗ I∞ ⊗σh I∞ ∞ ⊗I∞ ⊗σh I∞ ⊗I∞ O O

I ? ∞ ⊗I∞ ⊗eh I∞ ⊗I∞

? / I∞ ⊗eh I∞ ⊗ I∞ ⊗eh I∞ .

By [8, Theorem 4.2], [8, Theorem 5.3] and by duality, we deduce that the map I I I1 ⊗h I1 ⊗ I1 1 ⊗h I1 −→ I1 ⊗ 1 ⊗h I1 ⊗ a ⊗ b ⊗ c ⊗ d −→ a ⊗ c ⊗ b ⊗ d is well-deﬁned and completely contractive. Composing this map and (4.1), we deduce a completely contractive map I I 1 ⊗h I1 −→ I1 ⊗h I1 1 ⊗h I1 ⊗ a ⊗ b ⊗ c ⊗ d −→ P (a ⊗ c) ⊗ P (b ⊗ d). With the identiﬁcation RI1,cb = I1 ⊗h I1 , we obtain a completely contractive map I1,cb −→ RI1,cb RI1,cb ⊗R A ⊗ B −→ A ∗ B.

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This means that RI1,cb equipped with the Schur product is a completely con I1 completely tractive Banach algebra. Now, recall that we have RI2,cb = I1 ⊗ isometrically. Then, by a similar argument, RI2,cb equipped with the Schur product is also a completely contractive Banach algebra. We will use a strategy similar to that used in the proof of Theorem 3.7 for other values of p. We deﬁne the Schur multiplier ME : SpI (SpI ) → SpI (SpI ) associated with the matrix E = [δrt δsu ](t,r),(u,s)∈I×I ∈ MI×I . It is not difﬁcult to see that ME is a completely positive contraction. Note that, for all i, j, k, l ∈ I, we have ME (eij ⊗ ekl ) = [δrt δsu ](t,r),(u,s)∈I×I ∗ [δit δju δkr δls ](t,r),(u,s)∈I×I = δik δjl [δit δju δkr δls ](t,r),(u,s)∈I×I = δik δjl eij ⊗ ekl . Now, we deﬁne the linear map η : MI −→ MI×I A −→ [ars δrt δsu ](t,r),(u,s)∈I×I . Proposition 4.2. Let 1 p ∞. Suppose that MA : SpI → SpI is a completely bounded Schur multiplier on SpI associated with a matrix A. Then the map ME (MA ⊗ IdSpI )ME is a bounded Schur multiplier on SpI (SpI ). Its associated matrix is η(A). Proof. If i, j, k, l ∈ I and MA ∈ MIp,cb , we have Mη(A) (eij ⊗ ekl ) = [ars δrt δsu ](t,r),(u,s)∈I×I ∗ [δit δju δkr δls ](t,r),(u,s)∈I×I = δik δjl aij [δit δju δkr δls ](t,r),(u,s)∈I×I = δik δjl aij eij ⊗ ekl

(4.2)

and ME (MA ⊗ IdSpI )ME (eij ⊗ ekl ) = δik δjl ME (MA ⊗ IdSpI )(eij ⊗ ekl ) = δik δjl aij eij ⊗ ekl . Theorem 4.3. Suppose 1 p < ∞. The space RIp,cb equipped with the Schur product is a commutative completely contractive Banach algebra. Proof. We have already seen that it sufﬁces to prove the theorem with 1 < p < ∞. If MA ∈ MIp,cb , by Proposition 4.2, we have the following commutative diagram

506

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IEOT / SpI SpI O

Mη(A)

ME

ME

SpI SpI

/ SpI SpI .

MA ⊗IdS I

p

We have already seen that the map MA → MA ⊗ IdSpI is completely con I I tractive from MIp,cb into MI×I p,cb and w*-continuous. Since Sp Sp is reﬂexive, by Lemma 2.1 and by composition, the map MA → Mη(A) from MIp,cb into MI×I p,cb is a complete contraction and is w*-continuous. I We denote by η∗ : RI×I p,cb → Rp,cb its preadjoint. Now, by Lemma 2.5, we have for all i, j ∈ I and for all matrices A, B of Mﬁn I

[η∗ (A ⊗ B)]ij = Meij , η∗ (A ⊗ B) MI ,RI p,cb p,cb

= Mη(eij ) , A ⊗ B MI×I ,RI×I p,cb p,cb = M[δir δjs δrt δsu ](t,r),(u,s)∈I×I , [atu brs ](t,r),(u,s)∈I×I I×I I×I Mp,cb ,Rp,cb

= aij bij = [A ∗ B]ij . Thus we conclude that if A, B ∈ Mﬁn we have η∗ (A ⊗ B) = A ∗ B. By I ﬁn I I ⊗ M Proposition 3.1 and by density of Mﬁn I I in Rp,cb ⊗Rp,cb , we deduce that the map η∗

I×I ﬁn I Mﬁn I ⊗ MI −→ Rp,cb −→ Rp,cb

A ⊗ B −→ A ⊗ B −→ A ∗ B Ip,cb into RIp,cb . Moreover, admits a unique bounded extension from RIp,cb ⊗R this map is completely contractive. Finally, we complete the proof by a straightforward approximation argument with Lemma 2.5. Now, we will give a more simple proof of this theorem. It is easy to see that η induces a completely isometric map η : SpI → SpI SpI . Moreover, by the computation (4.2), its range is clearly 1-completely complemented by ME : SpI SpI → SpI SpI . We denote by η −1 : η SpI SpI → SpI the inverse map of η. For all B ∈ η SpI (SpI ) , we have η −1 (B) = b(r,r),(s,s) r,s∈I . Finally, for all i, j, k, l ∈ I we have ηMA η −1 ME (eij ⊗ ekl ) = δik δjl ηMA η −1 (eij ⊗ ekl ) = δik δjl ηMA η −1 [δit δju δkr δls ](t,r),(u,s)∈I×I = δik δjl ηMA ([δir δjs δkr δls ]r,s∈I ) = δik δjl aij η ([δir δjs δkr δls ]r,s∈I )

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= δik δjl aij eij ⊗ ekl = Mη(A) (eij ⊗ ekl ) where we have used the computation (4.2) in the last equality. Hence we have the following commutative diagram Mη(A) / SpI SpI SpI SpI O ME

η SpI (SpI )

η

η −1

SpI

/ SpI .

MA

We conclude with an argument similar to that used in the proof of Theorem 4.3.

5. Isometric Multipliers The next result is the noncommutative version of a theorem of Parrott [17] and Strichartz [27] which states that every isometric Fourier multiplier on Lp (G) for 1 p ∞, p = 2, is a scalar multiple of an operator induced by a translation. Theorem 5.1. Suppose 1 p ∞, p = 2. An isometric Schur multiplier on SpI is defined by a matrix [ai bj ] with ai , bj ∈ T. Proof. Suppose that MC is an isometric Schur multiplier on the Banach space SpI deﬁned by a matrix C. First, we observe that MC is onto. Indeed, for all i, j ∈ I, we have MC (eij ) = cij eij . Then cij = 0 since MC is one-to-one. Consequently eij belongs to the range of MC . By density, we conclude that MC is onto. Now we use the theorem of Arazy [1] which describes the onto isome tries on SpI . Then there exists two unitaries U = [uij ] and V = [vij ] of B I2 satisfying for all A ∈ SpI C ∗ A = U AV

or

C ∗ A = U AT V.

Examine the ﬁrst case, we have for all k, l ∈ I U ekl V = C ∗ ekl . Hence, for all i, j ∈ I, we have the equality [U ekl V ]ij = [C ∗ ekl ]ij .

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Since [U ekl V ]ij = uik vlj we have

uik vlj =

ckl if i = k and j = l 0 if i = k or if j = l.

Then ukk vll = ckl . Each ckl is non null since the image of each ekl by the map MC cannot be null. Then, for all k and all l, we have ukk = 0 and vll = 0. And for i = k, we have uik vll = 0. Then if i = k, we have uik = 0. Now if j = l, we have ukk vlj = 0. Then if j = l, we have vlj = 0. Finally, for all i, j ∈ I, we deﬁne the complex numbers ai = uii and bj = vjj . Since the diagonal matrices U and V are unitaries, we have ai , bj ∈ T. Thus we have the required form. Examine the second case. We have for all k, l ∈ I U elk V = C ∗ ekl . We deduce that, for all i, j, k, l ∈ I, we have [U elk V ]ij = [C ∗ ekl ]ij . Since [U elk V ]ij = uil vkj we obtain ukl vkl = ckl and uil vkj = 0 if i = k or if j = l. Each ckl is non null since the image of each ekl by the map MC cannot be null. Then for all k, l we have ukl = 0 and vkl = 0. Thus the second case is absurd [if card(I) > 1]. The converse is straightforward. Remark 5.2. It is easy to see that an isometric Schur multiplier on S2I is deﬁned by a matrix [aij ] with aij ∈ T. The next result is the noncommutative version of a theorem of Fig` a-Talamanca [10] which states that the space of bounded Fourier multipliers is the closure in the weak operator topology of the span of translation operators. Theorem 5.3. Suppose 1 p < ∞. 1.

2.

The space MIp,cb of completely bounded Schur multipliers on SpI is the closure of the span of isometric Schur multipliers in the weak* topology and in the weak operator topology. The space MIp of bounded Schur multipliers on SpI is the closure of the span of isometric Schur multipliers in the weak* topology and in the weak operator topology.

Proof. We will only prove the part 1. The proof of the part 2 is similar. It is easy to see that an isometric Schur multiplier on SpI is completely isometric. This fact allows us to consider the span of isometric Schur multipliers in MIp,cb . Let C be a matrix of RIp,cb . Suppose that C belongs to the

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orthogonal of the set of isometric Schur multipliers. Thus, we have for any isometric multiplier M[ai bj ] (with ai , bj ∈ T)

0 = M[ai bj ] , C MI ,RI p,cb p,cb = lim ai bj cij . J

i,j∈J

Let i0 , j0 be elements of I. Now, we choose the ai ’s, bj ’s, ai ’s and bj ’s such that ai = bj = 1 for all i, j ∈ I, ai = −1 if i = i0 , ai0 = 1, bj = −1 if j = j0 and bj0 = 1. Then, we have ai bj cij + lim ai bj cij + lim ai bj cij + lim ai bj cij 0 = lim J

= lim J

J

i,j∈J

(ai +

ai )(bj

i,j∈J

+

J

i,j∈J

J

i,j∈J

bj )cij

i,j∈J

= 4ci0 j0 . Hence ci0 j0 = 0. It follows that C = 0. Then, we deduce that the space MIp,cb of completely bounded Schur multipliers is the closure of the span of isometric Schur multipliers in the weak* topology. Moreover, this topology is more ﬁner that the weak operator topology. Thus, we have proved the theorem. Acknowledgements The author is grateful to his thesis adviser Christian Le Merdy for support and advice. He also thanks Jean-Christophe Bourin, Pierre Fima and Eric Ricard for fruitful discussions and the anonymous referee for useful comments.

References [1] Arazy, J.: The isometries of Cp . Israel J. Math. 22, 247–256 (1975) [2] Bergh, J., L¨ ofstr¨ om, J.: Interpolation spaces. Springer, Berlin (1976) [3] Blecher, D., Le Merdy, C.: Operator algebras and their modules-an operator space approach. Oxford University Press, Oxford (2004) [4] Blecher, D., Paulsen, V.: Tensor products of operator spaces. J. Funct. Anal. 99, 262–292 (1991) [5] Daws, M.: p-Operator spaces and Fig` a-Talamanca–Herz algebras. J. Oper. Theor. 63, 47–83 (2010) [6] Daws, M.: Representing multipliers of the Fourier algebra on non-commutative Lp spaces (2009). arXiv:0906.5128v2[math.FA] [7] Eﬀros, E., Ruan, Z.-J.: Operator spaces. Oxford University Press, Oxford (2000) [8] Eﬀros, E., Ruan, Z.-J.: Operator space tensor products and Hopf convolution algebras. J. Oper. Theor. 50, 131–156 (2003) [9] Eymard, P.: Alg´ebres Ap et convoluteurs de Lp . In: S´eminaire Bourbaki, vol. 1969/1970, Expos´es 364–381. Springer, Berlin (1971)

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[10] Fig` a-Talamanca, A.: Translation invariant operators in Lp . Duke Math. 32, 495–501 (1965) [11] Harcharras, A.: Fourier analysis, Schur multipliers on S p and noncommutative Λ(p)-sets. Studia Math. 137, 203–260 (1999) [12] Herz, C.: The theory of p-spaces with an application to convolution Operators. Trans. Amer. Math. Soc. 154, 69–82 (1971) [13] Hladnik, M.: Compact Schur multipliers. Proc. Amer. Math. Soc. 128, 2585– 2591 (2000) [14] Lambert, A., Neufang, M., Runde, V.: Operator space structure and amenability for Fig` a-Talamanca–Herz algebras. J. Funct. Anal. 211, 245–269 (2004) [15] Larsen, R.: An introduction to the theory of multipliers. Springer, Berlin (1971) [16] Neuwirth, S.: Cycles and 1-unconditional matrices. Proc. Lond. Math. Soc. 93, 761–790 (2006) [17] Parott, S.K.: Isometric multipliers. Paciﬁc J. Math. 25, 159–166 (1968) [18] Paulsen, V.: Completely bounded maps and operator algebras. Cambridge University Press, Cambridge (2002) [19] Pier, J.-P.: Amenable locally compact groups. Wiley-Interscience, New york (1984) [20] Pisier, G.: The operator Hilbert space OH, complex interpolation and tensor norms. Mem. Amer. Math. Soc. 122 (1996) [21] Pisier, G.: Non-commutative vector valued Lp -spaces and completely p-summing maps. Ast´erisque 247 (1998) [22] Pisier, G.: Similarity problems and completely bounded maps. Lecture notes in mathematics, Expanded edition 1618. Springer, Berlin (2001) [23] Pisier, G.: Introduction to operator space theory. Cambridge University Press, Cambridge (2003) [24] Runde, V.: Operator Fig` a-Talamanca–Herz algebras. Studia Math. 155, 153–170 (2003) [25] Runde, V.: Representations of locally compact groups on QSLp -spaces and a p-analog of the Fourier-Stieltjes algebra. Paciﬁc J. Math. 221, 379–397 (2005) [26] Spronk, N.: Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras. Proc. Lond. Math. Soc. 89, 161–192 (2004) [27] Strichartz, R.S.: Isomorphisms of group algebras. Proc. Amer. Math. Soc. 17, 858–862 (1966) [28] Xu, Q.: Interpolation of Schur multiplier spaces. Math. Z. 235, 707–715 (2000) C´edric Arhancet (B) Laboratoire de Math´ematiques Universit´e de Franche-Comt´e 25030 Besan¸con Cedex France e-mail: [email protected] Received: October 9, 2010. Revised: February 19, 2011.

Integr. Equ. Oper. Theory 70 (2011), 511–539 DOI 10.1007/s00020-011-1880-5 Published online April 19, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Higher Order Riesz Transforms in the Ultraspherical Setting as Principal Value Integral Operators Jorge J. Betancor, Juan C. Fari˜ na, Lourdes Rodr´ıguez-Mesa and Ricardo Testoni Abstract. In this paper we represent the kth Riesz transform in the ultraspherical setting as a principal value integral operator for every k ∈ N. We also measure the speed of convergence of the limit by proving Lp -boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators. Mathematics Subject Classification (2010). Primary 42C05; Secondary 42C15. Keywords. Riesz transforms, ultraspherical expansions, oscillation and variation operators.

1. Introduction Muckenhoupt and Stein [9] introduced a notion of conjugate functions associated with ultraspherical expansions. In this setting the conjugate function appears as a boundary value of a conjugate harmonic extension associated with a suitable Cauchy-Riemann type equation. Assume that λ > 0. For every n ∈ N, we denote by Pnλ the ultraspherical polynomial of degree n [13]. These polynomials are deﬁned by the generating relation ∞ wk Pkλ (t). (1 − 2tw + w2 )−λ = k=0

{Pnλ (cos θ)}n∈N

The sequence is orthogonal and complete in the Lebesgue space L2 ((0, π), dmλ (θ)), where dmλ (θ) = (sin θ)2λ dθ. When 2λ = k − 2, with k ∈ N, the λ-ultraspherical polynomial Pnλ , n ∈ N, arises in the Fourier This paper is partially supported by MTM2010/17974.

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analysis of functions in the surface of the n-Euclidean space sphere that are invariant under the rotations leaving a given axis ﬁxed. For every n ∈ N, Pnλ (cos θ) is an eigenfunction of the operator d2 d + λ2 , − 2λ cot θ dθ2 dθ associated with the eigenvalue μn = (n+λ)2 . The operator Lλ can be written as follows ∗ d d + λ2 , Lλ = − dθ dθ Lλ = −

d ∗ d d ) = dθ + 2λ cot θ denotes the formal adjoint of dθ in the space where ( dθ 2 L ((0, π), dmλ (θ)). In Buraczewski et al. [1] deﬁned a Riesz transform in the ultraspherical setting associated to Lλ . Note that this operator Lλ is slightly different from the one considered by Muckenhoupt and Stein (see [9, p. 23]). In [1] the authors follow the ideas developed in the monography of Stein [12]. Suppose that f ∈ L2 ((0, π), dmλ (θ)). The ultraspherical expansion of f is ∞ P λ (cos θ) aλn (f ) nλ , f (θ) = Pn (cos ·) L2 ((0,π),dmλ (θ)) n=0

where for every n ∈ N, π λ an (f ) = f (θ) 0

Pnλ (cos θ) dmλ (θ). Pnλ (cos ·)L2 ((0,π),dmλ (θ))

The Poisson integral Ptλ (f ), t > 0, is given by √

Ptλ (f )(θ) = e−t

Lλ

f (θ) =

∞ n=0

aλn (f )

e−t(n+λ) Pnλ (cos θ) , λ Pn (cos ·)L2 ((0,π),dmλ (θ))

According to [9, (2.12)] we can write π λ Pt f (θ) = rλ Pλ (e−t , θ, ϕ)f (ϕ)dmλ (ϕ),

t > 0,

t > 0.

(1.1)

0

where, for each 0 < r < 1 and θ, ϕ ∈ (0, π), π (sin t)2λ−1 λ 2 Pλ (r, θ, ϕ) = (1 − r ) dt. π (1 − 2r(cos θ cos ϕ + sin θ sin ϕ cos t) + r2 )λ+1 0

p

The L -boundedness properties for these Poisson integrals and the corresponding maximal operator were established in [1, Theorem 2.4] (see also [9, Theorem 2]). For every α > 0, the fractional power L−α of the operator λ Lλ is deﬁned by ∞ √ 1 −α e−t Lλ f (θ)t2α−1 dt, f ∈ L2 ((0, π), dmλ (θ)). Lλ f (θ) = Γ(2α) 0

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By using (1.1) we get, for every α > 0 and f ∈ L2 ((0, π), dmλ (θ)), L−α λ f (θ)

1 = Γ(2α) π = 0

∞ π 0

P λ (e−t , θ, ϕ)f (ϕ)dmλ (ϕ)t2α−1 dt

0

1 f (ϕ) Γ(2α)

1 0

2α−1 1 1 drdmλ (ϕ). P λ (r, θ, ϕ) log r r

Following [12] the Riesz transform of order k ∈ N, Rλk , is deﬁned as Rλk f (θ) =

dk − k2 L f (θ), dθk λ

when f is a nice function (for instance, f ∈ span{Pnλ (cos θ)}n∈N or f is a smooth function with compact support on (0, π)). It was proved in [1, Theorem 2.14] (when k = 1) and [2, Theorem 1.4] (when k > 1) that the operator Rλk can be extended to Lp ((0, π), w(θ)dmλ (θ)) as a bounded operator from Lp ((0, π), w(θ)dmλ (θ)) into itself, when 1 < p < ∞ and w ∈ Apλ , and as a bounded operator from L1 ((0, π), w(θ)dmλ (θ)) into L1,∞ ((0, π), w(θ)dmλ (θ)), when w ∈ A1λ . Here, for every 1 ≤ p < ∞, by Apλ we denote the Muckenhoupt class of weights associated with the doubling measure dmλ (θ) on (0, π). In this paper we prove that the kth Riesz transform Rλk is a principal value integral operator, for every k ∈ N. We extend [1, Theorem 2.13] where the result is shown for k = 1. Theorem 1.1. Let λ > 0 and k ∈ N. For every 1 ≤ p < ∞ and ω ∈ Apλ , we have that if f ∈ Lp ((0, π), w(θ)dmλ (θ)), then π Rλk f (θ)

= lim

ε→0+ 0,|θ−ϕ|>ε

Rλk (θ, ϕ)f (ϕ)dmλ (ϕ) + γk f (θ),

a.e. θ ∈ (0, π), (1.2)

where Rλk (θ, ϕ)

1 = Γ(k)

1 0

k−1 ∂k 1 Pλ (r, θ, ϕ) log rλ−1 dr, ∂θk r

θ, ϕ ∈ (0, π),

k

and γk = 0, when k is odd, and γk = (−1) 2 , when k is even. The complete proof of this theorem is presented in Sect. 2. The estimates established in Lemma 2.1 below are crucial in the proof. In this lemma we establish that in the local region, that is, close to the diagonal {θ = ϕ}, the kernel Rλk (θ, ϕ) differs from the kernel of the kth Euclidean Riesz transform by an integrable function. Also, we show that far from the diagonal Rλk (θ, ϕ) is bounded by Hardy type kernels. Suppose that {Tε }ε>0 is a family of operators deﬁned on Lp (Ω, μ), for some measure space (Ω, μ) and 1 ≤ p < ∞, such that for every f ∈ Lp (Ω, μ)

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there exists limε→0+ Tε f (x), μ-a.e. x ∈ Ω. It is an interesting question to measure the speed of that convergence. In order to do this it is usual to analyze expressions involving differences like |Tε f −Tη f |, ε, η > 0. The oscillation and variation operators deﬁned as follows have been used for this purpose. The oscillation operator associated with {Tε }ε>0 is deﬁned by O ({Tε }) (f )(x) =

12

∞

sup

i=0 ti+1 ≤εi+1 <εi
|Tεi+1 f (x) − Tεi f (x)|2

,

for a ﬁxed real sequence {ti }i∈N decreasing to zero. For every ρ > 2 the ρ-variation operator for {Tε }ε>0 is given as follows Vρ ({Tε }) (f )(x) = sup

∞

{εi }i∈N

ρ1 |Tεi+1 f (x) − Tεi f (x)|

ρ

,

i=0

where the supremum is taken over all real sequences {εi }i∈N decreasing to zero. These operators appear in an ergodic context. In [3,4,7] the Lp -boundedness properties for the oscillation and variation operators were studied when Tε , ε > 0, represents the truncated Hilbert and higher dimensional Riesz transform, and Euclidean Poisson semigroup (see also [5] and the references therein). The corresponding results for the truncated ultraspherical Riesz transform were established in [1, Theorem 8.3]. We also measure the speed of convergence in (1.2) in terms of variation and oscillation operators for the corresponding truncated operators. Next result is an extension of [1, Theorem 8.3] for the higher Riesz transform Rλk . k Theorem 1.2. Let λ > 0 and k ∈ N. For every ε > 0 we define by Rλ,ε the k ε-truncation of Rλ as follows

π k Rλ,ε (f )(θ)

Rλk (θ, ϕ)f (ϕ)dmλ (ϕ).

= 0,|θ−ϕ|>ε

If {ti }i∈N is a real decreasing sequence that converges to zero, the oscillation k }) is a bounded operator from Lp ((0, π), dmλ (ϕ)) into itself, operator O({Rλ,ε for every 1 2, the variation operator Vρ ({Rλ,ε p L ((0, π), dmλ (ϕ)) into itself, when 1 < p < ∞, and from L1 ((0, π), dmλ (ϕ)) to L1,∞ ((0, π), dmλ (ϕ)). We remark that the representation of the kth Riesz transform Rλk as a principal value integral operator will allow us to investigate weighted norm inequalities for Rλk involving a class of weights wider than the Muckenhoupt class considered in [2]. This question will be studied in a forthcoming paper. Throughout this paper by C we always denote a positive constant that can change from one line to the other one and i, j represent nonnegative integers.

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2. Proof of Theorem 1.1 In [2, Theorem 1.5] it was established that, for every k ∈ N, the kth Riesz transform Rλk is a Calder´on-Zygmund operator in the homogeneous type space ((0, π), |.|, dmλ (θ)). Then, according to [6, Theorem 9.4.5] the maxik deﬁned by mal operator Rλ,∗ k k Rλ,∗ (f ) = sup |Rλ,ε (f )|, ε>0

k is given as in Theorem 1.2, is bounded from Lp ((0, π), w(θ)dmλ (θ)) where Rλ,ε into itself, when 1 < p < ∞ and w ∈ Apλ , and from L1 ((0, π), w(θ)dmλ (θ)) into L1,∞ ((0, π), w(θ)dmλ (θ)), when w ∈ A1λ . Suppose we have proved that, for every f ∈ Cc∞ (0, π), there exists the limit

π Tλk (f )(θ)

Rλk (θ, ϕ)f (ϕ)dmλ (ϕ),

= lim+ ε→0

a.e. θ ∈ (0, π), (2.1)

0,|θ−ϕ|>ε

and that Tλk f = Rλk f −γk f . Then, Lp -boundedness properties of the maximal k operator Rλ,∗ imply that the limit in (2.1) exists for almost all θ ∈ (0, π), for every f ∈ Lp ((0, π), w(θ)dmλ (θ)), 1 ≤ p < ∞, and w ∈ Apλ . Moreover, by deﬁning Tλk in the obvious way on Lp ((0, π), w(θ)dmλ (θ)), 1 ≤ p < ∞, Tλk is a bounded operator from Lp ((0, π), w(θ)dmλ (θ)) into itself, when 1 < p < ∞ and w ∈ Apλ , and from L1 ((0, π), w(θ)dmλ (θ)) into L1,∞ ((0, π), w(θ)dmλ (θ)), when w ∈ A1λ . Hence, by [2, Theorem 1.4], we conclude that, for each f ∈ Lp ((0, π), w(θ)dmλ (θ)), 1 ≤ p < ∞, π Rλk (f )(θ)

= lim

ε→0+ 0,|θ−ϕ|>ε

Rλk (θ, ϕ)f (ϕ)dmλ (ϕ) + γk f (θ),

a.e. θ ∈ (0, π),

and the proof of this theorem would be ﬁnished. Let now f ∈ Cc∞ (0, π) and k ∈ N. We can write −k

Lλ 2 f (θ) =

∞

(n + λ)−k aλn (f )

n=0

Pnλ (cos θ) , Pnλ (cos ·) L2 ((0,π),dmλ (θ))

−k

θ ∈ (0, π).

Then, since f ∈ Cc∞ (0, π), Lλ 2 f ∈ C ∞ (0, π) (see [8, (2.4) and (2.6)]). We will see that π dk − k2 L f (θ) = lim f (ϕ)Rλk (θ, ϕ) dmλ (ϕ) + γk f (θ), a.e. θ ∈ (0, π), dθk λ ε→0+ 0,|θ−ϕ|>ε

where for each θ, ϕ ∈ (0, π), ⎛ ⎞ k−1 1 k 1 ∂ 1 k ⎝ Rλ (θ, ϕ) = r λ−1 log Pλ (r, θ, ϕ) dr⎠ , ∂θk Γ(k) r 0

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Figure 1. Estimation regions

Pλ (r, θ, ϕ) =

λ π

π 0

(1 − r 2 )(sin t)2λ−1 dt, (1 − 2r(cos θ cos ϕ + sin θ sin ϕ cos t) + r 2 )λ+1

r ∈ (0, 1),

k

and γk = 0, when k is odd, and γk = (−1) 2 , when k is even. As in [2] we introduce the following useful notation: σ = sin θ sin ϕ, a = cos θ cos ϕ + σ cos t = cos(θ − ϕ) − σ(1 − cos t), ∂ b = ∂θ a = − sin θ cos ϕ + cos θ sin ϕ cos t = − sin(θ − ϕ) − cos θ sin ϕ(1 − cos t), Δr = 1 − 2r cos(θ − ϕ) + r2 = (1 − r)2 + 2r(1 − cos(θ − ϕ)), Δ = Δ1 , Dr = 1 − 2ra + r2 = Δr + 2rσ(1 − cos t). We divide the proof in several steps. Step 1. We prove in the following that, for every = 0, . . . , k − 1, π d − k2 L f (θ) = f (ϕ)Rλk, (θ, ϕ) dmλ (ϕ), θ ∈ (0, π), dθ λ

(2.2)

0

where Rλk, (θ, ϕ)

1 = Γ(k)

1 r 0

λ−1

1 log r

k−1

∂ Pλ (r, θ, ϕ) dr, ∂θ

Let ∈ N, 0 ≤ ≤ k − 1. Our objective is ⎧ (sin ϕ)−2λ−1 , ⎪ ⎨ 1 k, Rλ (θ, ϕ) ≤ C (sin θ sin ϕ)λ √|θ−ϕ| , ⎪ ⎩ (sin θ)−2λ−1 , where Ai , i = 1, 2, 3, are the sets in the Fig. 1:

θ, ϕ ∈ (0, π).

to establish that (θ, ϕ) ∈ A1 ; (θ, ϕ) ∈ A2 , θ = ϕ; (θ, ϕ) ∈ A3 ;

(2.3)

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According to [2, Lemma 3.5] we have that 1 ∂ ri+j ai bj c , = ,s,i,j ∂θ Drλ+1 Drλ+1+s s,i,j

517

(2.4)

where c,s,i,j = 0 only if s = 1, . . . , , j ≥ 2s − ,

and i + j = s.

(2.5)

Moreover, by using the Fa` a di Bruno’s formula [10, Theorem 2], we can see that, for every ∈ N, s = 1, . . . , , and i + j = s, c,s,i,j = 2s !s!

(−1)s+j+α(k1 ,...,k ) , k1 ! · · · k !1!k1 2!k2 · · · !k

(2.6)

where the sum is over all different solutions in nonnegative integers k1 , . . . , k of the system ⎫ k1 + k2 + · · · + k = s ⎪ ⎪ ⎬ k 1 + 2k2 + · · · + k = k =i ⎪ ⎪ r par r ⎭ r impar kr = j and

α(k1 , . . . , k ) =

[2]

(r − 1)(k2r−1 + k2r ) + m ,

r=2 where m = 0, if is even, and m = (−1)k , when is odd. 2 We deﬁne, for every s, i, j satisfying (2.5),

1 π M,s,i,j (θ, ϕ) =

r 0

0

i+j+λ−1

1 log r

k−1

(1 − r2 )

ai bj (sin t)2λ−1 dtdr, Drλ+1+s

for θ, ϕ ∈ (0, π). In order to obtain (2.3) it is then sufﬁcient to see that ⎧ (sin ϕ)−2λ−1 , ⎪ ⎨ 1 |M,s,i,j (θ, ϕ)| ≤ C (sin θ sin ϕ)λ √|θ−ϕ| , ⎪ ⎩ (sin θ)−2λ−1 ,

(θ, ϕ) ∈ A1 ; (θ, ϕ) ∈ A2 , θ = ϕ; (θ, ϕ) ∈ A3 ;

for each s, i, j satisfying (2.5). Moreover, by the symmetry of the Fig. 1 and since M,s,i,j (π − θ, π − ϕ) = (−1)j M,s,i,j (θ, ϕ),

θ, ϕ ∈ (0, π),

when s, i, j are as in (2.5), we can assume that (θ, ϕ) ∈ (0, π2 ) × (0, π).

(2.7)

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Let us ﬁx s, i, j verifying (2.5), θ ∈ (0, π2 ) and ϕ ∈ (0, π). By proceeding as in [2, Lemma 3.6], and using that log 1r ∼ 1 − r, as r → 1− , and Dr ≥ C, r ∈ (0, 12 ),we get ⎛ 1 ⎞ 2 π 1 π ⎜ ⎟ i+j+λ−1 1 k−1 ⎟r |M,s,i,j (θ, ϕ)| ≤ ⎜ + (1 − r 2 ) log ⎝ ⎠ r 0 0

×

0

1 2

i

j

|a| |b| (sin t)2λ−1 Drλ+1+s

dtdr

⎛ 1 ⎞ k−1 2 1 π j 2λ−1 ⎜ ⎟ 1 |b| (sin t) λ−1 ≤ C⎜ log dr + (1 − r)k dtdr⎟ λ+1+s ⎝ r ⎠ r Dr ⎛

0

0

1 2

⎜ ≤ C⎜ ⎝1 +

⎞ 1 π j 2λ−1 ⎟ |b| (sin t) (1 − r)k dtdr⎟ λ+1+s ⎠. Dr 1 2

0

It can be seen that, if (α, β) ∈ [0, π] × [0, π], z ∈ (0, 1) and α ≤ zβ, then there exists C > 0 such that sin(β − α) ≥ min{sin β, sin((1 − z)β)} ≥ C sin β, and that, if(α, β) ∈ [0, π/2] × [0, π] and α2 ≤ β ≤ 3α 2 , then sin |β − α| ≤ sin α and sin α ∼ sin β. These considerations allow us to write |b|j ≤ C(| sin(θ − ϕ)|j + (sin ϕ)j ) ⎧ ⎨ | sin(θ − ϕ)|j , ϕ ≤ θ2 or ϕ ≥ ≤C ⎩ (sin ϕ)j , θ ≤ ϕ ≤ 3θ . 2 2

3θ 2,

(2.8)

Then, since 1 − cos α ≥ (sin α)2 /π, α ∈ [0, π], we obtain, when ϕ ≤ ϕ ≥ 3θ 2 , ⎛ ⎞ 1 k (1 − r) ⎜ ⎟ |M,s,i,j (θ, ϕ)| ≤ C ⎝1 + (sin |θ − ϕ|)j dr⎠ λ+s+1 Δr 1 2

⎛ ⎜ ≤ C ⎝1 + (sin |θ − ϕ|)j

(sin |θ − ϕ|) ⎜ ≤ C ⎝1 + j+1 Δλ+ 2

j

≤C ≤C

2s−j

(1 − r) ⎟ dr⎠ (Δ + (1 − r)2 )λ+s+1 1 √ Δ

2

0

1 (sin |θ − ϕ|)2λ+1 (sin ϕ)−2λ−1 , ϕ ≥ (sin θ)−2λ−1 ,

or

⎞

1 1 2

⎛

θ 2

⎞ 2s−j

u ⎟ du⎠ (1 + u2 )λ+s+1

3θ 2 ,

ϕ ≤ θ2 .

(2.9)

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Ultraspherical Riesz Transforms

Suppose now that 1 π (1 − r)k 1 2

0

⎛

⎜ ≤C⎝

θ 2

≤ϕ≤

(1 − r)k 0 π 2

1

(1 − r)k

+ 1 2

0

(1 − r)k 1 2

(sin |θ − ϕ|)j (sin t)2λ−1 dtdr Drλ+1+s

(sin ϕ(1 − cos t))j (sin t)2λ−1 dtdr Drλ+1+s ⎞

1 π +

= ϕ. One can write

|b|j (sin t)2λ−1 dtdr Drλ+1+s

1 π 1 2

3θ 2 ,θ

519

π 2

2λ−1

j

(sin ϕ) (sin t) Drλ+1+s

3 ⎟ dtdr⎠ = Iβ (θ, ϕ). β=1

We analyze the ﬁrst integral. By making two changes of variables, as in [2, p. 1235], and by taking into account that 2s − j ≤ ≤ k − 1 and that Δ = 2(1 − cos(θ − ϕ)) ∼ (sin |θ − ϕ|)2 ∼ (θ − ϕ)2 we get

I1 (θ, ϕ) ≤ C(sin |θ − ϕ|)j

≤ C(sin |θ − ϕ|)j

≤ C(sin |θ − ϕ|)j

1

(1 − r)k

1 2

0

1

2

(1 − r)k

1 2

0

1

2

(sin t)2λ−1 dtdr (Δr + 2rσ(1 − cos t))λ+1+s

π

(1 − r)k

1 1 2

(sin |θ − ϕ|)j ≤C σλ

(sin t)2λ−1 dtdr (Δr + 2rσ(1 − cos t))λ+1+s

π

0

1 2

≤ C(sin |θ − ϕ|)j

π

(1 − r)k

t2λ−1 dtdr (Δr + σt2 )λ+1+s

Δλ+s+1 r

1 1 2

(1 − r)k Δ1+s r

Δr σ

2λ

√ (sin |θ − ϕ|)j ( Δ)2s−j+3/2 ≤C σ λ Δ1+s ≤C

(sin |θ − ϕ|) 1

∞

j

σλ Δ 4 + 2

0

0

σ Δr

u2λ−1 dudr (1 + u2 )λ+s+1

(sin |θ − ϕ|)j dr ≤ C σλ 1 √ 2 Δ

j

π 2

0

2s−j+1/2

1 1 2

u2s−j+1/2 du (1 + u2 )1+s

u 1 . du ≤ C (1 + u2 )1+s σ λ |θ − ϕ|

(1 − r)2s−j+1/2 dr (Δ + (1 − r)2 )1+s

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For the second integral we write π

1

2

I2 (θ, ϕ) ≤ C(sin ϕ)j

(1 − r)k 1 2

0

1

2

π

≤ C(sin ϕ)j

(1 − r)k 0

1 2

≤C

≤C

1

(sin ϕ)j σ

λ+ 2j

σ

(1 − r)k 1+s− 2j

1

λ+ 2j

t2λ+j−1 dtdr (Δr + σt2 )λ+1+s

dr

Δr

1 2

(sin ϕ)j

t2λ+2j−1 dtdr (Δr + σt2 )λ+1+s

1

(1 − r)2s−j+ 2 j

(Δ + (1 − r)2 )1+s− 2

1 2

dr

1

≤C ≤C

√ 2 Δ √ 3 (sin ϕ)j ( Δ)2s−j+ 2 j

j

j

σ λ+ 2 Δ1+s− 2 (sin ϕ)j j

≤C

1

σ λ+ 2 Δ 4

1

u2s−j+ 2

0

(1 + u2 )1+s− 2

du

1

σλ

, |θ − ϕ|

because sin θ ∼ sin ϕ. Finally it has 1 π I3 (θ, ϕ) ≤ C(sin ϕ)

(1 − r)k

j π 2

1 2

≤C

(sin ϕ)j σ

1

λ+ 2j

(1 − r)k 1+s− 2j Δr

1 2

(sin t)2λ−1 dtdr (Δr + σ)λ+1+s

dr ≤ C

σλ

1 |θ − ϕ|

.

≤ ϕ ≤ 3θ 2 , θ = ϕ, ⎛ ⎞ 1 π j 2λ−1 |b| (sin t) ⎜ ⎟ |M,s,i,j (θ, ϕ)| ≤ C ⎝1 + (1 − r)k dtdr⎠ Drλ+1+s

Then we conclude that if

θ 2

1 2

≤C

σλ

0

1 |θ − ϕ|

,

that, jointly with (2.9), gives (2.7). We have proved that the integral in (2.2) is absolutely convergent. By analyzing carefully the above estimates we can also see that, for every = 0, 1, . . . , k − 2, Rλk, is a continuous function on (0, π) × (0, π). Then, we conclude (2.2), for every = 0, 1, . . . , k − 1. Note that when = k − 1 to prove (2.2) we need to use distributional arguments (see Lemma 4.2 in Appendix).

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Step 2. We now study the kernel ⎡ 1 ⎤ k−1 π 2λ−1 (sin t) λ ∂ ⎣ 1 Rλk (θ, ϕ) = rλ−1 log (1 − r2 ) dtdr⎦ , πΓ(k) ∂θk r Drλ+1 k

0

0

for θ, ϕ ∈ (0, π). We get estimates which are better than the ones obtained in (2.3). Lemma 2.1. Let λ > 0 and k ∈ N. Then, ⎧ O((sin ϕ)−(2λ+1)), ⎪ ⎪ ⎪ ⎨ sin ϕ 1+ |θ−ϕ| Rk (θ,ϕ) k Rλ (θ, ϕ) = (sin θ sin ϕ)λ + O (sin ϕ)2λ+1 , ⎪ ⎪ ⎪ ⎩ O((sin θ)−(2λ+1) ),

(θ, ϕ) ∈ A1 ; (θ, ϕ) ∈ A2 , θ = ϕ; (θ, ϕ) ∈ A3 ;

where ∂k 1 R (θ, ϕ) = 2πΓ(k) ∂θk

1

k

log 0

1 r

k−1

dr 1 − r2 . − 1 2 1 − 2r cos(θ − ϕ) + r r

Proof. Since Rλk (θ, ϕ) = (−1)k Rλk (π − θ, π − ϕ) and Rk (θ, ϕ) = (−1)k Rk (π − θ, π − ϕ), θ, ϕ ∈ (0, π), we can assume (θ, ϕ) ∈ (0, π2 ) × (0, π). When (θ, ϕ) ∈ A1 ∪ A3 , we can argue as in the proof of (2.9), for = k, and thus we get (sin ϕ)−2λ−1 , ϕ ≥ 3θ λ 2 , k ck,s,i,j Mk,s,i,j (θ, ϕ) ≤ C |Rλ (θ, ϕ)| = (sin θ)−2λ−1 , ϕ ≤ θ2 . πΓ(k) s=1,...,k j≥2s−k i+j=s We now consider

θ 2

≤ϕ≤

3θ 2

and ϕ = θ. First we write

⎤ ⎡ √ σ π π k−1 1 π 1− 2 2 1 2 ⎥ λ−1 λ ⎢ 1 ⎥r ⎢ Rλk (θ, ϕ) = + + log ⎦ πΓ(k) ⎣ r √ 0

×(1 − r2 ) =

π 2

0

0

1−

σ 2

0

∂ k (sin t)2λ−1 dtdr ∂θk Drλ+1

λ [I k,1 (θ, ϕ) + Iλk,2 (θ, ϕ) + Iλk,3 (θ, ϕ)]. πΓ(k) λ

(2.10)

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J. J. Betancor et al.

IEOT

Let us decompose Iλk,3 (θ, ϕ) as follows, k−1 1 rλ−1 log (1 − r2 ) r

1 Iλk,3 (θ, ϕ)

= 1−

√

σ 2 π

×

2

k

∂ ∂θk

0

dtdr

π

+ √ σ 2

0

Jλk (θ, ϕ)

=

2 k−1 k t2λ−1 1 λ−1 2 ∂ r (1 − r ) k dtdr log r ∂θ (Δr + rσt2 )λ+1

1 1−

(sin t)2λ−1 t2λ−1 − λ+1 (Δr + rσt2 )λ+1 Dr

+

Kλk (θ, ϕ).

(2.11)

Moreover, we observe that by making the change of variable u = ⎞ ∞ ∞ ⎟ ⎜ dt = ⎝ − ⎠

0

rσ Δr t,

⎛

π

2

2λ−1

t (Δr + rσt2 )λ+1

π 2

0

1 = (rσ)λ Δr

t2λ−1 dt (Δr + rσt2 )λ+1

∞ 0

u2λ−1 du − (1 + u2 )λ+1

1 = − 2λ(rσ)λ Δr

∞ π 2

∞ π 2

t2λ−1 dt (Δr + rσt2 )λ+1

t2λ−1 dt, (Δr + rσt2 )λ+1

r ∈ (0, 1).

Then, Leibniz’s rule leads to ⎡ Kλk (θ, ϕ) =

1 ⎢ ⎢ 2λ ⎣

1

log

1−

√

−

=

∂k ∂θk

σ 2

1 1−

1 r

k−1

r

λ−1

√ σ 2

⎡

1 ⎢ ⎢ 2λσ λ ⎣

1 log r

1

1−

log √ σ 2

k−1

1 r

k−1

1 (1 − r2 ) σ λ Δr

∂k (1 − r ) k ∂θ 2

∞ π 2

⎤ dr ⎥ ⎥ r ⎦

t2λ−1 dtdr (Δr + rσt2 )λ+1 ⎤

∂ k (1 − r2 ) dr ⎥ ⎥ ∂θk Δr r ⎦

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Ultraspherical Riesz Transforms

523

⎡ ⎤ 1 k−1 k−1 1 ⎢ 1 k ∂ k−n ∂ n (1 − r2 ) dr⎥ 1 ⎢ ⎥ + log 2λ n=0 n ∂θk−n σ λ ⎣ √ r ∂θn Δr r⎦ 1−

1 − 1−

√ σ 2

σ 2

k−1 ∞ t2λ−1 ∂k 1 rλ−1 log (1 − r2 ) k dtdr. r ∂θ (Δr + rσt2 )λ+1 π 2

We observe that 3

Kλk (θ, ϕ) =

k,β πΓ(k) k R (θ, ϕ) + Kλ (θ, ϕ), λ λσ

(2.12)

β=1

where

⎡

k,1 Kλ (θ, ϕ) = −

k,2 (θ, ϕ) = Kλ

1−

1 ⎢ ⎢ 2λσ λ ⎣ k−1

1 2λ

n=0

√ σ 2

log

0

k n

∂ k−n ∂θk−n

1 r

k−1

1 σλ

⎤ ∂ (1 − r ) dr ⎥ ⎥, ∂θk Δr r ⎦ k

⎡

2

1

⎢ ⎢ ⎣

1 r

log

1−

√ σ 2

k−1

⎤ ∂ n (1 − r 2 ) dr ⎥ ⎥, ∂θn Δr r ⎦

and

1 Kλk,3 (θ, ϕ)

=−

r

1−

λ−1

√ σ 2

1 log r

k−1

∂k (1 − r ) k ∂θ 2

∞ π 2

t2λ−1 dtdr. (Δr + rσt2 )λ+1

Thus, according to (2.10), (2.11) and (2.12), to establish our result we must analyze Iλk,β (θ, ϕ), β = 1, 2, Jλk (θ, ϕ) and Kλk,β (θ, ϕ), β = 1, 2, 3. Let us consider ﬁrst Iλk,1 (θ, ϕ). We will see that ! sin ϕ C k,1 |Iλ (θ, ϕ)| ≤ . (2.13) 1+ (sin ϕ)2λ+1 |θ − ϕ| Let s = 1, . . . , k, j ≥ 2s − k and i + j = s. We deﬁne

1 k,1 Iλ,s,i,j (θ, ϕ)

=

r

i+j+λ−1

0

1 log r

k−1

2

π

(1 − r ) π 2

ai bj (sin t)2λ−1 dtdr. Drλ+s+1

k,1 (θ, ϕ) According to (2.4) it is then sufﬁcient to obtain (2.13) when Iλ,s,i,j

replaces to Iλk,1 (θ, ϕ). By proceeding as in Step 1, using (2.8), since Dr ≥ C, for 0 < r < 12 , and Dr ≥ (Δr + σ), for 12 < r < 1 and t ∈ ( π2 , π), we have

1 k,1 |Iλ,s,i,j (θ, ϕ)|

≤C

r 0

λ−1

1 log r

k−1

2

π

(1 − r ) π 2

|b|j (sin t)2λ−1 dtdr Drλ+s+1

524

J. J. Betancor et al. ⎛

(1 − r) ⎟ dr⎠ (Δr + σ)λ+s+1 k

1 2

⎛

1

j

(sin ϕ) ⎜ ≤ C ⎝1 + j 1 σ λ+ 4 + 2 ≤ C ⎝1 +

s+ 3 − j Δr 4 2

1

⎟ dr⎠ ⎞

u2s−j j

3

σ λ+ 4 Δ 4

0

1+

(1 − r)

∞

1 1

⎞ 2s−j

1 2

⎛

≤C

⎞

1

⎜ ≤ C ⎝1 + (sin ϕ)j

IEOT

(1 + u2 )s+ 4 − 2

1

(sin ϕ)2λ+ 2

1

≤C

|θ − ϕ|

du⎠ 1+

sin ϕ |θ−ϕ|

(sin ϕ)2λ+1

.

For Iλk,2 (θ, ϕ) we proceed in a similar way. Consider s = 1, . . . , k, j ≥ 2s − k, and i + j = s, and 1−

k,2 (θ, ϕ) = Iλ,s,i,j

√ σ 2

ri+j+λ−1

0

1 log r

k−1

π

(1 − r2 )

2 0

ai bj (sin t)2λ−1 dtdr. Drλ+s+1

We have that √ ⎛ 1 ⎞ 1− 2σ k−1 2 (1 − r)k ⎟ 1 ⎜ k,2 (θ, ϕ)| ≤ C ⎝ rλ−1 log dr + (sin ϕ)j dr⎠ |Iλ,s,i,j r Δλ+s+1 r

⎛

0

1 2

⎜ ≤ C ⎝1 + (sin ϕ)j

1−

⎞

√

1 2

σ 2

2s−j

(1 − r) C ⎟ dr⎠ ≤ . (1 − r)2λ+2s+2 (sin ϕ)2λ+1

Hence, |Iλk,2 (θ, ϕ)| ≤

C . (sin ϕ)2λ+1

(2.14)

To estimate Jλk (θ, ϕ) (see (2.11)), we write Jλk (θ, ϕ) = Jλk,1 (θ, ϕ) + Jλk,2 (θ, ϕ), where 1 Jλk,1 (θ, ϕ) = 1−

√ σ 2

π

k−1 2 (sin t)2λ−1 − t2λ−1 ∂k 1 rλ−1 log (1 − r2 ) k dtdr, r ∂θ Drλ+1 0

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Ultraspherical Riesz Transforms

525

and k−1 1 rλ−1 log (1 − r2 ) r

1 Jλk,2 (θ, ϕ)

= 1−

√ σ 2

π

×

k

∂ ∂θk

2

t2λ−1

0

1 Drλ+1

−

1 (Δr + rσt2 )λ+1

dtdr.

To analyze Jλk,1 (θ, ϕ) assume, as above, s = 1, . . . , k, j ≥ 2s − k, and i + j = s and consider 1 k,1 Jλ,s,i,j (θ, ϕ)

= 1−

√ σ 2

k−1 1 ri+j+λ−1 log (1 − r2 ) r

π

2 × 0

ai bj [(sin t)2λ−1 − t2λ−1 ] dtdr. Drλ+s+1

By using the mean value theorem and that |b|j ≤ C(|θ−ϕ|j +(t2 sin ϕ)j ), t ∈ (0, π2 ), we have π

1

2

k,1 |Jλ,s,i,j (θ, ϕ)| ≤ C

(1 − r)k 1−

√ σ 2

0 π

1

2 (1 − r)k

≤C 1−

=

|b|j t2λ+1 dtdr Drλ+s+1

√ σ 2

0

k,1,1 C(Jλ,s,i,j (θ, ϕ)

[|θ − ϕ|j + (t2 sin ϕ)j ]t2λ+1 dtdr (Δr + σt2 )λ+s+1

k,1,2 + Jλ,s,i,j (θ, ϕ)).

(2.15)

We can obtain for each term in the last sum the following estimates. Firstly, √σ π 2 Δr 1 (1 − r)k u2λ+1 |θ − ϕ|j k,1,1 |Jλ,s,i,j (θ, ϕ)| ≤ C λ+1 dudr s σ Δr (1 + u2 )λ+s+1 √ 1−

|θ − ϕ|j ≤ C λ+1 σ

≤C

1

σ 2

1

0

(1 − r)k j

√ 1− 2σ

1 (1 − r)k−2s+j dr ≤ C

σ λ+1 1−

√ σ 2

j

(Δ + (1 − r)2 )s− 2 + 2 σ

dr

k+j 2 −s 1

σ λ+ 2

≤

C . (sin ϕ)2λ+1

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J. J. Betancor et al.

IEOT

We also have π

1

2

k,1,2 (θ, ϕ)| ≤ C(sin ϕ)j |Jλ,s,i,j

(1 − r)k 1−

≤C

≤C

√ σ 2

1

σ λ+ 2 Thus we get by (2.4)

k−2s+j− 12

(1 − r)

j 3 σ λ+ 2 + 4

k+j 2 −s

π

1

(sin ϕ)j

σ

0

t2λ+2j+1 dtdr ((1 − r)2 + σt2 )λ+s+1

1−

√ σ 2

≤

C . (sin ϕ)2λ+1

2

1

tj− 2 dt

dr 0

|Jλk,1 (θ, ϕ)| ≤

C . (sin ϕ)2λ+1

(2.16)

In the same way, and according to (2.4), to see the estimate for Jλk,2 (θ, ϕ) we analyze k−1 1 1 k,2 rλ+i+j−1 log (1 − r2 ) Jλ,s,i,j (θ, ϕ) = r √ 1−

σ 2 π

2 ×

t2λ−1

0

ai bj Drλ+s+1

−

Ai B j (Δr + rσt2 )λ+s+1

dtdr, 2

for every s = 1, . . . , k, j ≥ 2s − k, and i + j = s. Here A = cos(θ − ϕ) − σt2 and B = ∂A ∂θ . By using the mean value theorem we obtain, for t ∈ (0, π2 ), i j ab Ai B j − Dλ+s+1 2 λ+s+1 (Δr + rσt ) r i (a − Ai )bj + Ai (bj − B j ) 1 1 + ≤ ai bj − (Δr + rσt2 )λ+s+1 (Δr + rσt2 )λ+s+1 Drλ+s+1 √ |b|j σt4 |b|j−1 σt4 ≤C + 2 λ+s+2 (Δr + rσt ) (Δr + rσt2 )λ+s+1 √ |b|j t2 |b|j−1 σt3 ≤C + , (Δr + σt2 )λ+s+1 (Δr + σt2 )λ+s+1 where the second term in the two last sums does not appear when j = 0. Then, we write k,2 k,2,1 k,2,2 |Jλ,s,i,j (θ, ϕ)| ≤ C(Jλ,s,i,j (θ, ϕ) + Jλ,s,i,j (θ, ϕ)),

where π

1 k,2,1 (θ, ϕ) = Jλ,s,i,j

2 (1 − r)k

1−

√ σ 2

0

|b|j t2λ+1 dtdr, (Δr + σt2 )λ+s+1

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Ultraspherical Riesz Transforms

527

and k,2,2 Jλ,s,i,j (θ, ϕ) =

√

π

1

2

1−

|b|j−1 t2λ+2 dtdr, (Δr + σt2 )λ+s+1

(1 − r)k

σ √

0

σ 2

when j ≥ 1.

k,2,1 We observe that Jλ,s,i,j (θ, ϕ) was already analyzed in (2.15). On the other hand, when j ≥ 1, we can use that |b|j−1 ≤ C(|θ − ϕ|j−1 + (t2 sin ϕ)j−1 ), t ∈ (0, π2 ), and proceed as in the estimation of (2.15) to study k,2,2 . Thus we get that Jλ,s,i,j

C . (sin ϕ)2λ+1

|Jλk,2 (θ, ϕ)| ≤

(2.17)

By combining (2.16) and (2.17) we conclude that |Jλk (θ, ϕ)| ≤

C . (sin ϕ)2λ+1

(2.18)

Finally we deal with Kλk,β (θ, ϕ), β = 1, 2, 3 (see (2.12)). By invoking (2.4) with t = 0 and λ = 0, it has that 1 ∂ ri+j (cos(θ − ϕ))i (− sin(θ − ϕ))j c,s,i,j , = ∂θ Δr Δ1+s r s,i,j

∈ N,

where c,s,i,j = 0 only if s = 1, . . . , , j ≥ 2s − and i + j = s. Also, for every n = 0, . . . , k − 1, k−n ∂ k−n 1 −λ−k+n (sin ϕ)−λ ≤ Cσ −λ− 2 . ∂θk−n σ λ ≤ C(sin θ)

(2.19)

Hence, by proceeding as above, to estimate Kλk,β (θ, ϕ), β = 1, 2, 3, it is sufﬁcient to study the following integrals: 1− k,1 Kλ,s,i,j (θ, ϕ) =

1 σλ

√

σ 2

0

k−1 1 ri+j−1 log r

(1 − r2 )(cos(θ − ϕ))i (− sin(θ − ϕ))j × dr, Δs+1 r when s = 1, . . . , k, j ≥ 2s − k and i + j = s; Kλk,2,0 (θ, ϕ)

=σ

−k/2−λ

1

1− k,2,n Kλ,s,i,j (θ, ϕ)

∂ k−n = ∂θk−n

√ σ 2

1 σλ

1 1−

×

1 log r

√ σ 2

k−1

(1 − r2 ) dr ; Δr r

k−1 1 ri+j−1 log r

(1 − r2 )(cos(θ − ϕ))i (− sin(θ − ϕ))j dr, Δs+1 r

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J. J. Betancor et al.

IEOT

for each n = 1, . . . , k − 1, s = 1, . . . , n, j ≥ 2s − n, and i + j = s; and k−1 1 ri+j+λ−1 log (1 − r2 ) r

1 k,3 (θ, ϕ) Kλ,s,i,j

= 1−

√

σ 2

∞ × π 2

Ai B j t2λ−1 dtdr, (Δr + rσt2 )λ+s+1

when s = 1, . . . , k, j ≥ 2s − k, and i + j = s. Here, as before, A = cos(θ − 2 ϕ) − σt2 and B = ∂A ∂θ . Consider s = 1, . . . , k, j ≥ 2s − k, and i + j = s. We can write ⎛ 1 k−1 2 1 ⎜ 1 k,1 i+j−1 |Kλ,s,i,j (θ, ϕ)| ≤ C λ ⎝ r dr log σ r 0

√ 1− 2σ

(1 − r) |θ − ϕ| ⎟ dr⎠ (Δ + (1 − r)2 )s+1

+ 1 2

≤C

≤C

⎞ k

j

⎛

1 ⎜ |θ − ϕ|j ⎝1 + j λ σ Δ2 1 σλ

1+

σ

k+j 2 −s

√

1−

⎟ (1 − r)k−2s+j−2 dr⎠

1 2

≤

σ

⎞

√ σ 2

C . (sin ϕ)2λ+1

On the other hand, by using (2.19), for each n = 1, . . . , k − 1, s = 1, . . . , n, j ≥ 2s − n, and i + j = s, we obtain k,2,n (θ, ϕ)| |Kλ,s,i,j

≤C

1

|θ − ϕ|j σ

λ+ k−n 2 1−

≤C

√ σ 2

|θ − ϕ|j

(1 − r)k dr (Δ + (1 − r)2 )s+1 1 1−

≤C

σ

n+j 2 −s

1 1 σ λ+ 4 Δ 4

√

σ 2

!

C ≤ (sin ϕ)2λ+1

In a similar way we obtain |Kλk,2,0 (θ, ϕ)|

3

(1 − r)k−2s+j− 2 dr

j k−n 1 σ λ+ 2 Δ 2 + 4

sin ϕ . |θ − ϕ|

! C ≤ (sin ϕ)2λ+1

sin ϕ . |θ − ϕ|

Finally, assume s = 1, . . . , k, j ≥ 2s − k, and i + j = s. By taking into account that |B|j ≤ C(t2 sin ϕ)j and |A|i ≤ C(1 + σ i t2i ), t ≥ π2 , and the last

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Ultraspherical Riesz Transforms

529

formula in [9, p. 37], we obtain 1 k,3 (θ, ϕ) Kλ,s,i,j

≤C

∞ (1 − r)

k

1−

√ σ 2

π 2

⎛

1

(sin ϕ)j ⎜ ≤ C λ+j ⎜ ⎝ σ

1−

|A|i |B|j t2λ−1 dtdr (Δr + σt2 )λ+s+1 ∞

(1 − r)k Δs+1−j r

√ σ 2

1

π 2

√

u2λ+2j−1 dudr (1 + u2 )λ+s+1

σ Δr

⎞

∞

⎟ (1 − r)k u2λ+2j+2i−1 ⎟ dudr ⎠ Δr (1 + u2 )λ+s+1 √ √ π σ 1− 2σ 2 Δr ⎛ ⎞ 1 1 ⎟ (1 − r)k (sin ϕ)j ⎜ k ⎟ ≤ C λ+j+1 ⎜ dr + (1 − r) dr ⎝ ⎠ σ (Δ + (1 − r)2 )s−j √ √ +

1−

1−

(1 − r)k−2s+2j dr 1−

(sin ϕ)j σ

√ σ 2

k+j 2 −s

≤

j 1 σ λ+ 2 + 2

C . (sin ϕ)2λ+1

Thus, we have obtained that 3

σ 2

1

(sin ϕ)j ≤ C λ+j+1 σ ≤C

σ 2

|Kλk,β (θ, ϕ)|

β=1

C ≤ (sin ϕ)2λ+1

1+

! sin ϕ |θ − ϕ|

.

(2.20)

By considering (2.10), (2.11), (2.12) and the estimations (2.13), (2.14), (2.18) and (2.20) we conclude that, when θ2 ≤ ϕ ≤ 3θ 2 , θ = ϕ, ! k k C (θ, ϕ) sin ϕ R Rλ (θ, ϕ) − ≤ , 1+ σ λ (sin ϕ)2λ+1 |θ − ϕ| and the proof of Lemma 2.1 is ﬁnished.

Step 3. We now establish that the kth Riesz transform in the circle is a principal value integral operator, that is, dk dθk

1

π f (ϕ) 0

log 0

1 r

k−1

π

= 2πΓ(k) lim+ ε→0

dr 1 − r2 dϕ − 1 2 1 − 2r cos(θ − ϕ) + r r

0,|θ−ϕ|>ε

f (ϕ)Rk (θ, ϕ)dϕ + βk f (θ),

(2.21)

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J. J. Betancor et al.

IEOT k

for every θ ∈ (0, π), and where βk = 0 when k is odd, and βk = 2π(−1) 2 Γ(k), when k is even. Let us consider the function k−1 k−1 1 dr 1 − r2 ∂ 1 k H (ω) = , − 1 log k−1 2 r ∂ω 1 − 2r cos ω + r r 0

for every ω ∈ R\{2kπ : k ∈ Z}. Firstly we are going to analyze the behavior of H k (w) when w → 0+ . We have that 1 dr 1 − r2 1 − 1 H (w) = 2 1 − 2r cos w + r r 0

= − log(2(1 − cos w)),

w ∈ R\{2kπ : k ∈ Z}.

Then, limw→0 wH 1 (w) = 0. Assume that k ∈ N, k ≥ 2. According to (2.4), it has ⎛ 1 ⎞ 1 2 ⎜ ⎟ ri+j (cos w)i (− sin w)j ck−1,s,i,j ⎝ + ⎠ H k (w) = (1 − 2r cos w + r2 )s+1 s=1,...,k−1

0

j≥2s−k+1 i+j=s

1 2

k−1 dr 1 ×(1 − r2 ) log r r " 0 # 1 = ck−1,s,i,j Ik,s,i,j (w) + Ik,s,i,j (w) , w ∈ R\{2kπ : k ∈ Z}. s=1,...,k−1 j≥2s−k+1 i+j=s

Let s = 1, . . . , k − 1, i + j = s, j ≥ 2s − k + 1. By using the dominated convergence theorem we obtain 0, j ≥ 1; 0 # 2 s−1 lim Ik,s,i,j (w) = $ 1/2 " 1 k−1 (1−r )r w→0 log r dr, j = 0; 0 (1−r)2(s+1) and, when 2s − k + 1 < 0, lim

w→0

1 Ik,s,i,j (w)

=

0, $1 " 1/2

log

# 2 s−1 1 k−1 (1−r )r r (1−r)2(s+1)

dr,

j ≥ 1; j = 0.

Assume now 2s − k + 1 ≥ 0. We have that, for each w ∈ (−π, π)\{0}, 1

1 Ik,s,i,j (w) ≤ C|w|j+k−2s−1

2|w| 0

3 uk du ≤ C|w|j+k−2s− 2 . (1 + u2 )s+1

Then, 1 (w) = 0, lim wIk,s,i,j

w→0

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and if j > 2s − k + 1, 1 (w) = 0. lim Ik,s,i,j

w→0

Also, if j = 2s − k + 1 > 0 (as will be the case if k is even), by using mean value theorem it follows lim

w→0+

1 Ik,s,i,j (w)

= −2 lim

w→0+

∞ = −2 0

1

sin w w

2s−k+1 2w 0

uk du = −B (1 + u2 )s+1

uk du (1 + u2 )s+1

k+1 k−1 ,s − 2 2

,

where B(x, y), x, y > 0, represents the Euler’s Beta function. By combining the above estimates we conclude that limw→0 wH k (w) = 0, when k is odd. Assume now that k is even. In this case we obtain that k k−1 1 2 −1 (1 − r2 )rs−1 1 k lim H (w) = ck−1,s,s,0 dr log r w→0+ (1 − r)2(s+1) s=1 0

−

k−1

ck−1,s,k−s−1,2s−k+1 B

s= k 2

k+1 k−1 ,s − 2 2

.

By taking into account (2.6) and the duplication formula for the Euler’s Gamma function, we can write k−1 k+1 k−1 (−1)s+1 (k − 1)!s! B , s − lim H k (w) = − 2k−2s−1 (2s − k + 1)!(k − s − 1)! 2 2 w→0+ k s= 2

=

k−1 (−1)s π(Γ(k))2 2k−2 Γ( k2 ) k (2s − k + 1)(k − s − 1)!(s − k2 )! s= 2

k k 2 −1 1 (−1) 2 π(Γ(k))2 r 2 −1 (−1) = 2r + 1 r 2k−2 (Γ( k2 ))2 r=0 k

k

(−1) 2 π(Γ(k))2 = 2k−2 (Γ( k2 ))2

1

k

k

(1 − t2 ) 2 −1 dt = (−1) 2 πΓ(k).

0

By proceeding as in Step 1 we can see that, for every θ ∈ (0, π), dk−1 dθk−1

1

π f (ϕ) 0

=

0

1

π f (ϕ) 0

0

1 log r

1 log r

k−1

k−1

dr 1 − r2 dϕ −1 1 − 2r cos(θ − ϕ) + r2 r

∂ k−1 ∂θk−1

dr 1 − r2 dϕ. −1 1 − 2r cos(θ − ϕ) + r2 r (2.22)

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J. J. Betancor et al.

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It is clear that dk−1 dθk−1

1

π f (ϕ) 0

log 0

π

1 r

k−1

f˜(ϕ)H k (θ − ϕ) dϕ,

=

dr 1 − r2 dϕ − 1 1 − 2r cos(θ − ϕ) + r2 r

θ ∈ (0, π),

−π

% where f˜(ϕ) =

f (ϕ), ϕ ∈ [0, π) , and f˜(ϕ) = f˜(ϕ + 2π), ϕ ∈ R. 0, ϕ ∈ (−π, 0)

Also, since H k ∈ L1 (−π, π), we have that d dθ

π −π

∂ f˜(ϕ)H k (θ − ϕ)dϕ = ∂θ

π

= −π

π f˜(θ − u)H k (u) du −π

π

∂ ˜ f (θ − u)H k (u) du = − lim ∂θ ε→0+ ⎛

−π,|u|>ε

⎤π

d ˜ [f (θ − u)]H k (u) du du

π

d f˜(θ − u) H k (u) du du ε ε ⎞ −ε &−ε d + f˜(θ − u)H k (u) − f˜(θ − u) H k (u) du⎠ du −π −π ⎛

= − lim ⎝f˜(θ − u)H k (u)⎦ − ε→0+

= − lim+ ⎝f˜(θ − π)H k (π) − f˜(θ − ε)H k (ε) + f˜(θ + ε)H k (−ε) ε→0

⎞

π −f˜(θ + π)H k (−π) − −π,|u|>ε

d ⎟ f˜(θ − u) H k (u) du⎠ , du

θ ∈ (0, π).

Since the function H k is even when k is odd and H k is odd when k is even, we conclude that d dθ

π

π f˜(ϕ)

f (ϕ)H (θ − ϕ)dϕ = lim+ k

0

ε→0

−π,|θ−ϕ|>ε

d k H (θ − ϕ)dϕ du

− lim+ (f (θ + ε) − f (θ − ε))H k (ε) ε→0

π

= lim+ ε→0

0,|θ−ϕ|>ε

f (ϕ)

∂ k H (θ − ϕ)dϕ, θ ∈ (0, π), ∂θ

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when k is odd, and d dθ

π f (ϕ)H k (θ − ϕ)dϕ 0

π

= lim (f (θ + ε) + f (θ − ε))H k (ε) + lim ε→0+

ε→0+ 0,|θ−ϕ|>ε

π

k 2

= 2f (θ)(−1) πΓ(k) + lim+ ε→0

f (ϕ)

0,|θ−ϕ|>ε

f (ϕ)

∂ k H (θ − ϕ)dϕ ∂θ

∂ k H (θ − ϕ)dϕ, θ ∈ (0, π), ∂θ

when k is even. Step 4. We now ﬁnish the proof of Theorem 1.1. We ﬁrstly write, according to (2.2), dk−1 − k2 L f (θ) = dθk−1 λ

π 0

k,k−1 f (ϕ)Rλ (θ, ϕ)dmλ (ϕ) =

π

+

π f (ϕ) 0

k,k−1 f (ϕ) Rλ (θ, ϕ) −

0

k,k−1

Rk,k−1 (θ, ϕ) dmλ (ϕ) (sin θ sin ϕ)λ

(θ, ϕ) R (sin θ sin ϕ)λ

dmλ (ϕ),

θ ∈ (0, π),

where, for every θ, ϕ ∈ (0, π), Rk,k−1 (θ, ϕ) =

1 2πΓ(k)

1 1 k−1 ∂ k−1 1 − r2 dr log . − 1 r ∂θk−1 1 − 2r cos(θ − ϕ) + r 2 r 0

Moreover, by (2.21) and (2.22) we have that, for every θ ∈ (0, π), ⎛ π π d ⎝ 1 dk−1 f (ϕ) d Rk,k−1 (θ, ϕ) 1 f (ϕ) dmλ (ϕ) = λ λ k−1 dθ (sin θ sin ϕ) 2πΓ(k) dθ (sin θ) dθ (sin ϕ)λ 0 0 ⎞ k−1 1 dr 1 − r2 1 dmλ (ϕ)⎠ × −1 log r 1 − 2r cos(θ − ϕ) + r2 r 0

λ cos θ =− (sin θ)λ+1

π 0

f (ϕ) Rk,k−1 (θ, ϕ)dmλ (ϕ) (sin ϕ)λ

1 1 dk + 2πΓ(k) (sin θ)λ dθk 1 × 0

1 log r

k−1

π 0

f (ϕ) (sin ϕ)λ

dr 1 − r2 dmλ (ϕ) −1 2 1 − 2r cos(θ − ϕ) + r r

534

J. J. Betancor et al. λ cos θ =− (sin θ)λ+1

π

IEOT

f (ϕ) Rk,k−1 (θ, ϕ)dmλ (ϕ) (sin ϕ)λ

0

π

1 + lim (sin θ)λ ε→0+

f (ϕ) Rk (θ, ϕ)dmλ (ϕ) + γk f (θ), (sin ϕ)λ

0,|θ−ϕ|>ε

(2.23)

where the integral after the last equal sign is absolutely convergent and γk = k 0, if k is odd and γk = (−1) 2 , when k is even. A careful study of Lemma 2.1 and again a distributional argument allow us to justify the differentiation under the integral sign (see Lemma 4.2 in Appendix) to get d dθ

π 0

Rk,k−1 (θ, ϕ) f (ϕ) Rλk,k−1 (θ, ϕ) − dmλ (ϕ) (sin θ sin ϕ)λ

λ cos θ = (sin θ)λ+1

f (ϕ) Rk,k−1 (θ, ϕ)dmλ (ϕ) (sin ϕ)λ

0

π +

π

Rλk (θ, ϕ)

f (ϕ) 0

Rk (θ, ϕ) − (sin θ sin ϕ)λ

dmλ (ϕ),

a.e. θ ∈ (0, π), (2.24)

where all the integrals are absolutely convergent. By combining (2.23) and (2.24) we conclude that dk − k2 d L f (θ) = dθk λ dθ +

d dθ

π f (ϕ) 0

π 0

Rk,k−1 (θ, ϕ) f (ϕ) Rλk,k−1 (θ, ϕ) − dmλ (ϕ) (sin θ sin ϕ)λ

Rk,k−1 (θ, ϕ) dmλ (ϕ) (sin θ sin ϕ)λ

π

= lim

ε→0+ 0,|θ−ϕ|>ε

f (ϕ)Rλk (θ, ϕ)dmλ (ϕ)

π − lim+ ε→0

f (ϕ)

0,|θ−ϕ|>ε

cos θ +λ (sin θ)λ+1 +

d dθ

π f (ϕ) 0

π 0

Rk (θ, ϕ) dmλ (ϕ) (sin θ sin ϕ)λ

f (ϕ) Rk,k−1 (θ, ϕ)dmλ (ϕ) (sin ϕ)λ

Rk,k−1 (θ, ϕ) dmλ (ϕ) (sin θ sin ϕ)λ

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π = lim

ε→0+ 0,|θ−ϕ|>ε

a.e. θ ∈ (0, π).

f (ϕ)Rλk (θ, ϕ)dmλ (ϕ) + γk f (θ),

Thus the proof of Theorem 1.1 is complete.

3. Proof of Theorem 1.2 In order to show Theorem 1.2 we need to improve Lemma 2.1 as follows: Lemma 3.1. Let k ∈ N. If Rk and A2 are defined as in Lemma 2.1, then ! 1 1 k +O , (θ, ϕ) ∈ A2 , θ = ϕ, R (θ, ϕ) = Mk sin(θ − ϕ) |θ − ϕ| for a certain Mk ∈ R. Moreover, Mk = 0 provided that k is even. Proof. According to (2.4) with λ = t = 0 we have that 1 ri+j ai bj ∂k c , = k,s,i,j ∂θk Δr Δ1+s r s,i,j where a = cos(θ −ϕ), b = − sin(θ −ϕ), and ck,s,i,j = 0 only if s = 1, . . . , k, j ≥ 2s − k and i + j = s. Note ﬁrstly that 1

2 0

1 log r

k−1

k ∂ dr 1 ≤ C, θ, ϕ ∈ (0, π). (1 − r ) k ∂θ 1 + r2 − 2r cos(θ − ϕ) r 2

Also, we have 1 1 2

1 log r

k−1

ri+j |a|i |b|j dr ≤C (1 − r ) r Δ1+s r 2

≤C

≤C

1

(1 − r)k |b|j dr ((1 − r)2 + Δ)1+s

1 2

1 √ Δ

2

|b|j k

1

Δs− 2 + 2

|b|j j

1

Δ2+4

≤ C

0

1 √ 2 Δ

0

1 |θ − ϕ|

provided that s = 1, . . . , k, i + j = s, j > 2s − k.

,

uk du (1 + u)2+2s 1

u2s−j+ 2 du (1 + u)2+2s (θ, ϕ) ∈ A2 , θ = ϕ,

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Assume now s = 1, . . . , k, i + j = s, j = 2s − k. By using the mean value theorem we get 1 1 2

1 log r 1

=2 1 2

k−1

(1 − r2 )

ri+j ai bj dr r Δ1+s r

(1 − r)k drai bj + O ((1 − r)2 + Δ)1+s

Moreover, since 2s − k ≥ 0, 1 1 2

1 |θ − ϕ|

, (θ, ϕ) ∈ A2 , θ = ϕ.

⎛ ⎞ ∞ ∞ ⎜ ⎟ uk du uk du ⎜ ⎟ − ⎝ (1 + u2 )1+s ⎠ 2 1+s (1 + u )

(1 − r)k 1 dr = k 1 2 1+s s− ((1 − r) + Δ) Δ 2+2

0

2

1 √ Δ

and ∞

|a|i |b|j k

1

Δs− 2 + 2 2

1 √ Δ

∞

uk du 1 ≤ C 1/4 2 1+s (1 + u ) Δ 2

≤

C Δ

≤

1 4

1 √ Δ

1

uk+ 2 du (1 + u)2+s

C 1

|θ − ϕ| 2

,

(θ, ϕ) ∈ A2 , θ = ϕ.

Also, if k is odd, we have, for every (θ, ϕ) ∈ A2 , θ = ϕ, ai bj k

1

Δs− 2 + 2

=

(cos(θ − ϕ))i (− sin(θ − ϕ))j j

1

(2(1 − cos(θ − ϕ))) 2 + 2

1 +O =− sin(θ − ϕ)

1 1

|θ − ϕ| 2

.

By combining the above estimates we conclude that 1 1 Rk (θ, ϕ) = Mk +O , (θ, ϕ) ∈ A2 , θ = ϕ, 1 sin(θ − ϕ) |θ − ϕ| 2 for a certain Mk ∈ R, for every k odd. Assume now that k is even. We get 1 ai bj 1 + O = , 1 k 1 | sin(θ − ϕ)| |θ − ϕ| 2 Δs− 2 + 2

(θ, ϕ) ∈ A2 , θ = ϕ.

Hence, from Lemma 2.1 we deduce that, for every (θ, ϕ) ∈ A2 , θ = ϕ, ⎞ ⎛ sin θ 1 + |θ−ϕ| 1 ⎠. +O⎝ Rλk (θ, ϕ) = Mk | sin(θ − ϕ)|(sin θ sin ϕ)λ (sin θ sin ϕ)λ+1/2 By virtue of Theorem 1.1, Mk = 0 because 3θ/2

lim

ε→0+ θ/2,|θ−ϕ|>ε

1 (sin ϕ)λ dϕ, | sin(θ − ϕ)|

does not exist for every θ ∈ (0, π/2).

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From Lemmas 2.1 and 3.1 we deduce that,

# ⎧ " O (sin ϕ)−(2λ+1) , ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ sin ϕ ⎨ 1 + |θ−ϕ| M k k ⎠, ⎝ Rλ (θ, ϕ) = +O ⎪ (sin θ sin ϕ)λ sin(θ − ϕ) (sin ϕ)2λ+1 ⎪ ⎪ ⎪ # ⎩ " O (sin θ)−(2λ+1) ,

(θ, ϕ) ∈ A1 ; (θ, ϕ) ∈ A2 , θ = ϕ; (θ, ϕ) ∈ A3 . (3.1)

By using (3.1) we can prove Theorem 1.2 by proceeding as in the proof of [1, Proposition 8.1].

4. Appendix In this appendix we present the results we need about differentiation under the integral sign. We look for conditions on a function f deﬁned on R × R in order that the formula ∂ ∂ f (x, y)dy, a.e. x ∈ R, f (x, y) dy = ∂x ∂x R

R

holds. In the following we establish conditions on a function f in order that distributional and classical derivatives of f coincide. Lemma 4.1. Let −∞ ≤ a < b ≤ +∞. Assume that f is a continuous function on I × I, where I = (a, b), such that ∂ (i) For every y ∈ I, the function ∂x f (x, y) is continuous on I\{y}, where the derivative is understood in the classical sense. $ (ii) For every y ∈ I and every compact subset K of I, |f (x, y)|dx < +∞, K

and

∂f (x, y) dx < +∞. ∂x K

Then, Dx f (x, y) = for every y ∈ I. Here, as above, Dx f (x, y) denotes the distributional derivative with respect to x of f . ∂ ∂x f (x, y),

Proof. Let g ∈ Cc∞ (I). We can write ⎛ y−ε ⎞ b ⎠ g (x)f (x, y)dx + Dx f (x, y), g(x) = − lim ⎝ ε→0+

⎡

a

y+ε

= lim+ ⎣ − g(y − ε)f (y − ε, y) ε→0

⎤ ⎞ ⎛ y−ε b ⎠ g(x) ∂f (x, y)dx⎦ + + g(y + ε)f (y + ε, y) +⎝ ∂x a

y+ε

538

J. J. Betancor et al. b g(x)

=

∂f (x, y)dx, ∂x

IEOT

y ∈ I.

a

Then, Dx f (x, y) =

∂f ∂x (x, y), y

∈ I.

The differentiations under the integral sign that we have made in the proof of our results can be justiﬁed by using the following one. Lemma 4.2. Suppose that f is a measurable function defined on R × R that satisfies the following conditions: $ $ (i) for every compact subset K of R, K R |f (x, y)|dydx < ∞, and (ii) there exists a measurable $ $ function g on R × R such that, for every compact subset K of R, K R |g(x, y)|dydx < ∞, and that the distributional derivative Dx f (·, y) is represented by g(·, y), for every y ∈ R. Then, ∂ ∂ f (x, y)dy, a.e. x ∈ R, f (x, y)dy = ∂x ∂x R

R

where the derivatives are understood in the classical sense. $ Proof. We deﬁne the function h(x) = R f (x, y)dy, x ∈ R. By (i) h deﬁnes a regular distribution that we continue denoting by h. According to [11, Chap. 2, §5, Theorem V], we have that ∂ f (x, y) = g(x, y), a.e. (x, y) ∈ R × R, ∂x where the derivative is understood in the classical sense. Moreover, if F ∈ Cc∞ (R), then ∂f (x, y)dydx. Dx h, F = F (x) ∂x R

$

R

∂f (x, y)dy R ∂x

Hence, Dx h(x) = in the distributional sense. By using again [11, Chap. 2, §5, Theorem V] we conclude that ∂ ∂ h(x) = f (x, y)dy, a.e. x ∈ R. ∂x ∂x R

Thus the proof is completed.

References [1] Buraczewski, D., Mart´ınez, T., Torrea, J.L., Urban, R.: On the Riesz transform associated with the ultraspherical polynomials. J. Anal. Math. 98, 113– 143 (2006) [2] Buraczewski, D., Mart´ınez, T., Torrea, J.L.: Calder´ on-Zygmund operators associated to ultraspherical expansions. Canad. J. Math. 59(6), 1223–1244 (2007)

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[3] Campbell, J.T., Jones, R.L., Reinhold, K., Wierdl, M.: Oscillation and variation for the Hilbert transform. Duke Math. J. 105(1), 59–83 (2000) [4] Campbell, J.T., Jones, R.L., Reinhold, K., Wierdl, M.: Oscillation and variation for singular integrals in higher dimensions. Trans. Am. Math. Soc. 355(5), 2115–2137 (2003) [5] Gillespie, T.A., Torrea, J.L.: Dimension free estimates for the oscillation of Riesz transforms. Israel J. Math. 141, 125–144 (2004) [6] Grafakos, L.: Classical and modern Fourier analysis. Pearson Education, Upper Saddle River (2004) [7] Jones, R.L., Wang, G.: Variation inequalities for the Fej´er and Poisson kernels. Trans. Am. Math. Soc. 356(11), 4493–4518 (2004) [8] Muckenhoupt B.: Transplantation theorems and multiplier theorems for Jacobi series. Mem. Am. Math. Soc. 64(356) (1986) [9] Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965) [10] Roman, S.: The formula of Fa` a di Bruno. Am. Math. Monthly 87, 805–809 (1980) [11] Schwartz, L.: Th´eorie des Distributions. Hermann, Par´ıs (1973) [12] Stein, E.M.: Topics in Harmonic Analysis Related to the LittlewoodPaley Theory. Annals of Mathematics Studies, vol. 63. Princeton University Press, Princeton (1970) [13] Szeg˝ o, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Providence, RI, American Mathematical Society, Colloquium Publications (1975) Jorge J. Betancor, Juan C. Fari˜ na and Lourdes Rodr´ıguez-Mesa (B) Departamento de An´ alisis Matem´ atico Universidad de la Laguna Campus de Anchieta, Avda. Astrof´ısico Francisco S´ anchez, s/n 38271 La Laguna (Sta. Cruz de Tenerife) Spain e-mail: [email protected]; [email protected]; [email protected] Ricardo Testoni Departamento de Matem´ atica Universidad Nacional del Sur Avda. Alem 1253-2 Piso 8000 Bah´ıa Blanca Buenos Aires, Argentina e-mail: [email protected] Received: October 14, 2010. Revised: March 31, 2011.

Integr. Equ. Oper. Theory 70 (2011), 541–559 DOI 10.1007/s00020-011-1887-y Published online June 2, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Toeplitz Operators from One Fock Space to Another Zhangjian Hu and Xiaofen Lv Abstract. In this paper, we study Toeplitz operators Tμ from one Fock space Fαp to another Fαq for 1 < p, q < ∞ with positive Borel measures µ as symbols. We characterize the boundedness (and compactness) of r and the t-Berezin Tμ : Fαp → Fαq in terms of the averaging function µ transform µ t respectively. Quite differently from the Bergman space case, we show that Tμ is bounded (or compact) from Fαp to Fαq for some p ≤ q if and only if Tμ is bounded (or compact) from Fαp to Fαq for all p ≤ q. In order to prove our main results on Tμ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on Cn . Mathematics Subject Classification (2010). Primary 47B35; Secondary 47B10; 32A36. Keywords. Toeplitz operator, Fock space, Carleson measure.

1. Introduction Let Cn be the n-dimensional complex Euclidean space. For any z = (z1 , . . . , zn )and w = (w1 , . . . , wn ) ∈ Cn , we write z, w = z1 w1 + · · · + zn wn , and |z| = z, z. Given any α > 0, we consider the Gaussian probability measure α n 2 e−α|z| dv(z) dvα (z) = π on Cn , where dv(z) is ordinary Lebesgue volume measure on Cn . For 0 < p < ∞, the Lebesgue space Lpα consists of all measurable functions f for which ⎧ ⎫ p1 p ⎨ ⎬ α|z|2 f (z)e− 2 dv(z) < ∞. f p,α = ⎩ ⎭ Cn

This work was completed with the support of National Natural Science Foundation of China (10771064), Natural Science Foundation of Zhejiang province (Y7080197, Y6090036, Y6100219) and Foundation of Creative Group in Colleges and Universities of Zhejiang Province (T200924).

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Z. Hu and X. Lv

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Let H(Cn ) be the family of all entire functions on Cn . We deﬁne the Fock space Fαp = Lpα H(Cn ) for 0 < p < ∞. It is clear that, for p ≥ 1, Lpα is a Banach space under the norm · p,α , Fαp is a closed subspace of Lpα . And Fα2 is a Hilbert space with the inner product

2 f (z)g(z)e−α|z| dv(z). (f, g) = Cn

For 0 < p < 1, Fαp is an F-space under d(f, g) = f − gpp,α . The Fock space has been studied in [3,4,9,20] and by some other authors. Let Kα (z, w) be the reproducing kernel of the Fock space Fα2 , and (w,z) be the normalized kernel. It is well known that let kz (w) = KKαα(·,z) 2,α

Kα (z, w) = eαz,w , see [8]. The orthogonal projection P : L2α → Fα2 is deﬁned as

Kα (z, w)f (w) dvα (w). Pα f (z) = Cn

See [2–4,12] for details. Given ϕ ∈ L∞ , we can deﬁne a linear operator Tϕ : Fα2 → Fα2 by Tϕ (f ) = Pα (ϕf ), which is called the Toeplitz operator with symbol ϕ, see [3,13]. More generally, if μ is a positive Borel measure on Cn (simply write μ ≥ 0), we deﬁne the Toeplitz operator Tμ with symbol μ as

2 Tμ f (z) = Kα (z, w)f (w)e−α|w| dμ(w), z ∈ Cn . Cn

As mentioned in [11], Tμ is very loosely deﬁned here, because it is not clear when the integrals above will converge, even if the measure μ is ﬁnite, as the kernel function Kα (z, w) is unbounded for any ﬁxed z = 0. Suppose the given Borel measure μ satisﬁes the condition

2 |Kα (z, w)|e−α|w| d|μ|(w) < ∞ (1.1) Cn

for all z ∈ C . Because of the exponential form of Kα , condition (1.1) is equivalent to

2 |Kα (z, w)|2 e−α|w| d|μ|(w) < ∞ (1.2) n

Cn

for all z ∈ C . Set n

K = span {kz : z ∈ Cn } .

(1.3)

It is easy to check that K is dense in Fαp for all p > 1. If μ satisﬁes condition (1.1), then by (1.2) and the Cauchy–Schwarz inequality one knows that Tμ (f )

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is well-deﬁned on K. That is, under hypothesis (1.2), Tμ is well deﬁned on some dense subset of Fαp for each 1 0. Toeplitz operators with symbol ϕ ∈ L∞ , acting on Fα2 , have been well studied. The characterizations on ϕ, for which the induced Toeplitz operators Tϕ is bounded (or compact) on Fα2 , have been considered in [2,3,5,10,17–19]. Recently, Isralowitz and Zhu [11] obtained the boundedness, compactness and Schatten class membership of Toeplitz operators with μ ≥ 0 on the Fock space Fα2 . On the other hand, compared with the Toeplitz operators on Fock spaces, the Toeplitz operators on the Bergman spaces have been studied much more extensively, see for example [21,23] and the reference therein. Given a ﬁnite positive Borel measure ν on the unit disc D (or the unit ball B of Cn ), the behavior of the Toeplitz operator Tν on the Bergman spaces has been well understood. Bounded and compact Toeplitz operators Tν are completely characterized in terms of Carleson type measure as in [14,22,24]. Recently, the analogous characterizations on harmonic Bergman spaces have also been well discussed, see [7,16]. The purpose of this work is to characterize those positive Borel measures μ on Cn , for which the Toeplitz operators Tμ are bounded (or compact) from one Fock space Fαp to another Fαq . To our knowledge, little has been known about Tμ on Fαp with p = 2, not to mention the mapping property of Tμ between different Fock spaces. We will establish our main theorems in Sect. 4. In [6,7], it is proved that the boundedness of the Toeplitz operators with positive Borel measure symbols from the pth harmonic Bergman space to the qth harmonic Bergman space depends on the ratio pq . And the approach in [6,7] can be easily adapted for holomorphic Bergman spaces. To our surprising, we will see in Sect. 4 that Tμ is bounded from Fαp to Fαq for some q ≥ p if and only if Tμ is bounded from Fαp to Fαq for all q ≥ p. This is quite different from the Bergman space case. Before obtaining our main results, we need some preparations which are of their own interest. In Sect. 2, among other estimates we will prove that the t Berezin transform (see the definition in Sect. 2) is bounded on the Lebesgue spaces Lp (dv) for 1 ≤ p ≤ ∞, this operator has a very close connection with the heat ﬂow as mentioned in [1]. Section 3 is devoted to give the definition of (vanishing) Carleson measures, and we characterize this measure in terms of the averaging function and the Berezin transform. In the following sections, we use C to denote positive constants whose value may change from line to line but does not depend on the functions being considered. The expression “A B” means there exists some C such that C −1 A ≤ B ≤ CA.

2. Some Preliminary Results In this section, we are going to obtain some preliminary results, which will be used in the subsequent sections. Given some point z ∈ Cn and r > 0, write B(z, r) = {w ∈ Cn : |w − z| < r}

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for the Euclidean ball centered at z with radius r. For a Borel measure μ μ(B(z,r)) . Since the Lebesgue volume on Cn , the average of μ on B(z, r) is vol(B(z,r)) vol(B(z, r)) = B(z,r) dv is a constant when z varies in Cn , we simply write the average function of μ as μ r (z) = μ(B(z, r)). For a Borel measure μ on Cn , t > 0, deﬁne the t-Berezin transform of μ on Cn to be t

2 αt Kα (z, w) μ t (z) = e− 2 |z−w| dμ(w). (2.1) dμ(w) = Kα (z, z)Kα (w, w) n n C

C

Notice that μ 2 is just the Berezin transform μ deﬁned in [11]. Also, it is valuable to study μ t for general t > 0, because the t-Berezin transform is closely connected with the heat ﬂow. By the reproducing property (see Lemma 3 in [8]), we have α

Kα (·, z)p,α e 2 |z|

2

for all 0 0. Hence,

2 t α μ t (z) kz (w)e− 2 |w| dμ(w).

(2.2)

Cn

r and Given a measurable function f , we set dμ = f dv, and write fr (z) = μ ft = μ t for simplicity. Lemma 2.1. Let 1 ≤ p ≤ ∞, r > 0, t > 0. Then both operators f → fr and f → ft are bounded on Lp (dv). Proof. The conclusions are trivial if p = ∞. For p = 1, by the Fubini’s theorem and the fact χB(z,r) (w) = χB(w,r) (z) for all z, w ∈ Cn , we get

≤C dv(z) |f (w)| dv(w) fr L1 (dv)

Cn

B(z,r)

|f (w)| dv(w)

=C Cn

χB(w,r) (z) dv(z)

Cn

≤ C f L1 (dv) ; and

ft

L1 (dv)

=

|ft (z)| dv(z)

Cn

≤

dv(w) Cn

e−

Cn

2 αt 2 |z−w|

|f (z)| dv(z)

= Cn

= Cf L1 (dv) .

Cn

e−

|f (z)| dv(z)

2 αt 2 |z−w|

dv(w)

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The boundedness for the range 1 0, 0 0. Then there exists some constant C such that

2 p 2 p α α f (z)e− 2 |z| dμ(z) ≤ C f (z)e− 2 |z| μ (B(z, δ)) dv(z) Cn

Cn

for all f ∈ H(C ). n

Proof. By Lemma 2.1 of [11], for any f ∈ H(Cn ), we have

2 p 2 p α α f (z)e− 2 |z| ≤ C f (w)e− 2 |w| dv(w).

(2.3)

B(z,δ)

Hence,

p −α |z|2 2 f (z)e dμ(z) ≤ C Cn

2 p α f (w)e− 2 |w| dv(w) dμ(z)

Cn B(z,δ)

p −α |w|2 2 =C f (w)e dv(w) χB(w,δ) (z) dμ(z) Cn

n

C

2 p −α |w| =C f (w)e 2 μ (B(w, δ)) dv(w). Cn

This completes the proof.

For our next lemma, we need the concept of lattice. For r > 0, a sequence {ak } in Cn is called an 2r -lattice if the following conditions are satisﬁed: ∞ (1) B(ak , 2r ) = Cn ; k=1

(2) {B(ak , 4r )}∞ k=1 are pairwise disjoint. With these two hypotheses, it is easy to check that (3) For any δ > 0 there exists a positive integer m (depending only on r and δ) such that every point in Cn belongs to at most m of the sets B(ak , δ). → → → − For example, take − r1 = ( 2r , 0, . . . , 0), − r2 = ( ri 2 , 0, . . . , 0), r3 = → −−−→ −→ r r4 = (0, ri (0, 2r , 0, . . . , 0), − 2 , 0, . . . , 0), . . . , r2n−1 = (0, . . . , 0, 2 ), r2n = ri (0, . . . , 0, 2 ), set → − r + ··· + b r→, k = 1, 2, . . . , a =b − k

k,1 1

k,2n 2n

where bk,1 , . . ., bk,2n run over all the integers. Then the set of points {ak } is an 2r −lattice. The sequence {ak } is reserved for this lattice throughout the paper. Note that ak → ∞ as k → ∞. For t = 2 and r = δ, the following lemma, Lemma 2.3, was ﬁrst obtained in [11]. Lemma 2.3. Let 1 ≤ p ≤ ∞ and let μ ≥ 0. Then the following statements are equivalent:

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(1) μ t ∈ Lp (dv) for some (or any) t > 0; (2) μ(B(·, δ)) ∈ Lp (dv) for some (or any) δ > 0; (3) {μ(B(ak , r))} ∈ lp for some (or any) r > 0. Moreover, μt Lp (dv) μ(B(·, δ))Lp (dv) {μ(B(ak , r))}lp .

(2.4)

Proof. (1)⇒(2). For 0 < t < ∞, r > 0 and μ ≥ 0, (2.1) yields μ(B(z, r)) ≤ C μ t (z) for all z ∈ C , since

αt 2 μ(B(z, r)) = dμ(w) ≤ e 2 r

(2.5)

n

B(z,r)

e−

2 αt 2 |z−w|

dμ(w) ≤ e

αt 2 2 r

μ t (z).

B(z,r)

This means μ(B(·, δ))Lp (dv) ≤ C μt Lp (dv) < ∞. (2)⇒(1). By (2.2) and Lemma 2.2, we obtain

2 t α δ))] (z). μ t (z) ≤ C kz (w)e− 2 |w| μ(B(w, δ))dv(w) = C [μ(B(·, t Cn

This, together with Lemma 2.1, gives that, if μ(B(·, δ)) ∈ Lp (dv), δ))]t p μt Lp (dv) ≤ C [μ(B(·, ≤ C μ(B(·, δ))Lp (dv) < ∞. L (dv)

(2)⇔(3). Let μ ≥ 0, r, δ > 0. First, for any a ∈ Cn , we claim that

μ (B(z, δ)) dv(z). (2.6) μ (B(a, r)) ≤ C B(a,r)

In fact,

μ (B(z, δ)) dv(z) =

χB(a,r) (z) dv(z)

Cn

B(a,r)

=

dμ(w)

Cn

χB(z,δ) (w) dμ(w)

Cn

χB(z,δ) (w) dv(z).

B(a,r)

Since χB(z,δ) (w) = χB(w,δ) (z) for all z, w ∈ Cn , then

μ (B(z, δ)) dv(z) = dμ(w) χB(w,δ) (z) dv(z) Cn

B(a,r)

≥

B(a,r)

v (B(a, r) ∩ B(w, δ)) dμ(w) B(a,r)

≥ μ(B(a, r))

inf

w∈B(a,r)

v (B(a, r) ∩ B(w, δ)) .

For w ∈ B(a, r), there then exists a Euclidean ball with diameter 1 2 min{r, δ} contained in B(a, r) ∩ B(w, δ). Therefore, (2.6) holds. Setting

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f (w)=μ(B(w, δ)), the estimate (2.6) tells us μ(B(z, r)) ≤ C fr (z). Then, Lemma 2.1 gives μ(B(·, r))Lp (dv) ≤ Cfr Lp (dv) ≤ Cf Lp (dv) = Cμ(B(·, δ))Lp (dv) . Therefore, μ(B(·, r)) ∈ Lp (dv) for all r > 0 if μ(B(·, δ)) ∈ Lp (dv) for some δ > 0. Similar to the proof of Theorem 4.4 in [11], for r > 0, 1 ≤ p < ∞, we know μ(B(·, r))Lp (dv) {μ(B(ak , r))}lp ; and, which remains true by Theorem 2.3 of [11] if p = ∞. Checking the proof above carefully, we have estimate (2.4). This completes the proof. p Lemma 2.4. Let 1 ≤ p ≤ ∞. For λ = {λk }∞ k=1 ∈ l , set

S(λ)(z) =

∞

∞

λk kak (z) =

k=1

α

2

λk eαz,ak − 2 |ak | ,

z ∈ Cn .

(2.7)

k=1

Then S is a bounded operator from lp to Fαp . Proof. Recall from (2.1) that f1 (z) =

2

α

e− 2 |z−w| f (w) dv(w).

Cn

Given λ = {λk } ∈ lp , deﬁne f (w) = tells us f1 (z) =

∞

=

|λk |χB(ak ,r) (w). Lemma 2.1 of [11] 2

α

e− 2 |z−w| dv(w)

|λk | B(ak ,r)

α

2

e− 2 |w| dv(w)

|λk |

k=1

≥C

k=1

k=1 ∞

∞

B(ak −z,r)

∞

α

2

|λk |e− 2 |z−ak | .

k=1

∞ 2 2 α α Thus, S(λ)(z)e− 2 |z| ≤ |λk | e− 2 |z−ak | ≤ C f1 (z). By Lemma 2.1 we obtain

k=1

S(λ)pp,α

2 p α = S(λ)(z)e− 2 |z| dv(z) Cn

≤ Cf1 pLp (dv) ≤ Cf pLp (dv) ≤ Cλplp .

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The last estimate remains valid for p = ∞, because ∞ f L∞ = sup |λk |χB(ak ,r) (z) z∈Cn

k=1

≤ sup |λk | · sup k

z∈Cn

∞

χB(ak ,r) (z)

k=1

≤ mλl∞ . This means S is bounded from lp to Lpα for 1 ≤ p ≤ ∞. Since series (2.7) converges in norm and each term is holomorphic, and it converges uniformly on every compact subset of Cn , then S(lp ) ⊂ Lpα H(Cn )). Therefore, S : lp → Fαp is bounded. This completes the proof. Remark. Theorem 8.3 in [12] contains a special case of our Lemma 2.4.

3. (Vanishing) Carleson Measure In this section, for μ ≥ 0, we will give the definition of (vanishing) (p, q)-Fock Carleson measures, and then characterize them in terms of the t-Berezin r . First, we deﬁne vanishing transform μ t and the averaging function μ (p, q)-Fock Carleson measures as follows. Definition. Let 0 < p, q < ∞ and let μ ≥ 0. We call μ a (p, q)-Fock Carleson measure, if the embedding operator i : Fαp → Lqα (dμ) is bounded, i.e. there exists some constant C such that for all f ∈ Fαp , ⎛ ⎞ q1

q 2 α ⎝ f (z)e− 2 |z| dμ(z)⎠ ≤ Cf p,α . Cn

We will write μ for the operator norm of i from Fαp to Lqα (dμ). And also, we call μ a vanishing (p, q)-Fock Carleson measure if

2 q α lim fj (z)e− 2 |z| dμ(z) = 0 j→∞ Cn

whenever {fj } is a bounded sequence in Fαp that converges to 0 uniformly on compact subsets of Cn as j → ∞. The following three theorems, Theorems 3.1–3.3, characterize the (vanishing) (p, q)-Fock Carleson measures for all possible 0 < p, q < ∞. When p = q = 2, it was ﬁrst obtained by Isralowitz and Zhu in [11]. Theorem 3.1. Let 0 0; μ(B(·, δ)) is bounded on Cn for some (or any) δ > 0;

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For some (or any) r > 0, the sequence {μ(B(ak , r))}∞ k=1 is bounded. Furthermore, 1 q 1 1 q q} μ(B(·, δ)) {μ(B(a μ , r)) (3.1) μ ∞. k t ∞ L∞ l L

Proof. The equivalence among the statements (2), (3) and (4) follows from Lemma 2.3 with p = ∞. We prove (1) implies (2). For this purpose, ﬁx any z ∈ Cn , set fz (w) = kz (w), w ∈ Cn . It is easy to verify that fz ∈ Fαp with fz p,α = 1. Hence, statement (1) and estimate (2.2) imply

2 q α (3.2) μ q (z) = fz (w)e− 2 |w| dμ(w) ≤ Cfz qp,α = C. Cn

Lemma 2.3 tells us that μ t is bounded on Cn for any t > 0, and 1 1 q q q μ ∞ ≤ Cμ. t ∞ μ L

L

We now prove the implication “(4)⇒(1)”. For any f ∈ Fαp and a ∈ Cn , by (2.3),

p 2 p α −α |z|2 2 sup f (z)e ≤C f (w)e− 2 |w| dv(w). z∈B(a,r)

Since

q p

B(a,2r)

≥ 1, using the fact that ∞ l ∞ l bk ≤ bk if 1 ≤ l < ∞ and bk ≥ 0 for all k, k=1

k=1

we obtain

2 q α f (z)e− 2 |z| dμ(z) Cn

≤

∞

2 q α f (z)e− 2 |z| dμ(w)

k=1 B(ak ,r)

≤

∞

μ(B(ak , r))

k=1

≤C

sup z∈B(ak ,r)

⎛

∞

⎜ μ(B(ak , r)) ⎝

k=1

2 p α f (z)e− 2 |z|

⎞ pq 2 p α ⎟ f (w)e− 2 |w| dv(w)⎠

B(ak ,2r)

⎛ ⎜ ≤ C sup μ(B(ak , r)) ⎝ k

∞

k=1B(a ,2r) k

q p

≤ Cm sup μ(B(ak , r))f qp,α . k

pq

⎞ pq 2 p α ⎟ f (w)e− 2 |w| dv(w)⎠

(3.3)

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Therefore, μ is a (p, q)-Fock Carleson measure. Estimate (3.1) follows from the implication above and Lemma 2.3. This completes the proof.

Theorem 3.2. Let 0 0; μ(B(z, δ)) → 0 as z → ∞ for some (or any) δ > 0; μ(B(ak , r)) → 0 as k → ∞ for some (or any) r > 0.

Proof. It is easy to check that {kz : z ∈ Cn } in bounded in Fαp and that kz (w) → 0 uniformly on any compact subset of Cn as |z| → ∞. Then, (3.2) and statement (1) imply the statement (2) for t = q. This gives (2) for the special case t = q. The implication “(2)⇒(3)” for some t comes from (2.5) and that “(3)⇒(4)” follows from (2.6). To prove“(4)⇒(1)”, for any ε > 0, we then have some positive integer K0 such that μ(B(ak , r)) < ε whenever k > K0 . Let {fj } be any bounded sequence in Fαp and fj → 0 uniformly on each compact subset of Cn as j → ∞. We claim that

2 q α lim fj (z)e− 2 |z| dμ(z) = 0.

j→∞ Cn

In fact,

K0 k=1

B(ak , 2r) is a compact subset of Cn , so ⎛

K0

⎜ μ(B(ak , r)) ⎝

k=1

⎞ pq 2 p α ⎟ fj (w)e− 2 |w| dv(w)⎠ < ε

B(ak ,2r))

if j is sufﬁciently large. On the other hand, with statement (4) and estimate (3.3), ⎛ ∞

⎜ μ(B(ak , r)) ⎝

k=K0 +1

B(ak ,2r)) q p

< Cεm fj qp,α ≤ Cε,

⎞ pq

2 p α ⎟ fj (w)e− 2 |w| dv(w)⎠

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where C is independent of ε. Therefore, similar to the proof for Theorem 3.1, by (3.3), for j large enough, we get

2 q α fj (z)e− 2 |z| dμ(z) Cn

⎛ ≤C

K0

⎜ μ(B(ak , r)) ⎝

k=1

⎞ pq 2 p α ⎟ fj (w)e− 2 |w| dv(w)⎠

B(ak ,2r))

⎛

+C

∞

⎜ μ(B(ak , r)) ⎝

k=K0 +1

⎞ pq 2 p α ⎟ fj (w)e− 2 |w| dv(w)⎠

B(ak ,2r))

⎧ ⎛ ⎞ pq ⎫ ⎪ ⎪ ⎪ ⎪

∞ ⎬ ⎨ 2 p ⎜ ⎟ −α |w| < C ε + sup μ(B(ak , r)) ⎝ fj (w)e 2 dv(w)⎠ ⎪ ⎪ k≥K0 +1 ⎪ ⎪ k=1 ⎭ ⎩ B(ak ,2r)) q < C ε + m p fj qp,α ε = Cε. This means that μ is a vanishing (p, q)-Fock Carleson measure. Moreover, this tells us that (1) is equivalent to (3). Thus, if μ is a vanishing (p, q)-Fock Carleson measure, then μ is a vanishing ( pt q , t)-Fock Carleson measure, which gives the implication (1)⇒(2) for all possible t > 0. This completes the proof. In order to characterize the (p, q)-Fock Carleson measure for the case 0 < q < p < ∞, we will borrow Luecking’s idea in [15]. So, we ﬁrst introduce Khinchine’s inequality. Recall that Rademacher functions ψk are deﬁned by 1, if 0 ≤ t − [t] < 12 ψ0 (t) = −1, if 12 ≤ t − [t] < 1 and ψk (t) = ψ0 (2k t) for k = 1, 2, . . ., where [t] denotes the largest integer not greater than t. For 0 < l < ∞, Khinchine’s inequality holds. That is, there exists some positive constants C1 and C2 depending only on l such that l m 2l m 2l

1 m 2 2 C1 |bk | ≤ bk ψk (t) dt ≤ C2 |bk | k=1

0

k=1

k=1

for all m ≥ 1 and complex numbers b1 , b2 , . . . , bm . Now, we are ready to characterize the (p, q)-Fock Carleson measure for 0 < q < p < ∞. Theorem 3.3. Let 0 < q < p < ∞, and let μ ≥ 0. Set s = pq and s to be the conjugate exponent of s. Then the following statements are equivalent: (1) (2)

μ is a (p, q)-Fock Carleson measure; μ is a vanishing (p, q)-Fock Carleson measure;

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(3) μ t ∈ Ls (dv) for some (or any) t > 0; (4) μ(B(·, δ)) ∈ Ls (dv) for some (or any) δ > 0; ∞ s (5) k=1 μ(B(ak , r)) < ∞ for some (or any) r > 0. Furthermore, 1

1

1

μ μt Lq s (dv) μ(B(·, δ))Lq s (dv) {μ(B(ak , r))}lqs .

(3.4)

Proof. By Lemma 2.3, we know that (3), (4) and (5) are equivalent, moreover, the corresponding norms in (3.4) are equivalent. lp , put f as (2.7). Then We now prove (1) implies (5). For each ∞ {λk } ∈ p p p by Lemma 2.4, f ∈ Fα and f p,α ≤ C k=1 |λk | . Since μ is a (p, q)-Fock Carleson measure, we know q

∞ 2 2 α α λk eαz,ak − 2 |z| − 2 |ak | dμ(z) ≤ μq f qp,α Cn

k=1

≤ Cμq

∞

pq |λk |p

.

k=1

In the above inequality, replace λk with ψk (t)λk , where ψk (t) is deﬁned as in Khinchine’s inequality, and then integrate with respect to t from 0 to 1. After making use of Khinchine’s inequality and Fubini’s theorem, we obtain q2

∞ 2 −α|z−ak |2 |λk | e dμ(z) k=1

Cn

⎧ q ⎫

⎨ 1 ∞ ⎬ 2 2 α α λk eαz,ak − 2 |z| − 2 |ak | ψk (t) dt dμ(z) ≤C ⎭ ⎩ 0 k=1 Cn ⎫ ⎧ q

1 ⎨ ∞ ⎬ 2 2 α α =C [ψk (t)λk ] eαz,ak − 2 |z| − 2 |ak | dμ(z) dt ⎭ ⎩ 0

≤ Cμq

Cn

k=1

∞

pq

|λk |p

.

k=1

Therefore, applying estimate (3.3) if 2q ≤ 1 and Holder’s inequality if we get

∞ ∞ |λk |q μ(B(ak , r)) = |λk |q χB(ak ,r) (z) dμ(z) k=1

Cn k=1 1− q2

≤ max{m

, 1}

∞

Cn

k=1

2 q

> 1,

q2 |λk |2 χB(ak ,r) (z)

dμ(z)

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Toeplitz Operators

∞

αr 2

≤ Ce

Cn

≤ Cμ

q

2 −α|z−ak |2

q2

|λk | e

k=1 ∞

553

dμ(z)

pq

|λk |

p

.

k=1

Set ck = |λk |q , then {ck } ∈ ls and ∞

ck μ(B(ak , r)) ≤ Cμ

q

k=1

∞

1s |ck |

s

.

k=1

A duality argument implies statement (5) and 1

{μ(B(ak , r))}lqs ≤ Cμ.

(3.5)

(4)⇒(1). Since s > 1, Lemma 2.2 and Holder’s inequality yield

2 q 2 q α α f (z)e− 2 |z| dμ(z) ≤ C f (z)e− 2 |z| μ(B(z, δ)) dv(z) Cn

Cn

≤ Cf qp,α μ(B(·, δ))Ls (dv)

(3.6)

for any f ∈ Fαp . This means that μ is a (p, q)-Fock Carleson measure. Together 1

with (3.5), we know that μ and {μ(B(ak , r))}lqs in (3.4) are equivalent. (1)⇔(2). Let μ satisfy (2). That is, μ is a vanishing (p, q)-Fock Carleson measure. By Montel’s Theorem on the normal family of holomorphic functions we know that the inclusion map i : Fαp → Lqα (dμ) is compact. Hence, i must be bounded from Fαp to Lqα (dμ). Equivalently, μ is a (p, q)-Fock Carleson measure. This gives the implication (2)⇒(1). For (1)⇒(2), let {fj } be any bounded sequence in Fαp and fj → 0 uniformly on each compact subset of Cn as j → ∞. For R > δ > 0, denote ER = {z ∈ Cn : |z| ≤ R − δ}, and μR the restriction of μ to Cn \ER . Then ⎞ ⎛

q 2 2 q α ⎟ ⎜ −α 2 |z| f (z)e dμ(z) fj (z)e− 2 |z| dμ(z) = ⎝ + ⎠ j Cn

ER

Cn \ER

= I1 + I2 . Since fj → 0

I1 =

uniformly on each compact subset of Cn , we obtain 2 q α q fj (z)e− 2 |z| dμ(z) ≤ C sup |fj (z)| → 0, j → ∞. z∈ER

ER

By (3.6) we have

2 q α I2 = fj (z)e− 2 |z| dμR (z) Cn

≤ Cfj qp,α μR (B(·, δ))Ls

≤ CμR (B(·, δ))Ls .

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For any ε > 0, since μ(B(·, δ)) ∈ Ls (dv), then ⎛ μR (B(·, δ))Ls

⎜ ≤⎝

⎞ 1 ⎟ μ(B(z, δ))s dv(z)⎠

s

<ε

|z|>R

if R large enough. Therefore,

2 q α lim fj (z)e− 2 |z| dμ(z) = 0, j→∞ Cn

which means that μ is a vanishing (p, q)-Fock Carleson measure. This completes the proof.

4. Toeplitz Operators By the theorems in Sect. 3, the notion of (vanishing) (p, q)-Fock Carleson measures does not depend on the particular value of p, q, but depends only on the ratio pq in the case 0 < q < p < ∞. Thus, in what follows, (vanishing) (p, q)-Fock Carleson measures will be simply called vanishing s-Fock Carleson measures, where s = pq . For 0 < s < ∞, let W s be the class of all s-Fock Carleson measures; and let W0s be the class of all vanishing s-Fock Carleson measures. When 0 < s ≤ 1 (equivalently, p ≤ q), we simply write W and W0 for W s and W0s respectively. That is W = {μ ≥ 0 : μ(B(·, δ)) ∈ L∞ for some δ > 0} and W0 =

! μ ≥ 0 : lim μ(B(z, δ)) = 0 for some δ > 0 . |z|→∞

Notice that W s ⊂ W and W0s ⊂ W0 for all s > 0. In this section, under hypothesis (1.1), we characterize those μ ≥ 0 for which the induced Toeplitz operators Tμ are bounded (or compact) from one Fock space to another. Our characterizations will be carried out in terms of (vanishing) s-Fock Carleson measures. Before we do that, we need the following lemma. Lemma 4.1. Suppose μ ∈ W . Then for f, g ∈ K as (1.3), we have

(Tμ f, g) = Cn

2

f (z)g(z)e−α|z| dμ(z).

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Proof. With μ(B(·, δ)) ∈ L∞ , we claim that Tμ (f ) is well-deﬁned on K. In fact, for any δ > 0 and z ∈ Cn , Lemmas 2.2 and Lemma 3 of [8] yield that

α2 w,z − α2 |w|2 2 −α|w|2 |Kα (z, w)| e dμ(w) ≤ C e e μ(B(w, δ)) dv(w) Cn

Cn

αz,w −α|w|2 dv(w) e e

≤C

Cn

≤ Ce

α|z|2 4

< ∞.

By the deﬁnition of K, to prove the conclusion, we only need to prove

2 α·,a1 α·,a2 Tμ e = ,e eαz,a1 eαz,a2 e−α|z| dμ(z) (4.1) Cn

for a1 , a2 ∈ Cn . This can be done with the help of Fubini’s theorem. For μ ∈ W , we pick some δ > 0 , then Lemma 2.2 and Lemma 3 of [8] tell us ⎛ ⎞

2 2 αa2 ,z −α|z| ⎝ αw,a1 αz,w −α|w| e dμ(w)⎠ dv(z) e e e e Cn

Cn

⎛ ⎞

2 2 αa2 ,z −α|z| ⎝ αw,a1 +z −α|w| μ (B(w, δ)) dv(w)⎠ dv(z) ≤C e e e e Cn

Cn

⎛ ⎞

αa2 ,z −α|z|2 ⎝ αw,a1 +z −α|w|2 dv(w)⎠ dv(z) ≤C e e e e Cn

Cn

≤C

e

α|a1 +z|2 4

Cn

= Ce

α|a1 |2 4

αa2 ,z −α|z|2 dv(z) e e

2 2 ,z − 3α|z| αa1 +2a 2 4 dv(z) e e Cn

≤ Ce

α(3|a1 |2 +|a1 +2a2 |2 ) 12

< ∞.

This gives (4.1). This completes the proof.

Theorem 4.2. Let 1 < p ≤ q < ∞. For μ ≥ 0 satisfying (1.1), the following statements are equivalent: (1) (2)

Tμ : Fαp → Fαq is bounded; μ ∈ W.

Proof. (1)⇒(2). By (2.3), for any z ⎧ ⎪ ⎨

2 −α |z| Tμ kz (z)e 2 ≤C ⎪ ⎩

∈ Cn and δ > 0, we get

B(z,δ)

⎫ q1 ⎪ ⎬ 2 q −α |w| . Tμ kz (w)e 2 dv(w) ⎪ ⎭

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Estimate (2.5) implies that

α|z|2 μ(B(z, r)) ≤ C μ 2 (z) = C Tμ kz (z)e− 2 ≤ CTμ kz q,α .

(4.2)

By the boundedness of Tμ from Fαp to Fαq , we know μ < ∞. (2)⇒(1). Suppose μ ∈ W . For 1 < p ≤ q < ∞, Theorem 3.1 implies that μ is a (p, q)-Fock Carleson measure, and μ is also a (q , q )-Fock Carleson measure, where q is the conjugate exponent of q. Therefore, for f, g ∈ K, Lemma 4.1 and Holder’s inequality imply

2 |(Tμ f, g)| ≤ f (z)g(z) e−α|z| dμ(z) Cn

⎛

⎞ q1 ⎛ ⎞ 1 q

q q 2 2 α α ≤ ⎝ f (z)e− 2 |z| dμ(z)⎠ · ⎝ g(z)e− 2 |z| dμ(z)⎠ Cn

Cn

≤ Cf p,α gq ,α .

Notice that (Fαq )∗ = Fαq under the pairing (·, ·), see [12]. Because K is dense in Fαp , the duality argument shows that Tμ : Fαp → Fαq is bounded. Remark. The fact that Tμ can be extended to a bounded operator from Fαp to Fαq means that Tμ is itself bounded. To see this, Lemma 3 in [8] and Lemma 2.2 above tell us that

2 eαz,w f (w)e−α|w| dμ(w) Tμ f (z) = Cn

makes sense for each z ∈ C if f ∈ Fαp and μ ∈ W . Then the duality argument implies that Tμ itself is bounded from Fαp to Fαq . Furthermore, ⎛ ⎞ q1

q 2 α Tμ f q,α ≤ Cμ ⎝ f (z)e− 2 |z| dμ(z)⎠ (4.3) n

Cn

This estimate can also be deduced from the last inequalities for any f ∈ in the proof of Theorem 4.2. Fαp .

Theorem 4.3. Let 1 < p ≤ q < ∞. For μ ≥ 0 satisfying (1.1), then the following statements are equivalent: (1) Tμ : Fαp → Fαq is compact; (2) μ ∈ W0 . Proof. (1)⇒(2). It can be easily justiﬁed by (4.2) and the fact that kz → 0 weakly in Fαp as z → ∞. (2)⇒(1). Suppose 1 < p ≤ q < ∞ and that μ ∈ W0 , then by Theorem 3.2 we know that μ is a vanishing (p, q)-Fock Carleson measure. Let {fj } be a sequence of functions such that fj → 0 weakly in Fαp as j → ∞. Then

2 q α lim fj (z)e− 2 |z| dμ(z) = 0. j→∞ Cn

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Estimate (4.3) yields ⎛

Tμ fj q,α

⎞ q1

q 2 α ≤ C ⎝ fj (z)e− 2 |z| dμ(z)⎠ → 0 Cn

as j → ∞. Therefore Tμ :

Fαp

→ Fαq is compact, completing the proof.

Remark. Theorems 4.2 and 4.3 tell us that Tμ is bounded (or compact) from Fαp to Fαq for some 1 < p ≤ q < ∞ if and only if Tμ is bounded (or compact) from Fαp to Fαq for all 1 < p ≤ q < ∞. This is quite diﬀerent from the Bergman space case. Theorem 4.4. Let 1 < q < p < ∞, and let 1s = 1 − 1q + p1 . For μ ≥ 0 satisfies (1.1), then the following statements are equivalent: (1) (2)

Tμ : Fαp → Fαq is bounded; Tμ : Fαp → Fαq is compact;

(3)

Tμ : Fα2s → Fα2s−1 is bounded;

(4) (5) (6)

Tμ : Fα2s → Fα2s−1 is compact; μ ∈ W s; μ ∈ W0s .

2s 2s

Proof. We will prove by the order (1)⇒(3)⇒(5)⇒(1), (5)⇔(6), (2)⇒(4)⇒(3) and (6)⇒(2). The implication (4)⇒(3) is trivial. (5)⇔(6) follows from Theorem 3.3. Analogous to the proof of (2)⇒(1) for Theorems 4.2 and 4.3, we have (5)⇒(1) and (6)⇒(2), respectively. The implications (1)⇒(3) and (2)⇒(4) come from the complex interpolation with respect to the Fock spaces. In fact, by Lemma 4.2 and Theorem 7.1 in [12], (Tμ )∗ = Tμ which is also bounded (or compact) from Fαq to Fαp . Applying Theorem 9.3 in [12] we know that the linear operator Tμ is bounded 2s

(or compact) from Fα2s to Fα2s−1 with

1 s

=1−

1 q

+ p1 .

2s (3)⇒(5). Set s1 = 2s−1 > 1, then s1 = 2s. Applying Lemma 4.1 and Holder’s inequality, we have

2 2 α f (z)e− 2 |z| dμ(z) = |(Tμ f, f )| Cn

≤ Tμ f s1 ,α f s ,α 1

≤ Tμ

2s Fα2s →Fα2s−1

f 22s,α ,

or equivalently, f 2L2α (dμ) ≤ Tμ

2s

Fα2s →Fα2s−1

f 22s,α

(4.4)

for any f ∈ K. Since K is dense in Fα2s , Fatou’s Lemma tells us the estimate (4.4) holds for all f ∈ Fα2s . Thus, μ is a (2s, 2)-Fock Carleson measure, so μ ∈ W s by Theorem 3.3. This completes the proof.

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Acknowledgment The authors express their thanks to the referees for their careful reading and valuable suggestions.

References [1] Bauer, W., Coburn, L.A., Isralowitz, J.: Heat ﬂow, BMO, and the compactness of Toeplitz operators. J. Funct. Anal. 259, 57–78 (2010) [2] Berger, C.A., Coburn, L.A.: Toeplitz operators and quantum mechanics. J. Funct. Anal. 68, 273–299 (1986) [3] Berger, C.A., Coburn, L.A.: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301(2), 813–829 (1987) [4] Berger, C.A., Coburn, L.A.: Heat Flow and Berezin-Toeplitz estimates. Am. J. Math. 116(3), 563–590 (1994) [5] Coburn, L.A., Li, B., Isralowitz, J.: Toeplitz operators with BMO symbols on the Segal-Bargmann space. Trans. Am. Math. Soc. 363, 3015–3030 (2011) [6] Choe, B.R., Koo, H., Yi, H.: Positive Toeplitz operators between the harmonic Bergman spaces. Potential Anal. 17, 307–335 (2002) [7] Choe, B.R., Lee, Y.J., Na, K.: Positive Toeplitz operators from a harmonic Bergman space into another. Tohoku Math. J. 56(2), 255–270 (2004) [8] Dostani´c, M., Zhu, K.H.: Integral operators induced by the Fock kernel. Integr. Equ. Oper. Theory 60, 217–236 (2008) [9] Guillemin, V.: Toeplitz operators in n-dimensions. Integr. Equ. Oper. Theory 7, 145–205 (1984) [10] Grudsky, S., Vasilevski, N.: Toeplitz operators on the Fock space: radial component eﬀects. Integr. Equ. Oper. Theory 44(1), 10–37 (2002) [11] Isralowitz, J., Zhu, K.H.: Toeplitz operators on the Fock space. Integr. Equ. Oper. Theory 66(4), 593–611 (2010) [12] Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Revista Mat. Iberoamer. 3, 61–138 (1987) [13] Kr¨ otz, B., Olafsson, G., Stanton, R.: The image of the heat kernel transform on Riemannian symmetric spaces of the non-compact type. Int. Math. Res. Not. 22, 1307–1329 (2005) [14] Luecking, D.H.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73(2), 345–368 (1987) [15] Luecking, D.H.: Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Mich. Math. J. 40(2), 333–358 (1993) [16] Miao, J.: Toeplitz operators on harmonic Bergman spaces. Integr. Equ. Oper. Theory 27(4), 426–438 (1997) [17] Ram´ırez De Arellano, E., Vasilevski, N.L.: Toeplitz operators on the Fock space with presymbols discontinuous on a thick set. Math. Nachr. 180, 299–315 (1996) [18] Sangadji Stroethoﬀ, K.: Compact Toeplitz operators on generalized Fock spaces. Acta Sci. Math. (Szeged) 64(3–4), 657–669 (1998) [19] Stroethoﬀ, K.: Hankel and Toeplitz operators on the Fock space. Mich. Math. J. 39(1), 3–16 (1992) [20] Tung, J.: Fock Spaces, Ph. D. thesis, University of Michigan (2005)

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[21] Vasilevski, N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Operator Theory: Advances and Applications, vol. 185. Birkhauser, Basel (2008) [22] Zhu, K.H.: Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Oper. Theory 20, 329–357 (1988) [23] Zhu, K.H.: Spaces of Holomorphic Functions in the Unit Ball. Springer, Berlin (2005) [24] Zhu, K.H.: Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball. N. Y. J. Math. 13, 299–316 (2007) Zhangjian Hu (B) Huzhou Teachers College Huzhou 313000, Zhejiang People’s Republic of China e-mail: [email protected] Xiaofen Lv Huzhou Teachers College Huzhou 313000, Zhejiang People’s Republic of China e-mail: [email protected] Received: October 28, 2010. Revised: May 11, 2011.

Integr. Equ. Oper. Theory 70 (2011), 561–568 DOI 10.1007/s00020-011-1875-2 Published online March 27, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

On the Dunford Property (C) for Bounded Linear Operators RS and SR Pietro Aiena and Manuel Gonz´alez Abstract. In this paper we show that if S ∈ L(X, Y ) and R ∈ L(Y, X), X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and SR ∈ L(X) in the case that R and S satisfies the operator equations RSR = R2 and SRS = S 2 . Mathematics Subject Classification (2000). Primary 47A10, 47A11; Secondary 47A53, 47A55. Keywords. Glocal spectral subspaces, Dunford’s property (C), SVEP.

1. Introduction and Definitions For a bounded linear operator T defined on a complex Banach space X the local resolvent set of T at the point x ∈ X is defined as the union of all open subsets U of C such that there exists an analytic function f : U → X which satisfies (λI − T )f (λ) = x for all λ ∈ U.

(1)

The local spectrum σT (x) of T at x is the set defined by σT (x) := C\ρT (x) and, obviously, we have σT (x) ⊆ σ(T ), where σ(T ) denotes the spectrum of T . For every subset F of C, the analytic spectral subspace of T associated with F is the set XT (F) := {x ∈ X : σT (x) ⊆ F}. It is easily seen from the definition that XT (F) is a linear subspace T -invariant of X. Furthermore, for every closed F ⊆ C we have (λI − T )XT (F) = XT (F)

for all λ ∈ C\F

(2)

see [7, Proposition 1.2.16]. An operator T ∈ L(X) is said to have the single valued extension property at λo ∈ C (abbreviated SVEP at λo ), if for every Supported in part by MICINN (Spain), Grant MTM2010-20190.

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open disc Dλo centered at λo the only analytic function f : Dλo → X which satisfies the equation (λI − T )f (λ) = 0

(3)

is the function f ≡ 0. An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every point λ ∈ C. Clearly, the SVEP is inherited by the restrictions on invariant subspaces. A variant of the local spectral subspaces, which is more useful for operators without SVEP, is given by the glocal spectral subspace XT (F). This subspace is defined, for an operator T ∈ L(X) and a closed subset F of C, as the set of all x ∈ X for which there exists an analytic function f : C\F → X which satisfies the identity (λI − T )f (λ) = x for all λ ∈ C\F. In general, XT (F) ⊆ XT (F) for every closed subset F ⊆ C, but the two concepts of glocal spectral subspace and analytic spectral subspace coincide if T has SVEP, i.e., if T has SVEP then XT (F) = XT (F) for all closed subsets F ⊆ C, see [7, Proposition 3.3.2]. Note that XT (F), as well as XT (F), in general is not closed. Definition 1.1. An operator T ∈ L(X) is said to have Dunford’s property (C) (abbreviated property (C)) if, for each closed set F ⊆ C, XT (F) is closed. Let us consider the particular case that F is a singleton set, say F := {λ}. The glocal spectral subspace XT ({λ}) coincides with the quasi-nilpotent part H0 (λI − T ) of λI − T defined by H0 (λI − T ) := {x ∈ X : lim sup (λI − T )n x1/n = 0}, n→∞

see [1, Theorem 2.20]. In general H0 (λI − T ) is not closed and T ∈ L(X) is said to have property (Q) if H0 (λI − T ) is closed for every λ ∈ C. It is known that if H0 (λI − T ) is closed then T has SVEP at λ [2]. Consequently, Property (C) ⇒ property (Q) ⇒ SVEP. Now, let S ∈ L(X, Y ) and R ∈ L(Y, X), where X and Y are Banach spaces. It is known that SR ∈ L(Y ) and RS ∈ L(X) share many spectral properties [4] and local spectral properties, see [5,8], as property (β) and decomposability (see [7] for definitions and details). Decomposability entails property (β) and this property is stronger than property (C). Examples of operators which satisfy property (C) but not property (β) may be found in among unilateral weighted shifts, see section 1.6 of [7]. Also property (Q) is strictly weaker than property (C). For instance, a multiplier T of a semi-simple commutative Banach algebra always has property (Q), since H0 (λI − T ) coincides with the kernel of λI − T for all λ ∈ C [1, Theorem 4.33], while property (C) can fail for a multiplier, see [1, Theorem 6.72]. In this paper we show that RS has property (C) (respectively, property (Q)) if and only if SR has property (C) (respectively, property (Q)). In the

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third section we consider Dunford’s property (C) in the case that R and S satisfy the operator equations RSR = R2 and SRS = S 2 .

2. Property (C) for SR and RS We first begin with some preliminary results: Lemma 2.1. [5] Let S ∈ L(X, Y ) and R ∈ L(Y, X). Then we have: (i) For every x ∈ X the following inclusions hold: σSR (Sx) ⊆ σRS (x) ⊆ σSR (Sx) ∪ {0}. (ii) For every y ∈ Y the following inclusions hold: σRS (Ry) ⊆ σSR (y) ⊆ σRS (Ry) ∪ {0}. The SVEP is transferred from SR to RS and viceversa: Lemma 2.2. [5] Let S ∈ L(X, Y ) and R ∈ L(Y, X). Then SR has SVEP if and only if RS has SVEP. In the sequel we need the following result: Lemma 2.3. Let F be a closed subset of C and T ∈ L(X). If 0 ∈ F and T x ∈ XT (F) then x ∈ XT (F). Proof. We have to show that σT (x) ⊆ F, or equivalently C\F ⊆ ρT (x). By assumption T x ∈ XT (F), so σT (T x) ⊆ F, or equivalently C\F ⊆ ρT (T x). Let λ0 ∈ C\F be arbitrary. Then there exists an analytic function f : / U0 and U0 → X defined on some open neighborhood U0 of λ0 such that 0 ∈ (λI − T )f (λ) = T x for all λ ∈ U0 . Define x − f (λ) for all λ ∈ U0 . λ It is easily seen that (λI − T )g(λ) = x for all λ ∈ U0 , so that λ0 ∈ ρT (x), as desired. g(λ) :=

For every subspace Z of X we shall denote by T |Z the restriction of T ∈ L(X) to Z. Lemma 2.4. Suppose that T ∈ L(X) has SVEP and let F be a closed subset of C such that Z := XT (F) is closed. If A := T |XT (F) then XT (K) = ZA (K) for all closed K ⊆ F. Proof. Note first that A has SVEP, so ZA (K) = ZA (K), and XT (K) ⊆ XT (F) = Z. The inclusion ZA (K) ⊆ XT (K) is immediate. In order to prove the opposite inclusion, suppose that x ∈ XT (K) = XT (K). Then σT (x) ⊆ K and there is an analytic function f : C\K → X such that (μI − T )f (μ) = x for all μ ∈ C\K. By [7, Lemma 1.2.14] we have σT (f (μ)) = σT (x) ⊆ K

for all μ ∈ C\K,

thus f (μ) ∈ XT (K) ⊆ Z. Therefore, f is a Z-valued function and hence (μI − T )f (μ) = (μI − A)f (μ) = x for all μ ∈ C\K, i.e. x ∈ ZA (K) = ZA (K).

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In the sequel by IY and IX we denote the identity operator on Y and X, respectively. Theorem 2.5. Let F a closed subset of C and let S ∈ L(X, Y ), R ∈ L(Y, X) such that RS has SVEP. Then we have (i) If XRS (F) is closed then YSR (F) is closed. (ii) If YSR (F) is closed then XRS (F) is closed. Proof. (i)

Observe first that since both RS and SR have SVEP, we have YSR (F) = YSR (F) and XRS (F) = XRS (F).

Suppose that XRS (F) is closed and let (yn ) be a sequence in YSR (F) = YSR (F) which converges to some y ∈ Y . To show that YSR (F) is closed we need to prove that y ∈ YSR (F). Since yn ∈ YSR (F), for every n ∈ N there exists an analytic function fn : C\F → Y such that (λIY − SR)fn (λ) = yn

for all λ ∈ C\F.

From this we obtain R(λIY − SR)fn (λ) = (λR − RSR)fn (λ) = (λIX − RS)Rfn (λ) = Ryn

for all λ ∈ C\F.

Obviously, gn (λ) : = Rfn (λ) : C\F → X is analytic, so Ryn ∈ XRS (F) = XRS (F) and since by assumption XRS (F) is closed then Ry ∈ XRS (F). Hence σRS (Ry) ⊆ F. By part (i) of Lemma 2.1 applied to the element x = Ry, we have σSR (SRy) ⊆ σRS (Ry), so σSR (SRy) ⊆ F and hence SRy ∈ YSR (F). Finally, to show that YSR (F) is closed, we consider the following two different cases (a) and (b): (a) 0 ∈ F. In this case y ∈ YSR (F) immediately follows from Lemma 2.3, so YSR (F) is closed. (b) 0 ∈ / F. Let Z := YSR (F ∪ {0}). The subspace Z is closed by part (a). If we define A := SR|Z then A has SVEP and by Proposition 1.2.20 of [7] we have σ(A) ⊆ F ∪ {0}. If 0 ∈ / σ(A) then σ(A) ⊆ F and, since σA (z) ⊆ σ(A) ⊂ F for all z ∈ Z, we then conclude that Z = ZA (F). Taking into account Lemma 3.1, we then have Z = YSR (F), so YSR (F) is closed. Suppose that 0 ∈ σ(A) and let F0 := σ(A) ∩ F. Then σ(A) = F0 ∪ {0}. Since 0 ∈ / F0 , from [7, Proposition 3.3.3] we know that the decomposition Z = ZA (F0 ) ⊕ ZA ({0}) holds, with both subspaces ZA (F0 ) and ZA ({0}) closed (indeed, they coincide with the range of the spectral projections of A associated to the spectral subsets F0 and {0}, respectively). By Lemma 3.1 we also have ZA (F0 ) = ZA (F ∩ σ(A)) = ZA (F) = YSR (F), thus YSR (F) is closed also in this case. (ii) The proof is similar, just use part (ii) of Lemma 2.1.

We have already observed that property (C), as well as property (Q), entails SVEP. The following result is an immediate consequence of Theorem 2.5.

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Corollary 2.6. Let S ∈ L(X, Y ) and R ∈ L(Y, X). Then RS has property (C) if and only if SR has property (C). Analogously, RS has property (Q) if and only if SR has property (Q). Corollary 2.6 improves [8, Theorem 3], where the equivalence of property (C) for RS and SR was proved under the assumption that both R and S are injective. In the sequel we give some applications of Corollary 2.6. Let H be an infinite dimensional separable Hilbert space and T ∈ L(H). Let T = U |T | denote the polar decomposition of T and set T (s, t) := |T |s U |T |t for s, t ≥ 0. Theorem 2.7. Let T ∈ L(H) and s1 , t1 , s2 , t2 ≥ 0 be such that s1 +t1 = s2 +t2 . Then T (s1 , t1 ) has property (C) if and only if T (s2 , t2 ) has property (C). Proof. Denote by T = U |T | the polar decomposition of T . Let R := |T |s1 and S := U |T |t1 . Clearly, RS = T (s1 , t1 ) and SR = T (0, s1 +t1 ). By Corollary 2.6 then T (s1 , t1 ) has property (C) if and only if T (0, s1 + t1 ) has property (C). Similarly, T (s2 , t2 ) has property (C) if and only if T(0, s2 + t2 ) = T (0, s1 + t1 ) has property (C). Obviously, T = T (0, 1). The operator Tˆ := T ( 12 , 12 ) is called the Aluthge transform of T . Corollary 2.8. Suppose that T ∈ L(H). Then T has property (C) if and only if its Aluthge transform Tˆ has property (C). We conclude this section with another application. In the sequel suppose that X and Y are Banach spaces, with X a proper subspace of Y , and assume that the embedding of X into Y is continuous, i.e. there is a constant c > 0 such that xY ≤ cxX for all x ∈ X. Suppose that T ∈ L(X) admits an extension T ∈ L(Y ) and set M(X) := {T ∈ L(X) : T (Y ) ⊆ X}. The relationships between the spectral properties of T ∈ M(X) and those of T have been studied in [3]. Let S ∈ L(X, Y ) denote the canonical embedding of X into Y and define R ∈ L(Y, X) by Ry := T y for all y ∈ Y . It is easily seen that T = RS and T = SR. Corollary 2.9. If T ∈ M(X) then T has property (C) if and only if T has property (C).

3. The Case RSR = R2 and SRS = S 2 The case that S, R ∈ L(X) satisfy the operator equations RSR = R2 and SRS = S 2 has been first studied in [11], and more recently it has been

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investigated by some other authors. In this case R, S, SR and RS share many spectral properties [9,10], and local spectral properties as decomposability, property (β) and SVEP [6]. In this section we consider the permanence of property (C) in this context. It is easily seen that if 0 ∈ / σ(R) ∩ σ(S) then R = S = I, so this case is without interest. For this reason we shall assume that 0 ∈ σ(R) ∩ σ(S). Evidently, the operator equations RSR = R2 and SRS = S 2 imply that (SR)2 = SR2

and

(RS)2 = RS 2 .

Lemma 3.1. Let S, R ∈ L(X) be such that RSR = R2 and SRS = S 2 . For each x ∈ X we have (i) σSR (SRx) ⊆ σR (x) ⊆ σSR (SRx) ∪ {0}. (ii) σRS (RSx) ⊆ σS (x) ⊆ σRS (RSx) ∪ {0}. Proof. Let λ0 ∈ / σR (x). Then λ0 ∈ ρR (x), so that there exist an open neighborhood U0 of λ0 and an analytic function f : Uo → X such that (λI − R) f (λ) = x for all λ ∈ U0 .Then SR(λI − R)f (λ) = (λI − SR)SRf (λ) = SRx for all λ ∈ U0 . / σSR (SRx). Since g(λ) := SRf (λ) is analytic on U0 it then follows that λ0 ∈ Hence σSR (SRx) ⊆ σR (x). To show the second inclusion suppose that λ0 ∈ / σSR (SRx) ∪ {0}. Then there is a neighborhood U0 of λ0 and an analytic function f : U0 → X such that (λI − SR)f (λ) = x for all λ ∈ U0 . / U0 . We have Since λ0 = 0 we may assume 0 ∈ R(λI − SR)f (λ) = (λR − RSR)(f (λ) = (λR − R2 )f (λ) = (λI − R)Rf (λ) = Rx, for all λ ∈ U0 . Define g : U0 → X as follows 1 g(λ) := [x + Rf (λ)] for all λ ∈ U0 . λ Evidently, g is analytic and it is easily seen that g satisfies the identity (λI − R)g(λ) = x for all λ ∈ U0 . Thus λ0 ∈ ρR (x) and hence λ0 ∈ / σR (x). Therefore also the inclusion σR (x) ⊆ σSR (SRx) ∪ {0} is proved. Theorem 3.2. Let S, R ∈ L(X) be such that RSR = R2 and SRS = S 2 . Suppose that F is a closed subset of C (i) If R has SVEP and XR (F) is closed then also XSR (F) is closed. (ii) If S has SVEP and XS (F) is closed then also XSR (F) is closed. Proof. (i) As observed before if any one of R, S, SR and RS has SVEP then all have SVEP. Hence, if T is one of these operators the local spectral subspace XT (F) coincide with the glocal spectral subspace XT (F). Suppose that XR (F) is closed and let (xn ) be a sequence of XSR (F) = XSR (F) which converges to x ∈ X. We need to show that x ∈ XSR (F). For every n ∈ N let fn : C\F → X be an analytic function for which

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for all λ ∈ C\F.

Then R(λI − SR)fn (λ) = (λR − RSR)fn (λ) = (λI − RS)Rfn (λ) = Rxn

for all λ ∈ C\F,

so Rxn ∈ XR (F) = XR (F). Obviously, Rxn → Rx and since XR (F) is closed it then follows that Rx ∈ XR (F) and hence σR (Rx) ⊆ F. Now, to show that x ∈ XSR (F), consider the two different cases 0 ∈ F and 0 ∈ / F. If 0 ∈ F from Rx ∈ XR (F) we obtain, by Lemma 2.3, that x ∈ XR (F). Consequently, σR (x) ⊆ F and by part (i) of Lemma 3.1 it then follows that σSR (SRx) ⊆ σR (x) ⊆ F, which shows that SRx ∈ XSR (F). Applying again Lemma 2.3 to the operator SR we then conclude that x ∈ XSR (F). Consider the other case 0 ∈ / F. Proceed exactly as in the proof of Theorem 2.5 : if Z = XSR (F ∪ {0}) then Z is closed, by the first part of the proof. If A := SR|Z then XSR (F) = ZA (F). Again, if 0 ∈ σ(A) then / σ(A), denoting by F0 := F ∩σ(A) ZA (F) = Z is closed. In the other case, 0 ∈ we then have XSR (F) = ZA (F) = ZA (F0 ) and the last subspace is closed. (ii) The proof is similar to that of part (i). Theorem 3.3. Let S, R ∈ L(X) be such that RSR = R2 and SRS = S 2 . Then any one of R, SRS, and SR has property (C) [respectively, property (Q)] implies they all have property (C) [respectively, property (Q)]. Proof. The equivalence of property (C) for RS and SR has been proved in Corollary 2.6. By Theorem 3.2 property (C) for R implies property (C) for SR. We show the converse. Suppose that SR has property (C). To show that R has property (C), let (xn ) be a sequence of XR (F) = XR (F) which converges to x and, for every n ∈ N, let fn : C\F → X be an analytic function such that (λIX − R)fn (λ) = xn

for all λ ∈ C\F.

Then SR(λIX − R)fn (λ) = (λSR − SR2 )fn (λ) = (λSR − (SR)2 )fn (λ) = (λIX − SR)SRfn (λ) = SRxn , for all λ ∈ C\F. This implies that SRxn ∈ XSR (F), and since XSR (F) is closed it then follows that SRx ∈ XSR (F), or equivalently σSR (SRx) ⊆ F. Also here we consider the two different cases 0 ∈ F and 0 ∈ / F. If 0 ∈ F then, by the second inclusion of part (i) of Lemma 3.1, we have σR (x) ⊆ σSR (SRx) ∪ {0} ⊆ F ∪ {0} = F thus, x ∈ XR (F) and hence XR (F) is closed. Finally, consider the case 0 ∈ / F and again proceed as in the proof of Theorem 2.5, if Z := XR (F ∪ {0}) then Z is closed by the first part of the proof. If A := R|Z then XR (F) = ZA (F). Again, if 0 ∈ σ(A) then

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/ σ(A) if F0 = F ∩ σ(A) XR (F) = ZA (F) = Z is closed. In the other case, 0 ∈ then XR (F) = ZA (F) = ZA (F0 ) and the last subspace is closed. The equivalence S has property (C) if and only if RS has property (C) may be proved in a similar way. The proof relative to property (Q) is analogous to that of property (C).

References [1] Aiena, P.: Fredholm and Local Spectral Theory, with Application to Multipliers. Kluwer, Dordrecht (2004) [2] Aiena, P., Colasante, M.L., Gonz´ alez, M.: Operators which have a closed quasinilpotent part. Proc. Am. Math. Soc. 130, 2701–2710 (2002) [3] Barnes, B.A.: The spectral and Fredholm theory of extension of bounded linear operators. Proc. Am. Math. Soc. 105(4), 941–949 (1989) [4] Barnes, B.: Common operator properties of the linear operators RS and SR. Proc. Am. Math. Soc. 126, 1055–1061 (1998) [5] Benhida, C., Zerouali, E.H.: Local spectral theory of linear operators RS and SR. Integr. Equ. Oper. Theory 54, 1–8 (2006) [6] Duggal, B.P.: Operator equations ABA = A2 and BAB = B 2 . (2010, preprint) [7] Laursen, K.B., Neumann, M.M.: Introduction to Local Spectral Theory. Clarendon Press, Oxford (2000) [8] Lin, C., Yan, Z., Ruan, Y.: Common properties of operators RS and SR and p-hiponormal operators. Integr. Equ. Oper. Theory 43, 313–325 (2002) [9] Schmoeger, C.: On the operator equations ABA = A2 and BAB = B 2 . Publ. de L’Inst. Math (NS) 78(92), 127–133 (2006) [10] Schmoeger, C.: Common spectral properties of linear operators A and B such that ABA = A2 and BAB = B 2 . Publ. de L’Inst. Math (NS) 79(93), 109–114 (2006) [11] Vidav, I.: On idempotent operators in a Hilbert space. Publ. de L’Inst. Math (NS) 4(18), 157–163 (1964) Pietro Aiena (B) Dipartimento di Metodi e Modelli Matematici Facolt` a di Ingegneria Universit` a di Palermo Palermo, Italy e-mail: [email protected] Manuel Gonz´ alez Departamento de Matem´ aticas Facultad de Ciencias Universidad de Cantabria, 39071 Santander, Spain e-mail: [email protected] Received: November 3, 2010. Revised: March 8, 2011.

Integr. Equ. Oper. Theory 70 (2011), 569–582 DOI 10.1007/s00020-011-1885-0 Published online May 21, 2011 c The Author(s) This article is published with open access at Springerlink.com 2011

Integral Equations and Operator Theory

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains Hassan Issa Abstract. Let Ω ⊂ Cd be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators Tgν with symbol g acting on the standard weighted Bergman space Hν2 over Ω with weight ν. Under some conditions on the weights ν and ν0 we show that there exists C(ν, ν0 ) > 0, such that the Berezin transform g˜ν0 of g with respect to Hν20 satisﬁes: ˜ gν0 ∞ ≤ C(ν, ν0 )Tgν , for all g in a suitable class of symbols containing L∞ (Ω). As a consequence we apply a result in Engliˇs (Integr Equ Oper theory 33:426– 455, 1999), to prove that the compactness of Tgν is independent of the weight ν, whenever g ∈ L∞ (Ω) and ν > C where C is a constant depending on (r, a, b). Mathematics Subject Classification (2010). 32A36, 32M15, 53C35. Keywords. Berezin transform, Reproducing kernel, Jordan triple determinant.

1. Introduction Let Ω ⊂ Cd be an irreducible bounded symmetric domain of multiplicities a and b, rank r and genus p. For each ν > p − 1, we denote by μν the normalized standard measure on Ω with weight ν, and we write ·, ·ν for the usual inner product on L2ν := L2 (Ω, dμν ). The weighted Bergman space Hν2 := Hν2 (Ω, dμν ) is deﬁned to be the space of all holomorphic functions in L2ν . It is well known, that Hν2 is closed in L2ν and forms a reproducing kernel Hilbert space. Via the orthogonal projection Pν from L2ν onto Hν2 and for a measurable symbol f , the Toeplitz operator Tfν is deﬁned on a suitable domain in Hν2 as the product Tfν := Pν Mf where Mf is the multiplication H. Issa has been partially supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft).

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by f . Moreover, for a suitable function g : Ω −→ C the weighted Berezin transform gν is the real analytic map on Ω given by gν (z) = Tgν kν,[z] , kν,[z] ν where kν,[z] denotes the normalized Bergman kernel. In the analysis of Toeplitz operators the Berezin transform is an important tool related to various results in operator theory (cf. [2–5,7,8,14,15]). For example, it was shown in [8] that under some condition on the weight ν the Toeplitz operator Tgν with bounded symbol g is compact on Hν2 if and only if gν ∈ C0 (Ω). In case of the unit disc Ω = D in the complex plane, this result was generalized from bounded symbols to symbols of bounded mean oscillation BMO1 (D) (cf. [15]). The corresponding compactness characterization also holds true for weighted Bergman spaces over the n-dimensional unit ball Ω = Bn and g ∈ BMO1 (Bn ) (cf. [3]). Theorems of the above type have been obtained for unbounded domains, as well. In case of the Segal–Bargmann space H 2 (Cn , dμt ) of all entire functions square integrable with respect to a t-dependent family of Gaussian measures μt , (here t > 0 plays the role of the time parameter in the heat ﬂow), it was proved in [3] that for symbols f ∈ BMO1 (Cn ) the Toeplitz operator Tft is bounded (respectively compact) if and only if the heat transform t f( 2 ) at time 2t is bounded (respectively vanishing at inﬁnity) (cf. [3,7]). A natural question which arises in the study of Toeplitz operators with a ﬁxed symbol acting on a family of weighted Bergman spaces is whether their compactness is independent of the weight parameter. By essentially using the previously mentioned results it was shown in [3] that independence in fact holds in the case of the Segal–Bargmann space H 2 (Cn , dμt ) and the weighted Bergman spaces Hν2 over the unit ball Bn under the assumption that g ∈ BMO1 (Cn ) and g ∈ BMO1 (Bn ), respectively. However, there are counter examples for general symbols (cf. Section 6 in [3]). As an application it follows that for functions g ∈ BMO1 (Cn ) the heat transform g(t) ∈ C(Cn ) of g vanishes at inﬁnity for a certain time t0 > 0 if and only if g(t) vanishes at inﬁnity for each time t > 0. This, roughly speaking, gives some information on the heat ﬂow “backwards in time”. In the case of a bounded symmetric domain, the weight parameter ν replaces the time parameter t in the Segal–Bargmann space construction and the Berezin transform gν replaces the heat transform. The aim of this paper is to prove that the compactness of Tgν is uniform with respect to the weight ν, whenever g ∈ L∞ (Ω) and ν > C where C is a constant depending on (r, a, b). The ﬁrst attempt to solve such a problem was given in [3]. One important ingredient was to obtain an estimation between the sup-norm of the heat transform of the symbol and the norm of the Toeplitz operator (cf. Theorem 11 [5], Proposition 1 and Theorem 10 in [3]) and to use the above mentioned result in [15]. In this paper, we employ a similar technique to investigate the case of a general bounded symmetric domain and we use a compactness characterization in [8] which holds for bounded symbols. We point out that under the assumption of boundedness our result is a generalization to that in [3] since we deal with a wider class of domains. However, we are not able to generalize the statement to functions of bounded mean oscillation. This

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is due to the fact that the equivalence between the vanishing of the Berezin transform at the boundary of Ω and the compactness of Toeplitz operators is not clear for such a space of symbols. Here we state two of our main results: ˜ = Theorem A. Let ν > p − 1, ν0 > d with |ν − ν0 | > r−1 2 a and write ν min{ν, ν0 }. Then there exists C(ν, ν0 ) > 0 such that for all g ∈ τν (Ω) ∩ L1 (Ω, dμν˜ ) (cf. (6)) we have gν0 ∞ ≤ C(ν, ν0 )Tgν .

(1)

Theorem B. Let Ω ⊂ Cd be an irreducible bounded symmetric domain and suppose that ⎧ ⎫ 2 ⎬ ⎨ r−1 r−1 r−1 r−1 ν, ν0 > max d, p − 1 + a+ a + a a+p−1 . ⎩ ⎭ 4 4 2 2 Then for any g ∈ L∞ (Ω) we have the equivalence: Tgν0 is compact on Hν20 if and only if Tgν is compact on Hν2 . The paper is organized as follows: In Sect. 2 we set up notations and present some of the standard results concerning irreducible bounded symmetric domains. In Sect. 3 we prove the inequality in Theorem A by a technique similar to that in [5]. An essential idea is to construct for each pair (ν, ν0 ) of weights a certain trace class operator on Hν2 and represent the Berezin transform of a function as an operator trace. Section 4 is devoted to the proof of Theorem B, where we use a result in [8]. Finally, we present some open problems which are motivated by our results.

2. Preliminaries Let Ω ⊂ Cd be a bounded symmetric domain in its (Harish–Chandra) realization with multiplicities a and b and with rank r. In particular, Ω contains the origin and it is invariant under the natural S1 -action (circular). The triple (r, a, b) is called the type of Ω and it determines the domain up to biholomorphic equivalence (cf. [6,11]). E. Cartan proved that there exist only six types of irreducible bounded symmetric domains, the so called four classical domains and two exceptional domains of dimensions 16 and 27, respectively (cf. [6]). Moreover, the genus p of Ω and the complex dimension d are given by: ar(r − 1) + rb. (2) 2 The group Aut(Ω) of all biholomorphic automorphisms on Ω acts transitively on Ω. Moreover, if G is the connected component of Aut(Ω) containing the identity (with respect to the topology of uniform convergence on compact subsets of Ω), then G is a semi-simple Lie group. Let K denote the isotropy subgroup of G ﬁxing the origin. Then K is a maximal compact subgroup of G and G/K ∼ = Ω (cf. [10,12]). p = 2 + (r − 1)a + b;

d := dim Ω = r +

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To each such Ω there is attached a unique function h(z, w) (z, w ∈ Cd ) which is a polynomial in z and w, ¯ called the Jordan triple determinant and satisﬁes the following properties (cf. [11,13]): 1. 2.

h(z, 0) = 1 and h(z, w) = h(w, z), ∀z, w ∈ Cd . h(z, z) > 0, ∀z ∈ Ω and h(z, z) = 0, ∀z ∈ ∂Ω.

Thus for any λ ∈ R, we ﬁx a branch of h(z, w)λ for z, w ∈ Ω. It is well known that for any ν ∈ R (cf. [13]): h(z, z)ν−p dv(z) < ∞ ⇐⇒ ν > p − 1, (3) Ω

where dv(z) is the Euclidean measure on Ω. In this case, we consider the normalized weighted measure: dμν = cν h(z, z)ν−p dv. The weighted Bergman space Hν2 is the space of holomorphic functions in L2ν . We write Kν : Ω × Ω −→ C for the reproducing kernel of Hν2 , i.e. for any f ∈ Hν2 , w ∈ Ω we have: f (w) = [Pν f ](w) = f, Kν (·, w)ν = f, Kν,[w] ν , where Pν denotes the projection of L2ν onto Hν2 and Kν,[w] := Kν (·, w). The kernel Kν (z, w) is related to the Jordan triple determinant h(z, w) via (cf. [9]): Kν (z, w) = h(z, w)−ν ,

∀ν > p − 1.

(4)

For a measurable symbol g : Ω −→ C the Toeplitz operator Tgν is given by:

Tgν : D(Tgν ) := h ∈ Hν2 | gh ∈ L2ν ⊂ Hν2 −→ Hν2 : h → Pν (gh). (5) The Berezin transform T of an operator T on Hν2 with the domain of T containing all the normalized kernels kν,[z] := Kν,[z] /Kν,[z] ν where z runs through Ω, is the complex valued map deﬁned on Ω by: T(z) = T kν,[z] , kν,[z] ν ,

z ∈ Ω.

In the rest of the paper we will frequently use the symbol space:

τν (Ω) := g : Ω −→ C | gKν,[a] ∈ L2 (Ω, dμν ) for all a ∈ Ω .

(6)

Note that for g ∈ τv (Ω) the operator Tgν is densely deﬁned on Hν2 and its Berezin transform is deﬁned to be: ν gν := T g.

Let P be the space of all holomorphic polynomials on Cd . For f, g ∈ P we ∂ deﬁne g ∗ (z) := g(z) and ∂f := f ( ∂z ). Let dv(z) denote the Lebesgue volume measure, and equip P with the Fischer inner product: 2 f, gF := ∂f (g ∗ )(0) = π −d f (z)g(z)e−|z| dv(z). Cd

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Under the action of K on P which is given by: π(k)f := f ◦ k (each k ∈ K can be extended to a linear map on Cd (cf. [12])), P admits a Peter Weyl decomposition: P = ⊕Pm , Nr0

where m = (m1 , . . . , mr ) ∈ with m1 ≥ · · · ≥ mr ≥ 0 (in all what follows the multi-indices m will always have this ordering). Note that each Pm is a subspace of P|m| , the space all homogeneous polynomials on Cd of degree |m| (cf. [9]). The importance of this decomposition is given by the following theorem: Theorem 1. [1] The spaces Pm are K-invariant irreducible and orthogonal under ·, ·F . Moreover, if H is a Hilbert space of analytic functions on Ω with K-invariant inner product ·, ·, then Pm is orthogonal to Pn under ·, · whenever m = n. Moreover, ·, · is proportional to ·, ·F on each Pm . In particular, this holds for the space Hν2 and the constant of proportionality was calculated in [9]: f, gF = (ν)m f, gν , where (ν)m is the generalized Pochhammer symbol: r Γ mj + ν − j−1 2 a . (ν)m := Γ ν − j−1 2 a j=1 Since each Pm is closed in P (Pm is ﬁnite dimensional), it admits a reproducing kernel K m (z, w), i.e. f (w) = f, K m (·, w)F ,

∀f ∈ Pm .

The relation between these reproducing kernels and the Bergman kernel was given in [9]: Theorem 2. [9] For all ν ∈ C and all z, w ∈ Ω, we have: h(z, w)−ν = (ν)m K m (z, w).

(7)

m1 ≥···≥mr ≥0

The series converges uniformly and absolutely on compact subsets of Ω × Ω. Finally, we relate the orthonormal basis of the Fischer norm to the orthonormal basis of Hν2 : Proposition 1. Let ν > p − 1, dm = dim Pm , and {ψjm }j=1,...,dm be an orthonormal basis of (Pm , ·, ·F ). Then 1 m 2 B := em (8) j := (ν)m ψj | m and j = 1, . . . , dm is an orthonormal basis of (Hν2 , ·, ·ν ). Here m runs through the previously mentioned ordering.

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Proof. Let j = k then ψjm , ψkm ν = m = n (cf. Theorem 1) ψjm , ψin ν = 0;

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1 m m (ν)m ψj , ψk F

= 0. Moreover, for

∀j ≤ dm , i ≤ dn .

Also, note that: 1 1 2 2 (ν)m ψjm , (ν)m ψjm = (ν)m ψjm , ψjm ν = ψjm , ψjm F = 1. ν

Now, since the polynomials are dense in Hν2 it follows that the functions in B form a complete orthonormal system of Hν2 .

3. Upper Estimation of the Berezin Transform In this section we prove Theorem A using a similar technique to that in [5]. Under the assumptions on the weights ν and ν0 and for each z ∈ Ω, we ﬁnd a trace class operator Tzν,ν0 on Hν2 where Tzν,ν0 tr is independent of z and one has tr(Tgν Tzν,ν0 ) = ccνν gν0 (z). We then apply a standard estimate for the 0 trace norm to obtain the inequality (1). For X ∈ L(Hν2 ) deﬁne the function KνX (z, w) := (X ∗ Kν (·, w))(z) on Ω × Ω. By slightly modifying the arguments in [5], one can prove the following (cf. [5] corollary to Theorem 6): Proposition 2. Let g ∈ τν (Ω) and X ∈ L(Hν2 ) with ν > p − 1. Suppose the following conditions hold: 1. Tgν ∈ L(Hν2 ). 2 2. Tgν X is of trace class on XHν . 3. Ω Ω |g(z)| |Kν (w, z)| Kν (w, z) dμν (z)dμν (w) < ∞. Then tr(Tgν X) = g(z)KνX (z, z)dμν (z). (9) Ω

The basic idea in the proof of Theorem A is to construct a trace class operator X = T0ν,ν0 on Hν2 satisfying KνX (z, w) = h(z, w)ν0 −ν . In order to do that, we ﬁx {ψjm }j=1,...,dm as an orthonormal basis of (Pm , ·, ·F ). Then, the reproducing kernel K m (z, w) of Pm is given by: K m (z, w) =

dm

ψjm (z)ψjm (w).

(10)

j=1

Now, for each m ∈ Nr0 such that m1 ≥ · · · ≥ mr ≥ 0 and each j = 1, . . . , dm , we denote by Pjm the orthogonal projection from Hν2 to the one-dimensional space spanned by em j : m Pjm f := f, em j ν ej

f ∈ Hν2 .

Hence: Pm

m Kν j (z, w) = [Pjm Kν (·, z)](w) = Kν (·, z), em j ν ej (w)

= (ν)m ψjm (z)ψjm (w).

(11)

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Now, deﬁne the operator T0ν,ν0 on Hν2 as the inﬁnite sum:

T0ν,ν0 :=

m1 ≥···≥mr ≥0

dm (ν − ν0 )m P m. (ν)m j=1 j

Theorem 3. For ν > p − 1 and ν0 > d the operator T0ν,ν0 is of trace class on Hν2 and ν,ν0

KνT0

(z, w) = h(z, w)ν0 −ν .

(12)

Proof. For each l ∈ N, consider the operator

Slν,ν0 :=

m1 ≥···≥mr ≥0,|m|≤l

dm (ν − ν0 )m P m. (ν)m j=1 j

We write: r Γ(mj + ν − ν0 − j−1 Γ(ν − j−1 (ν − ν0 )m 2 a) 2 a) . = j−1 (ν)m Γ(ν − ν0 − 2 a) Γ(mj + ν − j−1 2 a) j=1

Therefore, by Stirling’s and as mj −→ ∞ we have: Γ(mj + ν − ν0 −

j−1 2 a) j−1 2 a)

Γ(mj + ν −

0 ∼ m−ν . j

Since ν0 > p − 1 ≥ 1, it follows that the operators Slν,ν0 converge to T0ν,ν0 in the norm of L(Hν2 ). Moreover, by an application of the Cauchy–Schwarz Theorem we know that: S

ν,ν0

Kν l

ν,ν0

(z, w) −→ KνT0

(z, w),

as l −→ ∞ uniformly on compact subsets of Ω × Ω (cf. Proposition 3-(7) in [5]). Together with Theorem 2 and the equalities (7), (10), and (11) we obtain: ν,ν0

KνT0

(z, w) =

dm (ν − ν0 )m Pm Kν j (z, w) (ν)m j=1

m1 ≥···≥mr ≥0

=

dm (ν − ν0 )m (ν)m ψjm (z)ψjm (w) (ν)m j=1

m1 ≥···≥mr ≥0

=

(ν − ν0 )m K m (z, w)

m1 ≥···≥mr ≥0

= h(z, w)ν0 −ν .

dm m In order to prove that T0ν,ν0 is of trace class, note that j=1 Pj tr = dm = dim Pm . One has the inclusion Pm ⊂ P|m| and P|m| admits {cα z α | |α| = |m|, α ∈ Nd0 } (cα are normalized constants) as an orthonormal basis with respect to the Fischer inner product. Hence it follows that dm ≤

(|m| + d − 1)! . |m|!(d − 1)!

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Therefore, T0ν,ν0

r j−1 j−1 Γ(m + ν − ν − a) a) Γ(ν − j 0 2 2 tr = dm j−1 j−1 Γ(ν − ν − a) Γ(m + ν − a) 0 j m1 ≥···≥mr ≥0 2 2 j=1 r j−1 j−1 (|m|+d−1)! Γ(mj +ν −ν0 − 2 a) Γ(ν − 2 a) ≤ . |m|!(d−1)! j=1 Γ(ν −ν0 − j−1 Γ(mj +ν − j−1 2 a) 2 a) m∈Nr

Now, as |m| → ∞, and by Stirling´s formula again we have: ⎫d−1 ⎧ r ⎬ ⎨ (|m| + d − 1)! |m|d−1 1 ∼ = mj |m|!(d − 1)! (d − 1)! (d − 1)! ⎩ j=1 ⎭ =

|γ|=d−1

r 1 γ m ≤ c (1 + mi )d−1 , γ! i=1

where c > 0 is a suitable constant independent of m. Therefore, as mj −→ ∞ it follows that: r Γ(mj + ν − ν0 − j−1 a) (|m| + d − 1)! j=1

c

Γ(mj + ν −

2 j−1 2 a)

(|m|)!(d − 1)!

r Γ(mj + ν − ν0 − j=1

Γ(mj + ν −

Since ν0 > d this shows that

j−1 2 a) (1 j−1 2 a)

T0ν,ν0

+ mj )d−1 ∼ c

r

(1 + mj )−ν0 +d−1 .

j=1

is of trace class.

We want to apply Proposition 2 to X = T0ν,ν0 and in order to check condition (3) therein, we need the generalized Forelli–Rudin inequalities which can be found in [8,9]: Lemma 1. Consider the integral −(λ+γ) λ−p Jλ,γ (z) = |h(z, w)| h(w, w) dv(w), Ω

where z ∈ Ω, γ ∈ R and λ > p − 1. Let Aλ,γ = {α ∈ R | ∃C > 0 s.t. |Jλ,γ (z)| ≤ Ch(z, z)−α ,

for all z ∈ Ω},

then: 1. Aλ,γ = [γ, ∞), if γ > r−1 2 a. 2. Aλ,γ = [0, ∞), if γ < − r−1 2 a. According to Theorem 3 and to the above lemma, we are able now to establish the following theorem: Theorem 4. Let ν > p−1, ν0 > d with |ν −ν0 | > r−1 ˜ = min{ν, ν0 }. 2 a, and write ν Moreover, suppose that g ∈ τν (Ω) ∩ L1 (Ω, dμν˜ ) and that Tgν is bounded. Then cν gν (0). (13) tr(Tgν T0ν,ν0 ) = cν0 0

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Proof. The ﬁrst step is to verify condition (3) of Proposition 2 i.e. to prove the convergence of the following integral ν,ν0 |g(z)||Kν (w, z)||KνT0 (w, z)|dμν (z)dμν (w). Ω Ω

Using (4) and (12) we obtain: ν,ν0 |g(z)||Kν (w, z)||KνT0 (w, z)|dμν (z)dμν (w) Ω Ω

= cν,ν0

|h(w, z)|−(ν+(ν−ν0 )) h(w, w)ν−p dv(w)|g(z)|h(z, z)ν−p dv(z).

Ω Ω

Now, we apply Lemma 1 for λ = ν and γ = ν − ν0 . Since |γ| > r−1 2 a, we have two case: (a) In case ν > ν0 , Lemma 1 (1) implies that ν − ν0 ∈ Aλ,γ , which means that: |h(w, z)|−(ν+(ν−ν0 )) h(w, w)ν−p dv(w) ≤ Ch(z, z)ν0 −ν . Ω

So, we have to check if 1 ν0 −ν ν−p h(z, z) h(z, z) |g(z)|dv(z) = |g(z)|dμν0 (z) < ∞. cν0 Ω

Ω

1

(b)

This is true since g ∈ L (Ω, dμν0 ). In case of ν0 > ν, we have γ < − r−1 2 a. By Lemma 1 (2), we know that 0 ∈ Aλ,γ i.e. |h(w, z)|−(ν+(ν−ν0 )) h(w, w)ν−p dv(w) ≤ C. Ω

Therefore, we have to investigate |g(z)|h(z, z)ν−p dv(z) < ∞. Ω

But this is also true since in this case we have g ∈ L1 (Ω, dμν ). Hence, we can apply Proposition 2 with X = T0ν,ν0 to obtain: ν,ν T 0 ν ν,ν0 tr(Tg T0 ) = g(z)Kν 0 (z, z)dμν (z) Ω

g(z)h(z, z)ν0 −ν h(z, z)ν−p dv(z)

= cν Ω

cν = cν0

g(z)dμν0 (z) = Ω

cν g˜ν (0). cν0 0

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Our aim now is to extend the above Theorem from z = 0 to an arbitrary z ∈ Ω. More precisely, under the assumptions of Theorem 4 and for each z ∈ Ω we ﬁnd a trace class operator Tzν,ν0 such that tr(Tgν Tzν,ν0 ) = ccνν gν0 (z) 0 and the trace norm Tzν,ν0 tr = T0ν,ν0 tr is independent of z ∈ Ω. For each z ∈ Ω we consider the automorphism φz which interchanges z and zero. We deﬁne the linear operator Uν,z on Hν2 by ν

Uν,z f := (f ◦ φz ) · Jφz p ,

∀f ∈ Hν2 ,

where Jφz denotes the complex Jacobian of φz . Using the transformation formula of the unweighted Bergman kernel, one can prove: Proposition 3. For all z ∈ Ω, the operator Uν,z is self-adjoint and unitary i.e. −1 ∗ = Uν,z = Uν,z . Uν,z

(14)

For each S ∈ L(Hν2 ), and each z ∈ Ω we deﬁne Sν,z ∈ L(Hν2 ) by: Sν,z := Uν,z SUν,z . The following lemma is found in ([8], Lemma 4): Lemma 2. For all a, z ∈ Ω, the Berezin transform S ν,a of Sν,a is given by: S ν,a (z) = S(φa (z)).

(15)

One can easily check a slightly more general version of Lemma 6 in [8]: Lemma 3. Let ν > p − 1 then for any g ∈ τν (Ω) and any a ∈ Ω we have: ν Uν,a Tgν Uν,a = Tg◦φ , a

(16)

where both sides are interpreted as operators with domain

ν D(Tg◦φ ) a

(cf. (5)).

According to (14), we know that for any z ∈ Ω the operator Tzν,ν0 := Uν,z T0ν,ν0 Uν,z Tzν,ν0 tr

T0ν,ν0 tr .

is of trace class and satisﬁes = Theorem 4 together with (15) and (16) we obtain:

(17) Moreover, according to

Corollary 1. Let ν > p − 1 and ν0 > d with |ν − ν0 | > r−1 2 a. Moreover, suppose that g ∈ τν (Ω) ∩ L1 (Ω, dμν˜ ), where ν˜ = min {ν, ν0 }, and that Tgν is bounded. Then we have: cν tr(Tgν Tzν,ν0 ) = gν (z) ∀z ∈ Ω. (18) cν0 0 Proof. We apply Theorem 4: tr(Tgν Tzν,ν0 ) = tr(Tgν Uν,z T0ν,ν0 Uν,z ) = tr(Uν,z Tgν Uν,z T0ν,ν0 ) cν ν T ν,ν0 ) = (g ◦ φz )ν0 (0) = tr(Tg◦φ z 0 cν0 cν cν = gν (φz (0)) = gν (z). cν0 0 cν0 0

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Proof of Theorem A: Let Tgν be bounded, then by Corollary 1 and by a standard estimate for the trace norm we have for all z ∈ Ω: |˜ gν0 (z)| ≤ C1 (ν, ν0 )|tr(Tgν Tzν,ν0 )| ≤ C1 (ν, ν0 )Tgν Tzν,ν0 tr = C1 (ν, ν0 )Tgν T0ν,ν0 tr ≤ C(ν, ν0 )Tgν , where C1 (ν, ν0 ) and C(ν, ν0 ) are suitable constants independent of g and z. Therefore: ˜ gν0 ∞ ≤ C(ν, ν0 )Tgν . Clearly, the trace norm T0ν,ν0 tr also depends on r and a. However, we omit indicating this fact in the constant C(ν, ν0 ).

4. Compactness Criteria In the following, we establish the equivalence stated in Theorem B. For this, we need a density result for a class of Toeplitz operators together with Theorem A and the main result in [8]. Similar to the corresponding result in [5] on Toeplitz operators acting on the Segal–Bargmann space, and by purely functional analytic methods one can prove the following: Theorem 5. Let ν > p − 1, then

ν Tg | g continuous with compact support in Ω is norm dense in the space of all compact operators acting on H 2 (Ω, dμν ). In order to prove Theorem B, we use the equivalence between the compactness of the Toeplitz operator and the vanishing of the Berezin transform at the boundary of a bounded symmetric domain (cf. Theorem A in [8]): Theorem 6. [8] Let r−1 ν >p−1+ a+ 4

2 r−1 r−1 r−1 a + a a+p−1 . 4 2 2

(19)

Denote by C0 (Ω) the space of functions vanishing on ∂Ω. Then for any g ∈ L∞ (Ω) the following two conditions are equivalent: 1. 2.

Tgν is compact on Hν2 . gν ∈ C0 (Ω).

In fact, the above theorem is generalized to all ﬁnite sums of Toeplitz operators with bounded symbols and other equivalent conditions can be given. In our proof, the conditions on the weights ν and ν0 in Theorem B is

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owned to an application of Theorem 6. The key of proving Theorem B is the following: Theorem 7. Let Ω ⊂ Cd be an irreducible bounded symmetric domain and suppose that ⎫ ⎧ 2 ⎬ ⎨ r−1 r−1 r−1 r−1 a+ a + a a+p−1 . ν, ν0 > max d, p−1+ ⎭ ⎩ 4 4 2 2 (20) Then for any g ∈ L∞ (Ω) the following are equivalent: (1) (2) (3) (4) (5) (6)

gν0 ∈ C0 (Ω). gν ∈ C0 (Ω), for all ν such that |ν − ν0 | > r−1 2 a. Tgν0 is compact on Hν20 . Tgν is compact on Hν2 , for all ν such that |ν − ν0 | > r−1 2 a. r−1 gν ∈ C0 (Ω), for some ν such that |ν − ν0 | > 2 a. Tgν is compact on Hν2 , for some ν such that |ν − ν0 | > r−1 2 a.

Proof. (i) We prove the implication (6 =⇒ 1): Suppose that Tgν is compact for some ν such that |ν − ν0 | > r−1 2 a. Then by Theorem A there exists C(ν, ν0 ) > 0 independent of g such that: gν0 ∞ ≤ C(ν, ν0 )Tgν . Since Tgν is compact, Theorem 5 tells us that there is a sequence (gk )k∈N of continuous functions with compact support such that: k→∞

Tgνk − Tgν −−−−→ 0, which shows that k→∞ gν0 − (g −−−→ 0. k )ν0 ∞ −

ν0 ∈ C0 (Ω). But each (g k )ν0 ∈ C0 (Ω), hence g By the same argument it is easy to show the implication (3 =⇒ 2) using the inequality ˜ gν ∞ ≤ C(ν0 , ν)Tgν0 ν . Finally, we remark that by Theorem 6 we have (1 ⇐⇒ 3), (2 ⇐⇒ 4), and (5 ⇐⇒ 6). The implications (4 =⇒ 6) and (2 =⇒ 5) are trivial and we obtain the proof.

(ii)

Proof of Theorem B Let (ν, ν0 ) be two weights satisfying (20) and suppose that Tgν0 is compact. Then by the previous theorem it is sufﬁcient to prove that Tgν is compact for |ν − ν0 | ≤ r−1 2 a. Set γ := max {ν0 , ν} and let a. Then we have ν1 = γ + 1 + r−1 2 min {ν1 − ν0 , ν1 − ν} >

r−1 a =⇒ Tgν1 is compact =⇒ Tgν is compact . 2

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5. Open Problems Finally, we would like to collect some open problems which are motivated by our results. Question 1: Is the obstruction |ν − ν0 | > r−1 2 a in Theorem A necessary to obtain the inequality (1)? Question 2: In [3] it was proved that in the case of the unit ball Ω = Bn ⊂ Cn Theorem B holds for symbols g ∈ BMO1 (Bn ) of bounded mean oscillation. This class of symbols strictly contains the bounded measurable functions. So one may ask if it is possible to extend Theorem B to the case of BMO1 (Ω) symbols for an arbitrary bounded symmetric domain Ω ⊂ Cn . Question 3: Let Sp (Hν2 ) denote the Schatten-p-class over Hν2 and assume that: ⎫ ⎧ 2 ⎬ ⎨ r−1 r−1 r−1 r−1 a+ a + a a+p−1 . ν, ν0 > max d, p − 1 + ⎭ ⎩ 4 4 2 2 For every g ∈ L∞ (Ω), and all 1 ≤ p < ∞, is it true that: Tgν0 ∈ Sp (Hν20 ) if and only if Tgν ∈ Sp (Hν2 )? Acknowledgments I would like to take this opportunity to thank Prof. Wolfram Bauer who has been most generous with his time and ideas, calling my attention to the above problems. It is my pleasure to dedicate this work to Ibn L. Hassan Al-Moaammal and to my wife Aliye. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References [1] Arazy, J., Upmeier, H.: Invariant inner product in spaces of holomorphic functions on bounded symmetric domains. Doc. Math. 2, 213–261 (1997) [2] Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47(2), 387–400 (1998) [3] Bauer, W., Coburn, L.A., Isralowitz, J.: Heat ﬂow, BMO, and the compactness of Toeplitz operators. J. Funct. Anal. 259, 57–78 (2010) [4] Bauer, W., Furutani, K.: Compact operators and the pluriharmupperonic Berezin transform. Int. J. Math. 19(6), 645–669 (2008) [5] Berger, C., Coburn, L.: Heat ﬂow and Berezin–Toeplitz estimates. Am. J. Math. 116(3), 563–590 (1994) [6] Cartan, E.: Sur les domaines born´es homog`enes de l’ espace de n-variables complexes. Abh. Math. Semin. Univ. Hamburg 11, 116–162 (1935)

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[7] Coburn, L.A., Isralowitz, J., Li, B.: Toeplitz operators with BMO symbols on the Segal–Bargmann space. Trans. Am. Math. Soc. 363, 3015–3030 (2011) [8] Engliˇs, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999) [9] Faraut, J., Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88(1), 64–89 (1990) [10] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. American Mathematical Society, Providence (2001) [11] Loos, O.: Bounded symmetric domains and Jordan pairs. Mathematical Lectures. University of California at Irvine, Irvine (1977) [12] Narasimhan, R.: Several Complex Variables. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1971) [13] Upmeier, H.: Toeplitz Operators and Index Theory in Several Complex Variables. In: Operator Theory: Advances and Applications, vol. 81, Birkh¨ auser, Basel (1996) [14] Zhu, K.: Operator Theory in Function Spaces. In: Mathematical Surveys and Monographs, vol. 138, 2nd edn. The American Mathematical Society, Providence (2007) [15] Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Math. Sci. 46, 2929–2945 (2003) Hassan Issa (B) Mathematisches Institut Georg-August-Universit¨ at Bunsen-str. 3-5 37073 G¨ ottingen Germany e-mail: [email protected]; [email protected] Received: December 16, 2010. Revised: May 5, 2011.

Integr. Equ. Oper. Theory 70 (2011), 583–600 DOI 10.1007/s00020-011-1883-2 Published online May 20, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

On Boundedness of Calder´ on–Toeplitz Operators Ondrej Hutn´ık Abstract. We study the boundedness of Toeplitz-type operators defined in the context of the Calder´ on reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calder´ on–Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given. Mathematics Subject Classification (2010). Primary 47B35, 42C40; Secondary 47G30, 47L80. Keywords. Wavelet, admissibility condition, continuous wavelet transform, Calder´ on reproducing formula, Toeplitz operator, Laguerre polynomial, boundedness, operator algebra.

1. Introduction Calder´ on–Toeplitz operators are integral operators which arise in the context of wavelet analysis [3] in connection with the Calder´ on reproducing formula, cf. [2]. This formula gives rise to a class of Hilbert spaces with reproducing kernels, the so-called spaces of Calder´ on (or, wavelet) transforms. These operators were formally deﬁned in [14] as a wavelet counterpart of Toeplitz operators deﬁned on Hilbert spaces of holomorphic functions. Therefore the name “Calder´ on–Toeplitz” reﬂects the close relationship with the Calder´ on reproducing formula on one side and, on the other side, it emphasizes the fact that this operator is unitarily equivalent to the Toeplitz-type operator Pψ Ma : Wψ (L2 (R)) → Wψ (L2 (R)), where Ma is the multiplication operator by a and Pψ is the orthogonal projection from L2 (R × R+ , v −2 du dv) onto the space of wavelet transforms Wψ (L2 (R)) = {Wψ f (u, v) = f, ψu,v ; f ∈ L2 (R)}, This research was partially supported by Grant VVGS 45/10-11.

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where ψ ∈ L2 (R) is an admissible wavelet and ψu,v , (u, v) ∈ R × R+ , are the shifted and scaled versions of ψ (see Sect. 2). These operators are also a useful localization tool which enables to localize a signal both in time and frequency. For further details about localization operators and wavelets transforms we refer to the Wong book [19]. In [6] we have described the structure of the space of Calder´ on transforms Wψ (L2 (R)) inside L2 (R × R+ , v −2 du dv). This representation was further used to study Calder´ on–Toeplitz operators acting on spaces of Calder´ on transforms for general (admissible) wavelets in [7]. Considering certain speciﬁc wavelets in [8] we were able to simplify these general results and discover the interesting connection between the spaces of Calder´ on transforms and poly-analytic Bergman spaces, cf. [17]. This connection was further investigated in [1] where also some interesting applications were given. The speciﬁcity of this choice of wavelets ψ (k) , whose Fourier transforms are related to Laguerre polynomials, has enabled us to study in [9] in more detail the (k) family of Calder´ on–Toeplitz operators Ta acting on wavelet subspaces A(k) (with parameter k = 0, 1, 2, . . . being the degree of Laguerre polynomial Lk ) with symbols depending only on vertical variable in the upper half-plane Π. In this speciﬁc case of wavelets the corresponding Calder´on–Toeplitz operators generalize classical Toeplitz operators acting on the Bergman space in an interesting way which differs from the case of Toeplitz operators acting on the weighted Bergman spaces studied in [5]. On the other hand, the classical Toeplitz operators and Calder´ on–Toeplitz operators share many features in common. In this paper we continue the detailed study of Calder´ on–Toeplitz oper(k) ators Ta acting on wavelet subspace A(k) , which was initiated in [9]. For a bounded symbol a on the aﬃne group G the Calder´ on–Toeplitz operator (k) Ta is clearly bounded on A(k) . However, an interesting and important feature of these operators on wavelet subspaces is that they can be bounded for symbols that are unbounded near the boundary. Therefore the aim of this paper is to study in detail the boundedness properties of Calder´ on–Toeplitz operators with such unbounded symbols and to give sufﬁcient conditions for their simultaneous boundedness on all wavelet subspaces. The main tool in our study is the result, which we have shown in [9], that the Calder´ on–Toep(k) (k) litz operator Ta acting on A with a symbol a = a(ζ), ζ = (u, v) ∈ G, is unitarily equivalent to the multiplication operator γa,k I acting on L2 (R+ ) with v γa,k (ξ) = χ+ (ξ) a 2k (v) dv, 2ξ R+

where the functions k (x) = e−x/2 Lk (x) forms an orthonormal basis in L2 (R+ ) with Lk (x) being the Laguerre polynomial of order k = 0, 1, . . . Thus, the boundedness of function γa,k is responsible also for the bound(k) edness of operator Ta . We will also show that for unbounded symbols a = a(v), v ∈ R+ , the behavior of iterated means

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v =

a(t) dt, 0

Ca(m) (v) =

v

Ca(m−1) (t) dt,

m = 2, 3, . . . ,

0

rather than the behavior of symbol a itself plays a crucial role in the boundedness properties. Contrary to the case of Toeplitz operators on weighted Bergman spaces studied in [5] these means do not depend on a weighted parameter k. In our case k = 0, 1, . . . may be viewed as a level of consideration of time-scale analysis of a signal. We present a number of examples and construct wide families of unbounded symbols for which the Calder´ on– Toeplitz operator is not only bounded, but also belongs to the algebra of bounded Calder´ on–Toeplitz operators generated by bounded symbols on R+ having limits at the endpoints of [0, +∞]. This extends results for bounded symbols from [9] to certain unbounded ones. In the last Sect. 4 we show how Calder´on–Toeplitz operators with unbounded symbols can appear as uniform limits of Calder´ on–Toeplitz operators with bounded symbols.

2. Preliminaries Here we brieﬂy recall some necessary notations and results from our previous works, mainly from [9]. As usual, R (C, N) is the set of all real (complex, natural) numbers, R = R ∪ {−∞, +∞} is the two-point compactification of R, and R+ is the positive half-line with χ+ being its characteristic function. Let L2 (G, dν) be the space of all square-integrable functions on G with respect to measure dν, where G = {ζ = (u, v); u ∈ R, v > 0} is the locally compact “ax + b”-group with the left invariant Haar measure dν(ζ) = v −2 du dv. In what follows we identify the group G with the upper half-plane Π = {ζ = u + iv; u ∈ R, v > 0} in the complex plane C. Then the square-integrable representation ρ of G on L2 (R) is given by x−u 1 (ρζ f )(x) = fζ (x) = √ f , f ∈ L2 (R), v v with ζ = (u, v) ∈ G. A function ψ ∈ L2 (R) is called an admissible wavelet if it satisﬁes the so-called admissibility condition 2 dξ ˆ =1 |ψ(xξ)| ξ R+

for almost every x ∈ R, where ψˆ stands for the unitary Fourier transform F : L2 (R) → L2 (R) given by F{g}(ξ) = gˆ(ξ) = g(x)e−2πixξ dx. R

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of degree n ∈ Z+ = N ∪ {0}, and type α

The Laguerre polynomials Ln are given by L(α) n (y)

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n y −α ey dn −y n+α n + α (−y)k , e y = = n−k n! dy n k!

y ∈ R+ ,

k=0

cf. [4, formula 8.970.1]. For α = 0 we simply write Ln (y). Recall that the system of functions n (y) = e−y/2 Ln (y),

y ∈ R+ ,

n ∈ Z+ ,

forms an orthonormal basis in the space L2 (R+ , dy). For appropriate parameters introduce the (non-negative) function p −x (α) Λ(α,β) |Lm (x)L(β) p,m,n (x) = x e n (x)|,

x ∈ R+ .

Then for each α, β ≥ − 21 , p > −1, x ∈ R+ , and m, n ∈ Z+ we have m n (α + 1)m−i (β + 1)n−j

Λ(α,β) p,m,n (x) ≤

(m − i)! i! (n − j)! j!

i=0 j=0

Γ(x+n) Γ(x)

cf. [9, Appendix], where (x)n =

Λ(α,β) p,m,n (x) dx ≤

(2.1)

is the Pochhammer symbol. Thus,

m n (α + 1)m−i (β + 1)n−j

(m − i)! i! (n − j)! j!

i=0 j=0

R+

xp+i+j e−x ,

Γ(p + i + j + 1)

:= const(α,β) p,m,n .

(2.2)

Recall also the following closely related exact formula, cf. [18, formula (16), p. 330], (β) xp e−x L(α) m (x)Ln (x) dx R+ min{m,n}

= Γ(p + 1)

i=0

m+n

(−1)

p−α p−β p+i , m−i n−i i

(2.3)

where p > −1, α, β > −1, m, n ∈ Z+ , and a Γ(a + 1) . = b Γ(b + 1)Γ(a − b + 1) Further, for k ∈ Z+ we consider the functions (admissible wavelets) ψ (k) and ψ¯(k) on R, whose Fourier transforms are given by ¯(k) (ξ) = ψˆ(k) (−ξ), ψˆ(k) (ξ) = χ+ (ξ) 2ξ k (2ξ) and ψˆ respectively. Let A(k) , resp. A¯(k) , be the spaces of wavelet transforms of functions f ∈ H2+ (R), resp. f ∈ H2− (R), with respect to wavelets ψ (k) , resp. ψ¯(k) , where

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H2+ (R) = f ∈ L2 (R); supp fˆ ⊆ [0, +∞) ,

H2− (R) = f ∈ L2 (R); supp fˆ ⊆ (−∞, 0] , are the Hardy spaces, respectively. Consider the unitary operators U1 = (F ⊗ I) : L2 (G, dν(ζ)) → L2 (R, du) ⊗ L2 (R+ , v −2 dv) with ζ = (u, v) ∈ G, and U2 : L2 (R, du) ⊗ L2 (R+ , v −2 dv) → L2 (R, dx) ⊗ L2 (R+ , dy) given by

2|x| y F x, . U2 : F (u, v) → y 2|x|

In [8] we have proved the following important result describing the structure of wavelet subspaces A(k) inside L2 (G, dν). Theorem 2.1. The unitary operator U = U2 U1 gives an isometrical isomorphism of the space L2 (G, dν) onto L2 (R, dx) ⊗ L2 (R+ , dy) under which the wavelet subspace A(k) is mapped onto L2 (R+ ) ⊗ Lk , where Lk is the rank-one space generated by function k (y) = e−y/2 Lk (y). This result is a “wavelet” analog of results obtained for the Bergman and poly-Bergman spaces, cf. [16], and enables to study an interesting connection between wavelet subspaces related to Laguerre polynomials and poly-Bergman spaces in more detail, which is done in paper [1]. Following the general scheme presented in [12], let us introduce the isometric imbedding Qk : L2 (R+ ) → L2 (R) ⊗ L2 (R+ ) given by (Qk f )(x, y) = χ+ (x)f (x)k (y). Here the function f is extended to an element of L2 (R) by setting f (x) ≡ 0 for x < 0. The adjoint operator Q∗k : L2 (R) ⊗ L2 (R+ ) → L2 (R+ ) is given by (Q∗k F )(x) = χ+ (x)

F (x, τ )k (τ ) dτ,

R+

and we have Theorem 2.2. The operator Rk : A(k) → L2 (R+ ), where dudv , (Rk F )(ξ) = χ+ (ξ) 2ξ F (u, v)k (2vξ)e−2πiξu v

(2.4)

R×R+

is an isometrical isomorphism admitting the decomposition Rk = Q∗k U2 (F⊗I).

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Corollary 2.3. The inverse isomorphism Rk∗ : L2 (R+ ) → A(k) given by √ (2.5) (Rk∗ f ) (u, v) = 2v f (ξ)k (2ξv)e2πiξu ξ dξ R+

admits the decomposition = (F −1 ⊗ I)U2−1 Qk with F −1 : L2 (R) → L2 (R) being the inverse Fourier transform. Rk∗

The above representation of wavelet subspaces is especially important in the study of Toeplitz-type operators related to wavelets whose symbols depend only on vertical variable v = ζ in the upper half-plane Π of the complex plane C. For a given L∞ (G, dν)-function a(ζ) = a(v) depending only (k) on v = ζ, ζ ∈ G, deﬁne the Calder´ on–Toeplitz operator Ta : A(k) → A(k) with symbol a as Ta(k) = P (k) Ma , where Ma is the operator of pointwise multiplication by a and P (k) is the orthogonal projection from L2 (G, dν) to the wavelet subspace A(k) . Formally, these operators were introduced in [14] and further studied in general, e.g. in papers [7,10,11,15]. The following important result, which enables to reduce Calder´ on–Toeplitz operator to a certain multiplication operator, was proved in [9]. In fact, it is the main tool of our study. Theorem 2.4. Let a = a(v), v ∈ R+ , be a measurable symbol on G. Then (k) the Calder´ on–Toeplitz operator Ta acting on A(k) is unitarily equivalent to the multiplication operator γa,k I = Rk Ta(k) Rk∗ acting on L2 (R+ ), where Rk and Rk∗ are given by (2.4) and (2.5), respectively. The function γa,k is given by v γa,k (ξ) = a (2.6) 2 (v) dv, ξ ∈ R+ . 2ξ k R+

Note that all the above results and definitions may be stated analogously for the space A¯(k) . In what follows we restrict our attention only to (k) wavelet subspaces A(k) and operators Ta acting on them. For more results (k) on properties of Calder´ on–Toeplitz operators Ta with symbols a = a(v) and properties of the corresponding function γa,k (ξ) responsible for many interesting features of these operators and their algebras (see the recent paper [9]).

3. Boundedness of Calder´ on–Toeplitz Operator (k)

Clearly, if a = a(v) is a bounded symbol on G, then the operator Ta bounded on A(k) , and for its operator norm holds

is

Ta(k) ≤ ess-sup |a(v)|. Thus all spaces A(k) , k ∈ Z+ , are naturally appropriate for Calder´ on–Toeplitz operators with bounded symbols. However, we may observe that the result

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of Theorem 2.4 suggests considering not only L∞ (G, dν)-symbols, but also unbounded ones. In this case we obviously have (k)

Corollary 3.1. The Calder´ on–Toeplitz operator Ta with a measurable symbol a = a(v), v ∈ R+ , is bounded on A(k) if and only if the function γa,k (ξ) is bounded on R+ , and Ta(k) = sup |γa,k (ξ)|. ξ∈R+

From this result we immediately have that the Calder´ on–Toeplitz (1) operator Ta with unbounded symbol 1 1 a(v) = √ sin , v ∈ R+ , (3.1) v v is bounded on A(1) because the corresponding function √ √ √ sin 2 ξ 2π −2√ξ cos 2 ξ e γa,1 (ξ) = + (3 − 2 ξ) √ , (2 ξ − 8ξ) √ 4 2 ξ 2 ξ

ξ ∈ R+ ,

is bounded (see [9, Example 4.4]). However, due to computational limitations (to ﬁnd an explicit form of the function γa,k (ξ)) we can not say any(k) thing about the boundedness of Ta for arbitrary k. Fortunately, according to Theorem 3.3 we will be able to show much more for a more general class of unbounded symbols including that symbol given by (3.1). Example 1. For oscillating symbol a(v) = e2vi (with i2 = −1) we have 2 k k (−1)k j (−1) ξ 2j+1 , ξ ∈ R+ , γa,k (ξ) = 2k+1 j (ξ − i) j=0 (k)

see [9, Example 4.5]. Thus, the Calder´ on–Toeplitz operator Ta acting on A(k) is bounded for each k ∈ Z+ , and moreover γa,k (ξ) ∈ C[0, +∞]. In both the above mentioned examples (more precisely, in the ﬁrst one just for the case k = 1, but we will show later that also for all k ∈ Z+ ) we (k) are in situation

that the Calder´ on–Toeplitz operator Ta belongs to the {0,+∞} C ∗ -algebra Tk L∞ (R+ ) generated by (bounded) Calder´ on–Toeplitz operators with L∞ (R+ )-symbols having limits at the points 0 and +∞, cf. (k) [9, Section 4]. Recall that the Calder´ Ta with a symbol

on–Toeplitz operator {0,+∞} a = a(v) belongs to the algebra Tk L∞ (R+ ) if and only if the corresponding functionγa,k (ξ) belongs to C[0, +∞]. This means that the algebra

{0,+∞} Tk L∞ (R+ ) contains many more Calder´ on–Toeplitz operators than was described in [9], because it also contains (bounded) Calder´ on–Toeplitz operators whose (generally unbounded) symbols a(v) need not have limits at the endpoints 0 and +∞. Example 2. An easy example of unbounded symbol, for which the Calder´ on– Toeplitz operator is unbounded for each k ∈ Z+ , is the function a = a(v) = v p

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with p > −1 and p = 0. The explicit formula for γa,k has the form v 1 (0,0) 2 γa,k (ξ) = a Λp,k,k (v) dv, ξ ∈ R+ . (v) dv = 2ξ k (2ξ)p R+

R+

Since by the formula (2.3) the last integral is a constant, the function γa,k (ξ) (k) is clearly unbounded on R+ . Thus the operator Ta is not bounded on A(k) {0,+∞} and does not belong to the algebra Tk L∞ (R+ ) . This case of symbols is subsumed in Theorem 3.7. As we have shown in [9, Theorem 4.2], the behavior of a bounded function a(v) near the point 0, or +∞, determines the behavior of function γa,k (ξ) near the point +∞, or 0, respectively. The existence of limits of a(v) in these endpoints guarantees the continuity of γa,k (ξ) on the whole R+ , however this condition is not necessary even for bounded symbols (see [9, Remark 4.3]). Continuity of function γa,k on the whole R+ then guarantees its boundedness, and therefore by Corollary 3.1 the boundedness of the corresponding (k) Calder´ on–Toeplitz operator Ta on wavelet subspace A(k) . However, as example of symbol (3.1) shows (we will do it exactly and (k) more generally in Example 3) the Calder´

on–Toeplitz operator Ta can be {0,+∞}

bounded and belong to the algebra Tk L∞ (R+ ) for each k ∈ Z+ even for unbounded symbols a = a(v). Thus, in what follows we study this phenomena in more detail and will be interested in unbounded symbols to have a sufﬁciently large class of them common to all admissible k. For this purpose denote by L1 (R+ , 0) the class of functions a = a(v) such that a(v)e−εv ∈ L1 (R+ ),

for any ε > 0.

We give some conditions on the behavior of L1 (R+ , 0)-symbols (in fact on the behavior of certain means of these symbols) which guarantees the boundedness of function γa,k (ξ). Remark 3.2. Using several formulas, more precisely [4, formula 8.976.3], [13, formula (5), p. 209] and [4, formula 8.976.1], we may rewrite the function γa,k (ξ) as follows k 2k j 2i 1 2k − 2i 2i j γa,k (ξ) = 2k+1 k − i j r i 2 i=0 j=0 r=0 ×

(−1)r (1 − 4ξ)2i−j (4ξ)j+1 Ir (ξ), r!

where

Ir (ξ) =

a(v)v r e−2vξ dv.

R+

Then the last integral is, in fact, the integral in the formula of function γa,λ (x) for Toeplitz operators on the upper half-plane (the so-called parabolic case) (see [17, formula (13.1.1), p. 329], or [5, formula (2.6)]). Therefore it would

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be natural to consider certain means of symbols depending on parameter k as it is done therein, but we will not do it this way and our means of symbols will not depend on weight parameter. For any L1 (R+ , 0)-symbol a(v) deﬁne the following averaging functions Ca(1) (v)

v =

a(t) dt, 0

Ca(m) (v) =

v

Ca(m−1) (t) dt,

m = 2, 3, . . .

0 (m)

The functions Ca

constitute a “sequence of iterated integrals” of symbol a.

Theorem 3.3. Let a = a(v) ∈ L1 (R+ , 0) and for any m ∈ N suppose that the (m) function Ca has the following asymptotic behavior Ca(m) (v) = O(v m ),

as

v → 0,

(3.2)

v → +∞.

(3.3)

and Ca(m) (v) = O(v m ),

as

Then for each k ∈ Z+ we have sup |γa,k (ξ)| < +∞.

ξ∈R+

(k)

Consequently, the corresponding Calder´ on–Toeplitz operator Ta on A(k) for every k ∈ Z+ .

is bounded

Proof. Let m ≥ 1. The condition (3.2) together with the condition (3.3) imply that for all v ∈ R+ the estimate |Ca(m) (v)| ≤ const v m

(3.4)

holds, where “const” does not depend on v ∈ R+ . Integrating by parts m-times we have for all ξ ∈ R+ that γa,k (ξ) = 2ξ 2k (2vξ) dCa(1) (v) R+

= −2ξ R+

d 2 (2vξ) dCa(2) (v) dv k

= (−1)m 2ξ

.. .

R+

= (−1)m 2ξ

R+

dm 2 (2vξ) dCa(m+1) (v) dv m k

Ca(m) (v)

dm 2 (2vξ) dv. dv m k

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Using the estimates (2.2) and (3.4) we then get |γa,k (ξ)| ≤ (2ξ)m+1

i m m i

i

i=0 j=0

≤ const

m i i=0 j=0

= const

i m i=0 j=0

≤ const

m i i=0 j=0

j

(i−j)

(j)

|Ca(m) (v)|e−2vξ |Lk−i+j (2vξ)Lk−j (2vξ)| dv

R+

m i (i−j) (j) (2ξv)m e−2vξ |Lk−i+j (2vξ)Lk−j (2vξ)| 2ξ dv i j R+

m i (i−j,j) Λm,k−i+j,k−j (x) dx i j R+

m i (i−j,j) constm,k−i+j,k−j i j

< +∞, thus the function γa,k is bounded for each k ∈ Z+ , and by Corollary 3.1 the (k) Calder´ on–Toeplitz operator Ta is bounded as well. Remark 3.4. The condition (3.2) guarantees the boundedness of the function γa,k (ξ) at a neighborhood of ξ = +∞, while the condition (3.3) guarantees its boundedness at a neighborhood of ξ = 0. Observe that if the conditions (3.2) and (3.3) hold for some m = m0 , then according to (3.4) hold also for m = m0 + 1. Indeed, v v (m0 +1) (m0 ) (v)| ≤ |Ca (t)| dt ≤ const tm0 dt ≤ const v m0 +1 . |Ca 0

0

The main advantage of Theorem 3.3 is that we need not have an explicit form of the corresponding function γa,k for an unbounded symbol a = a(v) to decide about its boundedness, as it is e.g. in Example 1. Theorem 3.5. Let a = a(v) ∈ L1 (R+ , 0). If for any m, n ∈ N, any λ1 ∈ R+ and any λ2 ∈ (0, n + 1) holds (3.5) Ca(m) (v) = O v m+λ1 , as v → 0, and

Ca(n) (v) = O v n−λ2 ,

v → +∞,

as

(3.6)

then for each k ∈ Z+ we have lim γa,k (ξ) = 0 = lim γa,k (ξ).

ξ→+∞

ξ→0

Proof. Analogously as in the proof of Theorem 3.3 using the condition (3.5) we have m i m i 1 (i−j,j) const Λm+λ1 ,k−i+j,k−j (x) dx, |γa,k (ξ)| ≤ λ 1 i j (2ξ) i=0 j=0 R+

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where “const” does not depend on v ∈ R+ . Thus, lim γa,k (ξ) = 0. ξ→+∞

Similarly, integrating by parts n-times and using the condition (3.6) we get |γa,k (ξ)| ≤ (2ξ)λ2 const

n i n i i=0 j=0

i

j

(i−j,j)

Λn−λ2 ,k−i+j,k−j (x) dx,

R+

where (the different) “const” does not depend on v ∈ R+ . Letting ξ → 0 we have again the desired result. In other words, Theorem 3.5 gives the condition on the behavior of L1 (R+ , 0)-symbols such that the function γa,k (ξ) ∈ C[0, +∞], and thus (k) the corresponding Calder´ on–Toeplitz operator Ta belongs to the algebra {0,+∞} Tk L∞ (R+ ) . In the next example we present a wide class of oscillat(k)

ing symbols a = a(v) on–Toeplitz operator Ta

for which the Calder´ {0,+∞} to the algebra Tk L∞ (R+ ) for each k ∈ Z+ .

belongs

Example 3. For α > 0 and β ∈ (0, 1) consider the unbounded symbol a(v) = v −β sin v −α ,

v ∈ R+ .

However, the function a(v) is continuous at v = +∞ for all admissible values of parameters, and therefore γa,k (0) = a(+∞) = 0. On the other side, it is difﬁcult to verify the behavior of function γa,k (ξ) at the endpoint +∞ by a direct computation. However, according to [17, Example 13.1.4] we have v α−β+1 cos v −α + O(v 2α−β+1 ), as v → 0. α From it follows that for α > β the ﬁrst condition in (3.5) holds for m = 1 and λ1 = α−β. By Theorem 3.5 the function γa,k (ξ) is bounded, and therefore the (k) corresponding Calder´ on–Toeplitz operator Ta is bounded for each k ∈ Z+ . Observe that this is in accordance with the obtained result of a special case for the symbol (3.1) and k = 1. Here we have extended it for much more general class of unbounded symbols and the whole range of parameters k. If α ≤ β, then Ca(1) (v) =

Ca(m) (v) = O(v mα−β+m ),

as

v → 0.

Thus for each α ≤ β there exists m0 ∈ N such that m0 α > β, and therefore the ﬁrst condition in (3.5) holds for m = m0 and λ1 = m0 α − β, which guarantees that γa,k (ξ) is continuous at ξ = 0. Since for all parameters α > 0 on–Toeplitz and β ∈ (0, 1) the function γa,k (ξ) ∈ C[0,

+∞], then each Calder´ (k)

operator Ta

{0,+∞}

belongs to the algebra Tk L∞

(R+ ) for each k ∈ Z+ .

In fact, Theorem 3.5 partially extends the result [9, Theorem 4.2] stated for bounded symbols to certain unbounded ones. In Example 1 and Example 3 we have provided such oscillating symbols a = a(v) for which the Calder´ on– (k) {0,+∞} (R+ ) for the whole Toeplitz operator Ta belongs to the algebra Tk L∞ range of parameters k. Now we give an example of a bounded oscillating

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(k)

symbol such thatthe bounded operator Ta

{0,+∞} (R+ ) . Tk L∞

does not belong to the algebra

Example 4. The function a(v) = v i = ei ln v ,

v ∈ R+ ,

is oscillating near the endpoints 0 and +∞, but it is bounded on R+ , and (k) therefore the Calder´ on–Toeplitz operator Ta is bounded for each k ∈ R+ . Changing the variable x = 2vξ yields γa,k (ξ) = 2ξ v i 2k (2vξ) dv = (2ξ)−i xi 2k (x) dx R+

= (2ξ)−i

R+ (0,0)

Λi,k,k (x) dx.

R+

Since by the formula (2.3) the last integral is a constant depending on k, the function γa,k (ξ) oscillates and has no limit when ξ → 0 as well as when (k) ξ → +∞. Thus the bounded Calder´ on–Toeplitz operator Ta does not belong {0,+∞} to the algebra Tk L∞ (R+ ) . Hence not all oscillating symbols (even

{0,+∞} bounded and continuous) generate an operator from Tk L∞ (R+ ) . In the following theorem we show that the boundedness of a Toeplitz operator on the Bergman space with non-negativity of symbol or its means guarantees the boundedness of Calder´ on–Toeplitz operator on each wavelet subspace as well. Theorem 3.6. (i) Let a = a(v) ∈ L1 (R+ , 0) be non-negative almost every(0) (k) where. If Ta is bounded on A(0) , then the operator Ta is bounded on (k) A for each k ∈ Z+ . (m) (ii) Let Ca be non-negative almost everywhere for a certain m = m0 . If (0) (k) Ta is bounded on A(0) , then the operator Ta is bounded on A(k) for each k ∈ Z+ . Proof. (i)

From assumptions we have

γa,0 (ξ) = 2ξ

a(v)e−2vξ dv ≥ 2ξ

−1 (2ξ)

a(v)e−2vξ dv ≥

0

R+

Putting (2ξ)−1 = v we get Ca(1) (v)

≤

2ξ (1) C (2ξ)−1 . e a

e sup |γa,0 (ξ)|

v = const v,

ξ∈R+

(k)

which by Theorem 3.3 means that Ta k ∈ Z+ .

is bounded on A(k) for each

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Integrating by parts m0 -times we obtain dm0 γa,0 (ξ) = (−1)m0 2ξ Ca(m0 ) (v) m0 e−2vξ dv dv R+

= (2ξ)m0 +1

Ca(m0 ) (v)e−2vξ dv

R+

≥ (2ξ)m0 +1

−1 (2ξ)

Ca(m0 ) (v)e−2vξ dv

0 m0 +1 −1

≥ (2ξ)

e

Ca(m0 +1) (2ξ)−1 .

Again putting (2ξ)−1 = v we have Ca(m0 +1) (v)

≤

e sup |γa,0 (ξ)| ξ∈R+

(k)

and by Theorem 3.3 the boundedness of Ta follows.

v m0 +1 on A(k) for each k ∈ Z+

According to the presented examples an unbounded symbol must have a sufﬁciently sophisticated oscillating behavior at neighborhoods of the points 0 and +∞ to generate a bounded Calder´ on–Toeplitz operator. In what follows we show that inﬁnitely growing positive symbols (as in the case of identity, or its powers) cannot generate bounded Calder´ on–Toeplitz operators in general. For this purpose for a non-negative function a = a(v) put θa (v) = inf a(t) and

Θa (v) =

t∈(0,v)

inf

t∈(v/2,v)

a(t).

Theorem 3.7. For a given non-negative symbol a = a(v) if either lim θa (v) = +∞

(3.7)

lim Θa (v) = +∞,

(3.8)

v→0

or v→+∞

(k)

then the Calder´ on–Toeplitz operator Ta

is unbounded on each A(k) , k ∈ Z+ .

Proof. If the condition (3.7) holds, then v Ca(1) (v) = a(t) dt ≥ v θa (v), 0 −1

(1) Ca (v)

which yields v → +∞, as v → 0. If the condition (3.8) holds, then v 1 −1 (1) −1 v Ca (v) > v a(t) dt ≥ Θa (v), 2 v/2

which again yields

(1) v −1 Ca (v)

→ +∞, as v → +∞.

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Example 5. For the family of non-negative symbols on R+ in the form a(v) = v −β ln2 v −α ,

β ∈ [0, 1],

α > 0,

we have that for all admissible parameters holds lim θa (v) = +∞, and thus v→0 (k)

by Theorem 3.7 the Calder´ on–Toeplitz operator Ta for each k ∈ Z+ .

is unbounded on A(k)

In the following example we use the result of Theorem 3.7 to study boundedness of a Calder´ on–Toeplitz operator with unbounded symbol as a product of two symbols for which the corresponding Calder´ on–Toeplitz operators are bounded on each wavelet subspace. Example 6. Let us consider two symbols on R+ in the form a(v) = v −β sin v −α ,

β ∈ (0, 1),

α ≥ β,

and b(v) = v τ sin v −α ,

τ ∈ (0, β).

According to Example 3, for the unbounded symbol a(v) the Calder´ on–Toep(k) litz operator Ta is bounded for each k ∈ Z+ . Since the symbol b(v) ∈ (k) C[0, +∞], then the Calder´ on–Toeplitz operator Tb is bounded for each k ∈ Z+ as well. Put v −δ v −δ − cos 2v −α = c1 (v) + c2 (v), 2 2 where δ = β − τ ∈ (0, 1). Clearly, c(v) is an unbounded symbol. However, (k) Tc2 is bounded for each k ∈ Z+ (analogously as in Example 3 replacing sin by cos). Since c(v) = a(v)b(v) =

1 1 = δ → +∞, δ 2v t∈(0,v) 2t

θc1 (v) = inf

as

v → 0, (k)

then by Theorem 3.7 the Calder´on–Toeplitz operator Tc1 is unbounded for (k) on–Toeplitz operator Tab is unbounded on each k ∈ Z+ . Thus, the Calder´ (k) A for each k ∈ Z+ . Moreover, this result shows that the semi-commutator (k) (k) (k) Ta(k) , Tb = Ta(k) Tb − Tab is not compact. This interesting feature and the algebras of Calder´ on–Toeplitz operators will be considered elsewhere.

4. Calder´ on–Toeplitz Operators with Unbounded Symbols as Uniform Limits of Calder´ on–Toeplitz Operators with Bounded Symbols In connection with the obtained results we now show how Calder´ on– Toeplitz operators with unbounded symbols given in Example 3 can appear as uniform limits of Calder´ on–Toeplitz operators with bounded symbols.

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Theorem 4.1. The Calder´ on–Toeplitz operator Ta a(v) = v

−β

sin v

−α

,

597

with symbol

v ∈ R+ , ∗

where β ∈ (0, 1) and α > β, belongs to the C -algebra generated by Calder´ on– Toeplitz operators with smooth bounded symbols on each wavelet subspace A(k) . Proof. Consider the sequence ϑn = (πn)−1/α , n ∈ N, of zeros of function a = a(v) and deﬁne the sequence a(v), v ∈ [ϑn , +∞), an (v) = 0, v ∈ [0, ϑn ). Each symbol an (v) is bounded and continuous. Further each an (v) can be uniformly approximated by smooth symbols, and thus belongs to the on–Toeplitz operators with smooth bounded C ∗ -algebra generated by Calder´ (k) symbols. According to Theorem 2.4 the Calder´on–Toeplitz operator Ta act(k) ing on A is unitarily equivalent to the multiplication operator γa,k I acting on L2 (R+ ), where the function γa,k is given by (2.6). Thus, (k)

= Ta−an = sup |γ(a−an ),k (ξ)| Ta(k) − Ta(k) n ξ∈R+

ϑ n 2 = sup 2ξ a(v)k (2vξ) dv ξ∈R+ 0 ϑn (1) 2 2 = sup 2ξCa (ϑn )k (2ϑn ξ) + 4ξ Ca(1) (v)2k (2vξ) dv ξ∈R+ 0 ϑn (1) + 8ξ 2 Ca(1) (v)e−2vξ Lk (2vξ)Lk−1 (2vξ) dv , 0

where integration by parts has been used. Since v v α−β+1 (1) Ca (v) = a(t) dt = cos v −α + O(v 2α−β+1 ), α

as

v → 0,

0

see Example 3, then |Ca(1) (v)| ≤ const v α−β+1 , where “const” does not depend on v ∈ (0, 1), and thus the Calder´ on–Toeplitz (k) (k) operator Ta is bounded on A for each k ∈ Z+ . Then ⎛ ϑn (k) (1) 2 2 Ta−an ≤ const sup ⎝2ξ|Ca (ϑn )|k (2ϑn ξ) + 4ξ |Ca(1) (v)|2k (2vξ) dv ξ∈R+

+ 8ξ 2

ϑn 0

0

⎞

(1) |Ca(1) (v)|e−2vξ |Lk (2vξ)Lk−1 (2vξ)| dv ⎠

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(2ϑn ξ)2k (2ϑn ξ) ≤ const sup ϑα−β n ξ∈R+

+ const sup 4ξ

2

ξ∈R+

ϑn

v α−β+1 2k (2vξ) dv

0

+ const sup 8ξ 2 ξ∈R+

ϑn

(1)

v α−β+1 e−2vξ |Lk (2vξ)Lk−1 (2vξ)| dv

0

= I1 + I2 + I3 . . To evaluate I2 we Since sup (2ϑn ξ)2k (2ϑn ξ) < +∞, then I1 ≤ q1 (k)ϑα−β n ξ∈R+

use the estimate (2.2), and we have ϑn I2 = const sup

ξ∈R+

≤

=

v α−β (2vξ)2k (2vξ)2ξ dv

0

ϑn sup (2vξ)2k (2vξ)2ξ dv

const ϑα−β n

ξ∈R+

0 2ϑ nξ

const ϑα−β n

x2k (x) dx

sup

ξ∈R+

≤ const ϑα−β n

0

(0,0)

Λ1,k,k (x) dx R+

≤

. q2 (k)ϑα−β n

Similarly for I3 we get ϑn I3 = 2 const sup

ξ∈R+

≤

=

2 const ϑα−β n

2 const ϑα−β n

≤ 2 const ϑα−β n

0

ϑn (1) sup (2vξ)e−2vξ |Lk (2vξ)Lk−1 (2vξ)|2ξ dv

ξ∈R+

≤

0 2ϑ nξ

ξ∈R+

(1)

xe−x |Lk (x)Lk−1 (x)| dx

sup

0

(0,1)

Λ1,k,k−1 (x) dx R+

. q3 (k)ϑα−β n

(1)

v α−β (2vξ)e−2vξ |Lk (2vξ)Lk−1 (2vξ)|2ξ dv

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On Boundedness of Calder´ on–Toeplitz Operators

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Thus, ≤ q(k)ϑα−β , Ta(k) − Ta(k) n n where the constant q(k) depends on k, but does not depend on n, and ϑn → 0 whenever n → +∞. It seems to be natural to ask whether the boundedness of Calder´ on– Toeplitz operator (and by Corollary 3.1 the boundedness of corresponding function γ· ) is equivalent to the boundedness of its Wick symbol. According to the result of Nowak [10] it is true for non-negative symbols a and sufﬁciently smooth wavelets. Thus, we immediately have the following result. Corollary 4.2. For a non-negative symbol a = a(v) the following statements are equivalent: (i) (ii) (iii)

(k)

the Calder´ on–Toeplitz operator Ta is bounded; the function γa,k is bounded; (k) the Wick symbol ak of Ta is bounded.

In connection with it we also mention another Nowak’s result, cf. [10], for compactness of Calder´on–Toeplitz operator: for a non-negative symbol the Calder´ on–Toeplitz operator is compact if and only if its Wick symbol tends to 0 at infinity. In our case of symbol a = a(v) we are in a different situation (k) because according to Theorem 2.4 the operator Ta is unitarily equivalent to a multiplication operator, and thus never compact. Acknowledgments Author has been on a postdoctoral stay at the Departamento de Matem´ aticas, CINVESTAV del IPN (M´exico), when writing this paper and investigating the topics presented herein. He therefore gratefully acknowledges the hospitality of the mathematics department of CINVESTAV on this occasion. Author wishes especially to thank Nikolai L. Vasilevski for having read previous versions of the present work and for having generously shared with him his comments and suggestions.

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