Integral Equations and Operator Theory - Volume 59

Integr. equ. oper. theory 59 (2007), 1–17 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-17, published ...

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Integr. equ. oper. theory 59 (2007), 1–17 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-17, published online June 27, 2007 DOI 10.1007/s00020-007-1513-1

Integral Equations and Operator Theory

Hankel Operators on Weighted Fock Spaces H´el`ene Bommier-Hato and El Hassan Youssfi Abstract. We consider Hankel operators Hf¯ with antiholomorphic symbol f¯ on the generalized Fock space A 2 (µm ), where µm is the measure with weight m e−|z| , m > 0 with respect to the Lebesgue measure in Cn . We prove that . We show Hf¯ is bounded if and only if f is a polynomial of degree at most m 2 that Hf¯ is compact if and only if f is a polynomial of degree strictly smaller . We also establish that Hf¯ is in the Schatten class Sp if and only if that m 2 p > 2n and f is a polynomial of degree strictly smaller than m (p−2n) . 2p Mathematics Subject Classification (2000). Primary 47B35, 32A36, 32A37. Keywords. Hankel operator, Fock space, Bergman kernel.

1. Introduction and statement of the main results We consider the Fock type space A2 (µm ) consisting of those holomorphic functions m which are square integrable with respect to the measure dµm (z) = e−|z| dV (z), where dV (z) is the Lebesgue measure on Cn and m > 0 is a positive parameter. When m = 2 the space A2 (µ2 ) is the Fock space, called also the SegalBargmann space. Let I be the identity operator and P2 is the orthogonal projection from L2 (µ2 ) onto A2 (µ2 ). Let T(Cn ) be the subspace of L2 (µ2 ) consisting of those functions f that satisfy f (·+a) ∈ L2 (µ2 ) for all a ∈ Cn . We recall that if f ∈ T(Cn ), then the Hankel operator Hf with symbol f is deﬁned by Hf (ϕ) = (I − P2 )(f ϕ), for all ϕ in the dense subspace of A2 (µ2 ) spanned by {K2 (·, a), a ∈ Cn } where K2 (z, a) := ez,a , z, a ∈ Cn , is the Bergman kernel. In this case, the study of compactness of Hankel operators with bounded symbols was considered in the works of Berger and Coburn [3] and Stroethoﬀ [14]. In the more general case f ∈ T(Cn ), the simultaneous membership of Hf and Hf¯ to the Schatten classes was characterized by Xia and Zheng [15] and by Bauer [1] in the Hilbert-Schmidt setting. A necessary and suﬃcient condition for simultaneous boundedness of Hf and Hf¯ was given recently by Bauer [2]. The tools used in these works use heavily

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the translation action of the group Cn and related properties to the Bergman kernel. We also mention that in the one dimensional case n = 1 the study of Hankel operators in the setting m > 0 was considered by Schneider [Sc] when the symbol is a monomial. His method is direct and relies on an approximation process. In this paper we consider the general case m > 0. We begin by clarifying the appropriate deﬁnition of densely deﬁned Hankel operators. Indeed, if f ∈ L2 (µm ) is a function of polynomial growth, then the Hankel operator Hf with symbol f is deﬁned by Hf (ϕ) = (I − Pm )(f ϕ), where Pm is the orthogonal projection from L2 (µm ) onto A2 (µm ) given by Km (z, w)g(w)dµm (w), for g ∈ L2 (µm ) Pm (g)(z) := Cn

where Km is the Bergman kernel given in Section 5 below. We shall show in Section 5 that the righthand side of the latter equality is well-deﬁned for functions g of the form g = f ϕ for all f ∈ L2 (µm ) and ϕ in the space P of holomorphic polynomials. This allows us to extend the deﬁnition of Pm on such functions and, using this, we see that Hf is deﬁned on holomorphic polynomials. In particular, it is densely deﬁned. We ﬁrst point out that the techniques used the case m = 2 to study Hankel operators do not apply to the case m = 2. Our goal herein is to develop new methods which are adequate to the setting m > 0 in the case of anti-analytic symbols f. Our ﬁrst main result is the following Theorem A. Let f be an entire function in A2 (µm ), where m is a positive real number. 1) Then the Hankel operator Hf¯ is bounded on the Fock space A2 (µm ) if and only if f is a polynomial of degree at most m 2. 2) The Hankel operator Hf¯ is compact on the Fock space A2 (µm ) if and only if f is a polynomial of degree smaller than m 2. We observe that when m is odd, then all bounded Hankel operators with anti-analytic symbols are also compact. This is not the case for m even. In the particular case m = 2, Theorem A was established in a recent work by [2] using a technique which does not work at all when m = 2. We recall that an operator T is in the Schatten class Sp (A2 (µm ), L2 (µm )) p ∗ if (T T ) 2 is in the trace class of A2 (µm ). Our second result characterizes such a class of operators. Theorem B. Let f be an entire function in A2 (µm ), where m is a positive real number. Then the Hankel operator Hf is in the Schatten class Sp (A2 (µm ), L2 (µm )) if and only if p > 2n and f is a polynomial of degree smaller than

m(p−2n) . 2p

We discovered recently that a weaker version of our results was established by Knirsch and Schneider [8] in the one dimensional particular case.

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We ﬁnally mention in passing that Hankel operators with antiholomorphic symbols are intimately related to the ∂-canonical solution operator (see [5], [6] and [7]).

2. Preparatory lemmas We recall some facts about Hankel operators with respect to certain rotation invariant measures, see [10]. Let Ω be a rotation invariant open set in Cn and let µ be a rotation invariant measure on Ω. We suppose that µ has moments of every order; that is, mk =

Ω

|z|2k dµ(z) < +∞,

for all k ∈ N0 .

We consider the Hilbert space L2 (Ω, µ) of square integrable complex-valued functions on Ω with respect to the measure µ and A2 (Ω, µ) its subspace consisting of holomorphic elements. We assume that for each set compact K ⊂ Ω there exists C = C(K) > 0 such that supz∈K |f (z)| ≤ C f L2 (Ω,µ) 2

for all f ∈ A (Ω, µ). Thus A2 (Ω, µ) is a closed space of L2 (Ω, µ). The corresponding orthogonal projection Pµ will be called the Bergman projection. We also assume that the subspace consisting of all holomorphic polynomials is dense in A2 (Ω, µ). Therefore, if f ∈ A2 (Ω, µ) has polynomial growth, then the Hankel operator Hf given by Hf (ϕ) = (I − Pµ )(f ϕ) is well deﬁned for all holomorphic polynomials ϕ. In particular, Hf is densely deﬁned. We ﬁrst ﬁx some notations. Let Nn0 denote the set of all n-tuples with components in the set N0 of all nonnegative integers. If α = (α1 , · · · , αn ) ∈ Nn0 , we let |α| := α1 + · · · + αn denote the length of α. If β = (β1 , · · · , βn ) ∈ Nn0 satisﬁes αj ≥ βj for all j = 1, · · · , n, then we write α ≥ β. Otherwise, set α ≥ β. The space of polynomials P is endowed with the Fischer inner product [12] , F , deﬁned on the monomials by α! if α = β α β z , z F = 0 if α = β. Finally, if A and B are two quantities, we use the symbol A ≈ B whenever A ≤ C1 B and B ≤ C2 A, where C1 and C2 are positive constants independent of the varying parameters. In this section we shall express the operators Hzk and Hzk Hzl on holomorphic homogeneous polynomials. Lemma 2.1. Suppose that β, k ∈ Nn0 and d ∈ N0 . Then k

(Hzk f ) (ξ) = ξ f (ξ) −

md Γ(n + d − |k|) ∂ |k| f (ξ) md−|k| Γ(n + d) ∂ξ k

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for all holomorphic polynomials f of degree d. In particular, if f = ξ α , then  m|α| Γ(n + |α| − |k|) α! k   ξ ξα − ξ α−k , if α ≥ k m Γ(n + |α|) (α − k)! |α|−|k| (Hz k f )(ξ) =   k α ξ ξ , otherwise. Proof. It suﬃces to prove the lemma for f (ξ) = ξ α , where α ∈ Nn0 . Let g be a homogeneous polynomial in P. If g is a monomial of the form g(ξ) = ξ β , where β ∈ Nn0 , then using the properties of Pµ , we see that Pµ (z k f ), g L2 (Ω,µ) = f, z k g L2 (Ω,µ) and hence Pµ (z k f ), g L2 (Ω,µ) = 0 as long as α = k + β. Now let α = k + β. By Lemma 2.1 in [10], we have the following identities (n − 1)! m|α| α! z α z α dµ(z) = (2.1) and z α , z α F = α! , (n + |α| − 1)! Ω from which we obtain Pµ (z k f ), g L2 (Ω,µ) =

(n − 1)!m|α| f, z k g F . (n + |α| − 1)!

Since the multiplication operator and the corresponding diﬀerentiation operator are adjoint to each other with respect to the Fischer inner product, this implies that Pµ (z k f ), g L2 (Ω,µ) =

m|β|+|k| ∂ |k| (n − 1 + |β|)! k f, g L2 (Ω,µ) m|β| (n − 1 + |β| + |k|)! ∂z

for all holomorphic homogeneous polynomials g of degree |β|. Therefore, if f is a holomorphic homogeneous polynomial of degree d, we have Pµ (z k f ) =

md (n − 1 + d − |k|)! ∂ |k| f. md−|k| (n − 1 + d)! ∂z k

This completes the proof of the lemma.

Lemma 2.2. The domain Dom(Hz∗k ) of Hz∗k contains all polynomials in w and w. ¯ Proof. It suﬃces to show that, if α and β are ﬁxed in Nn0 , then the linear functional g → Hzk (g), z α z β L2 (Ω,µ) is bounded on A2 (Ω, µ). To do so, choose an integer d ≥ |α|+|β|+2|k| and consider the subspace Nd of A2 (Ω, µ) consisting of polynomials with degree smaller than or equal to d. We denote by πd the orthogonal projection from A2 (Ω, µ) onto Nd . If g ∈ P, then (I − πd )g is a sum of holomorphic homogeneous polynomials in P with degree at least d + 1. In view of Lemma 2.1, we can write Hzk ◦ (I − πd )g = z k f + h

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where f is a sum of holomorphic homogeneous polynomials of degree at least d + 1 and h is a sum of holomorphic homogeneous polynomials of degree at least d + 1 − |k|. Therefore, Hzk ◦ (I − πd )g, z α z β L2 (Ω,µ) = z k f, z α z β L2 (Ω,µ) + h, z α z β L2 (Ω,µ) = f z β , z α+k L2 (Ω,µ) + z β h, z α L2 (Ω,µ) . Since d+1+|β| ≥ 1+|α|+2|β|+2|k| > |α|+|k|, it follows that f z β , z α+k L2 (Ω,µ) = 0. Also, due to the fact that the degree of z β f is greater than |α| we see that z β f, z α L2 (Ω,µ) = 0. Thus Hzk ◦ (I − πd )g, z α z β L2 (Ω,µ) = 0 for all g ∈ P and consequently Hzk (g), z α z β L2 (Ω,µ) = Hzk ◦ πd (g), z α z β L2 (Ω,µ) . The lemma now follows from the fact that Hzk ◦ πd is of ﬁnite rank and hence bounded. We observe by Lemmas 2.1 and 2.2 that P is contained in the domain of the operator Hv∗ Hu for all holomorphic polynomials u and v. Lemma 2.3. Suppose that u, v and f are holomorphic polynomials. Then Hv∗ Hu f = Pµ (vuf ) − vPµ (uf ). Proof. A little computing shows that for all g ∈ A2 (Ω, µ) Hu f, Hv g L2 (Ω,µ) = uf − Pµ (uf ), vg − Pµ (vg) L2 (Ω,µ) = vuf, g L2 (Ω,µ) − Pµ (uf ), vg L2 (Ω,µ) + (Pµ − I)(uf ), Pµ (vg) L2 (Ω,µ) = vuf, g L2 (Ω,µ) − Pµ (uf ), vg L2 (Ω,µ) where the latter equality holds since Pµ (vg) ∈ A2 (Ω, µ) and (Pµ − I)(uf ) is or thogonal to A2 (Ω, µ). This completes the proof. Lemma 2.4. Assume that k and l are elements of Nn0 . If f is a holomorphic homogeneous polynomial of degree d, then Pµ (z l z k f ) =

md+|l| Γ(d + n − |k| + |l|) ∂ |k| f. md−|k|+|l| Γ(d + n + |l|) ∂z k

Proof. It is suﬃcient to establish the lemma for monomials f (z) = z α . If β is an arbitrary element of Nn0 , then, due to the properties of the Fischer product and (2.1), we have Pµ (z l z k f ), z β L2 (Ω,µ) = z l+α , z k+β L2 (Ω,µ) (n − 1)! m|α|+|l| ∂ |k| l+α β (z ), z F (n + |α| + |l| − 1)! ∂z k m|α|+|l| Γ(n + |α| + |l| − |k|) ∂ |k| l k (z f ), z β L2 (Ω,µ) = m|α|+|l|−|k| Γ(n + |α| + |l|) ∂z =

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This completes the proof.

In what follows we shall compute Hz∗l Hzk f for a holomorphic homogeneous polynomial f . Lemma 2.5. Suppose that k and l are in Nn0 . If f is a holomorphic homogeneous polynomial of degree d, then Hz∗l Hzk f =

md+|l| Γ(n + d + |l| − |k|) ∂ |k| l md Γ(d + n − |k|) l ∂ |k| z (z f )− f md+|l|−|k| Γ(n + d + |l|) ∂z k md−|k| Γ(d + n) ∂z k

In particular, Hz∗l Hzk f is a holomorphic homogeneous polynomial of degree d + |l| − |k|. Proof. Follows from Lemmas 2.1 and 2.4.

An immediate consequence of Lemma 2.5 gives the following Proposition 2.6. For each α in Nn0 , the monomial z α is an eigenvector for the operator Hz∗k Hzk and the corresponding eigenvalue λα is given by λα =

m|α| Γ(|α| + n − |k|) m|α|+|k| Γ(n + |α|) (α + k)! α! − m|α| Γ(n + |α| + |k|) α! m|α|−|k| Γ(|α| + n) (α − k)!

if α ≥ k and λα =

m|α|+|k| Γ(n + |α|) (α + k)! , m|α| Γ(n + |α| + |k|) α!

otherwise.

3. The Fock space In this section, we consider the Fock space A2 (µm ), for m > 0. In this case, the m moments of the measure dµm (z) := e−|z| dν(z) are given by 1 2s + 2n m 2s ). (3.1) ms = Cn |z| e−|z| dν(z) = Γ( m m If k is a multi-index we set T = Hz∗k Hz k . Then T is deﬁned on the dense subspace P of A2 (µm ). For each multi-index α, the eigenvalue λα of T corresponding to the eigenvector ξ α is given by Proposition 2.6. In what follows we shall study the asymptotic of these eigenvalues. We distinguish the two cases m = 2 and m = 2. Lemma 3.1. Suppose m = 2. Then for each j = 1, · · · , n, the operator Hzj is bounded but not compact on A2 (µm ). If |k| ≥ 2, Hzk is unbounded on A2 (µm ). Proof. In this case, µ2 is the Gaussian measure on Cn . Its moments reduce to ms = Γ(s + n). Moreover, if α ∈ Nn0 ,  (α + k)! α!   − if α ≥ k, α! (α − k)! λα =   (α + k)! if α ≥ k. α!

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We ﬁrst observe that if |k| = 1, then the eigenvalues of T are all equal to 1. Therefore, T is bounded but not compact on A2 (µm ). This proves the ﬁrst part of the lemma. Suppose now that |k| ≥ 2. Choose j0 in [1, n] so that kj0 = maxj kj . If d is a nonnegative integer, set α(k, d) = (k1 , · · · , kj0 −1 , kj0 + d, kj0 +1 , · · · , kn ). Then   n

(2kj )!  λα(k,d) =  [(d + kj0 + 1) · · · (d + 2kj0 )] − (d + 1) · · · (d + kj0 ) (kj )! j=j0

Therefore, lim λα(k,d) = +∞, showing that T is unbounded on A2 (µm ). This d→+∞

implies that Hzk is also unbounded.

Henceforth, we assume that m = 2, m > 0. From Proposition 2.6 and (3.1) we see that if α ∈ Nn0 , then the eigenvalue λα can be written in the form  (α + k)! α!   − B|α| if α ≥ k  A|α| α! (α − k)! λα = (3.2)    A|α| (α + k)! if α ≥ k α! where, for a nonnegative integer d,  2d + 2n 2|k|   + ) Γ(d + n) Γ(   m m  Ad := ),   2d + 2n  Γ(d + n + |k|   ) Γ( m (3.3)  2d + 2n   ) Γ(  Γ(d + n − |k|)  m  Bd := .   2d + 2n 2|k|  Γ(d + n)  − ) Γ( m m The asymptotic behaviour of the eigenvalues {λα } when |α| = d → +∞ is given by the following Lemma 3.2. The sequences (Ad ) and (Bd ) given by (3.3) have the asymptotic behavior 2

2|k| |k| −1 1 2 |k|2 (m − 2) m m (d + n) +O 1− Ad = m 2m(d + n) (d + n)2 2

2|k| 2 |k| −1 1 2 (m − 2) |k| m m (d + n) Bd = +O 1+ , m 2m(d + n) (d + n)2 as d → +∞. Proof. Follows from the property of the Gamma function [11]

1 (y − z)(y + z − 1) Γ(x + y) y−z =x + O( 2 ) as x → +∞, 1+ Γ(x + z) 2x x

(3.4)

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where y and z are real numbers. Lemma 3.3. The eigenvalues λα have the form

2 |k| m |k| αn + 1 2 α1 + 1 ,··· , λα = (d + n)2 m −1 fn + ε(α) , m d+n d+n 1 where ε(α) = O d and tk |k|2 k t + kj2 m tj j=1 n

fn (t1 , · · · , tn ) := −(m − 2) when α ≥ k and d = |α| → +∞. Proof. We recall by (3.2) that if α ≥ k, then λα = A|α|

(α + k)! α! − B|α| , α! (α − k)!

where (Ad ) and (Bd ) are given by (3.3). On the other hand, by (3.4) we see that (αj + kj )! = (1 + αj )kj + kj (kj − 1)(1 + αj )kj −1 + qj (1 + αj ) αj ! αj ! = (1 + αj )kj − kj (kj + 1)(1 + αj )kj −1 + rj (1 + αj ) (αj − kj )! where qj and rj are one variable polynomials of degree at most kj − 2. This implies that n n

(α + k)! kj = (1 + αj ) + kj (kj − 1)(1 + αj )kj −1 (1 + αl )kl + q(α) α! j=1 j=1 l=j

n

n

α! = (1 + αj )kj − (α − k)! j=1 j=1

kj (kj + 1)(1 + αj )kj −1 (1 + αl )kl + r(α) l=j

where q and r are polynomials of degree at most |k|−2. These equalities, combined with Lemma 3.2, give the lemma. Lemma 3.4. The eigenvalues λα have the estimate |k| λα = O (d + n)2 m −kj0 , as long as αj0 < kj0 and d = |α| → +∞. Proof. Let j0 = 1, . . . , n and suppose that kj0 ≥ 1. We recall by (3.2) that if αj0 < kj0 , then (α + k)! , λα = A|α| α!

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where (Ad ) is as before. Set α = (α1 , · · · , αj0 −1 , 0, αj0 +1 , · · · , αn ) and k = (k1 , · · · , kj0 −1 0, kj0 +1 , · · · , kn ). Arguing as in the proof of Lemma 3.3 we have (α + k)! α! ≤ (2kj0 )!

(α + k )!  α!

= (2kj0 )! 

n

(1 + αj )kj +

j=1,j=j0

n

kj (kj − 1)(1 + αj )kj −1

j=1,j=j0

 (1 + αs )ks 

s=j,l

+ q(α ), where q is a polynomial of degree at most |k | − 2. These estimates, combined with Lemma 3.2, give the lemma.

4. Spectral properties of the operator Hz k Theorem 4.1. The operator Hzk ∗ Hzk is bounded if and only if 2 compact if and only if 2

|k| − 1 < 0. m

|k| − 1 ≤ 0 and m

Proof. Let Σn be the simplex consisting of those t = (t1 , · · · , tn ) ∈ Rn such tj ≥ 0 and t1 + · · · + tn = 1. By Lemmas 3.3 and 3.4 we see that |k| α1 + 1 αn + 1 2 m −1 ,··· , sup fn sup |λα | ≈ (d + n) d+n d+n |α|=d

|α|=d

2 |k| m −1

≈ (d + n)

sup |fn (t)|

t∈Σn

as d = |α| → +∞. Now the lemma follows since the operator Hzk ∗ Hzk is bounded if and only if the sequence sup|α|=d |λα | is bounded and Hzk ∗ Hzk is compact if and only if the sequence sup|α|=d |λα | tends to 0 as d = |α| → +∞. Theorem 4.2. Let k ∈ Nn0 and m be a positive real number. 1) The Hankel operator Hzk is bounded on the Fock space A2 (µm ) if and only if m ≥ 2|k|. 2) The Hankel operator Hzk is compact on the Fock space A2 (µm ) if and only if m > 2|k|. Proof. We use that the operator Hz k is bounded if and only if T = Hzk ∗ Hzk is bounded and Hzk is compact if and only if T = Hzk ∗ Hz k is compact. |k| − 1 < 0 and let p > 0. We shall investigate the Next, assume that 2 m membership of the T to a Schatten class Sp . Recall that T is in Sp if and operator only if the series λpα is convergent.

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Let d be an integer. We shall estimate the sum sd = |α|=d λpα, when d → +∞. The calculations above lead to study the cases α ≥ k and its opposite case separately. Let Bd := { α ∈ Nn0 , |α| = d}. We partition Bd = B d ∪ Bd , where B d = { α ∈ B : α ≥ k} and Bd = Bd \ Bd . Thus sd can be written in the form sd = sd + sd , where sd = α∈B d λpα and sd = α∈B λpα . d We need to compare the cardinalities Bd , Bd , and Bd of these sets. Lemma 4.3. We have the estimates Bd ≈ B d ≈ d → +∞.

dn−1 and B d ≈ dn−2 as (n − 1)!

Proof. Let Pn,d the space of n variables holomorphic polynomials of degree d. We (d + n − 1)! 1 have Bd = dim Pn,d = n−1+d . Therefore, Bd ∼ dn−1 = d (n − 1)!d! (n − 1)! as d → +∞. On the other hand, for j = 1, · · · , n, let Bd,j = {α ∈ Bd , αj < kj }. Since Bd = ∪1≤j≤n Bd,j , we see that Bd ≤ nj=1 Bd,j . If kj ≥ 1, then k −1

j Bd,j = ∪l=0 { α = (α1 , · · · , αj−1 , l, αj+1 , · · · , αn ), |α| = d } kj −1 = ∪l=0 {α = (α1 , · · · , αj−1 , l, αj+1 , · · · , αn ), αi = d − l}.

i=j

Therefore, Bd,j =

kj −1 l=0

dn−2 . (n − 2)! as d → +∞. The lemma now follows from the

dim Pn−1,d−l , and when d → +∞, Bd,j ∼ kj

This shows that, when Bd ≈ dn−2 observation Bd = Bd + Bd .

Lemma 4.4. Suppose that n ≥ 2 and g is a continuous function on Rn−1 . Consider n−1 the open set Ω := {(t1 , · · · , tn−1 ) ∈ Rn−1 + , j=1 tj < 1 }. For a multi-index n−1 γ = (γ1 , · · · , γn−1 ) in N0 , set

γ1 + 1 γn−1 + 1 cγ,d := ,··· , d d   n−1  

γj γj + 1 , . : Jd := γ ∈ Nn−1 ⊂ Ω 0   d d j=1 1 Then lim n−1 g(cγ,d ) = g(t)dt. d→+∞ d Ω γ∈Jd

#n−1 γj γj + 1 , . It is clear that Ωd ⊂ Ω. j=1 d d Next, we show that limd→+∞ χΩd = χΩ . If s is a real number, let [s] denote the largest integer smaller than or equal to s. If t = (t1 , · · · , tn−1 ) ∈ Ω and

Proof. For d ∈ N0 , let Ωd = ∪γ∈Jd

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1 ≤ j ≤ n − 1, then

[dtj ] [dtj ] 1 ≤ tj < + . Therefore, d d d n−1 j=1

11

n−1 n−1 [dtj ] n − 1 [dtj ] ≤ + . tj < d d d j=1 j=1

n−1

Since all d > d0 we have j=1 tj < 1, there is an integer d0 such that for n−1 [dtj ] #n−1 [dtj ] [dtj ] + 1 n−1 + < 1. Thus, t ∈ j=1 , and hence t ∈ Ωd j=1 d d d d for all d > d0 . Thus lim χΩd = χΩ . Therefore, d→+∞

1 dn−1



g(cγ,d ) −

γ∈Jd

g(t)dt = Ω

 1  n−1 g(cγ,d) − d

γ∈Jd

+ Ωd

g(t)dt −



# n−1 j=1

 γj γj + 1 g(t)dt , d d

g(t)dt. Ω

Since g is a bounded continuous function on the compact set Ω, we have, by Lebesgue’s theorem, limd→+∞ Ωd g(t)dt = Ω g(t)dt. On the other hand, by continuity of g on the compact set Ω, we see that    1   n−1 g(cγ,d ) − # γj γj + 1 g(t)dt n−1 d , γ∈Jd j=1 d d [g(c = γ,d ) − g(t)] dt # n−1 γj γj + 1 , γ∈Jd j=1 d d also tends to 0 as d → +∞. This shows that 1 lim g(cγ,d ) = g(t)dt. d→+∞ dn−1 Ω

γ∈Jd

The above result enables us to estimate sd when d = |α| → +∞. Lemma 4.5. If p ≥ 1, then |k|

sd ≈ dn−1 dp(2 m −1) .

Proof. Recall that sd = α∈B λpα . By Lemma 3.3, we know that the sequence d {λα }α∈Bd has the following expansion when d → +∞

λα =

2 m

2 |k| m

2

(d + n)

|k| m −1

α1 + 1 αn + 1 ,··· , fn + ε(α) , d+n d+n

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where ε(α) = O

1 d

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and tk |k|2 k t + kj2 m tj j=1 n

fn (t1 , · · · , tn ) := −(m − 2)

Using the properties of the function x → xp , we see that there exists a constant M > 0, such that p p fn α1 + 1 , · · · , αn + 1 + ε(α) − fn α1 + 1 , · · · , αn + 1 d+n d+n d+n d+n M . d Therefore, ≤

λα ≈

2 m

2 |k| m

|k| α1 + 1 αn + 1 ,··· , (d + n)2 m −1 fn , d+n d+n

as d = |α| → +∞. Applying Lemmas 4.3 we see that

2p |k| p m α1 + 1 αn + 1 2 p(2 |k| −1 ) fn , · · · , d m sd ≈ m d+n d+n

≈

2 m

2p |k| m

α∈Bd

|k|

dn−1 dp(2 m −1)

Ω

|fn (t)|p dt

so that the lemma follows from Lemma 4.4.

We recall that an operator T is in the Schatten class Sp (A2 (µm ), L2 (µm )) p if (T T ) 2 is in the trace class of A2 (µm ). Our second result characterizes such a class of operators. ∗

Theorem 4.6. Let k ∈ Nn0 and m be a positive real number. Then the Hankel operator Hzk is in the Schatten class Sp (A2 (µm ), L2 (µm )) if and only if p > 2n and m(p − 2n) > 2p|k|. Proof. We use that the operator Hzk is in Sp (A2 (µm ), L2 (µm )) if and only if T = Hzk ∗ Hzk is S p2 (A2 (µm )). Therefore, the theorem follows from Lemma 4.5

5. Boundedness of Hankel operators on the Fock space We ﬁrst study the behavior of the Bergman kernel Km (z, w) corresponding to A2 (µm ). Let E m2 , 2n be the generalized Mittag-Leﬄer’s function. This is the entire m function deﬁned by E

2 2n m, m

(λ) :=

+∞

λd +

Γ( 2d m d=0

2n m)

, λ ∈ C.

We shall express the Bergman kernel in terms of this function. Namely,

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Lemma 5.1. The Bergman kernel Km (z, w) of A2 (µm ) is given by m (n−1) E 2 2n (z, w ) , Km (z, w) = (n − 1)! m , m (n−1)

where E 2 , 2n is the derivatives of E m2 , 2n with order n − 1. m m

m

Proof. The monomials z α , α ∈ Nn0 , form an orthogonal basis of A2 (µm ). Since

2|α| + 2n α! (n − 1)! α 2 z L2 (µm ) = Γ m (|α| + n − 1)! m it follows that the Bergman kernel is zα wα Km (z, w) = z α L2 (µm ) wα L2 (µm ) n α∈N0

+∞

(d + n − 1)! m d (z, w ) 2d 2n (n − 1)! d!Γ( + ) m m d=0 m (n−1) E 2 2n (z, w ) . = (n − 1)! m , m =

This completes the proof of the lemma. The Bergman projection Pm is given by Pm (f )(z) := Cn Km (z, w)f (w)dµm (w), for f ∈ L2 (µm ).

(5.1)

This deﬁnition can be extended to functions of the form f g where f ∈ L2 (µm ) and g ∈ P. Indeed, Lemma 5.2. If g ∈ P and z ∈ Cn , then gKm (z, ·) is in L2 (µm ). Proof. It follows from Theorem 2, p. 6 in [9] that the generalized Mittag-Leﬄer’s function is E m2 , 2n is an entire function of ﬁnite order m 2 and type 1. Therefore m (n−1)

E 2 , 2n is also an entire function of ﬁnite order m 2 and type 1 and hence for any m m

> 0, there is a positive constant C that m+ (n−1) 2 , λ ∈ C. E 2 , 2n (λ) ≤ Ce|λ| m

m

This shows that for all z, w ∈ Cn , |Km (z, w)| ≤ Ce|z,w|

m+ 2

≤ Ce(|z||w|)

m+ 2

,

showing that for all g ∈ P and z ﬁxed in C , the function w → g(w)Km (z, w) is in L2 (µm ) as long as 0 < < m. n

It follows from Lemma 5.2 that if f ∈ L2 (µm ), then the Hankel operator Hf¯ with symbol f¯ is well-deﬁned on P by Hf (g)(z) := (f (z) − f (w)) Km (z, w)g(w)dµm (w), g ∈ P. Cn

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We point out that the measurable function z → Hf¯(g)(z) is not necessarly an element of L2 (µm ). Denote by M the subspace of those functions f ∈ A2 (µm ) such that Hf¯(g) ∈ 2 L (µm ) for all g ∈ P, and the densely deﬁned operator Hf¯ is bounded on A2 (µm ). We equip M with seminorm f := Hf¯ + |f (0)|. The subspace of M consisting of functions f such that Hf¯ is a compact operator will be denoted by M∞ . Then is not hard to see that M∞ is a closed subspace of M. If p ≥ 1, we denote by Mp the subspace of those functions f ∈ M such that the Hankel operator Hf¯ is the Schatten class Sp (A2 (µm ), L2 (µm )). We equip Mp with seminorm f := Hf¯Sp + |f (0)|. Lemma 5.3. The spaces M and Mp are Banach spaces. Proof. We prove the lemma for M, the proof for Mp is similar. Let (fn )n∈N0 be a Cauchy sequence in M. Without loss of generality we may assume that fn (0) = 0 for all n. The sequence (Hf n )n∈N0 is a Cauchy sequence of bounded operators on A2 (µm ). Therefore, there is an operator T in A2 (µm ) such that (Hf )n∈N0 n

converges to T in the norm operator. Let f := T (1) be the conjugate of the image T (1) of the constant function 1 under T . Since Hf n (1) = f n , it follows that fn − f L2 (µm ) = f¯n − T (1)L2(µm ) = Hf n (1) − T (1)L2(µm ) ≤ Hf n − T showing that lim fn − f L2 (µm ) = 0.

n→∞

(5.2)

Thus f ∈ A2 (µm ). We shall show that the Hankel operator Hf with symbol f is bounded. It is well deﬁned on P. We shall prove that Hf is equal to T on P. Let g be a holomorphic polynomial. We ﬁrst observe by (5.1), (5.2) and Lemma 5.2 that for all z ∈ Cn we have P (f − fn )g (z) ≤ fn − f L2 (µ ) gKm (z, ·)L2 (µ ) m m showing that limn→+∞ P (f − fn )g (z) = 0. Since again by (5.2) we have that limn→+∞ f − fn g(z) = 0, it follows that lim Hfn − Hf (g)(z) = 0. n→+∞

This proves that T g = Hf (g) and hence T = Hf . Therefore M is a Banach space. The proof of that Mp is a Banach space is similar.

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For θ = (θ1 , · · · , θn ) ∈ Rn , let Rθ be the unitary linear transformation in Cn deﬁned by Rθ (z) = (eiθ1 z1 , · · · , eiθn zn ), for all z = (z1 , · · · , zn ) ∈ Cn . Lemma 5.4. Let θ ∈ Rn . Then the operator Rθ f := f ◦ Rθ is a unitary isometry from L2 (µm ) onto itself and from A2 (µm ) onto itself. Moreover the following assertions hold. 1) If f ∈ M, then Rθ f ∈ M and Rθ f M = f M . 2) If f ∈ M∞ , then Rθ f ∈ M∞ . 3) If f ∈ Mp , then Rθ f ∈ Mp and Rθ f Mp = f Mp . Proof. It is clear that the operator Rθ is a unitary isometry from L2 (µm ) onto itself and from A2 (µm ) onto itself. Let f be in M and θ ∈ Rn . Then Rθ f is clearly in A2 (µm ). Moveover, if g is an element of P, then by a change of variable we see that & ' HRθ f (g)(z) = Km (Rθ z, w)g(R−θ w) Rθ f (z) − f (w) dµm (w) n C & ' = Km (Rθ z, w)(R−θ g)(w) f (Rθ z) − f (w) dµm (w) Cn

= Hf (R−θ g)(Rθ z) = (Rθ Hf R−θ )(g)(z). ∗ Since the adjoint of R−θ is R−θ = Rθ , it follows that

HRθ f = Hf , showing that f ◦ Rθ M = f M . This proves part (1) of the lemma. The proof of parts (2) and (3) of the lemma are similar. Lemma 5.5. Let f ∈ A2 (µm ). k

1) If f ∈ M, then for any multi-index k that satisfies ∂∂zfk (0) = 0, the monomial z k is in M. k 2) If f ∈ M∞ , then for any multi-index k that satisfies ∂∂zkf (0) = 0, the monomial z k is in M∞ . k 3) If p ≥ 1 and f ∈ Mp , then for any multi-index k that satisfies ∂∂zfk (0) = 0, the monomial z k is in Mp . Proof. To prove (1), suppose that f ∈ M. By the Cauchy formula we have 2π 2π f (Rθ z) ∂kf 1 k (0)z = ··· dθ, k n ik θ 1 ∂z (2π) 0 e 1 · · · eikn θn 0 where dθ := dθ1 · · · dθn for θ = (θ1 , · · · , θn ). By Lemmas 5.3 and 5.4 we see k k that ∂∂zkf (0)z k ∈ M. Therefore, z k ∈ M as long as ∂∂zfk (0) = 0. The proof of the remaining statements of the lemma is similar.

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Proof of Theorems A and B. We ﬁrst prove Theorem A. Let f ∈ A2 (µm ). Suppose k that Hf is bounded and let k be a multi-index that satisﬁes ∂∂zkf (0) = 0. By Lemma 5.5 we see that the monomial z k is in M. Now Theorem 4.2 implies that m ≥ 2|k|. Hence f is a polynomial of degree at most m 2. If Hf is compact then a similar argument shows that f is a polynomial of degree strictly smaller than m 2 . The converse follows from Theorem 4.2. The proof of Theorem B is similar to that of Theorem A.

References [1] W. Bauer, Mean oscillation and Hankel operators on the Segal-Bargmann space Integral Equations and Operator Theory 52 (2005), 1–15. [2] W. Bauer, Hilbert-Schmidt Hankel operators on the Segal-Bargmann space Proc. Amer. Math. Soc 132 (2005), 2989–2996. [3] C. A. Berger and L. Coburn Toeplitz operators on the Segal-Bargmann space Trans. Amer. math. Soc. 301 (1987), 813–829 [4] C. A. Berger, L. Coburn and K. Zhu Toeplitz operators and function theory in ndimensions, Lecture Notes in Mathematics 1256, Springer, 1987. [5] F. Haslinger, The canonical solution operator to ∂ restricted to Bergman spaces and spaces of entire functions Annales de la Facult´e des Sci. Toulouse Math. 11 (2002), 57–70. [6] F. Haslinger, Schr¨ odinger operators with magnetic fields and the canonical solution operator to ∂ J. Math. Kyoto Univ.46 (2006), 249–257. [7] F. Haslinger and B. Helfer, Compactness of the solution operator to ∂ in weighted L2 -spaces ESRI-preprint ( 2006). [8] W. Knirsch and G. Schneider, Continuity and Schatten-von Neumann p-class membership of Hankel operators with antiholomorphic symbols on (generalized) Fock spaces J. Math. Anal. Appl. 320 (2006), 403–414. [9] B. Y. Levin, Lectures on entire functions Translations of Math. Monographs 150, American Math. Soc., Providence, Rhode Island, 1996. [10] S. Lovera and E. H. Youssﬁ, Spectral properties of the ∂-canonical solution operator Journal of Functional Analysis 208 (2004), 360–376. [11] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathemetical Physics, Springer Verlag, 1966. [12] D. J. Newman and H. S Shapiro, Fischer spaces of entire functions Proc. Symp. Pure Math. II (1968), 360–369. [13] G. Schneider, Hankel operators with antiholomorphic symbols on the Fock space Proc. Amer. Math. Soc. 132 (2004), 2399–2409. [14] K. Stroethoﬀ, Hankel operators in the Fock space Michigan Math. J. 39 (1992), 3–16. [15] Jingbo Xi and Dechao Zheng, Standard deviation and Schatten class Hankel operators on the Segal-Bargmann space Indiana University Math. J. 53 (2004), 1381–1399. [16] K. H. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball Amer. J. Math. 113 (1991), 147–167.

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[17] K. H. Zhu, Hilbert-Schmidt Hankel operators on the Bergman space Proc. Amer. Math. Soc. 109 (1990), 721–730 H´el`ene Bommier-Hato and El Hassan Youssﬁ LATP, U.M.R. C.N.R.S. 6632 CMI, Universit´e de Provence 39, Rue F. Joliot-Curie F-13453 Marseille Cedex 13 France e-mail: [email protected] [email protected] Submitted: July 10, 2006 Revised: January 31, 2007

Integr. equ. oper. theory 59 (2007), 19–34 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010019-16, published online June 27, 2007 DOI 10.1007/s00020-007-1511-3

Integral Equations and Operator Theory

Quadratic Inequalities for Hilbert Space Operators V. A. Khatskevich, M. I. Ostrovskii and V. S. Shulman Abstract. Properties of sets of solutions to inequalities of the form X ∗ AX + B ∗ X + X ∗ B + C ≤ 0 are studied, where A, B, C are bounded Hilbert space operators, A and C are self-adjoint. Properties under consideration: closeness and interior points in standard operator topologies, convexity, non-emptiness. Mathematics Subject Classification (2000). Primary 47A56; Secondary 47B50. Keywords. Hilbert space, bounded linear operator, weak operator topology, operator inequalities.

1. Introduction We shall use the terminology from [2], [5], [6], where the reader can also ﬁnd basic facts about the strong operator topology (SOT) and the weak operator topology (WOT), see [2, §8] and [5, Chapter VI]. Let H1 and H2 be separable Hilbert spaces, B ∈ L(H1 , H2 ), A ∈ L(H2 ), A = A∗ , C ∈ L(H1 ), C = C ∗ (where L(H1 , H2 ), L(H1 ), and L(H2 ) are the corresponding spaces of bounded linear operators). There is a simple, popular and useful theory of linear operator equations of the form AX = B or XC = B which describes the properties of the sets of solutions X ∈ L(H1 , H2 ) and, in particular, the conditions for solvability (existence of such X). On the other hand, many problems from diﬀerent areas (extensions of operators [16], indeﬁnite metric spaces [12], linear fractional relations [10], operator The second named author was supported by St. John’s University Summer 2006 Support of Research Program. The authors thank the referee for helping them to make this paper more readable and error-free.

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functional equations [3], control theory and systems theory [4], [8], [13], [14], [15], and others) lead to quadratic equations and inequalities of the forms: X ∗ AX = C,

(1)

X ∗ AX ≤ C,

(2)

X ∗ AX + B ∗ X + X ∗ B + C = 0,

(3)

and, more generally,

X ∗ AX + B ∗ X + X ∗ B + C ≤ 0.

(4) ∗

∗

The equation (3) is very close to the equation XAX + B X + X B + C = 0 with self-adjoint A and C, which is usually called a continuous algebraic Riccati equation. It is clear that a self-adjoint solution of (3) is also a solution of the corresponding continuous algebraic Riccati equation. Since self-adjoint solutions are the most important in systems theory, our study can be of interest for inﬁnitedimensional systems theory. In control theory and systems theory there are many results stating that under certain assumptions the continuous algebraic Riccati equation has a non-negative solution (see, e.g. [15, Theorem 2.2.1], [4, Theorem 6.2.7]). Many other results on continuous algebraic Riccati equations can be found in [4], [8], and [14]. Another object studied in systems theory is an inequality of the form XAX + B ∗ X + X ∗ B + C ≤ 0 (A ≥ 0), called a Riccati inequality, see [7], [14, Section 9.1], [17]. The mentioned sources contain results on conditions of solvability of Riccati inequality and comparison of the sets of self-adjoint solutions of the continuous algebraic Riccati equation and the corresponding Riccati inequality. Scalar versions of the inequality (4) for an inﬁnite dimensional Hilbert space were studied in [19]. This survey of the literature shows that information about properties of sets M (A, B, C) can be of interest from diﬀerent points of view. Our interest in this topic was motivated by the fact that such sets appear in the study of linear fractional relations (LFR), and we consider the present paper as a continuation of [9], in which we started systematic study of properties of LFR. Namely, in terms of M (A, B, C) one can express the domains of LFR and the images of balls under LFR. The applications of the results of the present paper to LFR will be published in a separate paper. Relatively few results are known on topological and geometric properties of sets of solutions of the inequalities (2) and (4). We will denote by E(A, C), N (A, C) and M (A, B, C) the sets of all solutions for (1), (2) and (4), respectively. It is known, that M (A, B, C) = L(H1 , H2 ) if and only if A, C are non-positive and B = (−A)1/2 T (−C)1/2 for some operator T ∈ L(H1 , H2 ) with T ≤ 1. If the operators A, C are invertible, this condition is equivalent to C − B ∗ A−1 B ≤ 0. It is the so-called generalized Sylvester criterion (see [1, p. 321], [11, pp. 732– 733], and [18, §1]). If A is invertible and positive, then the sets E(A, C) and N (A, C) are non-void only for C ≥ 0 and in this case E(A, C) = A−1/2 UC 1/2 , N (A, C) = A−1/2 RC 1/2 where U is the set of all isometric operators from CH1

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into H2 , and R is the unit ball of L(CH1 , H2 ). Hence N (A, C) is convex and WOTcompact. If A is non-negative but not invertible, then N (A, C) is still convex (see, for example, [10]), and hence WOT-closed, because it is always SOT-closed. The analysis of M (A, B, C), and even of N (A, C) for an arbitrary self-adjoint operator A, is much more complicated. The purpose of this paper is to ﬁnd conditions under which the sets N (A, C) and M (A, B, C) are non-empty, closed, or convex, and to characterize their interior points. In Section 3 we consider conditions under which M (A, B, C) and N (A, C) are closed and sequentially closed in the weak operator topology. They turned out to be diﬀerent. Removing trivial exceptions one can say that M (A, B, C) is sequentially closed in WOT if and only if A is essentially non-negative (Theorem 1). For N (A, C) to be closed in WOT the condition rank(A− ) < ∞ is necessary and suﬃcient (Theorem 2), where A− is the negative part of A, see the deﬁnition below. The condition is also suﬃcient for WOT-closeness of M (A, B, C), but we do not know if it is necessary in this case. In Section 4 we show (Theorem 3 and Corollary 4) that N (A, C) is convex if and only if A ≥ 0. For M (A, B, C) the same is proved under the condition that the spectrum of A does not intersect some interval (−ε, 0), ε > 0. In Section 5 of the paper we obtain some conditions under which the sets M (A, B, C) and N (A, C) are non-empty. We give only a partial answer for sets M (A, B, C) and a complete answer for N (A, C) (see Theorem 4). Note on separability assumption. We consider only separable Hilbert spaces. In the general case some of our results (for example, Theorem 4 and Lemma 2) must be stated diﬀerently, while some others (e.g., Lemma 4) remain valid without modiﬁcations in formulation and proof.

2. Notation Throughout the paper inner products will be denoted by ·, ·. We consider G(X) = X ∗ AX + B ∗ X + X ∗ B + C

(5)

as a function of X ∈ L(H1 , H2 ). So M (A, B, C) = {X : G(X) ≤ 0} (and N (A, C) = M (A, 0, −C)). Let E C be the spectral measure of C. Set H1+ = E C (0, ∞)H1 , H10 = E ({0})H1 , H1− = E C (−∞, 0)H1 , then C

H1 = H1+ ⊕ H10 ⊕ H1− .

(6)

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Let H2 = H2+ ⊕ H20 ⊕ H2−

(7)

be the similar decomposition of H2 , corresponding to A. Let C+ , C0 and C− be the restrictions of C to the subspaces H1+ , H10 , and H1− ; A+ , A0 , and A− be the restrictions of A to the subspaces H2+ , H20 , and H2− . If H1− = 0 (H1+ = 0), we say that C is non-negative (non-positive). We say that C is essentially non-negative if C− is compact (this is equivalent to the condition that C is a sum of a non-negative operator and a compact operator). Similar terminology is used for A. Let P+ , P0 , and P− be the orthogonal projections on H2 corresponding to the decomposition (7). If X is an operator with range in H2 , then by X+ , X0 , X− , and X∼ we denote the operators P+ X, P0 X, P− X, and (P+ + P0 )X, respectively. Some of the results of this paper are proved for sets of the form N (A, C) rather than for M (A, B, C). The following observation shows that in the case, when B can be factored through A, such results have immediate consequences for sets of the form M (A, B, C). We leave exact statements of such consequences as exercises for interested readers. Observation. If B = AD for some D ∈ L(H1 , H2 ), then G(X) = X ∗ AX + D∗ AX + X ∗ AD + C = (X + D)∗ A(X + D) + (C − D∗ AD), and hence M (A, B, C) = N (A, D∗ AD − C) − D. In particular, if A is invertible, then M (A, B, C) = N (A, B ∗ A−1 B − C) − A−1 B.

3. Topological Properties It should be mentioned that the set M (A, B, C) (hence N (A, C) and E(A, C) = N (A, C) ∩ N (−A, −C) as well) is always closed in the strong operator topology (SOT). In fact, let {Xα }α ⊂ M (A, B, C) be a strongly convergent net, Xα → X. Then, for all x ∈ H1 , 0 ≥ G(Xα )x, x = AXα x, Xα x+Xα x, Bx+Bx, Xα x+ Cx, x → G(X)x, x, hence X ∈ M (A, B, C).

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3.1. Weak operator topology, sequential properties Theorem 1. The set M = M (A, B, C) is sequentially closed in WOT if and only if one of the following conditions is satisfied. (a) A is essentially non-negative; (b) A, C are non-positive and B = (−A)1/2 T (−C)1/2 for some operator T ∈ L(H1 , H2 ) with T ≤ 1. In the case (b) the set M (A, B, C) coincides with L(H1 , H2 ). The theorem immediately follows from the generalized Sylvester criterion (see the introduction) and the following two lemmas. Lemma 1. If A− is compact, then M (A, B, C) is sequentially closed in WOT. Proof. As is well-known (and easy to check), the map X → X ∗ A+ Xx, x (x ∈ H1 ) is lower semi-continuous in the WOT, in the sense that lim infXα∗ A+ Xα x, x ≥ X0∗ A+ X0 x, x as Xα → X0 in the WOT. The map X → (B ∗ X + X ∗ B)x+ Cx, x is clearly WOT-continuous. Hence it suﬃces to prove that if K is compact, then the map X → X ∗ KX is sequentially continuous in WOT. Let {Xn }∞ n=1 be a sequence converging to X in the WOT, then, by compactness of K, the sequence ∗ {KXn }∞ n=1 is convergent to KX in the SOT. Hence Xn (KXn − KX) → 0 in ∗ ∗ the SOT (because {Xn } is bounded) and (Xn − X) KX → 0 in the WOT, so Xn∗ KXn − X ∗ KX → 0 in the WOT. Lemma 2. If A− is non-compact and M (A, B, C) = L(H1 , H2 ), then M (A, B, C) is not sequentially closed in WOT. Proof. Let X ∈ / M (A, B, C). It suﬃces to ﬁnd a sequence {Xn } ∈ L(H1 , H2 ) such that Xn WOT-converges to X and Xn ∈ M (A, B, C). Since A− is non-compact, then there exists a subspace N ⊂ H2− and ε > 0 such that N is inﬁnite dimensional and Ax, x ≤ −ε||x||2 for each x ∈ N . Let Un : H1 → N be a sequence of isometries which WOT-converges to 0 (we can claim the existence of such isometries because H1 is assumed to be separable). We let Xn = X + kUn , the real number k ≥ 1 will be speciﬁed later. It is clear that Xn WOT-converges to X. It remains to show that we can select k in such a way that Xn ∈ M (A, B, C) for each n ∈ N. We have G(Xn ) = k 2 Un∗ AUn + Yn , where Yn ≤ ck for some c > 0 which does not depend on k or n. Therefore G(Xn )x, x ≤ −k 2 εx2 + ck||x||2 . Hence, if k is chosen large enough, Xn ∈ M (A, B, C) for each n ∈ N. Remark. Our proof of Theorem 1 implies that non-weakly-sequentially-closed sets of the form M (A, B, C) are weakly sequentially dense in L(H1 , H2 ).

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3.2. Weak operator topology Theorem 2. (a) If A− has finite rank, then M = M (A, B, C) is WOT-closed; (b) N (A, C) is WOT-closed if and only if either it is empty, or it coincides with L(H1 , H2 ), or A− has finite rank. Combining Theorem 2 with the observation from Section 2 we get Corollary 1. If B = AD for some D ∈ L(H1 , H2 ), then M (A, B, C) is WOTclosed if and only if either it is empty, or it coincides with L(H1 , H2 ), or A− has finite rank. Proof of (a). If we analyze our proof of Lemma 1, we see that it is enough to prove that if K has ﬁnite rank, then the map X → X ∗ KX is WOT-continuous. Decomposing K into a sum of rank-one operators we see that X ∗ KXx, y is the sum of functions of the form X → Xx, f · e, Xy. Since such functions are WOT-continuous, we are done. To prove part (b) we need several auxiliary results, which can be of independent interest. Lemma 3. If A is a positive operator of infinite rank and C is a self-adjoint operator of finite rank, then the WOT-closure of the set L of all operators X satisfying X ∗ AX ≥ C

(8)

contains 0. Proof. Indeed, if 0 is not in the WOT-closure of L, then there is a WOT-neighborhood U = {X : |Xxi , yi | ≤ ε, i = 1, . . . , n} of 0, which does not intersect L. Let Y be the linear span of {yi }ni=1 . Since E A (0, ∞)(H2 ) = A(H2 ), we have dim E A [λ, ∞)(H2 ) = ∞ λ>0

(we use the notation introduced in Section 2). Hence there exists λ > 0 such that the dimension of the subspace E A [λ, ∞)(H2 ) exceeds (n + rankC). It follows that dim(E A [λ, ∞)(H2 ) ∩ Y ⊥ ) ≥ rankC. Since C is a self-adjoint operator of ﬁnite rank, it is representable in the k k k form Cx = i=1 βi x, zi zi , where {βi }i=1 are real numbers and {zi }i=1 is an orthonormal basis in Z = CH1 . We introduce X ∈ L(H1 , H2 ) as an operator satisfying X|Z ⊥ = 0 and Xzi = ui , where {ui }ki=1 is an orthonormal set in E A [λ, ∞)(H2 ) ∩ Y ⊥ . Then, for each z ∈ Z we have ||Xz|| = ||z|| and Xz ∈ E A [λ, ∞)(H2 ), therefore X ∗ AXz, z = AXz, Xz ≥ λ||Xz||2 = λ||z||2 .

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Now let x ∈ H1 , then x = z + v, where z ∈ Z and v ∈ Z ⊥ . We have Cx, x =

k i=1

βi |x, zi |2 ≤ max |βi |||z||2 i

and X ∗ AXx, x = AXx, Xx = AXz, Xz ≥ λ||z||2 ≥ where

1 Cx, x, t2

1 λ = t2 maxi |βi | (of course, we may assume that maxi |βi | > 0). Thus t2 X ∗ AX ≥ C. On the other hand, the deﬁnition of X implies that X(H1 ) ⊂ Y ⊥ , therefore tX ∈ U . This is a contradiction.

Lemma 4. Let P be a finite dimensional projection on a Hilbert space H. Let S ∈ L(H) be a non-negative operator with an infinite dimensional range. Then, for each ε > 0, there exists an operator E ∈ L(H), such that ||P EP || ≤ ε and E ∗ SE ≥ S. Proof. First we suppose that S is injective. For λ > 0 let Cλ be the spectral measure of the interval [λ, ∞) (see [2, 9.1 and 10.2] or [6, Section X.1]). Since S is injective and non-negative, then the limit limλ→0 Cλ in the strong operator topology is the identity operator (see [6, Corollary X.2.3 and Lemma X.3.3]). Therefore there exists λ = λ(ε) > 0 such that the orthogonal projection Cλ ∈ L(H) satisﬁes SCλ = Cλ S, √ ||(I − Cλ )P || ≤ ε, √ ||(I − Cλ )P ∗ || ≤ ε, (9) dim(im(Cλ )) ≥ 2 dim(im(P )). Also, the spectral measure Cλ satisﬁes Sx, x ≥ λ||x||2 ∀x ∈ im(Cλ ).

(10)

The condition (9) implies that there exists an orthogonal (unitary) operator U on im(Cλ ) such that im(U Cλ P ∗ ) is orthogonal to im(Cλ P ). Let E ∈ L(H) be deﬁned by Ex = αU Cλ x + (I − Cλ )x, where α > 0 is chosen in such a way that E ∗ SE ≥ S. To see that such α exists, observe that E ∗ SEx, x = SEx, Ex = αSU Cλ x + S(I − Cλ )x, αU Cλ x + (I − Cλ )x = since S commutes with Cλ α2 SU Cλ x, U Cλ x + S(I − Cλ )x, (I − Cλ )x. On the other hand, Sx, x = SCλ , Cλ x + S(I − Cλ )x, (I − Cλ )x.

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Hence the only condition which has to be satisﬁed by α is α2 SU Cλ x, U Cλ x ≥ SCλ x, Cλ x. This condition is satisﬁed by suﬃciently large α because of (10). To show that ||P EP || ≤ ε observe that ||P EP || = sup{P EP x, y : x, y ∈ H, ||x|| = ||y|| = 1}. Let ||x|| = ||y|| = 1. We have P EP x, y = EP x, P ∗ y = Cλ EP x, Cλ P ∗ y + (I − Cλ )EP x, (I − Cλ )P ∗ y = αU Cλ P x, Cλ P ∗ y + (I − Cλ )P x, (I − Cλ )P ∗ y = 0 + (I − Cλ )P x, (I − Cλ )P ∗ y ≤ ||(I − Cλ )P || · ||(I − Cλ )P ∗ || ≤ ε. Now suppose that S has a kernel L. Decompose H into an orthogonal direct sum H = H1 ⊕L and let Q be the orthogonal projection onto H1 . Denote by S1 the restriction of S to H1 . Let Y = lin(QP H ∪ QP ∗ H) and let P1 be the orthogonal projection in H1 onto Y . Applying the ﬁrst part of the proof to the operator S1 and the projection P1 we ﬁnd an operator E1 ∈ L(H1 ) with ||P1 E1 P1 || ≤ ε/(||P ||2 ) and E1∗ S1 E1 ≥ S1 . Let E = E1 ⊕ 0, so E = QEQ. Then E ∗ SE ≥ S. It remains to show that ||P EP || ≤ ε. We have ||P EP || = =

sup

P EP x, y =

||x||=||y||=1

sup

sup

QEQP x, P ∗ y =

||x||=||y||=1

≤

sup

E1 x, y ≤ ||P ||2

||x||,||y||≤||P || x,y∈Y

EP x, P ∗ y

||x||=||y||=1

sup

EQP x, QP ∗ y

||x||=||y||=1

sup

E1 P1 x, P1 y

||x||=||y||=1 x,y∈H1

= ||P ||2 ||P1 E1 P1 || ≤ ε, where we used the fact that Q and P1 are self-adjoint.

Lemma 5. If S is a positive operator of infinite rank on a Hilbert space H, then there is a net Eλ of operators such that Eλ → 0 in WOT and Eλ∗ SEλ ≥ S for all λ. Proof. We take as an index set the set of all pairs λ = (P, ε) where P is a ﬁnite dimensional orthogonal projection on H, ε > 0. Set (P1 , ε1 ) ≺ (P2 , ε2 ) if P1 ≤ P2 and ε1 > ε2 . By Lemma 4, for each λ there exists Eλ ∈ L(H), such that ||P Eλ P || ≤ ε and Eλ∗ SEλ ≥ S. Each λ deﬁnes a WOT-neighborhood of 0 by Uλ = {X ∈ L(H) : P XP < ε}, and it is clear that we obtain in this way a basis of WOT-neighborhoods of 0. It remains to note that for each λ0 , all operators Eλ with λ0 < λ belong to Uλ0 .

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Proposition 1. If A is a positive operator of infinite rank and C is a self-adjoint operator, then the WOT-closure of the set L of all operators X ∈ L(H1 , H2 ), for which X ∗ AX ≥ C, (11) either is empty or contains 0. Proof. Let C = C+ + C− be the decomposition of C described in Section 2, and let R+ be the orthogonal projection onto H1+ . Let us consider also the condition Y ∗ AY ≥ C+ ,

(12)

If X satisﬁes (11) then Y = XR+ satisﬁes (12). On the other hand any solution of (12) satisﬁes also (11). Therefore it is enough to consider the case C ≥ 0. Due to Lemma 3 we may suppose that C has inﬁnite rank. Let X satisfy (11). Then the operator S = X ∗ AX has inﬁnite rank. Applying Lemma 5 to it, we ﬁnd a WOT-convergent to 0 net {Eλ }λ such that Eλ∗ SEλ ≥ S. It follows that XEλ satisﬁes (11). Since the net {Eλ }λ is WOT-convergent to 0, we are done. Proof of Theorem 2(b). We have to establish that if N = N (A, C) is WOT-closed, then either rankA− < ∞, or N = ∅ or N = L(H1 , H2 ). (The last possibility corresponds to the case A+ = 0, C ≥ 0). We use the decomposition (7) from Section 2 and the notation introduced there. The inequality X ∗ AX ≤ C can be rewritten as (X+ )∗ A+ X+ + (X− )∗ A− X− ≤ C.

(13)

By Lemma 2, it is enough to show that the assumptions • A− is a compact operator of inﬁnite rank; • N = ∅; imply that N = L(H1 , H2 ). It is clear that X ∈ N implies X− ∈ N . Since N is WOT-closed, then the set L = {X− : X ∈ N } is also WOT-closed. Observe that each Z ∈ L satisﬁes Z ∗ (−A− )Z ≥ (−C). By Proposition 1, the set L contains 0. Hence 0 ≥ −C or C ≥ 0. If A+ = 0, then, clearly, N = L(H1 , H2 ). So in what follows we assume that A+ = 0. Let us show that X+ ∈ N for X ∈ N , that is, ∗ A+ X+ − C ≤ 0. X+

Since N is WOT-closed, it is easy to check that the set of all operators Z ∈ L(H1 , H2 ) satisfying ∗ A+ X+ − C ≤ Z ∗ (−A− )Z X+

is also WOT-closed. It is also non-empty (it contains X− ). By Proposition 1 it contains 0. Hence X+ ∈ N .

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Let {en } be a (ﬁnite or inﬁnite) basis in H1 . Let {fn }∞ n=0 be a basis in H2− consisting of eigenvectors for −A− . We denote the corresponding eigenvalues by an (so an > 0). Choose y ∈ H2+ with A+ y, y = λ > C, and set Y = e1 ⊗y (this means that Y x = e1 , xy for all x ∈ H1 ). Then Y ∗ A+ Y − C = λe1 ⊗e1 − C is not ≤ 0, therefore Y ∈ / N. Let t be a number with t2 > λ/a1 . Deﬁne Z ∈ L(H1 , H2− ) by Ze1 = tf1 , Zei = fi for 2 ≤ i ≤ dim(H1 ). Then Z ∗ (A− )Ze1 = −t2 a1 e1 , Z ∗ (A− )Zei = −ai ei for i ≥ 2. Therefore Z ∗ A− Z + Y ∗ A+ Y ≤ 0. Since C ≤ 0, then Z + Y ∈ N . By the observation above Y ∈ N . We get a contradiction. Problem 1. Whether the set M (A, B, C) is WOT-closed if and only if it is either empty, or coincides with L(H1 , H2 ), or A− has ﬁnite rank? Another natural problem about the topological properties of M (A, B, C) is the characterization of its inner points. Since a point of M (A, B, C) can be “shifted” by changing the coeﬃcients in (4), it suﬃces to ﬁnd conditions under which 0 is an interior point. For r > 0, denote by K(r) the cone of all x⊕y ∈ H1 ⊕H2 such that x ≤ ry. Proposition 2. The point 0 is interior for M (A, B, C) if and only if the matrix A B B∗ C is non-positive on some cone K(r). Proof. Denote the matrix

A B B∗ C by T . It is easy to check that G(X)y, y = T (Xy⊕y), Xy⊕y. If X ≤ r, then Xy⊕y ∈ K(r). Hence if T is non-positive on K(r), then G(X)y, y ≤ 0 for each y, so M (A, B, C) contains the ball of radius r centered at 0. Conversely, if M (A, B, C) contains the ball of radius r, then T is non-positive on K(r) because each vector in K(r) can be written in the form Xy⊕y for some y and X with X ≤ r. We write C < 0 if Cy, y < 0 for all y = 0. We write C 0 if there is ε > 0 with C + εIH1 ≤ 0. Corollary 2. Suppose that M (A, B, C) = L(H1 , H2 ). (i) If C 0, then 0 is an interior point of M (A, B, C).

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(ii) If 0 is an interior point of M (A, B, C), then C < 0. Proof. Since C = G(0), part (i) follows from the continuity of G in norm topology. Suppose that 0 is an interior point for M (A, B, C). By Proposition 2 there is r > 0 such that (14) t2 Ax, x + 2tRex, By + Cy, y ≤ 0 for all x, y with x = y = 1 and |t| ≤ r. Letting t = 0, we get C ≤ 0. Hence, if there is a unit vector v such that Cv, v = 0, then Cv = 0. Setting y = v in (14), we obtain that Bv = 0 and A ≤ 0. Let now y = λv + µz, where z is an arbitrary unit vector, and µ, λ are such that y = 1. We get from (14): t2 Ax, x + 2tµRex, Bz + µ2 Cz, z ≤ 0.

(15)

This holds for all t ∈ (−r, r) and all suﬃciently small µ. Setting s = t/µ we see that s2 Ax, x + 2sRex, Bz + Cz, z ≤ 0 (16) for all s ∈ (−∞, ∞). Hence |x, Bz|2 ≤ −Ax, x−Cz, z.

(17)

Since x, z are arbitrary unit vectors, this is a contradiction because (17), as is easy to see, is equivalent to M (A, B, C) = L(H1 , H2 ). Corollary 3. Suppose that the space H1 is finite dimensional and M = L(H1 , H2 ). A point X0 is in the interior of M (A, B, C) if and only if G(X0 ) < 0. Proof. Substituting X − X0 for X, we see that it suﬃces to consider the case X0 = 0. Since for an operator in a ﬁnite dimensional space the conditions C < 0 and C 0 are equivalent, the result follows from the previous corollary.

4. Convexity We begin with an auxiliary result on operator quadratic forms with non-negative leading coeﬃcients. Lemma 6. Let G(X) be given by (5). Suppose that G(X) is non-constant, A ≥ 0, and that there is an operator X0 with G(X0 ) ≤ 0. Then there is an operator X1 such that G(X1 ) is not ≤ 0 but G(X1 ) ≤ K for some rank one operator K. Proof. Changing X by X0 + Y we may assume that X0 = 0, and hence C ≤ 0. Suppose that B = 0. Choose a positive rank-one orthogonal projection P ∈ L(H2 ) with P B = 0 and set X1 = tP B for t > 0. Since P AP = λP for some λ ≥ 0, we have G(X1 ) = C + (λt2 + 2t)B ∗ P B. For suﬃciently large t > 0, the operator G(X1 ) will not be ≤ 0. On the other hand, since C ≤ 0, then G(X1 ) ≤ (λt2 + 2t)B ∗ P B, which is of rank one. If B = 0 then A = 0 (otherwise G(X) is constant). Let P be a positive one-dimensional orthogonal projection, such that P AP = 0 (so λ > 0) and set

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X1 = tP . Then G(X1 ) = C + λt2 P and again F (X1 ) is not ≤ 0 for suﬃciently large t, but is ≤ λt2 P . Now we consider the conditions under which the set M = M (A, B, C) is convex. A well-known suﬃcient condition is A ≥ 0 (see, e.g. [10, Lemma 1.1]. It appears that under mild spectral assumptions this condition is also necessary. Theorem 3. Let M = M (A, B, C) be non-empty, convex, and not equal to the space L(H1 , H2 ). If there exists ε > 0, such that σ(A) does not contain points of an interval (−ε, 0), then A ≥ 0. Proof. Suppose the contrary, then A− is invertible. We use the notation introduced in Section 2. Let A∼ be the restriction of A to H2+ ⊕ H20 . We have X ∈ M if and only if ∗ ∗ ∗ ∗ ∗ ∗ X− A− X− + X∼ A∼ X∼ + B∼ X∼ + B− X− + X∼ B∼ + X− B− + C ≤ 0.

Let Y = (−A− )1/2 X− and D = (−A− )−1/2 B− . We get: X ∈ M if and only if

∗ ∗ ∗ −Y ∗ Y + D∗ Y + Y ∗ D + X∼ A∼ X∼ + B∼ X∼ + X∼ B∼ + C ≤ 0. Let Z = Y − D. The inequality (18) can be rewritten in the form

(18)

∗ ∗ ∗ −Z ∗ Z + X∼ A∼ X∼ + B∼ X∼ + X∼ B∼ + (C + D∗ D) ≤ 0.

Let

∗ ∗ ∗ F (X∼ ) = X∼ A∼ X∼ + B∼ X∼ + X∼ B∼ + (C + D∗ D). Then M (A, B, C) is equal to

{(−A− )−1/2 (Z + D) + X∼ : X∼ ∈ L(H1 , (H2+ ⊕ H20 )) and Z ∗ Z ≥ F (X∼ )}. If F (X∼ ) is constant (that is, if A∼ = 0 and B∼ = 0), then M = {(−A− )−1/2 (Z+D)+X∼ : X∼ ∈ L(H1 , (H2+ ⊕H20 )) and Z ∗ Z ≥ C+D∗ D}. It is easy to see that this set is convex if and only if C + D∗ D ≤ 0, and in such a case M = L(H1 , H2 ). Consider the case when F (X∼ ) is not constant. Since M is non-empty, then 0 0 there exists a pair Z 0 , X∼ , such that (−A− )−1/2 (Z 0 + D) + X∼ ∈ M . It is clear 0 that this implies (−A− )−1/2 (−Z 0 + D) + X∼ ∈ M . Hence, by convexity of M , we 0 ∈ M . Hence have (−A− )−1/2 D + X∼ 0 ) ≤ 0. F (X∼

(19)

Applying Lemma 6 to the function F (X∼ ) we get that there exists an operator X∼ such that F (X∼ ) ≤ K, where K is an operator of rank one, but F (X∼ ) is not ≤ 0. It is clear that there exists Z ∈ L(H1 , H2− ) with Z ∗ Z ≥ K. Hence (−A− )−1/2 (Z + D) + X∼ ∈ M and (−A− )−1/2 D + X∼ ∈ / M . This contradicts the observation above. Corollary 4. The set N (A, C) is convex if and only if it is either empty, or coincides with L(H1 , H2 ), or A ≥ 0.

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Proof. Since N (A, C) is SOT-closed and convex it is WOT-closed. By Theorem 2(b), A− is of ﬁnite rank. Hence the assumptions of Theorem 3 are satisﬁed. Remark. If the answer to Problem 1 is positive, then the restriction on σ(A) in Theorem 3 can be removed. Indeed, if M (A, B, C) is convex then, being SOTclosed, it is WOT-closed. So (under our assumptions) A− has ﬁnite rank and σ(A) ∩ R− is ﬁnite.

5. Non-triviality The purpose of this section is to describe the conditions under which the sets N (A, C) and M (A, B, C) are non-empty. We use the decompositions (6) and (7), and the notation introduced in Section 2. The dimension of a subspace can be a non-negative integer or ∞. (i) (ii) (iii) (iv)

Consider the following (pairwise exclusive) conditions: dim H1+ < dim H2− . dim H1+ = dim H2− < ∞. dim H1+ = dim H2− = ∞ and A− is non-compact. dim H1+ = dim H2− = ∞, both A− and C+ are compact, and ci sup < ∞, 1≤i<∞ ai ∞ where {ci }∞ i=1 are the eigenvalues of C+ in non-increasing order, and {ai }i=1 are the eigenvalues of (−A− ) in non-increasing order.

Theorem 4. N (A, −C) = ∅ if and only if one of the conditions (i)–(iv) is satisfied. Proof. Suppose N (A, −C) = ∅. Let X ∈ N (A, −C). Then (X ∗ AX + C)x, x ≤ 0 ∀x ∈ H1 , that is, AXx, Xx ≤ −Cx, x ∀x ∈ H1 . The inequality (20) can be rewritten as AX+ x, X+ x + AX0 x, X0 x + AX− x, X− x ≤ −Cx, x.

(20) (21)

The inequality (21) implies AX− x, X− x ≤ −Cx, x.

(22)

Therefore the restriction of the operator X− to H1+ is injective, and it maps H1+ into H2− . Hence dim H2− ≥ dim H1+ . Hence either we are in cases (i) or (ii), or dim H1+ = dim H2− = ∞. In the latter case, if A− is compact, the inequality (22) implies that the condition (iv) is

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satisﬁed. In fact, let R be the restriction of X− to H1+ . Then, for each x in H1+ , we have (−A− )Rx, Rx ≥ Cx, x.

(23)

By the minimax principle (see [6, Theorem X.4.3]) there exists an i-codimensional subspace Li ⊂ H2 such that ai+1 = max (−A− )x, x ≥ x∈Li ||x||=1

(−A− )Rx, Rx Cx, x ≥ max 2 2 −1 ||R|| (Li ) x∈R (Li ) ||R||

max −1

x∈R ||x||=1

1 ≥ ||R||2

||x||=1

min

max Cx, x =

Mi ⊂H1 x∈Mi codim(Mi )=i ||x||=1

1 ci+1 . ||R||2

The last inequality follows from the minimax principle, in this chain of inequalities we also used the fact that codimR−1 (Li ) ≤ i. Hence the condition (iv) is satisﬁed. If one of the conditions (i)–(iii) is satisﬁed, then there exists ε > 0 and a subspace N ⊂ H2− such that dim N = dim H1+ and Ax, x ≤ −ε2 ||x||2 for x ∈ N . We use the existence of such ε and N to ﬁnd X ∈ L(H1 , H2 ) such that G(X) ≤ 0. We write X as a 3 × 3 matrix according to the decompositions (6) and (7):   X++ X+0 X+− X =  X0+ X00 X0−  . X−+ X−0 X−− Let U : H1+ → H2− be an isometry between H1+ and N . Let   0 0 0 0 0 . X= 0 1 ||C||U 0 0 ε

(24)

For x ∈ H1 let x = x+ + x0 + x− be the decomposition of x according to (6). Then (X ∗ AX + C)x, x = AXx, Xx + Cx+ , x+ + Cx− , x− + Cx0 , x0 1

1

||C||U x+ , ||C||U x+ + ||C||||x+ ||2 ≤ A ε ε 1 ≤ −ε2 2 ||C||||x+ ||2 + ||C||||x+ ||2 ≤ 0. ε Suppose that the condition (iv) is satisﬁed. Observe that the following is an equivalent form of the condition (iv): there exists an operator Z : H1+ → H2− such that C+ = −Z ∗ A− Z. Set   0 0 0 X =  0 0 0 . (25) Z 0 0

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We get (X ∗ AX + C)x, x = AXx, Xx + Cx+ , x+ + Cx− , x− + Cx0 , x0 ≤ Z ∗ A− Zx+ , x+ + Cx+ , x+ = 0.

Combining Theorem 4 and the observation from Section 2 we get a criterion for non-emptiness of M (A, B, C) in the case, when B = AD for some D.

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[16] M. M. Malamud, On some classes of extensions of sectorial operators and dual pairs of contractions, in: Recent advances in operator theory, 401–449, Operator Theory: Advances and Applications, 124, Birkh¨ auser, Basel, 2001. [17] C. Scherer, The solution set of the algebraic Riccati equation and the algebraic Riccati inequality, Linear Algebra Appl., 153 (1991), 99–122. [18] Yu. L. Shmulyan, A Hellinger operator integral (Russian), Mat. Sb. (N.S.) 49 (91) (1959), 381–430; English transl. in Amer. Math. Soc. Transl., Ser. 2, vol. 22 (1962), 289-337. [19] V. A. Yakubovich, Conditions for semiboundedness of a quadratic functional on a subspace of a Hilbert space, Vestn. Leningr. Univ., Math. 14 (1982), 321–325. V. A. Khatskevich Department of Mathematics ORT Braude College College Campus, P.O. Box 78 Karmiel 21982 Israel e-mail: victor [email protected] M. I. Ostrovskii Department of Mathematics and Computer Science St. John’s University 8000 Utopia Parkway Queens, NY 11439 USA e-mail: [email protected] V. S. Shulman Department of Mathematics Vologda State Technical University 15 Lenina str. Vologda 160000 Russia e-mail: shulman [email protected] Submitted: June 6, 2006 Revised: April 4, 2007

Integr. equ. oper. theory 59 (2007), 35–51 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010035-17, published online June 27, 2007 DOI 10.1007/s00020-007-1507-z

Integral Equations and Operator Theory

On the Fredholm and Weyl Spectra of Several Commuting Operators R. Levy Abstract. In the paper the local structure of the Fredholm joint spectrum of commuting n-tuples of operators is considered. A connection between the spatial characteristics of operators and the algebraic invariants of the corresponding coherent sheaves is investigated. A new notion of Weyl joint spectrum of commuting n-tuple is introduced. Mathematics Subject Classification (2000). 47A13, 47A11. Keywords. Essential spectrum, Fredholm spectrum, commuting operators, coherent sheaf.

0. Introduction Let T = (T1 , . . . , Tn ), where T1 , . . . , Tn are mutually commuting linear bounded n operators acting in the Banach space X. By σ(T ) ⊂ C we denote the joint Taylor n spectrum of T . Recall that σ(T ) consist of all points λ = (λ1 , . . . , λn ) in C such that the Koszul complex K∗ (T − λ, X) of the operators (T1 − λ1 , . . . , Tn − λn ) is not exact. Suppose that for given λ ∈ σ(T ) all the homology spaces Hi (T, λ) := Hi (K∗ (T − λ, X)) are ﬁnite-dimensional: we call such a point Fredholm point for T and write λ ∈ σF (T ), the Fredholm spectrum of T . The remaining part is called essential spectrum and is denoted by σe (T ) := σ(T )\σF (T ). It is a simple observation (see [15]) that any ﬁnite Fredholm complex of Banach spaces with diﬀerentials, holomorphically depending on the parameters, is locally holomorphically quasi-isomorphic to a holomorphic complex of ﬁnitedimensional spaces. Therefore, the homology sheaves Hi (T ) of the complex of germs of holomorphic functions with values in K∗ (T − λ, X) are coherent on Cn \σe (T ). Fixing a point λ0 ∈ σF (T ), one can consider the stalk of the homology sheaf Hi (T )λ0 as a module over the Noetherian local ring Oλ0 of germs of The author was partially supported by the contract MM-1401/04 with the Bulgarian Ministry of Science and Education and by the contract 22/2006 with Sofia University.

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holomorphic functions at this point. The Fredholm spectrum σF (T ) is a complexn analytic subspace of C \σe (T ), its dimension near λ0 is well deﬁned, and it is an n integer not exceeding the dimension of the ambient space C . The existing results investigate mainly the case when the Fredholm spectrum is of maximal dimension, especially for sub- and factor-modules of spaces of analytic functions – see [4], [11], [9], [10], [6], [7], [8] etc. In the present paper we are interested mainly in the case when this dimension is strictly less than n. To every coherent sheaf, or ﬁnitely generated Oλ0 -module, one can attach an element of the cycle group of Oλ0 , i.e. a formal sum of prime ideals of Oλ0 . Taking the alternated sum of cycles of the modules Hi (T )λ0 for i = 0, . . . , n, one obtains the cycle of the Koszul complex of the n-tuple. Roughly speaking, all irreducible component of σF (T ) containing λ0 ”participate” in the homology sheaves multiplied by corresponding integer coeﬃcients, and we obtain a set of integers characterizing the homology sheaves of the Koszul complex of T . (In the case of a single operator T the corresponding invariant is the index of T − λ0 .) One of the purposes of the paper is to establish some connections between the algebraic characteristics of the homology sheaves, and the action of then operators of T in X. In the ﬁrst section of the paper we recall some necessary facts from the commutative algebra and the theory of analytic local rings. As a source, we use the books [17], [12] and [5]. Especially, we recall notions as isolated prime ideals in the support of a module, cycle of a module, and functoriality of the cycle map under ﬁnite morphisms. We recall also the notions of Hilbert-Samuel polynomial and multiplicity of a module with respect to a prime ideal. The second and third sections contain the main results of the paper. In the subsection 2.1 we deﬁne the cycle zλ0 (T ) in a point λ0 of the Fredholm spectrum of T as a formal linear combination with integer coeﬃcients of the irreducible components of σF (T ) passing through λ0 . More precisely, to every irreducible with corresponding prime ideal pP we attach component P of σF (T ) containing λ0 an integer lP (T ) such that zλ0 (T ) = lP (T ) pP . In the case of Fredholm spectrum of maximal dimension, this is simply the index of the n-tuple T near λ0 (multiplied by the zero ideal). Theorem 2.3 proves the functoriality of the maximal component of the cycle under the holomorphic functional calculus (in particular, the projection property for it). The subsection 2.4 is devoted to the connection between the algebraic invariant zλ0 (T ) introduced above and some index-type characteristics of the n-tuple T which in some particular cases can be used for the calculation of this invariant. The usual deﬁnition of the index of a commuting n-tuple is the Euler characteristic of the corresponding Koszul complex. However, in the case when the dimension of the Fredholm spectrum is strictly less than n this index is automatically zero. In order to obtain a non-trivial invariant, it is natural to choose a suitable k-sub-tuple of T , where k is the dimension of the Fredholm spectrum, and to investigate its index. Unfortunately, this is not always possible which can be seen in the following

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simple example. Suppose that the essential spectrum of T coincides with the unit n sphere of C and its Fredholm spectrum is contained in the interior of this sphere. Then every sub-tuple of T has empty Fredholm spectrum. This follows by the projection property and by the fact that any ﬁber of the corresponding coordinate projection, if non-empty, meets the essential spectrum. To avoid this situation, in [15] it was introduced the notion of local Euler characteristic (with respect to a given point in σF (T ) and a suitable coordinate projection) – see proposition 2.4 below. The main result of subsection 2.4 – theorem 2.5 – states that the local Euler characteristic coincides with the alternated multiplicity of homology sheaves of T in the given point with respect to the corresponding coordinate projection. We note that the connection between the index and the algebraic properties of the corresponding analytic modules, especially in the case of sub- and factor-modules of free Hilbert modules, has been studied in several papers, but mainly in the case of maximal dimension – see the recent papers [9] and [10]. In the non-maximal case, the connection between the analytic and index invariants has been studied in dimensions 2 and 3 in the second part of the paper [14]. In the subsection we are dealing with the highest homology sheaf Hn (T ). 2.5 n In the case when X/ i=1 Ti − λ0i X is ﬁnite-dimensional, this sheaf is coherent near the point λ0 . Under this condition in [4] R. Douglas and K. Yan deﬁned the notion of Hilbert-Samuel polynomial of the corresponding Hilbert module. The connection between the generic dimension of X/ ni=1 Ti − λ0i X and the HilbertSamuel polynomial, was studied in the above mentioned papers of Fang, as well as in the series of preprints [6], [7], [8]. Again, the results of the cited papers relies to the case when the Fredholm spectrum contains a neighborhood of the given point, i.e. is of dimension n. In this case, the Hilbert-Samuel polynomial is of dimension n as well, and the authors investigate the so called Hilbert-Samuel multiplicity – its coeﬃcient ito the power n, multiplied by n!. We prove a similar result in the case of Fredholm spectrum of arbitrary dimension. Namely, theorem 2.8 shows that the Hilbert-Samuel polynomial, deﬁned by Douglas and Yan, coincides with the Hilbert-Samuel polynomial of the local module Hn (T )λ0 , and therefore its degree is equal to the dimension of σF (T ) near λ0 . In subsection 2.6 this result is applied to the theory of multidimensional row contractions, developed by Arveson ([1], [2]). In particular, we prove that the degree of a graded ﬁnite-rank row contraction is equal to the dimension of its Fredholm spectrum near the origin (proposition 2.15). The connection between the characteristics of a multidimensional contraction and the geometry of its Fredholm spectrum will be considered in more details elsewhere. The third section of the paper is devoted to the Weyl spectrum of the commuting n-tuple T of operators. In [16], M. Putinar deﬁnes the Weyl spectrum ω(T ) of T as the complement in σ(T ) of the set of points λ ∈ σF (T ) such that ind (T − λ) = 0. However, in the case when dim σF (T ) < n, it is easy to see that ind (T − λ) = 0 for any λ ∈ / σe (T ), and therefore the Weyl spectrum will coincide with the essential spectrum of the n-tuple. We propose here an alternative

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deﬁnition: a Fredholm point λ is not in the Weyl spectrum iﬀ the cycle zλ (T ) is zero, i.e. all lP (T ) = 0. In the case when all the components of σF (T ) are of (maximal) dimension n, this deﬁnition coincides with the deﬁnition adopted in [16]. Note that even in the case of a single operator there is a certain discrepancy between our approach and the standard one: in our deﬁnition, all the isolated point of the Fredholm spectrum of T belong to its Weyl spectrum. Further we give some basic properties of the newly introduced notion of Weyl spectrum: any ntuple with SVEP (Single Value Extension Property) has the ”Weyl property” (the Weyl spectrum coincides with the essential spectrum). We prove also the property of spectral inclusion under holomorphic functional calculus for this spectrum. In some particular cases we get a stronger result – the spectral mapping theorem (see propositions 3.4 and 3.5). The present paper continues the investigations on the structure of the Fredholm spectrum of a commuting n-tuple of operators started in [15], and systematically uses some basic result of that paper.

1. Some basic facts from the commutative algebra n

Consider a coherent analytic sheaf L, deﬁned on an open subset of C . Recall that its (geometric) support supp L consists of all points where L is non-zero; supp L n is a complex-analytic subset of C . If λ0 ∈ supp L, then the stalk of L at λ0 is a ﬁnitely generated module over the local Noetherian ring Oλ0 . Below we recall some basic notions from the commutative local algebra, concerning Noeterian rings and modules, necessary for our considerations. Let A be a commutative local Noetherian ring, and M be a ﬁnitely generated A-module with annihilator Ann(M). Let P rime (A) be the set of prime ideals of A. Denote by Supp (M) the (algebraic) support of M – the set of all prime ideals in A, containing Ann(M). The set Supp(M) has a natural ordering by inclusion and the minimal elements with respect of this ordering (called isolated associated primes) play a special role. They correspond to the irreducible components of the geometric support of M. The set Iso(M) of all such prime ideals is ﬁnite (see [5], chap. 3). For p ∈ P rim (A), denote by Mp the localization of the A-module M with respect to p. Then p ∈ Supp (M) if and only if Mp = 0. Moreover, p ∈ Iso (M) iﬀ lp (M) = dimAp Mp is ﬁnite and nonzero. Suppose that 0 = M0 ⊂ . . . ⊂ Mk = M is a composition series for M, i.e. Mi /Mi−1 = A/pi with pi ∈ P rim (A) for all i. If p ∈ Iso (M), then p appears exactly lp (M) times in the sequence {p1 , . . . , pk }. By Isomax (M) we denote the set of elements of Iso (M) of maximal dimension (equal to the dimension of M). Let M and A be as above, and let q be an ideal in A such that M/qM is of ﬁnite dimension. Then there exists a polynomial Pq,M (called the HilbertSamuel polynomial of the pair q, M) of degree r = dimA M such that for n suﬃciently big one has dim (M/q n M) = Pq,M (n). The leading term of Pq,M has the

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e (M)

form q r! nr , where eq (M) is a positive integer, called a multiplicity of q at M. If q = m, where m is the maximal ideal of A, then the corresponding integer is called multiplicity of M and is denoted by e (M). The formula (∗) in the page 1 on the book [17] shows how eq (M) can be calculated in homological terms. Suppose now that M = {M0 , . . . , M n } is an nordered ﬁnite set of ﬁnitely generated modules over A. Denote Ann M = i=0 Ann (M). Let Supp M be the set of all prime ideals of A, containing Ann M , Iso M – the set of minimal elements of Supp M , and Isomax M – the set of elements of Iso M of maximal dimension. Take i ∈ {0, . . . , n} and a prime ideal p ∈ Supp (Mi ). Since Supp (Mi ) ⊂ Supp M , then if p is not minimal in Supp (Mi ), it will be not minimal in Supp M as well. So, one obtains Proposition 1.1. For any p ∈ Iso M and i ∈ {0, . . . , n} one has one of the following two possibilities: 1) p ∈ Iso (Mi ), or 2) p ∈ / Supp (Mi ). Therefore for any prime p ∈ Iso M and integer i lp (Mi ) = dimAp (Mi )p is well-deﬁned (it is positive in the case 1) and zero in the case 2)). So for such a p one can deﬁne: n lp M := (−1)i lp (Mi ) . i=0

Let q be an ideal in A such that Mi /qMi is ﬁnite-dimensional for all i. One can introduce the Hilbert-Samuel polynomial of M : n Pq,M (n) := (−1)i Pq,M (n). i=0

If one denotes by r = dimA M the maximal dimension of the A-modules Mi , i = 0, . . . , n, then Pq,M (n) is a polynomial of dimension r and with leading term eq (M) r n , where r! n eq M := (−1)i eq (Mi ) . i=0

Denote by Z (A) the group of the cycles of A, i.e. of all formal linear combinations of elements of P rime (A) with integer coeﬃcients. to any tuple of Then max M of Z (A), deﬁned A-modules M one can attach the elements z M and z by the formulae z M = lp M p , z max M = lp M p p∈Iso(M) p∈Isomax (M) and one has

eq M =

p∈Isomax (M)

lp M eq (A/p)

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(see [17], p. 125-126). Finally, we will need a functoriality result. Let B be a subring of the analytic local ring A such that the monomorphism ϕ : B → A is ﬁnite in the sense of [12], II.2.2, i.e. A is ﬁnitely generated as a B-module. For any ﬁnitely generated A-module M, denote by MB the underlying B-module. Then MB is a ﬁnitely generated B-module again. For any p ∈ P rime (A), denote by ϕ∗ p the element z max (pB ) of Z (B). By linearity one can extend this mapping up to a morphism ϕ∗ : Z (A) → Z (B). Proposition 1.2. For any ﬁnitely generated A-module M one has z max (MB ) = ϕ∗ z max (M). Proof. Indeed, theorem 2 of [12], II.5.1 asserts that dimA L = dimB LB for any ﬁnitely generated A-module L. In particular, this is true for the modules of the type A/p, p ∈ P rime (A). Taking the composition series for M and using the additivity of the mapping z max (see [17], p. 125), we obtain z max (MB ) = max (M). p∈Isomax (M) lp (M) pB = ϕ∗ z

2. Local structure of the Fredholm spectrum of commuting n-tuple 2.1. Main definitions Consider a commuting n-tuple T = (T1 , . . . , Tn ) of operators acting in the Banach space X, and denote by K∗ (T − λ, X) = {Xk , dk (λ)}k=0,1,...,n the Koszul complex n of T in the point λ ∈ C . Recall that Xk is a direct sum of nk copies of the space X, and dk (λ) depend linearly on λ. Denote by OX the sheaf of germs of n X-valued holomorphic functions on C and the complex OK∗ (T −λ, X) of sheaves n of holomorphic sections on C of the complex K∗ (T − λ, X). Denote by Hi (T, λ) the i-th homology space of the complex K∗ (T − λ, X), and by Hi (T ) the i-th sheaf of homologies of the complex of sheaves OK∗ (T − λ, X). Let Hi (T )λ0 be the stalk of Hi (T ) at λ0 , considered as a local Oλ0 -module. Since the operators Ti commute with the diﬀerentials dk (λ) of the parameterized Koszul complex, one can deﬁne the action of operators Tk , k = 1, . . . , n on the spaces Hi (T, λ0 ) and Hi (T ). It is easy to see that the operator Tk acts on the space Hi (T, λ0 ) as multiplication by the number λ0k , and on the sheaf Hi (T ) – as multiplication by the variable λk . Fix a point λ0 ∈ σF (T ); this means that all the spaces Hi (T, λ0 ) are ﬁnitedimensional. As it was noted in [15], there exists a holomorphic complex of ﬁnitedimensional spaces, deﬁned near λ0 and holomorphically quasi-isomorphic to K∗ (T − λ, X). Therefore, all the sheaves Hi (T ) are coherent in a neighborhood of λ0 , and the stalks Hi (T )λ0 at λ0 are ﬁnitely generated modules over the local Noetherian ring Oλ0 . Denote by H(T )λ0 the n + 1-tuple {Hi (T )λ0 }i=0,...,n . Then of the the prime ideals in Iso H(T )λ0 correspond to the irreducible components complex set σF (T ), containing the point λ0 . Denote by r = r λ0 the maximal dimension of these components.

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Definition 2.1. Deﬁne

zλ0 (T ) = z H(T )λ0 , zλmax (T ) = z max H(T )λ0 . 0

The element zλ0 (T ) has the form zλ0 (T ) =

lp (T ) p

p∈Iso(H(T )λ0 )

where lp (T ) = lp H(T )λ0 are integers. This deﬁnition can be considered from the geometric point of view. Let P be an irreducible component of the complex set σF (T ); then in any point λ ∈ P it determines a prime ideal pλ in the ring Oλ . Proposition 2.2. The integer lpλ (T ) does not depend on the choice of the point λ ∈ P. Proof. It is suﬃcient to proof that the integers lpλ (T ) are locally constant on P. Choose λ0 ∈ P and a suﬃciently small ball U , centered at λ0 , such that U does not intersect any irreducible component of σF (T ), not containing λ0 . Then for any i ∈ {0, . . . , n} the OU -module Hi (T )U has the same set of isolated primes, and the same composition series, as Hi (T )λ0 . The irreducible complex set P determines a prime ideal pU in OU . For any λ ∈ U consider the Oλ -module Hi (T )λ . If λ ∈ P, the localization of pU at λ is a nontrivial prime ideal in Oλ coinciding with the ideal pλ . The invariance of the composition series under localization shows that lpλ0 (T ) = lpU (T ) = lpλ (T ). Now, we can attach to any irreducible component its local index lP (T ). The set of all irreducible components and the corresponding local indexes contains certain information about the homology sheaves of the Koszul complex of T and will be called a spectral picture of the commuting n-tuple T . In the rest of the paper we will turn back to the algebraic point of view on the local indexes and will give some facts allowing to compute it in some cases. 2.2. Particular cases: dimensions 1 and n Suppose that the dimension of σF (T ) at λ0 is the maximal one, i.e. equal to n. This means that the Fredholm spectrum σF (T ) contains a neighborhood U of λ0 . It is easy to see (or to derive from theorem 2.5 below) that in this case Iso H(T )λ0 = {[0]} and n z (T )λ0 = z max (T )λ0 = (−1) ind T − λ0 . [0] , where ind T − λ0 is the Euler characteristic of the complex K∗ T − λ0 , X and [0] is the zero ideal in Oλ0 . Recall that the n-tuple T has the single value extension property (SVEP) at λ0 if Hi (T )λ0 = 0 for all i = n. Suppose in addition that the n-tuple T has SVEP. Then from the theory of coherent sheaves it follows that the sheaf Hn (T ) is a free of U . (In fact, U is the complement of a O-module on the open dense subset U

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one complex subset of U of dimension < n.) Then for any λ = (λ1 , . . . , λn ) ∈ U has n 0 n (Ti − λi ) X . ind T − λ = ind (T − λ) = (−1) dim X/ i=1

Now consider the case when σF (T ) is one-dimensional near λ0 . In [15] it was proved that the modules Hi (T )λ0 are nonzero only for i = n, n − 1. Moreover, prop. 3.6 of [15] shows that Hn−1 (T ) ∼ Hn (T ∗ ) , where T ∗ is the n-tuple (T1∗ , . . . , Tn∗ ) acting in the dual space X ∗ and ∼ denotes the equivalence modulo sheaves with zero-dimensional support, i.e. concentrated in the point λ0 . So we get the equality z max (T )λ0 = (−1)n (z max (Hn (T )λ0 ) − z max (Hn (T ∗ )λ0 )) . Finally, note that if σF (T ) is of dimension zero at λ0 (i.e. λ0 is an isolated point of the Fredholm spectrum), then l{λ0 } (T ) coincides with the dimension of the spectral subspace of T corresponding to the point λ0 . 2.3. Functoriality Let f (λ) = (f1 (λ) , . . . , fk (λ)) be a k-tuple of holomorphic functions, deﬁned n in a neighborhood of the spectrum σ(T ) ⊂ C of the commuting n-tuple T of operators. It is well-known that σ(f (T )) = f (σ(T )) and σe (f (T )) = f (σe (T )) The ﬁrst equality (the functoriality of the Taylor spectrum) was proved in the Taylor‘s fundamental paper [18]. Its modiﬁcation for the essential spectrum was proved by several authors (see for example [15]). k Take the point µ0 ∈ C belonging to the Fredholm spectrum σF (f (T )) of the operator k-tuple f (T ). Then its preimage f −1 µ0 ∩ σ(T ) is a ﬁnite subset {λ1 , . . . , λp } of the Fredholm spectrum σF (T ) of T . For any λ ∈ f −1 µ0 ∩ σ(T ) the inverse image f ∗ by f provides an embedding Oµk 0 → Oλn and the induced monomorphism Oµk 0 → Oλn /Ann H(T )λ is ﬁnite. Denote by

f∗ : Z Oλn /Ann H(T )λ → Z Oµk 0 the morphism used in prop. 1.2. Theorem 2.3. The following equality holds: z max H(f (T ))µ0 =

f∗ z max H(T )λ

λ∈f −1 (µ0 )∩σ(T )

Proof. In proposition 3.2 of [15], the equality Hi (f (T ))µ0 =

Hi+n−k (T )λ ,

λ∈f −1 (µ0 )∩σ(T )

is established, where both sides are considered as Oµk 0 -modules. Applying the functor z max and taking into account the equality z max (MB ) = ϕ∗ z max (M) proved in the previous section, we get the theorem.

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2.4. zλmax (T ) and local Euler characteristics 0 max We will try to extract (T )λ0 , i.e. on the integers some information on the cycle z lp (T ) := lp H(T )λ0 for p ∈ Isomax H(T )λ0 , by using some properties of operators of T . Suppose that the coordinates λ = (λ1 , . . . , λr ) form a coordinate system for σF (T ) at λ0 . This means that λ0 is an isolated point of Π−1 λ0 ∩ σF (T ), where Π is thecoordinate projection onto the ﬁrst r coordinates, and λ0 = λ01 , . . . , λ0r , λ0 = λ0r+1 , . . . , λ0n . This can always be achieved by a small perturbation of the coordinate system. We will denote T = (T , T ), where T = (T1 , . . . , Tr ), T = (Tr+1 , . . . , Tn ). Consider the Koszul complex K∗ (T −λ0 , X) of the operators T1 −λ01 , . . . , Tr − 0 λr in X. In general its homology spaces Hi (T , λ0 ), i = 0, . . . , r are not Hausdorﬀ spaces, but due to the commutativity of T the action of the operators from T on it is correctly deﬁned. Moreover, one can form the parameterized Koszul complex of / Π−1 λ0 σ(T ). operators of T − λ in Hi (T , λ0 ); this complex is exact for λ ∈ The following proposition was proved in [15]: Proposition 2.4. In the conditions from above there exists a decomposition Hi (T , λ0 ) = Hi (T , λ0 ) Hi (T , λ0 ) into subspaces, invariant under T , such that: a) Hi (T , λ0 ) is ﬁnite-dimensional (or empty) and the joint spectrum of operators of T in Hi (T , λ0 ) (in the non-empty case) consists of the point λ0 . b) The joint spectrum of T in Hi (T , λ0 ) does not contain the point λ0 . ) of c) There exists a ﬁnite-dimensional subcomplex L holomorphic 0 ∗ (λ L λ coincides with H T K∗ (T − λ , X), such that H − λ0 , X and ∗ ∗ i H∗ K∗ T − λ0 , X /L∗ λ0 – with Hi T − λ0 , X . d) Suppose that Π−1 λ0 is contained in σF (T ) and therefore consists of ﬁnitely many points λj = λ0 , λj , j = 1, . . . , k. Denote by Hij (T , λ0 ) the joint root space of the operators T − λj in Hi (T , λ0 ). Then Hi (T , λ0 ) = j 0 j Hi (T , λ ). In other words, Hi (T , λ0 ) is ﬁnite-dimensional and coincides with the joint root space of the operators Tr+1 − λ0r+1 , . . . , Tn − λ0n acting in the linear space Hi (T , λ0 ). Note that in the case of essentially normal tuple T the results of [15] provide also a connection between the K-theory of the holomorphic complexes from above and the Brown-Douglas-Fillmore invariant of T . Denote r (−1)i dim Hi (T , λ0 ). χ T , λ0 = i=0

Let q be the ideal in Oλ0 , generated by the functions λ1 − λ01 , . . . , λr − λ0r . Then the following statement holds true: Theorem 2.5. χ T , λ0 = eq H(T )λ0 .

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Proof. Let Oλ0 X be, as above, the stalk at λ0 of the sheaf of germs of X-valued holomorphic functions, and let Oλ0 K∗ (T − λ, X) be the Koszul complex of the operators of T − λ in Oλ0 X. This is a complex of Oλ0 -modules. Consider, in any stage Oλ0 Xk , k = 0, .. . , n, of this complex, the action of the operators T − λ0 = T1 − λ01 , . . . , Tr − λ0r . This action commutes with the diﬀerential of the complex. One can form the Koszul complexes of the operators of T −λ0 in Oλ0 Xk , obtaining a bicomplex of sheaves with r + 1 rows and n + 1 columns. As it is shown below, the cohomology sheaves of its total complex are in fact supported at λ0 and ﬁnitedimensional. The Euler characteristic of the total complex of this bicomplex can be computed in two ways. One may consider the homologies Hi (T )λ0 , i = 0, . . . , n of the initial complex, and take the alternated sum of dimensions of the homologies of the Koszul complex of operators T −λ0 in it. The alternated sum of these integers for i = 0, . . . , n is equal to the alternated sum of the dimensions of the homologies of the total complex. Since the action of the operators Tj on Hi (T )λ0 coincides with the multiplication by the variable λj , we arrive to the Koszul complex of λ1 − λ01 , . . . , λr − λ0r in Hi (T )λ0 , and the formula (∗) on page 1 of [17] can be applied. It shows that r

(−1)j dim Hj λ − λ0 , Hi (T )λ0 = eq (Hi (T )λ0 )

j=0

for any i = 0, . . . , n. Therefore the Euler characteristic of the total complex is equal to eq H(T )λ0 . 0 On the other hand, take the complex K − λ , X with stages Xk and T ∗ 0 diﬀerentials dk = dk λ : Xk → Xk+1 , k = 0, . . . , r. The Euler characteristic of the total complex is equal to the alternated sum on j, j = 1, . . . , r of the integers cj =

n (−1)i dim Hi (T − λ, Oλ0 Hj (T − λ , X)) i=1

0 Hj (T − λ , X) we denote the factor Oλ0 -module where by O λ0 0 Oλ0 ker dj λ /dj−1 λ (Oλ0 Xj−1 ). Take the ﬁnite-dimensional subcomplex L∗ λ0 = {Lj , aj } as in point c) of prop. 3.2 of [15]; then the action of the operators T − λ on the module Oλ0 (Xj /Lj ) is regular for any j, and therefore in the formula above we can replace Oλ0 Hj (T − λ , X) by Oλ0 Lj /aj (Oλ0 Lj−1 ) = 0 Oλ0 Hi T − λ , X . It is easy to prove the following:

Lemma 2.6. Let H be a ﬁnite-dimensional space, Oλ0 H – the Oλ0 -module of germs n of H-valued holomorphic functions in the point λ0 ∈ C . Let T = (T1 , . . . , Tn ) be an n-tuple of operators acting in H of the form Ti = λ0i I + Ki , i = 1, . . . , n, where Ki are nilpotent operators in H. Then

0 if i < n dim Hi (T − λ, Oλ0 H) = dim H for i = n.

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Indeed, if all Ki are zero operators, one obtains the Koszul complex of the germs λ0i − λi in Oλ0 H exact in all terms except in the last one, where the module of homology is Oλ0 H/mλ0 H, and mλ0 is the maximal ideal in Oλ0 . The general case can be reduced to the case of zero operators by taking a ﬁltration of the linear space H such that all Ki are zero on the corresponding graduate space. Applying the assertion of the lemma to the space Hi (T , λ0 ) and the operators λ01 , . . . , λ0r , Tr+1 , . . . , Tn , we get cj = dim Hi (T , λ0 ) for j = 1, . . . , r, which proves the theorem. Corollary 2.7. Suppose that Isomax H(T )λ0 consists of a single prime ideal p, (i.e. there is only one irreducible component P of σF (T ) containing λ0 ), and the coordinates λ = (λ1 , . . . , λr ) form a coordinate system for σF (T ) at λ0 . Then χ T , λ0 = eq (Oλ0 /p) lp (T ) , where q is, as above, the ideal generated by λ1 − λ01 , . . . , λr − λ0r . 2.5. The highest homology sheaf Hn (T )λ0

n

In this section we assume that the point λ0 ∈ C satisﬁes the regularity n following condition: the linear subspace of all elements of the type i=1 Ti − λ0i xi with x1 , . . . , xn ∈ X is of ﬁnite codimension in X. Then, following the arguments used e.g. in [15], 4.3., it is easy to see that the last homology sheaf Hn (T ) of the Koszul complex of T is coherent near λ0 . For simplicity, denote by Hλ0 X the Oλ0 -module Hn (T )λ0 . Let m be the maximal ideal in the local ring Oλ0 . As above, one can form the Hilbert-Samuel polynomial Pm,Hλ0 X (k) with leading term e (Hn (T )λ0 ) k r /r!, where r = dim supp Hn (T )λ0 . On the other hand, in [4] R. Douglas and K. Yan deﬁned the notion of HilbertSamuel polynomial of a given Hilbert module (see also chapter 2 of [3]). More precisely, consider the operator algebra A = C [T1 , . . . , Tn ] ⊂ L(X) and denote by m(T ) the operator ideal in A generated by the operators T1 − λ01 , . . . , Tn − λ0n . Then, under the regularity assumption from above, Douglas and Yan proved that for any natural k the subspace mk (T )X is of ﬁnite codimension in X. For k large enough the codimension of mk (T )X is a polynomial of k of degree not exceeding n. This polynomial is called by Douglas and Yan the Hilbert-Samuel polynomial of the Hilbert C [λ1 , . . . , λn ]-module, determined by the space X and the operators T1 , . . . , Tn . The following assertion shows that the Hilbert-Samuel polynomials of the Hilbert module and of the last homology sheaf coincide: Theorem 2.8. Under the condition from above, dim X/mk (T )X = dim Hn (T )λ0 /mk Hn (T )λ0 and therefore for k large enough it is a polynomial of k with leading term e (Hn (T )λ0 ) k r /r!, where r = dim supp Hn (T )λ0 . Proof. Denote Y = mk (T )X and Z = X/Y . Then Z is ﬁnite-dimensional and the joint spectrum of the operators induced by T in Z coincides with the point λ0 . Therefore Hi (T − λ, Z)λ0 = 0 for all i < n. Denote, as above, by Hλ0 X the

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Oλ0 -module Hn (T )λ0 = Oλ0 X/ i (Ti − λi ) Oλ0 X, and by Hλ0 Y , Hλ0 Z – the corresponding homology modules for Y and Z respectively. The exact sequence 0 → Y → X → Z → 0 determines an exact sequence of the corresponding Koszul complexes, and therefore a long exact sequence of the corresponding homology sheaves, ending with the sequence 0 → Hλ0 Y → Hλ0 X → Hλ0 Z → 0. The next assertion shows that the image of Hλ0 Y in Hλ0 X coincides with mk Hλ0 X. Lemma 2.9. Let J be a ﬁnitely generated ideal in the algebra of polynomials

C [λ1 , . . . , λn ], and let J (T ) be the corresponding ideal in the operator algebra C [T1 , . . . , Tn ]. Suppose that Y = J (T )X is of ﬁnite codimension in X. Then the image of Hλ0 Y in Hλ0 X under the natural inclusion coincides with the module J Hλ0 X.

Proof. Note that for any germ x(λ) of a X-valued holomorphic function and for any polynomial p(λ) the germs of the functions p(T )x(λ) and p(λ)x(λ) deﬁne the same class in the factor-sheaf Hλ0 X. Now choose the polynomials gl (λ), l = 1, . . . , L, generating the ideal J . Then the row operator g(T ) = (g1 (T ), . . . , gL (T )) gives L an epimorphism ∈ Hλ0 Y can be represented by l=1 Xl → Y . Any element y a germ of an Y -valued holomorphic function y(λ), which can be lifted through the epimorphism g(T ). In other words, there exist germs of X-valued holomorphic gl (T )xl (λ). Then the class of the functions x1 (λ), . . . , xL (λ) such that y(λ) = gl (T )xl (λ) ∈ function y(λ) in Hλ0 X coincides with the class of the function J Hλ0 X. Thus the image of Hλ0 Y in Hλ0 X under the natural inclusion is contained in J Hλ0 X. Conversely, take an element x = l gl (λ) xl ∈ J Hλ0 X. Representing x l by of holomorphic X-valued functions, and denoting x(λ) = l gl (λ)xl (λ) germs xl (λ) and y(λ) = l gl (T )xl (λ), we obtain that both x(λ) and y(λ) represent the class x . However, y(λ) is Y -valued function, and therefore x belongs to the image of Hλ0 Y . To complete the proof of the theorem, it is suﬃcient to apply the lema in the case J = mk which yields that Hλ0 Y and mk Hλ0 X are isomorphic submodules of Hλ0 X. Taking the corresponding factor-modules we obtain an isomorphism between Hλ0 Z and Hλ0 X/mk Hλ0 X. Applying lemma 2.6, we get dim Z = dim Hn (T )λ0 /mk Hn (T )λ0 . A similar equality can be proved in a more general situation. Suppose that, as in th. 2.3, f (λ) = (f1 (λ) , . . . , fp (λ)) is a k-tuple of holomorphic functions, deﬁned in the neighborhood of σ(T ), µ0 ∈ σF (f (T )), and therefore the intersection f −1 µ0 ∩ σ(T ) is a ﬁnite subset λ1 , . . . , λJ of σF (T ). Let q be the ideal, generated by f1 (λ)−µ01 , . . . , fp (λ)−µ0k , and q(T ) – the operator ideal in L(X) generated by f1 (T ) − µ01, . . . , fk (T ) − µ0k . Let r = max dimOλ0 (Hn (T )λj ), j = 1, . . . , J. Then, applying the above theorem for the p-tuple f (T ) − µ0 and using the functoriality stated in 2.3, we get

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Corollary 2.10. In the conditions from above dim X/q k (T )X =

J

dim Hn (T )λj /q k Hn (T )λj

j=1

and therefore for k suﬃciently large it is a polynomial of degree r. In particular, the following corollary holds true: Corollary 2.11. Suppose that, under the conditions of 2.10, the n-tuple T has SVEP at the points λ1 , . . . , λJ (or, equivalently, f (T ) has SVEP at µ0 ). Then for k suﬃciently large dim X/q k (T )X is a polynomial of k with leading term (−1)n eq (T )k r /r!, where r is the maximal dimension of the modules Hn (T )λj for j = 1, . . . , J. 2.6. Application to Arveson‘s row contraction theory Suppose that the operators of T act in the Hilbert space H. as above, a nChoose, point λ0 such that the subspace of elements of the type i=1 Ti − λ0i xi is of ﬁnite codimension in H, and denote, as in the previous section, by m the maximal ideal in the local ring Oλ0 . Take the polynomials gl (λ), l = 1, . . . , L, generating the ideal mk , and denote by g(T ) : H L → H the row operator with entries gl (T ), l = 1, . . . , L. Then the factor-space Z = H/mk (T )H from the previous section is isomorphic to the kernel of the operator g(T ) ◦ g(T )∗ = l=1 gl (T ) ◦ gl (T )∗ . In particular, consider the case when T is a commuting n-contraction with n ﬁnite rank in the sense of Arveson (see [1]), i.e. the operator 1 − i=1 Ti Ti∗ is ﬁnite-dimensional and positive. Then the Fredholm spectrum of T is contained in the open unit ball Bn . Suppose in addition that that the essential spectrum σe (T ) does not contains the origin, i.e. one can take λ0 = 0 (this is obviously satisﬁed if all the operators Ti are essentially normal; in this case the essential spectrum n is contained in the unit sphere in C ). Let φ : B(H) → nB(H) be the Arveson completely positive map deﬁned by the formula φ(A) = i=1 Ti ATi∗ (see [2]). In the Arveson‘s theory an important role is played by the expression k! T k1 . . . Tnkn T1∗k1 . . . Tn∗kn . φk (1) = k1 ! . . . kn ! 1 k1 +...+kn =k

Since the functions λk11 . . . λknn with k1 + . . . + kn = k form a system of generators for the k-th power of the maximal ideal in the local ring O0 of germs n of holomorphic functions in the point 0 ∈ C , then from the the results of the previous section it follows that: Proposition 2.12. Let T be a commuting n-contraction of ﬁnite rank. Then one has dim ker φk (1) = dim H/mk (T )H = dim Hn (T )0 /mk Hn (T )0 and therefore dim ker φk (1) is a polynomial on k of degree, equal to the dimension of the O0 -module Hn (T )0 (i.e. to the dimension of σF (T ) near the origine).

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In [2], prop. 7.2, it is proved that rank 1 − φk (1) is a polynomial of k of degree ≤ n. The degree of this polynomial is called the degree of the module H and is denoted by deg H. Since dim ker φk (1) ≤ rank 1 − φk (1) , we get the following Corollary 2.13. Under the assumptions of prop. 2.12 deg H ≥ dim Hn (T )0 . Remark 2.14. Suppose that the essential spectrum of T is contained in the unit n sphere in C . Applying an appropriate M¨ obius transform to the contraction T (see [13]), we can replace in the statement above the point 0 by an arbitrary point in the open ball Bn . One can conjecture that the degree of a pure ﬁnite-rank contraction coincides with the maximal degree of the corresponding analytic modules, i.e. with the dimension of the support of Hn (T ); this problem is to be considered elsewhere. In [2] Arveson introduced the notion of a graded n-contraction as a contraction endowed with a suitable action of the circle group (for the precise deﬁnition see section 6 of [2]). Then, roughly speaking, the Fredholm spectrum and the corn responding homology sheaves are determined by homogeneous polynomials in C . In this case the inequality of the proposition above becomes an equality: Proposition 2.15. Let T be a pure graded ﬁnite-rank contraction in the Hilbert space H. Then deg H = dimO0 Hn (T )0 , and the leading terms of the polynomial rank 1 − φk (1) and the Hilbert-Samuel polynomial of Hn (T )0 coincide. Proof. The assertion follows almost immediately from the proof of theorem B of [2]. Indeed, Arveson constructs a submodule H0 of ﬁnite codimension in H such that 1H0 − φk0 (1H0 ) is a (ﬁnite-dimensional) projection for any k, φk0 (A) being the Arveson‘s completely positive map for the submodule H0 . Therefore rank 1H0 − φk0 (1H0 ) = dim ker φk0 (1H0 ) and the equality above is satisﬁed for the submodule H0 . On the other hand, corollary 1 of theorem C of [2] shows that for k suﬃciently big the polynomials rank 1H0 − φk0 (1H0 ) and rank 1H − φk (1H ) diﬀer by a polynomial of degree strictly less than deg H. (Formally, this is stated only in the case when deg H = n, but it is easy to see that the proof works in the general case as well.) A similar statement is true for the right hand side. Indeed, the arguments used in the proof of 2.8 lead to the exact sequence 0 → Hn (H0 )0 → Hn (H)0 → Hn (H/H0 )0 → 0. Since H/H0 is ﬁnite-dimensional, then the Hilbert-Samuel polynomial of Hn (H/H0 )0 is of degree zero and the additivity property (see prop. II.10 of [17])

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shows that the Hilbert-Samuel polynomials of Hn (H0 ) and Hn (H) diﬀer by a polynomial of lower degree and therefore have identical leading terms. Now, since the statement of the proposition is valid for H0 , then it is valid for H also.

3. Weyl spectrum for commuting n-tuples of operators Proposition 3.1. Let λ0 ∈ σF (T ). Then the following assertions are equivalent: 1) zλ0 (T ) = 0, 2) zλmax (T ) = 0 for λ suﬃciently close to λ0 . Proof. Suppose that zλ0 (T ) = 0, i.e. lp (T ) = 0 for any ideal p ∈ Iso H(T )λ0 . Then, as in the proof of 2.2, one can show that lP (T ) = 0 for any irreducible component of σF (T ), containing λ0 . Conversely, suppose that 2) is satisﬁed in a neighborhood U of λ0 and P is an arbitrary irreducible component of σF (T ) containing λ0 . One can choose a point λ ∈ P ∩ U such that no other irreducible component of σF (T ) contains λ. Then the condition 2) in the point λ implies that lP (T ) = 0, and 1) holds true. Definition 3.2. We call the point λ0 ∈ σF (T ) a Weyl point for the n-tuple of commuting operators T if the conditions of the proposition above are fulﬁlled. We will denote by ρω (T ) the set of all Weyl points of T , and by ω(T ) – the Weyl spectrum of T , i.e. the complement of ρω (T ) in σ(T ). Note that in this deﬁnition, unlike in the standard one, in the case of a single operator the isolated points of the Fredholm spectrum are not Weyl points. If the n-tuple T possesses SVEP, then zλ0 (T ) = (−1)n z (Hn (T )λ0 ) and therefore (−1)n lP (T ) = 0 for any irreducible components P of σF (T ). In this case ω(T ) = σe (T ) and T has ”Weyl property”. One can derive a criterion for Weyl points using the Banach-space characteristics of the tuple T considered above. Let λ0 ∈ σF (T ), and let P1 , . . . , Ps be the irreducible components of σF (T ) containing λ0 , with dimensions r1 , . . . , rs corr. Let U be a neighborhood of λ0 not intersecting other irreducible components. Take / Pj for j = i. Choose coordinate sysλ1 , . . . , λs ∈ U such that λi ∈ Pi and λi ∈ n tem in C such that the set of coordinates contains local coordinate systems for any of the components Pi , i = 1, . . . , s in U . Now from propositions 2.5 and 3.1 we obtain the following characterization of the Weyl points using the local Euler characteristics: Proposition 3.3. λ0 is a Weyl point iﬀ all χ T , λi = 0, i = 1, . . . , s. The functoriality of z(T ), stated in theorem 2.3, implies the functorial properties of ω(T ). Proposition 3.4. Let f (λ) = (f1 (λ) , . . . , fk (λ)) be a k-tuple of holomorphic functions, deﬁned in the neighborhood of the spectrum σ(T ) of T . Then ω (f (T )) ⊂ f (ω(T )).

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Proof. Take a point µ0 ∈ / f (ω(T )). Then f −1 µ0 , if non-empty, is a ﬁnite subset of ρω (T ) and the same is true for the points µ in a suﬃciently small neighborhood of µ0 . Then from theorems 2.3 and 3.1 it follows that µ0 ∈ ρω (f (T )). In some particular cases the spectral inclusion in the proposition above can be replaced by equality. Indeed, the following proposition follows immediately from 2.3 in the same way as above: Proposition 3.5. Suppose that one of the following two conditions is satisﬁed: a) f (z) is monomorphic on σ(T ), or b) the operator T has SVEP. Then ω(f (T )) = f (ω(T )).

References [1] W. Arveson, Subalgebras of C ∗ -algebras III – Multivariate operator theory, Acta Math. 181(2) (1998), 159–228. [2] W. Arveson, The curvature invariant of a Hilbert module over C [z1 , . . . , zd ], J. Reine Angew. Math. 522 (2000), 173–236. [3] X. Chen, K. Guo Analytic Hilbert modules, Chapman& Hall/CRC, 2003. [4] R.G.Douglas, K. Yan, Hilbert-Samuel polynomials for Hilbert modules, Indiana Univ. Math. J., 42(3) (1993), 811–820. [5] D. Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate texts in mathematics 150, Springer-Verlag. 1998. [6] J. Eschmeier, On the Hilbert-Samuel multiplicity of Fredholm tuples, Preprint Nr. 163, Universit¨ at des Saarlandes, Saarbr¨ ucken 2006. [7] J. Eschmeier, Samuel multiplicity for several commuting operators, Preprint Nr. 171, Universit¨ at des Saarlandes, Saarbr¨ ucken 2006. [8] J. Eschmeier, Fredholm spectrum and growth of cohomoly groups, Preprint Nr. 181, Universit¨ at des Saarlandes, Saarbr¨ ucken 2006. [9] X. Fang, The Fredholm index of quotient Hilbert modules, Math. Res. Letters, 12 (2005), 911–920. [10] X. Fang, The Fredholm index of a pair of commuting operators, Geometric And Functional Analysis, 16 (2) (2006), 367–402. [11] J. Gleason, S. Richter, C. Sundberg, On the index of invariant subspaces in spaces of analytic functions of several complex variables, J. Reine Angew. Math. 587 (2005), 49–76. [12] H. Grauert, R. Remmert, Analytic Local Algebras, Springer-Verlag, 1971. [13] D. Greene, Free resolutions in multivariate operator theory, J. Funct. An., 200(2)(2003), 429–450. [14] K. Guo, K. Wang, Essentially normal Hilbert modules and K-homology, Preprint. [15] R. Levy, Algebraic and topological K-functors of commuting n-tuple of operators, J. Operator Theory 21 (1989), 219–253.

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[16] M. Putinar, On Weyl spectrum in several variables, Math. Japon. 50, n.3 (1999),335– 357. [17] J.-P. Serre, Algebre Locale et Multiplicites, Springer-Verlag, 1975. [18] J.L. Taylor, The analytic functional calculus for several commuting operators, Acta Math., 125 (1970), 1–48 R. Levy Sofia University Faculty of Mathematics and Informatics Bd. J. Bourchier 5 Sofia 1164 Bulgaria e-mail: [email protected] Submitted: May 4, 2006 Revised: January 5, 2007

Integr. equ. oper. theory 59 (2007), 53–65 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010053-13, published online June 27, 2007 DOI 10.1007/s00020-007-1508-y

Integral Equations and Operator Theory

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball Jie Miao Abstract. We give a necessary and suﬃcient condition for Hankel operators Hf on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B32. Keywords. Hankel operator, Toeplitz operator, commutator, Schatten class operator.

1. Introduction Let Bn denote the open unit ball in Rn for n ≥ 2 and let V denote the usual Lebesgue measure on Bn , normalized so that V (Bn ) = 1. Let L2 (Bn ) denote the set of all measurable functions f on Bn such that 1/2 2 |f (x)| dV (x) < ∞. f = Bn

The harmonic Bergman space L2h (Bn ) is the set of all harmonic functions on Bn that are also in L2 (Bn ). It is easy to see that L2h (Bn ) is a closed subspace of L2 (Bn ). We denote the orthogonal projection from L2 (Bn ) onto L2h (Bn ) by Q. Let R(x, y) denote the reproducing kernel for L2h (Bn ). Thus for any f ∈ L2 (Bn ) Q(f )(x) = f (y)R(x, y) dV (y), x ∈ Bn . Bn

There is a formula in closed form for R(x, y) (see [2]). The reproducing kernel R is real valued, but it can vanish on Bn × Bn . For f ∈ L2 (Bn ), let Mf denote the multiplication operator deﬁned by Mf (g) = f g. The Hankel operator Hf is deﬁned I am thankful to the referee for references [5] and [6].

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on L2 (Bn ) by Hf = (I − Q)Mf Q, the Topeplitz operator Tf is deﬁned on L2h (Bn ) by Tf = QMf , and the commutator Cf is deﬁned on L2 (Bn ) by Cf = Mf Q−QMf . Boundedness, compactness, and trace ideal criteria for Hankel operators on the analytic Bergman space of the unit ball in Cn have been well studied. Because of similarities between the analytic and harmonic Bergman spaces, it is natural to ask whether similar criteria can be established for Hankel operators Hf on the harmonic Bergman space L2h (Bn ). Some analogous results have been established in [13], and in [17] for the case n = 2. In [13], a necessary and suﬃcient condition is given for Hf to be bounded or compact on L2 (Bn ), and a question is raised about a necessary and suﬃcient condition for Hf to belong to the Schatten p-class. In this paper, we answer that question for 2 ≤ p < ∞. We also consider the special case when f ∈ L2h (Bn ). This paper is organized as follows. The main results are introduced in this section. We will also compare our results with those from [6] and [18]. Most of the necessary lemmas are given in Section 2, and the proof for the main results is given in Section 3. For r ∈ (0, 1) and x ∈ Bn , let Br (x) = {y ∈ Bn : |y − x| < r(1 − |x|)}. Let |Br (x)| denote V (Br (x)). There is a constant Cn > 0 depending on n only such that |Br (x)| = Cn rn (1 − |x|)n . For f ∈ L2 (Bn ), let 1 |f (y)|2 dV (y) MVr (f, x) = |Br (x)| Br (x) 2 1 1 f (z) dV (z) dV (y). MOr (f, x) = f (y) − |Br (x)| Br (x) |Br (x)| Br (x) Let f˜ denote the Berezin transform of f with respect to the reproducing kernel R: f˜(x) = R(x, x)−1 f (y)|R(x, y)|2 dV (y) B

and let MO(f, x) = (|f |2 )˜(x) − |f˜(x)|2 . For 0 < p < ∞ and a Hilbert space H, let Sp (H) (it will be simply denoted by Sp later) denote the Schatten p-class that consists of all compact operators T on H such that the sequence of the singular numbers sn (T ) = inf{T − K : rank K ≤ n} p

belongs to l . As is well known, Sp (H) is a two-sided ideal in the set of all bounded linearoperators on H. If T ∈ S1 (H) and {ek }∞ k=1 is an orthonormal basis for H, ∞ then k=1 T ek , ek converges absolutely and is independent of the choice of {ek }. This value is called the trace of T and is denoted tr(T ). If T is a compact operator on H, then Tp ∈ Sp (H) if and only if (T ∗ T )p/2 ∈ S1 (H). Before introducing our main results, we should mention some related results from [6] and [18]. Let B2n denote the unit ball in Cn (i.e. B2n is the unit ball in R2n ) and let P denote the orthogonal projection from L2 (B2n ) onto the analytic

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Bergman space L2a (B2n ) (which is the set of all analytic functions on B2n that are in L2 (B2n )). The Hankel operator Hfa is deﬁned on L2 (B2n ) by Hfa = (I − P )Mf P and let MOa (f, z) (for z ∈ B2n ) be deﬁned exactly like MO(f, x) except using the reproducing kernel 1 K(z, w) = (1 − z, w)n+1 instead of R(x, y). The main results of [18] are the following two theorems. Theorem A. Suppose f ∈ L2 (B2n ) and 2 ≤ p < ∞. Then both Hfa and Hfa¯ are in Sp if and only if MOa (f, z)p/2 dV (z) < ∞. n+1 B2n (1 − |z|) Theorem B. Suppose f ∈ L2a (B2n ) and n ≥ 2. Then (1) For 0 < p ≤ 2n, Hfa¯ ∈ Sp if and only if f is a constant. (2) For 2n < p < ∞, Hfa¯ ∈ Sp if and only if ˜ |2 (z)|p/2 ||f dV (z) < ∞, n+1 B2n (1 − |z|) ˜ denotes the invariant Lapacian on B2n . where We point out that when n = 1, a result similar to Theorem B is given in [3], and in that case, the largest p such that Sp contains only zero Hankel operators Hf¯, if f is analytic, is 1 (instead of 2). With our results, it is more understandable why this gap occurs. Beatrous and Li [6] extend Theorem A from the unit ball to more general strictly pseudoconvex domains in Cn . In fact they obtain a suﬃcient condition for commutators to be in Sp for p ≥ 2 when the underlying domain can be identiﬁed as X × (0, ∞) via a diﬀeomorphism (assuming that X has a homogeneous structure). The unit ball Bn is clearly such a domain. If the underlying domain is Bn and if the kernel function is the reproducing kernel R(x, y) in Theorem 1.5 of [6], then their result becomes the following: Theorem C. Suppose f ∈ L2 (Bn ) and 2 ≤ p < ∞. If MO(f, x)p/2 dV (x) < ∞, n Bn (1 − |x|)

(1.1)

then Hf ∈ Sp . The following are our main results. Theorem 1.1. Suppose f ∈ L2 (Bn ), 2 ≤ p < ∞, and 0 < r < 1. Then Hf ∈ Sp if and only if MOr (f, x)p/2 dV (x) < ∞. (1.2) (1 − |x|)n Bn

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Theorem 1.2. Suppose f ∈ L2h (Bn ) and n ≥ 3. Then (1) For 0 < p ≤ n − 1, Hf ∈ Sp if and only if f is a constant. (2) For n − 1 < p < ∞, Hf ∈ Sp if and only if | f (x)|p (1 − |x|)p dV (x) < ∞. (1 − |x|)n Bn When n = 2, both (1) and (2) of Theorem 1.2 are also true according to Theorems 4.2 and 5.1 in [17]. Now we make some comments on the relevance of conditions (1.1) and (1.2). Since 1 |f (y) − f (z)|2 |R(x, y)|2 |R(x, z)|2 dV (y) dV (z), MO(f, x) = 2R(x, x)2 Bn Bn for each r ∈ (0, 1), by Lemmas 1 and 2 in [11], there is a constant C > 0 (depending on r) such that C MO(f, x) ≥ |f (y) − f (x)|2 dV (z) dV (y) = 2C MOr (f, x). |Br (x)|2 Br (x) Br (x) This shows that (1.1) implies (1.2). Unfortunately, we are not able to give a proof that (1.2) implies (1.1) for 2 < p < ∞. The case p = 2 (the case for Hilbert-Schmidt operators) is of course well known. In other words, we can not prove that (1.1) is also necessary for Hf ∈ Sp in general. A similar question was raised in [18] (see page 166). Zhu asked, if MOa (f, z) is replaced by MOar (f, z) (which is deﬁned in the same way as MOr (f, x) except using the Bergman metric balls instead of the usual Euclidean balls), whether the condition in Theorem A is also suﬃcient for both Hfa and Hfa¯ to be in Sp .

2. Preparations In this section, we introduce a variety of lemmas in order to prove Theorems 1.1 and 1.2. From now on, we use B to denote the unit ball Bn for simplicity. For f , g ∈ L2 (B), let f, g denote B f g¯ dV . Let (L2h (B))⊥ denote the set {f ∈ L2 (B) : f, g = 0 for all g ∈ L2h (B)} and let Cc2 (B) denote the set of all C 2 functions on B with compact support. We note that the range of Hf is in (L2h (B))⊥ . The following two lemmas are given in Section 4 of [11]. Lemma 2.1. (L2h (B))⊥ = L2 -closure of {h, h ∈ Cc2 (B)}. Lemma 2.2. There are constants C1 , C2 > 0 such that |h(x)|2 | h(x)|2 dV (x) ≤ C dV (x) ≤ C | h(x)|2 dV (x) 1 2 4 2 (1 − |x|) (1 − |x|) B B B for all h ∈ Cc2 (B). The following lemma comes from Green’s identity easily.

(2.1)

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Lemma 2.3. Suppose f ∈ C 1 (B), h ∈ Cc2 (B), and u ∈ L2h (B). Then f u, h = −u, f · h + f · u, h. Based on the identity Cf = Hf ⊕ (−Hf¯)∗ and the fact that the reproducing kernel R(x, y) is real valued, we have the following lemma. Lemma 2.4. Let p ∈ (0, ∞) and f ∈ L2 (B). Then Cf ∈ Sp if and only if Hf ∈ Sp . The following lemma is stated as Lemma 5 in [13]. Lemma 2.5. Let r ∈ (0, 1) and x ∈ B. If y ∈ Br/3 (x), then Br/3 (y) ⊂ Br (x) and Br/3 (x) ⊂ Br (y). In fact the proof of Lemma 5 in [13] indicates that if y ∈ Br/3 (x), then Br/3 (y) ⊂ Br (x) and Br/3 (x) ⊂ Br (y). Now we introduce some properties on MOr (f, x). If y ∈ Br (x), then it is easy to see that (1 − r)(1 − |x|) < 1 − |y| < (1 + r)(1 − |x|). Throughout the paper, all constants that do not depend on variables will be simply denoted by C, whose value may change from one place to another. Lemma 2.6. Let f ∈ L2 (B) and r, s ∈ (0, 1). Then there is a constant C > 0 such that MOr (f, x) ≤ C MOs (f, y) for all x, y ∈ B satisfying Br (x) ⊂ Bs (y). Proof. If Br (x) ⊂ Bs (y), then 1 − |x| ≥ (1 − s)(1 − |y|), hence |Br (x)| ≥ C|Bs (y)|. The following identity 1 MOr (f, x) = |f (y) − f (z)|2 dV (y) dV (z) 2|Br (x)|2 Br (x) Br (x) gives the conclusion of the lemma immediately.

In particular, Lemma 2.6 implies that MOr (f, x) ≤ C MOs (f, x) if r < s. The following lemma comes from inequality (4.2) in [13]. Lemma 2.7. Let f ∈ L2 (B). Then there are some small r ∈ (0, 1) and a constant C > 0 such that 2 χBr (x) , x ∈ B. MOr (f, x) ≤ C Cf 1/2 |Br (x)| The following lemma is given in the proof for Lemma 6 in [13]. Lemma 2.8. Let f ∈ L2 (B) and r ∈ (0, 1). Then (1) There are a constant C > 0 and functions f1 and f2 with f2 ∈ C 1 (B) such that f = f1 + f2 , and MVr/3 (f1 , x) ≤ C MOr (f, x), | f2 (x)|(1 − |x|) ≤ C MOr (f, x)1/2 , x ∈ B. (2) MOr (f, x) ≤ 4 MVr (f, x) for all x ∈ B.

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(3) MOr (f, x)1/2 ≤ (2r/(1 − r)) supz∈Br (x) | f (z)(1 − |z|) for all x ∈ B if f ∈ C 1 (B). Lemma 2.9. Let r ∈ (0, 1). Then there is a constant C > 0 such that |f (x)|2 | u(x)|2 (1 − |x|)2 dV (x) ≤ C MVr (f, x) |u(x)|2 dV (x) B

(2.2)

B

for all u ∈ L2h (B) and f ∈ L2 (B). Proof. If u is harmonic on B, then for y ∈ B C 2 |u(y)| ≤ |u(z)|2 dV (z). (1 − |y|)n Br/9 (y) If y ∈ Br/9 (x), then Br/9 (y) ⊂ Br/3 (x) by Lemma 2.5. Cauchy’s Estimate for harmonic functions gives C C 2 2 | u(x)| ≤ sup |u(y)| ≤ |u(z)|2 dV (z). (1 − |x|)2 y∈Br/9 (x) (1 − |x|)n+2 Br/3 (x) Let U (x) denote | u(x)|2 (1 − |x|)2 . Then by Fubini’s Theorem χBr/3 (x) (z)|f (x)|2 2 |f (x)| U (x) dV (x) ≤ C dV (x) |u(z)|2 dV (z). n (1 − |x|) B B B Lemma 2.5 implies that χBr/3 (x) (z) ≤ χBr (z) (x). Thus |f (x)|2 2 |f (x)| U (x) dV (x) ≤ C dV (x) |u(z)|2 dV (z). n (1 − |x|) B B Br (z) Since (1 − |x|)n ≥ C|Br (z)| for x ∈ Br (z), inequality (2.2) now follows.

2

Lemma 2.10. Let p ∈ [2, ∞). Suppose T1 , T2 ∈ Sp (L (B)) and T3 is a bounded linear operator on L2 (B). If there is a constant C > 0 such that T3 f ≤ C[T1 f + T2 f ] 2

for all f ∈ L (B), then T3 ∈ Sp (L2 (B)). Proof. It is easy to see that T3 is a compact operator on L2 (B). Let {ek }∞ k=1 be an orthonormal set in L2 (B). Then by Minkowski’s inequality  ∞ 1/p 1/p ∞ 1/p  ∞

. T3 ek p ≤C T1 ek p + T2 ek p k=1

k=1

k=1 2

It follows from Theorem 1.4.9 in [19] that T3 ∈ Sp (L (B)).

We will need to establish a discrete version of Theorem 1.1. In order to do that, we introduce a terminology that has been used in [7]. A sequence of points {xk }∞ k=1 in B is said to be a lattice if there is some δ ∈ (0, 1) such that (i) Bδ/3 (xk ) Bδ/3 (xj ) = ∅ when k = j (in this case, the sequence {xk } is said to be separated);

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(ii) ∞ k=1 Bδ (xk ) = B. We state the following fact as a lemma whose proof is similar to that given for Lemma on Coverings in [14] and will be omitted. Lemma 2.11. If {xk } is a lattice, then for each r ∈ (0, 1), there is a positive integer N that depends on r such that every Br (xj ) intersects at most N balls of {Br (xk )}. Let µ be a positive Borel measure on B and f ∈ L1 (B, µ). Lemma 2.11 has the following implication. If {xk } is a lattice (hence there is some δ ∈ (0, 1) such B (x that ∞ k=1 δ k ) = B), then for each r ∈ (0, 1), there is a constant C > 0 such that ∞ ∞

C |f (x)| dµ(x) ≤ |f (x)| dµ(x) ≤ |f (x)| dµ(x). k=1

Br (xk )

B

k=1

Bδ (xk )

Now we introduce a trace ideal criterion on the multiplication operator Mf |L2h (B) . Our approach here is analogous to that used in [10]. According to Theorem 1.4.6 in [19] and the following identity Mf∗ Mf = T|f |2 : L2h (B) → L2h (B) we see that Mf |L2h (B) ∈ Sp if and only if T|f |2 ∈ Sp/2 . Thus we can formulate Theorem 11 in [11] as the following lemma with a minor change in the statement. Lemma 2.12. Let p ∈ [2, ∞) and r ∈ (0, 1). Then Mf |L2h (B) ∈ Sp if and only if ∞

MVr (f, xk )p/2 < ∞ for all lattices {xk }. k=1

The lemma above indicates that the convergence of independent of r ∈ (0, 1).

∞

k=1

MVr (f, xk )p/2 is

3. Proof of Theorems In order to prove Theorem 1.1, we ﬁrst establish the following result. Theorem 3.1. Let 2 ≤ p < ∞. Then Hf ∈ Sp if and only if there is some r ∈ (0, 1) ∞

such that MOr (f, xk )p/2 < ∞ for all lattices {xk }. k=1

Proof. We ﬁrst assume Hf ∈ Sp . Then by Lemma 2.4, Cf ∈ Sp . According to Lemma 2.7, there is some r ∈ (0, 1) such that 2 χBr (x) . C MOr (f, x) ≤ C (3.1) f 1/2 |Br (x)| 2 Let {xk } be a lattice and let {ek }∞ k=1 be an orthonormal basis for L (B). Let A 2 be a linear operator deﬁned on L (B) by χBr (xk ) . A(ek ) = |Br (xk )|1/2

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For any sequence of numbers {αk } ∈ l2 , Lemma 2.11 implies that there is a positive 1/2 integer N such that for every x ∈ B, ∞ has at most k=1 αk χBr (xk ) (x)/|Br (xk )| N nonzero terms in the sum. Thus by Schwarz inequality, 2 ∞ 2 ∞ ∞

χBr (xk ) χBr (xk ) A . α e = α ≤ N |αk |2 k k k |Br (xk )| |Br (xk )|1/2 k=1

k=1

k=1

This implies that A(g) ≤ N 1/2 g for all g ∈ L2 (B), hence A is a bounded operator on L2 (B). Therefore A∗ (Cf∗ Cf )p/2 A ∈ S1 and tr(A∗ (Cf∗ Cf )p/2 A) =

∞

A∗ (Cf∗ Cf )p/2 A(ek ), ek < ∞.

k=1

Since p/2 ≥ 1, by inequality (6.4) in [3] we have tr(A∗ (Cf∗ Cf )p/2 A) ≥

∞

Cf∗ Cf A(ek ), A(ek )p/2 =

k=1

∞

Cf A(ek )p .

k=1

Now it follows from (3.1) that ∞

MOr (f, xk )p/2 ≤ C

k=1

∞

Cf A(ek )p ≤ C tr(A∗ (Cf∗ Cf )p/2 A) < ∞.

k=1

This proves one implication of the theorem. the other implication. Assume there is a r ∈ (0, 1) such that ∞ Next we prove p/2 MO (f, x ) < ∞ for any lattice {xk }. By Lemma 2.8 (1), there are funcr k k=1 tions f1 and f2 such that f = f1 + f2 (f2 ∈ C 1 (B)), and MVr/9 (f1 , x) ≤ C MOr/3 (f, x), | f2 (x)|(1 − |x|) ≤ C MOr/3 (f, x)1/2 .

(3.2)

It follows that for any lattice {xk } ∞

k=1

MVr/9 (f1 , xk )p/2 ≤ C

∞

MOr/3 (f, xk )p/2 ≤ C

k=1

∞

MOr (f, xk )p/2 < ∞,

k=1

where the second inequality comes from Lemma 2.6. Therefore by Lemma 2.12, Mf1 |L2h (B) ∈ Sp , hence Hf1 = (I − Q)Mf1 Q ∈ Sp . We now need to show that Hf2 ∈ Sp . For h ∈ Cc2 (B) and u ∈ L2h (B), it follows from Lemma 2.3 that |Hf2 (u), h| ≤ |u, f2 · h| + | f2 · u, h|. Let f3 (x) denote | f2 (x)|(1 − |x|) and let f4 (x) denote MVr/9 (f3 , x)1/2 . By Schwarz inequality, we have | f2 · h| ≤ | f2 | | h|, and H¨ older’s inequality now gives |u, f2 · h| ≤ Mf3 (u) h/(1 − |x|).

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Thus by (2.1), we have |u, f2 · h| ≤ CMf3 (u) h. By a similar argument, we can get | f2 · u, h| ≤ f3 u(1 − |x|) h/(1 − |x|)2 . Now applying Lemma 2.9 to the ﬁrst factor and (2.1) to the second one on the right-hand side of the inequality above, we obtain | f2 · u, h| ≤ CMf4 (u) h. By Lemma 2.1 and the inequalities above, we have Hf2 (u) ≤ C[Mf3 (u) + Mf4 (u)],

u ∈ L2h (B).

According to Lemma 2.10, our proof will be complete if we can show both Mf3 |L2h (B) and Mf4 |L2h (B) belong to Sp . For z ∈ Br/3 (x), by Lemma 2.5, Br/3 (z) ⊂ Br (x), thus by (3.2) f3 (z) ≤ C MOr/3 (f, z)1/2 ≤ C MOr (f, x)1/2 . This implies that MVr/3 (f3 , x) =

1

|Br/3 (x)|

Br/3 (x)

(3.3)

[f3 (z)]2 dV (z) ≤ C MOr (f, x).

Furthermore if y ∈ Br/9 (x), then Br/9 (y) ⊂ Br/3 (x) (again by Lemma 2.5), hence according to (3.3) 1 [f4 (y)]2 = [f3 (z)]2 dV (z) ≤ C MOr (f, x). |Br/9 (y)| Br/9 (y) This gives MVr/9 (f4 , x) =

1 |Br/9 (x)|

Br/9 (x)

[f4 (y)]2 dV (y) ≤ C MOr (f, x).

We now conclude that for any lattice {xk }, ∞

MVr/3 (f3 , xk )p/2 ≤ C

k=1 ∞

∞

MOr (f, xk )p/2 < ∞,

k=1

MVr/9 (f4 , xk )p/2 ≤ C

k=1

∞

MOr (f, xk )p/2 < ∞.

k=1

According to Lemma 2.12, both M | of the theorem is complete.

f3 L2h (B)

and Mf4 |L2h (B) are in Sp , and the proof

The proof of Theorem 1.1 will be complete after the following lemma is established. Throughout the rest of the paper, let dλ(x) denote (1 − |x|)−n dV (x).

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Lemma 3.2. Let 2 ≤ p < ∞. Then and only if

∞

p/2

MOr (f, xk )

B

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MOr (f, x)p/2 dλ(x) < ∞ for all r ∈ (0, 1) if

< ∞ for some r ∈ (0, 1) and for all lattices {xk }.

k=1

Proof. One implication of the lemma is easy to prove. Suppose MO1/2 (f, x)p/2 dλ(x) < ∞. B

Let {xk } be a lattice. If x ∈ B1/6 (xk ), then B1/6 (xk ) ⊂ B1/2 (x) (by Lemma 2.5), thus MO1/6 (f, xk ) ≤ C MO1/2 (f, x) (by Lemma 2.6). Therefore ∞ ∞

p/2 MO1/6 (f, xk ) ≤C MO1/2 (f, x)p/2 dλ(x) B1/6 (xk )

k=1

k=1 ≤C MO1/2 (f, x)p/2 dλ(x) < ∞. B

Now we prove the other implication. Suppose there is some r0 ∈ (0, 1) such ∞ that k=1 MOr0 (f, xk )p/2 < ∞ for all lattices {xk }. By Lemma 2.8 (1), there are f1 and f2 such that f = f1 + f2 , and MVr0 /3 (f1 , x) ≤ C MOr0 (f, x),

| f2 (x)|(1 − |x|) ≤ C MOr0 (f, x)1/2 .

Thus for any lattice {xk }, ∞

k=1

MVr0 /3 (f1 , xk )p/2 ≤ C

∞

MOr0 (f1 , xk )p/2 < ∞,

k=1

∞

hence by Lemma 2.12, k=1 MVr (f1 , xk )p/2 < ∞ for any r ∈ (0, 1). Now Lemma 11 of [11] gives B MVr (f1 , x)p/2 dλ(x) < ∞, therefore according to Lemma 2.8 (2) MOr (f1 , x)p/2 dλ(x) ≤ 2p MVr (f1 , x)p/2 dλ(x) < ∞. B Thus we only need to show that B MOr (f2 , x)p/2 dλ(x) < ∞ for any r ∈ (0, 1) because of the inequality B

MOr (f, x)1/2 ≤ MOr (f1 , x)1/2 + MOr (f2 , x)1/2 . For each r ∈ (0, 1), there is a s ∈ (0, 1) small enough so that y∈Bs (x) Br (y) ⊂ Bt (x) for some t ∈ (0, 1). Hence MOr (f2 , y)p/2 dλ(y) ≤ C MOt (f2 , x)p/2 . Bs (x)

Now let {xk } be a lattice such that ∪∞ k=1 Bs (xk ) = B. Then ∞ ∞

MOr (f2 , x)p/2 dλ(x) ≤ MOr (f2 , x)p/2 dλ(x) ≤ C MOt (f2 , xk )p/2 . B

k=1

Bs (xk )

k=1

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By Lemma 2.8 (3), for each xk , there is some yk ∈ Bt (xk ) such that MOt (f2 , xk )1/2 ≤ C| f2 (yk )|(1 − |yk |). By Lemma 2.11, it is easy to see that {yk } is a ﬁnite union of separated sequences, hence ∞ ∞ ∞

MOt (f2 , xk )p/2 ≤ C | f2 (yk )|p (1 − |yk |)p ≤ MOr0 (f, yk )p/2 < ∞. k=1

This proves that lemma.

k=1

B

k=1 p/2

MOr (f2 , x)

dλ(x) < ∞ and completes the proof of the

Part (2) of Theorem 1.2 follows from Theorem 1.1 and the following lemma. Lemma 3.3. Let f ∈ L2h (B), 2 ≤ p < ∞, and 0 < r < 1. Then the following conditions are equivalent: MOr (f, x)p/2 dλ(x) < ∞. (1) B (2) | f (x)|p (1 − |x|)p dλ(x) < ∞. B

Proof. (1) ⇒ (2). This direction follows immediately from the following inequality | f (x)|(1 − |x|) ≤ C MOr (f, x)1/2 , x ∈ B. (This inequality can be easily proved. See for example the proof of Lemma 7 in [13]). (2) ⇒ (1). Suppose f ∈ L2h (B) and B | f (x)|p (1 − |x|)p dλ(x) < ∞. By the mean-value inequality for harmonic functions, we have for each k = 1, 2, . . . , n, p p |fxk (y)| dλ(y) ≤ C | f (y)|p dλ(y) |fxk (x)| ≤ C Br (x)

Br (x)

for all x ∈ B. Therefore

p/2 | f (x)|p = |fx1 (x)|2 + |fx2 (x)|2 + · · · + |fxn (x)|2 ≤ np/2−1 (|fx1 (x)|p + |fx2 (x)|p + · · · + |fxn (x)|p ) ≤C | f (y)|p dλ(y). Br (x)

Thus for any separated sequence {xk }, ∞

| f (xk )|p (1 − |xk |)p ≤ C

k=1

k=1

hence ∞

k=1

∞

p

p

| f (xk )| (1 − |xk |) ≤ C

B

Br (xk )

| f (y)|p (1 − |y|)p dλ(y),

| f (y)|p (1 − |y|)p dλ(y) < ∞.

(3.4)

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Now let {yk } be a sequence in B such that B1/3 (yk ) ∩ B1/3 (yj ) = ∅ (k = j). By Lemma 2.8 (3), there is some xk ∈ B1/9 (yk ) such that MO1/9 (f, yk )p/2 ≤ C| f (xk )|p (1 − |xk |)p . The remark following Lemma 2.5 indicates that B1/9 (xk ) ⊂ B1/3 (yk ), thus {xk } is also separated. It follows from (3.4) that ∞

MO1/9 (f, yk )p/2 ≤ C

k=1

∞

| f (xk )|p (1 − |xk |)p < ∞.

k=1

It is easy to see that the inequality above holds for any lattice {yk } because it is a ﬁnite union of such sequences according to Lemma 2.11. Now Lemma 3.2 immediately implies that B MOr (f, x)p/2 dλ(x) < ∞. Finally let us prove part (1) of Theorem 1.2. Suppose n ≥ 3 and 0 < p ≤ n−1. We only need to show that if Hf ∈ Sp , then f is a constant. If Hf ∈ Sp and 0 < p ≤ n − 1, then Hf ∈ Sn−1 because Sp ⊂ Sn−1 for p < n − 1. Since n − 1 ≥ 2, according to Theorem 1.1 and Lemma 3.3, | f (x)|n−1 (1 − |x|)−1 dλ(x) < ∞. B

Let S denote the unit sphere and let σ denote the normalized surface-area measure on S. The polar coordinates formula now gives 1 n−1 r n−1 | f (rζ)| dσ(ζ) dr < ∞. (3.5) 0 1−r S Let M (r) = S | f (rζ)|n−1 dσ(ζ). Since f ∈ L2h (B), then | f (x)|2 =

n

|fxk (x)|2

k=1

is subharmonic on B. This implies that | f (x)|n−1 = (| f (x)|2 )(n−1)/2 is subharmonic on B (because (n−1)/2 ≥ 1). It follows that M (r) is a nondecreasing function of r ∈ [0, 1). Therefore (3.5) implies that M (r) = 0 for all r ∈ [0, 1), hence 1 | f (x)|n−1 dV (x) = n rn−1 M (r) dr = 0. B

0

This show that | f | = 0 on B, thus f is a constant, and we ﬁnally complete the proof of Theorem 1.2.

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[3] J. Arazy, S. Fisher, J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989–1054. [4] D. B´ekoll´e, C. A. Berger, L. A. Coburn, and K. H. Zhu, BMO and the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), 310–350. [5] F. Beatrous and S. Li, On the Boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (1993), 350–379. [6] F. Beatrous and S. Li, Trace ideal creteria for operators of Hankel Type, Illinois J. Math. 39 (1995), 723–754. [7] R.R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions, Ast´ erisque 77 (1980), 11–65. [8] M. Jovovi´c, Compact Hankel operators on harmonic Bergman spaces, Integral Equations and Operator Theory 22 (1995), 295–304. [9] D. Luecking, Trace Ideal Criteria for Toeplitz Operators, J. Funct. Anal. 73 (1987), 345–368. [10] D. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disc, J. Funct. Anal. 110 (1992), 247–271. [11] J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations and Operator Theory 27 (1997), 426–438. [12] J. Miao, Hankel type operators on the unit disk, Studia Mathematica 146 (2001), 55–67. [13] J. Miao, Hankel operators on harmonic Bergman spaces of the unit ball, Acta Sci. Math. (Szeged) 69 (2003), 391–408. [14] V.L. Oleinik, Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 2 (1974), 135–142. [15] K. Stroethoﬀ, Compact Hankel operators on weighted harmonic Bergman spaces, Glasgow Math. J. 39 (1997), 77–84. [16] K. Stroethoﬀ and D. Zheng, Toeplitz and Hankel operators on Bergman spaces, Trans. Amer. Soc. 329, (1992), 773–794. [17] Z. Wu, Operators on harmonic Bergman spaces, Integral Equations and Operator Theory 24 (1996), 352–371. [18] K. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math. 113 (1991), 147–167. [19] K. Zhu, Operator Theory in Function Spaces, Marcell-Dekker, New York, 1990. Jie Miao Department of Mathematics and Statistics P.O. Box 70 State University, AR 72467 U.S.A. e-mail: [email protected] Submitted: May 11, 2006 Revised: May 21, 2007

Integr. equ. oper. theory 59 (2007), 67–98 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010067-32, published online June 27, 2007 DOI 10.1007/s00020-007-1520-2

Integral Equations and Operator Theory

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains Raul Quiroga-Barranco and Nikolai Vasilevski Abstract. Let D be a bounded logarithmically convex complete Reinhardt domain in Cn centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the C ∗ -algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on |z1 |, |z2 |, . . . , |zn |) is commutative. We show that the natural action of the n-dimensional torus Tn deﬁnes (on a certain open full measure subset of D) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball. Mathematics Subject Classiﬁcation (2000). Primary 47B35; Secondary 32A07, 32A36. Keywords. Toeplitz operator, Bergman space, separately radial symbol, Reinhardt domain, commutative C ∗ -algebra.

1. Introduction A family of recently discovered commutative C ∗ -algebras of Toeplitz operators on the unit disk (see for details [12, 13]) can be classiﬁed as follows. Each pencil of hyperbolic geodesics determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to geodesics forming a pencil. The C ∗ -algebra generated by Toeplitz operators with such symbols turns out to be commutative. Moreover, these commutative properties do not depend at all on smoothness properties of symbols: the corresponding symbols can be merely measurable. The prime cause appears to be the geometric conﬁguration of level lines of symbols. Further it has been proved in [4] that, assuming some This work was partially supported by CONACYT Projects 46936 and 44620, M´exico.

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natural conditions on the “richness” of the symbol set, the above symbol sets are the only possible which gnerate commutative Toeplitz operator algebras on each (commonly considered) weighted Bergman space on the unit disk. Recall that there are three diﬀerent types of pencils of hyperbolic geodesics: an elliptic pencil, which is formed by geodesics intersecting in a single point, a parabolic pencil, which is formed by parallel geodesics, and a hyperbolic pencil, which is formed by disjoint geodesics, i.e., by all geodesics orthogonal to a given one. Note, that in all cases the cycles are equidistant in the hyperbolic metric. The model case for elliptic pencils is when the geodesics intersect at the origin. In this case the geodesics are diameters and the cycles are the concentric circles centered at the origin. All other elliptic pencils can be obtained from this model by means of M¨ obius transformations. The commutative Toeplitz C ∗ -algebra for the elliptic model case is generated by Toeplitz operators with radial symbols. As proved in [3], the C ∗ -algebras generated by Toeplitz operators with radial symbols, acting on the weighted Bergman spaces over the unit ball Bn , are commutative as well. In the present paper, we consider a more deep and natural multidimensional analog of the elliptic model pencil on the unit disk. We study Toeplitz operators on weighted Bergman spaces over bounded Reinhardt domains in Cn , and prove, in particular, that the C ∗ -algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on |z1 |, |z2 |, . . . , |zn |) is commutative. Note that this single result can be also obtained directly by just calculating the matrix elements Ta z p , z q , but we deliberately follow a more general procedure used in all model cases on the unit disk (see, for example, in [12]). This permits us to construct an analog of the Bargman transform (the operator R restricted on the (weighted) Bergman space), obtain the decomposition of the Bergman projection by means of R∗ and R, and prepare these operators for the future use. The second important question treated in the paper is the understanding of an adequate geometric description which generalizes geodesics and cycles of the unit disk to a multidimensional case. Each complete bounded Reinhardt domain D in Cn centered at the origin admits a natural action of the n-dimensional torus Tn , and this action is isometric with respect to the Bergman metric in D. On a certain open full measure subset of D this action deﬁnes a foliation whose leaves are all diﬀeomorphic to Tn . Furthermore, such foliation carries a transverse Riemannian structure having distinguished geometric features. First, the leaves are equidistant with respect to the Bergman metric, and second, the direction perpendicular to the leaves is totally geodesic, every geodesic which starts in the perpendicular direction to a leaf stays perpendicular to all other leaves. Now geometrically: the C ∗ -algebra generated by Toeplitz operators with bounded measurable symbols, which are constant on the leaves of the above foliation, is commutative. We prove also that the orthogonal complement to the tangent bundle of the Tn -orbits is integrable, thus providing a pair of natural orthogonal foliations to a Reinhardt domain. Moreover, it turns out that both foliations are Lagrangian.

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It is worth mentioning that the above geometric properties hold for each pencil of geodesics on the unit disk, but do not hold, for example, for the case of Toeplitz operators on the unit ball with radial symbols (the corresponding foliations are not Lagrangian). We show that the unit ball Bn in Cn is the only Reinhardt domain which is, at the same time, bounded symmetric and irreducible. Then, we provide a detailed description of the extrinsic geometry of the foliation by Tn -orbits in the unit ball. In particular, it is shown, through a computation of its second fundamental form, that certain geodesics in this foliation have geodesic curvatures with the same behavior found in the elliptic pencil of the unit disk. We then consider oneparameter families of weighted Bergman spaces in Bn , commonly used in operator theory, and specify the results obtained to this special case. Finally, using C. Feﬀerman’s expression for the Bergman kernel of strictly pseudoconvex domains, we show that for any bounded complete Reinhardt domain with such pseudoconvexity property, the extrinsic curvature of the foliation coming from the Tn -action has the same asymptotic behavior at (suitable) boundary points as the one observed in the unit ball.

2. Bergman space on the Reinhardt domains Denote by P (0, r), where r = (r1 , . . . , rn ) and each rk > 0, the closed polydisk in Cn centered at the origin: P (0, r) = {z = (z1 , . . . , zn ) : |zk | ≤ rk , k = 1, . . . , n}. Recall (see, for example, [11]) that an open domain D in Cn is called the complete Reinhardt domain centered at the origin if for every its point z the polydisk P (0, τ (z)), where τ (z) = (|z1 |, . . . , |zn |), belongs to D. The set τ (D) = {r = (r1 , . . . , rn ) = (|z1 |, . . . , |zn |) : z = (z1 , . . . , zn ) ∈ D}, which belongs to Rn+ = R+ × · · · × R+ , is called the base of the Reinhardt domain D. The Reinhardt domain D is called logarithmically convex if the set log τ (D) is convex. It is well known (see, for example, [11]) that the Reinhardt domain is logarithmically convex if and only if it is a domain of holomorphy, or if and only if it is a region of convergence of a power series. Let now D be a bounded logarithmically convex complete Reinhardt domain in Cn centered at the origin. Consider a positive measurable function (weight) µ(r) = µ(r1 , . . . , rn ), r ∈ τ (D), such that n µ(|z|)dv(z) = (2π) µ(r)rdr < ∞, D

τ (D)

where dv(z) = dx1 dy1 · · · dxn dyn is the usual Lebesgue measure in Cn , |z| = n (|z1 |, . . . , |zn |), and rdr = k=1 rk drk . We assume as well that the weight-function

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µ(r) is bounded in some neighborhood of the origin and does not vanish in this neighborhood. Introduce the weighted Hilbert space L2 (D, µ) with the scalar product f (z) g(z) µ(|z|) dv(z), f, g = D

and its subspace, the weighted Bergman space A2µ (D), which consists of all functions analytic in D. We denote as well by BD,µ the (orthogonal) Bergman projection of L2 (D, µ) onto A2µ (D). Passing to the polar coordinates zk = tk rk , where tk ∈ T = S 1 , k = 1, . . . , n, and under the identiﬁcation z = (z1 , . . . , zn ) = (t1 r1 , . . . , tn rn ) = (t, r), where t = (t1 , . . . , tn ) ∈ Tn = T × · · · × T, r = (r1 , . . . , rn ) ∈ τ (D), we have D = Tn × τ (D) and n n dtk dv(z) = rk drk . itk k=1

k=1

That is we have the following decomposition L2 (D, µ) = L2 (Tn ) ⊗ L2 (τ (D), µ), where n

L2 (T ) =

n

L2 (T,

k=1

dtk ) itk

and the measure dµ in L2 (τ (D), µ) is given by dµ = µ(r1 , . . . , rn )

n

rk drk .

k=1

We note that the Bergman space A2µ (D) can be alternatively deﬁned as the (closed) subspace of L2 (D, µ) which consists of all functions satisfying the equations ∂ ∂ 1 ∂ ϕ= +i ϕ = 0, k = 1, . . . , n, ∂z k 2 ∂xk ∂yk or, in the polar coordinates, ∂ tk tk ∂ ∂ ϕ= − ϕ = 0, ∂z k 2 ∂rk rk ∂tk

k = 1, . . . , n.

Deﬁne the discrete Fourier transform F : L2 (T) → l2 = l2 (Z) by dt 1 f (t) t−n , n ∈ Z. F : f −→ cn = √ it 2π S 1 The operator F is unitary and

1 F −1 = F ∗ : {cn }n∈Z −→ f = √ c n tn . 2π n∈Z

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It is easy to check (see, for example, [12], Subsection 4.1) that the operator t ∂ t ∂ − u = (F ⊗ I) (F −1 ⊗ I) : l2 ⊗ L2 ((0, 1), rdr) −→ l2 ⊗ L2 ((0, 1), rdr) 2 ∂r r ∂t acts as follows u : {ck (r)}k∈Z −→

1 ∂ k−1 − . ck−1 (r) 2 ∂r r k∈Z

Introduce the unitary operator U = F(n) ⊗ I : L2 (Tn ) ⊗ L2 (τ (D), µ) −→ l2 (Zn ) ⊗ L2 (τ (D), µ), where F(n) = F ⊗ · · · ⊗ F . Then the image A21 = U (A2µ (D)) of the Bergman space is the closed subspace of l2 (Zn ) ⊗ L2(τ (D), µ) which consists of all sequences {cp (r)}p∈Zn , r = (r1 , . . . , rn ) ∈ τ (D), satisfying the equations 1 ∂ pk − k = 1, . . . , n. c(p1 ,...,pk ) (r1 , . . . , rn ) = 0, 2 ∂rk rk These equations are easy to solve, and their general solutions have the form cp (r) = αp cp rp ,

p = (p1 , . . . , pn ) ∈ Zn ,

where cp ∈ C, rp = r1p1 · · · · · rnpn , and αp = α|p| ( |p| = (|p1 |, . . . , |pn |), in this occurrence) is given by

− 12 αp

r2|p| µ(r) rdr

= τ (D)

=

τ (D)

2|p | r1 1

· · · · · rn2|pn | µ(r1 , . . . , rn )

n

− 12 rk drk

.

(2.1)

k=1

Recall that each function cp (r) = αp cp rp has to be in L2 (τ (D), µ), which implies that cp = 0 for each p = (p1 , . . . , pn ) such that at least one of pk < 0, k = 1, . . . , n. That is the space A21 ⊂ l2 (Zn ) ⊗ L2 (τ (D), µ) coincides with the space of all sequences αp cp rp , p ∈ Zn+ = Z+ × · · · × Z+ , cp (r) = 0, p ∈ Zn \ Zn+ and furthermore {cp (r)}p∈Zn+ l2 (Zn )⊗L2 (τ (D),µ) = {cp }p∈Zn+ l2 (Zn ) . Introduce now the isometric embedding R0 : l2 (Zn+ ) −→ l2 (Zn ) ⊗ L2 (τ (D), µ) as follows

R0 : {cp }

p∈Zn +

−→ cp (r) =

αp cp rp , p ∈ Zn+ . 0, p ∈ Zn \ Zn+

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Then the adjoint operator R0∗ : l2 (Zn ) ⊗ L2 (τ (D), µ) −→ l2 (Zn ) is deﬁned by

n rp fp (r) µ(r1 , . . . , rn ) rk drk , R0∗ : {fp (r)}p∈Zn −→ αp τ (D)

k=1

p∈Zn +

and it is easy to check that R0∗ R0 = I R0 R0∗ = P1

: l2 (Zn+ ) −→ l2 (Zn+ ), : l2 (Zn ) ⊗ L2 (τ (D), µ) −→ A21 ,

where P1 is the orthogonal projection of l2 (Zn ) ⊗ L2 (τ (D), µ) onto A21 . Summarizing the above we have Theorem 2.1. The operator R = R0 U maps L2 (D, µ) onto l2 (Zn+ ), and the restriction R|A2µ (D) : A2µ (D) −→ l2 (Zn+ ) is an isometric isomorphism. The adjoint operator R∗ = U ∗ R0 : l2 (Zn+ ) −→ A2µ (D) ⊂ L2 (D, µ) is the isometrical isomorphism of l2 (Zn+ ) onto the subspace A2µ (D) of L2 (D, µ). Furthermore RR∗ = I R∗ R = BD,µ

: l2 (Zn+ ) −→ l2 (Zn+ ), : L2 (D, µ) −→ A2µ (D),

where BD,µ is the Bergman projection of L2 (D, µ) onto A2µ (D). Theorem 2.2. The isometric isomorphism R∗ = U ∗ R0 : l2 (Zn+ ) −→ A2µ (D) is given by n

R∗ : {cp }p∈Zn+ −→ (2π)− 2

αp cp z p .

(2.2)

p∈Zn +

Proof. Calculate R∗ = U ∗ R0

: =

{cp }p∈Zn+ −→ U ∗ ({αp cp rp }p∈Zn+ ) n n αp cp (tr)p = (2π)− 2 αp cp z p . (2π)− 2 p∈Zn +

p∈Zn +

Corollary 2.3. The inverse isomorphism R : A2µ (D) −→ l2 (Zn+ ) is given by

−n p 2 R : ϕ(z) −→ (2π) αp ϕ(z) z µ(|z|) dv(z) D

p∈Zn +

.

(2.3)

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3. Toeplitz operators with separately radial symbols We will call a function a(z), z ∈ D, separately radial if a(z) = a(r) = a(r1 , . . . , rn ), i.e., a depends only on the radial components of z = (z1 , . . . , zn ) = (t1 r1 , . . . , tn rn ). Theorem 3.1. Let a = a(r) be a bounded measurable separately radial function. Then the Toeplitz operator Ta acting on A2µ (D) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on l2 (Zn+ ), where R and R∗ are given by (2.3) and (2.2) respectively. The sequence γa = {γa (p)}p∈Zn+ is as follows n a(r) r2p µ(r1 , . . . , rn ) rk drk , p ∈ Zn+ , (3.1) γa (p) = α2p τ (D)

k=1

where αp is given by (2.1). Proof. The operator Ta is obviously unitary equivalent to the operator R Ta R ∗

= R BD,µ aBD,µ R∗ = R(R∗ R)a(R∗ R)R∗ = (RR∗ )RaR∗ (RR∗ ) = RaR∗ = R0∗ U a(r)U −1 R0

−1 = R0∗ (F(n) ⊗ I)a(r)(F(n) ⊗ I)R0

= R0∗ a(r)R0 . Now R0∗ a(r)R0 {cp }p∈Zn+

= R0∗ {a(r) αp cp rp }p∈Zn + =

αp

p

τ (D)

p

r a(r) αp cp r µ(r1 , . . . , rn )

n

rk drk

k=1

p∈Zn +

= {γa (p) · cp }p∈Zn+ , where γa (p) = α2p

τ (D)

a(r) r2p µ(r1 , . . . , rn )

n

rk drk ,

p ∈ Zn+ .

k=1

It is easy to see that the system of functions {ep }p∈Zn+ , where ep (z) = n (2π)− 2 αp z p , forms an orthonormal base in A2µ (D). Corollary 3.2. The Toeplitz operator Ta with bounded measurable separately radial symbol a(r) is diagonal with respect to the above orthonormal base: Ta ep = γa (p) · ep ,

p ∈ Zn+ .

(3.2)

We can easily extend the notion of the Toeplitz operator for measurable unbounded separately radial symbols. Indeed, given a symbol a = a(r) ∈ L1 (τ (D), µ), we still have equality (3.2). Then the densely deﬁned (on the ﬁnite linear combinations of the above base elements) Toeplitz operator can be extended to a

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bounded operator on a whole A2µ (D) if and only if the sequence γa = {γa (p)}p∈Zn+ is bounded. That is we have Corollary 3.3. The Toeplitz operator Ta with separately radial symbol a = a(r) ∈ L1 (τ (D), µ) is bounded on A2µ (D) if and only if γa = {γa (p)}p∈Zn+ ∈ l∞ , and Ta = sup |γa (p)|. p∈Zn +

The Toeplitz operator Ta is compact if and only if γa ∈ c0 that is lim γa (p) = 0.

p→∞

The spectrum of the bounded Toeplitz operator Ta is given by sp Ta = {γa (p) : p ∈ Zn+ }, and its essential spectrum ess − sp Ta coincides with the set of all limit points of the sequence {γa (p)}p∈Zn+ . Corollary 3.4. The C ∗ -algebra generated by Toeplitz operators with separately radial L∞ -symbols is commutative.

4. Foliations, transverse Riemannian structures, and bundle-like metrics In this section we will brieﬂy summarize some notions of foliations and their geometry. We refer to [8] for further details. A foliation on a manifold M is a partition of M into connected submanifolds of the same dimension that locally looks like a partition given by the ﬁbers of a submersion. The local picture is given by considering foliated charts and the partition as a global object is obtained by imposing a compatibility condition between the foliated charts. We make more precise this notion through the following deﬁnitions. Deﬁnition 4.1. On a smooth manifold M a codimension q foliated chart is a pair (ϕ, U ) given by an open subset U of M and a smooth submersion ϕ : U → V , where V is an open subset of Rq . For a foliated chart (ϕ, U ) the connected components of the ﬁbers of ϕ are called the plaques of the foliated chart. Two codimension q foliated charts (ϕ1 , U1 ) and (ϕ2 , U2 ) are called compatible if there exists a diﬀeomorphism ψ12 : ϕ1 (U1 ∩ U2 ) → ϕ2 (U1 ∩ U2 ) such that the following diagram commutes: U1 ∩ U2N NNN pp p NNϕN2 p pp NNN p p p N& p xp ψ12 / ϕ2 (U1 ∩ U2 ) ϕ1 (U1 ∩ U2 ) ϕ1

(4.1)

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A foliated atlas on a manifold M is a collection {(ϕ α , Uα )}α of foliated charts which are mutually compatible and that satisfy M = α Uα . It is straightforward to check that the compatibility of two foliated charts (ϕ1 , U1 ) and (ϕ2 , U2 ) ensures that, when restricted to U1 ∩ U2 , both submersions ϕ1 and ϕ2 have the same plaques. This in turn implies that, for any given foliated atlas, the following is an equivalence relation in M . x∼y

⇐⇒

there is a sequence of plaques (Pk )lk=0 for foliated charts

(ϕk , Uk )lk=0 of the foliated atlas, such that x ∈ P0 , y ∈ Pl , and Pk−1 ∩ Pk = φ for every k = 1, . . . , l We will refer to the latter as the equivalence relation of the foliated atlas. It is a simple matter to show that the equivalence classes are in fact submanifolds of M of dimension dim(M ) − q, where q is the (common) codimension of the foliated charts. Deﬁnition 4.2. A foliation F on a manifold M is a partition of M which can be described as the classes of the equivalence relation of a foliated atlas. The classes are called the leaves of the foliation. Suppose that M is a manifold carrying a smooth foliation F. We will denote with T F the vector subbundle of T M that consists of elements tangent to the leaves of F. We can consider the quotient bundle T M/T F which we will denote by T t F. The latter will be referred to as the transverse vector bundle of the foliation F. Since T t F is a smooth vector bundle, we can consider the associated linear frame bundle which we will denote with LT (F). More precisely, we have as a set: LT (F) = {A : A : Rq → Txt F = Tx M/TxF is an isomorphism and x ∈ M }, where q is the codimension of F in M . It is easily seen that LT (F) is a principal ﬁber bundle with structure group GLq (R), we refer to [8] for the details of the proof. The principal bundle LT (F) is called the transverse frame bundle since it allows us to study the geometry transverse to the foliation F. When studying the transverse geometry of a foliation F it is useful to consider a certain natural foliation in LT (F), which is deﬁned as follows. Suppose that for a foliation F on a manifold M we choose a foliated atlas {(ϕα , Uα )}α that determines the foliation as in Deﬁnition 4.2. For any foliated chart (ϕα , Uα ) and every x ∈ Uα we have a linear map d(ϕα )x : Tx M → Rq whose kernel is Tx F. This induces a linear isomorphism d(ϕα )tx : Txt F = Tx M/Tx F → Rq . The latter allows us to deﬁne the smooth map: ϕ(1) α : LT (F|Uα ) → L(Vα )

A → d(ϕα )tx ◦ A,

where LT (F|Uα ) is the open subset of LT (F) given by inverse image of Uα under the natural projection LT (F) → M , A is mapped to x under such projection and Vα is the target of ϕα . Next we observe that, since Vα is open in Rq , the manifold

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L(Vα ) is open in Rq × GLq (R) and so it is open in Rq+q as well. Furthermore, from our choices it is easy to check that the commutative diagram (4.1) and the compatibility of charts in a foliated atlas induce a corresponding commutative diagram given by: LT (F|U1 ∩U2 ) QQQ mm QQQϕ(1) m m m QQαQ2 m m m QQQ m m m Q( (1) vmm ψα 1 α2 / L(ϕα2 (U1 ∩ U2 )) L(ϕα1 (U1 ∩ U2 )) ϕ(1) α1

(1)

where ψα1 α2 is deﬁned as above for the diﬀeomorphism ψα1 α2 for which we have (1) ϕα2 = ψα1 α2 ◦ϕα1 , as in diagram (4.1). This shows that the set {(ϕα , LT (F|Uα ))}α deﬁnes a foliated atlas. The corresponding foliation in LT (F) is called the lifted foliation. We state without proof the following result which can be found in [8]. Theorem 4.3. Let F be a foliation on a smooth manifold M . Then, the natural projection LT (F) → M maps the leaves of the lifted foliation of LT (F) locally diﬀeomorphically onto the leaves of F. From its construction, the principal ﬁber bundle LT (F) → M models some aspects of the transverse geometry of the foliation F. At the same time, Theorem 4.3 shows that the lifted foliation in LT (F) is needed to fully capture the foliated nature of the transverse geometry of F. In order to deﬁne transverse geometric structures for a given foliation F we consider now reductions of LT (F) compatible with the lifted foliation. More precisely, we have the following deﬁnition which also introduces the notion of a Riemannian foliation. Deﬁnition 4.4. Let M be a manifold carrying a smooth foliation F of codimension q, and let H be a Lie subgroup of GLq (R). A transverse geometric H-structure is a reduction Q of LT (F) to the subgroup H which is saturated with respect to the lifted foliation, i.e. such that Q ∩ L = φ implies L ⊂ Q for every leaf L of the lifted foliation. A transverse geometric O(q)-structure is also called a transverse Riemannian structure. A foliation endowed with a transverse Riemannian structure is called a Riemannian foliation. From the deﬁnition, it is easy to see that a transverse Riemannian structure deﬁnes a Riemannian metric on the bundle T M/T F = T t F. However, a transverse Riemannian structure is more than a simple Riemannian metric on T t F. By requiring the O(q)-reduction that deﬁnes a transverse Riemannian structure to be saturated with respect to the lifted foliation, as in Deﬁnition 4.4, we ensure the invariance of the metric as we move along the leaves in M . This is a well known property of Riemannian foliations whose further discussion can be found in [8] and other books on the subject. Here we observe that, since a Riemannian metric on a manifold deﬁnes a distance, the invariance of a transverse Riemannian structure as we move along the leaves can be interpreted as the leaves of the foliation in M to be equidistant while we move along them. Again, this sort of remark is

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well known in the theory of foliations and shows that a Riemannian foliation has a distinguished geometry. In particular, not every foliation admits a Riemannian structure, a standard example is given by the Reeb foliation of the sphere S 3 (see [8]). A fundamental way to construct transverse Riemannian structures for a foliation is to consider suitable Riemannian metrics on the manifold that carries the foliation. To describe such construction we will need some additional notions. Deﬁnition 4.5. Let F be a smooth foliation on a manifold M . A vector ﬁeld X on M is called foliate if for every vector ﬁeld Y tangent to the leaves of F the vector ﬁeld [X, Y ] is tangent to the leaves as well. From the previous deﬁnition, we observe that the set of foliate vector ﬁelds is the normalizer of the ﬁelds tangent to the leaves of F in the Lie algebra of all vector ﬁelds on M . Deﬁnition 4.6. Let F be a smooth foliation on a manifold M . A Riemannian metric h in M is called bundle-like for the foliation F if the real-valued function h(X, Y ) is constant along the leaves of F for every pair of vector ﬁelds X, Y which are foliate and perpendicular to T F with respect to h. Suppose that h is a Riemannian metric on a manifold M and that F is a foliation on M . Then, the canonical projection T M → T t F allows us to induce a Riemannian metric on the bundle T t F, which in turn provides an O(q)-reduction of the transverse frame bundle LT (F) (where q is the codimension of F). Nevertheless, such reduction does not necessarily deﬁnes a transverse Riemannian structure. The next result states that bundle-like metrics are precisely those that deﬁne transverse Riemannian structures. The proof of this theorem can be found in [8]. Theorem 4.7. Let M be a manifold carrying a smooth foliation F of codimension q. For every Riemannian metric h on M , denote by OT (M, h) the O(q)-reduction of LT (F) given by the Riemannian metric on T t (F) coming from h and the natural projection T M → T t F. If h is a bundle-like metric, then OT (M, h) deﬁnes a transverse Riemannian structure on F. Conversely, for every transverse Riemannian structure given by a reduction Q as in Deﬁnition 4.4, there is a bundle-like metric h on M such that Q = OT (M, h). Based on this result, we give the following deﬁnition. Deﬁnition 4.8. Let F be a Riemannian foliation on a manifold M . We will say that a bundle-like metric h on M is compatible with the Riemannian foliation if OT (M, h) is the reduction which deﬁnes the corresponding transverse Riemannian structure. A fundamental property of Riemannian foliations is that, with respect to compatible bundle-like metrics, geodesics which start perpendicular to a leaf of the foliation stay perpendicular to all leaves.

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Theorem 4.9. Let F be a Riemannian foliation on a manifold M and let h be a compatible bundle-like metric. If γ is a geodesic of h such that γ (t0 ) ∈ (Tγ(t0 ) F)⊥ , for some t0 , then γ (t) ∈ (Tγ(t) F)⊥ for every t. This theorem is fundamental in the theory of Riemannian foliations and its proof can be found in [8]. We can provide its geometric interpretation as follows. Let M , F and h be as in Theorem 4.9, and denote with T F⊥ the orthogonal complement of T F in T M ; in particular, T M = T F ⊕ T F⊥ . Hence, Theorem 4.9 states that every geodesic with an initial velocity vector in T F⊥ has velocity vector contained in T F⊥ for all time. In a sense, the above states that the orthogonal complement T F⊥ contains all geodesics perpendicular to T F. If the codimension of F is 1, then T F⊥ is onedimensional and it can be integrated to a smooth one-dimensional foliation F⊥ whose leaves are perpendicular to those of F. In such case, Theorem 4.9 ensures that the leaves of F⊥ are geodesics with respect to the bundle-like metric h. If F has codimension greater than 1, then we can still consider the possibility of T F⊥ to be integrable, e.g. to satisfy the hypothesis of Frobenius theorem (see [14]). In such case, we do have a foliation F⊥ whose leaves are orthogonal to those of F. Again, in this case, Theorem 4.9 implies that the leaves of F⊥ are totally geodesic. At the same time, the vector bundle T F⊥ is not always integrable. Nevertheless, the above discussion shows that T F⊥ can be thought of as being totally geodesic from a broader viewpoint. Alternatively, we can say that, from a geometric point of view, the foliation F is transversely totally geodesic. It is worth mentioning that the integrability of the bundle T F⊥ given by a Riemannian foliation and a bundle-like metric is not at all trivial and requires strong restrictions on the geometry of the foliation or its leaves. As an example, we refer to [10], where the integrability of the corresponding T F⊥ is only obtained for leaves carrying a suitable nonpositively curved Riemannian metric. At the same time, we will prove in the following sections that the orthogonal complement to the tangent bundle of the Tn -orbits in a Reinhardt domain is integrable, which will then imply the presence of strong geometric features on such domains.

5. Extrinsic geometry of foliations For a submanifold of any Riemannian manifold one can measure the obstruction for the submanifold to be a totally geodesic in the ambient. This also measures the extrinsic curvature of the submanifold, which is determined by the particular embedding and not just the inherited metric. We now brieﬂy discuss some well known methods to study this extrinsic curvature and refer to [7] and [9] for further details. For our purposes it will be convenient and natural to discuss these notions for foliations. be a Riemannian manifold and F be a foliation of M having codiLet M mension q and with p-dimensional leaves. We will denote by ∇ the Levi-Civita and by ∇ the connection of the bundle T F obtained by pasting connection of M

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together the Levi-Civita connections of the leaves of F for the metric inherited . Let us also denote by V and H the orthogonal projections of from that of M T M onto T F and T F⊥ , respectively. These projections are respectively called the vertical and horizontal projections with respect to F. Then the following holds (see [7]): More precisely, we Lemma 5.1. The connection ∇ is the vertical projection of ∇. have: X Y ), ∇X Y = V(∇ everywhere tangent to the leaves of F. for every pair of vector ﬁelds in M We recall that the Levi-Civita connection is the diﬀerential operator that allows to deﬁne geodesics. Hence, the previous result shows that the obstruction and for the leaves of F to be totally geodesic is precisely the diﬀerence between ∇ its vertical projection as above, in other words, the horizontal projection of ∇. This suggests to introduce the following classical deﬁnition (see [7]). . The second Deﬁnition 5.2. Let F be a foliation of a Riemannian manifold M fundamental form II of the leaves of F is given at every x ∈ M by: II x : Tx F × Tx F → (u, v) →

T x F⊥ X Y )x , H(∇

everywhere where X, Y are vector ﬁelds deﬁned in a neighborhood of x in M tangent to F and such that Xx = v and Yx = v. It is very well known that the deﬁnition of II x as above does not depend on the choice of the vector ﬁelds X and Y . It is also known that the second fundamental form at every every point is a symmetric bilinear form that deﬁnes a tensor which is a section of the bundle T F∗ ⊗ T F∗ ⊗ T F⊥. As it occurs with any tensor, it is easier to describe some of the properties of II by introducing local bases for the bundles involved and computing the components with respect to such bases. This is particularly useful if one has global bases for the bundles. These are deﬁned more precisely as follows. be a subbundle of the tangent bundle of the RiemannDeﬁnition 5.3. Let E → M . Then, a collection (V1 , . . . , Vk ) of sections of E deﬁned on all of ian manifold M , the set of tangent vectors M is called a global framing of E if for every x ∈ M (V1 (x), . . . , Vk (x)) is a basis for the ﬁber Ex . The next result is an obvious consequence of the symmetry of II . It will allow us to simplify the computation of the values for II . be a Riemannian manifold with a foliation F as above. Proposition 5.4. Let M Suppose that (Vk )pk=1 is a global framing of T F. Then II as a tensor is completely determined by the vector ﬁelds II (Vk + Vl , Vk + Vl ) for k, l = 1, . . . , p.

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Proof. It is enough to use the relation: 1 II (Vk , Vl ) = (II (Vk + Vl , Vk + Vl ) − II (Vk , Vk ) − II (Vl , Vl )) 2 which is satisﬁed by the symmetry of II .

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6. Isometric actions of Lie groups In this section we will consider some general notions about actions of Lie on a manifold preserving a Riemannian metric. In what follows M will a smooth manifold and G a connected Lie group acting smoothly on the M . For the next deﬁnition, we recall that the stabilizer of a point x ∈ M G-action is the set Gx = {g ∈ G : gx = x}.

groups denote left on for the

Deﬁnition 6.1. The action of G on M is called free (locally free) if for every x ∈ M the stabilizer Gx is trivial (respectively discrete). A straightforward application of Frobenius theorem on the integrability of vector subbundles of a tangent bundle (see [14]) allows us to obtain the following result. Proposition 6.2. If G acts locally freely on M , then the G-orbits deﬁne a smooth foliation on M . Proof. Denote by g the Lie algebra of G. Then for every X ∈ g we can deﬁne the transformations of M given by the maps: ϕt : M x

→ M → exp(tX)x

for every t ∈ R. This family of maps is in fact a one-parameter group of diﬀeomorphism of M , in other words, we have: ϕt1 +t2 = ϕt1 ◦ ϕt2 for every t1 , t2 ∈ R. Hence, there is a smooth vector ﬁeld X ∗ on M given by: d Xx∗ = (exp(tX)x). dt t=0 Also, it is easy to check that the global ﬂow of X ∗ is given by (ϕt )t . Furthermore, since the Lie group G acts locally freely, the condition Xx∗ = 0 for some x ∈ M implies X = 0; otherwise the subgroup (exp(tX))t would be nondiscrete and contained in Gx for some x ∈ M . From the above remarks it follows that the map: M × g → TM (x, X) → Xx∗ is a smooth vector bundle inclusion which thus deﬁnes a subbundle T O of T M .

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On the other hand, by using the results in [6], the following relation holds for every X, Y ∈ g: [X ∗ , Y ∗ ] = −[X, Y ]∗ From this it is easy to conclude that the smooth sections of T O are closed under the Lie brackets of smooth vector ﬁelds. By Frobenius theorem (see [14]) the vector subbundle T O induces a smooth foliation whose leaves have the ﬁbers of T O as tangent spaces. Since G is connected it is generated by the set exp g, and so one can conclude that the leaves of such foliation are precisely the G-orbits. In the proof of the previous result it is shown that the tangent bundle of the foliation by G-orbits is T O. Whenever G acts locally freely we will use T O to denote such tangent bundle. We will now consider the case where G acts locally freely preserving a Riemannian metric on M . Theorem 6.3. If G acts locally freely on M preserving a Riemannian metric h, then the G-orbits deﬁne a smooth Riemannian foliation for which h is a compatible bundle-like metric. Proof. By Theorem 4.7 it is enough to show that h is bundle-like with respect to the foliation by G-orbits given by Proposition 6.2. Choose X and Y foliate vector ﬁelds perpendicular to the G-orbits. We need to prove that v(h(X, Y )) = 0, for every v ∈ T O. By the proof of Proposition 6.2 there exists Z ∈ g, the Lie algebra of G, such that Zx∗ = v, where x is the basepoint of v. Hence, it suﬃces to prove that Z ∗ (h(X, Y )) = 0 for every Z ∈ g. For any Z ∗ as above, we denote with LZ ∗ the Lie derivative with respect to ∗ Z and refer to [6] for the deﬁnition. In fact, from [6] it follows that LZ ∗ when applied to h yields a bilinear form that satisﬁes: (LZ ∗ h)(X, Y ) = Z ∗ (h(X, Y )) − h([Z ∗ , X], Y ) − h(X, [Z ∗ , Y ]).

(6.1)

On the other hand, since the one-parameter group (exp(tX))t acts by isometries on (M, h), i.e. preserving h, it follows that Z ∗ is a Killing ﬁeld for h and so it satisﬁes: (LZ ∗ h)(X, Y ) = 0, (6.2) we refer to [6] for this fact and the deﬁnitions involved. From equations (6.1) and (6.2) we obtain: Z ∗ (h(X, Y )) = h([Z ∗ , X], Y ) + h(X, [Z ∗ , Y ]). Then we observe that, since X and Y are foliate, the vector ﬁelds [Z ∗ , X] and [Z ∗ , Y ] are tangent to the G-orbits, and so the terms on the right-hand side of the last equation vanish since X and Y are also perpendicular to the G-orbits. This shows that Z ∗ (h(X, Y )) = 0 thus concluding the proof. From the last result and Theorem 4.9 we obtain the following consequence.

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Theorem 6.4. If G acts locally freely on M preserving a Riemannian metric h and γ is a geodesic (with respect to h) perpendicular at some point to a G-orbit, then γ intersects every G-orbit perpendicularly. The previous result and the remarks following Theorem 4.9 allows us to say that, from a geometric point of view, every locally free action of a group G preserving a Riemannian metric h deﬁnes (through its orbits) a foliation which is transversely totally geodesic.

7. Lagrangian foliations associated with a Reinhardt domain We now proceed to study the geometry of Reinhardt domains. For this we will obtain some properties of its Bergman metric and apply the foliation theory considered in the previous sections. As before, in this section D ⊂ Cn denotes a bounded logarithmically convex complete Reinhardt domain centered at the origin. Using the monomial orthonormal base {ep }p∈Zn+ of A2µ (D), mentioned in Section 3, we have obviously Lemma 7.1. The Bergman kernel KD of the domain D admits the following representation KD (z, ζ) = (2π)−n α2p z p ζ¯p , p∈Zn +

where the coeﬃcients αp , p ∈ Zn+ , are given by (2.1). In particular, the function KD (z, z) depends only on r. In this section we will use the polar coordinates zk = rk tk = rk eiθk , k = 1, . . . , n, for points z = (z1 , . . . , zn ) ∈ D. Theorem 7.2. Let ds2D be the Bergman metric of D considered as a Hermitian metric and hD = Re (ds2D ) the associated Riemannian metric. Then: hD =

n

Fkl (r)(drk ⊗ drl + rk rl dθk ⊗ dθl ),

k,l=1

where the functions Fkl are given by: ∂2 1 δkl ∂ Fkl (r) = + log KD (z, z), 4 ∂rk ∂rl rk ∂rk and depend only on r. Proof. For the Bergman kernel KD , the associated Bergman metric considered as a Hermitian metric is given by: ds2D =

n ∂ 2 log KD (z, z) k dz ⊗ d¯ zl. ∂z k ∂ z¯l

k,l=1

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Let F (r) = F (z) = log KD (z, z), which by Lemma 7.1 depends only on r. Then a straightforward computation shows that: ∂2F 1 z¯k zl ∂ 2 F δkl ∂F (z) = (z) + (z) . ∂zk ∂ z¯l 4 rk rl ∂rk ∂rl rk ∂rk The required identity is then obtained by replacing these expressions into that of ds2D , using the relations zk = rk eiθk and computing the real part of the expression thus obtained. Consider the following action of the n-dimensional torus Tn on D Tn × D

→ D

(t, z) → tz, which being biholomorphic yields the following immediate consequence. Theorem 7.3. Let hD be the Riemannian metric of D deﬁned by its Bergman metric. Then Tn acts isometrically on (D, hD ). Note that the action of Tn is not locally free at all points of an n-dimensional Reinhardt domain, but it is almost so as the following obvious result states. We recall that in a measure space, a subset is called conull if its complement has zero measure. Lemma 7.4. For D as before, the set: = {z ∈ D : zk = 0 for every k = 1, . . . , n} D is the set of points whose stabilizers with respect to the action of Tn are discrete. is an open conull subset of D on which Tn acts freely. Furthermore, D As a consequence of Theorems 6.3 and 7.3 and Lemma 7.4 we obtain the following. the subset of D deﬁned in Lemma 7.4 and hD Theorem 7.5. Let D be as before, D the Riemannian metric deﬁned by the Bergman metric of D. Then, the Tn -orbits deﬁne a Riemannian foliation O for which h is a compatible bundle-like in D metric. Given such result we now obtain the following statement which makes use of Theorem 6.4 as well. the subset of D deﬁned in Lemma 7.4 and hD Theorem 7.6. Let D be as before, D the Riemannian metric deﬁned by the Bergman metric of D. If γ is a geodesic in (with respect to h) perpendicular at some point to a Tn -orbit, then γ intersects D every Tn -orbit perpendicularly. We now prove that the Riemannian foliation O obtained in the previous result is Lagrangian.

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the subset of D deﬁned in Lemma 7.4, ds2 Theorem 7.7. Let D be as before, D D the Bergman metric of D as a Hermitian metric and O the Riemannian foliation Then O is Lagrangian with respect to the Riemannian metric of Tn -orbits in D. 2 are Lagrangian with respect to hD = Re (dsD ), in other words, the Tn -orbits in D hD . Proof. We need to prove that Tz O and iTz O are perpendicular with respect to Since such condition is the Riemannian metric hD = Re (ds2D ) at every z ∈ D. invariant under the Tn -action we can assume that z = x ∈ Rn+ . we have Tx O = iRn . Hence the result follows We observe that for every x ∈ D by using Theorem 7.2 together with the fact that iRn and Rn are perpendicular with respect to the elements drk ⊗ drl + rk rl dθk ⊗ dθl for every k, l. We now prove that the normal bundle to O is integrable. the subset of D deﬁned in Lemma 7.4 and Theorem 7.8. Let D be as before, D hD the Riemannian metric deﬁned by the Bergman metric of D. If we denote of tangent vectors perpendicular to O, then with T O⊥ the vector subbundle of T D ⊥ T O is integrable to a foliation P. Furthermore, P is a Lagrangian totally geodesic foliation of D. Proof. If we let M0 = D∩Rn+ , then by the proof of Theorem 7.7 the tangent bundle to M0 coincides with T O⊥ restricted to M0 , and so M0 is an integral submanifold of T O⊥ . Since T O⊥ is invariant under the Tn -action and such action preserves the metric, it follows that for every t ∈ Tn the manifold: Mt = tM0 ⊥

is an integral submanifold of T O , thus showing the integrability of such bundle to some foliation P. By Lemma 7.1 we have T P = T O⊥ = iT O which implies that P is Lagrangian. Finally P is totally geodesic by Theorem 7.6. We now state the following easy corollary of the previous discussion. Corollary 7.9. The sets of vector ﬁelds: n ∂ and ∂θk k=1

∂ ∂rk

n , k=1

deﬁne global framings for the bundles T O and T O⊥ = T P, respectively, on D.

8. The unit ball An important class of domains in complex analysis is given by those which are bounded and symmetric. The next result shows that each irreducible bounded symmetric domain which is also Reinhardt has to be a unit ball. As usual, we will denote by Bn the unit ball in Cn .

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Theorem 8.1. Let D be an irreducible bounded symmetric domain. Then D is also a Reinhardt domain if and only if D = Bn for some n ∈ Z+ . Proof. First, the unit ball centered at the origin in a complex vector space is obviously a Reinhardt domain. Conversely, let us assume that D is an irreducible bounded symmetric domain which is also Reinhardt. We show that it is a unit ball centered at the origin of some complex vector space. For this we use Cartan’s classiﬁcation of irreducible bounded symmetric domains and the description of their biholomorphisms as found in [5]. We present the needed basic properties in Table 1, which recollects some of the information found in Table V in page 518 from [5]. Every irreducible bounded symmetric domain D in Table 1 is identiﬁed by its type in the ﬁrst column (following the notation from [5]) and is explicitly given as the quotient G0 /K for the groups in the second and third column. The group G0 is, up to a ﬁnite covering, the group of biholomorphisms of D and K is the subgroup of G0 consisting of those transformations that ﬁx the origin. For the exceptional bounded symmetric domains of type EIII and EVII we write down the Lie algebras of the corresponding groups, which is enough for our purposes; again, we follow here the notation from [5] to identify real forms of exceptional complex Lie algebras. The last two columns permit us to compare the complex dimension of D and the dimension of a maximal torus T in K. This last dimension is well known from the basic properties of the compact groups K that appear in Table 1. We recall from the basic theory of symmetric spaces that the universal covering of the group G0 completely determines the bounded symmetric domain: in other words, two bounded symmetric domains whose corresponding groups G0 in Table 1 have the same universal covering group are biholomorphic. Through out Table 1, the symbols p, q and n are assumed to be positive integers. The additional conditions on the types BDI(2,q) and DIII are required for the corresponding quotient G0 /K to actually deﬁne an irreducible bounded symmetric domain. Table 1. Irreducible bounded symmetric domains D

G0

K

AIII

SU (p, q)

S(U (p) × U (q)) pq

BDI(2,q) (q = 2) SO0 (2, q) SO(2) × SO(q)

dimC (D) q n(n−1) 2 n(n+1) 2

dim(T ) p+q−1 q 2 +1

DIII (n ≥ 2)

∗

SO (2n)

U (n)

CI

Sp(n, R)

U (n)

EIII

e6(−14)

so(10) ⊕ R

16

6

EVII

e7(−25)

e6 ⊕ R

27

7

n n

For D in Table 1 to be a Reinhardt domain, we clearly have as a necessary condition the inequality: dim(T ) ≥ dimC (D). (8.1)

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Let us now consider the cases where this might occur in Table 1. AIII. The condition (8.1) holds if and only if min(p, q) = 1, which clearly corresponds to the unit ball of dimension max(p, q). BDI(2,q). In this case the condition (8.1) holds if and only if q = 1. This corresponds to the bounded symmetric domain whose group of biholomorphisms is, up to a ﬁnite covering, SO0 (2, 1). Since the Lie algebras so(2, 1) and su(1, 1) are isomorphic, the bounded symmetric domain of type BDI(2,1) is the unit disc in the complex plane. DIII. In this case the condition (8.1) holds only for n = 2 or 3. The Lie algebras of the corresponding groups G0 are so∗ (4) and so∗ (6). There are well known isomorphisms so∗ (4) ∼ = su(2) × su(1, 1) and so∗ (6) ∼ = su(3, 1) (see [5]). We ∼ also recall that u(n) = su(n) ⊕ R, for every n. Hence, we conclude that type DIII for n = 2 and 3 deﬁnes the unit disc in the complex plane and the unit ball in C3 , respectively. CI. In this case the condition (8.1) holds only for n = 1, which yields the unit disk with an argument as above using the fact that sp(1, R) is isomorphic to su(1, 1) (see [5]). EIII, EVII. A simple inspection shows that in these cases the condition (8.1) cannot hold.

This completes the proof of Theorem 8.1.

Now the results of the previous sections lead directly to the following statements: n the Tn -action deﬁnes a Lagrangian foliation O. 1. On the subset B n to a foliation totally 2. The orthogonal complement T O⊥ is integrable in B geodesic Lagrangian foliation P. n , which in 3. The pair of foliations O and P deﬁne the polar coordinates in B ∗ turn yields the commutative C - algebra of Toeplitz operators whose symbols are constant on the leaves of O. In what follows we will normalize the (Hermitian) Bergman metric on the unit ball to the following expression:   n n l k z z dz ⊗ dz 4 k l  . n n dz k ⊗ dz k + ds2 = 1 − k=1 |zk |2 1 − k=1 |zk |2 k=1

k,l=1

which diﬀers from the usual Bergman metric as considered in the proof of Theorem 7.2 by a factor of (n+1)/4. The advantage of this normalization is that the sectional curvature varies in the interval [−1, −1/4], while with the metric as deﬁned in the proof of Theorem 7.2 the sectional curvature varies in the interval [−4/(n + 1), −1/(n + 1)]. We will now compute some values of the second fundamental form for the foliation O of the unit ball. First, we recall the notion of complex geodesic and some of its properties.

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Deﬁnition 8.2. A complex geodesic in Bn is a biholomorphic map ϕ : D → D where D is the unit disc and D = Bn ∩ L for some complex aﬃne line L in Cn . It is well known that complex geodesics are always totally geodesic maps. Furthermore, the images of complex geodesics are precisely the closed totally geodesic complex submanifolds of (complex) dimension 1 in Bn (see [2]). n The next result shows that some of the orbits ofn the T -action on the unit ∂ ball integrate the vector ﬁelds of the framing ∂θk from Corollary 7.9. Its k=1 proof is a straightforward computation. Lemma 8.3. For every k, 1 = 1, . . . , n with k = l, the curves: γz,k (s)

= (z1 , . . . , zk−1 , eis zk , zk+1 , . . . , zn )

γz,kl (s)

= (z1 , . . . , zk−1 , eis zk , zk+1 , . . . , zl−1 , eis zl , zl+1 , . . . , zn )

are integral curves of the vector ﬁelds

∂ ∂θk

and

∂ ∂θk

+

∂ ∂θl ,

respectively.

Proof. By the deﬁnition of polar coordinates it is clear that the ﬂows that integrate ∂ ∂ ∂ ∂θk and ∂θk + ∂θl are given by: z

→ (z1 , . . . , zk−1 , eis zk , zk+1 , . . . , zn ),

z

→ (z1 , . . . , zk−1 , eis zk , zk+1 , . . . , zl−1 , eis zl , zl+1 , . . . , zn )

from which the conclusion is clear. n: Let us deﬁne the following vector ﬁelds on B ∂ ∂ , Qk = II ∂θk ∂θk ∂ ∂ ∂ ∂ Qkl = II + , + , ∂θk ∂θl ∂θk ∂θl

then, by Proposition 5.4 and Corollary 7.9, such vector ﬁelds completely determine the second fundamental form II . We will compute Qk and Qkl using the curves n we deﬁne the following deﬁned in Lemma 8.3. To achieve this, for every z ∈ B complex geodesics: φz,k (w)

= (z1 , . . . , zk−1 , Rk w, zk+1 , . . . , zn )

φz,kl (w)

= (z1 , . . . , zk−1 , Rkl w, zk+1 , . . . , zl−1 ,

Rkl zl w, zl+1 , . . . , zn ) zk

where k, l = 1, . . . , n with k = l and: 1− |zj |2 Rk = j =k

Rkl

=

|zk | 1 − j =k,l |zj |2 . |zk |2 + |zl |2

Then we have the following easy to prove result.

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n the complex geodesics φz,k , φz,kl satisfy: Lemma 8.4. For every z ∈ B 1. φz,k (zk /Rk ) = φz,kl (zk /Rkl ) = z for every k, l = 1, . . . , n with k = l, 2. γz,k (R) ⊂ φz,k (D) and γz,kl (R) ⊂ φz,kl (D), in other words, they pass through z and contain the curves from Lemma 8.3 with the same indices. We now use the above to compute the value of the vector ﬁelds Qk and Qkl . n and k, l = 1, . . . , n with k = l we have the following Lemma 8.5. For every z ∈ B relations: 1. Qk (z) = γz,k (0) and Qkl (z) = γz,kl (0), where the acceleration is computed for the complex hyperbolic geometry of Bn , 2. γz,k (s) ∈ Riγz,k (s) and γz,kl (s) ∈ Riγz,kl (s) for every s ∈ R; in particular: ∂ ∂ ∂ Qk (z) ∈ R , Qkl (z) ∈ R + , ∂rk z ∂rk z ∂rl z 3. the norms of Qk and Qkl are given by: Qk (z) = Ck (z)γz,k (0)2 ,

Qkl (z) = Ckl (z)γz,kl (0)2 ,

where Ck (z) and Ckl (z) are the geodesic curvatures of γz,k and γz,kl , respectively, considered as curves in the images of the complex geodesics φz,k and φz,kl , respectively, endowed with the metric inherited from Bn . n the leaves of the foliation O by Tn -orbits are Proof. First we observe that in B n diﬀeomorphic to T under the action map. In particular, with respect to such diﬀeomorphisms, the metric of Bn restricted to any such Tn -orbit is left invariant. For such metrics on abelian Lie groups it is well known that the geodesics are precisely the one parameter subgroups and their translations (see [5]). Since the curves γz,k , γz,kl correspond to one parameter groups in Tn it follows that they deﬁne geodesics in the leaf of O through z. Then, by well known results on the geometry of Riemannian submanifolds (see [9]) it follows that the accelerations , γz,kl as computed in Bn are everywhere perpendicular to the leaves of O, in γz,k other words they are everywhere horizontal. By the remarks in Section 5 and the deﬁnition of Qk and Qkl we have: γ (0) γ ) = H(γ (0)) = γ (0) Qk (z) = H(∇ z,k

z,k

z,k

z,k

γ (0) γz,kl ) = H(γz,kl (0)) = γz,kl (0), Qkl (z) = H(∇ z,kl

the connection of Bn , and where the last identities follow from the remarks for ∇ in the previous paragraph. This proves (1). Next observe that since the curves γz,k , γz,kl are geodesics in some leaf of O it follows that they are up to a constant parameterized by arc-length. On the other hand, by Lemma 8.4 the curves γz,k , γz,kl lie in complex geodesics which, as we observed before, deﬁne totally geodesic submanifolds of Bn . In particular, their accelerations γz,k , γz,kl as computed in Bn are the same as computed in

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the (images) of the complex geodesics that contain them. Complex geodesics are isometric to the unit disk, and for the latter any curve γ which is parameterized up to a constant by arc length satisﬁes γ (s) ∈ Riγ (s). This implies the ﬁrst part of (2). For the second part of (2) it is enough to note that: ∂ ∂ (s) = i ∈R iγz,k ∂θk γz,k (s) ∂rk γz,k (s) ∂ ∂ ∂ ∂ iγz,kl (s) = i + + ∈R , ∂θk γz,kl (s) ∂θl γz,kl (s) ∂rk γz,kl (s) ∂rl γz,kl (s) and so we obtain (2) by applying (1). By the deﬁnition of the geodesic curvature (see [4] and its references) we have: Ck (z) = Ckl (z) =

(0) γz,k γz,k (0)2 (0) γz,kl , γz,kl (0)2

where we have used the fact that the norm of vectors and the acceleration of curves in a (image of a) complex geodesic in Bn computed in Bn or the complex geodesic yield the same result. Given the above identities, (3) follows from (1). The next result computes speciﬁc values for the second fundamental form of the foliation O for the unit ball. By Proposition 5.4 such values completely determine the second fundamental form. We observe that in the ﬁrst two parts of the statement we obtain a very explicit expression for the second fundamental form of the foliation O. Note that by part (3) of Lemma 8.5, the geodesic curvatures Ck and Ckl correspond to the norms of the values of Qk and Qkl , respectively, ∂ , renormalized so that they only depend on the direction of ∂θ∂k and ∂θ∂k + ∂θ l spectively. In view of this, the last part of the statement allows us to understand the asymptotic behavior of the curvature of the leaves of O as they move towards n . We observe as well that this result generalizes the origin or the boundary of B our geometric description of the elliptic model case in the unit disk found in [4]. n , let r = (r1 , . . . , rn ) = (|z1 |, . . . , |zn |), and consider Theorem 8.6. For every z ∈ B the curves γz,k , γz,kl and the complex geodesics φz,k , φz,kl deﬁned above. Then: 1. The vector ﬁelds Qk and Qkl are given by: ∂ 2 ∂ −1 ∂ Qk (z) = −Ck (z) ∂θk z ∂rk z ∂rk z 2 −1 ∂ ∂ ∂ ∂ ∂ ∂ Qkl (z) = −Ckl (z) + + + . ∂θk z ∂θl z ∂rk z ∂rl z ∂rk z ∂rl z

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2. The geodesic curvatures Ck (z) and Ckl (z) at z deﬁned in Lemma 8.5 are given by: rk2 + 1 − j =k rj2 Ck (z) = 2rk 1 − j =k rj2 rk2 + rl2 + 1 − j =k,l rj2 Ckl (z) = , 2 rk2 + rl2 1 − j =k,l rj2 in particular, such geodesic curvatures lie in the interval (1, +∞) and achieve all values therein. 3. The geodesic curvatures Ck (z) and Ckl (z) have the following asymptotic behavior: as |z| → 0, Ck (z), Ckl (z) → +∞, as z → u, Ck (z) → 1, Ckl (z) → 1,

as z → v,

n

for any u, v ∈ ∂B such that uk = 0 and |vk |2 + |vl |2 = 0, respectively. Proof. Up to a sign, (1) essentially follows from (2) and (3) in Lemma 8.5. The negative sign comes from the fact that, in the proof of Lemma 8.5, the accelerations of γz,k , γz,kl point towards the origin in the complex geodesics that contain them and the vector ﬁelds ∂ ∂ ∂ , + ∂rk ∂rk ∂rl point away from the origin. To prove (2), let φz,k , φz,kl be the complex geodesics considered before. Then the inverse images of the curves γz,k , γz,kl with respect to such maps are easily seen to be circles in D centered at the origin with Euclidean radius: rk2 + rl2 rk rk rk sk = = and s = = , kl Rk Rkl 1− r2 1− r2 j =k j

j =k,l j

respectively. Next, we observe that the geodesic curvature C(s) of the circle with Euclidean radius s in the unit disk D with the metric: 4(dx2 + dy 2 ) (1 − (x2 + y 2 ))2 is given by the formula:

1 + s2 . 2s This follows from two facts found in [2]. The hyperbolic radius ρ of a circle centered at the origin satisﬁes cosh2 (ρ/2) = 1/(1 − s2 ), where s is the Euclidean radius. And the geodesic curvature of such a circle is given by coth(ρ). The ﬁrst can be C(s) =

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deduce from the expression for the hyperbolic distance found in subsection 1.4.1 of [2] and the second is stated in subsection 1.4.2 of the same reference. Given the above formula for C(s) a simple substitution provides the required expressions for Ck and Ckl . Finally, (3) is a consequence of these expressions. We recall that the velocity and acceleration of curves in a manifold do not change when we renormalize the metric of the manifold by a constant multiple. More generally, the second fundamental form of a submanifold does not change either by such renormalizations (see [9]). However, the geodesic curvatures as deﬁned above involve the metric and so they are rescaled when we renormalize the metric by a constant. In particular, for the Bergman metric on Bn (i.e. without normalizing to have sectional curvature in the interval [−1, −1/4]) which is given by   n n l k z z dz ⊗ dz n + 1 k l  . n n ds2Bn = dz k ⊗ dz k + 1 − k=1 |zk |2 1 − k=1 |zk |2 k=1 k,l=1 (0), γz,k (0) and Qk (z) = II ∂θ∂k , ∂θ∂k as deﬁned above the tangent vectors γz,k z z have the same values, but computing the geodesic curvatures of γz,k involve applying a renormalized metric and the corresponding values are rescaled. In the next result we write down the geodesic curvatures of the curves γz,k for the Bergman metric and describe its asymptotic behavior. We also express such curvatures in terms of the second fundamental form II . These facts will allow us to compare our present situation with a more general asymptotic behavior discussed in the next section. Theorem 8.7. Let hBn be the Riemannian metric associated to the Bergman metric of Bn given as above and denote with · Bn the norm that it deﬁnes on tangent k (z) of the curve γz,k at n the geodesic curvature C vectors. Then, for every z ∈ B z for the metric hBn satisﬁes the relations: −2 −1 k (z) = − 1. C ∂θ∂k ∂r∂k hBn Qk (z), ∂r∂k , k (z) = 2. C

z Bn √ 2 Ck (z), n+1

z Bn

z

where Ck is given as in Theorem 8.6.

(z) is, up to a sign, the norm of the orthogonal projection of the In particular, C −2k vector ∂θ∂k Qk (z) onto ∂r∂k with respect to hBn . And we also have: z Bn

z

k (z) → √ 2 , C n+1 n for any u ∈ ∂B such that uk = 0.

as z → u.

Proof. The ﬁrst relation follows from the deﬁnition of the geodesic curvature as above applied to the new metric hBn , the fact that: ∂ ∂ Qk (z) = II , ∂θk z ∂θk z

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k . and the corresponding relation of (1) in Theorem 8.6 for C The second relation is a consequence of the fact that hBn = n+1 4 h for h the Riemannian metric on Bn rescaled so that its sectional curvature lies in [−1, −1/4]. We specify now the results obtained in Sections 2 and 3 for the unit ball Bn . The base τ (Bn ) of Bn has obviously the form τ (Bn ) = {r = (r1 , . . . , rn ) : r2 = r12 + · · · + rn2 ∈ [0, 1)}. As a custom in operator theory (see, for example, [15]), introduce the family of weights µλ (|z|) = cλ (1 − |z|2 )λ , where the normalizing constant cλ =

Γ(n + λ + 1) π n Γ(λ + 1)

is chosen so that µλ (|z|)dv(z) is a probability measure in Bn . Introduce L2 (Bn , µλ ) and its Bergman subspace A2λ (Bn ) = A2µλ (Bn ). It is well known (see, for example, [15]), that the Bergman projection Bλ of L2 (Bn , µλ ) onto A2λ (Bn ) has the form (Bλ ϕ)(z) = ϕ(ζ) Kλ (z, ζ) µλ (|ζ|)dv(ζ), Bn

where the (weighted) Bergman kernel is given by Kλ (z, ζ) =

(1 −

1 . n+1+λ k=1 zk ζ k )

n

To calculate the constant αp , p = (p1 , . . . , pn ) ∈ Zn+ , see (2.1), consider the integral p 2 |z | µλ (|z|)dv(z) = |z1 |2p1 · · · · · |zn |2pn µλ (r)dv(z) Bn

Bn

= =

n n dtk r12p1 · · · · · rn2pn µλ (r) rk drk itk τ (Bn)

Tn k=1 (2π)n α−2 p .

From the other hand side, by [15], Lemma 1.11, we have p! Γ(n + λ + 1) , |z p |2 µλ (|z|)dv(z) = Γ(n + |p| + λ + 1) n B that is,

1/2 (2π)n Γ(n + |p| + λ + 1) αp = . p! Γ(n + λ + 1) Now Theorem 3.1 for the case of the unit ball reads as follows.

k=1

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Theorem 8.8. Let a = a(r) be a bounded measurable separately radial function. Then the Toeplitz operator Ta acting on A2λ (Bn ) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on l2 (Zn+ ), where R and R∗ are given by (2.3) and (2.2) respectively. The sequence γa,λ = {γa,λ (p)}p∈Zn+ is given by γa,λ (p) = =

n 2n Γ(n + |p| + λ + 1) 2p 2 λ a(r) r (1 − r ) rk drk p! Γ(λ + 1) τ (Bn ) k=1 √ Γ(n + |p| + λ + 1) a( r) rp (1 − (r1 + · · · + rn ))λ dr, p ∈ Zn+ , p! Γ(λ + 1) ∆(Bn )

where ∆(Bn ) = {r = √ (r1 , . . . , rn ) : r1 + · · · + rn ∈ [0, 1), rk ≥ 0, k = 1, . . . , n}, √ √ dr = dr1 · · · drn , and r = ( r1 , . . . , rn ).

9. Asymptotic geometric behavior of the Tn -orbits in Reinhardt domains As before, let D be a bounded logarithmically convex complete Reinhardt domain with Bergman metric ds2D , associated Riemannian metric hD and with · D denoting the norm deﬁned by hD on tangent vectors. Also, we will continue denoting with II the second fundamental form of the As in the case of the unit ball, by Proposition 5.4, foliation O by Tn -orbits in D. II is completely determined by the vector ﬁelds: ∂ ∂ , Qk = II ∂θk ∂θk ∂ ∂ ∂ ∂ Qkl = II + , + . ∂θk ∂θl ∂θk ∂θl The norm of such vector ﬁelds was computed in the previous section for the unit ball and such norm was related to the geodesic curvature of suitable circles contained in complex geodesics. In this section we will study the asymptotic behavior towards the boundary of similar values for a more general Reinhardt domain. As in the case of the unit ball, on our given Reinhardt domain, we will and k = 1, . . . , n the curve: consider for every z ∈ D γz,k (s) = (z1 , . . . , zk−1 , eis zk , zk+1 , . . . , zn ). Then, the proofs of Lemmas 8.3 and 8.5 apply to our current more general setup without change to conclude that γz,k is an integral curve of ∂θ∂k and that we can write: Qk (z) = II (γz,k (0), γz,k (0)) = γz,k (0).

As in the case of the unit ball, to better understand the asymptotic behavior of the values of Qk one considers its normalized value obtained by dividing by

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γz,k (0)2D = ∂θ∂k |z 2D , i.e.: −2 ∂ −2 (0) γz,k ∂ ∂ Qk (z) = ∂ II , , = ∂θk z ∂θk z ∂θk z ∂θk z γz,k (0)2D D D (0) = ∂θ∂k |z and not on which now depends only on the direction associated to γz,k −2 its magnitude. Moreover, the above identities show that ∂θ∂k Qk (z) measures z D both the extrinsic curvature of the foliation O, given by II , and the curvature of γz,k , given by its acceleration. For the unit ball it was proved that such vector ﬁeld is collinear with ∂r∂k , and so to measure its magnitude in that case it was enough to consider the norm of its orthogonal projection onto ∂r∂k . In our more general −2 setup, ∂θ∂k Qk may not be collinear with ∂r∂k , but we can still consider the D properties of the orthogonal projection of the ﬁrst onto the latter. and k = 1, . . . , n: The previous discussion suggests to deﬁne for every z ∈ D ∂ −2 ∂ −1 ∂ Ck (z) = − hD Qk (z), , ∂θk z ∂rk z ∂rk z D

D

which thus provides a measure of both the extrinsic curvature of the foliation O and the curvature of γz,k . on D Note that for the unit ball endowed with the Bergman metric, Theorem 8.7 k (z) is precisely the geodesic curvature of γz,k in the complex geodesic shows that C φz,k considered in the previous section. Moreover, such Theorem 8.7 describes the k towards the boundary in the case of the unit ball. The asymptotic behavior of C main goal of this section is to prove that such asymptotic behavior remains valid for suitable domains. More precisely, we have the following result. We recall that D is said to have δ as a deﬁning function if D = {z ∈ Cn : δ(z) < 0}. Theorem 9.1. Let D be a bounded strictly pseudoconvex complete Reinhardt domain with smooth boundary and with a smooth deﬁning function δ. Then: k (z) → √ 2 , C as z → u, n+1 for any u ∈ ∂D such that uk = 0 and

∂δ ∂rk (u)

= 0.

We observe that for the unit ball we can take δ(z) = −1 + nj=1 |zj |2 = n ∂δ −1 + j=1 rj2 , and so the conditions uk = 0 and ∂r (u) = 0 are equivalent in this k case. Note that Theorems 9.1 and 8.7 together show that, under suitable convexity and smoothness conditions on the domain D, the extrinsic geometry of the foliation has exactly the same asymptotic behavior towards the boundary as the O in D one found for the unit ball, at least with respect to the values of Qk . To prove Theorem 9.1 we will use the expression of the metric hD in terms of the Bergman kernel KD from Theorem 7.2 and the following celebrated result by

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C. Feﬀerman that describes the Bergman kernel of strictly pseudoconvex domains with smooth boundary. This result appears as a Corollary in page 45 of [1]. Theorem 9.2 (C. Feﬀerman). If D is a strictly pseudoconvex domain with smooth boundary and smooth deﬁning function δ, then there exists ϕ, ψ ∈ C ∞ (D) with ϕ nonvanishing in ∂D, such that: KD (z, z) = ϕ(z)(−δ(z))−(n+1) + ψ(z) log(−δ(z)) for every z ∈ D. k in terms of the Bergman kernel KD . We ﬁrst express the value of C Lemma 9.3. Let D be a bounded logarithmically convex complete Reinhardt domain with Bergman kernel KD . Then: k = C , C C

where C

= +

C

=

2

∂KD ∂rk

3 − 3KD

3 ∂KD ∂ 2 KD 3KD 2 ∂ KD + KD − 2 ∂rk ∂rk ∂rk3 rk

∂KD ∂rk

2

2 ∂ 2 KD 3KD K 2 ∂KD + 2D , 2 rk ∂rk rk ∂rk

32 2 ∂KD ∂ 2 KD KD ∂KD − + KD + , ∂rk ∂rk2 rk ∂rk

and where KD and its partial derivatives are computed for the function z → KD (z, z). r

Proof. Let us denote with Γθjk θl , . . . , the Schwarz-Christoﬀel symbols for the LeviCivita connection of the Riemannian manifold (D, hD ) and with hθk θl , . . . , the coordinate functions of the metric. By Corollary 7.9 and the deﬁnition of II it follows that: n ∂ Γrθlk θk . Qk (z) = ∂rl l=1

By using the well known formula that expresses the Schwarz-Christoﬀel symbols in terms of the functions hθk θl , . . . , and its partial derivatives (see [9]) we have: Γrθlk θk = −

n

1 rl rj ∂hθk θk h , 2 j=1 ∂rj

where, as usual, hrl rj denotes the entries of the inverse of the matrix (hrl rj )lj . We have used here that, by Theorem 7.2, the functions hθk rl = 0. Hence, it follows

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∂ −2 ∂ −1 hD Qk (z), ∂ = − ∂θk z ∂rk z ∂rk z D D n 1 −1 −1/2 ∂hθk θk = h h hrk rl hrl rj 2 θ k θ k rk rk ∂rj l,j=1

1 −1 −1/2 ∂hθk θk h . h 2 θk θk rk rk ∂rk

=

Again by Theorem 7.2 we have hrk rk = Fkk and hθk θk = rk2 Fkk , where: 1 ∂2 1 ∂ log KD (z, z), + Fkk (z) = 4 ∂rk2 rk ∂rk from which we obtain: k = C

1 3/2

2Fkk

∂Fkk 2 + Fkk . ∂rk rk

(9.1)

(9.2)

kk Then the result follows by computing Fkk and ∂F ∂rk in terms of KD with the use of equation (9.1) and replacing into equation (9.2).

The following result can be proved easily using induction. Lemma 9.4. Let D be a strictly pseudoconvex domain with smooth boundary and smooth deﬁning function δ. Let ϕ, ψ ∈ C ∞ (D) be the smooth functions from Theorem 9.2. Then, for every k = 1, . . . , n and j ≥ 0 we have: ∂ j KD ∂rkj

=

j

−(n+1+l)

ϕjl (−δ)

l=0

+

j

ψjl δ −l + ψj0 log(−δ),

l=1

where the partial derivatives are computed for the function z → KD (z, z), and the functions ϕjl , ψjl are given inductively by the following conditions: 1. ϕ00 = ϕ, ψ00 = ψ, 2. ϕjl = ψjl = 0 if either j or l is negative, 3. for j ≥ 1: ϕjj ψjj

∂δ ∂rk ∂δ = −(j − 1)ψj−1,j−1 , ∂rk = (n + j)ϕj−1,j−1

4. for 0 ≤ l < j: ϕjl

=

ψjl

=

∂δ ∂ϕj−1,l + (n + l)ϕj−1,l−1 ∂rk ∂rk ∂δ ∂ψj−1,l − (l − 1)ψj−1,l−1 , ∂rk ∂rk

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k in terms of the deﬁning function The next result provides an expression of C δ. We observe that Theorem 9.1 is now an easy consequence of such expression. Theorem 9.5. Let D be a bounded strictly pseudoconvex complete Reinhardt domain with smooth boundary and with a smooth deﬁning function δ. If ϕ is the function given by Theorem 9.2, then: k = C

a(−δ)−(3n+6) + bδ −(3n+5) (c(−δ)−(2n+4) + dδ −(2n+3) )3/2

where a, b, c, d ∈ C ∞ (D) satisfy: on D, 3 ∂δ 1. a = 2(n + 1)ϕ3 ∂r , k 2 ∂δ , 2. c = (n + 1)ϕ2 ∂r k 3. b, d extend continuously to {z ∈ D : zk = 0}. k in terms of log(−δ) and powers of Proof. We use Lemmas 9.3 and 9.4 to express C Then the result is simply a matter of identifying δ with smooth coeﬃcients in D. the coeﬃcients in such expression. The functions a, c correspond to the lowest powers of δ in the numerator and the denominator, respectively. The functions b, d involve terms of the form δ l and δ l log(−δ) with l ≥ 1 and powers of 1/rk , all of which can be extended continuously to {z ∈ D : zk = 0}.

References [1] C. Feﬀerman. The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math., 26:1–65, 1974. [2] W.M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. [3] S. Grudsky, A. Karapetyants, and N. Vasilevski. Toeplitz operators on the unit ball in Cn with radial symbols. J. Operator Theory, 49:325–346, 2003. [4] S. Grudsky, R. Quiroga-Barranco, and N. Vasilevski. Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal., 234(1):1–44, 2006. [5] S. Helgason. Diﬀerential geometry, Lie groups, and symmetric spaces. American Mathematical Society, Providence, RI, 2001. [6] S. Kobayashi and K. Nomizu. Foundation of Diﬀerential Geometry, volume I. John Wiley & Sons, Inc., New York, 1996. [7] S. Kobayashi and K. Nomizu. Foundations of diﬀerential Geometry, volume II. John Wiley & Sons, Inc., New York, 1996. [8] P. Molino. Riemannian foliations. Birkhauser Boston, Inc., Boston, MA, 1988. [9] B. O’Neill. Semi-Riemannian geometry. With applications to relativity. Academic Press, Inc., New York, 1983.

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[10] R. Quiroga-Barranco. Isometric actions of simple Lie groups on pseudoRiemannian manifolds. Ann. of Math., 164:941-969, 2006. [11] R. M. Range. Holomorphic functions and integral representations in several complex variables. Springer-Verlag, New York, 1986. [12] N. L. Vasilevski. Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemp. Math., 289:79–146, 2001. [13] N. L. Vasilevski. Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry. Integr. Equat. Oper. Th., 46:235–251, 2003. [14] F. W. Warner. Foundations of diﬀerentiable manifolds and Lie groups. SpringerVerlag, New York-Berlin, 1983. [15] K. Zhu. Spaces of Holomorphic Functions in the Unit Ball. Springer Verlag, 2005. Raul Quiroga-Barranco Centro de Investigaci´ on en Matem´ aticas Apartado Postal 402 36000, Guanajuato, Gto. M´exico e-mail: [email protected] Nikolai Vasilevski Departamento de Matem´ aticas CINVESTAV Apartado Postal 14-740 07000, M´exico, D.F. M´exico e-mail: [email protected] Submitted: March 14, 2006 Revised: March 3, 2007

Integr. equ. oper. theory 59 (2007), 99–128 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010099-30, published online June 27, 2007 DOI 10.1007/s00020-007-1504-2

Integral Equations and Operator Theory

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols Patrik Wahlberg Abstract. We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modiﬁcations of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an Lp,q space and M ∞ , respectively. The ﬁrst result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space. Mathematics Subject Classification (2000). Primary 47G30, 42B35; Secondary 47B38, 35S99. Keywords. Time-frequency analysis, vector-valued modulation spaces, localization operators, pseudodiﬀerential operators.

1. Introduction The theory of the short-time Fourier transform (STFT) and modulation spaces of scalar-valued functions and tempered distributions is a very well developed theory of representation of tempered distibutions in the time-frequency (phase) space [22]. Deﬁne the modulation and translation operators by (Mξ f )(x) := ei2πξx f (x) and (Tt f )(x) := f (x− t), respectively. For a ﬁxed so called window function g ∈ S(Rd ), the STFT f → Vg f is the localized Fourier transform deﬁned by Vg f (t, ξ) = f, Mξ Tt g = f (x)ei2πxξ g(x − t)dx. (1.1) Rd

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The STFT is unitary L2 (Rd ) → L2 (R2d ) provided gL2 = 1, a topological isomorphism S(Rd ) → S(R2d ), and extends to a topological isomorphism S (Rd ) → S (R2d ). The inverse mapping is given by ∗ Vγ h = h(t, ξ)Mξ Tt γdtdξ, γ ∈ S(Rd ), γ, g = 1, (1.2) R2d

where Vγ∗ h in general is interpreted as the functional h(t, ξ)Mξ Tt γ, ϕdtdξ, ϕ ∈ S(Rd ). Vγ∗ h, ϕ = R2d

p,q The map (1.1) is also invertible on the family of modulation spaces Mm (Rd ), 1 ≤ p, q ≤ ∞, which were invented and developed 1983 by Feichtinger [12]. p,q (Rd ) = {f ∈ The weighted modulation spaces are Banach spaces deﬁned by Mm d p,q 2d p,q 2d S (R ); Vg f ∈ Lm (R )} where Lm (R ) is the weighted mix-normed space of all measurable h : R2d → C such that q/p 1/q hLp,q = |h(t, ξ)m(t, ξ)|p dt dξ < ∞. m Rd

Rd

The modulation spaces thus quantiﬁes the asymptotic decay of f ∈ S (Rd ) in the time and frequency variables. For the family of modulation spaces a discretized version of the map Vg is also invertible and there exists a so called Gabor frame expansion [6, 22] p,q f, Mβn Tαk gMβn Tαk γ, f ∈ Mm (Rd ), (1.3) f= k∈Zd n∈Zd

under certain conditions on the functions g, γ and the time-frequency lattice parameters α, β > 0 discovered by Feichtinger, Gr¨ ochenig, Leinert [15, 22, 24] and others. The inversion formula (1.2) suggests operators of the form g,γ (Aa f )(x) = a(t, ξ)Vg f (t, ξ)Mξ Tt γ(x)dtdξ R2d

which performs a multiplicative modiﬁcation in the time-frequency domain, using a symbol a, before the distribution is reconstructed by integration. Such operators are called (time-frequency) localization operators, sometimes also Toeplitz or Anti-Wick operators, and they are a special case of pseudodiﬀerential operators. The diﬀerent names reﬂect that they have been studied in the signal analysis, mathematical and physical literature, as signal ﬁltering operators [8, 45], pseudodiﬀerential operators [17, 45], and quantization rules [17], respectively. They have recently attracted much attention [5, 7, 43, 44, 45], not least with respect to continuity when they act on modulation spaces. Cordero, Gr¨ochenig [7] and Toft [43] have proved that the large space of symbols M ∞ (R2d ) corresponds to operators that are bounded on all modulation spaces M p,q (Rd ), 1 ≤ p, q ≤ ∞. Also the discrete expansion (1.3) can be used to deﬁne localization operators, using discrete symbols, called Gabor multiplier operators [16]. More information on the theory

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and applications (eg in pseudodiﬀerential calculus) of modulation spaces can be found in [14, 22, 23, 43, 44]. In this paper we generalize some of these results and replace scalar-valued functions and tempered distributions by vector-valued functions and tempered distributions taking values in a Banach space B or a Hilbert space H. First we discuss STFT and modulation space theory for vector-valued tempered distributions. It turns out that a large part of the theory of scalar-valued modulation spaces is possible to generalize to the H valued case, with certain modiﬁcation in the B valued case depending on the Banach space B. This is essentially due to the fact that Parseval’s formula is true for B-valued L2 spaces if and only if B is isomorphic to a Hilbert space. The identity L2 = M 2 which is true in the scalar-valued case generalizes to the vector-valued case if and only if the vector space is a Hilbert space. In the general Banach space case this identity is replaced by the embeddings Lp ∩ FLp → M p , 1 ≤ p ≤ r, which depends on the Fourier type r ≤ 2 of the Banach space B. There is a generalization of the duality result (M p,q ) = M p ,q , 1 ≤ p, q < ∞ ([22, Thm. 11.3.6]), which is true when B is a reﬂexive Banach space and p = q = 1 or 1 < p < ∞ and 1 ≤ q < ∞. For a corresponding discussion, of greater depth, on the properties of vector-valued Sobolev and Besov spaces we refer to [38]. Furthermore we state that vector-valued modulation spaces can be characterized as the Fourier transform of Wiener amalgam spaces with discrete global component [10, 12, 13], similarly to the scalar-valued case. We discuss the concept of weak stationarity (with application eg to stochastic processes) in the Hvalued case and obtain the result that each weakly stationary H-valued tempered ∞,1 . distribution belongs to a weighted modulation space of the type M1⊗m We also discuss Gabor frame expansions for H-valued functions and tempered distributions and conclude that the Gabor frame theory for scalar-valued modulation spaces is possible to generalize to H-valued modulation spaces almost without modiﬁcation. Finally we treat localization, Weyl and Gabor multiplier operators in this context, using an operator-valued symbol. We formulate generalizations of recent results on continuity of localization operators acting on modulation spaces, developed in the context of scalar-valued distributions and symbols. First we state that a result of Boggiatto’s [5] on continuity between certain modulation spaces of a localization operator, if the symbol belongs to Lp,q , generalizes to B-valued modulation spaces, regardless of B. By an example we show that there exists Banach spaces B1 , B2 and a symbol in L∞ such that the localization operator is not bounded from the B1 -valued L2 space to the B2 -valued L2 space, which can not happen in the scalar-valued case. Secondly we generalize the above mentioned result by Cordero and Gr¨ ochenig [7] on continuity of a localization operator, acting on any modulation space, provided the symbol belongs to M ∞ . We prove that the result is valid for H-valued modulation spaces. We also obtain a generalization of a result of Gr¨ ochenig and Heil [22, 23] on continuity of Weyl operators on all

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modulation spaces provided the symbol belongs to the Sj¨ ostrand space M ∞,1 , to the context of H-valued modulation spaces and operator-valued symbols. The motivating background of this work consists of two directions is contemporary analysis. On the one hand, pseudodiﬀerential calculus for scalar-valued functions, which classically is studied with symbols in Fr´echet spaces of smooth functions [17, 27, 40], has since the work of Sj¨ostrand [41] been extended to symbols in modulation spaces, which are Banach spaces of limited regularity (see eg [5, 7, 22, 23, 41, 43, 44]). One of Sj¨ostrand’s results says that symbols in M ∞,1 give rise to bounded operators on L2 , which was extended by Gr¨ ochenig and Heil who proved boundedness on all modulation spaces M p,q , 1 ≤ p, q ≤ ∞, for the same class of symbols. On the other hand, there has independently of this trend appeared several papers dealing with new versions of classical Fourier multiplier theorems, where the operator act on scalar-valued function spaces, to vector-valued function spaces and operator-valued symbols [2, 19, 20, 29]. For example, versions of Mihlin’s multiplier theorem on conditions on a symbol that are suﬃcient for continuity on Lp spaces, have been proved for vector-valued Besov spaces by Amann, Girardi, Weis and Hyt¨onen [2, 20, 29]. Fourier multiplier operators are a special case of pseudodiﬀerential operators, which also have been studied in the vector-valued context [30, 36]. These results have applications eg in PDE. Embeddings between scalar-valued Besov and modulation spaces have been proved by Toft [43]. It is natural to combine the above mentioned research directions and investigate pseudodiﬀerential operators with operator-valued symbols in modulation spaces, acting on vector-valued modulation spaces, which is the topic of the present study. For this purpose we ﬁrst need to examine the properties of vector-valued modulation spaces. There is also a motivation from the point of view of certain applications in engineering. In fact, pseudodiﬀerential operators are used as mathematical models of mobile radio channels [42]. Such channels are often assumed to be stochastic [34]. If one assumes that the signals to be transmitted also are stochastic processes, then the framework presented here may be a candidate for a model of signal transmission, since stochastic processes may be seen as vector-valued functions where the vector space is a space of stochastic variables. 1.1. Definitions and notation We denote Lebesgue measure by µ, the Schwartz space by S(Rd ) and the smooth functions of compact support by Cc∞ (Rd ). The Fourier transformation of f ∈ S(Rd ) is deﬁned by F f (ξ) = f(ξ) = f (t)e−i2πtξ dt. Rd

Partial Fourier transformation with respect to variable j is denoted Fj . The joint modulation-translation operator is sometimes denoted π(z) = Mξ Tt , z = (t, ξ) ∈ R2d . Discrete subgroups (lattices) of R2d of the form {(αk, βn)}k,n∈Zd , α, β > 0,

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will be denoted Λ. The (extended) Wigner distribution [17, 22] of a function f ∈ S(R2d ) is deﬁned by f (t + τ /2, t − τ /2)e−i2πξτ dτ = F2 (f ◦ κ)(t, ξ) (1.4) W (f )(t, ξ) = Rd

with κ(t, τ ) = (t+τ /2, t−τ /2). When f = g⊗g we write W (g⊗g) = W (g) (ambiguous but clear from context) which is the proper Wigner distribution [17, 22]. We denote coordinate reﬂection by f(x) = f (−x). In estimates we denote by C a positive constant which may change value over inequalities. The tempered distributions are denoted S (Rd ), and are in this paper assumed to be antilinear (ie conjugate linear), in order to be compatible with the L2 (Rd ) inner product. The bracket ·, · will denote, depending on context, (i) action of an antilinear scalar-valued tempered distribution, (ii) action of an antilinear Banach space valued tempered distribution, or (iii) the Lebesgue or Bochner integral f, g = Rd f (x)g(x)dx where in general f is vector-valued and g is scalar-valued. The consistent deﬁnition of the := f, ϕ, ϕ ∈ S(Rd ). SomeFourier transformation of f ∈ S (Rd ) is then f, ϕ times we emphasize the space on which a distribution acts by writing eg ·, ·S(Rd ) . The map (1.4) extends by duality to a continuous map W : S (R2d ) → S (R2d ). For the range space of vector-valued distributions we use B to denote a Banach space and H to denote a Hilbert space. The topological dual of a Banach (or more generally a linear topological) space B is denoted B , the duality (·, ·)B ,B = (·, ·)B , and the Hilbert space inner product (·, ·)H , linear in the ﬁrst and antilinear in the second argument. In order to be compatible with the Hilbert space inner product, (·, ·)B is deﬁned to be linear in the ﬁrst and antilinear in the second argument. For an exponent p, 1 ≤ p ≤ ∞, we denote the conjugate exponent p , which fulﬁlls 1/p + 1/p = 1. The set of bounded linear transformations from a Banach space B1 to another Banach space B2 is denoted L(B1 , B2 ) and L(B, B) := L(B). Capital letters will often denote Banach space valued

functions or distributions. A function F is simple if it is a ﬁnite sum F = j xj χAj where xj ∈ B, χA denotes indicator function, Aj ∈ B(Rd ) (the Borel σ-algebra) and µ(Aj ) < ∞ for all j. A function F is strongly measurable if there exists a sequence of simple functions Fn such that limn→∞ Fn (t) − F (t)B = 0 for almost all t ∈ Rd . It is weakly measurable if t → (x , F (t))B is measurable for each x ∈ B , and it is almost separably valued if there exists a null set NF ⊂ Rd such that the range space F (Rd \ NF ) ⊂ B is separable. According to Pettis’s measurability theorem [9] F is strongly measurable if and only if F is weakly measurable and almost separably valued. The Bochner integral of a B-valued function F is well deﬁned provided it is strongly measurable and Rd F (t)B dt < ∞ [9]. For each 1 ≤ p ≤ ∞ the Lebesgue-Bochner space Lp (Rd , B) [9] is deﬁned as the set of strongly measurable functions F : Rd → B such that F Lp(Rd ,B) := ( Rd F (t)pB dt)1/p < ∞, with the standard modiﬁcation when p = ∞. For F : Rd → B and G : Rd → B one deﬁnes the covariance function by (1.5) σF G (t, s) = F (t), G(s) B , t, s ∈ Rd .

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The Schwartz space of functions F : Rd → B such that supt∈Rd tα ∂ β F (t)B < ∞ for all multi-indices α and β is denoted S(Rd , B). There exists a theory of distributions which take values in a locally convex topological vector space, developed by L. Schwartz [39]. We will however restrict to tempered distributions taking values in a Banach space [1, 18, 28, 31, 32]. Such a distribution, denoted F ∈ S (Rd , B), is here deﬁned as a bounded antilinear map F : S(Rd ) → B, ie there exists positive constants C, N, M such that F, ϕB ≤ C ϕα,β (1.6) |α|≤N

|β|≤M

where α, β are multi-indices and ϕα,β := supt∈Rd |∂ α (tβ ϕ(t))| are seminorms on S(Rd ). We have the duality result S(Rd , B) = S (Rd , B ), where the duality is induced by f ⊗ x , ϕ ⊗ x := f, ϕ(x , x)B , f ∈ S (Rd ), ϕ ∈ S(Rd ), x ∈ B , x ∈ B [1]. If B is reﬂexive then S(Rd , B) is reﬂexive as well [1]. For F ∈ S (Rd , B ) and G ∈ S (Rd , B) the covariance distribution σF G is deﬁned by (1.7) σF G , ϕ ⊗ ψ = F, ϕ, G, ψ B , ϕ, ψ ∈ S(Rd ), for rank-one elements ϕ ⊗ ψ ∈ S(R2d ). According to the Schwartz kernel theorem [18, 37] one can extend σF G to a distribution σ ∈ S (R2d ). A function d F ∈ Lp (Rd , B), 1 ≤ p ≤ ∞, deﬁnes an element in S (R , B) by the Bochner integral F, ϕ = Rd F (t)ϕ(t)dt, which admits consistency between (1.5) and (1.7). The Fourier transform of a function F ∈ L1 (Rd , B) is deﬁned by a Bochner Fourier integral and the Fourier transform F ∈ L∞ (Rd , B). The Hausdorﬀ-Young inequality is however in general not valid [19, 20, 35]. A Banach space B is said to be of Fourier type r, 1 ≤ r ≤ 2, if the Fourier transform F : Lr (Rd , B) → Lr (Rd , B). This notion was introduced by Peetre [35]. It follows by interpolation that if B is of Fourier type r then it is also of Fourier type p for all p in the interval 1 ≤ p ≤ r. Every Banach space has Fourier type at least one. By a result of Kwapie´ n [33] a Banach space B has Fourier type 2 if and only if it is isomorphic to a Hilbert space. Hence Parseval’s formula holds for L2 (Rd , B) if and only if B is isomorphic to a Hilbert space.

2. The STFT of vector-valued tempered distributions

The STFT of F ∈ Lp (Rd , B) with respect to a window function g ∈ Lp (Rd ) is deﬁned by the Bochner integral F (s)ei2πξs g(s − t)ds = F (F Tt g)(ξ) = F, Mξ Tt g. (2.1) Vg F (t, ξ) = Rd

Following the deﬁnition of the STFT of scalar-valued tempered distributions [22], the STFT of a vector valued F ∈ S (Rd , B) is for g ∈ S(Rd ) deﬁned by the right hand side of (2.1) where ·, · denotes distribution action instead of Bochner

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integral. The following two lemmas generalize results for scalar-valued tempered distributions and will be useful in later sections. Lemma 2.1. If F ∈ S (Rd , B) and g ∈ S(Rd ) then Vg F ∈ C ∞ (R2d , B), Vg F is strongly measurable, and there exists an integer N > 0 and C > 0 such that Vg F (t, ξ)B ≤ C(1 + |t| + |ξ|)N , (t, ξ) ∈ R2d .

(2.2)

Proof. By [26, Thm. 2.1.3] (slightly modiﬁed) Vg F ∈ C ∞ (R2d , B), which implies that Vg F is strongly measurable. By (1.6) there exists positive C, M1 and M2 such that F, Mξ Tt gB ≤ C Mξ Tt gα,β . (2.3) |α|≤M1

|β|≤M2

Lemma 11.2.1 in [22] gives the bound Mξ Tt gα,β ≤ C(1 + |t| + |ξ|)2 max(|α|,|β|) , which in combination with (2.3) gives (2.2). Lemma 2.2. If F1 ∈ S(Rd , B ), F2 ∈ S(Rd , B) and g1 , g2 ∈ S(Rd ) then (F1 (x), F2 (x))B dx g1 , g2 L2 (Rd ) . Vg1 F1 (z), Vg2 F2 (z) B dz = R2d

Rd

Proof. We obtain from (2.1), since σF1 F2 ∈ S(R2d ), Vg1 F1 (z), Vg2 F2 (z) B dz R2d = ··· σF1 F2 (x, y)e−i2πξ(x−y) g1 (x − t)g2 (y − t)dxdydtdξ R4d = σF1 F2 (x, x)dx g1 , g2 L2 (Rd ) . Rd

Density arguments [1] now gives the following generalization of the orthogonality relations of the STFT [22] from scalar-valued L2 functions to H-valued L2 functions. The inner product in the Hilbert space L2 (Rd , H) is (F, G)L2 (Rd ,H) = Rd (F (t), G(t))H dt. Corollary 2.3. If F1 , F2 ∈ L2 (Rd , H) and g1 , g2 ∈ L2 (Rd ) then (Vg1 F1 , Vg2 F2 )L2 (R2d ,H) = (F1 , F2 )L2 (Rd ,H) g1 , g2 L2 (Rd ) . Given F ∈ L2 (R2d , H) and γ ∈ L2 (Rd ) we deﬁne for G ∈ L2 (Rd , H) (Uγ F, G) := (F (z), Vγ G(z))H dz. R2d

Then Uγ F is a continuous antilinear functional on L2 (Rd , H), since by Cor. 2.3 |(Uγ F, G)| ≤ F L2 (R2d ,H) GL2 (Rd ,H) γL2 (Rd ) . Since L2 (Rd , H) is a Hilbert space there exists an element, denoted R2d F (z)π(z)γdz ∈ L2 (Rd , H), such that (Uγ F, G) = F (z)π(z)γdz, G L2 (Rd ,H) . R2d

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The next result generalizes [22, Cor. 3.2.3] and says that F ∈ L2 (Rd , H) can be reconstructed from Vg F . Proposition 2.4. If g, γ ∈ L2 (Rd ) fulfill γ, g = 0 then −1 F = γ, g Vg F (z)π(z)γdz, F ∈ L2 (Rd , H).

(2.4)

R2d

Proof. Let G ∈ L2 (Rd , H). Then γ, g−1 Vg F (z)π(z)γdz, G L2 (Rd ,H) R2d = γ, g−1 (Vg F (z), Vγ G(z))H dz = (F, G)L2 (Rd ,H) R2d

2

by Cor. 2.3. Since G ∈ L (Rd , H) is arbitrary (2.4) follows.

Next we extend Prop. 2.4 and show that the STFT is invertible on S (Rd , B). The proof is similar to the corresponding result for S (Rd ) [22]. Deﬁne for a strongly measurable F : R2d → B and γ ∈ S(Rd ) the B-valued map F (z)π(z)γ, ϕdz, ϕ ∈ S(Rd ), (2.5) Vγ∗ F, ϕ := R2d

and denote this Vγ∗ F =

F (z)π(z)γdz. R2d

We can deﬁne F0 := γ, g−1 Vγ∗ Vg F , ie F0 , ϕ = γ, g−1 R2d Vg F (z)π(z)γ, ϕdz, since Vg F is strongly measurable by Lemma 2.1. In order to show that F0 ∈ S (Rd , B) we estimate using (2.2) Vg F (z)B |Vγ ϕ(z)| dz F0 , ϕB ≤ |γ, g|−1 R2d (2.6) ≤ C sup (1 + |z|)N +2d+1 |Vγ ϕ(z)| . z

According to [22, Cor. 11.2.6] the last expression is a seminorm on ϕ ∈ S(Rd ), and hence F0 ∈ S (Rd , B). Proposition 2.5. If g, γ ∈ S(Rd ) fulfill γ, g = 0 then F = F0 := γ, g−1 Vγ∗ Vg F, F ∈ S (Rd , B). (2.7) Proof. For ϕ ∈ S(Rd ) we have ϕ = g, γ−1 R2d Vγ ϕ(z)π(z)gdz. Thus, by a generalization of the tensor product rule f ⊗ 1 = 1 ⊗ f for f ∈ S (Rd ) [26] to f ∈ S (Rd , B) [38], we have −1 F, π(z)gVγ ϕ(z)dz F, ϕ = g, γ R2d = γ, g−1 Vg F (z)π(z)γ, ϕdz = F0 , ϕ. R2d

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p,q 3. Modulation spaces Mm (Rd , B)

The concept of weighted mix-normed spaces of scalar-valued functions [3] can be generalized to weighted mix-normed spaces of vector-valued functions F : R2d → 2d B. For 1 ≤ p, q ≤ ∞, F ∈ Lp,q m (R , B) if F is strongly measurable and q/p 1/q F Lp,q = F (t, ξ)pB m(t, ξ)p dt dξ < ∞, 2d ,B) m (R Rd

Rd

with the standard modiﬁcation when p or q equals inﬁnity. Here m is a positive weight function which is assumed to be v-moderate [22], ie m fulﬁlls m(x + y) ≤ Cv(x)m(y), x, y ∈ R2d , for a positive function v which is submultiplicative, ie v(x + y) ≤ v(x)v(y), x, y ∈ R2d . In this paper we restrict to polynomially bounded functions v. A standard class of polynomially increasing submultiplicative functions is vs (x) = (1 + |x|2 )s/2 , x ∈ R2d , s ≥ 0. In Section 4 we need also the corresponding class where the weight is constant in the ﬁrst variable, denoted τs (x) = (1 + |x2 |2 )s/2 , s ≥ 0, x1 ∈ Rd , x2 ∈ Rd , x = (x1 , x2 ). Analogously to the case of scalar-valued distributions [12, 22] we deﬁne modulation spaces, p,q 2d 1 ≤ p, q ≤ ∞, by F ∈ Mm,g (Rd , B), if F ∈ S (Rd , B) and Vg F ∈ Lp,q m (R , B) where g ∈ S. By Lemma 2.1 Vg F is always strongly measurable. The modulation space norm is d p,q F Mm,g 2d ,B) , g ∈ S(R ). (Rd ,B) := Vg F Lp,q m (R

(3.1)

Modulation spaces of Banach space valued distributions have already been considered by Toft [44]. 3.1. Retract property, admissible windows, Wiener amalgam space characterization, properties Prop. 2.5 says that γ, g−1 Vγ∗ Vg = idS (Rd ,B) . The following result corresponds to [22, Prop. 11.3.2] and [14, Cor. 4.6], has a similar proof, and shows that 2d p,q d p,q d Vγ∗ : Lp,q m (R , B) → Mm,g (R , B), ie Mm,g (R , B) is a Banach space retract of p,q 2d Lm (R , B) [4]. Proposition 3.1. Suppose g, γ ∈ S(Rd ) fulfill γ, g = 0, m is a v-moderate weight p,q p,q function, and 1 ≤ p, q ≤ ∞. Then Mm,g (Rd , B) → S (Rd , B), and Mm,g (Rd , B) is p,q 2d −1 ∗ p,q a Banach space retract of Lm (R , B), ie γ, g Vγ Vg = idMm,g (Rd ,B) and p,q 2d Vg : Mm,g (Rd , B) → Lp,q m (R , B), 2d p,q d Vγ∗ : Lp,q m (R , B) → Mm,g (R , B). p,q 2d Proof. By (3.1) Vg is an isometric linear map from Mm,g (Rd , B) to Lp,q m (R , B), −1 ∗ −1 ∗ p,q Vγ Vg = idS (Rd ,B) . If G ∈ and γ, g Vγ Vg = idMm,g (Rd ,B) follows from γ, g 2d (R , B) then G is strongly measurable and by H¨ o lder’s inequality Lp,q m Vγ∗ G, ϕB ≤ G(z)B |Vγ ϕ(z)| dz R2d (3.2) ≤ GLp,q sup (v (z) |V ϕ(z)|) , 2d ,B) v−N p ,q N γ (R 2d m L (R ) 1/m

z

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,q where N is chosen large enough to guarantee v−N ∈ Lp1/m (R2d ), which is possible since 1/m ≤ Cv [22] and v is polynomially bounded. Thus Vγ∗ G ∈ S (R2d , B). By Lemma 2.1 Vg (Vγ∗ G) is strongly measurable. By the commutation rule Tx My f = e−i2πxy My Tx f G(s, η)Vγ (Mξ Tt g)(s, η)dsdη Vg (Vγ∗ G)(t, ξ) = Vγ∗ G, Mξ Tt g = R2d (3.3) −i2πs(ξ−η) = G(s, η)Vg γ(t − s, ξ − η)e dsdη. R2d

Hence

Vg (Vγ∗ G)(z)B ≤ G(·)B ∗ |Vg γ| (z)

(3.4)

which, using an inequality of Young type for mix-normed spaces, [22, Prop. 11.1.3], admits the estimate ∗ p,q Vγ∗ GMm,g (Rd ,B) = Vg (Vγ G)(z)B Lp,q 2d ) m (R (3.5) ≤ C G(·)B Lp,q (R2d ) Vg γL1v (R2d ) . m

2d p,q d Thus Vγ∗ deﬁnes a bounded linear map from Lp,q m (R , B) to Mm,g (R , B). If we p,q put G = Vg F with F ∈ Mm,g (Rd , B) and use Prop. 2.5, we can also conclude from d p,q (3.2) that F, ϕB ≤ C(ϕ)F Mm,g (Rd ,B) where C(ϕ) is a seminorm on S(R ), p,q d d ie Mm,g (R , B) → S (R , B).

For later use we specialize the inequality (3.4) to G = Vγ F where F ∈ S (Rd , B), ie Vg F (z)B = γ−2 Vg Vγ∗ Vγ F (z)B ≤ γ−2 Vγ F (·)B ∗ |Vg γ| (z).

(3.6)

Likewise, with arguments similar to the proof of the corresponding result for scalar-valued modulation spaces [22, Thm. 11.3.7] (we omit the details), one can prove the following result. Proposition 3.2. Let m be a v-moderate weight function. p,q p,q (i) If g, γ ∈ S(Rd ) then · Mm,γ (Rd ,B) and · Mm,g (Rd ,B) are equivalent norms. d 1 d p,q p,q (ii) If g ∈ S(R ) and γ ∈ Mv (R ) then · Mm,γ (Rd ,B) and · Mm,g (Rd ,B) are equivalent norms. (iii) Prop. 3.1 holds with g, γ ∈ Mv1 (Rd ). p,q (Rd , B) are independent of g ∈ Mv1 (Rd ), and Consequently the spaces Mm,g for a ﬁxed but arbitrary 0 = g ∈ Mv1 (Rd ) and a v-moderate weight function p,q m we can deﬁne the modulation space Mm (Rd , B) ⊂ S (Rd , B) as the set of d p,q F ∈ S (R , B) such that F Mm,g (Rd ,B) < ∞. We follow the conventional notation p p,p (Rd , B) = Mm (Rd , B) and M p,q (Rd , B) for the unweighted case, ie m ≡ 1. Mm The fact that the Fourier transform of the modulation spaces of scalarvalued distributions equals certain Wiener amalgam spaces [12] is true also in

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the Banach space valued case. In fact, if F F ∈ M p,q (Rd , B) and g ∈ Cc∞ (Rd ), then F, Mξ T−t g = F (F T−t g)(ξ) by a generalization to the vector-valued case of a result valid for scalar-valued distributions of compact support [26, 38]. From gB = F, Mξ T−t gB = F (F T−t g)(ξ)B we obtain FF, Mt Tξ 1/q , F F M p,q (Rd ,B) = F T−t gqF Lp (Rd ,B) dt = F p d q d W F L (R ,B),L (R )

Rd

where W F Lp (Rd , B), Lq (Rd ) is the Wiener amalgam space of B-valued tempered distributions with local component F Lp (Rd , B) and global component Lq (Rd ) [10, 12]. The norm of a Wiener amalgam space is equivalent to a discrete norm deﬁned by a so called BUPU (bounded uniform partition of unity) [10, 12, 13]. A BUPU Ψ = {ψj }j∈J is a set of nonnegative functions required to fulﬁll, for a discrete set {xj }j∈J and a relatively compact set U with nonempty interior, ψj (x) ≡ 1, (i) j∈J

(ii) sup ψj F L1 (Rd ) < ∞, j∈J

(iii) supp ψj ⊂ xj + U ∀j ∈ J, (iv) sup |{j; (xi + U ) ∩ (xj + U ) = ∅}| < ∞. i∈J

Then if ψj ∈ M 1 (Rd ) for all j we have the norm equivalence for all 1 ≤ p, q ≤ ∞ [10, 13] 1/q F W (F Lp(Rd ,B),Lq (Rd )) F ψj qF Lp(Rd ,B) . j∈J

The proof of this result in the scalar-valued case [10, 13] extends to the Banach space valued case. The proof of the completeness of Wiener amalgam spaces W (X, Y ) under quite general hypotheses on the local norm X and the global p,q (Rd , B) are norm Y [10] also extends to W F Lp (Rd , B), Lq (Rd ) , and hence Mm Banach spaces for all 1 ≤ p, q ≤ ∞. The following proposition treats more generalizations of results valid for scalar-valued modulation spaces. We omit the proofs since they are almost identical to the proofs in [22], again after replacement of | · | with · B in the obvious places. Proposition 3.3. Suppose m is a v-moderate weight function. (i) If |m(z)| ≤ CvN (z), z ∈ R2d , for some N > 0 and 1 ≤ p, q < ∞, then p,q (Rd , B). S(Rd , B) is dense in Mm p1 ,q1 p2 ,q2 (ii) If p1 ≤ p2 , q1 ≤ q2 and m2 ≤ Cm1 then Mm (Rd , B) → Mm (Rd , B). 1 2 d ∞ d (iii) S (R , B) = s≥0 Mv−s (R , B). (iv) If m(ξ, −t) ≤ Cm(t, ξ) then the Fourier transformation is continuous on p (Rd , B). Mm

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(v) Let H be a Hilbert space. 2 If m depends only on t then L2m (Rd , H) = Mm (Rd , H) with equivalent norms. 2 d 2 If m depends only on ξ then F Lm (R , H) = Mm (Rd , H) with equivalent norms. In the following proposition we use the Fourier type of the Banach space B. It can be seen as a modiﬁcation of (v) in the last proposition. Proposition 3.4. Suppose B has Fourier type r, 1 ≤ r ≤ 2, let 1 ≤ p ≤ r and let m be a v-moderate weight function.

p (i) If m depends only on t then Lpm (Rd , B) → Mm (Rd , B). p d p (ii) If m depends only on ξ then F Lm (R , B) → Mm (Rd , B). p d p d p d (iii) Thus L (R , B) ∩ FL (R , B) → M (R , B).

Proof. (i). Let F ∈ S (Rd , B) and ϕ ∈ Cc∞ (R2d ). If p > 1, using the restricted Hausdorﬀ-Young inequality which is true by the deﬁnition of the Fourier type, p /p ≥ 1, Minkowski’s inequality and m(s − x) ≤ Cm(s)v(−x), we have F p p d = F (F Tt ϕ)(ξ)pB dξ m(t)p dt Mm (R ,B) d Rd R p /p ≤C (F Tt ϕ)(s)pB ds m(t)p dt d d R R p /p =C F (s)pB |ϕ(s − t)|p m(t)p ds dt Rd Rd (3.7) p/p p /p p p p ≤C F (s)B |ϕ(s − t)| m(t) dt ds d Rd R p /p ≤C F (s)pB m(s)p ds |ϕ(x)|p v(−x)p dx =

Rd CF pLpm (Rd ,B) .

Rd

The case p = 1 follows similarly. Result (ii) follows from the estimate (3.7) and B. F, Mξ Tt ϕB = FF, M−t Tξ ϕ p,q (Rd , B) when B is reflexive 3.2. The dual of Mm The following three lemmas have statements and proofs that are slight adaptations of results in [3] where B is a space of scalars.

Lemma 3.5. If 1 ≤ p, q ≤ ∞ and F : R2d → B is strongly measurable then

F Lp,q (R2d ,B) = sup G(z), F (z) B dz R2d

where the supremum is taken over all G : R2d → B such that: (i) z → G(z), F (z) B is measurable, (ii) z → G(z)B is measurable, and (iii) GLp ,q (R2d ,B ) = 1.

(3.8)

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If B = H is a Hilbert space then the supremum can be taken over G ∈ Lp ,q (R2d , H) such that GLp ,q (R2d ,H) = 1. Note that in the case of a general Banach space B the requirements (i)–(iii) do not imply G ∈ Lp ,q (R2d , B ) since G is not necessarily strongly measurable. Proof. Inequality ≥ in (3.8) follows from H¨ older’s inequality. To prove inequality ≤ we ﬁrst restrict to F Lp,q (R2d ,B) < ∞. For all (x, y) ∈ R2d there exists by the Hahn-Banach theorem a functional P (x, y) ∈ B such that P (x, y)B = 1 and P (x, y), F (x, y) B = F (x, y)B . When F (x, y) = 0 we deﬁne P (x, y) = 0. If p, q < ∞ we deﬁne q−p · F (x, y)p−1 · F (·, y)B Lp (Rd ) . G(x, y) := P (x, y) · F 1−q B Lp,q (R2d ,B) Then (i)–(iii) are fulﬁlled and R2d G(z), F (z) B dz = F Lp,q (R2d ,B) . Next suppose p = ∞, q < ∞ and supp(F ) is compact. Let > 0 and deﬁne q−1 G (x, y) := P (x, y) · F 1−q · F (·, y)B Lp (Rd ) · χU (x, y)/µ(U (·, y)) Lp,q (R2d ,B) if µ(U (·, y)) > 0 and G (x, y) := 0 otherwise, where U = {(x, y) ∈ R2d : F (x, y)B > (1 + )−1 F (·, y)B Lp (Rd ) } ⊂ supp(F ). Then (i)–(iii) are again fulﬁlled, and G (z), F (z) B dz ≥ (1 + )−1 F Lp,q (R2d ,B) . R2d

Also if supp(F ) is not compact the result follows since F Lp,q (R2d ,B) = lim F χKj Lp,q (R2d ,B) j→∞

for an exhausting nested sequence of compact sets Kj ⊂ R2d . The cases p < ∞, q = ∞ and p = q = ∞ are proved in a way similar to p = ∞, q < ∞ [3]. If F Lp,q (R2d ,B) = ∞ we deﬁne Fj := F χKj if F (z)B ≤ j and Fj := 0 otherwise. Then Fj ∈ Lp,q (R2d , B) and Fj (z)B → F (z)B monotoneously as j → ∞ for all z ∈ R2d . Hence ∞ = F Lp,q (R2d ,B) = lim Fj Lp,q (R2d ,B) j→∞

= lim sup G(z), Fj (z) B dz j→∞ G R2d

≤ sup G(z), F (z) dz . G

R2d

B

Finally, in the case B = H, P (z) = F (z)/F (z)H which means that P , and therefore also G, is strongly measurable.

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Remark 3.6. Not only in the Hilbert space case B = H the supremum in (3.8) can be taken over all (strongly measurable) G ∈ Lp ,q (R2d , B ) such that GLp ,q (R2d ,B ) = 1. In fact it can be checked that this is also true when B = lr (Z), 1 < r < ∞. Lemma 3.7. The mix–normed space Lp,q (R2d , B) is complete for 1 ≤ p, q ≤ ∞. Proof. Let {Fn }n≥1 be a Cauchy sequence in Lp,q (R2d , B) and let {Kj }j≥1 be an exhaustive nested sequence of compact sets in R2d . By H¨older’s inequality Fn (z) − Fm (z)B dz ≤ Fn − Fm Lp,q (R2d ,B) χKj Lp ,q (R2d ) . Kj

There exists F such that limn→∞ (F − Fn )χKj L1 (R2d ,B) = 0 for every j, by the completeness of L1 (R2d , B). F is strongly measurable since F χKj ∈ L1 (R2d , B) for each Kj . By a Cantor diagonal procedure we can extract a subsequence {Fnk } such that limk→∞ Fnk = F almost everywhere. Lemma 3.5 and Fatou’s lemma gives F − Fnk Lp,q (R2d ,B) ≤ sup lim G(z)B Fnj (z) − Fnk (z)B dz G R2d j→∞ G(z)B Fnj (z) − Fnk (z)B dz ≤ sup lim inf G

j→∞

R2d

≤ lim inf Fnj − Fnk Lp,q (R2d ,B) . j→∞

By the following lemma most mix–normed spaces may be looked upon as Lq –spaces of functions taking values in Lp (Rd , B). Lemma 3.8. If 1 ≤ p, q < ∞ then Lq (Rd , Lp (Rd , B)) ⊃ Lp,q (R2d , B) with equal norms. If 1 ≤ p, q ≤ ∞ then Lq (Rd , Lp (Rd , B)) ⊂ Lp,q (R2d , B) with equal norms. Proof. Suppose F ∈ Lp,q (R2d , B) and 1 ≤ p, q < ∞. Then there exists a sequence of simple functions ank χAnk (x)χBnk (y), ank ∈ B, Fn (x, y) = k

such that limn→∞ F − Fn Lp,q (R2d ,B) = 0 [3, 25]. Thus for a subsequence Fnk we have limk→∞ F (·, y) − Fnk (·, y)Lp (Rd ,B) = 0 for a.a. y. Hence y → F (·, y) ∈ Lp (Rd , B) is strongly measurable and F Lp,q (R2d ,B) = F Lq (Rd ,Lp (Rd ,B)) . Suppose on the other hand that F ∈ Lq (Rd , Lp (Rd , B)) and 1 ≤ p, q ≤ ∞. Since F is strongly measurable there exists a sequence of functions ∞ Fn (x, y) = fnk (x)χBnk (y), fnk ∈ Lp (Rd , B), Bnk ∈ B(Rd ), k=1

{Bnk }∞ k=1

where are pairwise disjoint, such that for a.a. y, limn→∞ Fn (·, y) − F (·, y)Lp (Rd ,B) = 0 uniformly [9]. Since fnk are almost separably valued in B for

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all k, Fn is almost separably valued in B, and since fnk is weakly measurable in B for all k, Fn is weakly measurable in B. Hence Fn is strongly measurable by Pettis’s theorem. Let Kj ⊂ Rd be a nested exhausting sequence of compact sets. Then χKj (y)(Fn (·, y) − Fm (·, y))Lp (Rd ,B) q d −→ 0, n, m −→ ∞, L (R )

and hence χ1×Kj Fn is a Cauchy sequence in L (R2d , B) for all j. By an argument borrowed from the proof of Lemma 3.7 there exists a strongly measurable function G : R2d → B such that Fnk → G for a.a. (x, y) for a subsequence Fnk . Thus we have for a.a. y, G(·, y) = limk→∞ Fnk (·, y) = F (·, y) in Lp (Rd , B). Therefore F = G in Lq (Rd , Lp (Rd , B)) and F Lq (Rd ,Lp (Rd ,B)) = GLp,q (R2d ,B) . p,q

If p or q is inﬁnite, the inclusion Lq (Rd , Lp (Rd , B)) ⊂ Lp,q (R2d , B) may be strict, since for F ∈ Lp,q (R2d , B) the function y → F (·, y) ∈ Lp (Rd , B) may fail to be strongly measurable [3]. Next we use the previous lemma to give a characterization of the dual of p,q Mm (Rd , B) when B is reﬂexive. In contrast to the corresponding result for scalar– valued modulation spaces, Thm. 11.3.6 in [22], we exclude the cases (p = 1, 1 < q < ∞). Here Lqm (Rd , Lp (Rd , B)) denotes the set of strongly measurable F : R2d → B such that F m ∈ Lq (Rd , Lp (Rd , B)). Proposition 3.9. Suppose p = q = 1 or 1 < p < ∞ and 1 ≤ q < ∞, B is a reflexive p,q Banach space, and let ϕ ∈ S(Rd ) fulfill ϕL2 (Rd ) = 1. Then (Mm (Rd , B)) =

p ,q p ,q (Rd , B ), in the sense that every G ∈ M1/m (Rd , B ) defines an element in M1/m p,q d p,q (Mm (R , B)) under the sesquilinear form (3.9), and for every u ∈ (Mm (Rd , B)) p ,q there is a G ∈ M1/m (Rd , B ) such that p,q p,q (u, F ) = (G, F )Mm (Rd ,B) = (Rd , B). (3.9) Vϕ G(z), Vϕ F (z) B dz, F ∈ Mm R2d

p ,q Proof. H¨ older’s inequality in Lp,q (R2d ) implies that every G ∈ M1/m (Rd , B ) p,q d deﬁnes an element in (Mm (R , B)) by (3.9). Suppose on the other hand that p,q p,q 2d u ∈ (Mm (Rd , B)) . The fact that Vϕ is isometric Mm (Rd , B) → Lp,q m (R , B) justiﬁes the factorization (u, F ) = (u1 , Vϕ F ) where u1 is an induced linear func2d tional which acts boundedly on a closed linear subspace of Lp,q m (R , B). By the p,q Hahn-Banach theorem u1 can be extended to the whole space Lm (R2d , B). In the next step we use the duality theory of the Lebesgue-Bochner spaces Lp (Rd , X) where X is a Banach space [1, 9, 21]. If 1 ≤ p < ∞ and X is reﬂexive, then to each ∈ (Lp (Rd , X)) there exists an F ∈ Lp (Rd , X ) such that (, Z) = (F (x), Z(x))X dx and = F Lp (Rd ,X ) . By identiﬁcation of and F we Rd

thus have (Lp (Rd , B)) = Lp (Rd , B ), and hence the reﬂexivity of B implies that X = Lp (Rd , B) is a reﬂexive Banach space if 1 < p < ∞. Thus for 1 < p < ∞ and 2d q d p d 1 ≤ q < ∞ we have, using Lemma 3.8, (Lp,q m (R , B)) = (Lm (R , L (R , B)) = q p ,q L1/m (Rd , Lp (Rd , B )) ⊂ L1/m (R2d , B ). Hence if 1 < p < ∞ and 1 ≤ q < ∞, u1

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,q 2d can be represented by an element Hu1 ∈ Lp1/m (R2d , B ) and acts on Lp,q m (R , B) by 2d (u1 , P )Lp,q 2d = Hu1 (z), P (z) B dz, P ∈ Lp,q ,B) m (R , B). m (R R2d

Use of (3.3) in the form Vϕ F (z) = Vϕ Vϕ∗ Vϕ F (z) = and Fubini’s theorem ﬁnally gives

R2d

Vϕ F (w)Vϕ (π(z)ϕ)(w)dw

2d ,B) (u, F ) = (u1 , Vϕ F )Lp,q m (R p,q Vϕ Vϕ∗ Hu1 (z), Vϕ F (z) B dz, F ∈ Mm = (Rd , B).

(3.10)

R2d

p ,q Thus we obtain (3.9) with G = Vϕ∗ Hu1 ∈ M1/m (Rd , B ) by Prop. 3.1. Concerning 1 2d 2d the case p = q = 1, the dual of Lm (R , B) is L∞ 1/m (R , B ). Thus in this case 2d ∗ ∞ d Hu1 ∈ L∞ 1/m (R , B ). Hence G = Vϕ Hu1 ∈ M1/m (R , B ) and (3.10) holds also in this case.

Remark 3.10. In the Hilbert space case B = H the sesquilinear form (3.9) is by Cor. 2.3 an extension of the L2 (Rd , H) inner product from S(Rd , H) × S(Rd , H) p ,q p,q (Rd , H) × Mm (Rd , H). It is independent of ϕ and unique except in the to M1/m cases (p, q) = (1, ∞) and (p, q) = (∞, 1) [43]. In Section 4 we will need the following lemma. We restrict to the Hilbert space valued case and denote M p ,q := M p ,q (Rd , H). Lemma 3.11. For any 1 ≤ p, q ≤ ∞ we have the norm equivalence

(G, F )M p,q (Rd ,H) . F M p,q (Rd ,H) sup G M p ,q ≤1

older’s inequality implies Proof. Let ϕ ∈ S fulﬁll ϕL2 (Rd ) = 1. If GM p ,q ≤ 1 H¨ that

(G, F )M p,q (Rd ,H) = Vϕ G(z), Vϕ F (z) H dz ≤ F M p,q (Rd ,H) . R2d

On the other hand, using Lemma 3.5 and Vϕ∗ P M p ,q ≤ CP Lp,q where Lp ,q := Lp ,q (R2d , H),

sup (Vϕ Vϕ∗ P (z), Vϕ F (z))H dz F M p,q (Rd ,H) = Vϕ F Lp,q (R2d ,H) = ≤C

sup G M p ,q ≤1

R2d

P Lp ,q ≤1

R2d

Vϕ G(z), Vϕ F (z) H dz .

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3.3. Weakly stationary Hilbert space valued tempered distributions In this subsection we discuss tempered distributions taking values in a Hilbert space H. Such a distribution F is said to be weakly stationary (with a term borrowed from generalized stochastic processes) if the covariance distribution σF F := σ is translation invariant in the sense of [18, 28, 31, 32] σ, ϕ ⊗ ψ = σ, Tx ϕ ⊗ Tx ψ, x ∈ Rd , ϕ, ψ ∈ S(Rd ). We denote the set of weakly stationary elements of S (Rd , H) by Ss (Rd , H). It can be shown that the covariance distribution of a weakly stationary distribution acting on a separable function ϕ ⊗ ψ equals σ, ϕ ⊗ ψ = σs , ϕ ∗ ψ ∗

(3.11)

where σs ∈ S (Rd ) and ψ ∗ (x) := ψ(−x) [18, 26]. Since σs is positive deﬁnite it has a Fourier transform which is a tempered, non-negative measure σ s [18], ie there exists u ≥ 0 such that v−u (ξ) σs (dξ) < ∞. Rd

2 Proposition 3.12. If F ∈ Ss (Rd , H) then Vϕ F (t, ξ)2H = σ s ∗ ϕ (ξ) and ∞,1 M1⊗v−u (Rd , H). Ss (Rd , H) ⊂ u≥0

Proof. Combination of (1.7) and (3.11) gives Vϕ F (t, ξ)2H = σs , Mξ Tt ϕ ∗ (Mξ Tt ϕ)∗

2 2 = σ s , |F (Mξ Tt ϕ)| = σ s , Tξ M−t ϕ

2 2 = σ s , s ∗ ϕ (ξ − ·) = σ ϕ (ξ). Thus Vϕ F (t, ξ)H is independent of t and by Lemma 2.1 there exists N > 0 such that Vϕ F (t, ξ)H ≤ CvN (ξ). Thus with u = N + d + 1 F M ∞,1 (Rd ,H) ≤ C vN −N −d−1 (ξ)dξ < ∞. 1⊗v−u

Rd

3.4. Gabor frame theory for Hilbert space valued modulation spaces A frame for L2 (Rd ) [6, 22] is a countable set {hk }k∈I ⊂ L2 (Rd ) such that for constants 0 < A ≤ B < ∞ |f, hk |2 ≤ Bf 2L2 (Rd ) , f ∈ L2 (Rd ). (3.12) Af 2L2 (Rd ) ≤ k∈I 2 d The frame

operator is a positive and invertible operator on L (R ) deﬁned by Sf = k∈I f, hk hk .

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Lemma 3.13. Let H be a Hilbert space and let {hk }k∈I be a frame for L2 (Rd ) with frame constants A and B. Then AF 2L2 (Rd ,H) ≤ F, hk 2H ≤ BF 2L2 (Rd ,H) , F ∈ L2 (Rd , H), (3.13) k∈I

where F, hk is a Bochner integral, and we have F = F, hk S −1 hk , F ∈ L2 (Rd , H),

(3.14)

k∈I

with unconditional convergence in L2 (Rd , H). Proof. For a given F ∈ L2 (Rd , H) there exists by Pettis’s theorem a null set NF such that {F (t); t ∈ Rd \ NF } ⊂ H is separable. Let {ej }∞ j=1 be an ONB for this space. Since fj (t) := (F (t), ej )H ∈ L2 (Rd ) for all j, we have ∞ ∞ F 2L2 (Rd ,H) = F (t)2H dt = |fj (t)|2 dt = fj 2L2 (Rd ) . Rd \NF

Rd \NF j=1

Likewise F, hk 2H =

Rd \NF

F (t)hk (t)dt2H

∞

= j=1

j=1

Rd \NF

∞

2 fj (t)hk (t)dt = |fj , hk |2 , j=1

and hence we obtain using the upper frame bound (3.12) k∈I

F, hk 2H ≤ B

∞

fj 2L2 (Rd ) = BF 2L2 (Rd ,H) ,

j=1

and likewise we obtain the lower bound in (3.13). If we deﬁne the Hilbert space

2 1/2 valued sequence norm cl2 (I,H) := , the right inequality (3.13) k∈I ck H says that the coeﬃcient operator CF = {F, hk }k∈I is bounded from L2 (Rd , H)

2 to l (I, H), and implies that the synthesis operator Dc = k∈I ck hk is bounded S can be extended to from l2 (I, H) to L2 (Rd , H) [22]. Thus the frame operator

act on L2 (Rd , H), by (Se F )(t) := (DCF )(t) = F, hk hk (t). By a slight k∈I modiﬁcation of the proof of [22, Cor. 5.1.2], replacing sequences in l2 (I, C) by

2 sequences in l (I, H), the synthesis operator Dc = k∈I ck hk can be proved to be unconditionally convergent in L2 (Rd , H) when c ∈ l2 (I, H). Since bounded operators acting on unconditionally convergent series can be applied inside the sum [22], we obtain F, hk hk , F = F, hk 2H . Se F, F L2 (Rd ,H) = k∈I

k∈I

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By condition (3.13) Se is a positive invertible operator on L2 (Rd , H). Finally we prove (3.14). If x ∈ H then by [22, Cor. 5.1.3] we have for all k Se (S −1 hk ⊗ x) =

∞ S −1 hk , h h ⊗ x = hk ⊗ x

(3.15)

=1

with unconditional convergence in L2 (Rd , H). Again by the commutativity of bounded linear operators and unconditionally convergent series summation, we obtain Se−1 F, hk hk = F, hk S −1 hk F = Se−1 Se F = k∈I

k∈I −1

in the last equality using (3.15). Since S hk is a frame for L2 (Rd ) [22] and {F, hk }k∈I ∈ l2 (I, H) the convergence is unconditional in L2 (Rd , H). p,q By a Gabor frame [15, 22] for the Banach space Mm (Rd , H) we understand a discrete set of functions {MβnTαk g}n,k∈Zd = {π(λ)g}λ∈Λ , where g ∈ Mv1 (Rd ) and α, β > 0, such that for constants 0 < A ≤ B < ∞ q/p 1/q p,q F, Mβn Tαk gpH m(αk, βn)p AF Mm (Rd ,H) ≤ (3.16) n∈Zd k∈Zd p,q ≤ BF Mm (Rd ,H) .

In the proof of the next proposition we need the amalgam space W (Lp,q m ) [22] which p,q (Z2d ), is deﬁned as the set of measurable functions f : R2d → C such that a ∈ lm where akn = ess supt,ξ∈[0,1]d |f (t + k, ξ + n)|, k, n ∈ Zd . Proposition 3.14. Let H be a Hilbert space, and suppose g ∈ Mv1 (Rd ), α, β > 0 and {π(λ)g}λ∈Λ is a Gabor frame for L2 (Rd ) with frame operator S. Then {π(λ)g}λ∈Λ p,q is also a Gabor frame for Mm (Rd , H), for all 1 ≤ p, q < ∞, and the expansion p,q F = F, π(λ)gπ(λ)γ, F ∈ Mm (Rd , H), (3.17) λ∈Λ

where γ = S −1 g, holds with unconditional convergence. Proof. By a fundamental result of Gr¨ ochenig and Leinert γ = S −1 g ∈ Mv1 (Rd ) [24]. By Lemma 3.13 {π(λ)g}λ∈Λ is a frame for L2 (Rd , H) and Dγ Cg = idL2 (R2 ,H) , where Cg is the coeﬃcient operator deﬁned by {π(λ)g}λ∈Λ and Dγ is the synthesis operator deﬁned by {S −1 π(λ)g}λ∈Λ = {π(λ)γ}λ∈Λ [22]. In the following we adapt p,q p,q → lm , where the proof of [22, Thm. 12.2.3] on the boundedness of Cg : Mm s p,q ms (k, n) = m(αk, βn), to the Hilbert space valued case. If F ∈ Mm (Rd , H) we have by (3.6) with γ = g Vg F (z)H ≤ g−2 |Vg g| ∗ Vg F (·)H (z), hence by an inequality of Young type for amalgam spaces [13], [22, Thm. 11.1.5], Vg F (·)H p,q 2d . (3.18) p,q ≤ CVg gW (L1 ) Vg F (·)H v W (L ) L (R ) m

m

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By [22, Prop. 12.1.11] Vg g ∈ W (L1v ) and thus Vg F (·)H ∈ W (Lp,q m ). Next we need to prove that Vg F is continuous. From the formula (3.3) and Prop. 2.5 with ϕ ∈ S(Rd ) and ϕL2 = 1 we get ∗ Vg F (t, ξ) = Vg (Vϕ Vϕ F )(t, ξ) = Vϕ F (s, η)Vϕ g(s − t, η − ξ)ei2πt(η−ξ) dsdη R2d

which gives Vg F (t + t1 , ξ + ξ1 ) − Vg F (t, ξ)H

≤ Vϕ F (s, η)H Vϕ g(s − t − t1 , η − ξ − ξ1 )ei2π(t1 (η−ξ−ξ1 )−tξ1 ) R2d

− Vϕ g(s − t, η − ξ) dsdη. Let (t, ξ) ∈ R2d be ﬁxed arbitrary. The integrand approaches zero everywhere as |t1 |, |ξ1 | → 0 since Vϕ g is continuous [22]. If we prove that the integrand is also bounded by an L1 (R2d ) function uniformly over t1 , ξ1 ∈ U where U ⊂ Rd is a compact neighbourhood of zero, then by the dominated convergence theorem Vg F is continuous at (t, ξ), and hence everywhere. Since Vϕ F (·)H m ∈ L∞ (R2d ), 1/m(s, η) ≤ Cv(s, η) ≤ Cv(s − t, η − ξ)v(t, ξ) [22] and (Vϕ g)v ∈ L1 (R2d ), it is by H¨older’s inequality enough to prove

sup Vϕ g(· − t1 , · − ξ1 ) v ∈ L1 (R2d ). (3.19) t ,ξ ∈U 1

1

The integral of (3.19) can be estimated from above, using the submultiplicativity of v and Q := [0, 1]d , by

sup v(s, η) sup Vϕ g(s − t1 , η − ξ1 ) t1 ,ξ1 ∈U

s∈k+Q

k,n∈Zd η∈n+Q

≤C

v(k, n) sup Vϕ g(k + t1 , n + ξ1 ) t1 ,ξ1 ∈K

k,n∈Zd

where K ⊂ Rd is a compact neighbourhood of zero. The inclusion K ⊂ where J is ﬁnite, implies the estimate of (3.20)

C v(k, n) sup Vϕ g(k + l + t1 , n + p + ξ1 ) k,n∈Zd

≤C

k,n∈Zd

(3.20)

|l|,|p|≤J

|l|≤J

l+Q

t1 ,ξ1 ∈Q

v(k, n) sup Vϕ g(k + t1 , n + ξ1 ) < ∞, t1 ,ξ1 ∈Q

where the last expression is ﬁnite due to Vϕ g ∈ W (L1v ) [22]. Thus by the dominated convergence theorem Vg F is continuous everywhere. Now we obtain by [22, Prop. 11.1.4] and (3.18)

Vg F H p,q Cg F lp,q 2d ,H) = ms (Z Λ lms ≤ C Vg F (·)H W (Lp,q ) m

p,q ≤ CF Mm (Rd ,H) .

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p,q p,q Thus Cg is continuous from Mm (Rd , H) to lm (Z2d , H). s We can also modify the proof of [22, Thm. 12.2.4] to prove that the synthep,q p,q sis operator Dγ : lm (Z2d , H) → Mm (Rd , H) is continuous, by replacement of s scalar-valued sequences by Hilbert space valued sequences and the sequence norm p,q p,q (Z2d , C) by lm (Z2d , H). Since p, q < ∞ the convergence is unconditional in lm s s p,q (Rd , H) [22]. As Dγ and Cg are bounded operators and Dγ Cg = idL2 (Rd ,H) Mm p,q we obtain by Prop. 3.3 (i) Dγ Cg = idMm (Rd ,H) , ie (3.17) and also (3.16), since p,q p,q p,q F Mm (Rd ,H) ≤ Dγ Cg F lms (Z2d ,H) ≤ Dγ Cg F Mm (Rd ,H) .

4. Localization operators, Weyl operators and Gabor multipliers with operator-valued symbols 4.1. Operators on Banach space valued function spaces with symbols in Lp,q The formula (2.7) of Prop. 2.5 suggests the deﬁnition of localization operators [5, 7, 8, 43, 44, 45], ie operators from B1 -valued distributions to B2 -valued distributions (B1 , B2 Banach spaces) of the form F )(s) := a(z)Vg F (z)(π(z)γ)(s)dz, F ∈ S (Rd , B1 ), (4.1) (Ag,γ a R2d

2d

where the symbol a : R → L(B1 , B2 ) is assumed to be strongly measurable. With an estimate similar to (2.6) in Section 2 one conﬁrms that Ag,γ a , deﬁned as before by action under the integral, is a bounded operator from S (Rd , B1 ) to S (Rd , B2 ) provided g, γ ∈ S(Rd ) and a is strongly measurable and polynomially bounded. First we notice that if we replace |a(z)Vg F (z)| by a(z)Vg F (z)B2 ≤ a(z) Vg F (z)B1 in the proof of [5, Prop. 3.2], then one obtains the following generalization of this result for scalar-valued modulation spaces to vector-valued spaces and operator-valued symbols. Proposition 4.1. Suppose g, γ ∈ Mv1 (Rd ), p0 , p1 , p2 , q0 , q1 , q2 ∈ [1, ∞] fulfill 1/p0 + 1/p1 = 1/p2 , 1/q0 + 1/q1 = 1/q2 , m1 and m2 are v-moderate weight functions, and 0 a ∈ Lpm02,q/m (R2d , L(B1 , B2 )). 1 p1 ,q1 d p2 ,q2 d Then Ag,γ a : Mm1 (R , B1 ) → Mm2 (R , B2 ) with operator norm bound p ,q Ag,γ a ≤ CgMv1 γMv1 aL 0 0

m2 /m1

(R2d ,L(B1 ,B2 )) .

The following two corollaries of Prop. 4.1, the second of which is proved with interpolation techniques, are stated and proved in [5]. Corollary 4.2. Suppose p0 , p1 , q0 , q1 ∈ [1, ∞] fulfill 1/p0 +1/p1 ≤ 1, 1/q0 +1/q1 ≤ 1, suppose a ∈ Lp0 ,q0 (R2d , L(B)) and m is v-moderate. Then Ag,γ is bounded on a p1 ,q1 (Rd , B) with operator norm bound Mm Ag,γ a ≤ CgMv1 γMv1 aLp0,q0 (R2d ,L(B)) .

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Corollary 4.3. Suppose 1 ≤ p, q, r ≤ ∞, m1 and m2 are v-moderate, and a ∈ p,q p,q Lrm2 (R2d , L(B1 , B2 )). Then Ag,γ : Mm (Rd , B1 ) → Mm (Rd , B2 ) with operator a 1 1 m2 g,γ norm bound Aa ≤ CgMv1 γMv1 aLrm (R2d ,L(B1 ,B2 )) . 2

Corollary 4.3 is not true if we replace the modulation spaces between which acts by L2 spaces. The following example shows that Ag,γ is not necessarAg,γ a a ily bounded from L2 (Rd , B1 ) to L2 (Rd , B2 ) when a ∈ L∞ (R2d , L(B1 , B2 )). This phenomenon can not occur in the theory of scalar-valued modulation spaces since L2 = M 2 then.

Example 4.4. Let the dimension d = 1, 1 < r < 2, B1 = lr (Z), B2 = lr (Z). Note that B1 and B2 both have Fourier type r [35]. Deﬁne the operator a0 acting on

ek where ek denotes the vector of zeros in all lr by a0 (β) = k βk |βk |r −2 β2−r lr positions except position k where it equals one. Then a0 L(lr ,lr ) = 1 and hence g ⊂ (−1/4, 1/4) a(z) ≡ a0 ∈ L∞ (R2 , L(lr , lr )). Let f, g ∈ S(R) fulﬁll supp f, supp and f L2 = gL2 = 1, let N > 0 be a given number, choose K such that

(2K + 1)1−2/r ≥ N . Deﬁne F (x) = |k|≤K Mk f (x)ek . Then F 2L2 (R,lr ) = (2K +

1)2/r and Vg F (t, ξ) = f ∗ M−t g∗ (ξ − k)ek , ie for ﬁxed t the summands |k|≤K

have non-overlapping support in the ξ variable due to supp g, f ⊂ (−1/4, 1/4).

Hence Vg F (t, ξ)2ls = |k|≤K |F((Mk f )(Tt g))(ξ)|2 independently of s ∈ [1, ∞). Parseval’s formula gives R2 Vg F (t, ξ)2ls dz = 2K + 1. Thus g,g Vg F (z)2lr dz ≥ N F 2L2(R,lr ) , (F, Aa F )L2 (R,lr ) = R2

which by Lemma 3.5 implies that Ag,g / L L2 (R, lr ), L2 (R, lr ) . a ∈ Large parts of the Weyl and the Kohn-Nirenberg pseudo-diﬀerential calculi can be modiﬁed to treat the case of functions taking values in a separable Hilbert space H and symbols with values in L(H) as outlined in [17, pp 135–37]. Many results are true also when B1 , B2 are reﬂexive Banach spaces and the symbol takes values in L(B1 , B2 ) [27, p 79]. If B1 , B2 are perfectly general Banach spaces and a ∈ S(R2d , L(B1 , B2 )) the localization operator Ag,γ can be formulated as a Weyl a operator [17, 27] s + t g,γ , ξ F (t)ei2πξ(s−t) dtdξ Aa F (s) = Lρ F (s) := ρ (4.2) 2 2d R provided g, γ ∈ M 1 (Rd ), where the Weyl symbol is [17, 40] ρ = a ∗ W (γ ⊗ g).

(4.3)

4.2. Operators on Hilbert space valued function spaces with symbols in M ∞ In this subsection we shall prove a version of one of the main results of [7] (see also [43]) for Hilbert space valued modulation spaces. This result says that Ag,γ a can be extended to a uniformly bounded operator on M µ1 ,µ2 (Rd ) for all 1 ≤ µ1 , µ2 ≤ ∞, ∞ (R2d ), g, γ ∈ Mv1s (Rd ) and s ≥ 0. provided the symbol a ∈ M1/τ s

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If F, G ∈ L2 (Rd , H) then, following [17], an operator-valued Wigner distribution can be deﬁned by (WF G (t, ξ)u, v)H = (F (t + τ /2), v)H (G(t − τ /2), u)H e−i2πτ ξ dτ, u, v ∈ H. Rd

Then WF G (t, ξ) ∈ L(H) for all (t, ξ) ∈ R2d follows from F, G ∈ L2 (Rd , H). It is also true that WF G (t, ξ) ∈ S1 (H) which denotes the set of trace-class operators, with norm [18, 37] ∞ T S1(H) = sup |(T ej , fj )H | j=1

where the supremum is taken over all pairs of orthonormal sequences {ej }j≥1 and {fj }j≥1 in H. In fact, ∞ ∞ |(WF G (t, ξ)ej , fj )H | ≤ |(F (t + τ /2), fj )H | |(G(t − τ /2), ej )H | dτ Rd j=1

j=1

≤

Rd d

F (t + τ /2)H G(t − τ /2)H dτ

≤ 2 F L2(Rd ,H) GL2 (Rd ,H) . If ψ1 , ψ2 ∈ S(Rd ) we have WF G (t, ξ)W (ψ1 ⊗ ψ2 )(t, ξ)dtdξ u, v)H R2d = (F (t + τ /2), v)H (G(t − τ /2), u)H ψ1 (t + τ /2)ψ2 (t − τ /2)dtdτ

(4.4)

R2d

= F, ψ1 , v H G, ψ2 , u H , u, v ∈ H, where F, ψ1 denotes a Bochner integral. The formula (4.4) can be generalized to F, G ∈ S (Rd , H) by the deﬁnition WF G , Φu, v H := F, ψ1 , v H G, ψ2 , u H , u, v ∈ H, if Φ = W (ψ1 ⊗ψ2 ). Thus WF G , W (ψ1 ⊗ψ2 ) is a rank-one operator with trace-class norm WF G , W (ψ1 ⊗ ψ2 )S1 (H) = F, ψ1 H G, ψ2 H (4.5) ≤C ψ1 α,β ψ2 α,β , |α|≤N1

|β|≤M1

|α|≤N2

|β|≤M2

where · α,β are seminorms on S(Rd ). The estimate (4.5) says that (ψ1 , ψ2 ) → WF G , W (ψ1 ⊗ ψ2 ) is a bilinear continuous map S(Rd ) × S(Rd ) → S1 (H). By the Schwartz kernel theorem [37] it can be extended to a continuous linear map S(R2d ) Ψ → WF G , W (Ψ) ∈ S1 (H). Since the partial Fourier transformation and the coordinate transformation which deﬁnes the Wigner distribution (1.4) are continuous operations on S(R2d ), we have WF G ∈ S (R2d , S1 (H)).

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The following lemma is needed in the main proposition. Its statement is analogous to [7, Lemma 2.2]. Lemma 4.5. Let H be a Hilbert space. If F, G ∈ S (Rd , H), ϕ ∈ S(Rd ) and Φ = W (ϕ), then with z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ R2d , VΦ WF G (z, ζ)S1 (H) = Vϕ F (z1 − ζ2 /2, z2 + ζ1 /2)H Vϕ G(z1 + ζ2 /2, z2 − ζ1 /2)H . Proof. Using W (ϕ)(t − z1 , ξ − z2 ) = W (Mz2 Tz1 ϕ)(t, ξ) it can be veriﬁed that Mζ Tz Φ(t, ξ) = W (ψ1 ⊗ ψ2 )(t, ξ) where ψ1 = eiπz2 ζ2 Mz2 +ζ1 /2 Tz1 −ζ2 /2 ϕ, ψ2 = e−iπz2 ζ2 Mz2 −ζ1 /2 Tz1 +ζ2 /2 ϕ.

(4.6)

Thus the result follows from VΦ WF G (z, ζ) = WF G , Mζ Tz Φ and insertion of (4.6) into (4.5). The deﬁnition of a Weyl operator (4.2) works for symbols ρ ∈ S(R2d , L(H)) but we need to extend the deﬁnition to more general symbols. Weyl operators can be deﬁned for ρ ∈ S (R2d , L(H)) as follows. Given any continuous bilinear multiplication with norm bounded by one, B1 × B2 (x, y) → x · y ∈ B3 , where Bj , j = 1, 2, 3 are Banach spaces, there exists according to [1, Thm. 1.7.2] a unique natural continuous bilinear extension of S(Rd , B1 ) × S(Rd , B2 ) (ρ, F ) → ρ(x) · F (x)dx ∈ B3 Rd

to (ρ, F ) ∈ S (Rd , B1 ) × S(Rd , B2 ). We use three such bilinear multiplications, (i) multiplication with scalars H × C → H, (ii) operator evaluation L(H) × H → H, and (iii) the inner product (·, ·)H H × H → C (which in fact is sesquilinear). Multiplication (i), modiﬁed with a conjugation in the second argument, has been used in the deﬁnition of S (Rd , H) in Section 1.1. Let now ρ ∈ S (R2d , L(H)) and F ∈ S(Rd , H) be ﬁxed. By Thm. 1.3.3 (op. cit.) there exists a sequence ρn ∈ S(R2d , L(H)) such that ρn −→ ρ in S (R2d , L(H)) as n −→ ∞. Then by (4.2) Lρn F ∈ S(Rd , H) and we deﬁne the H-valued functional Lρ F by Lρ F, ϕ := ρ, W (F ⊗ ϕ)SR (R2d ,H) , ϕ ∈ S(Rd ),

(4.7)

where the multiplication of the right hand side distribution action is extended from multiplication (ii). Here SR (R2d , H) indicates that the distribution ρ acts linearly (in contrast to the antilinear convention of this paper) on S(R2d , H). It is clear that (4.7)

extends (4.2) as a deﬁnition of Lρ F and it can be veriﬁed that Lρ F, ϕH ≤ C α,β ϕα,β , ie Lρ F ∈ S (Rd , H). The deﬁnition (4.1) can be extended to symbols a ∈ S (R2d , L(H)) by Ag,γ a F := Lρ F where ρ = a ∗ W (γ ⊗ g). By the continuity asserted by Thm. 1.7.2 (op. cit.) applied to the multiplication

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(ii), limn→∞ Lρn F = Lρ F in S (Rd , H). The same result can then be applied once again, with multiplication (iii), which yields Lρ F, GS(Rd ,H) = lim (Lρn F, G)L2 (Rd ,H) ∀G ∈ S(Rd , H). n→∞

(4.8)

Since F , G and Lρn F are strongly measurable there exists a closed separable subspace H0 ⊂ H such that the range spaces of F, G, Lρn F are contained in H0 for all n a.e. Since WF G (x) : H → H0 for a.a. x ∈ R2d , and x → ρn (x) is continuous for each n, we can enlarge H0 to a closed separable subspace, still denoted H0 , such that ρn (x)WF G (x) : H0 → H0 for all n and a.a. x. If {ej }∞ j=1 ⊂ H0 is an ONB for H0 we have ∞ Lρn F, G L2 (Rd ,H) = (Lρn F (s), ej )H (G(s), ej )H ds =

j=1

=

j=1

∞

∞ j=1

R3d

R2d

Rd

(F (t + τ /2), ρn (t, ξ)∗ ej )H (G(t − τ /2), ej )H e−i2πξτ dτ dtdξ

WF G (t, ξ)ej , ρn (t, ξ)∗ ej H dtdξ =

R2d

tr ρn (x)WF G (x) H0 dx. (4.9)

Let Φ ∈ S(R2d ) be real-valued and fulﬁll ΦL2(R2d ) = 1. By Lemma 2.1 (z, ζ) → VΦ ρ(z, ζ) is continuous, thus we can again enlarge H0 to a closed separable subspace still denoted H0 , such that VΦ ρ(z, ζ)VΦ WF G (z, −ζ) : H0 → H0 . Finally we obtain from (4.8), (4.9), the duality S1 (H0 ) = L(H0 ) and |tr(AB)| ≤ AL(H0 ) ·BS1 (H0 ) [37], Lemma 2.2, and the dominated convergence theorem

tr ρn (x)WF G (x) H0 dx Lρ F, GS(Rd ,H) = lim n→∞ R2d

tr VΦ ρn (z, ζ)VΦ WF G (z, −ζ) H0 dzdζ = lim (4.10) n→∞ R4d

= tr VΦ ρ(z, ζ)VΦ WF G (z, −ζ) H dzdζ. 0

R4d

We are now prepared to prove the main theorem. Proposition 4.6. Let H be a Hilbert space, s ≥ 0, γ ∈ Mvrs (Rd ) where 1 ≤ r < ∞, and g ∈ Mv1s (Rd ). Suppose 1 ≤ q ≤ p ≤ ∞, q ≤ r, t = qr/(r − q), µj , νj ∈ [1, ∞], vj ≤ p, p ≤ µj , 1/µj − 1/νj = 1/q − 1/p, for j = 1, 2, (ν1 , ν2 ) = (1, ∞), p,t (ν1 , ν2 ) = (∞, 1), and a ∈ M1/τ (R2d , L(H)). Then Ag,γ can be extended to a a s µ1 ,µ2 d ν1 ,ν2 d (R , H) to M (R , H) with operator norm depending bounded map from M only on g, γ and a. p,t (R2d , L(H)) and g, γ ∈ S(Rd ). Let Φ = W (ϕ), where Proof. Suppose a ∈ M1/τ s ϕ ∈ S(Rd ), fulﬁll ΦL2 (R2d ) = 1. By [7, Prop. 2.5]

W (γ ⊗ g)Mτ1,r (R2d ) ≤ CgMv1s (Rd ) γMvrs (Rd ) , s

(4.11)

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and by Prop. 2.4 (op. cit.), with integrals interpreted as Bochner integrals, ρM p,q (R2d ,L(H)) ≤ CaM p,t

1/τs

(R2d ,L(H)) W (γ

⊗ g)Mτ1,r (R2d ) .

(4.12)

s

Now we extend (4.10) from F, G ∈ S(Rd , H) to a sesquilinear form, still denoted µ1 ,µ2 (Ag,γ (Rd , H) × M ν1 ,ν2 (Rd , H). Since a F, G) = (Lρ F, G), acting on (F, G) ∈ M the range spaces of Vϕ F and Vϕ G are separable subspaces of H [37], there still exists a closed separable subspace H0 ⊂ H such that VΦ ρ(z, ζ)VΦ WF G (z, −ζ) : H0 → H0 . This H0 is used in the extension of (4.10). Next we use |tr(AB)| ≤ AL(H0 ) BS1 (H0 ) , H¨ older’s inequality for mix-normed spaces, Lemma 4.5, q /p ≥ 1 (which is a consequence of q ≤ p), p /µj + p /νj = 1 + p /q for j = 1, 2 (which follows from the assumption 1/µj − 1/νj = 1/q − 1/p), and Young’s inequality for mix-normed spaces [3]. Thus

g,γ

Aa F, G = Lρ F, G ≤ VΦ ρLp,q (R4d ,L(H)) ζ2 ζ1 ζ2 ζ1 q /p 1/q · Vϕ F (z1 + , z2 − )pH Vϕ G(z1 − , z2 + )pH dz dζ 2 2 2 2 R2d R2d q /p 1/q p Vϕ F (·)pH ∗ V = ρM p,q (R2d ,L(H)) dζ ϕ G(·)H (ζ2 , −ζ1 ) R2d

1/p 1/p ≤ ρM p,q (R2d ,L(H)) Vϕ F (·)pH Lµ1 /p ,µ2 /p (R2d ) Vϕ G(·)pH ν1 /p ,ν2 /p 2d L (R ) = ρM p,q (R2d ,L(H)) Vϕ F (·)H Lµ1 ,µ2 (R2d ) Vϕ G(·)H Lν1 ,ν2 (R2d ) ≤ CaM p,t

1/τs

(R2d ,L(H)) gMv1s (Rd ) γMvrs (Rd ) F M µ1 ,µ2 (Rd ,H) GM ν1 ,ν2 (Rd ,H) ,

where (4.11). Hence last inequality we have inserted the estimates (4.12) and g,γ in the Aa F, G deﬁnes a bounded sesquilinear form on M µ1 ,µ2 × M ν1 ,ν2 , where we denote M µ1 ,µ2 := M µ1 ,µ2 (Rd , H) for brevity, which extends the canonical sesquilinear form on S(Rd , H) × S(Rd , H) deﬁned by (·, ·)L2 (Rd ,H) . Since there is a unique extension of the latter form to M ν1 ,ν2 × M ν1 ,ν2 except in the given exceptional cases of ν1 , ν2 [43] (see Rem. 3.10), Ag,γ a F, G must equal this extended form (restricted if µj < νj ), denoted (·, ·)M p,q (Rd ,H) and deﬁned in (3.9). Now the result follows from Lemma 3.11. The operator norm is bounded according to Ag,γ a ≤ CaM p,t

1/τs

(R2d ,L(H)) gMv1s (Rd ) γMvrs (Rd ) .

Finally we extend the last inequality by the density of S(Rd ) in Mvrs (Rd ) to g ∈ Mv1s (Rd ) and γ ∈ Mvrs (Rd ). Remark 4.7. It seems to be an open question whether Prop. 4.6 is true when H is replaced by a Banach space of a suitable kind. If we choose r = q = 1 and p = ∞ we obtain the announced generalization of [7, Thm. 3.2].

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∞ Corollary 4.8. If a ∈ M1/τ (R2d , L(H)) and g, γ ∈ Mv1s (Rd ), then Ag,γ can be a s ν1 ,ν2 d (R , H) for all 1 ≤ ν1 , ν2 ≤ ∞ except extended to a bounded map on M (ν1 , ν2 ) = (1, ∞) and (ν1 , ν2 ) = (∞, 1).

We also obtain a result for Weyl operators which is a generalization of [22, Thm. 14.5.2], see also [23, 43]. The original, restricted (continuity on L2 ) theorem was proved by Sj¨ ostrand [41]. Corollary 4.9. If ρ ∈ M ∞,1 (R2d , L(H)) then Lρ can be extended to a bounded map on M ν1 ,ν2 (Rd , H) for all 1 ≤ ν1 , ν1 ≤ ∞ except (ν1 , ν2 ) = (1, ∞) and (ν1 , ν2 ) = (∞, 1). p,q (Rd , H) by The formula (3.17) suggests the deﬁnition of operators on Mm p,q (Ag,γ a(λ)Vg F (λ)(π(λ)γ)(s), F ∈ Mm (Rd , H), a F )(s) := λ∈Λ 2d

for a : Z → L(H). Such an operator is called a Gabor multiplier, and the scalarvalued case is treated in [16]. If g ∈ Mv1 (Rd ) and {π(λ)g}λ∈Λ is a Gabor frame 2d for L2 (Rd ), then one can prove (we omit the details), replacing Lp,q m (R ) norms p,q 2d by sequence norms lms (Z ) and using Prop. 3.14, the following discrete version. Here we again denote ms (k, n) = m(αk, βn). Proposition 4.10. Let H be a Hilbert space, g ∈ Mv1 (Rd ) and let {π(λ)g}λ∈Λ be a Gabor frame for L2 (Rd ). Suppose furthermore p1 , p2 , q1 , q2 ∈ [1, ∞) and p0 , q0 ∈ [1, ∞] fulfill 1/p0 + 1/p1 = 1/p2 , 1/q0 + 1/q1 = 1/q2 , m1 and m2 are v-moderate weight functions, m = m2 /m1 and p0 ,q0 (Z2d , L(H)). a ∈ lm s p1 ,q1 d p2 ,q2 d Then Ag,γ a : Mm1 (R , H) → Mm2 (R , H) with operator norm bound g,γ p ,q Aa ≤ CgMv1 γMv1 alm0s 0 (Z2d ,L(H)) .

Acknowledgment The author would like to thank Hartmut F¨ uhr, Franz Luef, Ghassem Narimani and in particular Hans Feichtinger for nice discussions, suggestions and useful remarks which have improved the paper. He would also like to express his gratitude to the organizers of the Special Semester on Modern Methods for Time-Frequency Analysis at the Erwin Schr¨odinger Institute in Vienna, spring 2005, H. G. Feichtinger and K. Gr¨ ochenig, for a very nice and stimulating event. The paper was worked out in the hospital environment of NuHAG, Faculty of Mathematics, University of Vienna, for which again H. Feichtinger is acknowledged. This project is supported by STINT, The Swedish Foundation for International Cooperation in Research and Higher Education.

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[22] K. Gr¨ ochenig, Foundations of time-frequency analysis. Birkh¨ auser, 2001. [23] K. Gr¨ ochenig and C. Heil, Modulation spaces as symbol classes for pseudodiﬀerential operators. In: Wavelets and their applications, Allied Publishers Pvt. Ltd. (2003), 151–170, Eds. M. Krishna, R. Radha and S. Thangavelu. [24] K. Gr¨ ochenig and M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1–18. [25] P. R. Halmos, Measure theory. Van Nostrand, 1950. [26] L. H¨ ormander, The analysis of linear partial diﬀerential operators I. 2nd Edition, Springer-Verlag, 1990. [27] L. H¨ ormander, The analysis of linear partial diﬀerential operators III. SpringerVerlag, 1985. [28] W. H¨ ormann, Generalized stochastic processes and Wigner distribution. PhD thesis, Universit¨ at Wien, 1989. [29] T. Hyt¨ onen, Fourier embeddings and Mihlin-type multiplier theorems, Math. Nachr. 274–275 (2004), 74–103. [30] T. Hyt¨ onen and P. Portal, Vector-valued multiparameter singular integrals and pseudodiﬀerential operators, preprint, October 2005. [31] K. Itˆ o, Stationary random distributions, Memoirs of the College of Science, Univ. Kyoto, Ser. A, XXVIII (1953), Math. No. 3, 209–223. [32] B. Keville, Multidimensional second order generalised stochastic processes on locally compact abelian groups. PhD thesis, University of Dublin, 2003. [33] S. Kwapie´ n, Isomorphic characterizations of inner product spaces by orthogonal series with vector-valued coeﬃcients, Studia Math. 44 (1972), 583–595. [34] G. Matz, On non-WSSUS wireless fading channels, IEEE Trans. Wireless Comm. 4 (2005), 2465–78. [35] J. Peetre, Sur la transformation de Fourier des fonctions a ` valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26. ˇ Strkalj, ˇ [36] P. Portal and Z. Pseudodiﬀerential operators on Bochner spaces and an application, Math. Z. 253 (2006), 805–819. [37] M. Reed and B. Simon, Methods of modern mathematical physics I. Wiley, 1975. [38] H.-J. Schmeisser, Vector-valued Sobolev and Besov spaces. In: Sem. Analysis of the Karl-Weierstrass-Institute 1985/86, Teubner Texte Math 96 (1987), 4–44. [39] L. Schwartz, Distributions a ` valeurs vectorielles, Ann. Inst Fourier I. 7 (1957), 1–141, II. 8 (1958), 1–209. [40] M. A. Shubin, Pseudodiﬀerential operators and spectral theory. 2nd Edition, Springer, 2001. [41] J. Sj¨ ostrand, An algebra of pseudodiﬀerential operators, Math. Res. L. 1 (1994), 185–192. [42] T. Strohmer, Pseudodiﬀerential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2006), 237–249. [43] J. Toft, Continuity properties for modulation spaces, with applications to pseudodiﬀerential calculus – I, J. Func. Anal. 207 (2004), 399–429.

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Integr. equ. oper. theory 59 (2007), 129–147 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010129-19, published online June 27, 2007 DOI 10.1007/s00020-007-1506-0

Integral Equations and Operator Theory

Noncommuting Domination in Krein Spaces Via Commutators of Block Operator Matrices Michal Wojtylak Abstract. The commutators of 2×2 block operator matrices with (unbounded) operator entries are investigated. The matrix representation of a symmetric operator in a Krein space is exploited. As a consequence, the domination result due to Cicho´ n, Stochel and Szafraniec is extended to the case of Krein spaces. Mathematics Subject Classification (2000). Primary 47B50; Secondary 47B25, 47B47. Keywords. Symmetric operator, selfadjoint operator, Krein space, block operator matrix.

Introduction The idea of domination of operators in Hilbert spaces is due to Nelson (see [16]). The topic was investigated many times, see e.g. [17, 18, 19, 3] for general theory and [18, 20, 4] for applications. The matter is closely related to the perturbation theory (cf. [9, 7]). So far the results concerned symmetric and selfadjoint operators [16, 18, 3, 4] or formally normal and normal operators [17, 19] in Hilbert spaces. In the present paper we deal with symmetric and selfadjoint operators in Krein spaces. The stimulant for the research was Theorem 3 of [3] (quoted below as Theorem 2.2). The authors of [3] gave criteria for a symmetric operator in a Hilbert space to be selfadjoint. On the other hand, it is a well known fact that a symmetric operator in a Pontriagin space admits a block operator matrix representation ∗ S00 −S10 , (0.1) S= S10 S11 The author was supported by the Stanislaw Estreicher grant out of the Jagiellonian University funds.

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where the operator S11 is unbounded and the remaining three are bounded. Selfadjointness (in the Pontriagin space sense) of S reduces to selfadjointness (in the Hilbert space sense) of S11 . It appears, that using the above representation one can prove an analogue of Theorem 2.2 in Pontriagin spaces. Moreover, the assump∗ tions that the operators S00 and S10 are ﬁnite dimensional and that S01 = −S10 are not essential in the calculations. Therefore we prove ﬁrst a more general result in Krein spaces (Theorem 8.2) and the Pontriagin space case is its simple consequence. Using the same methods we are also able to prove the invariance of the assumptions of Theorem 2.2 on some bounded perturbations of the dominating operator (Proposition 8.4). As was already mentioned, our main tool are 2 × 2 block operator matrices with three bounded and one unbounded entry. Such operators were considered by many authors (cf. [11, 1, 12, 13]). In the present paper we will usually assume that S00 , S01 and S10 are bounded and that ρ(S) = ∅. Such context is more general than is required for our purposes. The paper is organized as follows. Most of the paper (Sections 2–7) is devoted to block operator matrices. The main result of this part is Theorem 2.3, which is formulated in Section 2. In Sections 3–7 we develop the theory of commutators and sets of type ΩS for block operator matrices and we successively complete the proof of Theorem 2.3. The last section contains the main result of this paper – a domination result in Krein spaces.

1. Preliminaries If E and F are linear spaces (in practice, we consider only Hilbert and Krein spaces), then by an operator from E to F we understand a linear mapping A from a linear subspace D (A) of E, called the domain of A, to F . Denote by R (A) and N (A) the range and the kernel of A, respectively. If E = F then we will say that A is an operator in E. If H, K and L are linear spaces and A and B are operators from H to K and K to L respectively, then we deﬁne as usually the operator BA: D (BA) = {f ∈ D (A) : Af ∈ D (B)} ,

(BA)(f ) := B(Af ), f ∈ D (BA) . (1.1)

If, moreover, A is an operator from H to K (B is an operator from K to L), then the following remarkable property hold: (A + A )B = AB + A B,

A(B + B ) ⊇ AB + AB .

(1.2)

We also have C(BA) = (CB)A

(1.3)

for all operators A, B and C from H to K, K to L and L to L˜ respectively. For an operator A in H and m ∈ N := {0, 1, . . . } we deﬁne Am by induction: A0 := IH ,

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Am+1 := AAm = Am A. We also put D∞ (A) :=

∞

D Ak .

k=0

By [A, B] we mean AB − BA (deﬁned on the maximal possible domain D (AB) ∩ D (BA)). Let K be a Hilbert space and let X be an operator in K. We say that X is densely defined if its domain is dense in K. X is said to be bounded if Xf ≤ c f for some c ≥ 0 and for all f ∈ D (X). We would like to stress here that a bounded operator is not necessarily densely deﬁned. We write B(K, L) for the space of all bounded operators with domain equal K and range contained in a Hilbert space L. We abbreviate B(K) to B(K, K). As usual, σ(X) and ρ(X) stand for the spectrum and the resolvent set of an operator X in K, respectively. If X and Y are operators then the relation X ⊆ Y is understood as the inclusion of the graphs. We say that E ⊆ D (X) is a core for X if the graph of X is contained in the closure of the graph X|E . X ∗ stands for the adjoint of an operator X. We say that a symmetric operator X in K is essentially selfadjoint on E if E is a dense linear subspace of D (X) and (X|E )∗ = X|E . By maximality of selfadjoint operators, a symmetric operator X is selfadjoint on E if and only if X is selfadjoint and E is a core for X. Recall two important properties, the ﬁrst can be found in [3] as formula (2), the latter was communicated to me by D. Cicho´ n and J. Stochel. If N is a densely deﬁned, closed operator in K and ρ(N ) = ∅ then (1.4) (N − z)−1 D (N m ) ⊆ D N m+1 for all m ∈ N, z ∈ ρ(N ); D∞ (N ) is a core for N m for every m ∈ N.

(1.5)

2. Noncommuting domination in Hilbert spaces Let us consider two closed operators N, X in a Hilbert space K. We say that N dominates X (cf. [19]), or that X is N -bounded (e.g. [7], see also there for a stronger notion of N -completely boundedness) if D (N ) ⊆ D (X). Remark 2.1. By the closed graph theorem, if N and X are closed operators in K and N dominates X, then Xf ≤ c(f + N f )

(2.1)

for all f ∈ D (N ) and for some c ≥ 0. Observe, that if N and X are closable, E is a core for N and the inequality (2.1) holds for some c ≥ 0 and for all f ∈ E then N dominates X. Very often it is much easier to verify the inequality (2.1) on a core than to check the inclusion of the domains (see [19]).

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Let K be a Hilbert space and X be a densely deﬁned, closable operator in K. In [3] Cicho´ n, Stochel and Szafraniec introduced the sets Ωc,d (X) := {z ∈ C : |z| > d, dist(z, σ(X)) ≥ c|z|} ,

(2.2)

where c > 0, d ≥ 0. If X is selfadjoint then the set above is equal to Ωc,d (X) = z ∈ ρ(S) : |z| > d, |z| (X − z)−1 ≤ c−1 .

(2.3)

A set U ⊆ ρ(X) is called of type ΩX if it is unbounded and −1 ∃t>0,d≥0 ∀z∈U : |z| > d and |z| (X − z) ≤ t.

(2.4)

In other words, for X selfadjoint, U is of type ΩX if it is an unbounded subset of some Ωt−1 ,d (X). If X is selfadjoint and U is of type ΩX then the constant t in (2.4) must be greater than or equal to one; moreover, the set i[d, +∞) (d > 0) is always of type ΩX , see [3, Rem.6] for more on that. Of particular interest to us is the following theorem ([3, Thm.3], see also [6]). In [4] one can ﬁnd many examples showing its usefulness. For consistency we use here the deﬁnition of type ΩS and not the sets Ωc,d (S) as originally stated in [3]. Theorem 2.2. Assume that A is a symmetric operator in a Hilbert space K, S is a selfadjoint operator in K and U is a set of type ΩS . (i) Let m ∈ N. If D (S m ) ⊆ D (A) and (2.5) sup |z| [(S − z)−1 , A] < +∞ z∈U

then A is selfadjoint on any core of S m . (ii) If D∞ (S) ⊆ D (A), (2.5) holds and E is a core of S m for every m ∈ N, then A is essentially selfadjoint on E. Let N be an operator in the orthogonal sum H0 ⊕ H1 of two Hilbert spaces. We say that N is H0 -bounded if H0 ⊆ D (N ) and N00 := P0 N|H0 ∈ B(H0 ),

N01 := P0 N|H1 ∈ B(H1 , H0 ),

(2.6)

N10 := P1 N|H0 ∈ B(H0 , H1 ), N11 := P1 N|H1 is densely deﬁned, (2.7) where P0 (P1 ) denotes the orthogonal projection from H0 ⊕ H1 onto H0 (H1 ). Observe that in such case N is uniquely determined by the operators N00 ,N10 ,N01 ,N11 and can be identiﬁed with a block operator matrix N00 N01 N10 N11 in the sense that D (N ) = H0 ⊕ D (N11 ) ,

N

f0 f1

=

N00 f0 + N01 f1 N10 f0 + N11 f1

for

f0 f1

∈ D (N )

(we will not distinguish between the orthogonal sum H0 ⊕ H1 and the product H0 × H1 ). This identiﬁcation will be always used for all H0 -bounded operators. m Nij∗ (i, j = 0, 1) and N11 stands always for (Nij )∗ and (N11 )m respectively. Note

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that N m (which is always understood as a product of unbounded operators in the sense of (1.1)) need not to be H0 -bounded; it is enough to take m = 2, N11 unbounded and N01 = I. Our aim, for the next 5 sections, is to prove the following result. Theorem 2.3. Let H = H0 ⊕ H1 be an orthogonal sum of two Hilbert spaces. Let S and B be H0 -bounded operators such that S11 (resp. B11 ) is a selfadjoint (resp. a symmetric) operator in H1 . Assume also that U is of type ΩS . If for some m ∈ N \ {0} (2.8) H0 ⊆ D (S m ) ⊆ D (B) and

sup |z| [(S − z)−1 , B] < +∞

(2.9)

z∈U m . then B11 is essentially selfadjoint on any core of S11

We sketch here the proof, all the missing parts will be completed in the next sections. Proof. In Corollary 5.2 we will prove the existence of U ⊆ U of type ΩS11 . We show now that all the assumptions of Theorem 2.2(i) are satisﬁed with K = H1 , S = S11 , A = B11 and U instead of U . In Proposition 4.1(am ) ⇒ (cm ) (with N = S) we will prove that H0 ⊕ m m D (S11 ) = D (S m ) ⊆ D (B) = H0 ⊕ D (B11 ). Consequently D (S11 ) ⊆ D (B11 ). Proposition 6.2 (applied to N = S, X = B) will tell us that [(S11 − z)−1 , B11 ] is bounded for z ∈ ρ(S) ∩ ρ(S11 ). In Proposition 7.1 (with N , X as before and V = U ) we will show that −1 sup |z| [(S11 − z) , B11 ] < +∞. z∈U

By Theorem 2.2 the proof is (or rather will be) ﬁnished.

Observe that putting H0 := {0} in the above result we get Theorem 2.2(i). An analogue version of part (ii) of Theorem 2.2 is also possible (and easy). However, for applications we will need a slightly diﬀerent approach (see Section 8). As we have just seen, the proof of Theorem 2.3 is based on the formula m ) and on the fact that the inequality (2.9) for matrices S D (S m ) = H0 ⊕ D (S11 and B implies the inequality (2.5) for the right lower corners S11 and B11 (with U instead of U ). Thus the following open question arises. Suppose that (2.8) holds for some m ∈ N and sup |z| [(S11 − z)−1 , B11 ] < ∞, z∈V

where V is of type ΩS11 . Is it true that (2.9) holds for some U ⊆ V of type ΩS ? The main diﬃculty in answering it lies in inverting Propositions 6.2 and 7.1.

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3. H0 -bounded operators Recall that Nij∗ := (Nij )∗ for i, j = 0, 1. Lemma 3.1. Let N be H0 -bounded. The following three statements hold: ∗ ∗ N00 N10 ∗ ∗ (i) N is a H0 –bounded operator and N = . ∗ ∗ N01 N11 (ii) N is closable if and only if N11 is closable. In such case N00 N01 N= . N10 N11 (iii) N is closed if and only if N11 is closed.

(3.1)

∗ ∗ N00 N10 Proof. (i) Let M be the H0 -bounded operator given by a matrix . ∗ ∗ N01 N11 One can easily check that N h, k = h, M k for all h ∈ D (N ), k ∈ D (M ). Therefore M ⊆ N ∗ . Now let f ⊕ g ∈ D (N ∗ ). Then

φ f φ φ ∗ f N , = ,N , ∈ D (N ) . ψ g ψ g ψ This means that f f + ψ, P1 N ∗ −N00 φ, f −N01 ψ, f −N10 φ, g , N11 ψ, g = φ, P0 N ∗ g g and so the mapping D (N11 ) ψ → N11 ψ, g ∈ C is continuous. Hence, g ∈ ∗ D (N11 ) and f ⊕ g ∈ D (M ). Part (ii) results from (i) and the von Neumann theorem, because a linear subspace E of H1 is dense in H1 if and only if H0 ⊕ E is dense in H0 ⊕ H1 ; the ∗∗ = N11 . formula (3.1) follows from N ∗∗ = N and N11 Part (iii) is a consequence of (ii). In applications, the spaces H0 and H1 are diﬀerent (in the case of Pontriagin spaces H0 will be ﬁnite dimensional and H1 not). Thus we can not use the generalized determinants as in [8, Chapter 7]. Instead of it we apply the following well known result (see e.g. [1, 11, 12, 13] for various applications in the theory of block operator matrices, and [14, 15] for C0 -semigroups). The proof goes straightforwardly even in the case of Banach spaces. Lemma 3.2. (Schur’s complement) Let N be a H0 bounded, closed operator. Suppose that z ∈ ρ(N11 ) and put Nz := N11 − z, Rz := N00 − N01 Nz−1 N10 ∈ B(H0 ). Then the following are equivalent: (i) z ∈ ρ(N ); (ii) Rz is injective and Rz−1 ∈ B(H0 ). Moreover, if (i) holds, then Rz−1 −Rz−1 N01 Nz−1 −1 (N − z) = . (3.2) −Nz−1 N10 Rz−1 Nz−1 N10 Rz−1 N01 Nz−1 + Nz−1 Note that the matrix above has all entries bounded.

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4. Domains of powers of block operator matrices m Recall that N m is understood in the sense od (1.1) and N11 stands for (N11 )m .

Proposition 4.1. Let N be a H0 -bounded operator. For every m ∈ N \ {0} the following are equivalent: (am ) H0 ⊆ D (N m ); m−1 ; (bm ) R (N10 ) ⊆ D N11 m (cm ) D (N m ) = H0 ⊕ D (N11 ). Proof. We will prove the equivalences (am ) ⇔ (bm ) ⇔ (cm ) by induction on m = 1, 2 . . . . The case m = 1 is clear. Assume that (am ) ⇔ (bm ) ⇔ (cm ) for some m ∈ N \ {0}(1 ). (am+1 ) ⇒ (bm+1 ) Suppose that (am+1 ) holds. Since D N m+1 ⊆ D (N m ), By induction (cm ) is valid as well. Take we have H0 ⊆ D (N m ), i.e. (am ) holds.

f f ∈ H0 . It follows from (am+1 ) that ∈ D N m+1 . Thus 0

N00 f f =N ∈ D (N m ) . 0 N10 f m From (cm ) we obtain N10 f ∈ D (N11 ). Since f was an arbitrary vector from H0 , (bm+1 ) is proved. m−1 m we have ) ⊆ D N11 (bm+1 ) ⇒ (am+1 ) Assume (bm+1 ). Since D (N11 (bm ) and (by induction) (c ). Take f ∈ H . Observe that by (b ), the m 0 m+1 vec

f N00 f f m tor N = ∈ belongs to H0 ⊕ D (N11 ). Applying (cm ) we get N 0 0 N10 f

f ∈ D N m+1 . D (N m ) and consequently 0 ((am+1 ) ∧ (bm+1 )) ⇒ (cm+1 ) Assume that (am+1 ) and (bm+1 ) hold. Like before, we see that (a m ) holds and thus, by induction hypothesis, (cm ) holds as h well. Let us take ∈ D N m+1 . Then we have g

N00 h + N01 g h =N ∈ D (N m ) . g N10 h + N11 g m From (cm ) we deduce that N10 h + N11 g ∈ D (N11 ). Employing (bm+1 ) we get m+1 m m . This way we have ). Hence, N11 g ∈ D (N11 ), i.e. g ∈ D N11 N10 h ∈ D (N11 m+1 proved that D N m+1 ⊆ H0 ⊕ D N11 .

m+1 f We show now the opposite inclusion. Take ∈ H0 ⊕ D N11 . Since we g

m+1 f . Hence, it is enough to show ∈ D N11 have assumed (am+1 ), we obtain 0 1 Observe

that the whole induction step described below works also if m = 1.

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0 N01 g m = ∈ D N m+1 . Observe that N ). and N11 g ∈ D (N11 g N11 g

0 0 Now, owing to (cm ), we have N ∈ D (N m ). Consequently ∈ D N m+1 . g g The implication (cm+1 ) ⇒ (am+1 ) is trivial.

that

0 g

5. The sets of type ΩT In this section we will investigate the relation between the sets of type ΩN and ΩN11 for a H0 -bounded operator N . For this purpose we prove a more general result. Proposition 5.1. Let T be a closed operator with nonempty spectrum in a Hilbert space K and let K ∈ B(K). If U is a set of type ΩT , then there exists U ⊆ U of types ΩT +K and ΩT . Proof. Since U is of type ΩT , it is unbounded and |z| (T − z)−1 ≤ t for all z ∈ U and some t > 0 (see (2.4)). First note that the set

1 −1 U0 := z ∈ U : (T − z) < K is also unbounded. Indeed, if z ∈ U \ U0 then |z| ≤

t ≤ t · K (T − z)−1

and consequently U \ U0 is bounded. Now observe that U0 ⊆ ρ(T + K), since for z ∈ U0 the series S(z) :=

∞

(−K(T − z)−1 )j

j=0

is convergent in the norm topology on B(H) and (T − z)−1 S(z) is the inverse of T + K − z. If z ∈ U0 and |z| > (t + 1) K then K · t t K (T − z)−1 ≤ ≤ <1 |z| t+1 and consequently ∞ j −1 −1 K (T − z)−1 ≤ (T + K − z) ≤ (T − z) · j=0

≤

t 1 t(t + 1) · . = t |z| 1 − (t+1) |z|

Hence, the set U := {z ∈ U0 : |z| > (t + 1) K} is of type ΩT +K . As an unbounded subset of U it is also of type ΩT .

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N00 N01 in H0 ⊕H1 N10 N11 and set 0 0 N00 N01 N := . , M := 0 N11 N10 0 First assume that a set U ⊆ C is of type ΩN . We apply the above Proposition to the operators T = N , K = −M and get the set U ⊆ U of type ΩN . Note that U ⊆ ρ(N11 ) and, by the deﬁnition of type ΩN , for some t > 0 t −1 −1 , z ∈ U . (N11 − z) ≤ max |z|, (N11 − z) = (N − z)−1 ≤ |z| Let us consider now a H0 -bounded operator N :=

Consequently U is of type ΩN11 . This proves the following Corollary. Corollary 5.2. If N is closed, H0 -bounded and U is of type ΩN then there exists U ⊆ U of types ΩN11 and ΩN . / V and V ⊆ On the other hand, let V be of type ΩN11 . By deﬁnition, 0 ∈ ρ(N11 ). Hence, V ⊆ ρ(N ). Observe that for z ∈ V we have

(N − z)−1 = max 1 , (N11 − z)−1 ≤ max {1, t} , z ∈ V, |z| |z| where t is such that (N11 − z)−1 ≤ t/|z| for z ∈ V . Hence, V is of type ΩN . Applying Proposition 5.1 to T := N , K := M and U := V we get the following: Corollary 5.3. If N is H0 -bounded and V is of type ΩN11 then there exists V ⊆ V of types ΩN and ΩN11 .

6. Closability and boundedness of commutators In this section we consider four H0 -bounded operators X00 X01 N00 N01 , X= , (6.1) N= N10 N11 X10 X11 0 0 X00 X01 X := Y := . (6.2) 0 X11 X10 0 acting in the orthogonal sum H = H0 ⊕ H1 of two Hilbert spaces. It is clear that Y ∈ B(H) and X = X + Y . For z ∈ ρ(N11 ) we put Nz := N11 − z and Rz := N00 − N01 Nz−1 N10 . Lemma 6.1. The following equality holds: [X, (N − z)−1 ] = [X , (N − z)−1 ] + [Y, (N − z)−1 ].

(6.3)

−1

Moreover, [Y, (N − z) ] ∈ B(H) and −1 −1 D [X, (N − z) ] = D [X , (N − z) ] .

(6.4)

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Proof. By (1.2) we have −1

X (N − z)

−1

and since (N − z) −1

(N − z)

−1

= (X + Y ) (N − z)

−1

= X (N − z)

−1

+ Y (N − z)

∈ B(H) we get −1

X = (N − z)

−1

(X + Y ) = (N − z)

−1

X + (N − z)

Y,

which shows that (6.3) is true. The rest of the conclusion of Lemma 6.1 is obvious. Proposition 6.2. Let N be closed and let m ∈ N \ {0}. Assume that H0 ⊆ D (N m ) ⊆ D (X) (6.5) −1 and that z ∈ ρ(N ) ∩ ρ(N11 ). Then D (N m ) ⊆ D [X, (N − z) ] , in particular −1

[X, (N − z)

] is densely defined. Furthermore,2 −1

P0 [X , (N − z)

P1 [X , (N − and

]|H0 = 0H0 ,

−1

P1 [X , (N − z) ]|H0 = −Rz−1 N01 Nz−1 X11 , −1 P0 [X , (N − z) ]|H1 = −X11 Nz−1 N10 Rz−1 , z)−1 ]|H1 = [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] + [X11 , Nz−1 ],

D [X11 , Nz−1 ] ⊆ D [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] . −1

If, additionally, [X, (N − z)

(6.6) (6.7) (6.8) (6.9) (6.10)

] is bounded then so is [X11 , Nz−1 ].

Remark 6.3. We recall that N11 is closed, because N is closed (cf. Lemma 3.1). Since ρ(N ) = ∅, N m is densely deﬁned, for instance by (1.5). Observe that Propom m sition 4.1 implies that D (N m ) = H0 ⊕ D (N11 ) and consequently N11 is densely deﬁned. The above formula for D (N m ) will be frequently used in the proof without any comment. The inclusion D (N m ) ⊆ D (X) is equivalent to m ) ⊆ D (X11 ) . D (N11

(6.11)

Proof of Proposition 6.2. The into several steps. proof is divided −1 −1 m Step 1. D (N ) ⊆ D [X, (N − z) ] . Consequently D [X , (N − z) ] = H0 ⊕ E for some E ⊆ H1 . To see this, observe that by assumption (6.5), −1 D (N m ) ⊆ D (X) = D (N − z) X . According to (1.4) we have (N − z)−1 (D (N m )) ⊆ D (N m ). Hence, −1 . D (N m ) ⊆ D X (N − z) Consequently

2 Observe

(6.5) −1 H0 ⊆ D (N m ) ⊆ D [X, (N − z) ] ,

that the operators in (6.7) and (6.8) may be unbounded.

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and of Step 1. is shown. By (6.4) we also have H0 ⊆ so the ﬁrst statement −1 D [X , (N − z) ] and the “consequently” part of Step 1 follows. Step 2. D X11 Nz−1 N10 = H0 (3 ) by (6.5) and Proposition 4.1 ((am ) ⇔ (bm )) we see that R (N10 ) ⊆ Indeed, m−1 . Now, by (1.4) applied to N = N11 , we have D N11 (6.11) (1.4) m R Nz−1 N10 ⊆ D (N11 ) ⊆ D (X11 ) ,

(6.12)

which ﬁnishes the proof of Step 2. Step 3. E = D [X11 , Nz−1 ] . −1 Take g ∈ D [X11 , Nz ] . We intend show that g ∈ E which is equivalent to 0 −1 ∈ D [X , (N − z) ] . By Lemma 3.2 g

0 −Rz−1 N01 Nz−1 g (N − z)−1 . (6.13) = g Nz−1 N10 Rz−1 N01 Nz−1 g + Nz−1 g Moreover, −Rz−1 N01 Nz−1 g ∈ H0 ⊆ D (X ) and, by Step 2, we have −1 −1 Nz−1 N10 Rz−1 N01 Nz−1 g ∈ D (X

11 ). Since g ∈ D [X11 , Nz ] , we get Nz g ∈ 0 D (X11 ). Thus (N − z)−1 ∈ D (X ). Furthermore, by g ∈ D Nz−1 X11 = g

0 −1 ∈ D (X ) = D (N − z) X , which ﬁnishes the proof of D (X11 ), we have g the inclusion “⊇”.

0 −1 ∈ D [(N − z) , X ] , which means that Now let g ∈ E. In particular

g 0 0 ∈ D (X ) and (N − z)−1 ∈ D (X ). The ﬁrst fact gives g g g ∈ D (X11 ) ,

(6.14)

while the second and Lemma 3.2 provide Nz−1 N10 Rz−1 N01 Nz−1 g + Nz−1 g ∈ D (X11 ) (see (6.13) for details). Applying again Step 2, we get Nz−1 N10 Rz−1 N01 Nz−1 g ∈ Nz−1 g ∈ D (X11 ). This, together with (6.14), shows that D (X11) and consequently −1 g ∈ D [X11 , Nz ] . Step 4. The inclusion (6.10) holds. Note that D Nz−1 N10 Rz−1 N01 Nz−1 X11 = D (X11 ) and by Step 2 D X11 Nz−1 N10 Rz−1 N01 Nz−1 = H0 .

(6.15) (6.16)

that we do not assume here that X11 is closable, so we can not deduce that X1 Nz−1 N10 is in B(H0 ). It turns out that if we assume that [X, (N − z)−1 ] is bounded then X1 Nz−1 N10 ∈ B(H0 ) without any assumption on closability of X11 .

3 Note,

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Thus we have D [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] = D (X11 ). Moreover, D [X11 , Nz−1 ] = Nz (D (X11 )) ∩ D (X11 ) ⊆ D (X11 ) . which proves the claim. Step 5. The formulas (6.6)–(6.9) hold. −1 Take f ⊕ g in the linear space H0 ⊕ D [X11 , Nz ] which, by Steps 1 and 3, −1

equals D [X, (N − z)

] . Now

f −1 −1 P0 [X , (N − z) ]|H0 f = P0 [X , (N − z) ] 0

−1 −Rz f 0 −1 = P0 X − P0 (N − z) −Nz−1 N10 Rz−1 f 0

0 0 = P0 = 0 −X11 Nz−1 N10 Rz−1 f

and (6.6) is shown. The proof of (6.7)–(6.9) goes in the same way. −1

Step 6. If the operator [X, (N − z) ] is bounded then the operator [X11 , Nz−1 ] is bounded. −1 −1 Assume that [X, (N − z) ] is bounded. Then, by Lemma 6.1, [X , (N − z) ] is bounded. From formulas (6.7)– (6.9) we obtain that −Rz−1 N01 Nz−1 X11 is bounded,

(6.17)

−X11 Nz−1 N10 Rz−1 −1 [X11 , Nz N10 Rz−1 N01 Nz−1 ] +

is bounded, (6.18) [X11 , Nz−1 ] is bounded. (6.19) −1 Note that by (6.17) and (6.18) and by boundedness of N01 , N10 and Nz , the commutator [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] = (X11 Nz−1 N10 Rz−1 )N01 Nz−1 − Nz−1 N10 (Rz−1 N01 Nz−1 X11 ), is bounded as well. This together with (6.10) and (6.19) completes the proof.

In [4] it was shown that the commutator [(S − z)−1 , A] is closable for symmetric A and selfadjoint S such that D (S m ) ⊆ D (A) for some m ∈ N. Later on ∗ := (Xij )∗ we will need the following generalization of this fact. Recall that Xij m m (i, j = 0, 1), X11 := (X11 ) . Proposition 6.4. Let S and A be H0 -bounded operators and let m ∈ N \ {0} be such that (6.20) H0 ⊆ D (S m ) ⊆ D (A) . ∗ (i) If S11 is normal, D (A11 ) ⊆ D (A11 ) and H0 ⊆ D ((S ∗ )m ) , then the commutator [(S − w)

−1

(6.21)

, A] is closable for every w ∈ ρ(S) ∩ ρ(S11 ).

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(ii) If

141

∗ S00 qS10 , (6.22) S10 S11 where q ∈ C and S11 is selfadjoint and A11 is symmetric then the commutator −1 ¯ is closable for every w ∈ ρ(S) ∩ ρ(S11 ). [(S − w) , A] S=

m is densely deﬁned. Proof. (i) Take w ∈ ρ(S) ∩ ρ(S11 ). Since S11 is normal, S11 m m ), which is By (6.20) and Proposition 4.1 the domain of S equals H0 ⊕ D (S11 dense in H0 ⊕ H1 . Applying Proposition 6.2 to N = S, X = A and z = w we get [(S − w)−1 , A] densely deﬁned. Hence, to prove the ﬁrst part of the Proposition it is enough to show that [(S − w)−1 , A]∗ is densely deﬁned. First note that w ¯ ∈ ρ(S ∗ ) and

[(S − w)−1 , A]∗ ⊇ −[(S ∗ − w) ¯ −1 , A∗ ]. ∗

(6.23) ∗ m

By (6.21) and Proposition 4.1 (applied to N = S ), we have that D ((S ) ) = H0 ⊕ ∗ m ∗ m D ((S ∗ )m 11 ). By Lemma 3.1(i), D ((S )11 ) = D ((S11 ) ) and, since S11 is normal, m ∗ m m the latter equals D (S11 ). Consequently D ((S ) ) = H0 ⊕ D (S11 ) = D (S m ). The ∗ ∗ operator A is H0 -bounded and D (A) ⊆ D (A ) (Lemma 3.1(i)). Therefore, D ((S ∗ )m ) = D (S m ) ⊆ D (A) ⊆ D (A∗ ) . Now we can use Lemma 6.2 taking N = S ∗ , X = A∗ and z = w ¯ (note that w ¯ ∈ ρ(S ∗ ) ∩ ρ(S11 )) and get [(S ∗ − w) ¯ −1 , A∗ ] densely deﬁned. By (6.23), the ﬁrst part of the proof is ﬁnished. To prove (ii) it is enough to show that all the assumptions of (i) are satisﬁed ¯ 11 = A11 is symmetric. for S and A¯ (instead of A). Normality of S11 is obvious, (A) ¯ Since (6.20) holds for A, it holds for A as well. The thingthat remains is to only ∗ ∗ S00 S10 ∗ is a H0 -bounded show (6.21). Note that by Lemma 3.1(i), S = q¯S10 S11 operator. Moreover, m−1 R ((S ∗ )10 ) = R (¯ = D (S ∗ )m−1 , q S10 ) = R (S10 ) ⊆ D S11 11 where the inclusion holds by Proposition 4.1 (am ) ⇒ (bm ). By implication (bm ) ⇒ (am ) of the same proposition applied to N = S ∗ , we get (6.21). The operators of the form (6.22) with q = −1 appear in Section 8. See [11] for studies on the case of arbitrary q.

7. Estimation of the norm of the commutator In Proposition7.1 and in its proof we will use the operators N , X, X , Y , Nz and Rz appearing in Section 6. Proposition 7.1. Let N and X be H0 -bounded operators. Assume that N is closed, m ∈ N \ {0} and H0 ⊆ D (N m ) ⊆ D (X) .

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Let V be a set of types ΩN and ΩN11 (4 ). Suppose that for every z ∈ V the −1 commutator [(N − z) , X] is bounded and −1 (7.1) sup |z| [(N − z) , X] < +∞. z∈V

−1 sup |z| [(N11 − z) , X11 ] < +∞.

Then

(7.2)

z∈V

Proof. Since V is of types ΩN and ΩN11 it is, by deﬁnition, contained in the resolvent sets of both operators N and N11 and there exists t > 0 and d ≥ 0 such that −1 −1 |z| (N − z) < t, |z| (N11 − z) < t, and |z| > d for all z ∈ V. (7.3) Since [(N − z)−1 , X] is bounded for z ∈ V we deduce from Lemma 6.1 that −1 the commutator [(N − z) , X ] is bounded as well. Moreover, by the same lemma −1 −1 −1 [(N − z) , X ] ≤ [(N − z) , X] + [(N − z) , Y ] . According to (7.3) we have −1 −1 |z| [(N − z) , Y ] ≤ 2|z| (N − z) Y ≤ 2t Y and consequently sup |z| [(N − z)−1 , X ] ≤ sup |z| [(N − z)−1 , X ] + 2t Y < +∞. z∈V

z∈V

Using (6.7)–(6.9) we get

sup |z| Rz−1 N01 Nz−1 X11 < +∞,

(7.4)

z∈V

sup |z| X11 Nz−1 N10 Rz−1 < +∞,

(7.5) z∈V (7.6) sup |z| [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] + [X11 , Nz−1 ] < +∞, z∈V −1 −1 for Pi [(N − z) , X ]|Hj ≤ [(N − z) , X ] (i, j = 0, 1). Since D [X11 , Nz−1 ] ⊆ D [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] (see (6.10)), it is now enough to prove that (7.7) sup |z| [X11 , Nz−1 N10 Rz−1 N01 Nz−1 ] < +∞. z∈V

To show (7.7) notice that by (7.3), Nz−1 ≤ t|z|−1 ≤ td−1 for all z ∈ V . We estimate: sup |z| Nz−1 N10 Rz−1 N01 Nz−1 X11 − X11 Nz−1 N10 Rz−1 N01 Nz−1 z∈V

−1

≤ td

N10 sup |z| Rz−1 N01 Nz−1 X11 + sup |z| X11 Nz−1 N10 Rz−1 N01 td−1 , z∈V

4 cf.

Corollaries 5.2 and 5.3

z∈V

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which, together with (7.4) and (7.5), implies (7.2).

143

8. Applications The following result appeared in [3] as Proposition 1. Apparently, the assumption in [3] that the operator H (which plays the role of our S), is selfadjoint, is much too strong. The only thing that is needed for the proof is that D∞ (H) is a core for H and that all the powers of H are closed. For our purposes we need the following statement. The proof is only a slight modiﬁcation of the original one, see there for details and further study. Proposition 8.1. Let S and A be closed operators in a Hilbert space K, assume also that ρ(S) = ∅. Then D∞ (S) ⊆ D (A) if and only if D (S m ) ⊆ D (A) for some m ∈ N. Proof. Note that the “if” part of the conclusion does not require any proof. By the closed graph theorem there exists c ≥ 0 and m ∈ N such that m j S f , f ∈ D∞ (S) . Af ≤ c j=0

Since ρ(S) = ∅, S is closed for j ∈ N ([5, Thm.VII.9.7]). Let j ≤m. We have D S j ⊆ D (S m ). By Remark 2.1 there exists dj ≥ 0 such that S j f ≤ dj (f + S m f ) for f ∈ D (S m ). Hence, Af ≤ c m max dj (f + S m f ), f ∈ D∞ (S) . j

j≤m

By the second part of Remark 2.1 (with N = S m , A = X and E = D∞ (S), the latter being a core for S m by (1.5)), we obtain D (S m ) ⊆ D (A). In this section we will consider a Krein space (K, [·, −]). By · we understand any ﬁxed complete norm on K such that the inner product [·, −] is continuous. Recall that all such norms are equivalent ([10, Propositions I.1.2],[2]), all topological notions will refer to that topology. We say that a linear subspace L of K is uniformly positive (uniformly negative) if [f, f ] ≥ c f ([f, f ] ≤ −c f ) for some c > 0 and all f ∈ L. A uniformly positive linear subspace L of K will be called maximal uniformly positive (maximal uniformly negative) if it is not properly contained in any other uniformly positive (uniformly negative) linear subspace of K. We will call L maximal uniformly definite, if it is either maximal uniformly positive or maximal uniformly negative. Maximal uniformly deﬁnite linear subspaces of K are necessarily closed ([2, Lemma V.5.4]). The symmetric, selfadjoint and essentially selfadjoint operators in (K, [·, −]) are deﬁned in a standard way. If E ⊆ D (A) and A|E is essentially selfadjoint then we say that A is essentially selfadjoint on E. Note that selfadjoint operators in K are maximal (in the sense of inclusion of the graphs) among symmetric ones. The proof of this fact can be easily reduced to the Hilbert space case, since the

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transformation T → JT (where J is any fundamental symmetry of K) sends symmetric (selfadjoint) operators in the Krein space (K, [·, −]) onto symmetric (selfadjoint) operators in the Hilbert space (K, [J·, −]). Therefore, a symmetric operator A is essentially selfadjoint on E if and only if it is essentially selfadjoint and E is a core for A. If X is closable, then by a set of type ΩX we understand any unbounded subset U of C satisfying (2.4), where · is deﬁned as above. Note that the deﬁnition of type ΩX , as well as the condition (8.1) below, are independent from the choice of the norm ·. The ﬁrst application was the main motivation for the investigations in previous sections. Theorem 8.2. Assume that A and S are symmetric and selfadjoint respectively operators in a Krein space (K, [·, −]) and that U is a set of type ΩS . (i) Let m ∈ N. If D (S m ) contains a maximal uniformly definite linear subspace H0 of K, D (S m ) ⊆ D (A) and sup |z| [(S − z)−1 , A] < +∞ (8.1) z∈U

then A is selfadjoint on any core of S m . (ii) If D∞ (S) ⊆ D (A), (8.1) holds, for every m ∈ N the linear space E is a core of (m) S m and D (S m ) contains a maximal uniformly definite linear subspace H0 of K, then A is essentially selfadjoint on E. Proof. (i) By assumption ρ(S) is nonempty. Hence, S m is closed ([5, Thm.VII.9.7]). A is closable. By Remark 2.1 there exists c > 0 such that Af ≤ c (S m f + f ) ,

f ∈ D (S m ) .

(8.2)

Take a core E for S and put B := A|E . Consider the operators X = A|E and N = S m|E . By (8.2) and the second part of Remark 2.1 we obtain D N ⊆ D X , which means that D (S m ) ⊆ D (B). The case m = 0 is trivial, let m > 0. We assume that H0 is maximal uniformly negative, the proof for H0 maximal uniformly positive goes in the same way. The space H1 := {f ∈ K : [f, H0 ] = {0}} is a maximal uniformly positive linear subspace of K and K is a direct sum of H0 and H1 ([2, Thm.V.7.1]). If f = f0 + f1 (fj ∈ Hj , j = 0, 1) then for i = 0, 1 we put Pi f := fi and J := −P0 + P1 . The sesquilinear form f, g := [Jf, g] (f, g ∈ K) is a Hilbert space scalar product on K and the spaces H0 and H1 are orthogonal with respect to ·, −. S, B and A are H0 -bounded operators in (H0 ⊕ H1 , ·, −) and can be represented as block operator matrices: ∗ ∗ S00 −S10 B00 −B10 A00 −A∗10 S= , B= , A= , S10 S11 B10 B11 A10 A11 m

where S11 is selfadjoint, B11 and A11 are symmetric in the Hilbert space (H1 , ·, −) (cf. [2, Lemma VI.6.7]). We need to show that B is selfadjoint in (K, [·, −]), which is equivalent to selfadjointness of B11 in (H1 , ·, −) (cf. [2, Lemma VI.6.7]). There exists a set U ⊆ U ∩ ρ(S11 ) of type ΩS , see Corollary 5.2. Fix z ∈ U . The

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commutator [(S − z)−1 , A] is, by assumption, bounded. It is also densely deﬁned, ¯ being a by Proposition 6.2 applied to S = N and X = A. Thus [(S − z)−1 , A], closable (Proposition 6.4(ii)) extension of a bounded and densely deﬁned operator [(S − z)−1 , A] is bounded as well and both these operators have the same −1 norm (for instance, 2.1). by Remark Therefore, [(S − z) , B] is bounded and −1 −1 [(S − z) , B] ≤ [(S − z) , A]. Hence, all the assumptions of Theorem 2.3 are satisﬁed, possibly with U instead of U , and consequently B11 is selfadjoint in H1 . The point (ii) follows from (i) and Proposition 8.1. Remark 8.3. The assumption that D (S m ) contains a maximal uniformly deﬁnite subspace is fulﬁlled for every selfadjoint operator in a Pontriagin space [2, Theorem IX.1.4] or, even less, for a deﬁnitizable operator in a Krein space with a regular point at inﬁnity [10]. We are also able now to prove the following variation of Theorem 2.2. Theorem 8.4. Assume that A is a symmetric operator and T is a selfadjoint operator in the orthogonal sum of two a Hilbert spaces H = H0 ⊕ H1 . Moreover, let K ∈ B(H) be such that K(H1 ) ⊆ H0 and let U be a set of type ΩT +K . (i) Let m ∈ N. If H0 ⊆ D (T m ) ⊆ D (A), E is a core for T m and (8.3) sup |z| [(T + K − z)−1 , A|E ] < +∞ z∈U

then A is selfadjoint on E. (iii) If H0 ⊆ D∞ (T ) ⊆ D (A), E is a core of T m for every m ∈ N and (8.3) holds, then A is essentially selfadjoint on E. Proof. (i) The case m = 0 is trivial, we assume that m > 0. The operator S := T + K has the following matrix representation (with respect to the decomposition H = H0 ⊕ H1 ) T00 + K00 T01 + K01 S= , (8.4) T10 + K10 T11 in particular S11 = T11 is selfadjoint. The same arguments as in the proof of Theorem 8.2 allow us to apply Theorem 2.3 to S and B := A|E . Consequently B11 is selfadjoint. This ﬁnishes the proof (i). To show (ii) note that H0 ⊆ D (T m ) ⊆ D (A) for some m ∈ N (Proposition 8.1) and use (i). Remark 8.5. If a selfadjoint operator T in a Hilbert space H is given, K ∈ B(H) is such that R (K) ⊆ D (T m ) and we put H0 = R (K) and H1 := N (K ∗ ) then H0 ⊆ D (T m ) and K(H1 ) ⊆ H0 . Remark 8.6. The condition (8.3) appearing in Proposition 8.4 can be relaxed to supz∈U |z| [(T + K − z)−1 , A] < +∞, provided that the commutator [(T + K − ¯ is closable, see the proof of Theorem 8.2 for details. This happens for examz)−1 , A] m−1 ∗ ) ⊆ D T11 ple if, additionally, R (K ∗ ) ⊆ D T m−1 . Indeed, in such case R (K01 m−1 ∗ and consequently R (K01 . Hence, H0 ⊆ D (((T + K)∗ )m ) (see + T10 ) ⊆ D T11 ¯ is closable. Proposition 4.1). By Proposition 6.4(i) [(T + K − z)−1 , A]

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Remark 8.7. Note that for S as in Theorem 8.2 and in (8.4) then the sets of type ΩS exist. This follows from Corollary 5.3 and the fact that i[d, +∞) is a set of type ΩS11 for d > 0. Acknowledgment The author is indebted to Professor Jan Stochel for an inspiring discussion and many helpful comments.

References [1] V. Adamjan, H. Langer, Existence and Uniqueness of Contractive Solutions of Some Riccati Equations , J. Funct. Anal., 179, (2001), 448–473. [2] J. Bogn´ ar, Indefinite Inner Product Spaces, Springer-Verlag, 1974. [3] D. Cicho´ n, J. Stochel, F. H. Szafraniec, Noncommuting domination, Oper. Theory Adv. Appl. 154 (2004), 19–33 [4] D. Cicho´ n, J. Stochel, F. H. Szafraniec, Selfadjointness of integral and matrix operators, to appear in Journal of London Math. Soc. [5] N. Dunford, J. T. Schwartz, Linear Operators; I , Interscience Publishers, 1958. [6] W. Driessler, S. J. Summers, On commutators and selfadjointness, Lett. Math. Phys., 7 (1983), 319–326. [7] I. C. Gohberg, M. G. Krein, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspiehi Mat. Nauk 12, No. 2, (1957), 43–118, english translation: Amer. Math. Transl., (2) 13, (1960), 185–264. [8] P. R. Halmos, A Hilbert Space Problem Book Springer-Verlag , 1974. [9] T. Kato, Perturbation theory for linear operators, Springer-Verlag, 1966 [10] H. Langer, Spectral functions of deﬁnitizable operators in Krein spaces, Proc. Graduate School “Functional Analysis”, Dubrovnik 1981. Lecture Notes in Math. 948, Springer Verlag, Berlin, 1982, 1–46. [11] H. Langer, M. Langer, C. Tretter, Variational Principles for Eigenvalues of Block Operator Matrices, Indiana University Mathematics Journal, 51, No. 6 (2002), 1427– 1459. [12] H. Langer, A. Markus, V. Matsaev, C. Tretter, Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements, J. Funct. Anal., 199, (2003), 427–451. [13] H. Langer, C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices J. Operator Theory, 39 (1998), 339–350. [14] R. Nagel, Towards a “Matrix Theory” for Unbounded Operator Matrices, Mathematische Zeitschrift, 201 (1989), 57–68. [15] R. Nagel, The Spectrum of Unbounded Operator Matrices with Non-diagonal Domain, J. Funct. Anal., 89 (1990), 291–302. [16] E. Nelson, Analytic vectors, Ann. of Math., 70 (1959), 572–615. [17] A.E. Nussbaum, A commutativity theorem for unbounded operators in Hilbert space, Trans. Amer. Math. Soc., 140 (1969), 485–491.

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[18] M. Reed, B. Simon, Methods of Modern Mathematical Physics; II, Fourier analysis, selfadjointness, Academic Press, New York, 1975. [19] J. Stochel, F. H. Szafraniec, Domination of unbounded operators and commutativity, J. Math. Soc. Japan, 55 No. 2, (2003), 405–437. [20] M. Wojtylak, Algebras dense in L2 spaces: an operator approach, Glasgow Math. J., 47 (2005), 155–165. Michal Wojtylak Instytut Matematyki Uniwersytet Jagiello´ nski Reymonta 4 30-059 Krak´ ow Poland e-mail: [email protected] Submitted: April 24, 2006 Revised: March 16, 2007

Integr. equ. oper. theory 59 (2007), 149–164 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020149-16, published online June 27, 2007 DOI 10.1007/s00020-007-1512-2

Integral Equations and Operator Theory

Exotic Indecomposable Systems of Four Subspaces in a Hilbert Space Masatoshi Enomoto and Yasuo Watatani Dedicated to Professor Masahiro Nakamura on his 88th birthday

Abstract. We study the relative position of four (closed) subspaces in a Hilbert space. For any positive integer n, we give an example of exotic indecompos. By able system S of four subspaces in a Hilbert space whose defect is 2n+1 3 an exotic system, we mean a system which is not isomorphic to any closed operator system under any permutation of subspaces. We construct the examples using certain nice sequences construced by Jiang and Wang in their study of strongly irreducible operators. Mathematics Subject Classification (2000). 46C07, 47A15, 15A21, 16G20. Keywords. Subspace, Hilbert space, indecomposable system, defect, strongly irreducible operator.

1. Introduction Many problems of linear algebra can be reduced to the classiﬁcation of the systems of n subspaces in a ﬁnite-dimensional vector space. Nazarova [N] and GelfandPonomarev [GP] completely classiﬁed indecomposable systems of four subspaces in a ﬁnite dimensional vector space. On the other hand, in operator theory, Halmos initiated the study of transitive lattices of subspaces, see for example [Ha]. Transitive lattices give transitive systems of subspaces. Transitive system of subspaces in a ﬁnite dimensional space had been studied by Brenner in [B]. In [EW] we started to investigate systems of n subspaces in an inﬁnite dimensional Hilbert space considering an analogy with the subfactor theory invented by Jones [J]. As a building block, we investigate indecomposable systems of n subspaces in the sense that the system can not be isomorphic to a direct sum of two non-zero systems. Recently Moskaleva and Samoilenko [MS] studied a relation between systems of n-subspaces and representations of *-algebras generated by projections.

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Let H be a Hilbert space and E1 , . . . En n subspaces in H. Throughout the paper, a subspace always means a closed subspace. Then we say that S = (H; E1 , . . . , En ) is a system of n subspaces in H or a n-subspace system in H. A system S is said to be indecomposable if S is not be decomposed into a nontrivial direct sum. For any bounded linear operator A on a Hilbert space K, we associate an operator system SA of four subspaces in H = K ⊕ K by SA = (H; K ⊕ 0, 0 ⊕ K, graph A, {(x, x); x ∈ K}). Two such operator systems SA and SB are isomorphic if and only if the two operators A and B are similar. The direct sum of operator systems corresponds to the direct sum of the operators. In this sense the study of operators is included into the study of relative positions of four subspaces. In particular in a ﬁnite dimensional space, Jordan blocks correspond to indecomposable systems. Moreover in an inﬁnite dimensional Hilbert space, an operator system SA is indecomposable if and only if A is strongly irreducible. Recall that an operator A ∈ B(K) is said to be strongly irreducible if there are no non-trivial invariant subspaces M and N of A such that M ∩ N = 0 and M + N = K. A strongly irreducible operator is an inﬁnite-dimensional analog of a Jordan block. We refer to a good monograph [JW] by Jiang and Wang on strongly irreducible operators. In [EW] we discovered some examples of exotic indecomposable systems S of four subspaces in a Hilbert space. By an exotic system, we mean a system which is not isomorphic to any closed operator system SA under any permutation of subspaces. Gelfand and Ponomarev introduced an integer valued invariant ρ(S), called defect, for a system S = (H; E1 , E2 , E3 , E4 ) of four subspaces by ρ(S) =

4

dim Ei − 2 dim H.

i=1

They showed that if S is indecomposale, then the defect ρ(S) is one of {−2, −1, 0, 1, 2}. We extended the notion of defect to a certain class of systems of four subspaces in an inﬁnite dimensional Hilbert space by using the Fredholm index in [EW]. We showed that the defect for indecomposable systems of four subspaces takes any value in Z/3. These values are attained by bounded operator systems. In fact the exotic systems constructed in [EW] have the defect ρ(S) = 1. The aim of the paper is to give new examples of exotic indecomposable sysfor any tems S of four subspaces in a Hilbert space with the defect ρ(S) = 2n+1 3 positive integer n. We construct these examples using certain nice sequences constructed by Jiang and Wang in their study of strongly irreducible operators in [JW]. We do not know whether there exist exotic indecomposable systems of four subspaces with the defect 2n 3 (n ∈ N) or the negative defect.

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2. Relative position of subspaces We study the relative position of n subspaces in a separable Hilbert space. Firstly we recall some basic facts in [EW]. Let H be a Hilbert space and E1 , . . . , En be n subspaces in H. Then we say that S = (H; E1 , . . . , En ) is a system of n subspaces in H or an n-subspace system in H. Let T = (K; F1 , . . . , Fn ) be another system of n subspaces in a Hilbert space K. Then ϕ : S → T is said to be a homomorphism if ϕ : H → K is a bounded linear operator satisfying that ϕ(Ei ) ⊂ Fi for i = 1, . . . , n. And ϕ : S → T is called an isomorphism if ϕ : H → K is an invertible (i.e., bounded bijective) linear operator satisfying that ϕ(Ei ) = Fi for i = 1, . . . , n. We say that systems S and T are isomorphic if there is an isomorphism ϕ : S → T . This means that the relative positions of n subspaces (E1 , . . . , En ) in H and (F1 , . . . , Fn ) in K are the same disregarding angles. We say that systems S and T are unitarily equivalent if the above isomorphism ϕ : H → K can be chosen to be a unitary. This means that the relative positions of n subspaces (E1 , . . . , En ) in H and (F1 , . . . , Fn ) in K are the same, preserving the angles between the subspaces. We denote by Hom(S, T ) the set of homomorphisms of S to T and End(S) := Hom(S, S) the set of endomorphisms on S. For two systems S = (H; E1 , . . . , En ) and T = (K; F1 , . . . , Fn ) of n subspaces in H, their direct sum S ⊕ T is deﬁned by S ⊕ T := (H ⊕ K; E1 ⊕ F1 , . . . , En ⊕ Fn ). Definition. A system S = (H; E1 , . . . , En ) of n subspaces is said to be decomposable if the systems S is isomorphic to a direct sum of two non-zero systems. A system S = (H; E1 , . . . , En ) is said to be indecomposable if it is not decomposable. A system S is indecomposable if and only if Idem(S) := {V ∈ End(S); V 2 = V } = {0, I}. A system S is said to be transitive if End(S) = CI. Transitive systems in a ﬁnite dimensional space were studied by S. Brenner [B]. On the other hand, Halmos [Ha] initiated the study of transitive lattices of subspaces in Hilbert spaces, which give transitive systems. Some interesting examples were obtained by Harrison-Radjavi-Rosenthal [HRR] and Hadwin-LongstaﬀRosenthal [HLR]. We have a close relation between systems of subspaces and operators. In fact we can associate a system of four subspaces for any operator. Definition. We say that a system S = (H; E1 , . . . , E4 ) of four subspaces is a closed operator system if there exist Hilbert spaces K1 , K2 and closed operators T : K1 ⊃ D(T ) → K2 , S : K2 ⊃ D(S) → K1 such that H = K1 ⊕ K2 , E1 = K1 ⊕ 0, E2 = 0 ⊕ K2 , E3 = {(x, T x); x ∈ D(T )} and E4 = {(Sy, y); y ∈ D(S)}. Here D(T ) is the domain of T . In particular, if T and S are bounded operators with D(T ) = K1 and D(S) = K2 , then we say that S = (H; E1 , . . . , E4 ) is a bounded operator system. We denote it by ST,S . We put ST := ST,I and call it a bounded operator system associated with a single operator T . Two such operator systems SA and SB are isomorphic if and only if the two operators A and B are similar.

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Moreover, in an inﬁnite dimensional Hilbert space, a bounded operator system SA is indecomposable if and only if A is strongly irreducible. Definition. Let S = (H; E1 , E2 , E3 , E4 ) be a system of four subspaces. For any distinct i, j = 1, 2, 3, 4, deﬁne an adding operator Aij : Ei ⊕ Ej (x, y) → x + y ∈ H. Then Ker Aij = {(x, −x) ∈ Ei ⊕ Ej ; x ∈ Ei ∩ Ej } and Im Aij = Ei + Ej . We say S = (H; E1 , E2 , E3 , E4 ) is a Fredholm system if Aij is a Fredholm operator for any i, j = 1, 2, 3, 4 with i = j. Then Im Aij = Ei + Ej is closed and Index Aij = dim Ker Aij − dim Ker A∗ij = dim(Ei ∩ Ej ) − dim((Ei + Ej )⊥ ). Definition. We say S = (H; E1 , E2 , E3 , E4 ) is a quasi-Fredholm system if Ei ∩ Ej and (Ei + Ej )⊥ are ﬁnite-dimensional for any i = j. In this case we deﬁne the defect ρ(S) of S by ρ(S) :=

1 3

(dim(Ei ∩ Ej ) − dim(Ei + Ej )⊥ )

1≤i<j≤4

which coincides with the Gelfand-Ponomarev original defect if H is ﬁnite-dimensional. Moreover, if S is a Fredholm system, then it is a quasi-Fredholm system and 1 Index Aij . ρ(S) = 3 1≤i<j≤4

3. Construction of examples Consider a Hilbert space L = 2 (N). Let {e1 , e2 , e3 , . . . } be its canonical basis . For a bounded sequence w = (w(n))n , we deﬁne a backward weighted shift Bw ∈ B(2 (N)) of weight w by Bw en = w(n − 1)en−1 , (n ≥ 2) and Bw e1 = 0. Thus for x = (x(n))n ∈ 2 (N), we have (Bw x)(n) = w(n)x(n+1) for n = 1, 2, 3, . . . . We borrow a family of sequences a1 = (a1 (n))n , a2 = (a2 (n))n , a3 = (a3 (n))n , . . . used by Jiang and Wang in [JW, p.93-94] as follows: Deﬁne a sequence c = (c(n))n of positive numbers and an increasing sequence n1 < n2 < n3 < . . . of natural numbers as follows: Put c(1) = 2 = 1+1 1 > 1 and n1 = 1. There exists n2 ∈ N with n1 < n2 such that n2 1 k 1+1 < . 1 k+1 2 k=n1 +1

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for k = n1 + 1 = 2, . . . , n2 . There exists n3 ∈ N with n2 < n3 such 1+1 1

n2 k=n1 +1

k k+1

n3 k=n2 +1

k+1 > 3. k

Put c(k) = k+1 for k = n2 + 1, . . . , n3 . We continue in this fashion to obtain k an increasing sequence n1 < n2 < n3 < . . . of natural numbers and a sequence c = (c(n))n of positive numbers such that k+1 k , (k = n1 = 1, n2 + 1 ≤ k ≤ n3 , n4 + 1 ≤ k ≤ n5 , . . . ) c(k) = k k+1 , (n1 + 1 ≤ k ≤ n2 , n3 + 1 ≤ k ≤ n4 , n5 + 1 ≤ k ≤ n6 , . . . ), and

nj

c(k)

k=1

Then

2 3

> j (j is odd ) < 1j (j is even. ) 1

≤ c(k) ≤ 2. Deﬁne a1 (k) ≡ 1, a2 (k) = c(k) 2 and 1

1

1

1−i

ai (k) = c(k) 2 + 4 +···+ 2i−1 = c(k)1−2

for i ≥ 2

Then we have the following lemma: Lemma 3.1. (Jiang and Wang [JW]) There exists a family of sequences a1 = (a1 (n))n , a2 = (a2 (n))n , a3 = (a3 (n))n , . . . , of positive numbers satisfying 1. 23 ≤ ai (k) ≤ 2, n 2. limk→∞ ai (k) = 1, limn→∞ k=1 2ai (k) = ∞, n 3. lim supn→∞ k=1 aaji (k) (k) = ∞, (i = j), n ai (k) 4. lim inf n→∞ k=1 aj (k) = 0, (i = j). 5. the point spectrum σp (Bai ) contains {λ ∈ C : |λ| < 1}. We shall construct our examples. We ﬁx a family of sequences a1 = (a1 (n))n , a2 = (a2 (n))n , a3 = (a3 (n))n , . . . of positive numbers deﬁned in the above Lemma 3.1. Put wk = 2ak for k = 1, 2, . . . . Consider a sequence of backward weighted shifts Bw1 , Bw2 , Bw3 , . . . on L = 2 (N). Let S ∈ B(L) be a unilateral shift. For a ﬁxed natural number N , deﬁne K = L ⊕ · · · ⊕ L (N + 1 times) and H = K ⊕ K. In the sequel we sometimes use the symbol (x ⊕ y) ∈ K ⊕ K instead of (x, y) ∈ K ⊕ K for the sake of convenience. We consider the operator   Bw1 I O ... O  ..  ..  O . I .  Bw2    . . .. .. O  T = O  ∈ B(K) O     . .. ..  .. . BwN I  . O O ··· O S Let E1 = K ⊕ 0, E2 = 0 ⊕ K, E4 = {x ⊕ x ∈ K ⊕ K; x ∈ K} and E3 = {x ⊕ T x ∈ K ⊕ K; x ∈ K} + C((0, . . . , 0) ⊕ (0, . . . , 0, e1 )).

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Consider a system Sw,N = (H; E1 , E2 , E3 , E4 ). We shall show that Sw,N is indecomposable and is not isomorphic to any closed operator systems under any permutation. We could regard that the system Sw,N is a one-dimensional “deformation” of an operator system, since E3 = graph T + C((0, . . . , 0) ⊕ (0, . . . , 0, e1 )). Theorem 3.2. The above system Sw,N of four subspaces is indecomposable. Proof. In order to make the notation simple, we shall prove the theorem in case N = 3. The general N case will be proved similarly. Let V ∈ End(Sw,N ) satisfy V 2 = V . It is enough to show that V = O or V = I for Sw,N to be indecomposable. Since V (Ei ) ⊂ Ei for i = 1, 2, 4, we have A O V = ∈ B(H) for some A ∈ B(K). O A iT is suﬃcient to prove that A = O or A = I. We may write A = (Aij )ij as operator matrix , where Aij ∈ B(L) and i, j = 1, 2, 3, 4. Thus we have     I O O Bw1 A11 A12 A13 A14  O   Bw2 I O  and A = A21 A22 A23 A24  . T =  O A31 A32 A33 A34  O Bw3 I  O O O S A41 A42 A43 A44 Since E3 = graph T + C((0, 0, 0, 0) ⊕ (0, 0, 0, e1)), E3 is spanned by  (e1 , 0, 0, 0) ⊕ (0, 0, 0, 0), (en , 0, 0, 0) ⊕ (w1 (n − 1)en−1 , 0, 0, 0),     (0, e1 , 0, 0) ⊕ (e1 , 0, 0, 0), (0, en , 0, 0) ⊕ (en , w2 (n − 1)en−1 , 0, 0),  n ≥ 2, (0, 0, e1 , 0) ⊕ (0, e1 , 0, 0), (0, 0, en , 0) ⊕ (0, en , w3 (n − 1)en−1 , 0), ; k≥1   (0, 0, 0, ek ) ⊕ (0, 0, ek , ek+1 ),    (0, 0, 0, 0) ⊕ (0, 0, 0, e1 ), We may write  (x1 (n))n    (x2 (n))n E3 =   (x3 (n))n    (x4 (n))n





 (x1 (n + 1)w1 (n) + x2 (n))n 2   (x2 (n + 1)w2 (n) + x3 (n))n  x1 , x2 ∈ 2 (N) ⊕  ; x3 , x4 ∈ (N)   (x3 (n + 1)w3 (n) + x4 (n))n  y∈C (y, (x4 (n))n )

          

.

      

We shall continue to prove the theorem until the end of this section. We shall decompose the rest of the proof into several lemmas in the below. The proof will be completed by Lemma 3.12. Lemma 3.3. Let P ∈ B(2 (N)) be an operator of the form P = λI + N for some λ ∈ C and an upper (or lower) triangular matrix N ∈ B(2 (N)) with zero diagonal. Assume that P is an idempotent, then P = O or P = I. Proof. This is a known fact. See for example Lemma 10.1 in [EW].

Lemma 3.4. We have that A41 (k, n) = 0 for any k, n ≥ 1, A31 (k, n) = 0 for any k ≥ n + 1, A21 (k, n) = 0 for any k ≥ n + 2, and A11 (k, n) = 0 for any k ≥ n + 3. In particular A41 = O.

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Proof. Since u = (e1 , 0, 0, 0) ⊕ (0, 0, 0, 0) ∈ E3 , we have     A11 e1 0  A21 e1   0     Vu=  A31 e1  ⊕  0  A41 e1 0    (x1 (k))k (x1 (k + 1)w1 (k) + x2 (k))k  (x2 (k))k   (x2 (k + 1)w2 (k) + x3 (k))k   =  (x3 (k))k  ⊕  (x3 (k + 1)w3 (k) + x4 (k))k (x4 (k))k (y, (x4 (k))k )

155

   ∈ E3 . 

for some x1 , x2 , x3 , x4 ∈ 2 (N) and y ∈ C. Then x4 (k) = 0 for k ≥ 1. Thus A41 (k, 1) = (A41 e1 )(k) = x4 (k) = 0. Since x3 (k + 1)w3 (k) = x3 (k + 1)w3 (k) + x4 (k) = 0 and w3 (k) > 0, we have x3 (k + 1) = 0 for k ≥ 1, i.e., A31 (k, 1) = x3 (k) = 0 for k ≥ 2. Since x2 (k + 1)w2 (k) = x2 (k + 1)w2 (k) + x3 (k) = 0 for k ≥ 2 and w2 (k) > 0, we have x2 (k + 1) = 0 for k ≥ 2, i.e., A21 (k, 1) = x2 (k) = 0 for k ≥ 3. Since x1 (k + 1)w1 (k) = x1 (k + 1)w1 (k) + x2 (k) = 0 for k ≥ 3 and w1 (k) > 0, we have x1 (k + 1) = 0 for k ≥ 3, i.e., A11 (k, 1) = x1 (k) = 0 for k ≥ 4. Thus the statement of the lemma is proved for n = 1. Moreover, x2 (2)w2 (1)+ x3 (1) = 0 implies that A21 (2, 1)w2 (1)+ A31 (1, 1) = 0. And x1 (2)w1 (1) + x2 (1) = 0 implies that A11 (2, 1)w1 (1) + A21 (1, 1) = 0. And x1 (3)w1 (2) + x2 (2) = 0 implies that A11 (3, 1)w1 (2) + A21 (2, 1) = 0. We shall prove the lemma by induction on n. Assume that the statement of the Lemma holds for the n-th column of A11 , A21 , A31 , A41 . We shall prove it for n + 1. Since u = (en+1 , 0, 0, 0) ⊕ (w1 (n)en , 0, 0, 0) ∈ E3 , we have     A11 en+1 w1 (n)A11 en  A21 en+1   w1 (n)A21 en     Vu=  A31 en+1  ⊕  w1 (n)A31 en  A41 en+1 w1 (n)A41 en     (x1 (k + 1)w1 (k) + x2 (k))k (x1 (k))k  (x2 (k))k   (x2 (k + 1)w2 (k) + x3 (k))k     =  (x3 (k))k  ⊕  (x3 (k + 1)w3 (k) + x4 (k))k  ∈ E3 . (x4 (k))k (y, (x4 (k))k ) for some x1 , x2 , x3 , x4 ∈ 2 (N) and y ∈ C. Since (A41 en )(k) = A41 (k, n) = 0 for any k by the assumption of induction, (y, (x4 (k))k ) = w1 (n)A41 en = 0. Then A41 (k, n + 1) = (A41 en+1 )(k) = x4 (k) = 0. Since (A31 en )(k) = A31 (k, n) = 0 for any k ≥ n + 1 by the assumption of induction, x3 (k + 1)w3 (k) = x3 (k + 1)w3 (k) + x4 (k) = w1 (n)(A31 en )(k) = 0.

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Because w3 (k) > 0, we have x3 (k + 1) = 0 for k ≥ n + 1, i.e., A31 (k, n) = (A31 en )(k) = x3 (k) = 0 for k ≥ (n + 1) + 1. Since A21 (k, n) = 0 for any k ≥ n + 2 by the assumption of induction, x2 (k + 1)w2 (k) = x2 (k + 1)w2 (k) + x3 (k) = w1 (n)(A21 en )(k) = 0. Because w2 (k) > 0, we have x2 (k + 1) = 0 for k ≥ n + 2, i.e., A21 (k, n) = (A21 en )(k) = x2 (k) = 0 for k ≥ (n + 1) + 2. Since A11 (k, n) = 0 for any k ≥ n + 3 by the assumption of induction, x1 (k + 1)w1 (k) = x1 (k + 1)w1 (k) + x2 (k) = w1 (n)(A11 en )(k) = 0. Because w1 (k) > 0, we have x1 (k + 1) = 0 for k ≥ n + 3, i.e., A11 (k, n) = (A11 en )(k) = x1 (k) = 0 for k ≥ (n + 1) + 2. This ﬁnishes the proof by induction.

Lemma 3.5. A31 = O and A21 = O. Proof. From the proof in Lemma 3.4, A31 (k, n + 1) = x3 (k) and w1 (n)A31 (k, n) = x3 (k + 1)w3 (k) + x4 (k) = x3 (k + 1)w3 (k). Hence A31 (k + 1, n + 1) = Therefore for any j ≥ 1

w1 (n) A31 (k, n) for any n, k. w3 (k)

n w1 (j + k − 1) A31 (1, j). w3 (1 + k − 1) k=1 n w1 (k) = ∞ by Recall that 43 ≤ w1 (k) ≤ 4, 43 ≤ w3 (k) ≤ 4 and lim supn→∞ k=1 w 3 (k) Lemma 3.1. Since A31 < ∞, we have A31 (1, j) = 0. Furthermore A31 (1 + n, j + n) = 0 for any j, n, i.e., A31 (k, n) = 0 for any k ≤ n. By Lemma 3.4 A31 (k, n) = 0 for any k ≥ n + 1. Therefore A31 = O. Similarly we have w1 (n) A21 (k, n) for any n, k. A21 (k + 1, n + 1) = w2 (k) By a similar argument we also have A21 = O.

A31 (1 + n, j + n) =

Lemma 3.6. A11 = O or A11 = I. Proof. From the proof in Lemma 3.4 and the additional fact that x2 = A21 en+1 = 0, we have A11 (k, n + 1) = x1 (k) and w1 (n)A11 (k, n) = x1 (k + 1)w1 (k) + x2 (k) = x1 (k + 1)w1 (k). Hence A11 (k + 1, n + 1) =

w1 (n) A11 (k, n) for any n, k. w1 (k)

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Therefore for any j ≥ 1 A11 (j + n, 1 + n) =

n w1 (1 + k − 1) A11 (j, 1). w1 (j + k − 1)

k=1

And we also have A11 (1 + n, 1 + n) = A11 (n, n). Therefore the diagonal of A11 is a constant, say λ. From the proof in Lemma A21 (1,1) 3.4, we have A11 (3, 1) = − Aw211(2,1) (2) = 0 and A11 (2, 1) = − w1 (1) = 0, because A21 = O. Therefore for any n n w1 (1 + k − 1) A11 (3, 1) = 0. A11 (n + 2, n) = w1 (3 + k − 1) k=1

Similarly A11 (n + 1, n) = 0. We also have A11 (k, n) = 0 for any k ≥ n + 3 by Lemma 3.4. Therefore A11 = λI + N for some λ ∈ C and an upper triangular matrix N ∈ B(2 (N)) with zero diagonal. Since V is an idempotent, A is an idempotent. Hence A11 is also an idempotent, because A21 = A31 = A41 = O. Thus A11 = O or A11 = I by Lemma 3.3. In the below we shall show that if A11 = O(resp. A11 = I), then V = O (resp. V = I). Replacing V by I − V , it is enough to show that A11 = O implies V = O to prove Theorem 3.2. Lemma 3.7. Suppose that A11 = O. Then A42 (k, n) = 0 for any k, n ≥ 1, A32 (k, n) = 0 for any k ≥ n+1, A22 (k, n) = 0 for any k ≥ n+2, and A12 (k, n) = 0 for any k ≥ n + 3. In particular A42 = O. Proof. Since u = (0, e1 , 0, 0) ⊕ (e1 , 0, 0, 0) ∈ E3 and the ﬁrst column of A is 0,     0 A12 e1  A22 e1   0     Vu=  A32 e1  ⊕  0  ∈ E3 0 A42 e1 Since u = (0, en+1 , 0, 0) ⊕ (en+1 , w2 (n)en , 0, 0) ∈ E3 and the ﬁrst column of A is 0, we have     w2 (n)A12 en A12 en+1  A22 en+1   w2 (n)A22 en     Vu=  A32 en+1  ⊕  w2 (n)A32 en  ∈ E3 A42 en+1 w2 (n)A42 en Therefore the rest of the proof is as same as 3.4. Lemma 3.8. Suppose that A11 = O. Then A32 = O and A22 = λI + N for some λ ∈ C and an upper triangular matrix N ∈ B(2 (N)) with zero diagonal. Proof. Since w2 appears in V u instead of w1 , a diagonal block A22 plays a similar role of a diagonal block A11 in the argument of the proof in Lemma 3.6. The rest is similarly proved as the ﬁrst column of the operator matrix A = (Aij )ij is zero.

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Lemma 3.9. Suppose that A11 = O. Then A22 = O. Proof. Since A is an idempotent and A32 = A42 = O, A22 is also an idempotent. Thus A22 = O or A22 = I by 3.3. It is enough to show that A22 = I. On the contrary suppose that A22 = I. Then V ((0, e1 , 0, 0) ⊕ (e1 , 0, 0, 0)) = (A12 e1 , e1 , 0, 0) ⊕ (0, 0, 0, 0) ∈ E3 . This implies that A12 (21) = − w11(1) . Since V ((0, en+1 , 0, 0) ⊕ (en+1 , w2 (n)en , 0, 0)) ∈ E3 , we have     A12 en+1 w2 (n)A12 en  en+1   w2 (n)en   ⊕      0 0 0 0    (x1 (k + 1)w1 (k) + x2 (k))k (x1 (k))k  (x2 (k))k   (x2 (k + 1)w2 (k) + x3 (k))k   =  (x3 (k))k  ⊕  (x3 (k + 1)w3 (k) + x4 (k))k (x4 (k))k (y, (x4 (k))k )

   ∈ E3 . 

for some x1 , x2 , x3 , x4 ∈ 2 (N) and y ∈ C. Then A12 (n + 2, n + 1) = x1 (n + 2) and x2 (n + 1) = 1. We also have w2 (n)A12 (n + 1, n) = x1 (n + 2)w1 (n + 1) + x2 (n + 1). Therefore A12 (n + 2, n + 1) =

w2 (n) 1 A12 (n + 1, n) − . w1 (n + 1) w1 (n + 1)

Hence we have A12 (21) = − A12 (43) = −

1 , w1 (1)

A12 (32) = −

1 w2 (1) − , w1 (1)w1 (2) w1 (2)

w2 (2) 1 w2 (2)w2 (1) − − , ... w1 (3)w1 (2)w1 (1) w1 (3)w1 (2) w1 (3)

As w1 (n) > 0 and w2 (n) > 0, |A12 (n + 2, n + 1)| ≥

n

w2 (k) k=1 . n k=1 w1 (k)

w1 (n + 1) n w2 (k)

Since 1 < w1 (n) ≤ 4 and lim supn→∞ k=1 w1 (k) = ∞ by Lemma 3.1, we have lim supn→∞ |A12 (n + 2, n + 1)| = ∞. This contradicts the fact that ||A12 || < ∞. Therefore A22 = I. Hence A22 = O. Lemma 3.10. Suppose that A11 = O. Then A12 = O, A43 = O, A33 = O, A23 = O and A13 = O.

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Proof. Similar arguments before show that A12 = O, A43 = O. A33 is an idempotent and A33 = λI + N for some λ ∈ C and an upper triangular matrix N ∈ B(2 (N)) with zero diagonal. Thus A33 = O or A33 = I by Lemma 3.3. It is enough to show that A33 = I. On the contrary suppose that A33 = I. Then A23 (21) = −

1 , w2 (1)

A23 (n + 2, n + 1) =

w3 (n) 1 A23 (n + 1, n) − . w2 (n + 1) w2 (n + 1)

As in the proof of Lemma 3.9, we have lim supn→∞ |A23 (n + 2, n + 1)| = ∞. This contradicts the fact that ||A23 || < ∞. Therefore A33 = I. Hence A33 = O. The rest is similarly proved. Lemma 3.11. Suppose that A11 = O. Then A44 = O, A34 = O, A24 = O and A14 = O. Proof. Since the fourth column of operator matrix T = (Tij )ij has a diﬀerent form than the the other columns, we need to be careful to investigate. Since u = (0, 0, 0, 0) ⊕ (0, 0, 0, e1) ∈ E3 , we have     0 A14 e1  0   A24 e1     Vu=  0  ⊕  A34 e1  A44 e1 0     (x1 (k + 1)w1 (k) + x2 (k))k (x1 (k))k  (x2 (k))k   (x2 (k + 1)w2 (k) + x3 (k))k     =  (x3 (k))k  ⊕  (x3 (k + 1)w3 (k) + x4 (k))k  ∈ E3 . (x4 (k))k (y, (x4 (k))k ) for some x1 , x2 , x3 , x4 ∈ 2 (N) and y ∈ C. Then x1 = x2 = x3 = x4 = 0. Therefore A14 (k, 1) = A24 (k, 1) = A34 (k, 1) = 0 for any k ≥ 1. We also have A44 (k, 1) = 0 for any k ≥ 2. Since u = (0, 0, 0, en) ⊕ (0, 0, en , en+1 ) ∈ E3 , we have     A14 en A14 en+1  A24 en   A24 en+1     Vu=  A34 en  ⊕  A34 en+1  A44 en A44 en+1     (x1 (k))k (x1 (k + 1)w1 (k) + x2 (k))k  (x2 (k))k   (x2 (k + 1)w2 (k) + x3 (k)k     =  (x3 (k))k  ⊕  (x3 (k + 1)w3 (k) + x4 (k))k  ∈ E3 . (x4 (k))k (y, (x4 (k))k )) for some x1 , x2 , x3 , x4 ∈ 2 (N) and y ∈ C. Then A44 (k + 1, n + 1) = x4 (k) = A44 (k, n) for any k ≥ 1, n ≥ 1. Since A44 (1, 1) = y and A44 (k, 1) = 0 for k ≥ 2, A44 = yI + N for some y ∈ C and an upper triangular matrix N with zero diagonal. Since A44 is an idempotent,

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A44 = O or A44 = I. We shall show that A44 = I. On the contrary assume that A44 = I. Then x4 = A44 en = en . Moreover A34 (k, n + 1) = x3 (k + 1)w3 (k) + en (k) = A34 (k + 1, n)w3 (k) + en (k). This implies that A34 (en+1 ) = Bw3 A34 (en ) + en . Since A34 (e1 ) = 0, we have A34 (e2 ) = e1 , A34 (e3 ) = e2 , A34 (e4 ) = w3 (1)e1 + e3 , A34 (e5 ) = w3 (2)e2 + e4 , A34 (e6 ) = w3 (1)w3 (2)e1 + w3 (3)e3 + e5 , . . . . Therefore A34 =  0 w3 (1)w3 (2) 0 w3 (1)w3 (2)w3 (3) 0 1 0 w3 (1)  0 0 1 (2) 0 w (2)w (3) 0 0 w 3 3 3   0 0 0 1 0 w (3) 0 w (3)w 3 3 3 (4)   0 0 0 0 1 0 w (4) 0 3   0 0 0 0 0 1 0 w3 (5)  .. .. .. .. .. .. .. .. . . . . . . . .

.. .. .. .. .. .. .

        

In particular, we have A34 (1, 2n) =

n−1

w3 (k).

k=1

n Since limn→∞ k=1 w3 (k) = ∞ , limn→∞ A34 (1, 2n) = ∞. This contradicts that A34 is bounded. Therefore A44 = O. Moreover A34 (k, n + 1) = x3 (k + 1)w3 (k) + x4 (k) = x3 (k + 1)w3 (k) + A44 en = A34 (k + 1, n)w3 (k). Since A34 (k, 1) = 0, we have A34 = O. Similarly we have A24 = O and A14 = O. The proof of Theorem 3.2 will be completed by the following Lemma: Lemma 3.12. The system Sw,N of four subspaces is indecomposable. Proof. Let V ∈ End(Sw,N ) satisfy V 2 = V as in the beginning of the proof of Theorem 3.2. Then V = A ⊕ A and A11 = O or A11 = I by Lemma 3.6. If A11 = O, then A = O by the preceding lemmas so that V = O. If A11 = I, then (I − A)11 = 0. By replacing V by an idempotent I − V , the same argument implies I − A = O, so that V = I. This establishes that Sw,N is indecomposable.

4. Being exotic In this section we shall show that the indecomposable systems Sw,N constructed in the preceding section are exotic in the sense that Sw,N are not isomorphic to any closed operator system SA under any permutation of subspaces. We recall a necessary criterion in [EW].

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Definition (intersection diagram). Let S = (H; E1 , E2 , E3 , E4 ) be a system of four subspaces. The intersection diagram for a system S is an undirected graph ΓS = (Γ0S , Γ1S ) with the set of vertices Γ0S and the set of edges Γ1S deﬁned by Γ0S = {1, 2, 3, 4} and for i = j ∈ {1, 2, 3, 4} ◦j if and only if Ei ∩ Ej = 0.

◦i

Lemma 4.1. ([EW, Lemma 10.4]) Let S = ST,S = (H; E1 , E2 , E3 , E4 ) be a closed operator system. Then the intersection diagram ΓS for the system S contains ◦4

◦1

◦2

◦3 ,

that is, E4 ∩ E1 = 0, E1 ∩ E2 = 0 and E2 ∩ E3 = 0. In particular, then the intersection diagram ΓS is a connected graph. Proposition 4.2. The indecomposable systems Sw,N constructed in the preceding section are not isomorphic to any closed operator systems under any permutation of subspaces. Proof. It is clear that E4 ∩ E1 = 0, E1 ∩ E2 = 0 and E2 ∩ E4 = 0. Since (e1 , 0, . . . , 0) ⊕ (0, 0, . . . , 0) ∈ E1 ∩ E3 , we have E1 ∩ E3 = 0. Because (0, 0, . . . , 0) ⊕ (0, 0, . . . , e1 ) ∈ E2 ∩E3 , we have E2 ∩E3 = 0. By Lemma 3.1, there exists a non-zero vector x1 ∈ K = 2 (N) with Bw1 x1 = x1 . Then (x1 , 0, . . . , 0, 0) ⊕ (x1 , 0, . . . , 0, 0) ∈ E3 ∩ E4 , so that E3 ∩ E4 = 0. Therefore the vertex 3 is not connected to any other vertices 1, 2, 4. Thus the intersection diagram ΓSw,N is not a connected graph. This implies that Sw,N is not isomorphic to any closed operator system under any permutation of subspaces.

5. Defect computation We shall compute the defect of the indecomposable systems Sw,N constructed in section 3. Lemma 5.1. For ﬁxed j ∈ N and b ∈ 2 (N), consider an equation Bwj u + b = u for unknown sequence u ∈ 2 (N). Suppose that there exists a polynomial p(t) of degree r with positive coeﬃcients such that |b(n + 1)| ≤ p(n)( 34 )n for n ∈ N. For any c ∈ C, put u(1) = c and let n c b(m) n − for n ∈ N. k=1 wj (k) k=m wj (k) m=1

u(n + 1) = n

Then there exists a polynomial q(t) of degree r + 1 such that u := (u(n))n satisﬁes |u(n + 1)| ≤ q(n)( 34 )n for any n. Moreover u is in 2 (N) and a solution of the equation Bwj u + b = u. Conversely any solution u has this form. Proof. The equation Bwj u + b = u implies that wj (n)u(n + 1) + b(n) = u(n) for n ∈ N.

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Hence u(n + 1) =

u(n) wj (n)

−

b(n) wj (n) .

IEOT

Therefore any solution u has the desired form:

n c b(m) n − for n ∈ N. w (k) j k=1 k=m wj (k) m=1

u(n + 1) = n Since

4 3

≤ wj (n) ≤ 4

n 3 3 3 3 n p(m − 1)( )m−1 ( )n−m+1 ≤ q(n)( )n |u(n + 1)| ≤ |c|( ) + 4 4 4 4 m=1

for some polynomial q(t) of degree r + 1. It is easy to see that u satisﬁes the equation and is in 2 (N). Proposition 5.2. For any natural number N the indecomposable systems Sw,N have the defect ρ(Sw,N ) = 2N3+1 . Proof. We need to compute dim(Ei ∩ Ej ) and dim((Ei + Ej )⊥ ). It is obvious that dim(Ei ∩ Ej ) = 0 and dim((Ei + Ej )⊥ ) = 0 for any i, j = 1, 2, 4 with i = j. We consider E2 + E3 . Since E2 = 0 ⊕ K and E3 ⊃ {(x ⊕ T x); x ∈ K}, E2 + E3 ⊃ H. Thus dim((E2 + E3 )⊥ ) = 0. Next we investigate E2 ∩ E3 . We see that {(0, 0, . . . , 0, 0) ⊕ (0, 0, . . . , 0, αe1 ); α ∈ C} ⊂ E2 ∩ E3 . Conversely take any         0 x1 Bw1 x1 + x2 z1   ..   ..   ..   ..   .   .   .   . =  ∈ E2 ∩ E3 . ⊕  ⊕  0   zN   xN   BwN xN + y  Sy + αe1 zN +1 y 0 Since y = 0, x1 = x2 = · · · = xN = 0, z1 = · · · = 0 and E2 ∩ E3 = {(0, 0, . . . , 0, 0) ⊕ (0, 0, . . . , 0, αe1 ); α ∈ C}. Therefore dim(E2 ∩ E3 ) = 1. Next we shall show that E1 + E3 = H. Since E3 ⊃ graph T , E1 +E3 ⊃ 0⊕Im T . And Im T = (L, L, . . . , L, Im S), because Bwk is onto. Considering one dimensional perturbation by {(0, 0, . . . , 0, 0) ⊕ (0, 0, . . . , 0, αe1 ); α ∈ C}, E1 + E3 = H, so that dim((E1 + E3 )⊥ ) = 0. Consider E1 ∩ E3 . Take any         0 x1 Bw1 x1 + x2 x1   ..   ..   ..   ..   .   .   .   . ⊕ = ⊕  ∈ E1 ∩ E3 .   xN   0   xN   BwN xN + y  y 0 y Sy + αe1 Then y = 0, α = 0. Since BwN xN = 0, xN = (xN (1), 0, 0, 0, . . .). From BwN −1 xN −1 + xN = 0, we have xN −1 = (xN −1 (1), − wxNN−1(1) (1) , 0, 0, 0, . . .). We continue in this way to obtain xN −1 (1) xN (1) , , 0, 0, . . .), xN −2 = (xN −2 (1), − w N −2 (1) wN −2 (2)wN −1 (1) xN −2 (1) xN −1 (1) , , − wN −3 (3)wxNN−2(1) xN −3 = (xN −3 (1), − w (2)wN −1 (1) , 0, 0, . . .), N −3 (1) wN −3 (2)wN −2 (1)

x2 (1) x3 (1) x4 (1) . . . , and x1 = (x1 (1), (−1)1 w , (−1)2 w1 (2)w , (−1)3 w1 (3)w ,..., 1 (1) 2 (1) 2 (2)w3 (1)

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N (1) (−1)N −1 w1 (N −1)w2x(N −2)···wN −1 (1) , 0, 0, . . .) Conversely for any parameters x1 (1), x2 (1), . . . , xN (1) ∈ C, vectors x1 , x2 , . . . , xN with the above forms and y = 0, α = 0 give elements in E1 ∩ E3 . Therefore dim(E1 ∩ E3 ) = N. Next we investigate (E3 + E4 )⊥ . Since E4⊥ = {(−y, y) ∈ H; y ∈ K} and (graph T )⊥ = {(−T ∗z, z) ∈ H; z ∈ K}, we have

E3⊥ ∩ E4⊥ = {(−T ∗z, z) ∈ H; z ∈ K, T ∗ z = z, (z|(0, . . . , 0, e1 )) = 0}, ∗ Let z = (z1 , . . . , zN , w) ∈ K. Since Bw = Swk = Swk is a weighted shift, T ∗ z = z k implies that

(Sw1 z1 , z1 + Sw2 z2 , . . . , zN −1 + SwN zN , zN + S ∗ w) = (z1 , . . . , zN , w). From Sw1 z1 = z1 , we have z1 = 0. Then Sw2 z2 = z2 . Hence z2 = 0. We continue in this way to obtain z3 = · · · = zN = 0. Therefore 0 = (z|(0, . . . , 0, e1 )) = (w|e1 ). Furthermore S ∗ w = w. Hence w = 0. Thus z = 0. Hence E3⊥ ∩ E4⊥ = 0. Finally we investigate E3 ∩ E4 . Take any         x1 x1 Bw1 x1 + x2 x1   ..   ..   ..   ..   .   .   .   . =  ⊕ ⊕  ∈ E3 ∩ E4 .  xN   BwN xN + y   xN   xN  Sy + αe1 y y y Since Sy + αe1 = y, y = (α, α, α, . . . ), As y ∈ 2 (N), α = 0 and y = 0. Then BwN xN = xN . Hence wN (n)xN (n + 1) = xN (n) for n ∈ N. Therefore there exists a constant cN such that xN = cN (1,

1 1 1 , , , · · · ). wN (1) wN (2)wN (1) wN (3)wN (2)wN (1)

Then |xN (n+ 1)| = |cN | n 1wN (k) ≤ |cN |( 34 )n . Thus xN ∈ 2 (N). Apply Lemma 5 k=1 for the equations Bwj xj + xj+1 = xj for j = N − 1, N − 2, . . . , 1 step by step. There exist parameters cN −1 , cN −2 , . . . , c1 such that xj (1) = cj for j = N, N − 1, . . . , 1 and the other components xj (n) for n ≥ 2 are uniquely determined by these parameters. In fact, n cj x (m) nj+1 − for n ∈ N. k=1 wj (k) k=m wj (k) m=1

xj (n + 1) = n

Conversely any x1 , . . . , xN with this form gives an element of E3 ∩ E4 . Hence dim(E3 ∩ E4 ) = N . Therefore ρ(Sw,N ) = 2N3+1 . Finally we obtain the following theorem. Theorem 5.3. There exist exotic indecomposable systems of four subspaces with the defect 2n+1 3 (n ∈ N).

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Remark. We do not know whether there exist exotic indecomposable systems of four subspaces with the defect 2n 3 (n ∈ N). Moreover we do not know whether there exist exotic indecomposable systems of four subspaces with the negaitve defect. Let ST,S = (H; E1 , . . . , E4 ) be the closed operator system associated with ⊥ the densely deﬁned operators S and T . Then the orthogonal complement ST,S = ⊥ ⊥ (H; E1 , . . . , E4 ) is isomorphic to the closed operator system ST ∗ ,S ∗ . In fact, let ϕ : H → H be a bounded invertible operator deﬁned by ϕ(a, b) = (−b, a) for ⊥ → ST ∗ ,S ∗ is an isomorphism. Therefore the (a, b) ∈ K1 ⊕ K2 . Then ϕ : ST,S ⊥ orthogonal complement Sw,N is not isomorphic to any closed operator system associated with the densely deﬁned operators. But we do not know whether we can drop the condition of being densely deﬁned.

References S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967), 100–114. [EW] M. Enomoto and Y. Watatani, Relative position of four subspaces in a Hilbert space, Adv. Math. 201 (2006), 263–317. [GP] I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Coll. Math. Spc. Bolyai 5, Tihany (1970), 163–237. [HLR] D. W. Hadwin, W. E. Longstaﬀ and P. Rosenthal, Small transitive lattices, Proc. Amer. Math. Soc. 87 (1983), 121–124. [Ha] P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. [HRR] K. J. Harrison, H. Radjavi and P. Rosenthal, A transitive medial subspace lattice, Proc. Amer. Math. Soc. 28 (1971), 119–121. [JW] C. Jiang and Z. Wang, Strongly Irreducible Operators on Hilbert Space, Longman, 1998. [J] V. Jones, Index for subfactors, Inv. Math. 72 (1983), 1–25. [MS] Y. P. Moskaleva and Y. S. Samoilenko, Systems of n subspaces and representations of *-algebras generated by projections, preprint arXiv:math.OA/0603503. [N] L. A. Nazarova, Representations of a quadruple, Izv. AN. SSSR 31 (1967), 1361– 1377.

[B]

Masatoshi Enomoto College of Business Administration and Information Science, Koshien University, Takarazuka, Hyogo 665, Japan e-mail: [email protected] Yasuo Watatani Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan e-mail: [email protected] Submitted: June 19, 2006 Revised: December 26, 2006

Integr. equ. oper. theory 59 (2007), 165–172 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020165-8, published online June 27, 2007 DOI 10.1007/s00020-007-1521-1

Integral Equations and Operator Theory

Reflexivity for Subnormal Systems With Dominating Spectrum in Product Domains J¨org Eschmeier Abstract. The question whether every subnormal tuple S = (S1 , . . . , Sn ) on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain U = U1 × . . . × Un ⊂ Cn is reflexive. Mathematics Subject Classification (2000). Primary 47A13; Secondary 47A15, 47B20. Keywords. Subnormal systems, reflexivity, H ∞ -functional calculus.

1. Introduction Let H be a complex Hilbert space. For an arbitrary set S ⊂ L(H) of bounded linear operators on H, we denote by Lat(S) the lattice of all closed linear subspaces of H that are invariant under every operator S ∈ S. The set AlgLat(S) = {C ∈ L(H); Lat(C) ⊂ Lat(S)} is a unital subalgebra of L(H) which contains S and is closed in the weak operator topology (WOT). The family S is called reﬂexive if AlgLat(S) = WS , where WS is the smallest WOT-closed subalgebra of L(H) containing S.

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The notion of reﬂexivity was introduced by Sarason [14] who proved in 1966 that analytic Toeplitz operators on the Hardy space H 2 (D) and commuting families of normal operators are reﬂexive. In 1971 it was shown by Deddens [5] that every isometry on a Hilbert space is reﬂexive. All the previous single variable results were generalized in a paper of Olin and Thomson [13] from 1980 in which the Scott Brown technique was used to show that every subnormal operator on a Hilbert space is reﬂexive. The question whether, more generally, every subnormal system S = (S1 , . . . , Sn ) ∈ L(H)n , that is, every n-tuple S ∈ L(H)n that extends to a system N = (N1 , . . . , Nn ) ∈ L(K)n of commuting normal operators on a larger Hilbert space K, is reﬂexive, is one of the main open problems in this area of multivariable invariant subspace theory. The result of Deddens on the reﬂexivity of single isometries was extended to families of commuting isometries by Li and McCarthy [12] and Bercovici [2], and to spherical isometries by Didas [7]. Both, commuting tuples of isometries and spherical isometries, are examples of subnormal systems. To indicate some of the known reﬂexivity results for general subnormal systems, let us recall that a given compact set K ⊂ Cn is said to be dominating in an open set U ⊂ Cn if K ⊂ U and if the supremum-norms of any bounded analytic function f ∈ H ∞ (U ) on U and on U ∩ K satisfy

f U = f K∩U .

One way to prove the Olin and Thomson result on the reﬂexivity of single subnormal operators is to show that the spectrum of a pure subnormal operator S ∈ L(H) is dominating in a bounded open set U in C with simply connected components such that S possesses an isometric weak∗ -SOT continuous H ∞ -functional calculus on U . Then a canonical decomposition of the space H, together with the Riemann mapping theorem, allows the reduction of the general problem to the case of a subnormal contraction S ∈ L(H) of type C·0 with isometric H ∞ -functional calculus Φ : H ∞ (D) → L(H) over the open unit disc. In the multivariable case only parts of this programme have been realized so far. Let S = (S1 , . . . , Sn ) ∈ L(H)n be a subnormal tuple of pure subnormal operators Sν (1 ≤ ν ≤ n). It was shown in [8] and [9] that S is reﬂexive if its Taylor spectrum σ(S) is dominating in the open Euclidean unit ball Bn ⊂ Cn or in the open unit polydisc Dn ⊂ Cn . It was observed by Didas [7] that the Euclidean unit ball can be replaced by any strictly pseudoconvex bounded open set. In general, the question to decide for which classes of open sets U in Cn a result of the above type holds, seems to be diﬃcult. It is the purpose of the present note to show that the answer is positive for subnormal tuples S with pure components Sν and dominating spectrum in a bounded open polydomain U = U1 × . . . × Un in Cn .

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2. Preliminaries Let H be a complex Hilbert space, and let L(H) be the Banach algebra of all continuous linear operators on H. We regard L(H) as the norm-dual of the space C 1 (H) of all trace-class operators on H. Let T = (T1 , . . . , Tn ) ∈ L(H)n be a commuting tuple. Then the smallest weak∗ closed unital subalgebra AT of L(H) containing T1 . . . . , Tn is isometrically isomorphic to the norm-dual of the quotient space Q(T ) = C 1 (H)/⊥ AT . In this way AT becomes a dual algebra, that is, AT is the norm-dual of a suitable Banach space such that the multiplication in AT is separately weak∗ continuous. If A and B are dual algebras, then a dual algebra isomorphism ϕ : A → B is by deﬁnition an algebra homomorphism between A and B that is an isometric isomorphism and a weak∗ homeomorphism. For x, y ∈ H, let us denote by [x ⊗ y] ∈ Q(T ) the equivalence class of the rank-one operator H → H, ξ → ξ, y x. For a given real number r ≥ 1, the dual algebra AT is said to have property (A1 (r)) if, for every s > r and every element [L] ∈ Q(T ), there are vectors x, y ∈ H with [L] = [x ⊗ y] and x, y ≤ (s[L])1/2 . For a bounded open set U ⊂ Cn , the Banach algebra H ∞ (U ) of all bounded analytic functions on U is regarded as the norm-dual of the quotient space Q = L1 (U )/⊥ H ∞ (U ). Here we use that H ∞ (U ) is a weak∗ closed subspace of L∞ (U ) with respect to the duality L1 (U ), L∞ (U ) (formed with respect to the (2n)dimensional Lebesbue measure). A sequence (fk ) in H ∞ (U ) is weak∗ convergent to zero if and only if (fk ) is norm-bounded and converges to zero pointwise on U . Let K be a compact set in Cn . If there is a bounded open set U ⊂ Cn such that K is dominating in U in the sense explained in the introduction, then the union of all these open sets U is the largest bounded open set in which K is dominating. Let Φ : H ∞ (U ) → L(H) be a continuous algebra homomorphism over a bounded open set U ⊂ Cn . Then the map ˜ : H ∞ (U ∗ ) → L(H), Φ

f → Φ(f˜)∗

(f˜(z) = f (¯ z ))

is a continuous algebra homomorphism over the open set U ∗ = {¯ z ; z ∈ U }. The k

algebra homomorphism Φ is by deﬁnition of type C0· if (Φ(fk )) → 0 in the strong operator topology for every weak∗ zero sequence (fk ) in H ∞ (U ). We say that Φ ˜ is of type C0· . is of type C·0 if Φ For a commuting tuple T = (T1 , . . . , Tn ) ∈ L(H)n , we denote by σ(T ) its spectrum in the sense of J. L. Taylor. For the deﬁnition and basic properties of this notion of joint spectrum, we refer the reader to [10]. As usual we call a commuting tuple T pure if there is no non-zero reducing subspace M for T such that T |M is a commuting tuple of normal operators.

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3. Main result Let S = (S1 , . . . , Sn ) ∈ L(H)n be a subnormal tuple such that all components Sν (1 ≤ ν ≤ n) are pure subnormal operators on H. Then each component Sν of S possesses an isometric weak∗ continuous functional calculus Φν : H ∞ (Uν ) → L(H) of type C·0 over the largest bounded open set Uν ⊂ C in which the spectrum of Sν is dominating (Theorem 19.1.4 in [4]). Let us write each of the sets Uν as the disjoint union of its connected components (Cνk ; 0 ≤ k < Nν ) (ν = 1, . . . , n) Uν = with suitable Nν ∈ {1, 2, . . .} ∪ {∞}. Denote by eνk ∈ H ∞ (Uν ) (ν = 1, . . . , n, 0 ≤ k < Nν ) the characteristic functions of the sets Cνk . Since the maps Φν are contractive and since the polynomials are weak∗ dense in H ∞ (Uν ) (Theorem 19.1.2 in [4]), the operators Pνk = Φν (eνk ) are mutually commuting orthogonal projections on H such that the range spaces Hνk = Pνk H are reducing subspaces for Φ1 , . . . , Φn . Furthermore, for each ﬁxed ν = 1, . . . , n, the projections (Pνk )0≤k
ν=1

are reducing for Φ1 , . . . , Φn and yield an orthogonal decomposition H= Hi . i∈I

For a given function f ∈ H ∞ (Cνk ), we denote by f˜ ∈ H ∞ (Uν ) its trivial extension to the open set Uν . Since, as cited above, the polynomials are weak∗ dense in H ∞ (Uν ) and since the restriction mappings from H ∞ (Uν ) to H ∞ (Cνk ) are weak∗ continuous, it follows that the polynomials are weak∗ dense in each of the spaces H ∞ (Cνk ). Let us ﬁx an index tuple i = (i1 , . . . , in ) ∈ I. Then the mappings Φνi : H ∞ (Cν,iν ) → L(Hi ),

f → Φν (f˜)|Hi

(ν = 1, . . . , n)

are contractive weak∗ continuous algebra homomorphisms of type C·0 such that ImΦνi ⊂ A(Sν |Hi ) . Since the components of Uν are simply connected domains in

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C, the Riemann mapping theorem allows us to choose conformal mappings ϕν : Cν,iν → D

(ν = 1, . . . , n)

onto the open unit disc D in C. By composing the representations Φνi with the dual algebra isomorphisms H ∞ (D) → H ∞ (Cν,iν ),

f → f ◦ ϕν

(ν = 1, . . . , n),

we obtain contractive weak∗ continuous functional calculi Ψνi : H ∞ (D) → L(Hi ),

f → Φνi (f ◦ ϕν )

of type C·0 for the operators Sνi = Φνi (ϕν ) ∈ A(Sν |Hi ) . Since S ∈ L(H)n was supposed to be subnormal, every ﬁnite tuple of elements in AS is subnormal again (see the remarks following Theorem 1.8 in [8]). In particular, the tuple S (i) = (S1i , . . . , Sni ) ∈ L(Hi )n is subnormal and consists of C·0 -contractions. By a result of Apostol (Theorem 1.7 and Proposition 1.8 in [1]) the n-tuple S (i) possesses a contractive weak∗ continuous functional calculus Φ(i) : H ∞ (Dn ) → L(Hi ) n which is again of type C·0 . We deﬁne Ci = ν=1 Cν,iν ⊂ Cn and denote by ∞ ∞ n ρi : H (Ci ) → H (D ) the dual algebra isomorphism acting as −1 ρi (f )(z1 , . . . , zn ) = f ϕ−1 1 (z1 ), . . . , ϕn (zn ) . Then the composition ρi

Φ(i)

Φi : H ∞ (Ci ) −→ H ∞ (Dn ) −→ L(Hi ) is a contractive weak∗ continuous functional calculus of type C·0 for the subnormal n-tuple S|Hi . To check this, the reader should observe that Ψνi (f ) = Φ(i) (f ◦ πν )

(ν = 1, . . . , n, f ∈ H ∞ (D)),

since both sides deﬁne weak∗ continuous representations of H ∞ (D) that agree on polynomials. Here πν : Dn → D denotes the projection onto the ν-th coordinate. Deﬁne U = U1 × . . . × Un . It is elementary to check that the map Φ : H ∞ (U ) → L(H), Φ(f ) = Φi (f |Ci ) i∈I ∗

deﬁnes a contractive weak continuous functional calculus of type C·0 for S. Thus we have proved the following result.

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Theorem 3.1. Let S = (S1 , . . . , Sn ) ∈ L(H)n be a subnormal tuple of pure subnormal operators Sν . For ν = 1, . . . , n, let Uν ⊂ C be the largest bounded open set in C such that σ(Sν ) is dominating in Uν . Then S possesses a contractive weak∗ continuous functional calculus Φ : H ∞ (U ) → L(H) of type C·0 over the product set U = U1 × . . . × Un . Let us suppose in addition that the Taylor spectrum σ(S) of S is dominating in U . Then the functional calculus Φ constructed in Theorem 3.1 is isometric. For completeness sake we indicate the well-known argument. It suﬃces to show that the spectral inclusion f (σ(S) ∩ U ) ⊂ σ(Φ(f )) holds for all functions f ∈ H ∞ (U ). Indeed, in this case, we obtain that Φ(f ) ≥ sup{|f (z)|; z ∈ σ(S) ∩ U } = f U for every function f ∈ H ∞ (U ). To prove the above spectral inclusion property, ﬁx a point w ∈ U such that f (w) ∈ σ(Φ(f )). It is well known that, since U is a polydomain, there are functions f1 , . . . , fn ∈ H ∞ (U ) such that f (z) − f (w) =

n

(zν − wν )fν (z)

(z ∈ U ).

ν=1

By applying the functional calculus Φ to both sides, one ﬁnds that Φ(f ) − f (w) =

n

(Sν − wν )Φ(fν ).

ν=1

Since the left-hand side is invertible, there are operators R1 , . . . , Rn ∈ L(H) in the commutant of S which satisfy the identity I=

n

(Sν − wν )Rν .

ν=1

Since the Taylor spectrum σ(S) of S is contained in its commutant spectrum (see [10], Lemma 2.2.4), it follows that w ∈ σ(S). Lemma 3.2. Let S = (S1 , . . . , Sn ) ∈ L(H)n be a subnormal tuple consisting of pure subnormal operators Sν (1 ≤ ν ≤ n). Suppose that the functional calculus Φ : H ∞ (U ) → L(H) of S described in Theorem 3.1 is isometric. Then S is reflexive. Proof. We use the notations from the proof of Theorem 3.1. Since by hypothesis the representation Φi (f |Ci ) Φ : H ∞ (U ) → L(H), Φ(f ) = i∈I

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is isometric, all the functional calculi Φi : H ∞ (Ci ) → L(Hi ) are isometric. But then the subnormal tuples S (i) ∈ L(Hi )n (i ∈ I) are completely non-unitary and possess the isometric weak∗ continuous functional calculi Φ(i) : H ∞ (Dn ) → L(Hi ). As an application of Corollary 4.5 in [9] the subnormal tuples S (i) are reﬂexive, and according to Theorem 3.5 in the same paper, the dual algebras AS (i) generated by these tuples satisfy the factorization property (A1 (16)). In particular, these dual algebras are WOT-closed and the weak∗ and weak operator topologies coincide on AS (i) (i ∈ I) (see Proposition 2.09 in [3]). Since the representations Φi and Φ(i) are weak∗ continuous and isometric, we obtain that A(S|Hi ) ⊂ ImΦi = ImΦ(i) = AS (i) = Alg Lat(S (i) ) for every i ∈ I. Fix an operator C ∈ Alg Lat(S). Then it follows that C|Hi ∈ Alg Lat(S|Hi ) ⊂ Alg Lat(AS (i) ) = AS (i) for all index tuples i ∈ I. Hence Alg Lat(S) is contained in the direct sum i∈I AS (i) which by deﬁnition consists of all operators A ∈ L(H) such that Hi ∈ Lat(A) and A|Hi ∈ AS (i) for all i ∈ I. Since this direct sum inherits property (A1 (16)) (Proposition 2.055 in [3]), a standard argument (Proposition 2.5 in [11]) implies that S is reﬂexive and thus completes the proof. Specializing to the case of dominating spectrum, we obtain the following consequence. Corollary 3.3. Let S = (S1 , . . . , Sn ) ∈ L(H)n be a subnormal tuple consisting of pure subnormal operators Sν . If σ(S) is dominating in V = V1 × . . . × Vn for some bounded open sets Vν ⊂ C, then S is reflexive. Proof. As before, let us denote by U = U1 × . . . × Un the cartesian product of the largest bounded open sets Uν ⊂ C in which the spectra σ(Sν ) are dominating. Since σ(Sν ) = πν (σ(S)), an elementary argument shows that the sets σ(Sν ) are dominating in Vν . In particular, it follows that V ⊂ U. Fix a function f ∈ H ∞ (U ) and a point z = (z1 , . . . , zn ) ∈ U. Let ν ∈ {1, . . . , n} be arbitrary. We claim that there is a sequence (zν,j )j≥1 in Vν such that |f (z1 , . . . , zν,j , . . . , zn )| > |f (z)| −

1 j

(j ≥ 1).

Indeed, since σ(Sν ) is dominating in Uν , there is certainly such a sequence in Uν ∩ σ(Sν ) and since σ(Sν ) ⊂ V ν , we can also ﬁnd such a sequence in Vν . By

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applying this argument repeatedly, one obtains that |f (z)| ≤ f V for arbitrary z ∈ U. But then f U = f V = f V ∩σ(S) ≤ f U∩σ(S) . Hence σ(S) is dominating in U , and the assertion follows from Lemma 3.2.

References [1] C. Apostol, Functional calculus and invariant subspaces, J. Operator Theory 4 (1980), 159–190. [2] H. Bercovici, A factorization theorem with applications to invariant subspaces and the reflexivity of isometries, Math. Res. Lett. 1 (1994), 511–518. [3] H. Bercovici, C. Foias and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Series in Math., vol. 56, Amer. Math. Soc., Providence, RI 1985. [4] G. Dales, P. Aiena, J. Eschmeier, K. Laursen and G. Willis, Introduction to Banach algebras, operators, and harmonic analysis, London Mathematical Society, Student Texts, 57, Cambridge University Press, Cambridge 2003. [5] J.A. Deddens, Every isometry is reflexive, Proc. AMS 28 (1971), 509–512. [6] M. Didas, Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets, Dissertationes Math. 425, 2004. [7] M. Didas, Spherical isometries are reflexive, Integral Equations Operator Theory 52 (2005), 599–604. [8] J. Eschmeier, Algebras of subnormal operators on the unit ball, J. Operator Theory 42 (1999), 37–76. [9] J. Eschmeier, Algebras of subnormal operators on the unit polydisc, In: Recent Progress in Functional Analysis (eds. K. Bierstedt, J. Bonet, M. Maestre, J. Schmets), pp. 159–171, North-Holland, Amsterdam 2001. [10] J. Eschmeier and M. Putinar, Spectral decompositions and analytic sheaves, LMS Monograph Series, Vol. 10, Clarendon Press, Oxford 1996. [11] D. Hadwin and E. Nordgren, Subalgebras of reflexive algebras, J. Operator Theory 7 (1982), 3–23. [12] W.S. Li and J. McCarthy, Reflexivity of isometries, Studia Math. 124 (1997), 101– 105. [13] R. Olin and J.E. Thomson, Algebras of subnormal operators, J. Funct. Anal. 37 (1980), 271–301. [14] D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. J¨ org Eschmeier Fachrichtung Mathematik, Universit¨ at des Saarlandes, Postfach 15 11 50, D-66041 Saarbr¨ ucken, Germany e-mail: [email protected] Submitted: April 12, 2007 Revised: May 3, 2007

Integr. equ. oper. theory 59 (2007), 173–187 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020173-15, published online June 27, 2007 DOI 10.1007/s00020-007-1519-8

Integral Equations and Operator Theory

Kth Roots of p-Hyponormal Operators are Subscalar Operators of Order 4k Eungil Ko Abstract. In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T k is p-hyponormal and σ(T ) has nonempty interior in the plane, then T has a nontrivial invariant subspace. Mathematics Subject Classification (2000). 47B20, 47B38. Keywords. p-hyponormality, subscalarity, the property (β), invariant subspace.

1. Introduction Let H and K be separable, complex Hilbert spaces and L(H, K) denote the space of all bounded linear operators from H to K. If H = K, we write L(H) in place of L(H, K). If T ∈ L(H), we write σ(T ), σap (T ), and σe (T ) for the spectrum, the approximate point spectrum, and the essential spectrum of T , respectively. ¯ denote the closure of Let D be an open disc in C the complex plane. Let D ¯ denote the space of continuous complex valued functions D in C. And let C m (D) ¯ The notation f ∞ will be used to denote the sup norm of a function f on on D. ¯ Then deﬁne for f ∈ C m (D), ¯ D f ≡ f ∞ + fx ∞ + fy ∞ + · · · + fyy···y ∞ where for example fx denotes the partial with respect to the coordinate variable ¯ into a Banach space. Note that the pointwise product of x. This makes C m (D) ¯ is again in C m (D) ¯ (in fact C m (D) ¯ with this norm is a two functions in C m (D) topological algebra). The work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00461).

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A bounded linear operator S on H is called scalar of order m if for some open disc D in C there exists a map ¯ −→ L(H) Φ : C m (D) such that 1) Φ is an algebra homomorphism, and 2) Φ is continuous when the ¯ and the operator norm is placed on L(H), and 3) above norm is used on C m (D) Φ(z) = S, where z stands for identity function on C, and 4) Φ(1) = I. The map Φ is called a spectral resolution for S. An operator is called subscalar if it is similar to the restriction of a scalar operator to a closed invariant subspace. Recall that an operator T is called p-hyponormal, 0 < p ≤ 1, if (T ∗ T )p ≥ ∗ p (T T ) where T ∗ is the adjoint of T . If p = 1, T is called hyponormal and if p = 12 , T is called semihyponormal. p-Hyponormal operators were introduced by Aluthge (see [1]). There is a vast literature concerning p-hyponormal operators. In particular, Aluthge proved in [1] that if T = U |T | (polar decomposition) is 1 p-hyponormal with 0 < p < 12 where |T | = (T ∗ T ) 2 and U is the appropriate partial isometry satisfying kerU = ker|T | = kerT and ker U ∗ = kerT ∗ , then T˜ is ˜ is hyponormal where T˜ = |T | 21 U |T | 21 . Note that T˜ is (p + 12 )-hyponormal and T˜ called the Aluthge transform of T and we will use this notation throughout this paper. We say that an operator T ∈ L(H) is a kth root of a p-hyponormal operator √ if T k is p-hyponormal for some positive integer k. We denote this class by ( k P H) where k ≥ 2. If T is p-hyponormal, it is known that T k is kp -hyponormal (see [2]). Hence T k is not p-hyponormal. On the other hand, if T is a nilpotent operator of order k, then T k is p-hyponormal, but T is not p-hyponormal. Also, let T be a p-hyponormal operator. If 0 1H , A= T 0 then A is not p-hyponormal, but A2 = T ⊕ T is p-hyponormal. P.R. Halmos raised in [9] the following question. (P) If T ∈ L(H) and T 2 has a nontrivial invariant subspace, must T have a nontrivial invariant subspace too? In this paper, we will consider the special case of the question raised by Halmos. In particular, we show that if T k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T k is p-hyponormal and σ(T ) has interior in the plane, then T has a nontrivial invariant subspace.

2. Preliminaries An operator T ∈ L(H) is said to satisfy the single valued extension property if for any open set U in C, the function T − z : O(U, H) −→ O(U, H)

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deﬁned by the obvious pointwise multiplication is one-to-one where O(U, H) denote the Fr´echet space of H-valued analytic functions on U with respect to uniform topology. If T has the single valued extension property, then for any x ∈ H there exists a unique maximal open set ρT (x)(⊃ ρ(T ), the resolvent set) and a unique H-valued analytic function f deﬁned in ρT (x) such that (T − z)f (z) = x,

z ∈ ρT (x).

Moreover, if F ⊂ C is a closed set and σT (x) = C\ρT (x), then HT (F ) = {x ∈ H : σT (x) ⊂ F } is a linear subspace (not necessarily closed) of H and obviously HT (F ) = HT (F ∩ σ(T )). An operator T ∈ L(H) is said to satisfy the property (β) if for every open subset G of C and every sequence fn : G → H on H-valued analytic function such that (T − z)fn (z) converges uniformly to 0 in norm on compact subsets of G, fn (z) converges uniformly to 0 in norm on compact subsets of G. Let z be the coordinate in C and let dµ(z) denote the planar Lebesgue measure. Fix a separable, complex Hilbert space H and a bounded (connected) open subset U of C. We shall denote by L2 (U, H) the Hilbert space of measurable functions f : U → H, such that 1 f 2,U = { f (z)2 dµ(z)} 2 < ∞. U

The space of functions f ∈ L2 (U, H) which are analytic functions in U (i.e., ¯ = 0) is denoted by ∂f A2 (U, H) = L2 (U, H) ∩ O(U, H). A2 (U, H) is called the Bergman space for U . We will use the following version of Green’s formula for the plane, also known ¯ , H) in the exactly the same way as as the Cauchy-Pompeiu formula. Deﬁne C p (U p ¯ C (U ) except that the functions in the space are now H-valued. Cauchy-Pompeiu formula 2.1. Let D be an open disc in the plane, let z ∈ D and ¯ H). Then f ∈ C 2 (D, 1 f (ζ) ¯ ∗ (− 1 ) f (z) = dζ + ∂f 2πi ∂D ζ − z πz where ∗ denotes the convolution product.

Remark 2.2. The function

f (ζ) dζ ζ −z ∂D appearing in Cauchy-Pompeiu formula is analytic in D 2and extends continuously ¯ H). ¯ as can be seen by examining the term. So, g ∈ A (D, H) for f ∈ C 2 (D, to D D ¯ H) and F ∈ Remark 2.3. Let D be an open disc in C. Then for φ ∈ C ∞ (D, ∞ ¯ C (D, L(H)) we have the following relation. ¯ ∗ (− 1 ). ¯ ) · φ) ∗ (− 1 ) = F φ − F (∂φ) ((∂F πz πz g(z) =

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Let us deﬁne now a special Sobolev type space, called W m (D, H) where, as ¯ H), let before, D is a bounded open disc in C. For f ∈ C m (D, f 2W m =

m

∂¯i f 22,D

i=0

¯ H) under this norm. Note that Then let W m (D, H) be the completion of C m (D, m W (D, H) is a Hilbert space contained continuously in L2 (D, H). We next discuss the fact concerning the multiplication operator by z on W m (D, H). The linear operator M of multiplication by z on W m (D, H) is continuous and it has a spectral resolution, deﬁned by the relation ¯ −→ L(W m (D, H)), ΦM : C m (D)

ΦM (f ) = Mf .

Therefore, M is a scalar operator of order m. Moreover, let V : W m (D, H) → 2 ⊕m 0 L (D, H) be the operator deﬁned by ¯ . . . , ∂¯m f ). V (f ) = (f, ∂f, Since V f 2 = f 2W m =

m

∂¯i f 22,D ,

i=0

an operator V is an isometry such that V M = (⊕m 0 Nz )V , where Nz is the multiplication operator on L2 (D, H). Since ⊕m N is normal, M is a subnormal operator. z 0

3. Subscalarity In this section we show that every kth root of a p-hyponormal operator has a scalar extension. We begin with the following theorem. Theorem 3.1. For every bounded disk D in C there is a constant CD , such that for an arbitrary operator A ∈ L(H) and f ∈ W 2k (D, H) we have (I − P )f 2,D ≤ CD

2k

(A − z k )∗ ∂¯i f 2,D

i=k

where P denotes the orthogonal projection of L2 (D, H) onto the Bergman space A2 (D, H). ¯ H) such that si ≡ 1 on D − D for i = 1, 2, . . . , k. Let Proof. Let si be in C ∞ (D, ∞ ¯ fn ∈ C (D, H) be a sequence which approximates f in the norm W 2k . Then for

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a ﬁxed n we have 1 ∂¯k [fn + (A − z k )∗ ∂¯k fn ] k! k 1 k k ¯ = ∂ fn + ∂¯j (A − z k )∗ ∂¯2k−j fn k! j=0 j =

k−1 1 k ∂¯j (A − z k )∗ ∂¯2k−j fn . j k! j=0

(3.1)

By the Cauchy-Pompeiu formula and (3.1), we get

=

1 ∂¯k−1 [fn + (A − z k )∗ ∂¯k fn ] k! 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−1 [fn (ζ) + k! 1 dζ 2πi ∂D ζ −z k−1 1 k s1 +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ). k! j=0 j πz

Set 1 g1,n (z) = 2πi

∂¯k−1 [fn (ζ) +

∂D

(3.2)

1 k! (A

− ζ k )∗ ∂¯k fn (ζ)] dζ. ζ −z

Then g1,n ∈ A2 (D, H) by Remark 2.2. Thus

=

1 ∂¯k−1 [fn + (A − z k )∗ ∂¯k fn ] k! k−1 1 k s1 g1,n + [ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ). j k! πz j=0

In order to complete our proof, we need the following claims. Claim I. For t = 1, . . . , k we have the following relation.

=

1 ∂¯k−t [fn + (A − z k )∗ ∂¯k fn ] k! st st s2 gt,n + gt−1,n ∗ (− ) + · · · · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) πz πz πz k−1 1 st s1 k +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ) j k! j=0 πz πz

where gr,n (z) =

1 2πi

∂¯k−r [fn (ζ) + ∂D

if r > 0 and gr,n (z) = 0 if r ≤ 0.

1 k! (A

− ζ k )∗ ∂¯k fn (ζ)] dζ ζ−z

(3.3)

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Proof of Claim I. We prove this claim by induction. If t = 1 it is true from (3.3). Assume it holds when t = r − 1. If we apply the Cauchy-Pompeiu formula when t = r, then 1 ∂¯k−r [fn + (A − z k )∗ ∂¯k fn ] k! 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−r [fn (ζ) + k! 1 dζ = 2πi ∂D ζ −z 1 sr +∂¯k−(r−1) [fn + (A − z k )∗ ∂¯k fn ] ∗ (− ). k! πz Set 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−r [fn (ζ) + k! 1 gr,n (z) = dζ. 2πi ∂D ζ −z Then by induction assumption we have 1 ∂¯k−r [fn + (A − z k )∗ ∂¯k fn ] k! sr−1 sr−1 s2 ) + · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) +gr,n + {gr−1,n + gr−2,n ∗ (− πz πz πz k−1 1 k sr−1 sr s1 +[ )} ∗ (− ). ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− k! j=0 j πz πz πz So we complete the proof of Claim I.

By Claim I, we get 1 fn + (A − z k )∗ ∂¯k fn k! sk sk s2 = gk,n + gk−1,n ∗ (− ) + · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) πz πz πz k−1 1 k sk s1 +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ). j k! j=0 πz πz sk s2 sk Set gn = gk,n + gk−1,n ∗ (− πz ) + · · · · · · + g1,n ∗ (− πz ) ∗ · · · ∗ (− πz ). Then gn ∈ 2 A (D, H). Hence we have 1 fn + (A − z k )∗ ∂¯k fn k! k−1 sk 1 k s1 = gn + [ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ). j k! πz πz j=0

Claim II. For j = 0, 1, . . . , k − 1, the following relation holds. sk s1 [∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ) πz πz j sk sj−t+1 j t ) ∗ · · · ∗ (− )]. = [(−1) (A − z k )∗ ∂¯2k−(j−t) fn ∗ (− t πz πz t=0

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Proof of Claim II. We prove this claim by induction. If j = 0, it is trivial. Assume that Claim II holds when j = r. Then by applications of Remark 2.3, we obtain the following; sk s1 ) ∗ · · · ∗ (− ) πz πz s1 r k ∗ ¯2k−r−1 r k ∗ ¯2k−r ¯ ¯ [∂ (A − z ) ∂ fn − ∂ (A − z ) ∂ fn ∗ (− )] πz sk s2 ∗(− ) ∗ · · · ∗ (− ) πz πz r sk sr−t+2 r ) ∗ · · · ∗ (− )] [(−1)t (A − z k )∗ ∂¯2k−(r+1−t) fn ∗ (− t πz πz t=0 r sk sr−t+1 r ) ∗ · · · ∗ (− )] − [(−1)t (A − z k )∗ ∂¯2k−(r−t) fn ∗ (− t πz πz t=0 r+1 sk sr−t+2 r+1 ) ∗ · · · ∗ (− )]. [(−1)t (A − z k )∗ ∂¯2k−(r+1−t) fn ∗ (− t πz πz t=0 [∂¯r+1 (A − z k )∗ ∂¯2k−r−1 fn ] ∗ (−

=

=

=

So we complete the proof of Claim II. By Claim II, we get fn − gn

1 (A − z k )∗ ∂¯k fn k! k−1 j 1 k j + (−1)t (A − z k )∗ ∂¯2k−(j−t) fn j t k! j=0 t=0

= −

∗(−

sk sj−t+1 ) ∗ · · · ∗ (− ). πz πz

Taking the norm, we get fn − gn 2,D

≤

1 (A − z k )∗ ∂¯k fn 2,D k! k−1 j 1 k j + (A − z k )∗ ∂¯2k−(j−t) fn 2,D j t k! j=0 t=0

sj−t+1 sk (− ) ∗ · · · ∗ (− )2,D . πz πz Let CD be the maximum among coeﬃcients of {(A − z k )∗ ∂¯i fn 2,D }2k i=k . Then f − g2,D

≤ f − fn 2,D + fn − gn 2,D ≤ f − fn 2,D + CD

2k i=k

(A − z k )∗ ∂¯i fn 2,D .

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By passing to the limit we conclude f − P f 2,D ≤ CD

2k

(A − z k )∗ ∂¯i f 2,D .

i=k

So we complete our proof.

Corollary 3.2. Let T ∈ L(H) be a semihyponormal operator. If {fn } is a sequence in W 2k (D, H) such that limn→∞ (T − z k )fn W 2k = 0 for all z ∈ D, then there exists a sequence of analytic functions {gn } such that lim fn − gn 2,D = 0.

n→∞

Proof. From [10, Lemma 4.3] and the hypothesis we get that lim (T − z k )∗ ∂¯j fn 2,D = 0

n→∞

for j = 0, 1, 2, . . . , 2k. Hence Theorem 3.1 implies that lim (I − P )fn 2,D = 0.

n→∞

Set gn = P fn . Then {gn } is the desired sequence.

Recall that for bounded open disc D containing 0 in C the Bergman operator for D is the operator S deﬁned on A2 (D, H) by (Sf )(z) = zf (z). Then the Bergman operator for D has the following property. Lemma 3.3. ([7, Corollary 10.7]) If S is the Bergman operator for the bounded open disc D containing 0, then S is bounded below. Next we generalize [11, Lemma 3.4]. Lemma 3.4. Let T be any kth roots of a semihyponormal operator. Then for a bounded disk D which contains σ(T ), the operator V : H → H(D) defined by V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 2k (D, H)) is one-to-one and has closed range, where H(D) = W 2k (D, H)/(T − z)W 2k (D, H) and 1 ⊗ h denotes the constant function sending any z ∈ D to h. Proof. If hn ∈ H and fn ∈ W 2k (D, H) are sequences such that lim (T − z)fn + 1 ⊗ hn W 2k = 0,

n→∞

(3.4)

it suﬃces to show that limn→∞ hn = 0. Now by the deﬁnition of the norm of Sobolev space, (3.4) implies lim (T − z)∂¯j fn 2,D = 0

n→∞

for j = 1, . . . , 2k. From (3.5), we get lim (T k − z k )∂¯j fn 2,D = 0

→∞

(3.5)

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for j = 1, . . . , 2k. Since T k is semihyponormal, by [11, Lemma 4.3] lim (T k − z k )∗ ∂¯j fn 2,D = 0.

n→∞

(3.6)

for j = 1, . . . , 2k. Then by Theorem 3.1, we have lim (I − P )fn 2,D = 0

n→∞

(3.7)

where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). By (3.4) and (3.7), we have lim (T − z)P fn + 1 ⊗ hn 2,D = 0.

n→∞

Let Γ be a curve in D surrounding σ(T ). Then for z ∈ Γ lim P fn (z) + (T − z)−1 (1 ⊗ hn ) = 0

n→∞

uniformly. Hence, by Riesz-Dunford functional calculus, 1 P fn (z)dz + hn = 0. lim n→∞ 2πi Γ But since Γ P fn (z)dz = 0 by Cauchy’s theorem, limn→∞ hn = 0.

The next proposition is essential for the proof of our main theorem. √ Proposition 3.5. Let T be in ( k P H). Then for a bounded disk D which contains σ(T ) ∪ {0}, the operator V : H → H(D) defined by V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 4k (D, H)) is one-to-one and has closed range, where H(D) = W 4k (D, H)/(T − z)W 4k (D, H) and 1 ⊗ h denotes the constant function sending any z ∈ D to h. Proof. Let hn ∈ H and fn ∈ W 4k (D, H) be sequences such that lim (T − z)fn + 1 ⊗ hn W 4k = 0.

n→∞

(3.8)

Then by the deﬁnition of the norm of Sobolev space, the equation (3.8) implies (3.9) lim (T − z)∂¯i fn 2,D = 0 n→∞

for j = 1, 2, . . . , 4k. Hence from (3.9) we get lim (T k − z k )∂¯i fn 2,D = 0

n→∞

(3.10)

for j = 1, 2, . . . , 4k. a) If 12 ≤ p < 1, then T k is semihyponormal. Therefore, it is true from Lemma 3.4. b) If 0 < p < 12 , then T˜k is semihyponormal from [1]. Since T˜k |T k |1/2 = |T k |1/2 T k , from (3.10) we get 1 lim (T˜k − z k )∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.11)

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for j = 1, 2, . . . , 4k. By applications of [10, Lemma 4.3] and (3.11), we obtain 1 lim (T˜k − z k )∗ ∂¯i |T k | 2 fn 2,D = 0 (3.12) n→∞

for j = 1, 2, . . . , 4k. Then Theorem 3.1 and (3.12) imply 1

lim (I − P )∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.13)

for j = 1, 2, . . . , 2k where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). From (3.11) and (3.13) we get 1 (3.14) lim (T˜k − z k )P ∂¯i |T k | 2 fn 2,D = 0 n→∞

for j = 1, 2, . . . , 2k. Let T k = Uk |T k | be the polar decomposition of T k . Since Uk |T k |1/2 T˜k = T k Uk |T k |1/2 , from (3.14) we have 1 1 lim (T k − z k )Uk |T k | 2 P ∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.15)

for j = 1, 2, . . . , 2k. Since T k is p-hyponormal, it is known from [12] that T k has the property (β). Hence it is easy to show that 1 1 lim Uk |T k | 2 P ∂¯i |T k | 2 fn 2,D = 0

n→∞

for j = 1, 2, . . . , 2k. Since T k = Uk |T k |, from (3.16) 1 lim T˜k P ∂¯i |T k | 2 fn 2,D = 0 n→∞

(3.16)

(3.17)

for j = 1, 2, . . . , 2k. Then (3.14) and (3.17) imply 1 lim z k P ∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.18)

for j = 1, 2, . . . , 2k. By applications of Lemma 3.3, there exists a constant c > 0 such that 1 1 (3.19) z k P ∂¯i |T k | 2 fn 2,D ≥ cP ∂¯i |T k | 2 fn 2,D for j = 1, 2, . . . , 2k. Then from (3.18) and (3.19) we obtain 1

lim P ∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.20)

for j = 1, 2, . . . , 2k. Hence it follows from (3.13) and (3.20) that 1

lim ∂¯i |T k | 2 fn 2,D = 0

n→∞

(3.21)

for j = 1, 2, . . . , 2k. T k = Uk |T k |, from (3.21) we get lim T k ∂¯i fn 2,D = 0

(3.22)

for j = 1, 2, . . . , 2k. By (3.10) and (3.22), we have lim z k ∂¯i fn 2,D = 0

(3.23)

n→∞

n→∞

for j = 1, 2, . . . , 2k. Now applying Theorem 3.1 with A = (0), we obtain lim (I − P )fn 2,D = 0.

n→∞

(3.24)

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By (3.8) and (3.24), we get lim (T − z)P fn + 1 ⊗ hn 2,D = 0.

n→∞

(3.25)

Let Γ be a curve in D surrounding σ(T ). Then for z ∈ Γ lim P fn (z) + (T − z)−1 (1 ⊗ hn ) = 0

n→∞

uniformly. Hence, by Riesz-Dunford functional calculus, 1 lim P fn (z)dz + hn = 0. n→∞ 2πi Γ But since Γ P fn (z)dz = 0 by Cauchy’s theorem, limn→∞ hn = 0.

Now we are ready to prove our main theorem. √ Theorem 3.6. An arbitrary operator T in ( k P H) is a subscalar operator of order 4k. Proof. Consider an arbitrary bounded open disk D in C which contains σ(T ) ∪ {0} and the quotient space H(D) = W 4k (D, H)/(T − z)W 4k (D, H) endowed with the Hilbert space norm. The class of a vector f or an operator A on ˜ Let M (= Mz ) be the multiplication H(D) will be denoted by f˜, respectively A. 4k operator by z on W (D, H). Then M is a scalar operator of order 4k and its spectral resolution is ¯ −→ L(W 4k (D, H)), Φ : C 4k (D)

Φ(f ) = Mf ,

where Mf is the multiplication operator with f . Since M commutes with T − z, ˜ on H(D) is still a scalar operator of order 4k, with Φ ˜ as a spectral resolution. M Let V be the operator V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 4k (D, H)), ˜V. from H into H(D), denoting by 1 ⊗ h the constant function h. Then V T = M Since V is one-to-one and has closed range by Proposition 3.5, T is subscalar of order 4k. √ Corollary 3.7. Let T ∈ ( k P H). If N is a nilpotent operator of order k, then αT , V T V ∗ , and T ⊕ N are subscalar operators of order 4k, where α ∈ C and V is an isometry. √ Proof. Since αT and V T V ∗ are in ( k P H), they are trivial from Theorem 3.6. Since (T ⊕ N )k = T k ⊕ 0 ∈ (P H), T ⊕ N is a subscalar operator from Theorem 3.6.

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√ Next we can easily observe from Theorem 3.6 that the restriction of T ∈ ( k P H) is also subscalar. √ Corollary 3.8. If T ∈ ( k P H), then T |M is a subscalar operator of order 4k, where M is a nontrivial invariant subspace for T . Corollary 3.9. Let T ∈ L(H) be a unilateral weighted shift with positive weight sequence {αn }∞ n=0 . If αn−k . . . αn−1 ≤ αn · · · αn+k−1 for n = k, k + 1, . . ., then T is a subscalar operator of order 4k. k Proof. Let {en }∞ n=0 be an orthonormal basis of a Hilbert space H. Since T en = ∗k αn · · · αn+k−1 en+k and T en = αn−1 · · · αn−k en−k , it is easy to calculate that T k is p-hyponormal for n = k, k + 1, . . .. Hence the proof follows from Theorem 3.6.

Recall that if U is a non-empty open set in C and if Ω ⊂ U has the property that supλ∈Ω |f (λ)| = supβ∈U |f (β)| for every function f in H ∞ (U ) (i.e. for all f bounded and analytic on U ), then Ω is said to be dominating for U . The following corollary is an extension of S. Brown’s beautiful theorem (see [5]). √ Corollary 3.10. Let T be in ( k P H). If σ(T ) has the property that there exists some non-empty open set U such that σ(T ) ∩ U is dominating for U , then T has a nontrivial invariant subspace. Proof. This follows from Theorem 3.6 and [8].

Recall that an operator T ∈ L(H) is said to be power regular if 1 limn→∞ T nx n exists for every x ∈ H. √ Corollary 3.11. If T is in ( k P H), it is power regular. Proof. It is known from Theorem 3.6 that every kth root of a p-hyponormal operator is similar to the restriction of a scalar operator to one of its invariant subspace. Since a scalar operator is power regular and the restriction of power regular operators to their invariant subspaces clearly remains power regular, every kth root of a p-hyponormal operator is power regular. √ Corollary 3.12. If T is in ( k P H), it satisfies the property (β). Hence it satisfies the single valued extension property. Proof. Since every scalar operator satisﬁes the property (β) and the property (β) is transmitted from an operator to its restriction to closed invariant subspaces, it follows from Theorem 3.6 that every kth root of a p-hyponormal operator satisﬁes the property (β). Hence it satisﬁes the single valued extension property.

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Recall that an X ∈ L(H, K) is called a quasiaﬃnity if it has trivial kernel and dense range. An operator A ∈ L(H) is said to be a quasiaﬃne transform of an operator T ∈ L(K) there exists a quasiaﬃnity X ∈ L(H, K) such that XA = T X. Furthermore, operators A and T are said to be quasisimilar if there are quasiaﬃnities X and Y such that XA = T X and AY = Y T . √ Corollary 3.13. Let A and T be in ( k P H). If they are quasisimilar, then σ(A) = σ(T ) and σe (A) = σe (T ). Proof. Since A and T satisfy the property (β) by Corollary 3.12, the proof follows from [14]. √ Proposition 3.14. If T is in ( k P H), then for any bounded open disk D containing σ(T ) ∪ {0} and any sequence fn ∈ W 4k (D, H), we have limn→∞ fn 2,D = 0 whenever limn→∞ (T − z)fn W 4k = 0. Proof. If limn→∞ (T − z)fn W 4k = 0 for any sequence fn ∈ W 4k (D, H), by applications of the proof in Proposition 3.5 we get (cf, (3.24)) lim (I − P )fn 2,D = 0.

n→∞

Hence we have lim (T − z)P fn 2,D = 0.

n→∞

Since T satisﬁes the property (β) by Corollary 3.12, it is easy to show that lim P fn 2,D = 0.

n→∞

Hence limn→∞ fn 2,D = 0.

The following corollary is the special case of Proposition 3.14. √ Corollary 3.15. If T is in ( k P H), then for any bounded open disk D the operator T − z : W 4k (D, H) −→ W 4k (D, H) is one-to-one.

√ Corollary 3.16. Let T1 and T3 be in ( k P H). Then T1 − z T2 : ⊕W 4k (D, H) −→ ⊕W 4k (D, H) A−z = 0 T3 − z is one-to-one. Proof. Let f = f1 ⊕ f2 ∈ ⊕W 4k (D, H) be such that (A − z)f = 0. Then f1 (T1 − z)f1 + T2 f2 0 T1 − z T2 = = . 0 T3 − z f2 (T3 − z)f2 0 So we have (T1 − z)f1 + T2 f2 = 0 and

(3.26)

(T3 − z)f2 = 0.

(3.27)

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By Corollary 3.15 and (3.27), f2 = 0. Hence from (3.26) we have (T1 − z)f1 = 0. Again by Corollary 3.15, f1 = 0. Thus f = 0. √ Theorem 3.17. Let T be in ( k P H). If 0 1H A= , T 0 then A is not p-hyponormal, but is subscalar of order 8k. Proof. Since (A∗ A)p −(AA∗ )p = {(T ∗ T )p −1H }⊕{1H −(T T ∗)p } is not positive, A is not p-hyponormal. But since T k is p-hyponormal, A2k = T k ⊕ T k is p-hyponormal. By Theorem 3.6, A is a subscalar operator of order 8k. Since A2k is p-hyponormal, we remark that if σ(A) is rich, then A2k has a nontrivial invariant subspace from [12]. Next, we will consider the spcial case of the question raised by Halmos. Corollary 3.18. With the notation of Theorem 3.17, if σ(A) has the property that there exists some non-empty open set U such that σ(A) ∩ U is dominating for U , then A has a nontrivial invariant subspace. Proof. Since A is subscalar from Theorem 3.17, it follows from [8] that A has a nontrivial invariant subspace. Corollary 3.19. With the notation of Theorem 3.17, if σ(A) has the property that there exists some non-empty open set U such that σ(A) ∩ U is dominating for U , 0 Tm has a nontrivial invariant subspace for any positive integer then T m+1 0 m. Proof. Since

Tm = A T m+1 0 for any positive integer m and A has a nontrivial invariant subspace by Corollary 3.18, we get that A2m+1 has a nontrivial invariant subspace. So we complete the proof. 2m+1

0

References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Int. Eq. Op. Th. 13(1990), 307–315. [2] A. Aluthge and D. Wang, Powers of p-hyponormal operators, J. Inequality Appl. 3(1999), 279–284. [3] S. Brown and E. Ko, Operators of Putinar type, Op. Th. Adv. Appl. 104(1998), Birkh¨ auser Verlag, Boston, 49–57. [4] P. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44(1997), 345–353. [5] S. Brown, Hyponormal operators with thick spectrum have invariant subspaces, Ann. of Math. 125(1987), 93–103.

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[6] I. Colojoar˘ a and C. Foia¸s, Theory of generalized spectral operators, Gordon and Breach, New York, 1968. [7] J. Conway, Subnormal operators, Pitman, London, 1981. [8] J. Eschmeier, Invariant subspaces for subscalar operators, Arch. math. 52(1989), 562–570. [9] P. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), 887–933. [10] E. Ko, On w-hyponormal operators, Studia Math. 156(2003), 165–175. [11] E. Ko, Square roots of semihyponormal operators have scalar extensions, Bull. Sci. Math. 127(2003), 557–567. [12] E. Ko, w-Hyponormal operators have scalar extensions, Int. Eq. Op. Th. 53(2005), 363–372. [13] M. Putinar, Hyponormal operators are subscalar, J. Operator Th. 12(1984), 385–395. [14] M. Putinar, Quasisimilarity of tuples with Bishop’s property (β), Int. Eq. Op. Th. 15(1992), 1047–1052. [15] H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34(1971), 653–664. [16] D. Xia, Spectral theory of hyponormal operators, Op. Th. Adv. Appl. 10, Birkh¨ auser Verlag, Boston, 1983. Eungil Ko Department of Mathematics Ewha Women’s University Seoul 120-750 Korea e-mail: [email protected] Submitted: March 8, 2007 Revised: May 8, 2007

Integr. equ. oper. theory 59 (2007), 189–206 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020189-18, published online June 27, 2007 DOI 10.1007/s00020-007-1516-y

Integral Equations and Operator Theory

Lie and Jordan Ideals in Reflexive Algebras Fangyan Lu and Xiuping Yu Abstract. A CDCSL algebra is a reﬂexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace I of a CDCSL algebra A, that I is a Lie ideal if and only if AIA−1 ⊆ I for all invertibles A in A, and that I is a Jordan ideal if and only if it is an associative ideal. Mathematics Subject Classification (2000). 47L35, 47L35, 17B30, 17C65. Keywords. CDCSL algebras, Lie ideals, Jordan ideals.

1. Introduction Let A be an associative algebra. Then A becomes a Lie algebra and a Jordan algebra under the Lie product [A, B] = AB − BA and the Jordan product A ◦ B = 12 (AB + BA), respectively. A linear subspace L in A is called a Lie ideal if [A, X] ∈ L for all A ∈ A and X ∈ L. The Jordan ideals are deﬁned similarly. The purpose of this paper is to investigate the relationship between the Lie ideals, the Jordan ideals and the associative ideals in reﬂexive algebras with completely distributive and commutative subspace lattices. Let H be a complex Hilbert space. By B(H) we mean the set of all linear bounded operators on H. A subspace lattice L is a strongly closed lattice of orthogonal projections on H which contains the zero operator 0 and the identity operator I. A totally ordered subspace lattice is called a nest. A subspace lattice L is called a commutative subspace lattice, or a CSL, if each pair of projections in L commute. In this paper, we are mainly concerned with CSLs which are completely distributive. A subspace lattice L is called completely distributive if, for every family {Lγ,ω }γ∈Γ,ω∈Ω of elements of L, the inﬁnite distributive identity Lγ,ω = Lγ,f (γ) γ∈Γ ω∈Ω

f ∈ΩΓ γ∈Γ

This work was partly supported by NNSFC (No.10571054).

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and its dual hold, where ΩΓ denotes the set of all maps from Γ into Ω. See, for example, [14, 18]. For a subspace lattice L on H, the associated subspace lattice algebra AlgL is the set of operators on H that leave invariant every projection in L. Obviously, AlgL is a unital weakly closed subalgebra of B(H). Dually, if A is a subalgebra of B(H) we denote by LatA the collection of projections that are left invariant by all operators in A. An algebra A is reﬂexive if A = AlgLatA, and a lattice L is reﬂexive if L = LatAlgL. Every CSL is reﬂexive [1]. Clearly, every reﬂexive algebra is of the form AlgL for some subspace lattice L and vice versa. We shall call a reﬂexive algebra a CDCSL algebra if its lattice is completely distributive and commutative. A linear subspace M in an associative algebra A is called similarity invariant if AMA−1 ⊆ M for all invertible elements A in A. The relationship between Lie ideals and similarity invariant subspaces has been investigated for more than 30 years. For a few references, see [17, 24, 25, 16, 26, 7, 13, 12, 3] and the literature therein. For the history and the development in this area, we refer to a good introduction given by Hopenswasser and Paulsen [16]. In the same paper, they proved similarity invariance for Lie ideals in a Banach algebra when the invertibles are connected. In [25], Marcoux and Sourour showed that a weakly closed linear subspace in a nest algebra is a Lie ideal if and only it is similarity invariant. In Section 3, we shall show that this is also true for CDCSL algebras. It is obvious that associative ideals are Jordan ideals. The converse problem, which has been studied for more than 50 years, is to describe Jordan ideals in terms of associative ideals, and to ﬁnd conditions under which a non-zero Jordan ideal is in fact an associative ideal, or at least it contains a non-zero ideal. For a few references, see [2, 3, 27, 12, 13, 28, 29, 7] and the literature therein. Among other papers, Oliveira in [27] showed that a weakly closed Jordan ideal in a nest algebra, which satisﬁes certain conditions, is an associative ideal. In Section 4, we extend Oliveira’s result quite signiﬁcantly. More precisely, we shall prove that any weakly closed Jordan ideal in a CDCSL algebra is an associative ideal.

2. Preliminaries Erdos and Power in [6] described the structure of weakly closed ideals in nest algebras. Han in [11] extended this description to CDCSL algebras. If I is an associative ideal in the reﬂexive algebra AlgL, we associate with I a map φI of L deﬁned by φI (E) = [IE] for E ∈ L. Here [IE] denotes the norm closure of the linear span of {T x : T ∈ I, x ∈ E}. It is easily seen that φI is a left continuous order homomorphism of L into itself.

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Proposition 2.1 ([11]). Let I be an associative ideal in a CDCSL algebra AlgL. Then I = {T ∈ AlgL : (I − φI (E))T E = 0 for each E ∈ L}. Let L be a subspace lattice on a Hilbert space H and E be a projection in L. We deﬁne E− = ∨{F ∈ L : F ≥ E}. Let x and y be non-zero vectors in H. Then the rank one operator x ⊗ y is deﬁned as (x ⊗ y)z = (z, y)x for z ∈ H. The following lemma, due to Longstaﬀ, will get repeated use without explicit mention. Lemma 2.2 ([21]). Let L be a subspace lattice. Then the rank one operator x ⊗ y belongs to AlgL if and only if there is an element E ∈ L such that x ∈ E and ⊥ ⊥ . Here E− means (E− )⊥ . y ∈ E− The formal deﬁnition of completely distributivity given in Section 1 is, in practice, diﬃcult to use. We recall that a complete lattice is said to be strongly reflexive if it satisﬁes condition (3) of Theorem 2.3 below. From [20] we know that a subspace lattice is completely distributive if and only if it is strongly reﬂexive. Theorem 2.3 ([20]). Let L be a subspace lattice. Then the following are equivalent. (1) L is completely distributive. (2) E = ∨{F ∈ L : F− ≥ E} for every E ∈ L. (3) E = ∧{F− : F ∈ L and F ≤ E} for every E ∈ L. A projection E is invariant for an operator A exactly when I − E is invariant for A∗ . Thus it follows, for a lattice L, that (AlgL)∗ = AlgL⊥ , where L⊥ is the lattice {I − E : E ∈ L}. The diagonal (AlgL) ∩ (AlgL)∗ is a von Neumann algebra, which equals L , the commutant of L. If L is commutative, then the double commutant L of L is an abelian von Neumann algebra. Given a subspace lattice, by R1 (L) denote the linear span of rank one operators in AlgL. In general, R1 (L) is not necessarily equal to the set of all operators of ﬁnite rank in AlgL [15]. However, if L is commutative then the norm closure of R1 (L) contains all operators of ﬁnite rank [4]. Moreover, topological properties of R1 (L) may determine properties of the lattice L. For example, Longstaﬀ [21] showed that if R1 (L) is weakly dense in AlgL, then L is completely distributive (even if L is non-commutative). In [19], Laurie and Longstaﬀ proved the converse for commutative L. So, it is not a surprise that operators of rank one play an important role in the study of CDCSL algebras. Theorem 2.4 ([19]). Let L be a CSL. Then the following are equivalent. (1) L is completely distributive. (2) R1 (L) is weakly dense. Recall that a CSL algebra AlgL is irreducible if and only if the commutant is trivial, i.e. (AlgL) = CI, which is also equivalent to the condition that L ∩ L⊥ = {0, I}. It turns out that any CDCSL algebra can be decomposed into the direct sum of irreducible CDCSL algebras.

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Theorem 2.5 ([9]). Let AlgL be a CDCSL algebra on a separable Hilbert space H. Then there is a (finite) countable set {En : n ∈ Λ} of mutually orthogonal projections in L ∩ L⊥ with sum I such that AlgL = ⊕(AlgL)En , n

where each (AlgL)En viewed as a subalgebra of operators acting on the range of En is an irreducible CDCSL algebra. Finally, we shall need a structure theory of CDCSLs, which has proved eﬀective in the study of the Lie structure of CDCSL algebras [22, 23]. Theorem 2.6 ([22]). Let AlgL be a non-trivially irreducible CDCSL algebra. Then there is a projection E in L such that E(AlgL)E ⊥ is a faithful ideal in AlgL. (That is, for T ∈ AlgL, the condition T E(AlgL)E ⊥ = {0} implies that T E = 0 and the condition E(AlgL)E ⊥ T = {0} implies that E ⊥ T = 0.)

3. Lie ideals Throughout this section, L is a CDCSL on a Hilbert space H. For E, F ∈ L with E ≤ F , we call the projection EF ⊥ an interval of L. A minimal interval is called ⊥ an atom. If E is in L such that E− ≥ E, then EE− is an obvious atom. Conversely, each atom takes this form. To see this, suppose that P is an atom of L. Let E be the smallest projection in L which is greater than or equal to P , namely, E = ∧{L ∈ L : LP = P }. Since P is an atom, we know, for L ∈ L, that P L = 0 if and only if L ≥ E. Thus P E− = 0 by the deﬁnition of E− . So ⊥ ⊥ ⊥ P = EP = E(P E− ) = P EE− = EE− ⊥ since EE− is also an atom. Let Q be the sum of all atoms of L. Then the subalgebra QL Q of L is equal to Σ{P B(H)P : P is an atom of L}. We call it the atomic-diagonal of AlgL. A subset of AlgL is called atomic-diagonal-disjoint if the intersection with the atomic-diagonal is the trivial set {0}. For T ∈ AlgL, we deﬁne

(T ) = Σ{P T P : P is an atom of L}. (The sum in the right-hand converges in the weak operator topology.) Then it is not diﬃcult to verify that is an expectation of AlgL onto the atomic-diagonal QL Q, that is, (i) is surjective; (ii) 2 = and = 1; (iii) (AT B) = A (T )B for all T ∈ AlgL and A, B ∈ QL Q.

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Using this, one can verify that a weakly closed associative ideal in AlgL is atomicdiagonal-disjoint if and only if (T ) = 0 for all elements T in it. Lemma 3.1. Suppose that {Tα } is a net in AlgL which weakly converges to an operator T . If ∆(Tα ) = 0 for each α, then ∆(T ) = 0. Proof. Since Σ{P Tα P : P is an atom of L} = (Tα ) = 0 and each pair of atoms are orthogonal, it follows that P Tα P = 0 for each index α and each atom P . Therefore, P T P = 0 for all atoms P since P T P is the weak limit of P Tα P . Consequently, (T ) = 0 by the deﬁnition. Suppose that K is a Lie ideal in AlgL. We associate an atomic-diagonaldisjoint subspace ADS(K) deﬁned to be the weak closure of linear span of {ET E ⊥ : T ∈ K, E ∈ L}. Proposition 3.2. The following statements are true. (1) Suppose that K is a weakly closed Lie ideal in AlgL. Then ADS(K) is a weakly closed and atomic-diagonal-disjoint associative ideal in AlgL and contained in K. (2) Suppose that I is a weakly closed and atomic-diagonal-disjoint associative ideal in AlgL. Then I = ADS(I). (3) Suppose that I is a weakly closed associative ideal in AlgL. Then ADS(I) is a weakly closed associative ideal in AlgL and ADS(I) = {T ∈ I : (T ) = 0}. Proof. (1) It is obvious that ADS(K) is a weakly closed linear space. Since ET E ⊥ = [E, T ] ∈ K for all T ∈ K and E ∈ L, ADS(K) ⊆ K. To see that ADS(K) is an associative ideal, we let A and X be in AlgL and suppose that X = ET E ⊥ for some T ∈ K and E ∈ L. Then AX = AEX = EAEX = EAEX − EXE ⊥ EAE = [EAE, X] ∈ K, because of X ∈ K. Hence since AX = EAEXE ⊥ = E(AX)E ⊥ , we have that AX ∈ ADS(K). Similarly, XA ∈ ADS(K). So ADS(K) is an associative ideal. It remains to prove that ADS(K) is atomic-diagonal-disjoint. Let T be in K and E be in L. For an atom P of L, if P E = 0 then P E = P and hence P E ⊥ = 0. Thus we always have that P (ET E ⊥ )P = 0. So ∆(ET E ⊥ ) = 0. Hence (S) = 0 for all S ∈ ADS(K) by Lemma 3.1. Hence, ADS(K) is atomic-diagonal-disjoint since it is a weakly closed ideal. (2) By (1), we have ADS(I) ⊆ I. It is now suﬃcient to show that I ⊆ ADS(I). Suppose that x ⊗ y ∈ I. Then there exists E in L such that x ∈ E and ⊥ ⊥ ⊥ . Since I is atomic-diagonal-disjoint, EE− x ⊗ yEE− = 0. Therefore, either y ∈ E− ⊥ ⊥ ⊥ EE− x = 0 or EE− y = 0. If EE− x = 0, then x = Ex = EE− x = E− x and hence ⊥ ⊥ ⊥ ⊥ ∈ ADS(I). If EE− y = 0, then y = E− y = E ⊥ E− y = E⊥y x ⊗ y = E− (x ⊗ y)E− ⊥ and hence x ⊗ y = E(x ⊗ y)E ∈ ADS(I). Consequently, each rank one operator in I belongs to ADS(I). Noting that the linear span of rank one operators in an ideal in a CDCSL algebra is weakly dense in that ideal, we have that I ⊆ ADS(I).

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(3) Let M = {T ∈ I : (T ) = 0}. Then it is not diﬃcult to verify that M is an atomic-diagonal-disjoint associative ideal. Moreover, by Lemma 3.1, M is weakly closed. Thus by (2), M = ADS(M) ⊆ ADS(I). On the other hand, ADS(I) ⊆ M by (1). So, ADS(I) = M = {T ∈ I : (T ) = 0}. Recall that the diagonal of AlgL is L ; the commutant of L is L , the core of AlgL, which is an abelian von Neumann algebra. By [4, Theorem 8.3 and Lemma 8.4], there are expectations of AlgL onto L and L . Proposition 3.3. Suppose that K is a weakly closed Lie ideal in AlgL. Then K ⊆ ADS(K) + L . Proof. Let π be an expectation from AlgL onto L . Let K be in K. Then π(K) ∈ L . We show below that K − π(K) ∈ ADS(K). From Proposition 3.2 we know that ADS(K) is a weakly closed associative ideal. For simiplicity, we write I = ADS(K). It is now suﬃcient to show that (I − φI (E))(K − π(K))E = 0 for each E ∈ L. For E, F ∈ L, since F KF ⊥ ∈ I and π(K) ∈ L , we have F (I − φI (E))(K − π(K))E(I − F ) = (I − φI (E))F (K − π(K))(I − F )E = (I − φI (E))F K(I − F )E = 0. So (I − φI (E))(K − π(K))E is in AlgL⊥ . But it is also in AlgL. Therefore, (I − φI (E))(K − π(K))E ∈ L for all E ∈ L. Thus, for each E ∈ L, (I − φI (E))(K − π(K))E = π((I − φI (E))(K − π(K))E) = (I − φI (E))π(K − π(K))E = (I − φI (E))(π(K) − π(K))E = 0. So K − π(K) ∈ I, completing the proof.

Thus, for a weakly closed Lie ideal K in AlgL, ADS(K) ⊆ K ⊆ ADS(K) + L . Our aim is to prove that K is similarity invariant. However, there exist weakly closed subspaces M which is not similarity invariant but I ⊆ M ⊆ I + L for some atomic-diagonal-disjoint associative ideal I. For example, let H = C4 , L = {(0), span{e1 , e2 }, H}, I = P (AlgL)(I − P ), where P is the orthogonal projection onto span{e1 , e2 }. Then     ∗ ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ ∗ ∗ 0 0 ∗ ∗    AlgL =  0 0 ∗ ∗ and I = 0 0 0 0 . 0 0 ∗ ∗ 0 0 0 0

Let M=

A B : A, B ∈ M2 (C) . 0 A

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Then I ⊆ M ⊆ I + L (= AlgL). However if we let    0 1 0 0 1 0 0 0 0 0   M = 0 0 0 1 ∈ M and T = 0 0 0 0 0 0 then



T M T −1

0 0 = 0 0

1 0 0 0

0 0 0 0

195

0 1 0 0

0 0 2 0

 0 0  ∈ (AlgL)−1 , 0 1

 0 0  ∈ M. 2 0

This example is essentially due to [17]. So, we should add some atomic parts of R to ADS(K) so that it is enlarged. We do this as follows. Let I be a weakly closed associative ideal in AlgL which is atomic-diagonal-disjoint. Then φI (E) EE− . Indeed, since I is atomic-diagonal⊥ ⊥ ⊥ ⊥ ⊥ disjoint, E− IE = EE− IE− E = {0} and then E− φI (E) = E− [IE] = 0 by the deﬁnition. So φI (E) = φI (E)E− EE− . We associate with I a subset of atoms of L as follows: ⊥ atom(I) = {EE− = 0 :φI (E) = EE− and for each F > E ⊥ F = 0 or φI (F ) E}. either E ⊥ E−

For a subset A of atom(I), we deﬁne I ∨ A := w-cl{I + Σ{P B(H)P : P ∈ A}}. Here w-cl{·} denotes the weak closure of the set. It is not diﬃcult to verify that I ∨ A is also an associative ideal. Obviously, φI∨A (E) ≥ φI (E) for all E ∈ L; in ⊥ particular, φI∨A (E) = E if EE− ∈ A. Given an associative ideal J in AlgL, we associate following [25] a Lie ideal J◦ , called the zero-trace part of J, deﬁned by J◦ = {A ∈ J : trP AP = 0 for all ﬁnite-dimensional atoms P of L}. Then for the ideal I ∨ A just deﬁned in the paragraph above, we have {P B(H)P : P ∈ A, dimP = ∞} (I ∨ A)◦ = w-cl{I + { P (sln )P : P ∈ A, dimP = n < ∞}}, + n

where (sln ) denotes the Lie ideal of zero-trace matrices in Mn . Of course, we also should diminish the summand L . Given an associative ideal J in AlgL, we deﬁne a von Neumann algebra by CNA(J) := {D ∈ L : EφJ (E)⊥ DEφJ (E)⊥ is in the commutant of EφJ (E)⊥ (AlgL)EφJ (E)⊥ for all E ∈ L}.

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Finally, for the sake of technique, if A is an associative algebra and K is a Lie ideal in A, we deﬁne the Lie residual quotient of A by K as follows [A : K] := {A ∈ A : [A, X] ∈ K for all X ∈ A}. Lemma 3.4. [25, Lemma 3.2] Let K be a Lie ideal in an associative algebra A. (1) [A : K] is a Lie ideal that includes K. Furthermore, every linear subspace M satisfying K ⊆ M ⊆ [A : K] is a Lie ideal. (2) If, in addition, A is a weakly closed operator algebra and K is weakly closed in A, then [A : K] is weakly closed. Proposition 3.5. Let I be a weakly closed ideal in AlgL and let I◦ be the zero-trace part of I. Then [AlgL : I◦ ] = [AlgL : I] = I + CNA(I). Proof. The ﬁrst equality, namely that [AlgL : I◦ ] = [AlgL : I], is obvious. Let D be in CNA(I). For T ∈ AlgL and E ∈ L, since D is by deﬁnition in the commutant of EφI (E)⊥ (AlgL)EφI (E)⊥ , we have that EφI (E)⊥ [T, D]EφI (E)⊥ = [EφI (E)⊥ T EφI (E)⊥ , EφI (E)⊥ DEφI (E)⊥ ] = 0. So [T, D] ∈ I for all T ∈ AlgL by Proposition 2.1 and therefore D ∈ [AlgL : I]. Now the arbitrariness of D ∈ CNA(L) gives that CNA(L) ⊆ [AlgL, I]. This together with the fact that I ⊆ [AlgL : I] proves that I + CNA(I) ⊆ [AlgL : I]. It now remains to show that [AlgL : I] ⊆ I + CNA(I). To do this, let A be in [AlgL : I]. Then for T ∈ AlgL, [T, A] ∈ I. Hence for all E ∈ L, 0 = EφI (E)⊥ [T, A]EφI (E)⊥ = [EφI (E)⊥ T EφI (E)⊥ , EφI (E)⊥ AEφI (E)⊥ ]. This implies that

EφI (E)⊥ AEφI (E)⊥ ∈ (EφI (E)⊥ (AlgL)EφI (E)⊥ ) ⊆ L . Thus, if we let π be an expectation of AlgL onto L and let B = A − π(A), then the equations EφI (E)⊥ BEφI (E)⊥ = EφI (E)⊥ (A − π(A))EφI (E)⊥ = EφI (E)⊥ AEφI (E)⊥ − EφI (E)⊥ π(A)EφI (E)⊥ = EφI (E)⊥ AEφI (E)⊥ − π(EφI (E)⊥ AEφI (E)⊥ ) =0 are true for all E ∈ L. So B ∈ I. Hence for each E ∈ L, EφI (E)⊥ π(A)EφI (E)⊥ = EφI (E)⊥ AEφI (E)⊥

∈ (EφI (E)⊥ (AlgL)EφI (E)⊥ ) . This implies that π(A) ∈ CNA(I). So A = B + π(A) ∈ I + CNA(I) for each A ∈ [AlgL : I]. Namely, [AlgL : I] ⊆ I + CNA(I), completing the proof.

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The main result in this section follows. Theorem 3.6. Let AlgL be a CDCSL algebra on a separable Hilbert space H and suppose that K is a weakly closed linear subspace of AlgL. Then the following statements are equivalent. (1) K is similarity invariant. (2) K is a Lie ideal. (3) There exists an atomic-diagonal-disjoint ideal I and a subset A of atom(I) such that (I ∨ A)◦ ⊆ K ⊆ [AlgL : I ∨ A] = (I ∨ A) + CNA(I ∨ A). (4) There exists an associative ideal J such that (J)◦ ⊆ K ⊆ [AlgL : J] = J + CNA(J). Proof. That (1) implies (2) is an unpublished result of Topping [30], see also [26, 16, 25]. That (3) implies (4) is obvious. The proof that (4) implies (1) is completely analogous with the corresponding argument in [25, Theorem 3.5]. We now prove that (2) implies (3). For simplicity, we write I = ADS(K). Then I is a weakly closed and atomic-diagonal-disjoint associative ideal. Let A = {P ∈ atom(I) : P KP CP }. Then I ∨ A is a weakly closed ideal. From Proposition 3.2, we know that I ⊆ K. Let P be an atom in A. Then P KP is a weakly closed Lie ideal in P B(H)P . Since P KP CP , it follows from [7] that, P KP = P B(H)P if dimP = ∞ and, P KP ⊇ P (sln )P if dimP = n < ∞. Let A be in P B(H)P and B = P CP for some C ∈ K. From Proposition 3.3, we know that P CP + P ⊥ CP ⊥ ∈ K. It follows that [A, B] = [A, P CP + P ⊥ CP ⊥ ] ∈ K. So, if P is inﬁnite dimensional then P B(H)P ⊆ K since commutators span P B(H)P in this case [10, Problem 234]; if P is ﬁnite dimensional then P (sln )P ⊆ K since a zero-trace matrix is the commutator [A, B] with B being zero-trace. Consequently, (I ∨ A)◦ ⊆ K. It remains to show that K ⊆ [AlgL : I ∨ A], equivalently, to show that (I − φI∨A (E))[T, K]E = 0 for all K ∈ K, T ∈ AlgL and E ∈ L. In the sequel, we let K ∈ K and T ∈ AlgL. ⊥ ⊥ Claim 1. Let GG⊥ / A. Then GG⊥ − be an atom and suppose that GG− ∈ − [T, K]GG− = 0. We distinguish some cases.

Case 1: GG⊥ − ∈ atom(I). ⊥ ⊥ In this case, GG⊥ − KGG− ⊆ CGG− by the deﬁnition of A. Thus ⊥ ⊥ ⊥ ⊥ ⊥ GG⊥ − [T, K]GG− = [GG− T GG− , GG− KGG− ] = 0.

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Case 2: φI (G) = GG− but GG⊥ − ∈ atom(I). In this case, by the deﬁnition of atom(I), there exists F in L with F > G such that G⊥ G⊥ − F = 0 and φI (F ) G. The latter relation, together with the facts that ⊥ φI (F ) ≥ φI (G) = GG− and that GG⊥ − is an atom, yields that GG− φI (F ) = 0. ⊥ ⊥ Thus GG− K(F − G) = {0} since GG− K(F − G) is contained in I. ⊥ ⊥ Since G⊥ G⊥ − F = 0, we can take a non-zero vector y from G G− F . Then for ⊥ each x ∈ GG− , x⊗y ∈ AlgL and therefore (x⊗y)K −K(x⊗y) = GG⊥ − ((x⊗y)K − K(F − G) = {0} K(x ⊗ y))(F − G) ∈ K. It follows from the preceeding result GG⊥ − ⊥ ⊥ . This implies that GG that (x ⊗ y)K − K(x ⊗ y) = 0 for all x ∈ GG⊥ − − KGG− = ⊥ ⊥ ⊥ λGG− for some λ ∈ C. Therefore GG− (T K − KT )GG− = 0. Case 3: φI (G) < GG− . In this case, φI (G) < GG− < G. Let x be in GG− (φI (G))⊥ and y be in ⊥ GG− . Then x ⊗ y is in AlgL. Set B = (x ⊗ y)K − K(x ⊗ y). Then B ∈ K. Since K ⊆ I + L by Proposition 3.3, we can write B = R + D with R ∈ I and D ∈ L . Thus B = (G − φI (G))B(G − φI (G)) = (G − φI (G))D(G − φI (G)) ∈ L and hence B = (GG− − φI (G))B(G − GG− ) = (GG− − φI (G))(G − GG− )B = 0. So Kx ⊗ y = x ⊗ yK for all x ∈ GG− (φI (G))⊥ and y ∈ GG⊥ − . This implies that (GG− −φI (G))K(GG− −φI (G)) = λ(GG− −φI (G)) and (G−GG− )K(G−GG− ) = ⊥ λ(G − GG− ) for some λ ∈ C. So GG⊥ − (T K − KT )GG− = 0. Claim 2. Let G1 − G2 be a non-atomic interval of L. Suppose that (G1 − G2 )(AlgL)(G1 − G2 ) is irreducible and that (G1 − G2 )I(G1 − G2 ) = {0}. Then (G1 − G2 )[T, K](G1 − G2 ) = 0. Since the algebra (G1 − G2 )(AlgL)(G1 − G2 ) is a non-trivially irreducible CDCSL algebra, by Theorem 2.6 there exists an element L with G2 < L < G1 in L such that (L − G2 )(AlgL)(G1 − L) is a faithful associative ideal in (G1 − G2 )(AlgL)(G1 − G2 ). For any A in (L − G2 )(AlgL)(G1 − L), AK − KA ∈ K and hence [A, K] = (L − G2 )[A, K](G1 − L) ∈ I. So [A, K] = (G1 − G2 )[A, K](G1 − G2 ) = 0 for all A in (L−G2 )(AlgL)(G1 −L). Hence (L−G2 )K(L−G2 )+(G1 −L)K(G1 −L) is in the center of (G1 − G2 )(AlgL)(G1 − G2 ) since (L − G2 )(AlgL)(G1 − L) is faithful. Thus (G1 − G2 )[T, K](G1 − G2 ) = (G1 − G2 )(L − G2 )[T, K](G1 − L)(G1 − G2 ) ∈ I and hence it is 0, establishing the claim. Now let E be in L. Let R = (I − φI∨A (E))(T K − KT )E.

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We want to prove that R = 0. If φI (E) = E, then φI∨A (E) = E and therefore R = 0. Suppose now that φI (E) < E. Since (E − φI (E))(AlgL)(E − φI (E)) = Alg((E − φI (E))L) is a CDCSL algebra on (E − φI (E))H, by Theorem 2.5 there are no more than countably many Gn in L which satisfy: (a) for each n, φI (E) ≤ Gn ≤ E; (b) each pair of {Gn − φI (E)} are orthogonal and (Gn − φI (E)) = E − φI (E); (c) for each n, the algebra (Gn − φI (E))(AlgL)(Gn − φI (E)) is irreducible; (d) (E − φI (E))(AlgL)(E − φI (E)) = Σn (Gn − φI (E))(AlgL)(Gn − φI (E)). Fix an index n. We show below that (I − φI∨A (E))(T K − KT )Gn = 0.

(3.1)

First suppose that Gn − φI (E) is non-atomic. Since (Gn − φI (E))(AlgL)(Gn − φI (E)) is irreducible and (Gn −φI (E))I(Gn −φI (E)) ⊆ (E−φI (E))I(E−φI (E)) = {0}, it follows from Claim 2 that (I −φI (E))(T K −KT )Gn = 0. Hence the equality (3.1) holds since φI∨A (E) ≥ φI (E). Now suppose that Gn − φI (E) is atomic. Then ⊥ there exists G such that Gn − φI (E) = GG⊥ − . If GG− ∈ A, then φI∨A (G) = G. ⊥ Hence φI∨A (E) ≥ G ≥ GG− = Gn − φI (E) since E ≥ Gn ≥ G. Thus φI∨A1 (E) ≥ / A, then by Claim φI (E)+(Gn −φI (E)) = Gn . This obviously gives (3.1). If GG⊥ − ∈ ⊥ 1, GG⊥ (T K − KT )GG = 0. That is, (G − φ (E))(T K − KT )(G n I n − φI (E)) = 0. − − This obviously yields (3.1). Consequently, (I − φI∨A (E))(T K − KT )Gn = 0 for all n. By (b), (I − φI∨A (E))(T K − KT )E = 0, completing the proof. As mentioned in Introduction, Hopenswasser and Paulsen in [16] proved that if the group of invertible elements of a Banach algebra A is connected then every (norm) closed Lie ideal in A is similarity invariant. It is this case when AlgL is a nest algebra of inﬁnite multiplicity[5]. However, it is unknown whether the invertibles in inﬁnite multiplicity CDCSL algebras are connected. We mention in passing that the problem whether the invertibles in nest algebras are connected is open.

4. Jordan ideals In this section, we shall investigate conditions that assure a Jordan ideal in a CDCSL algebra is an associative ideal. First we give an algebraic condition. Theorem 4.1. Let J be a Jordan ideal in a CDCSL algebra AlgL. If J ◦ J = {0}, then J is an associative ideal. Proof. We ﬁrst show that J ⊆ E(AlgL)E ⊥ for some E ∈ L. Let E be the biggest projection in L on which each operator in J vanishes, that is E = ∨{L ∈ L : AL = 0 for each A ∈ J}. For F ∈ L with F ≤ E, there exist B ∈ J and x ∈ F such that Bx = 0. Then for y ∈ F−⊥ , C = Bx ⊗ y + x ⊗ yB ∈ J.

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Hence, since B 2 = B ◦ B = 0 we have Bx ⊗ yB = B ◦ C = 0. ∗

So B y = 0 since Bx = 0. Thus C = Bx ⊗ y ∈ J. Now let A be in J. If ABx = 0, then Bx ⊗ yA = CA = CA + AC = 0. This yields that A∗ y = 0. If ABx = 0, then ABx ⊗ yA = ACA = A ◦ (A ◦ C) − C ◦ A2 = 0. This also yields that A∗ y = 0. So A∗ y = 0 for all y ∈ F−⊥ . Namely A∗ F−⊥ = 0, equivalently, F− A = A. But E = ∧{F− : F ∈ L, F ≤ E}. It follows that A = EA. Therefore, A = EAE ⊥ for all A ∈ J. Now with A ∈ J and T ∈ AlgL, we have T A = 2A ◦ (ET E) ∈ J and AT = 2A ◦ (E ⊥ T E ⊥ ) ∈ J. So J is an associative ideal, completing the proof. Herstein in [12] proved that a Jordan ideal J in an algebra contains the ideal generated by J◦J. This together with Theorem 4.1 immediately gives the following conclusion. Corollary 4.2. Let J be a non-zero Jordan ideal in a CDCSL algebra. Then J contains a non-zero ideal. The central result in this section reads as follows. Theorem 4.3. Let AlgL be a CDCSL algebra. Let J be a weakly closed Jordan ideal in AlgL. Then J is an associative ideal. To prove this theorem, we need some lemmas. In those lemmas we shall assume that L be a CDCSL on a Hilbert space H. For a vector x ∈ H, we deﬁne ˆx and Eˇx be the smallest projection in L containing x and the biggest projection E in L annihilating x respectively, that is, ˆx = ∧{E ∈ L : Ex = x} and E ˇx = ∨{E ∈ L : Ex = 0}. E Lemma 4.4. Let J be a norm closed Jordan ideal of AlgL and suppose that the rank ˆx . one operator x ⊗ y belongs to J. Then u ⊗ y belongs to J for every u ∈ E Proof. We ﬁrst prove a claim. ˆx . Then u ⊗ y belongs to J for every u ∈ E. Claim. Let E be in L such that E− ≥ E ˆx , by the deﬁnition of E ˆx we know that (I − Let u be in E. Since E− ≥ E (I−E− )x E− )x = 0. Let v = (I−E− )x2 . Then u ⊗ v is in AlgL. A direct computation gives (x ⊗ y) ◦ (u ⊗ v) = 12 ((u, y)x ⊗ v + (x, v)u ⊗ y) = 12 (u, y)x ⊗ v + 12 u ⊗ y.

(4.1)

If (u, y) = 0, then we see from the equation above that u ⊗ y belongs to J. In the following we assume that (u, y) = 0. ˆx , it follows from Theorem 2.3 that E ≤ E ˆx . Since x ⊗ y ∈ J ⊆ Since E− ≥ E ⊥ ˆ AlgL, it follows from Lemma 2.2 that y ∈ (Ex )− . Again by Lemma 2.2, we have that u ⊗ y ∈ AlgL.

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If (x, y) = 0, by (4.1) we have that ((x ⊗ y) ◦ (u ⊗ v)) ◦ (u ⊗ y) = 12 (u, y)(u, v)x ⊗ y + (u, y)u ⊗ y, from which we see that u ⊗ y ∈ J. If (x, y) = 0, from (u, y)(x, y)u ⊗ y = (u ⊗ y)(x ⊗ y)(u ⊗ y) = 2(u ⊗ y) ◦ ((u ⊗ y) ◦ (x ⊗ y)) − (u ⊗ y)2 ◦ (x ⊗ y) we see that u ⊗ y is in J. The proof of the claim is complete. ˆx be arbitrary. Since E ˆx = ∨{E ∈ L : E− ≥ E ˆx }, there is a Now let u ∈ E sequence un in span{E ∈ L : E− ≥ Eˆx } which converges to u in norm. Suppose n ˆx . By the above un = kk=1 αn,k un,k , where αn,k ∈ C, un,k ∈ En,k , (En,k )− ≥ E claim each un,k ⊗ y is in J. Hence un ⊗ y is in J. Therefore u ⊗ y = limn→∞ un ⊗ y belongs to J. This completes the proof. Dually, we have Lemma 4.5. Let J be a norm closed Jordan ideal of AlgL and suppose that the rank ˇy )⊥ . one operator x ⊗ y belongs to J. Then x ⊗ v belongs to J for every v ∈ (E Proof. Recall that L⊥ = {E ⊥ : E ∈ L} and AlgL⊥ =(AlgL)∗ . So J∗ is a Jordan ˇy )⊥ is the smallest element in L⊥ ideal of AlgL⊥ and y ⊗ x ∈ J∗ . Note that (E containing y. It follows from Lemma 4.4 that v ⊗ x belongs to J∗ for every v ∈ ˇy )⊥ . ˇy )⊥ . So x ⊗ v belongs to J for every v ∈ (E (E Lemma 4.6. Let J be a norm closed Jordan ideal of AlgL and suppose that the rank ˆx and one operator x ⊗ y belongs to J. Then u ⊗ v belongs to J for every u ∈ E ˇy )⊥ . v ∈ (E ˆx . Hence by Lemma 4.5, u ⊗ v ∈ J Proof. By Lemma 4.4, u ⊗ y ∈ J for every u ∈ E ˇy )⊥ . for every v ∈ (E The following lemma gives a characterization of operators of rank one in a Jordan ideal. Lemma 4.7. Let J be a norm closed Jordan ideal of AlgL. Define a map τ : L → L by ˆx : x ⊗ y ∈ J and E ˇy ≥ E}, E ∈ L. τ (E) = ∨{E Then the rank one operator x ⊗ y belongs to J if and only if (I − τ (E))x ⊗ yE = 0 for every E ∈ L. ˇy , then E ˆx ≤ τ (E). Proof. Necessity. Suppose x ⊗ y ∈ J. Let E be in L. If E ≤ E ˇ ˇ Thus (I − τ (E))x = 0. So in both cases E ≤ Ey and E ≤ Ey , we always have that (I − τ (E))x ⊗ yE = 0. Suﬃciency. Suppose that the rank one operator x ⊗ y in AlgL satisﬁes (I − τ (E))x ⊗ yE = 0 for every E ∈ L. We want to prove x ⊗ y ∈ J. First we prove a claim.

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ˇy and yE be a vector in E ⊥ . Then x⊗yE ∈ J. Claim. Let E be in L satisfying E ≤ E − ˇy and (I − τ (E))x ⊗ yE = 0, it follows that (I − τ (E))x = 0. Since E ≤ E ˆu : u ⊗ v ∈ J and E ˇv ≥ E}. Therefore there So, by the deﬁnition, x ∈ ∨{E ˆ ˇ exists a sequence xn in span{Eu : u ⊗ v ∈ J and Ev ≥ E} which converges to x. kn ˆu , un,k ⊗ vn,k ∈ J and Suppose xn = k=1 αn,k xn,k , where αn,k ∈ C, xn,k ∈ E n,k ˇ ˇ ˇ ˇv ). By Evn,k ≥ E. Since Evn,k ≥ E, it follows that Evn,k ≤ E− . So yE ∈ (I − E n,k Lemma 4.6, we know that each xn,k ⊗ yE belongs to J. Hence xn ⊗ yE is in J and then x ⊗ yE = limn xn ⊗ yE ∈ J. This proves the claim. ⊥ ˇy } by Theorem 2.3, there is a sequence yn in ˇy⊥ = ∨{E− : E ≤ E Now since E kn ⊥ βn,k yn,k , where βn,k ∈ C, {E− : E ≤ Eˇy } such that limn yn = y. Write yn = k=1 ⊥ ˇ yn,k ∈ (En,k )− , En,k ≤ Ey . By the claim, each x ⊗ yn,k ∈ J. Hence x ⊗ yn is in J and then x ⊗ y ∈ J. The proof is complete. We are now in a position to prove the main result in this section. Proof of Theorem 4.3. Let the mapping τ : L → L be as in Lemma 4.7. Let A = {A ∈ AlgL : (I − τ (E))AE = 0 for all E ∈ L}. It is easy to verify that A is a weakly closed associative ideal. So, in order to prove that J is an associative ideal, it suﬃces to show J = A. Recall that R1 (L) is the linear submanifold of AlgL spanned by operators of rank one in AlgL. By Theorem 2.4, R1 (L) is weakly dense in AlgL. Thus we can α take a net Tα in R1 (L) which weakly converges to I . Suppose Tα = kk=1 Tα,k , where Tα,k is a rank one operator in AlgL. Let A be in A. Since ATα,k is in A, it follows from the deﬁnition that (I − τ (E))ATα,k E = 0 for every E ∈ L. Hence by Lemma 4.7, each ATα,k belongs to J. Thus ATα is in J. So A = limα ATα is in J. Consequently, A ⊆ J. To prove the other inclusion, we need a claim. Claim. Let B be in J and x ⊗ y be in AlgL. Then Bx ⊗ y + x ⊗ yB is in A. Assume on the contrary that Bx ⊗ y + x ⊗ yB ∈ / A. Then there is a projection E in L such that (I − τ (E))(Bx ⊗ y + x ⊗ yB)E = 0. Note that if one of S, T from AlgL is in J then ST S belongs to J (ST S = 2S ◦ (S ◦ T ) − S 2 ◦ T ). So, by Lemma 4.7, (I − τ (E))B(x ⊗ y)BE = 0. Therefore, either (I − τ (E))Bx = 0 or EB ∗ y = 0. Without loss of generalization, suppose that EB ∗ y = 0. Thus we would have (I − τ (E))Bx ⊗ yE = (I − τ (E))(Bx ⊗ y + x ⊗ yB)E = 0. On the other hand, since Bx ⊗ y + x ⊗ yB is in J and (I − τ (E))(Bx ⊗ y + x ⊗ yB)E = (I − τ (E))E(Bx ⊗ y + x ⊗ yB)E(I − τ (E)), it follows that (I −τ (E))Bx⊗yE is in J. Hence by Lemma 4.7, (I −τ (E))Bx⊗yE = 0. Thus we get a contradiction, which proves Claim.

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Now let B be in J. Then by Claim, each BTα + Tα B is in A. So B = limα 12 (BTα + Tα B) is in A. Therefore, J ⊆ A. Consequently, J = A, completing the proof. The question naturally rises: Is a norm closed Jordan ideal of a CDCSL algebra an associative ideal? We are unable to give an answer at present. However, we have the following conclusion. Proposition 4.8. Let A be a CDCSL algebra and J be a norm closed Jordan ideal. Then J is a bimodule of the norm closure of the algebra generated by operators of rank one in A. Proof. Let A be as in the proof of Theorem 4.3. Then J ⊆ A by the same argument. Thus T A and AT are in A for every A ∈ J and T ∈ A of rank one. But the set of operators of rank one in A coincides with that in J. The desired result follows.

5. Examples Marcoux and Sourour in [25] showed that if commutators of an operator algebra span the whole algebra then each Lie ideal is similarity invariant. Also, the authors in [2] showed that if the algebra generated by commutators of A is equal to A then each Jordan ideal is an associative ideal. In this section, we shall present some examples which shows that the Lie ideals and the Jordan ideals may be trivial even if the algebra does not satisfy those conditions. First note that any operator subspace in a ﬁnite-dimensional Hilbert space is weakly closed. Combining Theorem 3.6 and Theorem 4.3, we can conclude the following, the part of which concerning Lie ideals is covered by Hapenwasser and Paulsen [16]. Corollary 5.1. In a CDCSL algebra acting on a finite-dimensional Hilbert space, Lie ideals are similarity invraiant and Jordan ideals are associative ideals; in particular, the algebra of all upper triangular matrices is this case. We present below an inﬁnite-dimensional example. Example 5.2. Let L be a subspace lattice on a Hilbert space H and suppose that there exists a projection E in L such that E ∈ AlgL and both algebras E(AlgL)E and E ⊥ (AlgL)E ⊥ are commutative. Then each Jordan ideal in AlgL is an associative ideal, and each Lie ideal in AlgL is similarity invariant. Proof. For simplicity, we write A = AlgL. Let J be a Jordan ideal in A. Since ABA = 2A ◦ (A ◦ B) − B ◦ A2 , we know that EJE and E ⊥ JE ⊥ are both contained in J. Further, EJE ⊥ is also contained in J. Now let A be in A and T be in J. Since (EAE)(ET E) = (ET E)(EAE) and (E ⊥ AE ⊥ )(E ⊥ T E ⊥ ) = (E ⊥ T E ⊥ )(E ⊥ AE ⊥ ),

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we have AT =(EAE + EAE ⊥ + E ⊥ AE ⊥ )(ET E + ET E ⊥ + E ⊥ T E ⊥ ) =(EAE) ◦ (ET E) + 2(EAE) ◦ (ET E ⊥ ) + 2(EAE ⊥ ) ◦ (ET E ⊥ ) + (E ⊥ AE ⊥ ) ◦ (E ⊥ T E ⊥ ). So AT ∈ J since each summand in the above belongs to J. This proves that J is a left ideal. Similarly, we can verify that J is a right ideal. Let K be a Lie ideal. Let B be in K and S be an invertible element in A. With the decomposition H = E + E ⊥ , we can write −1 B1 B2 S S S2 −S1−1 S2 S3−1 B= , S= 1 and S −1 = 1 . 0 B3 0 S3 0 S3−1 Then

S1 B1 S1−1 (−S1 B1 S1−1 S2 S3−1 + S1 B2 + S2 B3 )S3−1 0 S3 B3 S3−1 B1 (S2 B3 − B1 S2 + S1 B2 )S3−1 = . 0 B3

SBS −1 =

In the last equality, we have used the commutativity of the algebras involved. By the proof of Proposition 3.2, EKE ⊥ is an ideal which is contained in K. So S1 B2 S3−1 ∈ K and B1 ⊕ B2 ∈ K. Moreover, S2 B3 − B1 S2 = [S2 , B] ∈ K. Consequently, SBS −1 ∈ K. So K is similarity invariant. We remark that the above example can apply to the so-called tridiagonal algebras A2n and A∞ . Those algebras have been found to be useful counterexamples to a number of plausible conjectures. In particular, they have non-trivial cohomology [8] and admit non-quasi-spatial automorphisms [9] and non-trivial Lie isomorphisms [22]. We refer the interested reader to those papers and references therein for further discussion of tridiagonal algebras.

Acknowledgment The authors thank the referee for the very thorough reading of the paper and valuable comments.

References [1] W. Arveson, Operator algebra and invariant subspace. Ann. Math. 100(1974), 433– 532. [2] M. Breˇsar, A. Foˇsner and M. Foˇser, Jordan ideals revisited. Monatsh. Math. 145(2005), 1–10. [3] P. Civin and B. Yood, Lie and Jordan structures in Banach algebras. Paciﬁc J. Math. 15(1965), 775–797.

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[4] K. Davidson, Nest algebras. Pitman Res. Notes, Math. Ser. 191, Longman Sci. Tech., New York, 1988. [5] K. Davidson and L. Orr, The invertibles are connected in infinite multiplicity nest algebras. Bull. London Math. Soc. 27 (1995), 155–161. [6] J. A. Erdos and S. C. Power, Weakly closed ideals of nest algebras. J. Operator Theory 7(1982), 219–235. [7] C. K. Fong, C. R. Miers, and A. R. Sourour, Lie and Jordan ideals of operators on Hilbert space. Proc. Amer. Math. Soc. 84(1982), 516–520. [8] F. Gilfeather, A. Hopenwasser and D. Larson, Reflexive algebras with finite width lattices: Tensor products, cohomology, compact pertubations. J. Funct. Anal. 55(1984), 176–199. [9] F. Gilfeather and R. L. Moore, Isomorphisms of certain CSL algebras. J. Funct. Anal. 67(1986), 264–291. [10] P. R. Halmos, A Hilbert space problem book. 2nd Edition, Springe, 1982. [11] D. Han, On A-submodules for reflexive operator algebras. Proc. Amer. Math. Soc. 104(1998), 253–266. [12] I. N. Herstein, On the Lie and Jordan rings of a simple, associative ring. Amer. J. Math. 77(1995), 279–285. [13] I. N. Herstein, Topics in ring theory. The University of Chicago Press, Chicago, 1969. [14] A. Hopenwasser, Complete distributivity. Proc. Symp. in Pure Math. 51(1990), 285– 305. [15] A. Hopenwasser and R. Moore, Finite rank operators in reflexive algebras. J. London Math. Soc. 27(1983), 331–338. [16] A. Hopenswasser and V. Paulsen, Lie ideals in operator algebras. J. Operator Theory 52(2004), 325–340. [17] T. D. Hudson, L. W. Marcoux and A. R. Sourour, Lie ideals in triagular operator algebras. Trans Amer. Math. Soc. 350(1998), 3321–3339. [18] M. S. Lambrou, Completely distributive lattices. Fundamenta Mathematica 119(1983), 227–240. [19] C. Laurie and W. E. Longstaﬀ, A note on rank one operators in reflexive algebras. Proc. Amer. Math. Soc. 89(1983), 293–297. [20] W. E. Longstaﬀ, Strongly reflexive lattices. J. London Math. Soc. 11(1975), 491–498. [21] W. E. Longstaﬀ, Operators of rank one in reflexive algebras. Canad. J. Math. 28(1976), 9–23. [22] F. Lu, Lie isomorphisms of reflexive algebras, J. Funct. Anal. 240(2006), 84–104. [23] F. Lu, Lie derivations of certain CSL algebras. Israel J. Math. 155(2006), 147–155. [24] L. W. Marcoux, On the closed Lie ideals of certain C*-algebras. Integral Equations Operator Theory 22(1995), 463–475. [25] L. W. Marcoux and A. R. Sourour, Conjugation-invariant subspaces and Lie ideals in non-selfadjoint operator algebras. J. London Math. Soc. 65(2002), 493–512. [26] C. R. Miers, Closed Lie ideals in operator algebras. Canad. J. Math. 33(1981), 1271– 1278.

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[27] L. Oliverira, Weak*-closed Jordan ideals of nest algebars. Math. Nachr. 248/249(2003), 129–143. [28] S. Shiral, On the Jordan structure of complex Banach *-algebras. Paciﬁc J. Math. 27(1968), 397–404. [29] E. Størmer, On the Jordan structure of C*-algebras. Trans. Amer. Math. Soc. 120(1965), 438–447. [30] D. Topping, The unitary invariant subspaces of B(H). preprint, 1970. Fangyan Lu and Xiuping Yu Department of Mathematics Suzhou University Suzhou 215006 P. R. China e-mail: [email protected] [email protected] Submitted: December 15, 2006 Revised: May 23, 2007

Integr. equ. oper. theory 59 (2007), 207–221 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020207-15, published online June 27, 2007 DOI 10.1007/s00020-007-1523-z

Integral Equations and Operator Theory

Schur Complements in Krein Spaces Alejandra Maestripieri and Francisco Mart´ınez Per´ıa To the memory of Professor Mischa Cotlar

Abstract. The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space H and a suitable closed subspace S of H, the Schur complement A/[S] of A to S is deﬁned. The basic properties of A/[S] are developed and diﬀerent characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. Mathematics Subject Classification (2000). Primary 46C20, 47B50; Secondary 46C50. Keywords. Krein spaces, Schur complements.

1. Introduction Let H be a Hilbert space, L(H) be the algebra of bounded linear operators on H and L(H)+ be the cone of positive operators in L(H). Given A ∈ L(H)+ and a closed subspace S of H, the Schur complement (or shorted operator) A/S was deﬁned by M. G. Krein [16] and W. N. Anderson and G. E. Trapp [2] as A/S = max{X ∈ L(H)+ : X ≤ A, R(X) ⊆ S ⊥ }, ≤

where the natural order ≤ in L(H)+ is considered. The notion of Schur complement was generalized to selfadjoint operators in Hilbert spaces, see [4], [9], [10], [17]. More generally, given Hilbert spaces H and K, J. Antezana et. al. [6] deﬁned the shorted operator for an arbitrary A ∈ L(H, K) with respect to a pair of suitable closed subspaces S and T of H ad K, respectively. If A is a positive operator, E. Pekarev [18] proved that A/S = A1/2 PM⊥ A1/2 ,

(1.1)

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where M = A1/2 (S) and PM⊥ is the orthogonal projection onto M⊥ . This paper is devoted to study the Schur complement of J-selfadjoint operators in Krein spaces, whose deﬁnition is inspired by Eq. (1.1). Let H be a Krein space with fundamental symmetry J. Bogn´ ar-Kramli’s theorem [8] states that, if A ∈ L(H) is J-selfadjoint then there exist a Krein space K and a bounded injective operator D ∈ L(K, H) such that A = DD# , where D# ∈ L(H, K) denotes the J-adjoint operator of D. However, this decomposition may not be unique (see [19]). A J-selfadjoint operator A ∈ L(H) has the unique factorization property if, for any pair of decompositions A = Di Di# , Di ∈ L(Ki , H), N (Di ) = {0} (i = 1, 2), there exists an isomorphism U ∈ L(K1 , K2 ) such that D1 = D2 U . Consider a J-selfadjoint operator A ∈ L(H) with the unique factorization property and suppose that M = D# (S) is a Krein subspace of K, then the Schur complement of A to S is deﬁned as A/[S] = DPM[⊥] //M D# ,

(1.2)

where M[⊥] is the J-orthogonal subspace to M in the Krein space K and PM[⊥] //M ∈ L(K) is the J-selfadjoint projection onto M[⊥] . The main properties of shorted operators in Hilbert spaces, which where proved by M. G. Krein [16], W. N. Anderson and G. E. Trapp [2] and E. Pekarev [18], have a natural counterpart for Schur complements in Krein spaces. The contents of the paper are the following: Section 2 introduces the basic notation and some known results in Krein spaces including topics such as Bogn´ ar-Kramli’s theorem, the unique factorization property, and J-contractive projections. It also contains the deﬁnition and a summary of the properties of the shorting operation in Hilbert spaces. In Section 3, the Schur complement of A to S, A/[S] , and the S-compression of A, A[S] , are deﬁned for a given J-selfadjoint operator A ∈ L(H) with the unique factorization property; also, the range and the nullspace of A/[S] and A[S] are characterized. Section 4 is devoted to study the Schur complement for deﬁnite subspaces. In particular, it is proved that, if M = D# (S) is a J-nonnegative subspace of K, then A/[S] = max{X ∈ I(A) : X ≤J A, R(X) ⊆ S [⊥] }, ≤J

where I(A) = {X = EE

#

: E ∈ L(K, H), R(E) ⊆ R(D)}. Also, it is shown that

A/[S] = inf {Q# AQ : Q ∈ Q(H), N (Q) = S}. ≤J

Finally, in Section 5 the Schur complement for J-positive operators is described in detail. In this case A/[S] is deﬁned for every closed subspace S of H and it always has both extremal characterizations. Furthermore, the shorting operation

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of a J-positive operator A in a Krein space H is intimately related to the shorted of JA in the Hilbert space |H|. This relationship allows to translate the classical results into the Krein space’s context.

2. Preliminaries Along this work H denotes either a (complex, separable) Hilbert space with inner product , or a (complex) Krein space with indeﬁnite metric [ , ], depending on the context. If S is a subspace of a Hilbert space H, S ⊥ is the orthogonal complement of S. Analogously, if S is a subspace of a Krein space H, the Jorthogonal subspace to S is the closed subspace of H deﬁned by S [⊥] = {x ∈ H : [ x, y ] = 0 for every y ∈ S}. Sometimes we use the notation S [⊥]H instead of S [⊥] to emphasize the Krein space considered. Given two Hilbert spaces H and K, L(H, K) is the algebra 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. Given a Hilbert space H, let L(H)+ be the cone of (semideﬁnite) positive operators in L(H) and denote by Q(H) the set of projections in L(H), i.e., Q(H) = {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 . If H = S T , the oblique projection onto S along T , PS//T , is the projection with R(PS//T ) = S and N (PS//T ) = T . In particular, PS = PS//S ⊥ is the orthogonal projection onto S. Krein spaces In what follows we give some basic results on Krein spaces. For a complete exposition of the subject and the proofs of the results below see the books by J. Bogn´ ar [7] and T. Ya. Azizov and I. S. Iokhvidov [15], the monographs by T. Ando [3] and by M. Dritschel and J. Rovnyak [12] and the paper by J. Rovnyak [19]. Given a Krein space H and a fundamental decomposition H = H+ ⊕ H− , the direct sum of the Hilbert spaces (H+ , [ , ]) and (H− , −[ , ]) is denoted by |H|. If H and K are Krein spaces then L(H, K) (respectively L(H)) stands for L(|H|, |K|) (respectively L(|H|)). Given T ∈ L(H, K), the J-adjoint operator of T is denoted by T # . An operator T ∈ L(H) is J-selfadjoint if T = T # . The following theorem is due to J. Bogn´ar and A. Kr´ amli [8]. See also Theorem 1.1 in [12]. Theorem 2.1 (Bogn´ar-Kr´amli). Let H be a Krein space with fundamental symmetry J. Any J-selfadjoint operator T ∈ L(H) can be written in the form T = W W #, where W ∈ L(K, H) for some Krein space K and N (W ) = {0}. While factorizations as in Theorem 2.1 always exist, they are not in general unique.

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Definition 2.2. Let H be a Krein space with fundamental symmetry J. A Jselfadjoint operator T ∈ L(H) has the unique factorization property (UFP) if for any two factorizations T = Wi Wi# ,

Wi ∈ L(Ki , H),

N (Wi ) = {0},

i = 1, 2,

there is an isomorphism U ∈ L(K1 , K2 ) such that W1 = W2 U . Remark 2.3. Let T ∈ L(H) be a J-selfadjoint operator satisfying the UFP and suppose that T = W W # where W ∈ L(K, H), N (W ) = {0} and K is a Krein space. Then, 1. if T = DD# is another factorization of T as in Theorem 2.1 then R(D) = R(W ); 2. if R(T ) is closed then R(D# ) = K. An operator T ∈ L(H) is J-positive if [ T x, x ] ≥ 0 for every x ∈ H. We denote it by T ≥J 0. If T1 and T2 are J-selfadjoint operators, we say that T1 ≥J T2 if T1 − T2 ≥J 0. It is easy to show that ≥J is a partial order in the real vector space of J-selfadjoint operators. The following theorem provides some examples of classes of operators with the UFP. Theorem 2.4. Let H be a Krein space with fundamental symmetry J, and let T ∈ L(H) be a J-selfadjoint operator. Each of the following conditions is suﬃcient for T to have the unique factorization property: 1. T ≥J 0; 2. T 2 ≤J T . Given a Krein space H, an operator T ∈ L(H) is J-contractive if [ T x, T x ] ≤ [ x, x ] for every x ∈ H. Therefore, T is J-contractive if and only if T # T ≤J I. Analogously, an operator T ∈ L(H) is J-expansive if [ T x, T x ] ≥ [ x, x ] for every x ∈ H (i.e. T # T ≥J I). We say that S is a Krein subspace of H if it is a Krein space with the indeﬁnite metric of H. It is well known that S is a Krein subspace of H if and only if S = R(Q) for some J-selfadjoint Q ∈ Q(H). Also, a subspace S of H is J-nonnegative (respectively J-nonpositive) if [ x, x ] ≥ 0 (respectively [ x, x ] ≤ 0) for every x ∈ S. S. Hassi and K. Nordstr¨ om proved the following result, which characterizes those projections which are J-contractive (see [14, §3, Proposition 5]). A similar result holds for J-expansive projections. Proposition 2.5. If Q ∈ Q(H) then the following conditions are equivalent: 1. Q is J-contractive; 2. Q is J-selfadjoint and N (Q) is J-nonnegative; 3. I − Q is J-positive. Hassi and Nordstr¨om [14, §4, Theorem 2] also proved that every J-selfadjoint projection Q can be factored as follows.

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Theorem 2.6. Let Q be a J-selfadjoint projection in a Krein space H. Then, Q can be represented as Q = Q+ Q− where Q+ and Q− are two commuting projections such that Q+ is J-contractive and Q− is J-expansive. Shorted operators in Hilbert spaces Definition 2.7 (Krein [16], Anderson-Trapp [1], [2]). Let H be a Hilbert space. Given A ∈ L(H)+ and a closed subspace S of H, the shorted operator of A to S is deﬁned by A/S = max{X ∈ L(H)+ : X ≤ A, R(X) ⊆ S ⊥ }, ≤

where ≤ is the natural order given by the cone L(H)+ . The following theorem collects many well known results about shorted operators. See [2], [18], [9], [10] for the proof of these facts. Theorem 2.8. Let S be a closed subspace of a Hilbert space H and let A ∈ L(H)+ . Then: 1. 2. 3. 4. 5.

If M = A1/2 (S) then A/S = A1/2 PM⊥ A1/2 . R(A) ∩ S ⊥ ⊆ R(A/S ) ⊆ R(A1/2 ) ∩ S ⊥ and N (A/S ) = A−1/2 (M). R((A/S )1/2 ) = R(A1/2 ) ∩ S ⊥ . A/S = inf{Q∗ AQ : Q ∈ Q(H), N (Q) = S}. If T is a closed subspace of H such that S + T is closed then A/S+T = (A/S )/T = (A/T )/S .

If H is a Hilbert space and (An )n∈N is a sequence in L(H) we say that (An )n∈N SOT converges in the SOT topology to A ∈ L(H) (and denote it by An −−−−→ A) if n→∞

An x − Ax −−−−→ 0 for every x ∈ H. Moreover, if (An )n∈N and A are selfadjoint n→∞

SOT

SOT

operators, we say that An A if An −−−−→ A and An ≥ An+1 (≥ A) for every n→∞ n ∈ N. The following are some results about the continuity of the shorting operation, see [2], [5]. SOT

Proposition 2.9. Let An (n ∈ N) and A be operators in L(H)+ such that An A SOT

as n → ∞. Then, (An )/S A/S as n → ∞, for every closed subspace S of H. SOT

Proposition 2.10. Let Sn (n ∈ N) and S be closed subspaces such that PSn PS SOT

as n → ∞. Then, A/Sn A/S as n → ∞, for every A ∈ L(H)+ . SOT

The following example shows that PSn PS is not a suﬃcient condition to imply the convergence of the sequence (A/Sn )n∈N to A/S .

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Example 2.11. Let A ∈ L(H)+ such that N (A) = {0} and R(A) is not closed. Consider a dense subspace T of H such that T ∩ R(A1/2 ) = {0} and let {en }n∈N be an orthonormal basis of H contained in T . SOT

Let Sn = span{ek : k ≥ n} for n ≥ 1. Then, PSn 0. Furthermore, A/Sn = 0 because R((A/Sn )1/2 ) = R(A1/2 ) ∩ Sn⊥ = R(A1/2 ) ∩ span{e1 , . . . , en } = {0}. But A/{0} = A = 0.

3. Schur complements in Krein spaces Let H be a Krein space with fundamental symmetry J and A ∈ L(H) be a Jselfadjoint operator satisfying the UFP. Suppose that A = DD# , where K is a Krein space and D ∈ L(K, H) with N (D) = {0}. Given a closed subspace S of H, consider M = D# (S) and suppose that M is a Krein subspace of K. Definition 3.1. Under the above hypothesis, the Schur complement of A to S is deﬁned by A/[S] = DPM[⊥] //M D# , and the S-compression of A is A[S] = DPM//M[⊥] D# . The operators A[S] and A/[S] are well deﬁned: by the UFP of A, if A = Di Di# where Di ∈ L(Ki , H) and N (Di ) = {0} for i = 1, 2, there exists an isomorphism U ∈ L(K1 , K2 ) such that D1 = D2 U . Given the subspaces Mi = Di# (S), for i = 1, 2, observe that M1 is a Krein subspace of K1 if and only if M2 = U (M1 ) is a Krein subspace of K2 , and in this case U PM1 //M[⊥] U # = PM2 //M[⊥] . Then, 1

D1 PM1 //M[⊥] D1# 1

= D2 (U PM1 //M[⊥] U 1

#

)D2#

2

= D2 PM2 //M[⊥] D2# . 2

Also, the following properties hold for the Schur complement A/[S] and the Scompression A[S] : i. A[S] , A/[S] ∈ L(H), ii. A[S] , A/[S] are JH -selfadjoint operators (because PM//M[⊥] and PM[⊥] //M are JK -selfadjoint), iii. A[S] + A/[S] = A. Let us characterize the range and the nullspace of A[S] and A/[S] . The lemma below is well known and its proof is straightforward. Lemma 3.2. Let H and K be Krein spaces. If T ∈ L(H, K) then, 1. N (T # ) = R(T )[⊥]K . 2. T # (S) [⊥]H = T −1 (S [⊥]K ) for every subspace S of K.

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Proposition 3.3. Let A = DD# ∈ L(H) be a J-selfadjoint operator satisfying the UFP and S a closed subspace of H such that M = D# (S) is a Krein subspace of K. Then, 1. A(S) ⊆ R(A[S] ) ⊆ A(S); 2. N (A[S] ) = A(S)[⊥] ; 3. R(A) ∩ S [⊥] ⊆ R(A/[S] ) ⊆ R(D) ∩ S [⊥] ; 4. N (A/[S] ) = (D# )−1 (M). Proof. 1. It is easy to see that A(S) = D(D# (S)) = A[S] (S)

⊆ R(A[S] ) ⊆ D(M) = D(D# (S)) ⊆ DD# (S) = A(S).

2. Since N (D) = {0}, it follows that N (A[S] ) = N (PM//M[⊥] D# ) = (D# )−1 (M[⊥] ) = A−1 (S [⊥] ) = A(S)[⊥] . 3. First of all observe that, by Remark 2.3, R(D) does not depend on the factorization. If y ∈ R(A) ∩ S [⊥] then there exists x ∈ H such that y = Ax ∈ S [⊥] . Note that D# x ∈ M[⊥] and A/[S] x = DPM[⊥] //M (D# x) = DD# x = y. Thus, R(A) ∩ S [⊥] ⊆ R(A/[S] ). On the other hand, R(A/[S] ) ⊆ D(M[⊥] ) = D(D−1 (S [⊥] )) = S [⊥] ∩ R(D). 4. As in item 2., notice that N (A/[S] ) = N (PM[⊥] //M D# ) = (D# )−1 (M).

In general, the inclusions in items 1. and 3. of the above proposition are strict. See the examples in [2] and [10]. Proposition 3.4. Let A ∈ L(H) be a J-selfadjoint operator satisfying the UFP, A = DD# , D ∈ L(K, H) with N (D) = {0}, and S a closed subspace of H such that M = D# (S) is a Krein subspace of K. If T is a closed subspace of H such that S ⊆ T ⊆ (D# )−1 (M) then D# (T ) = M and A/[T ] = A/[S] . Proof. Let T be a closed subspace of H such that S ⊆ T ⊆ (D# )−1 (M), then applying D# it follows that D# (S) ⊆ D# (T ) ⊆ D# ((D# )−1 (M)) ⊆ M. Therefore, D# (T ) = M and A/[T ] = A/[S] .

4. Extremal properties for definite subspaces The main results in this section are stated for both J-nonnegative and J-nonpositive subspaces, but we only give the proofs for J-nonnegative ones. The proofs in the nonpositive case are similar. Let A ∈ L(H) be a J-selfadjoint operator satisfying the UFP. If A = DD# where K is a Krein space and D ∈ L(K, H) with N (D) = {0}, consider the set I(A) = {X = EE # : E ∈ L(K, H), R(E) ⊆ R(D)}.

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By Remark 2.3, the subspace R(D) only depends on A, so that, the same is true for the set I(A). If S is a closed subspace of H, consider the subsets M− (A, S [⊥] ) = +

M (A, S

[⊥]

) =

{X ∈ I(A) : X ≤J A, R(X) ⊆ S [⊥] }, {X ∈ I(A) : A ≤J X, R(X) ⊆ S [⊥] }.

Observe that these sets can be empty. First of all, consider the particular case A = I. Observe that I ∈ L(H) has the UFP because it satisﬁes a suﬃcient condition: I 2 = I ≤J I (see Theorem 2.4). Furthermore, the unique factorization (up to isomorphism) is I = DD# , where D = I ∈ L(H) and therefore M− (I, S [⊥] ) = {X ∈ L(H) : X ≤J I, R(X) ⊆ S [⊥] } and M+ (I, S [⊥] ) = {X ∈ L(H) : I ≤J X, R(X) ⊆ S [⊥] }. Lemma 4.1. Let S be a Krein subspace of H and Q = PS [⊥] //S . Then, 1. Q = max M− (I, S [⊥] ) if S is J-nonnegative. ≤J

2. Q = min M+ (I, S [⊥] ) if S is J-nonpositive. ≤J

Proof. Suppose that S is a J-nonnegative Krein subspace of H. Then, Q is Jcontractive (see Proposition 2.5) and R(Q) = S [⊥] . Therefore, Q ∈ M− (I, S [⊥] ). Moreover, if X ∈ M− (I, S [⊥] ) then X ≤J Q: R(X) ⊆ S [⊥] implies that QX = X, and QXQ = (QX)Q = XQ = QX = X because X and Q are Jselfadjoint. Then, if x ∈ H, [ (Q − X)x, x ] = [ Q(I − X)Qx, x ] = [ (I − X)Qx, Qx ] ≥ 0, i.e. X ≤J Q. Therefore, Q = max≤J M− (I, S [⊥] ).

Corollary 4.2. Let S be a Krein subspace of H. If Q = PS [⊥] //S then there exist two Krein subspaces S+ and S− of H such that S = S+ S− and [⊥]

[⊥]

Q = max M− (I, S+ ) min M+ (I, S− ). ≤J

≤J

Proof. If S is a Krein subspace of H then, by Theorem 2.6, Q = Q+ Q− , where Q+ and Q− are commuting projections such that Q+ is J-contractive and Q− is J-expansive. Also, (I − Q+ )(I − Q− ) = 0 (see the proof in [14]) so that I − Q = (I − Q+ ) + (I − Q− ) and S = N (Q) = N (Q+ ) N (Q− ). By Lemma 4.1, Q+ = max M− (I, R(Q+ )) and Q− = min M+ (I, R(Q− )). ≤J

Therefore, taking S± = N (Q± ), the proof is complete.

≤J

The following theorem is an extremal characterization of the Schur complement similar to the one given by Anderson-Trapp [2, Theorem 1]. Theorem 4.3. Let M = D# (S) be a Krein subspace of K. Then: 1. A/[S] = max M− (A, S [⊥] ) if M is J-nonnegative. ≤J

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2. A/[S] = min M+ (A, S [⊥] ) if M is J-nonpositive. ≤J

Proof. Let Q = PM[⊥] //M and suppose that M is J-nonnegative (i.e. Q is Jcontractive). Notice that A/[S] = (DQ)(DQ)# and R(DQ) ⊆ R(D), then A/[S] ∈ I(A). Since Q ≤J I we have that A/[S] = DQD# ≤J DD# = A and, by Proposition 3.3, R(A/[S] ) ⊆ S [⊥] . Therefore, A/[S] ∈ M− (A, S [⊥] ). Moreover, A/[S] = max M− (A, S [⊥] ). Indeed, if X = EE # ∈ M− (A, S [⊥] ) ≤J

then R(E) ⊆ R(D) and, by Douglas’ theorem [11, Theorem 1], the equation DY = E admits a bounded solution in L(K). If Z ∈ L(K) is a solution of the above equation, then X = DZZ # D# . Since X ≤J A, given x ∈ H, [ (IK − ZZ # )D# x, D# x ]K = [ D(I − ZZ # )D# x, x ]H = [ (A − X)x, x ]H ≥ 0, so [ (IK − ZZ # )y, y ]K ≥ 0 for every y ∈ R(D# ) = N (D)[⊥]K = K. Hence, ZZ # ≤J IK . Since R(X) ⊆ S [⊥] we have that R(ZZ # D# ) ⊆ D−1 (S [⊥] ) = M[⊥] . Moreover, R(ZZ # ) = ZZ # (R(D# )) ⊆ R(ZZ # D# ) ⊆ M[⊥] . Therefore, ZZ # ∈ M− (I, M[⊥] ) and, by Lemma 4.1, ZZ # ≤J Q (notice that the Krein space considered here is K). Then, X = DZZ # D# ≤J DQD# = A/[S] , i.e. A/[S] = max M− (A, S [⊥] ).

≤J

Corollary 4.4. Let H be a Krein space and A ∈ L(H) a J-selfadjoint operator with the UFP. Consider a factorization A = DD# where K is a Krein space and D ∈ L(K, H) with N (D) = {0}. If A has closed range and S is a closed subspace of H such that M = D# (S) is a Krein subspace of K, then there exist two closed subspaces S+ and S− of H such that S+ S− = (D# )−1 (M) and [⊥]

[⊥]

A/[S] = max M− (A, S+ ) + min M+ (A, S− ) − A. ≤J

≤J

Proof. Suppose that M is a Krein subspace of K and let Q = PM[⊥] //M . By Theorem 2.6, there exist commuting projections Q+ and Q− such that Q = Q+ Q− , where Q+ is J-contractive, Q− is J-expansive and N (Q) = N (Q+ ) N (Q− ) (see the proof in [14]). Let S± = (D# )−1 (N (Q± )) and deﬁne M± = D# (S± ). Since R(D# ) = K (see Remark 2.3), it follows that M± = D# (S± ) = N (Q± ) ∩ R(D# ) = N (Q± ). Therefore, A/[S± ] = DQ± D# and A[S] = D(I − Q)D# = D((I − Q+ ) + (I − Q− ))D# = A[S+ ] + A[S− ] . As a consequence of Proposition 2.5, the subspaces M+ and M− are J-nonnegative and J-nonpositive, respectively. Then, by Theorem 4.3, A/[S]

=

A − A[S] = A − (A[S+ ] + A[S− ] ) = A/[S+ ] + A/[S− ] − A =

=

max M− (A, S+ ) + min M+ (A, S− ) − A.

[⊥]

≤J

[⊥]

≤J

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Theorem 4.5. Let S be a closed subspace of H. Suppose that A ∈ L(H) is Jselfadjoint and satisﬁes the UFP. If A = DD# with D ∈ L(K, H), N (D) = {0}, suppose that M = D# (S) is a Krein subspace of K. Then: 1. A/[S] = inf {Q#AQ : Q ∈ Q(H), N (Q) = S} if M is J-nonnegative. ≤J

2. A/[S] = sup{Q#AQ : Q ∈ Q(H), N (Q) = S} if M is J-nonpositive. ≤J

Proof. Suppose that M is J-nonnegative and consider P = PM[⊥] //M . Then, for every x ∈ K, [ P x, P x ]K = min [ x − m, x − m ]K . m∈M

Indeed, given x ∈ K and m ∈ M, [ x − m, x − m ] = =

[ P x + (I − P )x − m, P x + (I − P )x − m ] = [ P x, P x ] + [ (I − P )x − m, (I − P )x − m ] ≥ [ P x, P x ].

Furthermore, observe that D# (S) is dense in M. Then, if y ∈ H, = [ P D# y, P D# y ]K = min [ D# y − m, D# y − m ]K =

[ A/[S] y, y ]H

m∈M

=

inf [ D# (y − s), D# (y − s) ]K = inf [ A(y − s), y − s ]H .

s∈S

s∈S

If Q ∈ Q(H) with N (Q) = S, given x ∈ H, [ Q# AQx, x ]H

= ≥

[ AQx, Qx ]H = [ A(x − (I − Q)x), x − (I − Q)x ]H ≥ [ A/[S] x, x ]H

because (I − Q)x ∈ S. Then, A/[S] ≤J Q# AQ for every Q ∈ Q(H) with N (Q) = S i.e. A/[S] is a lower bound of the set {Q# AQ : Q ∈ Q(H), N (Q) = S}. Let C be any lower bound of the set {Q# AQ : Q ∈ Q(H), N (Q) = S}, we are going to show that C ≤J A/[S] . Fixed x ∈ H, if x ∈ S, observe that for every s ∈ S there exists Q ∈ Q(H) with N (Q) = S such that (I − Q)x = s. Therefore, [ A(x − s), x − s ]H = [ AQx, Qx ]H ≥ [ Cx, x ]H for every s ∈ S. Then, [ A/[S] x, x ]H ≥ [ Cx, x ]H . On the other hand, if x ∈ S then Q# AQx = 0 for every Q ∈ Q(H) with N (Q) = S. Therefore, [ Cx, x ]H ≤ [ Q# AQx, x ]H = 0. But A/[S] x = DPM[⊥] //M D# x = 0 because D# x ∈ M. Thus, [ A/[S] x, x ]H = 0 ≥ [ Cx, x ]H . Since x ∈ H was arbitrary, A/[S] ≥J C. So, A/[S] = inf {Q# AQ : Q ∈ Q(H), N (Q) = S}. ≤J

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5. Schur complements of J-positive operators in Krein spaces By Theorem 2.4, J-positive operators have the unique factorization property. Furthermore, it is easy to see that, given a factorization as in Theorem 2.1, the vector space K acting as the domain of the factor can be chosen to be a Hilbert space (see Theorem 1.1 in [12]). Let H be a Krein space and A ∈ L(H) be J-positive. Along this section, we are going to use the following factorization of A: if |A| = JA ∈ L(|H|)+ , consider the Hilbert space K = J(N (A)⊥ ) and D = J|A|1/2 J|K ∈ L(K, H). Then, N (D) = {0}, D# = J|A|1/2 ∈ L(H, K) and DD# = A. Observe that, if K is a Hilbert space and S is any closed subspace of H, then the subspace M = D# (S) is a closed subspace of K and therefore a “Krein subspace” of K. Thus, the Schur complement A/[S] is well deﬁned for every closed subspace S of H and A/[S]

= DPM⊥ D# = (J|A|1/2 J)PM⊥ (J|A|1/2 ) = J|A|1/2 (JPM⊥ J)|A|1/2 = = J|A|1/2 PJ(M⊥ ) |A|1/2 ,

(5.1)

where PJ(M⊥ ) ∈ L(K) is the orthogonal projection onto J(M⊥ ). Therefore, A/[S] is J-positive. Furthermore, notice that the operator E ∈ L(M⊥ , H) deﬁned by Ex = Dx = J|A|1/2 Jx, x ∈ M⊥ satisﬁes A/[S] = EE # ,

and

N (E) = {0}.

Therefore, it is the unique factorization (up to isomorphism) of A/[S] . ⊥

Remark 5.1. Observe that J(M⊥ ) = JD# (S) = (|A|1/2 (S))⊥ . Thus, from Eq. (5.1) and item 1. of Theorem 2.8 follows that, if A ∈ L(H) is J-positive then A/[S] = J (|A|/S ),

(5.2)

where |A|/S is the shorted operator (in the Hilbert space sense) of |A| to S. Therefore, the shorting operation of a J-positive operator A in a Krein space H is intimately related to the shorted of the positive operator JA in the Hilbert space |H|. The following propositions translate the classical results of Schur complements into Krein space’s context. First of all, we state Douglas’ theorem for J-positive operators in Krein spaces. Theorem 5.2. Let H be a Krein space and consider J-positive operators A, B ∈ L(H). If A = DD# , D ∈ L(K1 , H), N (D) = {0} is any factorization of A as in Theorem 2.1 (resp. B = EE # , E ∈ L(K2 , H), N (E) = {0}) then the following conditions are equivalent: 1. equation DX = E has a solution in L(K2 , K1 ); 2. R(E) ⊆ R(D); 3. there exists λ > 0 such that B ≤J λA. In this case, there exists a unique X ∈ L(K2 , K1 ) such that DX = E. Moreover, N (X) = N (E) and X = inf{λ > 0 : B ≤J λA}.

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Proof. Observe that if A (resp. B) is J-positive then K1 (resp. K2 ) is a Hilbert space. Therefore, D# = D∗ J and E # = E ∗ J. So, equation A ≤J λB is equivalent to DD∗ ≤ λEE ∗ and the results follows by Douglas’ theorem [11]. Proposition 5.3. If S and T are closed subspaces of H and A, B ∈ L(H) are J-positive, then 1. A/[S] = max M− (A, S [⊥] ) = max{X ∈ L(H) : 0 ≤J X ≤J A, R(X) ⊆ ≤J

≤J

S [⊥] }; 2. A/[S] = inf {Q#AQ : Q ∈ Q(H), N (Q) = S}; ≤J

3. if A ≤J B then A/[S] ≤J B/[S] ; 4. if T ⊆ S then A/[S] ≤J A/[T ] . Proof. 1. Given A ∈ L(H) J-positive and S a closed subspace of H, A/[S] = max≤J M− (A, S [⊥] ) by Theorem 4.3 (recall that K is a Hilbert space). Furthermore, M− (A, S [⊥] ) = {X ∈ L(H) : 0 ≤J X ≤J A, R(X) ⊆ S [⊥] }. Let A = {X ∈ L(H) : 0 ≤J X ≤J A, R(X) ⊆ S [⊥] }. If X ∈ A then X ≥J 0 and it admits a factorization X = EE # , where E ∈ L(K1 , H), N (E) = {0} and K1 is a Hilbert space, but we can substitute K1 be the Hilbert space K appearing in the decomposition of A. Since X ≤J A it follows that R(E) ⊆ R(D) by Theorem 5.2. Thus X ∈ I(A), and the conditions X ≤J A and R(X) ⊆ S [⊥] implies that X ∈ M− (A, S [⊥] ). On the other hand, if X ∈ M− (A, S [⊥] ) then there exists E ∈ L(K, H) such that X = EE # = EE ∗ J because K is a Hilbert space. Then, X ≥J 0 and, by the remaining conditions on X, X ∈ A. Therefore, M− (A, S [⊥] ) ⊆ A. 3. If A ≤J B then |A| = JA ≤ JB = |B|. By Theorem 2.8, |A|/S ≤ |B|/S and therefore A/[S] = J(|A|/S ) ≤J J(|B|/S ) = B/[S] (see Eq. (5.2)). Items 2. and 4. follows analogously.

The following proposition generalizes item 3. of Theorem 2.8: Proposition 5.4. Let S be a subspace of H and A ∈ L(H) a J-positive operator. If A = DD# (with K a Hilbert space, D ∈ L(K, H), N (D) = {0}) and A/[S] = EE # (with E a Hilbert space, E ∈ L(E, H), N (E) = {0}) then R(E) = R(D) ∩ S [⊥] . Proof. If A = DD# with D ∈ L(K, H), N (D) = {0} then A/[S] = F F # where F ∈ L(M⊥ , H) is deﬁned by F x = Dx for x ∈ M⊥ . Thus, R(F ) = R(DPM⊥ ) = D(M⊥ ) = D(D−1 (S [⊥] )) = R(D) ∩ S [⊥] , and, by Remark 2.3, R(E) = R(F ) = R(D) ∩ S [⊥] .

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Proposition 5.5. Let H be a Krein space and A ∈ L(H) a J-positive operator. If S1 and S2 are closed subspaces of H such that S1 + S2 is closed then A/[S1 +S2 ] = (A/[S1 ] )/[S2 ] = (A/[S2 ] )/[S1 ] . Proof. Suppose that S1 and S2 are closed subspaces of H such that S1 +S2 is closed. Consider |A| = JA ∈ L(|H|)+ . Then, by item 5. of Theorem 2.8, |A|/S1 +S2 = (|A|/S1 )/S2 = (|A|/S2 )/S1 . Therefore, by Eq. (5.2), A/[S1 +S2 ] = J(|A|/S1 +S2 ) = J[(|A|/S1 )/S2 ] = (J(|A|/S1 ))/[S2 ] = (A/[S1 ] )/[S2 ] . Analogously, A/[S1 +S2 ] = (A/[S2 ] )/[S1 ] .

In what follows, given a sequence (Tn )n∈N of J-positive operators, the notaJ-SOT

SOT

tion Tn T stands for Tn −−−−→ T and Tn ≥J Tn+1 (≥J T ) for every n ∈ N. n→∞

J-SOT

SOT

J-SOT

Observe that, Tn T if and only if JTn JT : Indeed, if Tn T SOT SOT then Tn −−−−→ T and Tn ≥J Tn+1 (≥J T ). Equivalently, JTn −−−−→ JT (because n→∞

SOT

n→∞

J is invertible) and JTn ≥ JTn+1 (≥ JT ), i.e. JTn JT . The next proposition follows easily using the remark above and Propositions 2.9 and 2.10. Proposition 5.6. Let H be a Krein space. J-SOT

1. If (An )n∈N is a sequence of J-positive operators in L(H) such that An A, then J-SOT

An /[S] A/[S] . 2. If (Sn )n∈N and S are closed subspaces of H such that Sn ⊆ Sn+1 for every J-SOT n ∈ N and S = n∈N Sn , then A/[Sn ] A/[S] for every J-positive operator A ∈ L(H). Remark 5.7. Example 2.11 can be modiﬁed to provethat item 2 of Proposition 5.6 is not true if Sn ⊇ Sn+1 for every n ∈ N and S = n∈N Sn .

Acknowledgment The authors would like to acknowledge Jorge A. Antezana for fruitful comments concerning shorted operators in Hilbert spaces and the results herein. They also express their gratitude to the referee who helped them to improve this paper.

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References [1] W. N. Anderson Jr., Shorted operators, SIAM J. Appl. Math. 20 (1971) 520–525. [2] W. N. Anderson Jr. and G. E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28 (1975) 60–71. [3] T. Ando, Linear operators on Krein spaces, Hokkaido University, Sapporo, Japan, 1979. [4] T. Ando, Generalized Schur complements, Linear Algebra Appl. 27 (1979), 173–186. [5] J. Antezana, G. Corach and D. Stojanoﬀ, Spectral Shorted Operators, Integ. equ. oper. theory 55 (2006), 169–188. [6] J. Antezana, G. Corach and D. Stojanoﬀ, Bilateral shorted operators and parallel sums, Linear Algebra Appl. 414 (2006), no. 2-3, 570–588. [7] J. Bogn´ ar, Indefinite inner product spaces, Springer-Verlag, 1974. [8] J. Bogn´ ar and A. Kr´ amli, Operators of the form C ∗ C in indefinite inner product spaces, Acta Sci. Math. (Szeged) 29 (1968), 19–29. [9] G. Corach, A. Maestripieri and D. Stojanoﬀ, Oblique projections and Schur complements, Acta Sci. Math. (Szeged) 67 (2001), 337–256. [10] G. Corach, A. Maestripieri and D. Stojanoﬀ, Generalized Schur complements and oblique projections, Linear Algebra Appl. 341 (2002), 259–272. [11] R. G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–416. [12] M. A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, Fields Institute Monographs no. 3, Amer. Math. Soc. Edited by Peter Lancaster 1996, 3, 141–232. [13] M. A. Dritschel and J. Rovnyak, Extension theorems for contraction operators on Krein spaces, Oper. Theory Adv. Appl. 47 (1990), 221–305. [14] S. Hassi and K. Nordstr¨ om, On projections in a space with an indefinite metric, Linear Algebra Appl. 208-209 (1994), 401–417. [15] I. S. Iokhvidov, T. Ya. Azizov, Linear Operators in spaces with an indefinite metric, John Wiley and sons, 1989. [16] M. G. Krein, The theory of self-adjoint extensions of semibounded Hermitian operators and its applications, Mat. Sb. (N. S.) 20 (62) (1947), 431–495. [17] P. Massey and D. Stojanoﬀ, Generalized Schur Complements and P -Complementable Operators, Linear Algebra Appl. 393 (2004) 299–318. [18] E. L. Pekarev, Shorts of operators and some extremal problems, Acta Sci. Math. (Szeged) 56 (1992) 147–163. [19] J. Rovnyak, Methods on Krein space operator theory, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999), Oper. Theory Adv. Appl., 134 (2002), 31–66.

Vol. 59 (2007)

Schur Complements in Krein Spaces

Alejandra Maestripieri Departamento de Matem´ atica – Facultad de Ingenier´ıa Universidad de Buenos Aires and Instituto Argentino de Matem´ atica – CONICET Saavedra 15 – 3o Piso 1083 Buenos Aires Argentina e-mail: [email protected] Francisco Mart´ınez Per´ıa Departamento de Matem´ atica – Facultad de Ciencias Exactas Universidad Nacional de La Plata and Instituto Argentino de Matem´ atica – CONICET Saavedra 15 – 3o Piso 1083 Buenos Aires Argentina e-mail: [email protected] Submitted: November 15, 2006 Revised: June 8, 2007

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Integr. equ. oper. theory 59 (2007), 223–244 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020223-22, published online June 27, 2007 DOI 10.1007/s00020-007-1514-0

Integral Equations and Operator Theory

The Structure of the Closure of the Rational Functions in Lq (µ) Zhijian Qiu Abstract. Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K ◦ . Let µ be a positive ﬁnite measure with its support contained in K. For 1 ≤ q < ∞, let Aq (K, µ) denote the closure of A(K) in Lq (µ). The aim of this work is to study the structure of the space Aq (K, µ). We seek a necessary and suﬃcient condition on K so that a Thomson-type structure theorem for Aq (K, µ) can be established. Our theorem deduces J. Thomson’s structure theorem for P q (µ), the closure of polynomials in Lq (µ), as the special case when K is a closed disk containing the support of µ. Mathematics Subject Classification (2000). Primary 46E30; Secondary 30H05, 30E10, 46E15. Keywords. Analytic bounded point evaluations, rational approximation, function algebras, subnormal operator.

Introduction Let 1 ≤ q < ∞ and let µ be a positive ﬁnite (regular Borel) measure with compact support in the complex plane C. Let K be a compact subset that contains the support of µ. The purpose of this paper is to investigate the following problem: What is the closure of A(K) in Lq (µ)? This is a very diﬃcult question to get a complete answer. For a given measure µ, the answer depends on K. Let K be a closed disk that contains the support of µ. Then every f in A(K) can be uniformly approximated by polynomials, and hence Aq (K, µ) = P q (µ), which is the closure of polynomials in Lq (µ). In this case, J. Thomson proved a structure theorem for P q (µ) in [28, 1991]. Roughly speaking, Thomson’s theorem says that there exists a Borel partition {∆n }n=0 of the support of µ such that P q (µ) = Lq (µ|∆0 ) ⊕ P q (µ|∆1 ) ⊕ ... ⊕ P q (µ|∆n ) ⊕ ...,

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where each P q (µ|∆n ) is identiﬁed with a space consisting of analytic functions on a simply connected domain Un via a so-called evalaution map. The union, ∪Un , is known as the set of analytic bounded point evaluations (abpes) for P q (µ). In this special case, since A(K) is the uniform closure of polynomials, Thomson’s theorem shows that A(K) is dense in Lq (µ) if and only if P q (µ) (= Aq (K, µ) has no abpes (this special result answered an open question raised by D. Sarason in 1972 [27]). For an arbitrary compact subset K, the author shows in [20] that it is still true: A(K) is dense in Lq (µ) if and only if Aq (K, µ) has no abpes. However, the corresponding result for Aq (K, µ) is not always possible even the set of abpes for Aq (K, µ) is not empty. J. Conway and N. Elias give an example in [6] that shows the set of abpes for Aq (K, µ) is a simply connected domain U , but Aq (K, µ)∩L∞ (µ) can not be identiﬁed with H ∞ (U ) via the evaluation map. For Rq (K, µ), the closure in Lq (µ) of R(K) (which is the uniform closure of the rational functions with poles oﬀ K), the situation is worse since the set of abpes may be empty even R(K) is not dense in Lq (µ) (see [4]). Assuming the existence of abpes for Rq (K, µ), Conway and Elias proved a structure-type theorem for Rq (K, µ) under additional conditions. But their result does not imply Thomson’s theorem. Then, can we have a structure theorem for Aq (K, µ) or Rq (K, µ) that is beyond the polynomial case and that also covers Thomson’s theorem? Prior our work, it was unknown whether such a structure theorem for Aq (K, µ) is possible. Thomson was unable to oﬀer any result for Rq (K, µ) that is beyond P q (µ) (that is, the disk case in our setting) (see [28, p. 505]). To tackle the problem, we ﬁrst need to restrict our eﬀort on those K such that the components of K ◦ are ﬁnitely connected. In fact, the author shows in [21, Theorem 2] even when K is a simplest kind of inﬁnitely connected domains, such as a “road-runner”, our main theorem (Theorem 2.1) for Aq (K, µ) could fail. In this paper, we seek a necessary and suﬃcient condition on K so that a Thomson type of structure theorem holds for Aq (K, µ). A domain is called a circular domain if its boundary consists of ﬁnitely many disjoint circles. We call a domain U multi-nicely connected if there is a circular domain W and a conformal map α from W onto U such that α is almost 1-1 on ∂W with respect to the arclength measure. Our main theorem, Theorem 2.1, extends Thomson’s theorem to Aq (K, µ) in the case when the components of K ◦ are multi-nicely connected and the harmonic measures of the components of K ◦ are mutually singular. We also show that the condition of Theorem 2.1 is necessary. If every f in H ∞ (K ◦ ) a pointwise limit of a bounded sequence in A(K), then K satisﬁes the condition of Theorem 2.1. In particular, when K is such that R(K) is a hypo-Dirichlet algebra [1, 8], K satisﬁes the condition of Theorem 2.1 (in this case, Rq (K, µ) = Aq (K, µ)). If the complement of K has only ﬁnitely many components (note, K ◦ may still has inﬁnitely many components in this case), then R(K) is a hypo-Dirichlet algebra and hence K satisﬁes the hypothesis

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of Theorem 2.1. Since a quite large of class of Rq (K, µ) satisﬁes our conditions, Theorem 2.1 is also a theorem for the rational Rq (K, µ). Since the polynomial case is just the disk case in our setting and since a general compact subset K is much more complicated than a disk in nature, one can expect the extension needs much more work. To get a structure theorem for Aq (K, µ) (here K is an arbitrary compact subset), we need more than Thomson’s technic and method. In fact, we need a new Thomson-type approximation scheme as developed in [20] that takes all of what is used in Thomson’s paper [28]. In addition, we need what was not involved in the case of P q (µ): we make extensive use of results and technics related to uniform algebra or rational approximation: such as peak points, harmonic measures, hypo-Dirichlet algebra, multi-nicely connected domains, representing measures, pointwise bounded approximation, etc (these concepts are not needed for P q (µ)). So, besides Thomson’s technic and method, we need a signiﬁcant part of the theories from the uniform algebras and the rational approximation to get the work done. Combination for Thomson’s technic and uniform rational approximation theory is the key to prove Theorem 2.1. However, not every thing we need in uniform algebra theory is ready for us. We have to prove some results in that theory by ourself. In doing so, we ﬁrst introduce the concept of multi-nicely connected domains, then we prove Proposition 3.1 and an interesting result in uniform algebra, Lemma 3.5, which is crucial for us to prove Lemma 3.7. That is one of our key lemmas and it is needed to prove another key lemma, Lemma 3.10. The rest of paper is to use these two lemmas and results in [20] and [21] to prove the main theorem and extend those lemmas and results that were proved for the polynomial case in [28]. So far, we are unable to oﬀer any other proof that is less involved with the theory of uniform algebra. Actually, due to the nature of this problem, we believe the rational approximation theory is the right tool in study this type of problems. Now we would like to point out the relation between this paper and some other related papers. This paper is the sequel of the author’s work [20, 21]. Thomson’s paper consist of two parts of important results. One is to give a suﬃcient and necessary condition on when ∇P q (µ) is not empty and another one is to have a structure theorem for P q (µ). In [20], we only study the problem of when ∇Aq (K, µ) (∇Rq (K, µ)) is not empty and we show it is empty if and only if A(K) is dense in Lq (µ). In [21], our eﬀort was primarily to establish the result that is a part of 4) and 5) of Theorem 2.1 in this paper and to solve a problem in [6]. In this paper, our eﬀort is to establish a full version of Thomson’s theorem for Aq (K, µ) (Rq (K, µ)) and this paper is based on [20, 21]. The readers may notice that our theorem (Theorem 2.1) not only completely covers Thomson’s theorem, it also has more important consequences, such as 4) and 5) (which are not in [28] and are important facts. One needs to have them when applying it to operator theory. For example, see [17] and [24]). In [6], Conway and Elias studied the same problem as that in this paper. However, since their theorem is based on the assumption that K is the closure of ∇Rq (K, µ) and Rq (K, µ) is pure, so their result does not cover

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Thomson’s theorem. For a given measure µ, one can not tell when their conditions are satisﬁed. In contrast, our paper deals with arbitrary measures just as [28] does. In 1972, D. Sarason [27] established a structure theorem for P ∞ (µ), the weak star closure of polynomials in L∞ (µ), which has a similar form to that of our theorem.

1. Preliminaries For a compact subset K in the complex plane C. Let C(K) denote the algebra of continuous functions on K. For an open subset G in the sphere C∞ whose boundary does not contains ∞, let A(G) be the closed subalgebra of C(G) that consists of functions continuous on G and analytic on G. Notice that A(Ω) = A(Ω) in general. A point w in C is called an analytic bounded point evaluation (abpe) for Aq (K, µ) if there is a neighborhood G of w and c > 0 such that for all λ ∈ G |f (λ)| ≤ c f Lq (µ) for all f ∈ A(K). So the map, f → f (λ), extends to a functional in Aq (K, µ)∗ . Thus, there is a (kernel) function kλ in Aq (K, µ)∗ such that f (λ) = f kλ dµ, f ∈ A(K). Clearly, the set of abpes is open. For each f ∈ Aq (K, µ), let fˆ(λ) = f kλ dµ. Then fˆ(λ) is analytic on the set of abpes. The abpes for Rq (K, µ) are deﬁne similarly. We shall use ∇Aq (K, µ) to denote the set of abpes for Aq (K, µ). The following is one of the main results in [20] which our main theorem relies on: Theorem 1.1. Let K be compact subset of C and let µ be a positive ﬁnite measure supported on K. Then A(K) is dense in Lq (µ) if and only if ∇Aq (K, µ) = ∅. For the case when K is the polynomially convex hull of the support of µ, the above theorem is a consequence of Thomson’s theorem. However, since it was known long before Thomson’s paper [28] that there is a compact K and a measure µ on K such that R(K) is not dense in Lq (µ) but Rq (K, µ) has no abpe, and since P q (µ) ⊂ Rq (K, µ) ⊂ Aq (K, µ) always holds, Theorem 1.1 was unexpected before [20]. Somehow, it was a surprise that the theorem is true for the spaces on the both sides of the inequality above, but fails for the spaces between. Nicely connected domains. Following Glicksburg [10], we call a domain Ω nicely connected if it is multi-nicely connected and if it is simply connected. Harmonic measures. Let Ω be a domain in the extended plane C∞ such that it is solvable for the Dirichlet problem and ∞ is not in ∂Ω. For u ∈ C(∂Ω), let u ˆ = sup {f : f is subharmonic on Ω and lim supz→a f (z) ≤ u(a), a ∈ ∂Ω}. The function uˆ turns out to be harmonic on Ω and continuous on Ω, and the map u → u ˆ(z) deﬁnes a positive linear functional on C(∂Ω) with norm one, so the

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Riesz representing theorem implies that there is a probability measure ωz on ∂Ω such that uˆ(z) = udωz , u ∈ C(∂Ω). ∂Ω

The measure ωz is called the harmonic measure of Ω evaluated at z. The harmonic measures evaluated at two diﬀerent points are boundedly equivalent. We shall use ωΩ to denote a harmonic measure of Ω. Hypodirichlet algebras. A closed subalgebra B of C(K) is said to be a hypo-Dirichlet algebra, if the uniform closure of Re(B) = {Re(f ) : f ∈ B} has ﬁnite codimension in CR (K) = {f : f ∈ C(K) and f is real} and the linear span of log|B −1| is uniformly dense in CR (K), where B −1 is the subset in B consisting of invertible elements. A function algebra B is called a Dirichlet algebra if Re(B) is uniformly dense in CR (K). Clearly, a Dirichlet algebra is also a hypo-Dirichlet algebra. An good example is that R(K) is hypo-Dirichlet if C \K has only ﬁnitely many components. This covers a large class of domains that have been studied. If R(K) is a hypo-Dirichlet algebra, then A(K) = R(K) [8, p. 116]. Peak points. A point a ∈ K is a peak point for a function algebra B ⊂ C(K) if there is a function in B such that f (a) = 1 at z = a and |f (z)| < 1 for z = a. Pure and irreducible spaces. The space Aq (K, µ) is called pure if there is no Borel subset ∆ of supp(µ) such that the restriction of A(K) on ∆ is dense in Lq (µ|∆). An observation is that for any Aq (K, µ), there is a Berel partition {∆0 , ∆1 } of the support of µ such that Aq (K, µ|∆1 ) is pure and Aq (K, µ) = Lq (µ|∆0 ) ⊕ Aq (K, µ|∆1 ). The space Aq (K, µ) is said to be irreducible if it contains no nontrivial characteristic functions. So an irreducible space must be pure. Nontangential limits. Let G be a bounded domain that is conformally equivalent to a circular domain W in the plane and let u be a conformal map from W onto Ω. Then u has well-deﬁned boundary values on ∂W , which are equal to the nontangential limits of u for almost every point on ∂W with respect to ωW (the harmonic measure of W ). We still use u to denote the boundary value function. Now, if E is a Borel subset of ∂W such that u is 1-1 on E a.e. [ωW ], then each f ∈ H ∞ (G) has nontangential limits almost everywhere on u(E) with respect to ωG . That is, f (a) =

lim

z→u−1 (a)

f ◦ u(z) a.e. on u(E) with respect to ωG .

So, if µ is such that µ ωG on on u(E), then each f ∈ H ∞ (G) has nontangential limits on u(E) almost everywhere with respect to µ.

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2. The Main Result In this section, we introduce our main result, Theorem 2.1. Recall that the connectivity of a ﬁnitely connected domain is deﬁned to be the number of the components of its complement. Theorem 2.1. Let K be a compact subset and let µ be a ﬁnite positive measure supported on K. If each of the components of K ◦ is multi-nicely connected and the harmonic measures of the components of K ◦ are mutually singular, then there exists a Borel partition {∆n }∞ n=0 of supp(µ) such that Aq (K, µ) = Lq (µ|∆0 ) ⊕ Aq (K, µ|∆1 ) ⊕ ... ⊕ Aq (K, µ|∆n ) ⊕ ... and for each n ≥ 1, if Un denotes ∇Aq (K, µ|∆n ), then 1) U n ⊃ ∆n and Aq (K, µ|∆n ) = Aq (U n , µ|∆n ); 2) each Un is a ﬁnitely connected domain that is conformally equivalent to a circular domain Wn ; the connectivity of Un does not exceed the connectivity of the component of K ◦ that contains Un ; 3) the map e, deﬁned by e(f ) = fˆ, is an isometrical isomorphism and a weak star homeomorphism from Aq (Kn , µ|∆n ) L∞ (µ|∆n ) onto H ∞ (Un ); 4) µ|∂Un ωUn ; and if un is a conformal map from Wn onto Un , then for each f ∈ H ∞ (Un ) has nontangential limits on ∂Un a.e. [µ] and e−1 (f )(a) =

lim

z→u−1 n (a)

f ◦ un (z) a.e. on ∂Un with respect to µ|∂Un ;

5) for each f ∈ H ∞ (Un ), if let f ∗ be equal to its nontangential limit values on ∂Un and let f ∗ = fˆ on Un , then the map m, deﬁned by m(f ) = f ∗ |∆n , is the inverse of the map e. Remark 2.1. Thomson proved 1), 2) and 3) of Theorem 2.1 in the case when Aq (K, µ) = P q (µ). For the polynomial case, 4) is the main result in [16]. The author proved 4) for Aq (K, µ) with a diﬀerent method in [21]. Remark 2.2. 3) clearly implies that each Aq (K, µn ) is irreducible. Remark 2.3. The condition on K is the best possible one. What we mean here is that in order to have Theorem 2.1 holds for any positive ﬁnite measure supported on K, it is necessary and suﬃcient that each component of K ◦ is multi-nicely connected and the harmonic measures of the components of K ◦ are mutually singular. Now we outline a proof for this fact. Let Ω be a component of K ◦ and let µ be a harmonic for Ω. By Theorem 3 of [21], the map f → fˆ, ∞ measure ∞ q from A (Ω, µ) L (µ) onto H (Ω) is surjective if and only if Ω is a multi-nicely connected domain. Hence we know that Ω must be multi-nicely connected. Now let Ω1 and Ω2 be two components of K ◦ . We want to show that ωΩ1 and ωΩ2 are mutually singular. Set µ = ωΩ1 + ωΩ2 . Then it is easy to see that Aq (K, µ) is pure. Since the harmonic measure at a given point is a representing

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measure, it follows by the deﬁnition of abpe and the Harnack’s inequality that ∇Aq (K, µ) ⊃ Ω1 ∪ Ω2 . Since each Ωi has no boundary slit, it follows clearly that ∇Aq (K, µ) = Ω1 ∪ Ω2 . If Theorem 2.1 holds for Aq (K, µ), then Aq (K, µ) = Aq (Ω1 , µ1 ) ⊕ Aq (Ω2 , µ2 ), where µi , i = 1, 2, are as in Theorem 2.1. Let v be a conformal map of Ω1 onto a circular domain W . Theorem 2.1 implies that there exists ve ∈ Aq (Ω1 , µ1 ) such that vˆe = v. Set η = µ1 ◦ ve−1 . According to Lemma 2 in [21], η is a measure on ∂W such that Aq (W , η) is irreducible and ∇Aq (W , η) = W . Moreover, η ωW by Lemma 3 of [21]. On the other hand, since W is circular and since ∇Aq (W , η) = W , it is easy to see that A∞ (W , η), the weak-star closure of A(W ) in L∞ (η), is equal to ∞ (W ), which is the image of the map f → f˜ from H ∞ (W ) into Lq (ωW ) (where H ˜ f is the boundary value function of f on ∂W ). Since the support of µ1 ⊂ ∂Ω1 , it follows by a classical result that [ωW ] = [η]. Now, applying Lemma 3 in [21], we conclude that [ωΩ1 ] = [µ1 ]. Similarly, we have [µ2 ] = [ωΩ2 ]. But µ1 and µ2 are mutually singular, therefore ωΩ1 and ωΩ2 must be mutually singular.

3. The proof of the main result Lemma 3.1. For each f ∈ Aq (K, µ), fˆ = f on ∇Aq (K, µ) a.e. [µ]. Proof. Let a be an abpe and choose a sequence {fn } in A(K) such that fn → f in Lq (µ). Since fn → fˆ uniformly in a neighborhood of a, it follows (by passing a sequence if necessary) that fn → f a.e. [µ] and consequently fˆ(a) = lim fˆn = lim fn = f (a) a.e. [µ]. n→∞

n→∞

The following lemma is elementary too. Lemma 3.2. If f ∈ L∞ (µ) ∩ Aq (K, µ) and g ∈ Aq (K, µ), then f g ∈ Aq (K, µ) and fg = fˆgˆ. The next lemma is proved in [21]. Lemma 3.3. Let Ω = ∇Aq (K, µ). If Ω is ﬁnitely connected, then every component of (C \Ω) has nonempty interior. Representing measures. Let B be a closed subalgebra of C(K). A complex representing measure of B for a ∈ K is a ﬁnite measure ν on K such that f (a) = f dν, f ∈ B. A representing measure for a is a probability measure that satisﬁes the above condition. Note, if a is a peak point, then the only representing measure for a is the point mass δa . The sweep of a measure. Let G be a domain that is regular for the Dirichlet problem ˜ and let µ be a measure on G. The sweep of µ is the unique positive measure µ

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on ∂G that satisﬁes G u ˜ dµ = ∂G ud˜ µ, u ∈ C(∂G), where u ˜ is the solution of the Dirichlet problem for u. A simple fact is that if µ is a measure on G, then µ ˜ = µ|∂G + µ|G. Lemma 3.4. Let K be a compact subset such that the components of K ◦ are multinicely connected and the harmonic measures of the components of K ◦ are mutually singular. Let Ω be a component of K ◦ . If Aq (K, µ) is pure and if K ◦ is dense in K, then µ|∂Ω ωΩ . ◦ Proof. Let {Ωj }∞ j=0 be the collection of the components of K . Fix an integer j ≥ 0 and let E be a component of ∂Ωj . Let Gj be the unique simply connected domain in the sphere C∞ that has E as its boundary and contains Ωj . Since Ωj is multinicely connected, Gj must be nicely connected. For i = j, let Gi be the bounded simply connected domain that contains Ωi and whose boundary is a component of ∂Ωi . Clearly, Gi is also nicely connected. Now let Ω be the union of those Gi ’s for which Gi ∩ Gj = ∅ (diﬀerent Gi ’s are either disjoint or one contains other). Set G = Ω Gj . Then each component of G is equal to some Gi . Let {Gik } be the collection of all the components of G. Then our hypothesis on K implies that the harmonic measures of the components of G are mutually singular. It follows from [7] that A(G) is a Dirichlet algebra on ∂G. Hence, every point in ∂G is a peak point for A(G) and every trivial Gleason part of A(G) consists of a single point. A(G). Therefore, {Gik } is the collection of all the nontrivial Gleason parts of Let η ⊥ A(G). By the Abstract F. and M. Riesz Theorem [8], η = m≥0 ηm , where each ηm ⊥ A(G), ηm vm for a representing measure vm at some point am in G, vm ’s are mutually singular. Let a ∈ ∂G. Then a is a peak point. Let f ∈ A(G) be a peak function for a. Then f n (z) → χ{a} pointwise, and thus 0 = limn→∞ f n dηm = ηm ({a}). Hence am ∈ G (otherwise, νm is the point mass at am and hence νm (G−{am }) = 0. So we conclude that ηm (G) = ηm (G−{am })+ ηm ({am }) = 0 + 0 = 0, a contradiction). Let Gikm be the component that contains am and let v˜m be the sweep of vm on ∂Gikm . Then for each g ∈ A(G) g(z)d˜ vm = g(z)d˜ vm = g(z)dvm = g(z)dωam , ∂G

∂Gik

m

Gik

m

∂Gik

m

where ωam is the harmonic measure of Gikm evaluated at am . It follows by the uniqueness that v˜m = ωam . Hence, we have η|∂G ( vam )|∂G v˜am |∂G = ωam |∂G In particular, we have that η|E ωΩj |E. Finally, suppose that g ∈ Lp (µ) such that f gdµ = 0, for f ∈ A(K), where 1 1 q + p = 1. Then g ⊥ A(G) as well. Hence, gµ|E ωΩj |E. This implies that (gµ)s |E = 0, where (gµ)s is the singular part of the Lebesgue decomposition of gµ with respect to ωΩj |E. Consequently, we have g ⊥ χ∆∩E , where ∆ is the carrier of µs |∂Ωj and µs is the singular part of the Lebesgue decomposition of µ with respect

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to ωΩj . Now an application of Hahn-Banach theorem yields χ∆∩E ∈ Aq (K, µ). By purity, χ∆∩E = 0 a.e. [µ] and therefore µ|E ωΩj |E. Since E is an arbitrary component of ∂Ωj , it follows that µ|∂Ωj ωΩj . Proposition 3.1. Let K be a compact subset such that the components of K ◦ are multi-nicely connected and the harmonic measures of K ◦ are mutually ∞singular. 1 ◦ Let {Ωj }∞ denote the collection of the components of K . Set ω = j=0 j=0 2j ωΩj . If Aq (K, µ) is pure, then µ|∂K ω. Proof. By Lemma 17.10 in [5, p. 246], there exists a function g ∈ Aq (K, µ)⊥ such that |f |µ |g|µ for each f ∈ Aq (K, µ)⊥ . Since Aq (K, µ) is pure, we see that g = 0 on supp(µ) a.e. [µ]. Thus [|g|µ] = [µ] a.e. [µ]. Set ν = |g|µ. d|ν| < ∞ and let f be a peak function for Let a ∈ K − K ◦ be such that |z−a| a. For each integer n ≥ 1,

1−f n (z) z−a

∈ A(K), so we have 1 − f n (z) dν = 0 νˆ(a) = lim n→∞ z−a

d|ν| Since the set {a : |z−a| < ∞} has full area measure in the plane, it follows by a well-known fact (see Theorem 3.1 or the comments after it) that ν is the zero measure oﬀ K ◦ . Finally, since µ|∂Ωj ωΩj for each j (by Lemma 3.4), the conclusion follows. The proof of the next lemma can be found in [26]. Lemma 3.5. Let Ω be an open subset whose boundary does not contain ∞. Suppose the components of Ω are multi-nicely connected and the harmonic measures of the components of Ω are mutually singular. Let {Ωj }∞ j=0 be the collection of the components of Ω. If all but ﬁnitely many components of Ω are simply connected, then A(Ω) is boundedly pointwise dense in H ∞ (Ω). For a ﬁnite positive measure µ, let A∞ (K, µ) denote the weak-star closure of A(K) in L∞ (µ). Lemma 3.6. Let Ω be a multi-nicely connected domain. Let W be a circular domain that is conformally equivalent to Ω and let φ be a conformal map of W onto Ω. Then the boundary value function φ˜ has a well-deﬁned inverse, φ˜−1 , on ∂Ω. Moreover, φ˜−1 ∈ A∞ (Ω, ωΩ ). Proof. Since φ is almost 1-1 on ∂W with respect to ωW , it is apparent that φ˜ has a well-deﬁned inverse function on a set of full ωΩ measure. By Lemma 3.5 A(Ω) is boundedly pointwise dense in H ∞ (Ω). Choose a bounded sequence {fn } in A(Ω) such that fn → φ−1 on Ω. Then one can show that fn → φ˜−1 in the weak-star topology of L∞ (ωΩ ). Hence φ˜−1 ∈ A∞ (W , ωΩ ). The following is one of our key lemmas.

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Lemma 3.7. Suppose that each component of K ◦ is multi-nicely connected and the harmonic measures of the components of K ◦ are mutually singular. Let U be a component of ∇Aq (K, µ) and let Ω be the component of K ◦ that contains U . Set τ = µ|Ω. Then Aq (Ω, τ ) ⊂ Aq (K, µ) and U ⊂ ∇Aq (Ω, τ ). Proof. We ﬁrst assume that Aq (K, µ) is pure. Let {Ωi }∞ i=0 be the collection of all the components of K ◦ . Without loss of generality, let Ω0 = Ω. Suppose h ∈ Aq (Ω, τ ) and choose a sequence {rn } in A(Ω) such that rn → h in Lq (τ ). Let ω = i 21i ωΩi . Fix a function rn . Extend both h and rn to be functions on the whole plane by deﬁning their values to be zero oﬀ Ω. We claim that there exists a sequence {qn } in A(K ◦ ) such that it weak-star converges to rn in L∞ (ω). For each i ≥ 1, let Gi be the bounded simply connected domain that contains Ωi and whose boundary is a component of ∂Ωi . Let G be the union of Ω with those domains Gi that do not intersect Ω. Then all but ﬁnitely many components of G are simply connected domains and each component of G is multi-nicely connected. By Lemma 3.5, A(G) is boundedly pointwise dense in H ∞ (G). Thus, there exists a bounded sequence {qm } in A(G) so that it pointwise converges to rn on G. Now for given > 0, let f ∈ L1 (ω). Then | f (rn − qm )dω| ≤ | f (rn − qm )dω| + for all m 2 k ∂K ∪o ∂Ωi whenever k is suﬃciently large. Observe that {qm } weak-star converges to rn in L∞ (ωΩi ) for each i. Thus, f (rn − qm )dω| ≤ + when m is suﬃciently large. | 2 2 ∂K Hence, for each f ∈ L1 (ω) we have f (rn − qm )dω = 0. lim n→∞

∂K

That is, {qm } weak-star converges to rn in L∞ (ω). This proves the claim. Next we show that rn belongs to the weak-star closure of A(K ◦ ) in L∞ (µ). By Proposition 3.1, µ|∂K ω. Thus f qm dµ = f rn dµ, f ∈ L1 (µ). lim m→

∂K

∂K

Since {qm } is bounded and pointwise converges to rn on K ◦ , it follows by the bounded convergence theorem that lim f qm dµ = f rn dµ, f ∈ L1 (µ). n→

Therefore, rn belongs the weak-star closure of A(K ◦ ) in L∞ (µ). But this closure is contained in the weak closure of A(K ◦ ) in Lq (µ). Because a convex set is normclosed if and only if it is weakly closed in Lq (µ), we have rn is in Aq (K ◦ , µ).

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Now for each n ≥ 1, choose xn in A(K ◦ ) such that rn − xn Aq (K ◦ ,µ) | ≤

1 . n

Then xn − h Lq (µ) = xn − rn Lq (µ) + rn − h Lq (µ) 1 ≤ + rn − h Lq (τ ) → 0, as → ∞. n Thus we have that h ∈ Aq (K ◦ , µ). Because h is an arbitrary element in A(Ω, τ ), we conclude that Aq (Ω, τ ) ⊂ Aq (K ◦ , µ). Since Aq (K, µ) is pure and since µ is supported on K ◦ (for µ|∂K ω and ω is supported on ∂K ◦ ), we have that Aq (K, µ) = Aq (K ◦ , µ). Hence Aq (Ω, τ ) ⊂ Aq (K, µ). Now let b ∈ U . Then b is an abpe for Aq (K, µ) and thus there exists d > 0 and a small open disk Db ⊂ U such that for all r ∈ A(K) 1 |r(a)| ≤ d{ |r|q dµ} q , a ∈ Db . Let y ∈ A(Ω) and extend y to be zero oﬀ Ω. Then y ∈ Aq (K, µ) and so there is a sequence {qn } in A(K) so that it converges to y in Lq (µ). Then {qn } converges to y uniformly on Db . Hence, it follows by the expression above that for all y ∈ A(Ω) 1 |y(a)| ≤ d{ |y|q dτ } q , a ∈ Db . Thus a ∈ ∇Aq (Ω, τ ). By the deﬁnition of abpe, U ⊂ ∇Aq (Ω, τ ). If Aq (K, µ) is not pure, let µ = µ0 + µ1 be the decomposition so that q A (K, µ) = Lq (µ0 ) ⊕ Aq (K, µ1 ) and Aq (K, µ1 ) is pure. Then ∇Aq (K, µ) ⊃ ∇Aq (K, µ1 ) ⊃ U.

So the conclusion of the lemma follows.

A function f analytic at ∞ can be written as a power series of the local 1 coordinate z−z at ∞: 0 f (z) = f (∞) +

a2 a1 + + .... z − z0 (z − z0 )2

The coeﬃcient a1 is called the derivative of f at ∞ and is denoted by f (∞). It is easy to see that f (∞) = limz→∞ z(f (z) − f (∞)). Deﬁne β(f, z0 ) = a2 . The next lemma is elementary. Lemma 3.8. Let δ > 0 and let a in C. Let B(a, δ) = {z : |z − a| ≤ δ}. If f is a bounded analytic function on C∞ \ B(a, δ), then |f (∞)| ≤ δ f ∞ and |β(f, a)| ≤ δ 2 f ∞ .

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Thomson’s Scheme. 1 Now, we introduce an approximation scheme originally developed by J. Thomson in [28]. For an integer k ≥ 1, let {Skp }∞ p=1 be the collection of all open squares −k with sides 2 , parallel to the coordinate axes and corners at the points whose coordinates are both integral multiples of 2−k . A ﬁnite sequence {Si }ni=1 of squares is called a path of squares if the interior of ∪S i is connected. In this case we say S1 and Sn are joined by a path of squares. The collection of {Skp }∞ p=1 is called the k-th generation of squares. 1 An open square S is said to be Let φ be a nonnegative function in L (Area). light with respect to φ if S φ d Area ≤ [Area(S)]2 . Now we begin with the scheme. Let a ∈ C and let S be a square in {Skp }∞ p=1 such that a ∈ S. Color S yellow and let Γk = ∂S. We then move to the squares in the next generation. First, color green every light square in {S(k+1)p }∞ p=1 that lies outside Γk and has a side on Γk . Second, color green every light square that can be joined to a green square in the ﬁrst step by a path of light squares in {S(k+1)p }∞ p=1 . Now if there is an unbounded green path (that is made up by inﬁnitely many squares), then this coloring process ends. Otherwise, let γk+1 be the boundary of the polynomially convex hull 2 of the union of Γk and the closure of the green squares. We then color red every square S in {S(k+1)p }∞ p=1 if S is outside γk+1 and S has a side on γk+1 . After that, color a square T yellow if T is outside γk+1 and T has no side lying on γk+1 and the distance from T to some red square in 2 −(k+1) {S(k+1)p }∞ . Now let Γk+1 be the boundary of p=1 is less or equal to (k + 1) 2 the polynomially convex hull of the union and the closure of the colored squares in the (k + 1)-th generation. To this step, the coloring process in (k + 1)-th generation of squares is completed. Next we continue this process to the (k +2)-th generation of squares and keep this process to all higher generations unless there is an unbounded green path in the coloring scheme in some (m + l)-th generation (l ≥ 1). We use (φ, a, k) to denote this colored scheme. Light and heavy points. For a nonnegative function φ ∈ L1 (Area), a point λ in C is called light (with respect to φ) if there exists δ > 0 such that for each δ0 ≤ δ {z : |z − a| = δ0 } ∩ {all colored squares in (φ, k, a)} = ∅, whenever k is a suﬃciently large integer. If a point is not light, then it is called a heavy point. Remark 3.1. The construction of our colored scheme is exactly the same as that in [28]. But the light and heavy points improved ’light route to ∞’ and ’heavy barrier’ in Thomson’s original work. Let us explain the diﬀerence: For a given φ, if there is an unbounded green path in the colored scheme (φ, k, a) for every k, it is said that 1 In

this paper, we don’t directly use this scheme. But we need the concept of light and heavy points and results related to them (Lemma 3.9 and Theorem 3.1). 2 The polynomially convex hull of a compact subset K in the plane is defined as the union of K and all the bounded components of C\K.

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there is a sequence of light routes from a to ∞. This is essentially the deﬁnition of ’light’ points in Thomson’s paper. Because most of the light points for a given φ in our deﬁnition don’t have a sequence of light routes from a to ∞, the set of the light points is much larger than the set of points that have a sequence of light routes from ∞. The Cauchy transform of a measure (with compact support) µ is deﬁned as 1 µ ˆ (z) = dµ(w) w−z . Because z is local integrable with respect to the area measure, it follows that µ ˆ(z) is deﬁned everywhere except a subset of zero area. The following is a practically useful result coming out of our light point concept [20, Theorem 2.4]. Theorem 3.1. Let µ be a ﬁnite measure with compact support. Let V be an open subset in C. If every point in V is light with respect to |ˆ µ|, then |µ|(V ) = 0. The above theorem generalizes a well-known result in the theory of uniform approximation: if µ ˆ = 0 a.e. on an open subset with respect to the area measure, then the restriction of µ on the open subset is zero. The following is the key lemma in [20, Lemma 2.3]. νi (z)| : 1 ≤ j ≤ k}. Lemma 3.9. Let ν1 , . . . , νk be ﬁnite measures. Let φ(z) = max{|ˆ If a is a light point with respect to φ, then there is an arbitrarily small positive number δ such that for any > 0 and α, β ∈ {z : |z − 1| ≤ 1}, there is a function in C(C∞ ) that has the following properties: 1) f ∞ ≤ C (a universal constant), 2) f is analytic on {z : |z − a| > δ}, 3) f (∞) = 0, 4) f (∞) = αδ, 2 5) β(f, a) = βδ , 6) | f dνj | ≤ for all 1 ≤ j ≤ k. Vitushkin covering. For a natural number k, let {Skl }∞ l=1 is the k-th generation of squares with sides of length 2−k . For each Skl , let Fkl be the square obtained by enlarging Skl 54 times. The collection {Fkl } is called a regular Vitushkin covering of the plane. We suppress k and let zl be the center of Fl . Then there exists a C 1 partition of unity {φl } subordinate to {Fl } with gradφl ≤ 100 2k such that ∞ 2−3k 2−k }, z ∈ C. min(1, ) ≤ C min{1, 3 |z − zl | dist(z, ∪l Fl ) l

One may consult [5] for a proof of the inequality. Lemma 3.10. Suppose that each component of K ◦ is multi-nicely connected and the harmonic measures of the components are mutually singular. Let U be a component of ∇Aq (K, µ) and let f ∈ H ∞ (U ). Then there exists a function h ∈ Aq (K, µ) ∩ L∞ (µ) such that ˆ h(z) = f (z) on U and h = 0 oﬀ U .

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Proof. First, we assume that K is ﬁnitely connected, K = K and K ◦ is connected. Let {xj } be a countable dense subset of Aq (K, µ)⊥ . For an integer k ≥ 1, let Φ(z) = max{|(

xj µ)(z)| : j ≤ k}. Let {Fl } be the regular Vitushkin covering of squares with sides of length 54 2−k and center zl . Then there is a C 1 partition {φl } subordinate f (z)−f (w) ∂φl to the covering {Fl }. For each l, let fl = Tφl f = π1 z−w ∂z d Area. Then fl is analytic oﬀ Fl , fl (∞) = 0, and fl ∞ ≤ 2 gradφl diam[supp(φl ] sup{|f (z) − f (w)| : z, w ∈ supp(φl } ≤ C0 , where C0 is a positive universal constant. Let l be such that Fl ∩ ∂U = ∅ and let a ∈ ∂U ∩ Fl . We claim that a is a light point with respect to Φ. In fact, ﬁrst let a ∈ K ◦ . Set V = K ◦ \ ∂U , it follows by Lemma 3.3 in [20] that |(

xj µ)(z)| = 0 on V for each j and thus Φ = 0 on V . By Lemma 3.7 in [20] we see a is light.3 Now suppose that a ∈ ∂K. Since νj ⊥ A(K) ⊃ R(K), we have that νˆj = 0 oﬀ K. Thus, again it follows from Lemma 3.7 in [20] that a is also a light point. This proves the claim. Next let dl = 12 2−k and let B(a, dl ) be open disk having radius dl and the center at a. Applying Lemma 3.8 to B(a, dl ), it follows that |fl (∞)| ≤ C0 dl and

f (∞)

l ,zl ) . Then |α| ≤ 1 and |β| ≤ 1. Let n β(fl , zl ) ≤ C0 d2l . Let α = Cl 0 dl and β = β(f C0 d2l be the number of those Fl ’s for which Fl ∩ ∂U = ∅. Then n is a positive integer. 1 , then there exists Because a is a light point, applying Lemma 3.9 with α, β, nkC 0 a function gl in C(C∞ ) that is analytic oﬀ B(a, dl ) and satisﬁes:

1) 2) 3) 4) 5)

gl ≤ C1 (C1 is a positive universal constant), gl (∞) = 0, gl (∞) = αdl , 2 β(g l , zl ) = βdl , 1 | gl xj dµ| ≤ nkC0 for all 1 ≤ j ≤ k.

Set hl = C0 gl . Then hl has the following properties: 1) 2) 3) 4)

hl ≤ C0 C1 , hl is analytic oﬀ B(a, dl ), 1 for j ≤ k, | hl xj dµ| ≤ nk hl − fl has a triple zero at ∞, that is, (hl − fl )(∞) = 0, (hl − fl ) (∞) = 0 and β(hl − fl , zl ) = 0.

Let δl be the length of a side of Fl . Since a ∈ Fl , it is evident that B(a, dl ) is contained in the square with center zl and sides of length 2δl . fl − hl is clearly analytic oﬀ {z : |z −zl | ≤ 2δl }. Since fl −hl has a triple zeros at ∞, (z −zl )3 (fl −hl ) is also analytic oﬀ {z : |z − zl | ≤ 2δl }. So the maximum principle implies that |(z − zl )3 (fl − hl )| ≤ 23 δl3 fl − hl ∞ ≤ C0 (C1 + 1)23 δl3 3 By

whenever |z − zl | ≥ 2δl .

combining the proof Lemma 3.3 in [20] and that of Theorem 4.8 in [28], we can extend Lemma 4.8 in [28] has (so it can be applied 3.3 in [20] so that it has the conclusion that Theorem to j≤k |xj µ|)). From this, we also see any point in K ◦ that is not an abpe must be light.

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C2 δ Let C2 = 8C0 (C1 + 1). Then, for each z |fl (z) − hl (z)| ≤ min(C2 , |z−z 3 ). So it l| follows that for every z (the ﬁrst sum is taken over those l’s for which Fl ∩∂U = ∅), ∞ ∞ C2 ( 54 )3 2−3k C2 δ 3 min(C2 , ) = min(C , ) |fl − hl | ≤ 2 |z − zl |3 |z − zl |3 l

l

5 2−k }. ≤ C2 ( )3 min{1, 4 dist(z, ∪Fl ) Notice that f is analytic on those Fl ’s for which Fl ∩ ∂U = ∅. It follows that 1 f (z) − f (w) ∂φl (z) f l = T φl f = d Area π z−w ∂z 1 ∂f (z) φl (z) d Area = 0 = − π ∂z z − w for those l’s. So there are only ﬁnitely many fl ’s that are not zero. Now deﬁne hl = 0 if l is such that fl = 0 and set yk = f + l (hl − fl ). Then yk = l hl . For any z oﬀ ∂U , it is clear that dist(z, ∪Fl ) → dist(z, ∂U ) as k → ∞, and hence it follows from the above inequalities that yk → f (z) for each z in C \∂U . According to 3), we have | yk xj dµ| ≤ k1 , for 1 ≤ j ≤ k. Notice that 5 |yk | ≤ |f | + | (hl − fl )| ≤ f ∞ + C2 ( )3 , 4 l

so {yk } is a bounded sequence. Since the weak-star topology on the unit ball of the dual space of a separable Banach space is metrizable, it follows by Alaoglu’s theorem that there exists a subsequence {ykj } that weak-star converges to some ∞ h ∈ L (µ). According to the last inequality above, we have that hxj dµ = 0 for all j ≥ 1. Consequently, we have that h ∈ Aq (K, µ). Because ykj → f pointwise oﬀ ∂U and f = 0 oﬀ U , we have h = 0 oﬀ U. Finally, we show that ˆ h = f on U . If supp(µ) contains an open subset of G, then this is easy to see this is true (since yk → f pointwise on U and hence ˆ a.e. [µ] on G. Because both f and h ˆ are analytic on U , hence f = h ˆ f =h=h on U ). Otherwise, let G be open so that G ⊂ U and let ρ = µ + Area|U . Then · µ and · ρ are equivalent norms. By the deﬁnition of abpe, we see that U ⊂ ∇Aq (K, µ) = ∇Aq (K, ρ). Therefore, there exists f1 ∈ Aq (K, ρ) ∩ L∞ (ρ) such that fˆ1 = f and f1 = 0 oﬀ ∂U . Now set h = f1 |supp(µ). We show h is the desired function. Let {fn } ⊂ A(K) such that fn → f1 in Lq (ρ). Then fn → h in Lq (µ). ˆ uniformly on G. Since fn → f uniformly on G as well, it follows Thus, fn → h ˆ that h = f on G. Because U is the union of such open subsets G, we conclude that ˆ = f on U . h Now we consider a general K that satisﬁes the hypothesis of this lemma. Let Ω be the component of K ◦ that contains U . Then the multi-nicely connectivity of Ω insures that there is circular domain W and a conformal map v from Ω onto W such that v is almost 1-1 on ∂Ω with respect the harmonic measure of Ω. Let µ = µ0 + τ be the decomposition such that Aq (K, τ ) is pure and Aq (K, µ) = Lq (µ0 )⊕Aq (K, τ ).

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By Proposition 3.1, τ |∂Ω is absolutely continuous with respect to the harmonic measure. Extend v to Ω by deﬁning its boundary values as its nontangential limits and set ν = τ ◦ v −1 . It is easy to check that v(U ) is a component of ∇Aq (W , ν) and ˆ 1 = f ◦v −1 . Set h = h1 ◦v −1 . Then, h ∈ Aq (Ω, τ ) there is h1 ∈ Aq (W , ν) such that h and it is straightforward to verify that h = f . Extend h to be a function on K by deﬁning h = 0 oﬀ Ω. By Lemma 3.7 h ∈ Aq (Ω, µ|Ω) ⊂ Aq (K, µ). Clearly, h does the job. Lemma 3.11. If a ∈ ∇Aq (K, µ), then

f (z)−fˆ(a) z−a

∈ Aq (K, µ) for each f ∈ Aq (K, µ).

Proof. Let W = ∇Aq (K, µ). Then there exists {fn } ⊂ A(K) such that fn → f n (a) → in Lq (µ) and so fn → fˆ uniformly on compact subset of W . Thus fn (z)−f z−a f (z)−fˆ(a) z−a

uniform on a small closed disk B(a, δ) ⊂ W . Note, fn (z) − fn (a) f (z) − fˆ(a) q − | dµ ≤ M | |fn (z) − f (z) − (fn (a) − fˆ(a))|q dµ z−a z−a K K fn (z) − fn (a) f (z) − fˆ(a) q − | dµ, | + z−a z−a B(a,δ)

1 q | . Thus, where M = supz∈K\B(a,δ) | z−a fn (z)−fn (a) z−a

fn (z)−fn (a) z−a

→

fˆ(z)−fˆ(a) z−a

in Lq (µ). Since

∈ A(K) for each n, the conclusion of the lemma follows.

Lemma 3.12. Suppose that ν ⊥ A(K) and supp(ν) ⊂ K. Let U be a component of K ◦ \ supp(ν). If νˆ(a) = 0 at some a ∈ U , then U ⊂ ∇A1 (K, |ν|). 1 Proof. Clearly νˆ(z) = z−w dν(w) is analytic on U . Observe that for f ∈ A(K), f (z)−f (a) z−a

∈ A(K) for every a ∈ K ◦ . Suppose that νˆ(a) = 0 for some a ∈ U . Then there exists a small closed disk B(a, δ) ⊂ U so that νˆ(z) = 0 on B(a, δ). For each (λ) dν = 0 and hence λ ∈ B(a, δ), f (z)−f z−λ 1 f (z) dν, for every f ∈ A(K). f (λ) = νˆ(λ) z−λ Since B(a, δ) does not interest supp(ν), we see that |f (λ)| ≤ c f L1 (|ν|) for some c > 0 on B(a, δ). Hence, a ∈ ∇A1 (K, |ν|). Since the zeros of νˆ is isolated on U , it follows by Lemma 3.3 that U ⊂ ∇A1 (K, |ν|). Lemma 3.13. Let h ∈ Aq (K, µ)⊥ and set ν = hµ. Then ∇A1 (K, |v|) ⊂ ∇Aq (K, µ). Proof. For f ∈ A(K), by H¨ older’s inequality f L1(|ν|) ≤ h Lp(|ν|) f Lq (µ) , where 1q + p1 = 1. The the conclusion of the lemma clearly follows. Proposition 3.2. Let µ be a positive ﬁnite measure with supp(µ) ⊂ K. Let U be a component of K ◦ \ supp(µ). If U ∩ ∇Aq (K, µ) = ∅, then U ⊂ ∇Aq (K, µ).

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Proof. Let a ∈ U ∩ ∇Aq (K, µ). Then there is g ∈ Aq (K, µ)⊥ such that Clearly, gµ ⊥ A(K) and U ⊂ K ◦ \ supp(gµ). So it follows from the previous lemmas that U ⊂ ∇A1 (K, |gµ|) ⊂ ∇Aq (K, µ).

239

gdµ z−a

= 0.

Lemma 3.14. Let Ω = ∇Aq (K, µ) and let U be a component of ∇Aq (K, µ). Suppose that Aq (K, µ) is pure. Let f ∈ Aq (K, µ). If fˆ = 0 and f = 0 oﬀ ∂U a.e. [µ], then f = 0. Proof. Since Aq (K, µ) is pure, as we argued in the proof of Proposition 3.1, there exists h ∈ Aq (K, µ)⊥ such that h = 0 a.e. [µ] and h = 0 oﬀ K ◦ . So h ⊥ A(K ◦ ) as well. Let ν = hf µ. Then ν is a measure such that it is perpendicular to A(K) and supp(ν) ⊂ ∂U . We show that νˆ(a) = 0 oﬀ ∂U . Let W be a component of K ◦ \ supp(ν). We claim that νˆ(z) = 0 on W . Suppose νˆ(a) = 0 for some a ∈ W . According to Lemma 3.12, W ⊂ ∇A1 (K, |ν|) ⊂ ∇Aq (K, µ). f By Lemma 3.11 and the hypothesis, we have that z−a ∈ Aq (K, µ). But h ⊥ A(K). So we conclude that νˆ(a) = 0, which contradicting our assumption above. Therefore, νˆ = 0 on W . In particular, νˆ = 0 on K ◦ \ ∂U . It is easy to see that K ◦ \ ∂U ⊃ K ◦ . So, by the continuity we have that νˆ = 0 on K ◦ − ∂U . Because h ⊥ A(K ◦ ) ⊃ R(K ◦ ), νˆ = 0 oﬀ K ◦ . Hence, we conclude that νˆ = 0 oﬀ ∂U . Now, according to our deﬁnition, it is apparent that every point oﬀ ∂U is a light point with respect to |ˆ ν |. So it follows from Lemma 3.7 in [20] that every point in ∂U is light as well. Consequently, every point in the plane C is a light point. Applying Theorem 3.1, we conclude that v = hf µ = 0. Since h = 0 a.e. on K, f must be the zero function in Lq (µ). So we are done.

Lemma 3.15. Let K be a compact subset in C such that each component of K ◦ is ﬁnitely connected. Let µ be a positive ﬁnite measure supported on K. Then each component U of ∇Aq (K, µ) is a ﬁnitely connected domains conformally equivalent to a circular domain in the plane. Moreover, the connectivity of U does not exceed the connectivity of the component of K ◦ that contains U . Proof. Suppose ∇Aq (K, µ) = ∅. Let U be a component of ∇Aq (K, µ) and let Ω be the component of K ◦ that contains U . Let M be the connectivity of Ω. Now suppose F is a component of C\U . We claim that F ∩ ( C \ Ω) = ∅. First, if F is unbounded, this is obvious. So we assume F is a bounded subset in the plane and assume that F ∩ ( C \ Ω) = ∅. Then F ⊂ Ω. Since U is a connected domain, F is polynomially convex (this means the complement of F is connected). Since U is ﬁnitely connected, there exists a Jordan curve γ in U such that F is contained in V , the bounded Jordan domain enclosed by γ. Let f ∈ Aq (K, µ) and choose a sequence of functions {rn } in A(K) such that rn → f in Lq (µ). Since γ is contained in U ⊂ ∇Aq (K, µ), it follows that rn → f uniformly on γ.

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Also, it is clear that we can choose γ such that dist(F, γ) small enough that the closure of V is contained in Ω ⊂ K ◦ . Then each rn is analytic on V and thus the maximum principle implies that {rn } uniformly converges to a function h near F . By the deﬁnition of abpes, we have that F ⊂ ∇Aq (K, µ). But F ∩ ∂U = ∅, and hence we conclude that ∂U ∩ ∇Aq (K, µ) = ∅, which is a contradiction. Hence F ∩ ( C \ Ω) = ∅. Now let {Ei } be the collection of all the components of (C \Ω) that intersect F . Then F ∪ (∪Ei ) is connected compact subset and [F ∪ (∪Ei )] ∩ U = ∅. So F ∪(∪Ei )) is contained a component of C \U that contains F . Hence F ∪(∪Ei ) = F and therefore Ei ⊂ F for each i. Consequently, each component of C \U contains at least a component of (C \Ω). Since the number of the components of C \Ω is M , we see that the number of the components of C \U is less than or equal to M . Finally, since U is ﬁnitely connected and since ∂U contains no single-point component (by Lemma 3.3), it follows by a classical result [30, Tsuji, p. 424] that U is conformally equivalent to a circular domain. The next two propositions are Theorem 1 and Theorem 3 in [21], respectively. We include them for readers convenience and self contained. Proposition 3.3. Let Aq (K, µ) be irreducible. Let U = ∇Aq (K, µ) be a ﬁnitely connected domain. If the map e, deﬁned by e(f ) = fˆ, from Aq (K, µ) ∩ L∞ (µ) to H ∞ (U ) is surjective, then µ|∂U ωU , the harmonic measure of U . Proposition 3.4. Let Aq (K, µ) be irreducible. Let U = ∇Aq (K, µ) be a ﬁnitely connected domain and let u be a conformal map from a circular domain W onto U . If the map e, deﬁned by e(f ) = fˆ, from Aq (K, µ) ∩ L∞ (µ) to H ∞ (U ) is surjective, then for each f ∈ H ∞ (U ) e−1 (f )(a) =

lim

z→u−1 (a)

f ◦ u(z) a.e. on ∂U with respect to µ|∂U.

Moreover, A(U ) ⊂ Aq (K, µ). The proof of Theorem 2.1. Let µ = µ0 +τ be the decomposition such that Aq (K, τ ) is pure and Aq (K, µ) = Lq (µ0 ) ⊕ Aq (K, τ ). Suppose A(K) is not dense in Lq (µ). Then τ = 0 in the decomposition. According to Theorem 1.1, ∇Aq (K, τ ) = ∅. q Let {Un }∞ n=1 be the components of ∇A (K, τ ). For each n ≥ 1, by Lemma 3.10 q ∞ there exists fn in A (K, τ ) ∩ L (τ ) such that fˆn = χUn and fn = 0 oﬀ Un . Since ˆ ˆ Un ’s are pairwise disjoint, we have f n fm = fn fm = 0. It follows by Lemma 3.14 2 ˆ ˆ that fn fm = 0. Similarly, since fn = fn , we get that fn2 = fn . Therefore, we conclude that fn = χ∆n for some Borel subset ∆n . But τ (∆n ∩ ∆m ) = 0 (because fn fm = 0). Thus ∆n ’s can be chosen to be pairwise disjoint. Moreover, since fn = 0 oﬀ U n , we can also require that ∆n ⊂ U n .

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For each n ≥ 1, let Kn = U n and let µn = τ |∆n . We claim that Un = ∇Aq (Kn , µn ). Let Ω be the component of K ◦ that contains Un . By Lemma 3.7, Aq (Ω, τ |Ω) ⊂ Aq (K, τ ). Note, every function f in Aq (Ω, τ |Ω) has zero values oﬀ Ω. Clearly, fn (χ∆n ) belongs to Aq (Ω, τ |Ω) also. Set Fn = Un ∪ (∆n \ Un ). Because ∆n ⊂ U n , we see that χFn = fn ∈ Aq (Ω, τ |Ω). So by Lemma 3.2, χFn f ⊂ Aq (Ω, τ |Ω) for each f ∈ A(Ω). This implies that Aq (Ω, µn ) = Aq (Ω, τ |Fn ) ⊂ Aq (Ω, τ ). Let a ∈ Un . Then there exists c > 0 and an open open disk Da ⊂ Un such that for all f ∈ A(Ω) 1 |f (a)| ≤ c{ |f |q dτ } q , a ∈ Da . Let g ∈ A(Ω). Then there is a sequence {qi }∞ i=1 in A(Ω) so that qi → gχFn in Lq (τ ). Then qi → g uniformly on Da . Hence, it follows that for a ∈ Da 1 1 1 |g(a)| = lim |qi (a)| ≤ c lim{ |qi |q dτ } q = c{ gχFn dτ } q = c{ |g|q dµn } q . Thus a ∈ ∇Aq (Ω, µn ). Therefore, Un ⊂ ∇Aq (Ω, µn ). By Lemma 3.7, ∇Aq (Ω, µn ) ⊂ ∇Aq (K, τ ), so we see (notice that Un is a component of ∇Aq (K, τ )) that Un = ∇Aq (Ω, µn ). The hypothesis and Lemma 3.15 together imply that Un is a ﬁnitely connected domain. It is also easy to see that Aq (Ω, µn ) is irreducible. Applying Proposition 3.4, we have A(Un ) ⊂ Aq (Ω, µn ). Consequently, Aq (U n , µn ) ⊂ Aq (Ω, µn ). Therefore, we conclude that Aq (Ω, µn ) = Aq (U n , µn ). Hence, Un = ∇Aq (Kn , µn ). This proves the claim. Since each Aq (Kn , µn ) is contained in Aq (K, µ) and since {Aq (Kn , µn )} are pairwise othrogonal, we have Aq (K, µ) ⊃ Lq (µ0 ) ⊕ Aq (K1 , µ|∆1 ) ⊕ ... ⊕ Aq (Kn , µ|∆n ) ⊕ .... n For the other direction of the equality, let f be the pointwise limit of { i=1 fi }∞ n=1 . Then the bounded convergence theorem implies that f ∈ Aq (K, τ ). We show that 1 − f = 0 a.e. [τ ]. Otherwise, there exists a Borel subset E of the support of τ such that 0 = χE = 1 − f . Since both 1 and f are in Aq (K, τ ), we have that χE ∈ Aq (K, τ ). By the purity, we have Lq (τ |E) = Aq (K, τ |E). So it follows by Theorem 1.1 that ∇Aq (K, τ |E) = ∅. But χE fn = 0 for each n ≥ 1, thus we have ∇Aq (K, τ |E) ∩ (∪Un ) = ∅. But by the deﬁnition of abpes, ∇Aq (K, τ |E) ⊂ ∇Aq (K, µ) = ∪Un . This is a contradiction, and hence f − 1 = 0. Therefore, {∆n } is a Borel partition. Let g ∈ Aq (K, τ ). By the Lebesgue dominated convergence theorem, we conclude that g = limn→∞ ni=1 fi g in Lq (τ ). Therefore, Aq (K, µ) ⊂ Lq (µ0 ) ⊕ Aq (K1 , µ|∆1 ) ⊕ ... ⊕ Aq (Kn , µ|∆n ) ⊕ .... Consequently, Aq (K, µ) = Lq (µ0 ) ⊕ Aq (K1 , µ|∆1 ) ⊕ ... ⊕ Aq (Kn , µ|∆n ) ⊕ .... Now we prove the rest of Theorem 2.1: For 1), since we have already proved U n ⊂ ∆n above, we only need to show that Aq (K, µn ) = Aq (U n , µn ). Because

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A(K) ⊂ A(U n ), it follows by the deﬁnition of abpe that Un ⊂ ∇Aq (U n , µn ) ⊂ ∇Aq (K, µn ). Notice that Aq (K, µn ) ⊂ Aq (U n , µn ) and the latter is irreducible. So we see that Aq (K, µn ) is irreducible also. This implies that ∇Aq (K, µn ) have only one component. Also, it is clear that ∇Aq (K, µn ) ⊂ ∇Aq (K, τ ). So we conclude that ∇Aq (K, µn ) = Un . Thus, there is hn ∈ Aq (K, µn ) such that ˆhn = χUn and hn = 0 oﬀ U n . By the uniqueness, we get that hn = fn = χ∆n . As we proved above, ∞ f = n=1 f hn for each f ∈ Aq (K, τ ). Hence, we conclude that Aq (K, τ ) ⊂ Aq (K, µ1 ) ⊕ ... ⊕ Aq (K, µn ) ⊕ ... ⊂ Aq (U 1 , µ1 ) ⊕ ... ⊕ Aq (U n , µn ) ⊕ ... = Aq (K, τ ). Consequently, Aq (K, µn ) = Aq (U n , µn ) for each n ≥ 1. 2) follows from Lemma 3.15. For 3), let e be the map, f → fˆ, from L∞ (µn ) ∩ Aq (Kn , µn ) into H ∞ (Un ). Then e is surjective by Lemma 3.10 and is injective by Lemma 3.14. Since e(f g) = e(f )e(g), e is an algebraic isomorphism between two commutative Banach algebras and thus e is an isometry. Next we need to show that e is a weak-star homeomorphism. To do this, we will argue as in [6]. Using Krein-Smulian theorem it suﬃces to show that e is weak-star sequentially continuous. Recall that a sequence of functions in H ∞ (Un ) is weak-star Cauchy sequence if and only if it is uniformly bounded on Un and it is a Cauchy sequence in the topology of pointwise convergence. Let {hi } be a sequence in Aq (Kn , µn )∩L∞ (µn ) that converges to zero in the weak star topology. By the uniform boundedness, {hi } is bounded and hence {e(hi )} is also bounded. Let a ∈ Un and let ka be kernel function. Then ˆ lim e(hi ) = lim hi (a) = lim hi ka dµn = 0. i→∞

i→∞

i→∞

So e(hi ) weak-star converges to zero. Therefore, e is a weak-star homeomorphism. 4) follows from Proposition 3.3 and Proposition 3.4. For 5), by Proposition 3.4, each f ∈ H ∞ (Un ) has nontangential limits almost everywhere on ∂Un with respect to µ|∂Un and the nontangential limits are equal to e−1 (f ) a.e. [µ|∂Un ]. Since fˆe = f = f ∗ a.e. [µ] on Un , we see that f ∗ = e−1 (f )|∆n a.e. [µ]. Evidently fˆ on Un m(e(f )) = m(fˆ) = −1 ˆ e (f ) on ∂Un f on Un = f on ∂Un = f,

for each f ∈ Aq (Kn , µn ).

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Therefore, m is the inverse map of e. So the proof of Theorem 2.1 is complete.

References [1] P. Ahern, D. Sarason, The H q spaces of a class of function algebras, Acta Math., 117 (1967), 123–163. [2] P. Ahern, D. Sarason, On some hypo-Dirichlet algebras of analytic functions, Amer. J. Math., 89 (1967), 932–941. [3] J. Akeroyd, E. Saleeby, A class of P t (dµ) spaces whose point evaluations very with t, Proc. Amer. Math. Soc., 127, No. 2 (1999), 537–542. [4] J. Brenann, Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–301. [5] J. Conway, The theory of subnormal operators, Math. Surveys and Monographs Vol. 36, Amer. Math. Soc. 1991. [6] J. Conway, N. Elias, Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis 117 (1993), 1–24. [7] A. Davie, Dirichlet algebras of analytic functions, J. Functional Analysis 6 (1967), 348–356. [8] T. Gamelin, Uniform Algebras, Prentice Hall, Englewood Cliﬀs, N.J., 1969. [9] T. Gamelin, J.B. Garnett, Constructive techniques in rational approximation, Trans. Amer. Math. Soc. 143 (1969), 187–200. [10] I. Glicksburg, The abstract F. and M. Riese theorem, (1967), 109–122.

J. Functional Analysis 1

[11] S. Mergeljan, On the completeness of system of analytic functions, Trans. Amer. Math. Soc. 19 (1962), 109–166. [12] J. McCarthy, Analytic structure for subnormal operators, Integral Equation and Operator Theory, 13 (1990), 251–270. [13] J. McCarthy, L. Yang, Bounded point evaluations on the boundaries of L regions, Indiana Univ. Math. J., 43, No. 3 (1994). [14] T. Miller, R. Smith, Nontangential limits of functions in some P 2 (µ) spaces, Indiana Univ. Math. J., 39, No. 1 (1990), 19–26. [15] T. Miller, W. Smith, L. Yang, Bounded point evaluations for certain P t (µ) spaces, Illinois J. Math., 43, No. 1 (1999). [16] R. Olin, L. Yang, The commutant of multiplication by z on the closure of polynomials in Lq (µ), J. Functional analysis, 134 (1995), 297–320. [17] Zhijian Qiu, Equivalence classes of subnormal operators, J. Operator Theory, 32 (1994), 47–75. [18] Zhijian Qiu, Density of polynomials, Houston J. Math., 21, No. 1 (1995), 109–118. [19] Zhijian Qiu, Boundary values of analytic functions in the Banach spaces P t (τ ) on crescents, Illinois Journal of Math., 39, No. 2 (1995), 305–322. [20] Zhijian Qiu, Approximation in the mean by rational functions, Integral Equation and Operator Theory, 25 (1996), 235–252.

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[21] Zhijian Qiu, The commutant of rationally cyclic subnormal operators and rational approximation, Integral Equation and Operator Theory, 27, No. 3 (1997), 334–346. [22] Zhijian Qiu, Carleson measures on circular domains, Houston J. Math., 31, No. 4 (2005), 1199–1206. [23] Zhijian Qiu, A class of operators similar to the shift on H 2 (G), Integral Equation and Operator Theory, 56, No. 3 (2006), 415–429. [24] Zhijian Qiu, On quasisimilarity of subnormal operators, Science in China, Series A Math., 50, No. 2 (2007), 145–154. [25] Zhijian Qiu, Carleson measure and polynomial approximation, Chinese Ann. of Math., Series A, 28 (2007), 221–228. [26] Zhijian Qiu, On pointwise bounded approximation, to appear. [27] D. Sarason, Weak-star density of polynomials, J. Reine Angew. Math., 252 (1972), 1–15. [28] J. Thomson, Approximation in the mean by polynomials, Ann. of Math. 133 (1991), 477–507. [29] T. Trent, H 2 (µ) spaces and bounded point evaluations, Paciﬁc J. Math. 80 (1979), 279–292 [30] M. Tsuji, Potential theory in modern function theory, Chelsa, New York, 1975. [31] A. Vitushkin, Analytic capacity of sets in problems of rational approximation, Uspehi Mat. Nauk 22 (1967), No. 6 (138), 141–199; Russian Math. Surveys 22 (1967), 139– 200. [32] J. Walsh, The approximation of harmonic functions by harmonic polynomials and harmonic rational functions, Bull. Amer. Math. Soc. 3, 276–277. [33] J. Wermer, Analytic disks in maximal ideal spaces, American J. of Mathematics, 86 (1964), 161–170. Zhijian Qiu Department of Mathematics Southwestern University of Finance and Economics Chengdu, 610074 China e-mail: [email protected] Submitted: August 30, 2006 Revised: September 25, 2006

Integr. equ. oper. theory 59 (2007), 245–256 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020245-12, published online June 27, 2007 DOI 10.1007/s00020-007-1522-0

Integral Equations and Operator Theory

An Operator Corona Theorem for Some Subspaces of H ∞ Amol Sasane Abstract. Let E, E∗ be separable Hilbert spaces. If S is an open subset of T, then AS (L(E, E∗ )) denotes the space of all functions f : D ∪ S → L(E, E∗ ) that are holomorphic in D, and bounded and continuous on D ∪ S. In this article we prove the following results: 1. A theorem concerning the approximation of f ∈ AS (L(E, E∗ )) by a function F that is holomorphic in a neighbourhood of D ∪ S and such that the error F − f is uniformly bounded in the disk D. 2. The corona theorem for AS (L(E, E∗ )) when dim(E) < ∞: If there exists a δ > 0 such that for all z ∈ D ∪ S, f (z)∗ f (z) ≥ δ 2 I, then there exists a g ∈ AS (L(E∗ , E)) such that for all z ∈ D ∪ S, g(z)f (z) = I. 3. The problem of complementing to an isomorphism for AS (L(E, E∗ )) when dim(E) < ∞ (Tolokonnikov’s lemma): f ∈ AS (L(E, E∗ )) has a left inverse g ∈ AS (L(E∗ , E)) iﬀ it is a ‘part’ of an invertible element F in AS (L(E∗ )). Mathematics Subject Classification (2000). Primary 30H05; Secondary 46J15, 47A56. Keywords. Operator-valued holomorphic functions, Tolokonnikov’s lemma, corona theorem, control theory.

1. Notation and introduction This paper is devoted to proving a corona theorem for a class of subspaces of operator-valued H ∞ functions. We use the following standard notation: D = {z | |z| < 1}, T = {z | |z| = 1}. Let X be a Banach space. If Ω is a domain in C, then by H ∞ (Ω, X) we mean the space of all bounded holomorphic functions in Ω, equipped with the supremum norm: f ∞ = sup f (z), f ∈ H ∞ (Ω, X). z∈Ω

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If Ω = D, then we denote the space H ∞ (D, X) simply by H ∞ (X). We refer the reader to Nikolski [8, §3.11] for preliminaries on vector- and operator-valued holomorphic functions. Definition 1.1. Let X be a Banach space. If S ⊂ T, then AS (X) denotes the Banach space of functions f : D ∪ S → X that are holomorphic in D, and continuous and bounded on D ∪ S, equipped with the supremum norm: f ∞ = sup f (z), f ∈ AS (X).

z∈D∪S

The reason behind using ‘A’ in the notation above is that the symbol A is usually used to denote the disk algebra (S = T and X = C). If S = ∅, then we get the other extreme, H ∞ (X). Note that if X is a Banach algebra (e.g. X = L (E), where E is a Hilbert space), then AS (X), with pointwise multiplication, is also a Banach algebra. In this article, except for §2, we consider the case X = L (E, E∗ ), where E, E∗ are separable Hilbert spaces and dim(E) < ∞. The space L (E, E∗ ) is equipped with the operator norm. We now give the motivation from Control Theory for considering the function spaces AS . A standard class of transfer functions of inﬁnite-dimensional systems (for example for systems with delays) is the Callier-Desoer class consisting of Fourier transforms of signed measures without singular continuous part; see [2]. In this case the stable transfer functions are in H ∞ of the half plane, and are continuous on the imaginary axis, but they have a discontinuity at ∞. By the standard linear fractional transformation taking the right half plane to the unit disk D (and iR ∪ {∞} to T), we then arrive at the class AS , where S is just the circle without a point (corresponding to ∞). Our class AS , with an arbitrary S, can then be considered as a generalization of this class, when one has transfer functions of inﬁnite-dimensional systems that are not exponentially stable, but stable in a weaker sense. In this case the spectrum of the inﬁnitesimal generator typically intersects the imaginary axis in some closed subset F . Then the corresponding transfer function is continuous at all points of the open set S = (iR ∪ {∞}) \ F . By passing over to the disk (so that S is taken to S ⊂ T), one gets an element of AS ; see [10]. In this paper we will prove an operator corona theorem and Tolokonnikov’s lemma for the space AS (L (E, E∗ )): Theorem 1.2. Let E ⊂ E∗ be separable Hilbert spaces and dim(E) < ∞. Suppose that S is an open subset of T, and that f ∈ AS (L (E, E∗ )). Then the following are equivalent: 1. (Corona condition) There exists a δ > 0 such that for all z ∈ D ∪ S, f (z)∗ f (z) ≥ δ 2 I. 2. (Left invertibility) There exists g ∈ AS (L (E∗ , E)) such that for all z ∈ D∪S, g(z)f (z) = I. 3. (Complementing to an isomorphism) There exists an invertible element F in AS (L (E∗ )) such that for all z ∈ D ∪ S, F (z)|E = f (z).

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The equivalence of 1 and 2 is referred to as the Operator Corona Theorem, while the equivalence of 2 and 3 is called Tolokonnikov’s Lemma. We note that the left invertability or corona condition in the corona theorem implies that f (z) is one-to-one for each z, and so dim(E) ≤ dim(E∗ ). Without loss of generality, we may assume that E ⊂ E∗ . The problem of complementing to an isomorphism is then that of describing those functions f ∈ AS (L (E, E∗ )) for which there exists an invertible F ∈ AS (L (E∗ )) such that F |E = f . We now list some instances when the equivalences in Theorem 1.2 are known in the case of H ∞ (L (E, E∗ )) (that is, for AS (L (E, E∗ )) where S = ∅): 1. Operator Corona Theorem for H ∞ (L (E, E∗ )). The equivalence of the corona condition with left invertability was proved by L. Carleson [1] for dim(E) = 1 and dim(E∗ ) < ∞, and this is the famous Carleson Corona Theorem. By a simple linear algebraic argument and the Carleson Corona Theorem, it was shown by P. Fuhrmann [4] that the operator corona theorem for H ∞ (L (E, E∗ )) holds if E, E∗ are both ﬁnite-dimensional. Later V. Vasyunin extended this result to the case dim(E∗ ) = ∞, still with dim(E) < ∞. (Vasyunin’s result was published, with attribution, in Tolokonnikov [12].) If E is not ﬁnite-dimensional, then this equivalence is not true, and this was shown by S. Treil [14]. 2. The problem of complementing to an isomorphism for H ∞ (L (E, E∗ )). This was shown by Tolokonnikov in the case when dim(E) < ∞. (See [12], [13]; this is called Tolokonnikov’s Lemma in [7, §10, Appendix, p. 293]). Tolokonnikov’s Lemma was generalized to the case when dim(E) = ∞ by S. Treil in [15]. We refer the reader to §9.2 of the book by Nikolski [8] and the article by S. Treil [15] for an account of these results. We generalize the operator corona theorem and Tolokonnikov’s lemma for H ∞ (L (E, E∗ )) (dim(E) < ∞) to the case of AS (L (E, E∗ )) (dim(E) < ∞), where S is an arbitrary open subset of T, that is, we prove Theorem 1.2. The main idea is to approximate the corona data by a function that can be extended analytically through intervals of continuity, solve the corona problem (or the problem of complementing to an isomorphism) for this new function in a bigger domain, and then to get the solution for the original function, which is continuous on the set S. Thus, we will use the corresponding theorems for H ∞ (L (E, E∗ )) together with an approximation result, which we prove ﬁrst in §2. Subsequently we prove the operator corona theorem (§3) and Tolokonnikov’s lemma (§4) for the space AS (L (E, E∗ )).

2. An approximation result In order to prove the corona theorem and Tolokonnikov’s lemma for our class AS , we will use the H ∞ versions of these theorems together with a key approximation result (Theorem 2.2 below). This result is a consequence of the following lemma,

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which we prove following Range [9]. We will use the notation: ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ = −i = +i , . ∂z 2 ∂x ∂y ∂z 2 ∂x ∂y Lemma 2.1. Let Ω be an open bounded subset of C containing 0 and with boundary ∂Ω that has a continuous polar parameterization r = ρ(θ). Suppose that C is a closed subarc in ∂Ω, and K is an open (in ∂Ω) set containing C. Let R := {rζ : r ≥ 0, ζ = ρ(θ) ∈ K}, the sector corresponding to K. Let X be a Banach space. Suppose that f : Ω → X is bounded and holomorphic in Ω, it extends continuously to K. Then given any > 0, there exists a function F : Ω ∪ R → X with the following properties: (S1) F |Ω − f ∞ < . (S2) There exists a neighbourhood O of C in C such that F is holomorphic in Ω ∪ O. Proof. Deﬁne a (trivial radial) continuous extension of f (denoted by the same letter) to Ω ∪ R by f (rz) = f (z), z ∈ ∂Ω, r > 1.

R ∂Ω

Ω

W

U

C K

0

Figure 1. Continuous extension of f across K. Support of the cut-oﬀ function ϕ is contained in W . Let ϕ ∈ D(R2 ) be a test function such that 0 ≤ ϕ ≤ 1, ϕ = 1 on a neighbourhood U of C (in C) and 0 oﬀ a slightly larger neighbourhood W ; see Figure 1. Deﬁne a function h (with values in X) by f (z) 1 dxdy + ϕ(ζ)f (ζ) =: u + ϕf (2.1) h(ζ) = (∂ϕ(z)) π z−ζ Note that the function h is well-deﬁned for all z ∈ C, if we put ϕf = (∂ϕ)f = 0 outside of Ω ∪ R, where f is not deﬁned.

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Moreover h is continuous on C. Indeed, the integral u is continuous since the convolution of the locally integrable function z → z1 with the compactly supported continuous function (∂ϕ)f is continuous, and trivially ϕf is continuous. Using Green’s Theorem one can see that for a given continuous compactly supported ψ, the formula ψ(z) 1 dxdy u(ζ) = π z−ζ gives a solution u of a ∂-equation ∂u = ψ, where the derivative is understood in the sense of distributions; see for instance §1 in Chapter VIII of Garnett [6]. Hence, u satisﬁes the ∂-equation ∂u = (∂ϕ)f.

(2.2)

We claim that h is holomorphic in Ω. Indeed, since f is holomorphic in Ω, the ∂-equation (2.2) implies ∂h = ∂(u − ϕf ) = (∂ϕ)f − (∂ϕ)f = 0. Furthermore, we show that f − h is holomorphic in U . Using again (2.2) and recalling that ϕ ≡ 1 in U , we get ∂u ≡ 0, ∂ϕ ≡ 0 on U , so ∂h = ∂(u − ϕf ) = ϕ∂f = ∂f in U . But that exactly means f − h is analytic in U . We observe that if we take the function F to be f − h, then it is holomorphic in Ω∪U , but it does not necessarily satisfy condition (S1). We rectify this situation by adding a shifted version of h (which is close to h). For 0 < r < 1 deﬁne hr (z) := h(rz). Since h is continuous, h(rz) → h(z) as r → 1, uniformly on compact subsets of C. Therefore, we can ﬁnd r < 1 suﬃciently close to 1 so that (2.3) (hr − h)|Ω ∞ ≤ . Deﬁne F = f − h + hr on Ω ∪ R. The condition (S1) is satisﬁed since F − f ∞ = hr − h∞ < on Ω. Moreover, F is holomorphic in 1 1 (Ω ∪ U ) ∩ Ω = Ω ∪ (U ∩ Ω) = Ω ∪ O r r because f, h, hr are all holomorphic in Ω, f − h is holomorphic in U , and hr is holomorphic in 1r Ω. Using the result above, we now prove the following result concerning uniform holomorphic approximation of functions in AS (X). In Lemma 2.1, we produced an approximate extension of a function across a compact arc, but in the following theorem we construct an approximate extension across an open arc. In order to do this, we decompose the open arc into disjoint open intervals, and furthermore, we will write each open interval as a union of closed intervals, and these closed intervals will serve as the compact arcs of Lemma 2.1: this lemma will then be used recursively in order to construct the desired extension.

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Theorem 2.2. Let S be an open subset of T, X be a Banach space, and f ∈ AS (X). Then given any > 0, there exists a neighbourhood O of S in C and a holomorphic function F : D ∪ O → X such that F |D − f ∞ < . Proof. Let In , n ∈ N, be pairwise disjoint open intervals such that S = ∪∞ n=1 In . Each In can be written as a union of closed intervals as follows: In = (∪∞ Q m=1 nm )∪ P ), where Q , Q , Q , . . . are pairwise disjoint closed intervals, Pn1 , (∪∞ n1 n2 n3 m=1 nm Pn2 , Pn3 , . . . are pairwise disjoint closed intervals, each Pnk joins the endpoints of two of the Qnl s, and each Qnk joins the endpoints of two of the Pnl s; see Figure 2.

Qn2

Pn1

Qn1

Pn2

Qn3

In Figure 2. The interlaced closed intervals. ∞ We can renumber these sets so that S = (∪∞ n=1 Qn )∪(∪n=1 Pn ), where Q1 , Q2 , Q3 , . . . are pairwise disjoint closed intervals, P1 , P2 , P3 , . . . are pairwise disjoint closed intervals, each Pn joins the endpoints of two of the Qk s, and each Qn joins the endpoints of two of the Pk s.

Step 1. We construct open sets On (in C) and functions ϕn with the following properties: 1. On is an open bounded neighbourhood of Qn , On ∩ O m = ∅ if n = m, Ωn := Ωn−1 ∪ On is a domain with a boundary having a continuous polar parameterization. 2. ϕn : Ωn → X is holomorphic, ϕn is bounded in Ωn and extends continuously to S and the boundary of On , and ϕn |Ωn−1 − ϕn−1 ∞ < /2n+1 . We do this construction inductively as follows. Let O0 := ∅, Ω0 = O0 ∪ D, and let ϕ0 : Ω0 → X be f . Assuming that we have already constructed O0 , . . . , Ok−1 and ϕ0 , . . . ϕk−1 , the existence of Ok and ϕk follows from Lemma 2.1, applied to ϕk−1 and the closed subarc C = Qk . The property that the boundary has a continuous polar parameterization can be preserved at each step by choosing the boundary of the open neighbourhood of Qk to be a circular arc Ck ; see Figure 3. ...

Ck1 Qk1

Ck2 Qk2

Figure 3. Neighbourhoods of Qk s.

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We observe that ∞ k=1 (ϕk −ϕk−1 )+f converges uniformly on compact subsets of D to a function Φ which is bounded and holomorphic in D. Also for each n, ∞ ∞ (ϕk − ϕk−1 ) + f = (ϕk − ϕk−1 ) + ϕn , Φ= k=1

k=n+1

and so Φ has a holomorphic extension to each Ωn . Finally, we also observe that Φ|D − f ∞ < /2. Step 2. Let Ω = D ∪ (∪∞ n=1 On ), and consider Φ : Ω → X. Then Ω has a boundary with a continuous polar parameterization, Φ is bounded on Ω and has a continuous ∞ extension to the neighbourhood (in ∂Ω) ∪∞ n=1 Cn ∪ (∪n=1 Pn ) of each closed subarc ∞ Pk \ (∪n=1 On ), k ∈ N. See Figure 4. Cn2

Cn1 On1

On2 Qn1

Pn

Qn2

Figure 4. Neighbourhoods of Qn s. Repeating the argument in Step 1 above with Φ instead of f , we can ﬁnd a neighbourhood Ωe of Ω ∪ (∪∞ n=1 Pn ) and a F : Ωe → X which is holomorphic in Ωe and such that F |Ω − Φ∞ < /2. So we have F |D − f ∞ < , and this completes the proof. The above Theorem 2.2 can be found in Stray [11] and Gamelin and Garnett [5] for the case of complex-valued functions. We will use Theorem 2.2 to prove the operator corona theorem and Tolokonnikov’s lemma for AS . Indeed, given the corona data from AS , we will approximate it by a function bounded and holomorphic in a somewhat larger set, and use the corona result for H ∞ (Ω), where Ω is a simply connected domain with a boundary that is close to T.

3. An operator corona theorem In this section we will prove an Operator Corona Theorem for AS (L (E, E∗ )). In order to prove this (Theorem 3.3 below), we will use the approximation result from the previous section (Theorem 2.2) and the following corona theorem in the H ∞ case. Proposition 3.1. (Fuhrmann-Vasyunin) Let E, E∗ be separable Hilbert spaces and n := dim(E) < ∞. Suppose that Ω is a simply connected domain not equal to C, and f ∈ H ∞ (Ω, L (E, E∗ )). Let δ > 0. Then there exists a constant C(n, δ) such that for all f ∈ H ∞ (Ω, L (E, E∗ )) satisfying ∀z ∈ Ω,

I ≥ f (z)∗ f (z) ≥ δ 2 I,

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there exists a g ∈ H ∞ (Ω, L (E∗ , E)) such that ∀z ∈ Ω,

g(z)f (z) = I,

and

g∞ < C(n, δ).

Proof. For Ω = D, this is precisely the statement of the Fuhrmann-Vasyunin theorem (see [7, §11, Appendix, p. 293]). The general case can be seen as follows. By the Riemann mapping theorem, there exists a one-to-one holomorphic map ϕ from Ω onto D. Hence f0 := f ◦ ϕ−1 ∈ H ∞ (L (E∗ , E)), and I ≥ f0 (z)∗ f0 (z) ≥ δ 2 I for all z ∈ D. From the Fuhrmann-Vasyunin theorem in the case of D, it follows that there exists a g0 ∈ H ∞ (L (E∗ , E)) such that for all z ∈ D, g0 (z)f0 (z) = I, and g0 (z)∞ ≤ C(n, δ). Deﬁning g := g0 ◦ ϕ, we see that g ∈ H ∞ (Ω, L (E∗ , E)), and the result follows. Remark 3.2. The original bound C(n, δ) (see §11, page 293, [7]) was improved a few years ago by T.Trent [18], and also in a paper by Treil and Wick [17]. This bound, given by √ √ M 1 1 C(n, δ) = n+1 log 2n + (where M = 1 + e2 + e + 2e ≈ 8.39), (3.1) δ δ δ is almost optimal, and the gap between it and the lower bound in Treil [16] is very small. We are now ready to prove our new corona theorem for AS (L (E, E∗ )). Theorem 3.3. Let E, E∗ be separable Hilbert spaces and n := dim(E) < ∞. Suppose that S is an open subset of T, and that δ > 0. Let C(n, δ) denote a constant as in Proposition 3.1, and let f ∈ AS (L (E, E∗ )) satisfy ∀z ∈ D ∪ S,

I ≥ f (z)∗ f (z) ≥ δ 2 I.

(3.2)

Then given any > 0, there exists a g ∈ AS (L (E∗ , E)) such that for all z ∈ D ∪ S g(z)f (z) = I,

and

g∞ ≤ (1 + )C(n, δ − ).

(3.3)

Remark 3.4. Estimates on solutions. In Proposition 3.1, we did not assume that the function δ → C(δ, n) is continuous, so we need C(n, δ−) in (3.3). If we assume that δ → C(n, δ) is continuous (for example, as in the C(n, δ) given by (3.1)), then we can get the estimate g∞ ≤ (1 + )C(n, δ) in (3.3). Proof. Given f ∈ AS (L (E, E∗ )) and 1 > 0, by Theorem 2.2 there exists a neighbourhood O of S and a L (E, E∗ )-valued holomorphic function fe deﬁned on Ω := D ∪ O such that fe |D − f ∞ < 1 . As fe is continuous in O, given any 2 > 0, we can shrink O suitably so as to ensure that for the new Ω = D ∪ O, we have that for all z ∈ Ω \ D, there exists a z∗ ∈ S such that fe (z) − fe (z∗ ) < 2 . For all z ∈ D ∪ S and x ∈ E, fe (z)x ≥ f (z)x−(fe(z)−f (z))x ≥ f (z)x−fe|D −f ∞ x > (δ −1 )x. For all z ∈ Ω \ D and x ∈ E, fe (z)x ≥ fe (z∗ )x−(fe (z∗ )−fe (z))x ≥ (δ −1 )x−2 x = (δ −1 −2 )x. Consequently for all z ∈ Ω, and all x ∈ E, fe (z)x ≥ (δ − 1 − 2 )x.

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For all z ∈ D ∪ S, fe (z) ≤ fe (z) − f (z) + f (z) < 1 + 1, and for all z ∈ Ω \ D, fe (z) ≤ fe (z) − fe (z∗ ) + fe (z∗ ) < 2 + 1 + 1. Thus 1 ∀z ∈ Ω, 1 ≥ fe (z) , and 1 + 1 + 2 δ − 1 − 2 1 ∀x ∈ E, 1 + 1 + 2 fe (z)x ≥ 1 + 1 + 2 x. We choose 1 , 2 small enough so that δ − 1 − 2 > δ − . 1 + 1 + 2 Hence by Proposition 3.1, it follows that there exists a ge ∈ H ∞ (Ω, L (E∗ , E)) such that for all z ∈ Ω, ge (z)fe (z) = I, and ge ∞ ≤

C(n, δ − ) ≤ C(n, δ − ). 1 + 1 + 2

Consequently for all z ∈ D ∪ S, ge (z)fe (z) = I, and so ge (z)f (z) = I − ge (z)[fe (z) − f (z)]. By choosing 1 at the beginning also to have satisﬁed 1 1 , 1 < 1 − 1 + C(n, δ − ) we see that I − ge [fe − f ] is invertible as an element of the Banach algebra AS (L (E)), with the norm of the inverse bounded by 1 + . Deﬁne g(z) = (I − ge (z)[fe (z) − f (z)])−1 ge (z),

z ∈ D ∪ S.

Then g ∈ AS (L (E∗ , E)), g(z)f (z) = I for all z ∈ D ∪ S, and g∞ ≤ (1 + )C(n, δ − ).

Remark 3.5. 1. Scalar case. The result in Theorem 3.3 in the scalar case when E = C and dim(E∗ ) < ∞ was shown in Theorem 2 of D´etraz [3] using algebraic tools. Arne Stray [11] gave another proof in this scalar case, and the proof of Theorem 3.3 follows his approach. But no bounds on solutions were obtained in [3] or [11]. 2. Application to control theory. Coprimeness plays an important role in the factorization approach to solving stabilization problems in control theory, and using the corona theorem 3.3, one can give a necessary and suﬃcient condition for the coprimeness of a matrix pair in AS ; see Chapter 8 of Vidyasagar [19]).

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4. Complementing to an isomorphism In this section, we will prove the equivalence of the operator corona problem with the problem of completing an embedding to an isomorphism. We will use the classical result of Tolokonnikov for H ∞ (Proposition 4.1 below), together with the approximation result (Theorem 2.2) in order to prove the corresponding version for AS (Theorem 4.2). Proposition 4.1. (Tolokonnikov’s lemma) Let E ⊂ E∗ be separable Hilbert spaces and dim(E) < ∞. Suppose that Ω is a simply connected domain not equal to C, and that f ∈ H ∞ (Ω, L (E, E∗ )). Then the following statements are equivalent: 1. There exists a g ∈ H ∞ (Ω, L (E∗ , E)) such that for all z ∈ Ω, g(z)f (z) = I. 2. There exists an invertible F ∈ H ∞ (Ω, L (E∗ )) such that F (z)|E = f (z) for all z ∈ Ω. Moreover, the F can be so chosen that it satisfies F −1 ∞ ≤ g∞ (1 + f ∞ ) + 1. Proof. When Ω = D, this is precisely the statement of Tolokonnikov’s lemma (see [7, §10, Appendix, p.293], and also [12], [13]). The general case is a trivial consequence using the Riemann mapping theorem in the same way as in the proof of Proposition 3.1. We now give the main result in this section. Theorem 4.2. Let E ⊂ E∗ be separable Hilbert spaces and n := dim(E) < ∞. Suppose that S is an open subset of T, and that f ∈ AS (L (E, E∗ )). Then the following are equivalent: 1. There exists g ∈ AS (L (E∗ , E)) such that for all z ∈ D ∪ S, g(z)f (z) = I. 2. There exists an invertible F ∈ AS (L (E∗ )) such that for all z ∈ D ∪ S, F (z)|E = f (z). Proof. Given f ∈ AS (L (E, E∗ )) and 1 > 0, by Theorem 2.2 there exists a neighbourhood O of S and a L (E, E∗ )-valued holomorphic function fe deﬁned on Ω := D ∪ O such that fe |D − f ∞ < 1 . As fe is continuous in O, given any 2 > 0, we can shrink O suitably so as to ensure that for the new Ω = D ∪ O, we have that for all z ∈ Ω \ D, there exists a z∗ ∈ S such that fe (z) − fe (z∗ ) < 2 . Let δ > 0 be such that for all z ∈ D ∪ S, f (z)∗ f (z) ≥ δ 2 I. Proceeding as in the proof of Theorem 3.3, we obtain ∀z ∈ Ω, and ∀x ∈ E,

fe (z)x ≥ (δ − 1 − 2 )x, and fe (z) ≤ 1 + 2 + f ∞ .

Choose 1 , 2 small enough so that δ − 1 − 2 > δ/2 and 1 + 2 < 1. By applying Proposition 3.1 to αfe , where α := (1 + f ∞ )−1 , it follows that there exists a ge ∈ H ∞ (Ω, L (E∗ , E)) such that for all z ∈ Ω, ge (z)fe (z) = I, and ge ∞ ≤ αC(n, α(δ/2)) =: β, where C(n, ·) denotes a constant from Proposition 3.1.

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By Proposition 4.1, there exists an invertible Fe ∈ H ∞ (Ω, L (E∗ )) such that for all z ∈ Ω, Fe (z)|E = fe (z), and Fe−1 ∞

≤ ge ∞ (1 + fe ∞ ) + 1 ≤ β(1 + 1 + 2 + f ∞ ) + 1 ≤ β(2 + f ∞ ) + 1 =: γ.

Let P ∈ L (E∗ , E) denote the projection onto E. Consider H : D ∪ S → L (E∗ ) deﬁned by H(z) = Fe (z)−1 (f (z) − fe (z))P ∈ L (E∗ ),

z ∈ D ∪ S.

It is clear that H ∈ AS (L (E∗ )). Furthermore, we have that for all z ∈ D ∪ S, 1 H(z) ≤ Fe (z)−1 f (z) − fe (z)P ≤ γ1 < 2 provided that we choose 1 < γ −1 /2 at the outset. So I + H is invertible in AS (L (E∗ )). Deﬁne F : D ∪ S → L (E∗ ) by F (z) = Fe (z)(I + H(z)),

z ∈ D ∪ S.

Then we have that F ∈ AS (L (E∗ ) is invertible, and if x ∈ E, then F (z)x = =

Fe (z)x + Fe (z)H(z)x = fe (z)x + (f (z) − fe (z))P x fe (z)x + (f (z) − fe (z))x = f (z)x,

and so F |E = f . This completes the proof.

Remark 4.3. Tolokonnikov’s lemma plays an important role in stabilization of linear systems in control theory. Indeed, Tolokonnikov’s lemma implies that if a transfer function G has a right (or left) coprime factorization, then G has a doubly coprime factorization, and the standard Youla parameterization yields all stabilizing controllers for G. For background on the relevance of Tolokonnikov’s lemma in control theory, see Vidyasagar [19]. Acknowledgment The author would like to thank Sergei Treil (Mathematics Department, Brown University) for many useful discussions.

References [1] L. Carleson, Interpolations by bounded analytic functions and the corona problem. Annals of Mathematics, 76 (1962), 547–559. [2] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Systems Theory. Springer, 1995. ´ [3] J. D´etraz. Etude du spectre d’alg`ebres de fonctions analytiques sur le disque unit´e. (French) Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences. S´eries A et B, 269 (1969), A833–A835. [4] P. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space. Transactions of the American Mathematical Society, 132 (1968), 55– 66.

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[5] T.W. Gamelin and J. Garnett, Uniform approximation to bounded analytic functions. Revista de la Uni´ on Matem´ atica Argentina, 25 (1970), 87–94. [6] J.B. Garnett, Bounded analytic functions. Academic Press, 1981. [7] N.K. Nikolski˘ı, Treatise on the shift operator. Grundlehren der Mathematischen Wissenschaften, vol. 273, Springer-Verlag, Berlin, 1986, Spectral function theory, With an appendix by S.V. Khrushch¨ev and V.V. Peller, Translated from the Russian by Jaak Peetre. [8] N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading. Volume I: Hardy, Hankel, and Toeplitz. American Mathematical Society, 2002. [9] R.M. Range, Approximation to bounded holomorphic functions on strictly pseudoconvex domains. Paciﬁc Journal of Mathematics, 41 (1972), 203–213. [10] A.J. Sasane, Irrational transfer function classes, coprime factorization and stabilization. CDAM Research Report 10, London School of Economics, 2005. [11] A. Stray, An approximation theorem for subalgebras of H ∞ . Paciﬁc Journal of Mathematics, 35 (1970), 511–515. [12] V.A. Tolokonnikov, Estimates in the Carleson corona theorem, ideals of the algebra H ∞ , a problem of Sz.-Nagy. (Russian. English summary) Investigations on linear operators and the theory of functions, XI. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI), 267, 113 (1981), 178–198. [13] V.A. Tolokonnikov, Extension problem to an invertible matrix. Proceedings of the American Mathematical Society, 117 (1993), 1023–1030. [14] S.R. Treil, Angles between coinvariant subspaces and an operator valued corona problem. A question of Sz¨ okefalvi-Nagy. Doklady Akademii Nauk SSSR, 302 (1988), 1063–1068, (Russian); English translation: Soviet Mathematics. Doklady, 38 (1989), 394–399. [15] S.R. Treil, An operator corona theorem. Indiana University Mathematics Journal, 53 (2004), 1763–1780. [16] S.R. Treil, Lower bounds in the matrix corona theorem and the codimension one conjecture. Geometric and Functional Analysis, 14 (2004), 1118–1133. [17] S.R. Treil and B.D. Wick, The matrix-valued H p corona problem in the disk and polydisk. Journal of Functional Analysis, 226 (2005), 138–172. [18] T.T. Trent, A new estimate for the vector valued corona problem. Journal of Functional Analysis, 189 (2002), 267–282. [19] M. Vidyasagar, Control System Synthesis: a Factorization Approach. MIT Press, 1985. Amol Sasane Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom e-mail: [email protected] Submitted: October 30, 2006 Revised: May 22, 2007

Integr. equ. oper. theory 59 (2007), 257–267 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020257-11, published online June 27, 2007 DOI 10.1007/s00020-007-1510-4

Integral Equations and Operator Theory

CBMO Estimates for Commutators of Multilinear Fractional Integral Operators on Herz Spaces Canqin Tang Abstract. The CBMO estimates for commutators of fractional integral and Multilinear fractional integral operators with rough kernel are established. Mathematics Subject Classification (2000). Primary 41B20; Secondary 47B47. Keywords. Multilinear fractional integral, commutator, CBMO, rough kernel, Herz space.

1. Introduction and main results Let TΩ,α be a fractional integral operator with rough convolution kernel: Ω(x − y) TΩ,α f (x) = p.v. f (y)dy, (1.1) |x − y|n−α n R where Ω ∈ Ls (S), 1 ≤ s < ∞, is homogeneous of degree zero. It was shown that TΩ,α is of (p, q) type, i.e., (1.2) TΩ,α f q ≤ Cf p if s ≥ n/(n−α), 1 < p < q < ∞ and 1/p−1/q = α/n; see e.g. [9]. When Ω ≡ 1, one sees that TΩ,α is just the Riesz potential and (1.2) is the Hardy-Littlewood-Sobolev inequality. The commutators of linear operators is of great interest in the study of harmonic analysis for its applications. The commutators generated by a function b and the fractional integral TΩ,α is deﬁned by [b, TΩ,α ]f (x) = b(x)TΩ,α f (x) − TΩ,α (bf )(x)

(1.3)

This work was completed with the support of Hunan Provincial Natural Science Foundation of China 06A0074.

258

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for suitable functions f . It is well known that the properties of such a commutator are always close related to the smoothness of b. Classical results includes the BMO estimates and Lipschitz estimates, e.g. [1],[4], [8]. In this paper, we are interested in the CBMO estimates. The central BMO space is introduced by Lu and Yang in [5]. Definition 1.1. let 1 ≤ q < ∞, CBM Oq (Rn ) is the space of all functions f ∈ Lqloc (Rn ) such that 1/q 1 f CBMOq = sup |f (x) − fB(0,r) |q dx < ∞, |B(0, r)| B(0,r) r>0 where B(0, r) = {x ∈ Rn : |x| < r} and fB(0,r) is the mean value of f on B(0, r). A result of Chanillo ([1]) states that when Ω is smooth, [b, TΩ,α ] is bounded from Lp (Rn ) to Lq (Rn ) if and only if b ∈ BM O, where 1 < p < q < ∞ and 1/p − 1/q = α/n. Since it is obvious that BM O(Rn ) CBM Oq (Rn ) for all 1 ≤ q < ∞ by the deﬁnition, we know the (Lp , Lq ) boundedness fails with only the assumption b ∈ CBM Oq (Rn ). Instead, certain boundedness properties on Herz spaces can be proved. Recall the deﬁnitions of the Herz spaces. Definition 1.2. Let Bk = {x ∈ Rn : |x| < 2k }, Ek = Bk \Bk−1 and χk = χEk be the characteristic function of the set Ek for k ∈ Z. For α ∈ R and 0 < p, q ≤ ∞, the homogeneous Herz space K˙ qα,p (Rn ) = f : f ∈ Lqloc (Rn \{0})andf K˙ qα,p < ∞ , where

f K˙ qα,p =

∞

1/p 2

kαp

f χk pq

k=−∞

with usual modiﬁcation made when p = ∞. In the following, let s be the conjugate index of s whenever s ≥ 1, i. e., 1/s + 1/s = 1. Denote by C a constant which may vary from line to line. One of main results in this paper is as follows. Theorem 1.3. Let 1 < s ≤ ∞, 1 < q < ∞, b ∈ CBM Oq (Rn ), and [b, TΩ,α ] be n , 1 < q2 < ∞. If defined as in (1.3) with Ω ∈ Ls (S), 0 < α < n, 1 < q1 < α 1 1 1 α 1 1 1 1 1 α 0 u and α − n(1/q1 − 1/s) − 1/s < α1 < n(1 − 1/q1 ), and α2 = α1 − n/q, then [b, TΩ,α ]f K˙ qα2 ,p ≤ CbCBMOq f K˙ qα1,p . 2

1

(1.4)

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259

We also consider the following multilinear fractional operator with rough kernel Ω(x − y) A TΩ,α f (x) = Rm (A; x, y)f (y)dy, (1.5) |x − y|n−α+m−1 n R where m ∈ N, Ω ∈ Ls (S), 1 ≤ s < ∞, A has derivatives of order up to m − 1 and Rm (A; x, y) = A(x) − Dγ A(y)(x − y)γ . |γ|≤m−1 A It was ﬁrst introduced in [3] and further studied in [7],[10]. When m = 1, TΩ,α reduced to the classical commutators [A, TΩ,α ]. When m ≥ 2, it can be regarded as a generalization of commutator. However, this generalization was shown to be nontrivial in [7] by the fact that it performs better in endpoint estimates . For this operator, we have the following result. A be defined as in (1.5), 0 < α < n, 1 < q1 < αn , and Theorem 1.4. Let m ≥ 2, TΩ,α 1 < q2 < ∞. Suppose that Ω ∈ Ls (S) with 1 < s ≤ ∞ and A has derivatives of order m − 1 in CBM Oq (Rn ), n < q < ∞. If 0 < p ≤ ∞, q12 = q11 + 1q − α n , and 1 1 1 1 1 α = + < 1, = − , α satisfies either of the following two conditions: 1 t q1 q u q1 n

(i) 1 ≤ s ≤ t and α − n/q1 < α1 < n(1/s − 1/q1 ) + 1/s; (ii) s ≥ q2 and α − n(1/q1 − 1/s) − 1/s < α1 < n(1 − 1/q1 ), and α2 = α1 − n/q, then A TΩ,α f K˙ qα2 ,p ≤ C Dγ ACBMOq f K˙ qα1 ,p . 2

|γ|=m−1

1

(1.6)

Comparing Theorem 1.3 with Theorem 1.4, we see the restriction on the A indices in Theorem 1.4 is weaker: s ≥ u implies s ≥ q2 . This also shows that TΩ,α performs better than commutators when m ≥ 2, as is a further evidence for the fact of nontrivial generalization.

2. Boundedness of commutators In this section, we will give the proof Theorem 1.3. We need the following lemmas. Lemma 2.1. (see [6]) Suppose that f ∈ CBM Oq (Rn ), 1 ≤ q < ∞ and r1 , r2 > 0. Then 1/q r1 1 q f CBMOq . |f (x) − fB(0,r2 ) | dx ≤ C 1 + log |B(0, r1 )| B(0,r1 ) r2 Lemma 2.2. (see [6]) Suppose that Ω ∈ Ls (S). Then the following propositions hold: |Ω(x − y)|s dy ≤ C2k(n−1)+l ; (1) if l ≤ k − 2, x ∈ Ek , then El

260

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(2) if l ≥ k + 2, x ∈ Ek , then (3) if l ≤ k − 2, y ∈ El , then

El Ek

(4) if l ≥ k + 2, y ∈ El , then

Ek

IEOT

|Ω(x − y)|s dy ≤ C2ln ; |Ω(x − y)|s dy ≤ C2kn ; |Ω(x − y)|s dy ≤ C2l(n−1)+k .

Proof of Theorem 1.3. We only consider the case 0 < p < ∞. Two cases will be proved separately. Case (i): 1 ≤ s ≤ t and α − n/q1 < α1 < n(1/s − 1/q1 ) + 1/s. In this case, we aim at proving estimate [b, TΩ,α ](f χl )χk q2 ≤ C2−kα2 2lα1 M (k, l)bCBMOq f χl q1 ,

(2.1)

where

 (k−l)[α1 −n(1/s −1/q1 )−1/s]  , when l ≤ k − 2; (k − l)2 M (k, l) = 1, when k − 1 ≤ l ≤ k + 1;   (l − k)2(k−l)(γ1 +n/q1 −γ) , when l ≥ k + 2.

Once (2.1) is proved, one can easily check that ∞ ∞ p 1/p kα p 2 [b, TΩ,α ]f K˙ qα2 ,p ≤ C 2 [b, TΩ,α ](f χl )χk q2 2

k=−∞

≤ CbCBMOq

l=−∞

∞ l=−∞

2

= CbCBMOq f K˙ qγ1 ,p

lγ1 p

f χl pq1

∞

1/p min(p,1)

[M (k, l)]

k=−∞

1

and draw the boundness. Therefore, to prove Theorem 1.1 under condition (i), it is enough to show (2.1). To this end, write [b, TΩ,α ](f χl )χk q2 ≤ ((b − bBk )TΩ,α (f χl )) χk q2 + TΩ,α ((b − bBk )f χl ) χk q2 =: J1 + J2 If k − 1 ≤ l ≤ k + 1, by (1.2) and the deﬁnition of CBMO norm we obtain J1 ≤ C2kn/q bCBMOq TΩ,α (f χl )u ≤ C2kn/q bCBMOq f χl q1 and

J2 ≤ C(b − bBk )f χl t ≤ C2ln/q bCBMOq f χl q1 .

We see (2.1) follows easily from the fact 2kn/q ∼ 2ln/q ∼ 2−kγ2 2lγ1 for γ2 = γ1 − n/q. For other cases, we have |k − l| ≥ 2. Denote k ∨ l = max(k, l). If x ∈ Ek , y ∈ El , we have |x − y| ∼ C2(k∨l) . Moreover, the estimates in Lemma 2.2 (1) and (2) tells that if x ∈ Ek , there holds |Ω(x − y)|s dy ≤ C2(k∨l)(n−1)+l . (2.2) El

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261

By H¨older’s inequality and (2.2), we have J1 ≤ (b − bBk )χk q (TΩ,α (f χl )) χk u ≤ C2kn/q bCBMOq 2−(k∨l)(n−α) ×f χl q1 |Ek |1/u

El

|Ω(x − y)|s dy

1/s

|El |1−1/s−1/q1

2(k−l)[α1 −n(1/s −1/q1 )−1/s] , when l ≤ k − 2 2(k−l)(α1 +n/q1 −α) , when l ≥ k + 2 ≤ C2−kα2 2lα1 M (k, l)bCBMOq f χl q1 .

= C2

−kα2 lα1

2

bCBMOq f χl q1

By H¨older’s inequality, Lemma 2.1, and (2.2), |TΩ,α ((b − bBk f χl )(x)| ≤ 2−(k∨l)(n−α) |Ω(x − y)(b(y) − bBk )f (y)|dy El 1/s s −(k∨l)(n−α) 1−1/s−1/q−1/q1 ≤2 |El | |Ω(x − y)| dy l 1/qE 1/q1 q q1 × |b(y) − bBk | dy |f (y)| dy El

El

≤ C|k − l|2(k∨l)[−(n−α)+(n−1)/s]+l[n−(n−1)/s−n/q1 ] ×bCBMOq f χl q1 . This implies J2 ≤ C2−kα2 2lα1 M (k, l)bCBMOq f χl q1 . Combining the estimates for J1 and J2 proves (2.1) when |k − l| ≥ 2, which completes the proof of case (i). Case (ii): s > u and α − n(1/q1 − 1/s) − 1/s < α1 < n(1 − 1/q1 ). Similar as in case (i), it is enough to prove the following estimate: [b, TΩ,α ](f χl )χk q2 ≤ C2−kα2 2lα1 N (k, l)bCBMOq f χl q1

(2.3)

where  (k−l)[α1 −n(1−1/q1 )]  , when l ≤ k − 2; (k − l)2 N (k, l) = 1, when k − 1 ≤ l ≤ k + 1;   (l − k)2(k−l)(α1 −α+n(1/q1 −1/s)+1/s) , when l ≥ k + 2. If k − 1 ≤ l ≤ k + 1, (2.3) has been proved in the case (i). Otherwise, we have |k − l| ≥ 2. Making use of the notations above, Lemma (2.2) (3) and (4) tell that if y ∈ El , |Ω(x − y)|s dy ≤ C2(k∨l)(n−1)+k . (2.4) Ek

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By Minkowski’s inequality, one may check that J1 ≤ (b − bBk )χk q (TΩ,α (f χl )) χk u u 1/u kn/q −(k∨l)(n−α) ≤ C2 bCBMOq 2 Ω(x − y)f (y)dy dx El k 1/u E kn/q −(k∨l)(n−α) ≤ C2 bCBMOq 2 |Ω(x − y)|u dx |f (y)|dy El

Ek

≤ C2kn/q bCBMOq 2−(k∨l)(n−α) 2[(k∨l)(n−1)+k]/s |Ek |1/u−1/s |El |1−1/q1 f χl q1 2kn(1/q−1+1/u+γ/n)+ln(1−1/q1 ) , if l ≤ k − 2 = CbCBMOq f χl q1 2k(n/q+n/u−n/s+1/s)+l(γ+n/s−n/q1 −1/s) , if l ≥ k + 2 ≤ C2−kα2 2lα1 N (k, l)bCBMOq f χl q1 , and

q2 1/q2 Ω(x − y) dx (b(y) − b )f (y)dy Bk n−α Ek El |x − y| 1/q2 ≤ C2−(k∨l)(n−α) |Ω(x − y)|q2 dx |b(y) − bBk ||f (y)|dy El Ek 1/q q −(k∨l)(n−α) [(k∨l)(n−1)+k]/s 1/q2 −1/s ≤ C2 2 |Ek | |b(y) − bBk | dy

J2 =

El

1 ×f χl q1 |El |1−1/q−1/q 2k[−(n−α)+n/q2 ] 2ln(1−1/q1 ) , ≤ C|k − l|bCBMOq f χl q1 2k(n/q2 −n/s+1/s) 2l[α+(n−1)/s−n/q1 ] , −kα2 lα1 = C2 2 N (k, l)bCBMOq f χl q1 .

if l ≤ k − 2 if l ≥ k + 2

This proves (2.3) and ﬁnishes the proof for the case (ii) and hence, the proof of Theorem1.3.

3. Boundedness of fractional multilinear operators Now we turn to prove Theorem 1.4. Lemma 3.1. (see [2]) Let b be a function on Rn with m-th order derivatives in Lqloc (Rn ) for some q > n. Then 1/q 1 |Rm (b; x, y)| ≤ Cm.n |x − y|m |Dγ b(z)|q dz , ˜ y)| Q(x,y) ˜ | Q(x, |γ|=m ˜ y) is the cube centered at x and having diameter 5√n|x − y|. where Q(x, Lemma 3.2. (see [3]) Suppose that Ω ∈ Ls (S) with s ≥ n/(n−γ), A has derivatives of order m − 1 in Lq (Rn ), and 1 < q ≤ ∞. If 0 < γ < n, q12 = q11 + 1q − α n , and 1 1 1 t = q1 + q < 1, then the operator Ω(x − y) A SΩ,α f (x) = Rm−1 (A; x, y)f (y)dy |x − y|n−α+m−1 n R

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has the following boundedness property: A SΩ,α f q2 ≤ C

263

Dγ Aq f q1 ,

|γ|=m−1

where c > 0 is independent of f . Proof of Theorem 1.4. We only consider the case 0 < p < ∞ while the case p = ∞ follows after slight modiﬁcations. Write ∞ k−2 p 1/p kα p A A 2 TΩ,α f K˙ qα2 ,p ≤ C 2 TΩ,α (f χl )χk q2 2

k=−∞

+C

+C

∞

k=−∞ ∞

l=−∞

2

2

k+1

kα2 p

kα2 p

k=−∞

l=k−1 ∞

p 1/p A TΩ,α (f χl )χk q2

p 1/p A TΩ,α (f χl )χk q2

l=k+2

:= I1 + I2 + I3 . For ﬁxed k, let Ak (x) = A(x) −

|γ|=m−1

1 (Dγ A)Bk xγ . γ!

It is easy to see that Rm (A; x, y) = Rm (Ak ; x, y). We ﬁrst estimate I1 . Note that l ≤ k − 2, so if x ∈ Ek , y ∈ El , we have |x − y| ∼ 2k . By Lemma 3.1, same as the proof in [6], we have |Rm (Ak ; x, y)| ≤ C|x − y|m−1 Dγ ACBMOq + |Dγ Ak (y)| . |γ|=m−1

When 1 ≤ s ≤ t, by the H¨ older inequality, Lemma 2.1 and Lemma 2.2 (1), we obtain

A T (f χl )(x) ≤ Ω,α

2−k(n−α)

|γ|=m−1

× ≤

|Ω(x − y)| Dγ ACBMOq + |Dγ Ak (y)| |f (y)|dy El 1/s −k(n−α) 1−1/s−1/q−1/q1 s 2 |El | |Ω(x − y)| dy

|γ|=m−1

×

El

q γ D ACBMOq +|Dγ Ak (y)| dy

El

1/q

≤ C(k − l)2k[−(n−α)+(n−1)/s]+l[n−(n−1)/s−n/q1 ] × Dγ ACBMOq f χl q1 . |γ|=m−1

El

q1

|f (y)| dy

1/q1

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Thus, A (f χl )χk q2 ≤ C(k − l)2k[−(n−α−n/q2 )+(n−1)/s]+l[n−(n−1)/s−n/q1 ] TΩ,α × Dγ ACBMOq f χl q1 |γ|=m−1

= C(k − l)2−kα2 2lα1 2(k−l)[α1 −n(1/s −1/q1 )−1/s] × Dγ ACBMOq f χl q1 . |γ|=m−1

When s ≥ q2 , using Minkowski’s inequality and Lemma 2.2 (3), we have A (f χl )χk q2 TΩ,α

≤ C2k[−(n−α)+n/q2 ] ≤ C(k − l)2

γ D ACBMOq + Dγ Ak (y)| |f (y)|dy

|γ|=m−1 El k(−n+α+n/q2 )+l(n−n/q1 )

Dγ ACBMOq f χl q1

|γ|=m−1

= C(k − l)2−kα2 2lα1 2(k−l)[α1 −n(1−1/q1 )]

Dγ ACBMOq f χl q1 .

|γ|=m−1

Deﬁne

W (k, l) =

(k − l)2(k−l)[α1 −n(1/s −1/q1 )−1/s] when 1 ≤ s ≤ t, (k − l)2(k−l)[α1 −n(1−1/q1 )] when s ≥ q2 .

Then, I1 ≤ C

γ

D ACBMOq

|γ|=m−1

≤C

γ

D ACBMOq

|γ|=m−1

k=−∞ ∞

2

k−2

p 1/p 2

lα1

W (k, l)f χl q1

l=−∞ lα1 p

f χl pq1

l=−∞

|γ|=m−1

≤C

∞

∞

1/p min(p,1)

W (k, l)

k=l+2

Dγ ACBMOq f K˙ qα1 ,p . 1

Next, we will estimate I3 . For x ∈ Ek , y ∈ El and l ≥ k + 2, then |x − y| ∼ 2l . By Lemma 2.1 and Lemma 2.2, we have (l − k)Dγ ACBMOq + |Dγ Ak (y)| . |Rm (Ak ; x, y)| ≤ C|x − y|m−1 |γ|=m−1

older inequality and Lemma 2.2 (2), we obtain When 1 ≤ s ≤ t, using the H¨ A (f χl )χk q2 TΩ,α ≤ C(l − k)2kn/q2 +l(α−n/q1 )

|γ|=m−1

= C(k − l)2−kα2 2lα1 2(k−l)(α2 +n/q2 )

Dγ ACBMOq f χl q1 |γ|=m−1

Dγ ACBMOq f χl q1 .

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265

When s ≥ q2 , using Minkowski’s inequality and Lemma 2.2 (4), we get A TΩ,α (f χl )χk q2 ≤ C(l − k)2k[n/q2 −(n−1)/s]+l[α+(n−1)/s−n/q1 ]

Dγ ACBMOq f χl q1

|γ|=m−1

= C(l − k)2−kα2 2lα1 2(k−l)[α2 +n(1/q2 −1/s)−1/s]

Dγ ACBMOq f χl q1 .

|γ|=m−1

Deﬁne

V (k, l) =

(l − k)2(k−l)(α2 +n/q2 ) when 1 ≤ s ≤ t, (l − k)2(k−l)[α2 +n(1/q2 −1/s)−1/s] when s ≥ q2 .

Then, similar to I1 , we have

I3 ≤ C

|γ|=m−1

Dγ ACBMOq f K˙ qα1 ,p . 1

Finally, let us turn to estimate I2 . Let φ ∈ C0∞ (Rn ) satisfying supp φ ⊂ B(0, 4) and φ ≡ 1 in B(0, 2). Set M = max{Dγ φ∞ , |γ| ≤ m − 1}. Choose y0 ∈ Ek+4 and let Aφk (x) = Rm−1 (Ak ; x, y0 )φ(2−k x). It was veriﬁed in [6] that Rm (A; x, y) = Rm (Aφk ; x, y) for x ∈ Ek and y ∈ El if k − 1 ≤ l ≤ k + 1. Thus, Aφ

A (f χl )(x)χk (x) = TΩ,γk (f χl )(x)χk (x). But TΩ,γ Ω(x − y) Aφ k R (Aφk ; x, y)f (y)dy, TΩ,α f (x) = n−α+m−1 m |x − y| n R 1 Ω(x − y)(x − y)γ + Dγ Aφk (y)f (y)dy γ! Rn |x − y|n−α+m−1 |γ|=m−1 1 Aφ k Tγ Dγ Aφk f (x), = SΩ,α f (x) + γ! |γ|=m−1

x−y γ where Tγ is a fractional integral operator with kernel Ω(x − y)( |x−y| ) . Since · γ s s Ω ∈ L (S) impliesΩ(·)( |·| ) ∈ L (S), by (1.2), Tγ is bounded from Lt to Lq2 ) bounded. Together with Lemma 3.2, we have Aφ

A k TΩ,α (f χl )χk q2 ≤ TΩ,α (f χl )q2 Aφ

k ≤ SΩ,α (f χl )q2 + C

|γ|=m−1

Aφ

k ≤ SΩ,α (f χl )q2 + C

≤C

D

γ

Tγ

Dγ Aφk f χl q2

Dγ Aφk f χl t

|γ|=m−1 φ Ak q f χl q1

|γ|=m−1 kn/q

≤ C2

|γ|=m−1

Dγ ACBMOq f χl q1 ,

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where in the last step we used the argument in the proof of Theorem 1.5 in [6]. This implies I2 ≤ C Dγ ACBMOq f K˙ qα1 ,p . |γ|=m−1

1

Combining the estimates for of I1 , I2 and I3 completes the proof of Theorem 1.4.

4. Some remarks on maximal operators Deﬁne a variant operator of the commutator of fractional integral as Ω(x − y) TΩ,γ,b f (x) = |b(x) − b(y)|f (y)dy. n−γ Rn |x − y| It is easy to see that our proof for Theorem 1.3 is valid to show the (K˙ qγ11 ,p , K˙ qγ22 ,p ) boundedness of TΩ,γ,b under assumptions of Theorem 1.1. This in turn implies the (K˙ qγ11 ,p , K˙ qγ22 ,p ) boundedness of the commutator of maximal fractional integral operator 1 |Ω(x − y)(b(x) − b(y))f (y)|dy MΩ,α,b f (x) = sup n−α r>0 r |x−y|
2 due to the fact MΩ,α,b f ≤ 1−2 α−n T|Ω|,α,b |f |; see [[4], Lemma 5]. Similarly, the conclusion in Theorem 1.4 can be extended to the operator 1 A MΩ,α f (x) = sup n−α+m−1 |Ω(x − y)Rm (A; x, y)f (y)|dy r>0 r |x−y|
by the same technique. Acknowledgment The author expresses her deep thanks to Dr. Qiang Wu for some valuable discussions.

References [1] S. Chanillo, A note on commutators, Indiana Univ. J., 31 (1982), 7–16. [2] J. Cohen and J. Gosselin, A BMO estimate for multilinear singular integrals, Illinois J. of Math., 30 (1986), 445–464. [3] Y. Ding, A note on multilinear fractional integrals with rough kernel, Adances in Math.(China), 30 (2001), 238–246. [4] Y. Ding and S. Z. Lu , Higher order commutators for a class of rough operators, Art. Math. , 37 (1999), 33–44. [5] S. Z. Lu and D. C. Yang, The central BMO spaces and littlewood-Paley operators, Approx. Theory Appl., 11 (1995), 72–94. [6] S. Z. Lu and Q. Wu, CBMO estimates for commutators and multilinear singular integrals, Math. Nachr., 276 (2004), 75–88.

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[7] S. Z. Lu and Q. Wu, Endpoint estimates for certain commutators of fraction and singular integrals, Proc. Amer. Math. Soc., 131 (2003), 467–477. [8] S. Z. Lu, Q. Wu, and D. C. Yang, Boundedness of commutators on Hardy type spaces, Sci. in China (Ser. A) 45 (2002), 984–997. [9] B. Muckenhoupt and R. L. Wheeden,Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971) 249–258. [10] H. X. WU, Boundedness of multilinear fractional integrals with rough kernel on weighted Herz spaces, Adances in Math.(China), 32 (2003), 489–497. [11] H. X. WU, Bounbedness of higher order commutators for a class of rough operators on weighted Herz spaces, J. Beijing Normal University (Natural Science)(China), 37 (2001), 299–306. Canqin Tang Department of Mathematics Dalian Maritime University Dalian 116026 People’s Republic of China e-mail: [email protected] Submitted: May 24, 2006 Revised: March 17, 2007

Integr. equ. oper. theory 59 (2007), 269–280 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020269-12, published online June 27, 2007 DOI 10.1007/s00020-007-1518-9

Integral Equations and Operator Theory

The Reduced Minimum Modulus in C ∗-Algebras Yifeng Xue Abstract. We deﬁne the reduced minimum modulus γ A (a) of a nonzero element a in a unital C ∗ –algebra A by γ A (a) = inf{a−b | AL (a) AL (b), b ∈ 1 A}. We prove that γ A (a) = inf{λ | λ ∈ σ((a∗ a) 2 )\{0}}. Applying this result to A and its closed two side ideal I, we get that dist (a, Φcl (A)) = min{λ | λ ∈ 1 σ(π((a∗a) 2 ))}, ∀ a ∈ A\{0} and γ B (π(a)) = sup{γ A (a + k) | k ∈ I} for any a ∈ A\I if RR (A) = 0, where B = A/I and π : A → B is the quotient homomorphism and Φcl (A) = {a ∈ A | π(a) is not left invertible in B}. These results generalize corresponding results in Hilbert spaces. Mathematics Subject Classification (2000). 46L05. Keywords. Reduced minimum modulus, polar decomposition, Moore–Penrose inverse, real rank zero .

1. Introduction Let B(X, Y ) denote the Banach space of all bounded linear operators T from Banach space X to Banach space Y . Set B(X) = B(X, X). Let T ∈ B(X, Y )\{0}. The reduced minimum modulus γ (T ) (resp. minimum modulus m(T )) of T is deﬁned by γ (T ) = inf{T x | dist (x, Ker T ) = 1},

m(T ) = inf{T x | x = 1}.

respectively, where Ker T is the null space of T , dist (x, Ker T ) =

inf

y∈Ker T

x − y

is the distance from x to Ker T . Let H be a Hilbert space and T ∈ B(H)\{0}. The essential minimum modulus me (T ) of T is given by me (T ) = inf{λ| λ ∈ 1 σe ((T ∗ T ) 2 )}. The reduced minimum modulus (or minimum modulus) and the essential minimum modulus and their connections with perturbation theory of generalized inverse, Fredholm perturbation theory and best approximation are extensively Research supported by Natural Science Foundation of China.

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studied in [1, 3, 4, 6, 14]. For example, if T ∈ B(X, Y )\{0} with Ran (T ) (the range of T ) closed, then γ (T ) = inf{T −B | Ran (B) Ran (T ), Ker (B) ⊃ Ker T, B ∈ B(X, Y )} (1.1) ([4, Theorem 2.3]); If T ∈ B(H)\{0}, then me (T ) = dist (T, B(H)\M+ (H)),

(1.2)

where M+ (H) is the set of all operators T ∈ B(H) with dim Ker T < +∞ and Ran (T ) closed ([16, Theorem]). In [9], Harte and Mbekhta introduced the left (resp. right) conorm of an element in a Banach algebra as follows. Let a be a nonzero element in the Banach algebra A. Then the left (resp. right) conorm of a is given by γ (a) = γ left (a) = inf{ax | dist (x, AL (a)) = 1} A

right

γA

(a) = inf{xa | dist (x, AR (a) = 1}

respectively, where AL (a) = {x ∈ A | ax = 0}, AR (a) = {x ∈ A | xa = 0}. They proved that if A is a C ∗ –algebra, then left (a) = γ right (a) = inf{λ | λ ∈ σ((a∗ a) 12 )\{0}}. γA A Thus the conorm of the element in a C ∗ –algebra can be viewed as the reduced minimum modulus of this element. Let A be a unital C ∗ –algebra and I be a closed ideal of A. Let π : A → A/I = B be the canonical homomorphism. For a ∈ A, set mI (a) = inf{λ|λ ∈ σ(π(|a|)}, ∗

m(a) = inf{λ|λ ∈ σ(|a|)},

1 2

where |a| = (a a) . When A is a von Neumann algebra, Gopalraj and Str¨ oh established some relations between m(a) and mI (a) in [7]. Some of them generalized corresponding results about minimum modulus and essential minimum modulus of operators on Hilbert spaces. In this paper, we ﬁrst deﬁne the reduced minimum modulus of a nonzero element a in the unital C ∗ –algebra A by γ A (a) = inf{a − b|AL (a) AL (b)}. Then we prove that γ A (a) = inf{λ|λ ∈ σ(|a|)\{0}}. Using this formula, we deduce that dist (a, A\Φl (A)) = mI (a), where Φl (A) is the set of all elements a in A such that π(a) is left invertible in B. This result generalizes Zem´anek’s Theorem in [16]. Finally we discuss the relation between γ A (·) and mI (·). We show that if A is of real rank zero, then sup{γ A (a + k)| k ∈ I} = mI (a), ∀ a ∈ Φl (A).

2. The reduced minimum modulus in C ∗ –algebras In the this section and the later, we always assume that A is a unial C ∗ –algebra, I is a closed two–side ideal of A and π : A → A/I = B is the canonical homomorphism. Let La denote the left regular representation of a ∈ A\{0}, i.e., La x = ax, x ∈ A. Harte and Mbekhta have deﬁned the number γ (La ) as the

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reduced minimum modulus of a (see [9]). Inspired by (1.1), we will give a new deﬁnition of the reduced minimum modulus of a nonzero element in a unital C ∗ – algebras as follows. Definition 2.1. Let a ∈ A\{0}. The reduced minimum modulus of a is deﬁned by γ A (a) = inf{a − b | b ∈ A, AL (a) AL (b)}. We may denote γ A (a) by γ (a) if there is no confusion. Recall from [8] that a nonzero element a in A is Moore–Penrose invertible, if there is b ∈ A such that aba = a, bab = b, (ab)∗ = ab, (ba)∗ = ba.

(2.1)

The b in (2.1) is unique and called the Moore–Penrose inverse of a, denoted by a+ . It is known from [8] that a is Moore–Penrose invertible iﬀ 0 ∈ σ(|a|) or 0 ∈ σ(|a|) is an isolated point iﬀ |a| is Moore–Penrose invertible. Lemma 2.2. Let a be a nonzero element in A.

1 ; b 2. If 0 ∈ σ(|a|) is not an isolated point, then γ A (a) = γ A (|a|) = 0.

1. If there is b ∈ A such that aba = a, then γ A (a) ≥

Proof. (1) Set p = ba. Then 1 − p ∈ AL (a). Thus, for any c ∈ A with AL (a) AL (c), we have c(1 − ba) = 0 and there is c0 ∈ AL (c) such that ac0 = 0. Since ac0 = (a − c)bac0 ≤ a − cbac0, 1 . b (2) Let a = u|a| be the polar decomposition in A (A is the enveloping von Neumann algebra of A), where u is the partial isometry in A with au∗ u = a and ux ∈ |a|A|a|, ∀ x ∈ |a|A|a| (cf. [10, Lemma 3.5.1] or [15, Lemma 2.2]). For any ∈ (0, a), deﬁne continuous functions f and g on [0, a] by   0 ≤ t < 2  0 0 ≤ t < 2 2 2 − t f (t) = 2 t − 2 (t) = , g ≤ t < 2  0  2 ≤ t ≤ a t ≤ t ≤ a and moveover, AL (|a|) = and set b = f (|a|), y = g (|a|). Then |a| − b ≤ 2 AL (a) ⊂ AL (b ) and y ∈ AL (b ), y ∈ AL (|a|). Then γ A (|a|) ≤ |a| − b ≤ 2 and γ A (|a|) = 0. Noting that b , ub ∈ |a|A|a| and AL (a) AL (ub ), we have γ A (a) ≤ a − ub = |a| − b ≤ . 2 Therefore, γ A (a) = γ A (|a|) = 0. it follows that γ A (a) ≥

Lemma 2.3. Let a be a nonzero element in A. Then γ A (a) = γ A (|a|).

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Proof. When 0 ∈ σ(|a|) is not an isolated point, the assertion is true by Lemma 2.2. Suppose 0 ∈ σ(|a|) or 0 ∈ σ(|a|) is an isolated point. Put u = a|a|+ . Then u ∈ A is unitary or partial isometry and a = u|a| (cf. [15]). So for any c ∈ A with AL (a) = AL (|a|) AL (c), we have AL (|a|) AL (u∗ c), AL (a) AL (uc). Thus, γ A (|a|) ≤ |a| − u∗ c = u∗ (a − c) ≤ a − c; γ A (a) ≤ a − uc = u(|a| − c) ≤ |a| − c and consequently, γ A (|a|) ≤ γ A (a) ≤ γ A (|a|).

Now we present the main result in this section as follows. Theorem 2.4. Let a be a nonzero element in a unital C ∗ –algebra A. Then γ A (a) = inf{λ | λ ∈ σ(|a|)\{0}}. Proof. If 0 is not an isolated point of σ(|a|), then inf{λ | λ ∈ σ(|a|)\{0}} = 0 and γ A (a) = γ A (|a|) = 0 by Lemma 3.2 (2). In the following, we assume that 0 is an isolated point of σ(|a|) or 0 ∈ σ(|a|). Let a = u|a| be the polar decomposition in A with |a|+ |a| = u∗ u = p and u∗ u|a| = 1 . By Theorem |a|. By Lemma 2.2 (1) and Lemma 2.3, γ A (a) = γ A (|a|) ≥ |a|+ 1 = inf{λ| λ ∈ σ(|a|)\{0}}. Thus, 2 and Theorem 3 of [9], |a|+ γ A (a) ≥ inf{λ| λ ∈ σ(|a|)\{0}}. We prove γ A (a) ≤ inf{λ | λ ∈ σ(|a|)\{0}}. Let µ = min{λ | λ ∈ σ(|a|)\{0}}. If µ = a, then |a| = µp. Thus, γ A (a) ≤ µp − 0 = µ. Suppose that µ < a. If µ is an isolated point of σ(|a|). Then |a| can be written as |a| = a1 + µp1 , where p1 is a projection in A and a1 is a positive element in (1 − p1 )A(1 − p1 ), it follows that AL (|a|) AL (a1 ) and hence γ A (a) ≤ |a| − a1 = µ. Assume that µ is an accumulation point of σ(|a|). Let ∈ (0, a|| − µ) and deﬁne continuous functions h , k on [0, a] by  t 0 ≤ t ≤ µ2     µ−t µ 2
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and put a = h (|a|), d = k(|a|). Then it is easy to verify that AL (|a|) ⊂ AL (a ) and d ∈ AL (a ), d ∈ AL (|a|) and γ A (a) ≤ |a| − a < µ + 2 so that γ A (a) ≤ µ. This completes the proof. Recall from [9] that an element a in A is called a left (resp. right) topological zero divisor relative to I if there exists a normalized sequence {π(bn )} in B such that lim π(abn ) = 0 (resp. lim π(bn a) = 0). We denote the set of these n→∞

n→∞

elements Zl (A, I) (resp. Zr (A, I)). Put Z(A, I) = Zl (A, I) ∩ Zr (A, I). Let Φl (A) (resp. Φr (A)) denote the set of all elements a in A such that π(a) is left (resp. right) invertible in B. Put Φ(A) = Φl (A) ∩ Φr (A). Φcl (A) = A\Φl (A), Φcr (A) = A\Φr (A) and Φc (A) = A\Φ(A) = Φcl ∪ Φr (A). Proposition 2.5. Zl (A, I) = Φcl (A),

Zr (A, I) = Φcr (A).

Proof. Let a ∈ Zl (A, I). Then there is a sequence {π(bn )} in B such that π(bn ) = 1, ∀ n ≥ 1 and π(abn ) → 0 as n → ∞. Thus π(a) is not left invertible in B, i.e., a ∈ Φcl (A) and Zl (A, I) ⊂ Φcl (A). Now let a ∈ Φcl (A). Then π(a∗ a) is not invertible in B. Deﬁne a sequence of continuous functions {fn } on [0, π(a∗ a)] by  1  0 ≤ t ≤ 2n 1 1 1 fn (t) = 2n n1 − t 2n ≤ t ≤ n   1 ∗ 0 n ≤ t ≤ π(a a) for n large enough. Set bn = fn (π(a∗ a)) ≥ 0. Then bn ∈ B, π(a∗ a)bn = bn π(a∗ a), bn = 1 and 1 1 1 1 . π(a)bn2 2 = bn2 π(a∗ a)bn2 = π(a∗ a)bn ≤ 2n Therefore, a ∈ Zl (A, I). Noting that a ∈ Zl (A, I) iﬀ a∗ ∈ Zr (A, I) and a ∈ Φcl (A) iﬀ a∗ ∈ Φcr (A), we get that Zr (A, I) = Φcr (A). Proposition 2.6. Let a be a nonzero element in A. Then dist (a, Φcl (A)) = mI (a), dist (a, Φcr (A)) = mI (a∗ ) dist (a, Φc (A)) = min{mI (a), mI (a∗ )}

Proof. We prove ﬁrst equality. If a ∈ Φcl (A), then π(|a|) is not invertible in B. So in this case, the statement holds. Now suppose that π(a) is left invertible. By Theorem 2.4, γ B (π(a)) = min{λ | λ ∈ σ(π(|a|))} = mI (a). By the deﬁnition of γ B (π(a)), we can ﬁnd b ∈ B for any > 0, such that γ B (π(a)) > π(a) − b − ,

AL (π(a)) AL (b ).

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Since AL (π(a)) = 0, b is not left invertible in B. Choose c ∈ A such that π(c ) = b and k ∈ I such that π(a)−b > a−c −k −. Then c +k ∈ Φcl (A). Thus γ B (π(a)) > π(a) − π(c ) − > a − c − k − 2 ≥ dist (a, Φcl (A)) − 2 and so that mI (a) = γ B (π(a)) ≥ dist (a, Φcl (A)). Since π(a) is left invertible, it follows that π(a∗ a) is invertible in B. Put b = (π(a∗ a))−1 π(a∗ ). It is easy to check that b = (π(a))+ and b2 = bb∗ = (π(a∗ a))−1 = (π(|a|))−1 2 , so that b = (π(|a|))−1 = max{λ−1 |λ ∈ σ(π(|a|))}. We now show that a − z ≥ mI (A), ∀ z ∈ Φcl (A). If there exists c0 ∈ Φcl (A), such that a − c0 < mI (a), then 1 − bπ(c0 ) ≤ bπ(a − c0 ) < (π(|a|))−1 mI (a) < 1. Thus bπ(c0 ) is invertible in B and hence c0 ∈ Φl (A), a contrary. So, mI (a) ≤ dist (a, Φcl (A)). Since a ∈ Φr (A) iﬀ a∗ ∈ Φl (A), we get that dist (a, Φcr (A)) = mI (a∗ ) by above argument. Similarly, we can obtain third equality. From Proposition 2.5 and Proposition 2.6, we have Corollary 2.7. Let a ∈ A\{0}. Then a ∈ Zl (A, I) iﬀ mI (a) = 0 and a ∈ Z(A, I) iﬀ mI (a) = mI (a∗ ) = 0. Remark 2.8. Take A = B(H) and I = K(H) (the algebra of compact operators on H) in Proposition 2.6. Then B is the Calkin algebra. In this situation, mK(H) (T ) = me (T ) for T ∈ B(H). Proposition 2.6 shows that dist (T, Φcl (B(H))) = me (T ), ∀ T ∈ B(H). This result was proved by Zem´anek in [16]. Corollary 2.7 also generalizes Theorem 4.2 and Corollary 4.3 of [7]. Let GL(A) denote the group of all invertible elements in A. When I = {0}, GL(A) = Φ(A). So we have by Proposition 2.6 Corollary 2.9. Let a ∈ GL(A). Then dist (a, A\GL(A)) =

1 . a−1

3. Some further results about reduced minimum modulus In Hilbert spaces, there are two important properties about reduced minimum modulus of operators. These are: (1) A ∈ Φl (B(H)) ∪ Φr (B(H)) implies that γ (A + K) > 0, ∀ K ∈ K(H); (2) For any A ∈ B(H)\K(H), there exists K ∈ K(H) such that γ (π(A)) = γ (A + K).

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Property (2) was generalized to the set of von Neumann algebras in [7]. In this section, we will extend above two properties to the set of C ∗ –algebras with real rank zero. To do this, we ﬁrst introduce some concepts used in this section as follows. Let E be a C ∗ –algebra. (a) Denote by Esa the set {a ∈ E | a = a∗ }; (b) E is said to be σ–unital if E admits a countable approximate identity (cf. [10, Deﬁnition 1.5.7]). (c) Suppose that E is non-unital. Denote by M (E) = {x ∈ E | xa, ax ∈ E, ∀ a ∈ E} the multiplier algebra. (d) Let I be a closed ideal of E. If any x ∈ E with xa = ax = 0 for all a ∈ J implies that x = 0, we call J essential. (e) E is said to have real rank zero (denoted by RR (E) = 0) if every element in Esa can be approximated by an element in Esa with ﬁnite spectrum (cf. [2]). It is well–known that RR (B(H)) = RR (B(H)/K(H)) = 0 and every von Neumann algebra has real rank zero. From [2], we know that if RR (E) = 0, then RR (J ) = RR (E/J ) = RR (eEe) = 0 for any closed two–side ideal J of E and any projection e ∈ E moreover for any projection q ∈ E\J there is a projection p ∈ E such that q = π(p). Proposition 3.1. The following conditions are equivalent: (1) γ A (a) > 0, for any a ∈ Φl (A) ∪ Φr (A); (2) γ A (1 + c) > 0, for any c ∈ Isa ; (3) σ(c) is countable and 0 is the only accumulation point of σ(c) for any c ∈ Isa if σ(c) is inﬁnite; (4) I is ∗–isomorphic to a C ∗ –subalgebra of K(H). Proof. (1)⇒(2) is Obvious. (2)⇒(3) Let c be a self–adjoint element in I and let λ ∈ σ(c)\{0}. By our assumption we have γ A (1−λ−1 c) > 0. Thus, 0 is an isolated point of σ(|1−λ−1 c|) by Theorem 2.4 and hence 1 is an isolated point of σ(λ−1 c). So λ is an isolated point of σ(c)\{0}. This indicates that σ(c)\{0} is countable and so is σ(c). The equivalence of (3) and (4) was shown in [5, Addendum 4.7.20]. (3)⇒(1) We assume that a ∈ Φl (A). Then π(|a|) is invertible and there are b ∈ A and k ∈ I such that b|a| = 1 + k. Set h = k ∗ k + kk ∗ . If 0 ∈ σ(h) is an isolated point, we obtain a projection q ∈ I such that h(1 − q) = 0. If 0 ∈ σ(h) is an accumulation point, then σ(h) = {0} ∪ {λn| n ≥ 1} and lim λn = 0, according n→∞

to the assumption. Let pi be the spectral projection of h corresponding to {λi } in ∞ n λi pi . Put en = pi ∈ I. Then I, i = 1, 2, · · · . Then h = i=1

i=1

hen − h ≤ max |λi | → 0 as n → ∞. i≥n+1

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Now choose n0 such that k(1 − en0 ) < 1 (when 0 is an isolated point of σ(h), take en0 = q) so that (1 − en0 )b|a|(1 − en0 ) − (1 − en0 ) < 1 and hence there is x ∈ (1 − en0 )A(1 − en0 ) such that xb|a|(1 − en0 ) = 1 − en0 , that is, a11 = (1 − en0 )(a∗ a)(1 − en0 ) is invertible in (1 − en0 )A(1 − en0 ). From now on a−1 11 denotes the inverse of a11 in (1 − en0 )A(1 − en0 ). Put a12 = (1 − en0 )(a∗ a)e n0 , a21 = en0 (a∗ a)(1 − en0 ) and a22 = en0 (a∗ a)en0 . a a12 Write a∗ a as a∗ a = 11 . Then we have a21 a22

1 − en0 0 a11 0 a12 1 − en0 a−1 ∗ 11 a a= . a21 en0 0 en0 0 a22 − a21 a−1 11 a12 ∗ Put m = a22 − a21 a−1 11 a12 , z0 = en0 − m ∈ I and z = 1 − z0 . Sine z0 + z0 − ∗ ∗ ∗ = 1 − z z ∈ I, we have 1 ∈ σ(1 − z z) or 1 ∈ σ(1 − z z) is an isolated point. Thus z + exists. Put

0 a−1 1 − en0 −a−1 11 11 a12 z + y= . 0 en0 −a21 a−1 en0 11

z0∗ z0

Then a∗ aya∗ a = a∗ a and hence γ A (a) > 0 by Lemma 2.2. If a ∈ Φr (A), then a∗ ∈ Φl (A). So γ A (a) = γ A (a∗ ) > 0.

Lemma 3.2. Suppose that RR (A) = 0. 1. Let u be a partial isometry in B. Then there exists a partial isometry w in A such that π(w) = u. 2. Let q be a projection in B and p be a projection in A with π(p) = q. Assume that there are mutually orthogonal projections q1 , · · · , qn in B such that q = n qi . Then there are mutually orthogonal projections p1 , · · · , pn in A such i=1

that p =

n i=1

pi and π(pi ) = qi , i = 1, · · · , n.

Proof. (1) Denote q = u∗ u and choose a projection p1 in A and an element v ∈ A such that π(p1 ) = q and π(v) = u. Then v ∗ v = p1 + f for some f ∈ I and hence (vp1 )∗ (vp1 ) = p1 + p1 f p1 . By [2, Proposition 2.9], for any ∈ 0, 1), we can ﬁnd a projection r ∈ p1 Ip1 such that (p1 − r)p1 f p1 < . Consequently, (p1 − r)v ∗ v(p1 − r) − (p1 − r) < . Set z = (p1 − r) +

∞

((p1 − r) − (p1 − r)v ∗ v(p1 − r))n ∈ (p1 − r)A(p1 − r).

n=1

Then Put w = vz

zv ∗ v(p1 − r) = (p1 − r)v ∗ vz = p1 − r 1 2

∗

and π(z) = q.

and p = p1 − r. Then w w = p and π(w) = u.

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(2) Pick a projection p1 ∈ pAp such that π(p1 ) = q1 and a projection p2 ∈ (p − p1 )A(p − p1 ) such that π(p2 ) = q2 ∈ (q − q1 )B(q − q1 ). In this way, we can ﬁnd mutually orthogonal projections p1 , · · · , pn−1 , pn ∈ pAp such that π(pi ) = qi , i = 1, · · · , n − 1 and π(pn ) = qn . Thus, π(p1 + · · · + pn−1 + pn ) = π(p). Put p0 = p − p1 − · · · − pn−1 − pn . Then p0 is a projection in pIp. Take pn = p0 + pn . n Then p1 , · · · , pn are mutually orthogonal and pi = p. i=1

Theorem 3.3. Let a ∈ A\I and suppose that A is of real rank zero. Then γ B (π(a)) = sup{γ A (a + k) | k ∈ I}. In addition, if I is a σ–unital essential ideal of A with RR (M (I)) = 0, then there is k ∈ I such that γ B (π(a)) = γ A (a + k). Proof. Since σ(π(|a|) ⊂ σ(|a + k|), ∀ k ∈ I, it follows from Theorem 2.4 that γ A (a + k) ≤ γ B (π(a)), ∀ k ∈ I. Thus γ B (π(a)) ≥ sup{γ A (a + k) | k ∈ I}. Put b = π(a). If 0 ∈ σ(|b|) is not an isolated point, then γ B (b) = 0 and γ A (a + k) = 0 ∀ k ∈ I. In the following, we assume that 0 ∈ σ(|b|) or 0 ∈ σ(|b|) is an isolated point. Let b = u|b| be the polar decomposition in B. Set q = u∗ u. Then q|b| = |b| 1 and |b| ∈ GL(qBq). Let ∈ 0, . Since RR (B) = 0, we can ﬁnd real numbers 3 µ1 , · · · , µm and mutually orthogonal projections q1 , · · · , qm in qBq such that m

qi = q,

|b| −

i=1

m

, |b|+

µi qi <

i=1

(3.1)

where |b|+ is the inverse of |b| in qBq. By Lemma 3.2, there are a partial isometry w in A with π(w) = u and mutually orthogonal projections s1 , · · · , sm in pAp, m where p = w∗ w, such that si = p and π(si ) = qi , i = 1, · · · , m. i=1

Choose a positive invertible element d ∈ pAp such that π(d) = |b| and set m a0 = wd. Then π(a0 ) = π(a) = u|b|. From (3.1), we have π(d − µi si ) < i=1

. Take k0 ∈ pIp such that |b|+ d + k0 −

m

µi si < π(d −

i=1

By (3.1), |b|+

m i=1

m

µi si ) +

i=1

2 < . |b|+ |b|+

m µi qi − q < . Thus µi qi is invertible in qBq and i=1

max |µ−1 i |=

1≤i≤m

m i=1

µ−1 i qi <

|b|+ . 1−

(3.2)

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It follows from (3.2) that m −1 (d + k0 ) µi si − p < i=1

m

−1 2 2 < 1. µi si < + |b| i=1 1−

Thus, d + k0 is invertible in pAp and (d + k0 )−1 <

m 1 − −1 |b|+ µi si < . 1 − 3 i=1 1 − 3

Now put x = w(d + k0 ), y = (d + k0 )−1 w∗ , k = k1 + wk0 , where k1 = a0 − a ∈ I. Then xyx = x, a + k = a0 + wk0 = x. Therefore, by Lemma 2.2 (1), γ A (a + k ) ≥

1 1 − 3 1 ≥ > . −1 y (d + k0 ) |b|+

On the other hand, by Lemma 2.3 and Theorem 2.4, γ B (π(a)) = γ B (|b|) =

1 = min{λ| λ ∈ σ(|b|)\{0}}. |b|+

So γ A (a + k ) > (1 − 3)γ B (π(a)). This proves that γ B (π(a)) = sup{γ A (a + k) | k ∈ I}. Now we assume that RR (M (I)) = 0. Let a, b, u, q be as above. By Lemma 3.2, there is a partial isometry w ∈ A such that π(w) = u. Let p = w∗ w. Set B0 = C ∗ (|b|) (the C ∗ –algebra generated by |b| and q in qBq) and A0 = π −1 (B0 ) ⊂ pAp. Consider the commutative diagram of two short exact sequences 0 −−−−→ pIp −−−−→

A0   i

π|A

−−−−0→

B0  τ

−−−−→ 0

π ˆ

0 −−−−→ pIp −−−−→ M (pIp) −−−−→ M (pIp)/pIp −−−−→ 0 Since I is essential in A, we get that pIp is essential in pAp so that A0 can be viewed as a C ∗ –subalgebra of M (pIp) (cf. [10, Proposition 5.1.5]). Note that M (pIp) = p M (I)p. So RR (M (pIp)) = 0. Since σ(|b|) is a compact subset in real line, we have by [11, Corollary 1.10, Proposition 1.5], there are an approximate identity {en } of pIp consisting of increasing projections and a dense sequence {ξn } in σ(|b|)\{0} with isolated points repeated inﬁnitely often such that τ (y) = π ˆ

∞

φn (y)(en − en−1 ) , (e0 = 0) ∀ y ∈ B0 ,

n=1

here φn is the character on B0 with φn (|b|) = ξn , n ≥ 1. Put a ˆ0 =

∞ n=1

ξn (en − en−1 ) ∈ M (pIp).

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Then ˆ (p|a|p) and π ˆ (ˆ a0 ) = τ (|b|) = τ ◦ π|A0 (p|a|p) = π

(3.3)

σ(ˆ a0 ) = {0} ∪ {ξn | n ≥ 1} = {0} ∪ σ(|b|) .

(3.4)

By (3.3), a ˆ0 − p|a|p = kˆ ∈ pIp and hence a ˆ0 ∈ pAp. Put a ˆ = wˆ a0 . Then |ˆ a| = 1 ∗ (ˆ a0 w wˆ a0 ) 2 = a ˆ0 and π(ˆ a) = π(a) = b for q|b| = |b|q. Set k = a ˆ − a ∈ I. We have by Theorem 2.4 and (3.4), a) = γ A (|ˆ a|) = γ A (ˆ a0 ) γ A (a + k) = γ A (ˆ = min{λ | λ ∈ σ(ˆ a0 )\{0}} = min{λ | λ ∈ σ(|b|)\{0}} = γ B (π(b)) . The proof is completed.

Remark 3.4. When the σ–unital C ∗ –algebra I is of real rank zero and stable rank one or purely inﬁnite simple with trivial K1 –group, then RR (M (I)) = 0 (see [12], [17]). Acknowledgement. The author is grateful to the referee for his (or her) helpful comments and kindly pointing out many typos in the paper

References [1] R.H. Bouldin, The essential minimum modulus, Indiana Univ. Math. J., 30 (1981), 513–517. [2] L.G. Brown and G.K. Pedersen, C ∗ –alegbras of real rank zero, J. Funct. Anal., 99 (1991), 131–149. [3] G. Chen, M. Wei and Y. Xue, Perturbation analysis of the least squares solution in Hilbert spaces, Linear Algebra Appl., 244 (1996), 69–80. [4] G. Chen, Y. Wei and Y. Xue, The generalized condition numbers of bounded linear operators in Banach spaces, J. Aust. Math. Soc., 76 (2004), 281–290. [5] J. Dixmier, C ∗ –algebras, Gauthier–Villars, Paris, 1964; North–Holland, Amsterdam, 1977. [6] F. Galaz–Fontes, Approximation by semi–Fredholm operators, Proc. Amer. Math. Soc., 120 (1994), 1219–1222. [7] P. Gopalraj and A. Str¨ oh, On the essential bound of elements in von Neumann algebras, Integr. Equ. Oper. Theory, 49 (2004), 379–386. [8] R. Harte and M. Mbekhta, Generalized inverses inverses in C ∗ –algebras, Studia Math., 103 (1) (1992), 71–77. [9] R. Harte and M. Mbekhta, Generalized inverses inverses in C ∗ –algebras, II, Studia Math., 106 (1993), 129–138. [10] H. Lin, An Introduction to the Classification of Amenable C ∗ –algebras, World Scientiﬁc, 2001. [11] H. Lin, C ∗ –algebra Extensions of C(X), Mem. Amer. Math. Soc., 115 (1995), no. 550.

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[12] H. Lin, Generalized Weyl–von Neumann theorems II, Math. Scand., 77 (1995), 129– 147. [13] G.K. Pedersen, C ∗ –algebras and their Automorphism Groups, Academic Press, London/ New York, 1979. [14] P.Y. Wu, Approximation by invertible and noninvertible operators, J. Approx. Theory, 56 (1989), 267–276. [15] Y. Xue, (APD)–property of C ∗ –algebras by extensions of C ∗ –algebras with (APD), Proc. Amer. Math. Soc., 135 (3) (2007), 705–711. [16] J. Zem´ aneck, Geometric interpretation of the essential minimum modulus, Invariant Subspaces and Other Topics, 6-th International Conference on Operator Theory. Birkh¨ auser, Verlag, Basel, 1982. [17] S. Zhang, C ∗ –algebras with real rank zero and their corona and multiplier algebras, Part III, Cand. J. Math., 41 (1989), 721–742. Yifeng Xue Department of Mathematics East China Normal University Shanghai 200062 P.R. China e-mail: [email protected] [email protected] Submitted: December 13, 2006 Revised: February 8, 2007

Integr. equ. oper. theory 59 (2007), 281–298 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020281-18, published online June 27, 2007 DOI 10.1007/s00020-007-1509-x

Integral Equations and Operator Theory

Disintegration of Measures and Contractive 2-Variable Weighted Shifts Jasang Yoon Abstract. In this paper we give a new proof of the existence of disintegration measures using the Hausdorﬀ Moment Problem on a Borel measurable space X × Y , where X ≡ Y is the unit interval. Using this new tool, we can give an abstract solution, moreover, and a concrete necessary condition for the Lifting Problem for contractive 2-variable weighted shifts. In addition, we have a new, computable, and suﬃcient condition for the Lifting Problem for 2-variable weighted shifts, and an improved version of the Curto-Muhly-Xia conjecture [8] for 2-variable weighted shifts. Mathematics Subject Classification (2000). Primary 47B20, 47B37, 47A13, 28A50; Secondary 44A60, 47-04, 47A20. Keywords. Disintegration of measures, Hausdorﬀ Moment Problem, subnormal pair, contractive 2-variable weighted shifts.

1. Introduction The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and suﬃcient conditions on commuting pairs of subnormal operators on Hilbert space to admit a joint normal extension. In previous work we gave an abstract solution [5, Theorem 3.1] and a concrete necessary condition, albeit not suﬃcient, for the lifting ([11, Theorem 3.3 and Proposition 4.8]). In this paper we give a new proof of the existence of disintegration of measures using a new tool, that is, the Hausdorﬀ Moment Problem (Theorem 2.6). Comparing our proof to the proof in [3, Theorem 2.11] (see Theorem 1.2 below) and Helson’s proof in [15], we can give an abstract solution of the Lifting Problem for contractive 2-variable weighted shifts. We do this by establishing two concrete relations between the Hausdorﬀ moment sequence {αk1 } (k1 ≥ 0) and the moments γ(k1 ,0) (k1 ≥ 0) of the 0-th horizontal 1-variable weighted shift Wα(0) := shif t(α00 , α10 , · · · ) of T : = (T1 , T2 ), and also between a probability measure µY on Y and the Berger measure ηk1 of the k1 -th vertical 1-variable weighted shift Wβ (k1 ) := shif t(βk1 0 , βk1 1 , · · · ) of T

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with ηk1 ≈ µY (all k1 ≥ 0), where µY (G) := µ(X × G) (all G ⊆ Y ) (Theorem 2.7). From Theorem 2.7, we can infer a concrete necessary condition which is not suﬃcient (Theorem 2.10 and Example 2.12). Next, we give in Theorem 2.14 a computable suﬃcient condition for the lifting for 2-variable weighted shifts. We show that the conditions ξk2 +1 ≤ ξk2 and ηk1 +1 ≤ ηk1 imply ξk2 +1 = ξk2 and ηk1 +1 = ηk1 (all k1 , k2 ∈ Z+ ). Hence T ∼ = (I ⊗ Wα(0) , Wβ (0) ⊗ I), where ξk2 (k2 ≥ 0) is the Berger measure of the associated k2 -th horizontal 1-variable weighted shift Wα(k2 ) := shif t(α0k2 , α1k2 , · · · ) of T (Theorem 2.14). We also give an example of a commuting hyponormal 2-variable weighted shift T ≡ (T1 , T2 ) with subnormal components and mutually absolutely continuous Berger measures for all horizontal and vertical slices. This weighted shift fails to admit a lifting (Theorem 3.9). Finally, we give an improved version of the Curto-Muhly-Xia conjecture [8] for 2-variable weighted shifts, that is, whether joint hyponormality is suﬃcient to guarantee the lifting (Problem 3.10). We devote the rest of this section to establishing our basic terminology and notation. Let H be a complex Hilbert space and let B(H) denote the algebra of bounded linear operators on H. For S, T ∈ B(H), let [S, T ] := ST − T S. We say that an n-tuple T : = (T1 , · · · , Tn ) of operators on H is (jointly) hyponormal if the operator matrix   [T1∗ , T1 ] [T2∗ , T1 ] · · · [Tn∗ , T1 ]  [T1∗ , T2 ] [T2∗ , T2 ] · · · [Tn∗ , T2 ]    [T∗ , T] :=   .. .. .. ..   . . . . [T1∗ , Tn ] [T2∗ , Tn ] · · ·

[Tn∗ , Tn ]

is positive on the direct sum of n copies of H (cf. [1], [8]). The n-tuple T is said to be normal if T is commuting and each Ti is normal, and T is subnormal if T is the restriction of a normal n-tuple to a common invariant subspace. We say that T is k-hyponormal if T(k) := (T1 , T2 , T12 , T2 T1 , T22 , · · · , T1k , T2 T1k−1 , · · · , T2k ) is hyponormal (k ≥ 1) [5]. The Bram-Halmos criterion [2] states that an operator T ∈ B(H) is subnormal if and only the k-tuple (T, T 2, · · · , T k ) is hyponormal for all k ≥ 1. For α ≡{αn }∞ n=0 a bounded sequence of positive real numbers (called weights), let Wα : 2 (Z+ ) → 2 (Z+ ) be the associated unilateral weighted shift, deﬁned by Wα en := αn en+1 (all n ≥ 0), where {en }∞ n=0 is the canonical orthonormal basis in 2 (Z+ ). The moments of α are given by 1 if k = 0 γk ≡ γk (α) := α20 · ... · α2k−1 if k > 0. Similarly, consider double-indexed positive bounded sequences αk , βk ∈ ∞ (Z2+ ), k ≡ (k1 , k2 ) ∈ Z2+ := Z+ × Z+ and let 2 (Z2+ ) be the Hilbert space of squaresummable complex sequences indexed Z2+ . (Recall that 2 (Z2+ ) is canonically by 2 2 isometrically isomorphic to (Z+ ) (Z+ ).) We deﬁne the 2-variable weighted

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shift T := (T1 , T2 ) by T1 ek := αk ek+ε1 T2 ek := βk ek+ε2 , where ε1 := (1, 0) and ε2 := (0, 1). Clearly, T1 T2 = T2 T1 ⇐⇒ βk+ε1 αk = αk+ε2 βk (all k).

(1.1)

Trivially, a pair of unilateral weighted shifts Wα and Wβ gives rise to a 2variable weighted shift T, if we let α(k1 ,k2 ) := αk1 and β(k1 ,k2 ) := βk2 (all k1 , k2 ∈ Z+ ). In this case, T is subnormal (resp. hyponormal) if and only if so are T1 2 2 2 and T2 ; in fact, under the canonical identiﬁcation of (Z ) and (Z ) 2 (Z+ ), + + ∼ ∼ I, and T is also doubly commuting. For this reason, Wα and T2 = Wβ T1 = I we do not focus attention on shifts of this type, and use them only when the above mentioned triviality is desirable or needed. Given k ∈ Z2+ , the moment of (α, β) of order k is   1 if k = 0    α2 · · · α2 if k1 ≥ 1 and k2 = 0 (0,0) (k1 −1,0) γk ≡ γk (α, β) := 2 2 β(0,0) · · · β(0,k2 −1) if k1 = 0 and k2 ≥ 1    α2 · · · α2 β2 · · · β2 if k ≥ 1 and k ≥ 1. (0,0)

(k1 −1,0) (k1 ,0)

(k1 ,k2 −1)

1

2

We remark that, due to the commutativity condition (1.1), γk can be computed using any nondecreasing path from (0, 0) to (k1 , k2 ). We now recall a well known characterization of subnormality for multivariable weighted shifts [16], due to C. Berger (cf. [3, III.8.16]) and independently established by Gellar and Wallen [14]): T ≡ (T1 , T2 ) admits a commuting normal extension if and only if there is a probability measure µ (which we call the Berger measure of T) deﬁned on 2 the 2-dimensional

k rectangle R k= k[0, a1 ] × [0, a2 ] (where2ai := Ti ) such that γk (α, β) = R t dµ(s, t) := R s 1 t 2 dµ(s, t), for all k ∈ Z+ (called Berger’s Theorem). In the single variablecase, if Wα is subnormal with Berger measure ξα and h ≥ 1, and if we let Lh := {en : n ≥ h} denote the invariant subspace obtained by removing the ﬁrst h vectors in the canonical orthonormal basis of 2 (Z+ ), then h the Berger measure of Wα |Lh is γsh dξ(s). Let X and Z be compact metric spaces and let µ be a positive regular Borel measure on Z and the space of all Borel measures on Z is denoted by M (Z). For φ : Z → X a Borel mapping, let ν be the Borel measure µ ◦ φ−1 on X; that is, ν(A) := µ(φ−1 (A)) for all Borel sets A contained in X. Let £1 (µ) := {f : f is Borel function on Z such that |f | dµ < ∞}, and let L1 (µ) := {[f ] : f ∈ £1 (µ)}, where [f ] := {g ∈

1 £ (µ) : |f − g| dµ = 0}. The map ψ → (ψ ◦ φ)f dµ Z

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deﬁnes a bounded linear functional on L∞ (ν). If attention is restricted to characteristic functions χA in L∞ (ν), f dµ A → (χA ◦ φ)f dµ = φ−1 (A)

Z

is a countably additive measure deﬁned on Borel sets in X, that is absolutely continuous with respect to ν. Hence there is a unique element E(f ) in L1 (ν) such that (χA ◦ φ)f dµ = χA E(f )dν Z

X

for all Borel subsets A of X. By an approximation argument one can show that (ψ ◦ φ)f dµ = ψE(f )dν Z

X

for all ψ in L∞ (ν). This deﬁnes a map E : £1 (µ) → L1 (ν) called the expectation operator. Definition 1.1. A disintegration of the measure µ with respect to φ is a function t → Φt from X to M (Z), such that measure; (i) for each t in X, Φt is a probability

(ii) if f ∈ £1 (µ), E(f )(t) = Z f dΦt a.e. [ν]. We now list the theorem of existence and uniqueness of disintegration of measures. Theorem 1.2. ([3, Theorem VII.2.11]) Given a regular Borel measure µ on a compact metric space Z, and a Borel function φ from Z into a compact metric space X, there is a disintegration t → Φt of µ with respect to φ. If t → Φt is another disintegration of µ with respect to φ, then Φt = Φt a.e. [ν].

2. Main Results We begin with several lemmas. We ﬁrst recall that given two positive regular Borel measures µ and ν, µ is said to be absolutely continuous with respect to ν (in symbols, µ ν) if for every Borel set E, ν(E) = 0 ⇒ µ(E) = 0. We say that µ and ν are mutually absolutely continuous relative each other (in symbols, µ ≈ ν) if µ ν and ν µ. It follows at once that µ ν ⇒ suppµ ⊆ suppν and µ ≈ ν ⇒ suppµ = suppν, where suppµ denotes the support of µ. For notational convenience, we will often write shif t(α0 , α1 , · · · ) to denote Wα . In particular, we shall let U+ := shif t(1, 1, · · · ) (U+ is the (unweighted) unilateral shift) and Sa := shif t(a, 1, 1, · · · ). We let X ≡ Y be the unit interval [0, 1], C(Y ) the set of all continuous functions on Y and C[s] the set of all polynimials over C. Moreover, let γ(k1 ,0) be the moments of the 0-th horizontal 1-variable weighted shift Wα(0) ≡ shif t(α00 , α10 , · · · ), and ηk1 (ξk2 ) the Berger measure of the k1 -th

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vertical (k2 -th horizontal) 1-variable weighted shift Wβ (k1 ) ≡ shif t(βk1 0 , βk1 1 , · · · ) (k2 ≥ 0) (Wα(k2 ) ≡ shif t(α0k2 , α1k2 , · · · ) (k1 ≥ 0)) of T ≡ (T1 , T2 ), respectively. (see Figure 1 (i)). Lemma 2.1. Let µ be a probability measure on the Borel sets contained in X×Y . For all s ∈ X, g ∈ C(Y ) and k1 ≥ 0, deﬁne sk1 g(t)dµ(s, t). Fk1 (g) := X×Y

Then Fk1 (all k1 ≥ 0) is a positive linear functional on C(Y ). Therefore, there exists a positive Borel measure ζk1 (called Riesz measure) on Y such that Fk1 (g) = g(t)dζk1 (t). Proof. This is straightforward from the Riesz Representation Theorem.

Lemma 2.2. Let µ and ζk1 be as in Lemma 2.1. Suppose η is a probability in Y . If ζk1 η (all k1 ≥ 0) then αk1 :=

dζk1 ∈ L1 (Y, η) dη

(2.1)

and satisﬁes the condition fulﬁlled by the diﬀerences of Hausdorﬀ moment sequences, a.e. [η], that is, if 0 αk1 (t) := αk1 (t), 1 αk1 (t) := αk1 (t) − αk1 +1 (t) and αk1 +1 (t) + αk1 +2 (t) − · · · + (−1) αk1 + (t) αk1 (t) := αk1 (t) − 1 2 (2.2) then αk1 ≥ 0, a.e. [η] (all , k1 ≥ 0).

Proof. That αk1 ∈ L1 (Y, η) is clear from the Radon-Nikodym Theorem. Let G be any Borel set in Y . Then we have χG αk1 (t)dη(t) = αk1 (t)dη(t) = dζk1 (t) ≥ 0, Y

G

G

because ζk1 is a positive Borel measure on Y . Since η is a positive measure on Y and αk1 ∈ L1 (Y, η), we have αk1 ≥ 0, a.e. [η]. Therefore 0 αk1 ≥ 0, a.e. [η] (all k1 ≥ 0).

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For 1 αk1 (t), because s ∈ [0, 1] and µ is a positive measure, we observe that

(αk1 (t) − αk1 +1 (t))dη(t) =

dζk1 (t) − dζk1 +1 (t)

= Fk1 (1) − Fk1 +1 (1) (by Lemma 2.1) = =

X×Y

X×Y

sk1 − sk1 +1 dµ(s, t) sk1 (1 − s)dµ(s, t) ≥ 0.

Thus we have αk1 ≥ αk1 +1 a.e. [η] (all k1 ≥ 0). Therefore 1 αk1 ≥ 0, a.e. [η] (all k1 ≥ 0). Let be an arbitrary positive integer. Note that

αk1 +1 (t) + αk1 +2 (t) − · · · + (−1) (αk1 (t) − αk1 + (t))dη(t) 1 2 =

dζk1 (t) −

dζk1 + (t) dζk1 +1 (t) + dζk1 +2 (t) − · · · (−1) 1 2

Fk1 +1 (1) + Fk1 +2 (1) − · · · (−1) Fk1 + (1) = Fk1 (1) − 1 2 = =

X×Y

X×Y

k1 +1 k1 +2 k1 k1 + s + s − · · · + (−1) s − s dµ(s, t) 1 2 sk1 (1 − s) dµ(s, t) ≥ 0, because s ∈ [0, 1].

Therefore, we have αk1 (t) ≥ 0, a.e. [η] (all , k1 ≥ 0), as desired.

Lemma 2.3. Let µ and ζk1 as in Lemma 2.1 and Lemma 2.2. Deﬁne µY (G) := µ(X × G) (all G ⊆ Y ). Then ζk1 µY (all k1 ≥ 0). Proof. Suppose µY (G) = 0, for some G ⊆ Y . We then have µ(X × G) = 0. Thus for all k1 ≥ 0 sk1 dµ(s, t) = 0 ⇒ χX×G · sk1 dµ(s, t) = 0 ⇒ χG (t) · sk1 dµ(s, t) = 0, X×G

X×Y

because (s, t) ∈ X × G ⇔ t ∈ G. Since χG ≥ 0, there exists a sequence of functions {fn } ⊆ C(Y ) (with 0 ≤ fn ≤ 1) such that fn → χG (in L1 (Y, µY )). Let f G (s, t) := χG (t), then f G (in L1 (X × Y, µ)) which n (s, t) := fn (t) and χ n → χ

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implies that there exists a subsequence {f G , a.e. [µ]. n } ⊆ {fn } such that fn → χ Thus for all k1 ≥ 0 we have

G (s, t) · sk1 dµ(s, t) = 0 X×Y χ ⇒

X×Y

sk1 limn →∞ f n dµ(s, t) = 0

⇒ limn →∞ X×Y sk1 f n dµ(s, t) = 0 (by the Lebesgue Dominated Convergence Theorem) ⇒ limn →∞ Fk1 (fn ) = 0 ⇒ limn →∞ ⇒ ⇒

Y

Y

Y

fn (t)dζk1 (t) = 0 (by Lemma 2.1)

limn →∞ fn (t)dζk1 (t) = 0 χG dζk1 (t) = 0 ⇒ ζk1 (G) = 0.

Therefore, we have ζk1 µY (all k1 ≥ 0).

Lemma 2.4. [17] (Hausdorﬀ Moment Problem) A necessary and suﬃcient condition that the one-dimensional Hausdorﬀ moment problem 1 sk1 dΦ(s) (k1 ≥ 0) (2.3) αk1 = 0

has a solution Φ (positive measure on X) is that all diﬀerences αk1 ≥ 0 (all ≥ 0 and k1 ≥ 0). Remark 2.5. Let αk1 (all k1 ≥ 0) be as in Lemma 2.2. Recall that for a given t ∈ Y , the sequence {αk1 (t)}∞ k1 =0 in (2.1) satisﬁes the conditions (2.2). Thus there exists a positive measure Φt on X which satisﬁes (2.3), that is, 1 αk1 (t) = sk1 dΦt (s) (k1 ≥ 0) (2.4) 0

We now give a new proof of the existence of disintegration of measures on X ×Y. Theorem 2.6. Let µ, µY and Φt be as in Lemma 2.3 and Remark 2.5. Then for almost every t in Y (with respect to µY ) and all continuous functions φ(s, t) on the product space X × Y, the measures µY and Φt satisfy φdµ(s, t) = ( φ(s, t)dΦt (s))dµY (t). (2.5) X×Y

Y

X

Proof. Let Λ be an algebra generated by the products pg, where p ∈ C[s] and g ∈ C[Y ]. Then Λ is a self-adjoint subalgebra of C(X × Y ), which separates the points of X × Y and contains the constant function 1. Thus the closure of Λ

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with respect to the norm · ∞ is C(X × Y ) by the Stone-Weierstrass Theorem. Therefore, it suﬃces to show that µY and Φt satisfy sk1 g(t)dµ(s, t) = ( sk1 dΦt (s))g(t)dµY (t) (all g ∈ C[Y ] and k1 ≥ 0). X×Y

Y

X

Consider the functional Fk1 : C[Y ] → C given by sk1 g(t)dµ(s, t) (all g ∈ C[Y ] and k1 ≥ 0). Fk1 (g) = X×Y

Then Fk1 is positive linear functional on C(Y ) (by Lemma 2.1). Thus the positive Borel measure ζk1 (k1 ≥ 0) on Y satisﬁes g(t)ζk1 (t) = Fk1 (g) (by Lemma 2.1). Since µY is a probability measure on Y such that ζk1 µY (by Lemma 2.3), we can write dζk1 = αk1 dµY . Thus the positive measure Φt on X satisﬁes αk1 (t) =

1 k Y 1 0 s dΦt , a.e. [µ ] (by Lemma 2.4 and Remark 2.5). Therefore, we have sk1 g(t)dµ(s, t) = Fk1 (g). X×Y

Since Fk1 (g) = Y g(t)dζk1 (t) = Y g(t)αk1 (t)dµY (t), we then have the desired result sk1 g(t)dµ(s, t) = ( sk1 dΦt (s))g(t)dµY (t) (all g ∈ C[Y ]). X×Y

Y

Thus the proof is complete.

X

We next give an abstract solution of the Lifting Problem for contractive 2variable weighted shifts. We say that T ≡ (T1 , T2 ) is contractive if Ti ≤ 1, i ∈ {1, 2}. Theorem 2.7. Let µ, µY , αk1 and Φt be as in Remark 2.5 and Theorem 2.6, and let T ≡ (T1 , T2 ) a contractive 2-variable weighted shift (see Figure 1 (i)). Then T is a subnormal if and only if αk1 µY = γ(k1 ,0) (ξ0 )ηk1 (all k1 ≥ 0), where γ(k1 ,0) (ξ0 ) is the moment of the 0-th horizontal 1-variable weighted shift shif t(α00 , α00 , · · · ) of T. Proof. (⇒) Suppose T is a subnormal with Berger measure µ(s, t) on X × Y . Then µ(s, t) is a probability measure on the Borel sets contained in X×Y . Thus by Berger’s Theorem we have sk1 tk2 du(s, t) = γk (α, β) (all k1 , k2 ∈ Z+ ). (2.6) X×Y

Given k1 ≥ 0 and k2 ≥ 0, let φ(k1 ,k2 ) (s, t) := sk1 tk2 . Then φ(k1 ,k2 ) (s, t) (all k1 , k2 ∈ Z+ ) is a continuous function on the product space X × Y . Thus the

Vol. 59 (2007) . . (0, 3) .

Disintegration of Measures and Contractive Shifts . ..

···

γ

1,3

β02

γ

1,2

γ

2,3

α22

γ

1,1

β00

(1, 0)

1

α21

···

···

1

···

1

···

1 1

1

···

y

γ

2,1

x

γ2,0

α20 (2, 0)

. ..

···

T2

2,2

α10

α00 (0, 0)

y

γ2,1

γ1,0

. .. 1

···

γ

α11

···

1

γ2,2

γ1,1

α01

. ..

···

α12

β01 (0, 1)

. ..

γ1,2

α02

(0, 2)

T2

···

289

y

··· (3, 0)

1 1

x (0, 0)

y 1

(1, 0)

···

1 (2, 0)

T1

(3, 0)

T1

(i)

(ii)

Figure 1. Weight diagrams of the 2-variable weighted shifts in Proposition 2.11 and Example 2.12 probability measures µY and Φt (on Y and X, respectively) a.e. [µY ] satisfy

sk1 tk2 du(s, t) X×Y =

Since

=

1 0

1

X×Y

0

1 tk2 ( 0 sk1 dΦt (s))dµY (t) (all k1 , k2 ∈ Z+ ) (by Theorem 2.6) tk2 (αk1 (t))dµY (t) (all k1 , k2 ∈ Z+ ) (by Remark 2.5).

2 2 sk1 tk2 du(s, t) = α2(0,0) · ... · α2(k1 −1,0) · β(k · ... · β(k , we have 1 ,0) 1 ,k2 −1) 1 1 tk2 (αk1 (t))dµY (t) = γ(k1 ,0) (ξ0 ) tk2 dηk1 (t) 0

and

0

Thus

1

0

tk2 (αk1 (t))dµY (t) =

0

1

tk2 dζk1 (t) (all k1 , k2 ∈ Z+ ) (by Lemma 2.1).

T ≡ (T1 , T2 ) is subnormal ⇒

1 0

tk2 d(ζk1 (t) − γ(k1 ,0) (ξ0 )ηk1 (t)) = 0 (all k1 , k2 ∈ Z+ )

⇔ ζk1 = γ(k1 ,0) (ξ0 )ηk1 (all k1 ∈ Z+ ). ⇔ αk1 µY = γ(k1 ,0) (ξ0 )ηk1 (all k1 ∈ Z+ ) (by Lemma 2.2).

(2.7)

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Therefore, we have the desired result. (⇐) αk1 µY = γ(k1 ,0) (ξ0 )ηk1 (all k1 ∈ Z+ ), we then have

1 k Y 2 2 2 2 2 0 t (αk1 (t))dµ (t) = α(0,0) · ... · α(k1 −1,0) · β(k1 ,0) · ... · β(k1 ,k2 −1) (by 2.7) and

1 0

1

tk2 ( 0 sk1 dΦt (s))dµY (t) = X×Y sk1 tk2 dµ(s, t) (all k1 , k2 ∈ Z+ ).

Let µ(s, t) be the Berger measure of T. Then, by Berger’s Theorem, T is a subnormal. From Theorem 2.7 we can establish our new computable necessary condition for the existence of a lifting for contractive commuting subnormal 2-variable weighted shifts. Corollary 2.8. Let T ≡ (T1 , T2 ) be as in Theorem 2.7. If T is subnormal, then ηk1 µY (all k1 ≥ 0).

(2.8)

Proof. Clear from Theorem 2.7. In an entirely similar way, we can show that

Remark 2.9. Let T ≡ (T1 , T2 ) be a contractive 2-variable weighted shift in Figure 1 (ii). If T is subnormal, then ξk2 µX (all k2 ≥ 0).

(2.9)

We summarize these results in Corollary 2.8 and Remark 2.9 as follows. Theorem 2.10. Let T ≡ (T1 , T2 ) be a contractive 2-variable weighted shift in Figure 1 (ii). If T is subnormal, then ηk1 µY and ξk2 µX (all k1 , k2 ≥ 0).

(2.10)

We shall now construct an example of commuting pairs of subnormal weighted shifts, satisfying condition (2.10) in Theorem 2.10, but without admitting a lifting to a commuting pair of normal operators. We begin with a review of the theory of subnormal backward extensions. We ﬁrst recall some deﬁnitions given in ([10], [12]): (i) Let µ and ν be two positive measures on R+ . We say that µ ≤ ν on X := R + , if µ(E)

≤ ν(E) for all Borel subsets E ⊆ R+ ; equivalently, µ ≤ ν if and only if f dµ ≤ f dν for all f ∈ C(X), where f ≥ 0 on R+ , (ii) given a probability measure µ on X ×Y ≡ R+ ×R+ , with 1t ∈ L1 (µ), the extremal measure µext (which is also a probability measure) on X × Y is given by dµext (s, t) := t 1 1 dµ(s, t); t L1 (µ) and (iii) given a measure µ on X × Y , the marginal measure µX is given by −1 , where πX : X × Y → X is the canonical projection onto X. Thus µX := µ ◦ πX µX (E) = µ(E × Y ), for every E ⊆ X. Observe that if µ is a probability measure, then so is µX . For example, 1 d(ξ × η)ext (s, t) = dξ(s)dη(t) (2.11) 1 t t L1 (η)

Vol. 59 (2007)

Disintegration of Measures and Contractive Shifts

291

and (ξ × η)X = ξ. Proposition 2.11. [10, Proposition 3.10] (Subnormal backward extension of a 2variable weighted shift) Consider the following 2-variable weighted shift T ≡ (T1 , T2 ) (see Figure 1 (i)), and let M be the subspace associated to indices k with k2 ≥ 1. Assume that TM is subnormal with Berger measure µM and that W0 := shif t(α00 , α10 , · · · ) is subnormal with Berger measure ξ0 . Then T is subnormal if and only if (i) 1t ∈ L1 (µ M ); 2 ≤ ( 1t L1 (µM ) )−1 ; and (ii) β00 2 1 X (iii) β00 t L1 (µM ) (µM )ext ≤ ξ0 . 2 1 X Moreover, if β00 t L1 (µM ) = 1, then (µM )ext = ξ0 . In the case when T is subnormal, Berger measure µ of T is given by 2 1 dµ(s, t) = β00 d(µM )ext (s, t) t 1 L (µM ) 2 1 +(dξ0 (s) − β00 d(µM )X ext (s))dδ0 (t). t L1 (µM ) Example 2.12. Let T ≡ (T1 , T2 ) be the 2-variable weighted shift in Figure 1 (ii) (with x and y in (0, 1]). Then (i) T is hyponormal ⇔ 1 − 2x2 + y 2 ≥ 0; (ii) T is subnormal ⇔ 1 − 2x2 + x2 y 2 ≥ 0; (iii) there exist a probability measure µ on the Borel sets contained in X×Y with ηk1 ≈ µY and ξk2 ≈ µX (all k1 , k2 ≥ 0); and (iv) ηk1 +1 ≈ ηk1 and ξk2 +1 ≈ ξk2 (all k1 , k2 ≥ 0). Proof. For (i) and (ii), see the proof in [11, Proposition 4.10]. That (iv) is clear from (iii). For (iii), consider a probability measure µ on the Borel sets contained in X×Y , that is, µ := x2 (1 − y 2 )δ0 + y 2 δ1 × δ1 + ((1 − x2 )δ0 + x2 δ1 − x2 (1 − y 2 )δ0 + y 2 δ1 ) × δ0 . Then µX = µY = (1−x2 )δ0 +x2 δ1 . Therefore, we have ηk1 ≈ (1−x2 )δ0 +x2 δ1 ≈ ξk2 (all k1 , k2 ≥ 0), as desired. More concretely, if (x, y) ∈ A := {(x, y) ∈ R2+ : 1 − 2x2 + x2 y 2 < 0, 1 − 2x2 + y 2 ≥ 0}, then the 2-variable weighted shift T in Example 2.12 is hyponormal but not subnormal with ηk1 +1 ≈ ηk1 and ξk2 +1 ≈ ξk2 (all k1 , k2 ≥ 0). (2.12)

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Remark 2.13. Let T ≡ (T1 , T2 ) be the 2-variable weighted shift as in Example 2.12. If 1 − 2x2 + x2 y 2 > 0, for (x, y) ∈ R2+ , then T is subnormal. We also observe by Proposition 2.11 and (cf. [6], [7]) that the Berger measure µ of T is µ = x2 (1 − y 2 )δ0 + y 2 δ1 × δ1 + ((1 − x2 )δ0 + x2 δ1 − x2 (1 − y 2 )δ0 + y 2 δ1 ) × δ0 . We remark that µ ≤ ν ⇒ µ ν, but µ ν doesn’t imply µ ≤ ν. We now observe a new suﬃcient condition for the lifting for commuting subnormal 2-variable weighted shifts. Theorem 2.14. Consider a 2-variable weighted shift T ≡ (T1 , T2 ) which has commuting subnormal components (see Figure 1 (i)) If for every k1 , k2 ≥ 0 ξk2 +1 ≤ ξk2

(2.13)

and ηk1 +1 ≤ ηk1 . ∼ then T is a subnormal and T = (I ⊗ Wα(0) , Wβ (0) ⊗ I).

(2.14)

Proof. We observe that Berger measure is a positive regular Borel measure on R+ . Suppose ξk2 +1 = ξk2 for some k2 ≥ 0. Then there exists a Borel set E in R+ such that ξk2 +1 (E) < ξk2 (E). Let E c := R+ − E we then have ξk2 (E c ) = 1 − ξk2 (E) < 1 − ξk2 +1 (E) = ξk2 +1 (E c ). Since E c is a Borel set in R+ , ξk2 (E c ) < ξk2 +1 (E c ) which cannot occur with ξk2 +1 ≤ ξk2 . Therefore ξk2 +1 = ξk2 . Thus we have ξk2 +1 = ξk2 and ηk1 +1 = ηk1 (all k1 , k2 ≥ 0). Let us put Wα(0) (Wβ (0) ) in all horizontal slices (in all vertical slices) respectively, then a direct calculation shows that T is doubly commuting and T ∼ = (I ⊗ Wα(0) , Wβ (0) ⊗ I). Therefore T is a subnormal with Berger measure ξ0 × η0 . We remark that the subnormality of T ≡ (T1 , T2 ) does not imply the conditions ξk2 +1 ≤ ξk2 and ηk1 +1 ≤ ηk1 (all k1 , k2 ≥ 0). For consider the 2variable weighted shift T in Example 2.12. We observe that T is subnormal with T (I ⊗ shif t(x, 1, 1, · · · ), shif t(x, 1, 1, · · · ) ⊗ I), if 1 − 2x2 + x2 y 2 > 0, for (x, y) ∈ R2+ .

3. Examples and a Problem In this section we give several striking examples for 2-variable weighted shifts and an improved version of the Curto-Muhly-Xia conjecture [8], that is, whether joint hyponormality is suﬃcient to guarantee a lifting for subnormal 2-variable weighted shifts. We recall some deﬁnitions. Let µ be a regular Borel measure on R+ := [0, ∞). A positive regular Borel measure µ is called discrete ∞ if there is a countable set {x } ⊂ R and positive numbers {a } such that + =1 a < ∞ and ∞ µ = =1 a δx , where δx is a point mass at x . On the other hand, µ is called continuous if µ{x} = 0 for all x ∈ R+ .

Vol. 59 (2007)

Disintegration of Measures and Contractive Shifts

293

We showed in Example 2.12 that a commuting hyponormal 2-variable weighted shift T with subnormalities T1 and T2 , with mutually absolutely continuous Berger measures for all horizontal and vertical slices, with all ﬁnitely atomic (2-atomic) measures, does not admit a lifting. The 2-variable weighted shift in Example 3.1 shows that the Berger measures of all horizontal slices Wα(k2 ) (k2 ≥ 1) and all vertical slices Wβ (k1 ) (k1 ≥ 0) are mutually absolutely continuous with continuous Berger measures of all horizontal and vertical slices. Even under these conditions, the 2-variable weighted shift T ≡ (T1 , T2 ) may fail to be subnormal. Example 3.1. Consider the following 2-variable weighted shift T ≡ (T1 , T2 ) (see Figure 2 (i)). Then T is commuting, hyponormal, with each of T1 and T2 subnormal, and ξk2 +1 ≈ ξk2 (all k2 ≥ 1) and suppηk1 = {0, 1} (all k1 ≥ 0), but T is not subnormal.

Proof. See [11, Proposition 4.19]. .. (0, 3) .

···

.. .

1

1

(0, 2)

···

1 2

.. .

···

.. .

···

.. .

···

.. .

···

1

1

1

1

2 3

3 4

···

T2

1 2

2 3

···

3 4

···

3 4

···

5 6

···

T2

1 (0, 1)

1 2

1 2

1 (0, 0)

1

1 4

2 3

1

3 2

(1, 0)

(2, 0)

1 9

1 3 4

y 5 3

···

(3, 0)

(0, 0)

1 1 2

y 1 2

(1, 0)

T1 (i)

···

1 2 3

3 4

8y 2 9

(2, 0)

(3, 0)

T1 (ii)

Figure 2. Weight diagrams of the 2-variable weighted shifts in Examples 3.1 and 3.5, and Propositions 3.7 and 3.8 We show in Example 3.5 that if we extend the mutually absolute continuity of measures to the 0-th level, that is, ξk2 +1 ≈ ξk2 (all k2 ≥ 0) in Example 3.1, then T is subnormal. We begin with several lemmas which are of independent interest. Lemma 3.2. If Wα := shif t(α0 , α1 , · · · ) be subnormal with Berger measure dµ(t) 2 on [0, Wα ], then cWα := shif t(cα0 , cα1 , · · · ), (c > 0) is also subnormal with 2 Berger measure dµ( ct2 ) on [0, cWα ].

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Proof. Since subnormality of cWα is obvious, we want to ﬁnd the Berger measure of cWα . Let υ be the Berger measure of cWα , then we have that

W 2

cWα 2 k t dυ(t) = γk (cWα ) = c2k α20 · · · α2k−1 = c2k 0 α tk dµ(t) (all k > 0). 0 Letting x = c2 t and replacing t by x in the last step, we have that

cWα 2 k

cW 2

cW 2 t dυ(t) = 0 α xk dµ( cx2 ) = 0 α tk dµ( ct2 ) (all k > 0). 0

Thus we have dυ(t) = dµα ( ct2 ), as desired. Lemma 3.3. shif t(1, √ ds π 2s−s2

3 2,

5 3,

7 4, · · ·)

is subnormal with Berger measure dµ(s) =

on [0, 2].

Proof. See ([13] and [9]). Lemma 3.4. shif t( 12 , 34 , 56 , · · · ) is subnormal with Berger measure dµ(s) = √ds π s−s2

on [0, 1].

[0, 2], let c = √12

3 2,

5 3,

7 4, · · · )

ds has Berger measure dµ(s) = π√2s−s on 2 1 3 5 7 in Lemma 3.2. We then observe that shif t( 2 , 4 , 6 , 8 , · · · )

Proof. Since shif t(1,

is subnormal with Berger measure

√ 2ds π 2·2s−4s2

=

√ds π s−s2

on [0, 1].

Example 3.5. The 2-variable weighted shift T ≡ (T1 , T2 ) given by Figure 2 (ii) (where y = 12 ) is subnormal with ξk2 +1 ≈ ξk2 and ηk1 +1 ≈ ηk1 (all k1 , k2 ≥ 0). Proof. Berger measure of shif t(α00 , α10 , · · · ) is dξ0 (s) = π√ds on [0, 1], and s−s2 of course dξk2 (s) = ds on [0, 1] (all k2 ≥ 1). Thus ξk2 +1 ≈ ξk2 for all k2 ≥ 0. Moreover, suppηk1 = {0, 1} (all k1 ≥ 0) implies ηk1 +1 ≈ ηk1 , and T1 and T2 are subnormal. XTo check the subnormality of T, it is enough to show that 2 1 (µM )ext ≤ ξ0 in Proposition 2.11. A direct calculation shows that β00 t L1 (µM ) 2 1 β = 1 and dµM (s, t) = dsdδ1 (t). Observe that 1 00

t

2

L (µM )

1 (dµM )X dµM )X = ds. ext = ((1 − dδ0 (t)) 1 t t L1 (µ) Thus we have 2 1 β00 t 1

L (µM )

(µM )X ext ≤ ξ0 ⇔

Therefore T is subnormal.

ds 1 ds 1 ≤ √ ⇔ ≤ √ on [0, 1] . 2 2 π s − s2 π s − s2

Vol. 59 (2007)

Disintegration of Measures and Contractive Shifts

295

As a consequence of Example 3.5, one might guess that if T is a hyponormal 2variable weighted shift with commuting subnormal components with ξk2 +1 ≈ ξk2 , ηk1 +1 ≈ ηk1 (all k1 , k2 ≥ 0), and either ηk1 (all k1 ≥ 0) or ξk2 (all k2 ≥ 0) is continuous measure then T must be subnormal. However even under these conditions, the 2-variable weighted shift may fail to be subnormal. First we recall a simple criterion detecting hyponormality for 2-variable weighted shifts (called the Six-point Test ). Lemma 3.6. [4] (Six-point Test) Let T ≡ (T1 , T2 ) be a 2-variable weighted shift, with weight sequences α and β. Then for all k ∈ Z2+

∗

[T , T] ≥ 0 ⇔ H(k) :=

α2k+ε1 − α2k αk+ε2 βk+ε1 − αk βk

αk+ε2 βk+ε1 − αk βk 2 βk+ε − βk2 2

≥ 0.

(k1 , k2 + 2) βk1 ,k2 +1 αk1 ,k2 +1

(k1 , k2 + 1)

(k1 + 1, k2 + 1) βk1 +1,k2

βk1 ,k2 αk1 ,k2 (k1 , k2 )

(k1 + 1, k2 )

αk1 +1,k2 (k1 + 2, k2 )

Figure 3. Weight diagram used in the Six-point Test

Proposition 3.7. The 2-variable weighted shift T ≡ (T1 , T2 ) given by Figure 2 (ii) 9 is hyponormal if and only if y ≤ 10 . Proof. We use the Six-point Test (Lemma 3.6) to show that T is hyponormal. In this case α2(1,0) − α2(0,0) α(0,1) β(1,0) − α(0,0) β(0,0) H((0, 0)) = 2 2 α(0,1) β(1,0) − α(0,0) β(0,0) β(0,1) − β(0,0) 1 0 4 = ≥ 0. 0 1 − y2

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For n ≥ 1,

H((n, 0))

α2(n+1,0) − α2(n,0) = α(n,1) β(n+1,0) − α(n,0) β(n,0) a b =: , b c

α(n,1) β(n+1,0) − α(n,0) β(n,0) 2 2 β(n,1) − β(n,0)

where a : b : c

:

1 2(n + 1)(n + 2) (2n)!! · 2(n + 1)2 (2n)!! · (2n + 1) =y − (2n + 1)!! · (n + 2)2 (2n − 1)!! · 2(n + 1)2 =

= 1 − y2

(2n)!! . (n + 1)(2n − 1)!!

By direct computation, using Mathematica [18], we have det H((n, 0)) ≥ 0 (all n ≥ 1) ⇔

1 2

1 n2 +3n+2

⇔y≤ ⇔y≤

+

4y 2 (2n)!! (2n−1)!!·(n+2)

(n2 +3n+2)

3

2

(2n−1)!!+(2n +7n+6)(2n+1)!!) − y 2 (2n)!!·(4(n+1) (n+1)(n+2)2 ((2n−1)!!·(2n+1))!! 2

(2n)!!·(4(n+1)3 (2n−1)!!+(2n2 +7n+6)(2n+1)!!) (2n)!! −4 (2n−1)!!·(n+2) (n+1)(n+2)2 ((2n−1)!!·(2n+1))!!

(n+2)·(2n+1)!! (3n+2)(2n)!!

(all n ≥ 1) ⇔ y ≤

≥0

(all n ≥ 1)

9 f (1) ⇔ y ≤ 10 ,

because

(n + 2) · (2n + 1)!! (3n + 2)(2n)!! is increasing function on N and f (n) > 1 (all n ≥ 4). It follow that T is hyponormal. f (n) :=

Proposition 3.8. The 2-variable weighted shift T ≡ (T1 , T2 ) given by Figure 2 (ii) is subnormal if and only if y ≤ π2 . Proof. From Figure 2 (ii), it is obvious that 1 2 3 ∼ TM = (I ⊗ shif t( , , , · · · ), U+ ⊗ 1) 2 3 4 (recall that U+ is the (unweighted) unilateral shift). Observe that TM is subnormal with Berger measure µM := ds × δ1 . By Proposition 2.11, 2 ds 2 1 X 2 T is subnormal ⇔ β00 , (µM )ext ≤ ν ⇔ y ds ≤ √ ⇔y≤ 2 t L1 (µM ) π π s−s

Vol. 59 (2007)

Disintegration of Measures and Contractive Shifts

because f (s) := 2 is the tangent line of

√ 1 s−s2

at s = 12 .

297

We summarize these results in Propositions 3.7 and 3.8 as follows. Theorem 3.9. The 2-variable weighted shift T ≡ (T1 , T2 ) given by Figure 2 (ii) is 2 9 hyponormal and not subnormal if and only if π < y ≤ 10 . If we assume that ηk1 (all k1 ≥ 0) and ξk2 (all k2 ≥ 0) are all continuous measures (see 1 (i)), then it is not clear that T ≡ (T1 , T2 ) is sub Figure normal because 1t L1 (µM ) is not known in general. We might guess that the 2 1 hyponormality condition of T makes the value β00 t L1 (µM ) smaller which forces 2 1 X β00 t L1 (µM ) (µM )ext ≤ ξ0 in Proposition 2.11, if ηk1 (all k1 ≥ 0) and ξk2 (all k2 ≥ 0) are all continuous measures. We conclude this section with a problem of independent interest. Problem 3.10. Let T ≡ (T1 , T2 ) be a 2-variable weighted shifts with commuting subnormal components, ξk2 +1 ≈ ξk2 , ηk1 +1 ≈ ηk1 (all k1 , k2 ≥ 0), and ηk1 (all k1 ≥ 0) and ξk2 (all k2 ≥ 0) are continuous measures. Then T is subnormal if and only if T is hyponormal. Acknowledgment Some of the discussion in Introduction was taken from [3, VII.2, pp. 317-319] and [11].

References [1] A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103 (1988), 417–423. [2] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75–94; MR 16:835a. [3] J. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, 1991. [4] R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symposia Pure Math. 51 (1990), 69–91. [5] R. Curto, S.H. Lee and J. Yoon, k-hyponormality of multivariable weighted shifts, J. Funct. Anal. 15 (2005), 462–480. [6] R. Curto, S.H. Lee and J. Yoon, Hyponormality and subnormality for powers of commuting pairs of subnormal operators, J. Funct. Anal., to appear. [7] R. Curto, S.H. Lee and J. Yoon, Reconstruction of the Berger measure when the core is of tensor form, Oper. Theory Adv. Appl., to appear. [8] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Operator Theory: Adv. Appl. 35 (1988), 1–22. [9] R. Curto, Yiu Poon and J. Yoon, The class of the Bergman-like weighted shifts, J. Math. Anal. Appl. 308 (2005) 334–342. [10] R. Curto and J. Yoon, Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006), 5139–5159.

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[11] R. Curto and J. Yoon, Disintegration-of-measure techniques for commuting multivariable weighted shifts, Proc. London Math. Soc., 93 (2006), 381–402. [12] R. Curto and J. Yoon, Propagation phenomena for hyponormal 2-variable weighted shifts, J. Operator Theory, to appear. [13] R. Curto and J. Yoon, Spectral picture of 2-variable weighted shifts, C. R. Acad. Sci. Paris, 343 (2006), 579–584. [14] R. Gellar and L.J. Wallen, Subnormal weighted shifts and the Halmos-Bram criterion, Proc. Japan Acad., 46 (1970), 375–378 [15] H. Helson, Disintegration of measures, Harmonic analysis and hypergroups (Delhi, 1995), 47–50, Trends Math., Birkh¨ auser Verlag, Boston, MA, 1998. [16] N.P. Jewell and A.R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1 (1979), 207–223. [17] J.A. Shohat and J.D. Tamarkin, The Problem of Moments, Math. Surveys I, American Math. Soc., Providence, 1943. [18] Wolfram Research, Inc. Mathematica, Version 4.2, Wolfram Research Inc., Champaign, IL, 2002. Jasang Yoon Department of Mathematics The University of Texas-Pan American Edinburg, Texas 78539 USA e-mail: [email protected] Submitted: May 16, 2006 Revised: May 25, 2007

Integr. equ. oper. theory 59 (2007), 299–307 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030299-9, published online October 18, 2007 DOI 10.1007/s00020-007-1532-y

Integral Equations and Operator Theory

On the Spectrum of Invertible Semi-hyponormal Operators Ariyadasa Aluthge Abstract. It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a diﬀerent proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A10. Keywords. p-hyponormal operators, semi-hyponormal operators.

1. Introduction Let H be a separable Hilbert space and let L(H) denote algebra of all bounded linear operators in H. An operator T ∈ L(H) is called p-hyponormal if (T ∗ T )p ≥ (T T ∗)p , 0 < p ≤ 1, where T ∗ is the adjoint operator of T . If p = 1, T is called hyponormal and if p = 12 , T is called semi-hyponormal. The well-known L¨owner’s theorem implies that any p-hyponormal operator is q-hyponormal for q ≤ p . Therefore, any semi-hyponormal operator is p -hyponormal for 0 < p < 12 . But the converse of the above statement is not true in general. Hyponormal operators have been studied by many authors. See Martin and Putinar [6] and Xia [8] for basic properties of hyponormal operators. Semi-hyponormal operators were introduced in Xia [7]. See Xia [8] also for properties of semihyponormal operators. This author ﬁrst studied p-hyponormal operators for 0 < p < 1 in depth. See Aluthge [1], [2], and [3] for details. Since then many authors have studied p-hyponormal operators extensively. Let T = U |T | be the polar decomposition of an invertible semi–hyponormal 1 operator, where |T | = (T ∗ T ) 2 . Therefore, the partial isometry U in the polar decomposition of T is a unitary operator.

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1 1 In [1], Aluthge introduced the operator transform T˜ = |T | 2 U |T | 2 to study the properties of p-hyponormal operators. Operators T˜ and T have the same spectrum. Throughout this paper, the spectrum, the point spectrum, the approximate point spectrum, and the joint point spectrum of an operator T are denoted by σ(T ), σp (T ), σa (T ), and σjp (T ), respectively. The spectral radius of T is denoted by rsp (T ). Note that some authors use the term normal point spectrum in place of joint point spectrum. The resolvent set of T is given by ρ(T ). Following results were obtained in [1]. Also, see [3] for more details.

Theorem 1.1. Let T = U |T | be a p-hyponormal operator,0 < p < 1 . Then the 1 1 operator T˜ = |T | 2 U |T | 2 is hyponormal if 1 ≤ p ≤ 1 , and (p + 1 )-hyponormal if 2

0 < p < 12 .

2

Theorem 1.2. Let T = U |T | be an invertible p-hyponormal operator for 0 < p < 12 . Then σjp (T ) = σp (T ); that is, for any f ∈ H, f = 0, U |T |f = reiθ f ⇔ U f = eiθ f and |T |f = rf.

(1.1)

Theorem 1.3. Let T = U |T | be an invertible p-hyponormal operator for 0 < p < 12 . Then the polar symbols of T , T+ = SU+ (T ) = st − lim U ∗ n T U n and T− = SU− (T ) = st − lim U n T U ∗ n n→∞

n→∞

exist. Theorem 1.4. Let T = U |T | be a p-hyponormal operator for 0 < p < 12 , and U be unitary. Then the eigenspaces of U reduce T . Theorem 1.5. Let T = U |T | be a p-hyponormal operator for 0 < p < 12 , and U be unitary. If σ(U ) = {z; |z| = 1}, then the eigenspaces of |T | reduce T .

2. The Main Result The general polar symbols of T = U |T | are deﬁned by, for k ∈ [0, 1] , Tk = kT+ + (1 − k)T− . It can be easily shown that T+ = U |T |+ and T− = U |T |− , where |T |+ = SU+ (|T |) = st − lim (U ∗ )n |T |U n and |T |− = SU− (|T |) = st − lim U n |T |(U ∗ )n. n→∞

n→∞

Also, we deﬁne the operator Q = Q+ + Q− , where Q+ =

∞ n=1

U ∗ n (|T | − U |T |U ∗ )U n and Q− =

0

U ∗ n (|T | − U |T |U ∗)U n .

n=−∞

We have Q+ = |T |+ − |T |, Q− = |T | − |T |− , Q = |T |+ − |T |− , and 0 ≤ Q± ≤ Q. One of the most fascinating results about semi-hyponormal operators is that the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. That is, Xia [7] proved the following.

Vol. 59 (2007) On the Spectrum of Invertible Semi-hyponormal Operators

Theorem 2.1. Let T be a semi-hyponormal operator, then σ(Tk ). σ(T ) =

301

(2.1)

0≤k≤1

The above theorem allows one to evaluate the spectrum of a semi-hyponormal operator by means of the spectra of the general polar symbols of the operator, which in most cases can be evaluated by using the corresponding singular integral model. Xia [7] used the so-called singular integral model to prove the above theorem. The purpose of this paper is to give a diﬀerent proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model. See Li[5] for a similar proof for the case of hyponormal operators. We divide our proof into several lemmas. Throughout this paper T = U |T | is an invertible semi-hyponormal operator and hence satisﬁes U ∗ |T |U ≥ |T | ≥ U |T |U ∗.

(2.2)

First we quote the following lemma from Xia [8] without a proof. Lemma 2.2. Let T ∈ L(H). There exists a Hilbert space R ⊃ H, and a map Π : L(H) → L(R) such that for any S, T ∈ L(H), and complex numbers α and β, 1. Π(T ∗ ) = [Π(T )]∗ , Π(αT + βS) = αΠ(T ) + βΠ(S), and Π(ST ) = Π(S)Π(T ), 2. Π(S) ≤ Π(T ) whenever T ≤ S, 3. σ(T ) = σ(Π(T )) , and σa (T ) = σa (Π(T )) = σp (Π(T )). Lemma 2.3. The operator

Tk (n) = U kU ∗ n |T |U n + (1 − k)U n |T |U ∗ n

(2.3)

is semi-hyponormal and σp (T ) = σp (Tk (n)). Proof. It is evident that |Tk (n)| = kU ∗ n |T |U n + (1 − k)U n |T |U ∗ n , and hence 1 [Tk (n)]∗ Tk (n) 2

1 − Tk (n)[Tk (n)]∗ 2 = |Tk (n)| − U |Tk (n)|U ∗

= kU ∗ n (|T | − U |T |U ∗ )U n + (1 − k)U n (|T | − U |T |U ∗ )U ∗ n ≥ 0. (2.4) Thus, Tk (n) is semi-hyponormal. To prove σp (T ) ⊂ σp Tk (n) , let reiθ ∈ σp (T ). Then there exists f ∈ H, f = 0, such that (2.5) U f = eiθ f and |T |f = rf iθ iθ byTheorem 1.2. Then a simple calculation gives Tk (n)f = re f and hence re ∈ σp Tk (n) as desired.

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On the other hand, if reiθ ∈ σp Tk (n) , since Tk (n) is semi-hyponormal, again by Theorem 1.2 there exists f ∈ H, f = 0 such that U f = eiθ f, and kU ∗ n |T |U n + (1 − k)U n |T |U ∗ n f = rf. (2.6) iθ The eigenspaces of U reduce |T |; that is, if U f = eiθ f , then U |T |f = e |T|f . iθ Therefore, (2.6) implies |T |f = rf and hence re f ∈ σp (T ). Thus, σp Tk (n) ⊂ σp (T ), and ﬁnally σp (T ) = σp Tk (n) .

Note that (2.4) implies that Tk = st − limn→∞ Tk (n) is semi-hyponormal. We also need the following lemma from Xia[8] for the next step of our proof. Lemma 2.4. Let R be a subset of the complex plane C, and t ∈ [0, 1], τt : R → τt (R) be a bijective mapping such that for any fixed z ∈ R, τt (z) is a continuous function of t, and τ0 is the identity function. Let T (t) be an operator-valued function which is continuous in the norm topology. Suppose that σa (T (t)) ∩ τt (R) = τ (σa (T (0)) ∩ R) for all t ∈ [0, 1]. Then σ(T (t)) ∩ τt (R) = τ (σ(T (0)) ∩ R) for all t ∈ [0, 1]. Lemma 2.5. The operators T and Tk (n) have the same spectrum. That is, σ(T ) = σ Tk (n) . (2.7) Proof. First we use Lemma 2.2 to prove σa (T )= σa T k (n) . Let Π be as in Lemma 2.2. Then it is evident that both Π(T ) and Π Tk (n) are semi-hyponormal and Π Tk (n) = Π(T ) k (n). From Lemma 2.3 we have σp Π(T ) = σp Π(T ) k (n) = σp Π Tk (n) , and hence

σa (T ) = σp Π(T ) = σp Π Tk (n) = σa Tk (n)

from Lemma 2.2. Now we use Lemma 2.4 to prove σ(T ) = σ Tk (n) . Deﬁne the operator valued function T (t) = U tU ∗ n |T |U n + (1 − t)U n |T |U ∗ n , (2.8) and the bijective mapping, τt : C → C, τt reiθ = eiθ te−inθ reinθ + (1 − t)einθ re−inθ = reinθ ,

(2.9)

for t ∈ [0, 1]. Then both T (t) and τt satisfy the conditions of Lemma 2.4, T (0) = U n T U ∗ n , and τ0 is the identity function. Thus, σa T (0) = σa (T ), and σ T (0) = σ(T ), (2.10)

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and hence from the ﬁrst part of the proof σa T (t) = σa (T ) = σa T (0) ,

(2.11)

Since τt is the identity function, we have from (2.11) σa T (t) ∩ τt (R) = τt σa T (0) ∩ R ,

(2.12)

for any R ⊂ C. Hence by Lemma 2.4, σ T (t) ∩ τt (R) = τt σ T (0) ∩ R , which implies

(2.13)

σ(Tt (n)) = σ T (t) = σ(T ),

by taking R = C.

Lemma 2.6. Let T = U |T | be an invertible semi-hyponormal operator. If z ∈ / σ(T ) and η = dist(z, σ(T )), then

(T − zI)f ≥

η f

1 |T |− 2

1

|T | 2

,

for any f ∈ H. Proof. First suppose S is a hyponormal operator. Then for any z ∈ / σ(S) and g ∈ H,

(S − zI)−1 g

≤

(S − zI)−1 g

=

max{|ω|; ω ∈ σ[(S − zI)−1 ]} g

= =

g /min{|ω| ∈ σ(S − zI)}

g /min{|ω − z| ∈ σ(S)}

=

g /dist(z, σ(S)).

Replacing g by (S − zI)g,

(S − zI)g ≥ dist(z, σ(T )) g .

(2.14) 1 2

1 2

1 2

1

Now for the semi-hyponormal operator T , let S = |T | U |T | = |T | T |T |− 2 . 1 Then S is hyponormal by Theorem 1.1, σ(S) = σ(T ), and (S − zI) = |T | 2 (T − 1 / σ(T ), zI)|T |− 2 . Hence from (2.14) for z ∈ 1

1

|T | 2 (T − zI)|T |− 2 g ≥ η g , 1

Taking |T |− 2 g = f , we get 1

1

|T | 2 (T − zI)f ≥ η |T | 2 f . and hence

1

(T − zI)f ≥ η

|T | 2 f

1

|T | 2

.

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1

Since f ≤ |T |− 2 |T | 2 f ,

(T − zI)f ≥

η f

1 2

1

|T | |T |− 2

.

Proof of Theorem 2.1. First we prove σ(Tk ) ⊂ σ(T ). 0≤k≤1

Let z ∈ / σ(T ), and η = dist(z, σ(T )). Then Lemma 2.5 implies z∈ / σ Tk (n) , and η = dist z, σ Tk (n) , and by Lemma 2.6

(Tk (n)) − zI)f ≥

η f

|Tk

1 (n)|− 2

1

|Tk (n)| 2

.

(2.15)

From the deﬁnition of Tk (n), |T |− ≤ U n |T |U ∗ n ≤ |Tk (n)| ≤ U ∗ n |T |U n ≤ |T |+ and hence, 1

1

1

1

|Tk (n)| 2 ≤ (|T |+ ) 2 and (|Tk (n)|− 2 ≤ (|T |− )− 2 . Thus, by (2.15)

(Tk (n)) − zI)f ≥

η f

1

1

(|T |− )|− 2 (|T |+ ) 2

.

By letting n → ∞,

(Tk − zI)f ≥

η f

(|T |−

1 )|− 2

and hence z ∈ / σ(Tk ). Therefore,

σ(Tk ) ⊂ σ(T ) and hence

1

(|T |+ ) 2

,

σ(Tk ) ⊂ σ(T ).

0≤k≤1

It remains to show σ(T ) ⊂

σ(Tk ).

0≤k≤1

It is shown in Xia [8] that 0≤k≤1 σ(Tk ) is closed. Suppose / σ(Tk ), and η = dist reiθ , σ(Tk ) . z = reiθ ∈ 0≤k≤1

0≤k≤1

Then for k ∈ [0, 1], dist reiθ , σ(Tk ) ≥ η > 0. That is, reiθ ∈ / σ(U (|T |− + kQ), iθ iθ where Q = |T |+ − |T |− ≥ 0. Suppose e ∈ / σ(U ). Then re ∈ / σ(T ) which would imply the desired result. So assume eiθ ∈ σ(U ) and choose δ > 0 suﬃciently small such that

(U − eiθ I)f ≤ δ f .

(2.16)

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Since reiθ ∈ / σ(Tk ), Lemma 2.3 and Lemma 2.6 imply

(U (|T |− + kQ) − reiθ )f

≥ ≥

η f

1

1

(|T |k )| 2 (|T |k )− 2

η f

1

1

(|T |+ )| 2 (|T |− )− 2

= ηˆ f.

because |T |− ≤ |T |k ≤ |T |+ , where ηˆ =

1

η

1

(|T |+ )| 2 (|T |− )− 2

(2.17)

. Now for k ∈ [0, 1],

(2.16) and (2.17) imply that

(U (|T |− + kQ) − reiθ )f

≥

ηˆ f

⇒ (U (|T |− + kQ) − rU + rU − reiθ )f

≥

ηˆ f

⇒ (U (|T |− + kQ − r)f +r (U − re )f

≥

⇒ U (|T |− + kQ − r)f

≥

⇒ (|T |− + kQ − r)f

≥

ηˆ f

ˆ f

(ˆ η − δ) ˆ f , (ˆ η − δ)

iθ

(2.18)

where δˆ = rδ. ˆ and 0 ≤ k ≤ 1 there exists η0 > 0 such that Then for 0 ≤ s ≤ δ, 1

(|T |− + k(Q 2 + s)2 − r)f

1

= (|T |− + k(Q + 2sQ 2 + s2 ) − r)f

1

= (|T |− + kQ − r)f + (s2 + 2sQ 2 )f

1

≥ (|T |− + kQ − r)f − (s2 + 2sQ 2 )f

≥ η0 f ,

(2.19)

as δ can be chosen arbitrarily small. The equality 1

(Q 2 + s)−1 (|T |−

+ =

1 1 12 (Q + s)2 − r)(Q 2 + s)−1 − λ 2 1 1 1 1 (Q 2 + s)−1 (|T |− + ( − λ)(Q 2 + s)2 − r)(Q 2 + s)−1 2

and (2.19) imply 1 1 1 ˆ 1 ˆ 1 1 ∪ ( + δ), ∞ . σ (Q 2 +s)−1 (|T |− + (Q 2 +s)2 −r)(Q 2 +s)−1 ⊂ −∞, −( + δ) 2 2 2 Therefore, 1 1 1 1 1 ˆ

(Q 2 + s)−1 (|T |− + (Q 2 + s)2 − r)(Q 2 + s)−1 ≥ + δ, 2 2

and hence 1 1 1 1 2 < 2.

(Q 2 + s)(|T |− + (Q 2 + s)2 − r)−1 (Q 2 + s) ≤ 2 1 + 2δˆ

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By letting s → 0,

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1 1 1

Q 2 (|T |− + Q − r)−1 Q 2 < 2 2

or equivalently

1

1

Q 2 (|T | 12 − r)−1 Q 2 < 2.

(2.20)

−1

exists, since it would imply reiθ ∈ / Our goal is to show that (|T | − r) σ(T ),the desired result. Not that |T |+ = |T | + Q+ and |T |− = |T | − Q− . Thus, 1 1 |T | 21 = |T | + (Q+ − Q− ) and (|T | 12 − r) = (|T | − r) + (Q+ − Q− ). 2 2 Since (|T | 21 − r)−1 exists, we can write 1 (|T | 21 − r)−1 (|T | − r) = I − (|T | 21 − r)−1 (Q+ − Q− ). 2 It suﬃces to show that (|T | 12 − r)−1 (|T | − r) is invertible, or equivalently 2 ∈ ρ (|T | 21 − r)−1 (Q+ − Q− ) . Since 0 ≤ Q± ≤ Q, there exist contractions R± such that (see Douglas [4]) 1

1

1

1

1

Q± 2 = R± Q 2 = Q 2 R± ∗ and hence Q+ − Q− = Q 2 (R+ ∗ R+ − R− ∗ R− )Q 2 . 1 1 So it suﬃces to show that 2 ∈ ρ (|T | 21 − r)−1 Q 2 (R+ ∗ R+ − R− ∗ R− )Q 2 . Since −R− ∗ R− f, f ≤ (R+ ∗ R+ − R− ∗ R− )f, f ≤ R+ ∗ R+ f, f , we have Thus,

R+ ∗ R+ − R− ∗ R− ≤ 1. 1 1 1 rsp Q 2 (|T | 12 − r)−1 Q 2 (R+ ∗ R+ − R− ∗ R− )Q 2 1 1 = rsp Q 2 (|T | 21 − r)−1 Q 2 (R+ ∗ R+ − R− ∗ R− ) 1

1

≤ Q 2 (|T | 21 − r)−1 Q 2 R+ ∗ R+ − R− ∗ R−

< 2, 1 1 and hence 2 ∈ ρ (|T | 21 − r)−1 Q 2 (R+ ∗ R+ − R− ∗ R− )Q 2 as required. This completes the proof.

References [1] A. Aluthge, On p-Hyponormal operators for 0 < p < 1, Integral Equations and Operator Theory 13 (1990), 307–315. [2] A. Aluthge, Some generalized theorems on p-hyponormal operators, Integral Equations and Operator Theory 24 (1996), 497–501 [3] A. Aluthge, Properties of p-hyponormal operators, Ph D. Dissertation, Vanderbilt University, Nashville, TN, 1990. [4] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.

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[5] S. Li, On the spectrum of hyponormal operators, Integral Equations and Operator Theory 11 (1988), 536–556. [6] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Birkh¨ auser Verlag, Boston, 1989. [7] D. Xia, On the nonnormal operators-semihyponormal operators, Sci. Sinica 23 (1980), 700–713. [8] D. Xia, Spectral Theory of Hyponormal Operators, Birkh¨ auser Verlag, 1983. Ariyadasa Aluthge Department of Mathematics Marshall University Huntington, WV 25755 USA e-mail: [email protected] Submitted: August 2, 2006 Revised: June 28, 2007

Integr. equ. oper. theory 59 (2007), 309–327 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030309-19, published online October 18, 2007 DOI 10.1007/s00020-007-1529-6

Integral Equations and Operator Theory

Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces Jussi Behrndt and Hans-Christian Kreusler Abstract. The classical Krein-Naimark formula establishes a one-to-one correspondence between the generalized resolvents of a closed symmetric operator in a Hilbert space and the class of Nevanlinna families in a parameter space. Recently it was shown by V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo that these parameter families can be interpreted as so-called Weyl families of boundary relations, and a new proof of the Krein-Naimark formula in the Hilbert space setting was given with the help of a coupling method. The main objective of this paper is to adapt the notion of boundary relations and their Weyl families to the Krein space case and to prove some variants of the Krein-Naimark formula in an indeﬁnite setting. Mathematics Subject Classification (2000). Primary 47B50, 47A20, 47B25; Secondary 46C20, 47A06. Keywords. Symmetric operator, self-adjoint extension, Krein-Naimark formula, generalized resolvent, boundary relation, boundary triplet, (locally) deﬁnitizable operator, Krein space.

1. Introduction Let A be a densely deﬁned closed symmetric operator with equal (possibly inﬁnite) deﬁciency indices in a Hilbert space K and let {G, Γ0 , Γ1 } be a boundary triplet for the adjoint operator A∗ . Let A0 be the self-adjoint extension of A in K corresponding to the boundary mapping Γ0 , A0 = A∗ ker Γ0 , and denote the γ-ﬁeld and Weyl function corresponding to {G, Γ0 , Γ1 } by γ and M , respectively. Here the Weyl function M is an L(G)-valued Nevanlinna function with the additional property 0 ∈ ρ(Im M (λ)), λ ∈ C\R. It is well known that in this case the Krein-Naimark formula − λ −1 K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)∗ PK A (1.1)

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Behrndt and Kreusler

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establishes a bijective correspondence between the class of Nevanlinna families τ in the parameter space G and the compressed resolvents of self-adjoint extensions of A in K × H, where H is a Hilbert space. This description of the generalA ized resolvents of a symmetric operator was originally given by M.G. Krein and M.A. Naimark in [25, 26, 31] for the case that A is densely deﬁned and has ﬁnite deﬁciency indices; see [8, 10, 12, 13, 14, 29, 30] for our more general situation. Various generalizations of the Krein-Naimark formula in an indeﬁnite setting have been proved in the last decades. E.g., the case that A is a symmetric operator in a Pontryagin space K and H is a Hilbert space was investigated by M.G. Krein and H. Langer in [23]. Later V.A. Derkach considered both K and H to be Pontryagin or even Krein spaces, cf. [7]. Under additional assumptions other variants of (1.1) were proved in [3, 4, 5, 6, 7, 27]. Recently a very interesting new proof of the Krein-Naimark formula in the Hilbert space case was given in [10, 12] by V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo with the help of a coupling method which allows to interpret the parameter family τ as a so-called Weyl family associated to a boundary relation of a symmetric operator or relation in the Hilbert space H. The concept of boundary relations is a generalization of the notion of boundary triplets which has the essential advantage that every Nevanlinna family can be realized as the Weyl family associated to a boundary relation, see [9, 11]. The basic aim of this paper is to introduce the concept of boundary relations for symmetric operators and relations in Krein spaces, and to prove some variants of (1.1) in the Krein space case with a similar method as in [8, 10, 12]. Roughly speaking, if A is a symmetric relation in a Krein space K which possesses a self-adjoint extension A0 in K with a nonempty resolvent set, then we show in Theorem 3.1 that formula (1.1) gives a correspondence between compressed re in K × H, where H is a Krein space, and solvents of self-adjoint extensions A the Weyl families τ corresponding to boundary relations of symmetric relations in H. In contrast to the Hilbert space case where formula (1.1) makes sense for all λ ∈ C\R it is not immediately clear in our setting for which λ ∈ ρ(A0 ) the and the inverse of M + τ are bounded operators on compressed resolvent of A K and G, respectively, cf. assertion (a) in Theorem 3.1 and Theorem 3.3. In the special situation that A has ﬁnite defect, the ﬁxed canonical extension A0 locally (with the possible exception of a discrete set) has the same spectral properties as a self-adjoint operator in a Hilbert space and H is a Hilbert space we study the in Theorem 3.5, see also [3] for a similar situation. local spectral properties of A The paper is organized as follows. Following the lines of [9, 11] we introduce the concept of boundary relations and associated Weyl families for symmetric relations in Krein spaces in Section 2. The special case of boundary triplets and corresponding Weyl functions is brieﬂy reviewed in Section 2.3. Section 3 contains our main results on Krein-Naimark type formulas in the Krein space setting discussed above. Finally, in Section 4 we show that certain classes of relation-valued functions can be realized as Weyl families corresponding to boundary relations of

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Boundary Relations and Generalized Resolvents

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symmetric relations in Krein spaces. As a special case we obtain an alternative proof of the main result in [9, 11], that is, each Nevanlinna family can be realized as the Weyl family of a boundary relation of a symmetric relation in a Hilbert space.

2. Boundary relations of symmetric relations in Krein spaces The main objective of this section is to adapt the notion of boundary relations and associated Weyl families for symmetric relations in Hilbert spaces from [9, 11] to symmetric relations in Krein spaces. 2.1. Symmetric, self-adjoint, isometric and unitary relations in Krein spaces In the following let (K, [·, ·]K ) and (H, [·, ·]H ) be separable Krein spaces and let JK and JH be corresponding fundamental symmetries. The linear space of bounded linear operators deﬁned on K with values in H is denoted by L(K, H). If K = H we simply write L(K). We study linear relations from K to H, that is, linear subspaces H). of K × H. The set of all closed linear relations from K to H is denoted by C(K, If K = H we write C(K). Linear operators from K into H are viewed as linear relations via their graphs. For the usual deﬁnitions of the linear operations with relations, the inverse etc., we refer to [15]. The domain (kernel, range, multivalued part) of a linear relation S from K to H will be denoted by dom S (ker S, ran S, mul S, respectively). The resolvent set ρ(S) of a closed linear relation S ∈ C(K) is the set of all λ ∈ C such that (S − λ)−1 ∈ L(K), the spectrum σ(S) of S is the complement of ρ(S) in C. The extended spectrum σ (S) of S is deﬁned by σ (S) = σ(S) if S ∈ L(K) and σ (S) = σ(S) ∪ {∞} otherwise. The extended resolvent set ρ(S) of S is deﬁned by ρ(S) = C\ σ (S). A point λ ∈ C is an eigenvalue of S if ker(S − λ) = {0}; we write λ ∈ σp (S). We say that λ ∈ C belongs to the continuous spectrum σc (S) (the residual spectrum σr (S)) of S if ker(S − λ) = {0}, ran(S − λ) is dense in K and ran (S − λ) = K (resp. if ker(S − λ) = {0} and ran (S −λ) is not dense in K). gλ ˆλ,S := λg We set Nλ,S := ker(S − λ) and N | gλ ∈ Nλ,S . λ If U ⊂ K × H is a linear relation from K to H, then the adjoint relation U + ∈ C(H, K) is deﬁned by ˜ k} ˜ ∈ H × K | [h, ˜ h]H = [k, ˜ k]K for all {k, h} ∈ U . U + := {h, The linear relation U ⊂ K × H is said to be isometric (unitary) if U −1 ⊂ U + (resp. U −1 = U + ). If A ⊂ K2 is a linear relation in K, then A is said to be symmetric H) we (self-adjoint) if A ⊂ A+ (resp. A = A+ ). For a unitary relation U ∈ C(K, have dom U is closed if and only if ran U is closed (2.1) as well as ker U = (dom U )[⊥]K and mul U = (ran U )[⊥]H , (2.2) see, e.g., [11, Proposition 2.3].

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A symmetric relation A ∈ C(K) is said to be of defect m ∈ N ∪ {∞}, if both deﬁciency indices n± (JK A) = dim ker (JK A)∗ ∓ i of the symmetric relation JK A in the Hilbert space (K, [JK ·, ·]K ) are equal to m. Here ∗ denotes the adjoint with respect to the Hilbert scalar product [JK ·, ·]K . We note that the symmetric relation A ∈ C(K) is of defect m if and only if there exists a self-adjoint extension of A in K and each self-adjoint extension A of A in K satisﬁes dim(A /A) = m. We deﬁne an indeﬁnite inner product [[·, ·]]K2 on K2 (and analogously [[·, ·]]H2 2 on H ) by f g ˆ ˆ f , gˆ K2 = i [f, g ]K − [f , g]K , f = , gˆ = ∈ K2 . f g K ∈ L(K2 ) is a corresponding Then (K2 , [[·, ·]]K2 ) is a Krein space and iJ0K −iJ 0 fundamental symmetry. Observe that also in the special case when (K, [·, ·]) is a Hilbert space, [[·, ·]]K2 is an indeﬁnite inner product. In the following we will say that a linear relation Γ ⊂ K2 × H2 from K2 to H2 is [[·, ·]]-isometric ([[·, ·]]-unitary) if Γ is an isometric (resp. unitary) relation from (K2 , [[·, ·]]K2 ) to (H2 , [[·, ·]]H2 ). The adjoint of Γ will be denoted by Γ[[+]] . 2.2. Definition and basic properties of boundary relations and associated Weyl families The notions of boundary relations and associated Weyl families were introduced in [9, 11] for symmetric relations in Hilbert spaces. The deﬁnitions and some of the basic properties remain the same in the Krein space case. Definition 2.1. Let A ∈ C(K) be a symmetric relation in the Krein space K. A linear relation Γ ⊂ K2 × G 2 is called a boundary relation for A+ if G is a Hilbert space, T := dom Γ is dense in A+ and Γ is [[·, ·]]-unitary. 2 , G 2 ) be a boundary Let A ∈ C(K) be a symmetric relation and let Γ ∈ C(K + relation for A . Then the ﬁrst relation in (2.2) implies A = ker Γ. The elements in ˆ ∈ Γ, where fˆ = f ∈ K2 and h ˆ = h ∈ G 2 . Γ will be written in the form {fˆ, h} h f Associated with the boundary relation Γ are the relations ˆ h} ˆ ∈ Γ and Γ1 := {fˆ, h } | {f, ˆ h} ˆ ∈Γ . Γ0 := {fˆ, h} | {f, (2.3) We note that ker Γ0 and ker Γ1 are symmetric relations in K which in general are not closed. Definition 2.2. Let A be a closed symmetric relation in K and let Γ be a boundary relation for A+ , T := dom Γ. The γ-ﬁeld γ and the Weyl family τ of the boundary relation Γ are defined by ˆ h} ˆ ∈ Γ and fˆ ∈ N ˆλ,T , λ ∈ C, γ(λ) := {h, f } | {f, and

ˆ | {fˆ, ˆh} ∈ Γ and fˆ ∈ N ˆλ,T , τ (λ) := Γ Nˆλ,T = h

λ ∈ C.

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Remark 2.3. In general the values of the Weyl family and the γ-field corresponding ˆ with fˆ ∈ N ˆλ,T , to a boundary relation have nontrivial multivalued parts. For {fˆ, h} λ ∈ C, the [[·, ·]]-unitarity of Γ yields (h , h)G − (h, h )G = (λ − λ)[f, f ]K . In the special case where K is a Hilbert space this leads to ker(Γ0 |Nˆλ,T ) = {0} for λ ∈ C\R and hence, if A is a closed symmetric relation in a Hilbert space K and Γ is a boundary relation for A∗ , then the γ-field γ associated with Γ is an operator-valued function on C\R which maps dom τ (λ) onto Nλ,T , cf. [11, §4.2]. Let again K be a Krein space and let G be a Hilbert space. Then the bijective transformation

f h f f J : K2 × G 2 → (K × G)2 , , → , (2.4) f −h h h establishes via Γ → J (Γ) a one-to-one correspondence between the set of [[·, ·]]isometric ([[·, ·]]-unitary) relations Γ ⊂ K2 × G 2 and the set of symmetric (resp. self-adjoint) relations in (K × G)2 . The mapping (2.4) is called the main transform in [11]. Clearly, if A is closed and symmetric in K then a relation Γ ⊂ K2 × G 2 with the property A = ker Γ is a boundary relation for A+ if and only if J (Γ) is self-adjoint. This also implies that for a symmetric relation A ∈ C(K) a boundary relation always exists. In fact, if A is a closed symmetric relation in the Krein space (K, [·, ·]K ), then JK A is a closed symmetric relation in the Hilbert space (K, [JK ·, ·]K ) and hence there exists a Hilbert space G and a selfadjoint extension ∈ C(K × G) of JK A in the Hilbert space K × G such that JK A = B ∩ K2 holds, B see e.g. the construction in the proof of [11, Proposition 3.7]. It follows that

f f f JK f , A := , ∈B h h h h is selfadjoint in K × G when K is equipped with the Krein space inner product ∩ K2 implies that Γ := J −1 (A) is a boundary relation for A. [·, ·]K , and A = A The next lemma shows how the Weyl family τ of a boundary relation Γ is connected with the compressed resolvent of J (Γ) onto G. The proof is straightforward and essentially contained in [11, §3]. We leave the details to the reader. 2, G2) Lemma 2.4. Let A be a closed symmetric relation in K and let Γ ∈ C(K + be a boundary relation for A with corresponding Weyl family τ . Define J as in (2.4) and denote by PG the orthogonal projection from K × G onto G and by G the canonical embedding of G in K × G. Then τ satisfies (i)-(iii). (i) The formula (2.5) PG (J (Γ) − λ)−1 G = −(τ (λ) + λ)−1 holds for all λ ∈ C. (ii) If ρ(J (Γ)) is nonempty, then −(τ (λ) + λ)−1 ∈ L(G) for λ ∈ ρ(J (Γ)). (iii) The Weyl family is symmetric with respect to the real line, i.e., τ (λ) = τ (λ)∗ holds for all λ ∈ C\R.

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Remark 2.5. The class of Weyl families corresponding to boundary relations for symmetric relations in Hilbert spaces is completely described in [11, Theorem 3.9]. Namely, in the case that K is a Hilbert space it follows from Lemma 2.4 that the values τ (λ) of the Weyl family τ are maximal dissipative (maximal accumulative) relations for every λ ∈ C+ (resp. λ ∈ C− ), and τ (λ) = τ (λ)∗ and −λ ∈ ρ(τ (λ)) holds for all λ ∈ C\R, i.e., τ is a so-called Nevanlinna family; we write τ ∈ R(G). Conversely, by [11, Theorem 3.9] each Nevanlinna family τ ∈ R(G) can be realized as the Weyl family of a boundary relation for a symmetric relation in a Hilbert space (see also Corollary 4.4). 2.3. Boundary triplets for symmetric relations in Krein spaces The concept of boundary relations is an extension of the notion of boundary triplets for symmetric relations in Krein and Hilbert spaces, cf. [5, 6, 7] and, e.g., [13, 14, 16] for the Hilbert space case. Definition 2.6. Let A be a closed symmetric relation in a Krein space K and let 2 , G 2 ) be a boundary relation for A+ with Γ0 and Γ1 as in (2.3). If Γ is Γ ∈ C(K surjective, then {G, Γ0 , Γ1 } is said to be a boundary triplet for A+ . Deﬁnition 2.6 coincides with the usual deﬁnition of a boundary triplet for a symmetric relation in a Krein space since by (2.1) and (2.2) a surjective boundary 2 , G 2 ) is necessarily an operator deﬁned on A+ and therefore Γ0 relation Γ ∈ C(K and Γ1 are operators such that the mapping ΓΓ01 : A+ → G 2 is surjective and [f , g]K − [f, g ]K = (Γ1 fˆ, Γ0 gˆ)G − (Γ0 fˆ, Γ1 gˆ)G holds for all fˆ = {f, f }, gˆ = {g, g } ∈ A+ . We brieﬂy recall some important properties of boundary triplets which can be found in [5, 6, 7, 13, 14]. Let in the following A be a closed symmetric relation in K and let {G, Γ0 , Γ1 }, Γ = ΓΓ01 , be a boundary triplet for A+ . The mapping Γ induces, via (2.6) Θ → AΘ := Γ−1 Θ = fˆ ∈ A+ Γfˆ ∈ Θ , a bijective correspondence between the set of all closed linear relations Θ in G and the set of all closed extensions AΘ ⊂ A+ of A in K. Furthermore, (2.6) establishes a one-to-one correspondence between the closed symmetric (self-adjoint) relations in G and the closed symmetric (self-adjoint) extensions of A in K. Note, that in particular A0 := ker Γ0 and A1 := ker Γ1 are self-adjoint extensions of A. Assume now that ρ(A0 ) is nonempty. Then for each λ ∈ ρ(A0 ) the relation + A is the direct sum of A0 and Nˆλ,A+ and it follows from Deﬁnition 2.2 that for λ ∈ ρ(A0 ) the γ-ﬁeld γ and the Weyl function M of the boundary triplet {G, Γ0 , Γ1 } are given by ˆλ,A+ −1 ∈ L(G, K) γ(λ) = π1 Γ0 | N and

ˆλ,A+ −1 ∈ L(G). M (λ) = Γ1 Γ0 | N

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Here π1 denotes the orthogonal projection onto the ﬁrst component of K × K. The functions γ and M are holomorphic on ρ(A0 ) and satisfy the relations γ(λ) = I + (λ − µ)(A0 − λ)−1 γ(µ) and M (λ) − M (µ)∗ = (λ − µ)γ(µ)+ γ(λ) for all λ, µ ∈ ρ(A0 ). Moreover +

γ(λ) h = Γ1

(A0 − λ)−1 h (I + λ(A0 − λ)−1 h

(2.7)

holds for each h ∈ K and λ ∈ ρ(A0 ) With the help of the Weyl function the spectral properties of the closed extensions AΘ ⊂ A+ of A can be described. Namely, if Θ ∈ C(G) and AΘ is the corresponding extension of A via (2.6), then a point λ ∈ ρ(A0 ) belongs to ρ(AΘ ) (σi (AΘ ), i = p, c, r) if and only if 0 belongs to ρ(Θ − M (λ)) (resp. σi (Θ − M (λ)), i = p, c, r) and the well-known formula −1 γ(λ)+ (AΘ − λ)−1 = (A0 − λ)−1 + γ(λ) Θ − M (λ) holds for all λ ∈ ρ(A0 ) ∩ ρ(AΘ ), see, e.g., [7].

3. Generalized resolvents of symmetric relations in Krein spaces If A is a closed symmetric operator or relation with equal (possibly inﬁnite) deﬁciency indices in a Hilbert space K and {G, Γ0 , Γ1 }, A0 = ker Γ0 , is a boundary with corresponding γ-ﬁeld γ and Weyl function triplet for the adjoint A∗ ∈ C(K) M , then the Krein-Naimark formula − λ)−1 K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)∗ , λ ∈ C\R, (3.1) PK (A establishes a bijective correspondence between the compressed resolvents of min of A in K × H, where the exit space H is a Hilbert imal self-adjoint extensions A space, and the Nevanlinna families τ , i.e., the Weyl families of boundary relations of symmetric relations in Hilbert spaces (see, e.g., [8, 10, 12, 13, 14, 29, 30]). In this section we prove some variants of (3.1) for the case that K and H are Krein spaces. Other indeﬁnite generalizations of (3.1) can be found in [3, 4, 5, 6, 7, 23, 27]. 3.1. The case of a Krein space as exit space In the next theorem we show, roughly speaking, that a correspondence of the form (3.1) exists also between the compressed resolvents of the self-adjoint extensions of a symmetric relation in a Krein space K and the Weyl families of boundary relations of symmetric relations acting in Krein spaces H. The idea of the proof is based on the coupling method from [8, 10, 12].

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Theorem 3.1. Let K and H be Krein spaces, let A ∈ C(K) be a symmetric relation and let {G, Γ0 , Γ1 } be a boundary triplet for A+ with corresponding γ-field γ and Weyl function M . Let A0 = ker Γ0 and assume that ρ(A0 ) is nonempty. ∈ C(K × H) is a self-adjoint extension of A and for some λ0 ∈ ρ(A0 ) (i) If A 2, G2) PK (A−λ0 )−1 K ∈ L(K), then there exists a boundary relation Γ ∈ C(H such that the corresponding Weyl family τ satisfies (a) and (b). (a) If λ ∈ ρ(A0 ), then −1 ∈ L(G) M (λ) + τ (λ)

if and only if

− λ −1 K ∈ L(K). PK A

(b) The formula − λ −1 K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK A

(3.2)

holds for all λ ∈ ρ(A0 ) such that (M (λ) + τ (λ))−1 ∈ L(G). 2 , G 2 ) is a boundary relation with corresponding Weyl family τ (ii) If Γ ∈ C(H and (M (λ0 ) + τ (λ0 ))−1 ∈ L(G) for some λ0 ∈ ρ(A0 ), then there exists a ∈ C(K × H) of A such that (a) and (b) are satisfied. self-adjoint extension A Proof. (i) 1. The proof of assertion (i) is organized in 4 steps. Let H be a Krein be a self-adjoint extension of A in K × H. We do not exclude the space and let A that is, H = {0}. It is not diﬃcult to case of a canonical self-adjoint extension A, see that the closed linear relations

f0 f , f0 , f0 ∈ K2 , 0 ∈A S0 : = 0 0

0 0 f1 , f1 ∈ H2 S1 : = , ∈ A f1 f1 are symmetric in K and H, respectively. The same arguments as in the Hilbert space case (see [11, Proposition 2.12]) imply that the closures of the linear relations

f f0 , f0 , f0 ∈ K2 T0 : = , 0 ∈A f1 f1

f f 0 f1 , f1 ∈ H2 , 0 ∈A T1 : = f1 f1 coincide with S0+ and S1+ , i.e. Si ⊂ Ti ⊂ T i = Si+ holds for i = 1, 2. Note also that S0 is an extension of the symmetric relation A. 2. In this step we show that

Γ0 fˆ0 f0 f0 2 2 fˆ1 , × G Γ := ∈ H , ∈ A f1 f1 −Γ1 fˆ0

(3.3)

is a boundary relation for S1+ , cf. [10, Proposition 2.2 and Theorem 4.2] or [12, Theorems 2.13 and 6.3].

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From the deﬁnition of Γ we immediately get that dom Γ is dense in S1+ . In order to verify that the linear relation Γ ⊂ H2 × G 2 is [[·, ·]]-unitary let

Γ0 fˆ0 Γ0 gˆ0 −1 ˆ , f1 ∈ Γ ∈ Γ . and gˆ1 , −Γ1 gˆ0 −Γ1 fˆ0 is self-adjoint in K × H we have As A

f0 f0 g g = , 0 , 0 f1 g1 K×H f1 g1 K×H and hence

Γ0 fˆ0 Γ0 gˆ0 , = i [f0 , g0 ]K − [f0 , g0 ]K = fˆ1 , gˆ1 H2 ˆ 2 −Γ1 gˆ0 G −Γ1 f0

implies that Γ is [[·, ·]]-isometric, i.e. Γ −1 ⊂ Γ[[+]] . ˆ fˆ1 } ∈ Γ [[+]] and choose fˆ0 ∈ A+ such that ˆh = Conversely, let {h,

f0 f =A + , 0 ∈A f1 f1

Γ0 fˆ0 −Γ1 fˆ0

. Then

g0 g0 we have gˆ1 , Γ0 gˆ0 ∈A ∈ Γ , and hence since for an arbitrary g1 , g −Γ1 g ˆ0 1 ˆ fˆ1 } ∈ Γ [[+]] and the choice of h ˆ imply {h,

f0 f0 g g0 − = −i fˆ0 , gˆ0 K2 + fˆ1 , gˆ1 H2 , 0 , f1 g1 K×H f1 g1 K×H Γ0 fˆ0 Γ0 gˆ0 Γ0 gˆ0 ˆ , = −i + h, = 0. Γ1 gˆ0 G 2 −Γ1 gˆ0 G 2 Γ1 fˆ0 ˆ belongs to Γ and this gives Γ [[+]] ⊂ Γ −1 . We have shown that Therefore {fˆ1 , h} Γ is a boundary relation for S1+ . − λ)−1 K ∈ L(K) and denote the Weyl family of 3. Let λ ∈ ρ(A0 ) such that PK (A Γ by τ . We check that

−1 Γ1 fˆλ + h Γ0 fˆλ 2 ˆ ˆ + f M (λ) + τ (λ) = ∈ N and ∈ G ∈ τ (λ) λ λ,A h Γ0 fˆλ is a bounded operator deﬁned on G. If Γ1 fˆλ + h = 0 then

Γ0 fˆλ −Γ1 fˆλ

ˆλ,T1 such that ∈ τ (λ) and hence there exists fˆ1 ∈ N

Γ0 fˆλ ∈ Γ . fˆ1 , −Γ1 fˆλ

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fλ λfλ and this implies that {0, fλ } belongs to ∈A f1 , λf1 −1 f f 0 2 0 −λ ˆλ,T1 . and fˆ1 ∈ N PK A K = f0 − λf0 , f0 ∈ K , ∈A f1 f1

By (3.3)

Therefore fλ = 0, i.e., Γ0 fˆλ = 0 and (M (λ) + τ (λ))−1 is an operator. Next we show −1 = G. dom M (λ) + τ (λ) Let g ∈ G and choose fˆ0 ∈ A+ such that 0 Γ0 fˆ0 = . g −Γ1 fˆ0 − λ)−1 K ) = K there exists fˆ ∈ A+ and fˆ1 ∈ N ˆλ,T1 By our assumption dom (PK (A such that

f f and f − λf = f − λf0 . , ∈A 0 λf1 f1 Hence

Γ0 fˆ ˆ ˆ ˆ ˆ ˆ f1 , fλ := f − f0 ∈ Nλ,A+ and ∈ Γ , −Γ1 fˆ ˆ Γ0 fˆ ∈ τ (λ). Setting h := −Γ1 fˆ we ﬁnd Γ0 f λ ∈ τ (λ) and i.e., −Γ fˆ h

1

Γ1 fˆλ + h = Γ1 (fˆ − fˆ0 ) + h = −Γ1 fˆ0 = g, that is, g ∈ dom (M (λ) + τ (λ))−1 . Finally M (λ) ∈ L(G) and the fact that τ (λ) is closed imply that (M (λ)+τ (λ))−1 is closed and therefore (M (λ)+τ (λ))−1 ∈ L(G). 4. Let now λ ∈ ρ(A0 ) such that (M (λ) + τ (λ))−1 ∈ L(G) holds. We prove in this − λ)−1 K ∈ L(K) has the form (3.2). step that PK (A −1 Let k ∈ K and fλ := −γ(λ) (M (λ) + τ (λ) γ(λ)+ k ∈ Nλ,A+ and deﬁne fˆ0 ∈ A+ by f0 fλ (A0 − λ)−1 k fˆ0 = + := . (3.4) k + λ(A0 − λ)−1 k f0 λfλ We have −1 γ(λ)+ k. f0 − λf0 = k and f0 = (A0 − λ)−1 k − γ(λ) (M (λ) + τ (λ) − λ)−1 K it remains to show that there exists Therefore, by the form of PK (A f0 f0 First of all ∈ A. fˆ1 ∈ Nˆλ,T1 with f1 , f 1

Γ0 fˆλ = −(M (λ) + τ (λ))−1 γ(λ)+ k implies

Γ0 fˆλ −γ(λ)+ k

∈ M (λ) + τ (λ) =

h M (λ)h + h

ˆ h ∈ τ (λ)

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Γ0 fˆλ ˆλ,T1 . ∈ τ (λ) = Γ N + ˆ −γ(λ) k − Γ1 fλ ˆ ˆλ,T1 such that Hence there exists f1 ∈ N

Γ0 fˆλ ˆ ∈ Γ f1 , −γ(λ)+ k − Γ1 fˆλ

and from (3.4) and (2.7) we obtain Γ0 fˆ0 = Γ0 fˆλ and Γ1 fˆ0 = γ(λ)+ k + Γ1 fˆλ . Γ0 fˆ0 f0 f0 This Therefore fˆ1 , −Γ belongs to Γ and thus by (3.3) ∈ A. ˆ f1 , f1 f 1 0 completes the proof of assertion (i). 2 , G 2 ) be a boundary relation with correWe prove assertion (ii). Let Γ ∈ C(H sponding Weyl family τ . We claim that

f0 Γ0 fˆ0 f0 2 ˆ A := , ∈ (K × H) f1 , ∈Γ (3.5) f1 f1 −Γ1 fˆ0 f0 f0 g0 g0 we is a self-adjoint extension of A. In fact, for , ∈ A g1 , g f1 , f1 1 have

Γ0 fˆ0 Γ0 gˆ0 , , g ˆ ∈ Γ fˆ1 , 1 −Γ0 gˆ1 −Γ1 fˆ0 and the [[·, ·]]-isometry of Γ implies

f0 f0 g g0 = −i fˆ0 , gˆ0 K2 + fˆ1 , gˆ1 H2 , 0 K×H − , f1 g1 f1 g1 K×H Γ fˆ Γ gˆ Γ0 fˆ0 Γ0 gˆ0 0 0 0 0 = −i , , + = 0, Γ1 gˆ0 G 2 −Γ1 gˆ0 G 2 Γ1 fˆ0 −Γ1 fˆ0 f0 f0 + . We show is symmetric. Let now ∈A that is, A f1 , f1

f0 f , 0 ∈ A. (3.6) f1 f1 First of all we have fˆ0 , gˆ0 K2 = − fˆ1 , gˆ1 H2

g0 g for all , 0 ∈ A. (3.7) g1 g1 g0 g from {0, 0} ∈ Γ and 0 ∈A If gˆ0 ∈ A = ker Γ0 ∩ ker Γ1 we conclude 0 , 0 Γ0 fˆ0 ∈ Γ . (3.7) yields in this case fˆ0 ∈ A+ . By (3.5) it remains to check fˆ1 , −Γ ˆ 1 f0 ˆ ∈ Γ and choose gˆ0 ∈ A+ such that kˆ = Γ0 gˆ0 . By (3.5) For this let {ˆ g1 , k} −Γ1 g ˆ0 g0 g0 belongs to A and (3.7) implies g1 , g1

fˆ1 , gˆ1

H2

Γ fˆ Γ gˆ Γ fˆ 0 0 0 0 , 0 0 , kˆ = − fˆ0 , gˆ0 K2 = − = . ˆ Γ1 gˆ0 G 2 G2 Γ1 f 0 −Γ1 fˆ0

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Γ0 fˆ0 ˆ , f1 ∈ Γ [[+]] = Γ −1 −Γ1 fˆ0 is self-adjoint. Moreover it is not and this gives (3.6). We have shown that A diﬃcult to see that A ∈ C(K × H) is an extension of A. is deﬁned by (3.5) then the boundary relation Γ coincides with the right If A hand side of (3.3). It was shown in step 2 of the proof of (i) that Γ is a boundary relation for S1+ (see step 1) and by step 3 and 4 the assertions (a) and (b) of (i) hold. We have proved Theorem 3.1.

Remark 3.2. The boundary relation Γ in (3.3) is called an induced boundary relation in [10, (4.6)] and [12, Section 6.2], where it appears as a composition of a boundary triplet and a [[·, ·]]-unitary relation. Then the statement in step 2 of the proof of Theorem 3.1 follows immediately from general properties of isometric and unitary relations in Krein spaces, cf. [10, Proposition 2.2 and Theorem 4.2] or [12, Theorems 2.13 and 6.3]. 3.2. The case of a Hilbert space as exit space We are now concerned with the situation that the exit space H is a Hilbert space. Under this additional assumption assertion (a) of the previous theorem can be improved. We note that the statements in Theorem 3.3 below are known. A more general result of very similar type has already been proved by V. Derkach with diﬀerent methods in [7, Theorem 4.1]. Theorem 3.3. Let K be a Krein space and let H be a Hilbert space, let A ∈ C(K) + be a symmetric relation and let {G, Γ0 , Γ1 } be a boundary triplet for A with corresponding γ-field γ and Weyl function M . Let A0 = ker Γ0 and assume that ρ(A0 ) is nonempty. ∈ C(K × H) is a self-adjoint extension of A and PK (A − λ0 )−1 K ∈ L(K) (i) If A holds for some λ0 ∈ ρ(A0 ), then there exists a Nevanlinna family τ ∈ R(G) such that (a) and (b) hold. (a) If λ ∈ ρ(A0 ), then (M (λ) + τ (λ))−1 ∈ L(G) if and only if λ ∈ ρ(A). (b) The formula − λ −1 K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK A holds for all λ ∈ ρ(A0 ) ∩ ρ(A). (ii) If τ ∈ R(G) is a Nevanlinna family and (M (λ0 ) + τ (λ0 ))−1 ∈ L(G) for some ∈ C(K × H) of A such λ0 ∈ ρ(A0 ), then there exists a self-adjoint extension A that (a) and (b) are satisfied. Proof. Since the class of Nevanlinna families R(G) coincides with the class of Weyl families of boundary relations in Hilbert spaces (see [11, Theorem 3.9], Remark 2.5 and Corollary 4.4) the assertions of Theorem 3.3 follow immediately

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from Theorem 3.1 if we show that each point λ ∈ ρ(A0 ) with the property (M (λ) + τ (λ))−1 ∈ L(G) belongs to ρ(A). 2 , G 2 ) as in (3.3), let τ and γ be the corresponding Weyl famDeﬁne Γ ∈ C(H ily and γ-ﬁeld, respectively, and assume that λ ∈ ρ(A0 ) is chosen such that − λ is injective, let (M (λ) + τ (λ))−1 ∈ L(G). In order to show that A

f f0 such that f − λf0 = 0 and f − λf1 = 0, , 0 ∈A 0 1 f1 f1 in the form (3.5) and setting h := Γ0 fˆ0 i.e., fˆ0 ∈ Nλ,A+ and fˆ1 ∈ Nλ,T1 . Writing A we conclude h Γ0 fˆ0 = ∈ Γ (Nλ,T1 ) = τ (λ) −M (λ)h −Γ1 fˆ0 and therefore h0 ∈ M (λ) + τ (λ). From (M (λ) + τ (λ))−1 ∈ L(G) we now get h = 0 and since both γ(λ) and γ (λ) are operators (cf. Remark 2.3) here we obtain − λ) = {0}. f0 = γ(λ)h = 0 and f1 = γ (λ)h = 0, that is, ker(A − λ we construct elements ˆf0 ∈ A+ and In order to show the surjectivity of A ˆf1 ∈ T1 with

ˆ f0 − λf0 g ˆf1 , Γ0 f0 and ∈ Γ = 0 (3.8) f1 − λf1 g1 −Γ1ˆf0 for an arbitrary gg01 ∈ K × H. First of all choose fˆ0 ∈ A0 such that f0 − λf0 = g0 and set x := Γ1 fˆ0 . Since J (Γ ) (cf. Section 2.2) is self-adjoint in the Hilbert space H × G there exists fˆ1 , ˆ h ∈ Γ with f1 − λf1 = g1 . Let now fˆλ ∈ Nλ,A+ such that Γ0 fˆλ = h (and hence Γ1 fˆλ = M (λ)h) holds and since M (λ) + τ (λ) is surjective by assumption there exists ˆl ∈ τ (λ) with M (λ)l + l = −(h + M (λ)h + x). Therefore there are fˆ1λ ∈ Nλ,T1 and fˆ0λ ∈ Nλ,A+ such that fˆ1λ , ˆl ∈ Γ and l = Γ0 fˆ0λ (and hence Γ1 fˆ0λ = M (λ)l). Setting ˆf0 := fˆ0 + fˆλ + fˆ0λ ∈ A+ and ˆf1 := fˆ1 + fˆ1λ ∈ T1 , ˆ + ˆl} ∈ Γ and from we have {ˆf1 , h ˆ Γ0 f0 g0 f0 − λf0 ˆ + ˆl = and =h f1 − λf1 g1 −Γ1ˆf0 − λ is surjective. we conclude that (3.8) holds, that is, A

In Theorem 3.5 below we will impose additional conditions on the symmetric relation A ∈ C(K) and the ﬁxed canonical self-adjoint extension A0 = ker Γ0 in order to get more information on the (local) spectral properties of the extensions For this we brieﬂy recall the notion of locally deﬁnitizable self-adjoint relations. A. For a detailed study of (locally) deﬁnitizable self-adjoint operators and relations we refer to the papers [18, 21, 28] of P. Jonas and H. Langer.

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If A0 is a self-adjoint relation in a Krein space K, then λ ∈ C belongs to the approximate point spectrum of A0 , denoted by σap (A0 ), if there exists a sequence xn , ∈ A yn 0 n = 1, 2, . . . , such that xn = 1 and limn→∞ yn − λxn = 0. The extended approximate point spectrum σ ap (A0 ) of A0 is deﬁned by σap (A0 ) if −1 0 ∈ σap (A0 ) and by σap (A0 ) ∪ {∞} otherwise. A point λ ∈ σap (A0 ) is said to be of positive type (negative type) with respect to A0 , if for every sequence xynn ∈ A0 , n = 1, 2 . . . , with xn = 1, limn→∞ yn − λxn = 0 we have lim inf [xn , xn ]K > 0 resp. lim sup [xn , xn ]K < 0, . n→∞

n→∞

If ∞ ∈ σ ap (A0 ), ∞ is said to be of positive type (negative type) with respect to A0 if 0 is of positive type (negative type, respectively) with respect to A−1 0 . The set of all spectral points of positive type (negative type) with respect to A0 will be denoted by σ++ (A0 ) (resp. σ−− (A0 )). An open subset ∆ of R is said to be of positive type (negative type) with respect to A0 if ∆ ∩ σ (A0 ) ⊂ σ++ (A0 ) (∆ ∩ σ (A0 ) ⊂ σ−− (A0 ), respectively) holds. Let in the following Ω be some domain in C symmetric with respect to the real axis such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. Definition 3.4. A self-adjoint relation A0 in a Krein space K is said to be deﬁnitizable over Ω if σ(A0 ) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0 , no point of Ω∩R is an accumulation point of the nonreal spectrum of A0 in Ω and the following holds. (i) For every finite union ∆, ∆ ⊂ Ω ∩ R, of open connected subsets there exists m ≥ 1, M > 0 and an open neighborhood U of ∆ in Ω such that

(A0 − λ)−1 ≤ M (1 + |λ|)2m−2 |Im λ|−m holds for all λ ∈ U\R. (ii) Every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that each component of Iµ \{µ} is either of positive or of negative type with respect to A0 . Let A0 be deﬁnitizable over Ω and let e be a discrete (possibly empty) set of points in Ω ∩ R. Then the property that (Ω ∩ R)\e is of positive type with respect to A0 is equivalent to the fact that σ−− (A0 ) is discrete in Ω. Theorem 3.5. Let K be a Krein space and let H be a Hilbert space, let A ∈ C(K) be a symmetric relation of finite defect and let {G, Γ0 , Γ1 } be a boundary triplet for A+ with corresponding γ-field γ and Weyl function M . Assume that A0 = ker Γ0 is definitizable over Ω and that σ−− (A0 ) is discrete in Ω. − λ0 )−1 K ∈ L(K) ∈ C(K × H) is a self-adjoint extension of A and PK (A (i) If A holds for some λ0 ∈ ρ(A0 )∩Ω, then there exists a Nevanlinna family τ ∈ R(G) such that (a)–(c) are satisfied. (a) If λ ∈ ρ(A0 ) ∩ Ω, then (M (λ) + τ (λ))−1 ∈ L(G) if and only if λ ∈ ρ(A).

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(b) The formula − λ −1 K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK A ∩ Ω. holds for all λ ∈ ρ(A0 ) ∩ ρ(A) is discrete ∩ Ω = ∅, then A is definitizable over Ω and σ−− (A) (c) If ρ(A) in Ω. (ii) If τ ∈ R(G) is a Nevanlinna family and (M (λ0 ) + τ (λ0 ))−1 ∈ L(G) for some ∈ C(K × H) of A λ0 ∈ ρ(A0 ) ∩ Ω, then there exists a self-adjoint extension A such that (a)–(c) are satisfied. Proof. The statement of Theorem 3.5 follows from Theorem 3.3 if we show that in (i) satisﬁes assertion (c). the extension A For this, let S0 and S1 be the symmetric relations in the Krein space K and the Hilbert space H, respectively, deﬁned in step 1 of the proof of Theorem 3.1. As A is of ﬁnite defect the deﬁciency indices n± (JK A) of the symmetric relation JK A in the Hilbert space (K, [JK ·, ·]) are both equal to n < ∞ and hence the deﬁciency indices n± (JK S0 ) of the symmetric relation JK S0 are both equal to m ≤ n. Considerations very similar to those in [11, Lemma 2.14] show that the deﬁciency indices n± (S1 ) of S1 coincide and are also equal to m. Let B0 be a self-adjoint extension of S1 in the Hilbert space H. We claim × H) is deﬁnitizable over Ω and that the self-adjoint relation A0 × B0 ∈ C(K σ−− (A0 × B0 ) is discrete in Ω. In fact, ﬁrst of all σ(A0 × B0 ) ∩ (Ω\R) coincides with σ(A0 ) ∩ (Ω\R) and the growth properties of the resolvent of A0 and B0 ,

(B0 − λ)−1 ≤ |Im λ|−1 ,

λ ∈ R,

imply that condition (i) in Deﬁnition 3.4 holds for the relation A0 × B0 . Moreover R ⊂ σ++ (B0 ) ∪ ρ(B0 ) and the assumptions that A0 is deﬁnitizable over Ω and σ−− (A0 ) is discrete in Ω imply that Ω ∩ R with the possible exception of a discrete set belongs to σ++ (A0 × B0 ) ∪ ρ(A0 × B0 ). Therefore A0 × B0 is deﬁnitizable over Ω and σ−− (A0 × B0 ) is discrete in Ω. and A0 × B0 are self-adjoint extensions of the symmetric relation Since A ∩ Ω is nonempty we conclude that S0 × S1 in K × H and ρ(A) −1 −1 ∩ ρ(A0 × B0 ) ∩ Ω, −λ − (A0 × B0 ) − λ , λ ∈ ρ(A) A is a ﬁnite rank operator. Hence we can apply [2, Theorem 2.2] and it follows that is deﬁnitizable over Ω and σ−− (A) is discrete in Ω. A

4. Realization of relation-valued functions as Weyl families We show in Theorem 4.1 that certain classes of C(G)-valued functions can be 2, G2) realized as Weyl families corresponding to boundary relations Γ ∈ C(K

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of symmetric relations in Krein spaces K. Lemma 2.4 (i) and the proof of [11, function τ the function λ → Theorem 3.9] suggest that for a given C(G)-valued −1 −(τ (λ) + λ) has to be realized as the compressed resolvent of some self-adjoint relation J (Γ) in K × G. We brieﬂy recall the notion of (locally) deﬁnitizable functions introduced and studied by P. Jonas in [19, 20, 21, 22]. Let, as in Deﬁnition 3.4, Ω be a domain which is symmetric with respect to the real axis such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. For an L(G)-valued function G meromorphic in Ω\R we denote by h(G) the union of the set of all points of holomorphy of G in Ω\R and the set of all points in Ω ∩ R into which G can be analytically continued in a unique way. An L(G)-valued function G meromorphic in C\R satisfying G(λ) = G(λ)∗ for all λ ∈ C\R is called definitizable if there exists a scalar rational function r such that rG is the sum of a Nevanlinna function N and an L(G)-valued rational function n whose poles belong to h(G), r(λ)G(λ) = N (λ) + n(λ) for all points λ ∈ C\R of holomorphy of rG, cf. [20, §3]. If Ω is a domain as above, then an L(G)-valued function G meromorphic in Ω\R and satisfying G(λ) = G(λ)∗ for all λ ∈ Ω\R is said to be locally definitizable in Ω, if for every domain Ω with the same properties as Ω, Ω ⊂ Ω, G can be written as the sum Gd + Gh of a deﬁnitizable function Gd and a function Gh locally holomorphic on Ω (see [22]). Theorem 4.1. Let G be a Hilbert space, let τ be an C(G)-valued family and assume that the function λ → G(λ) := −(τ (λ) + λ)−1 is an L(G)-valued locally definitizable function in Ω. Then for every domain Ω with the same properties as Ω, Ω ⊂ Ω, there exists a Krein space K, a closed 2 , G 2 ) for A+ such symmetric relation A ∈ C(K) and a boundary relation Γ ∈ C(K that the corresponding Weyl family coincides with τ in Ω ∩ h(G). Proof. Let us ﬁx some domain Ω , Ω ⊂ Ω, and a point λ0 ∈ Ω ∩ h(G). Since G is a deﬁnitizable function in Ω by [1] the same holds for the function G1 (λ) := λ − Re λ0 + (λ − λ0 )(λ − λ0 )G(λ),

λ ∈ Ω ∩ h(G).

(4.1)

a self-adjoint Hence [22, Theorem 3.8] implies that there exists a Krein space K, ∈ C( K) deﬁnitizable over Ω, and a mapping γ ∈ L(G, K) such that relation B Ω ∩ h(G1 ) = Ω ∩ ρ(B) and − λ)−1 γ (4.2) G1 (λ) = Re G1 (λ0 ) + γ + λ − Re λ0 + (λ − λ0 )(λ − λ0 )(B holds for all λ ∈ Ω ∩ h(G1 ). By (4.1) we have G1 (λ0 ) = iIm λ0 and Re G1 (λ0 ) = 0 which together with (4.2) yields γ + γ = IG . Since γγ + is a self-adjoint projection we can identify G with the Hilbert subspace ran γ in K. Then in the Krein space K [⊥] ˙ the orthogonal companion K := G of G in K is a Krein space and K = K[+]G

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and γ + the projection PG in K holds. Moreover γ is the embedding of G into K onto G. Hence (4.2) can be rewritten as − λ)−1 G G1 (λ) = λ − Re λ0 + (λ − λ0 )(λ − λ0 )PG (B and taking into account (4.1) we conclude − λ)−1 G , −(τ (λ) + λ)−1 = G(λ) = PG (B

λ ∈ Ω ∩ h(G).

Then Γ is a boundary relation for Let J be as in (2.4) and deﬁne Γ := J −1 (B). + A , A := ker Γ, and by (2.5) the associated Weyl family is τ . = J (Γ) corresponding to the boundary Remark 4.2. The self-adjoint relation B relation Γ (with Weyl family τ ) in the proof of Theorem 4.1 is definitizable over Ω . A converse statement also holds, that is, if Γ is a boundary relation such that J (Γ) is definitizable over Ω, then the corresponding Weyl family meets the assumptions of Theorem 4.1. By virtue of [1, Theorem 2.5] we immediately obtain the following corollary for matrix-valued locally deﬁnitizable functions. Corollary 4.3. Let τ be a matrix-valued definitizable function in Ω and assume that det (τ (λ) + λ) is not identically equal to zero. Then for every domain Ω as Ω, Ω ⊂ Ω, there exists a Krein space K, a closed symmetric relation A in K and 2 , G 2 ) such that the corresponding Weyl family coincides boundary relation Γ ∈ C(K with τ in Ω . The argument in the proof of Theorem 4.1 and a well-known representation result for L(G)-valued Nevanlinna functions yield an alternative proof of the main realization theorem in [11]. Corollary 4.4. Every Nevanlinna family τ ∈ R(G) can be realized as the Weyl 2 2 family of a boundary relation Γ ∈ C(K , G ), where K is a Hilbert space. Proof. As the sum and the negative inverse of a Nevanlinna family are Nevanlinna families G1 (λ) : = λ − (λ2 + 1)(τ (λ) + λ)−1 −1 −1 −1 = − − λ − τ (λ)−1 − τ (λ) − λ−1 , λ ∈ C\R, is an L(G)-valued Nevanlinna function. Hence there exists a Hilbert a self-adjoint relation B ∈ C( K) and an operator γ ∈ L(G, K) such that a space K, representation of the form (4.2) with λ0 = i holds, see e.g. [24] or [17, Theorem 4.2]. Now the same reasoning as in the proof of Theorem 4.1 shows that Γ = J −1 (B) is a boundary relation with Weyl family τ . Acknowledgement The authors are grateful to P. Jonas for fruitful discussions and valuable remarks.

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References [1] T.Ya. Azizov, P. Jonas: On locally deﬁnitizable matrix functions, Preprint 21 (2005), Preprint Series TU Berlin. [2] J. Behrndt: Finite rank perturbations of locally deﬁnitizable self-adjoint operators in Krein spaces, to appear in J. Operator Theory. [3] J. Behrndt, A. Luger, C. Trunk: Generalized resolvents of a class of symmetric operators in Krein spaces, Operator Theory: Advances and Applications 175, Birkh¨ auser Verlag Basel (2007), 13–32. [4] J. Behrndt, C. Trunk: On generalized resolvents of symmetric operators of defect one with ﬁnitely many negative squares, Proceedings AIT Conference, University of Vaasa, Finland, 124 (2005), 21–30. [5] V.A. Derkach: On Weyl function and generalized resolvents of a hermitian operator in a Krein space, Integral Equations Operator Theory 23 (1995), 387–415. [6] V.A. Derkach: On Krein space symmetric linear relations with gaps, Methods Funct. Anal. Topology 4 (1998), 16–40. [7] V.A. Derkach: On generalized resolvents of hermitian relations in Krein spaces, J. Math. Sci. (New York) 97 (1999), 4420–4460. [8] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo: Generalized resolvents of symmetric operators and admissibility, Methods Funct. Anal. Topology 6 (2000), 24–53. [9] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo: Boundary relations and their Weyl families, Rus. Doklady Acad. Sci. 399, No. 2 (2004), 151–156. [10] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo: Boundary relations and orthogonal coupling of symmetric operators, Proceedings AIT Conference, University of Vaasa, Finland, 124 (2005), 41–56. [11] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo: Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358, No. 12 (2006), 5351–5400. [12] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo: Boundary relations and generalized resolvents of symmetric operators, published in Mathematics ArXiv: http://arxiv.org/abs/math.SP/0610299 (2006). [13] V.A. Derkach, M.M. Malamud: Generalized resolvents and the boundary value problems for hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1–95. [14] V.A. Derkach, M.M. Malamud: The extension theory of hermitian operators and the moment problem, J. Math. Sci. (New York) 73 (1995), 141–242. [15] A. Dijksma, H.S.V. de Snoo: Symmetric and self-adjoint relations in Krein spaces I, Operator Theory: Advances and Applications 24, Birkh¨ auser Verlag Basel (1987), 145–166. [16] V.I. Gorbachuk, M.L. Gorbachuk: Boundary Value Problems for Operator Diﬀerential Equations, Kluwer Academic Publishers, Dordrecht (1991). [17] S. Hassi, H.S.V. de Snoo, H. Woracek: Some interpolation problems of NevanlinnaPick type. The Krein-Langer method, Operator Theory: Advances and Applications 106, Birkh¨ auser Verlag Basel (1998), 201–216.

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[18] P. Jonas: On a class of unitary operators in Krein space. Advances in Invariant Subspaces and Other Results of Operator Theory, Operator Theory: Advances and Applications 17, Birkh¨ auser Verlag Basel (1986), 151–172. [19] P. Jonas: A class of operator-valued meromorphic functions on the unit disc, Ann. Acad. Scie. Fenn. Math. 17 (1992), 257–284. [20] P. Jonas: Operator representations of deﬁnitizable functions, Ann. Acad. Scie. Fenn. Math. 25 (2000), 41–72. [21] P. Jonas: On locally deﬁnite operators in Krein spaces, in: Spectral Theory and Applications, Theta Foundation (2003), 95–127. [22] P. Jonas: On operator representations of locally deﬁnitizable functions, Operator Theory: Advances and Applications 162, Birkh¨ auser Verlag Basel (2005), 165–190. [23] M.G. Krein, H. Langer: On defect subspaces and generalized resolvents of Hermitian operators in Pontryagin spaces, Funktsional. Anal. i Prilozhen. 5 No. 2 (1971) 59–71; 5 No. 3 (1971) 54–69 (Russian); English transl.: Funct. Anal. Appl. 5 (1971/1972), 139–146, 217–228. ¨ [24] M.G. Krein, H. Langer: Uber einige Fortsetzungsprobleme, die eng mit der Theorie angen. I. Einige Funktionenhermitescher Operatoren im Raume Πκ zusammenh¨ klassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236. [25] M.G. Krein: On hermitian operators with defect-indices equal to unity, Dokl. Akad. Nauk SSSR, 43 (1944), 339–342. [26] M.G. Krein: On the resolvents of an hermitian operator with defect-index (m, m), Dokl. Akad. Nauk SSSR, 52 (1946), 657–660. [27] H. Langer: Verallgemeinerte Resolventen eines J-nichtnegativen Operators mit endlichem Defekt, J. Funct. Anal. 8 (1971), 287–320. [28] H. Langer: Spectral functions of deﬁnitizable operators in Krein spaces, Functional Analysis Proceedings of a Conference held at Dubrovnik, Yugoslavia, November 214, 1981, Lecture Notes in Mathematics 948, Springer Verlag Berlin-Heidelberg-New York (1982), 1–46. [29] H. Langer, B. Textorius: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space, Paciﬁc J. Math. 72 (1977), 135–165. [30] M.M. Malamud: On a formula for the generalized resolvents of a non-densely deﬁned Hermitian Operator, Ukrain. Mat. Zh. 44 (1992), 1658–1688 (Russian); English transl.: Ukrainian Math. J. 44 (1993), 1522–1547. [31] M.A. Naimark: On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR, Ser. Matem. 7 (1943), 373–375. Jussi Behrndt and Hans-Christian Kreusler Institut f¨ ur Mathematik TU Berlin Straße des 17. Juni 136, D-10623 Berlin, Germany e-mail: [email protected] [email protected] Submitted: September 12, 2006 Revised: August 31, 2007

Integr. equ. oper. theory 59 (2007), 329–343 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030329-15, published online October 18, 2007 DOI 10.1007/s00020-007-1534-9

Integral Equations and Operator Theory

Some New Observations on Interpolation in the Spectral Unit Ball Gautam Bharali Abstract. We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ωn , n ≥ 2. We begin by showing that a known necessary condition for the existence of a O(D; Ωn )-interpolant (D here being the unit disc in C), given that the matricial data are non-derogatory, is not suﬃcient. We provide next a new necessary condition for the solvability of the two-point interpolation problem – one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ωn , n ≥ 2. Mathematics Subject Classification (2000). Primary 30E05, 47A56; Secondary 32F45. Keywords. Complex geometry, Carath´eodory metric, minimial polynomial, Schwarz lemma, spectral radius, spectral unit ball.

1. Introduction and Statement of Results The interpolation problem referred to in the title, and which links the assorted results of this paper, is the following (D here will denote the open unit disc centered at 0 ∈ C): (*) Given M distinct points ζ1 , . . . , ζM ∈ D and matrices W1 , . . . , WM in the spectral unit ball Ωn := {W ∈ Mn (C) : r(W ) < 1}, ﬁnd conditions on {ζ1 , . . . , ζM } and {W1 , . . . , WM } such that there exists a holomorphic map F : D −→ Ωn satisfying F (ζj ) = Wj , j = 1, . . . , M . In the above statement, r(W ) denotes the spectral radius of the n × n matrix W . This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV.

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Under a very slight simpliﬁcation – i.e., that the interpolant F in (*) is required to satisfy supζ∈D r(F (ζ)) < 1 – the paper [5] provides a characterisation of the interpolation data ((ζ1 , W1 ), . . . , (ζM , WM )) that admit an interpolant of the type described. However, this characterisation involves a non-trivial search over a region 2 in Cn M . There is, therefore, interest in ﬁnding alternative characterisations that either: a) circumvent the need to perform a search; or b) reduce the dimension of the search-region. In this regard, a new idea idea was introduced by Agler & Young in the paper [1]. This idea was further developed over several works – notably in [2], in the papers [7] and [8] by Costara, and in David Ogle’s thesis [13]. It can be summarised in two steps as follows: • If the matrices W1 , . . . , WM are all non-derogatory, then (*) is equivalent to an interpolation problem in the symmetrized polydisc Gn , n ≥ 2, which is deﬁned as n (−1)j sj z n−j = 0 lie in D . Gn := (s1 , . . . , sn ) ∈ Cn : all roots of z n + j=1

• The Gn -interpolation problem is shown to share certain aspects of the classical Nevanlinna-Pick problems, either by establishing conditions for a von Neumann inequality for Gn – note that Gn is compact – or through function theory. It would be useful, at this stage, to recall the following Definition 1.1. A matrix A ∈ Mn (C) is said to be non-derogatory if the geometric multiplicity of each eigenvalue of A is 1 (regardless of its algebraic multiplicity). The matrix A being non-derogatory is equivalent to A being similar to the companion matrix of its characteristic polynomial – i.e., if z n + nj=1 sj z n−j is the characteristic polynomial, then   0 −sn  1 0 −sn−1    A is non-derogatory ⇐⇒ A is similar to  . .. .. ..   . . . 0 1 −s1 n×n The Agler-Young papers treat the case n = 2, while the last two works cited above consider the higher-dimensional problem. The reader is referred to [2] for a proof of the equivalence of (*), given non-derogatory matricial data, and the appropriate Gn -interpolation problem. The similarity condition given in Deﬁnition 1.1 is central to establishing this equivalence. Before presenting the ﬁrst result of this paper, we need to examine what is known about (*) from the perspective of the Gn -interpolation problem. Since we would like to focus on the matricial interpolation problem, we will paraphrase the results from [13] and [8] in the language of non-derogatory matrices. Given

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an matrix W , let its characteristic polynomial χW (z) = z n + nn × n jcomplexn−j , and deﬁne the rational function j=1 (−1) sj (W )z n j j−1 j=1 jsj (W )(−1) z . f(z; W ) := n−1 j j j=0 (n − j)sj (W )(−1) z Then, the most general statement that is known about (*) is: Result 1.2 (paraphrased from [13] and [8]). Let ζ1 , . . . , ζM be M distinct points in D and let W1 , . . . , WM ∈ Ωn be non-derogatory matrices. If there exists a map F ∈ O(D, Ωn ) such that F (ζj ) = Wj , j = 1, . . . , M , then the matrices M 1 − f(z; Wj )f(z; Wk ) ≥ 0 for each z ∈ D. (1.1) 1 − ζj ζk j,k=1

Here, and elsewhere in this paper, given two complex domains X and Y , O(X; Y ) will denote the class of all holomorphic maps from X into Y . Remark 1.3. The matrices in (1.1) may appear diﬀerent from those in [13, Corollary 5.2.2], but the latter are, in fact, ∗-congruent to the matrices above. Even though Result 1.2 provides only a necessary condition, (1.1) is more tractable for small values of M than the Bercovici-Foias-Tannenbaum condition. Its viability as a suﬃcient condition, at least for small M , has been discussed in both [13] and [8]. This is reasonable because the latter condition is suﬃcient when n = 2 and M = 2 (and the given matrices are, of course, non-derogatory); see [4]. Given all these developments, it seems appropriate to begin with the following: Observation 1.4. When n ≥ 3, the condition (1.1) is not suﬃcient for the existence of a O(D; Ωn )-interpolant for the prescribed data ((ζ1 , W1 ), . . . , (ζM , WM )), where each Wj ∈ Ωn , j = 1, . . . , M , is non-derogatory. The above observation relies on ideas from complex geometry; speciﬁcally – estimates for invariant metrics on the symmetrized polydisc Gn , n ≥ 3. Our argument follows from a recent study [11] of the Carath´eodory metric on Gn , n ≥ 3. This argument is presented in the next section. Observation 1.4 takes us back to the drawing board when it comes to realising goals of the type (a) or (b) (as in the opening paragraph) to determine whether a O(D; Ωn )-interpolant exists for a given data-set. Thus, new conditions that are inequivalent to (1.1) are desirable for the same reasons as those oﬀered in [2] and [3]. To wit: all extant approaches to implementing the Bercovici-FoiasTannenbaum solution of (*) are computational, and rely upon various search algorithms. Rigorous analytical results, even if they only indicate when a data-set ((ζ1 , W1 ), . . . , (ζM , WM )) does not admit an O(D; Ωn ) interpolant – i.e., necessary conditions – provide tests of existing algorithms/software and illustrate the complexities of (*). We will say more about this; but ﬁrst – notations for our

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next result. Given z1 , z2 ∈ D, the pseudohyperbolic distance between these points, written MD (z1 , z2 ), is deﬁned as:

z1 − z2

∀z1 , z2 ∈ D. MD (z1 , z2 ) := 1 − z2 z1 We can now state our next result. Theorem 1.5. Let F ∈ O(D; Ωn ), n ≥ 2, and let ζ1 , ζ2 ∈ D. Write Wj = F (ζj ), and let σ(Wj ) := the set of eigenvalues of Wj , j = 1, 2 (i.e., elements of σ(Wj ) are not repeated according to multiplicity). If λ ∈ σ(Wj ), then let m(λ) denote the multiplicity of λ as a zero of the minimal polynomial of Wj . Then:     MD (µ, λ)m(λ) , max MD (λ, µ)m(µ) max max  µ∈σ(W2 ) λ∈σ(W1 ) λ∈σ(W1 ) µ∈σ(W2 )

ζ1 − ζ2

. (1.2)

≤ 1 − ζ2 ζ1 Referring back to our previous paragraph: one could ask whether Theorem 1.5 is able to highlight any complexities of (*) that Result 1.2 misses. There are two parts to the answer: 1) The Jordan structure of the data-set ((ζ1 , W1 ), (ζ2 , W2 )): Several well-known examples from [6] and [2] reveal that the existence of a O(D; Ωn )-interpolant, n ≥ 2, is sensitive to the Jordan structure of the matrices W1 , . . . , WM . However, to the best of our knowledge, there are no results in the literature to date that incorporate information on the Jordan structures or the minimal polynomials of W1 , . . . , WM . In contrast, the following example shows that information on minimal polynomials is vital – i.e., that with the correct information about the minimal polynomials of F (ζ1 ) and F (ζ2 ), condition (1.2) is sharp. Example 1.6. For n ≥ 3 and d Fd : D −→ Ωn by  0 1 0   .. ..  . . Fd (ζ) :=   1  

0

= 2, . . . , n − 1, deﬁne the holomorphic map ζ 0 .. .



0

0 ζIn−d

       

,

ζ ∈ D,

n×n

where In−d denotes the identity matrix of dimension n − d for 1 < d < n. Let ζ1 = 0 and ζ2 = ζ. One easily computes – in the notation of Theorem 1.5 –

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that: max

µ∈σ(W2 )

max

λ∈σ(W1 )

λ∈σ(W1 )

µ∈σ(W2 )

333

MD (µ, λ)m(λ) = |ζ|, MD (λ, µ)m(µ) = |ζ|2 ,

where the ﬁrst equality holds because W1 is nilpotent of order d. So, (1.2) is satisﬁed as an equality for the given choice of ζ1 and ζ2 – which is what was meant above by saying that (1.2) is sharp. 2) Comparison with (1.1): Theorem 1.5 would not be eﬀective in testing any of the existing algorithms used in the implementation of the Bercovici-FoiasTannenbaum solution to (*) if (1.1) were a universally stronger necessary condition than (1.2). However, (1.1) is devised with non-derogatory data in mind, whereas no simple interpolation condition was hitherto known for pairs of arbitrary matrices in Ωn . Hence, by choosing any one of W1 and W2 to be derogatory, one would like to examine how (1.1) and (1.2) compare. This leads to our next observation. Observation 1.7. For each n ≥ 3, we can ﬁnd a data-set ((ζ1 , W1 ), (ζ2 , W2 )) for which (1.2) implies that it cannot admit any O(D; Ωn )-interpolant, whereas (1.1) provides no information. An example pertinent to this observation is presented at the end of Section 3. As for Theorem 1.5, it may be viewed as a Schwarz lemma for mappings between D and the spectral unit ball. Note that the inequality (1.2) is preserved under automorphisms of D and under the “obvious” automorphisms of Ωn (the full automorphism group Aut(Ωn ), n ≥ 2, is not known). The proof of Theorem 1.5 is presented in Section 3. The key new idea in the proof of Theorem 1.5 – i.e., to focus on the minimal polynomial of certain crucial matrices that lie in the range of F – pays oﬀ in obtaining a result that is somewhat removed from the our main theme. The result in question is a generalisation of the following theorem of Ransford and White [14, Theorem 2]: G ∈ O(Ωn ; Ωn ) and G(0) = 0 =⇒ r(G(X)) ≤ r(X) ∀X ∈ Ωn .

(1.3)

One would like to generalise (1.3) in the way the Schwarz-Pick lemma generalises the Schwarz lemma for D – i.e., by formulating an inequality that is valid without assuming that the holomorphic mapping in question has a ﬁxed point. This generalisation is as follows: Theorem 1.8. Let G ∈ O(Ωn ; Ωn ), n ≥ 2, and deﬁne dG := the degree of the minimal polynomial of G(0). Then: r(G(X)) ≤

r(X)1/dG + r(G(0)) 1 + r(G(0))r(X)1/dG

∀X ∈ Ωn .

(1.4)

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Furthermore, the inequality (1.4) is sharp in the sense that there exists a nonempty set Sn ⊂ Ωn such that given any A ∈ Sn and d = 1, . . . , n, we can ﬁnd a GA,d ∈ O(Ωn ; Ωn ) such that dGA,d = d, and r(GA,d (A)) =

r(A)1/d + r(GA,d (0)) . 1 + r(GA,d (0))r(A)1/d

(1.5)

2. A Discussion of Observation 1.4 We begin this discussion with a couple of deﬁnitions from complex geometry. Given a domain Ω ⊂ Cn , the Carath´eodory pseudodistance between two points z1 , z2 ∈ Ω is deﬁned as cΩ (z1 , z2 ) := sup {pD (f (z1 ), f (z2 )) : f ∈ O(Ω; D)} , where pD is the Poincar´e distance on D (and pD is given by pD (ζ1 , ζ2 ) = tanh−1 (MD (ζ1 , ζ2 )) for ζ1 , ζ2 ∈ D). In the same setting, the Lempert functional on Ω × Ω, is deﬁned as κ Ω (z1 , z2 ) := inf {pD (ζ1 , ζ2 ) : ∃ψ ∈ O(D; Ω) and ζ1 , ζ2 ∈ D so that ψ(ζj ) = zj , j = 1, 2.} . It is not hard to show that the set on the right-hand side above is non-empty. The reader is referred to Chapter III of [10] for details. Next, we examine a few technical objects. For the remainder of this section, S = (s1 , . . . , sn ) will denote a point in Cn , n ≥ 2. For z ∈ D deﬁne the rational map fn (z; S) := ( s1 (z; S), . . . , sn−1 (z; S)), n ≥ 2, by sj (z; S) :=

(n − j)sj − z(j + 1)sj+1 , n − zs1

S ∈ Cn s.t. n − zs1 = 0, j = 1, . . . , (n − 1).

Next, deﬁne F (Z; ·) := f2 (z1 ; ·) ◦ · · · ◦ fn (zn−1 ; ·) ∀Z = (z1 , . . . , zn−1 ) ∈ D

n−1

,

where the second argument varies through that region in C where the right-hand side above is deﬁned. The connection of these objects with our earlier discussions is established via n j j−1 j=1 jsj (−1) z , z ∈ D, f (z; S) := n−1 j j j=0 (n − j)sj (−1) z n

and S varies through that region in Cn where the right-hand side above is deﬁned. Note the resemblance of f (z; S) to f(z; W ) deﬁned earlier. From Theorem 3.5 of [8], we excerpt:

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Result 2.1. Let S = (s1 , . . . , sn ) denote a point in Cn . Then: 1) f (z; ·) = F (z, . . . , z; ·) ∀z ∈ D, wherever deﬁned. 2) S ∈ Gn if and only if supz∈D |f (z; S)| < 1, n ≥ 2. 3) If S ∈ Gn , n ≥ 2, then sup |f (z; S)| = z∈D

sup |F (Z; S)|. Z∈D

n−1

For convenience, let us refer to the Carath´eodory pseudodistance on Gn , n ≥ 2, by cn . Next, deﬁne – here we refer to Section 2 of [11] – the following distance function on Gn pn (S, T ) :=

max

Z∈(∂D)n−1

pD (F (Z; S), F (Z; T )) ∀S, T ∈ Gn .

(2.1)

This is the distance function – whose properties have been studied in [11] – we shall exploit to support Observation 1.4. The well-deﬁnedness of the right-hand side above follows from parts (2) and (3) of Result 2.1 above. Furthermore, since n−1 F (Z; S), F (Z; T ) ∈ D for each Z ∈ D whenever S, T ∈ Gn , n ≥ 2, it follows simply from the deﬁnition that cn (S, T ) ≥ pn (S, T ) ∀S, T ∈ Gn .

(2.2)

Since we have now adopted certain notations from [11], we must make the following Note. We have opted to rely on the notation of [8]. This leads to a slight discrepancy between our deﬁnition of pn in (2.1) and that in [11]. This discrepancy is easily reconciled by the observation that F (·; S) used here and in [8] will have to be read as F (·; −s1 , s2 , . . . , (−1)n sn ) in [11]. This is harmless because S ∈ Gn ⇐⇒ (−s1 , s2 , . . . , (−1)n sn ) ∈ Gn . Let us now refer back to the condition (1.1) with M = 2. An easy calculation involving 2 × 2 matrices reveals that

f(z; W ) − f(z; W )

ζ − ζ

1 2 1 2 When M = 2, (1.1) ⇐⇒ sup

≤

.

1 − ζ ζ 1 − f(z; W )f(z; W ) 2 1 2 1 z∈D If W1 is nilpotent of order n (recall that all matrices occuring in (1.1) are nonderogatory), then f(·; W1 ) ≡ 0. Of course, W2 ∈ Ωn implies (s1 (W2 ), . . . , sn (W2 )) ∈ Gn . By part (2) of Result 2.1, f(z; W2 ) ∈ D ∀z ∈ D. This leads to the following key fact: When M = 2 and W1 is nilpotent of order n, (1.1) ⇐⇒ sup tanh−1 |f(z; W2 )| = sup pD (0, f(z; W2 )) ≤ pD (ζ1 , ζ2 ). (2.3) z∈D

z∈D

We now appeal to Proposition 2 in [11], i.e., pn (0, ·) = cn (0, ·) for each n ≥ 3. Let us now ﬁx n ≥ 3. Let S0 ∈ Gn \ {0} be such that cn (0, S0 ) > pn (0, S0 ). Let

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ε0 > 0 be such that cn (0, S0 ) = pn (0, S0 ) + 2ε0 . Let us write S0 = (s0,1 , . . . , s0,n ) and choose two matrices W1 , W2 ∈ Ωn as follows:   0 (−1)n−1 s0,n  1 0 (−1)n−2 s0,n−1    W1 = a nilpotent of order n, W2 =  ,  .. .. ..   . . . 0 1 s0,1 n×n n i.e., W2 is the companion matrix of the polynomial z n + j=1 (−1)j s0,j z n−j . We emphasize the following facts that follow from this choice of W1 and W2 f(·, W1 ) = f (·; 0, . . . , 0) ≡ 0, f(·, W2 ) = f (·; S0 ),

(2.4)

W1 and W2 are, by construction, non-derogatory. The relations in (2.4) are cases of a general correspondence between matrices in Ωn and points in Gn , given by the surjective, holomorphic map Πn : Ωn −→ Gn , where Πn (W ) := (s1 (W ), . . . , sn (W )), and sj (W ), j = 1, . . . n, are as deﬁned in the beginning of this article. Let us pick two distinct points ζ1 , ζ2 ∈ D such that pD (ζ1 , ζ2 ) − ε0 < pn (0, S0 ) ≤ pD (ζ1 , ζ2 ).

(2.5)

Assume, now, that (1.1) is a suﬃcient condition for the existence of a O(D; Ωn )interpolant. Then, in view of the choices of W1 , W2 , the second inequality in (2.5), and (2.4) we get sup tanh−1 |f(z; W2 )| = sup tanh−1 |f(z; W2 )| = pn (0, S0 ) ≤ pD (ζ1 , ζ2 ). (2.6) z∈D

z∈∂D

The ﬁrst equality in (2.6) is a consequence of part (2) of Result 2.1: since S0 ∈ Gn , the rational function f(·; W2 ) = f (·; S0 ) ∈ O(D) C(D), whence the equality follows from the Maximum Modulus Theorem. But now, owing to the equivalence (2.3), the estimate (2.6) implies, by assumption, that there exists an interpolant F ∈ O(D; Ωn ) such that F (ζj ) = Wj , j = 1, 2. Then, Πn ◦ F : D −→ Gn satisﬁes Πn ◦ F (ζ1 ) = 0 and Πn ◦ F (ζ2 ) = S0 . Then, by the deﬁnition of the Lempert κn ) functional (for convenience, we denote the Lempert functional of Gn by κ n (0, S0 ) ≤ pD (ζ1 , ζ2 ) < pn (0, S0 ) + ε0 < cn (0, S0 ).

(from (2.5), 1st part) (by deﬁnition of ε0 )

But, for any domain Ω, the Carath´eodory pseudodistance and the Lempert function always satisfy cΩ ≤ κ Ω . Hence, we have just obtained a contradiction. Hence our assumption that (1.1) is suﬃcient for the existence of an O(D, Ωn )-interpolant, for n ≥ 3, must be false.

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3. The Proof of Theorem 1.5 The proofs in this section depend crucially on a theorem by Vesentini. The result is as follows: Result 3.1 (Vesentini, [15]). Let A be a complex, unital Banach algebra and let r(x) denote the spectral radius of any element x ∈ A. Let f ∈ O(D; A). Then, the function ζ −→ r(f (ζ)) is subharmonic on D. The following result is the key lemma of this section. The proof of Theorem 1.5 is reduced to a simple application of this lemma. The structure of this proof is reminiscent of [12, Theorem 1.1]. This stems from the manner in which Vesentini’s theorem is used. The essence of the trick below goes back to Globevnik [9]. The reader will notice that Theorem 1.5 specialises to Globevnik’s Schwarz lemma when W1 = 0. Lemma 3.2. Let F ∈ O(D; Ωn ). For each λ ∈ σ(F (0)), deﬁne m(λ) :=the multiplicity of λ as a zero of the minimal polynomial of F (0). Deﬁne the Blaschke product ζ − λ m(λ) B(ζ) := , ζ ∈ D. 1 − λζ λ∈σ(F (0)) Then |B(µ)| ≤ |ζ| ∀µ ∈ σ(F (ζ)). on Ωn : for any matrix Proof. The Blaschke product B induces a matrix function B A ∈ Ωn , we set B(A) := (I − λA)−m(λ) (A − λI)m(λ) , λ∈σ(F (0))

which is well-deﬁned on Ωn because whenever λ = 0, (I − λA) = λ(I/λ − A) ∈ GL(n, C). Furthermore, since ζ −→ (ζ−λ)/(1−λζ), |λ| < 1, has a power-series expansion that converges uniformly on compact subsets of D, it follows from standard arguments that σ(B(A)) = {B(µ) : µ ∈ σ(A)} for any A ∈ Ωn . (3.1) ◦ F (0) = 0. Since B ◦ F (0) = 0, By the deﬁnition of the minimal polynomial, B there exists a holomorphic map Φ ∈ O(D; Mn (C)) such that B ◦ F (ζ) = ζΦ(ζ). Note that ◦ F (ζ)) = σ(ζΦ(ζ)) = ζσ(Φ(ζ)) ∀ζ ∈ D. σ(B (3.2) ◦ F (ζ)) ⊂ D, the above equations give us: Since σ(B r(Φ(ζ)) < 1/R ∀ζ : |ζ| = R, R ∈ (0, 1).

(3.3)

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Taking A = Mn (C) in Vesentini’s theorem, we see that ζ −→ r(Φ(ζ)) is subharmonic on the unit disc. Applying the Maximum Principle to (3.3) and taking limits as R −→ 1− , we get r(Φ(ζ)) ≤ 1 ∀ζ ∈ D. (3.4) In view of (3.1), (3.2) and (3.4), we get |B(µ)| ≤ |ζ|r(Φ(ζ)) ≤ |ζ| ∀µ ∈ σ(F (ζ)).

We are now in a position to provide 3.3. The proof of Theorem 1.5. Deﬁne the disc automorphisms ζ − ζj , j = 1, 2, Mj (ζ) := 1 − ζj ζ and write Φj = F ◦ Mj−1 , j = 1, 2. Note that Φ1 (0) = W1 . For λ ∈ σ(W1 ), let m(λ) be as stated in the theorem. Deﬁne the Blaschke product ζ − λ m(λ) B1 (ζ) := , ζ ∈ D. 1 − λζ λ∈σ(W ) 1

Applying Lemma 3.2, we get

µ − λ m(λ)

ζ1 − ζ2

= |M1 (ζ2 )| ≥

1 − ζ ζ

1 − λµ 2 1 λ∈σ(W1 ) = MD (µ, λ)m(λ) ∀µ ∈ σ(Φ1 (M1 (ζ2 ))) = σ(W2 ). λ∈σ(W1 )

(3.5) Now, swapping the roles of ζ1 and ζ2 and applying the same argument to ζ − µ m(µ) , ζ ∈ D, B2 (ζ) := 1 − µζ µ∈σ(W2 )

we get

ζ1 − ζ2

1 − ζ ζ

2 1

≥

MD (λ, µ)m(µ)

Combining (3.5) and (3.6), we get   max MD (µ, λ)m(λ) , max µ∈σ(W2 ) λ∈σ(W1 )

∀λ ∈ σ(W1 ).

(3.6)

µ∈σ(W2 )

max

λ∈σ(W1 )

µ∈σ(W2 )

MD (λ, µ)m(µ)

 



ζ1 − ζ2 ≤

1−ζ ζ

2 1

.

We conclude this section with an example.

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Example 3.4. An illustration of Observation 1.7. We begin by pointing out that the phenomenon below is expected for n = 2. We want to consider n > 2 and show that there is no interpolant for the following data, but that this cannot be inferred from (1.1). First the matricial data: let n = 2m, m ≥ 2, and let W1 = any block-diagonal matrix with two m × m-blocks that are each nilpotent of order m.

(3.7)

Next, for an α ∈ D, α = 0, let W2 = the companion matrix of the polynomial (z 2m − αz m ). Note that, by construction, W2 is non-derogatory. We have the characteristic polynomials χW1 (z) = z m and χW2 (z) = z 2m − αz m . Hence f(·; W1 ) ≡ 0,

f(z; W2 ) =

−mαz m−1 . 2m − mαz m

We recall, from Section 2, the following equivalent form of (1.1):

f(z; W ) − f(z; W )

ζ − ζ

1 2 1 2 (3.8) When M = 2, (1.1) ⇐⇒ sup

≤

.

1 − ζ ζ 1 − f(z; W )f(z; W ) 2 1 2 1 z∈D Since, clearly, f(·; W2 ) ∈ O(D) C(D), by the Maximum Modulus Theorem

f(z; W ) − f(z; W ) m|α|

1 2 sup

= sup m

z∈∂D |2m − mαz | z∈D 1 − f(z; W2 )f(z; W1 ) =

m|α| < |α|. 2m − m|α|

(3.9)

Observe that σ(W1 ) = {0} and σ(W2 ) = {0, |α|1/m ei(2πj+Arg(α))/m , j = 1, . . . , m}. Therefore, max MD (µ, λ)m(λ) = |α|, λ∈σ(W1 ) µ∈σ(W2 ) max MD (λ, µ)m(µ) = 0. λ∈σ(W1 )

µ∈σ(W2 )

We set ζ1 = 0 and pick ζ2 ∈ D in such a way that

ζ1 − ζ2 m|α|

< |α|. < |ζ2 | =

2m − m|α| 1 − ζ2 ζ1

(3.10)

Such a choice of ζ2 is made possible by the inequality (3.9). In view of the last calculation above, we see that the data-set ((W1 , ζ1 ), (W2 , ζ2 )) constructed violates the inequality (1.2). Thus, there is no O(D, Ω2m )-interpolant for this data-set. In contrast, since the equivalent form (3.8) of (1.1) is satisﬁed, the latter does not yield any information about the existence of a O(D, Ω2m )-interpolant.

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4. The Proof of Theorem 1.8 In order to prove Theorem 1.8, we shall need the following elementary Lemma 4.1. Given a fractional-linear transformation T (z) := (az + b)/(cz + d), if T (∂D) C, then T (∂D) is a circle with centre(T (∂D)) =

bd − ac , |d|2 − |c|2

radius(T (∂D)) =

|ad − bc| . ||d|2 − |c|2 |

We are now in a position to present 4.2. The proof of Theorem 1.8. Let G ∈ O(Ωn ; Ωn ) and let λ1 , . . . , λs be the distinct eigenvalues of G(0). Deﬁne m(j) :=the multiplicity of the factor (λ − λj ) in the minimal polynomial of G(0). Deﬁne the Blaschke product m(j) s ζ − λj BG (ζ) := , ζ ∈ D. 1 − λj ζ j=1 BG induces the following matrix function which, by a mild abuse of notation, we shall also denote as BG s

BG (Y ) :=

(I − λj Y )−m(j) (Y − λj I)m(j)

∀Y ∈ Ωn ,

j=1

which is well-deﬁned on Ωn precisely as explained in the proof of Lemma 3.2. Once again, owing to the analyticity of BG on Ωn , σ(BG (Y )) = {BG (λ) : λ ∈ σ(Y )}

∀Y ∈ Ωn ,

whence BG : Ωn −→ Ωn . Therefore, if we deﬁne H(X) := BG ◦ G(X) ∀X ∈ Ωn , then H ∈ O(Ωn ; Ωn ) and, by construction, H(0) = 0. By the Ransford-White result, r(H(X)) ≤ r(X), or, more precisely  

s 

µ − λj m(j) 

≤ r(X) ∀X ∈ Ωn . max

1 − λ µ  µ∈σ(G(X))  j=1

In particular: max µ∈σ(G(X))

j

distM (µ; σ(G(0)))dG

≤ r(X) ∀X ∈ Ωn ,

where, for any compact K D and µ ∈ D, we deﬁne

distM (µ; K) := min (µ − ζ)(1 − ζµ)−1 . ζ∈K

For the moment, let us ﬁx X ∈ Ωn . For each µ ∈ σ(G(X)), let λ(µ) be an

eigenvalue of G(0) such that (µ − λ(µ) )(1 − λ(µ) µ)−1 = distM (µ; σ(G(0))). Now

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341

ﬁx µ ∈ σ(G(X)). The above inequality leads to

µ − λ(µ)

≤ r(X)1/dG .

1 − λ(µ) µ

(4.1)

Applying Lemma 4.1 to the M¨obius transformation T (z) = we deduce that

|µ|z − λ(µ) 1 − λ(µ) |µ|z

ζ − λ(µ) ||µ| − |λ(µ) ||

≥

1 − λ(µ) ζ 1 − |µ||λ(µ) |

,

∀ζ : |ζ| = |µ|.

Applying the above fact to (4.1), we get |µ| − |λ(µ) | ≤ r(X)1/dG 1 − |µ||λ(µ) | ⇒

|µ| ≤

r(X)1/dG + |λ(µ) | , 1 + |λ(µ) |r(X)1/dG

µ ∈ σ(G(X)).

(4.2)

Note that the function t −→

r(X)1/dG + t , 1 + r(X)1/dG t

t ≥ 0,

is an increasing function on [0, ∞). Combining this fact with (4.2), we get |µ| ≤

r(X)1/dG + r(G(0)) , 1 + r(G(0))r(X)1/dG

which holds ∀µ ∈ σ(G(X)), while the right-hand side is independent of µ. Since this is true for any arbitrary X ∈ Ωn , we conclude that r(G(X)) ≤

r(X)1/dG + r(G(0)) 1 + r(G(0))r(X)1/dG

∀X ∈ Ωn .

In order to prove the sharpness of (1.2), let us ﬁx an n ≥ 2, and deﬁne Sn := {A ∈ Ωn : A has a single eigenvalue of multiplicity n}. Pick any d = 1, . . . , n, and deﬁne   [tr(X)/n],       0   1 0  Md (X) :=    .   .. ...       1

tr(X)/n 0 .. . 0



if d = 1,

    

, if d ≥ 2, d×d

342

Bharali

IEOT

and, for the chosen d, deﬁne G(d) by the following block-diagonal matrix Md (X) (d) G (Y ) := ∀X ∈ Ωn . tr(X) In−d n For our purposes GA,d = G(d) for each A ∈ Sn ; i.e., the equality (1.5) will hold with the same function for each A ∈ Sn . To see this, note that • r(G(d) (X)) = |tr(X)/n|1/d ; and • G(d) (0) is nilpotent of degree d, whence dG(d) = d. Therefore, r(A)1/d + r(G(d) (0)) = r(A)1/d = r(G(d) (A)) 1 + r(G(d) (0))r(A)1/d which establishes (1.5)

∀A ∈ Sn ,

References [1] J. Agler and N.J. Young, A commutant lifting theorem for a domain in C2 and spectral interpolation, J. Funct. Anal. 161 (1999), 452–477. [2] J. Agler and N.J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Operator Theory 37 (2000), 375–385. [3] J. Agler and N.J. Young, The two-by-two spectral Nevanlinna-Pick problem, Trans. Amer. Math. Soc. 356 (2004), 573–585. [4] J. Agler and N.J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375–403. [5] H. Bercovici, C. Foias and A. Tannenbaum, Spectral variants of the Nevanlinna-Pick interpolation problem in Signal Processing, Scattering and Operator Theory, and Numerical Methods (Amsterdam, 1989), 23–45, Progr. Systems Control Theory 5, Birkh¨ auser Boston, Boston, MA, 1990. [6] H. Bercovici, C. Foias, A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), 741–763. [7] C. Costara, The 2 × 2 spectral Nevanlinna-Pick problem, J. London Math. Soc. (2) 71 (2005), 684–702. [8] C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), 23– 55. [9] J. Globevnik, Schwarz’s lemma for the spectral radius, Rev. Roumaine Math. Pures Appl. 19 (1974), 1009–1012. [10] M. Jarnicki and P. Pﬂug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Mathematics no. 9, Walter de Gruyter & Co., Berlin, 1993. [11] N. Nikolov, P. Pﬂug, P.J. Thomas and W. Zwonek, Estimates of the Carath´eodory metric on the symmetrized polydisc, in submission; also arXiv preprint arXiv:math.CV/0608496. [12] A. Nokrane and T. Ransford, Schwarz’s lemma for algebroid multifunctions, Complex Variables Theory Appl. 45 (2001), 183–196.

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[13] D.Ogle, Operator and Function Theory of the Symmetrized Polydisc, Thesis (1999), http://www.maths.leeds.ac.uk/ nicholas/. [14] T.J. Ransford and M.C. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), 256–262. [15] E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital. (4) 1 1968, 427–429. Gautam Bharali Department of Mathematics Indian Institute of Science Bangalore – 560 012 India e-mail: [email protected] Submitted: April 12, 2007 Revised: May 16, 2007

Integr. equ. oper. theory 59 (2007), 345–353 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030345-9, published online October 18, 2007 DOI 10.1007/s00020-007-1525-x

Integral Equations and Operator Theory

Finite Rank Perturbations of Toeplitz Operators ˇ ˇ ckovi´c Zeljko Cuˇ Abstract. We study ﬁnite rank perturbations of the Brown-Halmos type results involving products of Toeplitz operators acting on the Bergman space. Mathematics Subject Classification (2000). Primary 47B35. Keywords. Toeplitz operator, Bergman space, Berezin transform, ﬁnite rank operators.

ˇ ckovi´c [2] proved an analogue of the well-known BrownIn 2001, Ahern and Cuˇ Halmos theorem for the Bergman space Toeplitz operators with harmonic symbols. To state the result, we introduce the notation. Let D denote the open unit disk in the complex plane and let dA denote the normalized Lebesgue area measure on 2 valued functions f on D D. As usual, L (D)2 is the space of measurable complex such that D |f (z)| dA(z) < ∞. The Bergman space L2a (D) is the closed subspace of L2 (D) consisting of the analytic functions on D. Let P : L2 (D) → L2a (D) denote the orthogonal projection. For a bounded function u on D we have the Toeplitz operator Tu : L2a (D) → L2a (D) given by Tu f = P (uf ). We denote the Laplacian ∂2 ˜ = (1 − |z|2 )2 ∆. We can now state the and the invariant Laplacian by ∆ ∆ = ∂z∂z above mentioned theorem of Ahern and the author. Theorem A. Suppose f and g are bounded harmonic functions and h is a bounded ˜ is bounded on D. If Tf Tg = Th , then either f is conjugate C 2 function such that ∆h analytic or g is analytic. In either case, h = f g. Later on, Ahern [1] removed the assumptions on h and showed the theorem ˇ ckovi´c obtained a sequence is true for h ∈ L∞ (D). From Theorem A, Ahern and Cuˇ of results on products of Toeplitz operators that are parallel to the corollaries of the Brown-Halmos theorem for the Hardy space obtained in [5]. We list some of them. Corollary A. If f , g and h are bounded harmonic functions and Tf Tg = Th , then one of the following holds: 1. f and g are analytic.

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2. f and g are conjugate analytic. 3. f is constant. 4. g is constant. The next one resolved an open problem about zero products. Corollary B. If f and g are bounded harmonic functions and Tf Tg = 0, then either f = 0 or g = 0. Corollary C. If f and g are bounded and harmonic and Tf Tg = I, then either f and g are both analytic or they are both conjugate analytic. In either case f g = 1. Corollary D. If f is bounded and harmonic and Tf2 = Tf , then f ≡ 0 or f ≡ 1. Corollary E. If f and g are bounded harmonic and Tf Tg = Tf g , then either g is analytic or f is conjugate analytic. We would like to mention that this last corollary was proved earlier by Zheng [10] using a diﬀerent method. We would also like to point out that in [3] we constructed examples of Toeplitz operators with radial symbols that show that some of these corollaries do not hold in general. One of the main steps in the proof of Theorem A is the study of the range of the Berezin transform. For any integrable function f on D, the Berezin transform is deﬁned by f (w) 2 2 dA(w). Bf (z) = (1 − |z| ) |1 − zw|4 D 1 Let Kz (w) = (1−wz) 2 denote the Bergman kernel for z ∈ D. Then kz (w) denotes the normalized Bergman kernel:

kz (w) =

1 − |z|2 , w ∈ D. (1 − wz)2

The Berezin transform can then be expressed as Bf (z) = f kz , kz , where · , denotes the L2 (D) inner product. We can also deﬁne the Berezin transform of any bounded operator S as B(S)(z) = Skz , kz , for z ∈ D. Another important step in the proof of Theorem A is the proof of the fact that a rank 1 Toeplitz operator on L2a (D) must be 0. For any operator A, rank (A) = dim Ran(A). Compact Toeplitz operators on L2a (D) have been characterized by Axler and Zheng using the Berezin transform (see [4]). Surprisingly characterizing ﬁnite rank Toeplitz operators on L2a (D) is still an open problem. The common conjecture among the experts is that a ﬁnite rank Toeplitz operator on L2a (D) must be 0. In a recent paper, Guo, Sun and Zheng [6] have proved this conjecture in a special case. Theorem B. Suppose that f is in L∞ (D) and f = j=1 fj (z)gj (z) for finitely many functions fj and gj analytic on D. If Tf has finite rank, then f = 0. Using this theorem, they obtained an extension of Corollary B on the zero products of Toeplitz operators. More speciﬁcally, they proved that for two bounded harmonic functions f and g, if the product Tf Tg has ﬁnite rank, then either f = 0 or g = 0. We think of this product as Tf Tg = 0 + F , F ﬁnite rank, so the

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347

product is a ﬁnite rank perturbation of 0. Similarly, they also obtained a result characterizing ﬁnite rank semicommutators Tf g − Tf Tg of Toeplitz operators with harmonic symbols. This is a ﬁnite rank perturbation extension of Corollary E, where we considered the case Tf g − Tf Tg = 0. Inspired by these results of Guo, Sun and Zheng, we want to obtain results on ﬁnite rank perturbations of the products in the other corollaries listed above. They will follow from the following result. Before we state it, we recall two known results. First, an operator F of N ﬁnite rank N can be written as j=1 xj ⊗ yj , for some functions xj , yj in L2a (D) for j = 1, . . . , N . Here x ⊗ y is the rank one operator deﬁned by (x ⊗ y)h = h, yx, where x, y, h are in L2a (D). Second, if f is a bounded harmonic function on D, f can be written as f1 + f 2 , where f1 and f2 are analytic functions that belong to the Bloch space B = {f : f analytic on D and supz∈D (1 − |z|2 )|f (z)| < ∞}. Theorem 1. Suppose f = f1 + f 2 , g = g1 + g 2 and h = h1 + h2 are bounded harmonic functions on D such that h1 , h2 ∈ H ∞ (D). Suppose that Tf Tg = Thn +F , N where F = j=1 xj ⊗ yj is of finite rank N , xj , yj ∈ L2a (D) for j = 1, . . . , N and N , n ∈ N. Then: (i) g1 (z)f 2 (z) − hn (z) is harmonic, N (ii) f (z)g(z) = hn (z) + (1 − |z|2 )2 j=1 xj (z)yj (z), for z ∈ D. Conversely, suppose f = f1 + f 2 , g = g1 + g 2 and h = h1 + h2 are bounded harmonic functions on D such that (i) holds on D. If there exist nonzero vectors x1 , . . . , xN , y1 , . . . , yN in L2a (D), such that (ii) holds for z ∈ D, then Tf Tg = N Thn + F , where F = j=1 xj ⊗ yj is a finite rank operator. In particular, condition (ii) implies that f g = hn a.e. on ∂D. If in addition, Thn + F is an isometry, then h L∞ (D) = h L∞ (∂D) = f g L∞ (∂D) = 1. Proof. Suppose that Tf Tg = Thn + F . Then we have B(Tf Tg ) = B(hn ) + B(F ).

(1)

As in [2], B(Tf Tg )(z) = f1 (z)g1 (z) + f1 (z)g2 (z) + f2 (z) g2 (z) + B(f 2 g1 )(z), for z ∈ D. It is also easy to show that B(F )(z) =

N

2 2

B(xj ⊗ yj )(z) = (1 − |z| )

j=1

N

xj (z)yj (z).

j=1

Thus (1) can be written as f1 (z)g1 (z) + f1 (z)g2 (z) + f2 (z) g2 (z) + B(f 2 g1 ) − B(hn ) = (1 − |z|2 )2

N j=1

xj (z)yj (z)

(2)

ˇ ckovi´c Cuˇ

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IEOT

for z ∈ D. It is well known that the Berezin transform ﬁxes L1 -harmonic functions, i.e., B(u) = u if u is harmonic. Thus (2) can be written as n

2 2

B(f1 g1 + f 2 g2 + f 2 g1 − h )(z) = (1 − |z| )

N

xj (z)yj (z) − f1 (z)g2 (z).

j=1

˜ to both sides and use the fact that ∆ ˜ commutes Apply the invariant Laplacian ∆ with B (see [2]), to obtain   N ˜ 2 g1 − hn ) (z) = ∆ ˜ (1 − |z|2 )2 ˜ f1 (z)g2 (z) B ∆(f xj (z)yj (z) − ∆ (3) j=1

˜ 2 g1 − hn ). After cancelling (1 − |z|2 )2 on both sides of (3) for z ∈ D. Let σ = ∆(f we have   N σ(ξ) dA(ξ) = ∆ (1 − |z|2 )2 xj (z)yj (z) − f1 (z)g2 (z). (4) 4 |1 − ξz| D j=1 N N N Notice that (1 − |z|2 )2 j=1 xj (z)yj (z) = j=1 xj (z)yj (z) − 2 j=1 zxj (z)zyj (z) 3N N ˜j (z)˜ yj (z) with + j=1 z 2 xj (z)z 2 yj (z) for z ∈ D, which can be written as j=1 x 2 x ˜j , y˜j ∈ La (D). With this in mind, we can complexify (4) as was done in Lemma 2 of [2] to obtain

3N

σ(ξ) dA(ξ) = x ˜j (z)˜ yj (w) − f1 (z)g2 (w) 2 2 (1 − ξz) (1 − ξw) j=1

(5)

for all z, w ∈ D. If we diﬀerentiate (5) k times with respect to w and then let w = 0, we get 3N ξ k σ(ξ) dA(ξ) = akj x˜j (z) − ck f1 (z) (6) 2 (1 − ξz) D j=1 for some constants akj , ck , k = 1, 2, . . .. Then (6) tells us that for any k ∈ N, we have 3N ξ k σ(ξ) Tσ (ξ k ) = dA(ξ) = akj x ˜j (z) − ck f1 (z). 2 (1 − ξz) D j=1 Using the argument of Proposition 4 in [6] we have that Tσ has ﬁnite rank. 2 2 ˜ Notice that ∆(f 2 g1 ) = (1−|z| ) f 2 (z)g1 (z) is bounded since f2 and g1 belong

to the Bloch space. If n = 1, then ∆h = 0. For n > 1, n n−k n n n h = (h1 + h2 ) = hk1 · h2 k k=0

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Finite Rank Perturbations of Toeplitz Operators

so that ˜ n) = ∆(h

349

n (n−k−1) n · h1 · (n − k)h2 h2 (1 − |z|2 )2 khk−1 1 k

k=1

which is also bounded, since h1 and h2 are bounded by the assumption and they also belong to the Bloch space. 3n+3 Thus σ(z) is in L∞ (D) and it is of the form j=1 Fj (z)Gj (z) for some analytic functions Fj and Gj , j = 1, . . . , n. By Theorem B, σ ≡ 0, and hence f 2 g1 − hn is a harmonic function. Thus (i) holds. Now (2) gives f1 (z)g1 (z)+f1 (z)g 2 (z)+f2 (z) g2 (z)+f 2 (z)g1 (z)−hn (z) = (1−|z|2 )2

N

xj (z)yj (z)

j=1

N for all z ∈ D. In other words (f g)(z) − hn (z) = (1 − |z|2 )2 j=1 xj (z)yj (z) which n gives (ii). The expression on the right-hand side is equal to B( j=1 xj ⊗ yj )(z) N which goes to 0 as |z| → 1, since j=1 xj ⊗ yj is a ﬁnite rank operator and n therefore compact. Hence f g = h a.e. on ∂D. Also notice that (3) implies that N f1 (z)g2 (z) = (1 − |z|2 )2 j=1 xj (z)yj (z) + u(z) for some harmonic function u. If the operator F = 0, then this would imply f1 (z)g2 (z) = 0 on D. This means that f1 is constant or g2 is constant. In other words, f is conjugate analytic or g is analytic which is consistent with Theorem A from [2]. Conversely, suppose that g1 (z)f 2 (z)−hn (z) is harmonic on D, and f (z)g(z) = n h (z) + (1 − |z|2 )2 N j=1 xj (z)yj (z). As calculated earlier, B(Tf Tg − Thn )(z) = f1 (z)g1 (z) + f1 (z)g2 (z) + f2 (z) g2 (z) + B(f 2 g1 − hn )(z) = (f g)(z) − hn (z) = (1 − |z|2 )2  = B

N



N

xj (z)yj (z)

j=1

xj ⊗ yj  .

j=1

N Since the Berezin transform is one-to-one, it follows that Tf Tg = Thn + j=1 xj ⊗yj and the converse is proved. Assume, in addition, that Thn +F is an isometry. Then (Thn +F )∗ (Thn +F ) = I or Thn Thn + F ∗ Thn + Thn F + F ∗ F = I.

(7)

We will recall some classical results about the algebra of bounded analytic functions on D, denoted by H ∞ . Let M denote the maximal ideal space of H ∞ . Hoﬀman ([7], Lemma 4.4) has proved that the algebra C(M ) is identical to the sup norm closure of the algebra generated by the bounded harmonic functions. Thus

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hn ∈ C(M ). On the ideal M we can introduce an equivalence relation: m1 ∼ m2 if and only if ρ(m1 , m2 ) < 1, where ρ(m1 , m2 ) = sup{|fˆ(m2 )| : f ∈ H ∞ , f ≤ 1, fˆ(m1 ) = 0}. Here fˆ is the Gelfand transform of f deﬁned by fˆ(m) = m(f ), m ∈ M . The equivalence classes are called Gleason parts. Let M1 denote the set of one-point parts in M , and J = {ϕ ∈ C(M ) : ϕ = 0 on M1 }. Let τ (C(M )) be the closed subalgebra of the algebra of all bounded linear operators on L2a (D) generated by {Tϕ : ϕ ∈ C(M )} and let C be the commutator ideal of τ (C(M )). McDonald and Sundberg [9] have proved that C(M )/J is isomorphic to τ (C(M ))/C with the isomorphism ∧

ϕ + J −→Tϕ + C. It is also well known that C contains all compact operators. Let Π : τ (C(M )) −→ τ (C(M ))/C be the quotient map. Apply Π to the equation (7) and notice that F is ﬁnite rank and hence F is compact. Therefore F ∗ is also compact so (7) becomes Π(Thn )Π(Thn ) = Π(I). Applying the isomorphism above, we obtain n

(h + J)(hn + J) = 1 + J n

n

or h · hn − 1 ∈ J. This means h · hn − 1 = 0 on M1 . But the maximal ideal space of L∞ (∂D) is a subset of M1 . Hence n

ϕ(h ) · ϕ(hn ) = 1 or |ϕ(h)| = 1 for all ϕ ∈ M (L∞ (∂D)). Since h is a bounded harmonic function on D, we can identify it with its boundary value function, which we denote by h again. By Hoﬀman [8], p. 170 the Gelfand transform maps L∞ (∂D) isometrically and isomorphically onto C(M (L∞ )). Thus we have ˆ C(M(L∞ )) = sup{|ϕ(h)| : ϕ ∈ M (L∞ (∂D))} = 1. hL∞ (D) = hL∞ (∂D) = h Then clearly ||hn ||L∞ (∂D) = 1 and since f g = hn a.e. on ∂D, we have hL∞ (D) = hn L∞ (∂D) = f gL∞ (∂D) = 1.

Remark. It is clear from the proof that the theorem is valid if we replace Thn with the Toeplitz operator whose symbol is a product of n diﬀerent bounded harmonic functions on D. Remark. We do not have examples of functions that satisfy conditions (i) and (ii). The following corollary corresponds to Corollary A.

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Corollary 2. Suppose f = f1 + f 2 , g = g1 + g 2 and h = h1 + h2 are bounded harmonic functions on D and x1 , . . . , xN , y1 , . . . , yN are in L2a (D). Then Tf Tg = N Th + j=1 xj ⊗ yj if and only if the following two conditions hold: (i) either f is analytic or g is conjugate analytic, N (ii) (f g)(z) = h(z) + (1 − |z|2 )2 j=1 xj (z)yj (z), for z ∈ D. Proof. Apply Theorem 1 with n = 1. Then Tf Tg = Th + F implies that g1 f 2 is harmonic on D, so that ∆(g1 f 2 )(z) = g1 (z)f 2 (z) = 0. Hence g1 = constant or f2 is constant on D which means that either f is analytic or g is conjugate analytic. The other statements follow immediately from Theorem 1. If F = 0, then (3) implies f1 (z)g 2 (z) = 0 and hence either f is conjugate analytic or g is analytic. If f is analytic and f is conjugate analytic, then clearly f is constant. The same situation for g leads to the conclusion that g is constant. Otherwise, both f and g are analytic on D or both f and g are conjugate analytic and Corollary A follows. Conversely, if f is analytic, then f2 is constant so that g1 f 2 − h is harmonic on D. Apply Theorem 1 now with n = 1 and the converse follows. Similarly, the statement follows if g is conjugate analytic. A ﬁnite rank perturbation version of Corollary C is contained in the following corollary. Corollary 3. Suppose f and g are bounded and harmonic on D. Then Tf Tg = N I + j=1 xj ⊗yj and x1 , . . . , xN , y1 , . . . , yN are in L2a (D) if and only if the following two conditions hold: (i) either f is analytic or g is conjugate analytic, N (ii) f (z)g(z) = 1 + (1 − |z|2 )2 j=1 xj (z)yj (z), for z ∈ D. Corollary 4. If f is bounded and harmonic and Tf2 = Tf + F , then f = 0 or f = 1 on D. Proof. By Corollary 2, f is analytic or f is conjugate analytic and f 2 = f a.e. on ∂D. This means that f (f − 1) = 0 a.e. on ∂D. If f is analytic, then either f = 0 on ∂D (and hence f ≡ 0 on D) or f = 1 on ∂D (and hence f ≡ 1 on D). The same conclusion follows if f is conjugate analytic. If we slightly modify the argument in the proof of Theorem B, we get the following proposition. Proposition 5. Suppose E ⊂ D is a starlike with respect to 0 compact set. Let f (z) = χE (z)

fj (z)gj (z),

j=1

with fj , gj analytic on D for j = 1, . . . , . If Tf has finite rank N , then f = 0.

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Proof. Clearly f is bounded. As in the proof of Theorem B, for 0 < r < 1, deﬁne fr (z) = f (rz) and let gr (z) = f r . Then Tf gr = Tf χ

E

(rz)

j=1

fj (rz)gj (rz)

=

j=1

Tfj (rz) Tf χE (rz) Tgj (rz) .

(8)

But notice f (z)χE (rz) = χE (z)χE (rz) j=1 fj (z)gj (z). If z ∈ E, then rz ∈ E too since E is starlike. Thus χE (z)χE (rz) = 1. If z ∈ E, χE (z) = 0. Hence in both cases χE (z)χE (rz) = χE (z). Thus f (z)χE (rz) = f (z). Now (8) gives that Tf gr = j=1 Tfj (rz) Tf (z) Tgj (rz) and consequently rank Tf gr ≤ N , for all r. Thus lim sup rank Tf gr ≤ N . We continue r→1

as in Theorem B and conclude that T|f |2 has ﬁnite rank and therefore f ≡ 0 [see 6, p. 5]. Finally we would like to prove another zero product result involving two Toeplitz operators. Proposition 6. Suppose Dr = rD for some r ∈ (0, 1), h is an analytic function on D and g = g1 + g 2 is a bounded harmonic function. If f = χDr h and Tf Tg = 0, then either f = 0 or g = 0. Proof. Suppose f ∈ L∞ (D), g = g1 + g2 is a bounded and harmonic function, and Tf Tg = 0. Then B(Tf Tg )(z) =

Tf Tg kz , kz = (1 − |z|2 )2 Tf P (g1 + g2 )Kz , Kz

=

(1 − |z|2 )2 {f g1 Kz , Kz + f P (g2 Kz ), Kz }

=

B(f g1 )(z) + g2 (z)(Bf )(z) = 0.

Suppose now that f = χDr h, where h is analytic. Then (9) means h(ξ)g1 (ξ) h(ξ) dA(ξ) + g2 (z) dA(ξ) = 0. 4 |1 − zξ| |1 − zξ|4 Dr Dr

(9)

(10)

Let w = ξr ; then (10) becomes h(wr)g1 (wr) h(wr) 2 r2 dA(w) + r g (z) dA(w) = 0 2 4 |1 − zwr| |1 − zwr|4 D D or h(wr)g1 (wr) h(wr) 2 2 2 2 2 2 (1 − r |z| ) dA(w) + g2 (z) (1 − r |z| ) dA(w) = 0 4 4 D |1 − wzr| D |1 − wzr| so that B[(hg1 )r ](rz) + g2 (z)B(hr )(rz) = 0. Since the Berezin transform ﬁxes analytic functions, we have h(r2 z)g1 (r2 z) + g2 (z)h(r2 z) = 0, for z ∈ D which implies

h(r2 z)[g1 (r2 z) + g2 (z)] = 0.

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Then either h = 0 or g1 (r2 z) = −g2 (z) for z ∈ D. If h = 0, then f = 0. In the second case, an analytic function g1r2 is equal to the conjugate analytic function, so they both are constant functions; i.e., g1 = constant and g2 = constant. If g = constant, then Tf Tg = 0 implies g = constant = 0 or f = 0. Thus we have proved the proposition. Remark added after the acceptance of the paper. Daniel Luecking has recently proved the conjecture about ﬁnite rank Toeplitz operators.

References [1] P. Ahern, On the range of the Berezin transform, J. Funct. Anal. 215 (2004), 206–216. ˇ Cuˇ ˇ ckovi´c, A theorem of Brown-Halmos type for Bergman space [2] P. Ahern and Z. Toeplitz operators, J. Funct. Anal. 187 (2001), 200–210. ˇ Cuˇ ˇ ckovi´c, Some examples related to the Brown-Halmos theorem for [3] P. Ahern and Z. the Bergman space, Acta Sci. Math. (Szeged) 70 (2004), 373–378. [4] S. Axler and D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), 387–400. [5] A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math 213 (1964) 89–102. [6] K. Guo, S. Sun and D. Zheng, Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols, Illinois J. Math., to appear. (available at http: //www.math.vanderbilt.edu/∼ zheng/publication.html) [7] K. Hoﬀman, Bounded analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74–111. [8] K. Hoﬀman, Banach Spaces of Analytic Functions, Dover, New York 1962. [9] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595–611. [10] D. Zheng, Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 83 (1989), 98–120. ˇ ˇ ckovi´c Zeljko Cuˇ Department of Mathematics University of Toledo Toledo, OH 43606 USA e-mail: [email protected] Submitted: April 26, 2006 Revised: June 13, 2007

Integr. equ. oper. theory 59 (2007), 355–378 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030355-24, published online October 18, 2007 DOI 10.1007/s00020-007-1530-0

Integral Equations and Operator Theory

Wiener–Hopf Operators on Spaces of Functions on R+ with Values in a Hilbert Space Violeta Petkova Abstract. A Wiener–Hopf operator on a Banach space of functions on R+ is a bounded operator T such that P + S−a T Sa = T , a ≥ 0, where Sa is the operator of translation by a. We obtain a representation theorem for the Wiener–Hopf operators on a large class of functions on R+ with values in a separable Hilbert space. Mathematics Subject Classification (2000). Primary 47B38; Secondary 47B35. Keywords. Wiener–Hopf operators, symbol, Fourier transformation, spectrum of translation operators.

1. Introduction This paper deals with Wiener–Hopf operators on Banach spaces of functions on R+ with values in a separable Hilbert space H. Let E be a Banach space of functions on R+ such that E ⊂ L1loc (R+ ). For a ≥ 0, deﬁne the operator Sa : E −→ L1loc (R+ ), by the formula (Sa f )(x) = f (x−a), for almost every x ∈ [a, +∞[ and (Sa f )(x) = 0, for x ∈ [0, a[. For a ≥ 0, introduce S−a : E −→ L1loc (R+ ), deﬁned by the formula (S−a f )(x) = f (x+ a), for almost every x ∈ R+ . Notice that S−a Sa = I but Sa S−a = I. From now, we suppose that Sa E ⊂ E and S−a E ⊂ E, ∀a ∈ R+ . The Wiener–Hopf operators on E are the bounded operators T : E −→ E satisfying Denote by P + the operator

S−a T Sa = T, ∀a ∈ R+ .

P + : L1loc (R) −→ L1loc (R+ )

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deﬁned by (P + f )(x) = f (x), a.e. on R+ . The Wiener–Hopf operators which appear in theory of the signal and in control theory have been studied in a lot of papers. The problem we deal here is the existence of a symbol for operators of this type. It is well-known that if T is a Wiener–Hopf operator on L2 (R+ ) there exists h ∈ L∞ (R) such that T f = P + F −1 (hfˆ), ∀f ∈ L2 (R+ ). Here F denotes the usual Fourier transformation from L2 (R) into L2 (R). The function h is called the symbol of T . Despite of the extensive literature related to Wiener–Hopf operators, there are not analogous representation theorem for Wiener–Hopf operators on general Banach spaces of functions even if the functions are with values in C. Here we develop a theory of the existence of a L∞ symbol for every Wiener–Hopf operator in a very large class of spaces of functions on R+ with values in a separable Hilbert space. Moreover, we obtain a characterization of spec(S1 ) ∩ (spec(S−1 ))−1 . The determination of the spectrum of a translation operator is an open question in general spaces of functions on R+ and it plays an important role in the scattering theory. We are motivated by the results of [5] proving the existence of a symbol for every Wiener–Hopf operator on a weighted space L2ω (R+ ) (see Example 1 for the deﬁnition). On the other hand, the methods exposed in [6] and [2] show that the existence of the symbol of a multiplier (a bounded operator commuting with the translations) on spaces of scalar functions on R implies an analogous result for the multipliers on a space of functions on R with values in an Hilbert space. The arguments in [6] and [2] have been based on the link between the scalar and the vector-valued cases. However the results concerning the symbol of a multiplier do not imply analogous results about Wiener–Hopf operator in the general case. It is well-known that for every Wiener–Hopf operator T on L2 (R+ ), there exists a multiplier M on L2 (R) such that P + M = T . Unfortunately, a such result is not known even for Wiener–Hopf operators on a weighed space L2ω (R+ ). Despite some progress (see [2], [6]) in the study of the symbol of a multiplier on a space of functions on R with values in a Hilbert space, the analogous problem for Wiener–Hopf operators has been very few considered. Moreover, even in the case of the weighted spaces of functions on R+ with values in a Hilbert space the existence of the symbol of a Wiener–Hopf operator was an open problem still now. First in Section 2, we improve the results of [5] concerning the existence of the symbol of a Wiener–Hopf operator on L2ω (R+ ) replacing L2ω (R+ ) by a general Banach space of functions on R+ satisfying only three natural hypothesis given below. Next following the methods of [6] and [2] and using the results of Section 2, we obtain the existence of the symbol of a Wiener–Hopf operator on a very large class of spaces of functions on R+ with values in a separable Hilbert space. In Section 4 we explain how the setup considered here can by extended in several directions.

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Let E be a Banach space of functions on R+ with values in C satisfying the following three hypothesis. (H1) We have Cc∞ (R+ ) ⊂ E ⊂ L1loc (R+ ), the inclusions are continuous and Cc∞ (R+ ) is dense in E. (H2) For every x ∈ R, Sx E = E and supx∈K Sx < +∞, for every compact K of R. (H3) For all a ∈ R, the operator Γa deﬁned by (Γa f )(x) = eiax f (x), a.e. , ∀f ∈ E is bounded on E and sup Γa < +∞. a∈R

∞ (R+ ) be the Notice that (H3) is trivial, if we have f = |f | in E. Let CK space of C ∞ functions with a compact support included in K. For simplicity, we will write S instead of S1 . Since the norm of f given by supa∈R Γa f is equivalent to the norm of E, we will assume from now that Γa is an isometry for every a ∈ R. Denote by ρ(A) the spectral radius of a bounded operator A. Set IE = [− ln ρ(S−1 ), ln ρ(S)] and

UE = z ∈ C, z ∈ IE .

For f ∈ E, denote by (f )a the function deﬁned by (f )a (x) = eax f (x), a.e. on R+ . In Section 2 we obtain the following result which generalizes Theorem 1 in [5]. Theorem 1.1. Let T ∈ W (E). 1) For every a ∈ IE we have (T f )a ∈ L2 (R+ ), for f ∈ Cc∞ (R+ ). 2) For every a ∈ IE there exists a function νa ∈ L∞ (R) such that (T f )a = P + F −1 (νa (f )a ), f or f ∈ Cc∞ (R+ ), and we have νa ∞ ≤ C T , where C is a constant dependent only on E. ◦

◦

3) Moreover, if IE = ∅ (i.e. ρ(S1−1 ) < ρ(S)), there exists a function ν ∈ H∞ (UE ) ◦

such that for every a ∈ IE we have ν(x + ia) = νa (x), almost everywhere on R. ◦

Definition 1.2. If IE = ∅, ν is called the symbol of T , and if IE = {a}, then νa is the symbol of T . Using Theorem 1, we also obtain the following spectral result. Theorem 1.3. We have −1 spec(S) ∩ spec(S−1 ) = z ∈ C,

1 ≤ |z| ≤ ρ(S) . ρ(S−1 )

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This result is new even in the case of the spaces L2ω (R+ ). In particular, we conclude that if ρ(S) > ρ(S1−1 ) the spectrum of S contains a disk. The proof of Theorem 2 is based on the existence of a symbol for every Wiener–Hopf operator and the construction of suitable cut-oﬀ function f ∈ Cc∞ (R+ ). This application was one of the motivations to search a symbol of a Wiener–Hopf operator. Moreover, we extent below the same result for operators with values in a Hilbert space (see Theorem 4). The main result of this paper is an analogous result for Wiener–Hopf operators on spaces of functions on R+ with values in a separable Hilbert space. Denote by < u, v > the scalar product of u, v ∈ H and let u H be the norm of u ∈ H. Denote by L1loc (R+ , H) the space of functions F : R+ −→ H such that

R+ x −→ F (x) H ∈ L1loc (R+ ).

Let L(H) be the space of bounded operators on H. Introduce the vector space Cc∞ (R+ ) ⊗ H generated by f u for f ∈ Cc∞ (R+ ) and u ∈ H. Denote by C0 (R+ , H) the Banach space of all norm continuous functions Φ : R+ −→ H such that for every > 0, there exists a compact set K such that Φ(x) H = 0, ∀x ∈ R+ \ K . Let E be a Banach space of functions on R+ with values in C satisfying (H1), (H2) and (H3). Denote by E the Banach space of functions F : R+ −→ H such that

R+ x −→ F (x) H ∈ E.

We will see in Section 3 that Cc∞ (R+ ) ⊗ H is dense in E. For illustration, we give below some examples. Example 1. Let E = Lpω (R+ ), where ω is a weight on R+ and p ∈ [1, +∞[. We recall that ω is a weight on R+ if ω is a non-negative measurable function on R+ such that for all y ∈ R+ , 0 < sup x∈R+

and 0 < sup x∈R+

ω(x + y) < +∞ ω(x) ω(x) < +∞. ω(x + y)

The space Lpω (R+ ) is the set of measurable functions f from R+ into C such that |f (x)|p ω(x)p dx < +∞, R+

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equipped with the norm f ω,p =

R+

359

p1 |f (x)|p ω(x)p dx .

Lpω (R+ )

satisﬁes the hypothesis (H1), (H2) and (H3). For It is easy to see that the study of the Wiener–Hopf operators on L2ω (R+ ) the reader may consult [5]. The space E associated to Lpω (R+ ) is the space usually denoted by Lpω (R+ , H) of functions F : R+ −→ H such that

R+

F (x) pH ω(x)p dx < +∞.

Example 2. Let A be a real-valued continuous function on [0, +∞[, such that A(0) = 0 and let A(y) be non-decreasing for y > 0. Let LA (R+ ) be the set of all y complex-valued, measurable functions on R+ such that |f (x)| dx < +∞, A t R+ for some positive number t and let f A = inf t > 0 |

|f (x)| dx ≤ 1 , A t R+

for f ∈ LA (R+ ). Then LA (R+ ) is a Banach space called a Birnbaum-Orlicz space (see [1]). It is easy to check that LA (R+ ) satisﬁes (H1), (H2) and (H3). If E = LA (R+ ), the associated space E is the set LA (R+ , H) of measurable functions F : R+ −→ H such that for some t > 0, we have F (x) H dx < +∞. A t + R Example 3. Let A be a function satisfying the properties described in Example 2. Let ω be a weight on R+ . Deﬁne LA,ω (R+ ) as the space of measurable functions on R+ such that |f (x)| A ω(x)dx < +∞, t R+ for some positive number t and let |f (x)| f A,ω = inf t > 0 | ω(x)dx ≤ 1 A t R+ for f ∈ LA,ω (R+ ). Then LA,ω (R+ ) is a Banach space called a weighted Orlicz space. It is easy to check that LA,ω (R+ ) satisﬁes (H1), (H2) and (H3). If E = LA,ω (R+ ), the associated space E is the set LA,ω (R+ , H) of measurable functions F : R+ −→ H

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such that for some t > 0,

F (x) H ω(x)dx < +∞. A t R+

For a > 0, we deﬁne the operators Sa : E −→ E and S−a : E −→ E by (Sa F )(x) = F (x − a), a.e. on [a, +∞[, (Sa F )(x) = 0, ∀x ∈ [0, a[, (S−a F )(x) = F (x + a), a.e. on R+ . For simplicity, we will write S instead of S1 . For F ∈ E, we denote by F H the function F H : R+ x −→ F (x) H ∈ C. For ﬁxed a ∈ R+ , we see that for F ∈ E, F = 0, we have Sa ( F H ) Sa F = ≤ Sa . F F H We conclude that Sa is bounded and Sa ≤ Sa . If f = |f | , for every f ∈ E, obviously we get Sa = Sa . Introduce the operator P+ : L1loc (R, H) −→ L1loc (R+ , H) deﬁned by the formula (P+ F )(x) = F (x), a.e. on R+ . Definition 1.4. We call a Wiener–Hopf operator on E every bounded operator T on E such that TΦ = S−a TSa Φ, ∀a > 0, ∀Φ ∈ E. Denote by W (E) the set of the Wiener–Hopf operators on E. The main result of this paper is the following. Theorem 1.5. Let E be a Banach space satisfying (H1), (H2) and (H3). Let T ∈ W (E). 1) We have (TΦ)a ∈ L2 (R+ , H), ∀Φ ∈ Cc∞ (R+ ) ⊗ H, ∀a ∈ IE . 2) There exists Va ∈ L∞ (R+ , L(H)) such that a (.)]), ∀a ∈ IE , ∀Φ ∈ C ∞ (R+ ) ⊗ H. (TΦ)a = P+ F −1 (Va (.)[(Φ) c Moreover, ess supx∈R Va (x) ≤ C T , where C is a constant dependent only on E.

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◦

3) If UE = ∅, set ◦

V(x + ia) = Va (x), ∀a ∈ IE , f or almost every x ∈ R. Then for u, v ∈ H, the function z −→< u, V(z)[v] > is in

◦ H∞ (UE )

and sup

◦

z∈UE

V(z) ≤ C T .

Remark 1.6. We will see later that ρ(S) = ρ(S), ρ(S−1 ) = ρ(S−1) and IE = [− ln ρ(S−1 ), ln ρ(S)] = [− ln ρ(S−1 ), ln ρ(S)]. We also obtain the following. Theorem 1.7. We have −1 spec(S) ∩ spec(S−1 ) = z ∈ C,

1 ≤ |z| ≤ ρ(S) . ρ(S−1 )

The spectral characterization in Theorem 4 has not been known until now even in particular cases when E is a weighted Lp space with a simple weight.

2. Wiener–Hopf operators on Banach spaces of scalar functions on R+ In this section, we prove Theorem 1. We follow the arguments of [5] in our more general case. For the reader convenience we give the details of the steps which need some modiﬁcations. First, we show that every Wiener–Hopf operator is associated to a distribution. Denote by C0∞ (R+ ) the space of functions of C ∞ (R+ ) with support in ]0, +∞[. Set H 1 (R) = {f ∈ L2 (R) | f ∈ L2 (R)}, the derivative of f ∈ L2 (R) being computed in the sense of distributions. ∞ Lemma 2.1. If T ∈ W (E) and f ∈ CK (R+ ), then (T f ) = T (f ). ∞ Proof. Let f ∈ CK (R+ ) and let (hn )n≥0 ⊂ R be a sequence converging to 0. Since Shn f −f S f −f ∞ converges to f with respect to the topology of CK (R+ ), hnhn converges hn to f with respect to the topology of E. Then we have Shn f − f (x)φ(x)dx T (f )(x)φ(x)dx = T lim n→+∞ hn R+ R+ S Tf − Tf hn (x)φ(x)dx = = lim (T f )(x)φ (x)dx, ∀φ ∈ Cc∞ (R+ ). hn R+ n→+∞ R+ Consequently, T (f ) = (T f ) in the sense of distributions.

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Proposition 2.2. If T is a Wiener–Hopf operator, then there exists a distribution µT of order 1 such that T f = P + (µT ∗ f ), for f ∈ Cc∞ (R+ ). The proof of Proposition 1 follows the arguments of that of Theorem 2 in [5] and we omit it. We just give the deﬁnition of µT . We have < µT , f >= lim (T Sx f˜)(x), x→+∞

for f ∈ x ∈ R.

Cc∞ (R),

where f˜ is the function deﬁned by f˜(x) = f (−x), for f ∈ Cc∞ (R),

Definition 2.3. If φ ∈ Cc∞ (R), we denote by Tφ the Wiener–Hopf operator such that Tφ f = P + (φ ∗ f ), ∀f ∈ Cc∞ (R+ ). Proposition 2.4. If T ∈ W (E), then there exists a sequence (φn )n∈N ⊂ Cc∞ (R+ ) such that lim Tφn f − T f = 0, ∀f ∈ E n→+∞

and Tφn ≤ C T , ∀n ∈ N, where C is a constant depending only on E. Proof. The proof follows the idea of the proof of Theorem 3 in [5], but here we must work with Bochner integrals and this leads to some diﬃculties. For the convenience of the reader we give the details. Let T ∈ W (E) and set T (t) = Γt ◦T ◦Γ−t , ∀t ∈ R. For a > 0, and f ∈ E we have (S−a T (t)Sa f )(x) = (T (t)Sa f )(x + a) = eit(x+a) T (f (s − a)e−its ) (x + a) = eitx S−a T f (s − a)e−it(s−a) (x) = eitx (S−a T Sa (Γ−t f ))(x) = (T (t)f )(x), a.e. This shows that T (t) ∈ W (E). Moreover, we have T (t) = T , for t ∈ R and T (0) = T . The application T is continuous from R into W (E). For n ∈ N, η ∈ R, x ∈ R, set η gn (η) := 1 − χ[−n,n] (η) n and 1 − cos(nx) γn (x) = . πx2 n We have γ n (η) = gn (η), ∀η ∈ R, ∀n ∈ N. Clearly, γn L1 = 1 for all n and lim γn (x)dx = 0, ∀a > 0. n→+∞

|x|≥a

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Set Yn := (T ∗ γn )(0). Then for f ∈ E we obtain lim Yn f − T f = 0.

n→+∞

∞ We claim that for f ∈ CK (R+ ), we have (Yn f )(y) = (T (x)f )(y)γn (−x)dx, ∀y ∈ R+ .

(2.1)

R

From Lemma 1, we know that for ﬁxed x ∈ R the function R+ y −→ (T (x)f )(y) ∞ is C ∞ . Let K0 be a compact subset of R+ and let ψ ∈ CK (R+ ). We see that 0

|ψ(y)(T (x)f )(y)| = |ψ(y)(µT ∗ Γ−x (f ))(y)| = |ψ(y) < µT,z , f (z − y)e−ix(z−y) > | ≤ C(µ) ψ ∞ ( Sy Γ−x f ∞ + (Sy Γ−x f ) ∞ ) ≤ C(µ) ψ ∞ ( f ∞ + f ∞ ), ∀y ∈ K0 . Consequently,

R

and hence the integral

ψT (x)f ∞ γn (−x)dx < +∞ R

ψ(T (x)f )γn (−x)dx

∞ is a well-deﬁned Bochner integral with values in CK (R+ ). The map 0 ∞ CK (R+ ) g −→ g(x) ∈ C, 0

is a continuous linear form for every x ∈ R+ . Since Bochner integrals commute with continuous linear forms (see [3]) we have ψ(y)(Yn f )(y) = ψ(y) (T (x)f )(y)γn (−x)dx, ∀y ∈ R+ R

and the claim (2.1) is proved. It is clear that Yn f ≤ T (x)f γn (−x)dx ≤ T (x) f , ∀f ∈ E. R

Since T (x) = T , we get Yn ≤ T , ∀n ∈ N. Now consider the distribution associated to Yn . Let K be a compact subset of R and let zK ≥ 1 be such that K ⊂] − ∞, zK [. Choose g ∈ Cc∞ (R+ ) such that ∞ g is positive, suppg ⊂ [zK − 1, zK + 1] and g(zK ) = 1. For f ∈ CK (R), we have

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gT (SzK (f˜gn )) ∈ H 1 (R) and it follows from Sobolev’s lemma (see [7]) that |(T SzK (f˜gn ))(zK )| = |g(zK )(T SzK (f˜gn ))(zK )| 12 ≤C g(y)2 |(T SzK (f˜gn )(y)|2 dy |y−zK |≤1

+

|y−zK |≤1

|(g(T SzK (f˜gn )) (y)|2 dy

12

,

where C > 0 is a constant. Taking into account (H1), T may be considered as a bounded operator from Cc∞ (R+ ) into L1loc (R+ ) and we have |(T SzK (f˜gn ))(zK )| ˜ n ∞ + (f˜gn ) ∞ ) ≤ C(K) T ( fg ˜ ≤ C(K)( f ∞ + f ∞ ), ˜ where C(K) and C(K) are constants depending only on K. Therefore ∞ ˜ |(T Sz (f˜gn ))(z)| ≤ C(K)( f ∞ + f ∞ ), ∀z ≥ zK , ∀f ∈ CK (R)

and we conclude that µT gn deﬁned by < µT gn , f >= lim (T Sz (f˜gn ))(z) z→+∞

is a distribution of order 1. On the other hand, we have e−isy (T (Γs f ))(y)γn (s)ds (Yn f )(y) = (T (−s)f )(y)γn (s)ds = R R = < µT,x , f (y − x)e−isx > γn (s)ds =< µT,x , f (y − x) γn (s)e−isx ds > R

R

=< µT,x , f (y − x)gn (x) >= (µT gn ∗ f )(y), ∀y ≥ 0,

∀f ∈ Cc∞ (R+ ).

Finally, we obtain Yn f = P + (µT gn ∗ f ), ∀f ∈ Cc∞ (R+ ), ∀n ∈ N. Since suppµT gn ⊂ [−n, n], it is suﬃcient to obtain the Proposition 2 for T ∈ W (E) such that µT is a distribution with compact support. Without lost of generality we assume that µT is with compact support. Let (θn )n∈N ⊂ Cc∞ (R+ ) be a sequence such that suppθn ⊂ [0, n1 ], θn ≥ 0, θn (x)dx = 0, ∀a > 0 lim n→+∞

x≥a

and θn L1 = 1, for n ∈ N. For f ∈ E we have lim θn ∗ f − f = 0.

n→+∞

Set Tn f = T (θn ∗ f ), ∀f ∈ E.

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We conclude that (Tn )n∈N converges to T with respect to the strong operator topology and Tn = Tφn , where φn = µT ∗ θn ∈ Cc∞ (R+ ). For f ∈ E, we have n1 θn (y)Sy (µT ∗ f )dy Tn f = P + 0

≤

0

≤

1 n

1 n

θn (y)P + (µT ∗ Sy f )dy

θn (y) T Sy f dy,

0

Then we obtain Tn ≤

1 n

0

∀f ∈ Cc∞ (R+ ).

θn (y) Sy dy T , ∀n ∈ N

and this completes the proof of the proposition.

We need also the following lemma. Lemma 2.5. For every φ ∈ Cc∞ (R+ ), we have ˆ |φ(α)| ≤ Tφ , ∀α ∈ UE .

(2.2)

Proof. We use the fact that for a bounded operator A on E, there exists a sequence (fn )n∈N ⊂ E such that: lim Afn − ρ(A)fn = 0 and fn = 1, ∀n ∈ N.

n→+∞

Fix λ = ρ(S). Let (fn,1 )n∈N be a sequence of E such that lim Sfn,1 − ρ(S)fn,1 = 0

n→+∞

and fn,1 = 1, ∀n ∈ N. ∗

For p ∈ N , observe that

1

λ p = ρ(S p1 ). Let (fn, p1 )n∈N ⊂ E be a sequence such that lim S p1 fn, p1 − ρ(S p1 )fn, p1 = 0 n→+∞

fn, p1 = 1, ∀n ∈ N.

and

Notice that for all q ∈ N∗ , such that q ≤ p we have: p! p! 1 1 S 1q fn, p!1 − λ q fn, p!1 = (S p!1 ) q fn, p!1 − (λ p! ) q fn, p!1

1 1 ≤ S p!1 − uλ p! S p!1 fn, p!1 − λ p! fn, p!1 . u∈C, u

p! q

=1, u =1

(2.3)

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We have

1 S p!1 − uλ p! ≤ C,

u∈C, u

p! q

IEOT

=1, u =1

where C is a constant independent of n and hence we have 1 lim S q1 fn, p!1 − λ q fn, p!1 = 0. n→+∞

Consequently, by a diagonal extraction, we can construct (fn )n∈N such that : 1 lim S p1 fn − λ p fn = 0, ∀p ∈ N∗ n→+∞

and fn = 1, ∀n ∈ N. For all p ∈ N∗ and for all q ∈ N, we have q

1

S pq fn − λ p fn = Cq,p (S p1 − λ p I) fn , where Cq,p is a linear combination of translations. Then q

1

S pq fn − λ p fn ≤ Cq,p S p1 fn − λ p fn , ∀n ∈ N and

q

lim S pq fn − λ p fn = 0.

n→+∞

On the other hand, q

q

p

S− pq fn − λ− p fn ≤ |λ− p | S− qp λ q fn − S pq fn , ∀n ∈ N and

q

lim S− pq fn − λ− p fn = 0, ∀p ∈ N∗ , ∀q ∈ N.

n→+∞

Since Q is dense in R, we deduce that lim St fn − λt fn = 0, ∀t ∈ R.

n→+∞

Now, ﬁx φ ∈ Cc∞ (R+ ). Notice that R

φ(x)Sx fn dx

is a well-deﬁned Bochner interval on E and T φ fn = φ(x) (Sx fn ) dx.

(2.4)

R

∞ ∞ (R+ )) ⊂ CK+supp(φ) (R+ ) Indeed, let K be a compact subset of R+ . We have Tφ (CK ∞ (R+ ) can be considered as a Bochner and the restriction of R φ(x)Sx dx to CK ∞ + integral on CK (R ) with values in CK+supp(φ) (R+ ). It is clear that for x ∈ R+ , the map f −→ f (x)

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∞ is a continuous linear form on CK (R+ ), for every compact K0 . Since Bochner 0 integrals commute with continuous linear forms, we obtain, for g ∈ Cc∞ (R+ ), (Tφ g)(x) = (φ ∗ g)(x) = φ(y)g(x − y)dy = φ(y)(Sy g)(x)dy R supp(φ) = φ(y)Sy g (x), ∀x ∈ R+ supp(φ)

and the formula (2.4) follows from the density of Cc∞ (R+ ) in E. Then, for all n ∈ N, we get x φ(x)λx dx fn φ(x)λ dx = R R ≤ φ(x)λx fn dx − φ(x)Sx fn dx + φ(x)Sx fn dx R R R |φ(x)| λx fn − Sx fn dx + Tφ . ≤ R

Taking into account the properties of (fn )n∈N and the dominated convergence theorem, it follows that |φ(x)| λx fn − Sx fn dx = 0. lim n→+∞

R

Denote by Cr the circle of radius r and denote by Dr the line Dr = {z ∈ C | z = r}. We will write eia. for the function x −→ eiax . Since Tφ = Teia. φ , for all a ∈ R, we obtain φ(x)λx dx ≤ Tφ , ∀λ ∈ Cρ(S) . R

We conclude that ˆ |φ(α)| ≤ Tφ , ∀φ ∈ Cc∞ (R), ∀α ∈ Dln ρ(S) . Denote by E ∗ the dual space of E and denote by . ∗ the norm of E ∗ . For λ ∈ C ρ(S1 ) , applying the same methods in E ∗ , we obtain that there exists a sequence −1

(gn )n∈N ⊂ E ∗ such that

lim (Sx )∗ gn − λx gn ∗ = 0

n→+∞

and gn ∗ = 1, ∀n ∈ N. We notice that we have ∗ ∗ Tφ = φ(x)Sx dx = R+

R+

φ(x)(Sx )∗ dx, ∀φ ∈ Cc∞ (R),

(2.5)

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see [3]. Then we obtain as above that ˆ |φ(α)| ≤ Tφ∗ = Tφ , ∀φ ∈ Cc∞ (R), ∀α ∈ D− ln ρ(S−1 ) . From the Phragmen-Lindel¨of theorem, it follows that ˆ |φ(α)| ≤ Tφ , ∀φ ∈ C ∞ (R), ∀α ∈ UE . c

The proof of Theorem 1 follows from Proposition 2 and Lemma 2 exactly in the same way as in the proof of Theorem 1 in [5] and we omit the details. In the proof of Theorem 2, we need the following technical lemma. Lemma 2.6. Let > 0, η0 > 0 and V = {ξ ∈ R+ : |η0 − ξ| ≤ δ} ⊂ R+ be ﬁxed. Let C0 > 0 be a ﬁxed constant. For t0 > 0 suﬃciently large there exists a function f ∈ Cc∞ (R+ ) with the properties: |fˆ(ξ)|dξ ≤ /C0 . (2.6) R\V

R

√ |fˆ(ξ)|dξ ≤ 2 2π. |f (t0 )| = 1.

(2.7) (2.8)

Proof. Introduce the function g with Fourier transform 1 (ξ−η0 )2 gˆ(ξ) = e− 2a2 e−it0 ξ , a where a > 0 will be taken small enough below. We have (ξ−η0 )2 1 |ˆ g (ξ)|dξ = e− 2a2 dξ a |ξ−η0 |≥δ R\V (ξ−η0 )2 δ2 1 − 4a 2 ≤e e− 4a2 dξ ≤ a R 2C0 for a > 0 small enough. We ﬁx a > 0 with this property. Obviously, √ 2 |ˆ g (ξ)|dξ = e−µ /2 dµ = 2π. R

R

On the other hand, (ξ−η0 )2 1 g(t) = e− 2a2 ei(t−t0 )ξ dξ 2πa R a2 (t−t0 )2 2 1 i(t−t0 )η0 2 = e e−µ /2 ei(t−t0 )aµ dµ = e− ei(t−t0 )η0 2π R and |g(t0 )| = 1. Now we will take t0 > 2 suﬃciently large. Let ϕ ∈ Cc∞ (R) be a ﬁxed function such that ϕ(t) = 0 for t ≤ 1/2 and for t ≥ 2t0 − 1/2 and let ϕ(t) = 1 for 1 ≤ t ≤ 2t0 − 1, 0 ≤ ϕ ≤ 1. Introduce the function f = ϕg ∈ Cc∞ (R+ ). The property (2.8) is trivial. We will show that (2.6) is satisﬁed for t0 > 0 large enough depending on the choice of a > 0. The proof of (2.7) is similar and easier.

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The function F = (ϕ − 1)g has a small Fourier transform. Moreover, given > 0 we can take t0 > 0 large enough in order to have . (2.9) |(1 + ξ 2 )Fˆ (ξ)| ≤ 2πC0 Indeed, for ξ 2 Fˆ (ξ) we use an integration by parts with respect to t using the fact that ξ 2 e−itξ = −∂t2 e−itξ . On the support of (ϕ− 1) we have |t − t0 | > t0 − 1. Thus after the integration by parts in the integral R e−itξ (1 + ξ 2 )F (t)dt we are going to estimate an integral a2 (t−t0 )2 2 e− |P (t)|dt |t−t0 |≥t0 −1

with P a polynomial of degree not greater than 2. To get (2.9), remark that this integral is bounded by ∞ 1−t0

2 2 2 2 C y 2 e−a y /2 dy + y 2 e−a y /2 dy −∞

t0 −1

and taking t0 > 0 suﬃciently large we arrange (2.9). Next we obtain |fˆ(ξ)|dξ ≤ |ˆ g (ξ)|dξ + |Fˆ (ξ)|dξ R\V R\V R\V + (1 + ξ 2 )−1 dξ ≤ . ≤ 2C0 2πC0 R C0 The proof of the lemma is complete. Proof of Theorem 2. First, we show that 1 z ∈ C, ≤ |z| ≤ ρ(S) ⊂ spec(S). ρ(S−1 )

(2.10)

Fix λ ∈ / spec(S). Then the operator (S − λI)−1 is a Wiener–Hopf operator and following 2) of Theorem 1, we get (S − λI)−1 (f )a = P + F −1 (νa (f )a ), ∀a ∈ IE , ∀f ∈ Cc∞ (R+ ), where νa ∈ L∞ (R). Replacing, f by (S − λI)g, we obtain (g)a = P + F −1 νa F ((S − λI)g)a , ∀g ∈ Cc∞ (R+ ). Denote by ea+i. the function

x −→ ea+ix .

It is easy to see that F ((Sg)a )(t) = ea−it F ((g)a )(t), ∀a ∈ IE , ∀t ∈ R, ∀g ∈ Cc∞ (R+ ). Consequently, a )](t), ∀a ∈ IE , ∀t ∈ R+ , ∀g ∈ C ∞ (R+ ). (g)a (t) = F −1 [(ea−i. − λ)νa (g) c

(2.11)

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We have a ] ∞ ≤ |(ea−i. − λ)|νa (g) a L1 (R) , ∀a ∈ IE , ∀g ∈ C ∞ (R+ ). F −1 [(ea−i. − λ)νa (g) c (2.12) Now, suppose that |λ| = eb , for some b ∈ IE . Choose a small ∈]0, 1[. It is easy to ﬁnd an interval V ⊂ R+ such that |eb−it − λ| ≤ , ∀t ∈ V . 2 νb ∞ Taking into account Lemma 3, we can choose g ∈ Cc∞ (R+ ) satisfying the following three conditions: b (t)|dt ≤ b 1) R\V |(g) 4e νb ∞ . 2) |(g)b (t)|dt ≤ 1. V

3) There exists t0 ∈ R+ , such that |(g)b (t0 )| ≥ . Taking into account that (2.11) and (2.12) hold for g ∈ Cc∞ (R+ ),we get b−it b (t)|dt ≤ . |(g)b (t0 )| ≤ |e − λ| νb ∞ |(g)b (t)|dt + |eb−it − λ| νb ∞ |(g) V

R\V

a

Hence we obtain a contradiction so |λ| = e , ∀a ∈ IE and (2.10) follows. We will prove now that 1 z ∈ C, ≤ |z| ≤ ρ(S−1 ) ⊂ spec(S−1 ). ρ(S)

(2.13)

Let λ ∈ / spec(S−1 ). Then (S−1 −λI)−1 ∈ W (E). Indeed, for all x ∈ R+ , we observe that S−x (S−1 − λI)−1 Sx = (S−1 − λI)−1 (S−1 − λI)S−x (S−1 − λI)−1 Sx = (S−1 − λI)−1 S−x (S−1 − λI)(S−1 − λI)−1 Sx = (S−1 − λI)−1 . Hence, for all g ∈ Cc∞ (R+ ) and for each a ∈ IE , we have a ), ((S−1 − λI)−1 g)a = P + F −1 (ha (g) for some ha ∈ L∞ (R) and

(f )a = P + F −1 ha F (((S−1 − λI)f )a ) , ∀f ∈ Cc∞ (R+ ).

Then

F ((S−1 − λI)f )a (t) = (eit−a − λ)(f )a (t), a.e. on R+ ,

if we suppose that supp(f ) ⊂ [1, ∞[. Repeating the argument of the proof of (2.10), we get a contradiction if |λ| = e−a , for some a ∈ IE . We conclude that 1 ≤ |z| ≤ ρ(S−1 ) ⊂ spec(S−1 ). z ∈ C, ρ(S)

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It follows that, if z ∈ C is such that ρ(S1−1 ) ≤ |z| ≤ ρ(S) then z1 ∈ spec(S−1 ) and we deduce that −1 1 ≤ |z| ≤ ρ(S) ⊂ spec(S) ∩ spec(S−1 ) z ∈ C, . ρ(S−1 ) From the deﬁnition of the spectral radius we get immediately that −1 1 spec(S) ∩ spec(S−1 ) ≤ |z| ≤ ρ(S) ⊂ z ∈ C, ρ(S−1 ) and the proof of Theorem 2 is complete. Proposition 2.7. If φ ∈

Cc∞ (R),

then

E ) ⊂ spec(Tφ ). φ(U Proof. Fix λ ∈ / spec(Tφ ). Then (Tφ − λI)−1 is a Wiener–Hopf operator and we obtain as above a − λ](g) a ), ∀g ∈ C ∞ (R+ ), ∀a ∈ IE , (g)a = P + F −1 (νa [(φ) c where νa ∈ L∞ (R). Choosing a suitable g ∈ Cc∞ (R+ ), we obtain in the same way as in the proof of (2.10) a contradiction if a (t) = λ, (φ) for some a ∈ IE and some t ∈ R and the proposition follows immediately.

3. Wiener–Hopf operators on Banach spaces of functions on R+ with values in a Hilbert space H Now let H be a separable Hilbert space. Denote by < u, v > the scalar product of u, v ∈ H. Let u H be the norm of u ∈ H. In this section we prove Theorem 3. Let E be the Banach space of functions from R+ into H satisfying (H1), (H2) and (H3). Let E be a Banach space of functions F : R+ −→ H such that

R+ x −→ F (x) H ∈ E.

We have the following two lemmas. Lemma 3.1. The space Cc∞ (R+ ) ⊗ H is dense in E. Proof. Let Φ ∈ E. Then there exists a positive sequence (φn )n∈N ⊂ Cc∞ (R+ ) such that lim φn − Φ(.) H E = 0. n→+∞

For almost every x ∈ R+ , set Φn (x) = φn (x)

Φ(x) , if Φ(x) = 0, Φ(x)

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Φn (x) = 0, if Φ(x) = 0. We have Φn − Φ E = Φn (.) − Φ(.) H E = φn − Φ(.) H E and it is clear that lim Φn − Φ E = 0.

n→+∞

Since Cc∞ (R+ ) ⊗ H is dense in C0 (R+ , H), the space Cc∞ (R+ ) ⊗ H is dense in E. Lemma 3.2. If Φ ∈ E and u ∈ H, then the function deﬁned by R+ x −→< Φ(x), u >∈ C is an element of E. N Proof. Let n=1 φn un

N ≥0

Cc∞ (R+ ), ∀n ∈ N and

be a sequence in Cc∞ (R+ ) ⊗ H such that φn ∈

N φn un − Φ = 0. lim

N →+∞

n=1

E

Let u ∈ H. Then we have N lim < φn (.)un , u > − < Φ(.), u > N →+∞

≤

n=1

N

lim

N →+∞

n=1

E

φn (.)un − Φ(.) u = 0. H

E

Now, it is clear that x −→< Φ(x), u >∈ C

is a element of E. In the proof of Theorem 3 we will also use the following lemma. Lemma 3.3. Let G ∈ L2 (R, H) and v ∈ H. Then we have F (< G(.), v >)(x) =< F (G)(x), v >, for almost every x ∈ R.

The reader may ﬁnd the proof of Lemma 6 in [6]. Next we pass to the proof of our main result. Proof of Theorem 3. Let T ∈ W (E). Fix u, v ∈ H. Deﬁne Tu,v on E by the formula (Tu,v f )(x) =< T(f u)(x), v >, ∀f ∈ E, a.e. From Lemma 5, it follows that Tu,v is an operator from E into E. It is clear that S−x < T(Sx f u), v >=< S−x T(Sx f u), v >=< T (f u), v >, ∀x ∈ R+ .

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Then we see that Tu,v ∈ W (E). Following Theorem 1, for a ∈ IE there exists a function νa,u,v ∈ L∞ (R) such that )a ), ∀ f ∈ Cc∞ (R+ ). (Tu,v f )a = P + F −1 (νa,u,v (f Let B be an orthonormal basis of H and let O be the set of ﬁnite linear combinations of elements of B. We have |νa,u,v (x)| ≤ C Tu,v , ∀x ∈ R\Nu,v , where Nu,v is a set of measure zero. Without loss of generality, we can modify νa,u,v on N = ∪(u,v)∈O×O Nu,v in order to obtain |νa,u,v (x)| ≤ C Mu,v ≤ C T u v , ∀u, v ∈ O, a.e. For ﬁxed x ∈ R\N we observe that O × O (u, v) −→ νa,u,v (x) ∈ C is a sesquilinear and continuous form on O × O and since O is dense in H, we conclude that there exists an unique map H × H (u, v) −→ νa,u,v (x) ∈ C such that νa,u,v (x) = νa,u,v (x), ∀u, v ∈ O. Consequently, there exists an unique map Va : R −→ L(H) such that < Va (x)[u], v >= νa,u,v (x), ∀u, v ∈ H, a.e. It is clear that Va (x) =

sup

u =1, v =1

| < Va (x)[u], v > | ≤ C T , a.e.

Fix a ∈ IE and f ∈ Cc∞ (R+ ). It is obvious that we have (f )a (x)u ∈ H, ∀x ∈ R. + Next for almost every x ∈ R , we obtain F −1 < Va (.)[(f )a (.)u], v > (x) = F −1 < Va (.)[u], v > (f )a (.) (x) )a (.) (x) = (Tu,v f )a (x). = F −1 ν˜a,u,v (.)(f Consequently, F −1 (< Va (.)[(f )a (.)u], v >)(x) = (< T[f u](.), v >)a (x), +

+

(3.1)

for almost every x ∈ R . Now, consider the function Ψa on R deﬁned for almost every x ∈ R+ by the formula Ψa (x) = Va (x)[(f )a (x)u]

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and observe that Ψa ∈ L2 (R+ , H). Indeed, we have Va (x)[(f )a (x)u] 2 dx R+ ≤ Va (x) 2 (f )a (x)u 2 dx + R ≤ C 2 T 2 |(f )a (x)|2 u 2dx < +∞. R+

This makes possible to apply Lemma 6, and we get F −1 (< Va (.)[(f )a (.)u], v >)(x) =< F −1 (Va (.)[(f )a (.)u])(x), v >, for almost every x ∈ R+ . It follows from (3.1) that we have (T[f u])a (x) = F −1 (Va (.)[(f )a (.)u])(x), for almost every x ∈ R+ and this yields (T[f u])a ∈ L2 (R+ , H). This completes the proof of 1) and 2). The proof of 3) uses the same argument as the proof of the assertion 3) of Theorem 1. Proof of Theorem 4. Fix α ∈ C and suppose that α ∈ / spec(S). Then we have (S − αI)−1 ∈ W (E) and from Theorem 3, we get )a (.)]), ∀a ∈ IE , ∀F ∈ Cc∞ (R+ ) ⊗ H. ((S − αI)−1 F )a = F −1 (Va (.)[(F Replacing F by (S − αI)G, we get

a (.)] (x). (G)a (x) = F −1 (Va (.)F [(S − αI)G](.))(x) = F −1 Va (.)[(ea−i. − α)(G)

We have

a (.)] L1 (R) , ∀a ∈ IE . (G)a ∞ ≤ Va (.)[(ea−i. − α)(G)

Then if |α| = ea , for some a ∈ IE choosing a suitable G ∈ Cc∞ (R+ ) ⊗ H in the same way as in the proof of Theorem 2, we obtain a contradiction. Hence, 1 z ∈ C, ≤ |z| ≤ ρ(S) ⊂ spec(S). ρ(S−1 ) In the same way, we obtain 1 z ∈ C, ≤ |z| ≤ ρ(S) ⊂ (spec(S−1 ))−1 . ρ(S−1 ) It follows that z ∈ C,

−1 1 . ≤ |z| ≤ ρ(S) ⊂ spec(S) ∩ spec(S−1 ) ρ(S−1 )

Taking into account that −1 ⊂ z ∈ C, spec(S) ∩ spec(S−1 )

1 ≤ |z| ≤ ρ(S) ρ(S−1 )

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and S ≤ S , S−1 ≤ S−1 , (see Section 1), we observe that ρ(S) = ρ(S), ρ(S−1 ) = ρ(S−1 ). We deduce that

−1 spec(S) ∩ spec(S−1 ) = z ∈ C,

1 ≤ |z| ≤ ρ(S) ρ(S−1 )

and the proof of Theorem 4 is complete.

4. Generalizations In this section we ﬁrst deal with the Wiener–Hopf operators in a lager class of Banach spaces of functions on R+ with values in a separable Hilbert space. Let W be an operator-valued weight on R+ . It means that W : R+ −→ L(H) and W satisﬁes the property 0 < sup x∈R+

W (x + y) < +∞, ∀y ∈ R+ . W (x)

(4.1)

This implies (see [4], [5]) that for every compact K of R+ , we have sup W (x) < +∞.

x∈K

Notice that if H has a ﬁnite dimension, W is given by a matrix. We denote by LpW (R+ , H) the space of measurable functions F on R+ with values in H such that R+

W (x)[F (x)] pH dx < +∞.

For illustration we give a simple example. Example. If H is the space R5 , the operator-valued weight W deﬁned for x by the matrix   1 ex e3x 1 1  1+x x ex 1 e3x    x  e 1 1 x x+1     1 e2x 1  1 ex x2 x x 1 + x ex 2 is such that the condition (4.1) trivially holds. The space LpW (R+ , H) is equipped with the norm p1 W (x)[F (x)] pH dx . R+

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Let T be a Wiener–Hopf operator on LpW (R+ , H). We ﬁx u, v ∈ H. Notice that for u ∈ H and f ∈ Lp W (R+ ), we have f u ∈ LpW (R+ , H). Indeed, W (x)[f (x)u] p dx ≤ W (x) p |f (x)|p u p dx < +∞. R+

R+

Introduce the operator Tu,v deﬁned on Lp W (R+ ) by the formula (Tu,v f )(x) =< T(f u)(x), v >, a.e., ∀f ∈ Lp W (R+ ). It is easy to see that R+

W (x) p | < T(f u)(x), v > |p dx

≤

R+

W (x) p T p u p |f (x)|p v dx < +∞.

Consequently, Tu,v is a Wiener–Hopf operator on Lp W (R+ ). Therefore Tu,v has a symbol following Theorem 1. Applying the methods exposed in Section 3, we obtain that Theorem 3 holds also if we replace E by LpW (R+ , H), for 1 ≤ p < ∞. Denote by IW (resp. UW ) the set IE (resp. UE ) for E = Lp W (R+ ). We recall that IE and UE are deﬁned in the Introduction. We have the following. Theorem 4.1. Let T by a Wiener–Hopf operator on LpW (R+ , H), for 1 ≤ p < ∞. 1) We have (TΦ)a ∈ L2 (R+ , H), ∀Φ ∈ Cc∞ (R+ ) ⊗ H, ∀a ∈ IW . 2) There exists Va ∈ L∞ (R, L(H)) such that a ](.)), ∀a ∈ IW , ∀Φ ∈ C ∞ (R+ ) ⊗ H. (TΦ)a = P+ F −1 (Va (.)[(Φ) c Moreover, ess supx∈R Va (x) ≤ C T . ◦

3) If UW = ∅, set ◦

V(x + ia) = Va (x), ∀a ∈ IW , f or almost every x ∈ R V(z) ≤ C T and for u, v ∈ H, the function

We have sup

◦

is analytic on

◦ UW .

z∈UW

z −→< V(z)u, v >

The results of Section 3 and Section 4 hold if we replace H by a separable Banach space B satisfying the following conditions: 1) B has a countable basis. 2) The dual space of B denoted by B ∗ has a countable basis. For example these conditions are satisﬁed if B = lωp (Z), where ω is a weight on Z and 1 ≤ p < +∞. We recall that ω is a weight on Z, if ω is a positive sequence on Z satisfying ω(k + n) < +∞, ∀n ∈ Z. 0 < sup ω(k) k∈Z

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It is easy to see that B ∗ = lωq ∗ (Z), where q is such that is given by the formula 1 , ∀n ∈ Z. ω ∗ (n) = ω(−n)

377 1 p

+

1 q

= 1. The weight ω ∗

Denote by en the sequence deﬁned by en (k) = 0 if n = k and en (n) = 1. Considering the family {en }n∈Z included in lωp (Z) and in lωq ∗ (Z), it is trivial to see that the conditions 1) and 2) are satisﬁed. Let B be a Banach space satisfying 1) and 2). Let E by a Banach space of functions on R+ satisfying (H1)-(H3). Denote by < , >B the duality between B and B ∗ . Let E be the space of functions F : R+ −→ B such that F (.) B ∈ E. Let T be a Wiener–Hopf operator on E. Then using the operators Tu,v deﬁned by (Tu,v f )(x) =< T(f u)(x), v >B , ∀u ∈ B, ∀v ∈ B ∗ , a.e. and the arguments of the proof of Theorem 3, we obtain an extended version of Theorem 3 in the case of spaces of functions on R+ with values in B. For example Theorem 3 holds for the Wiener–Hopf operators on spaces of the form Lpω1 R+ , lωq 2 (Z) , for 1 ≤ p < ∞, 1 ≤ q < ∞, where ω1 (resp. ω2 ) is a weight on R+ (resp. Z). The arguments developed in this paper do not hold if we replace lωq 2 (Z) by Lqω2 (R), for q = 2. The existence of the symbol of a Wiener–Hopf operator

on the family of spaces Lpω1 R+ , Lqω2 (R) for q = 2 is an interesting direction of investigation.

References [1] I. M. Bund, Birnbaum-Orlicz spaces of functions on groups, Paciﬁc J. Math. 58 (1975), 351-359. [2] G.I. Gaudry, B.R.F. Jeﬀeries, W.J. Ricker, Vector-valued multipliers: convolution with operator-valued measures, Dissertations Math. 385 (2000). [3] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. (1957). [4] V. Petkova, Symbole d’un multiplicateur sur L2ω (R), Bull. Sci. Math. 128 (2004), 391415. [5] V. Petkova, Wiener–Hopf operators on L2ω (R+ ), Arch. Math. (Basel) 84 (2005), 311324. [6] V. Petkova, Multipliers on spaces of functions on a locally compact abelian group with values in a Hilbert space, Serdica Math. J. 32 (2006), 215-226. [7] G. Roos, Analyse et G´eom´etrie, M´ethodes hilbertiennes, Dunod, Paris, 2002.

378 Violeta Petkova Universit´e Paul S´ebatier UFR: MIG Laboratoire Emile Picard 118 route de Narbonne F-31062 Toulouse Cedex 4 France e-mail: [email protected] Submitted: October 6, 2006 Revised: June 13, 2007

Petkova

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Integr. equ. oper. theory 59 (2007), 379–419 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030379-41, published online October 18, 2007 DOI 10.1007/s00020-007-1537-6

Integral Equations and Operator Theory

Commutative C ∗-Algebras of Toeplitz Operators on the Unit Ball, I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators Raul Quiroga-Barranco and Nikolai Vasilevski Abstract. Extending known results for the unit disk, we prove that for the unit ball Bn there exist n + 2 diﬀerent cases of commutative C ∗ -algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47L80, 32A36. Keywords. Toeplitz operator, Bergman space, commutative C ∗ -algebra, unit ball, commutative groups of biholomorphisms.

1. Introduction The commutative C ∗ -algebras of Toeplitz operators acting on the (weighted) Bergman spaces over the unit disk as well as various properties of the operators from these algebras have been intensively studied recently (see, for example, [5, 6, 7, 8, 11, 15, 16]). It turned out that the smoothness properties of symbols do not play any essential role in order that the corresponding Toeplitz operators generate a commutative C ∗ -algebra. Surprisingly the deep reason lies in the geometry of the underlying manifold (the hyperbolic plane ≡ unit disk endowed with the standard hyperbolic metric, for the discussed case). The commutativity properties are governed only by the geometric conﬁguration of the level lines of symbols, while the symbols themselves can by merely measurable. As it turns out these level lines have to be the cycles of a pencil of hyperbolic geodesics. This work was partially supported by CONACYT Projects 46936 and 44620, M´exico.

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In this connection recall that there are three diﬀerent types of pencils of hyperbolic geodesics on the unit disk: an elliptic pencil, which is formed by geodesics intersecting in a single point, a parabolic pencil, which is formed by parallel geodesics, and a hyperbolic pencil, which is formed by disjoint geodesics, i.e., by all geodesics orthogonal to a given one. The orthogonal trajectories to geodesics forming a pencil are called cycles. The cycles are always equidistant in the hyperbolic metric. The main result of [8] states that assuming some natural conditions on the “richness” of the symbol set, the C ∗ -algebra generated by Toeplitz operators is commutative on each (commonly considered) weighted Bergman space if and only if there is a pencil of hyperbolic geodesics such that the symbols of the Toeplitz operators are constant on the cycles of this pencil. We mention that there is a natural one-to-one correspondence between the pencils of hyperbolic geodesics and the maximal commutative subgroups of the movements (conformal isometries) of the hyperbolic plane. Each such subgroup is just the one-parametric group generated by a (non identical) M¨obius transformation. Given any such subgroup, the cycles of the corresponding pencil are precisely the sets which remain invariant under the action of this subgroup. That is, the main result of [8] admits the following equivalent reformulation: assuming some natural conditions on the “richness” of the symbol set, the C ∗ -algebra generated by Toeplitz operators is commutative on each (commonly considered) weighted Bergman space if and only if there is a maximal commutative subgroup of the M¨ obius transformation such that the symbols of the Toeplitz operators are invariant under the action of this subgroup. The present paper is the ﬁrst part of a work aimed to extend the results from the unit disk in C to the unit ball in Cn . Our approach is based on the classiﬁcation of the maximal commutative subgroups of the biholomorphic automorphisms of the unit ball. In Section 3 of this Part I we list ﬁve diﬀerent types of commutative subgroups of the biholomorphisms of the unit ball Bn or its unbounded realization, the Siegel domain Dn . In the ﬁnal Section 10 we show that, given any such subgroup, the C ∗ -algebra, generated by Toeplitz operators with (bounded measurable) symbols which are invariant under the action of this subgroup, is commutative on each (commonly considered) weighted Bergman space. Moreover we show that in each case the corresponding Toeplitz operators Ta admit the spectral type representations, i.e., all of them are unitary equivalent to certain multiplication operators γa I. The ex plicit form of γa is given for each of the ﬁve cases under consideration. It is worth mentioning that such a spectral representation gives an easy access to the important properties of a Toeplitz operators: boundedness, compactness, spectral properties, invariant subspaces, etc. We note that one of the above types, namely the quasi-nilpotent, depends on a parameter k = 1, 2, . . . , n − 2. Thus for the unit ball Bn of (complex) dimension n we have in total n + 2 diﬀerent types of commutative C ∗ -algebras of Toeplitz

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operators. For n = 1 these algebras coincide exactly with the three known types of the commutative algebras on the unit disk. To achieve these results we construct in Sections 5 - 9 the analogues of the classical Bargmann transform and its inverse, which we use then as the unitary multiples in the representation R Ta R∗ = γa I. A general scheme to construct such analogues is presented in Section 4, while its concrete realizations for each of the above ﬁve diﬀerent cases constitute the content of Sections 5 - 9. In the forthcoming Part II of the work we will show that the above ﬁve commutative subgroups are maximal commutative ones, and that each maximal commutative subgroup of biholomorphisms is conjugate to one from our list, while neither two from the list are conjugate. That is, we will classify the maximal commutative subgroups of biholomorphisms of the unit ball Bn . Thus we will arrive to the following extension of the suﬃciency condition of the existence of commutative algebras of Toeplitz operators: given any maximal commutative subgroups of biholomorphisms of the unit ball Bn , the C ∗ -algebra, generated by Toeplitz operators with measurable bounded symbols which are invariant under the action of this group, is commutative. Another aim of Part II will be to describe the distinguished geometry of the level sets of such symbols (the orbits of maximal commutative subgroups of biholomorphisms of the unit ball), presenting thus the multidimensional generalization of a pencil of hyperbolic geodesics on the unit disk.

2. Weighted Bergman spaces and Bergman projections Denote by Bn the unit ball in Cn , that is, Bn = {z = (z1 , . . . , zn ) ∈ Cn : |z|2 = |z1 |2 + · · · + |zn |2 < 1}. Later on we will use the following notation for the points of Cn = Cn−1 × C: z = (z , zn ),

where z = (z1 , . . . , zn−1 ) ∈ Cn−1 , zn ∈ C.

Denote by Dn the following Siegel domain in Cn , Dn = {z = (z , zn ) ∈ Cn−1 × C : Im zn − |z |2 > 0}. It is well known (and easy to check directly) that the Cayley transform ζ = ω(z), where zk , k = 1, . . . , n − 1, ζk = i 1 + zn 1 − zn ζn = i , 1 + zn maps biholomorphically the unit ball Bn onto the Siegel domain Dn . The inverse transform z = ω −1 (ζ) is given by 2iζk zk = − , k = 1, . . . , n − 1, 1 − iζn 1 + iζn zn = . 1 − iζn

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Let D = Cn−1 × R × R+ . The mapping κ : (z , u, v) ∈ D −→ (z , u + iv + i|z |2 ) ∈ Dn ,

(2.1)

is obviously a diﬀeomorphism between D and Dn . Denote by dv(z) = dx1 dy1 · · · dxn dyn , where zk = xk + iyk , k = 1, . . . , n, the standard Lebesgue measure in Cn . We introduce the following one-parameter family of weights (see, for example, [17]), µλ (z) = cλ (1 − |z|2 )λ ,

λ > −1,

where the normalizing constant cλ =

Γ(n + λ + 1) π n Γ(λ + 1)

(2.2)

is chosen so that µλ (z)dv(z) is a probability measure in Bn . It is easy to see that under the inverse Cayley transform z = ω −1 (ζ) we have dv(z) = 1 − |z|2

=

1 + zn

=

22n dv(ζ), |1 − iζn |2n+2 22 (Im ζn − |z |2 ), |1 − iζn |2 2 . 1 − iζn

(2.3) (2.4) (2.5)

Given a function f ∈ L2 (Bn , µλ ), changing the variables z = ω −1 (ζ) and using (2.3), (2.4), we have f 2 = |f (z)|2 cλ (1 − |z|2 )λ dv(z) (2.6) Bn 22λ 22n = |f (ω −1 (ζ)|2 cλ (Im ζn − |ζ |2 )λ dv(ζ) 2λ |1 − iζ| |1 − iζ|2n+2 Dn 22n+2λ = |f (ω −1 (ζ)|2 cλ (Im ζn − |ζ |2 )λ dv(ζ). |1 − iζ|2n+2λ+2 Dn Introduce now the space L2 (Dn , µ λ ), where cλ (Im ζn − |ζ |2 )λ , µ λ (ζ) = 4 and the operator Uλ : L2 (Bn , µλ ) −→ L2 (Dn , µ λ ), which acts as

(Uλ f )(ζ) =

2 1 − iζn

n+λ+1

f (ω −1 (ζ)).

Then (2.6) can be rewritten as f 2L2 (Bn ,µλ ) = Uλ f 2L2 (Dn ,µλ ) .

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It is easy to check that the operator Uλ is unitary, and its inverse (and adjoint) operator λ ) −→ L2 (Bn , µλ ) Uλ−1 : L2 (Dn , µ has the form (Uλ−1 f )(z) =

1 f (ω(z)). (1 + zn )n+λ+1

Denote by A2λ (Bn ) and by A2λ (Dn ) the (weighted) Bergman subspaces of L2 (B , µλ ) and of L2 (Dn , µ λ ), respectively. Recall that, as always, the Bergman space is the subspace of the corresponding L2 -space which consists of all analytic functions. It is well known (see, for example, [17]) that the weighted Bergman kernel for the unit ball is given by n

KBn ,λ (z, ζ) =

1 1 = , n n+λ+1 (1 − z · ζ) (1 − k=1 zk ζ k )n+λ+1

and that the (weighted) Bergman projection BBn ,λ of L2 (Bn , µλ ) onto A2λ (Bn ) has the form (1 − |ζ|2 )λ (BBn ,λ f )(z) = f (ζ) cλ dv(ζ). (1 − z · ζ)n+λ+1 Bn Let z = ω −1 (w) and ζ = ω −1 (η), then 1−z·ζ =

22 (1 − iwn )(1 + iη n )

wn − η n − w · η . 2i

(2.7)

We note that the unitary operator Uλ , being the isomorphism between L2 (Bn , µλ ) λ ), maps isomorphically A2λ (Bn ) onto A2λ (Dn ) as well. Changing the and L2 (Dn , µ variables z = ω −1 (w), ζ = ω −1 (η) and using (2.3)–(2.5), (2.7), we have (Bf )(w)

= = =

(Uλ BBn ,λ Uλ−1 f )(w) n+λ+1 2 f (ω(ζ)) (1 − |ζ|2 )λ cλ dv(ζ) n+λ+1 1 − iwn (1 − ω −1 (w) · ζ)n+λ+1 Bn (1 + ζn ) n+λ+1 2 (1 − iηn )n+λ+1 f (η) 1 − iwn 2n+λ+1 Dn 22λ (Im ηn − |η |2 )λ (1 − iwn )n+λ+1 (1 + iη n )n+λ+1 |1 − iηn |2λ 22n+2λ+2 1 22n · dv(η) n+λ+1 |1 − iηn |2n+2 wn −η n · η − w 2i (Im ηn − |η |2 )λ cλ dv(η). f (η) n+λ+1 4 wn −η n Dn · η − w 2i · cλ

=

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That is, the (weighted) Bergman kernel for Dn has the form (see, for example, [2, 3, 9]) 1 KDn ,λ (z, ζ) = n+λ+1 , zn −ζ n · ζ − z 2i and the weighted Bergman projection BDn ,λ of L2 (Dn , µ λ ) onto the Bergman space A2λ (Dn ) is given by cλ (Im ζn − |ζ |2 )λ dv(ζ). (BDn ,λ f )(z) = f (ζ) n+λ+1 4 zn −ζ n Dn · ζ − z 2i Return now to the domain D = Cn−1 × R × R+ whose points we will denote by w = (z , u, v). Introduce the space L2 (D, ηλ ), where the weight ηλ is given by the formula cλ λ v , λ > −1, ηλ (w) = ηλ (v) = 4 and the constant cλ is given by (2.2). Introduce the operator U0 : L2 (Dn , µ λ ) −→ L2 (D, ηλ ) as follows, (U0 f )(w) = f (κ(w)), where the mapping κ is given by (2.1). The operator U0 is obviously unitary, and the inverse operator has the form (U0−1 f )(z) = f (κ−1 (z)). The (weighted) Bergman space A2λ (Dn ) on the Siegel domain Dn can be characterized alternatively as the (closed) subspace of L2 (Dn , µ λ ) which consists of all functions ϕ satisfying the equations ∂ ϕ = 0, k = 1, . . . , n. ∂z k Then the image A0 (D) = U0 (A2λ (Dn )) is the subspace of L2 (D, ηλ ) which consists of all functions ϕ satisfying the equations ∂ U −1 ϕ = 0, k = 1, . . . , n. U0 ∂z k 0 It is obvious that ∂ 1 ∂ ∂ −1 +i U = U0 . (2.8) ∂z n 0 2 ∂u ∂v While for k = 1, . . . , n − 1, we have ∂ ∂ U0−1 ϕ(z , u, v) = U0 ϕ(z , Re zn , Im zn − |z |2 ) U0 ∂z k ∂z k ∂ϕ ∂ϕ = U0 zk − ∂z k ∂v ∂ ∂ = zk ϕ. − ∂z k ∂v

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That is, ∂ ∂ ∂ zk , U −1 = − ∂z k 0 ∂z k ∂v ∂ in terms of ∂u using (2.8), U0

or, expressing

∂ ∂v

∂ ∂ ∂ zk . U −1 = −i ∂z k 0 ∂zk ∂u Thus the space A0 (D) coincides with the set of all L2 (D, ηλ )-functions which satisfy the equations ∂ ∂ 1 ∂ ∂ +i zk ϕ = 0, k = 1, . . . , n − 1, (2.9) − ϕ = 0 and 2 ∂u ∂v ∂z k ∂v U0

or the equations ∂ 1 ∂ +i ϕ = 0 and 2 ∂u ∂v

∂ ∂ zk ϕ = 0, −i ∂z k ∂u

k = 1, . . . , n − 1. (2.10)

3. Commutative subgroups of biholomorphisms We list here ﬁve essentially diﬀerent types of commutative subgroups of biholomorphisms of the unit ball Bn , or its unbounded realization, the Siegel domain Dn . In Part II of the paper we will show that, ﬁrst, these subgroups are maximal commutative subgroups of biholomorphisms, and second, each maximal commutative subgroup of biholomorphisms is conjugate to one from this list, while neither two from the list are conjugate. That is, in a sense, this list classiﬁes the maximal commutative subgroups of biholomorphisms of the unit ball Bn . Quasi-elliptic group of biholomorphisms of the unit ball Bn is isomorphic to n T with the following group action: t : z = (z1 , . . . , zn ) ∈ Bn −→ tz = (t1 z1 , . . . , tn zn ) ∈ Bn , for each t = (t1 , . . . , tn ) ∈ Tn . Quasi-parabolic group of biholomorphisms of the Siegel domain Dn is isomorphic to Tn−1 × R with the following group action: (t, h) : (z , zn ) ∈ Dn −→ (tz , zn + h) ∈ Dn , for each (t, h) ∈ Tn−1 × R. Quasi-hyperbolic group of biholomorphisms of the Siegel domain Dn is isomorphic to Tn−1 × R+ with the following group action: (t, r) : (z , zn ) ∈ Dn −→ (r1/2 tz , rzn ) ∈ Dn , for each (t, r) ∈ Tn−1 × R+ , Nilpotent group of biholomorphisms of the Siegel domain Dn is isomorphic to Rn−1 × R with the following group action: (b, h) : (z , zn ) ∈ Dn → (z + b, zn + h + 2iz · b + i|b|2 ) ∈ Dn , for each (b, h) ∈ Rn−1 × R;

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Quasi-nilpotent group of biholomorphisms of the Siegel domain Dn is isomorphic to Tk × Rn−k−1 × R, 0 < k < n − 1, with the following group action: (t, b, h) : (z , z , zn ) ∈ Dn −→ (tz , z + b, zn + h + 2iz · b + i|b|2 )) ∈ Dn , for each (t, b, h) ∈ Tk × Rn−k−1 × R. Note that setting in the quasi-nilpotent case k = n − 1 we obtain the quasiparabolic group, while for k = 0 we obtain the nilpotent group. At the same time we prefer to distinguish these three cases in order to make our calculations more transparent.

4. Bargmann type transform We describe here a scheme which has been already successfully used, for example, in [12, 13, 14, 15] and which will be used in the subsequent sections. Although the scheme is very simple, a considerable amount of work is required in each particular case in order to deﬁne and calculate all necessary data. The purpose of it is to give a description of the space of analytic functions under study in “real analysis terms”, i.e., as an appropriate L2 space, and to construct the operator R which being restricted onto the analytic space maps it isometrically onto the corresponding L2 space, and which together with its adjoint R∗ provide the factorization of the identity operator on the L2 space and the orthogonal projection of the initial Hilbert space onto the space of analytic functions. We note that the operator R restricted onto the space of analytic functions together with its inverse R∗ can be considered as the analogs of the classical Bargmann transform (and its inverse) [1], moreover they do give the classical Bargmann transform for the corresponding case, see [14]. Let H be a separable Hilbert space and A be its closed subspace. Denote by P the orthogonal projection of H onto A. We assume as well that there exist (i) measurable spaces X and Y with measures µ and η respectively, (ii) a unitary operator U : H −→ L2 (X, µ) ⊗ L2 (Y, η), (iii) a measurable subspace X1 of X and a function g0 = g0 (x, y) on X1 × Y , such that – for each x ∈ X1 the function g0 (x, ·) ∈ L2 (Y, η), and g0 (x, ·)L2 (Y,η) = 1, – the operator U maps A onto g0 L2 (X1 , µ) ⊂ L2 (X, µ) ⊗ L2 (Y, η): U : A −→ g0 L2 (X1 , µ). Then, for each ϕ = g0 f ∈ U (A) = g0 L2 (X1 , µ), where f ∈ L2 (X1 , µ), one has obviously ϕU(A) = f L2 (X1 ,µ) . We introduce now the isometric imbedding R0 : L2 (X1 , µ) −→ L2 (X, µ) ⊗ L2 (Y, η)

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by the formula R0 : f ∈ L2 (X1 , µ) −→ fg0 ∈ L2 (X, µ) ⊗ L2 (Y, η), where f =

f, x ∈ X1 . 0, x ∈ X \ X1

Then the adjoint operator R0∗ : L2 (X, µ) ⊗ L2 (Y, η) −→ L2 (X1 , µ) is given by (R0∗ ϕ)(x)

=

ϕ(x, y) g0 (x, y) dη,

x ∈ X1 .

Y

It is easy to check that R0∗ R = I R R0∗ = Q

: L2 (X1 , µ) −→ L2 (X1 , µ), : L2 (X, µ) ⊗ L2 (Y, η) −→ U (A),

where Q is the orthogonal projection of L2 (X, µ) ⊗ L2 (Y, η) onto the image of the space A under the unitary operator U , i.e., U (A) = g0 L2 (X1 , µ). Combining all the above we come to the following result. Theorem 4.1. The operator R = R0∗ U maps the Hilbert space H onto L2 (X1 , µ), and the restriction R|A : A −→ L2 (X1 , µ) is an isometric isomorphism. The adjoint operator R∗ = U ∗ R0 : L2 (X1 , µ) −→ A ⊂ H is an isometric isomorphism of L2 (X1 , µ) onto the subspace A of H. Furthermore RR∗ = I

: L2 (X1 , µ) −→ L2 (X1 , µ),

R∗ R = P

: H −→ A,

where P is the orthogonal projection of H onto A. In the subsequent ﬁve sections we will use this scheme in ﬁve diﬀerent cases deﬁning and calculating explicitly in each case all the necessary data. The nontrivial and essential part of the job is to ﬁnd the measurable spaces X, X1 , and Y and to construct the corresponding unitary operator U , appropriate for each speciﬁc case. The key idea here, as well as in [12, 13, 14, 15], is to make use of appropriate commutative subgroups of biholomorphisms of the domain under consideration, and doing the corresponding group Fourier transform, to reduce the system of partial diﬀerential equations, which deﬁne the Bergman space, to the ordinary ones, depending on certain parameters. Solving the last ordinary diﬀerential equations we come to the independent “real analysis type” description of the Bergman space.

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In what follows we will use the ﬁve diﬀerent types of commutative subgroups, described in Section 3, and we will name each of the ﬁve subsequent sections according to the above subgroups.

5. Quasi-elliptic case The results for this case are already known, see [10]. For the sake of completeness we present them here. Denote by τ (Bn ) the base of the unit ball Bn , considered as a Reinhard domain, i.e., τ (Bn ) = {r = (r1 , . . . , rn ) = (|z1 |, . . . , |zn |) : r2 = r12 + · · · + rn2 ∈ [0, 1)}, which belongs to Rn+ = R+ × · · · × R+ . Introduce in Cn the polar coordinates zk = tk rk , rk ∈ R+ , where tk ∈ T = S 1 , k = 1, . . . , n. Then under the identiﬁcation z = (z1 , . . . , zn ) = (t1 r1 , . . . , tn rn ) = (t, r), where t = (t1 , . . . , tn ) ∈ Tn = T × · · · × T, r = (r1 , . . . , rn ) ∈ τ (Bn ), we have Bn = Tn × τ (Bn ) and L2 (Bn , µλ ) = L2 (Tn ) ⊗ L2 (τ (Bn ), µ), where L2 (Tn ) =

n

L2 (T,

k=1

dtk ) itk

and the measure dµ in L2 (τ (Bn ), µ) is given by dµ = µλ (r)

n

2 λ

rk drk = cλ (1 − r )

k=1

n

rk drk .

k=1

We deﬁne the discrete Fourier transform F : L2 (T) → l2 = l2 (Z) by dt 1 F : f −→ cn = √ f (t) t−n , n ∈ Z. it 2π S 1 The operator F is unitary and

1 F −1 = F ∗ : {cn }n∈Z −→ f = √ c n tn . 2π n∈Z

In terms of the scheme of Section 4 we have here X = Zn , X1 = Zn+ , Y = τ (Bn ),

L2 (X, µ) = l2 (Zn ), L2 (X1 , µ) = l2 (Zn+ ), L2 (Y, η) = L2 (τ (Bn ),

the unitary operator U is deﬁned as follows, U = F(n) ⊗ I : L2 (Tn ) ⊗ L2 (τ (Bn ), µ) −→ l2 (Zn ) ⊗ L2 (τ (Bn ), µ),

(5.1)

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where F(n) = F ⊗ · · · ⊗ F, and the function g0 (sequence in this case) has the form 1/2 (2π)n Γ(n + |p| + λ + 1) rp , r ∈ τ (Bn ), g0 (r) = p! Γ(n + λ + 1) n p∈Z+

and Zn+ = Z+ × · · · × Z+ with Z+ = {0} ∪ N. Introduce the isometric embedding R0 : l2 (Zn+ ) −→ l2 (Zn ) ⊗ L2 (τ (Bn ), µ) where

R0 : {cp }p∈Zn+ −→ cp (r) =

(2π)n Γ(n+|p|+λ+1) p! Γ(n+λ+1)

1/2

0,

cp rp , p ∈ Zn+ . p ∈ Zn \ Zn+

Then the adjoint operator R0∗ : l2 (Zn ) ⊗ L2 (τ (Bn ), µ) −→ l2 (Zn ) is deﬁned by R∗ : {fp (r)}p∈Zn −→ 0

1/2 n

(2π)n Γ(n + |p| + λ + 1) rp fp (r) cλ (1 − r2 )λ rk drk p! Γ(n + λ + 1) τ (Bn ) k=1

. p∈Zn +

Theorem 5.1. The operator R = R0 U maps L2 (Bn , µλ ) onto l2 (Zn+ ), and the restriction R|A2λ (Bn ) : A2λ (Bn ) −→ l2 (Zn+ ) is an isometric isomorphism. The adjoint operator R∗ = U ∗ R0 : l2 (Zn+ ) −→ A2λ (Bn ) ⊂ L2 (Bn , µλ ) is the isometric isomorphism of l2 (Zn+ ) onto the subspace A2λ (Bn ) of L2 (Bn , µλ ). Furthermore RR∗ = I R∗ R = BBn ,λ

: l2 (Zn+ ) −→ l2 (Zn+ ), : L2 (Bn , µλ ) −→ A2λ (Bn ),

where BBn ,λ is the Bergman projection of L2 (Bn , µλ ) onto A2λ (Bn ). Theorem 5.2. The isometric isomorphism R∗ = U ∗ R0 : l2 (Zn+ ) −→ A2λ (Bn ) is given by ∗

−n 2

R : {cp }p∈Zn+ −→ (2π)

(2π)n Γ(n + |p| + λ + 1) 1/2 cp z p . p! Γ(n + λ + 1) n

p∈Z+

(5.2)

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Corollary 5.3. The inverse isomorphism R : A2λ (Bn ) −→ l2 (Zn+ ) is given by R : ϕ(z) −→ n

(2π)− 2

(2π)n Γ(n + |p| + λ + 1) p! Γ(n + λ + 1)

(5.3)

1/2

ϕ(z) z p µλ (z) dv(z) D

. p∈Zn +

6. Quasi-parabolic case We represent the space L2 (D, ηλ ) as the following tensor product, L2 (D, ηλ ) = L2 (Cn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ), and consider the unitary operator U1 = I ⊗ F ⊗ I acting on it. Here 1 (F f )(ξ) = √ f (u)e−iξu du 2π R

(6.1)

is the standard Fourier transform on L2 (R). For the operators in (2.10) we have obviously ∂ 1 ∂ i ∂ +i U1 U1−1 = ξ+ , 2 ∂u ∂v 2 ∂v ∂ ∂ ∂ zk U1−1 = −i + ξzk , k = 1, . . . , n − 1. U1 ∂z k ∂u ∂z k Thus the image A1 (D) = U1 (A0 (D)) is the subspace of L2 (D, ηλ ) which consists of all functions satisfying the equations ∂ i ∂ + ξzk ϕ = 0, k = 1, . . . , n − 1. (6.2) ξ+ ϕ = 0 and 2 ∂v ∂z k The ﬁrst equation is easy to solve, and its general solution has the form ϕ(z , ξ, v) = ψ0 (z , ξ)e−ξv . The function ϕ has to be in L2 (D, ηλ ), which implies that its support on the variable ξ has to be in R+ . That is, ϕ(z , ξ, v) = χR+ (ξ)ψ(z , ξ)e−ξv . Further, the function ϕ has to satisfy the second equations in (6.2), that is, ∂ + ξzk ψ(z , ξ) = 0, k = 1, . . . , n − 1. ∂zk

(6.3)

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391

Introduce in Cn−1 the polar coordinates, zk = rk tk , where rk ∈ R+ , tk ∈ S 1 = T, k = 1, . . . , n − 1. Then, it is easy to see that ∂ ∂ tk tk ∂ + ξzk = − + 2ξrk , k = 1, . . . , n − 1. ∂zk 2 ∂rk rk ∂tk Represent now L2 (D, ηλ )

= L2 (Cn−1 × R × R+ , ηλ ) n−1 = L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ), + , rdr) ⊗ L2 (T

where rdr =

n−1

rk drk ,

k=1

n−1

L2 (T

)=

n−1 k=1

1 dtk L2 S , . itk

Introduce the unitary operator U2 = I ⊗ F(n−1) ⊗ I ⊗ I acting from n−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) L2 (Rn−1 + , rdr) ⊗ L2 (T

onto n−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) L2 (Rn−1 + , rdr) ⊗ l2 (Z

=

l2 (Zn−1 , L2 (Rn−1 + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )),

where F(n−1) = F ⊗ · · · ⊗ F, and each F is the one-dimensional discrete Fourier transform (5.1). For ϕ of the form (6.3) we have U2 ϕ = χR+ (ξ)e−ξv (F ⊗ I)ψ = χR+ (ξ)e−ξv {cp (r, ξ)}p∈Zn−1 . Furthere, the sequence {dp }p∈Zn−1 = χR+ (ξ)e−ξv {cp (r, ξ)}p∈Zn−1 has to satisfy the equations ∂ tk tk ∂ U2 − + 2ξrk U2−1 {dp }p∈Zn−1 = 0, k = 1, . . . , n − 1, 2 ∂rk rk ∂tk or equivalently tk ∂ ∂ tk − + 2ξrk U2−1 {cp }p∈Zn−1 = 0, U2 2 ∂rk rk ∂tk The operator tk δ = U2 2

k = 1, . . . , n − 1.

∂ tk ∂ − + 2ξrk U2−1 ∂rk rk ∂tk

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acts as follows,

n−1 ∂ tk ∂ − + 2ξrk (2π)− 2 c p tp ∂rk rk ∂tk p∈Zn−1   p tk ∂cp p n−1 t k U2 (2π)− 2 tkk − pk cp tkPk −1 + 2ξrk tpkk cp  tˆk 2 ∂r r k k p∈Zn−1   p p +1 1 ∂ n−1 p k − + 2ξrk cp  tˆk tkk U2 (2π)− 2 2 ∂r r k k p∈Zn−1   n−1 ∂ 1 − 1 p k U2 (2π)− 2 tp − + 2ξrk cp−ek  2 ∂r r k k p∈Zn−1 1 ∂ pk − 1 − + 2ξrk cp−ek , 2 ∂rk rk p∈Zn−1

δ {cp }p∈Zn−1 = U2

=

=

= =

IEOT

tk 2

pn−1 where p = (p1 , . . . , pn−1 ), ek = (0, . . . , 0, 1, 0, . . . 0), tp = tp11 · · · tn−1 , and tˆpk is tp pk with the multiple tk omitted. That is, ∂ tk tk ∂ U2 − + 2ξrk U2−1 {cp }p∈Zn−1 2 ∂rk rk ∂tk 1 ∂ pk − 1 = − + 2ξrk cp−ek . 2 ∂rk rk p∈Zn−1

Now the space A2 (D) = U2 (A1 (D)) consists of all sequences {dp }p∈Zn−1 , where dp = χR+ (ξ)e−ξv cp (r, ξ), p ∈ Zn−1 , which satisfy the equations 1 ∂ pk − + 2ξrk cp = 0, 2 ∂rk rk

k = 1, . . . , n − 1.

These equations are easy to solve and their general solution has the form 2

cp = gp (ξ) rp e−ξ|r| ,

p ∈ Zn−1 ,

2 where |r|2 = r12 + · · · + rn−1 . Further, each function

dp = χR+ (ξ) gp (ξ) rp e−ξ(|r|

2

+v)

has to be in L2 (Rn−1 + ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ), and moreover we need that {dp }p∈Zn−1 ∈ l2 (Zn−1 , L2 (Rn−1 + ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )). In particular, this implies that dp ≡ 0 for all p ∈ Zn−1 \ Zn−1 + .

Vol. 59 (2007)

Toeplitz Operators on the Unit Ball, I.

We set

393

1 2n+1 (2ξ)|p|+λ+n 2 αp (ξ) = cp (ξ) , cλ p! Γ(λ + 1) where cp (ξ) ∈ L2 (R+ ) and is prolonged by zero to the negative half-axis, the constant cλ is given by (2.2), |p| = p1 + · · ·+ pn−1 , p! = p1 ! · · · pn−1 !, and p ∈ Zn−1 + . Then it is easy to see that dp L2 (Rn−1 )⊗L2 (R)⊗L2 (R+ ,ηλ ) = cp L2 (R+ ) , +

and {dp }p∈Z n−1 l2 (Zn−1 , L2 (Rn−1 )⊗L2 (R)⊗L2 (R+ ,ηλ )) = cp l2 (Zn−1 , L2 (R+ )) . +

Indeed, dp 2

+

=

Rn+1 +

+

+

2n+1 (2ξ)|p|+λ+n 2p −2ξ(|r|2 +v) cλ λ r e v rdrdξdv cλ p! Γ(λ + 1) 4

|cp (ξ)|2

ξ |p|+λ+n p −ξ(r1 +···+rn−1 +v) λ r e v drdξdv 2p! Γ(λ + 1) Rn+1 + ξ |p|+λ+n = dξ |cp (ξ/2)|2 v λ e−ξv dv 2p! Γ(λ + 1) R+ R+ rp e−ξ(r1 +···+rn−1 ) dr. · =

|cp (ξ/2)|2

Rn−1 +

By [4], formula 3.351.3, we have rp e−ξ(r1 +···+rn−1 ) dr = Rn−1 +

p! ξ |p|+n−1

,

and by [4], formula 3.381.4, we have Γ(λ + 1) v λ e−ξv dv = . ξ λ+1 R+ Thus dp 2L2 (Rn−1 )⊗L2 (R)⊗L2 (R+ ,η

λ)

+

=

Note that in terms of the X = Zn−1 × R, X1 = Zn−1 × R+ , + Y = Rn−1 × R+ , +

R+

|cp (ξ/2)|2

dξ = 2

R+

|cp (ξ)|2 dξ = cp 2L2 (R+ ) .

scheme of Section 4 we have here L2 (X, µ) = l2 (Zn−1 ) ⊗ L2 (R), L2 (X1 , µ) = l2 (Zn−1 + ) ⊗ L2 (R+ ), L2 (Y, η) = L2 (Rn−1 + , rdr) ⊗ L2 (R+ , ηλ ),

the unitary operator U is deﬁned as follows, U

= U2 U1 U0 : L2 (Dn , µ λ ) −→ l2 (Zn−1 ) ⊗ L2 (Rn−1 + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) = l2 (Zn−1 , L2 (Rn−1 + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )),

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and the function g0 (function-sequence in this case) has the form

1 2n+1 (2ξ)|p|+λ+n 2 p −ξ(|r|2 +v) r e g0 (p, r, ξ, v) = cλ p! Γ(λ + 1)

,

p∈Zn−1 +

where (r, ξ, v) ∈ Rn−1 × R+ × R+ . + Summarizing the above we come to the following statement. Lemma 6.1. The unitary operator U = U2 U1 U0 maps the Bergman space A2λ (Dn ) onto the space A2 (D) = g0 l2 (Zn−1 + , L2 (R+ )) which is the closed subspace of n−1 n−1 l2 (Z , L2 (R+ ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) and consisits of all sequences {dp (r, ξ, v)}p∈Zn−1 , where the functions dp (r, ξ, v), p ∈ Zn−1 + , have the form +

dp (r, ξ, v) =

2n+1 (2ξ)|p|+λ+n cλ p! Γ(λ + 1)

12

rp e−ξ(|r|

2

+v)

cp (ξ),

ξ ∈ R+ ,

with cp ∈ L2 (R+ ). Introduce now the isometric imbedding n−1 , L2 (Rn−1 R0 : l2 (Zn−1 + , L2 (R+ )) −→ l2 (Z + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ))

by the rule R0 : {cp (ξ)}p∈Zn−1 − → +

n+1 1 2 (2ξ)|p|+λ+n 2 p −ξ(|r|2 +v) χZn−1 (p)χR+ (ξ) r e cp (ξ) + cλ p! Γ(λ + 1)

, p∈Zn−1

where the function cp (ξ) is extended by zero for ξ ∈ R \ R+ for each p ∈ Zn−1 + . The adjoint operator n−1 R0∗ : l2 (Zn−1 , L2 (Rn−1 + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) −→ l2 (Z+ , L2 (R+ ))

has obviously the form R0∗ : {dp (r, ξ, v)}p∈Zn−1 −→

1 2n+1 (2ξ)|p|+λ+n 2 cλ v λ p −ξ(|r|2 +v) dv r e dp (r, ξ, v) rdr cλ p! Γ(λ + 1) 4 Rn +

. p∈Zn−1 +

Then we have R0∗ R0 = I R0 R0∗ = P2

n−1 : l2 (Zn−1 + , L2 (R+ )) −→ l2 (Z+ , L2 (R+ )), n−1 : l2 (Zn−1 + , L2 (R+ , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) −→ A2 (D),

where P2 is the orthogonal projection of l2 (Zn−1 , L2 (Rn−1 + , rdr) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) onto A2 (D). Thus ﬁnally we have

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395

Theorem 6.2. The operator R = R0∗ U maps L2 (Dn , µ λ ) onto l2 (Zn−1 + , L2 (R+ )), and the restriction R|A2λ (Dn ) : A2λ (Dn ) −→ l2 (Zn−1 + , L2 (R+ )) is an isometric isomorphism. The adjoint operator 2 λ ) R∗ = U ∗ R0 : l2 (Zn−1 + , L2 (R+ )) −→ Aλ (Dn ) ⊂ L2 (Dn , µ 2 is the isometric isomorphism of l2 (Zn−1 + , L2 (R+ )) onto the subspace Aλ (Dn ) of λ ). L2 (Dn , µ Furthermore

RR∗ = I

n−1 : l2 (Zn−1 + , L2 (R+ )) −→ l2 (Z+ , L2 (R+ )),

R∗ R = BDn ,λ

: L2 (Dn , µ λ ) −→ A2λ (Dn ),

where BDn ,λ is the Bergman projection of L2 (Dn , µ λ ) onto A2λ (Dn ). Theorem 6.3. The isometric isomorphism 2 R∗ = U ∗ R0 : l2 (Zn−1 + , L2 (R+ )) −→ Aλ (Dn )

is given by R∗ : {cp (ξ)}p∈Zn−1 −→ +

−n 2

(2π)

p∈Zn−1 +

R+

2n+1 (2ξ)|p|+λ+n cλ p! Γ(λ + 1)

12

cp (ξ) (z )p eiξzn dξ.

(6.4)

Proof. Calculate R∗ = U ∗ R0 : {cp (ξ)}p∈Zn−1 +

n+1 1 2 (2ξ)|p|+λ+n 2 p −ξ(|r|2 +v) ∗ −→ U r e cp (ξ) χR+ (ξ) cλ p! Γ(λ + 1) − n−1 2

−→ U0∗ U1∗ χR+ (ξ) (2π)

−→

U0∗

1 2n+1 (2ξ)|p|+λ+n 2 2 e−ξ(|r| +v) cp (ξ) (z )p cλ p! Γ(λ + 1) n−1

p∈Z+

−n 2

(2π)

p∈Zn−1 + −n 2

= (2π)

R+

p∈Zn−1 +

R+

p∈Zn−1 +

2n+1 (2ξ)|p|+λ+n cλ p! Γ(λ + 1)

2n+1 (2ξ)|p|+λ+n cλ p! Γ(λ + 1)

12

12

e

−ξ(|r|2 +v)

p −iuξ

cp (ξ) (z ) e

dξ

cp (ξ) (z )p eiξzn dξ.

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Corollary 6.4. The inverse isomorphism R : A2λ (Dn ) −→ l2 (Zn−1 + , L2 (R+ )) is given by

R : ϕ(z)

−n 2

−→

(2π) ·

1 (2ξ)|p|+λ+n 2 n−3 cλ 2 p! Γ(λ + 1)

ϕ(z) (z )p e−iξzn

(Im zn − |z | ) dv(z) 2 λ

. (6.5)

p∈Zn−1 +

Dn

7. Nilpotent case The ﬁrst step here will be the same as in the quasi-parabolic case. We consider the space L2 (D, ηλ ) = L2 (Cn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ), and the unitary operator U1 = I ⊗ F ⊗ I acting on it, where F is the Fourier transform on L2 (R), see (6.1). Then the image A1 (D) = U1 (A0 (D)) is the subspace of L2 (D, ηλ ) which consists of all functions satisfying the equations (6.2). The L2 (D, ηλ )-solution of the ﬁrst equation in (6.2) has the form ϕ(z , ξ, v) = χR+ (ξ)ψ(z , ξ)e−ξv ,

(7.1)

and the function ψ has to satisfy the equations ∂ + ξzk ψ(z , ξ) = 0, k = 1, . . . , n − 1. ∂zk Using the standard Cartesian coordinates x = (x1 , . . . , xn−1 ) and y = (y1 , . . . , yn−1 ), where zk = xk + iyk , in Cn−1 = Rn−1 × Rn−1 , we have L2 (D, ηλ ) = L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ). Consider the unitary operator U2 = F(n−1) ⊗ I ⊗ I ⊗ I, where F(n−1) = F ⊗ · · · ⊗ F is (n − 1)-dimensional Fourier transform, acting on above tensor decomposition. We have 1 ∂ ∂ ∂ i ∂ −1 U2 +i + yk . + ξ (xk + iyk ) U2 = + iξ ξk + 2 ∂xk ∂yk 2 ∂yk ∂ξk Thus the image A2 (D) = U2 (A1 (D)) is the subspace of L2 (D, ηλ ) which consists of all functions ϕ(ξ , y , ξ, v) = χR+ (ξ)ψ(ξ , y , ξ)e−ξv , where the function ψ has to satisfy the equations 1 ∂ ∂ i + yk +ξ ψ(ξ , y , ξ) = 0, ξk + 2 ∂yk ∂ξk

k = 1, . . . , n − 1.

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Introduce the following change of variables, 1 uk = √ ξk − ξyk , 2 ξ

1 vk = √ ξk + ξyk , 2 ξ

k = 1, . . . , n − 1,

1 yk = √ (−uk + vk ) , 2 ξ

k = 1, . . . , n − 1,

or ξk =

ξ (uk + vk ) ,

and the corresponding unitary operator U3 acting on L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) by the rule 1 (U3 ϕ)(u , v , ξ, v) = ϕ ξ (u + v ) , √ (−u + v ) , ξ, v , 2 ξ where u = (u1 , . . . , un−1 ) and v = (v1 , . . . , vn−1 ). We have obviously 1 ∂ ∂ ∂ −1 U3 i + yk U3 = i ξ + vk . +ξ ξk + 2 ∂yk ∂ξk ∂vk Thus the image A3 (D) = U3 (A2 (D)) is the subspace of L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) which consists of all functions , v , ξ)e−ξv , ϕ(u , v , ξ, v) = χR+ (ξ)ψ(u satisfying the equations ∂ i ξ + vk ϕ(u , v , ξ, v) = 0, ∂vk

k = 1, . . . , n − 1,

, v , ξ) have to satisfy the equations or the corresponding functions function ψ(u ∂ , v , ξ) = 0, + vk ψ(u k = 1, . . . , n − 1. ∂vk These equations are easy to solve and their general solution has the form , v , ξ) = π − n−1 4 ψ(u e−

|v |2 2

χR+ (ξ) e

−ξv

4(2ξ)λ+1 cλ Γ(λ + 1)

12

ψ(u , ξ),

where ψ(u , ξ) ∈ L2 (Rn−1 × R). Moreover, we have that ϕL2 (Rn−1 )⊗L2 (Rn−1 )⊗L2 (R)⊗L2 (R+ ,ηλ )

=

χR+ (ξ)ψ(u , ξ)L2 (Rn−1 ×R)

=

ψ(u , ξ)L2 (Rn−1 ×R+ ) .

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Indeed, ϕ2

π−

=

n−1 2

R2n−1 ×R+

2

e−|v | χR+ (ξ) e−2ξv

IEOT

4(2ξ)λ+1 |ψ(u , ξ)|2 cλ Γ(λ + 1)

cλ λ v du dv dξdv 4 n−1 2 (2ξ)λ+1 du dξ e−|v | dv |ψ(u , ξ)|2 π− 2 Γ(λ + 1) Rn−1 Rn−1 ×R+ v λ e−2ξv dv · R + |ψ(u , ξ)|2 du dξ = ψ(u , ξ)2L2 (Rn−1 ×R+ ) .

· =

=

Rn−1 ×R+

Calculating the integral over v ∈ R+ we have used formula 3.381.4 from [4]. Note that in terms of the scheme of Section 4 we have here X = Rn−1 × R, L2 (X, µ) = L2 (Zn−1 ) ⊗ L2 (R), X1 = Rn−1 × R+ , L2 (X1 , µ) = L2 (Rn−1 ) ⊗ L2 (R+ ), Y = Rn−1 × R+ , L2 (Y, η) = L2 (Rn−1 ) ⊗ L2 (R+ , ηλ ), the unitary operator U is deﬁned as follows, U = U3 U2 U1 U0 : L2 (Dn , µ λ ) → L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ), and the function g0 has the form 1 |v |2 4(2ξ)λ+1 2 − n−1 −ξv− 2 g0 (v , ξ, v) = π 4 e , cλ Γ(λ + 1)

(v , ξ, v) ∈ Rn−1 × R+ × R+ .

Summarizing the above we come to the following statement. Lemma 7.1. The unitary operator U = U3 U2 U1 U0 maps the Bergman space A2λ (Dn ) onto the space A3 (D) = g0 L2 (Rn−1 × R+ ) which is the closed subspace of L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) and consisits of all functions of the form 1 |v |2 4(2ξ)λ+1 2 − n−1 −ξv− 2 ψ(u , ξ), ϕ(u , v , ξ, v) = π 4 e cλ Γ(λ + 1) where ψ(u , ξ) ∈ L2 (Rn−1 × R+ ). Introduce now the isometric imbedding R0 : L2 (Rn−1 × R+ ) −→ L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) by the rule

R0 : ψ(u , ξ) −→ π

− n−1 4

2

e

−ξv− |v2|

χR+ (ξ)

4(2ξ)λ+1 cλ Γ(λ + 1)

12

ψ(u , ξ),

where the function ψ(u , ξ) is extended by zero for ξ ∈ R \ R+ for each u ∈ Rn−1 .

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The adjoint operator R0∗ : L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) −→ L2 (Rn−1 × R+ ) has obviously the form R0∗ : ϕ(u , v , ξ, v) −→ 1 |v |2 n−1 4(2ξ)λ+1 2 cλ λ v dv. π− 4 e−ξv− 2 f (u , v , ξ, v) dv c Γ(λ + 1) 4 n−1 λ R ×R+ Then we have R0∗ R0 = I

R0 R0∗

= P3

: L2 (Rn−1 × R+ ) −→ L2 (Rn−1 × R+ ), : L2 (Rn−1 ) ⊗ L2 (Rn−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) −→ A3 (D),

where P3 is the orthogonal projection of L2 (Rn−1 )⊗L2(Rn−1 )⊗L2(R)⊗L2 (R+ , ηλ ) onto A3 (D). Thus ﬁnally we have Theorem 7.2. The operator R = R0∗ U maps L2 (Dn , µ λ ) onto L2 (Rn−1 × R+ ), and the restriction R|A2λ (Dn ) : A2λ (Dn ) −→ L2 (Rn−1 × R+ ) is an isometric isomorphism. The adjoint operator λ ) R∗ = U ∗ R0 : L2 (Rn−1 × R+ ) −→ A2λ (Dn ) ⊂ L2 (Dn , µ is the isometric isomorphism of L2 (Rn−1 × R+ ) onto the subspace A2λ (Dn ) of L2 (Dn , µ λ ). Furthermore RR∗ = I ∗

R R = BDn ,λ

: L2 (Rn−1 × R+ ) −→ L2 (Rn−1 × R+ ), : L2 (Dn , µ λ ) −→ A2λ (Dn ),

where BDn ,λ is the Bergman projection of L2 (Dn , µ λ ) onto A2λ (Dn ). For this and the two remaining cases we will not give exact formulas for the operators R and R∗ . If needed, these formulas can be easily obtained by direct though rather lengthy calculations.

8. Quasi-nilpotent case This case is just a mixture of the two previous cases, quasi-parabolic and nilpotent. Given an integer 1 ≤ k ≤ n − 2, we will write the points of Dn as z = (z , w , zn ), where z ∈ Ck and w ∈ Cn−k−1 , and the points of D as (z , w , ζ), respectively. According to this notation we represent L2 (D, ηλ ) = L2 (Ck ) ⊗ L2 (Cn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ).

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Applying, as in previous two cases, the unitary operator U1 = I ⊗ I ⊗ F ⊗ I, we have that the image A1 (D) = U1 (A0 (D)) consists of all L2 -functions of the form ϕ(z , w , ξ, v) = χR+ (ξ)ψ(z , w , ξ) e−ξv , which satisfy the equations ∂ + ξzl ψ(z , w , ξ) ∂zl ∂ + ξwm ψ(z , w , ξ) ∂wm

= 0,

l = 1, . . . , k,

= 0,

m = 1, . . . , n − k − 1.

Now passing to the polar coordinates in Ck , zl = rl tl , where rl ∈ R+ , tl ∈ S 1 = T, l =, 1, . . . , k, and Cartesian coordinates in Cn−k−1 , x = (x1 , . . . , xn−k−1 ), y = (y1 , . . . , yn−k−1 ), where wm = xm + iym , m = 1, . . . , n − k − 1, we have that the space L2 (D, ηλ ) can be represented in the form L2 (Rk+ , rdr) ⊗ L2 (Tk ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ). Introduce the unitary operator U2 = I ⊗ F(k) ⊗ F(n−k−1) ⊗ I ⊗ I ⊗ I acting from L2 (D, ηλ ) onto L2 (Rk+ , rdr) ⊗ l2 (Zk ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ) = l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )), where F(k) = F ⊗ · · · ⊗ F is the k-dimensional discrete Fourier transform and F(n−k−1) = F ⊗ · · · ⊗ F is the (n − k − 1)-dimensional Fourier transform. Then, by the results of the previous two sections, the image A2 (D) = U2 (A1 (D)) consists of all sequences {dp (r, ξ , y , ξ, v)}p∈Zk+ , where the functions

dp (r, ξ , y , ξ, v) =

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1)

12

rp e−ξ(|r|

2

+v)

dp (ξ , y , ξ),

with (ξ , y , ξ) ∈ Rn−k−1 × Rn−k−1 × R+ , belong to the space L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ). Moreover, the corresponding functions dp (ξ , y , ξ) have to satisfy the equations 1 ∂ ∂ + ym dp (ξ , y , ξ) = 0, m = 1, . . . , n − k − 1. i +ξ ξm + 2 ∂ym ∂ξm Introduce the following change of variables 1 1 um = √ ξm − ξym , vm = √ ξm + ξym , 2 ξ 2 ξ or 1 ξm = ξ (um + vm ) , ym = √ (−um + vm ) , 2 ξ and the corresponding unitary operator U3 acting on

m = 1, . . . , n − k − 1,

m = 1, . . . , n − k − 1,

l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ ))

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by the rule U3 : {dp (r, ξ , y , ξ, v)}p∈Zk →

1 dp r, ξ (u + v ) , √ (−u + v ) , ξ, v , 2 ξ p∈Zk

where u = (u1 , . . . , un−k−1 ) and v = (v1 , . . . , vn−k−1 ). Combining the results of the previous two sections we have that in terms of the scheme of Section 4 our data now are as follows, X = Zk × Rn−k−1 × R, L2 (X, µ) = l2 (Zk ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R), k n−k−1 X1 = Z+ × R × R+ , L2 (X1 , µ) = l2 (Zk+ ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R+ ), k n−k−1 Y = R+ × R × R+ , L2 (Y, η) = L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R+ , ηλ ), the unitary operator U is deﬁned as follows, U

λ ) −→ U3 U2 U1 U0 : L2 (Dn , µ

=

l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )), and the function g0 (function-sequence in this case) has the form

g0 (p, r, v , ξ, v) = π

− n−k−1 4

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1)

12

rp e−ξ(|r|

2

2

+v)− |v2|

,

where (p, r, v , ξ, v) ∈ Zk+ × Rk+ × Rn−k−1 × R+ × R+ . Summarizing the above we come to the following statement. Lemma 8.1. The unitary operator U = U3 U2 U1 U0 maps the Bergman space A2λ (Dn ) onto the space A3 (D) = U3 (A2 (D)) = g0 l2 (Zk+ , L2 (Rn−k−1 × R+ )) which is the closed subspace of l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) and consisits of all sequences {dp (r, u , v , ξ, v)}p∈Zk+ , where the func-

tions dp (r, u , v , ξ, v), p ∈ Zk+ , have the form

dp (r, u , v , ξ, v) = π

− n−k−1 4

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1)

12

rp e−ξ(|r|

2

2

+v)− |v2|

cp (u , ξ),

with cp (u , ξ) ∈ L2 (Rn−k−1 × R+ ). Moreover, {dp }p∈Zk+ = {cp }p∈Zk+ l2 (Zk+ , L2 (Rn−k−1 ×R+ )) . We check now the above norm equality. Obviously it is suﬃcient to check only that dp L2 (Rk+ ,rdr)⊗L2 (Rn−k−1 )⊗L2 (Rn−k−1 )⊗L2 (R)⊗L2 (R+ ,ηλ ) = cp L2 (Rn−k−1 ×R+ ) .

402

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Calculate dp

2

IEOT

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1) n−k−1 ×Rn−k−1 ×R×R Rk + + ×R 2 2 cλ λ v dv · r2p e−2ξ(|r| +v)−|v | |cp (u , ξ)|2 rdr du dv dξ 4 2 2k (2ξ)|p|+k = |cp (u , ξ)|2 du dξ r2k e−2ξ|r| rdr p! Rn−k−1 ×R+ Rk + λ+1 n−k−1 2 (2ξ) · π− 2 e−|v | dv v λ e−2ξv dv. Γ(λ + 1) R+ Rn−k−1

= π

− n−k−1 2

By [4], formulas 3.351.3, 3.321.3, and 3.381.4, each of the last three integrals (with the corresponding multiple) is equal to 1, thus dp 2 = |cp (u , ξ)|2 du dξ = cp 2 . Rn−k−1 ×R+

Introduce the isometric imbedding R0 of the space l2 (Zk+ , L2 (Rn−k−1 × R+ )) into l2 (Zk , (L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) which maps the sequence {cp (u , ξ)}p∈Zk+ to π

− n−k−1 4

χZk+ (p)χR+ (ξ)

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1)

12

rp e−ξ(|r|

2

2

+v)− |v2|

cp (u , ξ)

p∈Zk

,

where the functions cp (u , ξ) is extended by zero for ξ ∈ R\R+ for each u ∈ Rn−k−1 and each p ∈ Zk . The adjoint operator R0∗ acts from l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) onto l2 (Zk+ , L2 (Rn−k−1 × R+ )) as follows, R0∗

: {dp (r, u , v , ξ, v)}p∈Zk −→ ·

n−k−1 ×R Rk + + ×R

rp e−ξ(|r|

2

π

− n−k−1 4

2

+v)− |v2|

2k+2 (2ξ)|p|+λ+k+1 cλ p! Γ(λ + 1)

dp (r, u , v , ξ, v) rdr dv

12

cλ v λ dv 4

. p∈Zk +

Then we have R0∗ R0

R0 R0∗

=

I : l2 (Zk+ , L2 (Rn−k−1 × R+ )) −→ l2 (Zk+ , L2 (Rn−k−1 × R+ ))

=

P3 ,

where P3 is the orthogonal projection of l2 (Zk , L2 (Rk+ , rdr) ⊗ L2 (Rn−k−1 ) ⊗ L2 (Rn−k−1 ) ⊗ L2 (R) ⊗ L2 (R+ , ηλ )) onto A3 (D).

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403

Thus ﬁnally we have λ ) onto l2 (Zk+ , L2 (Rn−k−1 × Theorem 8.2. The operator R = R0∗ U maps L2 (Dn , µ R+ )), and the restriction R|A2λ (Dn ) : A2λ (Dn ) −→ l2 (Zk+ , L2 (Rn−k−1 × R+ )) is an isometric isomorphism. The adjoint operator R∗ = U ∗ R0 : l2 (Zk+ , L2 (Rn−k−1 × R+ )) −→ A2λ (Dn ) ⊂ L2 (Dn , µ λ ) is the isometric isomorphism of l2 (Zk+ , L2 (Rn−k−1 × R+ )) onto the subspace A2λ (Dn ) of L2 (Dn , µ λ ). Furthermore RR∗ = I R∗ R = BDn ,λ

: l2 (Zk+ , L2 (Rn−k−1 × R+ )) −→ l2 (Zk+ , L2 (Rn−k−1 × R+ )), : L2 (Dn , µ λ ) −→ A2λ (Dn ),

λ ) onto A2λ (Dn ). where BDn ,λ is the Bergman projection of L2 (Dn , µ

9. Quasi-hyperbolic case We represent D = Cn−1 × R × R+ in the form Cn−1 × Π, where Π is the upper half-plane, and introduce in D the “non-isotropic” upper semi-sphere Ω = {(z , ζ) ∈ Cn−1 × Π : |z |2 + |ζ| = 1 }. The points of Ω admit the natural parameterization z k = s k tk , iθ

ζ = ρe ,

where sk ∈ [0, 1), tk ∈ S 1 , k = 1, . . . , n − 1, where ρ ∈ (0, 1], θ ∈ (0, π),

and

n−1

s2k + ρ = 1,

k=1

which in turn induces the following representation of the points (z , ζ) ∈ D = Cn−1 × Π 1 ζ = rρeiθ , zk = r 2 sk tk , k = 1, . . . , n − 1, where r ∈ R+ . We represent now D = τ (Bn−1 ) × Tn−1 × R+ × (0, π), where τ (Bn−1 ) = 2 : n−1 {s = (s1 , . . . , sn−1 ) ∈ Rn−1 + k=1 sk < 1} is the base (in the sense of a Reinn−1 , and Tn−1 = S 1 × · · · × S 1 is the n − 1 hardt domain) of the unit ball B dimensional torus. Introduce the new coordinate system (s, t, r, θ) in D, where s = (s1 , . . . , sn−1 ) ∈ τ (Bn−1 ), t = (t1 , . . . , tn−1 ) ∈ Tn−1 , r ∈ R+ , and θ ∈ (0, π), which is connected with the old one (z , ζ) by the formulas |zk | , sk = |z |2 + |ρ|

tk =

zk , |zk |

r = |z |2 + |ρ|,

θ = arg ζ,

(9.1)

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Quiroga-Barranco and Vasilevski

IEOT

or 1

z k = r 2 s k tk ,

ζ = r(1 − |s|2 )eiθ ,

where k = 1, . . . , n − 1. We pass now the operators in equations (2.10) to the new coordinate system. For a function f = f (s1 , . . . , sn−1 , t1 , . . . , tn−1 , r, θ), consider ∂ f ∂ζ

=

=

=

=

∂ 1 ∂ cos θ + i sin θ +i 2 ∂|ζ| |ζ| ∂θ |zn−1 | |z1 | ,..., , t1 , . . . , tn−1 , r, θ ·f |z |2 + |ρ| |z |2 + |ρ| n−1 1 ∂f |zl | 1 ∂f cos θ + i sin θ ∂f − +i 2 ∂r 2 ∂sl (|z |2 + |ζ|) 32 |ζ| ∂θ l=1 n−1 cos θ + i sin θ ∂f 1 sl ∂f ∂f 1 − +i 2 ∂r 2 r ∂sl r(1 − |s|2 ) ∂θ l=1 n−1 cos θ + i sin θ ∂f 1 ∂f ∂f 1 − r . sl +i 2r ∂r 2 ∂sl 1 − |s|2 ∂θ l=1

Further, ∂f tk ∂f tk ∂f sin θ ∂f ∂f ∂f = − − − izk − izk cos θ ∂z k ∂u 2 ∂|zk | |zk | ∂tk ∂|ζ| |ζ| ∂θ n−1 1 ∂f 2|zl ||zk | 1 ∂f ∂f tk ∂f tk − 2|zk | + − = 2 ∂r 2 ∂sl (|z |2 + |ζ|) 32 ∂sk |z |2 + |ζ| |zk | ∂tk l=1 n−1 1 ∂f |zl | ∂f sin θ ∂f − − izk cos θ − ∂r 2 ∂sl (|z |2 + |ζ|) 32 |ζ| ∂θ l=1 n−1 ∂f 1 ∂f |zl | ∂f t2 ∂f tk = tk |zk | − − k + 3 ∂r 2 ∂sl (|z |2 + |ζ|) 2 2|zk | ∂tk 2 |z |2 + |ζ| ∂sk l=1 n−1 1 ∂f |zl | ∂f sin θ ∂f − − itk |zk | cos θ − ∂r 2 ∂sl (|z |2 + |ζ|) 32 |ζ| ∂θ l=1 n−1 1 1 sl ∂f t2 ∂f ∂f tk ∂f = tk r 2 s k − − 1k + 1 ∂r 2 r ∂sl 2r 2 ∂sk 2r 2 sk ∂tk l=1 n−1 1 sl ∂f ∂f sin θ 1 ∂f − − itk r 2 sk cos θ − ∂r 2 r ∂sl r(1 − |s|2 ) ∂θ l=1

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Toeplitz Operators on the Unit Ball, I.

=

405

n−1 1 ∂f tk ∂f ∂f sk ∂f − sl − r + 1 ∂r 2 ∂sl 2 ∂sk 2 ∂tk r 2 sk l=1 n−1 1 ∂f ∂f sin θ ∂f 2 − . − isk cos θ r sl − ∂r 2 ∂sl 1 − |s|2 ∂θ tk

s2k

l=1

That is, the equations (2.10) are equivalent to n−1 ∂ 1 ∂ ∂ 1 r − f =0 sl +i ∂r 2 ∂sl 1 − |s|2 ∂θ l=1 n−1 1 ∂f tk ∂f ∂f sk ∂f 2 sk r − sl − + ∂r 2 ∂sl 2 ∂sk 2 ∂tk l=1 n−1 1 ∂f ∂f sin θ ∂f 2 − f = 0, −isk cos θ r sl − ∂r 2 ∂sl 1 − |s|2 ∂θ

(9.2)

l=1

where k = 1, . . . , n − 1. From the ﬁrst of these equations we have r

n−1 ∂f 1 ∂f ∂f 1 − , sl = −i 2 ∂r 2 ∂sl 1 − |s| ∂θ l=1

substituting into the second we obtain −i

s2k ∂f sk ∂f s2 sin θ ∂f tk ∂f s2 cos θ ∂f + +i k − − k 2 2 1 − |s| ∂θ 2 ∂sk 2 ∂tk 1 − |s| ∂θ 1 − |s|2 ∂θ sk ∂f tk ∂f s2k ∂f − +i (sin θ + i cos θ − 1) 2 ∂sk 2 ∂tk 1 − |s|2 ∂θ

= = 0.

From the last equation, for each k = 1, . . . , n − 1, we have sk ∂f tk ∂f s2k ∂f = −i (sin θ + i cos θ − 1) 2 ∂sk 2 ∂tk 1 − |s|2 ∂θ Summing up these equations for k = 1, . . . , n − 1 and substituting to (9.2), we have n−1 ∂f 1 ∂f ∂f |s|2 − = 0. r tl +i 1+ (sin θ + i cos θ) ∂r 2 ∂tl 1 − |s|2 ∂θ l=1

That is, ﬁnally the equations (2.10) are equivalent to n−1 ∂f ∂f 1 ∂f |s|2 r − tl +i 1+ (sin θ + i cos θ) ∂r 2 ∂tl 1 − |s|2 ∂θ

=

0

=

0,

l=1

sk

∂f ∂f 2s2k ∂f − tk +i (sin θ + i cos θ − 1) ∂sk ∂tk 1 − |s|2 ∂θ

where k = 1, . . . , n − 1.

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IEOT

The direct calculation shows that under the change of variables (9.1) we have dv(z , ζ) = rn (1 − |s|2 )

n−1

k=1

and

sk dsk

n−1

k=1

dtk drdθ, itk

cλ cλ λ r (1 − |s|2 )λ sinλ θ. 4 4 The intermediate result obtained we formulate in the following lemma. ηλ =

Lemma 9.1. The space A0 (D) = U0 (A2λ (Dn )) consists of all functions f = f (s, t, r, θ) which satisfy the equations n−1 ∂f 1 ∂f ∂f |s|2 − = 0 (9.3) r tl +i 1+ (sin θ + i cos θ) ∂r 2 ∂tl 1 − |s|2 ∂θ l=1

2s2k ∂f ∂f ∂f − tk +i (sin θ + i cos θ − 1) 2 ∂sk ∂tk 1 − |s| ∂θ where k = 1, . . . , n − 1, and belong to the space sk

=

0,

(9.4)

L2 (τ (Bn−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (Tn−1 ) cλ sinλ θdθ). ⊗ L2 (R+ , rλ+n dr) ⊗ L2 ((0, π), 4 Introduce the unitary operator U1 = I ⊗ F(n−1) ⊗ M ⊗ I which acts from the space cλ L2 (τ (Bn−1 ), (1−|s|2 )λ+1 sds)⊗L2 (Tn−1 )⊗L2 (R+ , rλ+n dr)⊗L2 ((0, π), sinλ θdθ) 4 onto the space cλ sinλ θdθ) L2 (τ (Bn−1 ), (1 − |s|2 )λ+1 sds) ⊗ l2 (Zn−1 ) ⊗ L2 (R) ⊗ L2 ((0, π), 4 cλ sinλ θdθ)), = l2 (Zn−1 , L2 (τ (Bn−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), 4 where the Mellin transform M : L2 (R+ , rλ+n dr) −→ L2 (R) is given by λ+n−1 1 (M ψ)(ξ) = √ r−iξ+ 2 ψ(r)dr, 2π R+ L2 (D, ηλ ) =

and F(n−1) = F ⊗ · · · ⊗ F is the (n − 1)-dimensional discrete Fourier transform and each F is given by (5.1). We note that λ+n+1 1 −1 r−iξ− 2 ψ(ξ)dξ, (M ψ)(r) = √ 2π R and λ+n+1 ∂ M −1 ψ = i ξ + i Mr ψ, ∂r 2 ∂ F −1 dpk = pk dpk , pk ∈ Z. F k tk ∂tk k

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Now the image A1 (D) = U1 (A0 (D)) consists of all sequences d = {dp }p∈Zn−1 the components cλ sinλ θdθ) dp = dp (s, ξ, θ) ∈ L2 (τ (Bn−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), 4 of which satisfy the equations n−1 ∂ 1 ∂ |s|2 ∂ − U1−1 dp U1 r tl +i 1+ (sin θ + i cos θ) ∂r 2 ∂tl 1 − |s|2 ∂θ l=1 ∂dp λ+n+1 |p| |s|2 = ξ+i (sin θ + i cos θ) dp + i dp + 1 + 2 2 1 − |s|2 ∂θ = 0, (9.5) where |p| = p1 + · · · + pn−1 , and ∂ ∂ 2s2k ∂ − tk +i (sin θ + i cos θ − 1) U 1 sk U1−1 dp ∂sk ∂tk 1 − |s|2 ∂θ ∂dp 2s2k ∂dp sk − pk dp + i (sin θ + i cos θ − 1) ∂sk 1 − |s|2 ∂θ

= = 0, (9.6)

where k = 1, . . . , n − 1, p = (p1 , . . . , pn−1 ) ∈ Zn−1 . Equation (9.5) is easy to solve. Using [4], formula 2.558.4, we have dp (s, ξ, θ) = dp (s, ξ) e

|s|2 −2(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

|s|2 θ 2 + 1−|s|2

.

Introduce the temporary notations E

=

α =

e

|s|2 −2(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

|s|2 θ 2 + 1−|s|2

,

|s|2 . 1 − |s|2

Then dp = dp E. By (9.5) we have ξ + i λ+n+|p|+1 ∂dp 2 =− dp E. ∂θ 1 + α(sin θ + i cos θ) Calculate ∂dp ∂sk

= =

where

∂E ∂ dp E + dp ∂sk ∂sk 1 − i tan θ2 ∂α ∂ dp λ + n + |p| + 1 E − dp E 2 ξ + i , 2 θ ∂sk 2 ∂s k 1 + (1 − iα) tan 2 + α ∂α (1 − |s|2 )2sk + |s|2 2sk = = 2sk (1 + α)2 . ∂sk (1 − |s|2 )2

408

Quiroga-Barranco and Vasilevski

Thus

IEOT

∂dp ∂ dp λ + n + |p| + 1 =E −E ξ+i 2sk (1 + α)2 A dp , ∂sk ∂sk 2

where A

= = = =

2(1 − i tan θ2 )

2 1 + (1 − iα) tan θ2 + α 2 cos2

cos2

θ θ θ 2 − 2i sin 2 cos 2 α2 ) sin2 θ2 + 2α(1 − iα) sin θ2

+ (1 − 2iα − 1 + cos θ − i sin θ 1 + α2 cos θ + α(1 − iα) sin θ − iα(1 − cos θ) 1 + cos θ − i sin θ . (1 − iα)[1 + α(sin θ + i cos θ)] θ 2

cos θ2 + α2 cos2

θ 2

Thus ﬁnally

∂dp ∂ dp λ + n + |p| + 1 2sk (1 + α)2 (1 + cos θ − i sin θ) dp . =E −E ξ+i ∂sk ∂sk 2 (1 − iα)[1 + α(sin θ + i cos θ)]

Substituting the above in (9.6) and canceling out E, we have ∂ dp λ + n + |p| + 1 2s2k (1 + α)2 (1 + cos θ − i sin θ) dp − ξ+i sk ∂sk 2 (1 − iα)[1 + α(sin θ + i cos θ)] λ + n + |p| + 1 2s2k (1 + α)(sin θ + i cos θ − 1) dp − pk dp − i ξ + i 2 1 + α(sin θ + i cos θ) 2s2k (1 + α) ∂ dp λ + n + |p| + 1 B dp = sk − pk dp − ξ + i ∂sk 2 (1 − iα)[1 + α(sin θ + i cos θ)] = 0, where B

= (1 + α)(1 + cos θ − i sin θ) + (i + α)(sin θ + i cos θ − 1) = (1 − i)[1 + α(sin θ + i cos θ)].

That is, we have ∂ dp λ + n + |p| + 1 2s2k (1 − i)(1 + α) sk dp = 0, − pk dp − ξ + i ∂sk 2 1 − iα where 1 1+α 1 = = . i|s|2 1 − iα 1 − (1 + i)|s|2 2 (1 − |s| ) 1 − 1−|s|2 Thus ﬁnally the equation (9.6) is reduced to 2(1 − i)s2k ∂ dp λ + n + |p| + 1 sk − pk dp − ξ + i dp = 0, ∂sk 2 1 − (1 + i)|s|2 where k = 1, . . . , n − 1, p = (p1 , . . . , pn−1 ) ∈ Zn−1 .

(9.7)

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The common general solution of (9.7) for k = 1, . . . , n − 1 is given by dp = cp (ξ) sp [1 − (1 + i)|s|2 ]−

λ+n+|p|+1 +iξ 2

.

Thus the general solution of the equations (9.5) and (9.6) has the form dp (s, ξ, θ)

= cp (ξ) sp [1 − (1 + i)|s|2 ]− ·e

λ+n+|p|+1 +iξ 2

|s|2 −2(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

|s|2 θ 2 + 1−|s|2

.

But, for each p ∈ Zn−1 , the function dp has to be in L2 (τ (Bn−1 ), (1−|s|2 )λ+1 sds)⊗ L2 (R)⊗L2 ((0, π), c4λ sinλ θdθ). This implies ﬁrst that dp ≡ 0 for all p ∈ Zn−1 \Zn−1 + . Second, introduce αp (ξ) = s2p (1 − |s|2 )λ+1 |1 − (1 − i)|s|2 )|−(λ+n+|p|+1)+2iξ τ (Bn−1 )×(0,π)

·e

|s|2 −4(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

|s|2 θ 2 + 1−|s|2 )

cλ sinλ θ sdsdθ 4

− 12 . (9.8)

Then, setting cp (ξ) = αp (ξ) cp (ξ) with cp ∈ L2 (R), we have that for each p ∈ Zn−1 + dp L2 (τ (Bn−1 ),(1−|s|2 )λ+1 sds)⊗L2 (R)⊗L2 ((0,π), cλ 4

sinλ θdθ)

= cp L2 (R) .

Note that in terms of the scheme of Section 4 we have here X = Zn−1 × R, X1 = Zn−1 × R, + Y = τ (Bn−1 ) × (0, π),

L2 (X, µ) = l2 (Zn−1 ) ⊗ L2 (R), L2 (X1 , µ) = l2 (Zn−1 + ) ⊗ L2 (R+ ), L2 (Y, η) = L2 (τ (Bn−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 ((0, π), c4λ sinλ θdθ),

the unitary operator U is deﬁned as follows, U = U1 U0 : L2 (Dn , µ λ ) −→ n−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), l2 (Zn−1 + , L2 (τ (B

cλ sinλ θdθ)), 4

and the function g0 (function-sequence in this case) has the form g0 (s, ξ, θ) = {g0 (p, s, ξ, θ)}p∈Zn−1 , +

where g0 (p, s, ξ, θ) = ·

αp (ξ) sp [1 − (1 + i)|s|2 ]− e

λ+n+|p|+1 +iξ 2

|s|2 −2(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

and (s, ξ, θ) ∈ τ (Bn−1 ) × R × (0, π). here p ∈ Zn−1 +

|s|2 θ 2 + 1−|s|2

,

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Summarizing the above we come to the following statement. Lemma 9.2. The unitary operator U = U1 U0 maps the Bergman space A2λ (Dn ) onto the space A1 (D) = g0 l2 (Zn−1 + , L2 (R)) which is the closed subspace of n−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), l2 (Zn−1 + , L2 (τ (B

cλ sinλ θdθ) 4

and consisits of all sequences {dp (s, ξ, θ)}p∈Zn−1 , where the functions dp = +

dp (s, ξ, θ), p ∈ Zn−1 + , have the form dp

=

cp (ξ) αp (ξ) sp [1 − (1 + i)|s|2 ]− ·e

−2(ξ+i λ+n+|p|+1 ) arctan 2

λ+n+|p|+1 +iξ 2 2

|s| 1−i 1−|s| 2

tan

|s|2 θ 2 + 1−|s|2

,

with cp ∈ L2 (R) and αp given by (9.8). Moreover, {dp }l2 (Zn−1 , L2 (τ (Bn−1 ),(1−|s|2 )λ+1 sds)⊗L2 (R)⊗L2 ((0,π), cλ +

=

4

sinλ θdθ))

{cp }l2 (Zn−1 , L2 (R)) . +

Introduce the isometric imbedding R0 of the space l2 (Zn−1 + , L2 (R)) into the space n−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), l2 (Zn−1 + , L2 (τ (B

cλ sinλ θdθ)) 4

by the rule R0 : {cp (ξ)}p∈Zn−1 −→ {cp (ξ) αp (ξ) βp (s, ξ, θ)}p∈Zn−1 , +

where the functions βp = βp (s, ξ, θ) are given by 2 − λ+n+|p|+1 +iξ 2

p

βp = s [1−(1+i)|s| ]

e

|s|2 −2(ξ+i λ+n+|p|+1 tan ) arctan 1−i 1−|s| 2 2

|s|2 θ 2 + 1−|s|2

. (9.9)

We note that αp (ξ) =

2

τ (Bn−1)×(0,π)

2 λ+1 cλ

|βp (s, ξ, θ)| (1 − |s| )

4

− 12 λ

sin θ sdsdθ

.

The adjoint operator R0∗ which acts from n−1 l2 (Zn−1 ), (1 − |s|2 )λ+1 sds) ⊗ L2 (R) ⊗ L2 ((0, π), + , L2 (τ (B

cλ sinλ θdθ)) 4

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onto the space l2 (Zn−1 + , L2 (R)) has obviously the form R0∗ : {dp (s, ξ, θ)}p∈Zn−1 −→

λ 2 λ+1 cλ αp (ξ) sin θ sdsdθ βp (s, ξ, θ) dp (s, ξ, θ) (1 − |s| ) 4 τ (Bn−1 )×(0,π)

. p∈Zn−1 +

Then we have R0∗ R0

R0 R0∗

=

n−1 I : l2 (Zn−1 + , L2 (R)) −→ l2 (Z+ , L2 (R)),

=

P1 ,

n−1 ), (1 − |s|2 )λ+1 sds) ⊗ where P1 is the orthogonal projection of l2 (Zn−1 + , L2 (τ (B λ cλ L2 (R) ⊗ L2 ((0, π), 4 sin θdθ)) onto A1 (D). Then ﬁnally we have

λ ) onto l2 (Zn−1 Theorem 9.3. The operator R = R0∗ U maps L2 (Dn , µ + , L2 (R)), and the restriction R|A2λ (Dn ) : A2λ (Dn ) −→ l2 (Zn−1 + , L2 (R)) is an isometric isomorphism. The adjoint operator 2 R∗ = U ∗ R0 : l2 (Zn−1 λ ) + , L2 (R)) −→ Aλ (Dn ) ⊂ L2 (Dn , µ 2 is the isometric isomorphism of l2 (Zn−1 + , L2 (R)) onto the subspace Aλ (Dn ) of L2 (Dn , µ λ ). Furthermore

RR∗ = I R∗ R = BDn ,λ

n−1 : l2 (Zn−1 + , L2 (R)) −→ l2 (Z+ , L2 (R)),

: L2 (Dn , µ λ ) −→ A2λ (Dn ),

where BDn ,λ is the Bergman projection of L2 (Dn , µ λ ) onto A2λ (Dn ).

10. Toeplitz operators with special symbols In this section we show that in each case of the previous ﬁve sections there exists a class of bounded measurable symbols a, such that the corresponding Toeplitz operators Ta are unitary equivalent to certain multiplication operators γa I. In each case the symbols are invariant with respect to the action of the corresponding commutative subgroup of Section 3. The speciﬁc form of γa and the space in which this multiplication operator acts depend essentially on the case under consideration. This fact implies an important joint feature, in each case the C ∗ -algebra generated by corresponding Toeplitz operators is commutative. Furthermore, being unitary equivalent to a multiplication operator γa I such a Toeplitz operator thus admits a spectral type representation, which gives an easy access to its important properties: boundedness, compactness, spectral properties, invariant subspaces, etc.

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10.1. Quasi-elliptic case We will call a function a(z), z ∈ Dn , quasi-elliptic if it is separately radial, i.e., a(z) = a(r) = a(r1 , . . . , rn ), or equivalently if a is invariant under the action of the quasi-elliptic group. The following result has been proved in [10]. Theorem 10.1. Let a = a(r) be a bounded measurable quasi-elliptic function. Then the Toeplitz operator Ta acting on A2λ (Bn ) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on l2 (Zn+ ), where R and R∗ are given by (5.3) and (5.2), respectively. The sequence γa,λ = {γa,λ (p)}p∈Zn+ is given by γa,λ (p) = =

n

2n Γ(n + |p| + λ + 1) a(r) r2p (1 − r2 )λ rk drk p! Γ(λ + 1) τ (Bn ) k=1 √ Γ(n + |p| + λ + 1) a( r) rp (1 − (r1 + · · · + rn ))λ dr, p! Γ(λ + 1) ∆(Bn )

where p ∈ Zn+ , ∆(Bn ) = {r = (r1 , . . . , rn ) : r1 + · · · + rn ∈ [0, 1), rk ≥ 0, k = √ √ √ 1, . . . , n}, dr = dr1 · · · drn , and r = ( r1 , . . . , rn ). 10.2. Quasi-parabolic case We will call a function a(z), z ∈ Dn , quasi-parabolic if a(z) = a(r, yn ) = a(r1 , . . . , rn−1 , Im zn ), i.e., a is invariant under the action of the quasi-parabolic group. Theorem 10.2. Let a = a(r, yn ) be a bounded measurable quasi-parabolic function. Then the Toeplitz operator Ta acting on A2λ (Dn ) is unitary equivalent to the mul∗ tiplication operator γa I = R Ta R∗ acting on l2 (Zn−1 + , L2 (R+ )), where R and R are given by (6.5) and (6.4), respectively. The sequence γa = {γa (p, ξ)}p∈Zn−1 , + ξ ∈ R+ , is given by √ (2ξ)|p|+λ+n γa (p, ξ) = a( r, v + r1 + · · · + rn−1 ) rp e−2ξ(v+r1 +···+rn−1 ) v λ drdv, p! Γ(λ + 1) Rn+ (10.1) √ √ √ where r = ( r 1 , . . . , r n−1 ). Proof. The operator Ta is obviously unitary equivalent to the operator R Ta R ∗

=

R BDn ,λ aBDn ,λ R∗ = R(R∗ R)a(R∗ R)R∗

=

(RR∗ )RaR∗ (RR∗ ) = RaR∗

=

R0∗ U2 U1 U0 a(r, yn )U0−1 U1−1 U2−1 R0

=

R0∗ U2 U1 a(r, v + |r|2 ) U1−1 U2−1 R0

=

R0∗ a(r, v + |r|2 )R0

=

T.

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413

Now, for c = {cp (ξ)}p∈Zn−1 , we have +

Tc =

R0∗

=

a(r, v + |r| )χR+ (ξ) 2

n+1

cλ · cp (ξ)

= ·e

2

|p|+λ+n

(2ξ) p! Γ(λ + 1)

cλ λ v rdrdv 4

(2ξ)|p|+λ+n p! Γ(λ + 1)

Rn +

Rn +

2n+1 (2ξ)|p|+λ+n cλ p! Γ(λ + 1)

12

p −ξ(|r|2 +v)

r e

a(r, v + |r|2 ) r2p e−2ξ(|r|

cp (ξ) p∈Zn−1

2

+v)

p∈Zn−1 +

√ a( r, v + r1 + · · · + rn−1 ) rp

−2ξ(v+r1 +···+rn−1 )

λ

cp (ξ) v drdv p∈Zn−1 +

= {γa (p, ξ) · cp (ξ)}p∈Zn−1 , +

with γa (p, ξ) =

(2ξ)|p|+λ+n p! Γ(λ + 1)

Rn +

√ a( r, v + r1 + · · · + rn−1 ) rp e−2ξ(v+r1 +···+rn−1 ) v λ drdv,

where p ∈ Zn−1 + , ξ ∈ R+ , and

√

√ √ r = ( r 1 , . . . , r n−1 ).

10.3. Nilpotent case Recall that the nilpotent group Rn−1 × R acts on Dn as follows. For (b, h) ∈ Rn−1 × R, τ(b,h) : (z , zn ) −→ (z + b, zn + h + 2iz · b + i|b|2 ). We note that both quantities y = Im z and Im zn − |z |2 are invariant under the action of this group. We will call a function a(z), z ∈ Dn , nilpotent if a(z) = a(y , Im zn − |z |2 ), i.e., a is invariant under the action of the nilpotent group. Theorem 10.3. Let a = a(y , Im zn − |z |2 ) be a bounded measurable nilpotent function. Then the Toeplitz operator Ta acting on A2λ (Dn ) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on L2 (Rn−1 × R+ ), where R and R∗ are given in Section 7. The function γa = γa (u , ξ), where u ∈ Rn−1 and ξ ∈ R+ , is given by 2 (2ξ)λ+1 1 γa (u , ξ) = n−1 a( √ (−u + v ), v) e−2ξv−|v | v λ dv dv. n−1 2 ξ 2 π Γ(λ + 1) R ×R+ (10.2)

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Proof. The operator Ta is obviously unitary equivalent to the operator R Ta R ∗

= R BDn ,λ aBDn ,λ R∗ = R(R∗ R)a(R∗ R)R∗ = (RR∗ )RaR∗ (RR∗ ) = RaR∗ = R0∗ U3 U2 U1 U0 a(y , Im zn − |z |2 )U0−1 U1−1 U2−1 U3−1 R0 = R0∗ U3 U2 U1 a(y , v) U1−1 U2−1 U3−1 R0 = R0∗ U3 a(y , v)U3−1 R0 1 = R0∗ a( √ (−u + v ), v)R0 2 ξ = T.

Now,

Tψ

|v |2 n−1 1 a( √ (−u + v ), v) π − 4 e−ξv− 2 χR+ (ξ) 2 ξ 1 4(2ξ)λ+1 2 · ψ(u , ξ) cλ Γ(λ + 1) n−1 2 1 4(2ξ)λ+1 = π− 2 a( √ (−u + v ), v) e−2ξv−|v | cλ Γ(λ + 1) 2 ξ Rn−1 ×R+ cλ λ v dv dv · ψ(u , ξ) 4 = γa (u , ξ) · ψ(u , ξ),

R0∗

=

with (2ξ)λ+1

γa (u , ξ) =

π

n−1 2

Γ(λ + 1)

Rn−1 ×R

2 1 a( √ (−u + v ), v) e−2ξv−|v | v λ dv dv, 2 ξ +

where u = (u1 , . . . , un−1 ) ∈ Rn−1 and ξ ∈ R+ . +

10.4. Quasi-nilpotent case For an integer 1 ≤ k ≤ n − 2, we keep using the notation z = (z , w , zn ) for points of Dn , where z ∈ Ck and w ∈ Cn−k−1 . Recall that the quasi-nilpotent group Tk × Rn−k−1 × R acts on Dn as follows. For (t, a, h) ∈ Tk × Rn−k−1 × R, τ(t,b,h) : (z , w , zn ) −→ (tz , w + b, zn + h + 2iw · b + i|b|2 ). We note that the quantities r, where r = (r1 , . . . , rk ) with rl = |zl |, y = Im w , and Im zn − |w |2 are invariant under the action of this group. We will call a function a(z), z ∈ Dn , quasi-nilpotent if a(z) = a(r, y , Im zn − 2 |w | ), i.e., a is invariant under the action of the quasi-nilpotent group, corresponding to the above parameter k.

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Theorem 10.4. Let a = a(r, y , Im zn − |w |2 ) be a bounded measurable quasinilpotent function. Then the Toeplitz operator Ta acting on A2λ (Dn ) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on l2 (Zk+ , L2 (Rn−k−1 × R+ )), where R and R∗ are given in Section 8. The sequence γa = {γa (p, u , ξ)}p∈Zk+ , (u , ξ) ∈ Rn−k−1 × R+ , is given by γa (p, u , ξ)

(2ξ)|p|+λ+k+1 p! Γ(λ + 1) √ 1 a( r, √ (−u + v ), v + r1 + · · · + rk ) n−k−1 ×R 2 ξ Rk + + ×R

= π− ·

n−k−1 2

2

· rp e−2ξ(v+r1 +···+rk )−|v | v λ drdv dv, √ √ √ where r = ( r 1 , . . . , r k ).

(10.3)

Proof. The operator Ta is obviously unitary equivalent to the operator R Ta R ∗

= R BDn ,λ aBDn ,λ R∗ = R(R∗ R)a(R∗ R)R∗ = (RR∗ )RaR∗ (RR∗ ) = RaR∗ = R0∗ U3 U2 U1 U0 aU0−1 U1−1 U2−1 U3−1 R0

= R0∗ U3 U2 U1 a(r, y , v + |r|2 ) U1−1 U2−1 U3−1 R0 = R0∗ U3 a(r, y , v + |r|2 ) U3−1 R0 1 = R0∗ a(r, √ (−u + v ), v + |r|2 )R0 2 ξ = T. We have

T {cp (u , ξ)}p∈Zk+

=

=

n−k−1 1 a(r, √ (−u + v ), v + |r|2 )π − 4 χZk+ (p)χR+ (ξ) 2 ξ

k+2 1 2 (2ξ)|p|+λ+k+1 2 p −ξ(|r|2 +v)− |v |2 2 · r e cp (u , ξ) cλ p! Γ(λ + 1) p∈Zk k+2 |p|+λ+k+1 n−k−1 2 (2ξ) π− 2 cλ p! Γ(λ + 1) 1 a(r, √ (−u + v ), v + |r|2 ) · k n−k−1 2 ξ R+ ×R ×R+

2p −2ξ(|r|2 +v)−|v |2 cλ λ v dv ·r e cp (u , ξ) rdr dv 4 k R0∗

p∈Z+

416

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π−

=

n−k−1 2

·

IEOT

(2ξ)|p|+λ+k+1 p! Γ(λ + 1)

n−k−1 ×R Rk + + ×R

√ 1 a( r, √ (−u + v ), v + r1 + · · · + rk ) 2 ξ

2

· rp e−2ξ(v+r1 +···+rk )−|v | cp (u , ξ) v λ drdv dv p∈Zk +

= {γa (p, u , ξ) · cp (u , ξ)}p∈Zk+ , with γa (p, u , ξ)

(2ξ)|p|+λ+k+1 p! Γ(λ + 1) √ 1 a( r, √ (−u + v ), v + r1 + · · · + rk ) k n−k−1 2 ξ R+ ×R ×R+

= π− ·

n−k−1 2

2

· rp e−2ξ(v+r1 +···+rk )−|v | v λ drdv dv √ √ √ where p ∈ Zk+ , u ∈ Rn−k−1 , ξ ∈ R+ , and r = ( r 1 , . . . , r k ). +

10.5. Quasi-hyperbolic case Recall that the quasi-hyperbolic group Tn−1 × R+ acts on Dn as follows. For (t, r) ∈ Tn−1 × R+ , 1

τ(t,r) : (z , zn ) −→ (r 2 tz , rzn ). We will call a function a(z), z ∈ Dn , quasi-hyperbolic if a is invariant under the action of this group. A convenient in our context way to describe such invariant functions is as follows. Consider the group of non-isotropic dilations {δr }, r ∈ R+ , acting on × Π by the rule Rn−1 + 1

1

δr : (q1 , . . . , qn−1 , ζ) −→ (r 2 q1 , . . . , r 2 qn−1 , rζ). Then each function a ˜ = a ˜(q1 , . . . , qn−1 , ζ) with is non-isotropic homogeneous of × Π depends only on its values on the non-isotropic upper half zero order on Rn−1 + sphere n−1 × Π : qk2 + |ζ| = 1 }, Ω+ = {(q1 , . . . , qn−1 , ζ) ∈ Rn−1 + k=1

and thus, passing to the polar coordinates in the upper half-plane Π, is a function of the form qn−1 ρ q1 ,..., , ,θ , a ˜=a ˜(q1 , . . . , qn−1 , ρ, θ) = a ˜ |q|2 + ρ |q|2 + ρ |q|2 + ρ n−1 where |q|2 = k=1 qk2 , ρ = |ζ|, and θ = arg ζ.

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Further, we parameterize the points of Ω+ by points s = (s1 , . . . , sn−1 , θ) of τ (Bn−1 ) × (0, π) as follows, ρ qk = sk , k = 1, . . . , n − 1, and θ = θ. = 1 − |s|2 , 2+ρ 2 |q| |q| + ρ Thus each function on Ω+ is of the form a(s, θ), where s = (s1 , . . . , sn−1 ) ∈ τ (Bn−1 ) and θ ∈ (0, π). Now each quasi-hyperbolic function, deﬁned in Dn , can be uniquely represented in the form |zn−1 | |z1 | 2 ,..., , arg(zn − i|z | ) , a=a |z |2 + |zn − i|z |2 | |z |2 + |zn − i|z |2 | (10.4) where a is a function, deﬁned in τ (Bn−1 ) × (0, π), and this correspondence is one to one. Theorem 10.5. Let a be a bounded measurable quasi-hyperbolic function of the form (10.4). Then the Toeplitz operator Ta acting on A2λ (Dn ) is unitary equivalent to the multiplication operator γa I = R Ta R∗ acting on l2 (Zn−1 + , L2 (R)), where R and R∗ are given in Section 9. The sequence γa = {γa (p, ξ)}p∈Zn−1 , ξ ∈ R, is given by + cλ sinλ θ sdsdθ, γa (p, ξ) = α2p (ξ) a(s, θ) |βp (s, ξ, θ)|2 (1 − |s|2 )λ+1 4 τ (Bn−1 )×(0,π) (10.5) where the functions αp (ξ) and βp (s, ξ, θ) are given by (9.8) and (9.9), respectively. Proof. The operator Ta is obviously unitary equivalent to the operator R Ta R ∗

= =

R BDn ,λ aBDn ,λ R∗ = R(R∗ R)a(R∗ R)R∗ (RR∗ )RaR∗ (RR∗ ) = RaR∗

=

R0∗ U1 U0 a U0−1 U1−1 R0

= =

R0∗ U1 a(s, θ) U1−1 R0 R0∗ a(s, θ)R0

=

T.

Now, T {cp (ξ)}

=

R0∗ {a(s, θ)) αp (ξ) βp (s, ξ, θ) cp (ξ)}p∈Zn−1 + α2p (ξ)

= · =

τ (Bn−1 )×(0,π)

a(s, θ) |βp (s, ξ, θ)|2 cp (ξ) (1 − |s|2 )λ+1

cλ sinλ θ sdsdθ 4 p∈Zn−1 +

{γa (p, ξ) · cp (ξ)}p∈Zn−1 , +

418

with γa (p, ξ) = α2p (ξ)

Quiroga-Barranco and Vasilevski

τ (Bn−1 )×(0,π)

a(s, θ) |βp (s, ξ, θ)|2 (1 − |s|2 )λ+1

IEOT

cλ sinλ θ sdsdθ, 4

where p = (p1 , . . . , pn−1 ) ∈ Zn−1 + , ξ ∈ R, and the functions αp (ξ) and βp (s, ξ, θ) are given by (9.8) and (9.9), respectively.

References [1] V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math., 3:187–214, 1961. [2] D. B´ekoll´e and A. Temgoua Kagou. Reproducing properties and Lp -estimates for Bergman projections in Siegel domains of type, II. Studia Math., 115(3):219–239, 1995. [3] S. G. Gindikin. Analysis on homogeneous domains. Russian Math. Surv., 4(2):1–89, 1964. [4] I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Series, and Products. Academic Press, New York, 1980. [5] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case. Bol. Soc. Mat. Mexicana, 10:119– 138, 2004. [6] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case. J. Operator Theory, 52(1):185– 204, 2004. [7] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators with radial symbols. Integr. Equat. Oper. Th., 20(2):217–253, 2004. [8] S. Grudsky, R. Quiroga-Barranco, and N. Vasilevski. Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal., 234(1):1–44, 2006. [9] A. Kor´ anyi and E. Stein. H 2 spaces of generazed half-spaces. Studia Math., 44:379– 388, 1972. [10] R. Quiroga-Barranco and N. Vasilevski. Commutative algebras of Toeplitz operators on the Reinhard domains. Integr. Equat. Oper. Th. (to appear). [11] N. L. Vasilevski. On Bergman-Toeplitz operators with commutative symbol algebras. Integr. Equat. Oper. Th., 34:107–126, 1999. [12] N. L. Vasilevski. On the structure of Bergman and poly–Bergman spaces. Integr. Equat. Oper. Th., 33:471–488, 1999. [13] N. L. Vasilevski. The Bergman space in tube domains, and commuting Toeplitz operators. Doklady, Mathematics, 61(3):9–12, 2000. [14] N. L. Vasilevski. Poly-Fock spaces. Operator Theory. Advances and Applications, 117:371–386, 2000. [15] N. L. Vasilevski. Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemp. Math., 289:79–146, 2001.

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[16] N. L. Vasilevski. Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry. Integr. Equat. Oper. Th., 46:235–251, 2003. [17] Kehe Zhu. Spaces of Holomorphic Functions in the Unit Ball. Springer Verlag, 2005. Raul Quiroga-Barranco Centro de Investigaci´ on en Matem´ aticas Apartado Postal 402 36000, Guanajuato, Gto. M´exico e-mail: [email protected] Nikolai Vasilevski Departamento de Matem´ aticas CINVESTAV Apartado Postal 14-740 07000, M´exico, D.F. M´exico e-mail: [email protected] Submitted: January 31, 2007 Revised: March 19, 2007

Integr. equ. oper. theory 59 (2007), 421–435 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030421-15, published online October 18, 2007 DOI 10.1007/s00020-007-1538-5

Integral Equations and Operator Theory

An Algorithm for Corona Solutions on H ∞ (D) Tavan T. Trent Abstract. We give an algorithm to find corona solutions in H ∞ (D) for polynomial input data. Keywords. Corona theorem, algorithm.

In 1962, Carleson [1] solved the corona problem for H ∞ (D) and provided estimates on the solutions. Using the Wolﬀ approach for solving the corona problem, Uchiyama [9] improved these estimates to get the best known estimates to date. For {fj }nj=1 ⊂ H ∞ (D) satisfying 0 < 2 ≤ there exists with

n

|fj (z)|2 ≤ 1 for all z ∈ D,

j=1 ∞ {uj }j=1 ⊂ H ∞ (D) n |uj (z)|2 } < sup { z∈D j=1

(

8 1 ln 2 )2 2

(1)

1 . e (For a recent account containing these estimates, see Trent [7] and Treil-Wick [6].) By giving a constructive proof of relevant ∂-equations, Jones [4] provided a constructive proof of the corona theorem for H ∞ (D). However, from this proof it is hard see how corona solutions can be explicitly given. for 0 < <

In this paper we wish to compute corona solutions explicitly and we will give an algorithm to do this. Our input data will be m polynomials {pj }m j=1 of maximal degree n. By this we mean that, if pj (z) = pj0 + · · · + pjk z kj where 1 ≤ j ≤ m and 0 ≤ kj ≤ n, we are given the string of numbers (pj0 , . . . , pjkj , 0 . . . 0). If the n+1

polynomials are given to us in the form pj (z) = Cj (z − αj1 )n1 . . . (z − αjl )nl , we Partially supported by NSF Grant DMS-0400307.

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compute the n + 1 coeﬃcients for pj (z). In addition, we assume that we are given > 0 satisfying 0 < 2 ≤

n

|pj (z)|2 ≤ 1

for all z ∈ D

(2)

j=1

with 0 < <

1 . e

We wish to give explicit formulas for rational functions {uj }m j=1 ⊂ A(D) satisfying (a)

m

pj (z)uj (z) = 1 for z ∈ D

j=1

(b) max {

m

|uj (z)|2 : z ∈ D} < (

j=1

8 1 ln 2 )2 2

and (c) max {order uj } ≤ n. Consider the following example: Suppose that p and q are nontrivial polynomials with no common 0’s in C and with max degree {p, q} n and normalized so that |p(z)|2 + |q(z)|2 ≤ 1, z ∈ D. Considering the resultant of p and q and, using linear algebra, we can explicitly ﬁnd polynomials u0 and v0 with max degree {u0 , v0 } ≤ n − 1 and pu0 + qv0 ≡ 1 in C. A general H ∞ (D) corona solution on D would be u = u0 − qh and v = v0 + ph where h ∈ H ∞ (D). If we let 2 min {|p(z)|2 + |q(z)|2 }, the corona theorem tells us that for some z∈D

h ∈ H ∞ (D), u and v will satisfy (a) and (b). Of course, there is no reason that the easily computed u0 , v0 should satisfy the norm estimate (b), and it does not in general. In particular, if for 1 > > 0 p (z) =

(2 − z)(1 − z) (2 + − z)z √ √ and q (z) = , 45 45

then if u and v are the explicit solutions computed as above, max |u (z)|2 + |v (z)|2 ≈ z∈D

1 → ∞ as ↓ 0. 2

So the problem is how to select a solution with an appropriate bound on its size on the unit disk. We will ﬁrst explain the steps of the algorithm. Then we will illustrate the algorithm with a simple computation. Next we prove that the algorithm works. Finally, we will give further directions and comments on stability. We remark that given (2), our algorithm leads to the best possible upper bound for corona solutions.

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Algorithm For ease of notation, we will illustrate the algorithm for two polynomials. The general ﬁnite case follows quite easily from this one. Assume that p(z) = p0 + p1 z + · · · + pn z n and q(z) = q0 + q1 z + · · · + qm z m are polynomials with pn = 0, m ≤ n and 0 < 2 ≤ |p(z)|2 + |q(z)|2 ≤ 1 for z ∈ D. We assume that 0 < <

1 e

and let δ=

8 1 ln 2 2

−1 .

def

Step 1 C(eit ) = |p(eit )|2 + |q(eit )|2 − δ 2 is a positive trigonometric polynomial of degree n on ∂D. Thus by the factorization theorem of Fejer-Riesz, there is a unique analytic polynomial of degree n, F (eit ), with |F (eit )|2 = C(eit ) on ∂ D, F (eit ) invertible on D, and F (0) > 0. It is important to note that F can be computed explicitly. Of course, we can say that for z ∈ D 1

F (z) = e 2

π eit +z −π eit −z

log |C(eit )| dσ(t)

,

but this does not seem to be a good way to compute the coeﬃcients of the polynomial F (z). Instead, we compute π 1 αj = e−ijt dσ(t) for j = 0, . . . , n. it ) C(e −π (Actually, if we allow rational solutions of order ≤ 3n satisfying (a) and (b), then we will see later that these coeﬃcients, {αj }nj=0 , need only be estimated with suﬃcient precision. See our ﬁnal remarks on stability.) Then we solve:



α0

  α1   ..  . αn

α1

···

α0 .. ···

.

 αn  F 2  1 0 ..   F F 0 1 0 .      .  =  . .  .    ..  .  0 Fn F0 α0

Now F (eit ) = F0 + F1 eit + · · · + Fn eint is an invertible analytic polynomial with |F (eit )|2 = C(eit ) on ∂D and F0 > 0. For details of this argument and extensions to the bidisk, see Geronimo-Woerdeman [2,3]. This completes Step 1.

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Form the n × n positive matrix, R, as follows:   F 0 · · · F n−1 ··· F0  ..   .. .. .. R =  ... . . .  . Fn−1 · · · F0 0 ···   pn · · · pn · · · p1    .. . . .. ..   ... .. −. . 0

···

pn

p1

···

qn  −  ... 0

··· .. . ···

q1 qn ..   .. .. q1 qn

··· .. . ···





 0 ..  .  F0 

0 ..  . pn  0 ..  . qn

Use a Cholesky L U decomposition and factor R = L L∗ , where L is lower triangular. def

Denote the n rows of L by Rj , j = 0, . . . , n − 1 and let R−1 = 0. Note that R depends on δ. The fact that R is positive is one of the main steps in the proof that our algorithm works. The largest δ for which R is positive gives the best bound ( δ12 ) for solutions to the corona problem. Step 3

Let

n pj p(z) zj = qj q(z) j=0 Pj

and P = (P 0 , . . . , P n ) be a 2 × (n + 1) matrix. Deﬁne an (n + 1) × (n + 1) grammian matrix, G, by T T n Rj−1 Rk−1 G= . , Pj Pk Cn+2 j,k=0 We will need to compute [GT ]−1 , which can be accomplished by L U methods. Step 4

Let T

T −1

A = [L , 0] [G ]

0 ∗ , LT

B = [LT , 0] [GT ]−1 P ∗ , 0 C = (1, 0) [GT ]−1 T ∗ , L and

D = (1, 0) [GT ]−1 P ∗ .

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Then for z ∈ D, deﬁne (u(z), v(z)) = D + z C(I − z A)−1 B.

(3)

Then (u(z), v(z)) solves our problem. Since A is an n × n contractive matrix, our construction shows that u(z) and v(z) are rational functions of order ≤ n. Also, u(z)p(z) + v(z)q(z) = 1 1 and |u(z)|2 + |v(z)|2 ≤ 2 , for all z ∈ D. δ These last two facts will follow from our proof, given later. We will also use that, from (3), u, v ∈ A(D), the disk algebra. Example. Let p(z) = 1 − z and q(z) = z. Clearly, p and q have no common 0’s in D (or, in fact, in C) and we will ﬁnd the best corona solution. Of course, u0 ≡ 1 and v0 ≡ 1 works and since the general corona solution has the form u(z) = 1 − z h(z) and v(z) = 1 + (1 − z)h(z) for h ∈ H ∞ (D), it is not hard to verify that (u0 , v0 ) is the best corona solution, i.e., sup |u0 (z)|2 + |v0 (z)|2 ≤ sup |u(z)|2 + |v(z)|2 ,

z∈D

z∈D

where (u, v) is another solution to up + vq ≡ 1 in D and u, v ∈ H ∞ (D). We illustrate how our algorithm produces this solution. Step 1 For δ > 0, so that δ 2 ≤ |1 − z|2 + |z|2 , we ﬁnd F (eit ) = F0 + F1 eit , invertible in H ∞ (D), so that |F0 + F1 eit |2 = |1 − eit |2 + |eit |2 − δ 2 = (3 − δ 2 ) − eit − e−it So |F0 |2 + |F1 |2 = 3 − δ 2 and F0 F 1 = −1. Since F is invertible in D, we choose F0 = |F0 | > |F1 |. Step 2

Form R and factor it. Then R = F02 − |p1 |2 − |q1 |2 = F02 − 2 ≥ 0. R = L L∗ , so L = F02 − 2

(“δ” will need to be chosen so that R ≥ 0).

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1 −1 and 0 1           0 0 0 L  1 , 1 1 , −1     0  0  0  1      G =    L 0 L L    −1 , 1 −1 , −1  1 0 1 1 1 −1 , since L2 + 2 = F02 . = −1 F02 2 F0 1 1 1 . Then [GT ]−1 = 2 F0 − 1

Step 3

Set P =

Step 4

Compute:

2 (F0 − 2) F2 − 2 F0 1 0 (1, 0) , = 02 2 1 1 1 F0 − 1 F0 − 1 2 F02 − 2 1 F0 1 1 0 B= (1, 0) ), = F02 − 2 (1, 2 1 1 −1 1 F02 − 1 F0 − 1 2 F02 − 2 F02 − 2 F0 1 0 (1, 0) , C= = 2 1 1 1 F0 − 1 F02 − 1 (1, 0) F02 1 1 1 0 ). and D = 2 = (1, 2 1 1 −1 1 F0 − 1 F0 − 1 A=

Now for z ∈ D, our corona solution is

−1 F02 − 2 1 1 F02 − 2 )+z 2 ) ) (1, 2 (u(z), v(z)) = (1, 2 1 − z( 2 F0 − 1 F0 − 1 F0 − 1 F0 − 1 1 (F02 − 2) 1 = (1, 2 )+z 2 (1, 2 ) 2 F −1 (F0 − 1) − z(F0 − 2) F0 − 1 0 F02 − 1 1 = , . 2 2 2 (F0 − 1) − z(F0 − 2) (F0 − 1) − z(F02 − 2)

These rational solutions of order ≤ 1 satisfy u(z)(1 − z) + v(z)z = 1 1 and |u(z)|2 + |v(z)|2 ≤ 2 , for z ∈ D. δ From Step 2, our δ must satisfy F02 − 2 ≥ 0; where F0 > |F1 | and δ 2 = 3 − (F02 + F12 ). To get the best solution, let F02 = 2 ⇒ (u(z), v(z)) = (1, 1); and 0

in this case, δ 2 = 12 .

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Verifying the algorithm We will need two lemmas. Suppose that we are given two Hilbert spaces H and K and collections of elements {xα }α∈A ⊂ H and {yα }α∈A ⊂ K, satisfying

xα , xβ H = yα , yβ K for all α, β ∈ A. Then it is easily shown that there exists a partial isometry, D ∈ B(H, K), with D xα = yα for all α ∈ A and D x = 0 if x ⊥H xα for all α ∈ A. For a ﬁnite indexing set, A, our ﬁrst lemma will give an explicit form for D, when the {xα }α∈A are linearly independent. Lemma 1. Let {xj }nj=1 ⊂ H and {yj }nj=1 ⊂ K with xj , xk H = yj , yk K for all k, j = 1, . . . , n. Assume that the {xj }nj=1 are linearly independent. Let G denote the invertible n × n grammian matrix, whose jk-entry is xj , xk H . Define D=

n n

[G−1 ]kj yj ⊗ xk .

j=1 k=1

Then D is a partial isometry satisfying: D xp = yp Dx = 0

and Proof.

D xp =

=

if x ⊥ xj , for j = 1, . . . , n.

n n j=1

=

for p = 1, . . . , n

xp , xk [G−1 ]kj

k=1

n n j=1 n

yj

[G]pk [G−1 ]kj

yj

k=1

δpj yj = yp .

j=1

Also, if x ⊥ xj j = 1, . . . , n, then, as above, n n Dx = 0 · [G−1 ]kj yj = 0. j=1

k=1

Our next lemma is the simple but powerful result of Sz. Nagy-Foias [5]. Lemma 2 (Sz. Nagy-Foias). Assume that D : H ⊕ K → H ⊕ K is a contraction. A B ) , where A ∈ B(H) satisfies A = P D P Write D = ( C H H and define B, C, D D similarly. For |z| < 1, define Φ(z) = D + z C(IH − z A)−1 B. Then

Φ(z)B(K) ≤ 1.

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Proof. Since A is a contraction and |z| < 1, (IH − z A)−1 ∈ B(H). Let k ∈ K. Then 2 A B z(IH − zA)−1 Bk = Φ(z)k2K + (IH − zA)−1 Bk2H C D k H⊕K ≤ k2K + |z|2 (IH − zA)−1 Bk2H , since D is a contraction. But then Φ(z)k2K ≤ k2K − (1 − |z|2 ) (IH − z A)−1 B k2H ≤ k2K , since |z| < 1.

2

We will also need the H (D)-corona theorem, which easily follows from the Carleson corona theorem and whose estimates follow from Uchiyama’s results. However, since the H 2 (D)-corona theorem is a Hilbert space theorem, there should be a direct proof based on Hilbert space principles, namely the Riesz representation theorem and the Cauchy-Schwarz inequality. See Trent [7] for such a proof. H 2 (D)-Corona Theorem. Assume that ∞ {fj }∞ j=1 ⊆ H (D) with

0 < 2 ≤

∞

|fj (z)|2 ≤ 1 for all z ∈ D

j=1

and that

Then

δ2I <

0<< ∞

1 . e

Tfj Tf∗j ≤ I,

j=1

where

δ2 =

8 1 ln 2 2

−2 .

We are now ready to justify our algorithm. Theorem. Assume that p(z) = p0 + p1 z + · · · + pn z n and q(z) = q0 + · · · + qm z m are polynomials with pn = 0, m ≤ n, and 0 < 2 ≤ |p(z)|2 + |q(z)|2 ≤ 1 for z ∈ D. "−2 ! . Assume that 0 < < 1e and let δ 2 = 82 ln 12 Then there exists an algorithm to compute rational functions u and v in A(D) satisfying (a) p(z)u(z) + q(z)v(z) = 1 for z ∈ D 1 (b) max{|u(z)|2 + |v(z)|2 } < 2 δ and (c) max{order(u), order(v)} ≤ n.

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Proof. This proof gives the justiﬁcation for our algorithm. By the H 2 (D)-corona theorem, Tp Tp∗ + Tq Tq∗ − δ 2 I > 0. Since Tp∗ Tp ≥ Tp Tp∗ and Tq∗ Tq ≥ Tq Tq∗ , we have T|p|2 +|q|2 −δ2 = Tp∗ Tp + Tq∗ Tq − δ 2 I > 0. Since T|p|2 +|q|2 −δ2 is an invertible Toeplitz operator, we see that |p(eit )|2 + |q(eit )|2 − δ 2 is an invertible trigonometric polynomial on the unit circle. By the theorem of Fejer-Riesz, we write |p(eit )|2 + |q(eit )|2 − δ 2 = |F (eit )|2 , where F (eit ) is a polynomial of degree n, which is invertible in H ∞ (D). Now Tp Tp∗ + Tq Tq∗ − δ 2 I = TF∗ TF − [(Tp∗ Tp − Tp Tp∗ ) + (Tq∗ Tq − Tq Tq∗ )].

(4)

(∗)

A simple computation shows that with respect to the standard orthonormal basis for H 2 , {eint }∞ n=0 , (∗) has the matrix form:        pn · · · p1 pn · · · 0 qn · · · 0 qn · · · q1  .. ..   .. .  . ..  +  .. . . ..  0 .. .. ..  . . . ..   .. . . .  . . . .   .  0 · · · pn  0 · · · qn p1 · · · pn q1 · · · qn 0 0 def

∗ ∗ It is clear that Mn = sp {eijt }n−1 j=0 is an invariant subspace for (Tp Tp −Tp Tp )+ ∗ ∗ (Tq Tq − Tq Tq ). Let Pn denote the orthogonal projection onto this space.

We wish to show that TF∗ Pn TF − (∗) ≥ 0.

(5)

This is the key point that makes our algorithm a ﬁnite one, and thus feasible for computer implementation. TF∗ TF − (∗) > 0.

Now TF is invertible and So Thus

I − T ∗1 (∗) T F1 > 0. F

∗

Pn − T 1 (∗) T F1 + Pn⊥ > 0. F

But Mn is an invariant subspace for Tz∗ and thus for T ∗1 . Hence F

∗

Pn − T 1 (∗) T F1 > 0 and thus F

TF∗ Pn TF

− (∗) > 0.

Let S = Tz ; then Pn = I − S n S n∗ . For F (eit ) = F0 + F1 eit + · · · + Fn eint , we have n Fj S ∗j . TF∗ = j=0

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Thus TF∗ S n = =

n j=0 n

Fj S ∗j S n =

n

Fj S n−j

j=0

Fn−k S k = TF# .

k=0

Here F# (eit ) = Fn + Fn−1 eit + · · · + F1 eint . We conclude that TF∗ Pn⊥ TF = TF∗ S n S n∗ TF = TF# TF∗# . Let R = TF∗ Pn TF − (∗). By (4) and the above calculations Tp Tp∗ + Tq Tq∗ − δ 2 I = R + TF# TF#∗ . For |w| < 1, kw (eit ) = 1−w1 eit , the reproducing kernel for H 2 (D), and H ∞ (D), it is easy to see that Tϕ∗ kw = ϕ(w) kw . Thus from (6),

(6) ϕ ∈

(Tp Tp∗ + Tq Tq∗ − δ 2 I)kw , kz = R kw , kz + TF# TF∗# kw , kz . Hence, p(z)p(w) + q(z)q(w) − δ 2 = Rkw , kz (1 − zw) + F# (z)F#(w) for all |z|, |w| < 1. Or p(z)p(w) + q(z)q(w) + z w Rkw , kz = δ 2 + F#(z)F#(w) + Rkw , kz .

(7)

for all |z|, |w| < 1.

!p " Recall that P j = qjj , P = [P 0 . . . P n ] and R = L L∗ , with the rows of L being denoted by R0 , . . . , Rn−1 . For j = 0, . . . , n, let T Rj−1 xj = ∈ Cn+2 . Pj  T  T  Rj R0 Also, deﬁne y 0 =  Fn  , y j = Fn−j  for j = 1, . . . , n − 1 δ 0   0 and y n = F0  . 0 Again {yj }nj=0 ⊆ Cn+2 . If X denotes the (n+2)×(n+1) matrix, X = [x0 , . . . , xn ], then T 0L X= . P

Algorithm for Corona Solutions on H ∞ (D)

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Similarly, if Y denotes the (n + 2) × (n + 1)  T L ··· Y =  Fn · · · δ ···

431

matrix, Y = [y0 , . . . , y n ], then  0 F0  . 0

Then since L is n × n and invertible and P 0 = ( 00 ), we know that X has rank (n + 1) so the columns {xj }n−1 j=0 are linearly independent. By (7), we have that for |w|, |z| < 1, n n n n

z j xj , wj xj Cn+2 = z j yj , wj yj Cn+2 . j=0

j=0

j=0

j=0

Let

M = {xj }nj=0 ≤ Cn+1

and

M= {

n

z j xj : z ∈ D} ≤ Cn+1 .

j=0

Clearly, M ≤ M . But M ≤ M as well, since if we let {zp }np=0 denote distinct elements of D and let n zpj xj ∈ M for p = 0, . . . , n, hp = j=0

    1 h0 x0  z0   ..  T −1  .  then  .  = [V ]  ..  , where V =  .. . xn hn z0n is a Vandermonde matrix. 

··· ··· ···

 1 zn     n zn

Thus any partial isometry taking n

z j xj →

j=0

n

z j yj

for all z ∈ D

j=0

is determined by the partial isometry D ∈ M (Cn+2 ) with D xj = yj j = 0, . . . , n − 1 and D x = 0 if x ⊥ xj for all j = 0, . . . , n − 1.

for all

By Lemma 1, with respect to the standard basis in Cn+2 D = Y [GT ]−1 X ∗ , where [G]j,k = xj , xk Cn+2

for j, k = 0, . . . , n.

Let Pj denote the orthogonal projection of Cn+2 onto the span of {ej : j = n+2 1, . . . , n}, where {ej }n+2 . Deﬁne C0 = j=1 denotes the standard basis of C (I − (Pn+1 Pn )) C and D0 = (I − (Pn+1 Pn )) D.

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We may apply Lemma 2 to D to get 1 Φ(z) = (D0 + z C0 (I − z A)−1 B), δ δ which satisﬁes (b) of the theorem. def

(u(z), v(z)) =

p(z) (u(z), v(z)) ≡1 q(z)

We claim that

for |z| < 1.   1 z T T A B 0 R0 · · · Rn−1   . C0 D0 P0 P1 ··· P n  ..  zn   1  z  T T T R0 R1 · · · Rn−1 0   = . δ 0 ··· 0 0  ..  zn  $n−1 k T  $n−1 k T z( k=0 zRk ) A B  k=0 z Rk .  = p(z) C0 D0 δ q(z) So

n−1

z A(

z k RTk ) + B

k=0

or

n−1 p(z) = z k RTk q(z) k=0

n−1 p(z) (I − z A)−1 B = z k RTk . q(z) k=0

Also

D0

p(z) + z C0 q(z) and thus

%n−1 k=0

Φ(z)

& z k RTk

=δ

p(z) = δ, q(z)

Φ(z) = D0 + z C0 (I − z A)−1 B and |z| < 1. 0 T T −1 A = Pn D Pn = [L 0] [G ] , LT ∗

where Here

B = Pn D(Pn+2 Pn ) = [LT 0] [GT ]−1 P ∗ , 0 C0 = (Pn+2 Pn+1 ) D Pn = δ(1, 0) [GT ]−1 T ∗ , L and

D0 = (Pn+2 Pn+1 ) D(Pn+2 − Pn ) = δ(1, 0) [GT ]−1 P ∗ .

This completes the proof of the algorithm.

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We note that if our corona input is m polynomials, {pj }m j=1 , (instead of two $m 2 2 polynomials), satisfying 0 < ≤ ≤ 1 and if max {degree j=1 |pj (z)| (pj ) : 1 ≤ j ≤ m} = n, our vectors {xj }nj=0 will now belong to Cn+m . Our R now becomes m [Tp∗j , Tpj ]. R = TF∗ Pn TF − j=1

The proof proceeds as before. Remarks. (1) Any δ > 0 which causes R to be positive gives a corona solution of size controlled by 1δ . Thus the best solution is given by the largest δ > 0 for which the matrix R remains positive. (2) A similar argument applies to a ﬁnite number of ﬁnite Blaschke products replacing the polynomial corona data. Blaschke product corona data has the advantage that Step 1 is trivial in this case. Combining with (1), this case should have further applications. (3) The case of matrix-valued rational functions on D as input data can be handled in a crude way using determinants and the previous algorithm. See TrentZhang [8]. However, to get a suitable computational algorithm in this case will require more investigation. (4) The main problem for an explicit computation of a corona solution is the Fejer-Riesz factorization of a positive trignometric polynomial on the unit circle. Suppose by some technique we compute an invertible analytic polynomial G with G − F ∞,D “small”, where F is the actual invertible polynomial of degree n satisfying F0 > 0 and |F (eit )|2 = |p(eit )|2 + |q(eit )|2 − δ 2 . Then we build R > 0. So factor R into its decomposition and form matrices X , Y similar to X and Y. U . We let U = Y (X ∗ X )−1 X ∗ and D = U ' ( A B and form Φ (z) = 1δ (D0 + z C0 (I − z A )−1 B ). Decompose D into C D Then for |z| < 1, let

0

0

Ψ(z) =

Φ (z) . p(z) Φ (z) q(z)

Then Ψ(z) has rational polynomial entries of order ≤ 3 n. Also, Φ (z) p(z) . Ψ(z) = 1 and Ψ(z) = q(z) p(z) 1 − (Φ(z) − Φ (z)) q(z) So

Ψ∞ ≤

1 δ

1 − Φ − Φ ∞

, provided Φ − Φ ∞,D < 1.

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So we must estimate Φ − Φ∞,D . (Here Φ is the theoretical solution gotten from our algorithm using the polynomial F (eit ).) For an n × m matrix A, we let Aop denote the operator norm of the matrix A. It is clear that we can explicitly connect an estimate of F − G∞,D with D − D op ; that is, given > 0, there is an explicit δ(, n) so that if

F − G∞,D ≤

n

Fj − Gj ∞,D ≤ δ(, n),

j=0

then

D − D op < .

But what can be said about Φ − Φ ∞,D ? First, note that σ(A) ⊂ D, i.e., ρ(A) < 1, where ρ(A) is 'the spectral radius of ( eit 0 ∗ A. Else there exists a unitary U ∈ Mn , so that U A U = 0 ∗ for some real t. Thus, ∗ U 0 A B U U 0 0 0 LT 0 I2 C D 0 I2 0 I2 P   T L 0 U 0  Fn · · · F0  = 0 I2 δ ··· 0  it    0 U B U∗ 0 e U LT T UL  0 ∗ · · · F1  . or  0 =  Fn P D C U∗ δ ··· 0 But this forces the top row of invertible.

U LT to consist of all 0’s, contradicting

We have A, A ∈ Mn (C), A − Aop ≤ D − D op < , Aop ≤ 1, A op ≤ 1, and

ρ(A) < 1.

(a) There exists an explicit constant depending only on n so that Cn (eit I − A)−1 op ≤ . (1 − ρ(A))n (b) If N is large enough, by the spectral radius theorem if

A − Aop ≤ ρ(A), then

ρ(A) ≤ (1 + 2) ρ(A ).

(c) Thus Φ − Φ ∞,D ≤ +

3 2 Cn2 . (1 − (1 + 2 )ρ(A))2

U LT

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Combining (a), (b), (c) and the above results, we conclude that for a computationally stable algorithm for corona solutions it suﬃces to estimate ρ(A) in terms of the input data.

References [1] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Annals of Math. 76 (1962), 547–559. [2] J.S. Geronimo and H.J. Woerdeman, Positive extensions and Riesz-Fejer factorizations for two-variable trignometric polynomials, CDSNS (2000), preprint. , Positive extensions, Fejer-Riesz factorization and autoregressive filters in [3] two variables, Annals of Math. 160 (2004), 839–906. [4] P.W. Jones, Estimates for the corona problem, J. Func. Anal. 39 (1980), 162–181. [5] B. Sz.-Nagy, Unitary dilations of Hilbert space operators and related topics, CBMS 19 (1974). [6] S. Treil and B.D. Wick, The matrix-valued H p corona theorem in the disk and polydisk, J. Func. Anal. 226 (2005), 138–172. [7] T.T. Trent, A new estimate for the vector-valued corona problem, J. Func. Anal. 189 (2002), 267–282. [8] T.T. Trent and X. Zhang, A matricial corona theorem, Proc. Amer. Math. Soc., to appear. [9] A. Uchiyama, Corona theorems for countably many functions and estimates for their solutions, 1980, preprint. Tavan T. Trent Department of Mathematics The University of Alabama Box 870350 Tuscaloosa, AL 35487-0350 USA e-mail: [email protected] Submitted: February 8, 2007 Revised: June 11, 2007

Integr. equ. oper. theory 59 (2007), 437–448 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030437-12, published online June 27, 2007 DOI 10.1007/s00020-007-1524-y

Integral Equations and Operator Theory

Structure on Powers of p-Hyponormal and log-Hyponormal Operators Jiangtao Yuan and Zongsheng Gao Abstract. Inspired by the problem of powers of hyponormal operators, this paper is to discuss the structure on powers of p-hyponormal and log-hyponormal operators. The structure on powers of operators consists of same-side structure and diﬀerent-side structure. The same-side structure means relations between n+m n n n+m T n+m and T ∗ T n (or T n T ∗ and T n+m T ∗ ), and the diﬀerent-side T∗ m n structure means relations between T ∗ T m and T n T ∗ where m, n are positive integers and T is a bounded linear operator on a Hilbert space. Thus, the original problem of powers of hyponormal operators belongs to diﬀerentside structure on powers of hyponormal operators. The structure on powers of p-hyponormal operators for p > 0 is emphasized. Also, some applications are obtained. Mathematics Subject Classiﬁcation (2000). 47B20, 47A63. Keywords. p-hyponormal operator, log-hyponormal operator, Furuta inequality.

1. Introduction A capital letter (such as T ) means a bounded linear operator on a Hilbert space. T ≥ 0 and T > 0 mean a positive operator and an invertible positive operator respectively. For p > 0, T is called a p-hyponormal operator if (T ∗ T )p ≥ (T T ∗)p , where T ∗ is the adjoint operator of T . If p = 1, T is called a hyponormal operator and if p = 1/2, T is called a semi-hyponormal operator. It is clear that every p-hyponormal operator is q-hyponormal for 0 < q ≤ p by the celebrated L¨owner-Heinz theorem (A ≥ B ≥ 0 ensures Aα ≥ B α for any 1 ≥ α ≥ 0). An invertible operator T is called a log-hyponormal operator if log(T ∗ T ) ≥ log(T T ∗ ). Each invertible phyponormal operator for p > 0 is log-hyponormal since log t : (0, ∞) → (−∞, ∞) This work is supported in part by the National Key Basic Research Project of China Grant No. 2005CB321902.

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is an operator monotone function. log-hyponormality is sometimes regarded as 0-hyponormal since limp→0 (X p − 1)/p = log X for X > 0. Hyponormal operators have been studied by many authors. See Halmos [13], Martin–Putinar [18] and Xia [24] for related topics and basic properties of hyponormal operators. The semi-hyponormal operators were introduced by Xia [23]. See Xia [24] for related topics of semi-hyponormal operators. Fujii–Nakatsu [8] introduced p-hyponormal operators for p ≥ 1. Aluthge [1] introduced p-hyponormal operators for 0 < p < 1/2. log-hyponormal operators were introduced by Tanahashi [20]. It is well known that if T is a hyponormal operator, T 2 is not hyponormal in general. Halmos ([13], Problem 209) gave an example of a hyponormal operator T whose square T 2 is not hyponormal, the operator T is constructed by Ito-Wong [17] by using unilateral shift. In 1999, Aluthge [2] showed the following results by using induction, Furuta inequality [9] and Hansen inequality [14]. Theorem 1.1 ([2]). If n is a positive integer and T is p-hyponormal for p ∈ (0, 1], then ∗n n p/n ∗ p ∗p n ∗n p/n ≥ T T ≥ TT ≥ T T . T T This result can be regarded as a jump from the original problem. It implies that the method (induction) and the tools (the two inequalities [9, 14]) are useful for the problem. Soon, some researches obtained more precise results. Theorem 1.2 ([11, 12, 15, 25]). Let k, m and n be positive integers and T be a p-hyponormal operator for p ∈ (k − 1, k]. (1) If p ∈ (0, 1], then ∗n n (p+1)/n 2 (p+1)/2 ∗ p+1 T T ≥ ··· ≥ T∗ T2 ≥ T T , (1.1) ∗ p+1 2 ∗2 (p+1)/2 n ∗n (p+1)/n TT ≥ T T ≥ ··· ≥ T T . (2) If p ∈ (k − 1, k], then ∗1+m 1+m (1+min{p,m})/(1+m) ∗ 1+min{p,m} T T ≥ T T , ∗1+min{p,m} 1+m ∗1+m (1+min{p,m})/(1+m) TT ≥ T T .

(1.2) (1.3)

(1.1) is a generalization of [11, 25], [25] also showed that (1.1) is valid for loghyponormal operator (p = 0) by using Furuta type operator functions. (1.2)–(1.3) is showed by [15] which is a sharpen of [11] by induction based on classiﬁcation in of m (m < p and m ≥ p). It is interesting that the outer exponent 1+min{p,m} 1+m (1.2)–(1.3) is the best possible, that is, it can not be improved in general. Theorem 1.2 implies that the structure (same-side and diﬀerent-side) on powers of p-hyponormal operators for p ≥ 0 always exists. The similar property is proved for more wide classes of operators, see [5, 16, 27, 30, 31]. Now, the discussion is focused on the best possibility of the structure. YangYuan [29] showed the following results on such discussion by operator functions

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and induction based on classiﬁcation of m (m < p and m ≥ p) and n (n < p and n ≥ p). Theorem 1.3 ([29]). Let k, m, n be positive integers and p ∈ (k − 1, k]. (1) If k ∈ {1, 2} and T is a p-hyponormal, then ∗n+m n+m (n+min{p,m})/(n+m) ∗n n (n+min{p,m})/n T T ≥ T T , n ∗n (n+min{p,m})/n n+m ∗n+m (n+min{p,m})/(n+m) ≥ T T . T T (2) If T is log-hyponormal operator, then ∗n+m n+m n/(n+m) n n n+m n/(n+m) T ≥ T ∗ T n , T n T ∗ ≥ T n+m T ∗ . T

(1.4)

(1.5)

Theorem 1.3 can not be improved in general. This implies that the structure on powers of p-hyponormal operators for 2 ≥ p ≥ 0 is clear (the diﬀerent-side structure always follows by the same-side structure). In this paper, we consider the best possible structure on powers of loghyponormal and p-hyponormal operators for p > 0. Some applications related to the structure are obtained.

2. Improvements of the Structure on Powers In this section, we shall show brieﬂy the following improvements by applying Furuta inequality only (without induction and operator functions). For convenience, let γ = min{p, m, n}. Theorem 2.1 (Diﬀerent-side structure). Let k, m, n be positive integers and p ∈ (k − 1, k]. (1) If n ≥ 2, m ≥ 2 and T is p-hyponormal, then m

γ

n

γ

(T ∗ T m ) m ≥ (T n T ∗ ) n . m

(T ∗ T m )

min{p,m} m

(2.1)

≥ (T T ∗ )min{p,m} . n

(T ∗ T )min{p,n} ≥ (T n T ∗ )

min{p,n} n

.

(2.2) (2.3)

(2) If T is log-hyponormal, then m

1

n

1

log(T ∗ T m ) m ≥ log(T n T ∗ ) n .

(2.4)

Theorem 2.1 (1) and (2) can be regarded as generalizations of ([15], Corollary 2) and ([25], Corollary 3) respectively. The outer exponents in Theorem 2.1 are the best possible (Theorem 3.2 below), too. Though (2.1)–(2.3) follow by (1.2)–(1.3), on the contrary, (1.2) and (1.3) follow by (2.2) and (2.3) respectively. Theorem 2.2 (Equivalence relations). For each positive integer k, n, m and p ∈ (k − 1, k]. If T is p-hyponormal, then (1.2), (1.3), (2.2) and (2.3) hold, and (1.2) and (1.3) are equivalent to (2.2) and (2.3) respectively.

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According to Theorem 1.3, we may expect that (1.3) is valid for p-hyponormal operators where p > 0. Here we extended it to the following. Theorem 2.3 (Same-side structure). Let k, m, n be positive integers and p ∈ (k − 1, k]. If n ≥ 2 and T is p-hyponormal, then ∗n+m n+m (n+γ)/(n+m) ∗n n (n+γ)/n T T ≥ T T , n ∗n (n+γ)/n n+m ∗n+m (n+γ)/(n+m) T T ≥ T T .

(2.5) (2.6)

Theorem 2.3 can be regarded as a parallel result to (2) of Theorem 1.3 (p = 0). By contrasting Theorem 2.3 with Theorem 1.2 (2), case n = 1 of the same-side structure is diﬀerent to the case n ≥ 2 of the same-side structure. The diﬀerence also implies that the restriction 2 ≥ p > 0 in Theorem 1.3 is necessary. To give proofs, we need Furuta inequality (call it FI in brief) which is an essential extension of the L¨ owner-Heinz inequality (call it LH in brief). Theorem 2.4 ([9]). If A ≥ B ≥ 0, then for each r ≥ 0, r/2 p r/2 q1 1 B A B ≥ B r/2 B p B r/2 q , (2.7) r/2 p r/2 q1 r/2 p r/2 1q ≥ A B A A A A (2.8) hold for p ≥ 0 and q ≥ 1 with (1+r)q ≥ p + r.

(1 + r)q = p + r

p

p=q

q=1

(1, 1)

q

(1, 0) (0, −r)

Domain of Furuta inequality.

FI yields the famous LH by putting r = 0 in (i) or (ii) of FI. It was shown by Tanahashi [20] that the domain for p, q and r in Theorem 2.4 is the best possible. See [10] for related topics. Proof of Theorem 2.1. We will prove (2.1)–(2.4) in order. (1) To prove (2.1)–(2.3). (i) To prove cases n ≥ k and m ≥ k of (2.1)–(2.3). p ∈ (0, 1] and kp ∈ (0, 1], we have In fact, by (1.2) and LH for 1+p m

p

(T ∗ T m ) m ≥ (T ∗ T )p ,

n

p

(T T ∗ )p ≥ (T n T ∗ ) n .

(2.9)

Hence, (2.1)–(2.3) follow by (2.9) and p-hyponormality of T . (ii) To prove cases n < k (or m < k) of (2.1)–(2.3). In fact, by (1.2) and LH for min{n,m} ∈ (0, 1] (or min{n,m} ∈ (0, 1]) and m n min{n,m} ∈ (0, 1], we have p m

(T ∗ T m )

min{n,m} m

n

≥ (T ∗ T )min{n,m} , (T T ∗ )min{n,m} ≥ (T n T ∗ )

min{n,m} n

Hence, (2.1)–(2.3) follow by (2.10), p-hyponormality of T and LH.

. (2.10)

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(2) To prove (2.4). It is obvious that (2.4) is a direct result of case n = 1 of (1.5) by LH and log-hyponormality of T . Proof of Theorem 2.2. We only need to prove (2.2) and (2.3) imply (1.2) and (1.3) respectively. To prove (2.2) ⇒ (1.2). In fact, if m ≥ p, (2.2) becomes m

p

(T ∗ T m ) m ≥ (T T ∗)p . On the other hand, (1.2) is equivalent to the following by the property of the polar decomposition of T ∗ ∗ m 2 ∗ (1+p)/(1+m) ∗ 2(1+p) T T T ≥ T . 2p 2p m 1 Therefore, applying (2.7) to |T m | m and T ∗ for (1 + p1 ) 1+m 1+p ≥ p + p , (1.2) follows. The proof of (2.3) ⇒ (1.3) is similar to the proof of (2.2) ⇒ (1.2), so we omit it here. Proof of Theorem 2.3. To prove (2.5) and (2.6). (i) To prove case n ≥ k and m ≥ k of (2.5). In fact, (2.5) is equivalent to the following by the property of the polar n decomposition of T ∗ ∗n m 2 ∗n (n+p)/(n+m) ∗n 2(n+p)/n T T T ≥ T . (2.11) 2p 2p n By Theorem 2.1 (1) and applying (2.7) to |T m | m and T ∗ n for (1 + np ) n+m n+p ≥ m n p + p , (2.11) follows. (ii) To prove case 2 ≤ m < k and n ≥ m of (2.5). Similarly, (2.5) is equivalent to the following ∗n m 2 ∗n ∗n 2(n+m)/n T T T ≥ T . (2.12) (2.12) is a direct result of (2.1). (iii) To prove case 2 ≤ n < k and m ≥ n of (2.5). Similarly, (2.5) is equivalent to the following ∗n m 2 ∗n (n+n)/(n+m) ∗n 2(n+n) T T T ≥ T .

(2.13)

2n n 2 By Theorem 2.1 (1) and applying (2.7) to T m m and T ∗ for (1 + 1) n+m 2n ≥ m + 1, (2.13) follows. n (iv) Proof of (2.6) is similar to the prove of (2.5). For convenience, we only give the proof of case n ≥ k and m ≥ k of (2.6). In fact, (2.6) is equivalent to the following by the property of the polar decomposition of T n n ∗m 2 n (n+p)/(n+m) n 2(n+p)/n T T T ≤ T . (2.14)

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m 2p 2p By Theorem 2.1 (1) and applying (2.8) to T ∗ m and T n n for (1 + np ) n+m n+p ≥ m n + , (2.14) follows. p p (2.5) and (2.6) can be rephrased to be the following result that is similar to (1.1). Corollary 2.5. Let k, m, n be positive integers such that n ≥ 2 and k ≥ 3, p ∈ (k − 1, k] and T be a p-hyponormal operator. (1) If n ≥ k, m ≥ k, then n+p

n+m

n+p

n+k

n

n+p

T n+m ) n+m ≥ · · · ≥ (T ∗ T n+k ) n+k ≥ (T ∗ T n ) n , (T ∗ n+k n+p n+m n+p n ∗n n+p (T T ) n ≥ (T n+k T ∗ ) n+k ≥ · · · ≥ (T n+m T ∗ ) n+m . (2) If 2 ≤ m < k, n ≥ m, then n+n

n+m

n+m

n+m

n

(T ∗ T n+n ) n+n ≥ · · · ≥ T ∗ T n+m ≥ (T ∗ T n ) n , n n+m n+m n+n n+m (T n T ∗ ) n ≥ T n+m T ∗ ≥ · · · ≥ (T n+n T ∗ ) n+n . (3) If 2 ≤ n < k and m > n, then 2n

n+m

n+n

n

(T ∗ T n+m ) n+m ≥ · · · ≥ T ∗ T n+n ≥ (T ∗ T n )2 , 2n n n+n n+m (T n T ∗ )2 ≥ T n+n T ∗ ≥ · · · ≥ (T n+m T ∗ ) n+m .

3. Best Possibility of the Improved Structure In this section, we consider the best possibility of the improved structure in section 2. For convenience, let γ = min{p, m, n}. [29] showed the following estimation on same-side structure which are generalizations of [11, 12, 15]. Theorem 3.1 ([29]). For each positive integer k, n, m and p ∈ (k − 1, k], α > 1, the following assertions hold. (1) There exists a p-hyponormal operator T such that (T ∗

n+m

n

T n+m )(n+min{p,m})α/(n+m) ≥ (T ∗ T n )(n+min{p,m})α/n .

(2) There exists a p-hyponormal operator T such that n

(T n T ∗ )(n+min{p,m})α/n ≥ (T n+m T ∗

n+m

)(n+min{p,m})α/n+m .

(3) There exists a log-hyponormal operator T such that (T ∗

n+m

n

T n+m )nα/(n+m) ≥ (T ∗ T n )α .

(4) There exists a log-hyponormal operator T such that n

(T n T ∗ )α ≥ (T n+m T ∗

n+m

)nα/(n+m) .

This estimation implies that the same-side structure (1.4)–(1.5), case n ≥ k, m ≥ k and case 2 ≤ m < k, n ≥ m of (2.5)–(2.6) is the best possible. Here, we show the best possibility of the structure Theorem 2.1.

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Theorem 3.2. For each positive integer k, n, m and p ∈ (k − 1, k], α > 1, the following assertions hold. (1) If n ≥ 2, m ≥ 2, there exists a p-hyponormal operator T such that m

n

(T ∗ T m )γα/m ≥ (T n T ∗ )γα/n .

(3.1)

(2) If m ≥ 2, there exists a p-hyponormal operator T such that m

(T ∗ T m )

min{p,m}α m

≥ (T T ∗)min{p,m}α .

(3.2)

(3) If n ≥ 2, there exists a p-hyponormal operator T such that n

(T ∗ T )min{p,n}α ≥ (T n T ∗ )

min{p,n}α n

.

(3.3)

(T ∗ T m )(α−1)/m ≥ (T n T ∗ )(α−1)/n .

(3.4)

(4) There exists a log-hyponormal operator T such that m

n

Remark 3.3. It is not known whether or not the case 2 ≤ n < k, m > n of (2.5)– (2.6) is the best possible. The simplest case is: Is the out exponent of the following inequality is the best possible in the sense of Theorem 3.2 under the condition that T is p-hyponormal for p ∈ (2, 3], (T ∗

2+3

2

T 2+3 )(2+2)/(2+3) ≥ (T ∗ T 2 )(2+2)/2 .

We need the following results to give proofs. Theorem 3.4 ([20, 26]). Let δ > 0, p > 0, r > 0 and q > 0. If 0 < q < 1 or (δ + r)q < p + r, then the following assertions hold: (i) There exist positive invertible operators A and B on R2 such that r/2 p r/2 1/q B A B ≥ B (p+r)/q . Aδ ≥ B δ ,

(3.5)

(ii) There exist positive invertible operators A and B on R2 such that 1/q A(p+r)/q ≥ Ar/2 B p Ar/2 . Aδ ≥ B δ ,

(3.6)

Lemma 3.5 ([5, operators A and B on H, define operators 12, 26, 31]). For positive ∼ H where H H as follow: U and D on ∞ = k k k=−∞   ..  .   ..   . 0      1 0   U = , 1 (0)     1 0     1 0   .. .. . .

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       D=     

..

IEOT

 .

           

1

B2

1

B2

1

(A 2 )

1

A2

1

A2 ..

.

where (·) shows the place of the (0, 0) matrix element, and T = U D. Then the following assertions hold: (i) T is p-hyponormal for p > 0 if and only if Ap ≥ B p . (ii) T is log-hyponormal if and only if A and B are invertible and log A ≥ log B. Furthermore, the following assertions hold for β > 0 and any positive integer n and m: β m n β (iii) (T ∗ T m ) m ≥ (T n T ∗ ) n if and only if Aβ ≥ B β holds l 2

l 2

(B Am−l B )

β m

j 2

and

≥ B β holds f or l = 1, 2, . . . , m. j 2

Aβ ≥ (A B n−j A )

β n

(3.7)

holds f or j = 1, 2, . . . , n.

Proof of Theorem 3.2. The proof is similar to that of [12, 15] which proved the case m = n of Theorem 3.2. (1) To prove (3.1). (i) To prove case n ≥ k and m ≥ k of (3.1). In fact, it is well known that there exist positive operators A and B on H such that Ap ≥ B p , Apα ≥ B pα (3.8) ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal pα m n pα by (3.8) and (i) of Lemma 3.5, and (T ∗ T m ) m ≥ (T n T ∗ ) n by (iii) of Lemma 3.5 since the ﬁrst inequality of (3.7) does not hold for β = pα by (3.8). (ii) To prove case 2 ≤ n < k or 2 ≤ m < k of (3.1). If m ≥ n, put p1 = n − 1 > 0, r1 = 1 > 0, q1 = α1 ∈ (0, 1) and δ = p. By (ii) of Theorem 3.4, there exist positive invertible operators A and B on H such that Aδ ≥ B δ ,

A

(p1 +r1 ) q1

≥ (A

r1 2

B p1 A

r1 2

1

) q1 ,

that is, 1

1

(3.9) Ap ≥ B p , Anα ≥ (A 2 B n−1 A 2 )α . ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal m n nα by (3.9) and (i) of Lemma 3.5, and (T ∗ T m ) m ≥ (T n T ∗ )α by (iii) of Lemma 3.5 since the case j = 1 of (3.7) does not hold for β = nα by (3.9).

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If m ≤ n, put p1 = m − 1 > 0, r1 = 1 > 0, q1 = α1 ∈ (0, 1) and δ = p. By (i) of Theorem 3.4, there exist positive invertible operators A and B on H such that (p1 +r1 ) r1 r1 1 Aδ ≥ B δ , (B 2 Ap1 B 2 ) q1 ≥ B q1 , that is, 1

1

Ap ≥ B p , (B 2 Am−1 B 2 )α ≥ B mα . (3.10) ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal m n mα by (3.10) and (i) of Lemma 3.5, and (T ∗ T m )α ≥ (T n T ∗ ) n by (iii) of Lemma 3.5 since the case l = 1 of (3.7) does not hold for β = mα by (3.10). (2) To prove (3.2). If m ≥ p, it is well known that there exist positive operators A and B on H such that (3.6) ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal pα m by (3.8) and (i) of Lemma 3.5, and (T ∗ T m ) m ≥ (T T ∗ )pα by (iii) of Lemma 3.5 since the ﬁrst inequality of (3.7) does not hold for β = pα by (3.8). If 2 ≤ m < p, put p1 = m − 1 > 0, r1 = 1 > 0, q1 = α1 ∈ (0, 1) and δ = p. By (i) of Theorem 3.4, there exist positive invertible operators A and B on H such that (3.10). ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal m by (3.10) and (i) of Lemma 3.5, and (T ∗ T m )α ≥ (T T ∗ )mα by (iii) of Lemma 3.5 since the case l = 1 of (3.7) does not hold for β = mα by (3.10). (3) To prove (3.3). Proof of (3.3) is similar to the proof of (3.2). If n ≥ p, it is well known that there exist positive operators A and B on H such that (3.8) ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is p-hyponormal n pα by (3.8) and (i) of Lemma 3.5, and (T ∗ T )pα ≥ (T n T ∗ ) n by (iii) of Lemma 3.5 since the ﬁrst inequality of (3.7) does not hold for β = pα by (3.8). If 2 ≤ n < p, put p1 = n − 1 > 0, r1 = 1 > 0, q1 = α1 ∈ (0, 1) and δ = p. By (ii) of Theorem 3.4, there exist positive invertible operators A and B on H such that (3.9). Deﬁne an operator T on ∞ k=−∞ H as Lemma 3.5. Then T is p-hyponormal n by (3.9) and (i) of Lemma 3.5, and (T ∗ T )nα ≥ (T n T ∗ )α by (iii) of Lemma 3.5 since the case j = 1 of (3.7) does not hold for β = nα by (3.9). (4) To prove (3.4). It is well known that there exist positive invertible operators A and B on H such that (3.11) log A ≥ log B, Aα−1 ≥ B α−1 ∞ Deﬁne an operator T on k=−∞ H as Lemma 3.5. Then T is log-hyponormal m n α α by (3.11) and (ii) of Lemma 3.5, and (T ∗ T m ) m ≥ (T n T ∗ ) n by (iii) of Lemma 3.5 since the ﬁrst inequality of (3.7) does not hold for β = α − 1 by (3.11).

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4. Applications [2] showed some applications of the results on powers of p-hyponormal operators for 0 0. It is well known that if T is p-hyponormal for p > 0, and r ≥ 0 with r2 ∈ ∗ σ(T T ), then there is a z ∈ σ(T ) such that z = r ([6]). This has been generalized to more wide classes of operators [3, 28]. Theorem 4.1 ([3, 28]). If T is a class wF (p, r, q) operator for p + r ≤ 1, q ≥ 1 and σ(T ) is connected, then σ(T ) ⊂ ρ(σ(T )), where ρ : C → R is defined by ρ(z) = |z|. Theorem 2.1 (2) can be utilized to give the following result. Theorem 4.2. Let T be a log-hyponormal operator and n be a positive integer. If n σ(T ) is connected and r > 0 with r2 ∈ σ(T ∗ T n ), then there is a z ∈ σ(T ) such n that z = r. n there is a Proof. By Theorem 2.1 (2), T is a log-hyponormal operator. Therefore, n w ∈ σ(T ) such that w = r. Thus, there is a z ∈ σ(T ) such that z n = w. Clearly, n z = r.

The well known Putnam’s area inequality for hyponormal operators [19] has been extended to the case that T is p-hyponormal for 0 0. If there is a positive integer m such that σ(T ) ∈ {reiθ : 0 ≤ θ < 2π/m} and n is a positive integer so that m ≥ n. Then ∗n n p/n np n (T T ) − (T n T ∗ )p/n ≤ r2p−1 drdθ. π σ(T ) We generalize it to the following results. Theorem 4.4. Let T be a p-hyponormal operator for p > 0. If there is a positive integer m such that σ(T ) ∈ {reiθ : 0 ≤ θ < 2π/m} and n is a positive integer so that m ≥ n ≥ 2. Then ∗n n min{p,n} n min{p,n} n min{p, n} (T T ) n − (T n T ∗ ) n ≤ r2 min{p,n}−1 drdθ. π σ(T ) Proof. Theorem 2.1 (1) implies that T n is min{p, n}-hyponormal. Therefore, the following follows by [6], ∗n n min{p,n} min{p, n} min{p,n} n ∗n min{p,n} (T T ) n n ≤ − (T T ) ρ2 n −1 dρdφ nπ σ(T n ) where σ(T n ) = {ρeiφ : 0 ≤ φ < 2π}. Since σ(T n ) = {rn einθ : reiθ ∈ σ(T )}, the proof is completed by substituting ρ = rn and φ = nθ.

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Theorem 4.5. Let T be a log-hyponormal operator. If there is a positive integer m such that σ(T ) ∈ {reiθ : 0 ≤ θ < 2π/m} and n is a positive integer so that m ≥ n. Then 2 log(T ∗n T n ) − log(T n T ∗n ) ≤ n r−1 drdθ. π σ(T ) Proof. Theorem 2.1 (2) implies that T n is log-hyponormal. Therefore, the following follows by [22], log(T ∗n T n ) − log(T n T ∗n ) ≤ 1 ρ−1 dρdφ π σ(T n ) where σ(T n ) = {ρeiφ : 0 ≤ φ < 2π}. Since σ(T n ) = {rn einθ : reiθ ∈ σ(T )}, the proof is completed by substituting ρ = rn and φ = nθ.

Acknowledgements We would like to express our cordial gratitude to Professor Daoxing Xia and the referee for their valuable advice and suggestions.

References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory, 13 (1990), 307–315. [2] A. Aluthge and D. Wang, Powers of p-hyponormal operators, J. Inequal. Appl., 3 (1999), 279–284. [3] A. Aluthge and D. Wang, Putnum’s theorems for w-hyponormal operators, Hokkaido Math. J., 29 (2000), 383–389. [4] A. Aluthge and D. Xia, A trace estimate of (T ∗ T )p − (T T ∗ )p , Integral Equations Operator Theory, 12 (1989),300–303. [5] M. Ch¯ o and T. Huruya, Square of the w-hyponormal operators, Integral Equations Operator Theory, 39 (2001), 413–420. [6] M. Ch¯ o and M. Itoh, Putnum’s inequality for p-hyponormal operators, Proc. Amer. Math. Soc., 123 (1995), 2435–2440. [7] R. Curto, P. Muhly and D. Xia, A trace estimate of the p-hyponormal operators, Integral Equations Operator Theory, 6 (1983), 507–514. [8] M. Fujii and Y. Nakatsu, On subclasses of hyponormal operators, Proc. Japan Acad. Ser. A Math. Sci., 51 (1975), 243–246. [9] T. Furuta, A ≥ B ≥ 0 assures (B r Ap B r )1/q ≥ B (p+2r)/q for r ≥ 0, p ≥ 0, q ≥ 1 with (1 + 2r)q ≥ p + 2r, Proc. Amer. Math. Soc., 101 (1987), 85–88. [10] T. Furuta, Invitation to Linear Operators–From Matrices to Bounded Linear Operators on a Hilbert Space, Taylor & Francis, London, 2001. [11] T. Furuta and M. Yanagida, On powers of p-hyponormal operators, Sci. Math. Jpn., 2 (1999), 279–284.

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[12] T. Furuta and M. Yanagida, On powers of p-hyponormal and log-hyponormal operators, J. Inequal. Appl., 5 (2000), 367–380. [13] P. R. Halmos, A Hilbert space problem book, 2nd ed. Springer, New York, 1982. [14] F. Hansen, An operator inequality, Math. Ann., 246 (1980), 249–250. [15] M. Ito, Generalizations of the results on powers of p-hyponormal operators, J. Inequal. Appl., 6 (2000), 1–15 r

r

r

[16] M. Ito and T. Yamazaki, Relations between two inequalities (B 2 Ap B 2 ) p+r ≥ B r p p p and (A 2 B r A 2 ) p+r ≤ Ap and its applications, Integral Equations Operator Theory, 44 (2002), 442–450. [17] T. Ito and T. K. Wong, Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc., 34 (1972), 157–164. [18] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Birkhauser Verlag, Boston, 1989. [19] C. R. Putnam, Spectra of polar factors of hyponormal operators, Trans. Amer. Math. Soc., 188 (1974), 419–428. [20] K. Tanahashi, Best possibility of Furuta inequality, Proc. Amer. Math. Soc., 124 (1996), 141–146. [21] K. Tanahashi, On log-hyponormal operators, Integral Equations Operator Theory, 34 (1999), 364–372. [22] K. Tanahashi, Putnum inequality for log-hyponormal operators, Integral Equations Operator Theory, 48 (2004), 103–114. [23] D. Xia, On the nonnormal operators-semihyponormal operators, Sci. China Ser. A, 23 (1980), 700–713. [24] D. Xia, Spectral Theory of Hyponormal Operators, BirkhauserVerlag, Boston, 1983. [25] T. Yamazaki, Extensions of the results on p-hyponormal and log-hyponormal operators by Aluthge and Wang, SUT J. Math., 35 (1999), 139–148. [26] M. Yanagida, Some applications of Tanahashi’s result on the best possibility of Furuta inequality, Math. Inequal. Appl., 2 (1999), 297–305. [27] M. Yanagida, Powers of class wA(s, t) operators associated with generalized Aluthge transformation, J. Inequal. Appl., 7(2) (2002), 143–168. [28] C. Yang and J. Yuan, Spectrum of class wF (p, r, q) operators for p + r ≤ 1 and q ≥ 1, Acta Sci. Math. (Szeged), 71 (2005), 767–779. [29] C. Yang and J. Yuan, Extensions of the results on powers of p-hyponormal and log-hyponormal operators, J. Inequal. Appl., 2006 (2006), Article ID 36919, 1–14. [30] C. Yang and J. Yuan, On class wF (p, r, q) operators, Acta Math. Sci. Ser. A Chin. Ed., to appear. [31] J. Yuan and C. Yang, Powers of class wF (p, r, q) operators, JIPAM. J. Inequal. Pure Appl. Math., 7 (1) (2006), Artical 32, 1–9. Jiangtao Yuan and Zongsheng Gao LMIB and Department of Mathematics, Beihang University, Beijing 100083, China e-mail: [email protected], [email protected] Submitted: September 6, 2006 Revised: March 16, 2007

Integr. equ. oper. theory 59 (2007), 449–489 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040449-41, published online October 18, 2007 DOI 10.1007/s00020-007-1542-9

Integral Equations and Operator Theory

Spectral Properties of Some Linear Matrix Diﬀerential Operators in Lp-spaces on R E. Albrecht and W. J. Ricker Abstract. A detailed study is made of matrix-valued, ordinary linear diﬀerential operators T in Lp (R, CN ) for 1 < p < ∞, which arise as the perturbation of a constant coeﬃcient diﬀerential order n ≥ 1 by a lower orn−1 operator of d j der diﬀerential operator S = j=0 Fj (x)(−i dx ) which has a factorisation S = AB for suitable operators A and B. Via techniques from Lp -harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisﬁes certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties of T . Mathematics Subject Classiﬁcation (2000). Primary 47A60, 47B40; Secondary 47F05. Keywords. Lp -theory, matrix diﬀerential operator, resolvent, perturbation, decomposable operator, functional calculus, spectral properties.

Introduction 2

d Consider the second order ordinary diﬀerential operator L = − dx 2 + v(x), where v : R → C is a Lebesgue measurable function. Aspects of the spectral theory of L when acting in L2 (R) have been studied by numerous authors; see for example [23, 29, 5, 6, 15, 25, 19, 20, 14, 31, 32] and the references therein. The case when v is R-valued (i.e., L is symmetric) is well understood, whereas for C-valued v so called “spectral singularities” may occur, thereby leading to non-selfadjoint operators. Under certain exponential decay conditions on v the number of such spectral singularities is at most ﬁnite, in which case it is possible to construct a bounded functional calculus for L (with an appropriate domain of deﬁnition) admitting partitions of unity; see for example [6, 14] and the references therein. For analogous but higher order diﬀerential operators, the idea (due to J.T. Schwartz, [29]) is to consider the coeﬃcient diﬀerngiven operator T as a constant d ) perturbed by a lower ential operator T0 = q(D) := j=0 aj Dj (with D = 1i dx

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j order diﬀerential operator S = n−1 j=0 bj (x)D , where a factorisation S = AB for suitable operators A and B is used; see for example [29, 25, 19, 20]. The spectrum of the normal operator T0 is contained in q(R) ∪ {∞}. A smallness condition for bn−1 and decay conditions for b0 , . . . , bn−1 imply that the spectra of T and T0 coincide outside of some compact set. In the case of exponential decay and some restrictions on q, with q an even function, this compact set turns out to be ﬁnite. In this case one obtains a ﬁnite order growth condition for the resolvent of T near q(R). Such resolvent estimates typically imply “nice” properties for T ; see for example [14, 19], and also [18, 30] for related problems on a half-line. The point of departure of this paper is two-fold. Firstly, the diﬀerential operators involved arise from linear systems, i.e., the coeﬃcients are matrix-valued. Secondly, the operators act in (vector-valued) Lp rather than L2 , for 1 < p < ∞. So, the unperturbed operator T0 is a linear ordinary diﬀerential operator with matrix coeﬃcients acting in the Banach space Lp (R, CN ) and the perturbation will be an appropriate lower order diﬀerential operator. Accordingly, Hilbert space techniques have to be replaced by methods from Lp -harmonic analysis and the spectrum of T0 is more involved when N > 1 as for N = 1. Also, the operators A and B used in the factorisation need to be deﬁned on suitable Sobolev spaces of CN -valued functions and are more complicated than in the N = 1 setting. Nevertheless, it is still possible to make a detailed study of the local spectral theory/ functional calculi properties of perturbed matrix diﬀerential operators acting in Lp (R, CN ). We now indicate brieﬂy the main features of the paper. The ﬁrst section presents an abstract Banach space version of the Schwartz perturbation method in a form suitable for later use in the vector-valued Lp -setting. ˆ := C ∪ {∞}, that a closed linear operator Recall, for a compact set S ⊆ C T : X ⊇ D(T ) → X with domain D(T ) in a Banach space X is said to be residually decomposable (in the sense of Vasilescu [33]) with residuum S (brieﬂy, ˆ with S ⊂ U0 S-decomposable), if for every ﬁnite open covering U0 , . . . , Um of C and Uj ∩ S = ∅ for 1 ≤ j ≤ m, there exist closed subspaces X0 , . . . , Xm of X with X = X0 + · · ·+ Xm that are invariant for T (in the sense that T (D(T )) ∩ Xj ⊆ Xj ) and satisfy σ(T |Xj ) ⊂ Uj , j = 0, . . . , m. If S = ∅, then T is said to be decomposable (in the sense of Foia¸s [13]). Section 2 treats certain aspects of the spectral theory of the maximal extension τ (Q), acting in Lp (R, CN ), of the formal matrix diﬀerential noperator n Q(D) = j=0 Aj Dj corresponding to the matrix polynomial Q(x) = j=0 Aj xj , where A0 , . . . , An ∈ MN ×N (C) are constant (N × N )-matrices over C (with An = 0). It is known, [2], that τ (Q) is decomposable if and only if its spectrum σ(τ (Q)) coincides with the natural spectrum Σ(Q) ∪ {∞}, where Σ(Q) := {λ ∈ C; ∃ x ∈ R with qQ (λ, x) = 0} and λ → qQ (λ, x) := det(λ − Q(x)), for λ ∈ C, is the characteristic polynomial of Q(x) for each x ∈ R. In this case, for each λ ∈ C \ Σ(Q), the entries of the matrix function x → (λ − Q(x))−1 are bounded rational functions and hence, are Fourier

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p-multipliers for each p ∈ (1, ∞). Accordingly, the resolvent operator (λ − τ (Q))−1 is a Fourier p-multiplier operator in Lp (R, CN ) whose representation as an integral operator can be explicitly calculated via the calculus of residues. In the case that Q is regular (i.e., An is invertible), the operator τ (Q) is decomposable and its resolvent satisﬁes ﬁnite order growth conditions near the spectrum. These estimates are essential for later use, as is the notion of the type T (Q) = (α, β) associated to Q. Here α ∈ Z and β ∈ N are integers related to the maximal orders of the poles and certain asymptotic properties of the functions z → mk,j (λ, z)z eiz(s−t) /qQ (λ, z), where mk,j (λ, z) are the minors of the (N × N )-matrix λ − Q(z). Section 3 is devoted to the perturbed operator τ (Q) + σ1 , where the perturbation σ1 isthe maximal closed extension of the (formal) matrix diﬀerential α operator σ := k=0 Fk Dk , with Fk : R → MN ×N (C) being suitable (measurable) functions and α coming from the type T (Q) = (α, β) of Q. Under certain natural restrictions on the {Fk }α k=0 , essentially due to E. Balslev and T.W. Gamelin, [4], at least for N = 1, it turns out that σ1 is τ (Q)-compact (in Lp (R, CN ) and for any ﬁxed 1 < p < ∞) and, most importantly, that there exists a factorisation σ1 |D(τ (Q)) = AB|D(τ (Q)) for suitable operators W ∞,α (R, CN ) → Lp (R, CN )α+1 → (L1 + Lp )(R, CN ), B

A

deﬁned in terms of the coeﬃcients {Fk }α k=0 and with B deﬁned on the vectorvalued Sobolev space W ∞,α (R, CN ). Accordingly, the perturbation method of J.T. Schwartz alluded to above is applicable (under further additional constraints on the {Fk }α k=0 ) to obtain the existence of a compact set H0 ⊂ C containing all accumulation points of σ(τ (Q) + σ1 ) ∩ ρ(τ (Q)) and satisfying σ(τ (Q) + σ1 ) \ H0 ⊆ σ(τ (Q)), together with local resolvent estimates (where η = max{N, 2β − 1}) of the form C , λ ∈ W \ Σ(Q), R(λ, τ (Q) + σ1 ) ≤ dist(λ, Σ(Q))η for each compact set H ⊂ Σ(Q) \ H0 and a suitable neighbourhood W of H. By considering the resolvent of the resolvent, such estimates are used to show that τ (Q) + σ1 has a local C η+2 -functional calculus and consequently, that τ (Q) + σ1 is H0 -decomposable. The operator theoretic properties of τ (Q) + σ1 and the nature of H0 , as described above, can be reﬁned if the coeﬃcients {Fk }α k=0 satisfy some further constraints; this is the topic of the ﬁnal section 4. Here, for for any 1 0, k = 0, . . . , α, sup s∈R\(−x−1,x)

s

for some constants C, δ > 0. If, for some λ0 ∈ C \ Σ(Q), the inverse of 1 − ˜ 0 , τ (Q) + σ1 )A exists in L(Lp (R, CN )), then it turns out that R(λ0 , τ (Q) + B R(λ σ1 ) = R(λ0 , τ (Q)) + K, for some compact operator K ∈ L(Lp (R, CN )) whose ap−δ

proximation numbers satisfy αk (K) = O(k max{1,δ}+p ) for k → ∞, where

1 1 p + p

= 1.

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These facts, together with results of B. Droste, [8, 9, 10], imply (modulo a certain “bad compact set” S1 ), that τ (Q) + σ1 has a functional calculus (extending its analytic functional calculus) based on the algebra of all functions that are analytic in (individual) neighbourhoods of S1 and ultradiﬀerentiable of a certain Gevrey class on the totally real, real analytic, 1-dimensional manifold σ(τ (Q) + σ1 ) \ S1 . In particular, τ (Q) + σ1 is S-decomposable relative to the set of all accumulation points of ρ(τ (Q))∩σ(τ (Q)+σ1 ). If the coeﬃcients Fk satisfy exponential estimates of the form ∞

−∞

Fk (t)eε|t| dt < ∞,

k = 0, . . . , α,

for some ε > 0, then better resolvent estimates are available which, in turn, lead to richer functional calculi for the perturbed operator τ (Q) + σ1 . In this case, if additionally all components of C \ Σ(Q) are unbounded, then τ (Q) + σ1 is decomposable.

1. An abstract perturbation method In this section we formulate an abstract perturbation method which, in some variants in Hilbert spaces, has also been applied by J. T. Schwartz [29] and G. E. Huige [18]–[20]. Throughout this section T : D(T ) → X denotes a closed linear operator with domain D(T ) ⊂ X in a Banach space (X, ·) and σ(T ) and ρ(T ) are the spectrum and the resolvent set of T , respectively. For λ ∈ ρ(T ) deﬁne R(λ, T ) := (λ − T )−1 . Then D(T ) is also a Banach space under the T -norm (i.e., graph norm) ·T given by xT := x + T x for x ∈ D(T ). Recall that an operator S : D(S) → X, with domain D(S) ⊂ X, is said to be T -bounded (T -compact) if D(T ) ⊂ D(S) and if the restriction S|D(T ) : D(T ) → X is a bounded (compact) linear operator with respect to the T -norm on D(T ). The space of all bounded linear operators from a Banach space X into a Banach space Y is denoted by L(X, Y ). If X = Y , then we simply write L(X). Lemma 1.1. Let T : X ⊃ D(T ) → X be a closed, densely deﬁned linear operator. Let S : X ⊃ D(S) → X be a T -compact linear operator in X which admits a ˜ and factorisation S|D(T ) = AB|D(T ) via bounded linear operators A : Y → X ˜ ˜ ˜ ˜ ˜ B : D → Y , for some Banach spaces D, X, Y , where X ⊂ X and D(T ) ⊂ D ⊂ X ˜ (D(T ), · T ) → D ˜ and D ˜ → X. with continuous inclusion maps, X → X, Suppose that λ is a point in ρ(T ) for which R(λ, T ) extends to a bounded linear ˜ into X, denoted by R(λ, ˜ T ), and satisfying R(λ, ˜ T )AY ⊂ D. ˜ If operator from X ˜ T )A)−1 exists in L(Y ) then λ ∈ ρ(T + S) and (1 − B R(λ, ˜ T )A(1 − B R(λ, ˜ T )A)−1 BR(λ, T ) . (1.1) R(λ, T + S) = R(λ) := R(λ, T ) + R(λ, Moreover, if Ω denotes that component of ρ(T ) containing λ, then σ(T + S) ∩ Ω is at most countable with any accumulation points belonging to ∂Ω ⊂ σ(T ).

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Proof. It is well known (cf. [22], Theorem 1.11 in Chapter IV, §1), that T + S is again a closed linear operator with domain D(T + S) = D(T ). Moreover, for all w ∈ ρ(T ), the operator w − T − S is Fredholm of index 0. Direct computation shows that R(λ)(λ − T − S)f = f

for all f ∈ D(T ) = D(T + S) .

This shows that R(λ) is a left inverse and hence, inverse of λ−T −S. The remaining statements now follow from some standard facts from Fredholm theory. We will apply this lemma later in its following form: Corollary 1.2. Suppose, in the setting of the preceding Lemma, that there exists a compact set K ⊂ C with the following properties. (i) For all λ in ρ(T ) \ K the resolvent operator R(λ, T ) extends to a bounded ˜ T ) from X ˜ into X satisfying R(λ, ˜ T )AY ⊂ D. ˜ linear operator R(λ, ˜ T )A is a bounded linear operator (ii) For all λ in ρ(T ) \ K the operator B R(λ, ˜ T )A < 1. on Y and a := supλ∈ρ(T )\K B R(λ, Then, for all λ ∈ ρ(T ) \ K, we have λ ∈ ρ(T + S) and R(λ, T + S) = R(λ, T ) +

∞

˜ T )A(B R(λ, ˜ T )A)k BR(λ, T ) R(λ,

(1.2)

˜ T )A · BR(λ, T ) R(λ, . 1−a

(1.3)

k=0

with the estimate R(λ, T + S) ≤ R(λ, T ) +

In particular, for each component Ω of ρ(T ) with Ω \ K = ∅ the set Ω ∩ σ(T + S) is at most countable with any accumulation points belonging to σ(T ) ∩ K.

2. Constant coeﬃcient linear diﬀerential operators in Lp (I, CN ) In the sequel, let I be an interval in the real line R. We shall be mainly interested in the cases I = R and I = [0, ∞). If F(I) is any space of C-valued functions or distributions on I, we shall denote by F(I, CN ) the space of all N -tuples f = (f1 , . . . , fN ) such that the components fj are in F(I) for j = 1, . . . , N . In particular, we write D(I, CN ) for the set of all functions in C ∞ (I, CN ) having compact support in int(I) and D (I, CN ) for its dual space, which may be identiﬁed with the space of all CN -valued distributions. As usual, the space of all n − 1 times continuously differentiable, CN -valued functions ϕ on I for which ϕ(n−1) is absolutely continuous on each compact subinterval of I will be denoted by An (I, CN ). Consider a matrix polynomial of the form Q(x) =

n j=0

Aj xj ,

(2.1)

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where Aj ∈ MN ×N (C) for j = 0, . . . , n and An = 0 (i.e., Q is of degree n). In the event that the leading coeﬃcient An is invertible nwe call Q regular. Let Q(D) denote the formal diﬀerential operator Q(D) := j=0 Aj Dj associated to d . For each 1 < p < ∞ one associates to Q(D) a minimal Q, where D := 1i dt and a maximal closed linear operator in Lp (I, CN ), denoted by τ0 (Q) and τ1 (Q), respectively. The operator τ1 (Q) has domain D1 (Q) := u ∈ Lp (I, CN ); Q(D)u ∈ Lp (I, CN ) and is given by τ1 (Q)u := Q(D)u

for u ∈ D1 (Q).

Here, the derivatives used in forming Q(D)u have to be taken in the sense of distributions. Since the natural embedding of Lp (I, CN ) into D (I, CN ) and the linear mapping Q(D) : D (I, CN ) → D (I, CN ) are both continuous, it follows that τ1 (Q) is a closed linear operator in Lp (I, CN ). Of course τ1 (Q) is an extension of the linear mapping Q(D) : D(I, CN ) → D(I, CN ). In particular, Q(D) : D(I, CN ) → D(I, CN ) is closable in Lp (I, CN ) and its closure is by deﬁnition the minimal operator τ0 (Q). Thus, the domain D0 (Q) of τ0 (Q) is the set of all those u ∈ N Lp (R, CN ) such that there exists a sequence (ϕk )∞ k=1 in D(I, C ) converging to u p N p in L (I, C ) and satisfying Q(D)ϕk → v for some v ∈ L (I, CN ) (which is then by deﬁnition τ0 (Q)u). Moreover, τ0 (Q) ⊂ τ1 (Q). We shall need the following fact which is well known for the scalar-valued case (cf. [27] or [16], Proof of Theorem VI.1.9). Lemma 2.1. For a regular matrix polynomial Q of degree n we have D1 (Q) = u ∈ An (I, CN ) ∩ Lp (I, CN ); Q(D)u ∈ Lp (I, CN ) .

(2.2)

Proof. Since the right hand side of (2.2) is obviously contained in D1 (Q), we only have to prove the reverse inclusion. Hence, ﬁx an arbitrary u ∈ Lp (I, CN ) with the property that v := Q(D)u ∈ Lp (I, CN ). Let I0 = [a, b] be an arbitrary compact subinterval of I with a < b. For all ϕ ∈ D(I, CN ) with supp ϕ ⊂ (a, b), we obtain by successive integration by parts that t (t − s)n−k−1 (n) (k) ϕ (s) ds for 0 ≤ k ≤ n − 1. (2.3) ϕ (t) = a (n − k − 1)! Since Q(D)u = v in the sense of distributions, we have b b v(t), ϕ(t)dt = u(t), (Q∗ (D)ϕ)(t)dt a

(2.4)

a

where ·, · denotes the usual scalar product in the N -dimensional Hilbert space CN and, if Q is given by (2.1), then Q∗ (x) =

n k=0

A∗k xk ,

(2.5)

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with A∗k denoting the adjoint matrix of Ak . Inserting (2.3) and (2.5) into (2.4), we obtain b t (t − s)n−1 v(t), ϕ(n) (s)ds dt (n − 1)! a a b = u(t), (−i)n A∗n ϕ(n) (t) dt+ a

+

n−1 b k=0

a

a

t

(t − s)n−k−1 u(t), (−i)k A∗k ϕ(n) (s)ds dt . (n − k − 1)!

By means of the Fubini Theorem we conclude b b (t − s)n−1 (n) v(t) dt, ϕ (s) ds (n − 1)! a s b = in An u(s), ϕ(n) (s) ds+ a b b n−1 (t − s)n−k−1 k (n) i Ak u(t) dt, ϕ (s) ds . + (n − k − 1)! a s k=0

Since this holds for all ϕ ∈ D((a, b), CN ) it follows that the n-th distributional derivative of the integrable CN -valued function (on I0 ) n−1 b (t − s)n−k−1 ik Ak u(t) dt s → F (s) := in An u(s) + (n − k − 1)! s k=0 b (t − s)n−1 − v(t) dt (n − 1)! s vanishes on (a, b). Hence, there exists a polynomial P of degree at most n − 1 with coeﬃcients in CN coinciding with F a.e. on I0 with respect to Lebesgue measure; see [17], Cor. 3.1.6. Therefore, after a change on a Lebesgue null set, the function u satisﬁes (on I0 ) in u(s) =A−1 n P (s) b

n−1 n−1 (t − s)n−k−1 (t − s) v(t) − ik Ak u(t) dt . + A−1 n (n − 1)! (n − k − 1)! s k=0

This implies that u is absolutely continuous on I0 . Taking the derivative, we obtain for almost all s ∈ I0 , n−1 −1 An An−1 u(s) in u (s) =A−1 n P (s) + i

b n−2 n−2 (t − s)n−k−2 (t − s) ik Ak u(t) − v(t) dt , + A−1 n (n − k − 2)! (n − 2)! s k=0

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which shows that u is absolutely continuous on I0 . Continuing this line of argument we ﬁnally see, after a change on a Lebesgue null set, that the function u is in C n−1 (I0 , CN ) and u(n−1) is absolutely continuous on I0 . Since this holds for all compact subintervals I0 = [a, b] of I with a < b, we conclude that u ∈ An (I, CN ). The following facts can be obtained as in the scalar-valued case (cf. [16], Theorem VI.6.2 and Lemma VI.8.2). Lemma 2.2. Let 1 < p < ∞ and I ⊂ R be an interval of length greater than 1. Let Q be a regular matrix polynomial of degree n. Then the following statements hold. (a) D1 (Q) = W p,n (I, CN ) := {f ∈ Lp (I, CN ); f (k) ∈ Lp (I, CN ) for 1 ≤ k ≤ n}. (b) There exists a constant K > 0 (only depending non-increasingly on the length of I) such that for all f ∈ D1 (Q) and k ∈ {1, . . . , n},

f (k) Lp (I,CN ) ≤ K f Lp(I,CN ) + τ1 (Q)f Lp(I,CN ) . (c) If B ∈ Lploc (I, MN ×N (C)), then for each ε > 0 there exists a constant C > 0 (only depending on ε, p, and the length of I) such that for all f ∈ D1 (Q), s+1 sup B(t)p dt. Bf pLp(I,CN ) ≤ εf pLp (I,CN ) + Cf pLp(I,CN ) [s,s+1]⊂I

s

Here, · denotes the operator norm with respect to the p norm on CN and C may be chosen to be non-increasing as a function of the length of I. In particular, estimates as in (b) and (c) hold for I = R and the half-lines I = [a, ∞), a ∈ R. For matrix polynomials Q of the form (2.1) we now investigate the behaviour of τ1 (Q) in the special case when I = R. Lemma 2.3. The minimal and the maximal closed operator for Q(D) in Lp (R, CN ) coincide. Hence, write τ (Q) for τ0 (Q) = τ1 (Q). Moreover, τ (Q) coincides with the closure in Lp (R, CN ) of the operator ϕ → Q(D)ϕ for ϕ in the Schwartz space S(R, CN ). Proof. The ﬁrst assertion can be proved as in the scalar-valued case (see e.g. [28], Theorem 4.2.1) with some straightforward modiﬁcations. The second statement is obvious, as Q(D) is continuous on S(R, CN ) and D(R, CN ) is dense in S(R, CN ). We write qQ for the characteristic polynomial of Q, i.e., qQ (λ, x) := det(λ − Q(x)),

λ ∈ C, x ∈ R,

and deﬁne Σ(Q) := {λ ∈ C; ∃ x ∈ R with qQ (λ, x) = 0}. If q is any polynomial in λ and x, we deﬁne its degree with respect to the variable x by ∂kq degx q := sup k ∈ N0 ; does not vanish identically . k ∂x

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Notice that degx q = supλ∈C deg q(λ, ·). Recall from [2], Prop. 4.5, that the following statements are equivalent for Q: (i) σ(τ (Q)) = Σ(Q) ∪ {∞}, (ii) τ (Q) is decomposable in the sense of C. Foia¸s, [13], (iii) for all λ ∈ C\Σ(Q) and all minors mj,k (λ, x) (j, k ∈ {1, . . . , N }) of the matrix λ − Q(x), the degree of the polynomial qQ (λ, ·) is larger than or equal to that of mj,k (λ, ·), and (iv) for all minors mj,k (λ, x) (j, k ∈ {1, . . . , N }) of the matrix λ − Q(x), we have degx qQ ≥ degx mj,k . If one (and hence all) of these conditions is satisﬁed then, for λ ∈ C \ Σ(Q), the p resolvent operator R(λ, τ (Q)) of τ (Q) is the continuous extension T(λ−Q) −1 to Lp (R, CN ) of the operator on S(R, CN ) given by ϕ → F−1 ((λ − Q)−1 ϕ)

for ϕ ∈ S(R, CN ).

(2.6)

Here, as usual, F(ϕ) = ϕ is the Fourier transform of a given C -valued function (or tempered distribution) ϕ on R. N

Lemma 2.4. Let Q be a regular matrix polynomial of degree n. Then τ (Q) is decomposable in Lp (R, CN ). In particular, σ(τ (Q)) = Σ(Q) ∪ {∞}. Moreover, Q actually satisﬁes the stronger degree condition (iii) For all λ ∈ C\Σ(Q) and all minors mj,k (λ, x) (j, k ∈ {1, . . . , N }) of λ−Q(x), we have deg qQ (λ, ·) ≥ deg mj,k (λ, ·) + n Proof. Let · be any unital algebra norm on MN ×N (C). Fix λ ∈ C \ Σ(Q) and notice, in the notation of (2.1), that

n−1 n k−n −1 −n −1 λ − Q(x) = −x An 1 + x An Ak − λx An . k=0

Hence,

−1 n−1 −1 −n k−n −1 −n −1 A |x| (λ − Q(x))−1 ≤ 1 + x An Ak − λx An n k=0

which tends to 0 as x → ∞. This shows that the entries of the matrix (λ − Q(x))−1 are bounded rational functions of x, vanishing at inﬁnity of order n. So, by Cramer’s rule, the degree condition (iii) is satisﬁed. Since (iii) is stronger than (iii) above, τ (Q) is decomposable. It then follows that σ(τ (Q)) = Σ(Q) ∪ {∞} by (i) above. Matrix diﬀerential operators τ (Q) can also be decomposable without Q being regular. For instance, this is the case for 2 x x3 1 0 2 0 1 3 Q(x) = x . = x + 0 1 0 x2 0 0

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In view of the above mentioned equivalences we deﬁne d(Q) := degx qQ −

max

j,k=1,...,N

degx mj,k

and note that d(Q) ≥ 0 if and only if τ (Q) is decomposable. Let us also remark that the previous proof shows that if Q is regular with degree n, then d(Q) = n. For d(Q) ≥ 0 and λ ∈ C \ Σ(Q), the entries of the matrix (λ − Q(x))−1 are bounded rational functions in x and hence, Fourier p-multiplier functions for all p ∈ (1, ∞). In particular, the operator given by (2.6) extends to a bounded p p N linear operator T(λ−Q) ) which coincides with R(λ, τ (Q)). In the −1 on L (R, C event that d(Q) ≥ 1, the entries of (λ − Q(x))−1 actually vanish at ∞. Thus, by inserting the deﬁnition of the Fourier transform and its inverse and using the fact that the Fourier transform and its inverse are unitary operators in L2 (R), we obtain (for ϕ ∈ S(R, CN )) ∞ (R(λ, τ (Q))ϕ)(s) = Kλ (s, t)ϕ(t) dt, s ∈ R , −∞

where the matrix-valued kernel Kλ is given by ∞ 1 (λ − Q(x))−1 eix(s−t) dx , Kλ (s, t) = 2π −∞

s, t ∈ R ,

and may be computed explicitly by means of the calculus of residues. This gives for the (j, k)-entry Kλ,j,k (s, t):  r m (λ, z)  k,j j+k iz(s−t)  i(−1) e Res ; z = µ (λ) s>t  +,ν   qQ (λ, z)  ν=1  Kλ,j,k (s, t) =  (degx qQ )−r  m (λ, z)   k,j  j+k  eiz(s−t) ; z = µ−,ν (λ) s < t, Res −i(−1) qQ (λ, z) ν=1 (2.7) where µ+,ν (λ), ν = 1, . . . , r, and µ−,ν (λ), ν = 1, . . . , (degx qQ ) − r are the zeros of qQ (λ, ·) with positive and negative imaginary parts, respectively. Notice, in the same way, that for 0 ≤ l < d(Q), the operator ϕ → Dl (R(λ, τ (Q))ϕ)(·) is the [l] [l] integral operator with kernel Kλ (s, t) = (Kλ,j,k (s, t))j,k=1,...,N , where the entries are given by:  r m (λ, z)z l  k,j j+k iz(s−t)  i(−1) e Res ; z = µ (λ) s>t  +,ν   qQ (λ, z)  ν=1  [l] Kλ,j,k (s, t) =  (degx qQ )−r  m (λ, z)z l   k,j  j+k  eiz(s−t) ; µ−,ν (λ) Res s < t, −i(−1) qQ (λ, z) ν=1 (2.8)

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In the special case that all zeros of the function x → qQ (λ, x) are simple, this turns out to be:  r mk,j (λ, µ+,ν (λ))µ+,ν (λ)l iµ+,ν (λ)(s−t)  j+k  i(−1) s>t e   ∂qQ  (λ, µ (λ))  +,ν ν=1 ∂z  [l] Kλ,j,k (s, t) =  (degx qQ )−r    mk,j (λ, µ−,ν (λ))µ−,ν (λ)l iµ−,ν (λ)(s−t)  j+k  −i(−1) e s < t.  ∂qQ ν=1 ∂z (λ, µ−,ν (λ)) (2.9) Let us denote the resultant (c.f. [34], Section 27) of the polynomials qQ (λ, ·) ∂q and ∂zQ (λ, ·) by RQ (λ). Since RQ is a polynomial in λ, we conclude that either RQ (λ) ≡ 0 or RQ (λ) = 0 outside of some ﬁnite set Λ. Thus, if RQ does not vanish identically then, by the theorem in [34], Section 27, (2.9) holds for all λ ∈ C \ (Σ(Q) ∪ Λ). Lemma 2.5 below gives a suﬃcient condition, quite useful in practice, which guarantees that RQ does not vanish identically. n So, let Q(x) = j=0 Aj xj be a regular matrix polynomial of degree n with coeﬃcients in MN ×N (C). Let µ(λ) be any one of the branches of the solution to qQ (λ, µ(λ)) ≡ 0. Then |µ(λ)| → ∞ as |λ| → ∞. Moreover,   n 1  λ Aj µ(λ)j  = − An + o(1) as |λ| → ∞ λ− n µ(λ) µ(λ)n j=0 and hence, qQ (λ, µ(λ)) 0≡ = det µ(λ)nN

λ − An µ(λ)n

+ o(1)

as |λ| → ∞.

(2.10)

This shows that the only possible accumulation points of λ/µ(λ)n for |λ| → ∞ are the eigenvalues of the invertible matrix An . In particular, we obtain λ ≤ max |z| + o(1) for |λ| → ∞ . (2.11) min |z| + o(1) ≤ µ(λ)n z∈σ(An ) z∈σ(An ) In the same way, if for µ ∈ C we denote by λ(µ) any solution of qQ (λ(µ), µ) = 0, then |λ(µ)| → ∞ as |µ| → ∞, the only possible accumulation points of λ(µ) µn for |µ| → ∞ are in σ(An ) and λ(µ) min |z| + o(1) ≤ n ≤ max |z| + o(1) for |µ| → ∞ . (2.12) µ z∈σ(An ) z∈σ(An ) n Lemma 2.5. Let Q(x) = j=0 Aj xj be a regular matrix polynomial of degree n with coeﬃcients in MN ×N (C) whose leading coeﬃcient has N pairwise distinct eigen∂q values. Then the resultant RQ , of qQ (λ, ·) and ∂zQ (λ, ·), does not vanish identically and (2.9) holds in C \ (Σ(Q) ∪ Λ) for some ﬁnite set Λ. Moreover, if µ(λ) is as

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above, then mk,j (λ, µ(λ)) ∂qQ ∂z (λ, µ(λ))

= O(|µ(λ)|1−n ) = O(|λ|

1−n n

)

as |λ| → ∞ .

(2.13)

Proof. We ﬁrst show that RQ (λ) does not vanish identically. Assume to the con∂q trary that RQ ≡ 0. Then, considering qQ and ∂zQ as polynomials in z with coefﬁcients in the unique factorisation domain C[λ], we see that the two polynomials have a common factor, which is not constant with respect to z (see e.g. [12], Anhang A.1.1). Since the leading coeﬃcient of qQ with respect to λ is 1, this common factor must be of the form k−1 cj (z)λj , λk + j=0

where k ≥ 1 and at least one of the coeﬃcients c1 , . . . , ck−1 ∈ C[z] is not constant. Therefore, for each λ ∈ C there is some µ(λ) ∈ C with ∂qQ (λ, µ(λ)) = 0 = qQ (λ, µ(λ)) . ∂z Consider a sequence (λν )∞ ν=1 with |λν | → ∞ for ν → ∞. By passing to some subsequence, if necessary, we may assume that λν /µ(λν )n converges to some eigenvalue, say κ1 , of the leading coeﬃcient An of Q. The polynomial qQ can be written as qQ (λ, z) = q0 (λ, z) + q1 (λ, z), where q0 (λ, z) :=

N

dj λj z (N −j)n = det(λ − An z n )

j=0

and q1 has the form q1 (λ, z) =

N −1

ej (z)λj with deg ej ≤ (N − j)n − 1 .

j=0

Hence, N −1 N −1 ∂qQ (λ, z) = (N − j)ndj λj z (N −j)n−1 + ej (z)λj , ∂z j=0 j=0

where deg ej ≤ (N − j)n − 2. Letting ν → ∞ we obtain ∂qQ (λν , µ(λν )) ∂z N −1 ej (µ(λν ) λν ∂q0 λjν = , 1 + · ∂z µ(λν )n µ(λν )(N −j)n−1 µ(λν )jn j=0

0 = µ(λν )1−N n

∂q0 (κ1 , 1) . ∂z 0 Hence, 1 is a common zero of q0 (κ1 , ·) and ∂q ∂z (κ1 , ·), i.e., a zero of order at least 2 of q0 (κ1 , ·). This, however, is impossible since the zeros of the polynomial q0 (κ1 , ·), →

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which has degree nN , are the pairwise distinct n-th roots of the pairwise distinct numbers κ1 /κ1 , . . . , κ1 /κN , where κ1 , . . . , κN are the distinct eigenvalues of An . Therefore, RQ (λ) cannot vanish identically. As noticed before, this implies that (2.9) holds in C \ (Σ(Q) ∪ Λ) for some ﬁnite set Λ. To prove (2.13), let us ﬁrst remark that the minors mj,k (λ, ξ) are of the form mj,k (λ, z) =

N −1

aj,k,ν (z)λν ,

ν=0

where the coeﬃcients aj,k,ν are polynomials in z of degree at most (N − ν − 1)n. It follows from this, using (2.11), that mj,k (λ, µ(λ)) = O(|λ|N −1 ) = O(|µ(λ)|(N −1)n ) for |λ| → ∞ . Thus, for the proof of (2.13), it suﬃces to show that ∂q −1 Q (λ, µ(λ)) = O(|µ(λ)|1−nN )for |λ| → ∞ . ∂z Assume to the contrary, that there exists a sequence (λr )∞ r=1 such that |λr | → ∞ and 1−nN ∂qQ (2.14) |µ(λr )| ∂z (λr , µ(λr )) → 0 for r → ∞ . As seen prior to Lemma 2.5, by passing to a subsequence we may assume that λr /µ(λr )n → κ for some eigenvalue κ of An , in which case q0 (κ, 1) = 0. But, it was argued above that in such a case ∂q0 1−nN ∂qQ |µ(λr )| ∂z (λr , µ(λr )) → ∂z (κ, 1) . Therefore, by (2.14), we have z

∂q0 ∂z (κ, 1)

= 0. An easy computation shows that

∂q0 ∂q0 (λ, z) + nλ (λ, z) ≡ N nq0 (λ, z). ∂z ∂λ

0 For λ = κ and z = 1 this yields nκ ∂q ∂λ (κ, 1) = 0. Since all eigenvalues of An are diﬀerent from 0 and of multiplicity 1, this gives a contradiction.

Example 2.6. In the case N = 1 we know that τ (Q) is always decomposable with Σ(Q) = Q(R) ∪ {∞}. Moreover, the resolvent set ρ(τ (Q)) may have bounded components only if n ≥ 3. For N ≥ 2 bounded components are possible already for n = 1. The polynomial x 2x − 2 1 2 0 −2 Q(x) = = x+ x+1 x 1 1 1 0 is regular and satisﬁes the conditions of the preceding lemma. For this example, Σ(Q) is the union of R with the ellipse {λ ∈ C; 2(Re λ)2 + (Im λ)2 = 2}. Hence, ρ(τ (Q)) consists of two bounded and two unbounded components.

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If the leading coeﬃcient An has multiple eigenvalues, then the estimates are typically weaker and (2.13) is in general no longer valid, even if the resultant RQ (λ) does not vanish identically. For N = 2, we demonstrate this via upper triangular matrix polynomials. Example 2.7. Let a, c ∈ C[Z] be two polynomials, both of degree n ≥ 1, and let b ∈ C[Z] be a polynomial of degree d ≤ n. The matrix polynomial a(z) b(z) Q(z) = 0 c(z) is clearly regular of degree n, with qQ (λ, z) = (λ − a(z))(λ − c(z)) and

∂qQ (λ, z) = (a (z) + c (z))(a(z) − λ) + a (z)(c(z) − a(z)) . ∂z The resultant condition is then satisﬁed if and only if a = c. Suppose now that this is the case and let µ(λ) be a branch of qQ (λ, µ(λ)) ≡ 0. Without loss of generality, we may assume that λ − a(µ(λ)) ≡ 0. It follows that ∂qQ (λ, µ(λ)) = a (µ(λ))(c(µ(λ)) − a(µ(λ))) . ∂z Therefore −1 ∂qQ 1−n−deg(c−a) ) for |λ| → ∞ . ∂z (λ, µ(λ)) = O(|µ(λ)| We obtain m2,2 (λ, µ(λ)) ≡ m1,2 (λ, µ(λ)) ≡ 0 and, for |λ| → ∞, m1,1 (λ, µ(λ)) ∂qQ ∂z (λ, µ(λ))

= O(|µ(λ)|1−n ) ,

m2,1 (λ, µ(λ)) ∂qQ ∂z (λ, µ(λ))

= O(|µ(λ)|1−n−deg(c−a)+d ) .

Thus, in this case, (2.13) is satisﬁed if and only if deg(c − a) ≥ deg(b). If c − a m (λ,µ(λ)) → ∞ for is a non-zero constant and deg(b) = n, then we even have ∂q2,1 Q ∂z

(λ,µ(λ))

|λ| → ∞. Note that An has a repeated eigenvalue precisely when the leading terms of c and a are equal, in which case deg(c − a) < n. For general regular matrix polynomials of degree n ≥ 1 one has to consider the prime factor decomposition of the characteristic polynomial qQ , namely qQ (λ, z) =

r

qν (λ, z)αν ,

(2.15)

ν=1

where r, α1 , . . . , αr ∈ N with α1 +· · ·+αr ≤ N and where the polynomials q1 , . . . , qr are irreducible and pairwise diﬀerent. The zeros of qQ coincide with the zeros of the polynomial q˜Q given by q˜Q (λ, z) :=

r ν=1

qν (λ, z) (λ, z ∈ C).

(2.16)

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Deﬁnition 2.8. Let Q be a regular matrix polynomial. Let the prime factor decomposition of qQ be given by (2.15) and q˜Q be as in (2.16). The set Λ(Q) of critical points for Q is deﬁned as Λ(Q) := Λ1 (Q) ∪ Λ2 (Q), where ∂ q˜Q (λ, z)} , ∂z ∂ q˜Q (λ, z)} . Λ2 (Q) := {λ ∈ C ; ∃ z ∈ C with q˜Q (λ, z) = 0 = ∂λ It follows from the B´ezout theorem that Λ(Q) is a ﬁnite set of points. Λ1 (Q) := {λ ∈ C ; ∃ z ∈ C with q˜Q (λ, z) = 0 =

For instance, for N = 2 and Q as in Example 2.6 we have Λ1 = {±i} and Λ2 = {±1}. For λ ∈ C \ (Σ(Q) ∪ Λ1 (Q)) the zeros of qQ (λ, ·) are all zeros of order 1 of / Σ(Q) ∪ Λ1 (Q) and ν = 1, . . . , r, precisely one of the factors qν (λ, ·). Hence, for λ ∈ [l] the residues Rj,k (µ(λ)) of the functions z →

mk,j (λ, z)z l iz(s−t) e qQ (λ, z)

(2.17)

(where 0 ≤ l < n = d(Q)) at a zero µ(λ) of qν (λ, ·), which is a pole of the functions (2.17) of order βνk,j ≤ αν , have to be computed (via the Leibniz rule) as 1

[l]

Rj,k (µ(λ)) =

k,j

lim

∂ βν

k,j

(βνk,j − 1)! z→µ(λ) ∂z βν

−1

βνk,j −1

= eiµ(λ)(s−t)

(i(s − t))γ γ! γ=0

(z − µ(λ))βνk,j m

−1

l

qQ (λ, z) k,j

lim

k,j (λ, z)z

∂ βν

−1−γ

(z − µ(λ))βνk,j m

eiz(s−t)

k,j (λ, z)z

l

. (βνk,j − 1 − γ)!qQ (λ, z) (2.18) As a result of these computations one will obtain an expression of the form [l] Rj,k (µ(λ))

z→µ(λ)

βνk,j −1

=e

iµ(λ)(s−t)

k,j

∂z βν

−1−γ

[l]

gk,j,γ,ν (λ, µ(λ))(s − t)γ ,

(2.19)

γ=0 [l]

where (λ, z) → gk,j,γ,ν (λ, z) are rational functions. Deﬁnition 2.9. The type T (Q) of a regular matrix polynomial Q is deﬁned as T (Q) := (α, β), where β := max βνk,j j,k,ν

(i.e., β is the maximal order of all occurring poles) and α ∈ Z is chosen maximal with respect to the property that [l]

gk,j,γ,ν (λ, µ(λ)) = O(|µ(λ)|l−α ) for |λ| → ∞ , holds for all ν = 1, . . . , r, all j, k = 1, . . . , N , all γ = 0 . . . , βν k,j , and all branches of qν (λ, µ(λ)) = 0.

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In particular, if a regular polynomial Q fulﬁls the hypothesis of Lemma 2.5, then T (Q) = (n − 1, 1). In the setting of Example 2.7 we have T (Q) = (α, 1), where all α ∈ {−1, 0, . . . , n − 1} may occur. Example 2.10. Let n ∈ N and a, b1 , . . . , bN −1 ∈ C[Z] be polynomials of degree at most n with deg(a) = n. Consider the regular (N × N )-matrix polynomial Q given by   0 0 ··· 0 a(z) b1 (z)  0 a(z) b2 (z) 0 ··· 0      .. .. .. .. .. ..   . . . . . . .  Q(z) :=  .  . . . . .. .. .. ..   ..    0 0 ··· 0 a(z) bN −1 (z) 0 0 ··· 0 0 a(z) In this situation we have qQ (λ, z) = q1 (z, λ)N = q˜(λ, z)N = (λ − a(z))N and Λ(Q) = Λ1 (Q) = Λa := {a(z) ; z ∈ C, a (z) = 0}. Moreover, mk,j ≡ 0 for k < j, mk,k (λ,z) 1 qQ (λ,z) = λ−a(z) for j = k, and k−1 mk,j (λ, z) = (λ − a(z))j−k−1 (−br (z)) qQ (λ, z) r=j

for j < k. Then Q is of type T (n − 1, β) with β = 1 if and only if br = 0 for all 1 ≤ r < N and β := max{k − j + 1 ; 1 ≤ j < k ≤ N and br = 0 for j ≤ r < k} ≥ 2, otherwise. Hence, all values for β in {1, . . . , N } may occur. The value α = n − 1 is obtained by direct computation from (2.18), by again using the Leibniz formula, and the following result. Lemma 2.11. Let a be a polynomial of degree n.Then, for m ∈ N, λ ∈ C \ Λa and all solutions µ(λ) of λ − a(z) = 0, we have dk z − µ(λ) m lim = O(µ(λ)−m(n−1)−k ) as |λ| → ∞. z→µ(λ) dz k λ − a(z) Proof. With f (w) := (−w)−m and ϕ(z) :=

n−1 a(j+1) (µ(λ)) a(z) − λ = (z − µ(λ))j z − µ(λ) (j + 1)! j=0

we apply the Fa` a di Bruno formula for higher derivatives of composite functions: (f ◦ ϕ)(k) (z) =

k (r) k! ϕ (z) βr f (j) (ϕ(z)) , β! r=1 r! j=1

k

β

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where

β

465

is the sum over all β = (β1 , . . . , βk ) ∈ Nk0 satisfying the equations |β| = β1 + · · · + βk = j 1 · β1 + . . . k · βk = k .

(2.20)

For our choice of f and ϕ we obtain dk z − µ(λ) m lim z→µ(λ) dz k λ − a(z) =

k j=1

Note that a

(r)

k (r+1) k! a (m − 1 + j)! (µ(λ)) βr . (m − 1)!(−a (µ(λ)))m+j β! r=1 (r + 1)! β

has the degree n − r. Hence, because of (2.20), the expression k (r+1) k! a (µ(λ)) βr β! r=1 (r + 1)! β

is a polynomial in µ(λ) of degree at most j(n − 1) − k and hence, the statement follows as (a )m+j has degree (m + j)(n − 1). Lemma 2.12. Let Q be a regular matrix polynomial and let K be a compact subset of Σ(Q) \ Λ(Q). Then there exists a compact neighbourhood W of K in C \ Λ(Q) and a constant C > 0 such that, for λ ∈ W \Σ(Q) and all µ ∈ C with qQ (λ, µ) = 0, we have C 1 ≤ . (2.21) | Im µ| dist(λ, Σ(Q)) Proof. It suﬃces to show, for each λ0 ∈ Σ(Q) \ Λ(Q), that such an estimate is valid in a compact neighbourhood of λ0 . Hence, ﬁx λ0 ∈ Σ(Q) \ Λ(Q). Then, ∂ q˜ because of ∂zQ (λ0 , z) = 0 for all z ∈ C with q˜Q (λ0 , z) = 0, the function z → q˜Q (λ0 , z) has ﬁnitely many pairwise distinct zeros z1 , . . . , zd˜, all of multiplicity 1. Hence, by the (complex version of the) implicit function theorem, there exist open ˜ with Ω ∩ Ω = ∅ neighbourhoods U ⊂ C \ Λ(Q) of λ0 and Ω of z , = 1, . . . , d, 1 2 for 1 = 2 , and analytic functions µ : U → Ω satisfying q˜Q (λ, µ(λ)) = 0. Moreover, if q˜Q (λ, z) = 0 for some (λ, z) ∈ U × Ω , then z = µ (λ). Because of U ⊂ C \ Λ(Q) we have ∂ q˜Q

µ (λ) = − ∂∂λ q˜Q ∂z

(λ, µ (λ)) (λ, µ (λ))

= 0

in some neighbourhood of λ0 . In particular, by possibly decreasing the neighbourhoods U of λ0 and Ω of z , we may assume that µ : U → Ω is biholomorphic. If z = µ (λ0 ) ∈ C \ R we have, for some compact neighbourhood V ⊂ U of λ0 , that inf λ∈V | Im(µ (λ))| =: η > 0. Let now ζ be real. Then Ω contains a square R = {z ∈ C ; | Im z| ≤ δ , | Re(z − z )| ≤ δ }. For each λ ∈ µ−1 (R ) both −1 µ (λ), Re µ (λ) ∈ R and µ (R ∩ R ) ⊂ Σ(Q). So, by the mean value theorem, dist(λ, Σ(Q)) ≤ |λ − µ−1 (Re µ (λ))| ≤ C · | Im µ (λ)|

(2.22)

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−1 dµ where C := supw∈V ; 0≤t≤1 dz (Re µ (w) + it Im µ (w)). Suppose there is no C > 0 such that the estimate (2.21) is satisﬁed on V := V1 ∩· · ·∩Vd˜. Thus, for each k ∈ N there exists some λk ∈ V with |λk −λ0 | < k1 and some zk ∈ C with q˜Q (λk , zk ) = 0 and 1 dist(λk , Σ(Q)) > | Im zk | . (2.23) k In particular, λk → λ0 and Im zk → 0 for k → ∞. Moreover, the sequence (zk )∞ k=1 is bounded by (2.12). Hence, we may pass to some subsequence (zkj )∞ j=1 converging to some z0 ∈ R. Because of 0 = limj→∞ q˜Q (λkj , zkj ) = q˜Q (λ0 , z0 ), the point z0 must be one of the real roots, say z1 , of z → q˜Q (λ0 , z). Hence, for j ≥ j0 , we have (λkj , zkj ) ∈ V1 × R1 and therefore zkj = µ1 (λkj ). By (2.22), it follows that dist(λ, Σ(Q)) ≤ C1 · | Im µ1 (λkj )| = C1 | Im zkj | ,

j ≥ j0 ,

in contradiction to (2.23).

We now show, for regular matrix polynomials Q, that the resolvent satisﬁes ﬁnite order growth conditions near the spectrum and at inﬁnity. For the proof, m j α ξ ∈ C[ξ] is a polynomial with we introduce some notation. If h(ξ) = j j=0 1 ˜ complex coeﬃcients, then h 1 is the -norm of its coeﬃcients and h denotes the m j non-negative function ξ → j=0 |αj ||ξ| , for ξ ∈ C. Proposition 2.13. Let Q be a regular matrix polynomial of degree n. Denote by | · | the operator norm in L(Lp (R, CN ), W p,n (R, CN )). For all λ ∈ C with dist(λ, Σ(Q)) ≤ 1 we have |(λ − τ (Q))−1 | ≤ M

(1 + |λ|)N , dist(λ, Σ(Q))N

where M > 0 is a constant depending only on Q, p, and N . Proof. The domain of τ (Q) coincides with W p,n (R, CN ) and the graph norm on W p,n (R, CN ) is equivalent to the usual norm · ; see Lemmas 2.2 and 2.3. Hence, it suﬃces to show that for all λ ∈ C \ Σ(Q) with dist(λ, Σ(Q)) ≤ 1, we have (for 0 ≤ α ≤ n) α D (λ − τ (Q))−1 p ≤K L(L (R,CN ))

(1 + |λ|)N dist(λ, Σ(Q))N

(2.24)

with a constant K > 0 that depends only on Q, p, and N . Fix λ ∈ C \ Σ(Q) with dist(λ, Σ(Q)) ≤ 1. The entries of the operator matrix Rα := Dα (λ − τ (Q))−1 ,

0 ≤ α ≤ n,

are Fourier multiplier operators in Lp (R) with bounded rational symbols ξ → rαj,k (ξ) = (−1)j+k

ξ α mk,j (λ, ξ) , qQ (λ, ξ)

j, k = 1, . . . , N .

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For the real and imaginary parts we have j,k uj,k α (ξ) := Re(rα (ξ))

= (−1)j+k

ξ α [Re(mk,j (λ, ξ)) Re(qQ (λ, ξ)) + Im(mk,j (λ, ξ)) Im(qQ (λ, ξ))] , |qQ (λ, ξ)|2

vαj,k (ξ) := Im(rαj,k (ξ)) = (−1)j+k

ξ α [Im(mk,j (λ, ξ)) Re(qQ (λ, ξ)) − Re(mk,j (λ, ξ)) Im(qQ (λ, ξ))] . |qQ (λ, ξ)|2

j,k Hence, uj,k α , vα are quotients of polynomials in R[ξ] with 2nN the degree of the denominator and with the degree of the numerator being at most 2nN −d(Q)+α = j,k 2nN + α − n. In particular, the R-valued rational functions uj,k α , vα are piecewise monotonic with at most 4nN intervals of monotonicity. Thus, they are of bounded variation and their BV-norm can be estimated by j,k j,k uj,k α BV ≤ 8nN uα L∞ ≤ 8nN rα L∞

and similarly vαj,k BV ≤ 8nN rαj,k L∞ , respectively. Hence, by Steˇckin’s theorem [11], p. 110, Rα L(Lp (R,CN )) ≤ C0 max rαj,k L∞ , 1≤j,k≤N

where the constant C0 depends only on p, n, and N . The minors mj,k (λ, ξ) are of the form N −1 aj,k,ν (ξ)λν , mj,k (λ, ξ) = ν=0

where the coeﬃcients aj,k,ν are polynomials in ξ of degree at most (N − ν − 1)n. Also qQ (λ, ξ) can be written as qQ (λ, ξ) =

N

bν (ξ)λν ,

ν=0

where the coeﬃcients bν are polynomials in ξ of degree at most (N −ν)n. Moreover, bN ≡ 1 and b0 is of the form b0 (ξ) = ± det(An )ξ nN +

nN −1

cκ ξ κ ,

κ=0

where the determinant det(An ) of the leading coeﬃcient of the matrix polynomial Q is diﬀerent from 0, as Q is regular. There exists η ∈ (0, 1) such that N ν=1

bν 1 η ν +

nN −1

|cκ |η nN −κ ≤

κ=0

Notice that η depends only on Q and N .

1 | det(An )| . 2

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For |ξ| ≥ max η −1 , (|λ|η −1 )1/n > 1 we have |qQ (λ, ξ)| ≥| det(An )||ξ|nN −

N

|bν (ξ)||λ|ν −

ν=1

≥ | det(An )||ξ|nN −

N

nN −1

|cκ ||ξ|κ

κ=0

bν 1 |ξ|(N −ν)n η ν |ξ|νn

ν=1

−

nN −1

|cκ ||ξ|κ (η|ξ|)nN −κ

κ=0

1 ≥ | det(An )||ξ|nN . 2 Therefore, in this situation, we obtain the estimate N −1 |ξ|α |mk,j (λ, ξ)| 2 ν=0 ak,j,ν 1 |ξ|(N −ν−1)n+α |λ|ν α |rj,k (ξ)| = |qQ (λ, ξ)| | det(An )||ξ|nN ≤

N −1 2 ak,j,ν 1 η ν =: cj,k ≤ C1 ≤ | det(An )| ν=0

≤ C1

(1 + |λ|)N , dist(λ, Σ(Q))N

where the constant C1 := maxj,k=1,...,N cj,k depends only on Q and N . For the case |ξ| ≤ (|λ|η −1 )1/n we obtain α/n N −1 a ˜k,j,ν (ξ)|λ|ν |λ| α . · ν=0 |rj,k (ξ)| ≤ η |qQ (λ, ξ)| Notice that |qQ (λ, ξ)| =

N

|λ − λj (ξ)|

j=1

where λ1 (ξ), . . . , λN (ξ) are the eigenvalues of Q(ξ). Thus, λ1 (ξ), . . . , λN (ξ) are in Σ(Q) and it follows that α/n N −1 a ˜k,j,ν ((|λ|/η)1/n )|λ|ν |λ| α (ξ)| ≤ · ν=0 |rj,k η dist(λ, Σ(Q))N ≤

N −1 (1 + |λ|)N −1+α/n a ˜k,j,ν (η −1/n ) η α/n dist(λ, Σ(Q))N ν=0

≤ C2

(1 + |λ|)N −1+α/n , dist(λ, Σ(Q))N

where C2 depends only on Q and N . Actually, this estimate can be improved to α |rj,k (ξ)| ≤ C2

(1 + |λ|)N −1+α/n dist(λ, Σ(Q))n0

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near branches λj (ξ) of multiplicity n0 ≤ N . Finally, for the remaining case |ξ| ≤ 1/η, we obtain in the same way N −1 a ˜k,j,ν (η −1 )|λ|ν (1 + |λ|)N −1 α |rj,k (ξ)| ≤η −α · ν=0 ≤ C3 , N dist(λ, Σ(Q)) dist(λ, Σ(Q))N where the constant N −1 a ˜j,k,ν (η −1 ) C3 = η −α max j,k=1,...,N

ν=0

depends only on Q and N . It now follows that (2.24) holds with K := C0 max{C1 , C2 , C3 }.

Corollary 2.14. Let Q be a regular matrix polynomial of degree n. Then, for λ0 ∈ C \ Σ(Q), we have σ((λ0 − τ (Q))−1 ) = {(λ0 − λ)−1 ; λ ∈ Σ(Q)}

(2.25) −1

and there exists a constant M > 0 such that for all µ ∈ ρ((λ0 − τ (Q)) −1 −1

(µ − (λ0 − τ (Q))

)

),

L(Lp (R,CN ))

≤M

−N (|µ| + |λ0 µ − 1|)N dist µ, σ((λ0 − τ (Q))−1 ) . |µ| dist(λ0 , Σ(Q)) (2.26)

Proof. By Lemma 2.4 and the spectral mapping theorem we have (2.25) and obtain for all µ ∈ ρ((λ0 − τ (Q))−1 ) and λ ∈ Σ(Q), that |µ|N |µ − (λ0 − λ)−1 |−N |λ0 − λ|N |µ|N ≤ |µ − (λ0 − λ)−1 |−N dist(λ0 , Σ(Q))N

|(λ0 − µ−1 ) − λ|−N =

and hence, dist(λ0 − µ−1 , Σ(Q))−N ≤

|µ|N |µ − (λ0 − λ)−1 |−N . dist(λ0 , Σ(Q))N

(2.27)

Because of the identity (µ − (λ0 − τ (Q))−1 )−1 =

−1 1 (λ0 − τ (Q)) (λ0 − µ−1 ) − τ (Q) µ

we obtain from Proposition 2.13, (µ − (λ0 −τ (Q))−1 )−1 L(Lp(R,CN ))

−1 1 ≤ λ0 − τ (Q)L(W p,n (R,CN ),Lp (R,CN )) (λ0 − µ−1 ) − τ (Q) |µ| M ≤ (1 + |λ0 − µ−1 |)N dist(λ0 − µ−1 , Σ(Q))−N . |µ| Together with (2.27) this implies (2.26).

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Deﬁnition 2.15. Let Q be a regular matrix polynomial of degree n. For λ0 ∈ Σ(Q) \ Λ(Q), the polynomial q˜Q (λ0 , ·) as given by (2.16) has only simple real zeros (because λ0 ∈ / Λ1 (Q)) say x1 , . . . , xm , for some positive integer m ≤ n. We call m the real multiplicity of λ0 with respect to q˜Q . By the implicit function theorem and because λ0 ∈ / Λ(Q), there exists a neighbourhood V of λ0 in C\Λ(Q), neighbourhoods U1 , . . . , Um in C of x1 , . . . , xm , respectively, with pairwise disjoint closures, and biholomorphic maps λj : Uj → V (j = 1, . . . , m) satisfying λj (xj ) = λ0 such that the only solutions of q˜Q (λ, z) = 0 in V × (U1 ∪ · · · ∪ Um ) are of the form (λj (z), z) for some j ∈ {1, . . . , m} and some z ∈ Uj . A point λ0 is called a proper crossing point of Σ(Q), if there exist distinct j, k ∈ {1, . . . , m}, such that for all open intervals Ij ⊆ Uj ∩ R, Ik ⊆ Uk ∩ R, with xj ∈ Ij , xk ∈ Ik , we have λj (Ij ) \ λk (Ik ) = ∅ and λk (Ik ) \ λj (Ij ) = ∅. We write CP (Q) for the set of all proper crossing points of Σ(Q) in Σ(Q) \ Λ(Q). The following fact may be known, but we could not ﬁnd any reference. Lemma 2.16. Let Q be a regular matrix polynomial of degree n. The set CP (Q) is discrete in Σ(Q) \ Λ(Q). Proof. Assume that λ0 ∈ Σ(Q) \ Λ(Q) is an accumulation point of CP (Q) and let (ων )∞ ν=1 be a sequence in CP (Q) converging to λ0 . In particular, for all ν ∈ N, there (1) (2) (1) (2) (1) (2) (1) (2) exist ξν , ξν ∈ R with ξν = ξν and open neighbourhoods Uν , Uν of ξν , ξν (in C), respectively, with disjoint closures, an open neighbourhood Vν of ων and (i) (i) biholomorphic maps ων : Ui → Vν satisfying q˜Q (ων (z), z) ≡ 0 such that for all (i) (i) (1) (2) open intervals Ii ⊆ Uν ∩ R, with ξν ∈ Ii , i = 1, 2, we have ων (I1 ) \ ων (I2 ) = ∅ (2) (1) (i) ∞ and ων (I2 ) \ ων (I1 ) = ∅. As the sequences (ξν )ν=1 (i = 1, 2) are bounded (c.f. (2.12)), by passing to some subsequence, we may assume that the limits (i) ξ (i) = limν→∞ ξν exist (i = 1, 2). Let m ∈ {1, . . . , n} be the real multiplicity of λ0 with respect to q˜Q . Then q˜Q (λ0 , ·) has m (pairwise distinct and simple) real zeros x1 , . . . , xm . Let also V, U1 , . . . , Um and λj : Uj → V be as in Deﬁnition 2.15. By continuity we have q˜Q (λ0 , ξ (i) ) = 0 for i = 1, 2 and hence, ξ (1) , ξ (2) ∈ {x1 , . . . , xm }. If ξ (1) = ξ (2) = xj for some j ∈ {1, . . . , m}, then for some ν0 and all ν ≥ ν0 (1) (2) (i) (i) (i) we have ων ∈ V , ξν , ξν ∈ Uj and hence, (ων , ξν ) = (λj (ξν ), ξν ) for i = 1, 2. It follows that for ν → ∞ we have (1)

0=

(1)

∂ q˜

(2)

λj (ξν ) − λj (ξν ) (2)

ξν − ξν

→

Q (λ0 , xj ) dλj (xj ) = − ∂∂z q ˜ Q dz (λ0 , xj )

∂λ

/ Λ(Q). Hence, we must have ξ (1) = ξ (2) , say ξ (1) = x1 in contradiction to λ0 ∈ (1) and ξ (2) = x2 . Then, for some ν0 > 0 and all ν ≥ ν0 we have ων ∈ V , ξν ∈ U1 (2) and ξν ∈ U2 . Note that f := λ−1 1 ◦ λ2 : U2 → U1 is biholomorphic and 0 ≡ (2) q˜Q (λ1 (f (z)), f (z)) = q˜Q (λ2 (z), f (z)) on U2 . In particular, q˜Q (ων , f (ξν )) = 0. As (1) (2) (1) ξν is the only solution of q˜Q (ων , z) = 0 in U1 , we must have f (ξν ) = ξν ∈ R for

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all ν ≥ ν0 . With u(x, y) := Re f (x + iy) and v(x, y) := Im f (x + iy) for all x, y ∈ R (2) with x + iy ∈ U2 we thus have v(ξν , 0) = 0 for all ν ≥ ν0 . Let J ⊆ U2 ∩ R be (2) an open interval containing x2 . As ξν → x2 for ν → ∞, an iterated application of Rolle’s Theorem to the R-valued C ∞ -function x → v(x, 0) on J implies that ∂kv (x2 , 0) = 0 for all k ∈ N0 . It follows that ∂xk dk f ∂kf ∂ku ∂kv (x ) = (x ) = (x , 0) + i (x2 , 0) ∈ R , 2 2 2 dz k ∂xk ∂xk ∂xk for all k ∈ N0 . Hence, f has to be R-valued on an open interval I2 contained in U2 ∩ R with x2 ∈ I2 and I1 := f (I2 ) must be an open interval contained in U1 ∩ R (j) with x1 ∈ I1 . It follows that there exists some ν1 ∈ N such that ξν ∈ Ij for all (j) ν ≤ ν1 , j = 1, 2. For all those ν the implicit function theorem implies ων ≡ λj on Uj , j = 1, 2, and hence, ων(1) (I1 ) = λ1 (I1 ) = λ1 (f (I2 )) = λ2 (I2 ) = ων(2) (I2 ) which is a contradiction to ω ∈ CP (Q). Therefore CP (Q) cannot have any accumulation points in Σ(Q) \ Λ(Q). In particular, S(Q) := Λ(Q) ∪ CP (Q) ∪ {∞} is compact and totally disˆ and has at most ﬁnitely many accumulation points. The study of connected in C examples suggests that S(Q) is actually ﬁnite and that Σ(Q) \ S(Q) is the union of pairwise disjoint analytic Jordan arcs and may be considered as a totally real, real analytic manifold. ˆ be a compact Let M ⊂ C be a totally real, real analytic manifold and K ⊂ C set such that M ∪K is compact. For k ∈ N0 ∪{∞}, (following [8, 10]) let Ck O(M, K) be the algebra consisting of all functions f that are k-times diﬀerentiable on M and are analytic in (individual) open neighbourhoods Uf of K. We endow such algebras with their natural inductive limit topology. If K is totally disconnected, then these algebras are normal and quasi-admissible in the sense of Vasilescu [33], Deﬁnition IV.9.2. If, in addition, M ∪ K ⊂ C then the algebra Ck O(M, K) is admissible in the sense of Colojoar˘ a and Foia¸s [7]. We also refer to [7] for the deﬁnition of A-generalised scalar operators for any admissible algebra A. From Corollary 2.14 above and Theorem 10 in [30] (with its proof) we obtain: Corollary 2.17. Let Q be a regular matrix polynomial of degree n and λ0 ∈ C\Σ(Q). Let Σ0 := {(λ0 − λ) ; λ ∈ Σ(Q)} and K = {(λ0 − z)−1 ; z ∈ S(Q)} with the convention that 1/∞ := 0. Then (λ0 − τ (Q))−1 is a CN +2 O(Σ0 \ K, K)generalised scalar operator. If Φ : CN +2 O(Σ0 \ K, K) → L(Lp (R, CN ) denotes the corresponding funcˆ →C ˆ is the map z → (λ0 − z)−1 , tional calculus for (λ0 − τ (Q))−1 and if ϕ : C N +2 then a C O(Σ(Q) \ S(Q), S(Q))-functional calculus Φ0 : CN +2 O(Σ(Q) \ S(Q), S(Q)) → L(Lp (R, CN ))

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for τ (Q) is given by Φ0 (f ) := Φ(f ◦ ϕ−1 ). For a slightly richer functional calculus, see [2], Section 5.

3. Local spectral properties in the perturbed situation

n Throughout this and the following section, Q(x) = k=0 Ak xk will always denote a regular matrix polynomial of degree n with coeﬃcients in MN ×N (C). Let F0 , . . . , Fn−1 be measurable MN ×N (C)-valued functions on an interval I ⊂ R and consider the formal diﬀerential operator σ :=

n−1

Fk Dk .

(3.1)

k=0

Deﬁne D(σ1 ) := {f ∈ An (I, CN ) ∩ Lp (I, CN ); σf ∈ Lp (I, CN )} and the operator σ1 , with domain D(σ1 ), by σ1 f := σf for f ∈ D(σ1 ). The following perturbation result is due Balslev and Gamelin [4] (see also [16], Theorem VI.8.1) for the scalar case on the half-line. The proof uses Lemma 2.1 and Lemma 2.2 and easily carries over to the case of CN -valued functions on both the half-line and on R. Lemma 3.1. Let I = [0, ∞) or I = R and 1 < p < ∞. (a) The operator σ1 is τ1 (Q)-bounded if and only if each Fk in (3.1) belongs to Lploc (I, MN ×N (C)) and s+1 sup Fk (t)p dt < ∞ for 0 ≤ k ≤ n − 1 , s∈I

s

where · is the operator norm deﬁned in Lemma 2.2. (b) The operator σ1 is τ1 (Q)-compact if and only if each Fk in (3.1) belongs to Lploc (I, MN ×N (C)) and (for s ∈ I) s+1 Fk (t)p dt = 0 , 0 ≤ k < n. (3.2) lim |s|→∞

s

In this case, τ1 (Q) and τ1 (Q) + σ1 have the same essential spectrum and, for λ ∈ ρe (τ (Q)), we have ind(λ − τ1 (Q) − σ1 ) = ind(λ − τ1 (Q)). Moreover, τ1 (Q) + σ1 coincides with the maximal operator of Q(D) + σ analogously deﬁned as in [16], pp. 169–170, for the scalar case. In particular, it is closed. Let LpN,Σ be the Banach space L1 (R, CN ) + Lp (R, CN ) endowed with the norm f Σ := inf{u11 + up p ; u1 + up = f, u1 ∈ L1 (R, CN ), up ∈ Lp (R, CN )} for all f ∈ LpN,Σ . Notice that LpN,Σ contains the Banach spaces L1 (R, CN ) and Lp (R, CN ) as continuously embedded dense subspaces. For each ν ∈ N0 , deﬁne a weight function ων by ων (t) := 1 + |t|ν for t ∈ R.

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Lemma 3.2. Let ν ∈ N0 and µ+ , µ− ∈ C with Im µ− < 0 < Im µ+ . Deﬁne h+ ν (t, µ+ ) :=

tν eiµ+ t 0

for t > 0 , for t < 0

h− ν (t, µ− ) :=

tν eiµ− t 0

for t < 0 . for t > 0

(a) For every 1 ≤ p ≤ ∞, the Lp (R)-valued function µ± → h± ν (·, µ± ) is analytic on the upper resp. lower half-plane and satisﬁes 1

h± ν (·, µ± )p

=

[Γ(pν + 1)] p

(1 ≤ p < ∞).

1

(| Im µ± |p)ν+ p

p 1 In particular, the operator of convolution with h± ν (·, µ± ) on L (R) ∩ L (R) p 1 p p extends to a bounded linear operator from L1,Σ = L (R) + L (R) into L (R) ± whose norm is at most max{h± ν (·, µ± )1 , hν (·, µ± )p } and which is analytic as an operator-valued function on the upper resp. lower half-plane. (b) Fix p ∈ (1, ∞) and let p1 + p1 = 1. Let a, b : R → C be measurable functions

such that aων ∈ Lp (R) and bων ∈ Lp (R). Then the integral operators T± (µ± ) with kernels kν± (s, t, µ± ) := b(s)h± ν (s−t, µ± )a(t) are compact linear operators on Lp (R) with T± (µ± ) ≤ 2ν aων p bων p (which does not depend on the special values of µ± ). Moreover, the operator-valued function T± (·) is analytic on the upper resp. lower half-plane.

Proof. (a) For all µ ∈ C with |µ − µ+ | < 12 Im µ+ and for all t > 0 we have h+ (t, µ ) − h+ (t, µ) i(µ−µ+ )t ν + ν ν+1 − Im µ+ t 1 − e − ih+ − 1 e ν+1 (t,µ+ ) = t µ+ − µ i(µ+ − µ)t ∞ 1 =|µ − µ+ |tν+2 e− Im µ+ t (i(µ − µ+ )t)j (j + 2)! j=0 ≤|µ − µ+ |t

ν+2 − Im µ+ t

e

∞ j=0

≤|µ − µ+ |tν+2 e

− 12

1 (|µ − µ+ |t)j (j + 2)!

Im µ+ t

and a similar estimate holds for h− ν (t, µ) for t < 0 in the lower half-plane. From this one easily obtains that ± h± ν (·, µ) − hν (·, µ± ) → ihν+1 (·, µ± ) µ − µ±

as µ → µ±

in Lp (R) for 1 ≤ p ≤ ∞. Hence, the Lp (R)-valued function h± ν is analytic on the upper resp. lower half-plane for 1 ≤ p ≤ ∞. ! ∞ νp −tp Im µ p + dt. The statement for h+ We have h+ ν (·, µ+ )p = 0 t e ν (·, µ+ ) then follows, via the substitution s := tp Im µ+ , from the deﬁnition of the Gamma function. The computation for h− ν (·, µ− ) is similar. p The boundedness of the operator of convolution with h± ν (·, µ± ) from L1,Σ p into L (R) follows from the deﬁnition of · Σ and the well known inequality

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f ∗ gp ≤ f 1 gp , whenever f ∈ L1 (R) and g ∈ Lp (R). The analyticity of this operator-valued function on the upper resp. lower half-plane follows easily from 1 p the analyticity of h± ν (·, µ± ) as a function with values in L (R) and in L (R). ν ν (b) Using |s − t| ≤ 2 ων (s)ων (t) and H¨ older’s inequality we obtain ∞ ∞ p p1 p T± (µ± ) ≤ |k± (s, t, µ± )|p dt ds ≤ 2ν aων p bων p . −∞

−∞

Thus, T± (µ± ) is a Hille–Tamarkin integral operator on Lp (R) and hence, compact (see [35], Chapter 11, 2, Example D or [21], Satz 11.6). We also have, for all f ∈ Lp (R), that T± (µ± )f = b · h± ν (·, µ± ) ∗ (f · a) ∞ where f · a ∈ L1 (R), b ∈ Lp (R) and µ± → h± ν (·, µ± ) is an analytic L (R)-valued function on the upper resp. lower half-plane by (a). Hence, T± is analytic there as well.

If the functions a and b in Lemma 3.2 (b) satisfy exponential decay conditions we get the following stronger statement. Lemma 3.3. Fix p ∈ (1, ∞) and let 1p + p1 = 1. Let a, b : R → C be measurable functions such that, for some ε > 0, we have ∞ ∞ |a(t)|p eε|t| dt < ∞ and |b(t)|p eε|t| dt < ∞ . (3.3) −∞

−∞

Then the conditions of Lemma 3.2(b) are satisﬁed for all ν ∈ N0 Moreover, for −1 each δ > 0 satisfying δ < ε min{p−1 , p }, the operator-valued functions T+ (·) and T− (·) have analytic extensions to the half-plane H+,δ := {z ∈ C ; Im z > −δ} resp. H−,δ := {z ∈ C ; Im z < δ} , with the extensions still assuming their values in the ideal K(Lp (R)) of all compact operators on Lp (R). Proof. That the conditions of Lemma 3.2(b) are satisﬁed for all ν ∈ N0 is obvious. −1 Fix δ > 0 with δ < ε min{p−1 , p }. Note, for all s, t ∈ R, that a(t) , b(s)h+ (s − t, µ+ )a(t) = ˜b(s)h+ (s − t, iδ + µ+ )˜ ν

ν

−δt

and ˜b(s) := b(s)eδs also satisfy the assumptions (for a and where a ˜(t) := a(t)e b) in Lemma 3.2(b) and µ+ → h+ ν (s − t, iδ + µ+ ) is analytic on H+,δ with values in L∞ (R). Hence, T+ (·) has an analytic extension to H+,δ with values in K(Lp (R)). The proof for T− is similar. The following result will be needed to obtain resolvent estimates for the perturbed diﬀerential operator τ (Q) + σ1 . Lemma 3.4. Let p ∈ (1, ∞) and Q have type T (Q) = (α, β) with 0 ≤ α ≤ n − 1. For k = 0, . . . , α let Fk be functions in L1 (R, MN ×N (C)) and deﬁne 1

1 p

ψk (t) := Fk (t) ,

Φk (t) :=

Fk (t)− p Fk (t) 0

if Fk (t) = 0 . if Fk (t) = 0

(3.4)

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(a) For all λ ∈ C \ Σ(Q), the resolvent R(λ, τ (Q)) extends to a bounded linear p N 1 N ˜ τ (Q)) from Lp operator R(λ, N,Σ into L (R, C ) which maps L (R, C ) into ∞,α N ˜ the Sobolev space W (R, C ). Moreover, λ → R(λ, τ (Q)) is analytic on C \ (Σ(Q) ∪ Λ(Q)). (b) The operators A : Lp (R, CN )α+1 → L1 (R, CN ) → LpN,Σ and

B : W ∞,α (R, CN ) → Lp (R, CN )α+1

deﬁned by A(g0 , . . . , gα ) :=

α

Φj g j

for g0 , . . . , gα ∈ Lp (R, CN )

j=0

and

α Bf := ψj Dj f j=0

for f ∈ W ∞,α (R, CN )

are bounded. For each compact subset H ⊂ Σ(Q) \ Λ(Q) there exist a neighbourhood W of H and constants CA , CB > 0 with the property that, for all λ ∈ W \ Σ(Q), we have the estimates ˜ τ (Q))A ≤ R(λ, and BR(λ, τ (Q)) ≤

CA dist(λ, Σ(Q))β−1+1/p

(3.5)

CB . dist(λ, Σ(Q))β−1+1/p

(3.6)

Proof. (a) For 0 ≤ l < n, by means of (2.9) and (2.19), we see that the entries of ˜ τ (Q)) are locally ﬁnite linear combinations of operathe operator matrix Dl R(λ, tors which are convolution with functions of the type h± ν (·, µ±,l (λ)), 1 ≤ ν ≤ β − 1, where µ±,l (λ) are locally analytic solutions to qQ (λ, µ) = 0 with positive or negative imaginary part, respectively. Note that the coeﬃcients in these linear combinations are locally analytic in the variable λ. Hence, by Lemma 3.2(a), R(·, τ (Q)) ˜ τ (Q)) : C \ (Σ(Q) ∪ Λ(Q)) → extends to an analytic operator-valued function R(·, p p N L(LN,Σ , L (R, C )). Notice, for each λ ∈ C \ (Σ(Q) ∪ Λ(Q)), that the functions h± ν (·, µ± (λ)) are bounded on R. Hence, because of ± sup |(h± ν (·, µ± (λ)) ∗ f )(s)| ≤ hν (·, µ± (λ))∞ f 1 ,

f ∈ L1 (R) ,

s∈R

˜ τ (Q))f ∈ L∞ (R, CN ) for all f ∈ L1 (R, CN ) and l = 0, 1, . . . , n− we see that Dl R(λ, ˜ τ (Q)) maps L1 (R, CN ) into the Sobolev space 1. This shows that the range of R(λ, ∞,n−1 N W (R, C ) and hence, into W ∞,α (R, CN ). ˜ τ (Q))A are (b) Because of (2.8) and (2.19), the entries of the matrix Dl R(λ, linear combinations of multiplication operators with the entries a∗ of Φ0 , . . . , Φα followed by convolutions with functions of the type h± ν (·, µ± (λ)) where 0 ≤ ν ≤ β−1. The coeﬃcients in these linear combinations depend on λ but are bounded on

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compact sets. By Lemma 3.2(a) and Lemma 2.12 we obtain, for all λ in W \ Σ(Q) for some compact neighbourhood W of H and all f ∈ Lp (R), that ± h± ν (·, µ± (λ)) ∗ (a∗ f )p ≤ C1 a∗ p hν (·, µ± (λ))p f p 1/p

≤ C2 | Im µ± (λ)|−ν−1/p max Fl 1 0≤l≤α

≤

f p

C3 f p . dist(λ, Σ(Q))β−1+1/p

This implies (3.5). In the same way the entries of BR(λ, τ (Q)) are linear combinations (with the coeﬃcients depending on λ, but being bounded on compact sets) of operators of convolution with functions of the type h± ν (·, µ± (λ)), 0 ≤ ν ≤ β − 1, followed by multiplication with ψl , 0 ≤ l ≤ α. Notice that for all f ∈ Lp (R), we have ∀s ∈ R :

± |(h± ν (·, µ± (λ)) ∗ f )(s)| ≤ hν (·, µ± (λ))p f p .

Using again Lemma 3.2(a) and Lemma 2.12, we obtain for λ ∈ W \ Σ(Q) the estimate (3.6). Theorem 3.5. Let p ∈ (1, ∞) and Q have type T (Q) = (α, β) with 0 ≤ α ≤ n − 1. For k = 0, . . . , α let Fk be functions in Lploc (R, MN ×N (C)) satisfying (3.2) and p0 ∈ L1 (R, MN ×N (C)) where p0 := max{p, p } and 1p + p1 = 1. Let σ be the Fk ωβ−1 formal diﬀerential operator α σ= Fk Dk , k=0 p0 and σ1 be deﬁned analogously as for (3.1). If ωβ−1 Fα 1 is suﬃciently small, then there exists a compact set H0 ⊂ C such that the set of all accumulation points of σ(τ (Q) + σ1 ) ∩ ρ(τ (Q)) is contained in H0 , the inclusion σ(τ (Q) + σ1 ) \ H0 ⊂ σ(τ (Q)) holds, and such that for each compact subset H ⊂ Σ(Q) \ H0 there exists a neighbourhood W of H with the property, for some constant C > 0, that

R(λ, τ (Q) + σ1 ) ≤

C , dist(λ, Σ(Q))η

λ ∈ W \ Σ(Q),

(3.7)

with η := max{N, 2β − 1}. Proof. By Lemma 3.1(b) the operator σ1 is τ (Q)-compact. With A and B as in Lemma 3.4 we have σ1 |D(τ (Q)) = AB|D(τ (Q)) . Note that, because of α < n, the space (D(τ (Q)), · τ (Q) ) = W p,n (R, CN ) is contained in W ∞,α (R, CN ) with a continuous inclusion map (by the proof of [4], Lemma 4). By Lemma 3.4(a), for each λ ∈ C \ Σ(Q), the resolvent extends to a bounded ˜ τ (Q)) from Lp into Lp (R, CN ) which maps L1 (R, CN ) into linear operator R(λ, N,Σ ˜ τ (Q))A is contained Lp (R, CN ) ∩ W ∞,α (R, CN ). In particular, the range of R(λ,

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in the domain W ∞,α (R, CN ) of B and the conditions of Lemma 1.1 are satisﬁed with ˜ := Lp , X X :=Lp (R, CN ), N,Σ ˜ := Lp (R, CN ) ∩ W ∞,α (R, CN ). Y :=Lp (R, CN )α+1 , D ˜ with the norm given by f ˜ := max{f p, f W ∞,α } for all Here we endow D D ˜ ˜ f ∈ D and we replace B by its restriction to D. ˜ τ (Q))A are By (2.8) and (2.19), the entries of the operator matrix B R(λ, locally (with respect to λ) linear combinations of integral operators with kernels of the form [l] ψl (s)g∗ (λ, µ±,ν (λ))h± (3.8) γ (s − t, µ±,ν (λ))a(t) , where a is an entry of one of the matrix functions Φ0 , . . . , Φα , 0 ≤ l ≤ α, and where [l] |g∗ (λ, µ±,ν (λ))| = O(|µ(λ)|l−α ) for |λ| → ∞. By Lemma 3.2(b), these integral operators are compact and depend locally analytically on λ. Moreover, for l < α, their norms tend to 0 as |λ| → ∞ and it will be small (independent of λ) in the case p0 ˜ τ (Q))A ≤ c < 1 Fα 1 is small enough. Hence, we will have B R(λ, l = α if ωβ−1 p0 if ωβ−1 Fα 1 is small enough and |λ| is large enough, i.e., for λ ∈ ρ(τ (Q)) \ H0 for some suﬃciently large compact set H0 ⊂ C. Thus we have shown that there exists some compact set H0 ⊂ C with the properties (i) and (ii) in Corollary 1.2. Hence, for all λ ∈ ρ(τ (Q)) \ H0 , we have the estimate ˜ τ (Q))A · BR(λ, τ (Q)) R(λ, . (3.9) R(λ, τ (Q) + σ1 ) ≤ R(λ, τ (Q)) + 1−c for all λ ∈ ρ(τ (Q)) \ H0 . We may choose H0 such that Λ(Q) ⊂ H0 . If H is now a compact subset of Σ(Q) \ H0 , then, by Lemma 3.4 there exists a neighbourhood W of H such that (3.5) and (3.6) hold for all λ ∈ W \ Σ(Q). Inserting these estimates into (3.9) and using the estimate of Lemma 2.13 for the ﬁrst term in (3.9), we obtain (3.7). ˜ τ (Q))A are Remark 3.6. As noticed in the above proof, the operators B R(λ, compact for all λ ∈ C \ Σ(Q). Hence, if Ω is a component of C \ Σ(Q), then ˜ τ (Q))A)−1 exists in Ω outside of some discrete subset (consisting of (1 − B R(λ, poles) of Ω, once it exists for some single point λ0 ∈ Ω. As the proof of the theorem shows, for an unbounded component and for all bounded components Ω satisfying p0 Fα 1 is suﬃciently Ω ∩ (C \ H0 ) = ∅ such a point λ0 always exists whenever ωβ−1 small. Hence, if in addition, all components of C \ Σ(Q) are unbounded, then σ(τ (Q) + σ1 ) = Σ(Q) ∪ D, where D is a discrete set in C \ Σ(Q). The resolvent estimates obtained in the preceding theorem may be used to obtain estimates for the resolvent of the resolvent. Corollary 3.7. In the setting of and with the notation of Theorem 3.5, ﬁx λ0 ∈ C \ (Σ(Q) ∪ H0 ) so that T := (λ0 − τ (Q) − σ1 )−1 exists in L(Lp (R, CN )). Let ˆ →C ˆ be the map given by ψ(z) := λ0 − 1/z for 0 = z ∈ C, and ψ(∞) := λ0 ψ:C and ψ(0) := ∞. Then σ(T ) \ ψ −1 (H0 ) ⊂ σ(R(λ0 , τ (Q)) = ψ −1 (Σ(Q) ∪ {∞}). The

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set M := σ(T ) \ ψ −1 (H0 ∪ S(Q)) is a totally real, real analytic manifold. For each compact subset K ⊂ M there exists a neighbourhood W of K and a constant C1 > 0 such that R(z, T ) ≤

C1 , dist(z, ψ(Σ(Q)))η

z ∈ W \ M,

(3.10)

where η := max{N, β − 1} with β coming from the type of Q. Proof. The ﬁrst part of the statement follows from Theorem 3.5 by means of the spectral mapping theorem. For the estimate (3.10), we use the following (easy to verify) formula for the resolvent of the resolvent 1 1 1 1 1 R(z, T ) = + 2 R λ0 − , τ (Q) + σ1 = + 2 R(ψ(z), τ (Q) + σ1 ) . (3.11) z z z z z By Theorem 3.5 there exists a bounded neighbourhood W0 of ψ(K) ⊂ σ(τ (Q) + / W0 . σ1 ) \ (H0 ∪ S(Q)), such that the resolvent estimate (3.7) holds in W0 and λ0 ∈ / W . For all Then W := ψ −1 (W0 ) is a bounded neighbourhood of K and 0 ∈ z ∈ W \ M we have the estimate R(ψ(z), τ (Q) + σ1 ) ≤

C dist(ψ(z), Σ(Q))η

If W0 is chosen appropriately, then the value dist(ψ(z), Σ(Q)) = min |ψ(z) − λ| λ∈Σ(Q)

will be attained at λ1 = ψ(z1 ) ∈ ψ(M ) for some z1 ∈ M . Because of |z| ≥ c0 := dist(0, W ) > 0 we obtain from (3.11) (as W is bounded) that C C = c−1 0 + 2 −1 c20 |ψ(z) − ψ(z1 )|η c0 |λ0 − z − λ0 + z1−1 |η η η C|z1 | · |z| C0 ≤ c−1 ≤ , 0 + 2 η c0 |z1 − z| dist(z, M )η

R(z, T ) ≤ c−1 0 +

where the constant C0 does not depend on z.

Corollary 3.8. In the setting of the preceding Corollary the analytic functional calculus for the operator T = R(λ0 , τ (Q)+σ1 ) has a unique extension to a continuous homomorphism Φ : Cη+2 O(M, ψ −1 (H0 ∪ S(Q))) → L(Lp (R, CN )), where η satisﬁes (3.10). In particular, the operator T is ψ −1 (H0 )-decomposable. Proof. The existence and uniqueness of Φ follows from Theorem 10 in [30] (with its proof) and [8], Satz 6.3. As ψ −1 (S(Q)) is totally disconnected (cf. Section 2), C∞ O(M, ψ −1 (H0 ∪ S(Q))) is a C \ ψ −1 (H0 )-basic algebra in the sense of [1], Def. 2.2. Hence, T is ψ −1 (H0 )-decomposable by [1], Folgerung 2.23.

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Corollary 3.9. In the setting of and with the notation of Theorem 3.5 and the previous two Corollaries, there exists a continuous homomorphism ΦQ : Cη+2 O(ψ(M ), H0 ∪ S(Q)) → L(Lp (R, CN )) extending the analytic functional calculus of the closed linear operator τ (Q) + σ1 such that, for all ϕ ∈ Cη+2 O(ψ(M ), H0 ∪ S(Q)) having compact support contained in C, the range of ΦQ (ϕ) is contained in D(τ (Q) + σ1 ) and (τ (Q) + σ1 )ΦQ (ϕ) = ΦQ (id · ϕ). In particular, the operator τ (Q) + σ1 is H0 -decomposable. Proof. As the function ψ is analytic, we have ϕ ◦ ψ ∈ Cη+2 O(M, ψ −1 (S(Q))) for all ϕ ∈ Cη+2 O(ψ(M ), S(Q)) and ϕ → ΦQ (ϕ) := Φ(ϕ ◦ ψ) (with Φ as in Corollary 3.8) deﬁnes a continuous unital homomorphism which is easily seen to extend the analytic functional calculus for τ (Q)+σ1 and which satisﬁes ran ΦQ (ϕ) ⊆ D(τ (Q)+σ1 ) and (τ (Q)+σ1 )ΦQ (ϕ) = ΦQ (id·ϕ) for all ϕ ∈ Cη+2 O(ψ(M ), S(Q)) having compact ˆ support contained in C. Let now U0 , . . . , Um be an arbitrary open covering of C with H0 ⊂ U0 and H0 ∩ Uj = ∅ for j = 1, . . . , m. As the set S(Q) is totally disconnected , there exist ϕ0 , . . . , ϕm ∈ Cη+2 O(ψ(M ), H0 ∪ S(Q)) with ϕ0 + · · · + ϕm ≡ 1 p N and suppϕj ⊂ Uj for j = 0, . . . , m. " It follows that L (R, C ) = E(suppϕ0 ) + · · · + E(suppϕm ), where E(suppϕj ) := {ker ΦQ (ψ) ; suppψ ∩suppϕj = ∅}. Direct computation shows that the spaces E(suppϕj ) are invariant for τ (Q) + σ1 and satisfy σ((τ (Q)+σ1 )|E(suppϕj ) ) ⊆ suppϕj ⊂ Uj . Hence, τ (Q)+σ1 is H0 -decomposable. In the event that the smallness condition for Fα is not satisﬁed or that some component of C \ Σ(Q) is contained in the set H0 of Theorem 3.5, some (or even all) of the components of C \ Σ(Q) could be completely contained in the point spectrum of the perturbed operator τ (Q) + σ1 . Therefore the following lemma may be of some interest. Lemma 3.10. Let p ∈ (1, ∞) and Q have type T (Q) = (α, β) with 0 ≤ α ≤ n − 1. For k = 0, . . . , α let Fk be functions in Lploc (R, MN ×N (C)) satisfying (3.2) and p0 Fk ωβ−1 ∈ L1 (R, MN ×N (C)) where p0 := max{p, p } and 1p + p1 = 1. Let σ be α the formal diﬀerential operator σ = k=0 Fk Dk with σ1 deﬁned analogously as for (3.1). Then, for each ε > 0, there exists some r ∈ (1 − ε, 1 + ε) such that σ(τ (Q) + rσ1 ) ∩ (C \ Σ(Q)) is discrete in C \ Σ(Q). Proof. The set C \ Σ(Q) has at most countably many components (Ω )m =1 (where 1 ≤ m ≤ ∞). For each of these components Ω we ﬁx some point λ ∈ Ω . By ˜ , τ (Q))A are compact. Hence, the the proof of Theorem 3.5, the operators B R(λ ˜ operator valued function z → (1 − zB R(λ , τ (Q))A)−1 is meromorphic on C and has only ﬁnitely many singularities z1, , . . . , zk , in (1 − ε, 1 + ε). As the set of all z1, , . . . , zk , , for 1 ≤ ≤ m, is countable and (1 − ε, 1 + ε) is uncountable, ˜ , τ (Q))A)−1 )−1 exists there exists some r ∈ (1 − ε, 1 + ε) such that (1 − rB R(λ p N α+1 in L(L (R, C ) ) for 1 ≤ ≤ m. By Lemma 1.1, for 1 ≤ ≤ m, the set σ(τ (Q) + rσ1 ) ∩ Ω is at most countable with any accumulation points belonging to ∂Ω ⊂ σ(τ (Q)).

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4. Reﬁnements for perturbations satisfying asymptotic decay conditions near inﬁnity. Recall that the n-th approximation number of a bounded linear operator A between two Banach spaces X and Y is deﬁned by an (A) :=

inf

K∈Fn (X,Y )

A − K ,

where Fn (X, Y ) denotes the set of all those K ∈ L(X, Y ) with dim ran(K) ≤ n. This deﬁnes a monotone decreasing sequence of non-negative real numbers which tends to 0 if and only if A is in the norm closure of the set of all ﬁnite rank operators in L(X, Y ). In this case we deﬁne 1 , nA (t) := min n − 1 ∈ N ; an (A) < t

t > 0.

Remark 4.1. If, for some δ > 0, we have an (A) = O(n−δ ) for n → ∞, then nA (t) = O(t1/δ ) as t → ∞. Indeed, suppose that an (A) ≤ Cn−δ for all n ∈ N and some positive constant C. Fix t ≥ 1 and choose n ∈ N minimal with Cn−δ < t−1 . ˜ 1/δ Then nA (t) ≤ n and C(n − 1)−δ ≥ t−1 . Hence, nA (t) ≤ n ≤ 1 + C 1/δ t1/δ ≤ Ct with C˜ := 1 + C 1/δ . Lemma 4.2. Let p ∈ (1, ∞) and δ > 0. Let q ∈ Lploc (R) satisfy s+1 1/p |q(t)|p dt ≤ Cx−δ , x > 0, sup s∈R\(−x−1,x)

(4.1)

s

for a constant C > 0. Let K : W p,1 (R) → Lp (R) be the linear operator (Kf )(x) := q(x)f (x),

f ∈ W p,1 (R).

Then K is a compact operator whose approximation numbers satisfy −δ ak (K) = O k max{1,δ}+p , k → ∞ .

(4.2)

−δ

Proof. Because of k 1+p → 0 for k → ∞, it suﬃces to prove (4.2). For k ∈ N and r > 1 we deﬁne a rank k operator Kk,r : W p,1 (R) → Lp (R) by (Kk,r f )(t) :=

0 q(t)k 2r

!

Ij f (ξ)dξ

for |t| > r for t ∈ Ij and j = 1, . . . , k,

2r for all f ∈ W p,1 (R), where Ij := [−r + (j − 1) 2r k , −r + j k ) for j = 1, . . . , k. Hence, p,1 for f ∈ W (R), we have

1/p

(K−Kk,r )f p ≤

R\[−r,r]

|q(x)f (x)|p dx

r

+ −r

1/p

|((K − Kk,r )f )(x)|p dx

.

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By (4.1) and Lemma 4 of [4] (which carries over from the half-line to R), we have for the ﬁrst term:

1/p p |q(x)f (x)| dx ≤ C1 r−δ f W p,1 (R) , R\[−r,r]

where C1 > 0 is a constant which is independent of r and k. For the second term we proceed as follows. If x ∈ [−r, r], then x ∈ Ij for some unique j and so, by the H¨older inequality, q(x)k (f (x) − f (ξ))dξ |((K − Kk,r )f )(x)| = 2r Ij x |q(x)|k ≤ f (s)ds dξ 2r ξ Ij |q(x)|k |x − ξ|1/p · f Lp (R) dξ ≤ 2r Ij 2r 1 +1 |q(x)|k p ≤ · f W p,1 (R) · 2r k r 1/p 1 ≤ 2 p · · |q(x)| · f W p,1 (R) . k Hence, by a straightforward estimate via (4.1), we have for 0 < δ ≤ 1, r 1/p 1−δ r 1/p 1 K − Kk,r ≤ C1 r−δ + 2 p · q|[−r,r]Lp ([−r,r]) ≤ C1 r−δ + C2 ·r , k k 1

where C2 > 0 does not depend on r or k. With the special choice r := k p +1 we get −δ

ak+1 (K) ≤ K − Kk,r ≤ (C1 + C2 )k p +1 which implies (4.2). In the case δ > 1, again with the help of (4.1), we obtain an estimate r 1/p r 1/p 1 K − Kk,r ≤ C1 r−δ + 2 p · q|[−r,r] Lp ([−r,r]) ≤ C1 r−δ + C3 , k k 1

where C3 > 0 does not depend on r or k. In this case we choose r := k δp +1 and obtain (36) with C = C1 + C3 . For the proof of the next result we shall use the following facts for bounded linear operators A : Y → Z, and B1 , B2 : X → Y , and C : W → X between Banach spaces W, X, Y, Z: ak (AB1 C) ≤ A · C · ak (B1 ),

k ∈ N,

(4.3)

and a2k (B1 + B2 ) ≤ ak (B1 ) + ak (B2 ) ,

k ∈ N,

(4.4)

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see [26], Section 8.1. Hence, if for some γ > 0 and constants cj > 0 we have ak (Bj ) ≤ cj k −γ for all k ∈ N and j = 1, 2 then, for all k ≥ 3, k −γ ak (B1 + B2 ) ≤ a2[k/2] (B1 + B2 ) ≤ a[k/2] (B1 ) + a[k/2] (B2 ) ≤ (c1 + c2 ) 2 k −γ −γ ≤ (c1 + c2 ) ≤ ck , 3 where c does not depend on k. By induction we obtain, for ﬁnitely many operators B1 . . . , Bm ∈ L(X, Y ), that m Bj = O(k −γ ) whenever ak (Bj ) = O(k −γ ) for j = 1, . . . , m as k → ∞ . ak j=1

(4.5) Proposition 4.3. Let p ∈ (1, ∞) and Q have type T (Q) = (α, β) with 0 ≤ α ≤ p0 ∈ n − 1. For k = 0, . . . , α let Fk be functions in Lploc (R, MN ×N (C)) with Fk ωβ−1 L1 (R, MN ×N (C)) where p0 := max{p, p } and p1 + p1 = 1. Let σ be the formal α diﬀerential operator σ = k=0 Fk Dk and σ1 be deﬁned analogously as for (3.1). Assume, in addition, that for some δ > 0 and C > 0 we have s+1 1/p Fk (t)p dt ≤ Cx−δ , x > 0, k = 0, . . . , α. (4.6) sup s∈R\(−x−1,x)

s

˜ 0 , τ (Q))A exists in If, for some λ0 ∈ C \ Σ(Q), the inverse of the operator 1 − B R(λ p N α+1 L(L (R, C ) ), then λ0 ∈ ρ(τ (Q)+σ1 ) and R(λ0 , τ (Q)+σ1 ) = R(λ0 , τ (Q))+K, where K ∈ L(Lp (R, CN )) is a compact operator satisfying −δ

as k → ∞ . ak (K) = O k max{1,δ}+p , In particular, by Remark 4.1, max{1,δ}+p δ nK (t) = O t

as t → ∞ .

Proof. Note that (4.6) implies (3.2). By the proof and notation of Theorem 3.5 ˜ := R(λ ˜ 0 , τ (Q)), that and using (1.1) we have, with R ˜ ˜ −1 BR(λ0 , τ (Q)) R(λ0 , τ (Q) + σ1 ) =R(λ0 , τ (Q)) + RA(1 − B RA) ˜ =R(λ0 , τ (Q)) + RABR(λ 0 , τ (Q))+ ˜ ˜ ˜ −1 B RABR(λ + RA(1 − B RA) 0 , τ (Q)) . Note that σ1 |W p,n (R,CN ) = AB|W p,n (R,CN ) and that ran R(λ0 , τ (Q)) = D(τ (Q)) = W p,n (R, CN ) is continuously embedded in W ∞,α (R, CN ). Moreover, both of the ˜ ˜ −1 B R ˜ : Lp (R, Cn ) → ˜ : Lp (R, Cn ) → Lp (R, Cn ) and RA(1 − B RA) operators R p n L (R, C ) are bounded. Hence, by (4.3), it suﬃces to show that K0 := σ1 R(λ0 , τ (Q)) =

α j=0

Fj Dj R(λ0 , τ (Q)) : Lp (R, Cn ) → Lp (R, Cn )

Vol. 59 (2007) Spectral Properties of Some Matrix Diﬀerential Operators

is a compact operator and satisﬁes −δ

ak (K0 ) = O k max{1,δ}+p ,

483

as k → ∞ .

(4.7)

Because of (4.6) the entries of the matrices Fj satisfy the conditions of Lemma 4.2. By that lemma, the operators Mj : W p,1 (R, Cn ) → Lp (R, Cn ) of multiplication by Fj are compact with −δ

akN 2 (Mj ) ≤ Cj k max{1,δ}+p ,

k ∈ N, for some constants Cj > 0. Hence, for k > N 2 , we have Nk2 ≥ ak (Mj ) ≤a[k/N 2 ]N 2 (Mj ) ≤ Cj

k N2

k 2N 2

and so

−δ max{1,δ}+p

−δ k −δ max{1,δ}+p ≤ Cj k max{1,δ}+p 2 2N where Cj does not depend on k. By continuity of the operators Dj R(λ0 , τ (Q)) : Lp (R, CN ) → W p,1 (R, Cn ) and the facts noted in (4.3) and (4.5) we now obtain (4.7).

≤Cj

Lemma 4.4. In the situation of the previous Proposition and with λ0 and δ as given there, let S be the set of all accumulation points of ρ(τ (Q)) ∩ σ(τ (Q) + σ1 ). Deﬁne S(λ0 ) := {(λ0 − λ)−1 ; λ ∈ S}. If Ω ⊂ ρ(R(λ0 , τ (Q))) is an open set with Ω∩(S(λ0 )∪{0}) = ∅, then Ω contains at most ﬁnitely many eigenvalues µ1 , . . . , µr of T := (λ0 −τ (Q)−σ1 )−1 . If G ⊆ Ω is any bounded, simply connected domain with Dini-smooth boundary then, for each ε > 0, there exists a constant C = C(ε) > 0 such that

µ ∈ G, (4.8) (µ − T )−1 ≤ C exp dist(µ, ∂G)−b ,

+ 1 + ε. where b = 2N max{1,δ}+p δ Proof. For simplicity of notation let R(λ0 ) := R(λ0 , τ (Q)) and K := T − R(λ0 ), in which case K is a compact operator by Proposition 4.3. Then σ(T ) ∩ ρ(R(λ0 )) consists of eigenvalues of T and S(λ0 ) is the set of all of its accumulation points. Hence, by the choice of Ω, the set Ω ∩ σ(T ) consists of ﬁnitely many eigenvalues µ1 , . . . , µr of T . Let G be as in the statement of the lemma. Since 0 ∈ / G and G is bounded, Corollary 2.14 implies that (µ − R(λ0 ))−1 ≤ C0 dist(µ, ∂G))−N ,

µ ∈ G,

for some constant C0 > 0 independent of µ. By Theorem 4.5 of [3] we obtain the following estimate for all µ ∈ G:

aN log+ log+ R(µ, T ) ≤ C1 + 2 log+ log+ C0 N dist(µ, ∂G)

(2a)N a + log+ , + 2 log nK 2C0 dist(µ, ∂G)N dist(µ, ∂G)

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where C1 > 0 and a ≥ 3 are constants, independent of µ. Using the estimate for nK from Proposition 4.3 and standard properties of log+ , we obtain (4.8) by direct computation. ˆ be a compact set and S be a closed subset of K such that Let K ⊂ C M := K \ S is a totally real, real analytic, 1-dimensional sub-manifold of C. Following Droste [8, 9, 10] we denote by E(M ) O(M, S) the algebra consisting of all those functions that are analytic in (individual) neighbourhoods of the compact set S and ultradiﬀerentiable of degree (M )∞ =0 , endowed with its natural inductive limit topology. In the situation of the previous Lemma and with the notation of its proof, let S0 := ρ(R(λ0 , τ (Q))) ∩ σ(T ) ∪ {(λ0 − z)−1 ; z ∈ CP (Q) ∪ Λ(Q)} ∪ {0}. Note that σ(T )\S0 may be considered as a totally real submanifold of C which is real analytic. 1+1/b ∞ In our special situation, we choose for (M )∞ )=0 =0 the Gevrey sequence ( ! (M ) with b as in (4.8). Note that E O(σ(T )\S0 , S0 ) is a C\S(λ0 )-basic algebra in the sense of [1], Def. 2.2, because S0 \ (S(λ0 ) ∪ {(λ0 − z)−1 ; z ∈ CP (Q) ∪ Λ(Q)} ∪ {0}) consists of isolated eigenvalues of T and {(λ0 − z)−1 ; z ∈ CP (Q) ∪ Λ(Q)} ∪ {0}) is totally disconnected. Theorem 4.5. In the setting of Proposition 4.3 and with b as in (4.8) let (M )∞ =0 := ( !1+1/b )∞ . Then there exists a unique continuous homomorphism =0 Φ : E(M ) O(σ(T ) \ S0 , S0 ) → L(Lp (R, CN )) extending the analytic functional calculus for T := (λ0 − τ (Q) − σ1 )−1 . In particular, the operator T is S(λ0 )-decomposable. If S(λ0 ) is totally disconnected, then E(Mp ) O(σ(T ) \ S0 , S0 ) is an admissible algebra (in the sense of [7]) and T is decomposable. Proof. If Γ is any compact Jordan arc contained in σ(T ) \ S0 , then there exist disjoint, simply connected domains G1 , G2 ⊂ ρ(T ) with Γ ⊂ ∂G1 ∩ ∂G2 satisfying the conditions that G satisﬁes in Lemma 4.4. Hence, if b = 2N max{1,δ}+p + δ 1 + ε (where ε > 0 is now ﬁxed), we have by that lemma that (µ − T )−1 ≤ C exp(dist(µ, σ(T ))−b ), for all µ ∈ G1 ∪G2 . Hence, by [8], Satz 6.1 (or [10], Theorem 3 ) in connection with Remark (ii) in Section 2 of [10], the analytic functional calculus for T admits a unique extension to a continuous unital homomorphism Φ : E(M ) O(σ(T ) \ S0 , S0 ) → L(Lp (R, CN )). Accordingly, T is S(λ0 )-decomposable by [1], Folgerung 2.23. Corollary 4.6. In the setting of the preceding Theorem and with S1 := ρ(τ (Q)) ∩ σ(τ (Q) + σ1 ) ∪ CP (Q) ∪ Λ(Q) ∪ {∞}, there exists a continuous unital homomorphism ΦQ : E(M ) O(σ(τ (Q)) \ S1 , S1 ) → L(Lp (R, CN ))

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extending the analytic functional calculus for the closed operator τ (Q) + σ1 such that, for all ϕ ∈ E(M ) O(σ(τ (Q)) \ S1 , S1 ) having compact support contained in C, the range of ΦQ (ϕ) is contained in D(τ (Q)+σ1 ) and (τ (Q)+σ1 )ΦQ (ϕ) = ΦQ (id·ϕ). In particular, the operator τ (Q) + σ1 is S-decomposable, where S is the set of all accumulation points of ρ(τ (Q)) ∩ σ(τ (Q) + σ1 ) ˆ →C ˆ be the map given by ψ(z) := λ0 − 1/z for 0 = z ∈ C and Proof. Let ψ : C ψ(∞) := λ0 , ψ(0) := ∞. Note that S1 = ψ(S0 ). As ψ is analytic, we have ϕ ◦ ψ ∈ E(M ) O(σ(T ) \ S0 , S0 ) for all ϕ ∈ E(M ) O(σ(τ (Q)) \ S1 , S1 ) and ϕ → ΦQ (ϕ) := Φ(ϕ ◦ ψ) (with Φ as in Theorem 4.5) deﬁnes a continuous unital homomorphism which is easily seen to extend the analytic functional calculus for τ (Q) + σ1 and which satisﬁes ran ΦQ (ϕ) ∈ D(τ (Q) + σ1 ) and (τ (Q) + σ1 )ΦQ (ϕ) = ΦQ (id · ϕ) for all ϕ ∈ E(M ) O(σ(τ (Q)) \ S1 , S1 ) having compact support contained in C. Let now ˆ such that S ⊂ U0 and S ∩ Uj = ∅ U0 , . . . , Um be an arbitrary open covering of C for j = 1, . . . , m. As the set CP (Q) ∪ Λ(Q) ∪ {∞} is totally disconnected and S1 \ (S ∪ CP (Q) ∪ Λ(Q) ∪ {∞}) consists of isolated eigenvalues of τ (Q) + σ1 , there exist ϕ0 , . . . , ϕm ∈ E(M ) O(σ(τ (Q)) \ S1 , S1 ) with ϕ0 + · · · + ϕm ≡ 1 and suppϕj ⊂ Uj for j = 0, . . . , m. It follows that Lp (R, CN ) = E(suppϕ0 ) + · · · + E(suppϕm ), where # E(suppϕj ) := {ker ΦQ (ψ) ; suppψ ∩ suppϕj = ∅}. Direct computation shows that the spaces E(suppϕj ) are invariant for τ (Q) + σ1 and satisfy σ((τ (Q) + σ1 )|E(suppϕj ) ) ⊆ suppϕj ⊂ Uj . Hence, τ (Q) + σ1 is Sdecomposable. p0 Fα 1 is Note that S will be a closed bounded set, by Theorem 3.5, if ωβ−1 suﬃciently small. We now show how exponential decay conditions imply better estimates for the resolvents and lead to richer functional calculi. As before, Q will be a regular matrix polynomial of degree nhaving type T (Q) = (α, β) with 0 ≤ α < n. We consider α perturbations by σ = j=0 Fj Dj , where the coeﬃcients Fj ∈ Lploc (R, MN ×N (C)) satisfy (3.2) for j = 0, . . . , α and, for some ε > 0, we have ∞ Fj (t)eε|t| dt < ∞ , j = 0, . . . , α. (4.9) −∞

In this situation, the entries of the matrix functions Φk and the scalar functions ψk in (3.4) satisfy the conditions for a and b in Lemma 3.3. Let Ω be any connected component of C \ Σ(Q) such that, for some λ0 ∈ Ω, the opera˜ 0 , τ (Q))A is invertible. Then, at points of ∂Ω \ Λ(Q), by the proof tor 1 − B R(λ ˜ τ (Q))A has locally of Theorem 3.5 and Lemma 3.3, the function λ → B R(λ, an analytic extension across the boundary of Ω with the extension taking values in the compact operators. Thus, the meromorphic operator-valued function ˜ τ (Q))A)−1 has locally meromorphic extensions across λ → V (λ) := (1 − B R(λ, the boundary of Ω and its singularities are discrete in Ω \ Λ(Q). Let now H be a

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compact subset of ∂Ω \ Λ(Q). For points λ ∈ Ω which are not a singularity of V (·) we have, by (1.1), that ˜ τ (Q))A · V (λ) · BR(λ, τ (Q)) . R(λ, τ (Q) + σ1 ) ≤ R(λ, τ (Q)) + R(λ, By Lemma 3.4 there exists a neighbourhood W of H such that the estimates (3.5) and (3.6) hold. By means of these estimates we obtain via Lemma 2.13 the estimate C R(λ, τ (Q) + σ1 ) ≤ , λ ∈ Ω ∩ W, dist(λ, Σ(Q))η+kH where η = max{N, 2β − 1}, C does not depend on λ, and kH is the maximal order of the ﬁnitely many poles of V (·) in H. Hence, we obtain the following result. Theorem 4.7. Let p ∈ (1, ∞) and Q have type T (Q) = (α, β) with 0 ≤ α ≤ n − 1. For k = 0, . . . , α let Fk be functions in Lploc (R, MN ×N (C)) satisfying (3.2) and α (4.9). Let σ be the formal diﬀerential operator σ = k=0 Fk Dk and σ1 be deﬁned analogously as for (3.1). Suppose that Ω ∩ ρ(τ (Q) + σ1 ) = ∅ for each component Ω of C \ Σ(Q). Then (C \ Σ(Q)) ∩ σ(τ (Q) + σ1 ) is countable and all accumulation points belong to Λ(Q) ∩ Σ(Q). For each compact set K ⊂ Σ(Q) \ Λ(Q) there exists a bounded neighbourhood W of K such that, for all λ ∈ W \ Σ(Q), the estimate C R(λ, τ (Q) + σ1 ) ≤ dist(λ, Σ(Q))η+kK holds, where η = max{N, 2β − 1}, C does not depend on λ, and kK is the maximal ˜ τ (Q))A)−1 in K order of the ﬁnitely many poles of (1 − B R(·, In particular, the set Kσ := σ(τ (Q) + σ1 ) ∩ (C \ Σ(Q)) will necessarily be totally disconnected. Corollary 4.8. In the setting of the preceding Theorem, ﬁx λ0 ∈ C \ (Σ(Q) ∪ Kσ ) so ˆ →C ˆ be the map that T := (λ0 − τ (Q) − σ1 )−1 exists in L(Lp (R, CN )). Let ψ : C given by ψ(z) := λ0 − 1/z for 0 = z ∈ C and ψ(∞) := λ0 , ψ(0) := ∞. The set M := σ(T ) \ ψ −1 (Kσ ∪ S(Q)) is a totally real, real analytic manifold. For each compact subset K ⊂ M there exists a neighbourhood W of K, a constant C > 0 and ηK ∈ N, such that C R(z, T ) ≤ , z ∈W \M. (4.10) dist(z, ψ(Σ(Q)))ηK The proof is analogous to that of Corollary 3.7. Notice, that C∞ O(M, ψ −1 (Kσ ∪ S(Q))) is an admissible algebra in the sense of [7]. Hence, we obtain the following result from (4.10) by means of Satz 6.3 in [8]. Corollary 4.9. In the setting of the preceding Corollary the analytic functional calculus for the operator T = R(λ0 , τ (Q)+σ1 ) has a unique extension to a continuous homomorphism Φ : C∞ O(M, ψ −1 (Kσ ∪ S(Q))) → L(Lp (R, CN )). In particular, T is C∞ O(M, ψ −1 (Kσ ∪ S(Q)))-scalar and thus decomposable.

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Corollary 4.10. In the setting of Theorem 4.7 and with the notation of the preceding Corollaries, there exists a continuous unital homomorphism ΦQ : C∞ O(ψ(M ), Kσ ∪ S(Q)) → L(Lp (R, CN )) extending the analytic functional calculus for the closed operator τ (Q) + σ1 such that, for all ϕ ∈ C∞ O(ψ(M ), Kσ ∪ S(Q)) having compact support contained in C, the range of ΦQ (ϕ) is contained in D(τ (Q)+σ1 ) and (τ (Q)+σ1 )ΦQ (ϕ) = ΦQ (id·ϕ). In particular, τ (Q) + σ1 is decomposable. Proof. As in earlier situations, ϕ → ΦQ (ϕ) := Φ(ϕ ◦ ψ) deﬁnes a continuous homomorphism with the desired properties. The proof proceeds as for Corollary 3.9 and Corollary 4.6. Notice, in the present situation, that C∞ O(ψ(M ), Kσ ∪ S(Q)) is quasi-admissible in the sense of [33], Deﬁnition IV.9.2. Example 4.11. Let A ∈ MN ×N (C) be invertible, n ∈ N and let F0 , . . . , Fn−1 be measurable functions satisfying (3.2) and (4.9). If Fn−1 1 is suﬃciently small, then the linear diﬀerential operator L := ADn +

n−1

Fk Dk ,

k=0

with domain W

p,n

(R, C ), is decomposable with N

σ(L) = {κxn ; κ ∈ σ(A), x ∈ R} ∪ S , where S is a null-sequence or ﬁnite point set in C. To see this, note that Q(x) = Axn is regular of type T (Q) = (n − 1, 1) with Λ(Q) = CP (Q) = {0} and S(Q) = {0, ∞}. The statement then follows from Theorem 3.5 and Corollary 4.10 together with Remark 3.6. For the special case that n = p = 2, A = 1 and F1 ≡ 0, see also [31] for further information.

References [1] E. Albrecht, E., Funktionalkalk¨ ule in mehreren Ver¨ anderlichen f¨ ur stetige lineare Operatoren auf Banachr¨ aumen, Manuscripta Math. 14 (1974) 1–40. [2] E. Albrecht, W.J. Ricker, Local spectral properties of certain matrix diﬀerential operators in Lp (RN )m , J. Operator Theory 35 (1996) 3–37. [3] E. Albrecht, W.J. Ricker, Local spectral theory for operators with thin spectrum, in: Spectral Analysis and its Applications, A. Gheondea and M. S ¸ abac (Eds.), pp. 1–25, Theta Foundation, Bucharest, 2003. [4] E. Balslev, T.W. Gamelin, The essential spectrum of a class of ordinary diﬀerential operators, Paciﬁc J. Math. 14 (1964) 755–776. [5] V.O. Blaˇsˇcak, On an expansion in terms of the principal functions of a nonselfadjoint second-order diﬀerential operator on the whole axis, (Ukrainian), Dopov¯ıd¯ı Akad. Nauk Ukra¨ın. RSR, (1965) 416–419.

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[6] V.O. Blaˇsˇcak, On a diﬀerential operator of the second order on the whole axis with spectral singularities, (Ukrainian), Dopov¯ıd¯ı Akad. Nauk Ukra¨ın. RSR (1966) 38–41. [7] I. Colojoar˘ a, C. Foia¸s, Theory of Generalized Spectral Operators, Gordon and Breach, New York–London–Paris, 1968. [8] B. Droste, Fortsetzung des holomorphen Funktionalkalk¨ uls auf Algebren mit Zerlegung der Eins, Ph.D. Dissertation, Mainz, 1980. [9] B. Droste, Holomorphic approximation of ultradiﬀerentiable functions, Math. Ann. 257 (1981) 293–316. [10] B. Droste, Extensions of analytic functional calculus mappings and duality by ∂closed forms with growth, Math. Ann. 261 (1982) 185–200. [11] R.E. Edwards, G.I. Gaudry, Littlewood–Paley and Multiplier Theory, SpringerVerlag, Berlin–Heidelberg–New York, 1977. [12] G. Fischer, Ebene algebraische Kurven, Vieweg-Verlag, Braunschweig–Wiesbaden, 1994. [13] C. Foia¸s, Spectral maximal spaces and and decomposable operators in Banach spaces, Arch. Math. (Basel) 14 (1963) 341–349. [14] G.B. Folland, Spectral analysis of a non-selfadjoint diﬀerential operator, J. Diﬀerential Equations 39 (1981) 151–185. [15] M.M. Gehtman, On the spectrum of a non-selfadjoint diﬀerential operator of even order. (Russian) Dokl. Akad. Nauk SSSR 150 (1963) 726–729. [16] S. Goldberg, Unbounded Linear Operators, Dover, New York, 1985. [17] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators I, SpringerVerlag, New York–Berlin–Heidelberg–Tokyo, 1983. [18] G.A. Huige, The theory of some non-selfadjoint diﬀerential operators, Comm. Pure Appl. Math. 21 (1968) 25–49. [19] G.A. Huige., Perturbation theory of some spectral operators, Comm. Pure Appl. Math. 24 (1971) 741–757. [20] G.A. Huige, The spectral resolution of some non-selfadjoint partial diﬀerential operators, Canad. J. Math. 27 (1975) 1316–1322. [21] K. J¨ orgens, K., Lineare Integraloperatoren, Teubner-Verlag, Stuttgart, 1970. [22] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York– Berlin–Heidelberg, 1976. [23] R.R.D. Kemp, A singular boundary value problem for a non-selfadjoint diﬀerential operator, Canad. J. Math. 10 (1958) 447–462. [24] Yu.I. Ljubiˇc, V.I. Macaev, On operators with a separable spectrum (in Russian), Mat. Sbornik, 56 (1962), 433–468, English translation: Amer. Math. Soc. Transl. (2), 47 (1965) 89–129. [25] R.M. Martirosjan, On the spectrum of certain non-selfadjoint operators. (Russian. Armenian summary), Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963) 677–700, English translation: Amer. Math. Soc. Transl. (2) 53 (1966), 609–639. [26] A. Pietsch, A., Nuclear Locally Convex Spaces, Springer-Verlag, Berlin–Heidelberg– New York, 1972. [27] G.C. Rota, Extension theory of diﬀerential operators I, Comm. Pure Appl. Math. 11 (1958) 23–65.

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[28] M. Schechter, Spectra of Partial Diﬀerential Operators, North-Holland, Amsterdam– New York–Oxford, Second edition, 1986. [29] J.T. Schwartz, Some non-selfadjoint operators I, Comm. Pure Appl. Math. 13 (1960) 609–636. [30] H.J. Sussmann, Generalized spectral theory and second order ordinary diﬀerential operators, Canad. J. Math. 25 (1973) 178–193. [31] I.-P.P. Syroid, Suﬃcient conditions for the absence of spectral singularities in a nonselfadjoint Schr¨ odinger operator in terms of potential. (Russian) Sibirsk. Mat. Zh. 22 (1981) 151–157, 230. English translation: Siberian Math. J. 22 (1981) 111–116 [32] G.B. Tunca, E. Bairamov, Discrete spectrum and principal functions of nonselfadjoint operators, Czech. Math. J. 49(124) (1999) 689–700. [33] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publ. Comp., Dordrecht, 1982. [34] B.L. van der Waerden, Modern Algebra, Vol. I, Frederick Ungar Publ. Comp., New York, Third printing, 1964. [35] A.C. Zaanen, Linear Analysis, North-Holland Publ. Comp., Amsterdam, Fourth Printing, 1964. E. Albrecht Fachrichtung 6.1 – Mathematik Universit¨ at des Saarlandes Postfach 15 11 50 D–66041 Saarbr¨ ucken Germany e-mail: [email protected] W. J. Ricker Mathematisch-Geographische Fakult¨ at Katholische Universit¨ at Eichst¨ att–Ingolstadt D–85072 Eichst¨ att Germany e-mail: [email protected] Submitted: August 21, 2007

Integr. equ. oper. theory 59 (2007), 491–500 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040491-10, published online October 18, 2007 DOI 10.1007/s00020-007-1533-x

Integral Equations and Operator Theory

The Strong Variant of a Barbashin Theorem on Stability of Solutions for Non-autonomous Diﬀerential Equations in Banach Spaces Constantin Bu¸se, Mihail Megan, Manuela-Suzy Prajea and Petre Preda Abstract. Among others we shall prove that an exponentially bounded evolution family U = {U (t, s)}t≥s≥0 of bounded linear operators acting on a Banach space X is uniformly exponentially stable if and only if there exists q ∈ [1, ∞) such that  q1  t sup  ||U (t, τ )∗ x∗ ||q dτ  = M (x∗ ) < ∞, ∀x∗ ∈ X ∗ . t≥0

0

This result seems to be new even in the ﬁnite dimensional case and it is the strong variant of an old result of E. A. Barbashin ([1]Theorem 5.1). Mathematics Subject Classiﬁcation (2000). Primary 47D06, Secondary 35B35. Keywords. Operator semigroups, evolution families of bounded linear operators, uniform exponential stability.

1. Introduction The well-known theorem of Datko says that for an exponentially bounded and strongly continuous evolution family U = {U (t, s)}t≥s≥0 of bounded linear operators acting on a Banach space X the following three statements are equivalent: (i) The family U is uniformly exponentially stable, that is, there exist constants ν > 0 and N > 0 such that ||U (t, s)|| ≤ N e−ν(t−s) ,

∀t ≥ s ≥ 0.

The first author was partially supported by the CNCSIS’s grant no. 546/2006.

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ii) Strong Datko’s Condition holds, that is, there exist p ≥ 1 and Mp > 0 such that ∞  p1 sup  ||U (t, s)x||p dt ≤ Mp ||x|| for all x ∈ X. (SDC) s≥0

s

(iii) Uniform Datko’s Condition holds, that is, the following inequality fulfils: ∞  p1 (U DC) sup  ||U (t, s)||p dt := Kp < ∞. s≥0

s

Even if the family U is not strongly continuous, the equivalence between (i) and (ii) can be stated under the more general assumption that for each x ∈ X the map t → ||U (t, s)x|| is measurable. The measurability of the map t → ||U (t, s)|| follows by the strong continuity of the family U. Moreover, either of the conditions (SDC) or (UDC) may be replaced by the following more general condition, originally given by S. Rolewicz. (iv) Strong Rolewicz’s Condition holds, i.e., there exists a nondecreasing function φ : R+ → R+ which φ(t) > 0 for all t > 0 such that ∞ φ(||U (t, s)x||)dt < ∞,

sup s≥0

for all x ∈ X with ||x|| ≤ 1.

s

Particularly this shows that (ii) and (iii) can be reformulated with p > 0 instead of p ≥ 1. For more details, proofs and other formulations of this type of theorems we refer to [5], [6], [11], [13], [15],[4], [3] and references therein. On the other hand, a reformulation of an old result of E.A. Barbashin ([[1], Theorem 5.1], [10]) reads as follows: Let U = {U (t, s) : t ≥ s ≥ 0} be an exponentially bounded evolution family of bounded linear operators acting on a Banach space X such that for each t > 0 the map s → ||U (t, s)|| : [0, t] → R+ is measurable. The following statements are equivalent: (i) The family U is uniformly exponentially stable. (ii) Uniform Barbashin’s Condition holds, that is, there exists 1 ≤ p < ∞ such that  t  p1 sup  ||U (t, s)||p ds < ∞. t≥0

0

A Rolewicz’s variant of a similar uniform result for families on the entire real line, can be found in [[3], Theorem 4.1].

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It is natural to ask what is the variant of the Strong Barbashin Condition equivalent with the uniform exponential stability property of the evolution family U? In this paper we shall give an answer to this question.

2. Notation and preliminary results Let X be a real or complex Banach space and X ∗ its dual space. By B(X) will denote the Banach algebra of all linear and bounded operators acting on X. The norms on X, X ∗ and B(X) shall be denoted by the symbol || · ||. Let R+ := [0, ∞) and J either of R or R+ . By ∆J will denote the set of all pairs (t, s) ∈ J × J with t ≥ s, and let ∆∗J := ∆J \ {(t, t) : t ∈ J}. By evolution family of bounded linear operators acting on X will mean a family U = {U (t, s) : (t, s) ∈ ∆J } ⊂ B(X) which veriﬁes the following two conditions: 1. U (t, t) = I-the identity of B(X)- for all t ∈ J, and 2. U (t, s)U (s, r) = U (t, r) for all t, s, r ∈ J with t ≥ s ≥ r. An evolution family is called strongly continuous if for each x ∈ X the maps τ → U (τ, s)x : [s, t] → X and s → U (t, s)x : [s, t] → X are continuous for any pair (t, s) ∈ ∆J . An evolution family is exponentially bounded if there exist ω ∈ R and Mω ≥ 1 such that ||U (t, s)|| ≤ Mω eω(t−s) for all (t, s) ∈ ∆J .

(2.1)

If the evolution family U is exponentially bounded, then we may choose a positive ω such that (2.1) holds. An evolution family is uniformly exponentially stable if there exists a negative ω such that the relation (2.1) is fulﬁlled. If an evolution family U satisﬁes the convolution condition U (t, s) = U (t − s, 0) for all (t, s) ∈ ∆J ,

(2.2)

then the one parameter evolution family T := {U (t, 0), t ≥ 0} is a semigroup of operators on X. If a semigroup T is strongly continuous, then it is exponentially bounded. The converse statement is not true. It is known that every strongly measurable semigroup is strongly continuous on (0, ∞), but the measurability property does not imply the continuity of semigroup at the origin. However, the norm of each trajectory of an exponentially bounded semigroup are measurable such as been stated in [15]. Precisely, we have: Proposition 1. If a one parameter semigroup {T (t)}t≥0 is exponentially bounded, then for each x ∈ X, the map t → ||T (t)x|| is measurable. Let {T (t)}t≥0 be a strongly continuous one parameter semigroup on a Banach space X and T∗ = {T (t)∗ }t≥0 the associated one parameter dual semigroup on X ∗ . It is known that the dual semigroup T∗ may be not strongly continuous but it is exponentially bounded because ||T (t)|| = ||T (t)∗ || for all t ≥ 0. Then as been stated in the above Proposition 1, for each x∗ ∈ X ∗ the map t → ||T (t)∗ x∗ ||

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is measurable. At this moment we do not know if a similar result for evolution families holds true. Throughout in as follows we shall suppose that for each x ∈ X each x∗ ∈ X ∗ and each (t, s) ∈ ∆J the maps τ → ||U (t, τ )∗ x∗ || : [s, t] → R+ and r → ||U (r, s)x|| : [s, t] → R+ are measurable.

3. Uniform Boundedness Let p ∈ (1, ∞) and q ∈ (1, ∞) such that p1 + 1q = 1. We are in the position to state our ﬁrst result on uniform boundedness of evolution families. Theorem 1. Let U = {U (t, s) : (t, s) ∈ ∆J } be an evolution families which verifies the measurability conditions stated above. The following two conditions are equivalent: (i) The evolution family is uniformly bounded, that is, sup (t,s)∈∆J

||U (t, s)|| = MJ < ∞.

(3.1)

(ii) There exist the positive constants Mp = Mp (J) and Mq∗ = Mq∗ (J) such that   q1 t 1 ||U (t, τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||, ∀x∗ ∈ X ∗ (3.2) sup  t − s (t,s)∈∆∗ J s

and

 sup

(t,s)∈∆∗ J

 1 t−s

t

 p1 ||U (τ, s)x||p dτ  ≤ Mp ||x||,

∀x ∈ X.

(3.3)

s

Proof. The implication (i) ⇒ (ii) is trivial because ||U (t, s)|| = ||U (t, s)∗ ||. In this case can choose Mp = Mq∗ = MJ . Now we are stating the proof of (ii) ⇒ (i). Let x ∈ X, x∗ ∈ X ∗ and (t, s) ∈ ∆∗J . Using the H¨older inequality we get: t

∗

(t − s)| x , U (t, s)x| =  1 ≤ t−s

| x∗ , U (t, τ )U (τ, s)x|dτ

s

t s

 1q 

1 ||U (t, τ )∗ x∗ ||q dτ  ·  t−s

≤ (t − s)Mp Mq∗ ||x|| · ||x∗ ||.

t

 p1 ||U (τ, s)x||p dτ  (t − s)

s

Finally we get: sup (t,s)∈∆J

||U (t, s)|| =

sup

sup

(t,s)∈∆J ||x||≤1,||x∗||≤1

that is, (3.1) holds true.

| x∗ , U (t, s)x| ≤ Mp Mq∗ := MJ < ∞,

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Corollary 1. Let T = {T (t)}t≥0 be an exponentially bounded one parameter semigroup on a Banach space X. The following two statements are equivalent: (i) The semigroup T is uniformly bounded, that is, sup ||T (t)|| = M < ∞. t≥0

(ii) There exist the positive constants Mp and Mq∗ such that  sup  t>0

and

 1 sup  t t>0

1 t

t

t

 p1 ||T (τ )x||p dτ  ≤ Mp ||x||,

∀x ∈ X

(3.4)

∀x∗ ∈ X ∗ .

(3.5)

0

 1q ||T (τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||,

0

Proof. The measurability of the maps τ → ||T (τ )x|| and τ → ||T (τ )∗ x∗ || follows by the fact that the semigroup T is exponentially bounded as been stated in the Proposition 1. See also the comments after its proof. The inequalities (3.4) and (3.5) can be easily obtained from (3.2) and respectively from (3.3) by making changes of variables in the integrals and using the convolution condition (2.2). Remark 1. The result from Corollary 1 in the particular case of strongly continuous semigroups was announced earlier by Hans Zwart ([17]), see also the Section 8 of the paper by Guo and Zwart, ([7]). We already know that if a semigroup T is exponentially bounded, then for each x ∈ X the map t → ||T (t)x|| is measurable. However this fact does not imply the measurability of the function t → ||T (t)|| and it is not clear if the measurability of this function implies the exponential boundedness of the semigroup T. This latter statement is true and well-known for groups of operators. See [8], Theorem 7.4.1, page 241. It is important to note that if X has a countable dense set V and the semigroup T is exponentially bounded, then the function t → ||T (t)|| is measurable, since ||T (t)|| = supx∈V ||T (t)x|| that the supremum of countable many measurable functions is measurable. Corollary 2. A one parameter semigroup {T (t)} on a Banach space X for which the map t → ||T (t)|| is measurable, is uniformly bounded if and only if 1 sup t>0 t

t 0

||T (τ )||2 dτ = M < ∞.

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Proof. As in the proof of the Theorem 1 one has: ||T (t)|| = ≤

1 t

t

1 t

t

||T (t − τ )||||T (τ )||dτ 12 t t 1 2 2 ||T (t − τ )|| dτ ||T (τ )|| dτ ≤ M. t 0

||T (t)||dτ ≤

0

0

0

Hence the semigroup T is uniformly bounded. Of course, one get the same result if assume that supt>0 and supt>0

1 t

t 0

||T (τ )||q dτ < ∞ with

1 p

+

1 q

1 t

t 0

||T (τ )||p dτ < ∞

= 1.

The implication (ii)⇒(i) from Corollary 1 cannot be preserved if either one of the relations (3.4) or (3.5) is removed such that the following example from [14] shows. 1 Let X = L2 (R) and g : R → R+ the function given by g(s) = (1 + |s|) 4 . For each t ≥ 0 and each f ∈ X let us consider the bounded linear operator T (t) deﬁned by: g(t + s) f (t + s), s ∈ R, t ≥ 0. (T (t)f )(s) = g(s) It is easily to check that the family T = {T (t)}t≥0 is a strongly continuous semi1 group of bounded linear operators on X and ||T (t)|| = (1 + t) 4 , that is, T is not uniformly stable. However, the relation (3.4), with p = 2, is veriﬁed by T.

4. Uniform exponential stability The result of this section reads as follows: Theorem 2. Let U = {U (t, s) : (t, s) ∈ ∆J } be an exponentially bounded evolution family on a Banach space X which verifies the measurability conditions stated in the end of the second section of our note. The following five statements are equivalent: (i) The family U is uniformly exponentially stable. (ii) There exists a function g : R+ → R+ such that inf g(t) < 1 and ||U (t, s)|| ≤ g(t − s), for all t ≥ s ∈ J.

t>0

(iii) There exist p, q ∈ (1, ∞) with p1 + 1q = 1 and the positive constants Mp = Mp (J) and Mq∗ = Mq∗ (J) such that  t  q1 ||U (t, τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||, ∀x∗ ∈ X ∗ if J = R (4.1) sup  t∈R

−∞

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or  sup  t≥0

t

 1q ||U (t, τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||,

∀x∗ ∈ X ∗ if J = R+

(4.2)

0

and  1 sup  t−s (t,s)∈∆∗ J

t

 p1 ||U (τ, s)x||p dτ  ≤ Mp ||x||,

s

(iv) There exist p, q ∈ (1, ∞) with Kq∗ such that

sup (t,s)∈∆J

∀x ∈ X.

1 p

+

1 q

= 1 and the positive constants Kp and

 t  p1  ||U (τ, s)x||p dτ  ≤ Kp ||x||,

∀x ∈ X

(4.3)

s

and  sup

(t,s)∈∆∗ J

 1 t−s

t

 q1 ||U (t, τ )∗ x∗ ||q dτ  ≤ Kq∗ ||x∗ ||,

∀x∗ ∈ X ∗ .

(4.4)

s

(v) There exist q ∈ [1, ∞) and a positive constant L∗q such that  sup 

t∈R

t

 1q ||U (t, τ )∗ x∗ ||q dτ  ≤ L∗q ||x∗ ||,

∀x∗ ∈ X ∗ if J = R

−∞

or  t  q1 sup  ||U (t, τ )∗ x∗ ||q dτ  ≤ L∗q ||x∗ ||, t≥0

∀x∗ ∈ X ∗ if J = R+ .

0

Proof. The equivalence between (i) and (ii) has been established in [[2], Lemma 4]. The proof of the implications (i)⇒(iii), (iv), (v) is trivial and the proof of the rest of implications is based on (ii)⇒(i). In order to prove the implication (iii)⇒(ii) we remark ﬁrst that if (4.1) or (4.2) are fulﬁlled, then the inequality  1 sup  t−s (t,s)∈∆∗ J

t s

 q1 ||U (t, τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||,

∀x∗ ∈ X ∗

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holds true as well. Indeed if t − s ≥ 1, then   1q  t  1q t  1 ||U (t, τ )∗ x∗ ||q dτ  ≤  ||U (t, τ )∗ x∗ ||q dτ  t−s s



 q1

t

≤

s

||U (t, τ )∗ x∗ ||q dτ  ≤ Mq∗ ||x∗ ||,

∀x∗ ∈ X ∗ .

−∞ or 0 If 0 < t − s < 1, we have that:   1q   1q t t 1  1 ||U (t, τ )∗ x∗ ||q )dτ  ≤  Mωq eωq(t−τ ) ||x∗ ||q dτ  ≤ Mω eω ||x∗ ||. t−s t−s s

s

Here ω has been considered to be positive. Thus as stated in the Theorem 1, the evolution family U is uniformly bounded, that is, there exists a positive constant N1 such that ||U (t, s)|| ≤ N1 for all (t, s) ∈ ∆J . (4.5) ∗ On the other hand, for each (t, s) ∈ ∆J we have that: t

∗

(t − s)| x , U (t, s)x| =

| x∗ , U (t, τ )U (τ, s)x|dτ

s

 t  1q  t  p1 ≤  ||U (t, τ )∗ x∗ ||q dτ   ||U (τ, s)x||p dτ   ≤

s

1 t−s

s

t

 p1  t  1q 1 ||U (τ, s)x||p dτ   ||U (t, τ )∗ x∗ ||q dτ  (t − s) p

s

 1 p

≤ Mp ||x||(t − s) 

s

t

−∞

≤

Mp Mq∗ ||x||

or

 1q

||U (t, τ )∗ x∗ ||q dτ  0 1

∗

· ||x ||(t − s) p .

As a consequence there exists a positive constant N2 such that 1

(t − s) q ||U (t, s)|| ≤ N2 ,

∀(t, s) ∈ ∆J .

(4.6)

From (4.5) and (4.6) shall obtain the following estimation for the norm of U (t, s) : N1 + N2 ||U (t, s)|| ≤ 1 . 1 + (t − s) q (iv)⇒(ii). Clearly ||U (t, s)∗ x∗ || ≤ Mω eω(t−s) ||x∗ || for all (t, s) ∈ ∆J and all x ∈ X ∗ . Therefore (3.3) is a consequence of (4.3). From Theorem 1 follows that ∗

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the evolution family U is uniformly bounded, i.e., (3.1) holds true. In order to recapture an inequality of type (4.6) remarking that:  t  1q  t  p1 (t − s)| x∗ , U (t, s)x| ≤  ||U (t, τ )∗ x∗ ||q dτ   ||U (τ, s)x||p dτ  s

s

 t  p1   1q t 1 1 ≤  ||U (τ, s)x||p dτ   ||U (t, τ )∗ x∗ ||q dτ  (t − s) q t−s s

≤

s

Mp Mq∗ ||x||

1 q

∗

· ||x ||(t − s) .

Then there exists a positive constant N3 such that 1

(t − s) p ||U (t, s)|| ≤ N3 ,

∀(t, s) ∈ ∆J .

(v)⇒(i). Let s ∈ J be ﬁxed. For t ≥ s + 1 we have that 1 0

Mω−q e−ωqu du| x∗ , U (t, s)x|q 

≤  ≤

t

 ||U (t, τ )∗ x∗ ||q · ||U (τ, s)||q Mω−q e−ω(τ −s)q dτ  ||x||q

s

t

 ||U (t, τ )∗ x∗ ||q dτ  · ||x||q ≤ (Mq∗ )q ||x∗ ||q ||x||q

s

while that for s ≤ t < s + 1 we get that ||U (t, s|| ≤ Mω eω by the uniform boundedness. Thus the family U is uniformly bounded, and have ﬁnished the proof in the case q ∈ (1, ∞). For q = 1 and J = R+ we have that: ∗

t

(t − s)| x , U (t, s)x| ≤ MR+

||U (t, τ )∗ x∗ ||dτ ||x|| for all (t, s) ∈ ∆R+ .

0

A similar estimation can be easily obtained in the case J = R.

Acknowledgement 1. The authors would like to express their gratitude to the anonymous referee for his/her helpful remarks and suggestions.

References [1] E. A. Barbashin, Introduction in the theory of stability Izd. Nauka, Moscow, 1967(Russian). [2] C. Bu¸se, On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces, New Zealand Journal of Mathematics, Volume 27(1998), pp.183-190.

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[3] C. Bu¸se, S.S. Dragomir, A theorem of Rolewicz’s type in solid function spaces, Glasgow Math. Journal, 44(2002), 125-135. [4] C. Chicone, Yuri Latushkin, Evolution Semigrous in Dynamical Systems and Diﬀerential Equations, Amer. Math. Soc., Math. Surv. and Monographs, 70, 1999. [5] Y. L. Daletkii, M. G. Krein, Stability of Solutions of Diﬀerential Equations in Banach Spaces, Izd. Nauka, Moskwa, 1970. [6] R. Datko, Uniform asymptotic stability of evolutionary processes in Banach space, SIAM J. Math. Analysis 3 (1973), 428-445. [7] B. Z. Guo and Hans Zwart, On the Relation between Stability of Continuous- and Discrete- Time Evolution equations via the Cayley Transform, Integr. equ. oper. theory, 54(2006), 349-383. [8] E. Hille and R. S. Philips, Functional Analisis and semi-groups, Amer. Math. Soc., Providence, 1957. [9] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diﬀerential Eq. 1(1983), 43-56. [10] M. Megan, Proprietes qualitatives des systems lineaires controles dans les espaces de dimension inﬁnie, Monographies Mathematiques (Timisoara, 1988). [11] A. Pazy, Semigroups of linear operators and applications to partial diﬀerential equations, (Springer-Verlag, 1983). [12] A. J. Pritchard, J. Zabczyk, Stability and stabilizability of inﬁnite dimensional systems, SIAM Rev.23(1983), 25-52. [13] S. Rolewicz, On uniform N-equistability, Journal of Math. Anal. Appl. 115 (1986), 434-441. [14] J. A. van Casteren, Operators similar to unitary or selfadjoint ones, Paciﬁc Journal of Mathematics, 104, 1983, pp.241-255. [15] J. M. A. M. van Neerven, Exponential stability of operators and operator semigroups, Journal of Functional Analysis, Volume 130, no. 2(1995), 293-309. [16] G. Weiss, Weak Lp - stability of linear semigroup on a Hilbert space implies exponential stability, J. Diﬀ. Eqns., 76, (1988), 269-285. [17] H. Zwart, Boundedness and Strongly Stability of C0 -semigroups on a Banach Spaces, Ulmer Seminare, 2003. Constantin Bu¸se, Mihail Megan and Petre Preda West University of Timisoara, Department of Mathematics, Bd. V. Parvan No. 4 300223-Timisoara, Romania e-mail: [email protected] [email protected] [email protected] Manuela-Suzy Prajea National College “Traian”, Drobeta Turnu-Severin, Bd. Carol I, No. 6, Romania e-mail: [email protected] Submitted: February 3, 2007 Revised: August 24, 2007

Integr. equ. oper. theory 59 (2007), 501–521 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040501-21, published online October 18, 2007 DOI 10.1007/s00020-007-1526-9

Integral Equations and Operator Theory

Toeplitz Operators and Herz Spaces on the Half-Space Boo Rim Choe and Kyesook Nam Dedicated to Professor Yoshihiro Mizuta on the occasion of his sixtieth birthday

Abstract. On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. Mathematics Subject Classification (2000). Primary 47B35; Secondary 46E30. Keywords. Toeplitz operator, Schatten-Herz class, Harmonic Bergman space, half-space.

1. Introduction For a positive integer n > 1, let H = Rn−1 × R+ be the upper half-space where R+ denotes the set of all positive real numbers. We will write point z ∈ H as z = (z , zn ) where z ∈ Rn−1 and zn > 0. Let V be the volume measure on H. Throughout the paper we write dw = dV (w) for simplicity. Also, we let Lp = Lp (V ) for 1 ≤ p ≤ ∞.

The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).

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The harmonic Bergman space b2 is the space of all complex-valued harmonic functions u on H satisfying 1/2 2 2 |f | dV < ∞. ||f ||b = H

The space b2 is a closed subspace of L2 and hence is a Hilbert space. Basic structures of general Lp -harmonic Bergman spaces are studied in [9]. For more general Banach space of harmonic functions on the half-spaces, see [10], [11] and references therein. By the mean value property of harmonic functions it is easily seen that each point evaluation is a bounded linear functional on b2 . Thus, to each z ∈ H, there corresponds a unique function Rz = R(z, ·) that has the reproducing property: f (z) = f Rz dV (1.1) H

for all f ∈ b2 . As is well known, the explicit formula of the kernel R(z, w) is given by R(z, w) =

4 n(zn + wn )2 − |z − w|2 , nωn |z − w|n+2

z, w ∈ H

(1.2)

where ωn is the volume measure of the unit ball of Rn and w = (w , −wn ). See [1] for more information and related facts. Note that the kernel is real. So, the complex conjugation in (1.1) can be removed. Let R be the Hilbert space orthogonal projection from L2 onto b2 . Then the reproducing property (1.1) leads us to the integral realization of the projection R as follows: ψ(w)R(z, w) dw, z∈H Rψ(z) = H

for functions ψ ∈ L2 . From the explicit formula (1.2) of R(z, w), it is easily seen that C |R(z, w)| ≤ , z, w ∈ H |z − w|n

(1.3)

for some constant C = C(n). So, Rz ∈ Lp for each 1 < p ≤ ∞ and thus we have R extends to an integral operator from Lp , 1 ≤ p < ∞, into the space all harmonic functions on H. One can easily see from (1.3) that R extends even to a class of positive Borel measures µ (we simply write µ ≥ 0) satisfying (1 + |z|n)−1 ∈ L1 (µ). Namely, for such µ, the integral R(z, w) dµ(w), z∈H Rµ(z) = H

deﬁnes a function harmonic on H. Let M+ be the set of all µ ≥ 0 such that (1 + |z|β )−1 ∈ L1 (µ) for some β > 0. For µ ∈ M+ , the Toeplitz operator with symbol µ is deﬁned by Tµ f = R(f dµ)

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for f ∈ b2 . In case dµ = ϕdV , we write Tϕ = Tµ . Let M be the set of all “formal” ﬁnite linear combinations of measures in M+ . For a measurable function ϕ on H, we say that ϕ ∈ M if ϕdV ∈ M. The class M turns out to contain most of interesting symbols: see [3] and (3.3) below. We extend the notion of Tµ for µ ∈ M by linearity. Note that Tϕ is clearly bounded on b2 for ϕ ∈ L∞ . In general, Tµ may not even be deﬁned on all of b2 , but it is always densely deﬁned by the fact ([9]) that, for each β > n, harmonic functions f on H such that f (z) = O (1 + |z|β )−1 form a dense subset of b2 . For positive Toeplitz operators on harmonic Bergman spaces, basic operator theoretic properties such as boundedness, compactness and the membership in the Schatten classes have been studied on various settings. See [3], [4], [5] and [8]. Motivated by an earlier work [7] in the holomorphic case on the unit disk, Choe et al. [2] have recently studied another aspect of positive Toeplitz operators on the ball. Roughly speaking, they decomposed a given operator into a family of local operators, introduced mixed norm spaces associated with Schatten classes and then gave characterizations of membership in those spaces in terms of the socalled Herz spaces. Compact Toeplitz operators with Herz symbols are also studied in [2]. Our contribution in this paper is to obtain results analogous to those in [2] on the unbounded domain H. While most of our result are parallel to those in [2], the unboundedness of our domain causes some diﬃculties and extra work. Nevertheless, our argument towards Theorem 4.4, which is one of our main results, much simpliﬁes that employed in [2] and some results even extend the corresponding results in [2]. For example, in Theorem 3.6 below, the range of parameters that ensures the boundedness of the Berezin transform now includes the case p = ∞, even with a much simpliﬁed proof. Also, we show some pathology caused by the unboundedness of H; see the example at the end of the paper. Our simpliﬁed argument has an advantage for a problem left open in [7] as follows. In [7] the holomorphic version (on the disk) of the Berezin transform characterization in Theorem 4.4 was proved only for the case p = 1 or p = q and the case q = p > 1 has been left open. Such remaining cases were ﬁrst settled in [2] for the harmonic Bergman space on the ball and one can modify the argument in [2] for the holomorphic version. However, the argument in [2] is rather complicated and the modiﬁcation seems to require some extra work. One can easily modify mutatis mutandis our argument in the present paper to get the holomorphic version on the disk (and probably on some other settings, too). In Section 2 we review some basic notions such as Berezin transform, averaging functions, Schatten class operators and Herz spaces. In Section 3 we establish boundedness of the Berezin transform on Herz spaces for a certain range of parameters, which will be the main tool in our argument later. In Section 4 we introduce the so-called Schatten-Herz classes of Toeplitz operators and obtain characterizations for a positive Toeplitz operator to be a member of such Schatten-Herz classes. Characterizations are given in terms of averaging functions and Berezin

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transforms of symbols. Finally, in Section 5, we make some observations concerning the boundedness and compactness of Toeplitz operators with Herz symbols. The authors thank Hyungwoon Koo for helpful discussions about the material in this paper. Notation. Throughout the paper we use the same letter C to denote various constants, often with parameters in a parenthesis indicating dependency, which may change at each occurrence. We will often abbreviate inessential constants involved in inequalities by writing A B for positive quantities A and B if the ratio A/B has a positive upper bound. We write A ≈ B if A B and A B. Also, for 1 ≤ p ≤ ∞, we use p to denote the conjugate exponent of p, i.e., 1/p + 1/p = 1.

2. Preliminaries In this section, we collect some basic notions and related facts which we need in later sections. The pseudohyperbolic distance between two points z, w ∈ H is deﬁned by ρ(z, w) =

|z − w| . |z − w|

This ρ is an actual distance. For z ∈ H and r ∈ (0, 1), let Er (z) denote the pseudohyperbolic ball of radius r centered at z. Then, by a straightforward calculation, we have 1 + r2 2r z, z z Er (z) = B , (2.1) n n 1 − r2 1 − r2 where B(a, t) denotes the euclidean ball of radius t centered at a. So, we have n 2r z . (2.2) |Er (z)| = ωn n 1 − r2 Here and elsewhere, the notation |E| stands for the volume V (E) of a Borel set E ⊂ H. Also, it is easily seen from (2.1) that 1+r 1−r zn < wn < zn , 1+r 1−r

w ∈ Er (z)

(2.3)

for each z ∈ H. We now recall the well-known notions of the averaging functions and the Berezin transform of µ ≥ 0 on H. Let r ∈ (0, 1). For z ∈ H, the averaging function µ r of µ is deﬁned by µ(Er (z)) µ r (z) = . |Er (z)| Also, the Berezin transform µ of µ is deﬁned by |Rz |2 µ (z) = 2 dµ. H Rz L2

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Also, for a measurable function ϕ on H (not necessarily positive), its averaging functions ϕ r and Berezin transform ϕ is deﬁned similarly, whenever they are well deﬁned. The following basic properties of the averaging functions and the Berezin transform are taken from [3]. By Proposition 3.2, Lemma 3.8, Lemma 3.10 of [3], we have the following pointwise inequalities. Proposition 2.1. Given δ, r ∈ (0, 1), there exists a constant C = C(r, δ) > 0 such

that µ r ≤ C ( µδ )r for µ ≥ 0. Proposition 2.2. Given r ∈ (0, 1), there exists a constant C = C(r) > 0 such that µ ≤ C ( µr ) for µ ≥ 0. Proposition 2.3. Given r ∈ (0, 1/6), there exists a constant C = C(r) > 0 such that µ r ≤ C µ for any µ ≥ 0. In what follows we let Vγ denote the weighted measure on H deﬁned by dVγ (z) = znγ dz. for γ real. The case γ = −n is of special interests, because dV−n (z) = cn R(z, z) dz where cn = n(n − 1)−1 ωn 2n−2 . For this reason we put λ = V−n . Also, we let ∞ for all γ. Lpγ = Lp (Vγ ). So, Lp = Lp0 and L∞ γ =L The following fact asserting that weighted Lp -norms of averaging functions are all equivalent is also a special case of Corollary 3.3 of [3]. Proposition 2.4. Let 1 ≤ p ≤ ∞, δ, r ∈ (0, 1) and α be real. Then µδ Lpα µr Lpα ≈ for µ ≥ 0. The constants suppressed above are independent of µ. We now turn to Schatten class operators. For a compact operator T on b2 , let {sm (T )} be the nonzero eigen values (listed by multiplicity) of |T | = (T ∗ T )1/2 arranged so that the sequence is non-increasing where T ∗ denotes the Hilbert space adjoint of T . For 1 ≤ p < ∞, we say that T is a Schatten p-class operator if 1/p

p |sm (T )| < ∞. T Sp = m

Let Sp be the space of all Schatten p-class operators on b2 . Then the space Sp is a Banach space with the above norm and is a two-sided ideal in S∞ , the space of all bounded linear operators on b2 . See [12], for example, for more information and related facts. For an operator T ∈ S∞ , we let T S∞ denote the operator norm T of T ∈ S∞ . The following characterization for a positive Toeplitz operator to be a member of the class Sp is taken from Theorem 6.4, Theorem 4.6 and Theorem 4.1 of [3]. Proposition 2.5. Assume 1 ≤ p ≤ ∞, r ∈ (0, 1) and µ ∈ M+ . Then the following conditions are equivalent.

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(a) Tµ ∈ Sp . (b) µ r ∈ Lp (λ). (c) µ ∈ Lp (λ). Moreover, the equivalences Tµ Sp ≈ µr Lp (λ) ≈ µLp (λ) hold. Now, we introduce the Herz spaces. We ﬁrst ﬁx some notation to be used for the rest of the paper. We decompose H into a family of horizontal stripes {Ak |k ∈ Z} where Ak = {z ∈ H| 2k−1 < zn ≤ 2k } and Z is the set all integers. Let χk = χAk for each k. Here and elsewhere, χE denotes the characteristic function of E. Also, given µ ∈ M, we let µχk stand for the restriction of µ to Ak for each k. For 1 ≤ p, q ≤ ∞ and α real, the Herz space Kqp,α consists of those functions ϕ ∈ Lploc (V ) for which kα p ϕKp,α = 2 ϕχ <∞ k L q q

where q = q (Z) denotes the two-sided sequence spaces. Equipped with this norm, the space Kqp,α is a Banach space. Also, we let K0p,α be the space of all functions p,α such that f ∈ K∞ kα 2 f χk Lp ∈ c0 where c0 denotes the subspace of ∞ consisting sequences vanishing at ±∞. Note that Kqp,α ⊂ K0p,α for all q < ∞. For more information on the Herz spaces, see [6] and references therein. Let 1 ≤ p < ∞ and α be real. Then, since zn ≈ 2k for z ∈ Ak and k ∈ Z, we have 2kαp |ϕ(z)|p dz ≈ |ϕ(z)|p znαp dz 2kαp ϕχk pLp = Ak

and thus

Ak

ϕKp,α = ϕχk Lpαp q

q

for 1 ≤ q ≤ ∞. In particular, we have

ϕKp,−n/p = ϕχk Lp (λ) q

q

and this estimate is valid even for p = ∞, because −n/p = 0 if p = ∞. For this reason, we put Kqp (λ) = Kqp,−n/p

p,−n/p

and K0p (λ) = K0

for the full range 1 ≤ p, q ≤ ∞. Note that Kpp (λ) ≈ Lp (λ)

(2.4)

for 1 ≤ p ≤ ∞. That is, these two spaces are the same as sets and have equivalent norms as Banach spaces.

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3. Berezin transform The key step in our proofs later is to establish the boundedness of the Berezin transform on Herz spaces Kqp (λ). In this section we ﬁnd a range of parameters p, q, α which ensures the boundedness of the Berezin transform on Kqp,α . We ﬁrst observe the Lp -boundedness of a certain operator related to the Berezin transform. Given α and β real, we let f (w)|R(z, w)|1+(α+β)/n dVβ (w), z∈H Φα,β f (z) = znα H

whenever the integral is well deﬁned. Note that the Berezin transform is equivalent to Φn,0 . We will obtain the precise range of parameters for which Φα,β is bounded on Lpγ when 1 ≤ p ≤ ∞. Associated with the operators Φα,β are the integrals Iα,β (z) = |R(z, w)|1+(α+β)/n dVβ (w) (3.1) H

for z ∈ H. We begin with estimates of these integrals. Lemma 3.1. Given α, β real, the following estimates hold: zn−α if α > 0 and β > −1 Iα,β (z) ≈ ∞ otherwise for z ∈ H. The suppressed constants in these estimates are independent of z. Proof. Let α and β be given. Fix an arbitrary point z ∈ H. We may assume z = (0 , zn ) by horizontal translation invariance. Note that R(z, w) =

4 n − (1 + |x|2 ) nωn (zn + wn )n (1 + |x|2 )(n+2)/2

for w ∈ H where x = w /(zn + wn ). It follows that 1+(α+β)/n ∞ |n − (1 + |x|2 )| wnβ dwn 4 · dx Iα,β (z) = nωn 0 (zn + wn )α+β+1 Rn−1 (1 + |x|2 )(n+2)/2 1+(α+β)/n ∞ ∞ |n − (1 + r2 )| tβ dt = C(n)zn−α · rn−2 dr (1 + t)α+β+1 0 (1 + r2 )(n+2)/2 0 ∞ ∞ tβ dt dr · . ≈ zn−α α+β+1 α+β+2 (1 + t) (1 + r) 0 0 Note that the ﬁrst integral above diverges for α ≤ 0 or β ≤ −1. For α > 0 and β > −1, both integrals converge. The proof is complete. A trivial modiﬁcation of the proof above yields the following integral identity, which we record here for easier references later.

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Lemma 3.2. Let A = {z ∈ H : 0 ≤ a < zn −1/n, there exists a constant C = C(n, α) such that b dt |R(z, w)|1+α dw = C(n, α) (z + t)nα+1 n a A for z ∈ H. Remark. For β > −1, it is well known that the weighted integral of (holomorphic) Bergman kernel on the ball, similar to (3.1), is of bounded growth for α < 0 and logarithmic growth for α = 0. See, for example, Theorem 1.12 of [13]. The analogous estimate for the harmonic Bergman kernel on the ball has been recently proved in Lemma 3.5 of [2]. So, Lemma 3.1 shows a quite diﬀerent nature caused by the unboundedness of the half-space. We now provide a complete range of parameters for the boundedness of the operators Φα,β on weighted Lebesgue spaces Lpγ . Theorem 3.3. Let 1 ≤ p ≤ ∞ and α, β, γ be real. Then Φα,β is bounded on Lpγ if and only if −α < (γ + 1)/p < β + 1. In particular, the Berezin transform is bounded on Lpγ if and only if −n < (γ + 1)/p < 1. The above theorem has been known on the ball with precisely the same range of parameters. See Theorem 2.10 of [13] for the holomorphic version and Proposition 3.6 of [2] for the harmonic version. The proof of the suﬃciency for 1 < p < ∞ below is also almost the same and included here for reader’s convenience. Proof. As mentioned earlier, the Berezin transform is equivalent to Φn,0 . So, we need to prove only the ﬁrst part of the theorem. Let K(z, w) = Kα,β,γ (z, w) be the kernel for the operator Φα,β against the measure dVγ (w). That is, K(z, w) = znα wnβ−γ |R(z, w)|1+(α+β)/n for z, w ∈ H. It then follows from duality between L1γ and L∞ that Φα,β is bounded on L1γ if and only if K(z, w) dVγ (z) < ∞, sup w∈H

H

which, in turn, holds if and only if β − γ > 0 and α + γ > −1 by Lemma 3.1. This completes the proof for p = 1. Also, by Lemma 3.1, it is easily seen that Φα,β is bounded on L∞ if and only if α > 0 and β > −1, as desired. We now consider the case 1 < p < ∞. We ﬁrst prove the suﬃciency. So, assume −α < (γ + 1)/p < β + 1. Since β+1 β−γ < p p α α+γ+1 < , −pα < γ + 1 ⇐⇒ − p p

γ + 1 < p(β + 1) ⇐⇒ −

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we have a nonempty intersection of intervals: α+γ+1 β−γ β+1 α , − − , = ∅. p p p p Choose a number s in this intersection so that β + p s > −1, α − p s > 0, α + ps + γ > −1, β − ps − γ > 0. Thus a routine calculation in conjunction with Lemma 3.1 yields wnp s K(z, w) dVγ (w) znp s H

and

H

znps K(z, w) dVγ (z) wnps

for z, w ∈ H. Having these estimates, we ﬁnally conclude the boundedness of Φα,β on Lpγ by Schur’s test (see, for example, Theorem 2.9 of [13]). We now prove the necessity. Assume Φα,β is bounded on Lpγ . We ﬁrst show the inequality (γ +1)/p < β +1. If (γ +1)/p ≥ β +1, or equivalently, (β −γ)p +γ ≤ −1, then we have by Lemma 3.1 |R(z, w)|[1+(α+β)/n]p wn(β−γ)p +γ dw = ∞ K(z, ·)p p = znαp Lγ

H

for every z ∈ H and thus Φα,β f fails to exist for some f ∈ Lpγ . This contradiction yields (γ + 1)/p < β + 1. It remains to prove the inequality −α < (γ + 1)/p. Let Φ∗α,β be the adjoint operator of Φα,β with respect to the dual action induced by the inner product of L2γ . A straightforward calculation yields the relation Φ∗α,β = Φβ−γ,α+γ . Since Φ∗α,β is bounded on Lpγ by duality, we have (γ + 1)/p < α + γ + 1, or equivalently, −α < (γ + 1)/p by what we have just proved. The proof is complete. We now proceed to investigate the Herz spaces on which the Berezin transform is bounded. However, it is not even clear on which Herz spaces the Berezin transform is well deﬁned, not only because Herz spaces are mixed-norm spaces, but also because the half-space is unbounded. With that in mind we prove certain Herz norm estimates of the kernel functions as in the next lemma. Lemma 3.4. Let 1 ≤ p, q ≤ ∞ and −1/p < α < nt − n/p. Then there exists a constant C = C(n, α, p, q, t) such that |Rz |t Kp,α ≤ q for z ∈ H.

C nt−n/p−α zn

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Proof. Let z ∈ H. We may assume z = (0 , zn ). Consider the case p < ∞ ﬁrst. Since ntp > n + αp and αp > −1 by assumption, we have ntp − (n − 1) > 0. So, we have by Lemma 3.2 2k dwn 2k ≈ |Rz |t χk pLp ≈ ntp−n+1 k (zn + 2 )ntp−n+1 2k−1 (zn + wn ) and thus |Rz |t χk Lp ≈

2k/p (zn + 2k )nt+(1−n)/p

(3.2)

for all k. Also, it is easily veriﬁed that this estimate continues to hold for p = ∞. Now, having the estimate (3.2), we proceed to the estimate of |Rz |t Kqp,α . First, consider the case q = ∞. In this case, since 0 < α + 1/p < nt + (1 − n)/p by assumption, we easily deduce from (3.2) that 2kα |Rz |t χk Lp ≈

2k(α+1/p) 1 ≤ nt−n/p−α k nt+(1−n)/p (zn + 2 ) zn

for all k, as desired. Next, consider the case q < ∞. Let 2d ≤ zn < 2d+1 for some integer d. Then we have by (3.2) ∞

|Rz |t qKqp,α =

2kαq |Rz |t χk qLp

k=−∞ d

=

+

k=−∞

∞

k=d+1 d

1

(nt−(n−1)/p)q zn k=−∞

2k(α+1/p)q +

∞

k=d+1

1 2k(nt−n/p−α)q

: = I + II. If d ≥ 0, then we have I=

1 (nt−(n−1)/p)q zn

−1

k=−∞

+

d

≈

1 + 2d(α+1/p)q

k=0

(nt−(n−1)/p)q zn

≈

and II ≈

1 1 ≈ (nt−n/p−α)q 2(d+1)(nt−n/p−α)q zn

so that we have the desired estimate. If d < 0, then we have I≈

2d(α+1/p)q (nt−(n−1)/p)q zn

≈

1 (nt−n/p−α)q zn

1 (nt−n/p−α)q zn

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and II =

0

+

k=d+1

∞

≈

k=1

1 1 + 1 ≈ (nt−n/p−α)q 2(d+1)(nt−n/p−α)q zn

so that we again have the desired estimate. The proof is complete.

Proposition 3.5. Let 1 ≤ p, q ≤ ∞ and α be real. If −2n+n/p < α < 1/p , then the Berezin transform is well defined on Kqp,α and there is a constant C = C(p, q, α) such that |ϕ(z)| ≤ Czn−n/p−α ϕKp,α , q

z∈H

for ϕ ∈ Kqp,α . In particular, ϕ L∞ ≤ CϕKpq (λ) for ϕ ∈ Kqp (λ). Proof. Assume −2n + n/p < α < 1/p and let ϕ ∈ Kqp,α . We may further assume ϕ ≥ 0. Let z ∈ H. Then, since Rz 2L2 = R(z, z) ≈ zn−n , we have by successive applications of H¨older’s inequality ϕ(z) ≈ znn ϕ(w)|Rz (w)|2 dw H

≤ znn

∞

2kα ϕχk Lp · 2−kα Rz2 χk Lp

k=−∞

≤ znn ϕKp,α Rz2 Kp ,−α . q q

Note that we have −1/p < −α < 2n − n/p by assumption. So, the ﬁrst part of the proposition follows from Lemma 3.4. By taking α = −n/p, we have the second part. The proof is complete. Remark. Note that 1 + |z| ≈ 1 + |z | + zn ≈ |z0 − z| for z ∈ H where z0 = (0 , 1). Thus, easily extending the proof of Proposition 3.5, one can check that ϕ(z) dz < ∞, β H (1 + |z|) p,α if ϕ ∈ K∞ and n/p − β < α < 1/p . So, we have p,α K∞ ⊂M

(3.3)

whenever 1 ≤ p ≤ ∞ and α < 1/p . We now provide the range of parameters for Herz spaces on which the Berezin transform is bounded. In case p = ∞ the next theorem is proved on the ball [2] only for α = 0 with an argument much more complicated than the one provided below.

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Theorem 3.6. Let 1 ≤ p ≤ ∞ and α be real such that −n < α + 1/p < 1. Then the Berezin transform is bounded on Kqp,α for 1 ≤ q ≤ ∞. Moreover, the Berezin transform takes K0p,α into itself. Proof. First, we show that there exists some constants δ = δ(p, α) > 0 and C = C(p, α) such that 2kα ϕχ k Lp ≤ C

∞

2mα ϕχm Lp 2δ|m−k| m=−∞

(3.4)

for all k ∈ Z and positive measurable functions ϕ. Consider the case p < ∞ ﬁrst. Pick ε > 0 such that αp ± ε + 1 < 1. −n < p Note that the Berezin transform is bounded on Lpαp±ε by Theorem 3.3. Let ϕ ≥ 0 ∞ be a given measurable function. Then, since ϕ = m=−∞ ϕχm , we have by the monotone convergence theorem ∞

ϕ =

ϕ χm .

m=−∞

Meanwhile, by the boundedness of the Berezin transform on Lpαp±ε , we have | ϕχm (z)|p znαp±ε dz |ϕ(z)|p znαp±ε dz H

Am

for all m. Thus, we have | ϕχm |p dVαp ≈ 2∓εk | ϕχm |p znαp±ε dz Ak A k ∓εk 2 |ϕ(z)|p znαp±ε dz Am ±ε(m−k) |ϕ|p dVαp ≈2 Am

so that ( ϕχm )χk Lpαp 2−ε|m−k|/p ϕχm Lpαp , or equivalently, 2mα ϕχm Lp 2ε|m−k|/p for all m and k. So, we obtain (3.4) with δ = ε/p by (3.5). Next, assume p = ∞. So, −n < α < 1. We have by (3.5) ϕχm )χk Lp 2kα (

ϕ ≤

∞

m=−∞

χ m ϕχm L∞

(3.5)

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and thus 2kα ϕχ k L∞ ≤

∞

2(k−m)α χ m χk L∞ · 2mα ϕχm L∞

m=−∞

for each k. Meanwhile, we have by Lemma 3.2 2m dwn 2m znn ≈ , z∈H χ m (z) ≈ znn n+1 (zn + 2m )n+1 2m−1 (zn + wn ) and thus 2m 2nk 2(1−α)(m−k) 2(k−m)α χ m χk L∞ ≈ 2(k−m)α k = (2 + 2m )n+1 (1 + 2m−k )n+1 for all m and k. If m ≥ k, then 2(1−α)(m−k) 2(1−α)(m−k) 1 ≈ = (n+α)|m−k| . m−k n+1 (n+1)(m−k) (1 + 2 ) 2 2 If m < k, then 2(1−α)(m−k) 1 ≈ 2(1−α)(m−k) = (1−α)|m−k| . m−k n+1 (1 + 2 ) 2 So, taking δ = min{1 − α, n + α} > 0 and combining all these estimates, we have (3.4) for p = ∞. Now, the boundedness for 1 ≤ q ≤ ∞ is a consequence of (3.4) and Young’s inequality. Assume ϕ ∈ K0p,α and let k ∈ Z be given. Then we have by (3.4) ∞

2mα ϕχm Lp 2δ|m−k| m=−∞

= +

k Lp 2kα ϕχ

|m|>d

|m|≤d

sup 2mα ϕχm Lp + ϕKp,α ∞ |m|>d

|m|≤d

1 2δ|m−k|

for each integer d ≥ 1. Now, taking the limit |k| → ∞ (with d ﬁxed), we obtain lim sup 2kα ϕχ k Lp sup 2mα ϕχm Lp |k|→∞

|m|>d

for all d. So, taking another limit d → ∞, we conclude ϕ ∈ K0p,α . The proof is complete.

4. Positive Toeplitz operators in Schatten-Herz class In this section we characterize prove Toeplitz operators that belong to the so-called Schatten-Herz classes. Given 1 ≤ p, q ≤ ∞, the Schatten-Herz class Sp,q is the class of all Toeplitz operators Tµ such that Tµχk ∈ Sp for each k and the sequence {Tµχk Sp } belongs

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to q ; recall that µχk denotes the restriction of µ to Ak . The norm of Tµ ∈ Sp,q is given by Tµ Sp,q = Tµχk Sp q . Also, we let Sp,0 be the space of all operators Tµ ∈ Sp,∞ such that Tµχm Sp ∈ c0 . Note that Sp,q ⊂ Sp,0 for all q < ∞. We begin with the following property of the Herz norm of averaging functions, which will lead us to the equivalence of the Schatten-Herz norm of Toeplitz operators and averaging functions. Theorem 4.1. Let 1 ≤ p, q ≤ ∞, r ∈ (0, 1) and α be real. Then the following statements hold for µ, τ ≥ 0. (a) 2kα µr χk Lp (τ ) q ≈ 2kα (µχ

k )r Lp (τ ) q . (b) 2kα µr χk Lp (τ ) ∈ c0 if and only if 2kα (µχ

k )r Lp (τ ) ∈ c0 . The constants suppressed above are independent of µ and τ . Before proceeding to the proof, we note the following covering property which is a simple consequence of (2.3): If r ∈ (0, 1) and N = N (r) is a positive integer such that 1+r < 2N , 2N −1 ≤ (4.1) 1−r then Er (z) ⊂

k+N

Aj

(4.2)

j=k−N

for z ∈ Ak and k ∈ Z. Proof. Let µ ≥ 0. We prove the proposition only for q < ∞; the case q = ∞ is simpler. Choose N = N (r) as in (4.1). We then have by (4.2) µ r χk ≤

k+N

(µχ

j )r

j=k−N

for all k ∈ Z. Since 2kα ≤ 2N |α| 2jα for k − N ≤ j ≤ k + N , it follows that µr χk Lp (τ ) 2kα

k+N

2jα (µχ

j )r Lp (τ )

j=k−N

for all k. This implies one direction of (b). Also, we obtain µr χk qLp(τ ) (2N + 1)q−1 2kαq

k+N

j=k−N

2jαq (µχ

j )r qLp (τ )

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for all k ∈ Z. Thus, summing these estimates, we have ∞

2kαq µr χk qLp (τ )

∞

2jαq (µχ

j )r qLp (τ ) ,

j=−∞

k=−∞

which gives one direction of (a). We now prove the other direction. Note that Er (z) can intersect Ak only when z ∈ ∪k+N j=k−N Aj by (4.2). Thus we have (µχ k )r =

k+N

(µχ k )r χj ≤

j=k−N

k+N

µ r χj .

j=k−N

Thus, a similar argument yields the other directions of (a) and (b). The proof is complete. As an immediate consequence of Proposition 2.4 (with α = 0) and Theorem 4.1 (with τ = V ), we have the following Herz space version of Proposition 2.4. Corollary 4.2. Let 1 ≤ p, q ≤ ∞, δ, r ∈ (0, 1) and α be real. Then the following statements holds for µ ≥ 0. (a) µr Kp,α ≈ µδ Kp,α . q q (b) µ r ∈ K0p,α if and only if µ δ ∈ K0p,α . The constants suppressed above are independent of µ. We now turn to the Herz norm equivalence of averaging functions and Berezin transforms. Theorem 4.3. Let 1 ≤ p, q ≤ ∞, r ∈ (0, 1) and α be real. Assume −n < α+1/p < 1. Then the following statements hold for µ ≥ 0. ≈ µKp,α . (a) µr Kp,α q q ∈ K0p,α . (b) µ r ∈ K0p,α if and only if µ The constants suppressed above are independent of µ. Proof. We may assume r ∈ (0, 1/6) by Corollary 4.2. Now, the theorem follows from Proposition 2.2, Proposition 2.3 and Theorem 3.6. The proof of the next theorem, which is our main result, is now simple applications of what we have proved so far. Theorem 4.4. Let 1 ≤ p ≤ ∞, r ∈ (0, 1) and µ ∈ M+ . Also, assume 1 ≤ q ≤ ∞ or q = 0. Then the following conditions are equivalent. (a) Tµ ∈ Sp,q . (b) µ ∈ Kqp (λ). (c) µ r ∈ Kqp (λ).

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Proof. We have

Tµ Sp,q = Tµχk Sp

and Tµ ∈ Sp,0 ⇐⇒

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k )r Lp (λ) ≈ (µχ

q

(µχ

k )r Lp (λ) ∈ c0

by Proposition 2.5. Thus the theorem follows from Theorem 4.1 (with τ = λ and α = 0) and Theorem 4.3. The following is an immediate consequence of (2.4), Proposition 2.5 and Theorem 4.4. Corollary 4.5. Let 1 ≤ p ≤ ∞ and µ ∈ M+ . Then Tµ ∈ Sp,p if and only if Tµ ∈ Sp . As another consequence of Theorem 4.4, we can construct examples asserting that the classes Sp,q are all diﬀerent. Corollary 4.6. Let 1 ≤ p, q, p1 , q1 ≤ ∞. Then Sp,q = Sp1 ,q1 if and only if p = p1 and q = q1 . Proof. Let f = R(·, z0 ) where z0 = (0 , 1). Given 0 ≤ a, b ≤ 1, put ϕ = ϕa,b =

∞

log(1 + |k|) ak(n−1) 2 f χk . (1 + |k|)b

k=−∞

Let r ∈ (0, 1) be a suﬃciently small number. For example, one may take r < 1/11. Note that harmonicity of f , together with (2.1), gives 1 fr (z) = f (w) dw = f (z ∗ ), z∈H (4.3) |Er (z)| Er (z) 2

where z ∗ = (z , 1+r 1−r 2 zn ). Let 1 ≤ s ≤ ∞. We ﬁrst prove if m ≥ 0, log(1 + |m|) 2am(n−1) 2−nm ϕ r χm Ls (λ) ≈ × (4.4) b m(n−1)(a−1/s) (1 + |m|) if m < 0. 2 Given m, put Bm = z ∈ Am : 4 · 2m−1 /3 ≤ zn < 5 · 2m−1 /3 . Since r < 1/11, it is easily checked that Er (z) ⊂ Am for z ∈ Bm by (2.3). So, (f χm )r χBm = fr χBm and (f χk ) χB = 0 for k = m. It follows from (4.3) that r

m

ϕ r χm Ls (λ) ≥ ϕ r χBm Ls (λ) log(1 + |m|) am(n−1) 2 fr χBm Ls (λ) (1 + |m|)b 1/s log(1 + |m|) am(n−1) ∗ s −n = 2 |f (z )| z dz . n (1 + |m|)b Bm =

(4.5)

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Meanwhile, setting am = 4(1+r2 )2m−1 /3(1−r2 ) and bm = 5(1+r2 )2m−1 /3(1−r2 ), we have by Lemma 3.2 bm dzn |f (z ∗ )|s zn−n dz ≈ 2−nm n(s−1)+1 Bm am (1 + zn ) 2−m(n−1) (1 + 2m )n(s−1)+1 2−mns if m ≥ 0, ≈ 2−m(n−1) if m < 0. ≈

(4.6)

Combining this and (4.5), we obtain the lower estimate of (4.4). Note that (f χk )r χm = 0 if |m − k| > 1 by (4.2) (with N = 1). Thus, we have ϕ r χm Ls (λ) ≤

m+1

k=m−1

log(1 + |k|) ak(n−1) 2 fr χm Ls (λ) . (1 + |k|)b

Note that the estimate (4.6) remains valid for fr χm sLs (λ) . So, the upper estimate of (4.4) holds. Now, assume that either p = p1 or q = q1 . We only need to consider two cases: (i) p < p1 ; (ii) p = p1 and q < q1 . By Theorem 4.4 it suﬃces to prove that ϕ r ∈ Kqp11 (λ),

ϕ r ∈ / Kqp (λ)

(4.7)

for some choice of a and b. First, consider Case (i). In this case we take a = 1/p and b = 0. Then, for 1 ≤ s, t ≤ ∞, it follows from (4.4) that log(1 + m) ϕ r ∈ Kts (λ) ⇐⇒ ∈ t ⇐⇒ p < s 2m(n−1)(1/p−1/s) m>0 and therefore (4.7) holds. Next, consider Case (ii). In this case we take a = 1/p and b = 1/q. Then, for 1 ≤ t ≤ ∞, we have by (4.4) log(1 + m) p ϕ r ∈ Kt (λ) ⇐⇒ ∈ t ⇐⇒ q < t (1 + m)1/q m>0 so that (4.7) holds. The proof is complete.

5. Toeplitz operators with Herz symbols In this section we apply our results obtained in earlier sections and observe some consequences concerning boundedness and compactness of Toeplitz operators with Herz symbols. We need the following characterizations for boundedness (compactness resp.) for positive Toeplitz operators, which are taken from Theorem 4.1 (Theorem 5.3)

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and Theorem 4.6 (Theorem 5.5) of [3]. Here, L0 denote the space of all bounded functions ϕ on H such that ϕ(z) → 0 as zn → 0 or |z| → ∞. Proposition 5.1. Let r ∈ (0, 1) and µ ∈ M+ . Then the following conditions are equivalent: (a) Tµ is bounded (compact) on b2 . (b) µ ∈ L∞ (L0 ). (c) µ r ∈ L∞ (L0 ). Moreover, the equivalences Tµ ≈ µL∞ ≈ µr L∞ hold. Recall that M is the set of all formal linear combinations of measures in M+ . Given µ ∈ M represented by µ = µ1 − µ2 + iµ3 − iµ4

(5.1)

where µ1 , µ2 , µ3 , µ4 ∈ M+ , we let |µ| = µ1 + µ2 + µ3 + µ4 . L∞ for each j, we see that if T is µj L∞ ≤ |µ| Note that |µ| ∈ M+ . Since |µ| bounded, then so is Tµ and Tµ T|µ| by Proposition 5.1. Similarly, if T|µ| is compact, then Tµ is compact. More generally, we have the following theorem. Theorem 5.2. Let 1 ≤ p, q ≤ ∞ and µ ∈ M . Then the following statements hold: (a) If T|µ| ∈ Sp,q , then Tµ ≤ CT|µ| Sp,q for some constant C = C(p, q). (b) If T|µ| ∈ Sp,0 for some p < ∞, then Tµ is compact on b2 . ∞ (λ) = L∞ , we Proof. To prove (a) we ﬁrst consider the case p = q = ∞. Since K∞ have by Theorem 4.4 and proposition 5.1 ≈ T|µ| . T|µ| S∞,∞ ≈ |µ| L∞

So, we have Tµ T|µ|

(5.2)

by the remark mentioned above. We now consider the general case. By (5.2) we may assume µ ∈ M+ . Fix r ∈ (0, 1). First, we have

µr L∞ ( µr )r L∞ Tµ ≈ by Proposition 2.5 and Proposition 2.1. Meantime, we have

µr )L∞ µr Kpq (λ) ≈ Tµ Sp,q ( µr )r L∞ ≈ ( by Proposition 2.5, Proposition 3.5 and Theorem 4.4. So, (a) holds.

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We now prove (b). Suppose that T|µ| ∈ Sp,0 and 1 ≤ p < ∞. Again by the remark mentioned above, it is suﬃcient to show that T|µ| is compact. Now, given an integer d ≥ 0, let |µ|d = |m|≤d |µ|χm . Then, by Proposition 2.5, we have d Lp (λ) ≤ (2d + 1) sup |µ|χ |µ| m Lp (λ) ≈ (2d + 1)T|µ| Sp,∞ < ∞ m

and thus T|µ|d ∈ Sp . In particular, each T|µ|d is compact on b2 . Also, since T|µ| − T|µ|d = T |m|>d |µ|χm , we have by (a) T|µ| − T|µ|d T|µ| − T|µ|d Sp,∞ = sup T|µ|χm Sp → 0, |m|>d

as d → ∞, because T|µ| ∈ Sp,0 . So, T|µ| is compact, as desired. The proof is complete. p (λ) are all well deﬁned by Note that Toeplitz operators with symbols in K∞ (3.3). As an application of Theorem 5.2 we prove the following boundedness and compactness properties of such Toeplitz operators.

Theorem 5.3. Let 1 ≤ p, q ≤ ∞. Then the following statements hold: (a) There exists a constant C = C(p, q) such that Tϕ ≤ CϕKpq (λ) for ϕ ∈ Kqp (λ). (b) Tϕ is compact on b2 for ϕ ∈ K0p (λ) with p < ∞. Proof. The assertion (a) follows from Theorem 5.2 and Theorem 4.4. Given a Borel set E ⊂ H, let E ∗ = {w ∈ H : E ∩ Er (w) = ∅}. Note that E ∗ = ∪z∈E Er (z). Before proceeding to the proof of (b), we ﬁrst show that there exists a constant C = C(p, α, r) such that ψr χE Lpα ≤ CψχE ∗ Lpα

(5.3)

for 1 ≤ p ≤ ∞, r ∈ (0, 1) and α real, Borel sets E ⊂ H and measurable functions ψ ≥ 0. The case p = ∞ is immediate from (4.2). So, assume p < ∞. Since |Er (z)| ≈ znn , we have by (2.2) and (2.3) znα dz ≈ wnα Er (w) |Er (z)| for w ∈ H. It follows that |ψr χE (z)|p znα dz ≤ H

1 |ψ(w)|p dw znα dz |Er (z)| Er (z) E znα p dz dw = |ψ(w)| E∗ E∩Er (w) |Er (z)| |ψ(w)|p wnα dw E∗

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and thus (5.3) holds. Now, we prove (b). Let ϕ ∈ K0p (λ) with p < ∞ and ﬁx any r ∈ (0, 1). Then we have by Proposition 2.5 and (5.3) ϕ r χm Lp(λ) ≤

m+N

ϕχk Lp (λ)

k=m−N

for all m ∈ Z where N = N (r) is the number as in (4.1). Note that the right side of the above tends to 0 as |m| → ∞. Thus, Tϕ ∈ Sp,0 by Theorem 4.4. So, (b) holds by Theorem 5.2. The proof is complete. Note that if a measure (in M) is supported in a compact set in H, then its averaging functions are also supported in compact sets in H. So, Toeplitz operators with symbols supported in compact sets in H are all compact by Proposition 5.1. This can be extended to symbols supported in a horizontal stripe as follows. Corollary 5.4. Let 1 ≤ p < ∞. If ϕ ∈ Lp is supported in a horizontal stripe in H, then Tϕ is compact on b2 . Proof. It is clear that ϕ ∈ K0p (λ) for ϕ ∈ Lp supported in a horizontal stripe in H. So, the corollary follows from Theorem 5.3. Extending Corollary 3.3 and Theorem 4.7 of [2], one can see that the ball versions of the second parts of Theorem 5.2 and Theorem 5.3 hold even for p = ∞. However, it is not the case on the half-space, as the next example shows. Such a pathology is of course caused by the unboundedness of the half-space. Example. Let A = {w ∈ H : 0 < a ≤ wn ≤ b}. It is clear that χA ∈ K0∞ (λ). Also, we have by Lemma 3.2 b dwn znn (b − a) a n χ ≤1− A (z) = C(n)zn n+1 n+1 (zn + b) b a (zn + wn ) ∞ and TχA ∈ S∞,0 . However, χ for z ∈ H. So, χ A ∈L A (z) cannot converge to 0 as |z | → ∞, because χ A (z) depends on zn only. Consequently, TχA is not compact by Proposition 5.1.

References [1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Springer-Verlag, New York, 1992. [2] B. Choe, H. Koo and K. Na, Positive Toeplitz operators of Shatten-Herz type, Nagoya Math. J. 185 (2007), 31–62. [3] B. Choe, H. Koo and H. Yi, Positive Toeplitz Operators between the Harmonic Bergman Spaces, Potential Analysis 17 (2002), 307–335. [4] B. Choe, Y. Lee and K. Na, Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J. 174 (2004), 165–186.

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[5] B. Choe, Y. Lee and K. Na, Positive Toeplitz operators from a harmonic Bergman space into another, Tohoku Math. J. 56 (2004), 255–270. [6] E. Heren´ andez and D. Yang, Interpolation of Herz spaces and applications, Math. Nachr. 205 (1999), 69–87. [7] M. Loaiza, M. L´ opez-Garc´ıa and S. P´erez-Esteva, Herz Classes and Toeplitz operators in the disk, Integr. Equ. Oper. Theory 53 (2005), 287–296. [8] J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), 25–35. [9] W. Ramey and H. Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc. 348 (1996), 633–660. [10] F. Ricci and M. Taibleson, Representation theorems for harmonic functions in mixed norm spaces on the half plane, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980). Rend. Circ. Mat. Palermo (2) 1981, suppl. 1, 121–127. [11] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 10 (1983), no. 1, 1–54. [12] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York and Basel, 1989. [13] K. Zhu, Spaces of holomorphic functions in the unit ball, Springer, New York, 2005. Boo Rim Choe Department of Mathematics Korea University Seoul 136–701 Korea e-mail: [email protected] Kyesook Nam Department of Mathematics Hanshin University Gyeonggi 447–791 Korea e-mail: [email protected] Submitted: June 6, 2006 Revised: August 21, 2007

Integr. equ. oper. theory 59 (2007), 523–554 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040523-32, published online October 18, 2007 DOI 10.1007/s00020-007-1536-7

Integral Equations and Operator Theory

Discretizations of Integral Operators and Atomic Decompositions in Vector-Valued Weighted Bergman Spaces Olivia Constantin Abstract. We prove a general atomic decomposition theorem for weighted vector-valued Bergman spaces, which applies to duality problems and to the study of compact Toeplitz type operators. Mathematics Subject Classification (2000). 32A36, 47B35, 47B38. Keywords. Atomic decomposition, vector-valued weighted Bergman spaces, Toeplitz operators.

1. Introduction Let Ω be a bounded domain in the complex plane and denote by dA the normalized Lebesgue area measure on Ω (that is dA(z) = dxdy/( Ω 1dxdy) with z = x + iy). For a Banach space X and a scalar function ω, integrable with respect to dA on Ω, we consider the weighted Bergman space Lpa (ω, X) consisting of X-valued analytic functions f on Ω with f Lpa(ω,X) =

Ω

p1 f (z)p ω(z)dA(z) < ∞,

where p > 0. When X = C, we shall simply denote this space by Lpa (ω). Throughout the paper we shall assume that ω is continuous with ω > 0 on Ω. This condition ensures that Lpa (ω, X) is a closed subspace of Lp (ω, X), and, conversely, if ω is any integrable weight on Ω such that Lpa (ω, X) is closed in Lp (ω, X), we can replace ω by a continuous strictly positive weight, without changing the topology of the space. For a ﬁnite complex Borel measure µ, compactly supported with

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¯ we consider the integral operator Tµ formally deﬁned on Lp (ω, X) by supp µ ⊆ Ω, a Tµ f (z) = f (ζ) Kζ (z) dµ(ζ), (1.1) Ω

where Kζ (z) is a function deﬁned on Ω × Ω, analytic in the z-variable, and antianalytic in ζ. When Kζ is the reproducing kernel for the space L2a (ω, X), these operators are usually called Toeplitz operators and occur naturally in many problems related to these spaces. Even in this particular case, basic questions like boundedness and compactness of these operators deﬁned for general complex measures, are not completely understood. We shall continue to use this terminology in our more general context. This paper is concerned with discretizations of the integral operators Tµ . The main applications of our results refer to the study of duality for vector-valued Bergman spaces and to the compactness of such Toeplitz operators induced by complex measures. A discretization of a Toeplitz operator Tµ means a stable representation of each element in the range of Tµ as a usually inﬁnite linear combination of simpler functions, or building blocks from the space Tµ is acting on. More precisely, we aim for a representation of the form Tµ f =

i

α,β

∂¯β Kλi Ci,α,β (f ) β , ∂¯ Kλi p L (ω)

f ∈ Lpa (ω, X),

(1.2)

a

such that the coeﬃcients involved form an p (X) sequence with Ci,α,β (f )p ≤ kf pLpa (ω,X) , i

α,β

for some constant k, depending on the norm of the operator Tµ . This concept is inspired by atomic decompositions of function spaces, which can be seen as discretizations of the identity operator. The idea of atomic decomposition, closely related to sampling problems, applies to the study of H p spaces, especially when p ≤ 1 and their real-variables generalizations. The study of atomic decompositions for weighted Bergman spaces was initiated by Coifman and Rochberg [7] for certain domains in Cn and for the standard weights (1 − |z|2 )α . Later Luecking [11] used functional analytic methods together with a powerful theorem of B´ekoll´e [5] to extend the results in [7] to more general weighted Bergman spaces. His methods are based on duality and therefore work only in the case p ≥ 1. Since our results deal with Banach space-valued functions and with all values of p > 0, the need of stable estimates leads to a somewhat diﬀerent approach which is essentially based on Taylor’s formula and a Whitney-type decomposition of the domain. Although the technique applies also to the case of several variables, especially to product domains, we prefer to state our results in the one variable setting, since most of the applications refer to this last situation. The decomposition technique appears also in the study of Toeplitz operators by Luecking in [13]. Apart from being interesting in their own right, they provide powerful tools for the study of Bergman spaces.

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For example, as we shall see below, they apply to duality problems and to weak product decompositions in operator-valued Bergman spaces (see [2]). Our general approach is described in Section 2. Here, we associate to the operator Tµ a discrete version TµD of the form given in the right-hand-side of (1.2), and which arises in a natural way from Taylor’s formula. The main step is to estimate the error term Tµ − TµD . This is done in Theorem 2.2 and leads to a representation of the form (1.2) for invertible Toeplitz operators. These ideas are closely related to atomic decompositions. In fact, we also show in this section an atomic decomposition theorem, provided the kernel K has a certain reproducing property for the space. The atomic decomposition then enables us to discretize any bounded Toeplitz operator with kernel K. We point out that a certain converse statement holds; given an arbitrary p (X) sequence (Cij ), the series in the righthand-side of (1.2) deﬁnes an Lpa (ω, X)-function with its norm dominated by a constant multiple of the p (X)-norm of the sequence. The suﬃcient conditions we use in order to prove these results are quite natural and they are inspired by B´ekoll´e’s work. We assume that there exists a sublinear integral operator of the form Pu f (z) = |f (w)| |Kz (w)|p/p0 u(w) dA(w), Ω

p0

which is bounded on L (ω), i.e., 1/p0 p0 |f (w)| |Kz (w)|p/p0 u(w) dA(w) ω dA ≤ Cf Lp0 (ω) , Ω

(1.3)

Ω

for some p0 > 1, C > 0 and all f ∈ Lp0 (ω). The further assumptions essentially ensure that there exists a Whitney type decomposition of the domain such that the function u, the kernel and its derivatives are almost constant on each cell. As a ﬁrst application, in Section 3, we consider Bergman spaces with B´ekoll´e weights and a standard decomposition of the disc in “c-adic squares”, for some c > 1, c close to 1. It is easy to see that, in this setting, all assumptions of the decomposition theorem are fulﬁlled. This yields an atomic decomposition for these spaces which extends the result of Luecking [11] to the case p < 1 and to vectorvalued Bergman spaces. We also obtain an atomic decomposition theorem for vector-valued weighted Bergman spaces deﬁned on the polydisc in Cn . For this topic we refer to Zhu’s recent book [18]. In Section 4 we use the atomic decomposition theorem from Section 3 to prove a decomposition theorem for vector-valued weighted Bergman spaces deﬁned on ﬁnitely connected domains with analytic boundary. As another natural application, we determine in Section 5 the dual of the Banach space-valued weighted Bergman spaces, in the case p < 1 and for weights of B´ekoll´e type. The duality for the scalar-valued Bergman spaces, for p ≥ 1 and B´ekoll´e weights was investigated by Luecking [11]. Duality problems for p < 1 have attracted a lot of attention. Duren, Romberg and Shields [9] investigated this

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problem for the Hardy spaces H p , p < 1. J. Shapiro [16] obtained the analogue results for the standard weighted Bergman spaces. In the case of the standard weighted vector-valued Bergman spaces Lpa ((1 − |z|2 )α , X) with p ≥ 1, a duality theorem was obtained by Arregui and Blasco [3]. For p = 1, the predual of these spaces is identiﬁed in [2]. The main result in this section identiﬁes the dual of Lpa (ω, X) with a space of X ∗ -valued holomorphic functions g which satisfy a growth restriction of the type

( Dλ,r ω(z) dA(z))1/p as |λ| → 1, g (γ+1) (λ)X ∗ = O (1 − |λ|)γ+2 where ω satisﬁes a B´ekoll´e type condition and the parameter γ ∈ Z depends only on ω and p. In Section 6 we turn to the study of compact Toeplitz operators. Note that, for arbitrary complex measures, a complete characterization of compactness is not available (see [17]). Quite recently, Axler and Zheng [4] have characterized the compactness of Toeplitz operators with L∞ symbols acting on unweighted Bergman spaces. Our approach is based on the discretization theorem from Section 2 and uses conditions on the measure of the cells involved in the corresponding Withney type decomposition. As it is to expect, results about compactness concern only the ﬁnite dimensional case, i.e., dim X < ∞. Using ideas from Section 2, we give a general criterion for the compactness of Tµ −TµD , where TµD is the discretized Toeplitz operator mentioned above. With this result at hand, we turn to the special case of Bergman spaces with B´ekoll´e weights on the unit disc, where, as in Section 3, we have a good understanding of the decomposition of the domain. Here we give a compactness criterion for Tµ , for every ﬁnite, complex, Borel measure µ, for which |µ| satisﬁes a a Carleson-type condition. It is important to note that the compactness condition obtained in Corollary 6.2 below does not involve |µ|, but only a ﬁnite number of moments of the complex measure µ on Qi .

2. The discretization of Toeplitz operators acting on vector-valued Bergman spaces Throughout our considerations k will stand for a generic positive constant whose dependence on parameters will be indicated whenever it is relevant. Moreover, X will be a Banach space and by “E1 ∼ E2 ” we denote two functions that are comparable, i.e., there is a constant k > 0 independent of the argument such that kE2 ≥ E1 ≥ 1/k E2 . As mentioned in the introduction, the aim of this section is to estimate the integral operator in (1.1) with the help of an integral of type f (ζ) Kζ (z) dµd (ζ), (2.4) Ω

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where µd is a ﬁnite linear combination of Dirac measures on Ω and their distributional derivatives, that is by sums of the form ∂¯β Kλi ci,α,β f (α) (λi ) β , ∂¯ Kλ p i

i

α,β

La (ω)

in such a manner that the coeﬃcients involved form an p (X) sequence, i.e., ci,α,β p f (α) (λi )p < ∞. i

α,β

Above we used the notation ∂¯β Kλ (z) = ∂ζβ¯ Kζ (z)

ζ=λ

, for β ≥ 0, z, λ ∈ Ω.

Our ﬁrst assumption is the following. 1) There exists p0 > 1, p0 ≥ p, and a function u ∈ L1 (Ω) with u > 0, such that the sublinear projection-type operator |f (w)| |Kz (w)|p/p0 u(w) dA(w), Pu f (z) = Ω

is bounded on Lp0 (ω), i.e., 1/p0 p0 p/p0 |f (w)| |Kz (w)| u(w) dA(w) ω dA ≤ Cf Lp0 (ω) , Ω

(2.5)

Ω

for some C > 0 and all f ∈ Lp0 (ω). The inﬁmum of these constants will be denoted by Pu . Our second assumption is 2) There exist a compact set E ⊂ Ω and a nonnegative integer depending on K denoted N (K), such that (2.6) ∂¯β Kζ (z) = 0, for any 0 ≤ β ≤ N (K), z ∈ Ω1 = Ω \ E, ζ ∈ Ω. Given a measurable set Q ⊆ Ω, and a function g > 0 on Ω, we introduce the following notation: g(ζ) . V (g, Q) = sup w,ζ∈Q g(w) We also denote |∂¯β Kζ (z)| V (∂¯β K, Q) = sup . ¯β z∈Ω1 ,ζ,w∈Q |∂ Kw (z)| We shall keep these notations throughout the paper. Our third assumption concerns the existence of a Whitney-type decomposition of Ω. Namely, we suppose that Ω can be decomposed in measurable regions {Qi }i≥1 , with the following properties. 3a) The Qi s are closures of pairwise disjoint open domains with piecewise “smooth” boundary, such that Ω = ∪i≥1 Qi , and Qi ∩ Qj has zero area for i = j, and 4 diam Qi < dist(Qi , ∂Ω), for i ≥ 1.

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3b) If F ⊆ Ω is compact, then ∪∞ i=M Qi ∩ F = ∅ for large M . For every i ≥ 1, consider the points λi ∈ intQi and the discs Di = {z : |z − λi | < ri }, where the radius ri satisﬁes 3c) 4 diam Qi < ri < dist(Qi , ∂Ω). To simplify the exposition, we shall also assume that 3d) Each Qi is starlike with respect to λi . This last condition is not really needed, but it improves our estimates below. The reason for using it is Taylor’s formula, i.e., if φ = φ(z, z¯) is a C ∞ , X-valued function near Qi and n ∈ N, then, for z ∈ Qi , we have α β ¯ i ) (z − λi ) (z − λi ) ∂zα ∂z¯β φ(λi , λ φ(z, z¯) − α! β! α+β
≤

sup ∂zα ∂z¯β φ

α+β=n

Qi

(diam Qi )n . α!β!

(2.7)

From now on, we denote by D a decomposition Ω = ∪i≥1 Qi with the above properties and such that the discs Di , and, implicitly, the points λi are ﬁxed. We also let (2.8) J(µ) = {i ∈ N : Qi ∩ supp(µ) = ∅, Qi ∈ D}, and denote by I = I(µ, D) (2.9) the maximal number of discs Di , i ∈ J(µ) that intersect a ﬁxed but arbitrary Di0 from the family {Di }i∈J(µ) (note that it I does not need to be ﬁnite). To avoid overcharging the exposition, as soon as the decomposition D and the measure are ﬁxed, we shall simply write I instead of I(µ, D). Our ﬁrst step is to use (2.5) in order to estimate the norm of a Toeplitz-type operator of the form Lpa (ω, X) f → f (α) (ζ)∂¯β Kζ (·) dµ(ζ). Ω

We begin with the following lemma. Lemma 2.1. Let λ ∈ Ω and for 0 < r < dist(λ, ∂Ω), consider the discs ∆ = {z : |z − λ| < r}, ∆ = {z : |z − λ| < r/2}. Assume that the kernel K is as above and let α, β be nonnegative integers. Then for any X-valued function f , analytic in Ω and s > 0, the following inequality holds: |∂¯β Kλ (z)|s f (α) (w)s dA(w) ∆ V 2s (K, ∆) ≤ k (α+β)s f (w)s |Kw (z)|s dA(w), z ∈ Ω1 , r ∆ where the constant k depends on α, β, s: e.g., one can choose k = (α!β!)s 4(α+β)s+3 .

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Atomic Decompositions in Weighted Bergman Spaces

529

Proof. By Cauchy’s formula together with the subharmonicity of the function ξ → f (α) (ξ) we obtain f (α) (w) s k f (ξ)s dA(ξ), w ∈ ∆ , ≤ 2+αs α! r ∆ where k depends on α, s (k = 4αs+2 ). Hence k f (α) (ξ)s dA(ξ) ≤ αs f (ξ)s dA(ξ), r ∆ ∆

(2.10)

where k depends on α, s (k = (α!)s 4αs+1 ). In the same way we have k kV s (K, ∆) β s ¯ |∂ Kλ (z)| ≤ 2+βs |Kξ (z)|s dA(ξ) ≤ |Kλ (z)|s , r rβs ∆

(2.11)

where k = (β!)s 4βs+2 . In view of (2.10), (2.11), we now get |∂¯β Kλ (z)|s

f (α) (ξ)s dA(ξ) ∆

kV s (K, ∆) s |K (z)| f (ξ)s dA(ξ) λ r(α+β)s ∆ kV 2s (K, ∆) f (ξ)s |Kξ (z)|s dA(ξ). ≤ r(α+β)s ∆

≤

Theorem 2.1. Assume p, p0 , ω, u, Pu , K are as above and let µ be a ﬁnite nonnegative Borel measure on Ω. Suppose also that Ω admits a decomposition D of the type described previously, and let g be a simple function of the form νi χQi , νi ≥ 0. g= i≥1

Then for any nonnegative integers α, β with β ≤ N (K) and f ∈ Lpa (ω, X), we have 1/p p (α) β ¯ f (ζ) g(ζ) |∂ Kζ (z)| dµ(ζ) ω(z) dA(z) Ω1

Ω

p0 /p ≤ kCD (α, β, µ) g p/p0 (λj )χDj f p/p0 p , L

j∈J(µ)

0 (ω)

where k = α!β! 4α+β+5p0 /p and µ(Qj )V (u, Dj )p0 /p

CD (0, 0, µ) = sup j∈J(µ)

2p0 /p

u(λj )p0 /p rj

Pu p0 /p ,

and for α + β > 0, CD (α, β, µ) = sup j∈J(µ)

µ(Qj )V (u, Dj )p0 /p V 2 (K, Dj ) α+β+2p0 /p

u(λj )p0 /p rj

Pu p0 /p .

(2.12)

530

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IEOT

Proof. For a ﬁxed z ∈ Ω1 , we can ﬁnd points ξj = ξj (z) ∈ Qj such that

p/p0 f (α) (ζ) g(ζ) |∂¯β Kζ (z)| dµ(ζ) Ω p/p0 ≤ f (α) (ζ) g(ζ) |∂¯β Kζ (z)| dµ(ζ) Qj

j∈J(µ)

≤

(f (α) (ξj ) g(λj ) |∂¯β Kξj (z)| µ(Qj ))p/p0 .

(2.13)

j∈J(µ)

By the subharmonicity of f (α) p/p0 , we have f

(α)

(ξj )

p/p0

16 ≤ 2 rj

|w−ξj |
f (α) (w)p/p0 dA(w),

and by Lemma 2.1 we obtain (f (α) (ξj ) |∂¯β Kξj (z)|)p/p0 V 2p/p0 (K, Dj ) ≤ k (α+β)p/p +2 f (w)p/p0 |Kw (z)|p/p0 dA(w), 0 Dj rj

(2.14)

with k = (α!β!)p/p0 4(α+β)p/p0 +5 . To introduce the weight u, note that f (w)p/p0 |Kw (z)|p/p0 dA(w) Dj

≤

V (u, Dj ) u(λj )

f (w)p/p0 |Kw (z)|p/p0 u(w) dA(w). (2.15) Dj

Now replace this in (2.13) to conclude that Ω

p/p0 f (α) (ζ) g(ζ) |∂¯β Kζ (z)| dµ(ζ) ≤ k sup

µ(Qj )p/p0 V (u, Dj )V 2p/p0 (K, Dj ) (α+β)p/p +2

0 u(λj )rj ×Pu g p/p0 (λj )χDj f p/p0 (z).

j∈J(µ)

j∈J(µ)

(2.16)

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Then

Atomic Decompositions in Weighted Bergman Spaces

Ω1

Ω

531

1/p p f (α) (ζ) g(ζ) |∂¯β Kζ (z)| dµ(ζ) ω(z) dA(z) µ(Qj )V (u, Dj )p0 /p V 2 (K, Dj )

≤ k sup

α+β+2p /p

0 u(λj )p0 /p rj p0 1/p × g p/p0 (λj )χDj f p/p0 ) ω(z) dA(z) Pu (

j∈J(µ)

Ω

≤ k sup

j∈J(µ)

µ(Qj )V (u, Dj )p0 /p V 2 (K, Dj ) α+β+2p /p

0 u(λj )p0 /p rj p0 /p ×Pu p0 /p g p/p0 (λj )χDj f p/p0 p ,

j∈J(µ)

L

j∈J(µ)

(2.17)

0 (ω)

where k = α!β! 4α+β+5p0 /p . This completes the proof for α + β > 0. Let α = β = 0. Then (2.13) holds and using the subharmonicity of w → f (w)Kw (z)p/p0 , instead of (2.14), we obtain 4 (f (ξj ) |Kξj (z)|)p/p0 ≤ 2 f (w)p/p0 |Kw (z)|p/p0 dA(w) rj |w−ξj |
µ(Qj )p/p0 V (u, Dj ) Pu g p/p0 (λj )χDj f p/p0 (z). 2 u(λj )rj j∈J(µ)

≤ 4 sup

j∈J(µ)

(2.18) Following the same steps as above we obtain 1/p p f (ζ) g(ζ) |Kζ (z)| dµ(ζ) ω(z) dA(z) Ω1

Ω

≤ k sup j∈J(µ)

with k = 4p0 /p .

µ(Qj )V (u, Dj )p0 /p 2p0 /p

u(λj )p0 /p rj

p0 /p Pu p0 /p g p/p0 (λj )χDj f p/p0 p , j∈J(µ)

L

0 (ω)

Corollary 2.1. Assume p, p0 , ω, u, K, Pu are as above and let µ be a ﬁnite complex ¯ Then Tµ is bounded whenever Borel measure, compactly supported with supp µ ⊆ Ω. I(µ, D) < ∞ and CD (0, 0, |µ|) < ∞.

532

Constantin

Proof. Let g =

i≥1

χQi , α = β = 0 in Theorem 2.1 to get

Ω1

IEOT

Ω

1/p p f (ζ) |Kζ (z)| d|µ|(ζ) ω(z) dA(z) p0 /p ≤ kCD (0, 0, |µ|) χDj f p/p0 p , (2.19) L

j∈J(µ)

0 (ω)

since g(λi ) = 1, i ≥ 1, and g = 1 a.e. on Ω. But j∈J(µ) χDj ≤ I(µ, D) and therefore p0 /p χDj f p/p0 p ≤ I(µ, D)p0 /p f Lpa (ω,X) . (2.20) 0 (ω)

L

j∈J(µ)

Now, since E is a compact subset of Ω, there exists a constant k > 0, independent of f , such that 1/p p Tµ f Lpa(ω,X) ≤ f (ζ) |Kζ (z)| d|µ|(ζ) ω(z) dA(z) Ω

≤k

Ω

Ω1

Ω

1/p p f (ζ) |Kζ (z)| d|µ|(ζ) ω(z) dA(z) .

In view of (2.19)–(2.20), we can now deduce that Tµ is bounded on Lpa (ω, X). A slightly more involved application is the following. Corollary 2.2. If Ω, D, p, p0 , u, K, Pu , ω are as above, and cj ≥ 0 for j ≥ 1, then |∂¯β Kλj (z)| p cj ¯β ω(z) dA(z) ≤ aD (β) |cj |p , β ∈ N, (2.21) ∂ Kλj Lpa (ω) Ω j≥1 j≥1 and aD (β) = sup j≥1

( Dj ω dA)1/p V (u, Dj )p0 /p V 2 (K, Dj ) β+2p0 /p ∂¯β Kλj Lpa (ω) u(λj )p0 /p rj

Pu p0 /p I(A, D)p0 +1 ,

where A denotes as usually the normalized area measure on Ω. Proof. Let J be a ﬁnite subset of N∗ = N \ {0}. Since E is a compact subset of Ω, we obtain by subharmonicity |∂¯β Kλj (z)| p cj ¯β ω(z) dA(z) ∂ Kλj Lpa (ω) Ω j∈J |∂¯β Kλj (z)| p cj ¯β ω(z) dA(z) , (2.22) ≤k ∂ Kλj Lpa (ω) Ω1 j∈J

Vol. 59 (2007)

Atomic Decompositions in Weighted Bergman Spaces

where k depends only on p, E, ω. Now put cj g= χQ , ( ω dA)1/p j Dj j≥1 ( Dj ω dA)1/p δλj , µ= ∂¯β Kλ

533

f ≡ 1, α = 0,

j

j∈J

in Theorem 2.1 to get Ω1

|∂¯β Kλj (z)| p cj ¯β ω(z) dA(z) ∂ Kλj Lpa (ω,X) j≥1 p0 p ≤ kCD (0, β, µ) g p/p0 (λj )χDj p L

j∈J

0 (ω)

.

(2.23)

Now since Qi ∩ Qj has zero area and Ω = ∪i≥1 Qi , we have p0 p0 = g p/p0 (λj )χDj ω dA j∈J g p/p0 (λj )χDj p L

0 (ω,X)

Ω j∈J

=

Ω

=

g p/p0 (λj )χDj ∩Qi

p0

i≥1 j∈J

Ω i≥1 j∈J i

g p/p0 (λj )χDj ∩Qi

ω dA

p0

ω dA,

where Ji = {j ∈ J : Qi ∩ Dj = ∅} contains at most I = I(A, D) elements. Therefore, we get p0 p0 p/p0 (λj )χDj p ≤I g p (λj )χDj ∩Qi ω dA j∈J g L

0 (ω)

Ω i≥1 j∈J i

=I

p0

g (λj )

i≥1 j∈Ji

= I p0

i≥1 j∈Ji

≤I

p0 +1

p

ω dA Dj ∩Qi

D ∩Q ω dA cpj j i Dj ω dA

cpi .

i≥1

Replace this in (2.23) and use also relation (2.22) to obtain Ω j∈J

p |∂¯β Kλj (z)| p p cj ¯β ω(z) dA(z) ≤ kI p0 +1 CD (0, β, µ) ci . ∂ Kλj Lpa (ω,X) i≥1

534

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If we now put J = JM = {1, . . . , M } in the above inequality and apply Fatou’s Lemma for M → ∞, we deduce that (2.21) holds with aD (β)

p = I p0 +1 CD (0, β, µ) ( Dj ω dA)1/p V (u, Dj )p0 /p V 2 (K, Dj ) = sup Pu p0 /p I p0 +1 . β+2p0 /p β p /p ¯ 0 j≥0 ∂ Kλj u(λj ) rj

We now turn to the main result of this section which describes our method of approximation of the operators Tµ . Theorem 2.2. Let µ be a ﬁnite complex Borel measure, compactly supported with ¯ Assume that p, p0 , ω, u, K, Pu are as above and that Ω admits a desupp µ ⊆ Ω. composition D as above with I(µ, D) < ∞, CD (0, 0, |µ|) < ∞ and bD (N, µ) =

(diam Q )N |µ|(Q )V (u, D )p0 /p V 2 (K, D )V (∂¯β K, Q ) j j j j j

sup

N +2p0 /p

u(λj )p0 /p rj

0≤β≤N,j∈J(µ)

× Pu p0 /p I p0 /p < ∞. Then: (i) For f ∈ Lpa (ω, X) and Ci,α,β (f ) = f

(α)

(λi ) ∂¯β Kλi Lp (ω) a

the series TµD f =

(ζ − λi )α (ζ − λi )β dµ(ζ), α! β! Qi

∂¯β Kλi Ci,α,β (f ) β ∂¯ Kλi p i≥1 α+β
(2.24)

a

converges in Lpa (ω, X) and deﬁnes a bounded linear operator on this space. Moreover, Tµ − TµD ≤ kbD (N, µ),

(2.25)

where k depends only on ω, E, N, p and p0 . (ii) The coeﬃcients Ci,α,β (f ) satisfy Ci,α,β (f )p ≤ kbD (N, µ)f Lpa (ω,X) ,

(2.26)

i≥1 α+β
where bD (N, µ) = IPu p0

sup i∈J(µ)

0≤α+β
(diam Qi )(α+β)p |µ|(Qi )p 2p0 +(α+β)p

ri

u(λi )p0

V 2p (K, Di )V p0 (u, Di ) .

Vol. 59 (2007)

Atomic Decompositions in Weighted Bergman Spaces

535

Proof. Let J be a ﬁnite subset of N. Let TµJ be the operator given by (1.1) corresponding to the measure µ|∪i∈J Qi . Consider also the operator TJf =

i∈J

∂¯β Kλi Ci,α,β (f ) β . ∂¯ Kλi p L (ω) α+β
(2.27)

a

Let z ∈ Ω1 and apply (2.7) on each Qi , i ∈ J to obtain TµJ f (z) − T J f (z) f (α) (w) |∂¯β Kw (z)| sup (diam Qi )N |µ|(Qi ). sup ≤ α! β! w∈Qi w∈Qi i∈J α+β=N

Note that there exists ζi = ζi (α) ∈ intQi with f (α) (ζi ) ≥

1 sup f (α) (w), 2 w∈Qi

and that sup |∂¯β Kw (z)| ≤ V (∂¯β K, Qi )|∂¯β Kζi (z)|, z ∈ Ω1 .

w∈Qi

If we set να,β =

(diam Qi )N |µ|(Qi )V (∂¯β K, Qi )δζi ,

i≥1

g=

χQi ,

i∈J

then, clearly, the number J(να,β ) given in (2.8) satisﬁes J(να,β ) = {i ∈ N : Qi ∩ supp(να,β ) = ∅, Qi ∈ D} = J(µ). Then we can apply Theorem 2.1 for να,β and g, to conclude 1/p TµJ f − T J f p ωdA Ω1

p 1/p 2(N + 1)p+1/p ≤ f (α) (ζ) g(ζ) |∂¯β Kζ (z)| dνα,β (ζ) ωdA α!β! Ω1 Ω α+β=N p0 /p p/p0 ≤k CD (α, β, να,β ) g (λj )χDj f p/p0 p0 , (2.28)

α+β=N

La (ω,X)

j∈J(µ)

with k = 4N +5p/p0 +1 (N + 1)p+1/p+1 . Note that g p/p0 (λj )χDj = χDj ≤ χ(∪j∈J Dj ) I, j∈J(µ)

j∈J ∩J(µ)

a.e. on Ω. This together with (2.28) imply 1/p TµJ f − T J f p ωdA Ω1

pp0 p ≤ k I p0 /p sup CD (α, β, να,β )χ(∪j∈J Dj ) f p0 p α+β=N

L

0 (ω)

.

536

Constantin

IEOT

Now sup CD (α, β, να,β ) =

α+β=N

=

να,β (Qj )V (u, Dj )

sup α+β=N, j∈J(µ)

(diam Qj )N |µ|(Qj )V (u, Dj )

sup 0≤β≤N, j∈J(µ)

and hence

u(λj )

Ω1

u(λj )

p0 p

p0 p

p0 p

p0 p

V 2 (K, Dj )

α+β+2 rj

p0 p

Pu

V 2 (K, Dj )V (∂¯β K, Qj )

p N +2 p0

p0 p

Pu

p0 p

,

rj

1/p TµJ f − T J f p ωdA ≤ k bD (N, µ)χ(∪j∈J Dj ) f Lp (ω,X) .

Since E is a compact subset of Ω, we obtain from above 1/p TµJ f − T J f p ωdA Ω 1/p = TµJ f − T J f p ωdA + TµJ f − T J f p ωdA Ω1 E 1/p J J p ≤k Tµ f − T f ωdA Ω1

≤ kbD (N, µ)χ(∪j∈J Dj ) f Lp (ω,X) ,

(2.29)

where k depends only on p, p0 , N, E. For M ∈ N we choose J = JM = {1, .., M }. We claim that (2.29) implies that {TµJM f − T JM f }M∈N is a Cauchy sequence, for any f ∈ Lpa (ω, X). Indeed, for M, P ∈ N, M < P , we have (TµJM − T JM )f − (TµJP − T JP )f Lpa (ω,X) = (TµJM,P − T JM,P )f Lpa (ω,X) ≤ k bD (N, µ)χ(∪j∈JM,P Dj ) f Lp (ω,X) → 0

as M, P → ∞,

where JM,P = {M + 1, . . . , P }. Since I p0 /p CD (0, 0, µ) < ∞ we deduce by Theorem 2.1 that Tµ is bounded and it is easy to see that TµJM f → Tµ f as M → ∞. Then from the above we deduce that the limit limM→∞ T JM f exists and denote it by TµD f . We now put J = JM in (2.29) and let M → ∞, to obtain that (2.25) holds and that TµD is bounded, which proves (i). It remains to prove (ii). Since E is a compact subset of Ω, we have β ¯β Kλ (z)|p ω(z) dA(z) ≤ k ∂¯ Kλi (z)p p = | ∂ |∂¯β Kλi (z)|p ω(z) dA(z), i L (ω) a

Ω

Ω1

where k depends on E, p, ω. Using similar arguments as above, namely subharmonicity together with Lemma 2.1, it follows that

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Atomic Decompositions in Weighted Bergman Spaces

f (α) (λi )p

537

|∂¯β Kλi (z)|p ω(z) dA(z) Ω kV 2p (K, Di ) f (ξ)p |Kξ (z)|p dA(ξ) ω(z) dA(z) ≤ 2+(α+β)p ri Ω |ξ−λ |< ri 1 i 2 p0 kV 2p (K, Di ) p/p0 p/p0 f (ξ) |Kξ (z)| dA(ξ) ω(z) dA(z). ≤ 2p +(α+β)p Ω Di ri 0 In the last relation we have used (again) the mean value inequality. Moreover, (α) p |∂¯β Kλi (z)|p ω(z) dA(z) f (λi ) Ω

≤k

V 2p (K, Di )V p0 (u, Di ) 2p0 +(α+β)p

ri

×

Di

V 2p (K, Di )V p0 (u, Di )

Pu (χDi f p/p0 )pL0p0 (ω) 2p0 +(α+β)p p 0 ri u(λi ) p0 2p Pu V (K, Di )V p0 (u, Di ) k (χDi f p/p0 )pL0p0 (ω) . 2p0 +(α+β)p p 0 ri u(λi )

≤k ≤

Ω

u(λi )p0

p0 u(ξ)f (ξ)p/p0 |Kξ (z)|p/p0 dA(ξ) ω(z) dA(z)

(2.30)

This last estimate implies that M

Ci,α,β (f )p

i=1 α+β
≤k

M

(diam Qi )(α+β)p |µ|(Qi )p 2p0 +(α+β)p

i=1 α+β
ri

2p

u(λi )p0

×Pu V (K, Di )V (u, Di ) f (z)p ω(z) dA(z), ≤ k bD (N, µ) p0

f (z)p ω(z) dA(z)

p0

Di

∪M i=1 Di

for all M ≥ 1. The result now follows if we let M tend to inﬁnity.

We now want to apply the above results in order to obtain a discretization of the operator Tµ . In the case of an invertible operator Tµ this follows immediately from Theorem 2.2. Corollary 2.3. Assume that p, p0 , ω, u, K, Pu are as above. Let µ be a ﬁnite complex ¯ and suppose that Tµ is bounded and invertible. Borel measure with supp µ ⊆ Ω Suppose there exist N ∈ N and a decomposition D of Ω such that kbD (N, µ) < Tµ−1 −1 ,

(2.31)

538

Constantin

IEOT

if k is the constant in (2.25). Then for any f ∈ Lpa (ω, X) we have Tµ f (z) =

∞ N −1 i=1 j=0

∂¯j Kλi (z) Cij (f ) ¯j , ∂ Kλi (z)Lpa (ω)

Cij (f ) ∈ X.

(2.32)

If, in addition, bD (N, µ0 ), sup0≤β
Cij (f )p ∼ f pLpa (ω,X) ,

(2.33)

i=0 j=0

where “∼” means that the involved constants are independent of f . Moreover, f → Cij (f ) are bounded linear operators from Lpa (ω, X) into X. Proof. Theorem 2.2, (i), together with (2.31) give Tµ − TµD <

1 Tµ−1

.

(2.34)

Then I − Tµ−1 TµD ≤ Tµ−1 Tµ − TµD < 1. and hence Tµ−1 TµD is invertible and therefore for any f ∈ Lpa (ω, X), there exists g ∈ Lpa (ω, X), such that Tµ f = TµD g. If bD (N, µ), aD (β) < ∞, then (2.33) follows immediately by Theorem 2.2 and Corollary 2.2 . There is an interesting special case, where we do not need the invertibility of the operator Tµ . Assume that there exists a function v ∈ C(Ω) such that K has the following reproducing property for the space Lpa (ω, X): f (z) = f (w)Kw (z)v(w) dA(w), z ∈ Ω, f ∈ Lpa (ω, X). (2.35) Ω

This means that the identity operator I on Lpa (ω, X) can be regarded as the Toeplitz-type operator Tµ0 , where dµ0 = v dA. Its discretization is described below. Usually this is called an atomic decomposition. Corollary 2.4. Suppose that K has the reproducing property (2.35) for the space Lpa (ω, X), and that there exist N ∈ N and a decomposition D of Ω with kbD (N, µ0 ) < 1, where k is the constant from relation (2.25). Then any f ∈ Lpa (ω, X) can be written as ∞ N −1 ∂¯j Kλi (z) Cij (f ) ¯j , Cij (f ) ∈ X. (2.36) f (z) = p ∂ K (z) λ L (ω) i a i=1 j=0 Also, if bD (N, µ0 ), sup0≤β
Cij (f )p ∼ f pLpa (ω,X) ,

(2.37)

Vol. 59 (2007)

Atomic Decompositions in Weighted Bergman Spaces

539

where “∼” means that the involved constants are independent of f . Moreover, f → Cij (f ) are bounded linear operators from Lpa (ω, X) into X. Remark 2.1. Such atomic decompositions can be used to obtain discretizations of general Toeplitz operators. Suppose that K has the reproducing property (2.35) and that we can discretize the identity operator on Lpa (ω, X) , i.e., the conclusion of Corollary 2.4 holds. Then, we claim that we can discretize any bounded Toeplitz operator Tµ with kernel K. Indeed, consider a bounded Toeplitz operator Tµ . Then, for s ∈ C with |s| < (2Tµ )−1 , the operator S = I + sTµ ,

(2.38)

is invertible. For any f ∈ Sf ∈ and hence, by our assumptions, we get ∞ N −1 ∂¯j Kλi (z) Cij (Sf ) ¯j , Cij (f ) ∈ X, Sf = p ∂ K (z) λi La (ω) i=1 j=0 Lpa (ω, X),

with

∞ N −1

Lpa (ω, X),

Cij (Sf )p ∼ Sf pLpa(ω,X) ∼ f pLpa(ω,X) .

i=0 j=0

Moreover, relations (2.36) and (2.37) hold for f . Then from (2.38) we get Tµ f =

∞ N −1 i=1 j=0

where

Dij (f )

=

j 1 s (Ci (Sf )

∂¯j Kλi (z) Dij (f ) ¯j , ∂ Kλi (z)Lpa (ω)

Dij (f ) ∈ X,

− Cij (f )), and clearly

∞ N −1

Dij (f )p ≤ kf pLpa(ω,X) .

i=0 j=0

3. Atomic decompositions for vector-valued weighted Bergman spaces on the polydisc In this section we focus on the simplest domain Ω. Throughout we assume that Ω = D and Ω1 = D \ {z : |z| ≤ 1/2}. For η > −1, we let dAη (z) = (η + 1)(1 − |z|2 )η dA(z), where dA(z) denotes the normalized Lebesgue area measure on D (that is dA(z) = π1 dxdy when z = x + iy). The most common reproducing kernel for the unit disc has the form 1 η Kw (z) = f or w, z ∈ D, (1 − wz)η+2 and it corresponds to the space L1a ((1 − |z|2 )η , X). The problem of characterizing the weights for which the Bergman projection f (ξ) f → Pη f (z) = (1 − |ξ|2 )η dA(ξ), η+2 (1 − zξ) D

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is bounded from Lp0 (ω(1 − |z|2 )η ) to Lp0 (ω(1 − |z|2 )η ) was solved by B´ekoll´e [5]. He found that these weights are precisely those ω ∈ L1 ((1 − |z|)η ), ω > 0 that satisfy the following condition called the B´ekoll´e condition Bp0 (η). Condition Bp0 (η). There is a constant k > 0 such that

p0 /q0 ω −q0 /p0 (z) dAη

ω dAη S(a)

≤ kAη (S(a))p0 ,

a ∈ D,

S(a)

where p0 > 1, q0 =

p0 p0 −1 ,

and S(a) denotes the set

a−z : Re(z¯ a) ≤ 0 . S(a) = 1−a ¯z

This is a very natural condition which is analogous to the famous Muckenhoupt condition Ap0 . For the standard weights ω(z) = (1 − |z|2 )a , a > −1, the condition Bp0 (η) is satisﬁed when −η − 1 < a < (η + 1)(p0 − 1). B´ekoll´e’s result is as follows. Theorem 3.1 (B´ekoll´e, [5]). Let p0 > 1, η > −1. For a weight function ω we consider on Lp0 (ω) the sublinear operator |f (ξ)| (1 − |ξ|2 )η dA(ξ). f → Pη f (z) = η+2 |1 − zξ| D Then Pη is bounded from Lp0 (ω) to Lp0 (ω) if and only if condition Bp0 (η).

ω (1−|z|2 )η

satisﬁes the

From now on we shall work with Bergman spaces Lpa (ω, X), p > 0 where satisﬁes the condition Bp0 (η), for some p0 > 1, p0 ≥ p, η > −1. The considerations in the previous section apply for the sublinear projection-type operator Pu = Pη , that is, for u(z) = (1 − |z|2 )η , and for the kernel K = K γ , where γ = (η + 2)p0 /p − 2. Let us also note that for such a weight ω, the space Lpa (ω, X) is continuously contained in L1a ((1 − |z|2 )γ , X). ω(z) (1−|z|2 )η

ω(z) Proposition 3.1. If (1−|z| 2 )η satisﬁes the condition Bp0 (η), then there exists k > 0, depending only on p, ω, such that 1/p p f (z) dAγ (z) ≤ k f (z) ω(z) dA(z) , (3.1) D

D

for any f ∈ Lpa (ω, X). Proof. If f ∈ Lpa (ω, X), then obviously f p/p0 ∈ Lp0 (ω), and by Theorem 3.1, p/p we have Pη (f p/p0 )(z) < ∞. This implies f ∈ La 0 ((1 − |z|2 )η , X). But p/p La 0 ((1−|z|2 )η , X) is continuously contained in L1a ((1−|z|2 )γ , X) (see [12], Theorem 2.2, for the proof in the scalar case; the proof in the vector-valued case follows with the same arguments).

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Now (3.1) implies that the function K γ has the reproducing property, i.e., (2.35), for Lpa (ω, X), γ f (w)Kw (z)dAγ (w), z ∈ Ω, f ∈ Lpa (ω, X). (3.2) f (z) = D

Thus the identity operator I on Lpa (ω, X) can be regarded as the Toeplitz type operator Tµ0 , where dµ0 = dAγ . In the example below we describe a decomposition D of the unit disc and estimate the constants considered in the previous section. Example 1. For c ∈ (1, 17 16 ), we consider a decomposition Dc of D in “curved” rectangles that we shall now describe. For each n = 0, 1, 2, 3, . . . we let ρn = 1 − c1n and we consider the annulus Acn = {z : ρn ≤ |z| ≤ ρn+1 }.

n+1

(3.3)

For every ﬁxed n ≥ 1 we divide the annulus Acn in M (n) = 2πc c−1 + 1 equally sized “curved” rectangles deﬁned by

2π(k + 1) 2πk ≤t≤ , 0 ≤ r < 1 ∩ Acn , Qcn,k = z = reit : (3.4) M (n) M (n) k = 0, 1, . . . , M (n) − 1. Taking into account our geometric construction it is easy to see that for c−1 2 1 c n ≥ 1, we have diam Qcn,k ≤ 2 cc−1 , dist(Qcn,k , ∂D) = cn+1 , n+1 , area Qn,k ≤ 4 cn+1 c and hence 3a) is satisﬁed. Obviously, 3b) holds as well. We let {Qi }i≥2 be an enumeration of all sets obtained in this way and put Qc1 = Ac0 . Choose λi ∈ int Qci such that Qci is starlike with respect to λi and let ri = (1 − |λi |)/2, i ≥ 1. Note that 3c)–3d) are also veriﬁed. Thus, this is a decomposition of the type described in the previous section. Our geometric construction also shows that the maximal number of discs Di that intersect a ﬁxed Di0 from the family is a ﬁnite number I = I(c), with I ∼ 1/ ln2 c.

(3.5)

It is a standard matter to check that sup V (∂¯β K γ , Di ) < ∞, sup V (u, Di ) = V ((1 − |z|2 )η , Di ) < ∞, i

i

where the upper bounds can be chosen independently of c ∈ (1, 17/16), and |µ0 |(Qci ) = Aγ (Qci ) ≤ k(area Qci )(1 − |λi |)γ . By these considerations it follows easily that there exists k > 0, independent of c, such that I ≤ k(c − 1)−2 , CD (0, 0, µ0 ) ≤ k(c − 1)2 , bDc (N, µ0 ) ≤ k(c − 1)N +2−2p0 /p , bDc (N, µ0 ) ≤ k(c − 1)2p−2 < ∞. Let us now estimate aD (β), 0 ≤ β < N . This will follow from an asymptotic estimate for Kλγ Lpa (ω) given below.

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Lemma 3.1. Let r ∈ (0, 1), β ∈ N, and assume that p, γ, K, ω are as above. Then there exists k > 1 independent of λ ∈ D such that 1/p ( Dλ,r ω(z) dA(z))1/p 1 ( Dλ,r ω(z) dA(z)) γ β ≤ ∂¯ Kλ Lpa (ω) ≤ k , k (1 − |λ|)γ+β+2 (1 − |λ|)γ+β+2 where Dλ,r = {z : |z − λ| < r(1 − |λ|)}. Proof. The ﬁrst inequality follows easily, since, for z, λ ∈ Dλ,r , we have (1 − |λ|) ≤ ¯ ≤ 4(1 − |λ|). To prove the second one, consider a decomposition Dc of D, |1 − λz| with λ = λi , for some i ≥ 1. Note that supλ∈D V (u, Dλ,r ), V (K, Dλ,r ) < ∞ and apply Theorem 2.1 with f = g ≡ 1, α = 0, µ = δλ . By the above we deduce that aDc (β) < k(c − 1)−2(p0 +1) < ∞, 0 ≤ β < N , with k independent of c. Now, for N > 2p0 /p − 2 and c close enough to 1, we get kbDc (N, µ0 ) < 1, where k is the constant from relation 2.25. Thus the hypotheses in Corollary 2.4 are satisﬁed and we obtain the following result. Corollary 3.1. Consider p > 0, p0 > 1, η > −1, and suppose that p0 ≥ p if p > 1. Assume that ω is a weight function such that ω(z)/(1 − |z|)η satisﬁes the condition Bp0 (η). Let N ∈ N with N > 2(p0 /p − 1) and set γ = pp0 (η + 2) − 2. Then any f ∈ Lpa (ω, X) can be written as f (z) =

∞ N −1 i=1 j=0

∂¯j K γ (z) Cij (f ) ¯j γ λi , ∂ Kλi (z)Lpa (ωdA)

λi ∈ D, Cij (f ) ∈ X,

(3.6)

where f → Cij (f ) are bounded linear operators from Lpa (ω, X) into X, with ∞ N −1

Cij (f )p ∼ f pLpa (ω,X) .

(3.7)

i=0 j=0

Here “∼” means that the involved constants are independent of f . The parameters p0 and η in the above corollary are, of course, determined by the B´ekoll´e condition. However, we do have some freedom in choosing these numbers since: (i) for p0 < p0 , condition Bp0 (η) implies condition Bp0 (η); (ii) if η < η and ω(z)/(1−|z|)η satisﬁes the condition Bp0 (η), then ω(z)/(1−|z|)η will satisfy the condition Bp0 (η ) as well. Since the Bergman kernel of a product of two domains is the product of the Bergman kernels for the domains, if an atomic decomposition as in Corollary 3.1 is true for two domains, it is true for their product (see [7]). In particular, from Corollary 3.1 we immediately obtain an atomic decomposition theorem for vector-valued weighted Bergman spaces deﬁned on the polydisc.

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Before stating this theorem let us introduce some useful notation. Let Dn denote the polydisc in Cn for n ∈ N. Suppose (λi )i≥0 is a sequence of points in D. For α = (α1 , . . . , αn ), β = (β1 , . . . , βn ), γ = (γ1 , . . . , γn ) ∈ Nn , we shall denote ∂¯β K γ (z) = ∂¯β1 K γ1 (z1 ) · ∂¯β2 K γ2 (z2 ) · · · ∂¯βn K γn (zn ), λα1

λα

λα2

λαn

where z = (z1 , . . . , zn ) ∈ D . For a positive function ω : D → R+ , we deﬁne the Bergman space Lpa (Dn , ω, X) that consists of analytic functions f : Dn → X with 1/p p f La (Dn ,ω,X) = f (z)p ω(z) dA(z1 ) · · · dA(zn ) < ∞, n

n

Dn

where z = (z1 , . . . , zn ) ∈ D . n

Corollary 3.2. Consider p > 0 and p0i > 1, ηi > −1 for 1 ≤ i ≤ n, and suppose that p0i ≥ p if p > 1. Assume that ω(z) = ω1 (z1 ) · · · ωn (zn ) is a weight function on Dn such that each ωi , 1 ≤ i ≤ n, is a continuous strictly positive function integrable on D with ωi (zi )/(1 − |zi |)ηi satisfying the condition Bp0i (ηi ). Let Ni ∈ N with Ni > 2(p0i /p − 1), set γi = pp0i (ηi + 2) − 2 for 1 ≤ i ≤ n and put γ = (γ1 , . . . , γn ). Then there exists a sequence (λi )i≥0 ⊂ D such that any f ∈ Lpa (Dn , ω, X) can be written as ∂¯β Kλγα (z) β Cα (f ) ¯β γ , Cαβ (f ) ∈ X, f (z) = ∂ K (z)Lp (Dn ,ω,C) α∈Nn β∈B

λα

a

where B = {β = (β1 , . . . , βn ) ∈ N : 0 ≤ βi ≤ Ni , 1 ≤ i ≤ n}. Moreover, f → Cαβ (f ) are bounded linear operators from Lpa (Dn , ω, X) into X, with Cαβ (f )p ∼ f pLpa (Dn ,ω,X) . n

α∈Nn β∈B

Here “∼” means that the involved constants are independent of f . Proof. The conclusion is obtained inductively from Corollary 3.1, taking into account ω Lpa (Dn , ω, X) = Lpa D, ω1 , Lpa Dn−1 , , X ω1 = Lpa (D, ω1 , Lpa (D, ω2 , . . . , Lpa (D, ωn , X), . . .)).

4. Atomic decompositions for vector-valued weighted Bergman spaces on domains of finite connectivity Let Ω be a bounded domain whose boundary consists of ﬁnitely many analytic simple closed curves. Denote by Γ0 , . . . , ΓN the boundary curves with ∂Ω = ∪N i=0 Γi , such that Γ0 is the boundary of the unbounded component of C \ Ω. Denote Ω0 = Int(Γ0 ) and Ωi = Ext(Γi ) ∪ {∞} for 1 ≤ i ≤ N . Let ε=

min

0≤i<j≤N

dist(Γi , Γj ) . 2

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For 1 ≤ i ≤ N consider the Riemann mappings gi : D → Ωi with hi (z) , hi (0) = 0, (4.8) z with hi : D → C analytic. Since the boundary curves are analytic, each function gi can be continued analytically to a one to one conformal map on a neighborhood of D with |gi (z)| > 0 on D. Let ω : Ω → R be a continuous, integrable and strictly positive function. For p0 > 1, p0 = p0 /(p0 − 1) and η > −1 we say that ω satisﬁes the condition Bp0 (η) on Ω if for any a ∈ ∂Ω and h ∈ (0, ε) we have p0 p p − 0 ω dAη ω p0 dAη 0 ≤ k Apη0 D(a, h) gi (z) =

Ω∩D(a,h)

Ω∩D(a,h)

where D(a, h) = {z : |a − z| < h} and dAη (z) = dist(z, ∂Ω)η dA(z). Given a Banach space X, we shall from now on denote Lpa (Ω, ω, X) = {f : Ω → X analytic with f (z)pω(z) dA(z) < ∞}. Ω

An application of the Cauchy integral formula (see [6]) shows that if f is analytic on Ω then we can write f = f0 + f1 + · · · + fN , (4.9) where each fi , 0 ≤ i ≤ N , is an X-valued analytic function on Ωi , and for 1 ≤ i ≤ N we have 1 a1i a2i near ∞. (4.10) a0i + + 2 + ··· fi (z) = z z z Given a continuous, integrable and strictly positive weight function ω : Ω → R, we can extend it to each Ωi in the standard way. Let ω0 : Ω0 → R be a continuous, integrable and strictly positive function such that ω0 (z) = ω(z) for z ∈ A0 = {w ∈ Ω : dist(w, ∂Ω0 ) < ε}. Since all fi ’s, i ≥ 1, are bounded near ∂Ω0 and p f ∈ Lpa (Ω, ω, X), we deduce that f0 = f − N i=1 fi ∈ La (Ω0 , ω0 , X). We are now going to deﬁne continuous, integrable and strictly positive functions ωi : Ωi → R with ωi (z) = ω(z) for z ∈ Ai = {w ∈ Ω : dist(w, ∂Ωi ) < ε}, i ≥ 1. Using (4.8) we deduce that we can deﬁne ωi on Ωi in such a way that ωi ◦ gi · |gi |2 becomes a positive continuous function on D. Note that, in this case, ωi (z) = O( |z|1 2 ) near ∞. We claim that fi ∈ Lpa (Ωi , ωi , X), i ≥ 1. Indeed, all fj with j = i are bounded near ∂Ωi and since f ∈ Lpa (Ω, ω, X) we obtain that |fi |p is integrable with the weight ω near ∂Ωi . On the other hand, by the decay of ωi at ∞ and relation (4.10), we deduce the integrability near ∞ and the claim follows. In view of the above considerations, Corollary 3.1 leads to the corresponding result for ﬁnitely connected domains. Theorem 4.1. Let p > 0, p0 ≥ p, p0 ≥ 1. Consider M ∈ N with M > 2(p0 /p − 1) and set γ = pp0 (η + 2) − 2. Assume Ω is as above and ω : Ω → R is a continuous, strictly positive and integrable function such that ω(z)/(dist(z, ∂Ω))η satisﬁes

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Bp0 (η) on Ω. Then any f ∈ Lpa (Ω, ω, X) can be written as f (z) =

∞ M−1 N

α Cij (f )

j=1 α=0 i=1

◦ gi−1 (z)

∂¯α Kgγ−1 (µ

ji )

i

∂¯α Kgγ−1 (µ i

ji

◦ gi−1 Lpa (Ω,ω,X) )

,

α Ci,j (f ) ∈ X,

where gi are as above, µji ∈ C and only ﬁnitely many µji ’s lie outside Ω. Moreover, α f → Cij (f ) are bounded linear operators from Lpa (Ω, ω, X) into X, with ∞ M−1 N

α Cij (f )p ∼ f pLpa(Ω,ω,X) .

j=1 α=0 i=1

Here “∼” means that the involved constants are independent of f . Proof. Let f ∈ Lpa (Ω, ω, X) and consider its decomposition given in relation (4.9). Let ωi be the above described extension of ω to Ωi , 0 ≤ i ≤ N . By a change of variables we obtain p |fi | ωi dA = |fi ◦ gi |p · |gi |2 · ωi ◦ gi dA, 1 ≤ i ≤ N. Ωi

Since

|gi |

D

is bounded below and above on gi−1 (Ai ), we obtain z, w ∈ gi−1 (Ai ),

|gi (z) − gi (w)| ∼ |z − w| for and

|gi−1 (z) − gi−1 (w)| ∼ |z − w| for z, w ∈ Ai , 0 ≤ i ≤ N. Using the above, it is easy to see that (ωi ◦ gi · |gi |2 )/(1 − |z|2 )η satisﬁes Bp0 (η) on D, since ω/dist(z, ∂Ω)η satisﬁes Bp0 (η) on Ω. Then by Corollary 3.1 applied to fi ◦ gi ∈ Lpa (D, ωi ◦ gi · |gi |2 , X) we obtain that there exist λj ∈ D and cα ij ∈ X such that ∞ M−1 ∂¯α Kλγj (z) cα , fi ◦ gi (z) = ij ¯α γ ∂ K Lp (D, ω ◦g ·|g |2 ) λj

j=1 α=0

with

i

a

i

i

p p cα . ij ∼ fi Lp a (Ωi ,ωi ,X)

j,α

Hence there are µji = gi (λj ) ∈ Ωi , out of which only ﬁnitely many lie outside Ai , with ∂¯α Kgγ−1 (µ ) ◦ gi−1 (z) ji α fi (z) = cij α γ i ¯ K −1 ∂ ◦ gi−1 Lpa (Ωi ,ωi ) j,α g (µ ) ji

i

=

j,α

α Cij

∂¯α Kgγ−1 (µ

ji )

i

∂¯α Kgγ−1 (µ i

ji )

◦ gi−1 (z)

◦ gi−1 Lpa (Ω,ω)

,

where α ¯α γ Cij = cα ij ∂ Kg−1 (µ i

ji )

◦ gi−1 Lpa (Ω,ω) /∂¯α Kgγ−1 (µ i

ji )

◦ gi−1 Lpa (Ωi ,ωi ) .

(4.11)

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Since only ﬁnitely many µji ’s lie outside Ai , we have ∂¯α Kgγ−1 (µ i

and hence

j,α

ji )

◦ gi−1 Lpa (Ωi ,ωi ) ∼ ∂¯α Kgγ−1 (µ

α |Cij |∼

i

ji )

p |cα ij | ∼ fi La (Ωi ,ωi ,X) ,

◦ gi−1 Lpa (Ω,ω)

0 ≤ i ≤ N.

(4.12)

j,α

By the deﬁnitions of fi , ωi it follows that f Lpa(Ω,ω,X) ∼

N

fi Lpa (Ωi ,ωi ,X) .

i=0

From the above relation together with (4.9) and (4.11)–(4.12) we can conclude. As in Section 3, the above theorem can be extended to products of ﬁnitely connected domains.

5. The dual of Lpa (ω, X), 0 < p ≤ 1 As an application of the atomic decomposition theorem, we shall determine the dual of the spaces Lpa (ω, X), 0 < p ≤ 1, where the domain Ω is again the unit disc in C. Of course, the atomic decomposition theorem provides a strong tool for this purpose since it implies that a bounded linear functional is completely determined by its action on the kernels Kλγi and their derivatives. As mentioned in the introduction, the duality of such spaces has attracted a lot of attention, see [16], [9]. As in the previous section, for 0 < r < 1, λ ∈ D, we denote Dλ,r = {z : |z − λ| < r(1 − |λ|)}. Theorem 5.1. Suppose that the hypotheses of Corollary 3.1 are satisﬁed for some p ≤ 1, and assume, without loss of generality, that γ, p0 /p ∈ Z. Then, the dual of Lpa (ω, X) can be identiﬁed with the space (1 − |λ|)γ+2 g (γ+1)(λ)X ∗ ∗ <∞ , M = g : D → X : g analytic with sup ( Dλ,r ω(z) dA(z))1/p λ∈D under the pairing

g, f = lim

r→1

0

2π

f (reit ), g(re−it )

dt , 2π

(5.13)

where g ∈ M and f is analytic in a neighbourhood of D. Proof. The fact that we can choose γ, p0 /p ∈ Z follows immediately from the remarks after Corollary 3.1. A direct consequence of Corollary 3.1, is that the Xvalued polynomials are dense in Lpa (ω, X). Let l be a linear bounded functional on Lpa (ω, X). For each n ≥ 0, consider the linear mapping φn deﬁned on X by x, φn = l(z n x),

x ∈ X.

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|x, φn | ≤ l∗ z n Lpa (ω) x ≤ l∗

D

1/p ω(z) dA(z) x,

547

n ≥ 0,

shows that φn are bounded linear functionals on X with supn≥0 φn X ∗ < ∞. Hence φn z n , g : D → X ∗ , g(z) = n≥0

is analytic in D. Now let f = n≥0 fˆ(n)z n be analytic in a neighbourhood of D. Since its power series converges to f uniformly on D, it will converge in the · Lpa (ω,X) norm as well. Using also the fact that supn≥0 φn X ∗ < ∞, we get l(f ) = l(fˆ(n) z n ) = fˆ(n), φn = lim fˆ(n), φn r2n n≥0

2π

= lim

r→1

r→1

n≥0

0

f (reit ), g(re−it )

n≥0

dt . 2π γ+1

In particular, for λ ∈ D, x ∈ X and fλ,x (z) = (1−zλz) ¯ γ+2 x, we obtain by Cauchy’s formula 2π ¯ x, g (γ+1) (λ) dt (reit )γ+1 = . (5.14) x, g(re−it ) l(fλ,x ) = lim ¯ it )γ+2 2π r→1 0 (γ + 1)! (1 − λre From (5.14) we now deduce |x, g (γ+1) (λ)| ≤ (γ + 1)! l∗ fλ,x Lpa (ω,X) ≤ (γ + 1)! l∗ Kλγ Lpa (ω) x, which implies g (γ+1) (λ)X ∗ < ∞. Kλγ Lpa (ω) λ∈D sup

By Lemma 3.1, this condition is equivalent to (1 − |λ|)γ+2 g (γ+1)(λ)X ∗ < ∞. ( Dλ,r ω(z) dA(z))1/p λ∈D sup

Thus we proved that any bounded linear functional on Lpa (ω, X) is given by a function in M. It remains to show that any function in M deﬁnes a bounded linear functional on Lpa (ω, X) via the pairing (5.13) . Take g ∈ M. We ﬁrst note that the condition g (γ+1) (λ)X ∗ sup < ∞, Kλγ Lpa (ω) λ∈D implies sup λ∈D

g (γ+j+1) (λ)X ∗ Kλγ+j Lpa (ω)

< ∞,

j ≥ 0.

(5.15)

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This follows by the standard Cauchy estimate and Lemma 3.1 . Indeed, we have g (γ+j+1) (z)X ∗ j!

≤

supDz,1/4 g γ+1 (ζ)X ∗ (1 − |z|)j

≤

k (1 − |z|)j

sup ζ∈Dz,1/4

Kζγ Lpa (ω) .

Hence, in order to prove (5.15), it suﬃces to verify that (1 − |z|)−j

sup ζ∈Dz,1/4

Kζγ Lpa (ω) ≤ kKζγ+j Lpa (ω) ,

which is, again, a direct application of Lemma 3.1. Let now x ∈ X, λ ∈ D and j ∈ N, j ≤ m for some ﬁxed m ∈ N. Note that (γ + j + 1)! 2π dt (reit )γ+j+1 lg (z γ+1 ∂¯j Kλγ (z)x) = lim x, g(re−it ) it γ+j+2 ¯ r→1 (γ + 1)! 2π (1 − λre ) 0 (γ + j + 1)! ¯ = x, g (γ+j+1) (λ), (γ + 1)!2 by the same formula used in (5.14). By the above relation together with (5.15) we obtain ∂¯j K γ (z) x | ≤ kx, λ ∈ D, 0 ≤ j ≤ m. (5.16) |lg z γ+1 ¯j γ λ ∂ Kλ (z)Lpa (ω) Take now h ∈ Lpa (ω, X). By Corollary 3.1, we have h(z) =

∞ N −1

Cij

i=0 j=1

with

∂¯j Kλγi (z)

∂¯j Kλγi (z)Lpa (ωdA) ∞ N −1

,

Cij = Cij (h) ∈ X, λi ∈ D,

Cij p ≤ khpLpa (ω,X) .

(5.17)

(5.18)

i=0 j=1

Hence, in view of (5.16) (with m = N − 1), we deduce |lg (z γ+1 h)| ≤ k

∞ N −1

Cij = k

i=0 j=1

∞ N −1

Cij 1−p Cij p ,

i=0 j=1

and by (5.18), we get |lg (z γ+1 h)| ≤ kh1−p Lp a (ω,X)

∞ N −1

Cij p ≤ khLpa (ω,X) .

i=0 j=1

Now let f be analytic in a neighbourdhood of D. For this choice of f we have 2π γ dt it −it lg (f ) = lim = f (re ), g(re ) fˆ(n), gˆ(n) + lg (z γ+1 h), (5.19) r→1 0 2π n=0 where h(z) = (f (z) − γn=0 fˆ(n)z n )/z γ+1. Clearly h ∈ Lpa (ω, X) and there exists k > 0 independent of f, h such that hLpa (ω,X) ≤ kf Lpa(ω,X) , and hence |lg (z γ+1 h)| ≤ kf Lpa(ω,X) .

(5.20)

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We can easily see that fˆ(n) ≤ kf Lpa (ω,X) , for 0 ≤ n ≤ γ, and therefore we get γ fˆ(n), gˆ(n) ≤ k (sup ˆ g(n)X ∗ ) f Lpa (ω,X) . (5.21) n≤γ

n=0

Relations (5.19)–(5.21) show that lg is a bounded linear functional on Lpa (ω, X) and thus the proof is complete. Remark 5.1. An equivalent norm on (Lpa (ω, X))∗ is given by (1 − |λ|)γ+2 g (γ+1) (λ)X ∗ , ( Dλ,r ω(z) dA(z))1/p λ∈D

g = sup |g (n) (0)| + sup n≤γ

in other words, the identiﬁcation of (Lpa (ω, X))∗ , given by Theorem 5.1 is an isomorphism. Since we have worked in full generality, the result has a number of interesting consequences. For instance, it can be applied to study the duality for Bergman spaces on product domains. An example of this type is given below. Example. Denote D2 = D × D and Lsa = Lsa (1, C). Consider for 0 1, p/s 2 |f (z, w)|s dA(w) dA(z) < ∞}. N = {f : D → C analytic with D

D

Note that N = Lpa (1, Lsa ). For ω = 1/(1 − |z|2 )η to satisfy Bp0 (η), one must have η > −1/p0, which, in terms of γ, means γ > (p0 + 1)/p − 2. Then ∗

2

N = {g : D → C analytic with : sup λ∈D

∂λγ+1 g(λ, ·)Lsa

(1 − |λ|)−γ−2+2/p

< ∞},

where 1/s + 1/s = 1 and γ is the smallest integer for which γ > (p0 + 1)/p − 2. In the case of the standard weighted vector-valued Bergman spaces, Theorem 5.1 completes the results obtained by Arregui and Blasco [3] for p ≥ 1.

6. Compactness of Tµ For investigating the compactness of Tµ we restrict our attention to the case when the target space X has ﬁnite dimension. In the inﬁnite dimensional case this operator fails to be compact for obvious reasons. For example, let us take a sequence for m = n and suppose there {en }n ⊂ X with en = 1 and en − em ≥ 1/2, exists a scalar-valued function φ ∈ Lpa (ω) with D φ(w)Kw (z) dµ(w) = 0, for some z ∈ D, in other words, the corresponding scalar-valued operator is not identically zero. The functions fn (z) = φ(z)en , z ∈ D, p belong to La (ω, X) and supn fn Lpa (ω,X) < ∞. On the other hand, the sequence Tµ fn (z) = en φ(w)Kw (z) dµ(w), z ∈ D, D

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does not contain any convergent subsequence in Lpa (ω, X). From now on we shall therefore assume that the target space is X = Cn . In the general setting considered in Section 2, we can prove the following result which is our main tool for the investigation of the compactness of Toeplitz operators. Theorem 6.1. Let dim X < ∞ and u, Pu , K, N be as in Section 2. Assume that ¯ and such that Tµ is µ is a ﬁnite complex, Borel measure, with supp µ ⊆ Ω, bounded. In addition, suppose there exists a decomposition D of Ω and a partition S1 , S2 , . . . , Sn , . . . of N such that bD (N, µ|∪j∈S Qj ) < ∞. (6.22) n

n≥1

Then the operator Tµ − TµD is compact. Proof. Let fn , n = 1, 2, . . . be a bounded sequence in Lpa (ω, X). We want to show that {(Tµ − TµD )fn }n contains a convergent subsequence in Lpa (ω, X). We have that {fn }n is a normal family, since our hypotheses on the weight ω imply k fn (z) ≤ fn Lpa (ω,X) , ( Dz ω dA)1/p

z ∈ Ω,

where Dz = {w : |w − z| < (1 − |z|)/4}. Thus, the family {fn }n is uniformly bounded on the compact subsets of Ω, and , by Montel’s theorem, {fn }n contains a subsequence {fnk }k uniformly convergent on the compact subsets of Ω. We redenote this sequence by {fn }n and we let f be its limit. Since supn fn Lpa (ω,X) < ∞, by Fatou’s lemma, we see that f ∈ Lpa (ω, X). Put gn = fn −f . We want to show that (Tµ −TµD )gn Lpa (ω,X) → 0 as n → ∞. With the notation introduced in (2.27), we have (Tµ −TµD )gn Lpa (ω,X) =

∞

(TµSl −T Sl )gn Lpa (ω,X) ≤

l=1

∞

(TµSl −T Sl )gn Lpa (ω,X) .

l=1

We now apply Theorem 2.2 to conclude that ∞ (Tµ − TµD )gn Lpa (ω,X) ≤ k bD (N, µ|∪j∈S l=1

l

Qj

) χ∪i∈Sl Di gn Lpa (ω,X) .

¯ i is a compact subset of Ω we have Since for every l ≥ 1, ∪i∈Sl D χ∪i∈Sl Di gn Lpa (ω,X) → 0 as n → ∞.

(6.23)

This together with (6.22) give (Tµ − TD )gn pLpa (ω,X) → 0 as n → ∞, and, with this, the proof is complete.

Let us now turn to a more concrete situation. For the remainder of this section, we shall assume that Ω = D and that X is a ﬁnite dimensional Banach ω(z) space. Consider the Bergman spaces Lpa (ω, X), p > 0 with (1−|z| 2 )η satisfying the

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condition Bp0 (η), for some p0 > 1, p0 ≥ p, η > −1. As we have already observed in Section 3, the associated sublinear projection Pu corresponds in this case to u(z) = (1 − |z|2 )η , K = K γ with γ = (η + 2)p0 /p − 2. We want to modify the natural decomposition Dc of D from Example 1 by varying the parameter c > 1. Based on this type of decomposition, we shall introduce a new decomposition of D, that is, in some sense, a “diagonalisation” of the previous decompositions when the parameter c → 1. We shall choose a convenient decreasing sequence cl → 1 as l → ∞. For each l = 1, . . . and some positive integers Ml ≤ Nl , we consider the annuli Acnl , where n = Ml , . . . , Nl , divided into the corresponding curved rectangles described in (3.3), (3.4). For each of these rectangles we also consider the corresponding disc Di described in Example 1. The integers Ml , Nl will be chosen such that the Acnl ’s do not overlap and at the same time cover the whole unit disc. For example, this is achieved when 1/2l c0 ∈ (1, 17/16), cl = c0 and Nl Ml

= 2l l = 2Nl−1 + 1 = 2l (l − 1) + 1,

l ≥ 0.

(6.24)

Then 1 1 ln c0 ∼ ln(1 + l ). l 2 2 With this choice of parameters we clearly have ln cl =

(6.25)

1) cl → 1 as l → ∞, 1 1 = l → 0 as l → ∞, 2) l c cN 0 l 1 1 3) = Ml+1 +1 , l ≥ 0. clNl +1 cl+1 From 2) and 3) it follows that the annuli Acnl do not overlap and at the same time cover the whole disc. We let {Qi }i be an enumeration of the curved rectangles obtained this way, and denote Sl = {i ∈ N : Qi ⊆ Acnl for some n ∈ N with Ml ≤ n ≤ Nl },

l ≥ 1.

Corollary 6.1. In the context described above, if µ is a ﬁnite complex, Borel mea¯ that satisﬁes sure, compactly supported with supp µ ⊆ Ω, |µ|(Qi ) ≤ k(1 − |λi |2 )γ+2 ,

(6.26)

for any rectangle Qi from the above described decomposition of D, then Tµ − TµD is compact. Proof. Taking into account (6.26), in the same way as in Example 1 we can estimate bD (N, µ|∪j∈S

l

Qj

) ≤ k(cl − 1)N I(µ|∪j∈S

l

Qj

)p0 /p ≤ k(ln cl )N −2p0 /p .

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Above we also used the fact that the number of intersections I(µ|∪j∈S

l

2

Qj

) ∼

1/(ln cl ) . Hence, for N > 2p0 /p, we obtain N −2p0 /p ∞ ∞ 1 bD (N, µ|∪j∈S Qj ) ≤ k ln c0 < ∞. l 2l l=1

l=1

Thus, by Theorem 6.1, Tµ −

TµD

is compact.

The next corollary provides a compactness criterion for the operators TµD and Tµ . Note that the suﬃcient condition for compactness that we obtain below (condition (6.28)), refers only to a ﬁnite number of moments of the complex measure µ on Qi , and does not involve |µ|. Corollary 6.2. Let η, γ, ω, p, p0 , K γ , X be like in Example 2 and denote by D, the “diagonal” decomposition considered there. Assume µ is a ﬁnite complex measure satisfying condition (6.26) and let N ∈ N, N > 2p0 /p. For α, β positive integers with α + β ≤ N − 1, put (ξ − λi )α (ξ − λi )β dµ(ξ) Q α,β . (6.27) Bi = i α!β!(1 − |λi |)α+β+γ+2 If lim Biα,β = 0,

i→∞

for α + β ≤ N − 1,

(6.28)

then TµD is compact, and hence Tµ is compact. Proof. For f ∈ Lpa (ω, X), we can write TµD f =

∞

i=0 α+β≤N −1

Biα,β f (α) (λi )∂¯β Kλγi (z)(1 − |λi |)α+β+γ+2 ,

and for n = 0, 1, . . . set (TµD )n f =

n

i=0 α+β≤N −1

Biα,β f (α) (λi )∂¯β Kλγi (z)(1 − |λi |)α+β+γ+2 .

Now let > 0. Then, there exists n0 = n0 () such that |Bnα,β | < , for all n ≥ n0 , α + β ≤ N − 1. Then TµD f − (TµD )n f ∞ ≤ i=n0 α+β≤N −1

(6.29) f (α) (λi ) × |∂¯β Kλγi (z)|(1 − |λi |)α+β+γ+2

Lp (ω)

Using the same arguments as in Theorem 2.1 one can show that ∞ f (α) (λi ) |∂¯β Kλγi (z)|(1 − |λi |)α+β+γ+2 ≤ kf Lpa (ω,X) . p i=n0 α+β≤N −1

L (ω)

.

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Return now to (6.29) to get TµD f − (TµD )n f ≤ kf Lpa(ω,X) ,

f ∈ Lpa (ω, X), n ≥ n0 .

The above relation shows that TµD can be approximated with the ﬁnite rank op erators (TµD )n , and hence it is compact. Recall from the introduction that, for the unweighted Bergman spaces with L∞ symbols, the compactness of the Hankel operator has been completely characterized by Axler and Zheng [4] in terms of the Berezin transform. Acknowledgements. This paper includes part of the author’s dissertation written under the supervision of Alexandru Aleman at Lund University. The author wishes to thank him for his guidance and encouragement. Useful suggestions made by the referee are also gratefully acknowledged.

References [1] A. Aleman and O. Constantin, Hankel operators on Bergman spaces and similarity to contractions, Int. Math. Res. Not. 35 (2004), 1785–1801. [2] O. Constantin, Weak product decompositions and Hankel operators on vector-valued Bergman spaces, J. Operator Theory, to appear. [3] J.L. Arregui and O. Blasco, Bergman and Bloch spaces of vector-valued functions, Math. Nachr. 261/262 (2003), 3–22. [4] S. Axler and D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), no. 2, 387–400. [5] D. B´ekoll´e, In´egalit´e ` a poids pour le projecteur de Bergman dans la boule unit´e de C n , Studia Math. 71 (1981/82), no. 3, 305–323. [6] K.C. Chan and A.L. Shields, Zero sets, interpolating sequences, and cyclic vectors for Dirichlet spaces, Michigan Math. J. 39 (1992), no. 2, 289–307. [7] R. Coifman, and R. Rochberg, Representation theorems for holomorphic and harmonic functions, Ast´erisque 77 (1980), 11–65. [8] J.B. Conway, Subnormal Operators, Research Notes in Mathematics, 51. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. [9] P.L. Duren, B.W. Romberg and A.L. Shields, Linear functionals on H p spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969), 32–60. [10] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000. [11] D. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), 319–336. [12] D. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), 85–111. [13] D. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345–368.

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[14] J. Nilsson, Embedding theorems for weighted Bergman spaces, Master’s thesis, Lund University, 2002. [15] V.L. Oleinik, Estimates of the widths of compact sets of analytic functions in Lp with a weight, Vestnik Leningrad Univ. Math. 8 (1980), 219–224. [16] J.H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43 (1976), no. 1, 187–202. [17] K. Zhu, Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics 139, Marcel Dekker, Inc., New York, 1990. [18] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics 226, Springer-Verlag, New York, 2005. Olivia Constantin Department of Mathematics Lund University Box 118 SE-221 00 Lund Sweden e-mail: [email protected] Submitted: January 25, 2007 Revised: April 13, 2007

Integr. equ. oper. theory 59 (2007), 555–578 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040555-24, published online October 18, 2007 DOI 10.1007/s00020-007-1527-8

Integral Equations and Operator Theory

On the Essential Commutant of Toeplitz Operators in Several Complex Variables Trieu Le Abstract. Using the joint local mean oscillation, Jingbo Xia [13] showed that the essential commutant of T(L), where L is the subalgebra of L∞ generated by all functions which are bounded and have at most one discontinuity, is T(QC). Even though Xia’s method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a diﬀerent approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra T is T(QC). Mathematics Subject Classification (2000). Primary 47B35. Keywords. Essential commutant, Toeplitz operators, Hardy spaces.

1. Introduction For an integer n ≥ 1, let Cn denote the Cartesian product of n copies of C. For z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ) in Cn , we use z, w = z1 w 1 + · · · + zn wn and |z| = |z1 |2 + · · · + |zn |2 for the inner product and the associated Euclidean norm. Let Bn denote the open unit ball which consists of points z ∈ Cn with |z| < 1. Let Sn denote the unit sphere which is the boundary of Bn . We denote by σ the rotation-invariant Borel measure on Sn so normalized that σ(Sn ) = 1. Let A(Bn ) be the algebra of all functions that are analytic in the open unit ball Bn and continuous in the closed unit ball B¯n . We write Lp for Lp (Sn , dσ) and H p for the Hardy subspace of Lp for 1 ≤ p ≤ ∞. For each z ∈ Bn , kz (ζ) = (1 − |z|2 )n/2 (1 − ζ, z)−n , ζ ∈ Sn , is a normalized reproducing kernel for H 2 . We have kz = 1 and ϕ, kz = (1 − |z|2 )n/2 ϕ(z) for all ϕ ∈ H 2 . Let P : L2 −→ H 2 denote the orthogonal projection. For any f ∈ L∞ , the Toeplitz operator Tf and the Hankel operator Hf are deﬁned by Tf ϕ = P (f ϕ) and Hf ϕ = f ϕ − P (f ϕ), respectively, for all ϕ ∈ H 2 . We have Tgf − Tg Tf = Hg∗ Hf for all f, g ∈ L∞ . Let B(H 2 ) denote the C ∗ −algebra of all bounded linear operators on H 2 . For any subset G of L∞ , we denote by T(G) the Banach subalgebra of B(H 2 ) generated

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by the set {Tf : f ∈ G}. If G is closed under complex conjugation, then T(G) is a C ∗ −algebra. The full Toeplitz algebra is T = T(L∞ ). Let K denote the ideal of all compact operators on H 2 . It is well-known that K ⊂ T and that Tηf − Tη Tf ∈ K for all η ∈ C(Sn ) and f ∈ L∞ . See [2] for more details. We use π to denote the canonical map from T onto the Calkin algebra B(H 2 )/K. For any ζ ∈ Sn , let Kζ be the closed two-sided ideal of T generated by the set {Tf : f is continuous on Sn and f (ζ) = 0}. Then each Kζ contains the Kζ = K. This follows from localization of T. See compact operators K and ζ∈Sn

[6, p. 176-177]. For any subset S of B(H 2 ), we denote by EssCom(S) the essential commutant of S, that is, EssCom(S) = {A ∈ B(H 2 ) : AT − T A is compact for all T ∈ S}. For any z, w ∈ B¯n , let d(z, w) = |1−z, w|1/2 . Then d is a metric on Sn , called the nonisotropic metric (see [11, Proposition 5.1.2].) Because of the denominators of the reproducing kernels, it is this metric, not the usual Euclidean metric, which is closely associated to the theory of Toeplitz and√Hankel operators on Sn . Note that for |ζ| = 1 and |z| ≤ 1, (d(ζ, z))2 ≤ |ζ − z| ≤ 2 d(ζ, z). For ζ ∈ Sn and r > 0, let Q(ζ, r) = {z ∈ Sn : d(z, ζ) < r} be the open ball of radius r centered at ζ with respect to the metric d. There is a constant A0 depending on n so that 2−n r2n ≤ σ(Q(ζ, r)) ≤ A0 r2n ,

(1.1) √ for all 0 < r ≤ 2. See [11, Proposition 5.1.4]. For any function f ∈ L1 and any ζ ∈ Sn , the mean oscillation of f at ζ is 1 |f − fQr | dσ : Qr ⊂ Q(ζ, δ), r < δ , LMO(f )(ζ) = lim sup δ↓0 σ(Qr ) Qr

where Qr denotes a ball of radius r centered at a point on Sn with respect to the 1 f dσ. metric d and fQr = σ(Qr ) Qr

A function f ∈ L1 is said to have vanishing mean oscillation if LMO(f )(ζ) = 0 for all ζ ∈ Sn . A function f ∈ L1 is said to have bounded mean oscillation if 1 |f − fQ(ζ,r) | dσ : ζ ∈ Sn , r > 0 < ∞. f BMO = sup σ(Q(ζ, r)) Q(ζ,r)

Deﬁne BMO = {f ∈ L1 : f has bounded mean oscillation}, VMO = {f ∈ L1 : f has vanishing mean oscillation}.

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Let QC = VMO ∩L∞ . Kenneth Davidson proved in [3] that for n = 1, EssCom(T) = T(QC). The key result from which Davidson derived the above identity is the one-dimensional version of the following theorem. Theorem 1.1. If S is a bounded operator on H 2 which is not the sum of a Toeplitz operator and a compact operator then there is a function h ∈ H ∞ such that Th S − STh is not compact. The function h can be taken to have at most one discontinuity. For any ζ ∈ Sn , let L(ζ) be the algebra of all bounded measurable functions ∞ on Sn which are continuous on Sn \{ζ} and let H(ζ) = H ∩ L(ζ). Let L denote ∞ the subalgebra of L generated by ζ∈Sn L(ζ) and let H denote the subalgebra of H ∞ generated by ζ∈Sn H(ζ). Xia proved in [13] the following theorem which is a local version of a result of Sarason and used this together with Theorem 1.1 to deduce that EssCom(T(L)) = T(QC) when n = 1. Note that T(L) is indeed smaller than T, which was also showed by Xia in the same paper. Theorem 1.2 (Xia, 2002). Let f ∈ L∞ and let ζ ∈ S1 . (a)

If [Tf , Th ] ∈ Kζ for all h ∈ H(ζ) then LMO(Qf )(ζ) = 0.

(b)

If LMO(Qf )(ζ) = 0, then [Tf , Tg ] ∈ Kζ for all g ∈ H ∞ .

Here Q = 1 − P is the orthogonal projection of L2 onto L2 H 2 . We would like to generalize Xia’s result to Toeplitz operators on the Hardy space of the unit sphere in higher dimensions. To do that, we need Theorem 1.1 and a higher dimensional version of Theorem 1.2. A result quite similar to Theorem 1.1 was proved by Xuanhao Ding and Shunhua Sun in [5] but there they did not have the continuity of h, which is essential for proving Xia’s result. In fact, because they used inner functions on the unit sphere, which are far from being continuous, to construct h directly, it seems that their method cannot give the continuity of h. By modifying their construction, we are able to get the continuity of h. Our proof of Theorem 1.1 for all n ≥ 1 is given in Section 2. One of the important ingredients needed to prove Theorem 1.2 is the identity 1 − Tξz Tξ∗z = kz ⊗ kz , where z ∈ B1 , ξz (τ ) = (z − τ )(1 − zτ )−1 for τ ∈ S1 and (kz ⊗ kz )(ϕ) = ϕ, kz kz for ϕ ∈ H 2 . We do not know if such a similar identity exists in higher dimensions where there is no continuous inner function (note that each ξz is a continuous inner function on S1 .) So we cannot use Xia’s method for n ≥ 2. Nevertheless, with a completely diﬀerent approach, we can generalize Theorem 1.2. Note that when n = 1, by checking directly, we can show that Hf kz = (Qf − (Qf )(z))kz and hence lim Hf kz = 0 if and only if |z|<1 z→ζ

LMO(Qf )(ζ) = 0 for any ζ ∈ S1 . See [14, Theorem 6] for more details. Therefore, the following theorem can be considered as a version of Theorem 1.2 for n ≥ 1.

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Theorem 1.3. Let f, g ∈ L∞ and ζ ∈ Sn . (a) If [Tf , Th ] ∈ Kζ for all h ∈ H(ζ), then lim Hf kz = 0. (b) If lim Hf kz Hg kz = 0, then |z|<1 z→ζ

|z|<1 z→ζ

Hg∗ Hf

= Tgf − Tg Tf ∈ Kζ .

A recent result due to Xia shows that in the case n ≥ 2, the compactness of Hg∗ Hf does not imply lim|z|↑1 Hf kz Hg kz = 0. Thus Theorem 1.3(a) is perhaps the best “partial converse” of Theorem 1.3(b) that one can hope for. Using Theorem 1.1 and Theorem 1.3, we obtain the following results about essential commutants of Toeplitz operators on the Hardy space of the unit sphere Sn for all n ≥ 1. Corollary 1.1. If f ∈ L∞ so that [Tf , Th ] is compact for all h ∈ H, then we have lim Hf kz = 0. As a consequence, the essential commutant of T(H) is T(C), |z|↑1

where C = {f ∈ L∞ : Hf is compact}. Corollary 1.2. The essential commutant of T(L) equals T(QC). Remark. That EssCom(T) = T(QC) was proved by Kunyu Guo and Shunhua Sun in [7]. What interesting about Corollary 1.2 is that T(L) is in fact strictly smaller than T. To see this, since there is a *-homomorphism s : T → L∞ with s(Tf ) = f for all f ∈ L∞ (see [4, Theorem 2.2]), we only need to show that L L∞ . This was showed by Xia in [13, Lemma 10] for the case n = 1 but his arguments can be easily generalized to all n ≥ 1. In [13], Xia proved the one-dimensional version of the following theorem. We will present in Section 4 a proof that works for all n ≥ 2. Theorem 1.4. Suppose S is a subset of T and suppose that S is separable in the operator-norm topology. Then there is a real-valued function f ∈ L∞ so that f is not in VMO and [Tf , S] is compact for all S ∈ S. Furthermore, given such an S, there is a ζ = ζ(S) such that there is an f ∈ L(ζ) which satisfies the above requirements. Theorem 1.4 shows that for any separable subalgebra S of T, EssCom(S) is strictly bigger than T(QC). In the rest of the paper, we will prove several lemmas before giving the proof of Theorem 1.1 in Section 2 and proofs of Theorem 1.3, Corollary 1.1 and Corollary 1.2 in Section 3. In Section 4, we construct a function f that satisﬁes the requirements of Theorem 1.4.

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2. Operators Essentially Commuting with Analytic Toeplitz Operators We begin this section by a lemma about pointwise approximation of a positive lower semi-continuous function on Sn by a sequence of functions in the ball algebra A(Bn ). For a proof, see [9, Theorem 3.5 and Remark 3.6]. Lemma 2.1. Suppose ϕ is a lower semi-continuous function on Sn and ϕ > 0. Then there is a sequence {ϕm }∞ m=1 of functions in A(Bn ) with the following properties: 1)

|ϕm | ≤ ϕ on Sn for all m ≥ 1, and

2)

{|ϕm |}∞ m=1 converges to ϕ a.e. on Sn .

The following lemma gives a construction of the holomorphic function which appears in Theorem 1.1. Lemma 2.2. Suppose {Em }∞ m=1 is a sequence of mutually disjoint measurable subsets of Sn which cluster at only one point ζ ∈ Sn . Suppose {ϕm }∞ m=1 is a sequence of functions in A(Bn ) so that ϕm (1 − χEm )∞ ≤ αm and ϕm ∞ ≤ β for all m, where β > 0 and 0 < α < 1. Suppose S is a bounded operator on H 2 so that [Tϕm , S] is compact and [Tϕm , S] > c > 0 for all m, where c is a fixed constant. Then there is a function h in H ∞ which is continuous on Sn \{ζ} so that π([Th , S]) ≥ c. Recall that π is the canonical quotient map from B(H 2 ) onto the Calkin algebra B(H 2 )/K. Proof. For each m, put Am = [Tϕm , S]. Then Am is a compact operator. ∞ ∞ ∞ Let E = Em . Then σ(E) = 0 since σ(Em ) ≤ σ(Sn ) = 1. m=1 ∞

k=1 m=k

For w ∈ Sn \E, there is a k so that w ∈ Sn \(

m=k

m

Em ) =

{ϕm }∞ m=1

∞ m=k

(Sn \Em ). Hence

|ϕm (w)| ≤ α for all m ≥ k. So the sequence converges to 0 almost everywhere. By the Lebesgue Dominated Convergence Theorem, it follows that the ∞ sequences {Tϕm }∞ m=1 and {Tϕm }m=1 converge to 0 in the strong operator topology. ∞ ∗ ∞ Therefore {Am }m=1 and {Am }m=1 converge to 0 in the strong operator topology. Now, since Am ≤ 2βS for all m, passing to a subsequence of {Am }∞ m=1 if converges. We necessary, we may assume that the numerical sequence {Am }∞ m=1 then have lim Am ≥ c. m→∞

By [8, Lemma 2.1], there is an increasing sequence of positive integers {m(k)} so that the sum ∞ N A= Am(k) = lim Am(k) k=1

N →∞

k=1

exists in the strong operator topology and ∞ π(A) = π( Am(k) ) = lim Am ≥ c. k=1

m→∞

560

Le

Now put h =

∞

IEOT

ϕm(k) . Since

k=1 ∞

ϕm(k) 1 ≤

k=1

∞

αm(k) + βσ(Em(k) ) ≤ 1/(1 − α) + β, k=1

the above series converges in H 1 . We conclude that h is in H 1 . If w ∈ Sn , then there is at most one k so that w ∈ Em(k) since the sets Em(k) ’s are mutually disjoint. Thus, |h(w)| ≤

∞

|ϕm(k) (w)| ≤

∞

k=1

αm(k) + β ≤ 1/(1 − α) + β.

k=1

This implies that h is bounded on Sn , hence h ∈ H ∞ . For any neighborhood U of ζ in Sn , there is a k(U ) so that Em(k) ⊂ U for all k ≥ k(U ). Thus, ∞ k=1

k(U)

ϕm(k) χSn \U ∞ ≤

k=1

∞

ϕm(k) χSn \U ∞ +

k(U)

≤

k=1

ϕm(k) χSn \U ∞

k=k(U)+1

ϕm(k) χSn \U ∞ +

∞

αm(k)

k=k(U)+1

≤ βk(U ) + 1/(1 − α). So {

N k=1

ϕm(k) }∞ N =1 converges uniformly to h on Sn \U . Because this holds

true for any neighborhood U of ζ, it follows that h is continuous on Sn \{ζ}. Now in the strong operator topology, A = lim

N →∞

= lim

N →∞

= [T

lim

N

Am(k)

k=1 N

[Tϕm(k) , S]

k=1 N

N →∞

k=1

ϕm(k) , S]

= [Th , S]. So π([Th , S]) = π(A) ≥ c > 0.

In what follows, by a characteristic function we mean the characteristic function of a Lebesgue measurable subset of Sn . If g is a characteristic function, we also use g to denote its support. We use w−lim to denote limit in the weak operator topology of B(H 2 ). The ﬁrst two conclusions in the following lemma were proved in [3] for functions on the unit circle. The high-dimensional case is similar and was

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proved in [5]. For our purpose of proving Theorem 1.1, we have added the third conclusion. Lemma 2.3. Suppose F : L∞ → B(H 2 ) is a linear map which has the following properties: (P1) If gm ’s are in L∞ with gm ∞ ≤ M and gm → g0 almost everywhere, then w− lim F (gm ) = F (g0 ). m→∞

(P2) If f1 , f2 are characteristic functions and have disjoint closed supports, then F (f1 )F (f2 ) and F (f1 )F (f2 )∗ are compact. (P3) There is a characteristic function g so that π(F (g)) > α > 0. Then there exists a sequence of characteristic functions of disjoint closed supports {χm }∞ m=1 so that F (χm g) > C(α)/2 > 0 for all m, where C(α) depends on α and the dimension n. These sets can be chosen to cluster at only one point. It then follows that there is a sequence of continuous functions {hm }∞ m=1 so that hm ≥ 0, hm ∞ ≤ 2, hm (1 − χm g)∞ ≤ 4−m−1 and F (hm ) > C(α)/2 for all m. Consequently, if F is of the form F (g) = Tg S − Tgf , where S is a bounded operator on H 2 and f is a bounded measurable function, then there exists a sequence −m {ϕm }∞ m=1 of functions in A(Bn ) so that ϕm ∞ ≤ 2, ϕm (1 − χm g)∞ ≤ 2 and F (ϕm ) > C(α)/8. ∞ Proof. The existence of the sequences {χm }∞ m=1 , {hm }m=1 and the constant C(α) was established in [5, Lemma 4]. Now ﬁx an integer m. Choose a positive number β so that β < 4−m−1 , and βF (1) < C(α)/4. Let gm = hm + β. Then gm is continuous, positive on Sn , gm ∞ ≤ 4 and |gm (1 − χm g)| ≤ 4−m . Also,

F (gm ) ≥ F (hm ) − F (β) = F (hm ) − βF (1) > C(α)/4. From Lemma 2.1, there is a sequence {ϑj }∞ j=1 of functions in A(Bn ) so that 2 |ϑj | ≤ gm on Sn , and |ϑj | → gm a.e. on Sn . Since F (|ϑj |2 ) → F (gm ) as j → ∞ in the weak operator topology by (P 1) and F (gm ) > C(α)/4, there is a j(m) so that F (|ϑj(m) |2 ) > C(α)/4. Let ϕm = ϑj(m) . Then ϕm ∈ A(Bn ), ϕm ∞ ≤ 2, |ϕm (1 − χm g)| ≤ 2−m and F (|ϕm |2 ) > C(α)/4. Now if F (|ϕm |2 ) = T|ϕm |2 S − T|ϕm |2 f , then 2

C(α)/4 < F (|ϕm |2 ) = Tϕm ϕm S − Tϕm ϕm f = Tϕm Tϕm S − Tϕm Tϕm f (since ϕm is analytic) = Tϕm (Tϕm S − Tϕm f ) ≤ Tϕm Tϕm S − Tϕm f ≤ 2F (ϕm ).

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Hence, F (ϕm ) > C(α)/8.

Now suppose S is a bounded operator on H 2 so that [Tϕ , S] is compact for all ϕ ∈ A(Bn ). So π(S) and π(Tϕ ) commute in the Calkin algebra. Since π(Tϕ ) is normal, Fuglede’s Theorem implies that π(S) commutes with π(Tϕ )∗ = π(Tϕ∗ ) = π(Tϕ ) for all ϕ ∈ A(Bn ). Thus S essentially commutes with the C ∗ −algebra generated by {Tϕ : ϕ ∈ A(Bn )}. This implies that [Tϕ , S] is compact for any continuous function ϕ on Sn . Now ﬁx a sequence {ξm }∞ m=1 of functions in A(Bn ) so that (C1) ξm ∞ ≤ 1 for all m ∈ N, (C2) |ξm | → 1, a.e. on Sn as m → ∞, (C3) Tξ → 0 in the weak operator topology as m → ∞. m

On the unit circle, we can choose ξm (w) = wm for w ∈ S1 and (C2) can be replaced by |ξm | ≡ 1. In higher dimensions, there is no function in A(Bn ) whose absolute values on the unit sphere are identically 1. This is the reason why we require condition (C2) which is much weaker. The proof of the existence of such a sequence uses inner functions in the ball. See [1, 9, 10] for more details about the existence of inner functions in the ball. The proof of the existence of a sequence satisfying (C1)–(C3). Suppose ϑ is an inner function in the ball Bn . We then also use ϑ to denote the radial limit of ϑ which is deﬁned almost everywhere on Sn . For each 0 < r < 1, put ϑr (z) = ϑ(rz), z ∈ Bn . Then ϑr ∈ A(Bn ), ϑr ∞ ≤ 1 and ϑr → ϑ a.e. on Sn when r → 1. Let δ > 0 be given. By Egoroﬀ’s Theorem, there is a measurable set Eδ ⊂ Sn with σ(Eδ ) < δ such that ϑr → ϑ uniformly on Sn \Eδ . Since |ϑ| = 1 a.e. on Sn , we can require that |ϑ(w)| = 1 for all w ∈ Sn \Eδ . Also for any > 0, there is an r so that |ϑ(w) − ϑr (w)| ≤ for all w ∈ Sn \Eδ . Take η to be any inner function with η(0) = 0. Then for any u ∈ L1 (Sn ), lim η m u dσ = 0, (2.1) m→∞ Sn

see [10, Lemma 5.1]. Applying the above argument to each η m , we get a measurable set Em with σ(Em ) < 2−m , |η m | = 1 on Sn \Em , and a function ξm ∈ A(Bn ) so that ξm ∞ ≤ 1 and |η m (w) − ξm (w)| ≤ m−1 for all w ∈ Sn \Em . Clearly the sequence {ξm }∞ m=1 satisﬁes (C1). We claim that it also satisﬁes (C2) and (C3). ∞ ∞ ∞ σ(Em ) < ∞. For any Let E = k=1 m=k Em . Then σ(E) = 0 because m=1

w ∈ Sn \E, there is a k so that w ∈ Sn \Em for all m ≥ k. Hence

1 − |ξm (w)| = |η m (w)| − |ξm (w)|

≤ η m (w) − ξm (w) ≤ m−1 ,

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563

for all m ≥ k. Thus lim |ξm (w)| = 1. So |ξm | → 1 a.e. on Sn as m → ∞. m→∞

In addition, for any f, g ∈ H 2 , |Tξm f, g| = |P (ξ m f ), g| = | ξm f g dσ| Sn

≤|

(ξm − η m )f g dσ| + |

Sn

≤ m−1 f g + 2

η m f g dσ|

Sn

|f g| dσ + |

Em

η m f g dσ|.

Sn

Letting m → ∞ and using (2.1) together with the fact that σ(Em ) < 2−m , we get lim |Tξ f, g| = 0. Hence {Tξ }∞ m=1 converges to 0 in the weak operator m m m→∞ topology as m → ∞. Put σm = Tξ STξm , for m = 1, 2, . . .. Then {σm }∞ m=1 is a bounded sequence m 2 of operators in B(H ). By passing to a subsequence if necessary, we can assume that the sequence {σm }∞ m=1 converges in the weak operator topology. Lemma 2.4. Let T = w− lim σm . Then the following statements hold true. m→∞ ∞

(a) There is an f ∈ L so that T = Tf . (b) For any continuous function h on Sn , w− lim Tξm Th STξm = Thf . m→∞

(c) For any continuous function h on Sn , w− lim Tξm (Thξm S − SThξm ) = Th S − Thf . m→∞

Consequently, if α < Th S − Thf , then there is an mα so that Thξmα S − SThξmα > α. Proof. (a) For any function b ∈ A(Bn ), we have T − Tb T Tb = w− lim (Tξ STξm − Tb Tξ STξm Tb ) m→∞

m

m

= w− lim Tξm (S − Tb STb )Tξm m→∞ = w− lim Tξm Tb (Tb S − STb )Tξm + Tξm T1−|b|2 STξm m→∞

= w− lim Tξm T1−|b|2 STξm , m→∞

because Tb (Tb S − STb ) is compact and Tξm converges to 0 in the weak operator topology so w− lim Tξm Tb (Tb S − STb )Tξm = 0. m→∞

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IEOT

From this, we have T−

n j=1

Tbj T Tbj = w−lim Tξm T(1− nj=1 |bj |2 ) STξm , n→∞

for any b1 , . . . , bn ∈ A(Bn ). In particular, for bj (z) = zj , j = 1, . . . , n, we get T−

n

Tzj T Tzj = 0.

j=1

By [4, Theorem 2.6], there is an f ∈ L∞ so that T = Tf . Thus, w− lim σm = m→∞ w− lim Tξm STξm = Tf . m→∞

(b) Since the span of {h1 h2 : h1 , h2 ∈ A(Bn )} is dense in C(Sn ), it suﬃces to prove the identity for h = h1 h2 with h1 , h2 ∈ A(Bn ). For each positive integer m, Tξm Th1 h2 STξm = Tξm Th1 Th2 STξm = Tξm Th1 [Th2 , S]Tξm + Tξm Th1 STh2 Tξm = Tξm Th1 [Th2 , S]Tξm + Th1 Tξm STξm Th2 . Since [Th2 , S] is compact, the ﬁrst term in the last sum goes to 0 in the weak operator topology when m → ∞. So w− lim Tξm Th1 h2 STξm = Th1 Tf Th2 = Th1 h2 f . m→∞

(c) Now for any continuous function h on Sn , Tξm (Thξm S − SThξm ) = Th|ξm |2 S − Tξm STh Tξm = (Th S − Th(1−|ξm |2 ) S) − (Tξ Th STξm + Tξ [S, Th ]Tξm ) m

m

= (Th S − Tξm Th STξm ) − (Th(1−|ξm |2 ) S + Tξm [S, Th ]Tξm ). Since |ξm | → 1 almost everywhere, w− lim Th(1−|ξm |2 ) S = 0. Since [S, Th ] is m→∞

compact and Tξ

m

→ 0 weakly, w− lim Tξ [S, Th ]Tξm = 0. Also from (b), w− m→∞

m

lim T Th STξm = Thf . Therefore, w− lim Tξm (Thξm S − SThξm ) = Th S − Thf . m→∞ ξ m m→∞ Since the norm is lower semi-continuous with respect to the weak operator topology, if α < Th S − Thf , there is an mα so that Tξm (Thξmα S − SThξmα ) > α. α

But Tξm (Thξmα S − SThξmα ) ≤ Thξmα S − SThξmα , hence α

Thξmα S − SThξmα > α. Now we are in the position of proving Theorem 1.1.

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Proof of Theorem 1.1. Suppose S is a bounded operator on H 2 which is not the sum of a Toeplitz operator and a compact operator. We need to ﬁnd an h in some H(ζ) so that [Th , S] = Th S − STh is not compact. If there is a function ϕ ∈ A(Bn ) so that [Tϕ , S] is not compact, take h = ϕ. Otherwise, suppose [Tϕ , S] is compact for all ϕ ∈ A(Bn ). Let f be the function in Lemma 2.4. Let F : L∞ → B(H 2 ) be deﬁned by F (g) = Tg S − Tgf , g ∈ L∞ . Then F satisﬁes properties (P 1) and (P 2) in Lemma 2.3 (see [3].) Since S is not the sum of a Toeplitz operator and a compact operator, F (1) = S − Tf is not compact. The last conclusion of Lemma 2.3 gives a sequence ∞ {ϕm }∞ m=1 of functions in A(Bn ) and a sequence {χm }m=1 of mutually disjoint −m closed sets so that ϕm ∞ < 2, |ϕm (1 − χm )| < 2 , and Tϕm S − Tϕm f = F (ϕm ) > C(α)/8 for all m ∈ N, where α = π(F (1))/2. Furthermore, the sets {χm } cluster at only one point of Sn , say ζ. For each k, apply part (c) of Lemma 2.4 to ϕk , we get a function ϑk = ϕk ξmk in A(Bn ) so that ϑk ∞ ≤ 2, ϑk (1 − χk )∞ ≤ 2−m , and [Tϑk , S] > C(α)/8. By Lemma 2.2, there is a function h in H ∞ such that h is continuous on Sn \{ζ} and [Th , S] is not compact. The proof of the theorem is thus completed.

3. Local commutativity and essential commutant In order to prove Theorem 1.3, we need some auxiliary results. The ﬁrst lemma gives a necessary condition for an operator to be in Kζ . Lemma 3.1. Let ζ be in Sn and A be an element in Kζ . Then for any > 0, there is an open neighborhood Vζ of ζ depending on A and so that for any function η ∈ L∞ , η∞ ≤ 1 and η vanishes almost everywhere on Sn \Vζ , we have π(Tη A) < . Proof. Let > 0 be given. There is an A˜ in the algebraic ideal of T generated by ˜ < /2. Now there are continuous {Tϕ : ϕ ∈ C(Sn ), ϕ(ζ) = 0} so that A − A functions ϕ1 , . . . , ϕm with ϕ1 (ζ) = · · · = ϕm (ζ) = 0 and operators B1 , . . . , Bm m and C1 , . . . , Cm in T so that A˜ = j=1 Bj Tϕj Cj . Since each ϕj is continuous, the m commutator [Bj , Tϕj ] is compact, so A˜ − j=1 Tϕj Bj Cj is compact. For any η ∈ L∞ (Sn ) and η∞ ≤ 1, the semi-commutator Tηϕj − Tη Tϕj is compact for each j, so we have ˜ π(Tη A) ≤ /2 + π(Tη A) m m = /2 + π( Tη Tϕj Bj Cj ) = /2 + π( Tηϕj Bj Cj ) j=1 m

≤ /2 + M η(

j=1

j=1

|ϕj |)∞ ,

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where M is a positive m constant, depending on Bj ’s, Cj ’s and m. Since ϕ = j=1 |ϕj | is a continuous function on Sn with ϕ(ζ) = 0, there is an open neighborhood Vζ of ζ so that for all ω ∈ Vζ , 0 ≤ ϕ(ω) ≤ (2M )−1 . If η ∈ L∞ , η∞ ≤ 1 and η vanishes almost everywhere on Sn \Vζ then ηϕ∞ ≤

(2M )−1 . Hence, π(Tη A) ≤ . The next lemma asserts that commutators of a certain kind cannot belong to Kζ unless they are compact. Lemma 3.2. Let ζ ∈ Sn be given. Suppose f ∈ L∞ and g ∈ L(ζ). Then [Tf , Tg ] is in Kζ if and only if it is in K. Proof. Because each g ∈ L(ζ) is continuous on Sn \{ζ}, for any continuous function η on Sn with η(ζ) = 1, we have π(Tη [Tf , Tg ]) = π([Tf , Tg ]). If [Tf , Tg ] is in Kζ , Lemma 3.1 and this identity show that π([Tf , Tg ]) < for all > 0. So [Tf , Tg ] ∈ K. The converse is obvious since K is contained in Kζ . Lemma 3.3. Suppose g is a function in L∞ with g∞ ≤ 1. Suppose ζ ∈ Sn and {zm }∞ m=1 in Bn so that lim |zm − ζ| = 0, hence lim d(zm , ζ) = 0 as well. Also m→∞

m→∞

suppose that Hg kzm ≥ a > 0 for all m ∈ N, where 0 < a < 1 is a constant. Then there exists a function h ∈ H ∞ which is continuous on Sn \{ζ} so that [Th , Tg ] is not compact. Proof. For each m, there is a δm > 0 so that |kzm |2 dσ < a4 /9, where Q(ζ,δm )

Q(ζ, δm ) = {ω ∈ Sn : d(ω, ζ) ≤ δm }. We may choose δm such that δm → 0 as m → ∞. For each m, let m = d(zm , ζ) + 3a−2 (1 − |zm |2 )1/4 . Then m → 0 as m → ∞. So we can choose a subsequence { ml }∞ l=1 so that ml+1 < ml , and ml+1 < δml for all l ∈ N. Let Bl = {ω ∈ Sn : ml+1 < d(ω, ζ) < ml }. Then these are mutually disjoint open sets of Sn . For each l, |gkzml |2 dσ ≤ |kzml |2 dσ + |kzml |2 dσ Sn \Bl

Q(ζ,ml+1 ) 4

≤ a /9 +

Sn \Q(ζ,ml ) 2

|kzml | dσ. Sn \Q(ζ,ml )

Now for ω ∈ Sn \Q(ζ, ml ), we have d(ω, zml ) ≥ d(ω, ζ) − d(zml , ζ) ≥ ml − d(zml , ζ) = 3a−2 (1 − |zml |2 )1/4 .

(3.1)

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Essential Commutant of Toeplitz Operators

567

So 3a−2 (1 − |zml |2 )1/4 ≤ d(ω, zml ) = |1 − ω, zml |1/2 . Hence, |kzml (ω)|2 = Thus,

(1 − |zml |2 )n ≤ (a2 /3)4n ≤ a4 /9. |1 − ω, zml |2n

|kzml |2 dσ ≤ a4 /9.

(3.2)

Sn \Q(ζ,ml )

Inequalities (3.1) and (3.2) give |gkzml |2 dσ ≤ a4 /9 + a4 /9 < 4a4 /9. Sn \Bl

So χSn \Bl gkzml < 2a2 /3.

(3.3)

Now, since Hg kzml ≥ a, we have Hg kzml , Hg kzml ≥ a2 . On the other hand, |Hg kzml , HgχSn \Bl kzml | ≤ HgχSn \Bl kzml ≤ gχSn \Bl kzml < 2a2 /3

by (3.3).

So |Hg kzml , HgχBl kzml | > a2 − 2a2 /3 = a2 /3.

Now write g = (f (1) − f (2) ) + i(f (3) − f (4) ), where 0 ≤ f (j) ≤ 1 for 1 ≤ j ≤ 4. For each l, the above inequality shows that there is a j = j(l) ∈ {1, 2, 3, 4} so that |Hg kzml , Hf (j) χB kzml | > a2 /12. By approximating f (j) χBl almost everywhere on l Sn by continuous functions with compact supports in Bl , we can ﬁnd a continuous function φl so that 0 ≤ φl ≤ 1, supp(φl ) ⊂ Bl and |Hg kzml , Hφl kzml | > a2 /12. Let α be any positive number less than a2 /24. For each l ∈ N, put ηl = max{φl , αl }. Then ηl is continuous, αl ≤ ηl ≤ 1 + αl , ηl (w) = αl for w ∈ Sn \Bl and ηl − φl ∞ = αl . So |Hg kzml , Hηl −φl kzml | ≤ Hηl −φl kzml ≤ ηl − φl ∞ = αl . Thus, for l ∈ N, |Hg kzml , Hηl kzml | > a2 /12 − αl > a2 /12 − a2 /24 = a2 /24. From Lemma 2.1, for each l, there is a function ϑl ∈ A(Bn ) so that (a) (b)

|ϑl (ω)|2 ≤ ηl (ω) for ω ∈ Sn , and |Hg kzml , H|ϑl |2 kzml | > a2 /24.

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√ Property (a) implies that ϑl ∞ ≤ 2, and |ϑl (1 − χBl )| ≤ ( α)l . Property (b) ∗ 2 implies that H|ϑl |2 Hg > a /24. Now, ∗ H|ϑ 2 Hg = T|ϑl |2 g − T|ϑl |2 Tg l|

= Tϑl Tg Tϑl − Tϑl Tϑl Tg = −Tϑl [Tϑl , Tg ]. Since Tϑl ≤ ϑl ≤ 2, we get [Tϑl , Tg ] > a2 /48. If there is an l so that [Tϑl , Tg ] is not compact, let h = ϑl . Otherwise, applying Lemma 2.2, we get a function h ∈ H ∞ so that h is continuous on Sn \{ζ} and π([Th , Tg ]) ≥ a2 /48. Before proceeding to the proof of Theorem 1.3, we need one lemma about functions in A(Bn ). This is a kind of Urysohn’s Lemma for A(Bn ). Lemma 3.4. Suppose ζ ∈ Sn and 0 < δ, < 1 are given. Then there exists a function η ∈ A(Bn ) so that η(ζ) = η∞ = 1 and |η(ω)| ≤ for ω ∈ Sn with d(ω, ζ) ≥ δ. Proof. Let ϕ be any continuous function on Sn so that 0 < ϕ ≤ 2 and for any ω ∈ Sn , ϕ(ω) = 2 if d(ω, ζ) ≤ δ/4 and ϕ(ω) ≤ if d(ω, ζ) ≥ δ/2. By Lemma 2.1, there is a sequence {ϕm }∞ m=1 ⊂ A(Bn ) such that |ϕm | < ϕ on Sn , and |ϕm | → ϕ a.e. σ. ˜ = So there exists m0 ∈ N so that ϕm0 ∞ ≥ 1. Let ζ˜ ∈ Sn such that |ϕm0 (ζ)| ˜ ϕm0 ∞ . Then d(ζ, ζ) ≤ δ/2. ϕm0 ˜ = ϑ∞ = 1 and |ϑ(ω)| ≤ ϕ(ω) ≤ if . Then ϑ(ζ) Put ϑ = ˜ ϕm0 ∞ ϕm0 (ζ) d(ω, ζ) ≥ δ/2. ˜ Put η = ϑ ◦ U . Then Now take U to be any rotation on Sn so that U ζ = ζ. ˜ = 1. For any ω ∈ Sn with d(ω, ζ) ≥ δ, η ∈ A(Bn ), η∞ = 1, η(ζ) = ϑ(U ζ) = ϑ(ζ) d(U ω, ζ) = d(ω, U −1 ζ) ≥ d(ω, ζ) − d(ζ, U −1 ζ) ˜ = d(ω, ζ) − d(ζ, ζ) ≥ δ − δ/2 = δ/2, so |η(ω)| = |ϑ(U ω)| < .

Proof of Theorem 1.3. (a) If it were not true that lim Hf kz = 0 then there {zm }∞ m=1

|z|<1 z→ζ

would be a sequence of points in Bn and a constant a > 0 such that lim |zm − ζ| = 0 and Hf kzm ≥ a for all m. By Lemma 3.3, there is a function

m→∞

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h ∈ H(ζ) so that [Tf , Th ] is not compact. Lemma 3.2 then implies that [Tf , Th ] is not in Kζ either, which is a contradiction. (b) Now suppose lim Hf kz Hg kz = 0. Let > 0 be given. There is a |z|<1 z→ζ

δ > 0 so that for all z ∈ Bn with d(z, ζ) < 3δ, Hf kz Hg kz < .

(3.4)

By Lemma 3.4, there is a function η ∈ A(Bn ) with η∞ = η(ζ) = 1 and

|η(w)| < = if w ∈ Sn with d(w, ζ) ≥ δ. We claim that g∞ f ∞ + 1 lim sup Hf η kz Hg kz ≤ .

(3.5)

|z|↑1

To prove this, it suﬃces to show that for any ω ˜ ∈ Sn , lim sup Hf η kz Hg kz ≤ . |z|<1 z→˜ ω

We have Hf η kz − ηHf kz = (f ηkz − Tf η kz ) − η(f kz − Tf kz ) = ηTf kz − Tf η kz = (Tη Tf − Tf η )kz (since η is holomorphic) = −Hη∗ Hf kz . So Hf η kz − ηHf kz = Hη∗ Hf kz . Since Hη∗ is compact (because η is continuous) and kz → 0 weakly as |z| → 1, we have lim Hη∗ Hf kz = 0. So for any ω ˜ ∈ Sn |z|→1

with d(˜ ω , ζ) < 3δ, lim sup Hf η kz Hg kz = lim sup ηHf kz Hg kz ≤ , |z|<1 z→˜ ω

|z|<1 z→˜ ω

by (3.4) and the fact that η∞ = 1. Now ﬁx an ω ˜ ∈ Sn with d(˜ ω , ζ) ≥ 2δ. For any z ∈ Bn , f ηkz 2 ≤ χSn \Q(ζ,δ) kz 2 + χQ(ζ,δ) kz 2 f ∞ ≤ + χQ(ζ,δ) kz 2 f ∞ . For w ∈ Sn with d(w, ζ) < δ and for z ∈ Bn with d(z, ω ˜ ) < δ/2, we have d(w, z) ≥ d(˜ ω , ζ) − d(w, ζ) − d(z, ω ˜) ≥ 2δ − δ − δ/2 = δ/2. Thus the function χQ(ζ,δ) kz is bounded by (2/δ)2n , which is independent of z when z → w. ˜ On the other hand, kz → 0 a.e. as z → ω ˜ . So by the Lebesgue Dominated Convergence Theorem, lim χQ(ζ,δ) kz 2 = 0. z→˜ ω

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It then follows that lim sup f ηkz 2 ≤ f ∞ . Hence lim sup Hf η kz ≤ |z|<1 z→˜ ω

f ∞ , and so

|z|<1 z→˜ ω

lim sup Hf η kz Hg kz ≤ g∞ f ∞ ≤ . |z|<1 z→˜ ω

Therefore, (3.5) has been proved. From the proof of [15, Theorem 3], for any γ > 0 there is a constant Cγ > 0 so that π(Hg∗ Hf η ) ≤ C(Cγ + γ),

(3.6)

where C depends only on the dimension. Now, Hg∗ Hf η = Tgf η − Tg Tf η = Tgf Tη − Tg Tf Tη

(since η is analytic)

= (Tgf − Tg Tf )Tη = (Tgf − Tg Tf ) + (Tgf − Tg Tf )Tη−1 . Since η − 1 is continuous on Sn and η(ζ) − 1 = 0, the second term is in Kζ . Let πζ denote the canonical map from T onto T/Kζ . Inequality (3.6) then gives πζ (Tf g − Tg Tf ) = πζ (Hg∗ Hf η ) ≤ π(Hg∗ Hf η ) ≤ C(Cγ + γ). Since and γ were arbitrary and C depends only on the dimension, we conclude that πζ (Tgf − Tg Tf ) = 0. Proof of Corollary 1.1. The ﬁrst conclusion follows directly from Theorem 1.3 (a). Now suppose S is a bounded operator that essentially commutes with all Toeplitz operators Th , h ∈ H. Theorem 1.1 implies that S = Tf + K where f ∈ L∞ and K is a compact operator. Now Tf essentially commutes with all Th , h ∈ H. Hence lim Hf kz = 0. By [15, Theorem 5], Hf is compact. So EssCom(T(H)) = T(C)

|z|↑1

where C = {f ∈ L∞ : Hf is compact}.

Proof of Corollary 1.2. Suppose S is a bounded operator that essentially commutes with all Toeplitz operators Tg , where g ∈ L. Then in particular, S essentially commutes with all Th , where h ∈ H. Theorem 1.1 implies that S = Tf + K, where K is a compact operator. Now Tf essentially commutes with all Tg , where g ∈ L. Since L is ∗−symmetric, Tf = Tf∗ also essentially commutes with all Tg , g ∈ L. As in the proof of Corollary 1.1, both Hf and Hf are compact. Hence f ∈ VMO, see [4, p. 365] or [12, p. 465].

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4. Essential Commutant of a Separable Subset of T In this section, we will prove Theorem 1.4 by constructing a real-valued function f that satisﬁes all the requirements. We begin with the following lemma. Lemma 4.1. Let 0 < a < 1 and let η be a continuous function on [a, 1] with η(a) = η(1). Define ϕ : [0, 1] −→ R by ϕ(0) = 0 and ϕ(t) = η(a−m t) if am+1 < t ≤ am , m = 0, 1 . . .. Then ϕ is continuous on (0, 1] and it satisfies the following properties: (a) For any m ≥ 0 and 0 < t ≤ am , we have ϕ(t) = ϕ(a−m t). (b) For any > 0, there is a δ > 0 so that for any s, t in (0, 1] with |1 − s−1t| ≤ δ, we have |ϕ(s) − ϕ(t)| ≤ . (c) Suppose n ≥ 2. For any complex number α and any 0 < r < 1,

(1 − |z|2 )n−2 ϕ(|1 − z|1/2 ) − α dA(z) ≥ cr2n ,

(4.1)

|1−z|≤r 2 |z|<1

√ where c = ( 2 − 1)n−2

a2n 1 − a2n

1

v 2n−1 |η(v) − α| dv and dA(z) = π −1 dxdy.

a

Proof. (a) Suppose as+1 < a−m t ≤ as for some s ≥ 0. Then as+m+1 < t ≤ as+m . So by the deﬁnition of ϕ, we have ϕ(t) = η(a−s−m t) = η(a−s (a−m t)) = ϕ(a−m t). (b) Let > 0 be given. Since ϕ is uniformly continuous on [a2 , 1], there is a δ0 > 0 so that |ϕ(u) − ϕ(v)| < for all u, v ∈ [a2 , 1] with |u − v| < δ0 . This δ0 depends only on η, a and . Let δ = min{δ0 , 1 − a}. For s, t ∈ (0, 1] with |1 − s−1 t| < δ, we have 1 − δ < s−1 t < 1 + δ. But a < 1 − δ and 1 + δ < 2 − a ≤ a−1 , so a < s−1 t < a−1 . This implies that there is an m ≥ 0 so that s, t ∈ (am+2 , am ]. Now a−m s, a−m t ∈ (a2 , 1] and |a−m s − a−m t| = a−m s|1 − s−1 t| ≤ |1 − s−1 t|

(since s ≤ am )

< δ0 . Hence |ϕ(a−m s) − ϕ(a−m t)| < . By (a), we have |ϕ(s) − ϕ(t)| = |ϕ(a−m s) − ϕ(a−m t)| < .

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(c) By the change-of-variable u = 1 − z, the integral on the left hand side of (4.1) becomes

n−2

ϕ(|u|1/2 ) − α dA(u) 1 − |1 − u|2 |u|≤r 2 |1−u|≤1

= π −1

n−2

ϕ(|t|1/2 ) − α t dtdθ 1 − |1 − teiθ |2

0≤t≤r 2 |1−teiθ |≤1

= π −1

n−2

ϕ(|t|1/2 ) − α t dtdθ. 2t cos θ − t2

0≤t≤r 2 0≤2 cos θ−t

For |θ| ≤

π and 0 ≤ t ≤ 1 we have 4 √ 2t cos θ − t2 ≥ 2t − t2

√ = (t − t2 ) + ( 2 − 1)t √ ≥ ( 2 − 1)t.

Hence the above integral is not less than 2

r π/4 √

−1 π ( 2 − 1)n−2 tn−2 ϕ(t1/2 ) − α t dθdt 0 −π/4 r2

√

= 2−1 ( 2 − 1)n−2 tn−1 ϕ(t1/2 ) − α dt √ = ( 2 − 1)n−2

0

r

s2n−1 |ϕ(s) − α| ds.

0

Now choose an m ≥ 0 so that a r

s2n−1 |ϕ(s) − α| ds ≥

0

m+1

< r ≤ am . Then

m+1 a

s2n−1 |ϕ(s) − α| ds

0

=

∞

k

a

s2n−1 |ϕ(s) − α| ds

k=m+1 k+1 a

=

∞ k=m+1

a

2nk

1 a

v 2n−1 |ϕ(v) − α| dv

Vol. 59 (2007)

Essential Commutant of Toeplitz Operators

a2n(m+1) = 1 − a2n ≥

=

√ With c = ( 2 − 1)n−2

a2n 1 − a2n

a2n r2n 1 − a2n a2n r2n 1 − a2n 1

1

573

v 2n−1 |ϕ(v) − α| dv

a

1

v 2n−1 |ϕ(v) − α| dv

a

1

v 2n−1 |η(v) − α| dv.

a

v 2n−1 |η(v) − α| dv, we have the required

a

inequality.

For each function f ∈ L1 (Sn ), we use f to denote the harmonic extension of f in Bn . So for any z ∈ Bn , f (z) = f (w)Pz (w) dσ(w), Sn

(1 − |z|2 )n is the Poisson’s kernel. |1 − w, z|2n For for ζ ∈ Sn , any f, g ∈ BMO and 1 ≤ p < ∞, deﬁne 1/p Pp (f, g)(ζ) = lim sup |f − f (z)|p Pz dσ |g − g(z)|p Pz dσ :

where Pz (w) =

δ↓0

Sn

Mp (f, g)(ζ) = lim sup δ↓0

1 × σ(Q(z, r))

Sn

|z| < 1, d(z, ζ) < δ , 1 |f − fQ(z,r) |p dσ σ(Q(z, r))

Q(z,r)

|g − gQ(z,r) |p dσ

1/p

: Q(z, r) ⊂ Q(ζ, δ) .

Q(z,r)

Then LMO(f, g)(ζ) = M1 (f, g)(ζ) is called the joint local mean oscillation of f and g at ζ. It was proved in [14] that for any f, g ∈ BMO, any ζ ∈ S1 and any 1 ≤ p < ∞, Mp (f, g)(ζ) = 0 if and only if Pp (f, g)(ζ) = 0. The proof can be carried over to the case n ≥ 2 with only some minor changes. In particular, for any f, g ∈ L∞ and ζ ∈ Sn , LMO(f, g)(ζ) = 0 implies that P1 (f, g)(ζ) = 0. It then gives P2 (f, g)(ζ) = 0 because f, g are bounded.

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For f ∈ L2 and z ∈ Bn , deﬁne

2

Var(f ; z) = f − f Pz dσ Pz dσ = |f − f (z)|2 Pz dσ. Sn

Sn

Sn

∞

Then for any f, g ∈ L and any ζ ∈ Sn , we have 2 P2 (f, g)(ζ) = lim sup Var(f ; z) Var(g; z) : |z| < 1, d(z, ζ) < δ δ↓0

= lim sup Var(f ; z) Var(g; z) |z|<1 z→ζ

≥ lim sup Hf kz 2 Hg kz 2 ,

(4.2)

|z|<1 z→ζ

since Hf kz 2 ≤ Var(f ; z) and Hg kz 2 ≤ Var(g; z). See [12, Inequality 6.4]. Inequality (4.2) shows that if P2 (f, g)(ζ) = 0 then lim Hf kz Hg kz = 0. |z|<1 z→ζ

It then follows that if LMO(f, g)(ζ) = 0 then lim Hf kz Hg kz = 0. |z|<1 z→ζ

Now ﬁx 0 < a < 1. Choose a function η as in the hypothesis of Lemma 4.1 1 so that t2n−1 |η(t) − α| dt ≥ c0 > 0 for all α, where c0 is a constant independent a

of α. Any continuous function η on [a, 1] with η(a) = η(1) = 0, η(t) = 1 if a1 < t < a2 , η(t) = 0 if a3 < t < a4 , where a < a1 < a2 < a3 < a4 < 1 will satisfy the requirements. Let ϕ be as in Lemma 4.1. Let ζ be a point on Sn . Deﬁne f (w) = ϕ(d(w, ζ)) for all w ∈ Sn . Then f is a continuous function on Sn \{ζ}. Proposition 4.1. Let f be as in the preceding paragraph. Then the following statements hold true. (a) LMO(f )(ζ) > 0. (b) For any function g ∈ L∞ so that ζ is a Lebesgue point of g, we have LMO(f, g)(ζ) = 0. Therefore, lim Hf kz Hg kz = 0. |z|<1 z→ζ

Proof. (a) Let 0 < r < 1 be given. Put αr =

1 σ(Q(ζ, r))

|f − fQ(ζ,r) | dσ Q(ζ,r)

= Q(ζ,r)

ϕ(|1 − w, ζ|1/2 ) − αr dσ(w)

f dσ. We have Q(ζ,r)

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Essential Commutant of Toeplitz Operators

575

(1 − |u|2 )n−2 |ϕ(|1 − u|1/2 ) − αr | dA(u)

= (n − 1) |u|≤1 |1−u|1/2 ≤r

(see [11, p. 15]) √ ≥ (n − 1)( 2 − 1)n−2

a2n c0 r2n (from Lemma 4.1.) 1 − a2n

Recall that there is a constant A0 so that σ(Q(ζ, r)) ≤ A0 r2n . Hence for all 0 < r < 1, 1 σ(Q(ζ, r))

√ |f − fQ(ζ,r) | dσ ≥ (n − 1)( 2 − 1)n−2

Q(ζ,r)

a2n c0 . 1 − a2n A0

This implies that LMO(f )(ζ) > 0. (b) Now let g be a function in L∞ so that τ is a Lebesgue point of g. Let > 0 be given. By Lemma 4.1(b), there is an M > 0 so that |ϕ(s)−ϕ(t)| <

for all s, t ∈ (0, 1] with |1 − s−1 t| < M −1 . Since ζ is a Lebesgue point of g, there is a δ0 > 0 so that for all 0 < r < δ0 , 1 σ(Q(ζ, r))

|g − g(ζ)| dσ < (M + 2)−2n .

(4.3)

Q(ζ,r)

Now for all 0 < r < (M + 2)−1 δ0 , we have 1 σ(Q(ζ, r))

|g − g(ζ)| dσ Q(ζ,(M+2)r)

≤

σ(Q(ζ, (M + 2)r)) (M + 2)−2n . σ(Q(ζ, r))

Since σ(Q(ζ, r)) ≥ 2−n r2n and σ(Q(ζ, (M + 2)r)) ≤ A0 (M + 2)2n r2n , the above inequality implies 1 σ(Q(ζ, r))

|g − g(ζ)| dσ ≤ 2n A0 .

(4.4)

Q(ζ,(M+2)r)

Let 0 < r < (M + 2)−1 δ0 and let Q(z, r) be any ball of radius r centered at z (with respect to the metric d) that is contained in Q(ζ, (M + 2)−1 δ0 ).

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There are two cases. The ﬁrst case, Q(z, r) ∩ Q(ζ, M r) = ∅. It then follows that Q(z, r) ⊂ Q(ζ, (M + 2)r). Therefore, 1 2 |g − gQ(z,r) | dσ ≤ |g − g(ζ)| dσ σ(Q(z, r)) σ(Q(z, r)) Q(z,r)

Q(z,r)

2 ≤ σ(Q(z, r))

|g − g(ζ)| dσ Q(ζ,(M+2)r)

(because of (4.4).) ≤ 2n+1 A0 1 2 Here we have used |g − gE | dσ ≤ |g − α| dσ, any α ∈ Cn , in the σ(E) σ(E) E

E

ﬁrst inequality. Hence, 1 1 |f − fQ(z,r) | dσ |g − gQ(z,r) | dσ ≤ f ∞ 2n+2 A0 . σ(Q(z, r)) σ(Q(z, r)) Q(z,r)

Q(z,r)

(4.5) The second case, Q(z, r) ∩ Q(ζ, M r) = ∅. For any w ∈ Q(z, r) we have d(w, ζ) ≥ M r. This gives |d(w, ζ) − d(z, ζ)| d(w, z) ≤ d(w, ζ) d(w, ζ) < r(M r)−1 = M −1 . By our choice of M , we then have |f (w) − f (z)| = |ϕ(d(w, ζ)) − ϕ(d(z, ζ))| < . Therefore, 1 σ(Q(z, r))

|f − fQ(z,r) | dσ ≤

Q(z,r)

2 σ(Q(z, r))

|f − f (z)| dσ < 2 . Q(z,r)

Hence for this case, 1 1 |f − fQ(z,r) | dσ |g − gQ(z,r) | dσ ≤ 2g∞ . σ(Q(z, r)) σ(Q(z, r)) Q(z,r)

(4.6)

Q(z,r)

Inequalities (4.5) and (4.6) give 1 1 |f − fQ(z,r) | dσ |g − gQ(z,r) | dσ σ(Q(z, r)) σ(Q(z, r)) Q(z,r)

Q(z,r)

≤ max{2 −1

for any Q(z, r) ⊂ Q(ζ, (M + 2)

n+1

A0 f ∞ , 2g∞} ,

δ0 ). Thus, LMO(f, g)(ζ) = 0.

Vol. 59 (2007)

Essential Commutant of Toeplitz Operators

577

Proof of Theorem 1.4. The separability of S implies that S is in the operator norm closure of a countable subset {A1 , A2 , . . .} of T. Now each Aj is the limit in the M operator norm of a sequence of operators of the form k=1 Tgk1 · · · TgkM , where gkl ∈ L∞ . Hence we can ﬁnd a countable set G = {g1 , g2 , . . .} of real-valued functions in L∞ so that S is contained in T(G). For each gj , almost every point of Sn is a Lebesgue point. Therefore there is a ζ ∈ Sn which is a Lebesgue point for all functions gj , for j = 1, 2 . . .. For this ζ, let f be the function in Proposition 4.1. Since f is continuous on Sn \{ζ}, LMO(f, gj )(w) = 0 for all w ∈ Sn \{ζ}. So lim Hf kz Hgj kz = 0 for |z|<1 z→w

all w ∈ Sn \{ζ}. Proposition 4.1 also gives lim Hf kz Hgj kz = 0. By Theorem |z|<1 z→ζ

1.3 (b), Tgj f − Tgj Tf ∈ Kw for all w ∈ Sn . Thus, Tgj f − Tgj Tf is compact. Since both f and gj are real-valued functions, it then follows that [Tf , Tgj ] is compact for all j. Hence [Tf , S] is compact for all S ∈ T(G). Proposition 4.1 also yields that LMO(f )(ζ) > 0, hence f ∈ / VMO. Acknowledgments This paper is part of the author’s Ph.D. dissertation, written at the State University of New York at Buﬀalo under the supervision of Professor Jingbo Xia. The author wishes to thank the referee for many valuable suggestions which improved the presentation of the paper.

References A. B. Aleksandrov, The existence of inner functions in a ball, Mat. Sb. (N.S.) 118(160) (1982), no. 2, 147–163, 287 (Russian). MR 658785 (83i:32002) [2] L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439. MR 0322595 (48 #957) [3] Kenneth R. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Functional Analysis 24 (1977), no. 3, 291–302. MR 0454715 (56 #12963) [4] A. M. Davie and N. P. Jewell, Toeplitz operators in several complex variables, J. Functional Analysis 26 (1977), no. 4, 356–368. MR 0461195 (57 #1180) [5] Xuanhao Ding and Shunhua Sun, Essential commutant of analytic Toeplitz operators, Chinese Sci. Bull. 42 (1997), no. 7, 548–552. MR 1454164 (98m:47033) [6] Ronald G. Douglas, Banach algebra techniques in operator theory, 2nd ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998. MR 1634900 (99c:47001) [7] Kunyu Guo and Shunhua Sun, The essential commutant of Toeplitz algebra on Hardy space H 2 (S n , dσ), Chinese J. Contemp. Math. 16 (1995), no. 4, 383–390. MR 1384131 (97d:47033) [8] Paul S. Muhly and Jingbo Xia, On automorphisms of the Toeplitz algebra, Amer. J. Math. 122 (2000), no. 6, 1121–1138. MR 1797658 (2001m:46139) [9] Walter Rudin, New constructions of functions holomorphic in the unit ball of Cn , CBMS Regional Conference Series in Mathematics, vol. 63, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. MR 840468 (87f:32013) , Inner functions in the unit ball of Cn , J. Funct. Anal. 50 (1983), no. 1, 100–126. [10] MR 690001 (84i:32007) [1]

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[12] [13]

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, Function theory in the unit ball of Cn , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York, 1980. MR 601594 (82i:32002) Jingbo Xia, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal. 209 (2004), no. 2, 444–467. MR 2044231 (2005d:46106) , Joint mean oscillation and local ideals in the Toeplitz algebra. II. Local commutivity and essential commutant, Canad. Math. Bull. 45 (2002), no. 2, 309–318. MR 1904095 (2003g:47052) , Joint mean oscillation and local ideals in the Toeplitz algebra, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2033–2042. MR 1664395 (2000m:47039) Dechao Zheng, Toeplitz operators and Hankel operators on the Hardy space of the unit sphere, J. Funct. Anal. 149 (1997), no. 1, 1–24. MR 1471097 (98m:47038)

Trieu Le Department of Mathematics University of Toronto Toronto, Ontario, M5S 2E4 Canada e-mail: [email protected] Submitted: September 1, 2006 Revised: August 21, 2007

Integr. equ. oper. theory 59 (2007), 579–583 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040579-5, published online October 18, 2007 DOI 10.1007/s00020-007-1531-z

Integral Equations and Operator Theory

Fredholmness of Linear Combinations of Two Idempotents Hwa-Long Gau and Pei Yuan Wu Abstract. Let E and F be idempotent operators on a complex Hilbert space, and let a and b be nonzero scalars with a + b = 0. We prove that aE + bF is Fredholm if and only if E + F is, thus answering aﬃrmatively a question asked by Koliha and Rakoˇcevi´c. Mathematics Subject Classification (2000). 47A53. Keywords. Idempotent operator, Fredholm operator.

In recent years, there has been some interest in the study of the invertibility and the Fredholmness of linear combinations of two idempotent operators on a complex Hilbert space. For example, if E and F are idempotents (E 2 = E and F 2 = F ) on a ﬁnite-dimensional Hilbert space and a and b are nonzero scalars with a + b = 0, then it was shown in [1] that the invertibility of aE + bF and E + F are equivalent. This is further strengthened in [5] to the equality of the nullities of aE + bF and E + F . For E and F on a not necessarily ﬁnite-dimensional space, the equivalence of the invertibility is also true as proved in [3]. In connection with this, Koliha and Rakoˇcevi´c asked in [5] whether the Fredholmness of aE + bF and E + F are equivalent. The purpose of this note is to give an aﬃrmative answer to this question. Recall that an operator T on a Hilbert space is Fredholm if the nullities of T and T ∗ are ﬁnite and the range of T is closed. For a Fredholm T , its index, ind T , is by deﬁnition nullity T − nullity T ∗ . It is known that the Fredholmness of T is preserved under compact perturbations and is equivalent to the existence of an operator T with T T − I and T T − I compact. An excellent reference for properties of Fredholm operators is [2, Chapter XI]. The main result of this note is the following theorem. Theorem 1. Let E and F be idempotents on a Hilbert space H, and a and b be nonzero scalars with a + b = 0. Then aE + bF is Fredholm if and only if E + F is. In this case, ind (aE + bF ) = ind (E + F ). For its proof, we need the next lemma.

580

Gau and Wu

Lemma 2. Let T =

A C

B D

IEOT

on H ⊕ K, where A is Fredholm with A on H

satisfying AA = I + K1 and A A = I + K2 for some compact operators K1 and K2 . Then T is Fredholm if and only if D − CA B is. In this case, ind T = ind A + ind (D − CA B). This is due to Zhang [6, Theorem 1]. Here we give a much simpliﬁed proof. A B Note that this is an analogue of the invertible case: if T = C D with A

invertible, then T is invertible if and only if D − CA−1 B, the Schur complement of A in T , is. In analogy to this, we may call, for the Fredholm case, D − CA B the essential Schur complement of A in T , which is, of course, determined only up to compact perturbations. Proof of Lemma 2. Since A 0 I A B I 0 0 I 0 D − CA B CA I A AA B = CA A CA AA B + D − CA B A (I + K1 )B = C(I + K2 ) C(I + K2 )A B + D − CA B A B 0 K1 B , = + CK2 CK2 A B C D where

are invertible and

I CA

0 I

0 CK2

and

I 0

K1 B CK2 A B

A B I

is compact, we infer that T is Fredholm if and only if A ⊕ (D − CA B) is. The latter is the case if and only if D − CA B is. In this case, we have ind T = ind (A ⊕ (D − CA B)) = ind A + ind (D − CA B), completing the proof.

Note that, in the preceding the Fredholmness of T does not imply lemma, 0 I that of A as the operator T = I 0 on l2 ⊕ l2 shows. However, if T is positive semideﬁnite, then we do have this implication. A B on H ⊕ K be positive semidefinite. Then T is Corollary 3. Let T = C D

Fredholm if and only if A and D − CA B are, where A is any operator on H with AA − I and A A − I compact.

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Proof. In light of Lemma 2, we need only prove that the Fredholmness of T implies that of A. Let T = A0 ⊕ 0 and P = 0 ⊕ I on ran T ⊕ ker T . Since T is Fredholm, A0 is invertible and hence so is T +P = A0 ⊕I. Thus W (T + P ) = σ(T +P )∧ ⊆ [ε, ∞) for some ε > 0, where W (T + P ) and σ(T +P )∧ denote the closure of the numerical range and the convex hull of the spectrum of T +P , respectively, and their equality is by [4, Problem 216]. Since A B P1 P2 A + P1 B + P2 T +P = + = on H ⊕ K, C D P3 P4 C + P3 D + P4 we infer that σ(A + P1 ) ⊆ W (A + P1 ) ⊆ W (T + P ) ⊆ [ε, ∞) (cf. [4, Problem 214]). This shows that A + P1 is also invertible. Note that P is of ﬁnite rank and thus so is P1 . Therefore, A is Fredholm as asserted. We are now ready to prove Theorem 1. The proof models after the one for the invertible case as given in [3, Theorem 1]. Proof of Theorem 1. Since the idempotency and the Fredholmness are preserved under the similarity of operators, we may assume that one of E and F , say, F is an (orthogonal) projection (F 2 = F = F ∗ ). We can express E and F on H = ran E ⊕ ker E ∗ as 1/2 1/2 I E1 F1 DF2 F1 , E= and F = 1/2 1/2 0 0 F2 D∗ F1 F2 where D is a contraction (D ≤ 1) from ker E ∗ to ran E. We further decompose F1 and F2 as F1 = 0 ⊕ I ⊕ F11 and F2 = F22 ⊕ I ⊕ 0 on ran E = ker F1 ⊕ ker (I − F1 ) ⊕ (ran E (ker F1 ⊕ ker (I − F1 ))) and ker E ∗ = (ker E ∗ (ker F2 ⊕ ker (I − F2 ))) ⊕ ker (I − F2 ) ⊕ ker F2 , respectively. Since F is positive semideﬁnite, we have 1/2 1/2 F11 D1 F22 F11 ⊕I ⊕0 F =0⊕I ⊕ 1/2 1/2 F22 D1∗ F11 F22 for some contraction D1 from ker E ∗ (ker F2 ⊕ ker (I − F2 )) to ran E (ker F1 ⊕ ker (I − F1 )). From 2 1/2 1/2 1/2 1/2 F11 F11 F11 D1 F22 F11 D1 F22 = , 1/2 1/2 1/2 1/2 F22 D1∗ F11 F22 F22 D1∗ F11 F22 we obtain

1/2

1/2

2 F11 + F11 D1 F22 D1∗ F11 = F11 , 3/2

1/2

1/2

3/2

1/2

1/2

F11 D1 F22 + F11 D1 F22 = F11 D1 F22 and

1/2

1/2

2 F22 D1∗ F11 D1 F22 + F22 = F22 . It can be derived using the injectivity of Fjj and I − Fjj , j = 1, 2, that

(∗)

D1 D1∗ = I, D1∗ D1 = I and D1∗ (I − F11 )D1 = F22

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(cf. [3, p. 1454]). Note that  aI  (∗∗) aE + bF =  

(a + b)I aI + bF11 1/2 1/2 bF22 D1∗ F11

aE31

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aE11 aE21 1/2 1/2 + bF11 D1 F22 bF22

aE12 aE22 aE32 bI

aE13 aE23 aE33

  . 

0

We claim that aE +bF is Fredholm if and only if I −F11 is invertible, I −E31 D1∗ (I − 1/2 F11 )−1/2 F11 is Fredholm and dim ker F2 < ∞. Indeed, if aE + bF is Fredholm, then, letting A be an operator on H such that K ≡ (aE + bF )A − I is compact, we have, with K1 K2 A1 A2 and K = on H = ran E ⊕ ker E ∗ , A= A3 A4 K3 K4 1/2 1/2 A1 A2 aI + bF1 I + K1 aE1 + bF1 DF2 K2 = . 1/2 1/2 A3 A4 K3 I + K4 bF2 D∗ F1 bF2 Carrying out the multiplication here yields 1/2

1/2

bF2 D∗ F1 A2 + bF2 A4 = I + K4 or

1/2

1/2

1/2

1/2

bF2 (D∗ F1 A2 + F2 A4 ) = I + K4 .

This shows that F2 is Fredholm and hence so is F2 . Therefore, F22 is invertible and thus so is I − F11 by (∗). From Lemma 2, we derive that the Fredholmness of aE + bF is equivalent to that of 1/2 1/2 aI + bF11 aE31 + bF11 D1 F22 1/2 1/2 bF22 D1∗ F11 bF22 together with the ﬁniteness of dim ker F2 . The Fredholmness of this latter operator is in turn equivalent to that of 1/2

1/2

1/2

1/2

−1 (F22 D1∗ F11 ) aI + bF11 − (aE31 + bF11 D1 F22 )F22

by Lemma 2. This is equal to 1/2

1/2

aI + bF11 − (aE31 + bF11 D1 D1∗ (I − F11 )1/2 D1 )D1∗ (I − F11 )−1/2 D1 D1∗ F11 , which can be further simpliﬁed to 1/2

a(I − E31 D1∗ (I − F11 )−1/2 F11 ) by (∗). This proves one direction. For the other, if I−F11 is invertible, I−E31 D1∗ (I− 1/2 F11 )−1/2 F11 is Fredholm and dim ker F2 < ∞, then we can reverse the above arguments to show that aE +bF is Fredholm. The equivalence of the Fredholmness of aE + bF and E + F follows easily. Finally, we also have 1/2

ind (aE + bF ) = ind (I − E31 D1∗ (I − F11 )−1/2 F11 ) = ind (E + F ), which completes the proof.

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The relation between the Fredholmness of aE+bF with a+b = 0 and a+b = 0 is given in the next ﬁnal corollary. Corollary 4. Let E and F be idempotents on H. (a) If E − F is Fredholm, then so is E + F and, in this case, ind (E − F ) = ind (E + F ) + dim (ran E ∩ ran F ). The converse is false. (b) If dim (ran E ∩ ran F ) < ∞, then E − F is Fredholm if and only if E + F is. Proof. As in the proof of Theorem 1, E +F (resp., E −F ) is Fredholm if and only if 1/2 I − F11 is invertible, I − E31 D1∗ (I − F11 )−1/2 F11 is Fredholm and dim ker F2 < ∞ (and, in addition, dim ker (I − F1 ) < ∞). Since ker (I − F1 ) = ran E ∩ ran F , the assertions in (a) and (b) follow easily. Acknowledgements. This research was partially supported by the National Science Council of the Republic of China under the research projects NSC 95-2115-M-008011 and NSC 95-2115-M-009-001 of the two authors. The second author was also supported by the MOE-ATU project.

References [1] J. K. Baksalary and O. M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004), 25–29. [2] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990. [3] H. Du, X. Yao and C. Deng, Invertibility of linear combinations of two idempotents, Proc. Amer. Math. Soc. 134 (2006), 1451–1457. [4] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. [5] J. J. Koliha and V. Rakoˇcevi´c, The nullity and rank of linear combinations of idempotent matrices, Linear Algebra Appl. 418 (2006), 11–14. [6] G. Zhang, Wiener–Hopf operators and factorization of operators (Chinese), Northeast. Math. J. 4 (1988), 357–362. Hwa-Long Gau Department of Mathematics National Central University Chungli 320 Taiwan e-mail: [email protected] Pei Yuan Wu Department of Applied Mathematics National Chiao Tung University Hsinchu 300 Taiwan e-mail: [email protected] Submitted: November 1, 2006 Revised: August 17, 2007

Integr. equ. oper. theory 59 (2007), 585–590 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040585-6, published online October 18, 2007 DOI 10.1007/s00020-007-1528-7

Integral Equations and Operator Theory

Remarks on the Structure of Complex Symmetric Operators T.M. Gilbreath and Warren R. Wogen Abstract. A conjugation C is antilinear isometric involution on a complex Hilbert space H, and T ∈ B(H) is called complex symmetric if T ∗ = CT C for some conjugation C. We use multiplicity theory to describe all conjugations commuting with a fixed positive operator. We expand upon a result of Garcia and Putinar to provide a factorization of complex symmetric operators which is based on the polar decomposition. Keywords. Complex symmetric operator, factorization, conjugation.

1. Introduction An operator T on a complex Hilbert space H is called complex symmetric if T has a symmetric matrix relative to some orthonormal basis for H. Complex symmetric operators have been studied for many years in the ﬁnite dimensional setting. Recently, S.R. Garcia and M. Putinar have proven interesting results for this class of operators, primarily in the inﬁnite dimensional case. They show that the class is surprisingly large and includes the normal operators, the Hankel operators, the compressed Toeplitz operators, and many integral operators. We refer the reader to [2, 3, 4] for details, including historical comments and references. The polar decomposition of a complex symmetric operator is described in Theorem 2 of [4]. This result gives an outline of a description, up to unitary equivalence, of all complex symmetric operators. Our main goal in this note is to use spectral multiplicity theory to complete this description. Our discussion will emphasize an operator algebraic point of view.

This paper is based in part on the first author’s Master’s Project.

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2. Terminology and preliminary results Let B(H) be the algebra of bounded linear operators on the separable complex Hilbert space H. Recall that a map A : H → H is antilinear if A(αx + βy) = αAx + βAy for x, y ∈ H and α, β ∈ C. Let Ba (H) denote the collection of all bounded antilinear operators on H. We will see in Proposition 2.2 a natural relation between B(H) and Ba (H). C is a conjugation if C ∈ Ba (H), C 2 = I, and Cx, Cy = y, x ∀x, y ∈ H. That is, C is an antilinear isometric involution. We say that T ∈ B(H) is Csymmetric if T ∗ = CT C. (Equivalently, T ∗ C = CT , or T C = CT ∗ .) Let SC (H) = {T ∈ B(H) : T ∗ = CT C} denote the collection of C-symmetric operators. T is complex symmetric provided T ∈ SC (H) for some conjugation C. We note that this deﬁnition agrees with the deﬁnition given in the introduction. In fact, (see there is a basis {en } that is ﬁxed by C; so [3]), C ∈ Ba (H) is a conjugation iﬀ C( αn en ) = αn en for (αn ) ∈ 2 . Thus, if T ∈ B(H) and (Tij ) is the matrix of T relative to this basis, then T ∈ SC (H) iﬀ Tij = Tji ∀i, j. Remark 2.1. If C1 and C2 are conjugations on H, then there is a unitary operator U ∈ B(H) so that the map T → U T U −1 carries SC1 (H) onto SC2 (H). (Thus SC1 (H) and SC2 (H) are spatially linearly isomorphic.) In fact, if {en } and {fn } are bases ﬁxed by C1 and C2 , respectively, one can take U to be deﬁned by U en = fn ∀n. We now list some elementary structural results for SC (H) and Ba (H). Some of these results appear at least implicitly in the work of Garcia and Putinar. Proposition 2.2. If C is a conjugation on H, then Ba (H) = B(H) C. Further, the map φ : T → T C is a linear isometry of B(H) onto Ba (H) with inverse given by φ−1 : A → AC for A ∈ Ba (H) . Proof. Since C is an isometric bijection, if follows that φ is isometric. Clearly T ∈ B(H) implies T C ∈ Ba (H). Also, if A ∈ Ba (H), then AC ∈ B(H) and (AC)C = A, so φ is onto. The linearity of φ is trivial. Remark 2.3. Similar arguments show that ψ : T → CT is an antilinear isometry of B(H) onto Ba (H). Also, note that while B(H) is an algebra, Ba (H) is only a Banach space. If A, B ∈ Ba (H) then AB ∈ B(H), not Ba (H) . Proposition 2.4. If C is a conjugation on H, then SC (H) is a weakly closed, *closed subspace of B(H) . Proof. If T ∗ = CT C, then CT ∗ C = C(CT C)C = T , so SC (H) is *-closed. Next, if S, T ∈ SC (H) and β ∈ C, then C(S + βT )C = CSC + βCT C = S ∗ + βT ∗ and SC (H) is a subspace. Now let {Tα } be a net in SC (H) so that Tα → T weakly. For x, y ∈ H, Tα∗ x, y = CTα Cx, y = Cy, Tα Cx. Taking limits, we get that T ∗ x, y = CT Cx, y = Cy, T Cx. Thus, T ∈ SC (H) and SC (H) is weakly closed.

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Complex Symmetric Operators

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Note that if dim H ≥ 2, then SC (H) is not an algebra. To see this fact, it suﬃces to choose two symmetric matrices whose product is not symmetric. However, the next proposition shows that SC (H) does contain many algebras. Given a family F of operators on B(H), let W(F ) denote the weakly closed unital algebra generated by F . Proposition 2.5. If F is a commuting family of operators in SC (H), then W(F ) ⊂ SC (H) . Proof. We may as well assume that I ∈ F. Then W(F ) is the weakly closed span of all ﬁnite products of elements in F , so it will suﬃce to show that if A and B are commuting operators in SC (H), then AB ∈ SC (H). This is easy. Suppose that AC = CA∗ , BC = CB ∗ and AB = BA. Then (BA)C = ABC = ACB ∗ = CA∗ B ∗ = C(BA)∗ ,

and AB ∈ SC (H).

We note that the conclusion of this proposition can be stengthened slightly. Suppose F is as in Proposition 2.5 and let F ∗ = {A∗ : A ∈ F }. Then the weak closure of the subspace W(F ) + W(F * ) lies in SC (H). That is, the weakly closed, *closed subspace generated by W(F ) is in SC (H). For the proof, apply Propositions 2.4 and 2.5.

3. Structure of complex symmetric operators In [4, Theorem 2], the polar decompostion of a complex symmetric operator is described. This theorem and the discussion following its proof yields the following structure theorem. Theorem 3.1. If C is a conjugation on H, then SC (H) = {CJP :

P is a positive operator and J is a conjugation commuting with P }.

The factorization in Theorem 3.1 is related to polar decomposition as follows. Garcia and Putinar show that if T ∈ SC (H), then T = CJP , where P is the positive factor in the polar decomposition of T . If the kernel of T is trivial, then the partially isometric factor in the polar decomposition of T is the unitary operator CJ, while in general CJ is a unitary extension of the partially isometric factor. Our goal now is to give a complete description of the conjugations that commute with a ﬁxed positive operator. We will use spectral multiplicity theory, so we begin with some brief remarks on normal operators. If N ∈ B(H) is normal, then there is a compactly supported measure ν and a function φ ∈ L∞ (ν) so that N ∼ = Mφ , where Mφ denotes multiplication of φ on L2 (ν) and ∼ = denotes unitary equivalence. As noted earlier, N is complex symmetric ([2]). In fact, if K is the conjugation on L2 (ν) given by Kf = f , then Mφ ∈ SK (L2 (ν)). The following is now immediate from Proposition 2.5.

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Corollary 3.2. If N is a normal operator in SC (H), then W({N, N ∗ , I}), the von Neumann algebra generated by N , lies in SC (H). Proof. Take F = {N, N ∗ , I} in Proposition 2.5.

Next we describe the antilinear operators that commute with a normal operator. Lemma 3.3. Suppose that K is a conjugation and that the normal operator N lies in SK (H). Then {A ∈ Ba (H) : AN = N A} = {T K : T ∈ B(H) and T N = N T ∗ }. Proof. We apply Proposition 2.2. If A ∈ Ba (H), then A = T K for some T ∈ B(H). If also AN = N A, then T (N ∗ K) = T (KN ) = N (T K), so T N ∗ = N T . Now, reverse that argument. If T ∈ B(H) and T N ∗ = N T , and if A = T K, then A ∈ Ba (H) and AN = T KN = T N ∗ K = N T K = N A. Proposition 3.4. Suppose that P is a positive operator, that K is a conjugation, and that P ∈ SK (H). Then J is a conjugation commuting with P iﬀ J ∈ {U K : U ∈ SK (H), U is unitary, and U P = P U }. Proof. Suppose that J is a conjugation and that JP = P J. By Lemma 3.3, J = U K for some U ∈ B(H) with U P = P U . But then U = JK, so U is unitary. Also, I = J 2 = (U K)2 = U (KU K), so U ∗ = KU K and U ∈ SK (H). It is easy to check that the argument can be reversed to ﬁnish the proof. The last proposition says that to ﬁnd the conjugations commuting with a positive operator P , ﬁx a conjugation K so that P ∈ SK (H) and then ﬁnd the unitary operators in SK (H) that commute with P . For this we apply spectral multiplicity theory to P (see Chapter IX, Section 10 of [1]). Suppose ﬁrst that P has multiplicity one. Then there is a Borel measure µ whose support is the spectrum of P so that P ∼ = Pµ on L2 (µ), where Pµ : f (t) → tf (t) is multiplication by the independent variable. Then {Pµ } , the commutant of Pµ , is {Mφ : φ ∈ L∞ (µ)} [1, Corollary 6.9]. Deﬁne the conjugation Kµ on L2 (µ) by Kµ f = f , f ∈ L2 (µ). Then Pµ ∈ SKµ (L2 (µ)). In fact, Corollary 3.2 gives that {Pµ } = W({Pµ }) ⊂ SKµ (L2 (µ)). Note that U ∈ {Pµ } is unitary iﬀ U = Mφ for some φ ∈ L∞ (µ) with |φ| = 1 µ a.e. Using Proposition 3.4 we see that {J : JPµ = Pµ J} = {Mφ Kµ : φ ∈ L∞ (µ) and |φ| = 1 µ a.e.}.

(1)

Next we suppose that P has uniform multiplicity n, 1 < n ≤ ∞. Thus there (n) (n) is a Borel measure µ so that P ∼ = Pµ , where Pµ is the direct sum of n copies of (n) Pµ , acting on L2 (µ)(n) , the direct sum of n copies of L2 (µ). Kµ is a conjugation (n) on L2 (µ)(n) , and Pµ ∈ SK (n) (L2 µ(n) ). It is well-known (see [1, Proposition 6.1]) µ

(n)

and easy to check that {Pµ } = Mn ({Pµ } ), the n × n matrices with entries in {Pµ } .

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(n)

Suppose that U ∈ {Pµ } is unitary. Then U = (Mφij ), an n × n operator matrix with φij ∈ L∞ (µ) ∀i, j. Also U ∗ = (Mφji ). An elementary matrix computa(n)

(n)

tion shows that Kµ U Kµ

= (Kµ Mφij Kµ ) = (Mφij ). Thus U ∈ SK (n) (L2 (µ)(n) ) µ

iﬀ U ∗ = (Mφij ) iﬀ φji = φij ∀i, j. That is, U has a symmetric operator matrix. If we now apply Proposition 3.4, we get the following result. (n)

Proposition 3.5. J is a conjugation commuting with Pµ if and only if J has (n) the form J = (Mφij )Kµ , where (Mφij ) is a unitary operator matrix such that ∞ φij = φji ∈ L (µ), ∀i, j. Now consider any positive operator P . Following [1, Theorem 10.20], there are mutually singular measures µ∞ , µ1 , µ2 , · · · so that, up to unitary equivalence, ∞ ∞ (n) (∞) 2 (n) P = Pµn ⊕ Pµ∞ on H = L (µn ) ⊕ L2 µ∞ (∞) (2) n=1

and {P }

=

∞

{Pµ(n) } n n=1

n=1

} . ⊕ {Pµ(∞) ∞

(3)

(n) (∞) ∞ ⊕Kµ∞ . Also, J is a conjugation K Then P ∈ SK (H), where K = µ n n=1 ∞ commuting with P iﬀ J = ( n=1 Jn ) ⊕ J∞ is a direct sum of conjugations. For (n) each n, Jn = Un Kµn where Un is a symmetric unitary operator matrix described in Proposition 3.5.. This completes the description, up to unitary equivalence, of the complex symmetric operators. We close with an observation of how the complex symmetric operators sit between B(H) and the set of normal operators. First note that each unitary U is the product of conjugations. This follows immediately from Theorem 3.1. So suppose T ∈ B(H) has polar decomposition T = U P and that U is unitary. (For example, suppose T is invertible.) Then T = CJP for some conjugations C and J. But if T is complex symmetric, Theorem 3.1 shows that the above factorization can be achieved with the additional condition that J commutes with P . If T is normal, we show that we can choose both C and J to commute with P. Proposition 3.6. {T ∈ B(H) : T is normal } = {CJP : P is positive and C and J are conjugations commuting with P }. Proof. Supose T is normal. If H0 = ker T , then H0 reduces T so if H1 = H⊥ 0 and if T1 = T|H1 , then T1 is normal, so we have T1 = C1 J1 P for conjugations C1 , J1 on H1 with J1 commuting with P . Clearly C1 J1 also commutes with P , so C1 P J1 = C1 J1 P = P C1 J1 , and C1 commutes with P . Now extend C1 and J1 to be conjugations on all of H.

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The other inclusion is trivial to check.

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Thus there is a natural sense in which the complex symmetric operators sit halfway between the normal operators and B(H).

References [1] J.B. Conway, A Course in Functional Analysis, Springer, New York, (1990). [2] S.R. Garcia, Conjugation and Clark Operators, Contemp. Math. 393 (2006), 67–112. [3] S.R. Garcia and M. Putinar, Complex Symmetric Operators and Applications, Trans. Amer. Math. Soc. 358 (2006), 1285–1315. [4] S.R. Garcia and M. Putinar, Complex Symmetric Operators and Applications II, Trans. Amer. Math. Soc., to appear. T.M. Gilbreath and Warren R. Wogen Department of Mathematics University of North Carolina 3250, Phillips Hall Chapel Hill, N.C., 27599 USA e-mail: [email protected] Submitted: September 8, 2006 Revised: March 26, 2007

Integr. equ. oper. theory 59 (2007), 591–596 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040591-6, published online October 18, 2007 DOI 10.1007/s00020-007-1535-8

Integral Equations and Operator Theory

Nonautonomous Exponential Dichotomy Cornelis van der Mee Abstract. In this note we generalize the strongly continuous bisemigroups generated by exponentially dichotomous operators to so-called bievolution families. These families are then related to strongly continuous bisemigroups on certain Banach spaces of continuous and measurable vector-valued functions. Keywords. Bisemigroup, dichotomous operator, bievolution family.

1. Introduction In recent years exponentially dichotomous operators S(X → X) deﬁned on a dense linear subspace of a complex Banach space X have been studied extensively [1, 7, 10, 12]. They can be deﬁned through the Laplace transform relation ∞ −1 e−λt E(t; x) dt, (λ − S) x = −∞

where, for each x ∈ X, E(·; x) : R → X is strongly measurable and satisﬁes ∞ eε|t| E(t; x)X dt ≤ const.xX , x ∈ X, −∞

for some constant ε > 0. Then there exists a strongly continuous function E : R → L(X), the so-called bisemigroup, having its values in the complex Banach algebra L(X) of bounded linear operators on X and having a strong jump discontinuity at t = 0 such that E(t)x = E(t; x) for 0 = t ∈ R. Also E(0+ ) − E(0− ) = IX , the identity operator on X. Further, ±E(0± ) are complementary projections reducing S. Exponentially bounded evolution families have been deﬁned as strongly continuous operator functions U : {(t, s) ∈ R2 : t ≥ s} → L(X) having the properties (i) U (t, r)U (r, s) = U (t, s) for t ≥ r ≥ s, and (ii) U (t, s)L(X) ≤ M eε(t−s) for The research leading to this note was supported by the Italian Ministry of Education and Research (MIUR) under PRIN grant no. 2006017542-003, and by INdAM-GNCS.

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certain constants M, ε > 0. They are the natural generalizations of strongly continuous semigroups when modeling nonautonomous ﬁrst order initial value problems (cf. [2] and references therein). In the context of [4, 3, 2] exponential dichotomy pertains to the existence of a projection-valued function P : R → L(X) such that (i) U (t, s)P (s) = P (t)U (t, s) for t ≥ s, (ii) there exists ε > 0 such that U (t, s)P (s)xX ≤ const.eε(t−s) P (s)xX for t ≥ s, and (iii) the restriction of U (t, s) to the kernel of P (s) is a boundedly invertible operator deﬁned on the kernel of P (t) with norm bounded above by const.e−ε(t−s) for some ε > 0. Exponential dichotomy of exponentially bounded evolution families can be proven equivalent to the hyperbolicity of the strongly continuous semigroup E : R → L(Lp (R; X)) deﬁned by [E(t)f ](τ ) = U (τ, τ − t)f (τ − t) for t ≥ 0 and τ ∈ R (cf. [8]). In this note we generalize the exponentially dichotomous operators as studied in [1, 12] to so-called bievolution families, mimicking the terminology of bisemigroups introduced in [1]. In Theorem 2.2 we prove that U is a bievolution family on X if and only E deﬁned by [E(t)f ](τ ) = U (τ, τ − t)f (τ − t) is a strongly continuous bisemigroup on C0 (R; X). In Proposition 2.1 we also show that EU is a strongly continuous bisemigroup on Lp (R; X) (1 ≤ p < ∞) if U is a bievolution family. Exponentially dichotomous operators have among their applications Riccati equations [10], transport equations [5], functional diﬀerential equations [9], and noncausal linear systems [6]. Some of these applications have nonautonomous counterparts conductive to treatment as bievolution systems, such as nonautonomous functional diﬀerential equations [9] and evolution equations in Banach spaces [11]. Let us introduce some notations. Given a complex Banach space X, we write IX for the identity operator on X, L(X) for the Banach algebra of bounded linear operators on X, C0 (R; X) for the Banach space of strongly continuous functions f : R → X such that f (t)X → 0 as t → ±∞. For 1 ≤ p < ∞ we mean by Lp (R; X) the Banach space of strongly measurable functions f : R → X for which the scalar function f (·)X ∈ Lp (R).

2. Bievolution Families and Main Theorem Letting ∆± = {(t, s) ∈ R2 : ±(t − s) ≥ 0}, the disjoint (set theoretical and topological) union ∆ = ∆+ ∪ ∆− represents the Euclidean plane R2 , where we distinguish between (t, t− ) ∈ ∆+ and (t, t+ ) ∈ ∆− . Letting X be a complex Banach space, by a bievolution family on X we mean a strongly continuous operator function U : ∆ → L(X) having the following properties: 1. For (t, r) and (r, s) in ∆± we have the product rule U (t, r)U (r, s) = ±U (t, s). 2. For (t, τ ) ∈ ∆+ and (s, σ) ∈ ∆− we have U (t, τ )U (s, σ) = U (s, σ)U (t, τ ) = 0.

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3. There exist positive constants M and ε such that U (t, s)L(X) ≤ M e−ε|t−s| ,

(t, s) ∈ ∆± .

4. We have U (t+ , t) − U (t− , t) = IX ,

t ∈ R.

Then U (t, t− ) and −U (t, t+ ) for (t, t∓ ) ∈ ∆± are bounded complementary projections on X which are strongly continuous in t ∈ R. When U (t, s) only depends on (t − s) ∈ ∆ and hence we may write E(t − s) = U (t, s) while distinguishing between E(0+ ) and E(0− ), we obtain a (strongly continuous) bisemigroup on X. The separating projection then no longer depends on t and is called the separating projection of the bisemigroup. For convenience we ˙ for the disjoint (set theoretical and topological) union of R˙ − = (−∞, 0] write R and R˙ + = [0, ∞), so that E can be viewed as a strongly continuous operator function E : R˙ → L(X). Given a bievolution family U : ∆ → X, we deﬁne the evolutionary bisemi˙ → L(Lp (R; X)) (1 ≤ p < ∞) or EU : R ˙ → L(C0 (R; X)) by group EU : R (EU (t)f )(τ ) = U (τ, τ − t)f (τ − t),

(τ, τ − t) ∈ ∆.

Proposition 2.1. Let 1 ≤ p < ∞ and let U : ∆ → L(X) be a bievolution family. Then EU : R˙ → Lp (R; X) is a strongly continuous bisemigroup. Proof. Let U : ∆ → L(X) be a bievolution family. For t ∈ R˙ we have EU (t)f Lp (R;X) =

∞

p

−∞

∞

1/p

(EU (t)f )(τ ) dτ p

= −∞

U (τ, τ − t)f (τ − t) dτ

1/p

≤ M e−ε|t| f Lp(R;X) ,

˙ as well as the exponential bound which implies the boundedness of EU (t) for t ∈ R ˙ of the same sign we estimate on its norm. Further, for t, s ∈ R p

EU (t)f − EU (s)f Lp (R;X) ∞ = U (τ, τ − t)f (τ − t) − U (τ, τ − s)f (τ − s)pX dτ −∞ ∞ [U (τ + t, τ ) − U (τ + t, τ + t − s)]f (τ )pX dτ ≤ −∞ ∞ f (τ ) − f (τ + t − s)pX dτ, + M e−ε|s| −∞

which vanishes as s → t as a result of the strong continuity of U .

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For t, s ∈ R˙ we have (EU (t)EU (s)f )(τ ) = U (τ, τ − t)(EU (s)f )(τ − t) = U (τ, τ − t)U (τ − t, τ − t − s)f (τ − t − s)    U (τ, τ − t − s)f (τ − t − s) = (EU (t + s)f )(τ ), = −U (τ, τ − t − s)f (τ − t − s) = −(EU (t + s)f )(τ ),   0,

t, s ≥ 0, t, s ≤ 0, ts < 0,

which implies the product rule. Next, (EU (0+ )f )(τ ) − (EU (0− )f )(τ ) = U (τ, τ − )f (τ ) − U (τ, τ + )f (τ ) = f (τ ), so that EU (0+ ) − EU (0− ) = ILp (R;X) . Thus, if U : ∆ → L(X) is a bievolution family on X, then EU is a strongly continuous bisemigroup on Lp (R; X). We now derive the main result of this note. Theorem 2.2. The operator function U : ∆ → L(X) is a bievolution family iﬀ ˙ → L(X) is a strongly continuous bisemigroup on C0 (R; X). EU : R Proof. For t ∈ R˙ we have EU (t)f C0 (R;X) ≤ sup U (τ, τ − t)L(X) f C0 (R;X) ≤ M e−ε|t| f C0(R;X) , τ ∈R

which yields the boundedness of EU (t) and the exponential bound on its norm. For t, s ∈ R˙ of the same sign we estimate EU (t)f − EU (s)f C0 (R;X) = sup U (τ, τ − t)f (τ − t) − U (τ, τ − s)f (τ − s)X τ ∈R

≤ sup [U (τ + t, τ ) − U (τ + t, τ + t − s)]f (τ )X τ ∈R

+ M e−ε|s| sup f (τ ) − f (τ + t − s)X , τ ∈R

˙ → C0 (R; L(X)). Thus, if U : ∆ → L(X) proving the strong continuity of EU : R is a bievolution family on X, then EU is a strongly continuous bisemigroup on C0 (R; X). ˙ → C0 (R; X) be a strongly continuous bisemigroup. Conversely, let EU : R For x ∈ X and φ ∈ C0 (R) we deﬁne F(φ,x) ∈ C0 (R; X) by [F(φ,x) ](t) = φ(t)x,

t ∈ R.

Then taking a function φ without zeros we see from the identity U (τ, τ − t)x =

[E(t)F(φ,x) ](τ ) φ(τ − t)

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that U : ∆ → L(X) is strongly continuous. For t, s ∈ R˙ we have on the one hand EU (t)EU (s)F(φ,x) (τ ) = U (τ, τ − t) EU (s)F(φ,x) (τ − t) = U (τ, τ − t)U (τ − t, τ − t − s) F(φ,x) (τ − t − s) = φ(τ − t − s)U (τ, τ − t)U (τ − t, τ − t − s)x and on the other hand EU (t + s)F(φ,x) (τ ) = U (τ, τ − t − s) F(φ,x) (τ − t − s) = φ(τ − t − s)U (τ, τ − t − s)x. Taking φ ∈ C0 (R) to be nonzero, we obtain from the product rule for EU the product properties 1 and 2 for U . Moreover,

φ(τ )x = EU (0+ ) − EU (0− ) F(φ,x) (τ ) = φ(τ ) U (τ, τ − )x − U (τ, τ + )x implies that U (τ, τ − ) − U (τ, τ + ) = IX . Finally, φC0 (R) U (τ, τ − t)xX = EU (t)F(φ,x) C (R;X) 0 −ε|t| F(φ,x) C (R;X) = M e−ε|t| φC0 (R) xX ≤ Me 0

implies the exponential decay condition 4 on U . Thus, U : ∆ → L(X) is a bievolution family.

References [1] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Funct. Anal. 68, 1–42 (1986). [2] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Diﬀerential Equations, Math. Surveys and Monographs 70, Amer. Math. Soc., Providence, RI, 1999. [3] Ju.L. Daleckii and M.G. Krein, Stability of Solutions of Diﬀerential Equations in Banach Space, Transl. Math. Monographs 43, Amer. Math. Soc., Providence, RI, 1974; also: Izdat. “Nauka,” Moscow, 1970 (in Russian). [4] H.O. Fattorini, The Cauchy Problem, Encycl. Math. Appl. 18, Addison Wesley, Reading, 1983. [5] W. Greenberg, C.V.M. van der Mee, and V. Protopopescu, Boundary Value Problems in Abstract Kinetic Theory, Birkh¨ auser OT 23, Basel and Boston, 1987. [6] M.A. Kaashoek, C.V.M. van der Mee, and A.C.M. Ran, Wiener-Hopf factorization of transfer functions of extended Pritchard-Salamon realizations, Math. Nachrichten 196, 71–102 (1998). [7] M.A. Kaashoek and S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Diﬀ. Eqs. 112, 374–406 (1994). [8] Yu. Latushkin and T. Randolph, Dichotomy of diﬀerential equations on Banach spaces and an algebra of weighted translation operators, Integral Equations and Operator Theory 23, 472–500 (1995).

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[9] J. Mallet-Paret and S.M. Verduyn Lunel, Exponential dichotomies and WienerHopf factorizations for mixed-type functional diﬀerential equations, J. Diﬀ. Eqs., to appear. [10] A.C.M. Ran and C. van der Mee, Perturbation results for exponentially dichotomous operators on general Banach spaces, J. Funct. Anal. 210, 193–213 (2004). [11] R.J. Sacker and G.R. Sell, Dichotomies for linear evolution equations in Banach spaces, J. Diﬀ. Eqs. 113, 17–67 (1994). [12] C. van der Mee, Exponentially Dichotomous Operators and Applications, Birkh¨ auser, Basel and Boston, in preparation. Cornelis van der Mee Dip. Matematica e Informatica Universit` a di Cagliari Viale Merello 92 I-09123 Cagliari Italy e-mail: [email protected] Submitted: April 24, 2007

Integr. equ. oper. theory 59 (2007), 597–601 c 2007 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040597-5, published online October 18, 2007 DOI 10.1007/s00020-007-1539-4

Integral Equations and Operator Theory

On a Maximum Principle for Bergman Spaces with Small Exponents Chunjie Wang Abstract. Let Ap (D)(0 < p < ∞) be the Bergman space over the open unit disk D in the complex plane. We know that Korenblum’s maximum principle holds when 1 ≤ p < ∞. In this note we prove that it is true for Bergman spaces with small exponents (0 < p < 1) under some supplementary conditions. Mathematics Subject Classification (2000). Primary 30C80; Secondary 30H05. Keywords. Bergman space, Korenblum’s maximum principle.

1. Introduction Let D be the open unit disk in the complex plane C. For 0 < p < ∞ the Bergman space Ap (D) consists of analytic functions f in D such that Z p1 1 kf kAp = |f (z)|p dA(z) < +∞, π D where dA(z) = dx dy = rdr dθ,

z = x + iy = reiθ

is the Lebesgue area measure. It is well known that the Bergman space Ap (D) is a Banach space when 1 ≤ p < ∞ and a complete metric space when 0 < p < 1. Korenblum’s maximum principle states that, for any real number p with p ≥ 1, there is a constant c ∈ (0, 1) (may depend on p) such that whenever |f (z)| ≤ |g(z)| (f, g ∈ Ap (D)) in the annulus c < |z| < 1, then kf kAp ≤ kgkAp . First conjectured by Korenblum [3] with p = 2, the maximum principle was proved by Hayman [1] when p = 2. Later Hinkkanen [2] proved that it is valid for the Bergman space Ap (D) (p ≥ 1) with an absolute constant c ≥ 0.15724 . . . . And Hinkkanen [2] asked whether Korenblum’s maximum principle holds when 0 < p < This work was supported by NNSF of China No. 10601025 and the Natural Science Foundation of Hebei Province of China (No. 07M001).

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1. In this note we prove that it is true under some supplementary conditions. Our main results are the following two theorems. Theorem 1.1. Let p be a real number with 0 < p < 1. Suppose that f (z) and g(z) are analytic functions in the unit disk D, and g(z) has only a simple zero at 0 in D. If |f (z)| ≤ |g(z)| for c < |z| < 1, where c = 0.1123. Then kf kAp ≤ kgkAp . The proof of Theorem 1.1 is presented in the next section. Once Theorem 1.1 is proved, we can easily obtain the following theorem. Theorem 1.2. Let p be a real number with 0 < p < 1. Suppose that f (z) and g(z) are analytic functions in the unit disk D, and g(z) has only a simple zero a with R−|a| < |z| < 1. Then |a| < R = 0.1123 in D. If |f (z)| ≤ |g(z)| in the annulus 1−|a|R p p kf kA ≤ kgkA . Proof. Note that a−z R − |a| <1 . < |z| < 1 ⊃ z : R < z: 1 − |a|R 1 − az Now the assumption of Theorem 1.2 implies that |f (z)| ≤ |g(z)| in the annulus a−z < 1. R< 1 − az So the desired result follows from Theorem 1.1 and a change of variables z → a−z 1−az . However, we can not prove Korenblum’s maximum principle for Bergman spaces with small exponents 0 < p < 1 without these supplementary conditions. If it is true, we can obtain an upper bound for the constant cp . Let f (z) = cp , g(z) = p1 2 . Then kf k = kgk and |f (z)| ≤ |g(z)| in cp < |z| < 1. It z, where cp = p+2 follows a simple calculation that cp is increasing on p. So 2 1 0 cp > √ = 0.6065 . . . , 3 e

2. Proof of Theorem 1.1 Let c ∈ (0, 1) be a constant to be determined later. Write D(c) = {z ∈ C : |z| ≤ c} and A(c, 1) = {z ∈ C : c < |z| < 1}. We want to show that Z Z (|f (z)|p − |g(z)|p )dA(z) ≤ (|g(z)|p − |f (z)|p )dA(z). (2.1) D(c)

A(c,1)

From the assumption of Theorem 1.1, we may assume that g(z) = zϕ(z), where ϕ(z) is analytic and nonvanishing in D. Notice that g is not identically equal to 0. Define ω(z) = f (z)/g(z) for c < |z| < 1, so that |ω| ≤ 1. We may assume that |ω(z)| < 1 in c < |z| < 1, since otherwise the result is trivial. Define ω0 = ω0 (ρ) = max{|ω(z)| : |z| = ρ}.

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On a Maximum Principle for Bergman Spaces

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Suppose that 0 < p < 1. Then for any x ≥ 0, y ≥ 0, we have pxp−1 (x − y) ≤ xp − y p ≤ py p−1 (x − y).

(2.2)

Inequality (2.2) can be proved by an easy calculus argument. So we get |f |p − |g|p ≤ p|g|p−1 (|f | − |g|) ≤ p|g|p−1 (|f | − |ω0 g|) ≤ p|g|p−1 |f − ω0 g|.

(2.3)

Using (2.3), and the fact that, for a subharmonic function h in D and 0 < R 2π R 2π r1 < r2 < 1, 0 h(r1 eiθ )dθ ≤ 0 h(r2 eiθ )dθ, we obtain for r < c < ρ < 1 Z 2π (|f (reiθ )|p − |g(reiθ )|p )dθ 0 2π

Z

p|g(reiθ )|p−1 |f (reiθ ) − ω0 g(reiθ )|dθ

≤ 0

= rp−1

Z

2π

p|ϕ(reiθ )|p−1 |f (reiθ ) − ω0 g(reiθ )|dθ

0

≤ rp−1 =

r ρ

Z

2π

p|ϕ(ρeiθ )|p−1 |f (ρeiθ ) − ω0 g(ρeiθ )|dθ

0 p−1

2π

Z

p|g(ρeiθ )|p−1 |f (ρeiθ ) − ω0 g(ρeiθ )|dθ

0

p−1 Z 2π r p|ω(ρeiθ ) − a| (|g(ρeiθ )|p − |f (ρeiθ )|p )dθ. ρ 1 − |ω(ρeiθ )|p 0

=

Taking x = 1 and y = |ω| in the left hand inequality in (2.2), we obtain p 1 1 + |ω| 2 ≤ = ≤ . p 2 1 − |ω| 1 − |ω| 1 − |ω| 1 − |ω|2 Write γ(ρ) = max Then Z 2π

2|ω(z) − a| : |z| = ρ . 1 − |ω(z)|2

p−1 Z 2π r (|f (re )| − |g(re )| )dθ ≤ γ(ρ) (|g(ρeiθ )|p − |f (ρeiθ )|p )dθ. ρ 0 0 (2.4) Let us keep ρ fixed and multiply both sides of (2.4) by r and then integrate with respect to r from 0 to c. We obtain Z ρp−1 (|f (z)|p − |g(z)|p )dA(z) γ(ρ) D(c) Z cp+1 2π ≤ (|g(ρeiθ )|p − |f (ρeiθ )|p )dθ. (2.5) p+1 0 iθ

p

iθ

p

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Wang

IEOT

Multiplying both sides of (2.5) by ρ and integrating with respect to ρ from c to 1 yields Z 1 p Z ρ dρ (|f (z)|p − |g(z)|p )dA(z) γ(ρ) c D(c) Z cp+1 (|g(z)|p − |f (z)|p )dA(z). ≤ (2.6) p + 1 A(c,1) To prove (2.1), we need only to choose c such that Z 1 p ρ cp+1 dρ ≥ , p+1 c γ(ρ) which is equivalent to Z Φ(p, c) = c

1

p + 1 ρ p dρ ≥ 1. cγ(ρ) c

(2.7)

It is easy to see that Φ(p, c) is an increasing function of p. So Z 1 1 Φ(p, c) ≥ Φ(0, c) = dρ. cγ(ρ) c Thus if we can choose c so that Z c

1

1 dρ ≥ 1, cγ(ρ)

(2.8)

then Theorem 1.1 is proved. Recall that the pseudohyperbolic distance d between two points α, β ∈ D is defined by α−β . d(α, β) = 1 − αβ Schuster [5](see also in [6]) proved that ω(z) − ω0 ≤K<1 d(ω(z), ω0 ) = 1 − ω0 ω(z) for |z| = ρ, where K = 2ρ(1 + ρ−2 c)

∞ Y (1 + ρ2 c2n−1 )(1 + ρ−2 c2n+1 )(1 + c2n )2 . (1 + ρ2 c2n−2 )(1 + ρ−2 c2n )(1 + c2n−1 )2 n=1

(2.9)

From Hinkkanen [2] or Schuster [5], we can choose γ(ρ) = √

2K . 1 − K2

Using Mathematica, we obtain that (2.8) holds with c = 0.1123. This completes the proof of Theorem 1.1.

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3. The case of several complex variables Korenblum’s maximum principle is true in the setting of several complex variables with arbitrary constant c > 0. Let n ≥ 2 be an integer, B denote the unit ball in Cn . For 0 < p < ∞ the Bergman space Ap (B) consists of holomorphic functions f in B ⊂ Cn such that Z p1 kf kAp = |f (z)|p dλ(z) < +∞, B

where dλ(z) is the normalized Lebesgue measure. If |f (z)| ≤ |g(z)| for c < |z| < 1, where c > 0 is fixed. Without loss of generality, we may assume that g is not identically equal to 0. Define ω(z) = f (z)/g(z) for c < |z| < 1, so that |ω| ≤ 1. We may assume that |ω(z)| < 1 in c < |z| < 1, since otherwise the result is trivial. Then it follows from Hartogs’ theorem (see [4], p.341) that ω(z) can extend to a holomorphic function in B. Applying the maximum modulus theorem we know |ω(z)| < 1 for all z ∈ B. That is, |f (z)| < |g(z)| for all z ∈ B. This shows kf kAp ≤ kgkAp .

References [1] W. K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19(1999), 195-205. [2] A. Hinkkanen, On a maximum principle in Bergman space, J. Anal. Math. 79(1999), 335-344. [3] B. Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35(1991), 479-486. [4] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980. [5] A. Schuster, The maximum principle for the Bergman space and the M¨ obius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134(2006), 3525-3530. [6] C. Wang, On Korenblum’s maximum principle, Proc. Amer. Math. Soc. 134(2006), 2061-2066. Chunjie Wang Department of Mathematics Hebei University of Technology Tianjin 300130 P.R. China Mathematics Research Center of Hebei Province Shijiazhuang 050016 P.R. China e-mail: [email protected] Submitted: January 31, 2007 Revised: June 19, 2007