Integral Equations and Operator Theory - Volume 58

Integr. equ. oper. theory 58 (2007), 1–33 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-33, published ...

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Integr. equ. oper. theory 58 (2007), 1–33 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-33, published online April 16, 2007 DOI 10.1007/s00020-007-1493-1

Integral Equations and Operator Theory

Toeplitz Operators on Arveson and Dirichlet Spaces Daniel Alpay and H. Turgay Kaptano˘glu Abstract. We define Toeplitz operators on all Dirichlet spaces on the unit ball of CN and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schattenclass weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. Mathematics Subject Classification (2000). Primary 47B35, 32A37; Secondary 47B07, 47B10, 47B37, 47B33, 46E22, 32A36, 32A35. Keywords. Toeplitz operator, weighted shift, m-isometry, unitary equivalence, Carleson measure, Berezin transform, Bergman metric, Bergman projection, weak convergence, Schatten-von Neumann ideal, Besov, Bergman, Dirichlet, Hardy, Arveson space.

1. Introduction The theory of Toeplitz operators on Bergman spaces on the unit ball in one and several variables is a well-established subject. Weighted Bergman spaces A2q with q > −1 are naturally imbedded in Lebesgue classes L2q by the inclusion i, and there are suﬃciently many Bergman projections from Lebesgue classes onto Bergman spaces. Then one deﬁnes the Toeplitz operator Tφ : A2q → A2q with symbol φ by Tφ = Pq Mφ i, where Mφ is the operator of multiplication by φ and Pq is the orthogonal projection from L2q onto A2q , a Bergman projection. Investigating the boundedness and compactness of these Toeplitz operators with symbols in various The research of the second author is partially supported by a Fulbright grant.

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classes of functions has been an active area of research. A good source, especially for positive φ, is [37, Chapter 6]. By contrast, there is not one single deﬁnition of a Toeplitz operator that is agreed upon even on the classical Dirichlet space of the disc. The papers [11], [12], [14], [20], [26], [32], [35], [36] discuss several diﬀerent kinds of Toeplitz operators on the Dirichlet space. The connections among them, and between them and the Toeplitz operators on Bergman spaces are not clear. Only [26] deals with the Dirichlet space on the ball, and only [32] and [35] can handle the more general Dirichlet spaces Dq but for limited values of q, those between the Dirichlet space and the Hardy space. To the best of our knowledge, there is no work on Toeplitz operators on the Arveson space, not to mention one that can encompass all Dirichlet spaces Dq on the unit ball. There are some diﬃculties with Toeplitz operators on Dirichlet spaces that are not Bergman spaces, and these are the causes for discrepancies in various deﬁnitions used. The ﬁrst is that inclusion does not imbed these spaces in the most appropriate Lebesgue classes. The second is to decide which projections to use from which Lebesgue classes. Thus one sees in literature Toeplitz operators Tφ f deﬁned via an integral that involve f or its derivatives, or φ or its derivatives, or the Bergman, Hardy, or Dirichlet kernels or their derivatives. A third diﬃculty is that reproducing kernels of Dq for a large range of q are bounded and their normalized forms are not weakly convergent. This makes them impossible to use for obtaining a Berezin transform and perhaps explains why this range of q is never touched upon. The diﬃculties are resolved by recognizing Dirichlet spaces Dq on the ball as the Besov spaces Bq2 , where q ∈ R is adjusted so that Dq = A2q when q > −1. These spaces are deﬁned by imbedding them into Lebesgue classes via the linear maps Ist f (z) = (1 − |z|2 )t Dst f (z), where Dst is a radial diﬀerential operator of suﬃciently high order t with q + 2t > −1. Extended Bergman projections Ps that map Lebesgue classes boundedly onto Dirichlet spaces can be precisely identiﬁed as in the case of weighted Bergman spaces by q + 1 < 2(s + 1). Then Ist is a right inverse to Ps . This is all done in [22]. Now for all q ∈ R, we deﬁne the Toeplitz operator s Tφ : Dq → Dq with symbol φ by s Tφ = Ps Mφ Is−q+s . When q > −1, the case of weighted Bergman spaces, s = q is classical, but when q ≤ −1, s must satisfy −q + 2s > −1, so s = q. It is possible to take s = q also when q > −1. So we have more general Toeplitz operators deﬁned via Is−q+s strictly on Bergman spaces too. It turns out that the properties of s Tφ studied in this paper are independent of s and q. The results we obtain on the boundedness, compactness, and membership in Schatten classes of s Tφ for φ ≥ 0 specialize to what is known for weighted Bergman spaces when s = q. Our main tools are Carleson measures and Berezin transforms. The ﬁrst is deﬁned via Ist rather than i; the second is deﬁned via weakly convergent families in all Dq that are actually Bergman reproducing kernels with diﬀerent normalizations. These Carleson measures and weakly convergent families for all Dq are studied ﬁrst in [23].

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More is true. The space D−1 is the Hardy space H 2 . Now s > −1 must hold, so s = −1, and hence s Tφ is not the classical Toeplitz operator on H 2 . However, as s → −1+ , we indeed recover the classical Toeplitz operators on H 2 . We thereby present a uniﬁed theory of Toeplitz operators on all Dirichlet and Bergman spaces, the Arveson space, and the Hardy space. The paper is organized as follows. The notation and some preliminary material are summarized in Section 2. Section 3 is for groundwork on Dirichlet spaces, Bergman projections on them, their imbeddings, and the diﬀerential operators between them, on which so much of this work rests. In Section 4, we deﬁne Toeplitz operators on all Dq and develop several of their elementary properties. An intertwining relation between Toeplitz operators on Dq and the classical ones on weighted Bergman spaces turns out to be versatile. We introduce the Berezin transforms in Section 5 and obtain some of their immediate consequences. We then explore the connection with the classical Hardy-space Toeplitz operators. Our main results are in Section 6. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols. We work more generally with Toeplitz operators whose symbols are positive measures. The results in Sections 4, 5, and 6 attest to the fact that the Toeplitz operators on general Dq are natural extensions of classical Bergman-space Toeplitz operators. Section 7 describes an important application of Toeplitz operators on Dq . We readily obtain characterizations of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces on the ball in terms of Carleson measures and Berezin transforms. The paper concludes with some remarks on the relationship between Toeplitz and shift operators in Section 8.

2. Notation and Preliminaries The unit ball of CN is denoted B, and the volume measure ν on it is normalized with ν(B) = 1. When N = 1, it is the unit disc D. For c ∈ R, we deﬁne on B also the measures dνc (z) = (1 − |z|2 )c dν(z), which are ﬁnite only for c > −1, where |z|2 = z, z and z, w = z1 w 1 +· · ·+zN w N . In particular, we set τ = ν−(N +1) . The associated Lebesgue classes are Lpc , and L∞ simply is the class of bounded measurable functions on B. If X is a set, then X denotes its closure and ∂X its boundary. We let C be the space of continuous functions on B and C0 its subspace whose members vanish on ∂B. If T is a Hilbert-space operator, then σ(T ) denotes its spectrum and σp (T ) its point spectrum. In multi-index notation, α = (α1 , . . . , αN ) ∈ NN is an N -tuple of nonnegative αN integers, |α| = α1 + · · · + αN , α! = α1 ! · · · αN !, z α = z1α1 · · · zN , and 00 = 1. The symbol δnm denotes the Kronecker delta.

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Constants in formulas are all denoted by unadorned C although each might have a diﬀerent value. They might depend on certain parameters, but are always independent of the functions that appear in the formulas. We use the convenient Pochhammer symbol deﬁned by (a)b =

Γ(a + b) Γ(a)

when a and a + b are oﬀ the pole set −N of the gamma function Γ. For ﬁxed a, b, Stirling formula gives Γ(c + a) ∼ ca−b Γ(c + b)

and

(a)c ∼ ca−b (b)c

(c → ∞),

(2.1)

where x ∼ y means that both |x| ≤ C |y| and |y| ≤ C |x|, and above such C are independent of c. The hypergeometric function is 2 F1 (a, b; c; x)

=

∞ (a)k (b)k xk (c)k k!

(|x| < 1).

k=0

The Bergman metric on B is d(z, w) =

1 + |ϕz (w)| 1 log = tanh−1 |ϕz (w)| 2 1 − |ϕz (w)|

(z, w ∈ B),

where ϕz (w) is the M¨ obius transformation on B that exchanges z and w; see [33, §2.2]. The ball centered at w with radius 0 < r < ∞ in the Bergman metric is denoted b(w, r). The Bergman ball b(0, r) is also the Euclidean ball with the same center and radius 0 < tanh r < 1. The Bergman metric is invariant under compositions with the automorphisms of B, hence ψ(b(w, r)) = b(ψ(w), r) for any ψ ∈ Aut(B). Bergman balls have the following properties, whose proofs can be found in [24, §2]. Lemma 2.1. Given c ∈ R and r, we have νc (b(w, r)) ∼ (1 − |w|2 )N +1+c

(w ∈ B).

Given also w ∈ B, we have 1 − |z|2 ∼ 1 − |w|2

and

|1 − z, w| ∼ 1 − |w|2

(z ∈ b(w, r)).

Lemma 2.2. Given c ∈ R and r, there is a constant C such that for all 0 < p < ∞, g ∈ H(B), and w ∈ B, we have C p |g(w)| ≤ |g|p dνc . νc (b(w, r)) b(w,r) Let’s note that the measure τ is also invariant under compositions with the members of Aut(B); see [33, Theorem 2.2.6]. Given 0 < r < ∞, we call a sequence {an } of points in B an r-lattice in B if the union of the balls {b(an , r)} cover B and d(an , am ) ≥ r/2 for n = m. The second condition controls the amount of cover so that any point in B belongs to

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at most M of the balls {b(an , 2r)} for some M that does not depend on anything. That r-lattices exist is proved for the unit disc in [7, Lemma 3.5]. A twice diﬀerentiable function f on B satisfying ∆(f ◦ ϕz )(0) = 0 for all z ∈ B is called M-harmonic, where ∆ is the usual Laplacian on R2N , and ϕz is the M¨ obius transformation of B mentioned above. If f is M-harmonic, so is f ◦ ψ for any ψ ∈ Aut(B). If f is M-harmonic, then the mean value of f on a sphere of radius less than 1 is equal to f (0); see [33, p. 52]. If additionally f ∈ L1c for c > −1, it follows that (1 + c)N f (ψ(0)) = (f ◦ ψ) dνc (ψ ∈ Aut(B)) N! B by polar coordinates. Now we pick ψ = ϕw , make a change of variables in the integral using formula [33, Theorem 2.2.6 (6)] for the Jacobian of φw , and use identity [33, Theorem 2.2.2 (iv)] to simplify. The result is (1 + c)N (1 − |z|2 )c f (w) = (1 − |w|2 )N +1+c f (z) dν(z). (2.2) (N +1+c)2 N! B |1 − z, w| The right hand side is seen to be a Berezin transform of f in Section 5.

3. Dirichlet Spaces Dirichlet spaces are Hilbert spaces of holomorphic functions on B. We give three equivalent deﬁnitions each of which has its use. The index q ∈ R is everywhere unrestricted. Definition 3.1a. The Dirichlet space Dq is the reproducing kernel Hilbert space on B with reproducing kernel  ∞  1 (N + 1 + q)k   z, wk , = if q > −(N + 1);   (1 − z, w)N +1+q k! k=0 Kq (z, w) = ∞  k! z, wk 2 F1 (1, 1; 1 − N − q; z, w)   = , if q ≤ −(N + 1).   −N − q (−N − q)k+1 k=0

Thus Dq for q > −1 are the weighted Bergman spaces A2q , D−1 is the Hardy space H 2 , D−N is the Arveson space A (see [1] and [4]), and D−(N +1) is the classical Dirichlet space D since K−(N +1) (z, w) =

1 1 log . z, w 1 − z, w

The hypergeometric kernels appear in [10, p. 13]. The kernels Kq are complete Nevanlinna-Pick kernels if and only if q ≤ −N as explained in [5]. Further, they are bounded if and only if q < −(N + 1).

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The reproducing kernel Kq is sesqui-holomorphic, Dq consists of functions in H(B), and monomials are dense in Dq . By (2.1), we have Kq (z, w) ∼

∞

k

N +q

k

z, w =

k=0

∞

k N +q

k=0

k! |α|N +q |α|! z α wα = z αwα α! α! α

|α|=k

for any q. Thus α! (α ∈ NN ) (3.1) |α|N +q |α|! by [6, Theorem 3.3.1]. The norms (3.1) lead to the second equivalent deﬁnition of Dirichlet spaces. Definition 3.1b. The Dirichlet space Dq is the space of f (z) = α cα z α in H(B) for which α! < ∞. |cα |2 N +q |α| |α|! z α 2Dq ∼

α=0

If N = 1, the growth rate of the norms in (3.1) is z n Dq ∼ n−(1+q)/2 . For this reason, the Dq deﬁned here is often named D−(1+q) or D−(1+q)/2 elsewhere. The third equivalent deﬁnition recognizes that the Dirichlet space Dq as the Besov space Bq2 as described in [21] and [22]. For comparison, it is also the holomorphic Sobolev space A21+q+2t,t of [10], but this must not be confused with the Bergman-space notation A2q of ours. But we need to introduce some radial derivatives ﬁrst. ∞ Let f ∈ H(B) be given by its homogeneous expansion f = k=0 fk , where fk is a homogeneous polynomial of degree k. Then its radial derivative at z is k f (z). In [22, Deﬁnition 3.1], Rf (z) = ∞ k k=1 fort any s, t, the radial diﬀerential operator Dst is deﬁned on H(B) by Dst f = ∞ k=0 (s dk )fk , where  (N + 1 + s + t)k   , if s > −(N +1), s+t > −(N +1);    (N + 1 + s)   (N +1+s+t) k(−(N +s))   k k+1  , if s ≤ −(N +1), s+t > −(N +1);  2 (k!) t 2 s dk = (k!)   , if s > −(N +1), s+t ≤ −(N +1);    (N +1+s) (−(N +s+t))k+1   (−(N + ks))  k+1   , if s ≤ −(N +1), s+t ≤ −(N +1).  (−(N + s + t))k+1 What is important is that t s dk

for any s, t.

= 0

(k = 0, 1, 2, . . .)

Clearly Ds0 is the identity u Dst = Dsu+t , Ds+t

for any s, t, u. It turns out that each t on H(B) with two-sided inverse

and

t s dk

∼ kt

(k → ∞)

for any s,

Dst

and

Dst (1) = st d0

(3.2)

is a continuous invertible operator of order

−t (Dst )−1 = Ds+t .

(3.3)

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1 Other useful properties are that D−N = I + R and Dst (z β ) = st d|β| z β . The parameters s and t can be complex numbers too; then we just need to replace them with their real parts in inequalities as done in [22]. A script Dq with only a lower index represents a Dirichlet space while an upper case Dst with a lower and an upper index represents a radial diﬀerential operator. They should not be confused. Another property of Dst we use without further mention is that it always acts on the holomorphic variable. Hence the series expansion of Kq shows that always

Dqt Kq (z, w) = Kq+t (z, w).

(3.4)

Now we deﬁne the linear transformations Ist that are essential to this work by Ist f (z) = (1 − |z|2 )t Dst f (z)

(f ∈ H(B)).

Definition 3.1c. The Dirichlet space Dq is the space of f ∈ H(B) for which the function Ist f belongs to L2q for some s and t satisfying q + 2t > −1.

(3.5)

The L2q norm of any such Ist f is an equivalent Dq norm of f . It is shown in [10, Theorem 5.12 (i)] and [22, Theorem 4.1] that Deﬁnition 3.1c is independent of s, t, and that the L2q norms of Ist f and Ist11 f are equivalent, both as long as (3.5) is satisﬁed by t and t1 . To obtain the equivalence of this deﬁnition to the ﬁrst two deﬁnitions of Dq , it suﬃces to compute the norm of z α in Dq in Deﬁnition 3.1c and to observe that it has the same growth rate as that of (3.1) as |α| → ∞; see also [10, pp. 13–14]. We use [22, Proposition 2.1] in such norm computations. Thus Ist : Dq → L2q with t satisfying (3.5) is an isometric imbedding modulo the equivalences of norms in Dq . Deﬁnition 3.1c yields explicit equivalent forms for the inner product of Dq as t Ist f Ist g dνq = [Ist f, Ist g]L2q (f, g ∈ Dq ) q [f, g]s = B

with t satisfying (3.5). The reproducing property q [f, Kq (·, w)]ts = C f (w) written explicitly takes the form Dst f (z) Dst Kq (z, w) dνq+2t (z) = C f (w) B

for the same t, which can be further simpliﬁed for s = q using (3.4). We need a constant C in order to accomodate the variation due to s, t. Let’s show the norm on Dq associated to q [·, ·]ts by q · ts . The following is easy to show, but a proof can be found in [25, §3]. Proposition 3.2. For any q, s, t, Dst (Dq ) = Dq+2t is an isometric isomorphism with appropriate norms on the two spaces; for example, when Dq has q · us and Dq+2t has q+2t · u−t s+t while (3.5) is satisﬁed with u in place of t.

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We would like to know the adjoint of Dst : Dq → Dq+2t . Because each Dirichlet space has several equivalent inner products, let’s state it explicitly by showing the particular inner products used. It is the operator (Dst )∗ : Dq+2t → Dq satisfying u−t t t ∗ u q+2t [Ds f, g]s+t = q [f, (Ds ) g]s with q + 2u > −1 for f ∈ Dq and g ∈ Dq+2t . Writing this out in integrals, by the uniqueness of the adjoint and using (3.3) and (3.2), we obtain the somewhat surprising result that −t (Dst )∗ = Ds+t = (Dst )−1 .

(3.6)

Bergman projections, as extended in [22], are the linear transformations Ps f (z) = Ks (z, w) f (w) dνs (w) (z ∈ B) B

deﬁned for all s with suitable f . The next result is contained in [22, Theorem 1.2]. Theorem 3.3. The operator Ps : L2q → Dq is bounded if and only if −q + 2s > −1.

(3.7)

Given an s satisfying (3.7), if t satisﬁes (3.5), then Ps Ist f =

N! 1 f =: f (1 + s + t)N Cs+t

(f ∈ Dq ).

The second statement clearly shows that Ps is onto whenever it is bounded. Note that (3.7) and (3.5) together imply s + t > −1 so that 1 + s + t does not hit a pole of Γ and Cs+t > 0. If q > −1, we can take t = 0, then Is0 = i, and Theorem 3.3 reduces to the classical result on Bergman spaces. The next result is proved in [25, §5]. Proposition 3.4. If Ps : L2q → Dq is bounded and the norm on Dq is q · ts , then N ! Γ(1 − q + 2s) Γ(1 + q + 2t) . Ps = Γ(N + 1 + s + t) We often write the inequalities (3.7) and (3.5) in the form q + 1 < p(s + 1) and q + pt > −1 when we consider the general family of Bqp or Apq spaces and Lebesgue classes Lpq . Theorem 3.3 states that the composition Ps Ist : Dq → Dq is a constant times the identity with s, t satisfying (3.7) and (3.5). The composition Ist Ps : L2q → L2q in reverse order is also important in our analysis of Toeplitz operators. Starting with diﬀerentiation under the integral sign and (3.4), the following result is compiled from [22, §5] and [19, Theorem 1.9]. Theorem 3.5. The operator Ist Ps : L2q → L2q is bounded if and only if s, t satisfy (3.7) and (3.5), and in that case, it is the operator (1 − |w|2 )s t 2 t f (w) dν(w) (f ∈ L2q ). Vs f (z) = (1 − |z| ) N +1+s+t B (1 − z, w)

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Note again that (3.7) and (3.5) together imply s + t > −1 so that Ks+t is binomial. Now we have the operator equalities Cs+t Ps Ist = I,

Ist Ps = Vst ,

Cs+t Vst Ist = Ist ,

and Cs+t Ps Vst = Ps .

(3.8)

Analogous equalities appear, for example, in [38, Lemma 20] for q > −1. The adjoint (Vst )∗ : L2q → L2q of Vst is computed using Fubini theorem and is −q+s (Vst )∗ = Vq+t .

Hence

Vst

is self-adjoint on

L2q

(3.9)

if and only if s − t = q.

(3.10)

Let q be given. If s satisﬁes (3.7), then the value of t obtained from (3.10) satisﬁes (3.5). Conversely, if t satisﬁes (3.5), then the value of s obtained from (3.10) satisﬁes (3.7). Notation 3.6. Henceforth given a q, we select s so as to satisfy (3.7), and put Q = −q + 2s

and

u = −q + s.

(3.11)

in the remaining part of the paper. Note that Q = s + u = q + 2u > −1 A2Q .

We use only the self-adjoint Vsu in order to have Toeplitz so that DQ = operators that are direct extensions of classical Bergman-space Toeplitz operators and to have exact equalities as much as possible. Also we use only the inner product [·, ·]Dq = q [·, ·]us and the corresponding norm 2 u u u 2 u 2 f Dq = [f, f ]Dq = [Is f, Is f ]L2q = Is f L2q = Ds f L2 = |Dsu f |2 dνQ (3.12) Q

B

in Dq . This is a genuine norm, that is, the only function whose norm is 0 is the one that is identically 0. If q > −1, it is standard to use u = 0. Finally, we redeﬁne the Bergman projections Ps : L2q → Dq by multiplying them by CQ as done in [16, (7)]. Then (3.8) takes the form Ps Isu = I,

Isu Ps = CQ Vsu ,

CQ Vsu Isu = Isu ,

and CQ Ps Vsu = Ps .

(3.13)

Lastly Ps = 1 now by Proposition 3.4. The adjoint Ps∗ : Dq → L2q of Ps can now be computed. If g ∈ L2q and f ∈ Dq , then [Ps g, f ]Dq = [Isu Ps g, Isu f ]L2q = CQ [Vsu g, Isu f ]L2q = CQ [g, Vsu Isu f ]L2q = [g, Isu f ]L2q by (3.12), (3.13), (3.9), and (3.10). Thus Ps∗ = Isu . The same computation read backwards shows that the adjoint (Isu )∗ : L2q → Dq of Isu is (Isu )∗ = Ps . More generally, the Banach space adjoints of Ps : Lpq → Bqp are computed with respect to more general asymmetric pairings in Besov spaces in [22, Theorem 5.3]. Summarizing, (Vsu )∗ = Vsu ,

Ps∗ = Isu ,

and

(Isu )∗ = Ps .

(3.14)

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In particular, with the inclusion i = Is0 : A2Q → L2Q , we have PQ∗ = i

and

i∗ = PQ .

(3.15)

This might seem unusual, but we remind that the target space of PQ here is A2Q , and not L2Q as it is commonly taken. Let Mφ : L2q → L2q be the operator of multiplication by a suitable measurable, say L∞ , function φ on B. Its adjoint Mφ∗ : L2q → L2q is clearly Mφ∗ = Mφ . What ∗ 2 2 is more interesting is that the adjoint M(1−|z| 2 )u : Lq → LQ of the particular multiplication operator M(1−|z|2 )u : L2Q → L2q turns out to be ∗ M(1−|z| 2 )u = M(1−|z|2 )−u

simply by writing out the deﬁnition of the adjoint. Now we have one more way to compute the adjoint of Isu = M(1−|z|2 )u iDsu : Dq → L2q , where Dsu : Dq → A2Q , i is the inclusion i : A2Q → L2Q , and the multiplication is as just discussed. Then by (3.6), (3.15), the above remarks, diﬀerentiating under the integral sign, and (3.4), we reobtain that −u ∗ (Isu )∗ f (z) = (Dsu )∗ i∗ M(1−|z| 2 )u f (z) = DQ PQ M(1−|z|2 )−u f (z) (1 − |w|2 )Q−u −u = CQ DQ f (w) dν(w) N +1+Q B (1 − z, w) = CQ Ks (z, w) f (w) dνs (w) = Ps f (z). B

Example 3.7. We repeat [24, Remark 4.8] in our notation. We need it when we deﬁne Berezin transforms in Section 5. Given a q, pick an s satisfying (3.7), recall that Q > −1, let w ∈ B, and put Ks (z, w) = CQ (1 − |w|2 )(N +1+Q)/2 Ks (z, w) (z ∈ B). q gw (z) = Ks (·, w)Dq Then obviously q gw Dq = 1 for all w ∈ B. Thus q gw is essentially a normalized reproducing kernel; but although the kernel Ks is that of Ds , the normalization is done with respect to the norm of Dq . The kernels Kq (·, w) and Ks (·, w) have the reproducing properties [f, Kq (·, w)]Dq = C f (w)

and

[f, Ks (·, w)]Dq =

1 Du f (w) CQ s

in Dq . The second property parallels the fact that q gw → 0 weakly in Dq by [24, Theorems 4.3 and 4.4], which relate weak convergence in Dq to convergence of certain derivatives. This relationship is further mirrored in Dsu (q gw )(z) = which deﬁnes

Q kw

KQ (z, w) (1 − |w|2 )(N +1+Q)/2 CQ = =: Q kw (z), (1 − z, w)N +1+Q KQ (·, w)DQ

∈ A2Q .

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When q > −1, then s = q satisﬁes (3.7), and q gw (z) is nothing but the normalized reproducing kernel of the Bergman space A2q . When q ≤ −1, we can use s = 0 or Q = 0 for simplicity in q gw (z).

4. Toeplitz Operators In this section, we deﬁne the Toeplitz operators on all Dq and obtain their several elementary properties. The main theme is that they extend and preserve the character of classical Toeplitz operators on weighted Bergman spaces. Theorem 3.3 forces us to deﬁne them as follows. Definition 4.1. Let q, an s satisfying (3.7), and a measurable function φ on B be given. We deﬁne the Toeplitz operator s Tφ : Dq → Dq with symbol φ as the composition s Tφ = Ps Mφ Isu of linear operators, where u is as in (3.11). When q > −1, a value of s satisfying (3.7) is s = q, whence u = 0. Then Iq0 is inclusion, and s Tφ reduces to the classical Toeplitz operator q Tφ = Pq Mφ i on the Bergman space A2q = Dq . We use the term classical to mean a Toeplitz operator with i = Iq0 . The value s = q does not work when q ≤ −1, but we can use s = 0 or Q = 0 for simplicity for any such q, and for the latter C0 = 1. So by introducing s Tφ in Deﬁnition 4.1, we not only are able to handle all Dirichlet spaces, but also study several generalized Toeplitz operators indexed by s even on a single Bergman space. One of our aims below is to show that the essential features of s Tφ are unaﬀected by any s satisfying (3.7). Hankel-Toeplitz operators with analytic symbols on weighted Bergman spaces of the unit disc that employ Cauchy-Riemann operators resembling Isu are investigated in [36]. Explicitly, Ks (z, w) φ(w) (1 − |w|2 )2u Dsu f (w) dνq (w) s Tφ f (z) = CQ B = CQ Ks (z, w) φ(w) Dsu f (w) dνQ (w) (f ∈ Dq ). B

We see that s Tφ f makes sense if φ ∈ L1Q and f is a polynomial. Hence s Tφ is a densely deﬁned possibly unbounded operator on Dq for such φ, because polynomials are dense in each Dq . It is also clear that the map φ → s Tφ is linear. Proposition 4.2. If φ ∈ L∞ , then s Tφ is bounded with s Tφ ≤ φL∞ . Proof. Taking f ∈ Dq and using Ps = 1, s Tφ f Dq = Ps Mφ Isu f Dq ≤ φ Isu f L2q ≤ φL∞ Isu f L2q = φL∞ f Dq , as desired.

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Remark 4.3. If f ∈ Dq , then Dsu f ∈ DQ = A2Q ⊂ L2Q by Proposition 3.2. If φ ∈ L∞ , from its integral form, we surmise that s Tφ f makes sense even when Dsu f belongs to the larger space L1Q since also φ Dsu f ∈ L1Q . This is typical of objects deﬁned through Bergman projections, because Ks (z, ·) is bounded for each z for any s. Having obtained the integral form for s Tφ , we can now deﬁne Toeplitz operators on Dq with symbols that are measures on B. If µ is Borel measure on B and u is as in (3.11), we let dκ(w) = (1 − |w|2 )2u dµ(w), and deﬁne

s Tµ f (z)

= CQ = CQ

B

B

Ks (z, w) (1 − |w|2 )2u Dsu f (w) dµ(w) Ks (z, w) Dsu f (w) dκ(w)

(f ∈ Dq ).

The operator s Tµ is more general and reduces to s Tφ when dµ = φ dνq . It makes sense when κ is ﬁnite and f is a polynomial. Like s Tφ , it is a densely deﬁned possibly unbounded operator on Dq for ﬁnite κ. Note that µ need not be ﬁnite in conformity with that q is unrestricted. We develop basic properties of s Tφ and s Tµ in this section. We can assume φ and µ are such that the corresponding Toeplitz operators are bounded. First, if φ ≡ λ, then s Tλ = λ I for any s by (3.13). Next, ∗ s Tφ

= (Isu )∗ Mφ∗ Ps∗ = Ps Mφ Isu = s Tφ

by (3.14). So s Tφ is self-adjoint if φ is real-valued a.e. in B. By (3.14) again, [s Tφ f, f ]Dq = [Ps Mφ Isu f, f ]Dq = [Mφ Isu f, Isu f ]L2q = φ |Dsu f |2 dνQ (f ∈ Dq ).

(4.1)

B

Also [s Tφ f, f ]Dq ≤ φL∞ f 2Dq if φ ∈ L∞ . Similarly, [s Tµ f, f ]Dq = |Dsu f |2 dκ (f ∈ Dq ).

(4.2)

B

Proposition 4.4. If φ ≥ 0 a.e. in B, then s Tφ is a positive operator. If µ is a positive measure, then s Tµ is a positive operator. We now present a very useful intertwining relation for transforming certain problems for Toeplitz operators on Besov spaces to similar problems for classical Toeplitz operators on Bergman spaces.

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Theorem 4.5. We have Dsu (s T φ ) = (Q Tφ )Dsu and Dsu (s Tµ ) = (Q Tκ )Dsu , where Q Tφ = PQ Mφ i and Q Tκ = CQ B KQ (z, w) f (w) dκ(w) are classical Toeplitz operators on A2Q . Consequently s Tφ

−u = DQ (Q Tφ )Dsu ,

s Tµ

−u = DQ (Q Tκ )Dsu ,

Q Tφ

−u = Dsu (s Tφ )DQ ,

Q Tκ

−u = Dsu (s Tµ )DQ ,

and where −u DQ = (Dsu )−1 = (Dsu )∗

by (3.6). In other words, s Tφ : Dq → Dq and Q Tφ : A2Q → A2Q are unitarily equivalent, and so are s Tµ and Q Tκ . Said diﬀerently, the following diagrams commute: Q Tφ

A2Q −−−−→  Dsu  s Tφ

A2Q Du  s

Dq −−−−→ Dq

Q Tκ

A2Q −−−−→  Dsu 

A2Q D u  s

s Tµ

Dq −−−−→ Dq

Proof. By diﬀerentiation under the integral sign and (3.4), if φ ∈ L∞ , then φ(w) Dsu f (w) dνQ (w) Dsu (s Tφ f )(z) = CQ N +1+Q (1 − z, w) B (f ∈ Dq ), = PQ Mφ (Dsu f )(z) because Q > −1 so that KQ is binomial. But Dsu f ∈ A2Q by Proposition 3.2, where t = u, which means that A2Q has norm · L2Q . This is the ﬁrst intertwining relation; the second is identical. −u For the second assertion, we note that (Dsu )−1 = DQ by (3.3). The third assertion follows from Proposition 3.2. Similar relations can be found in [36, §1] and [12, Lemma 3.1]. They are more limited than ours since N = 1 for both, the ﬁrst is only for Bergman spaces, and the second is only with ﬁrst-order derivatives. One property of classical Toeplitz operators on Bergman spaces is that if φ is holomorphic, then Q Tφ = Mφ . Theorem 4.5 shows that the corresponding relationship for Toeplitz operators on Besov spaces is not so simple; we have instead −u u aro operators s Tφ = DQ Mφ Ds when φ is holomorphic. These are related to Ces` and considered in [24, §11]. Here is an interesting consequence of Theorem 4.5. Recall s Tφ = (Isu )∗ Mφ Isu by deﬁnition, where Isu : Dq → L2q . A similar relationship holds for s Tµ too when the target space of Isu is chosen appropriately. Theorem 4.6. Let I˘su be the operator I˘su : Dq → L2 (µ) deﬁned by the same formula as Isu . Then s Tµ = (I˘su )∗ I˘su .

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Proof. Let f, g ∈ Dq . Then [(I˘su )∗ I˘su f, g]Dq = [I˘su f, I˘su g]L2 (µ) , and Dsu g ∈ A2Q by Proposition 3.2. On the other hand, Theorem 4.5, (3.7), Fubini theorem, and Theorem 3.3 with t = 0 yield −u [s Tµ f, g]Dq = [DQ (Q Tκ )Dsu f, g]Dq = [(Q Tκ )Dsu f, Dsu g]L2Q Dsu f (w) = CQ dκ(w) Dsu g(z) dνQ (z) N +1+Q B B (1 − z, w) Dsu g(z) = Dsu f (w) CQ dνQ (z) dκ(w) N +1+Q B B (1 − w, z) Dsu f (w) Dsu g(w) dκ(w) = [I˘su f, I˘su g]L2 (µ) . = B

By the uniqueness of the adjoint, we are done.

As a matter of fact, Carleson measures on Dq are deﬁned in [23] using this I˘su : Dq → L2 (µ), and we use those Carleson measures to characterize s Tµ with positive µ in Section 6. The classical Bergman-space version of Theorem 4.6 is in [27, §1], where the inclusion R : A20 → L2 (µ) is used in place of I˘su . The eﬀects of the choice for u are evident in the results obtained so far. Other t would not yield these expected properties. We see more eﬀects below. Every property of Toeplitz operators obtained above can also be derived from Theorem 4.5 and the corresponding property of classical Bergman-space Toeplitz operators. We prove several other properties employing the same instrument. Proposition 4.7. If ψ ∈ H(B), then (s Tφ )(s Tψ ) = s Tφψ and (s Tψ )(s Tφ ) = s Tψφ . Proof. By Theorem 4.5, a similar result on Bergman-space Toeplitz operators, and Theorem 4.5 again, s Tφ (s Tψ )

−u −u −u = DQ (Q Tφ )Dsu DQ (Q Tψ )Dsu = DQ (Q Tφψ )Dsu = s Tφψ .

The second identity follows by taking adjoints.

It also follows that (s Tψ )(s Tψ ) = s Tψ2 for ψ ∈ H(B) or ψ ∈ H(B). We are now in a position to prove a result about the commutants of Toeplitz operators with holomorphic symbols on the disc. Theorem 4.8. Suppose N = 1. If φ ∈ L∞ , ψ ∈ H ∞ is nonconstant, and s Tφ and ∞ s Tψ commute on Dq , then φ ∈ H . Proof. Let PQ (φ) = f ; then f ∈ A2Q ∩ H ∞ and φ = f + g with g in the orthogonal complement of A2Q in L2Q . We let k = 0, 1, 2, . . . and compute the successive actions of the given Toeplitz operators on 1 ∈ Dq ordered in two ways. By Theorem 4.5, (3.2), and the proof of [8, Theorem] which is equally valid for weighted Bergman spaces, we obtain s Tψ k (s Tφ )1

−u −u = DQ (Q Tψk )(Q Tφ )Dsu 1 = DQ (f ψ k )

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−u −u −u = DQ (Q Tφψk )1 = DQ (f ψ k ) + DQ PQ (g ψ k ).

Thus PQ (g ψ k ) = 0 by (3.3). Let h ∈ Dq . Then again by the proof of [8, Theorem], we have g = 0 and φ = f ∈ H ∞ . Obviously, if f ≡ 0, then s Tφ f = 0. And it is clear from the integral form of that if φ = 0 a.e. in B, then s Tφ is the zero operator. The converses are also true.

s Tφ

Proposition 4.9. If φ ∈ H(B) and φ ≡ 0, then s Tφ is one-to-one on Dq . The map φ → s Tφ is one-to-one. Proof. These follow from their classical Bergman-space counterparts, which are in [3], and Theorem 4.5. We have already shown that a bounded φ gives rise to a bounded s Tφ . It is reasonable to expect that a more restricted φ gives rise to a compact s Tφ . Proposition 4.10. If φ ∈ L∞ has compact support in B, then s Tφ is compact. Similarly, if µ is ﬁnite and has compact support in B, then s Tµ is compact. If φ ∈ C, then s Tφ is compact if and only if φ ∈ C0 . Proof. These all follow from the same classical Bergman-space results (see [37, §6.1], for example), Theorem 4.5, and the fact that a composition of a compact operator with a bounded one is compact.

5. Berezin Transforms To develop the theory of Toeplitz operators further, we need to introduce the Berezin transforms. Definition 5.1. Let {q gw } be the family of functions in Dq described in Example 3.7, and let T be a linear operator on Dq . We deﬁne the Berezin transform of T as the function T (w) = [T (q gw ), q gw ]Dq on B. It is clear that T∗ (w) = T (w), that |T (w)| ≤ T for all w ∈ B if T is bounded, and that T (w) is a continuous function of w since q gw depends on w continuously. When T is a Toeplitz operator, we also use the common notation s φ q for s T φ and s µ

q for s T µ , and call them the Berezin transforms of φ and µ. Equation (4.1), Example 3.7, and Theorem 4.5 yield the explicit forms

φ(z) |Q kw (z)|2 dνQ (z) s φq (w) = B (1 − |z|2 )2u 2 N +1+Q = CQ (1 − |w| ) φ(z) dνq (z) (N +1+Q)2 B |1 − z, w| (w ∈ B), = [Q Tφ (Q kw ), Q kw ]L2 = φ Q (w) Q

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which is valid for any φ ∈ L1Q , where φ Q is the classical Bergman-space Berezin transform of φ. Hence, when N = 1, s φ q = CQ BQ φ of [19, §2.1] since Q > −1. Analogously, by (4.2),

q (w) = |Q kw (z)|2 dκ(z) sµ B (1 − |z|2 )2u (5.1) 2 N +1+Q = CQ (1 − |w| ) dµ(z) (N +1+Q)2 B |1 − z, w|

Q (w) (w ∈ B) = [Q Tκ (Q kw ), Q kw ]L2Q = κ

q = CQ BQ for those µ for which the integral converges. Hence s µ q µ of [24, §5]. It is

now clear that if φ ≥ 0 a.e. in B, then s φq ≥ 0 on B, and if µ is a positive measure,

q ≥ 0 on B. then s µ Clearly, if s Tφ = 0 or φ = 0 a.e. in B, then s T φ = s φ q = 0 on B. The converse of this property justiﬁes Deﬁnition 5.1. Proposition 5.2. The maps s Tφ → s T φ and φ → s φ q are one-to-one. Proof. The ﬁrst claim is an obvious consequence of the second, which can be proved, because Q > −1, as in [19, Proposition 2.6] by taking more partial derivatives since now N is arbitrary. Deﬁnition 4.1, Example 3.7, and Deﬁnition 5.1 depend on the action on Dq of the reproducing kernel Ks with s satifying (3.7), which can be chosen as Kq if and only if q > −1. In other words, in many instances on Toeplitz operators on general Dq , the parameter s replaces the parameter q. Here’s one more result in this direction. Proposition 5.3. If φ ∈ H(B), then s Tφ∗ (q gw ) = φ(w) q gw . Proof. We have s Tφ (q gw )(z)

−u = DQ (Q Tφ )Dsu (q gw )(z) −u = CQ (1 − |w|2 )(N +1+Q)/2 DQ (Q Tφ )KQ (z, w) −u = φ(w) CQ (1 − |w|2 )(N +1+Q)/2 DQ KQ (z, w) 2 (N +1+Q)/2 Ks (z, w) = φ(w) q gw (z) = φ(w) CQ (1 − |w| )

by Theorem 4.5, Example 3.7, the classical Bergman-space result, and (3.4).

Therefore if φ ≡ λ, then λ is an eigenvalue for s Tλ with eigenvector q gw . As expected, this is the only possibility for the point spectrum of s Tφ as we show next, where we also determine the spectrum of s Tφ . Theorem 5.4. If φ ∈ H ∞ , then σ(s Tφ ) = φ(B), and σp (s Tφ ) = ∅ unless φ is identically constant.

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Proof. Again this is a straightforward consequence of the unitary equivalence stated in Theorem 4.5 and the well-known Bergman-space result which can be found in [37, Chapter 6]. We do not pursue spectral theory any further in this work. Let’s ﬁnally give some general equivalent conditions for the boundedness and compactness of s Tφ . Proposition 5.5. Suppose φ ∈ L1Q is M-harmonic. Then s Tφ is bounded if and only φ is bounded. And s Tφ is compact if and only if φ = 0 on B. Proof. The if part of the ﬁrst statement is Proposition 4.2, and the if part of the second statement is obvious. If s Tφ is bounded, then by (2.2) and Example 3.7, |φ(w)| = |s φ q (w)| = [s Tφ (q gw ), q gw ]Dq ≤ s Tφ (q gw )Dq q gw Dq ≤ s Tφ for all w ∈ B. Hence φ is bounded. If s Tφ is compact, then |φ(w)| ≤ s Tφ (q gw )Dq → 0

as

|w| → 1.

That is, the restriction of φ to ∂B vanishes. By the maximum principle, φ vanishes on all of B. We summarize the basic formulas for the Arveson space A = D−N . The parameter s is chosen so that Q = N + 2s > −1. Then s > −(N + 1)/2 > −(N + 1) and the kernel Ks is always binomial. Also u = N + s > 0, and thus a strictly positive-order derivative is required in all deﬁnitions and formulas. If f ∈ A, then 2 f D−N = (1 − |z|2 )N +2s |DsN +s f (z)|2 dν(z). B

We write only those formulas in which the symbol of the Toeplitz operator is a function; for the formulas when the symbol is a measure, we just substitute dµ(w) for (1 − |w|2 )−N dν(w). The Toeplitz operator is φ(w) (1 − |w|2 )N +2s N +s (N + 1 + 2s)N Ds f (w) dν(w). s Tφ f (z) = N +1+s N! B (1 − z, w) The weakly convergent family in A we use in deﬁning the Berezin transform is (N + 1 + 2s)N (1 − |w|2 )(2N +1+2s)/2 . q gw (z) = N! (1 − z, w)N +1+s The Berezin transform is (1 − |z|2 )N +2s (N + 1 + 2s)N 2 2N +1+2s

φ (1 − |w| (w) = ) φ(z) dν(z). s −N (2N +1+2s)2 N! B |1 − z, w| A value of Q that gives simpler formulas is Q = N + 2s = 0, because the factors (1 − | · |2 )N +2s disappear, and then s = −N/2 and u = N/2. Another case that might be of interest is s = 0 in which Q = u = N . When N = 1, the Arveson space becomes one with the Hardy space H 2 . Setting N = 1 above, it is clear that the Toeplitz operators studied in this paper are not the classical Toeplitz operators on H 2 . The ones here depend on an imbedding

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of H 2 in L2−1 by way of Isu rather than its usual imbedding in L2 (∂D) by way of inclusion, and require a radial derivative of positive order u. Remark 5.6. However, let’s take the limits as u → 0+ , that is, as s → −1+ , of the formulas for H 2 when N = 1. Let’s assume φ has boundary values on ∂D, also called φ, so that Hardy-space expressions make sense; f ∈ H 2 clearly has boundary values. It is known by weak-∗ convergence of measures that lim + 2(1 + s) f D−1 = f H 2 (f ∈ H 2 ), s→−1

where · H 2 is the classical norm on H 2 . For a detailed proof, [25, §3] can be consulted. With the same computation, we obtain 1 lim q gw (z) = kw (z), s→−1+ 2(1 + s) where kw is the classical normalized reproducing kernel of H 2 . Next we obtain lim

s→−1+

s Tφ f (z)

= Tφ f (z)

(f ∈ H 2 ),

where Tφ f = P(φ f ) is the classical Toeplitz operator on H 2 deﬁned via the Szeg˝ o projection P. We also obtain

lim (s φ −1 )(w) = Φ(w), s→−1+

is the classical Berezin transform on H 2 , which is the Poisson transform where Φ of the boundary values of φ. No extra factor is required for s Tφ or s φ −1 , because the factor CQ = 2(1 + s) is built into them. The same conclusions hold on D−1 also when N > 1; no change is necessary for s Tφ or s φ −1 ; in · D−1 and q gw we just replace 2(1 + s) by (2(1 + s))N /N !. Therefore the classical Toeplitz operators on H 2 are limiting cases of the Toeplitz operators on D−1 studied in this paper as the order of the radial derivative in their deﬁnition tends to 0.

6. Toeplitz Operators with Positive Symbols Throughout this section we assume φ ≥ 0 and µ ≥ 0 so that the resulting Toeplitz operators s Tφ and s Tµ on Dq are positive. We then give equivalent conditions for the boundedness, compactness, and membership in Schatten classes of these Toeplitz operators. Our main tools are the Berezin transform and Carleson measures. The only exception to positivity is Theorem 6.7, where φ is bounded instead. Definition 6.1. A positive Borel measure µ on B is called a q-Carleson measure if the ratio µ(b(w, r)) r (w) = qµ νq (b(w, r)) is bounded for w ∈ B for some 0 < r < ∞. The measure µ is called a vanishing q-Carleson measure if the same ratio tends to 0 as |w| → 1 for some 0 < r < ∞.

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The following characterization of q-Carleson and vanishing q-Carleson measures is given in [24, Theorem 5.9], actually in slightly more general form. Its corollary also appears in the same source. Theorem 6.2. Fix q. Let r, an r-lattice {an }, and s satisfying (3.7) be given. The following conditions are equivalent for a positive Borel measure µ on B. (i) The measure µ is a q-Carleson (resp. vanishing q-Carleson) measure. (ii) The sequence {q µ r (an )} is bounded (resp. has limit 0). ˘ (iii) The imbedding Isu : Dq → L2 (µ) is bounded (resp. compact).

q is bounded on B (resp. in C0 ). (iv) The Berezin transform s µ Thus the property of being a (vanishing) q-Carleson measure is independent of r, {an }, and s under (3.7), but depends on q. In accordance with that q is unrestricted, a (vanishing) q-Carleson measure need not be ﬁnite. Corollary 6.3. A positive Borel measure µ on B is a q-Carleson (resp. vanishing q-Carleson) measure if and only if κ is a Q-Carleson (resp. vanishing Q-Carleson) measure. Now we can state our main theorem. Theorem 6.4. Suppose µ is a positive Borel measure on B. Then s Tµ is bounded (resp. compact) on Dq if and only if µ is a q-Carleson (resp. vanishing q-Carleson) measure. Proof. With all the preparation done in earlier sections, we give two related very short proofs. By Theorem 4.6, s Tµ is bounded or compact on Dq if and only if I˘su has the same property. By Theorem 6.2, either property is equivalent to a q-Carlesonmeasure property for µ. Or, by Theorem 4.5, s Tµ is bounded or compact if and only if Q Tκ has the same property. By [37, Theorems 6.4.4 and 6.4.5], either property translates to a Q-Carleson-measure property for κ. By Corollary 6.3, we fall back to a q-Carlesonmeasure property for µ. It is among the consequences of Theorem 6.2 that if µ is a q-Carleson measure, then κ is ﬁnite; see [24, §1]. In the light of Theorem 6.4, the ﬁniteness of κ, which is stated for s Tµ to make sense when it is ﬁrst deﬁned in Section 4, is as natural a condition as possible. Corollary 6.5. Suppose φ ≥ 0 is a measurable function on B. Then s Tφ is bounded (resp. compact) on Dq if and only if φ dνq is a q-Carleson (resp. vanishing qCarleson) measure. It is clear from Theorem 6.2 that the results of Theorem 6.4 and Corollary 6.5 are independent of the particular value of s used in the deﬁnition of the Toeplitz operator or the particular weakly convergent family {q gw } used in the deﬁnition of its Berezin transform or the particular value of the radius r used in the deﬁnition

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of q µ r . We next show that the results are also independent of the Dirichlet space Dq that the Toeplitz operator acts on when the operator in question is s Tφ . So suppose dµ(z) = φ(z) dνq (z). Then by Lemma 2.1, 1 r (w) ∼ φ(z) (1 − |z|2 )q dν(z) qµ (1 − |w|2 )N +1+q b(w,r) 1 ∼ φ(z) dν(z) =: φr (w), ν(b(w, r)) b(w,r) which deﬁnes the averaging function φr on Bergman balls independently of q. Corollary 6.6. Suppose φ ≥ 0 is a measurable function on B. Let r, an r-lattice {an }, and s satisfying (3.7) be given. The following are equivalent. (i) The Toeplitz operator s Tφ : Dq → Dq is bounded (resp. compact). (ii) The Berezin transform s φ q is bounded on B (resp. in C0 ). (iii) The averaging function φr is bounded on B (resp. in C0 ). (iv) The sequence {φr (an )} is bounded (resp. has limit 0). We make an excursion from our main line of development to insert a result on the compactness of Toeplitz operators whose symbols are not necessarily positive. Theorem 6.7. Let N = 1 and φ ∈ L∞ . Then s Tφ on Dq is compact if and only if

s φq lies in C0 . Proof. Pick u so that Q = 0. By Theorem 4.5, s Tφ is compact if and only if the classical Toeplitz operator 0 Tφ on A20 is compact, which in turn holds if and only if 0 φ 0 is in C0 by [9, Corollary 2.5]. But s φ q = 0 φ 0 by our choice of Q. Unfortunately, the methods of [9] do not immediately generalize to dimensions N > 1 or to classical Toeplitz operators q Tφ = Pq Mφ i on weighted Bergman spaces A2q with q = 0. There are some extensions to non-Hilbert Bergman spaces Ap0 with p > 1 in [29], but with extra assumptions. Example 6.8. Let’s illustrate Corollaries 6.5 and 6.6 and Theorem 6.7 by picking Q = 0 and φ(z) = (1 − |z|2)c when N = 1. By Corollary 6.5, s Tφ is compact if and only if c > 0. Its Berezin transform is (1 − |z|2 )c 2 2

dν(z). q φs (w) = (1 − |w| ) 4 D |1 − z, w| By [33, Proposition 1.4.10], q φ s (w) ∼ (1 − |w|2 )b , where the power b depends on c but is always positive so that q φ s ∈ C0 in all cases. This is as predicted by Corollary 6.6 or Theorem 6.7. We return to positive symbols and now investigate the conditions under which the operators s Tφ or s Tµ belong to the Schatten-von Neumann ideal S p of Dq . For 0 < p < ∞, a compact operator T on a Hilbert space H with inner product [·, ·] is said to belong to to S p of H if its sequence of singular values lies in p . We refer to

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[18, Chapter III] for relevant deﬁnitions and basic properties of Schatten ideals. If T is a compact operator or an operator in S 1 , then the value of the sum j [T ej , ej ] is the same for any orthonormal basis {ej }j∈J in H, and is called the trace tr(T ) of T . The sum is ﬁnite in the latter case whence we call T a trace-class operator. If T is a positive compact operator on H, then T p is uniquely deﬁned, and T ∈ S p if and only if T p ∈ S 1 . An operator in S 2 is called a Hilbert-Schmidt operator. A compact operator T belongs to S p if and only if |T |p deﬁned as (T ∗ T )p/2 belongs to S 1 , which holds if and only if T ∗ T belongs to S p/2 . We have S 1 ⊂ S p ⊂ S ∞ for 1 < p < ∞. Further, for operators on H, T1 ≤ T2 means that [T1 f, f ] ≤ [T2 f, f ] for all f ∈ H. We are interested in H = Dq for any q ∈ R. We need a few lemmas before we characterize the Toeplitz operators with positive symbols that are in Schatten ideals S p of Dq for 1 ≤ p < ∞. Recall that φ, µ, s Tφ , and s Tµ are all positive in this section. Lemma 6.9. If T is a positive or a trace-class operator on Dq , then −u −u ∼ tr(T ) = tr(Dsu T DQ ) = CQ (Dsu T DQ ) dτ, B

−u ∼ where (Dsu T DQ ) is the classical Bergman-space Berezin transform of the oper−u u ator Ds T DQ : A2Q → A2Q .

Proof. Let { eα : α ∈ NN } be an orthonormal basis for Dq with respect to the inner product [·, ·]Dq . Put fα = Dsu eα . Then { fα : α ∈ NN } is an orthonormal basis for DQ = A2Q with respect to the inner product [·, ·]L2Q by Proposition 3.2. Then −u tr(T ) = (Dsu T DQ [T eα , eα ]Dq = [Dsu T eα , Dsu eα ]L2Q = )fα , fα L2 , α

α

Q

α

which proves the ﬁrst equality. The second equality follows by modifying the proof of [37, Proposition 6.3.2] for the ball and for weighted Bergman spaces. Lemma 6.10. We have tr(s Tµ ) = CQ s µ

q dτ = CQ KQ (z, z) dκ(z) = CQ B

and

tr(s Tφ ) = CQ

B

B

dτ = CQ

s φq

B

dµ(z) (1 − |z|2 )N +1+q

B

φ(z) KQ (z, z) dνQ (z) = CQ

Proof. By Lemma 6.9 and (5.1), we have tr(s Tµ ) = CQ Q T κ dτ = CQ B

B

q sµ

φ dτ. B

dτ.

The rest now follows by modifying the proof of the Corollary to [37, Proposition 6.3.2] to suit the weighted Bergman spaces and the ball. Lemma 6.11. If 1 ≤ p < ∞ and φ ∈ Lp (τ ), then s Tφ ∈ S p .

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Proof. Let {eα } be any orthonormal basis for Dq . By Lemma 6.9, we have tr(s Tφ ) = [s Tφ eα , eα ]Dq = [Q Tφ fα , fα ]L2Q = tr(Q Tφ ), α

α

where Q Tφ is a classical Bergman-space Toeplitz operator. So s Tφ ∈ S p if and only if Q Tφ ∈ S p . We are done by [37, Lemma 6.3.4]. Lemma 6.12. Given r, there is a C such that s Tµ ≤ C (s Tq µ r ). Proof. Let f ∈ Dq . We compute using (4.1), Lemma 2.1, Fubini theorem, Lemma 2.2, (4.2), and obtain µ(b(z, r)) |Dsu f (z)|2 dνQ (z) [s Tq µ r f, f ]Dq = B νq (b(z, r)) |Dsu f (z)|2 ∼ χb(z,r) (w) dµ(w) dν(z) 2 N +1−2u B (1 − |z| ) B |Dsu f (z)|2 dν(z) dµ(w) = 2 N +1−2u B b(w,r) (1 − |z| ) 1 ∼ (1 − |z|2 )2u |Dsu f (z)|2 dνq (z) dµ(w) ν (b(w, r)) q B b(w,r) 2 2u ≥ C (1 − |w| ) |Dsu f (w)|2 dµ(w) = [s Tµ f, f ]Dq , B

which is what is wanted.

The classical Bergman-space versions of Lemmas 6.9–6.12 can be found in [37, §6.3]. Now we are ready for a characterization of Toeplitz operators in S p . Theorem 6.13. Suppose µ is a positive Borel measure on B. Let 1 ≤ p < ∞, r, an r-lattice {an }, and s satisfying (3.7) be given. The following are equivalent. (i) (ii) (iii) (iv)

The The The The

Toeplitz operator s Tµ : Dq → Dq belongs to S p . Berezin transform s µ

q belongs to Lp (τ ). r belongs to Lp (τ ). averaging function q µ sequence {q µ r (an )} belongs to p .

Proof. (i) =⇒ (ii): By positivity, if s Tµ is in S p , then s Tµp is in S 1 so that tr(s Tµp ) is ﬁnite. Now by deﬁnition and [37, Proposition 6.3.3], p p µ

dτ = [ T ( g ), g ] dτ (w) ≤ [s Tµp (q gw ), q gw ]Dq dτ (w). s q s µ q w q w Dq B

B

But the last term is just

B

tr(s Tµp ).

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(ii) =⇒ (iii): Lemma 2.1 shows that 1 r (w) ∼ dµ qµ (1 − |w|2 )N +1+q b(w,r) (1 − |z|2 )2u ∼ (1 − |w|2 )N +1+Q dµ(z) (N +1+Q)2 b(w,r) |1 − z, w| dκ(z) ≤ (1 − |w|2 )N +1+Q = sµ

q (w). (N +1+Q)2 B |1 − z, w| p r ∈ L (τ ). Then s Tq µ r ∈ S p by Lemma 6.11. By (iii) =⇒ (i): Suppose q µ positivity and [37, Theorem 1.4.7], α [s Tq µ r eα , eα ]pDq < ∞ for any orthonormal set {eα } in Dq . Then α [s Tµ eα , eα ]pDq < ∞ too by Lemma 6.12. We are done by applying [37, Theorem 1.4.7] again. (iii) ⇐⇒ (iv): This is in [24, §5] and has an independent proof.

As observed above, the conclusions of Theorem 6.13 do not depend on s, r, {an }, or {q gw }, but do depend on q. When we specialize to s Tφ , that dependence disappears too in the same way as in Corollary 6.6. Corollary 6.14. Suppose φ ≥ 0 is a measurable function on B. Let 1 ≤ p < ∞, r, an r-lattice {an }, and s satisfying (3.7) be given. The following are equivalent. (i) (ii) (iii) (iv)

The The The The

Toeplitz operator s Tφ : Dq → Dq belongs to S p . Berezin transform s φ q belongs to Lp (τ ). averaging function φr belongs to Lp (τ ). sequence {φr (an )} belongs to p .

The classical Bergman-space versions (q > −1 with i = Iq0 ) of Theorems 6.4 and 6.13 on D can be found in [37, Chapter 6]. What is new here are that the results now hold for all Dirichlet spaces (q ∈ R), that they hold although Toeplitz operators here are deﬁned via Isu for all q rather than i, and thus they give a uniﬁed picture of Toeplitz operators on weighted Bergman and other Dirichlet spaces. Thus, when φ ≥ 0, the Toeplitz operator s Tφ on the Arveson space is bounded, compact, or in S p precisely when the classical Toeplitz operator 0 Tφ on the Bergman space A20 is bounded, compact, or in S p , which occurs precisely when the averaging function φr is bounded, in C0 , or in Lp (τ ), respectively. Remark 6.15. We continue Remark 5.6 by letting q = −1 and taking limits as s → −1+ in Corollary 6.5. We take N = 1 for simplicity, and recall that φ ≥ 0. We know s Tφ becomes the classical Toeplitz operator Tφ on H 2 in the limit. By Theorem 6.2 (iii), the condition that φ dν−1 is a (−1)-Carleson measure means (1 − |z|2 )1+2s |Ds1+s f (z)|2 φ(z) dν(z) ≤ C f 2D−1 (f ∈ H 2 ). D

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After multiplying both sides by 2(1 + s), as s → −1+ , it takes the form 2π 2π dθ dθ ≤C (f ∈ H 2 ) |f (eiθ )|2 φ(eiθ ) |f (eiθ )|2 2π 2π 0 0 in the same way as in Remark 5.6. Since this is true for all f ∈ H 2 , it is equivalent to that φ is bounded by C a.e. on ∂D. By Theorem 6.2 (iv), the condition that φ dν−1 is a vanishing (−1)-Carleson measure means that s φ q is in C0 . This is the same as having 2(1 + s) (s φ q ) in C0 . As s → −1+ , by Remark 5.6, it is equivalent

the Poisson transform of the boundary values of φ, in C0 . This holds to having Φ, if and only if φ = 0 a.e. in D, or equivalently, φ = 0 a.e. on ∂D. As in Remark 5.6, when N > 1, 2(1 + s) is replaced by (2(1 + s))N /N ! in intermediate steps with no eﬀect on conclusions. Thus we recover the characterizations of the boundedness and compactness of the classical Tφ on H 2 (see [37, Propositions 9.1.2 and 9.1.3]) in the limiting case s → −1+ of s Tφ on D−1 , supplying further evidence that s Tφ uniﬁes Toeplitz operators on Hardy, weighted Bergman, and Dirichlet spaces.

7. Weighted Composition Operators on Weighted Bergman Spaces Definition 7.1. Let f, η, ϕ ∈ H(B) and ϕ have range in B. The operator Mη Cϕ deﬁned by Mη Cϕ f = η(f ◦ ϕ) is called a weighted composition operator. We are interested in weighted composition operators Mη Cϕ : A2Q → A2Q for Q > −1. Suppose Q, q, and s are related as in (3.11). Consider Dsu : Dq → A2Q −u which is an isometry. We also know (Dsu )−1 = DQ . If f, g ∈ Dq2 , then F = Dsu f −u Mη Cϕ Dsu . and G = Dsu g are in A2Q . We now deﬁne η Eϕ : Dq → Dq by η Eϕ = DQ Operators resembling η Eϕ are used in [28] and [39] in similar contexts. Then (η Eϕ )∗ (η Eϕ )f, g Dq = (η Eϕ )f, (η Eϕ )g Dq −u −u = [Dsu DQ Mη Cϕ Dsu f, Dsu DQ Mη Cϕ Dsu g]L2Q

= [Mη Cϕ F, Mη Cϕ G]L2Q = F (ϕ(z)) G(ϕ(z)) |η(z)|2 (1 − |z|2 )Q dν(z) B = F (ϕ(z)) G(ϕ(z)) d(η νQ )(z) B = F (ζ) G(ζ) d(η νQ ◦ ϕ−1 )(ζ) B = (Dsu f )(ζ) (Dsu g)(ζ) dκ(ζ) = [s Tµ f, g]Dq , B

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as in the proof of Theorem 4.6. Thus s Tµ = (η Eϕ )∗ (η Eϕ ). Here κ = η νQ ◦ ϕ−1 is the pull-back measure that assigns the value κ(Ω) = |η(z)|2 (1 − |z|2 )Q dν(z) ϕ−1 (Ω)

to each Borel subset Ω of B. As in Section 4, dµ(ζ) = (1 − |ζ|2 )q−Q dκ(ζ). Note that both κ and µ are positive Borel measures. Theorem 7.2. The weighted composition operator Mη Cϕ is bounded (resp. compact) on the weighted Bergman space A2Q of B if and only if the Berezin transform s µ

q is bounded on B (resp. in C0 ). Proof. For compactness, we use the fact that a composition of a bounded operator and a compact one is compact. The operator Mη Cϕ is bounded (resp. compact) on A2Q if and only if η Eϕ is bounded (resp. compact) on Dq if and only if s Tµ is bounded (resp. compact) on Dq if and only if µ is a q-Carleson (resp. vanishing q-Carleson) measure by Theorem 6.4. By Theorem 6.2, these conditions are equivalent to the stated conditions on the Berezin transform. We can restate the equivalent conditions more explicitly in terms of the parameters η and ϕ of the operator Mη Cϕ . By (5.1) and the deﬁnition of µ above, 1 2 N +1+Q µ

(w) = C (1 − |w| ) d(η νQ ◦ ϕ−1 )(ζ) s q Q (N +1+Q)2 |1 − ζ, w| B |η(z)|2 2 N +1+Q = CQ (1 − |w| ) dνQ (z). (7.1) (N +1+Q)2 B |1 − ϕ(z), w| Thus Mη Cϕ is bounded (resp. compact) if and only if the quantity in (7.1) as a function of w is bounded in B (resp. in C0 ). When N = 1, this theorem is proved in [13, Proposition 2] using a characterization of Carleson measures via a derivative of disc automorphisms, a tool not readily available for N > 1. (Incidentally, the so-called weighted ϕ-Berezin transform Bϕ,α in [13] should have the measure dAα instead of dA in its deﬁnition.) Yet we are able to prove Theorem 7.2 with great ease once the theory of Carleson measures on Besov and Toeplitz operators on Dirichlet spaces are developed. Our next result on the Schatten-ideal membership of Mη Cϕ follows from Theorem 6.13 with a proof very similar to that of Theorem 7.2. Theorem 7.3. Let 2 ≤ p < ∞. The weighted composition operator Mη Cϕ belongs to S p of the weighted Bergman space A2Q of B if and only if the Berezin transform

q lies in Lp/2 (τ ). sµ For N = 1 and Q = 0, Theorem 7.3 is contained in [13, Theorem 3] with a similar proof. The following corollary follows similarly too.

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Corollary 7.4. The weighted composition operator Mη Cϕ : A2Q → A2Q on the ball is Hilbert-Schmidt if and only if |η(z)|2 dνQ (z) < ∞. 2 N +1+Q B (1 − |ϕ(z)| )

8. Shift Operators In this section, we always take N = 1, so our operators act on function spaces on the disc D, and the constant CQ is equal to 1 + Q. We need explicit orthonormal bases for Dq . Deﬁnition 3.1a and [6, Theorem 3.3.1] imply the following. On each Dq , there is an inner product [[·, ·]]Dq with respect to which {z k }k∈N is a complete orthogonal set, and the corresponding norm |z k |Dq of z k is the square root of the reciprocal of the coeﬃcient of (zw)k in the Taylor expansion of Kq (z, w). This inner product and its norm are equivalent to the ones in (3.12). Explicitly,  k!   , if q > −2; (8.1) |z k |2Dq = [[z k , z k ]]Dq = (2 + q)k   (−1 − q)k+1 , if q ≤ −2. k! On the other hand, by (3.12) and [22, Proposition 2.1],  (2 + Q)k k!   , if s > −2;  (1 + Q) (2 + s)2k k 2 z Dq = (2 + Q)k (−1 − s)2k+1    , if s ≤ −2. (1 + Q) (k!)3

(8.2)

In either case, none of the norms are 0 and the norm of z k is ∼ k (−1−q)/2 as k → ∞. Consequently, { q ek (z) = z k /z k Dq : k ∈ N } is an orthonormal basis for Dq with respect to the norm · Dq , and { q Ek (z) = z k /|z k |Dq : k ∈ N } is an orthonormal basis for Dq with respect to the norm | · |Dq . Note also that if q > −1 and u = 0, then Q = q = s and |z k |A2Q = CQ z k A2Q (k = 0, 1, . . .). (8.3) Definition 8.1. We call Mz : Dq → Dq the q-shift. So the (−1)-shift is the unilateral shift on the Hardy space H 2 , the 0-shift is the Bergman shift, and the (−2)-shift is the Dirichlet shift. There are intimate connections between Toeplitz operators, multiplication operators, and shift operators. Recall that Q Tφ is the classical Toeplitz operator (u = 0) on the weighted Bergman space A2Q , and for that, Q Tz = Mz . For the general Toeplitz operators deﬁned via Isu with u = 0 on general Dq considered in this paper, s Tz is not a

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ﬁxed multiple of Mz any more. To see how s Tz behaves on Dq , it suﬃces to check its action on z k . By [22, Proposition 2.1],  2 + s + k k+1   z , if s > −2;  2+Q+k k (k = 0, 1, 2, . . .). 2 s Tz (z ) = (1 + k)   z k+1 , if s ≤ −2;  (2 + Q + k)(−s + k)

Thus s Tz (q ek )

=

1+k (q ek+1 ) 2+Q+k

1+k and s Tz on Dq is a weighted shift operator with weight sequence Wk = 2+Q+k

with respect to the orthonormal basis {q ek }. No Wk is 0, {Wk } is bounded, but does not tend to 0; hence s Tz is one-to-one, bounded, but not compact. Noncompactness of s Tz can also be deduced via Theorem 6.7 by a laborious computation of s z q using the methods of [33, Proposition 1.4.10]. Hence by [34, Theorem 2 (b)], s Tz and Mz with respect to either orthonormal basis are similar operators. Moreover, by [34, Proposition 7], s Tz on Dq with respect to {q ek } is unitarily equivalent to Mz k acting on the space of holomorphic functions on D in which the norm of z is k! W0 · · · Wk−1 = (2+Q)k . Recalling (8.1) and that Q = −q + 2s > −1, this space is familiar. Moreover, we have (8.3). Let’s sum up. Theorem 8.2. The operator s Tz on the Dirichlet space Dq with respect to the orthonormal basis {q ek } is unitarily equivalent to the Q-shift Mz on the weighted Bergman space A2Q with the norm | · |DQ or the norm · DQ . This theorem also follows from Theorem 4.5 and the discussion following it. Let’s note that s Tz with respect to {q Ek } and Mz with respect to either orthonormal basis are also weighted shifts. The unilateral shift Mz has a special place for the classical Toeplitz operators Tφ = PMφ on H 2 , where P : L2 (∂D) → H 2 is the Szeg˝o projection. An operator T : H 2 → H 2 is the classical Toeplitz operator Tφ for some φ ∈ L∞ (∂D) if and only if Mz∗ T Mz = T . This property fails for classical Toeplitz operators on Bergman spaces. With s Tφ , the more relevant equation is (s Tz∗ )T (s Tz ) = T . If T = s Tφ satisﬁes this equation, then (s Tz∗ )(s Tφ )(s Tz ) = s Tφ|z|2 = s Tφ by Proposition 4.7. Then by linearity s Tφ(1−|z|2 ) = 0, and by Proposition 4.9, φ(z)(1 − |z|2 ) = 0 for almost every z ∈ D. Thus φ = 0 a.e. in D and s Tφ = 0. (We have promised to have N = 1 in this section, but this last result clearly holds for all N .) More is true. Theorem 8.3. The equation (s Tz∗ )T (s Tz ) = T has no bounded nonzero solution T : Dq → Dq . Proof. We adapt the proofs of [17, Theorems 3 and 5 (a)] to our situation and sketch the parts that are diﬀerent only in the case s > −2. The B and wk of [17] correspond to our T and |z k |2Dq .

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We set hk (z) =

zk 1+s+k ∈ Dq , k 1 + Q + k |z |2DQ

and deﬁne T on Dq by T (z k ) =

1+Q+k hk (z), 1+s+k

with the understanding that h0 = 1 and T (1) = 1. Also ∗ s Tz (hk+1 )

=

1+Q+k hk . 1+s+k

Combining these with the s Tz (z k ) computed above, we see that T satisﬁes the operator equation in the statement of the theorem. However, T is bounded if and only if 2 ∞ 1+Q+k 1 hk , g ≤ C |g|2Dq |z k |2Dq 1 + s + k Dq k=0

for any g ∈ Dq . Substituting in the details of the norm and the inner product (1+Q)2 } is bounded. But yields that T is bounded if and only if {|z k |−4 DQ ∼ k since Q > −1, this is impossible. Theorem 8.3 is a little surprising, because it is proved in [17, Theorem 5 (a)] that the similar operator equation Mz∗ T Mz = T has bounded solutions on Dq with q ≤ −1, that is, if Dq is not a Bergman space, and some solutions are of the form T (z k ) = k 1+q z k . However, s Tz is not a constant multiple of Mz on Dq with q ≤ −1, and the fact that a derivative is used within Isu in deﬁning s Tz eﬀectively sends the case into the Bergman space DQ . Note the |z k |2DQ in the deﬁnition of hk (z), for example. We do not know whether the solutions given above to Mz∗ T Mz = T are Toeplitz operators, but such an equation cannot be satisﬁed by all Toeplitz operators s Tφ on Dq , as our next result implies. Theorem 8.4. Let L, N be nonzero bounded operators on Dq . If L(s Tφ )N = s Tφ for all φ ∈ L∞ , then L and N are both scalar multiples of the identity. Proof. This time, we adapt the proof in [15] to our situation, and again give only a sketch in the case s > −2. Initially proceeding as in [15], and using additionally (3.14) and that Dsu is invertible, we conclude that N commutes with s Tz . −u k u 2 Next let h = N (1) = ∞ k=0 hk z ∈ Dq and H = Ds h ∈ AQ ; then h = DQ H. m We compute N (s Tz 1) in two ways. First, by Theorem 4.5, −u −u 1 = N DQ Mz m 1 = N (s Tzm 1) = N (s Tzm )DQ

(2 + s)m N (z m ). (2 + Q)m

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Second, by the commutativity just stated and Theorem 4.5, −u −u −u m N (s Tzm 1) = s Tzm N 1 = s Tzm h = s Tzm DQ H = DQ M z m H = DQ (z H)

=

∞ k=0

∞

(2 + Q)k −u (2 + Q)k −u m hk DQ Mzk (z m ) = hk s Tz k DQ (z ) (2 + s)k (2 + s)k k=0

∞ (2 + Q)k (2 + s)m m hk = s Tz k (z ). (2 + Q)m (2 + s)k k=0

Thus N (z m ) =

∞ k=0

hk

(2 + Q)k m m s Tz k (z ) = s TH (z ) (2 + s)k

at each z ∈ D. By the density of polynomials in Dq , we have N = s TH . That L = s TG for some G ∈ A2Q follows by taking adjoints. The rest of the proof is identical to that in [15] and omitted. The characterization by Mz∗ T Mz = T of the classical Toeplitz operators on H and its failure on A2q rely on the fact that Mz is an isometry on H 2 = D−1 while it is not on A2q , all with respect to the classical norms of the spaces. There is also the following weaker notion; see [2]. 2

Definition 8.5. Let m be a positive integer. A bounded linear operator T on a Hilbert space H with norm · is called an m-isometry if it satisﬁes either of the equivalent conditions m m j m ∗ j j j m (−1) or (−1) (T ) T = 0 T j f 2 = 0, j j j=0 j=0 the second for all f ∈ H. A 1-isometry is an isometry, and an (m − 1)-isometry is also an m-isometry. It is shown in [31, Theorem 3.7] that the Dirichlet shift, which is not an isometry, is a 2-isometry with respect to some norm on D−2 . Our last aim in this paper is to extend this result to other Dirichlet spaces Dq with q a negative integer and thus give concrete examples of natural spaces and norms for which the shift operator is an m-isometry and not an (m − 1)-isometry. Theorem 8.6. For a positive integer m, the (−m)-shift Mz on D−m is an misometry with respect to the norm | · |D−m , but not an (m − 1)-isometry. Proof. Considering the orthogonality of monomials and the series expansion of f in D−m , it suﬃces to check the second equality deﬁning an m-isometry only on {z k }. Here q = −m and Mzj z k = Mzj z k = z k+j . If m = 1, (8.1) gives |z k+j |D−1 = 1 for any k and j, and this means nothing but that Mz is an isometry on H 2 . If m = 2, 3, . . ., (8.1) gives m−1+k (m − 1)k+1 k 2 = (m − 1) . |z |D−m = m−1 k!

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Thus we need to know the value of n m−1+k+j j n (−1) j m−1 j=0

(8.4)

for all k when n = m and n = m−1. The formula [30, 4.2.5 (47)] with a = m−1+k and b = 1 says that (8.4) is equal to (−1)n δn,m−1 for 0 ≤ m − 1 ≤ n and for any k, which is 0 if n = m and nonzero if n = m − 1. This proves both assertions of the theorem. We have another similar partial result with · Dq . This time let q = −2m with m = 1, 2, . . . and Q = 0. Then s = −m, and (8.2) gives z k 2D−2 = k + 1 and 2 m−1+k z k 2D−2m = (m − 1)2 (k + 1) = (m − 1)2 (k + 1) (k + 1)2m m−1 for m = 2, 3, . . .. If 2 2m m−1+k+j (−1) =0 (k + j + 1) j m−1 j=0

2m

j

(8.5)

for all k, then Mz is a 2m-isometry on D−2m with respect to the norm ·D−2m . We have checked that it is true for q = −2, −4, −6, −8, and a computation of random cases on a computer algebra software gives results in the desired direction, but we do not know if (8.5) is true in general, nor do we know if Mz is not a (2m − 1)isometry. The corresponding result for q odd and negative seems to be wrong; for example, there is no Q for q = −1 or −3 that can make it true. Comparing the cases q = −2 of Theorem 8.6 and of the above computation with the case µ = ν of [31, Theorem 3.7], we see that

if f is in the classical Dirichlet space D−2 , then |f |2D−2 = f 2D−2 = f 2H 2 + D |f |2 dν, where · D−2 is with Q = 0. As a ﬁnal remark, we have considered whether Mz on, say, the Bergman space A21 or the Dirichlet space D−3/2 , could be a (−1)-isometry or a (3/2)-isometry with respect to one of the norms considered, where a c-isometry for c ∈ R is deﬁned appropriately through the inﬁnite binomial expansion of (1−x)c , but the few cases we have checked have not yielded a positive answer. Acknowledgments Parts of this work were done while the second author’s host institution was Middle East Technical University and during his sabbatical visit to the University of Virginia. He thanks Middle East Technical University for granting the sabbatical leave, the operator theory group and the Department of Mathematics of the University of Virginia for their hospitality, and particularly James Rovnyak also for several suggestions and bringing [31] and [2] to his attention.

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References [1] J. Agler and J. E. McCarthy, Complete Nevanlinna-Pick Kernels, J. Funct. Anal. 175 (2000), 111–124. [2] J. Agler and M. Stankus, m-Isometric Transformations of Hilbert Space III, Integral Equations Operator Theory 24 (1996), 379–421. ˇ Cuˇ ˇ ckovi´c. A Theorem of Brown-Halmos Type for Bergman Space [3] P. Ahern and Z. Toeplitz Operators, J. Funct. Anal. 187 (2001), 200–210. [4] D. Alpay and H. T. Kaptano˘ glu, Some Finite-Dimensional Backward-Shift-Invariant Subspaces in the Ball and a Related Interpolation Problem, Integral Equations Operator Theory 42 (2002), 1–21. [5] D. Alpay and H. T. Kaptano˘ glu, Gleason’s Problem and Homogeneous Interpolation on Hardy and Dirichlet-Type Spaces of the Ball, J. Math. Anal. Appl. 276 (2002), 654–672. [6] T. Ando, Reproducing Kernel Spaces and Quadratic Inequalities, Hokkaido Univ., Sapporo, 1987. [7] S. Axler, Bergman Spaces and Their Operators, Surveys of Some Recent Results in Operator Theory I (J. B. Conway and B. B. Morrel, eds.), Pitman Res. Notes Math. Ser., vol. 271, Longman, Harlow, 1988, pp. 1–50. ˇ Cuˇ ˇ ckovi´c, and N. V. Rao, Commutants of Analytic Toeplitz Operators [8] S. Axler, Z. on the Bergman Space, Proc. Amer. Math. Soc. 128 (2000), 1951–1953. [9] S. Axler and D. Zheng, Compact Operators via the Berezin Transform, Indiana Univ. Math. J. 47 (1998), 387–400. [10] F. Beatrous and J. Burbea, Holomorphic Sobolev Spaces on the Ball, Dissertationes Math. 276 (1986), 57 pp. [11] G. Cao, Fredholm Properties of Toeplitz Operators on Dirichlet Spaces, Pacific J. Math. 188 (1999), 209–223. [12] R. Chartrand, Toeplitz Operators on Dirichlet-Type Spaces, J. Operator Theory 48 (2002), 3–13. ˇ Cuˇ ˇ ckovi´c and R. Zhao, Weighted Composition Operators on the Bergman Space, [13] Z. J. London Math. Soc. 70 (2004), 499–511. [14] J. J. Duistermaat and Y. J. Lee, Toeplitz Operators on the Dirichlet Space, J. Math. Anal. Appl. 300 (2004), 54–67. [15] M. Engliˇs, A Note on Toeplitz Operators on Bergman Spaces Comment. Math. Univ. Carolin. 29 (1988), 217–219. [16] F. Forelli and W. Rudin, Projections on Spaces of Holomorphic Functions in Balls, Indiana Univ. Math. J. 24 (1974/75), 593–602. [17] R. Frankfurt, Operator Equations and Weighted Shifts, J. Math. Anal. Appl. 62 (1978), 610–619. [18] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, 1969. [19] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Grad. Texts in Math., vol. 199, Springer, New York, 2000.

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[20] Q. Jiang and L. Peng, Toeplitz-Hankel Type Operators on Dirichlet Spaces, Integral Equations Operator Theory 23 (1995), 336–352. [21] H. T. Kaptano˘ glu, Besov Spaces and Bergman Projections on the Ball, C. R. Math. Acad. Sci. Paris 335 (2002), 729–732. [22] H. T. Kaptano˘ glu, Bergman Projections on Besov Spaces on Balls, Illinois J. Math. 49 (2005), 385–403. [23] H. T. Kaptano˘ glu, Besov Spaces and Carleson Measures on the Ball, C. R. Math. Acad. Sci. Paris 343 (2006), 453–456. [24] H. T. Kaptano˘ glu, Carleson Measures for Besov Spaces on the Ball with Applications, J. Funct. Anal. (2007) (to appear). ¨ [25] H. T. Kaptano˘ glu and A. E. Ureyen, Analytic Properties of Besov Spaces via Bergman Projections, Contemp. Math. (2007) (to appear). [26] Y. F. Lu and S. H. Sun, Toeplitz Operators on Dirichlet Spaces, Acta Math. Sin. 17 (2001), 643–648. [27] D. H. Luecking, Trace Ideal Criteria for Toeplitz Operators, J. Funct. Anal. 73 (1987), 345–368. [28] D. H. Luecking and K. Zhu, Composition Operators Belonging to Schatten Ideals, Amer. J. Math. 114 (1992), 1127–1145. [29] J. Miao and D. Zheng, Compact Operators on Bergman Spaces, Integral Equations Operator Theory 48 (2004), 61–79. [30] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 1: Elementary Functions, Gordon and Breach, New York, 1986. [31] S. Richter, A Representation Theorem for Cyclic Analytic Two-Isometries, Trans. Amer. Math. Soc. 328 (1991), 325–349. [32] R. Rochberg and Z. Wu, Toeplitz Operators on Dirichlet Spaces, Integral Equations Operator Theory 15 (1992), 325–342. [33] W. Rudin, Function Theory in the Unit Ball of Cn , Grundlehren Math. Wiss., vol. 241, Springer, New York, 1980. [34] A. L. Shields, Weighted Shift Operators and Analytic Function Theory, Topics in Operator Theory (C. Pearcy, ed.), Math. Surveys Monogr. vol. 13, Amer. Math. Soc., Providence, 1974, pp. 49–128b. [35] Z. Wu, Hankel and Toeplitz Operators on Dirichlet Spaces, Integral Equations Operator Theory 15 (1992), 503–525. [36] G. Zhang, Ha-Plitz Operators Between Moebius Invariant Subspaces, Math. Scand. 71 (1992), 69–84. [37] K. Zhu, Operator Theory in Function Spaces, Monogr. Textbooks Pure Appl. Math., vol. 139, Dekker, New York, 1990. [38] K. Zhu, Holomorphic Besov Spaces on Bounded Symmetric Domains, Q. J. Math. 46 (1995), 239–256. [39] K. Zhu, Schatten Class Composition Operators on Weighted Bergman Spaces of the Disk, J. Operator Theory 46 (2001), 173–181.

Vol. 58 (2007)

Toeplitz Operators on Arveson and Dirichlet Spaces

Daniel Alpay Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel e-mail: [email protected] URL: http://www.math.bgu.ac.il/∼dany/ H. Turgay Kaptano˘ glu Department of Mathematics Bilkent University Ankara 06800 Turkey e-mail: [email protected] URL: http://www.fen.bilkent.edu.tr/∼kaptan/ Submitted: July 7, 2006 Revised: December 12, 2006

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Integral Equations and Operator Theory

Quasinilpotent Operators in Operator Lie Algebras Peng Cao and Shanli Sun Abstract. It is proved that the nilpotent Lie algebra generated by a family of decomposable operators generates an Engel- Banach algebra. We also proved that if a Lie algebra of quasinilpotent operators is essentially nilpotent, then the Banach algebra generated by this Lie algebra consists of quasinilpotent operators. Mathematics Subject Classification (2000). Primary 47B48; Secondary 47L70. Keywords. Engel Lie algebra; decomposable operator; quasinilpotent operator.

1. Introduction Let L be a normed Lie algebra. A Banach Lie algebra is a normed Lie algebra which is complete. If for every a ∈ L, the adjoint operator ad a : x → [a, x] is a quasinilpotent operator, then L is called an Engel Lie algebra. Similarly, for a normed algebra A, if for every a ∈ A, the adjoint operator ad a : x → [a, x] is a quasinilpotent operator, then A is called an Engel algebra. It is clear that nilpotent Lie algebras of operators are Engel Lie algebras. For the subset M of the Lie algebra L (resp., the algebra A), the Lie algebra (resp., associative algebra) generated by M means the least Lie algebra (resp., the least associative algebra) containing M . In the theory of operator Lie algebras, one of the important problems is the relations between the Lie algebra (algebra) generated by M and the property of M . For example, there are some open problems on it. For applications to the theory of Taylor spectrum [1], V. S. Shulman and Y. V. Turovskii in [2], A. A. Dosiev in [3] posed the following question: Question 1.1. Is a Banach algebra generated by a nilpotent Lie subalgebra an Engel algebra? Project supported by National Natural Science Foundation of China and the Innovation Foundation of BUAA for PhD Graduates Doctor’s Foundation of Beihang University.

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To generalize the classical Engel theorem, W. Wojtynski posed the following question in [4]: Question 1.2. (The generalized Engel Theorem) Let X be a Banach space and L be a Banach Lie subalgebra of B(X ) consisting of quasinilpotent operators. Does the associative subalgebra of B(X ) generated by L also consist of quasinilpotent operators? One should note that a Banach algebra generated by a Lie algebra which consists of quasinilpotent operators may not consist of quasinilpotent operators by [5]. Some special cases of the above problems have been solved, especially for compact operators. For example, V. S. Shulman and Y. V. Turovskii have proved the following result (see [6, Theorem 11.4]): Lemma 1.1. Suppose that a nonscalar Lie algebra L ⊂ B(X ) is the image of an Engel Banach Lie algebra under a bounded representation. If L contains a nonzero compact operator, then L has a nontrivial hyperinvariant subspace, where L means the norm-closure of L in B(X ). It is proved that any associative hull of a Volterra Lie algebra is a Volterra algebra by Lemma 1.1, where the Volterra Lie algebra (algebra) means the Lie algebra (algebra) consisting of Volterra operators which are quasinilpotent compact operators (cf.[6]). In this paper, we will give partial answers of the above questions. In section 2, we will gather some deﬁnitions and lemmas. In section 3, we will prove that the nilpotent Lie algebras generated by a family of decomposable operators generate Engel Banach algebras. In section 4, we will generalize the results in [6] to non-compact operators. It is proved that if a Lie algebra of quasinilpotent operators is essentially nilpotent, then the Banach algebra generated by the Lie algebra consists of quasinilpotent operators.

2. Notations and Lemmas Let X be a Banach space, and B(X ), K(X ) are the Banach algebras consisting of all the bounded linear operators and compact operators on X , respectively. On B(X ), we can deﬁne a Lie product [., .] : [T1 , T2 ] = T1 T2 −T2 T1 , for T1 , T2 ∈ B(X ). So B(X ) can be seen as an operator Lie algebra. For any Lie algebra L, we denote by ad|L the adjoint representation of L on L deﬁned by the formula (ad|L a)b = [a, b]. When L = B(X ), then L will be omitted. For T ∈ B(X ), r(T ) denotes the spectral radius of T . For a Banach algebra A, RadA denotes the Jacobson radical of A. The following lemma is well known [cf.7].

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Lemma 2.1. If A/RadA is commutative, then QA = Rad(A), where QA is the set of all quasinilpotent elements in A. Elementary spectral manifolds are introduced in [8], and for every S ∈ B(X ), λ ∈ C and r ≥ 0, the elementary spectral manifold ηλ,r (S) of S is deﬁned as follows: ηλ,r (S) := {x ∈ X| lim sup (S − λ)n x1/n ≤ r} and ηλ (S) := ηλ,0 (S). Some results on elementary spectral manifolds can be found in [8, Section 3]. One can ﬁnd the concepts of ‘decomposable operator’ and ’spectral maximal subspace’ in [9]. The following Lemma can be found in [10, § 13, Theorem 3]. Lemma 2.2. If T ∈ B(X ) is decomposable and X ∈ B(X ), then the following assertions are equivalent: (j) limn→∞ (adT )n X1/n = 0. (jj) Every spectral maximal subspace of T is invariant for X. If L is a Lie algebra, A(L) means the associative algebra generated by L, and A(L) means the Banach algebra generated by L. For every operator S ∈ L, it is clear that A(L) is an invariant subspace of adS. If M ⊂ B(X ), ε(M ) means the Lie algebra generated by M . L(1) = L, and L(k) = [L, L(k−1) ], for k ≥ 2. The following lemmas will be used frequently (see [11]). Lemma 2.3. If L ⊂ B(X ) is a nilpotent Lie algebra, then A(L)/Rad(A(L)) is commutative. It is known that Lemma 2.3 also holds for general Banach algebras. A family F of bounded linear operators on a Banach space X is triangularizable, if there is a chain C that is maximal as a chain of subspaces of X and has the property that every subspace in C is invariant under all operators in F . For I ⊂ B(X ), Lat I denotes the set of all invariant subspaces of I. Let Π ⊂ Lat I, we say that V is a gap-quotient of Π, if V = Y /Z, where Y, Z ∈ Π, Z ⊂ Y and there exist no subspaces in Π which are intermediate between Y and Z. For every T ∈ I, T |V denotes the operator induced by T on V , and M |V := {T |V |T ∈ M }.

3. Nilpotent Lie algebras Now we begin to solve the ﬁrst question. Proposition 3.1. Let L ⊂ B(X ) be an Engel Lie algebra. For every operator S ∈ L, if the elementary spectral manifold η0 (adS) of adjoint operator adS is closed, then adS : A(L) → A(L) is a quasinilpotent operator.

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Proof. Since L is an Engel Lie algebra, so adS : L → L is quasinilpotent for 1 every S ∈ L. That is limn→∞ adn S(x) n = 0, for any x ∈ L. So L ⊂ η0 (adS). η0 (adS) is an algebra by [8, Lemma 3.5], so A(L) ⊂ η0 (adS). By supposition, η0 (adS) is closed, so A(L) ⊂ η0 (adS), r(adS|η0 (adS)) = 0 by [8, Proposition 3.3]. So r(adS|A(L)) = 0, that is to say: adS : A(L) → A(L) is a quasinilpotent operator. For every S ∈ B(X ), recall two bounded linear operators LS : X → SX, RS : X → XS, for all X ∈ B(X ). Lemma 3.1. If L is a nilpotent Lie algebra, then L := span{LS , RS | S ∈ L} is a nilpotent Lie algebra on B(X ). Proof. For any S1 , S2 ∈ L, α, β ∈ C, αLS1 + βLS2 = LαS1 +βS2 , [LS1 , LS2 ] = L[S1 ,S2 ] . So {LS | S ∈ L} is a Lie algebra. Note that the map L → LL is a Lie morphism, and L is nilpotent, so {LS | S ∈ L} is nilpotent. Similarly, {RS | S ∈ L} is nilpotent too. Let L = span{LS , RS | S ∈ L}, L1 := {LS | S ∈ L}, L2 := {RS | S ∈ L}. So L = L1 + L2 . For every S1 , S2 ∈ L, [LS1 , RS2 ] = 0, so [L , L ] = [L1 , L1 ] + [L2 , L2 ]. Therefore, L is a Lie algebra. Since L1 , L2 are nilpotent and [L1 , L2 ] = {0}, so L is a nilpotent Lie algebra. Let L be a nilpotent Lie algebra, and B be the Banach algebra generated by L in B(B(X )). Note that A(L) is an invariant subspace of L , so is B. We denote the restriction of B on A(L) by BL . Lemma 3.2. Q = Rad(BL ), where Q is the set of quasinilpotent operators in BL . Proof. It follows from Lemma 3.1 that BL is a Banach algebra generated by a nilpotent Lie algebra. So BL /Rad(BL ) is commutative by Lemma 2.3. Then Q = Rad(BL ) follows from Lemma 2.1. From now on, for every S ∈ L, the operators LS , RS , adS mean their restriction on A(L), LS , RS , adS : A(L) → A(L), respectively. Theorem 3.1. Let L ⊂ B(X ) be a nilpotent Lie algebra. If for any operator S ∈ L, the adjoint operator adS : A(L) → A(L) is quasinilpotent, then the Banach algebra generated by L is an Engel algebra. Proof. For every S ∈ L, note that adS = LS − RS , so adS ∈ BL . Since adS is quasinilpotent, so adS ∈ Rad(BL ) by Lemma 3.2. Let E = {X ∈ A(L)| ad X ∈ Rad(BL )}. It is evident that E is linear. For every X, Y ∈ E, since Rad(BL ) is a closed two-side ideal, the equality ad XY = LX ad Y + RY ad X shows that E is an algebra. To see E is closed, if {Sn } ⊂ E and Sn → S ∈ BL , then adSn → adS ∈ Rad(BL ) because Rad(BL ) is closed. So E is a closed algebra. Note that L ⊂ E, so A(L) ⊂ E. That is, the Banach algebra generated by L is an Engel algebra.

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Lemma 3.3. Let S be a decomposable operator. The elementary spectral manifold η0 (adS) of adS is closed. Proof. Let {Xn } ⊂ η0 (adS) and Xn → X. For every spectral maximal subspace Y, Xn Y ⊂ Y by Lemma 2.2. Since Y is closed and Xn → X, so for every y ∈ Y, Xy = lim Xn y ∈ Y. Therefore Y is an invariant subspace of X. Then X ∈ η0 (adS) by Lemma 2.2. Let M = {Tα ∈ B(X ), α ∈ I}, where I is the index set. We have the following lemma. Lemma 3.4.

ε(M ) = span{M } + ε(2) (M ), where ε(2) (M ) := [ε(M ), ε(M )].

Proof. Let L := span{M }+ε2(M ), then span{M } ⊂ ε(M ), ε2 (M ) ⊂ ε(M ) by the deﬁnition of ε(M ), that is, L ⊂ ε(M ). On the other hand, [L, L] ⊂ [ε(M ), ε(M )] ⊂ L, and clearly, L is linear, so L is a Lie algebra. Note that span{M } ⊂ L, so ε(M ) ⊂ L. That is, L = ε(M ). Theorem 3.2. Suppose M is a family of decomposable operators. If ε(M ) is a nilpotent Lie algebra, then the Banach algebra generated by ε(M ) is an Engel algebra. Proof. Let L = ε(M ) be a nilpotent Lie algebra. It is suﬃcient to prove that for every S ∈ L, adS : A(L) → A(L) is a quasinilpotent operator by Theorem 3.1. Since L is nilpotent, [A(L), A(L)] ⊂ Rad(A(L)) by Lemma 2.3. So [L, L] ⊂ [A(L), A(L)] ⊂ Rad(A(L)) consists of quasinilpotent operators. For every S ∈ L, there exist T1 , T2 , ... , Tn ∈ M , such that adS ∈ span{adT1 , . . . , adTn } + ad([L, L]) by Lemma 3.3, where ad([L, L]) := {ad a|a ∈ [L, L]}. For all a ∈ [L, L], ad a is quasinilpotent by Rosenblum’s theorem. Note that ad a ∈ BL , so ad a ∈ Rad(BL ) by Lemma 3.2. For every Ti , adTi : A(L) → A(L) is quasinilpotent by Lemma 3.3 and Proposition 3.1, for i ∈ {1, ..., n}. Similarly, adTi ∈ Rad(BL ) by Lemma 3.2, for i ∈ {1, ..., n}. Therefore, adS ∈ span{adT1 , . . . , adTn } + ad([L, L]) ⊂ Rad(BL ). So adS : A(L) → A(L) is quasinilpotent.

4. Essentially nilpotent Lie algebras The famous Engel Theorem pointed out that, if Lie algebra L consists of nilpotent n × n complex matrices, then the associative algebra A generated by L also consists of nilpotent matrices. Some generalized results have been got. See [11], [12], [6]. Now we generalize it to essentially nilpotent Lie algebras. Let π : B(X ) → B(X )/K(X ) be the natural homomorphism. For a Lie algebra L, L is essentially nilpotent if π(L) is nilpotent. So all nilpotent Lie algebras,

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Lie algebras of compact operators are essentially nilpotent. T ∈ B(X ) is a Riesz operator if the spectrum of π(T ) is {0}. Recall that a set M ⊂ B(X ) is reducible if it has a nontrivial invariant subspace. Theorem 4.1. Let L be a Lie algebra of quasinilpotent operators on X . If L is an essentially nilpotent Lie algebra, then A(L) consists of quasinilpotent operators. Proof. For π(L) is a nilpotent Lie algebra, so there is an integer k, such that π(L(k) ) = (π(L))(k) = {0}. If L(k) = {0}, then L is nilpotent. By Lemma 2.1 and 2.3, A(L) consists of quasinilpotent operators. If L(k) = {0}, then there is a nonzero compact operator in L. We will show that L is an Engel Lie algebra. Claim. L is an Engel Lie algebra. For every T ∈ L, T is quasinilpotent, so π(T ) is quasinilpotent. Hence A(π(L)) consists of quasinilpotent operators by Lemma 2.1 and 2.3. For every S ∈ L, there is a sequence {Tn } in L such that limn→∞ Tn = S, so limn→∞ π(Tn ) = π(S). Therefore, π(S) ∈ A(π(L)) and then S is a Riesz operator. The spectrum of S is countable. So the spectrum of adS is countable by Rosenblum’s theorem. Note that limn→∞ adTn = adS, and the spectral radius is continued on adS(cf.[13]). Since r(adTn ) = 0 by Rosenblum’s theorem, so r(adS) = 0. Hence, L is an Engel Lie algebra. Therefore L is reducible by Lemma 1.1. Let V be a gap-quotient of Lat L. Then L|V consists of quasinilpotent operators and L(k) |V consists of compact operators. If L(k) |V is nonzero, L|V is reducible by Lemma 1.1. It is a contradiction with the choice of V . So L(k) |V = {0}. so L(k) ⊂ Rad(A(L)) by [8, Lemma 2.6]. Let γ be the canonical morphism of A(L) → A(L)/Rad(A(L)). So γ(L) is nilpotent. Note that L consists of quasinilpotent operators, so does γ(L). So γ(L) ⊂ Rad(A(L)/Rad(A(L))) = {0} by Lemma 2.3 and Lemma 2.1. So L ⊂ Rad(A(L)). Since the Jacobson radical is a closed ideal, A(L) = Rad(A(L)). It is clear that A(L) consists of quasinilpotent operators. Acknowledgment The authors would like to express their gratitude to Y. V. Turovskii for sending his useful paper and help! The authors are also grateful to referee for the useful comments.

References [1] A. S. Fainshtein, Taylor joint spectrum for families of operators generating nilpotent Lie algebras, J. Operator, Theory 29 (1993), 3-27. [2] Y. V. Turovskii and V. S. Shulman, Radicals in Banach Algebras and Some Problems in the Theory of Radical Banach Algebras, Func. Anal. Appl. 35 (2001), no. 4, 312314.

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[3] A. A. Dosiev, Cartan-Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems, J. Func. Anal. 230 (2006), 446-493. [4] W. Wojtynski, Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorﬀ, J. Funct. Anal. 153 (1998), 405-413. [5] D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal. A nil algebra of operators on Hilbert spaces with semisimple norm closure. Integr. Equ. and Oper. Theory 9 (1986), 729-743. [6] V. S. Shulman and Y. V. Turovskii, Joint Spectral Radius, Operator Semigroups, and a Problem of W. Wojtynski, J. Func. Anal. 177 (2000), 383-441. [7] A. Katavolos and C. Stamatopoulos, Commutators of Quasinilpotents and Invariant Subspaces, Studia. Math 128 (1998), no. 2, 159-169. [8] V. S. Shulman and Y. V. Turovskii, Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action, J. Func. Anal. 223 (2005), 425-508. [9] K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000. [10] D. Beltita and M. Sabac, Lie Algebras of Bounded Operators, Birkhauser, 2001. [11] Y. V. Turovskii, Spectral properties of certain Lie subalgebras and the spectral radius of subsets of a Banach algebra, in: Spectral Theory of Operators and its Applications, F. G. Maksudov (Ed.), vol. 6, Elm, Baku, 1985, pp. 144-181 (in Russian). [12] B. A. Barnes and A. Katavolos, Properties of Quasinilpotents in Some Operator Algebras, Proc. R. Ir. Acad. 93A (1993), 155-170. [13] J. Newburgh. The variation of spectra, Duck Math. J. 18 (1951), 165-176. Peng Cao and Shanli Sun LMIB & Department of Mathematics Beihang University Beijing 100083 China e-mail: [email protected] [email protected] Submitted: June 8, 2006 Revised: October 23, 2006

Integr. equ. oper. theory 58 (2007), 43–63 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010043-21, published online April 16, 2007 DOI 10.1007/s00020-007-1480-6

Integral Equations and Operator Theory

Characteristic Functions for Ergodic Tuples Santanu Dey and Rolf Gohm Abstract. Motivated by a result on weak Markov dilations, we deﬁne a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a deﬁnition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed. Mathematics Subject Classification (2000). Primary 47A20, 47A13; Secondary 46L53, 46L05. Keywords. Completely positive, dilation, conjugacy, ergodic, coisometric, row contraction, characteristic function, Cuntz algebra.

0. Introduction

d ∗ If Z = i=1 Ai · Ai is a normal, unital, ergodic, completely positive map on B(H), the bounded linear operators on a complex separable Hilbert space, and if there is a (necessarily unique) invariant vector state for Z, then we also say that A = (A1 , . . . , Ad ) is a coisometric, ergodic row contraction with a one-dimensional invariant subspace for the adjoints. Precise deﬁnitions are given below. This is the main setting to be investigated in this paper. In Section 1 we give a concise review of a result on the dilations of Z obtained by R. Gohm in [7] in a chapter called ‘Cocycles and Coboundaries’. There exists a conjugacy between a homomorphic dilation of Z and a tensor shift, and we emphasize an explicit inﬁnite product formula that can be obtained for the intertwining unitary. [7] may also be consulted for connections of this topic to a scattering theory for noncommutative Markov chains by B. K¨ ummerer and H. Maassen (cf. [9]) and more general for the relevance of this setting in applications. In this work we are concerned with its relevance in operator theory and correspondingly in Section 2 we shift our attention to the row contraction A = (A1 , . . . , Ad ). Our starting point has been the observation that the intertwining unitary mentioned above has many similarities with the notion of characteristic function occurring in the theory of functional models of contractions, as initiated

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by B. Sz.-Nagy and C. Foias (cf. [11, 6]). In fact, the center of our work is the commuting diagram 3.3 in Section 3, which connects the results in [7] mentioned above with the theory of minimal isometric dilations of row contractions by G. Popescu (cf. [12]) and shows that the intertwining unitary determines a multi-analytic inner function, in the sense introduced by G. Popescu in [14, 15]. We call this inner function the extended characteristic function of the tuple A, see Deﬁnition 3.3. Section 4 is concerned with an explicit computation of this inner function. In Section 5 we show that it is an extension of the characteristic function of the ◦ ∗-stable part A of A, the latter in the sense of Popescu’s generalization of the Sz.-Nagy-Foias theory to row contractions (cf. [13]). This explains why we call our inner function an extended characteristic function. The row contraction A is a ◦

one-dimensional extension of the ∗-stable row contraction A, and in our analysis we separate the new part of the characteristic function from the part already given by Popescu. G. Popescu has shown in [13] that for completely non-coisometric tuples, in particular for ∗-stable ones, his characteristic function is a complete invariant for unitary equivalence. In Section 6 we prove that our extended characteristic function does the same for the tuples A described above. In this sense it is characteristic. This is remarkable because the strength of Popescu’s deﬁnition lies in the completely non-coisometric situation while we always deal with a coisometric tuple A. The extended characteristic function also does not depend on the choice d of the decomposition i=1 Ai ·A∗i of the completely positive map Z and hence also characterizes Z up to conjugacy. We think that together with its nice properties established earlier this clearly indicates that the extended characteristic function is a valuable tool for classifying and investigating such tuples respectively such completely positive maps. Section 7 contains a worked example for the constructions in this paper.

1. Weak Markov dilations and conjugacy In this section we give a brief and condensed review of results in [7], Chapter 2, which will be used in the following and which, as described in the introduction, motivated the investigations documented in this paper. We also introduce notation. A theory of weak Markov dilations has been developed in [2]. For a (single) normal unital completely positive map Z : B(H) → B(H), where B(H) consists of the bounded linear operators on a (complex, separable) Hilbert space, it asks for ˆ → B(H), ˆ where H ˆ is a Hilbert space a normal unital ∗ −endomorphism Jˆ : B(H) containing H, such that for all n ∈ N and all x ∈ B(H) Z n (x) = pH Jˆn (xpH ) |H . Here pH is the orthogonal projection onto H. There are many ways to construct ˆ In [7], 2.3, we gave a construction analogous to the idea of ‘coupling to a shift’ J.

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used in [10] for describing quantum Markov processes. This gives rise to a number of interesting problems which remain hidden in other constructions. We proceed in two steps. First note that there is a Kraus decomposition d Z(x) = i=1 ai x a∗i with (ai )di=1 ⊂ B(H). Here d = ∞ is allowed in which case the sum should be interpreted as a limit in the strong operator topology. Let P be a d-dimensional Hilbert space with orthonormal basis {1 , . . . , d }, further K another Hilbert space with a distinguished unit vector ΩK ∈ K. We identify H with H ⊗ ΩK ⊂ H ⊗ K and again denote by pH the orthogonal projection onto H. For K large enough there exists an isometry u:H⊗P →H⊗K

s.t. pH u(h ⊗ i ) = ai (h),

for all h ∈ H, i = 1, . . . , d, or equivalently, u∗ (h ⊗ ΩK ) =

d

a∗i (h) ⊗ i .

i=1

Explicitly, one may take K = C

d+1

(resp. inﬁnite-dimensional) and identify

H ⊗ K (H ⊗ ΩK ) ⊕

d

H H⊕

1

d

H.

1

d Then, using isometries u1 , . . . , ud : H → H ⊕ 1 H with orthogonal ranges and such that ai = pH ui for all i (for example, such isometries are explicitly constructed in Popescu’s formula for isometric dilations, cf. [12] or Equation (3.2) in Section 3), we can deﬁne u(h ⊗ i ) := ui (h) for all h ∈ H, i = 1, . . . , d and check that u has the desired properties. Now we deﬁne a ∗ −homomorphism J : B(H) → B(H ⊗ K), x → u (x ⊗ 1P ) u∗ . It satisﬁes pH J(x)(h ⊗ ΩK ) = pH u (x ⊗ 1)u∗ (h ⊗ ΩK ) d d = pH u (x ⊗ 1) a∗i (h) ⊗ i = ai x a∗i (h) = Z(x)(h), i=1

i=1

which means that J is a kind of ﬁrst order dilation for Z. ˜ := ∞ K for an inﬁnite tensor product of For the second step we write K 1 Hilbert spaces along the sequence (ΩK ) of unit vectors in the copies of K. We have a distinguished unit vector ΩK˜ and a (kind of) tensor shift ˜ → B(P ⊗ K), ˜ R : B(K)

y˜ → 1P ⊗ y˜.

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˜ := H ⊗ K ˜ and we deﬁne a normal ∗ −endomorphism Finally H ˜ → B(H), ˜ J˜ : B(H) ˜ x ⊗ y˜ → J(x) ⊗ y˜ ∈ B(H ⊗ K) ⊗ B(K). ˜ B(H) ⊗ B(K) Here we used von Neumann tensor products and (on the right hand side) a shift ˜ K. ˜ We can also write J˜ in the form identiﬁcation K ⊗ K ˜ = u (IdH ⊗ R)(·) u∗ , J(·) ˜ leads where u is identiﬁed with u ⊗ 1K˜ . The natural embedding H H ⊗ ΩK˜ ⊂ H n ˜ ˜ ˆ with H ˆ := span ˜ to the restriction Jˆ := J| n≥0 J (pH )(H), which can be checked to H be a normal unital ∗ -endomorphism satisfying all the properties of a weak Markov dilation for Z described above. See [7], 2.3. A Kraus decomposition of Jˆ can be written as ˆ J(x) =

d

ti x t∗i ,

i=1

ˆ is obtained by linear extension of H ⊗ K ˜ h ⊗ k˜ → ui (h) ⊗ k˜ = where ti ∈ B(H) ˜ ˜ ˜ ˆ u(h⊗i )⊗ k ∈ (H⊗K)⊗ K H⊗ K. Because J is a normal unital ∗ −endomorphism ˆ which we called the (ti )di=1 generate a representation of the Cuntz algebra Od on H a coupling representation in [7], 2.4. Note that the tuple (t1 , . . . , td ) is an isometric dilation of the tuple (a1 , . . . , ad ), i.e., the ti are isometries with orthogonal ranges and pH tni |H = ani for all i = 1, . . . , d and n ∈ N. The following multi-index notation will be used frequently in this work. Let Λ denote the set {1, 2, . . . , d}. For operator tuples (a1 , . . . , ad ), given α = (α1 , . . . , αm ) in Λm , aα will stand for the operator aα1 aα2 . . . aαm , |α| := m. n 0 ˜ := ∪∞ Further Λ n=0 Λ , where Λ := {0} and a0 is the identity operator. If we write ∗ ∗ aα this always means (aα ) = a∗αm . . . a∗α1 . Back to our isometric dilation, it can be checked that ˜ = H, ˆ span{tα h : h ∈ H, α ∈ Λ} which means that we have a minimal isometric dilation, cf. [12] or the beginning of Section 3. For more details on the construction above see [7], 2.3 and 2.4. Assume now that there is an invariant vector state for Z : B(H) → B(H) given by a unit vector ΩH ∈ H. Equivalent: There is a unit vector ΩP = di=1 ω i i ∈ P such that u(ΩH ⊗ ΩP ) = ΩH ⊗ ΩK . Also equivalent: For i = 1, . . . , d we have d a∗i ΩH = ω i ΩH . Here ωi ∈ C with i=1 |ωi |2 = 1 and we used complex conjugation to get nice formulas later. See [7], A.5.1, for a proof of the equivalences. ∞ On P˜ := 1 P along the unit vectors (ΩP ) in the copies of P we have a tensor shift ˜ → B(P), ˜ S : B(P) y˜ → 1P ⊗ y˜. d ˜ = i ⊗ k˜ ˜ and si (k) Its Kraus decomposition is S(˜ y ) = i=1 si y˜ s∗i with si ∈ B(P) ˜ ˜ for k ∈ P and i = 1, . . . , d. In [7], 2.5, we obtained an interesting description of

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the situation when the dilation Jˆ is conjugate to the shift endomorphism S. This result will be further analyzed in this paper. We give a version suitable for our present needs but the reader should have no problems to obtain a proof of the following from [7], 2.5. Theorem 1.1. Let Z : B(H) → B(H) be a normal unital completely positive map with an invariant vector state ΩH , · ΩH . Notation as introduced above, d ≥ 2. The following assertions are equivalent: (a) Z is ergodic, i.e., the fixed point space of Z consists of multiples of the identity. (b) The vector state ΩH , · ΩH is absorbing for Z, i.e., if n → ∞ then φ(Z n (x)) → ΩH , xΩH for all normal states φ and all x ∈ B(H). (In particular, the invariant vector state is unique.) ˆ → P˜ such that (c) Jˆ and S are conjugate, i.e., there exists a unitary w : H ˆ x) = w∗ S(w xˆ w∗ ) w. J(ˆ (d) The Od −representations corresponding to Jˆ and S are unitarily equivalent, i.e., w ti = si w for i = 1, . . . , d. An explicit formula can be given for an intertwining unitary as occurring in (c) and (d). If any of the assertions above is valid then the following limit exists strongly, ˜ → H ⊗ P, ˜ ˜ = lim u∗0n . . . u∗01 : H ⊗ K w n→∞

where we used a leg notation, i.e., u0n = (IdH ⊗ R)n−1 (u). In other words u0n is ˜ is a partial isometry with u acting on H and on the n−th copy of P. Further w ˆ and final space P˜ ΩH ⊗ P˜ ⊂ H ⊗ P˜ and we can define w as the initial space H ˜ corresponding restriction of w. To illustrate the product formula for w, which will be our main interest in this work, we use it to derive (d). ˜ = w u(h ⊗ i ) ⊗ k˜ = lim u∗ . . . u∗ u01 (h ⊗ i ⊗ k) ˜ w ti (h ⊗ k) 0n 01 n→∞

=

lim u∗ n→∞ 0n

. . . u∗02 (h

˜ = si w(h ⊗ k). ˜ ⊗ i ⊗ k)

Let us ﬁnally note that Theorem 1.1 is related to the conjugacy results in [16] and [4]. Compare also Proposition 2.4.

2. Ergodic coisometric row contractions In the previous section we considered a map Z : B(H) → B(H) given by Z(x) = d ∗ d d i=1 Ai x Ai , where (Ai )i=1 ⊂ B(H). We can think of (Ai )i=1 as a d-tuple A = (A1 , . . . , Ad ) or (with the same notation) as a linear map A = (A1 , . . . , Ad ) :

d i=1

H → H.

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(Concentrating now on the tuple we have changed to capital letters A. We will sometimes return to lower case letters a when we want to emphasize that we are in the (tensor product) setting of Section 1.) We have the following dictionary. Z(1) ≤ 1

⇔

d

Ai A∗i ≤ 1

i=1

⇔ A is a contraction Z(1) = 1

⇔

d

Ai A∗i = 1

i=1

Z is called unital

A is called coisometric

ΩH , ·ΩH = ΩH , Z(·)ΩH ⇔ A∗i ΩH = ω i ΩH , ωi ∈ C,

invariant vector state

d

|ωi |2 = 1

i=1

common eigenvector for adjoints

Z ergodic ⇒ {Ai , A∗i } = C 1 trivial ﬁxed point space trivial commutant The converse of the implication at the end of the dictionary is not valid. This is related to the fact that the ﬁxed point space of a completely positive map is not always an algebra. Compare the detailed discussion of this phenomenon in [3]. By a slight abuse of language we call the tuple (or row contraction) A = (A1 , . . . , Ad ) ergodic if the corresponding map Z is ergodic. With this terminology we can interpret Theorem 1.1 as a result about ergodic coisometric row contractions A with a common eigenvector ΩH for the adjoints A∗i . This will be examined starting with Section 3. To represent these objects more explicitly let us write ◦

◦

H:= H C ΩH . With respect to the decomposition H = C ΩH ⊕ H we get 2 × 2− block matrices

ω i i | ωi 0 ∗ Ai = , Ai = . (2.1) ∗ |i ˚ Ai 0 ˚ Ai ◦

◦

Here ˚ Ai ∈ B(H) and i ∈H. For the oﬀ-diagonal terms we used a Dirac notation that should be clear without further comments. Note that the case d = 1 is rather uninteresting in this setting because if A

ω 0 is a coisometry with block matrix then because | ˚ A

|ω|2 ω | 1 0 = A A∗ = ∗ ˚A ˚ 0 1 ω | | | + A

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we always have = 0. But for d ≥ 2 there are many interesting examples arising from unital ergodic completely positive maps with invariant vector states. See Section 1 and also Section 7 for an explicit example. We always assume d ≥ 2. Proposition 2.1. A coisometric row contraction A = (A1 , . . . , Ad ) is ergodic with ◦

common eigenvector ΩH for the adjoints A∗1 , . . . , A∗d if and only if H is invariant ◦ for A1 , . . . , Ad and the restricted row contraction (˚ A1 , . . . , ˚ Ad ) on H is ∗-stable, ◦

i.e., for all h ∈H lim

n→∞

∗

˚ Aα h2 = 0 .

|α|=n

Here we used the multi-index notation introduced in Section 1. Note that ∗-stable tuples are also called pure, we prefer the terminology from [6]. ◦

Proof. It is clear that ΩH is a common eigenvector for the adjoints if and only if H is invariant for A1 , . . . , Ad . Let Z(·) = di=1 Ai · A∗i be the associated completely ◦

positive map. With q := 1 − |ΩH ΩH |, the orthogonal projection onto H, and by using q Ai q = Ai q ˚ Ai for all i, we get ∗ ˚ Aα q A∗α = Aα ˚ Aα Z n (q) = |α|=n

|α|=n

◦

and thus for all h ∈H

∗

˚ Aα h2 = h, Z n (q) h .

|α|=n

Now it is well known that ergodicity of Z is equivalent to Z n (q) → 0 for n → ∞ in the weak operator topology. See [8], Prop. 3.2. This completes the proof. Remark 2.2. Given a coisometric row contraction a = (a1 , . . . , ad ) we also have the isometry u : H ⊗ P → H ⊗ K from Section 1. We introduce the linear map a : P → B(H), k → ak deﬁned by a∗k (h) ⊗ k := (1H ⊗ |k k|) u∗ (h ⊗ ΩK ). Compare [7], A.3.3. In particular ai = ai for i = 1, . . . , d, where {1 , . . . , d } is the orthonormal basis of P used in the deﬁnition of u. Arveson’s metric operator spaces, cf. [1], give a conceptual foundation for basis transformations in the operator space linearly spanned by the ai . Similarly, in our formalism a unitary in B(P) transforms a = (a1 , . . . , ad ) into another tuple a = (a1 , . . . , ad ). If ΩH is a common eigenvector for the adjoints a∗i then ΩH is also a common eigenvector for the adjoints (ai )∗ but of course the eigenvalues are transformed to another tuple ω = (ω1 , . . . , ωd ). We should consider the tuples a and a to be essentially the same. This also means that the complex numbers ωi are not particularly important and they should not play a role in classiﬁcation. They just reﬂect a certain choice of orthonormal basis in the relevant metric operator space.

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Independent of basis transformations is the vector ΩP = di=1 ω i i ∈ P satisfying d u(ΩH ⊗ ΩP ) = ΩH ⊗ ΩK (see Section 1) and the operator aΩP = i=1 ω i ai . For later use we show Proposition 2.3. Let A = (A1 , . . . , Ad ) be an ergodic coisometric row contraction d such that A∗i ΩH = ω i ΩH for all i, further AΩP := i=1 ω i Ai . Then for n → ∞ in the strong operator topology (A∗ΩP )n → |ΩH ΩH |. Proof. We use the setting of Section 1 to be able to apply Theorem 1.1. From d u∗ (h ⊗ ΩK ) = i=1 a∗i (h) ⊗ i we obtain u∗ (h ⊗ ΩK ) = a∗ΩP (h) ⊗ ΩP ⊕ h ◦

∗ with h ∈ H ⊗ Ω⊥ P . Assume that h ∈H . Because u is isometric on H ⊗ ΩK we conclude that (2.2) u∗ (ΩH ⊗ ΩK ) = ΩH ⊗ ΩP ⊥ u∗ (h ⊗ ΩK ) ◦

and thus also a∗ΩP (h) ∈H. In other words, ◦

◦

a∗ΩP (H) ⊂ H .

n n Let qn be the orthogonal projection from H ⊗ 1 P onto ΩH ⊗ 1 P. From Theorem 1.1 it follows that n ΩK ) → 0 (n → ∞). (1 − qn )u∗0n . . . u∗01 (h ⊗ 1

On the other hand, by iterating the formula from the beginning, u∗0n . . . u∗01 (h

⊗

n 1

n ∗ n ΩK ) = (aΩP ) (h) ⊗ ΩP ⊕ h 1

n with h ∈ H ⊗ ( 1 ΩP )⊥ . It follows that also

n (1 − qn ) (a∗ΩP )n (h) ⊗ ΩP → 0. 1

n But from a∗ΩP (H) ⊂ H we have qn (a∗ΩP )n (h)⊗ 1 ΩP = 0 for all n. We conclude that (a∗ΩP )n (h) → 0 for n → ∞. Further ◦

◦

a∗ΩP ΩH

=

d i=1

and the proposition is proved.

ωi a∗i

ΩH =

d

ω i ω i Ω H = ΩH ,

i=1

The following proposition summarizes some well known properties of minimal isometric dilations and associated Cuntz algebra representations.

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Proposition 2.4. Suppose A is a coisometric tuple on H and V is its minimal isometric dilation. Assume ΩH is a distinguished unit vector in H and ω = 2 (ω1 , . . . , ωd ) ∈ Cd , i |ωi | = 1. Then the following are equivalent. 1. A is ergodic and A∗i ΩH = ω i ΩH for all i. 2. V is ergodic and Vi∗ ΩH = ω i ΩH for all i. 3. Vi∗ ΩH = ω i ΩH and V generates the GNS-representation of the Cuntz algebra Od = C ∗ {g1 , · · · , gd } (gi its abstract generators) with respect to the Cuntz state which maps ˜ gα g ∗ → ωα ωβ , ∀α, β ∈ Λ. β

Cuntz states are pure and the corresponding GNS-representations are irreducible. This Proposition clearly follows from Theorem 5.1 of [3], Theorem 3.3 and Theorem 4.1 of [4]. Note that in Theorem 1.1(d) we already saw a concrete version of the corresponding Cuntz algebra representation.

3. A new characteristic function First we recall some more details of the theory of minimal isometric dilations for row contractions (cf. [12]) and introduce further notation. The full Fock space over Cd (d ≥ 2) denoted by Γ(Cd ) is 2

Γ(Cd ) := C ⊕ Cd ⊕ (Cd )⊗ ⊕ · · · ⊕ (Cd )⊗ ⊕ · · · . m

1⊕0⊕· · · is called the vacuum vector. Let {e1 , . . . , ed } be the standard orthonormal basis of Cd . Recall that we include d = ∞ in which case Cd stands for a complex ˜ eα will denote the vector separable Hilbert space of inﬁnite dimension. For α ∈ Λ, eα1 ⊗ eα2 ⊗ · · · ⊗ eαm in the full Fock space Γ(Cd ) and e0 will denote the vacuum vector. Then the (left) creation operators Li on Γ(Cd ) are deﬁned by L i x = ei ⊗ x for 1 ≤ i ≤ d and x ∈ Γ(C ). The row contraction L = (L1 , . . . , Ld ) consists of isometries with orthogonal ranges. d

Let T = (T1 , · · · , Td ) be a row contraction on a Hilbert space H. Treating d 1 T as a row operator from H to H, deﬁne D∗ := (1 − T T ∗ ) 2 : H → H and i=1 1 d d D := (1 − T ∗ T ) 2 : i=1 H → i=1 H. This implies that D∗ = (1 −

d

1

1

2 Ti Ti∗ ) 2 , D = (δij 1 − Ti∗ Tj )d×d .

(3.1)

i=1

Observe that T D2 = D∗2 T and hence T D = D∗ T . Let D := Range D and D∗ := Range D∗ . Popescu in [12] gave the following explicit presentation of the minimal isometric dilation of T by V on H ⊕ (Γ(Cd ) ⊗ D), Vi (h ⊕ eα ⊗ dα ) = Ti h ⊕ [e0 ⊗ Di h + ei ⊗ eα ⊗ dα ] (3.2) ˜ α∈Λ

˜ α∈Λ

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for h ∈ H and dα ∈ D. Here Di h := D(0, . . . , 0, h, 0, . . . , 0) and h is embedded at the ith component. In other words, the Vi are isometries with orthogonal ranges such that Ti∗ = ˜ together span the Hilbert Vi∗ |H for i = 1, . . . , d and the spaces Vα H with α ∈ Λ space on which the Vi are deﬁned. It is an important fact, which we shall use repeatedly, that such minimal isometric dilations are unique up to unitary equivalence (cf. [12]). Now, as in Section 2, let A = (A1 , · · · , Ad ), Ai ∈ B(H), be an ergodic ∗ coisometric tuple with Ai ΩH = ωi ΩH for some unit vector ΩH ∈ H and some ω ∈ Cd , i |ωi |2 = 1. Let V = (V1 , · · · , Vd ) be the minimal isometric dilation of A d given by Popescu’s construction (see Equation (3.2)) on H⊕ Γ(C )⊗DA . Because A∗i = Vi∗ |H we also have Vi∗ ΩH = ωi ΩH and because V generates an irreducible Od −representation (Proposition 2.4), we see that V is also a minimal isometric dilation of ω : Cd → C. In fact, we can think of ω as the most elementary example of a tuple with all the properties stated for A. Let V˜ = (V˜1 , · · · , V˜d ) be the minimal isometric dilation of ω given by Popescu’s construction on C ⊕ (Γ(Cd ) ⊗ Dω ). Because A is coisometric it follows from Equation (3.1) that D is in fact a projection and hence D = (δij 1 − A∗i Aj )d×d . We infer that D(A∗1 , · · · , A∗d )T = 0, where T stands for transpose. Applied to ω instead of A this shows that Dω = (1 − |ω ω|) and Dω ⊕ C(ω 1 , · · · , ω d )T = Cd , where ω = (ω 1 , · · · , ωd ). ˜ we have Remark 3.1. Because ΩH is cyclic for {Vα , α ∈ Λ} ˜ = span{pH Vα ΩH : α ∈ Λ} ˜ = H. span{Aα ΩH : α ∈ Λ} Using the notation from Equation (2.1) this further implies that ◦

˜ 1 ≤ i ≤ d} = H . span{˚ Aα li : α ∈ Λ, As minimal isometric dilations of the tuple ω are unique up to unitary equivalence, there exists a unitary W : H ⊕ (Γ(Cd ) ⊗ DA ) → C ⊕ (Γ(Cd ) ⊗ Dω ), such that W Vi = V˜i W for all i. After showing the existence of W we now proceed to compute W explicitly. For A, by using Popescu’s construction, we have its minimal isometric dilation V on H ⊕ (Γ(Cd ) ⊗ DA ). Another way of constructing a minimal isometric dilation t ˆ (obtained by restricting to the of a was demonstrated in Section 1 on the space H ˜ minimal subspace of H ⊗ K with respect to t). Identifying A and a on the Hilbert ˆ → H ⊕ (Γ(Cd ) ⊗ DA ) which is the identity on space H there is a unitary ΓA : H H and satisﬁes Vi ΓA = ΓA ti . By Theorem 1.1(d) the tuple s on P˜ arising from the tensor shift is unitarily equivalent to t (resp. V ), explicitly w ti = si w for all i. An alternative viewpoint

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on the existence of w is to note that s is a minimal isometric dilation of ω. In fact, s∗i ΩP˜ = i , ΩP ΩP˜ = ω i ΩP˜ for all i. Hence there is also a unitary Γω : P˜ → C ⊕ (Γ(Cd ) ⊗ Dω ) with Γω ΩP˜ = 1 ∈ C which satisﬁes V˜i Γω = Γω si . in doing so Remark 3.2. It is possible to describe Γω in an explicit way and ∞ d to construct an interesting and natural (unitary) identiﬁcation of 1 C and ∞ C ⊕ (Γ(Cd ) ⊗ Cd−1 ). In fact, recall (from Section 1) that P˜ = 1 P and the space P is nothing but a d-dimensional Hilbert space. Hence we can identify ◦

Cd P = P ⊕ CΩP Dω ⊕ C ω T Cd−1 ⊕ C. d In this identiﬁcation the orthonormal basis ( i )i=1 of P goes to the canonical basis d d (ei )i=1 of C , in particular the vector ΩP = i ωi i goes to ω T = (ω 1 , · · · , ωd )T ◦

and we have P Dω . Then we can write Γω :

→ 1 ∈ C, → e0 ⊗ k

ΩP˜ k ⊗ ΩP˜

α ⊗ k ⊗ ΩP˜ → eα ⊗ k, ˜ α = α1 ⊗. . . αn ∈ n P (the ﬁrst n copies of P in the inﬁnite where k ∈P , α ∈ Λ, 1 ˜ eα = eα1 ⊗ . . . eαn ∈ Γ(Cd ) as usual. It is easily checked that tensor product P), Γω given in this way indeed satisﬁes the equation V˜i Γω = Γω si (for all i), which may thus be seen as the abstract characterization of this unitary map (together with Γω ΩP˜ = 1). ◦

Summarizing, for i = 1, . . . , d Vi ΓA = ΓA ti ,

w ti = si w,

V˜i Γω = Γω si

and we have the commuting diagram w

ˆ H

/ P˜

ΓA

H ⊕ (Γ(Cd ) ⊗ DA )

W

(3.3)

Γω

/ C ⊕ (Γ(Cd ) ⊗ Dω ).

From the diagram we get W = Γω wΓ−1 A . Combined with the equations above this yields W Vi = V˜i W and we see that W is nothing but the dilations-intertwining map which we have already introduced earlier. Hence w and W are essentially the same thing and for the study of certain problems it may be helpful to switch from one picture to the other. In the following we analyze W to arrive at an interpretation as a new kind of characteristic function. First we have an isometric embedding Cˆ := W |H : H → C ⊕ (Γ(Cd ) ⊗ Dω ).

(3.4)

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Note that Cˆ ΩH = W ΩH = 1 ∈ C. The remaining part is an isometry d d MΘ ˆ := W |Γ(Cd )⊗DA : Γ(C ) ⊗ DA → Γ(C ) ⊗ Dω ..

(3.5)

From equation (3.2) we get for all i Vi |Γ(Cd )⊗DA = (Li ⊗ 1DA ), V˜i |Γ(Cd )⊗Dω = (Li ⊗ 1Dω ), and we conclude that MΘ ˆ (Li ⊗ 1DA ) = (Li ⊗ 1Dω )MΘ ˆ , ∀1 ≤ i ≤ d.

(3.6)

In other words, MΘ ˆ is a multi-analytic inner function in the sense of [14, 15]. It is determined by its symbol θˆ := W |e0 ⊗DA : DA → Γ(Cd ) ⊗ Dω ,

(3.7)

where we have identiﬁed e0 ⊗ DA and DA . In other words, we think of the symbol θˆ as an isometric embedding of DA into Γ(Cd ) ⊗ Dω . ˆ Definition 3.3. We call MΘ ˆ (or θ) the extended characteristic function of the row contraction A. See Sections 5 and 6 for more explanation and justiﬁcation of this terminology.

4. Explicit computation of the extended characteristic function To express the extended characteristic function more explicitly in terms of the tuple A we start by deﬁning ◦

◦

ˆ ∗ : H= H CΩH → P = P CΩP Dω , D h → ΩH | ⊗ 1P u∗ (h ⊗ ΩK ),

(4.1)

where u : H ⊗ P → H ⊗ K is the isometry introduced in Section 1. That indeed ◦ ˆ ∗ is contained in P follows from Equation (2.2), i.e., u∗ (h ⊗ ΩK ) ⊥ the range of D ◦

ΩH ⊗ΩP for h ∈H. With notations from Equation (2.1) we can get a more concrete formula. ◦ ˆ ∗ (h) = d i , h i . Lemma 4.1. For all h ∈H we have D i=1 ∗ d d ∗ Proof. ΩH | ⊗ 1P u (h ⊗ ΩK ) = i=1 ΩH , ai h ⊗ i = i=1 i , h i . Proposition 4.2. The map Cˆ : H → C ⊕ (Γ(Cd ) ⊗ Dω ) from Equation (3.4) is given ◦ ˆ H = 1 and for h ∈H by explicitly by CΩ ∗ ˆ = ˆ ∗˚ Ch eα ⊗ D Aα h. ˜ α∈Λ

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◦

ˆ H = 1. Assume h ∈H. Then Proof. As W ΩH = 1 also CΩ u01 (h ⊗ ΩK˜ ) = a∗i h ⊗ i ⊗ ΩK˜ i

=

i , h ΩH ⊗ i ⊗ ΩK˜ +

i

◦∗

ai h ⊗ i ⊗ ΩK˜ .

i

∗

Because u (ΩH ⊗ ΩK ) = ΩH ⊗ ΩP we obtain (with Lemma 4.1) for the ﬁrst part lim u∗0n · · · u∗02 ( i , h ΩH ⊗ i ⊗ ΩK˜ ) n→∞

=

i

ˆ ∗h ⊗ Ω ˜ D ˆ ∗ h ⊗ Ω ˜ ∈ P. ˜ i , h ΩH ⊗ i ⊗ ΩP˜ = ΩH ⊗ D P P i

Using the product formula from Theorem 1.1 and iterating the argument above we get ˆ C(h) = W h = Γω wΓ−1 A (h) ◦∗ ˆ ∗ h ⊗ Ω ˜ ) + Γω lim u∗0n · · · u∗02 ai h ⊗ i ⊗ ΩK˜ = Γω (D P n→∞

ˆ ∗ h + Γω lim u∗ · · · u∗ = e0 ⊗ D 0n 03 n→∞

ˆ ∗h + = e0 ⊗ D

d

i

◦∗ ◦∗ ◦∗ j , ai h ΩH + aj ai h ⊗ i ⊗ j ⊗ ΩK˜ j,i

∗

ˆ ∗ a◦ i h + Γω lim u∗0n · · · u∗03 ei ⊗ D

n→∞

i=1

◦∗ ◦∗

aj ai h ⊗ i ⊗ j ⊗ ΩK˜

j,i

= ... ◦∗ ∗ ˆ ∗ a◦ h + Γω lim u∗ · · · u∗ aα h ⊗ α ⊗ ΩK˜ . eα ⊗ D = 0n 0,m+1 α |α|<m

n→∞

|α|=m

◦∗ From Proposition 2.1 we have |α|=m aα h2 → 0 for m → ∞ and we conclude that the last term converges to 0. It follows that the series converges and this proves Proposition 4.2. ◦

Remark 4.3. Another way to prove Proposition 4.2 for h ∈H consists in repeatedly applying the formula u∗ (h ⊗ ΩK ) = a∗ΩP h ⊗ ΩP + h ,

◦

h ∈ H⊗ P

to the u∗0n (h ⊗ ΩK ) and then using (a∗ΩP )n h → 0, see Proposition 2.3. This gives some insight how the inﬁnite product in Theorem 1.1 transforms into the inﬁnite sum in Proposition 4.2. Now we present an explicit computation of the extended characteristic function. One way of writing DA is DA = span{(Vi − Ai )h : i ∈ Λ, h ∈ H}.

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Let dih := (Vi − Ai )h. Then ˆ − CA ˆ i h. θˆ dih = W (Vi − Ai )h = V˜i Ch Case I: Take h = ΩH . ˆ H = V˜i 1 = ωi ⊕ [e0 ⊗ (1 − |ω ω|)i ], V˜i CΩ ∗ ˆ i ΩH = ω i ⊕ ˆ ∗˚ CA eα ⊗ D Aα li , α

and thus θˆ di

ΩH

ˆ ∗ li ] − = e0 ⊗ [(1 − |ω ω|)i − D = e0 ⊗ [i −

ω j ωi j −

j

= e0 ⊗ [i − = e0 ⊗ [i −

∗

ˆ ∗˚ eα ⊗ D Aα li

|α|≥1

lj , li j ] −

eα ⊗

|α|≥1

j

(ω j ωi + lj , li )j ] −

eα ⊗

|α|≥1

j

Aj ΩH , Ai ΩH j ] −

eα ⊗

|α|≥1

j

∗ lj , ˚ Aα li j

j

˚ Aα lj , li j

j

˚ Aα lj , li j .

(4.2)

j

◦

Case II: Now let h ∈ H. With i ∈ Λ, ˆ = (Li ⊗ 1)Ch ˆ = V˜i Ch ˆ ih = CA

∗ ˆ ∗˚ ei ⊗ eα ⊗ D Aα h,

α ∗ ˆ ∗˚ eβ ⊗ D Aβ ˚ Ai h.

β

Finally θˆ dih =

∗ ˆ ∗˚ ei ⊗ eα ⊗ D Aα h −

α

ˆ ∗˚ = −e0 ⊗ D Ai h+ei ⊗

β

∗ ∗ ˆ ∗˚ eα ⊗ D Ai ˚ Aα (1− ˚ Ai )h+

ej ⊗

j =i

α

ˆ ∗˚ = −e0 ⊗ D Ai h +

∗ ˆ ∗˚ eβ ⊗ D Aβ ˚ Ai h

d j=1

ej ⊗

∗ ∗ ˆ ∗˚ eα ⊗ D Aj ˚ Aα (−˚ Ai )h

α

∗ ∗ ˆ ∗˚ eα ⊗ D Aj ˚ Aα (δji 1 − ˚ Ai )h.

(4.3)

˜ α∈Λ

◦

5. Case II is the characteristic function of A In this section we show that case II in the previous section can be identiﬁed with ◦

the characteristic function of the ∗-stable tuple A, in the sense introduced by Popescu in [13]. This is the reason why we have called θˆ an extended characteristic ◦

function. All information about A beyond A must be contained in case I. First recall the theory of characteristic functions for row contractions, as developed by G. Popescu in [13], generalizing the theory of B. Sz.-Nagy and C. Foias

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(cf. [11]) for single contractions. We only need the results about a ∗-stable tuple ◦

◦

◦ ◦

◦

◦

◦

◦

1 A= (˚ A1 , . . . , ˚ Ad ) on H. In this case, with D∗ = (1− AA∗ ) 2 : H→H and D∗ its range, the map

◦

◦

◦

d C : H→ Γ(C )⊗ D∗ ◦ h → eα ⊗ D ∗ ˚ A∗α h

(5.1)

˜ α∈Λ ◦ ◦ 1 ◦ ◦ d is an isometry (Popescu’s Poisson kernel). If, as usual, D = (1− A∗ A) 2 : 1 H→ ◦ d ◦ 1 H, with D its range, and if Pj is the projection onto the j-th component, then ◦

the characteristic function θ˚ of A can be deﬁned as A ◦

◦

θ˚ : D→ Γ(Cd )⊗ D∗

(5.2)

A

f → −e0 ⊗

d

˚ Aj Pj f +

j=1

d

ej ⊗

◦

◦

eα ⊗ D ∗ ˚ A∗α Pj D f.

˜ α∈Λ

j=1

See [13] for details, in particular for the important result that θ˚ characterizes the A

◦

∗-stable tuple A up to unitary equivalence. Now consider again the tuple A of the previous section, with extended charˆ From Equation (2.1) acteristic function θ.

ω i i | ωi 0 ∗ Ai = ∗ ˚i , Ai = |i A 0 ˚ Ai and hence Ai A∗i

=

|ω i |2 ω i li | ∗ Ai ˚ Ai |ω i li |li li | + ˚

.

Recall that D∗2 = 1 − i Ai A∗i which is 0 as A is coisometric. Thus i ωi li = 0 ◦ ◦ ∗ and 1 − i ˚ Ai ˚ Ai = i |li li |. The ﬁrst equation means that A∗ΩP (H) ⊂H and that ˆ∗ h, ΩP = i , h i , D ω j j = ωi i , h = 0, i

j

i

which we already know (see 4.1). The second equation yields ◦2

D∗ = 1 −

∗ ˚ Ai ˚ Ai =

i

|li li |.

i ◦

◦

◦

Lemma 5.1. There exists an isometry γ : D ∗ → P Dω defined for h ∈H as ◦ ˆ ∗ h. li , h i = D D∗ h → i

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◦ ˆ ∗ (h) = d i , h i . Now we can Proof. Take h ∈H . By Lemma 4.1 we have D i=1 compute ◦2 ◦ ˆ ∗ h2 = li , h i , lj , h j = h, li li , h = h, D∗ h = D∗ h2 . D i

j

i

◦

ˆ ∗ h is isometric. Hence γ : D∗ h → D

tuple Theorem 5.2. Let A = (A1 , · · · , Ad ), Ai ∈ B(H), be an ergodic coisometric 2 with A∗i ΩH = ω i ΩH for some unit vector ΩH ∈ H and some ω ∈ Cd , i |ωi | = 1. Let θˆ be the extended characteristic function of A and let θ˚ be the characteristic A

◦

function of the (∗-stable) tuple ˚ A. For h ∈H ◦

γ D∗ h = ◦

ˆ ∗ h, D

ˆ (1 ⊗ γ) C h = Ch, (1 ⊗ γ) θ˚dih = θˆ dih for i ∈ Λ. A In other words, the part of θˆ described by case II in the previous section is equivalent to θ˚. A

Proof. We only have to use Lemma 5.1 and compare Proposition 4.2 and Equa◦

tion (5.1) as well as Equations (4.3) and (5.2). For the latter note that dih =D (0, . . . , 0, h, 0 . . . , 0), where h is embedded at the i-th position. Hence ◦ ◦ ◦ ˚ ˚ γ Aj Pj dih = γ Aj Pj D (0, . . . , 0, h, 0 . . . , 0) = γ A D (0, . . . , 0, h, 0 . . . , 0) j

j ◦

◦

ˆ∗ ˚ = γ D ∗ A (0, . . . , 0, h, 0 . . . , 0) = D Ai h and also ◦

◦2

∗

Pj D dih = Pj D (0, . . . , 0, h, 0 . . . , 0) = (δji 1 − ˚ Aj ˚ Ai )h.

Of course, Theorem 5.2 explains why we have called θˆ an extended characteristic function.

6. The extended characteristic function is a complete unitary invariant In this section we prove that the extended characteristic function is a complete invariant with respect to unitary equivalence for the row contractions investigated in this paper. Suppose that A = (A1 , . . . , Ad ) and B = (B1 , . . . , Bd ) are ergodic and coisometric row contractions on Hilbert spaces HA and HB such that A∗i ΩA = ω i ΩA and Bi∗ ΩB = ω i ΩB for i = 1, . . . , d, where ΩA ∈ HA and ΩB ∈ HB are unit vectors and ω = (ω1 , . . . , ωd ) is a tuple of complex numbers. Recall from Remark 2.2 that it is no serious restriction of generality to assume that it is the same

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tuple of complex numbers in both cases because this can always be achieved by a transformation with a unitary d × d−matrix (with scalar entries). We will use all the notations introduced earlier with subscripts A or B. Let us say that the extended characteristic functions θˆA and θˆB are equivalent if there exists a unitary V : DA → DB such that θˆA = θˆB V . Note that the ranges of θˆA and θˆB are both contained in Γ(Cd ) ⊗ Dω and thus this deﬁnition makes sense. Let us further say that A and B are unitarily equivalent if there exists a unitary U : HA → HB such that U Ai = Bi U for i = 1, . . . , d. By ergodicity the unit eigenvector ΩA (resp. ΩB ) is determined up to an unimodular constant (see Theorem 1.1(b)) and hence in the case of unitary equivalence we can always modify U to satisfy additionally U ΩA = ΩB . Theorem 6.1. The extended characteristic functions θˆA and θˆB are equivalent if and only if A and B are unitarily equivalent. Proof. If A and B are unitarily equivalent then all constructions diﬀer only by naming and it follows that θˆA and θˆB are equivalent. Conversely, assume that there is a unitary V : DA → DB such that θˆA = θˆB V . Now from the commuting diagram 3.3 and the deﬁnitions following it d WB HB = C ⊕ Γ(Cd ) ⊗ Dω MΘ ˆ B Γ(C ) ⊗ DB d = C ⊕ Γ(Cd ) ⊗ Dω MΘ ˆ B Γ(C ) ⊗ V DA d = C ⊕ Γ(Cd ) ⊗ Dω MΘ ˆ A Γ(C ) ⊗ DA =

WA HA ,

where we used Equation (3.6), i.e., MΘ ˆ (Li ⊗ 1D ) = (Li ⊗ 1Dω )MΘ ˆ , ∀1 ≤ i ≤ d, ˆ ˆ to deduce MΘ ˆ A = MΘ ˆ B (1 ⊗ V ) from θA = θB V . Now we deﬁne the unitary U by U := WB−1 WA |HA : HA → HB . Because WA ΩA = 1 = WB ΩB we have U ΩA = ΩB . Further for all i = 1, . . . , d and h ∈ HA , U Ai h = WB−1 WA Ai h = WB−1 WA PHA ViA h = PHB WB−1 WA ViA h = PHB WB−1 V˜i WA h = PHB ViB WB−1 WA h = Bi U h, i.e., A and B are unitarily equivalent.

Remark 6.2. An analogous result for completely non-coisometric tuples has been shown by G. Popescu in [13], Theorem 5.4. Note further that if we change A = (A1 , . . . , Ad ) into A = (A1 , . . . , Ad ) by applying a unitary d × d−matrix with scalar entries (as described in Remark 2.2), then θˆA = θˆA . In fact, this follows immediately from the deﬁnition of W as an intertwiner in Section 3, from which it is evident that W does not change if we take the same linear combinations on the left and on the right. This does not contradict Theorem 6.1 because ω and ω are now diﬀerent tuples of eigenvalues

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and Theorem 6.1 is only applicable when the same tuple of eigenvalues is used for A and B. For another interpretation, let Z be a normal, unital, ergodic, completely positive map with an invariant vector state ΩA , · ΩA . If we consider two minimal Kraus decompositions of Z, i.e., Z=

d

Ai · A∗i =

i=1

d

Ai · (Ai )∗ ,

i=1

with d minimal, then the tuples A = (A1 , . . . , Ad ) into A = (A1 , . . . , Ad ) are related in the way considered above (see for example [7], A.2). It follows that θˆA = θˆA does not depend on the decomposition but can be associated to Z itself. Hence we have the following reformulation of Theorem 6.1. Corollary 6.3. Let Z1 , Z2 be normal, unital, ergodic, completely positive maps on B(H1 ), B(H2 ) with invariant vector states Ω1 , · Ω1 and Ω2 , · Ω2 . Then the associated extended characteristic functions θˆ1 and θˆ2 are equivalent if and only if Z1 and Z2 are conjugate, i.e., there exists a unitary U : H1 → H2 such that Z1 (x) = U ∗ Z2 (U xU ∗ )U

for all x ∈ B(H1 ).

7. Example The following example illustrates some of the constructions in this paper. Consider H = C3 and     0 0 0 1 1 0 1  1 A1 = √ 1 0 0  , A2 = √  0 0 1  . 2 2 0 1 1 0 0 0 2 Then i=1 Ai A∗i = 1. Take the unital completely positive map Z : M3 → M3 2 by Z(x) = i=1 Ai xA∗i . It is shown in Section 5 of [8] (and not diﬃcult to verify directly) that this map is ergodic. We will use the same notations here as in previous sections. Observe that the vector ΩH := √13 (1, 1, 1)T gives an invariant vector state for Z as ΩH , Z(x)ΩH = ΩH , xΩH = A∗i ΩH =

√1 ΩH 2

and hence ω =

√1 (1, 1). 2

3 1 xij . 3 i,j=1 ◦

The orthogonal complement H of CΩH ◦

in C3 and the orthogonal projection Q onto H are given by     k1 2 −1 −1 ◦ 1  : k1 , k2 ∈ C}, k2 Q =  −1 2 −1  . H = { 3 −(k1 + k2 ) −1 −1 2

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Characteristic Functions for Ergodic Tuples

From this we get for ˚ Ai = QAi Q = Ai Q    0 0 0 1 1 1 1 ˚ A1 = √  2 −1 −1  , ˚ A2 = √  −1 −1 3 2 3 2 −2 1 1 0 0

61

 −2 2 . 0

◦

A1 , ˚ A2 ) is ∗-stable as (by induction) We notice that the tuple A= (˚   1 −1 0 1 ∗  −1 2 −1  → 0 (n → ∞). ˚ Aα ˚ Aα = 3 × 2n−1 0 −1 1 |α|=n ◦

Here P = C2 and P := P CΩP with ΩP = ◦

◦

√1 (1, 1)T . 2

Easy calculation shows

ˆ ∗ : H→P is given by that D  

k1 1 −1   k2 . → √ (2k1 + k2 ) 1 6 −(k1 + k2 )   1 0 −1 ◦ ◦ ◦ Moreover D∗ = √16  0 0 0  . There exists an isometry γ : D∗ →P such −1 0 1  

1 ◦ ◦ −1 ˆ ∗ h for h ∈H.   0 that → and γ(D∗ h) = D 1 −1 ◦ ˆ H ) = 1 and for h ∈H by The map Cˆ : H → Γ(Cd ) ⊗ Dω is given by C(Ω  

k1 (k1 + 2k2 ) (2k1 + k2 ) 1 −1 ˆ   √ √ k2 C = e0 ⊗ eα ⊗ ( √ )|α| + 1 6 2 6 α,α1 =1 −(k1 + k2 )

(k1 − k2 ) 1 −1 −1 × + eα ⊗ ( √ )|α| √ 1 1 2 6 α,α1 =2

˜ such that αi = αi+1 for all where the summations are taken over all 0 = α ∈ Λ 1 ≤ i ≤ |α| and ﬁxing α1 to 1 or 2 as indicated. This simpliﬁcation occurs because ˚ A2i = 0 for i = 1, 2. All the summations below in this section are also of the same kind. Now using the Equations (4.2) and (4.3) for θˆA : DA → Γ(Cd ) ⊗ Dω and simplifying we get

1 1 1 −1 −1 θˆA d1ΩH = −e0 ⊗ eα ⊗ ( √ )|α| + 1 1 6 6 2 α,α1 =1

1 1 −1 eα ⊗ ( √ )|α| + = −θˆA d2ΩH , 1 6 2 α,α1 =2

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◦

and for h ∈H, θˆA d1h

θˆA d2h

=

(k1 + k2 ) −1 √ e1 ⊗ eα + e1 ⊗ + 1 6 α,α1 =1

1 |α| (k1 + 2k2 ) 1 |α| k2 −1 −1 √ ⊗( √ ) e1 ⊗ eα ⊗ ( √ ) √ − 1 1 2 6 2 6 α,α1 =2

k1 1 −1 eα ⊗ ( √ )|α| √ + , 1 2 2 3 α,α1 =2 k1 −e0 ⊗ √ 2 3

=

−1 1

(k1 + k2 ) 1 |α| (k1 + k2 ) −1 −1 √ √ −e0 ⊗ eα ⊗ ( √ ) + 1 1 2 3 2 2 3 α,α1 =1

k2 k1 1 −1 −1 e2 ⊗ eα ⊗ ( √ )|α| √ +e2 ⊗ √ + 1 1 6 2 6 α,α1 =1

(k1 − k2 ) 1 −1 e2 ⊗ eα ⊗ ( √ )|α| √ + . 1 2 6 α,α1 =2 ◦

◦

From this we can easily obtain C and θ˚ for h ∈H by using the following A relations from Theorem 5.2, ◦

ˆ (1 ⊗ γ) C h = Ch, (1 ⊗ γ)θ˚dih = θˆA dih . A

Further

    −1 1 1 1  1 1 0  , l2 = A2 ΩH − √ ΩH = √  0  , l1 = A1 ΩH − √ ΩH = √ 2 6 2 6 1 −1   0 ◦ 1  ˜ as already  and clearly H= span{˚ ˚ Aα li : i = 1, 2 and α ∈ Λ}, −1 A1 l1 = 2√ 3 1 observed in Remark 3.1.

References [1] W. Arveson, Noncommutative Dynamics and E-Semigroups, Springer Monographs in Mathematics (2003). [2] B.V.R. Bhat and K.R. Parthasarathy, Kolmogorov’s existence theorem for Markov processes in C ∗ algebras, Proc. Indian Acad. Sci., Math. Sci., 104 (1994), 253–262. [3] O. Bratteli, P.E.T. Jorgensen, A. Kishimoto and R.F. Werner, Pure states on Od , J. Operator Theory, 43 (2000), 97–143.

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[4] O. Bratteli, P.E.T. Jorgensen and G.L. Price, Endomorphisms of B(H). Quantization, nonlinear partial diﬀerential equations, and operator algebra (Cambridge, MA, 1994), 93–138, Proc. Sympos. Pure Math., 59, Amer. Math. Soc., Providence, RI (1996). [5] J. Cuntz, Simple C ∗ -algebras generated by isometries, Commun. Math. Phys., 57 (1977), 173–185. [6] C. Foias and A.E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, 44, Birkh¨ auser Verlag, Basel (1990). [7] R. Gohm, Noncommutative stationary processes, Lecture Notes in Mathematics, 1839, Springer-Verlag, Berlin (2004). [8] R. Gohm, B. K¨ ummerer and T. Lang, Noncommutative symbolic coding, Ergod.Th. & Dynam.Sys., 26 (2006), 1521–1548. [9] B. K¨ ummerer and H. Maassen, A scattering theory for Markov chains. Inf. Dim. Analysis, Quantum Prob. and Related Topics, vol. 3 (2000), 161–176. [10] B. K¨ ummerer, Markov dilations on W ∗ -algebras, J. Funct. Anal., 63 (1985), 139–177. [11] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North Holland Publ., Amsterdam-Budapest (1970). [12] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc., 316 (1989), 523–536. [13] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory, 22 (1989), 51–71. [14] G. Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ.Math.J., 36 (1989), 693–710. [15] G. Popescu, Multi-analytic operators on Fock spaces J. Funct. Anal., 161 (1999), 27–61. [16] R.T. Powers, An index theory for semigroups of ∗-endomorphisms of B(H) and type II1 factors, Canad. J. Math., 40 (1988), no. 1, 86–114. Santanu Dey Institut f¨ ur Mathematik und Informatik Ernst-Moritz-Arndt-Universit¨ at Friedrich-Ludwig-Jahn-Str. 15a, 17487 Greifswald Germany e-mail: [email protected] Rolf Gohm Department of Mathematics University of Reading Whiteknights, P.O. Box 220, Reading, RG6 6AX England e-mail: [email protected] Submitted: June 30, 2005 Revised: December 7, 2006

Integr. equ. oper. theory 58 (2007), 65–86 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010065-22, published online April 16, 2007 DOI 10.1007/s00020-007-1491-3

Integral Equations and Operator Theory

Factorization in Weighted Wiener Matrix Algebras on Linearly Ordered Abelian Groups Torsten Ehrhardt, Cornelis van der Mee, Leiba Rodman and Ilya M. Spitkovsky Abstract. Factorizations of Wiener-Hopf type of elements of weighted Wiener algebras of continuous matrix-valued functions on a compact abelian group are studied. The factorizations are with respect to a ﬁxed linear order in the character group (considered with the discrete topology). Among other results, it is proved that if a matrix function has a canonical factorization in one such matrix Wiener algebra then it belongs to the connected component of the identity of the group of invertible elements in the algebra, and moreover, the factors of the canonical factorization depend continuously on the matrix function. In the scalar case, complete characterizations of canonical and noncanonical factorability are given in terms of abstract winding numbers. Wiener-Hopf equivalence of matrix functions with elements in weighted Wiener algebras is also discussed. Mathematics Subject Classification (2000). Primary 46J10; Secondary 43A20. Keywords. Wiener algebra, Wiener-Hopf factorization, compact abelian group.

1. Introduction Let G be a compact multiplicative abelian group and Γ its additive character group equipped with the discrete topology. We denote by j, g the action of the character j ∈ Γ on the group element g ∈ G or, by Pontryagin duality, of the character g ∈ G on the group element j ∈ Γ. It is well-known [29] that Γ can be made into a linearly ordered group if and only if G is connected; the latter hypothesis will be maintained throughout the The second author is supported by COFIN grant 2004015437 and by INdAM; the third and the fourth authors are partially supported by NSF grant DMS-0456625; the third author is also partially supported by the Faculty Research Assignment from the College of William and Mary.

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paper. We ﬁx a linear order on Γ such that Γ is an ordered group with respect to , i.e., x + z y + z if x, y, z ∈ Γ and x y. The notations ≺, , , max, min (with obvious meaning) will also be used. We introduce the additive semigroups Γ+ = {x ∈ Γ : x 0} and Γ− = {x ∈ Γ : x 0}. In applications, often Γ is an additive subgroup of Rk so that G is its Bohr compactiﬁcation, or Γ = Zd so that G = Td is the d-torus. In this paper we study factorization of Wiener-Hopf type of matrix valued functions, as shown in formula (1.1) below, where we use diag (x1 , . . . , xn ) to denote the n × n diagonal matrix with x1 , . . . , xn on the main diagonal, in that order: + (g) (diag (j1 , g, . . . , jn , g)) A − (g), A(g) =A

g ∈ G.

(1.1)

+ and (A + )−1 belong to the n × n matrix function algebra of abstract Here A − and (A − )−1 belong Fourier transforms of a weighted 1 space indexed by Γ+ , A to the n × n matrix function algebra of abstract Fourier transforms of a weighted 1 space indexed by Γ− , and j1 , . . . , jn ∈ Γ. Factorization of type (1.1), for 1 spaces without weights, is classical when G is the unit circle; it goes back to [14], and see also [10, 6], among many books on this subject. Factorization of type (1.1) (without weights) in the case when Γ is a subgroup of the additive group Rk (endowed with the discrete topology) and its numerous applications have been extensively studied in the literature. A very partial list of relevant references here include [30, 31, 17, 18, 19, 3, 26, 25, 27], and see also the recent book [4]. In the abstract setting, but still for 1 spaces without weights, the factorization (1.1) was studied in [22, 21, 24]. On the other hand, in the paper [8] the weighted case was studied for scalar valued functions when G is the d-dimensional torus. In the present paper we continue this line of investigation, and focus on the abstract compact multiplicative abelian connected group G and its additive ordered character group Γ. The factorization (1.1) will be considered in the matrix function algebras of abstract Fourier transforms of weighted 1 spaces, with arbitrary weights subject only to natural admissibility assumptions (see Section 3 for details). Let us discuss the contents of this article. Sections 2 and 3 are devoted to Wiener algebras of scalar functions, without and with weights, and contain a full characterization of Wiener-Hopf factorizations. In Section 4 the matrix analog is discussed, in particular the uniqueness of factorization indices, hereditary properties of factors with respect to subalgebras, and connectedness. In Section 5 we relate canonical factorability to the unique solvability of certain Toeplitz equations. In Section 6 we conclude the paper with a discussion of Wiener-Hopf equivalence. The following notation is used throughout the paper: N the set of positive integers, Z the set of integers, R the set of reals, T the unit circle, C the set of complex numbers.

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2. Unweighted Wiener Algebras Let G be a compact connected multiplicative abelian group and Γ its additive character group equipped with the discrete topology and the linear order . For any nonempty set M , let 1 (M ) stand for the complex Banach space of all complex-valued M -indexed sequences x = {xj }j∈M having at most countably many nonzero terms that are ﬁnite with respect to the norm x1 = |xj |. j∈M

Then it is clear that ˙ 1 (Γ− \ {0}) = 1 (Γ+ \ {0})+ ˙ 1 (Γ− ) 1 (Γ) = 1 (Γ+ )+ ˙ + ˙ 1 (Γ− \ {0}), = 1 (Γ+ \ {0})+C

(2.1)

where the projections involved all have unit norm. Moreover, 1 (Γ) is a commutative Banach algebra with unit element with respect to the convolution product xk yj−k . (x ∗ y)j = k∈Γ 1

1

Further, (Γ+ ) and (Γ− ) are closed subalgebras of 1 (Γ) containing the unit element. For every Banach algebra A with identity element we denote its group of invertible elements by G(A) and the connected component of G(A) containing the identity by G0 (A). It is well-known that G0 (A) = {exp(b1 ) · · · exp(bn ) : b1 , . . . , bn ∈ A, n ∈ N} for arbitrary Banach algebras with identity element and G0 (A) = {exp(b) : b ∈ A} for those that are commutative, see [7], for example. Given a = {aj }j∈Γ ∈ 1 (Γ), by the symbol of a we mean the complex-valued continuous function a on G deﬁned by a(g) = aj j, g, g ∈ G. (2.2) j∈Γ

The set {j ∈ Γ : aj = 0} will be called the Fourier spectrum of a given by (2.2). Since Γ is written additively and G multiplicatively, we have j + k, g = j, g · k, g, j, gh = j, g · j, h,

j, k ∈ Γ, g ∈ G, j ∈ Γ,

g, h ∈ G.

We will also use the shorthand notation ej for the function ej (g) = j, g, Thus, ej+k = ej ek , j, k ∈ Γ.

g ∈ G.

(2.3)

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The set of all symbols of elements a ∈ 1 (Γ) forms an algebra W (G) of continuous functions on G (with pointwise addition, scalar multiplication and multiplication). The algebra W (G) is made into a Banach algebra isomorphic to 1 (Γ) by letting Λ : a → a be an isometry. (This is possible since Λ is injective, which follows from [29, Sec. 1.3.6].) Standard Gelfand theory implies that the algebra W (G) is inverse closed in the algebra of all continuous functions on G (indeed, it is well-known that the maximal ideal space of 1 (Γ) can be identiﬁed with G, see, for example, [9, Section 21]). a we mean a Given a = {aj }j∈Γ ∈ 1 (Γ), by a canonical factorization of factorization of the symbol a of the form a(g) = a+ (g) a− (g),

g ∈ G,

(2.4)

where a+ ∈ G(1 (Γ+ )) and a− ∈ G(1 (Γ− )). In that case we obviously have a ∈ G(1 (Γ)) and a = a+ ∗a− = a− ∗a+ . Moreover, for any two canonical factorizations of a, say (2.4) and a(g) = b+ (g)b− (g) for g ∈ G, there exists a nonzero complex number c such that b+ = c a+ and a− = c b− . The following result has been established in [15] for the torus G = T2 . It has been generalized to compact connected groups G with ﬁnitely generated character group Γ and to weighted generalizations of the Banach algebra 1 (Γ) in [8]. Here we achieve full generality in the unweighted case. Theorem 2.1. Let G be a compact multiplicative abelian group with ordered character group (Γ, ), and let a ∈ 1 (Γ). Then the following statements are equivalent: (a) a has a canonical factorization. (b) a ∈ G0 (1 (Γ)). Proof. (b) ⇒ (a) Suppose a ∈ G0 (1 (Γ)). Then there exists b ∈ 1 (Γ) such that a = exp(b). We can now ﬁnd b+ ∈ 1 (Γ+ ) and b− ∈ 1 (Γ− ) such that b = b+ + b− . Then, by commutativity, we have the canonical factorization (2.4), where a+ = exp(b+ ) and a− = exp(b− ). (a) ⇒ (b) For a ∈ 1 (Γ), let a have a canonical factorization. Then a ∈ 1 G( (Γ)). Let b ∈ 1 (Γ) be such that b − a < a−1 −1 and {j ∈ Γ : bj = 0} is a ﬁnite subset of {j ∈ Γ : aj = 0}. Then the symbols (1 − t) a + tb (0 ≤ t ≤ 1) all 1 have a canonical factorization in (Γ), because [(1 − t)a + tb] − a = tb − a < a−1 −1 ,

0 ≤ t ≤ 1.

(This follows from a general result on factorization in decomposable Banach algebras, see for example [10, Lemma I.5.1] or [11, Theorem XXIX.9.1].) On the other hand, the canonical factorization of b with respect to 1 (Γ) is actually a canonical factorization of b with respect to 1 (Γ0 ), where Γ0 is the additive group generated by the ﬁnite set {j ∈ Γ : bj = 0}, see [24, Theorem 1]. According to [8], b ∈ G0 (1 (Γ0 )). Then a is continuously connected to b within G(1 (Γ)) and b ∈ G0 (1 (Γ)), and hence a ∈ G0 (1 (Γ)).

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For any commutative Banach algebra A with unit element we have the group isomorphisms G(A) G(C(M)) π 1 (M), (2.5) G0 (A) G0 (C(M)) where M stands for the maximal ideal space of A, C(M) is the Banach algebra of continuous functions on M, and π 1 (M) denotes the ﬁrst cohomotopy group of M (see, for example, [16]). The ﬁrst isomorphism follows from the Arens-Royden theorem [33] (also [1, 28]). The second isomorphism follows from [7, Theorem 2.18]. It is most instructive to spell out the equivalences in (2.5). For every m ∈ M we let φm stand for the unique multiplicative linear functional on A with kernel m. Then a ∈ G(A) is mapped onto the homotopy class [Fa ] of the continuous function Fa : M → T deﬁned by Fa (m) =

φm (a) , |φm (a)|

a ∈ A.

In the situation we are interested in, where A = 1 (Γ), we have M = G (see [9, Section 21]) and Fa becomes j∈Γ j, g aj , a = {aj }j∈Γ ∈ 1 (Γ). Fa (g) = (2.6) j∈Γ j, g aj Moreover, the groups in (2.5) can be identiﬁed explicitly. Since G is connected, we have the group isomorphism G(C(G)) ∼ (2.7) =Γ G0 (C(G)) (see [32, Proposition in Subsection 8.3.2]). More speciﬁcally, each continuous function a ∈ G(C(G)) can be written as a(g) = ej (g)b(g) with uniquely determined j ∈ Γ and b ∈ G0 (C(G)). As a result we obtain the group isomorphism G(1 (Γ)) Γ G0 (1 (Γ))

(2.8)

between the so-called abstract index group of 1 (Γ) on the left-hand side (cf. [7]) and the given discrete group Γ. Given a ∈ G(1 (Γ)), the element j ∈ Γ uniquely determined by the above isomorphism will be called the abstract winding number of a. a If Γ = Z, G = T, and a ∈ G(1 (Z)), the abstract winding number of coincides with the usual winding number of the function a ∈ G(C(T)). Clearly, π 1 (T) ∼ = Z. If Γ = R with discrete topology, G is the Bohr compactiﬁcation of R, and a ∈ G(C(G)) is known as the a ∈ G(1 (Γ)), the abstract winding number of mean motion of a (more precisely, of the function a restricted to R). In this case, π 1 (G) ∼ = R.

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Corollary 2.2. If a ∈ 1 (Γ) and a(g) = 0 for every g ∈ G, then a admits a factorization a(g) = a+ (g)ej (g) a− (g), g ∈ G, (2.9) 1 1 where a+ ∈ G( (Γ+ )), a− ∈ G( (Γ− )), and j ∈ Γ is the abstract winding number of a. Proof. Clearly, a ∈ G(1 (Γ)). The speciﬁc form of the isomorphisms (2.5) and (2.7) implies that a(g) = ej (g)b(g) with b ∈ G0 (1 (Γ)) and j ∈ Γ. It now remains to apply Theorem 2.1. For the particular case Γ = Rk , Corollary 2.2 was proved in [23] by elementary means. See also [26], where the case when Γ is a subgroup of Rk is treated. For Γ = Zd and G = Td Corollary 2.2 was proved in [8]. Note that either condition (a) or (b) of Theorem 2.1 is equivalent to a(g) = 0 for every g ∈ G and a(g) having winding number zero. We now prove that the abstract index groups of 1 (Γ+ ) and 1 (Γ− ) are trivial, as is immediate in the case Γ = Z. Proposition 2.3. The groups G(1 (Γ+ )) and G(1 (Γ− )) are connected. a admits a canonical factorization and hence, Proof. Let a ∈ G(1 (Γ+ )). Trivially, by Theorem 2.1, a ∈ G0 (1 (Γ)). This means that a = exp(b) for some b ∈ 1 (Γ). Writing b = b+ + b− with b+ ∈ 1 (Γ+ ) and b− ∈ 1 (Γ− \ {0}), we have a ∗ exp(−b+ ) = exp(b− ). Since the left-hand side belongs to 1 (Γ+ ) and the right-hand side to e + 1 (Γ− \ {0}), either side equals e and hence a = exp(b+ ). Consequently, a ∈ G0 (1 (Γ+ )). In a similar way we prove that G(1 (Γ− )) is connected.

3. Weighted Wiener Algebras Let us now introduce weighted Wiener algebras on an arbitrary ordered discrete abelian group (Γ, ). An admissible weight β = {βj }j∈Γ is deﬁned as a Γ-indexed sequence of positive numbers βj such that 1 ≤ βi+j ≤ βi βj ,

i, j ∈ Γ.

(3.1)

1β (M )

stand for the complex Banach space of all complexFor a subset M of Γ, let valued M -indexed sequences x = {xj }j∈M having at most countably many nonzero terms for which βj |xj | < ∞. x1,β := j∈M

A decomposition analogous to (2.1) holds, again with the projections having all unit norm. Further, 1β (Γ), 1β (Γ+ ) and 1β (Γ− ) are commutative Banach algebras with unit element. As sets, we clearly have 1β (M ) ⊆ 1 (M ) for every set M ⊆ Γ. We now introduce the weighted Wiener algebra W (G)β = { a : a ∈ 1β (Γ)} which inherits its norm from 1β (Γ) by natural isometry, and its subalgebras a : a ∈ 1β (Γ± )}. (W (G)β )± = {

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Clearly, W (G)β coincides with W (G) if βj ≡ 1. Moreover, due to Gelfand’s theorem, a ∈ G(W (G)β ) if and only if a ∈ W (G)β and 0 ∈ / { a(g) : g ∈ Mβ }, where Mβ stands for the maximal ideal space of W (G)β (or of 1β (Γ)). Here we observe that every multiplicative linear functional φ on 1β (Γ) corresponds to a Γ-indexed sequence of nonzero complex numbers φj such that φi+j = φi φj for all i, j ∈ Γ and supj∈Γ (|φj |/βj ) < +∞, while φ(x) = φj xj , x = {xj }j∈Γ ∈ 1β (Γ). (3.2) j∈Γ

Thus Mβ contains the maximal ideal space G of W (G). Before generalizing Theorem 2.1 to W (G)β , we derive two propositions on the unique extension of functions a ∈ W (G)β to the (generally larger) maximal ideal space Mβ of a weighted Wiener algebra. This justiﬁes our usage of the notation a also for the Gelfand transform of a ∈ 1β (Γ). Proposition 3.1. Let G be a compact abelian group with ordered character group a, b ∈ W (G)β (Γ, ), β = {βj }j∈Γ an admissible weight. Then two functions coincide on Mβ if and only if they coincide on G. Proof. Certainly a, b ∈ W (G). If they coincide on G, then they are Fourier transforms of sequences in 1 (Γ) which must coincide. Since these two sequences also belong to 1β (Γ), we may apply any multiplicative functional on 1β (Γ) and show that a, b coincide on Mβ , as claimed. This easy result generalizes the analytic continuation property for symbols on annuli in the case of Γ = Z. Proposition 3.1 is also true for the corresponding algebra of matrix functions. 1 A similar argument, where the maximal ideal space M± β of β (Γ± ) extends that of 1 (Γ), can be used to prove the following. Proposition 3.2. Let G be a compact abelian group with ordered character group (Γ, ), β = {βj }j∈Γ an admissible weight. Then two functions a, b ∈ (W (G)β )± coincide on M± if and only if they coincide on G. β The next theorem generalizes the main result of [8] from ﬁnitely generated discrete abelian groups Γ to arbitrary discrete ordered abelian groups. It also generalizes Theorem 2.1 to the setting of weighted Wiener algebras. Theorem 3.3. Let G be a compact abelian group with ordered character group (Γ, ), β = {βj }j∈Γ an admissible weight, and a ∈ 1β (Γ). Then the following statements are equivalent: (a) a has a canonical factorization of the form (2.4), where a+ ∈ G(1β (Γ+ )) and a− ∈ G(1β (Γ− )). (b) a ∈ G0 (1β (Γ)). a is zero. (c) a(φ) = 0 for every φ ∈ Mβ and the abstract winding number of

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Proof. The equivalence of (b) and (a) is proved as in Theorem 2.1. To show the equivalence of (b) and (c) we have to show that the natural group homomorphism G(1β (Γ)) G(1 (Γ)) → G0 (1β (Γ)) G0 (1 (Γ)) is an isomorphism. In view of the concrete form of the isomorphism in (2.5) this amounts to proving that the natural group homomorphism Inj : [f ] ∈ π 1 (Mβ ) → [f |G ] ∈ π 1 (G)

(3.3)

is an isomorphism. This natural group-isomorphism assigns to the homotopy class [f ] of a continuous function f : Mβ → T the homotopy class of the restriction of f onto G. (Notice that G ⊆ Mβ , and that f1 |G and f2 |G are homotopic whenever f1 and f2 are homotopic.) For the map Inj to be an isomorphism it is suﬃcient to show that G is a strong deformation retract of Mβ [16]. The latter means that there exists a continuous function (3.4) F : Mβ × [0, 1] → Mβ such that F (φ, 0) = φ for all φ ∈ Mβ , F (g, t) = g for all g ∈ G, t ∈ [0, 1], and F (φ, 1) ∈ G for all φ ∈ Mβ . Indeed, in order to see the suﬃciency consider the map Proj : [h] ∈ π 1 (G) → [ h] ∈ π 1 (Mβ ), h(φ) = h(F (φ, 1)). (Again, notice that if h1 and h2 are homotopic, then h1 and h2 are homotopic, too.) Since for h : G → T we have h|G (g) = h(F (g, 1)) = h(g), g ∈ G, it follows that Inj ◦ Proj = id. On the other hand, each continuous function f : Mβ → T, which equals f (F (φ, 0)), is homotopic to f (F (φ, 1)) with the connecting function f (F (φ, t)), t ∈ [0, 1], φ ∈ Mβ . This implies that Proj◦ Inj = id. Hence Inj is indeed an isomorphism once we have shown that a retracting deformation (3.4) exists. Recall that every multiplicative linear functional φ on 1β (Γ) corresponds to a Γ-indexed sequence of nonzero complex numbers φj such that φi+j = φi φj for all i, j ∈ Γ and |φj | = 1, (3.5) φ := sup j∈Γ βj by means of (3.2). (Every multiplicative linear functional has norm one.) The topology on the set of all such sequences φ = {φj }j∈Γ which is compatible with that on Mβ can be deﬁned as the weakest topology that makes each function (α) φ → φj ∈ C continuous, j ∈ Γ. Indeed, if φ(α) = {φj }j∈Γ is a net converging to φ = {φj }j∈Γ such that the continuity condition just mentioned holds, then (α) lim φj xj = φj xj for each x = {xj }j∈Γ ∈ 1β (Γ) α

j∈Γ

j∈Γ

by (3.5) and by the dominated convergence theorem.

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Clearly, when βi ≡ 1, then the above characterization applies to G in place of Mβ and expresses the fact that G, the group of characters on Γ, can be identiﬁed with the maximal ideal space of 1 (Γ). Notice that condition (3.5) then forces φi to have modulus one. We now deﬁne the mapping F as follows: F (φ, t) = ψ t

with

ψit =

φi |φi |t

(3.6)

It is easy to see that ψ t is a Γ-indexed sequence enjoying the multiplicativity property and (3.5). Moreover, ψ 0 = φ, ψ 1 ∈ G (since |ψi1 | = 1), and ψ t ∈ G whenever φ ∈ G (i.e., |φi | = 1). The continuity of F can be seen as follows. Let φ(α) be a net of such sequences converging φ, and consider a net tα of numbers in (α) [0, 1] converging to t. Then φi → φi for each i. Hence (α)

φi

(α) |φi |t(α)

→

φi |φi |t

for each i ∈ Γ, which means nothing but lim F (φ(α) , t(α) ) = F (φ, t). α

Elaborating on the maximal ideal space Mβ (= space of all multiplicative Γ-index sequences φ = {φj }j∈Γ satisfying (3.5)) it is easy to show that Mβ ∼ = Mβ,pos × G,

(3.7)

both topologically and algebraically, where Mβ,pos stands for the subset of all real positive valued sequences satisfying (3.5) and G can be identiﬁed with the set of all unimodular valued sequences. The multiplication of two such sequences is deﬁned pointwise, i.e., (φψ)j = φj ψj , j ∈ Γ. The set Mβ,pos can be shown to be a convex subset of the set M of all multiplicative nonzero Γ-indexed sequences φ not necessarily satisfying (3.5). The set M becomes a topological vector space when introducing the algebraic operations as (φ + ψ)i := φi ψi , (λφ)i = (φi )λ and the topology as the weakest topology that makes each mapping φ ∈ M → φj ∈ C \ {0}, j ∈ Γ, continuous. In fact, Mβ,pos is contractible to the neutral element in M, namely φ ≡ 1. The contracting deformation is given by F (φ, λ) = λφ (with the scalar multiplication as deﬁned above). This provides another argument for the fact that G is the strong deformation retract of Mβ . Moreover, this fact generalizes to some extent the explicit characterization of the maximal ideal space Mβ of 1β (Γ) given in [8] for the case Γ = Zd , G = Td (see also [11] if d = 1). Using the map z i xi ∈ Mβ , z = (z1 , . . . , zd ) ∈ Cd → x ∈ 1 (Γ) → i∈Zd

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(where z i = z1i1 · · · zdid ), Mβ corresponds exactly to the set |z d | <∞ Ωβ = z ∈ Cd : sup i∈Zd βi

= (t1 eξ1 , . . . , td eξd ) : (t1 , . . . , td ) ∈ Td , (ξ1 , . . . , ξd ) ∈ Kβ ,

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(3.8)

where Kβ is the compact convex subset of Rd given by Kβ = y ∈ Rd : sup (i, y − log(βi )) < ∞ . i∈Zd

In this case (3.7) has the form Mβ exp(Kβ ) × Td , both topologically and algebraically. Finally, let us mention the analogue of Corollary 2.2 in the weighted case. Corollary 3.4. Let G be a compact abelian group with ordered character group a(φ) = 0 for every (Γ, ), β = {βj }j∈Γ an admissible weight, and a ∈ 1β (Γ). If φ ∈ Mβ , then a admits a factorization a− (g), a(g) = a+ (g)ej (g) where a+ ∈ of a.

G(1β (Γ+ )),

a− ∈

G(1β (Γ− )),

g ∈ G,

(3.9)

and j ∈ Γ is the abstract winding number

Proof. As in the proof of Corollary 2.2 we can write a(g) = ej (g)b(g) with b ∈ G0 (1 (Γ)) and j ∈ Γ being the abstract winding number. Clearly, b ∈ G(1β (Γ)) and the abstract winding number of b is zero. It remains to apply Theorem 3.3.

4. Weighted Wiener Algebras of Matrix Valued Functions If A is a commutative Banach algebra, we denote by An×n the Banach algebra of n×n matrices with entries in A. Invertibility in the weighted algebra (W (G)β )n×n is characterized in terms of pointwise invertibility, as immediately follows from the natural matrix generalization of Gelfand’s theorem: Proposition 4.1. Let G be a compact abelian group with character group Γ, and let (W (G)β )n×n be the corresponding Wiener algebra of n × n matrix functions, ∈ G((W (G)β )n×n ) if and only if A(g) ∈ where the weight β satisfies (3.1). Then A n×n ) for every g ∈ Mβ , where Mβ is the maximal ideal space of W (G)β . G(C The concept of factorization as in Proposition 2.2 extends to n × n matrix ∈ (W (G)β )n×n is a reprefunctions in (W (G)β )n×n . A (left) factorization of A sentation of the form + (g) (diag (ej1 (g), . . . , ejn (g))) A − (g), g ∈ G, A(g) =A (4.1) − ∈ G((W (G)− )n×n ), and j1 , . . . , jn ∈ Γ. + ∈ G((W (G)+ )n×n ), A where A β β ∈ (W (G)β )n×n has the left factorization (4.1), where We remark that if A g ∈ G, then (4.1) holds automatically for all g ∈ Mβ , the maximal ideal space

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of W (G)β . Indeed, since (4.1) obviously is a left factorization in W (G)n×n , each factor is the Fourier transform of a Γ-indexed sequence of n × n matrices belonging to (1 (Γ))n×n . Since each sequence in fact belongs to (1β (Γ))n×n , we may apply any multiplicative functional to either side of the equation obtained from (4.1) by restricting it to the (i, j)-element and conclude, using Proposition 3.1, that (4.1) holds for each g ∈ Mβ . Proposition 4.2. If A(g) ∈ (W (G)β )n×n admits a factorization, then the elements jk are uniquely defined (if ordered j1 j2 . . . jn ). The elements j1 , . . . , jn in (4.1) are called the (left) factorization indices of A. For Γ = Z and βj ≡ 1 Proposition 4.2 is a classical result (see [10, Theorem VIII.1.1]). The same method can be used to prove Proposition 4.2, and this was done in [26] in the context of almost periodic matrix functions of several variables. We omit further details. ∈ (W (G)β )n×n we mean a repreAnalogously, by a right factorization of A sentation of the form − (g) (diag (ej1 (g), . . . , ejn (g))) A + (g), g ∈ G, A(g) =A (4.2) − ∈ G((W (G)− )n×n ), and j1 , . . . , jn ∈ Γ. We can + ∈ G((W (G)+ )n×n ), A where A β β prove as above that (4.2) in fact holds for g ∈ Mβ , the maximal ideal space of W (G)β . Unless stated otherwise, all notions involving factorization will pertain to left factorization. If all factorization indices are zero, the factorization is called canonical. If a exists, the function A is called factorable. For Γ = Z, G the unit factorization of A circle, and βj ≡ 1, the deﬁnitions and results are classical [14, 10, 6]. Many of these results have been generalized to unweighted Wiener algebras for the cases when Γ = Rk (see [4] and references there) and Γ a subgroup of Rk (see [25, 26]). The factorability and nonfactorability of certain block triangular matrix functions has been studied in [22, 21], generalizing respective results for Γ = R from [17, 18, 3], see also [4]. Wiener-Hopf factorization is hereditary (in the terminology of [26]) with respect to subgroups of Γ, with the induced order: ∈ (W (G)β )n×n be such that the Theorem 4.3. Let Γ be a subgroup of Γ, and let A admits a Wiener-Hopf factorization is contained in Γ . If A Fourier spectrum of A (4.1), then the factorization indices belong to Γ , and there exists a Wiener-Hopf ± ∈ G((W (G)± )n×n ) and their inverses factorization (4.1) in which the factors A β have their Fourier spectrum also contained in Γ . admits a canonical Wiener-Hopf factorization, then the In particular, if A Fourier spectra of its factors and of the inverses of the factors belong to Γ . Proof. Since a Wiener-Hopf factorization in the weighted Wiener algebra is automatically a Wiener-Hopf factorization in the corresponding unweighted Wiener

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algebra, it follows from [24, Theorem 1] that the factorization indices of (4.1) belong to Γ . Now repeat the arguments from the proof of [24, Theorem 3], working with the weighted Wiener algebra rather than with the unweighted one as in [24]. The statement about the canonical factorization follows from the uniqueness of the canonical factorization up to a constant invertible multiplier. The following result relates canonical factorizations and the connected component of the identity in the group G((1β (Γ))n×n ). Theorem 4.4. Let G be a compact abelian group with ordered character group (Γ, ), β = {βj }j∈Γ an admissible weight on Γ, Mβ the maximal ideal space of 1β (Γ), and a ∈ (1β (Γ))n×n . (a) If a ∈ G0 ((1β (Γ))n×n ) then a(g) is invertible for every g ∈ Mβ and the winding number of det a is zero. (b) If a has a canonical factorization with factors and their inverses belonging to (W (G)β )n×n , then a ∈ G0 ((1β (Γ))n×n ). Before proving Theorem 4.4 we recall some topology [16]. For any two topological spaces X and Y we denote by [X, Y ] the set of homotopy classes of continuous maps f : X → Y . We say that f0 , f1 : X → Y are homotopic if there exists a continuous map F : [0, 1] × X → Y such that F (0, z) = f0 (z) and F (1, z) = f1 (z). We deviate here from the usual deﬁnition, which requires ﬁxing base points x0 ∈ X, y0 ∈ Y0 and imposing the additional assumption that f (x0 ) = y0 , at the expense that [X, Y ] is not necessarily a group (cf. [16], where setups with and without ﬁxed base points are presented). Proof. For part (a) observe that the invertibility of a(g) is obvious, and the statement concerning the winding number follows from Theorem 3.3. To prove part (b), let us assume that a has a canonical factorization in (W (G)β )n×n . Then, using the argument for proving the implication (a) =⇒ (b) in the proof of Theorem 2.1, we connect a to another element a1 within (W (G)β )n×n which has ﬁnite Fourier spectrum. Then, replacing Γ by a ﬁnitely generated group containing the Fourier spectrum of a1 , in view of Theorem 4.3 we may assume that G Td and Γ Zd . Letting Γ Zd and G Td , we denote by M+ β the maximal ideal space 1 d of β (Γ+ ), which can be identiﬁed with a compact subset Ω+ β of C [8, Theorem 5.2]. Recall the generalized Arens’ theorem [33, 1] according to which for each commutative Banach unital algebra A the quotient group G(An×n )/G0 (An×n ) depends (up to an isomorphism) only on the maximal ideal space of A. Using the + obvious result that M+ β is the maximal ideal space of C(Mβ ), we have the group isomorphism n×n G(1 (Γ+ )n×n ) ) G(C(M+ β β) n×n + n×n . 1 G0 ( (Γ+ )β ) G0 (C(Mβ ) )

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Moreover, we have n×n G(C(M+ ) β)

n×n ) G0 (C(M+ β)

[M+ β , GL(C, n)].

This follows from the fact (see [7, Theorem 2.18], generalized to matrix valued n×n n×n functions) that f1 , f2 ∈ G(C(M+ ) are equivalent modulo G0 (C(M+ ) β) β) −1 if and only if f1 f2 is path-connected to the identity element e of the group n×n G(C(M+ ), which means that f1 f2−1 and e are homotopic. Clearly, this is β) equivalent to f1 and f2 being homotopic (as deﬁned in the paragraph after Theorem 4.4; see [7, Def. 2.17 and Th. 2.18]). In [8, Corollary 5.3] it is proved that M+ β is contractible to the trivial multiplicative functional φ0 on 1 (Γ+ ) that sends a = {aj }j∈Γ+ ∈ 1 (Γ+ ) to a0 . This + means that there exists a continuous function F : [0, 1] × M+ β → Mβ such that + F (0, z) = z and F (1, z) = φ0 . Now given a representative f : Mβ → GL(C, n) of some homotopy class belonging to [M+ β , GL(C, n)] we deﬁne fr (z) = f (F (r, z)), 0 ≤ r ≤ 1, which implies that f is homotopic to the constant map f1 (z) = f (φ0 ). Because GL(C, n) is arcwise connected, all constant maps are homotopic to each other. This proves that [M+ β , GL(C, n)] is trivial. In the same way we prove that G(1β (Γ− )n×n )

G0 (1β (Γ− )n×n )

[M− β , GL(C, n)],

1 where M− β is the space of multiplicative functionals on β (Γ− ), and that this group is trivial. We have proved that the groups G(1β (Γ± )n×n ) are connected. Now argue as in the proof of Proposition 2.3 to complete the proof.

5. Canonical Factorization and Toeplitz Operators In the classical case (G = T, βj ≡ 1) Wiener-Hopf factorization and Toeplitz operators are closely related to each other, see, for example, [10]. This relationship can be generalized to a more abstract context, e.g., Toeplitz operators acting on (1 (Γ+ ))n , the space of column vectors with entries in 1β (Γ+ ). As before the setting is that of a compact abelian group G with ordered character group (Γ, ) and an admissible weight β. The symbols of such Toeplitz operators are supposed to belong to (W (G)β )n×n (or, equivalently, to (1β (Γ))n×n ). Given A ∈ (1β (Γ))n×n we deﬁne TA by TA x = y, Ai−j xj = yi , i ∈ Γ+ , (5.1) j∈Γ+

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where x = {xi }i∈Γ and y = {yi }i∈Γ . This operator is well-deﬁned and bounded on (1β (Γ+ ))n since β is an admissible weight and the norm can be estimated by (W (G) )n×n . TA ≤ A β

(5.2)

This can be seen from the inequality βi yi ≤ βi−j βj Ai−j · xj ≤ βi Ai βj xj . i∈Γ+

i,j∈Γ+

i∈Γ

j∈Γ+

If A, B ∈ (1β (Γ))n×n , then y = TAB x − TA TB x evaluates to Ai−j Bj−k xk , i ∈ Γ+ , yi = j≺0 k0

which vanishes if the corresponding entries of either A or B vanish. More precisely, we have the formula TAB = TA TB (5.3) if A ∈ (1β (Γ− ))n×n or B ∈ (1β (Γ+ ))n×n . From this it follows easily that if A ∈ G(1β (Γ− ))n×n or A ∈ G(1β (Γ+ ))n×n , then the Toeplitz operator TA is invertible and the inverse is given by (TA )−1 = TA−1 . (5.4) 1 n×n possesses a right canonical factorand A Now assume that A ∈ (β (Γ)) ization A(g) = A− (g)A+ (g). Then (5.3) implies the factorization TA = TA− TA+ from which we can conclude that TA is invertible and its inverse is given by (TA )−1 = TA−1 TA−1 . +

(5.5)

−

∈ (W (G)β )n×n implies the The right canonical factorization of a symbol A invertibility of yet another Toeplitz operator. Given A ∈ (1β (Γ))n×n , we deﬁne A∗ = {AT−j }j∈Γ ,

where A = {Aj }j∈Γ

(5.6) ∗

AT−j refers n×n

to the matrix transpose of the matrix A−j . Clearly, A belongs to and 1 with the underlying weight being deﬁned by β∗ = {β−j }j∈Γ where (β∗ (Γ)) β = {βj }j∈Γ . − (g)A + (g) is a right canonical = A Indeed, if A ∈ (1β (Γ))n×n and A(g) factorization, then ∗ (g) = A ∗ (g)A ∗ (g) A + − is also a right canonical factorization. Hence the Toeplitz operator TA∗ acting on (1β∗ (Γ+ ))n is invertible and its inverse is given by (TA∗ )−1 = T(A∗− )−1 T(A∗+ )−1 .

(5.7)

Let us remark that the deﬁning equation for the Toeplitz operator TA∗ , TA∗ x = y, ATj−i xj = yi , i ∈ Γ+ , (5.8) j∈Γ+

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can be equivalently written as

i ∈ Γ+

uj Aj−i = vi , ,

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(5.9)

j∈Γ+

when passing to the transpose. Therein, uT = {uTj }j∈Γ+ and v T = {vjT }j∈Γ+ belong to (1β∗ (Γ+ ))n . The following result is well-known for G = T (see [10]) if βj ≡ 1. It represents to some extent the converse of the observations just made that the canonical implies the (unique) solvability of the Wiener-Hopf factorization of a symbol A equations (5.1) and (5.9). In fact, the solutions to these equations for particular right hand sides allow the construction of the right canonical Wiener-Hopf factorization. Let δi,j stand for the Kronecker symbol and In for the n × n identity matrix. Theorem 5.1. Let G be a compact abelian group with ordered character group (Γ, ), let β = {βj }j∈Γ be a Γ-indexed sequence of positive numbers satisfying (3.1), and let A ∈ (1β (Γ))n×n . If the convolution equations Ai−j Xj = δi,0 In , i ∈ Γ+ , (5.10) j∈Γ+

and

i ∈ Γ+ ,

Uj Aj−i = δi,0 In ,

j∈Γ+

each have a solution such that two conditions

i∈Γ+ (βi Xi

(5.11)

+ β−i Ui ) < ∞, and either of the

(a) det(X0 ) = 0, equivalently, det(U0 ) = 0, or (b) A ∈ G(1β (Γ))n×n ∈ (W (G)β )n×n has a right canonical factorization with factors is fulfilled, then A and their inverses belonging to (W (G)β )n×n . Before we give the proof, let us remark that if (5.10) and (5.11) hold, then necessarily X0 = U0 since Ui Ai−j Xj = U0 . X0 = i,j0

Proof. Given the solutions of the equations (5.10) and (5.11) we can deﬁne Y ∈ (1β (Γ− ))n×n and V ∈ (1β∗ (Γ− ))n×n ) by Yi = Ai−j Xj , i ∈ Γ, (5.12) j∈Γ+

Vi =

j∈Γ+

Uj Aj−i ,

i ∈ Γ.

(5.13)

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Clearly, Yi = Vi = 0 for i 0, and (5.12) represents a convolution of two sequences belonging to (1β (Γ))n×n while (5.13) represents the convolution of two sequences belonging to (1β∗ (Γ)))n×n , namely {Uj }j∈Γ+ and {A−j }j∈Γ . The ﬁrst equation can be rewritten in terms of the corresponding symbols as g ∈ Mβ , j, gYj = A(g) j, gXj , j0

j0

while the second one turns into j, gV−j = j, gU−j A(g), j0

g ∈ Mβ .

(5.14)

j0

All of the symbols encountered here belong to (W (G)β )n×n , while the previous two equations, which obviously hold for g ∈ G, are easily seen to be true for g ∈ Mβ as well (cf. Proposition 3.1). Using obvious abbreviations we rewrite (5.14) as X + (g), Y− (g) = A(g)

− (g)A(g) V+ (g) = U

(5.15)

and conclude that + (g) = U − (g)A(g) X + (g) = U − (g)Y− (g). V+ (g)X Inspecting the Fourier spectra of the products on either side it follows that they must be constant, and since V0 = Y0 = In we obtain that they are equal to X0 = U0 . Hence + (g) = U − (g)A(g) X + (g) = U − (g)Y− (g) = U0 . X0 = V+ (g)X (5.16) + ∈ G(W (G)β )n×n (a) If we assume that X0 = U0 is non-singular, then X + − ∈ G(W (G)β )n×n with and U − −1 + X (g) = X0−1 V+ (g),

−1 − U (g) = Y− (g)U0−1 ,

and the right canonical factorization is given by −1 (g). ˆ −1 (g)X0 X A(g) =U −

+

∈ (b) If we assume that A(g) is invertible on all of Mβ , then clearly det A as in GW (G)β ⊆ GW (G) and we can deﬁne the winding number γ ∈ Γ of det A Section 2. By Theorem 3.3 we have a factorization det A(g) = a− (g)eγ (g) a+ (g), g ∈ Mβ , (5.17) with a± ∈ G(W (G)β )± . Taking determinants in (5.15) and using this factorization we obtain + (g), a+ (g) det X (5.18) a− (g)−1 det Y− (g) = eγ (g) − (g) a+ (g)−1 = det U a− (g)eγ (g). det V+ (g)

(5.19)

This implies γ = 0 because otherwise in one of these equations the left and right hand sides must vanish identically, which contradicts the fact that [det V+ ]0 = det V0 = 1 and [det Y− ]0 = det Y0 = 1. Here [. . . ]0 stands for the zero-th Fourier

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coeﬃcient of the underlying function. We use the fact that the map c → [ c]0 is a multiplicative linear functional on both of (W (G)β )± . Thus, (5.17) is in fact a canonical factorization. With γ = 0 it follows that both sides of equations (5.18) and (5.19) must be constants, and for the same reasons as just pointed out these constants must + (g) and det U − (g) are nonzero on all of Mβ , which by be nonzero. Hence det X (5.16) implies that det X0 = det U0 = 0. Hence this case is reduced to case (a). ∈ (W (G)β )n×n possesses a right canonical factorization Suppose that A − (g)A + (g). Then the zero-th Fourier coeﬃcients of A − and A + must A(g) = A be non-singular matrices, which can be pulled out from those factors. Thus one arrives at what might be called a normalized right canonical factorization, − (g)C A+ (g), A(g) =A ± ∈ G(W (G)β )n×n with [A± ]0 = In and C ∈ Cn×n is invertible. The where A signiﬁcance of this normalized representation is that both left and right factors as well as C are uniquely determined. Moreover, the solutions of the equations (5.10) and (5.11) are given by −1 (g)C −1 , + (g) = A X +

−1 (g), − (g) = C −1 A U −

as follows easily from (5.15). In particular, X0 = C −1 = U0 , and hence condition (a) in Theorem 5.1 is also necessary for the existence of a right canonical factorization. In the case n = 1, the constant C can be interpreted as the geometric mean a well-known notion in the almost periodic case (see [4, Chapter 3]). In fact, of A, if a ∈ G0 (W (G)β ), then C = exp([log a]0 ). The results of this section allow us to easily obtain the continuity property of canonical factorizations. A proof of this property is known in many particular situations (see, for example, [27] for the case of almost periodic functions of several variables), and is presented here for completeness. It will be convenient to work with normalized right canonical factorizations, and with analogously deﬁned normalized left canonical factorizations, in the next theorem. ∈ (W (G)β )n×n which have a left Theorem 5.2. The set of all matrix functions A (resp., a right) canonical factorization with factors and their inverses belonging to (W (G)β )n×n , is open in (W (G)β )n×n . Further, the factors in a normalized left in (W (G)β )n×n . (resp., right) canonical factorization depend continuously on A Proof. Using the remarks made at the beginning of this section, namely, that existence of a right canonical factorization implies invertibility of the corresponding Toeplitz operator, and using Theorem 5.1, the result of Theorem 5.2 follows from the well-known continuity of inversion of invertible bounded operators in a Banach space.

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A diﬀerent proof of Theorem 5.2 may be given using the properties of decomposing algebras (see [5, 10]); note that (W (G)β )n×n is a decomposing algebra.

6. Wiener-Hopf Equivalence B ∈ (W (G)β )n×n are called left Wiener-Hopf equivalent Two matrix functions A, − ∈ G((W (G)− )n×n ) such that ) and C if there exist C+ ∈ G((W (G)+ )n×n β β + (g)B(g) C − (g), A(g) =C

g ∈ G.

(6.1)

It is easily seen that left Wiener-Hopf equivalence is indeed an equivalence relation on G((W (G)β )n×n ). Similarly, we deﬁne right Wiener-Hopf equivalence. Clearly, either notion depends essentially on the weight. For n = 1 the notions of left and right Wiener-Hopf equivalence obviously coincide, but this is not the case for n ≥ 2. The concept of Wiener-Hopf equivalence has been introduced and studied in [2, 13, 12] in the context of operator polynomials and analytic operator valued functions. For Γ = R the notion of Wiener-Hopf equivalence was implicitly discussed in [20, Section 2.3]. Observe that the Portuguese transformation, introduced in the setting of Γ = R in [3], christened in [4] and then further used in the setting of ordered abelian groups in [22], is in fact a convenient tool for establishing Wiener-Hopf equivalence of some block triangular matrix functions. B ∈ G((W (T))n×n ) are left WienerFor Γ = Z, two matrix functions A, Hopf equivalent if and only if up to rearrangement they have the same left partial indices, as it easily follows from the classical results of [10, 14]. For n = 1 it is clear from Theorem 2.2 that two nowhere zero scalar functions a, b ∈ W (G)β are (left and right) Wiener-Hopf equivalent if and only if they have the same (abstract) winding number. The following result serves primarily to illustrate the scope of the WienerHopf equivalence problem. We only state and prove it in the left Wiener-Hopf equivalence case. The right Wiener-Hopf equivalence class follows by reversing the order of the character group. Proposition 6.1. Let G be a compact abelian group with ordered character group (Γ, ) and β = {βj }j∈Γ an admissible weight. Then the left (or, right, resp.) Wiener-Hopf equivalence classes containing at least one diagonal n × n matrix ) are completely specified by the elements of the set function in G((W (G))n×n β {(γ1 , . . . , γd ) ∈ Γn : γ1 . . . γn } .

(6.2)

Above we have used the term “factorable” for those matrix functions which are left Wiener-Hopf equivalent to a diagonal matrix with entries ej . This proposition characterizes the left Wiener-Hopf equivalence classes of the factorable matrix functions.

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Proof. Given a left Wiener-Hopf equivalence class, let diag( a1 , . . . , an ), with a1 , . . . , an ∈ W (G)β , be one of its elements. Let us denote the (abstract) winding number of as by γ s (s = 1, . . . , n). Then the above element is obviously left Wiener-Hopf equivalent to diag(eγ1 , . . . , eγn ), where (γ1 , . . . , γn ) is the rearrangement of the n-tuple ( γ1 , . . . , γ n ) that satisﬁes γ1 . . . γn . Since, by Proposition 4.2, the diagonal factor in (4.1) is uniquely determined by the left Wiener-Hopf equivalence class up to rearrangement of diagonal entries, there exists a one-to-one correspondence between the left Wiener-Hopf equivalence classes containing at least one diagonal ) and the elements of the set given in (6.2). matrix function in G((W (G))n×n β ∈ G((W (G)β )n×n ) which are It is possible to construct matrix functions A not left Wiener-Hopf equivalent to a diagonal matrix function. Such examples have been constructed in the following cases: (1) Γ = R, n ≥ 2, and βj ≡ 1 (see [18] and the book [4]), (2) Γ not isomorphic to a subgroup of the additive group of the rational numbers, n ≥ 2, and βj ≡ 1 (see [22]). We mention here the case when

eλ 0 = A , (6.3) c−1 e−ν + c0 + c1 eα e−λ where α, ν ∈ Γ = R are positive with an irrational ratio, λ = α + ν, and the coeﬃcients cj ∈ C are such that |c−1 |α |c1 |ν = |c0 |λ = 0. According to [19], the Wiener-Hopf equivalence class of (6.3) does not contain any diagonal matrix functions at all. On the other hand, the proof of this result given in [19] (see also [4]) implies that this equivalence class contains a sequence of j of the same type (6.3) with the diagonal exponents triangular matrix functions A λj going to zero. One reduction of the problem is the following. A left Wiener-Hopf equivs ∈ alence class of functions in (W (G)β )n×n is called reducible if there exist A ns ×ns (W (G)β ) (s = 1, 2) with n1 , n2 ∈ N and n1 + n2 = n such that the di1 + 2 is contained in the class. It is then suﬃcient to characterize ˙A rect sum A the irreducible classes varying the matrix order n. For Γ = Z and n ≥ 2 all left Wiener-Hopf equivalence classes of everywhere invertible matrix functions are reducible. Obviously, the example of [18] belongs to an irreducible class. A suﬃcient is reducibility condition, in terms of the Toeplitz operators associated with eα A, given in [20, Theorem 2.6].

References [1] R. Arens, The group of invertible elements of a commutative Banach algebra, Studia Math. Zeszyt (Ser. Specjalna) 1, 21–23 (1963).

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[2] H. Bart, I. Gohberg, and M.A. Kaashoek, Invariants for Wiener-Hopf equivalence of analytic operator functions. In: I. Gohberg and M.A. Kaashoek (eds.), Constructive methods of Wiener-Hopf factorization, Birkh¨ auser OT 21, Basel and Boston, 1986; pp. 317–355. [3] M.A. Bastos, Yu.L. Karlovich, F.A. dos Santos, and P.M. Tishin, The Corona theorem and the existence of canonical factorization of triangular AP-matrix functions, J. Math. Anal. Appl. 223, 494–522 (1998). [4] A. B¨ ottcher, Yu.I. Karlovich, and I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkh¨ auser OT 131, Basel and Boston, 2002. [5] M.S. Budjanu and I.C. Gohberg, The factorization problem in abstract Banach algebras, Amer. Math. Soc. Transl. 110, 107–123 (1977). (Translation from Russian.) [6] K.F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkh¨ auser OT 3, Basel and Boston, 1981. [7] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Second Ed., Graduate Texts in Math. 179, Springer, Berlin etc., 1998. [8] T. Ehrhardt and C. van der Mee, Canonical factorization of continuous functions on the d-torus, Proc. Amer. Math. Soc. 131, 801–813 (2002). [9] I. Gelfand, D. Raikov, and G. Shilov, Commutative Normed Rings, Chelsea, Bronx, New York, 1964. (Translation from Russian). [10] I.C. Gohberg and I.A. Feldman, Convolution Equations and Projection Methods for their Solution, Transl. Math. Monographs 41, Amer. Math. Soc., Providence, R.I., 1974. [11] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Vol II, Birkh¨ auser OT 63, Basel and Boston, 1993. [12] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Partially Speciﬁed Matrices and Operators: Classiﬁcation, Completion, Applications, Birkh¨ auser OT 79, Basel and Boston, 1995. [13] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Similarity of operator blocks and canonical forms. II. Inﬁnite-dimensional case and Wiener-Hopf factorization. In: C. Apostol, R.G. Douglas, and B. Sz.-Nagy (eds.), Topics in Modern Operator Theory, Birkh¨ auser OT 2, Basel and Boston, 1981; pp. 121–170. [14] I.C. Gohberg and M.G. Krein, Systems of integral equations on a half line with kernels depending on the diﬀerence of arguments, Amer. Math. Soc. Transl. (2)14, 217–287 (1960). [15] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99, 165–202 (1958); part II: Acta Math. 106, 175–213 (1961). [16] S.-T. Hu, Homotopy Theory, Pure and Applied Mathematics 8, Academic Press, New York, 1959. [17] Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions and (semi) Fredholmness of some convolution type equations, VINITI, Moskow, 1985. (Russian). [18] Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix valued functions and the Noether theory for certain classes of equations of convolution type, Izv. Akad. Nauk. SSSR 53, 276–308 (1989) (Russian); English translation: Math. USSR - Izv. 34, 281–316 (1990).

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[19] Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions, J. Math. Anal. Appl. 193, 209–232 (1995). [20] Yu.I. Karlovich and I.M. Spitkovsky, Semi-Fredholm properties of certain singular integral operators. In: A. B¨ ottcher and I. Gohberg (eds.), Singular Integral Operators and Related Topics, Birkh¨ auser OT 90, Basel and Boston, 1996, pp. 264–287. [21] C.V.M. van der Mee, L. Rodman, and I.M. Spitkovsky, Factorization of block triangular matrix functions with oﬀ-diagonal binomials. In: M.A. Kaashoek, C. van der Mee, and S. Seatzu (eds.), Recent Advances in Operator Theory and its Applications, Birkh¨ auser OT 160, Basel and Boston, 2005; pp. 425–439. [22] C.V.M. van der Mee, L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Factorization of block triangular matrix functions in Wiener algebras on ordered abelian groups. In: J.A. Ball, J.W. Helton, M. Klaus, and L. Rodman (eds.), Current Trends in Operator Theory and its Applications, Birkh¨ auser OT 149, Basel and Boston, 2004, pp. 441–465. [23] A.I. Perov and A.V. Kibenko, A theorem on the argument of an almost periodic function of several variables, Litovskii Matematicheskii Sbornik 7, 505–508 (1967). (Russian) [24] L. Rodman and I.M. Spitkovsky, Factorization of matrix functions with subgroup supported Fourier coeﬃcients, J. Math. Anal. Appl., 323, 604–613 (2006). [25] L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Carath´eodory-Toeplitz and Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc. 350, 2185–2227 (1998). [26] L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Noncanonical factorizations of almost periodic multivariable matrix functions. In: Birkh¨auser OT 142, Basel and Boston, 2003, pp. 311–344. [27] L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Factorization of almost periodic matrix functions of several variables and Toeplitz operators. In: H. Bart, I. Gohberg, and A.C.M. Ran (eds.), Operator Theory and Analysis, Birkh¨ auser OT 122, Basel and Boston, 2001, pp. 385–416. [28] H.L. Royden, Function algebras, Bull. Amer. Math. Soc. 69, 281–298 (1963). [29] W. Rudin, Fourier Analysis on Groups, John Wiley, New York, 1962. [30] I.M. Spitkovsky, On the factorization of almost periodic functions, Mathematical Notes 45, 482–488 (1989). [31] I.M. Spitkovsky and H.J. Woerdeman, The Carath´eodory-Toeplitz problem for almost periodic functions, J. Func. Anal. 115, 281–293 (1993). [32] J.T. Taylor, Measure Algebras, CBMS Regional Conf. Series in Math 16, Amer. Math. Soc., Providence, R.I., 1973. [33] J.T. Taylor, Banach algebras and topology. In: Algebras in Analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973), Academic Press, London, 1975, pp. 118–186.

Torsten Ehrhardt Dept. of Mathematics University of California Santa Cruz, CA 95064 USA e-mail: [email protected]

86

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Cornelis van der Mee Dip. Matematica e Informatica Universit` a di Cagliari Viale Merello 92 09123 Cagliari Italy e-mail: [email protected] Leiba Rodman Dept. of Mathematics The College of William and Mary Williamsburg, VA 23187-8795 USA e-mail: [email protected] Ilya M. Spitkovsky Dept. of Mathematics The College of William and Mary Williamsburg, VA 23187-8795 USA e-mail: [email protected] Submitted: May 15, 2006 Revised: December 20, 2006

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Integr. equ. oper. theory 58 (2007), 87–98 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010087-12, published online December 26, 2006 DOI 10.1007/s00020-006-1475-8

Integral Equations and Operator Theory

Hwp −Lpw Boundedness of Marcinkiewicz Integral Chin-Cheng Lin and Ying-Chieh Lin Abstract. The Marcinkiewicz integral is essentially a Littlewood-Paley gfunction, which plays a very important role in harmonic analysis. In this paper p − Lpw we give weaker smoothness conditions assumed on Ω to imply the Hw boundedness of the Marcinkiewicz integral operator µΩ , where w belongs to the Muckenhoupt weight class. Mathematics Subject Classification (2000). 42B20, 42B30. Keywords. Ap weights, Dini-type condition, Marcinkiewicz integral, weighted Hardy spaces.

1. Introduction Let S n−1 denote the unit sphere in Rn (n ≥ 2) and dσ be the Lebesgue measure on S n−1 . Let Ω be a homogeneous function of degree zero on Rn which is locally integrable and satisﬁes Ω(x )dσ(x ) = 0,

(1.1)

S n−1

where x = x/|x| for any x = 0. For a function f on Rn , the Marcinkiewicz integral operator µΩ is deﬁned by ∞ 1/2 2 dt µΩ (f )(x) = |FΩ,t (f, x)| 3 , t 0 where Ω(x − y) FΩ,t (f, x) = f (y)dy. n−1 |x−y|≤t |x − y| The operator µΩ was originally introduced by Marcinkiewicz [11] in 1938 for n = 1 and Ω(t) = sign t. In 1958, Stein [12] deﬁned the Marcinkiewicz integral of higher dimensions and proved that if Ω ∈ Lipα (S n−1 ), 0 < α ≤ 1, then µΩ is of type (p, p) for 1 < p ≤ 2 and of weak type (1, 1). In 1962, Benedek, Calder´ on and Research supported by National Science Council, Republic of China under Grant #NSC 95-2115M-008-004.

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Panzone [1] showed that if Ω ∈ C 1 (S n−1 ), then µΩ is of type (p, p) for 1 < p < ∞. In 1990, Torchinsky and Wang [14] proved that if Ω ∈ Lipα (S n−1 ), 0 < α ≤ 1, then, for 1 < p < ∞ and w ∈ Ap (the Muckenhoupt weight class), µΩ is bounded on Lpw . It is worth pointing out that the results mentioned above were obtained when Ω satisﬁes some smoothness conditions. In 1999, Ding, Fan, and Pan [3] improved Torchinsky and Wang’s result by ridding of the smoothness condition assumed on Ω.

Theorem 1.1. Let Ω ∈ Lq (S n−1 ), 1 < q < ∞. If wq ∈ Ap , 1 0 independent of f such that µΩ (f )Lpw ≤ Cf Lpw . As to the H p − Lp boundedness, recently Ding, Lu, and Xue [5] showed that if Ω satisﬁes the L1 -Dini condition, then µΩ is bounded from H 1 to L1 . Later on Ding, Lee, and Lin [4] extended to the weighted case.

Theorem 1.2. Let Ω satisfy the Lq -Dini condition for q > 1. If wq ∈ A1 , then there exists a constant C > 0 independent of f such that µΩ (f )L1w ≤ Cf Hw1 . n Theorem 1.3. Let 0 < α ≤ 1, β = min{α, 1/2}, and n+β 0 independent of f such that n µΩ (f )Lpw ≤ Cf Hwp .

In this article, we will show that under weaker smoothness conditions assumed on Ω, which is called Dinqα (S n−1 ) and will be deﬁned in the next section, the Marcinkiewicz integral operator µΩ is bounded from Hwp to Lpw . Theorem 1.4. Let 0 < α ≤ 1, β = min{α, 12 }, and Ω ∈ Lq (S n−1 ) ∩ Din1α (S n−1 ) for 1 < q ≤ ∞. If (a) 1 < q ≤ (b)

1 p

1 p

and wq ∈ A

pβ n(1−p)

n n+β

 0 independent of f such that µΩ (f )Lpw ≤ Cf Hwp . Theorem 1.5. Let 0 < α ≤ 1, β = min{α, 12 }, and Ω∈

Dinqα (S n−1 )

for 1 < q < ∞. If w

q

n n+β

 0 independent of f such that µΩ (f )Lpw ≤ Cf Hwp .

Remark 1.6. It is worthy noting that Theorem 1.2 can be regarded as the limit case of Theorem 1.5 by choosing p = 1 and letting α → 0. We do have a substantial improvement of Theorems 1.3 and 1.2. n 0 independent Ω ∈ Dinα (S n of f such that µΩ (f )Lpw ≤ Cf Hwp .

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Theorem 1.8. Let Ω ∈ Lq (S n−1 ), q > 1, and wq ∈ A1 . If Ω satisfies Ω(x − y) Ω(x) w(x + h)dx ≤ C w(y + h) (∀y = 0, ∀h ∈ Rn ) (1.2) − n |x|n |x|≥2|y| |x − y| for certain absolute constant C , then there exists a constant C > 0 independent of f such that µΩ (f )L1w ≤ Cf Hw1 . Remark 1.9. If Ω satisﬁes the condition (1.5) in [9]; that is, there exist C > 0 and η > 1 such that C η |Ω(x) − Ω(y)| ≤ ∀ x, y ∈ S n−1 , 1 log |x−y| then it satisﬁes the Lq -Dini condition. Also, if Ω satisﬁes the Lq -Dini condition, then it satisﬁes (1.2) provided wq ∈ A1 . Throughout the paper, we always assume that Ω is homogeneous of degree zero and satisﬁes (1.1), and denote the conjugate exponent of q > 1 by q = q/(q − 1). We use a ≈ b to mean the equivalence of a and b; that is, there exists two positive constants C1 and C2 independent of a, b such that C1 a ≤ b ≤ C2 a. Moreover, C denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript.

2. Lq -Dini type condition For q ≥ 1 and 0 < α ≤ 1, we say that Ω satisﬁes the Lq -Dini type condition of order α (when α = 0, it is called the Lq -Dini condition) if Ω ∈ Lq (S n−1 ) is homogeneous of degree zero on Rn and 1 ωq (δ) dδ < ∞, 1+α 0 δ where ωq (δ) is the integral modulus of continuity of order q of Ω deﬁned by 1/q q ωq (δ) = sup |Ω(ρx ) − Ω(x )| dσ(x ) |ρ|<δ

S n−1

and ρ is a rotation in R with |ρ| = ρ − I. For 0 < β < α ≤ 1, if Ω satisﬁes the Lq -Dini type condition of order α, then it also satisﬁes the Lq -Dini type condition of order β. We thus denote by Dinqα (S n−1 ) the class of all functions which satisfy the Lq -Dini type conditions of all orders β < α. For 0 < α ≤ 1, we deﬁne n−1 Din∞ )= Dinqα (S n−1 ). α (S n

q≥1

It is easy to check that Dinpα (S n−1 ) ⊂ Dinqα (S n−1 )

if 1 ≤ q < p ≤ ∞,

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and Dinqα (S n−1 ) ⊂ Dinqβ (S n−1 ) We note that, on the unit sphere S

n−1

if 0 < β < α ≤ 1.

,

+ q q q 1 1 Lipα ⊂ Din∞ α ⊂ Dinα ⊂ {L -Dini condition} ⊂ L ⊂ L log L ⊂ H ⊂ L ,

where 0 < α ≤ 1 and q > 1. For q = 1, we still have Lipα ⊂ Din1α ⊂ {L1 -Dini condition}.

3. Ap weights We recall the deﬁnition and properties of Ap weights. For 1 0 such that, for every n dimensional cube Q with sides parallel to the coordinate axes, p−1 1 1 −1/(p−1) w(x)dx w(x) dx ≤ C, |Q| Q |Q| Q where |Q| denotes its Lebesgue measure. For the case p = 1, w ∈ A1 if there exists C > 0 such that, for every cube Q ⊂ Rn , 1 w(x)dx ≤ C ess inf w(x). x∈Q |Q| Q A function w ∈ A∞ if it satisﬁes the condition Ap for some p > 1. It is well-known that if w ∈ Ap for 1 ≤ p ≤ ∞, then wε ∈ Ap for all 0 < ε ≤ 1 and wη ∈ Ap for some η > 1. Also, if w ∈ Ap for 1 p and w ∈ Aq for some 1 < q < p. We thus use qw := inf{q > 1 : w ∈ Aq } to denote the critical index of w. A close relation to Ap is the reverse H¨older condition. If there exists r > 1 and a ﬁxed condition C > 0 such that, for every cube Q ⊂ Rn , 1/r 1 1 w(x)r dx ≤C w(x)dx , |Q| Q |Q| Q we say that w satisﬁes the reverse H¨ older condition of order r and write w ∈ RHr . Lemma 3.1 ([13]). Let r > 1. Then wr ∈ A∞ if and only if w ∈ RHr . For any cube Q and any λ > 0, we denote by λQ the cube concentric with Q which is λ times as long. We use w(E) to denote the weighted measure E w(x)dx. Lemma 3.2 ([7]). Let w ∈ Ap , p ≥ 1. Then, for any cube Q and λ > 1, w(λQ) ≤ Cλnp w(Q), where C does not depend on Q nor on λ.

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4. Proofs of theorems In order to show Hwp − Lpw boundedness of µΩ , we will use Garcia-Cuerva’s atomic decomposition theory (cf. [6, 10]) for weighted Hardy spaces. We characterize weighted Hardy spaces in terms of atom decomposition in the following way. Definition 4.1. Let 0 < p ≤ 1 ≤ q ≤ ∞ and p = q such that w ∈ Aq with critical index qw . Let [ · ] be the greatest integer function. For s ∈ Z satisfying s ≥ [n(qw /p − 1)], a real-valued function a is called w-(p, q, s)-atom centered at x0 if (i) a ∈ Lqw (Rn ) and is supported in a cube Q centered at x0 , (ii) aLqw ≤ w(Q)1/q−1/p , (iii) Rn a(x)xα dx = 0 for every multi-index α with |α| ≤ s.

∞ and f L∞ = f ∞ . When q = ∞, L∞ w will be taken to mean L w

Theorem 4.2 ([6, 10]). Let w ∈ Aq , 0 < p ≤ 1 ≤ q ≤ ∞, and p = q. For each f ∈ Hwp (Rn ), there exists a sequence {ai } of w-(p, q, [n(qw /p − 1)])-atoms, and a |λi |p ≤ Cf pHwp such that f = λi ai both sequence {λi } of real numbers with in the sense of distributions and in the Hwp norm. Using the same method as proving [8, Lemma 5], we may obtain an estimate Ω(x) about the kernel |x| n−γ . Lemma 4.3. Suppose that 0 ≤ γ < n, 1 ≤ q < ∞, and Ω satisfies the Lq -Dini condition. Then q 1/q Ω(x − y) Ω(x) − dx n−γ |x|n−γ R<|x|<2R |x − y|

|y|/R |y| ωq (δ) + dδ for all |y| ≤ R/2. ≤ CRn/q−n+γ R δ |y|/2R As usual we write Br = {x ∈ Rn : |x| < r} for r > 0. Lemma 4.4. For q ≥ 1 and 0 < α ≤ 1, let Ω satisfy the Lq -Dini type condition of order α, and β = min{α, 12 }. Let f ∈ L∞ (Rn ) with supp(f ) ⊂ Br satisfy f (x)dx = 0. Br

(a) If q > 1 and w ≥ 0, then, for R ≥ 2r,

1/q n/q r n+β |µΩ (f )(x)|w(x)dx ≤ Cf ∞ wq (B2R ) R . R R<|x|<2R (b) If q = 1 and w ≡ 1, then, for R ≥ 2r, r n+β |µΩ (f )(x)|dx ≤ Cf ∞ Rn . R R<|x|<2R

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Proof. We only √prove for case (a). The proof of case (b) is similar. Note that √ √ a + b ≤ a + b for a, b ≥ 0. We hence have |x|+r 1/2 2 dt |µΩ (f )(x)|w(x)dx ≤ |FΩ,t (f, x)| 3 w(x)dx t 0 R<|x|<2R R<|x|<2R ∞ 1/2 2 dt + |FΩ,t (f, x)| 3 w(x)dx t |x|+r R<|x|<2R :=I1 + I2 . Let K(x) =

Ω(x) |x|n−1 .

I1 ≤ R<|x|<2R

Using Minkowski’s inequality for integrals, we have

|K(x − y)||f (y)|

Br

≤ Cf ∞ R<|x|<2R

Br

|K(x − y)|

|x|+r |x−y|

dt t3

1/2 dyw(x)dx

1/2 1 1 dyw(x)dx. − 2 2 |x − y| (|x| + r)

For y ∈ Br and R < |x| < 2R, we have |x − y| ≈ |x| ≈ |x| + r and hence 1 1 r |x − y|2 − (|x| + r)2 ≤ C |x − y|3 . Apply this inequality, we obtain I1 ≤ Cf ∞ r1/2 Br

R<|x|<2R

|Ω(x − y)| w(x)dx dy := I1 . |x − y|n+1/2

H¨ older’s inequality gives, for y ∈ Br and R ≥ 2r, |Ω(x − y)| w(x)dx |x − y|n+1/2 R<|x|<2R 1/q 1/q |Ω(x − y)|q w(x)q ≤ dx dx n+1/2 n+1/2 R<|x|<2R |x − y| R<|x|<2R |x − y| 1/q q

1/q |Ω(z)|q (−n−1/2)/q w (B2R ) ≤ CR dz n+1/2 R−r<|z|<2R+r |z|

1/q ≤ CR−n/q −1/2 wq (B2R ) . Combining the above two inequalities, we get

1/q n/q r n+1/2 I1 ≤ I1 ≤ Cf ∞ wq (B2R ) R . R

(4.1)

To estimate I2 , we note that, for t ≥ |x| + r, Br is contained in the ball {y ∈ Rn : |x − y| ≤ t}. We use Minkowski’s inequality for integrals and Br f (x)dx = 0

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to obtain

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93

2 1/2 dt I2 = (K(x − y) − K(x))f (y)dy 3 w(x)dx t R<|x|<2R |x|+r |x−y|≤t ∞ 1/2 dt ≤ |K(x − y) − K(x)||f (y)| dyw(x)dx 3 R t R<|x|<2R Br |K(x − y) − K(x)|w(x)dx dy. ≤ Cf ∞ R−1

∞

Br

R<|x|<2R

Applying H¨ older’s inequality and Lemma 4.3, we have |K(x − y) − K(x)|w(x)dx R<|x|<2R

1/q |K(x − y) − K(x)|q dx

≤ R<|x|<2R

1/q w(x)q dx

R<|x|<2R

|y|/R

1/q |y| ωq (δ) + dδ wq (B2R ) ≤ CR R δ |y|/2R

q

1/q |y| |y| α |y|/R ωq (δ) n/q−n+1 ≤ CR + w (B2R ) dδ . 1+α R R |y|/2R δ n/q−n+1

Thus,

1/q n/q r n+1 r n+α 1 ωq (δ) I2 ≤ Cf ∞ wq (B2R ) R + dδ , 1+α R R 0 δ which completes the proof of Lemma 4.4.

Remark 4.5. We note that the estimate (4.1) about I1 and still works for Ω ∈ Lq (S n−1 ).

I1

in the previous proof

We are ready to show our main theorems. Proof of Theorem 1.4. By Theorem 4.2, it suﬃces to show that, for any w-(p, ∞, 0)atom a, there exists a constant C > 0 independent of a such that µΩ (a)Lpw ≤ C. Let a be a w-(p, ∞, 0)-atom √ centered at 0 with supp(a) ⊂ Q. Denote by d the side length of Q and Q∗ = 2 nQ. We note that, in either case (a) or (b), there pβ such that both wq ∈ Ar and w1/(1−p) ∈ Ar . By Theorem exists an 1 < r < n(1−p) 1.1 and Lemma 3.2, we have p/r |µΩ (a)(x)|p w(x)dx ≤ w(Q∗ )1−p/r |µΩ (a)(x)|r w(x)dx Q∗

Q∗

≤ Cw(Q∗ )1−p/r

p/r |a(x)|r w(x)dx

Q

≤ Cn w(Q)1−p/r ap∞ w(Q)p/r ≤ Cn .

(4.2)

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We now estimate the integral I := (Q∗ )c |µΩ (a)(x)|p w(x)dx. Deﬁne Ej = √ √ {x ∈ Rn : 2j nd < |x| < 2j+1 nd}. H¨older’s inequality yields ∞ |µΩ (a)(x)|p w(x)dx I≤ ≤

j=0 Ej ∞ j=0

1−p w(x)1/(1−p) dx

Ej

p |µΩ (a)(x)|dx .

Ej

By Lemmas 3.1 and 3.2, we get 1−p

1−p √ 1/(1−p) w(x) dx ≤ w1/(1−p) (2j+2 nQ) Ej

1−p ≤ Cn 2j(nr−nrp) w1/(1−p) (Q) ≤ Cn 2j(nr−nrp) w(Q)|Q|−p .

Since r < 1 2 }.

pβ n(1−p) ,

we may choose an α ˜ < α such that r < Din1α (S n−1 ),

pβ˜ n(1−p) , 1

where β˜ =

min{α ˜, By the assumption Ω ∈ Ω satisﬁes the L -Dini type condition of order α ˜ . Applying Lemma 4.4 (b), we obtain ˜ |µΩ (a)(x)|dx ≤ Ca∞ dn 2−j β . Ej

The above three inequalities yield I ≤ Cn

∞

˜

2j(nr−nrp−pβ) ≤ Cn ,

j=0 ˜

pβ . where the last inequality is due to r < n(1−p) If a is a w-(p, ∞, 0)-atom centered at x0 ∈ Rn , then τ−x0 a is a w1 -(p, ∞, 0)atom centered at 0, where w1 = τ−x0 w and τh f (x) = f (x − h). We have already shown µΩ (τ−x0 a)Lpw1 ≤ Cn , so

µΩ (a)Lpw = τx0 (µΩ (τ−x0 a))Lpw = µΩ (τ−x0 a)Lpw1 ≤ Cn

and the proof is completed.

Proof of Theorem 1.5. We only prove the case p < 1. The proof of the case p = 1 is similar and easier. Let a be a w-(p, ∞, 0)-atom centered at 0, with supp(a) ⊂ Q, √ and let Q∗ = 2 nQ. Denote by d the side length of Q. Since wq ∈ A(p+ pβ − 1 )q ,

n

q

1 q there exists an 1 < r < (p + pβ n − q )q such that w ∈ Ar . Like the estimate (4.2) in the proof of Theorem 1.4, we have |µΩ (a)(x)|p w(x)dx ≤ Cn . Q∗

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√ Let J = (Q∗ )c |µΩ (a)(x)|p w(x)dx and Ej = {x ∈ Rn : 2j nd < |x| < √ 2j+1 nd}. H¨older’s inequality implies ∞ J≤ |µΩ (a)(x)|p w(x)dx ≤

j=0 Ej ∞ j=0

1−p w(x)dx

Ej

p |µΩ (a)(x)|w(x)dx .

Ej

Use H¨older’s inequality again to obtain 1/q w(x)dx ≤ 1dx Ej

Ej

1/q w(x) dx q

Ej

1/q √ ≤ Cn 2jn/q dn/q wq (2j+2 nQ) . ˜

1 Since r < (p + pβ ˜ < α such that r < (p + pnβ − 1q )q , n − q )q , there exists an α where β˜ = min{α ˜ , 12 }. By the assumption Ω ∈ Dinqα (S n−1 ), Ω satisﬁes the Lq -Dini type condition of order α ˜ . Applying Lemma 4.4 (a), we have

1/q n/q j(n/q−n−β) √ ˜ |µΩ (a)(x)|w(x)dx ≤ Ca∞ wq (2j+2 nQ) d 2 . Ej

Combining the above three inequalities and applying Lemmas 3.1 and 3.2, we have J ≤ Cn ap∞ dn/q

∞

1/q √ ˜ 2j(n/q−np−pβ) wq (2j+2 nQ)

j=0 ∞

1/q ˜ ≤ Cn ap∞ dn/q wq (Q) 2j(nr/q +n/q−np−pβ) j=0

≤ Cn

∞

˜

2j(nr/q +n/q−np−pβ)

j=0

≤ Cn , ˜

where the last inequality is due to r < (p + pnβ − 1q )q . Given a w-(p, ∞, 0)-atom a centered at x0 ∈ Rn , τ−x0 a is a w1 -(p, ∞, 0)atom centered at 0, where w1 = τ−x0 w. Then µΩ (τ−x0 a)Lpw1 ≤ Cn implies µΩ (a)Lpw ≤ Cn . By Theorem 4.2, the Hwp − Lpw boundedness of µΩ follows. Proof of Corollary 1.7. For w ∈ Ap+ pβ , there exists η > 1 such that wη ∈ Ap+ pβ . Since q ↓ 1 as q ↑ ∞ and (p +

pβ n

n

− 1q )q > p +

pβ n

for p >

n n+β ,

n

we may choose q

large enough such that wq ∈ Ap+ pβ ⊂ A(p+ pβ − 1 )q . Thus, Corollary 1.7 follows n n q from Theorem 1.5.

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Proof of Theorem √ 1.8. Let a be a w-(1, ∞, 0)-atom centered at 0, with supp(a) ⊂ Q, and let Q∗ = 2 nQ. Denote by d the side length of Q. Like the estimate (4.2) in the proof of Theorem 1.4, we have Q∗

|µΩ (a)(x)|w(x)dx ≤ C.

√ Let K = (Q∗ )c |µΩ (a)(x)|w(x)dx and Ej = {x ∈ Rn : 2j nd < |x| < √ √ √ √ 2j+1 nd}. Since a + b ≤ a + b for a, b ≥ 0, we have ∞

K≤

j=0

|µΩ (a)(x)|w(x)dx

Ej

∞

≤

j=0

+

0

Ej

∞ j=0

√ |x|+ nd/2

Ej

dt |FΩ,t (a, x)|2 3 t

∞

√ |x|+ nd/2

|FΩ,t (a, x)|2

1/2

dt t3

w(x)dx 1/2 w(x)dx

:=K1 + K2 . By Remark 4.5, we have the same estimate as (4.1) for K1 : K1 ≤ Ca∞

∞ q

1/q j √ w (B2j+1 √nd ) (2 nd)n/q 2−j(n+1/2) . j=0

Using Lemmas 3.1 and 3.2, we get K1 ≤ Ca∞ dn/q

∞

1/q √ 2j(n/q−n−1/2) wq (2j+2 nQ)

j=0 ∞

1/q ≤ Ca∞ dn/q wq (Q) 2−j/2 j=0

≤ C. As to the estimate of K2 , for t ≥ |x| + and hence K2 =

∞ j=0

Ej

√

nd/2, we have Q ⊂ {y : |x − y| ≤ t}

2 1/2 dt Ω(x − y) Ω(x) − w(x)dx. a(y)dy t3 n−1 n−1 √ |x − y| |x| |x|+ nd/2 Q

∞

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√ Applying Minkowski’s inequality for integrals and using |x| + nd/2 ≈ |x| for x ∈ Ej , we have ∞ 1/2 ∞ Ω(x − y) Ω(x) 1 |a(y)| − dt dyw(x)dx K2 ≤C |x − y|n−1 √ 3 |x|n−1 |x|+ nd/2 t j=0 Ej Q ∞ Ω(x − y) Ω(x) |a(y)| ≤C |x − y|n−1 − |x|n−1 |x| dyw(x)dx j=0 Ej Q ∞ Ω(x − y) Ω(x − y) w(x)dx|a(y)|dy =C − n−1 |x − y|n Q j=0 Ej |x| · |x − y| ∞ Ω(x − y) Ω(x) w(x)dx|a(y)|dy +C − n |x|n Q j=0 Ej |x − y| :=K2 + K2 . For y ∈ Q and x ∈ Ej , we have |x| ≈ |x − y| and |y| ≤ |x − y|. Thus 1 d1/2 1 |x| − |x − y| ≤ C |x − y|3/2 , which implies K2

1/2

≤ Ca∞ d

∞ Q

j=0

Ej

|Ω(x − y)| w(x)dx dy. |x − y|n+1/2

By Remark 4.5 and the same argument as estimating K1 , we have K2 ≤ C. Finally, it follows from (1.2) that Ω(x − y) Ω(x) w(x)dx|a(y)|dy − K2 = n √ |x|n Q |x|> nd |x − y| Ω(x − y) Ω(x) w(x)dx|a(y)|dy ≤C − n |x|n Q |x|≥2|y| |x − y| ≤C |a(y)|w(y)dy Q

≤ Ca∞ w(Q) ≤ C. Given a w-(1, ∞, 0)-atom a centered at x0 ∈ Rn , τ−x0 a is a w1 -(1, ∞, 0)-atom centered at 0, where w1 = τ−x0 w. By (1.2), Ω satisﬁes Ω(x − y) Ω(x) w1 (x)dx ≤ C w1 (y) − (∀y = 0). n |x|n |x|≥2|y| |x − y| Thus µΩ (τ−x0 a)L1w ≤ C implies µΩ (a)L1w ≤ C. By Theorem 4.2, the Hw1 − L1w 1 boundedness of µΩ follows.

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References [1] A. Benedek, A. P. Calder´ on, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. USA 48 (1962), 356-365. [2] A. P. Calder´ on, M. Weiss, and A. Zygmund, On the existence of singular integrals, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp.5673. [3] Y. Ding, D. Fan, and Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J. 48 (1999), 1037-1055. [4] Y. Ding, M.-Y. Lee, and C.-C. Lin, Marcinkiewicz integral on weighted Hardy spaces, Arch. Math. (Basel) 80 (2003), 620-629. [5] Y. Ding, S. Lu, and Q. Xue, Marcinkiewicz integral on Hardy spaces, Integral Equations and Operator Theory 42 (2002), 174-182. [6] J. Garcia-Cuerva, Weighted H p spaces, Dissertations Math. 162 (1979), 1-63. [7] J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland, 1985. [8] D. S. Kurtz and R. L. Wheeden, Results on Weighted Norm Inequalities for Multipliers, Trans. Amer. Math. Soc. 255 (1979), 343-362. [9] J. Lee and K. S. Rim, Estimates of Marcinkiewicz Integrals with Bounded Homogeneous Kernels of Degree Zero, Integral Equations and Operator Theory 48 (2004), 213-223. [10] M.-Y. Lee and C.-C. Lin, The molecular characterization of weighted Hardy spaces, J. Funct. Anal. 188 (2002), 442-460. [11] J. Marcinkiewicz, Sur quelques int´egrales du type de Dini, Annales de la Soci´et´e Polonaise de Math´ematiques 17 (1938), 42-50. [12] E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. [13] J.-O. Str¨ omberg and R. L. Wheeden, Fractional integrals on weighted H p and Lp spaces, Trans. Amer. Math. Soc. 287 (1985), 293-321. [14] A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235-243. Chin-Cheng Lin and Ying-Chieh Lin Department of Mathematics National Central University Chung-Li, Taiwan 320 Republic of China e-mail: [email protected] [email protected] Submitted: June 14, 2006 Revised: November 15, 2006

Integr. equ. oper. theory 58 (2007), 99–110 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010099-12, published online April 16, 2007 DOI 10.1007/s00020-007-1497-x

Integral Equations and Operator Theory

Polar Wavelet Transforms and Localization Operators Yu Liu and M. W. Wong Abstract. The notion of a polar wavelet transform is introduced. The underlying non-unimodular Lie group, the associated square-integrable representations and admissible wavelets are studied. The resolution of the identity formula for the polar wavelet transform is then formulated and proved. Localization operators corresponding to the polar wavelet transforms are then deﬁned. It is proved that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators, paraproducts and Fourier multipliers. Mathematics Subject Classiﬁcation (2000). Primary 47G10, 47G30; Secondary 42C40. Keywords. Aﬃne groups, wavelet transforms, square-integrable representations, resolution of the identity, localization operators, paracommutators, paraproducts, Fourier multipliers.

1. Wavelet Transforms in a Nutshell Let U be the upper half plane given by U = {(b, a) : b ∈ R, a > 0}. Then with respect to the binary operation · on U deﬁned by (b1 , a1 ) · (b2 , a2 ) = (b1 + a1 b2 , a1 a2 ),

(b1 , a1 ), (b2 , a2 ) ∈ U,

U becomes a non-abelian groupin which (0,1) is the identity element and the inverse element of (b, a) is − ab , a1 for all (b, a) in U . In fact, it is a non-unimodular This research was supported by the Natural Sciences and Engineering Research Council of Canada.

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Lie group on which the left Haar measure and the right Haar measure are given, respectively, by db da dµ = a2 and db da . dν = a It is commonly known as the aﬃne group in the literature. 2 (R) be the subspace of L2 (R) deﬁned by Let H+ 2 H+ (R) = {f ∈ L2 (R) : supp(fˆ) ⊆ [0, ∞)},

where supp(fˆ) is the support of the Fourier transform fˆ. The Fourier transform F of a function F in L2 (Rn ) to be used in this paper is the one deﬁned by F (ξ) = lim (2π)−n/2 e−ix·ξ χR (x)F (x) dx, R→∞

Rn

where x · ξ is the inner product of x and ξ in Rn , χR is the characteristic function on the ball {x ∈ Rn : |x| ≤ R}, and the convergence is understood to occur in 2 L2 (Rn ). We deﬁne H− (R) to be the subspace of L2 (R) by 2 H− (R) = {f ∈ L2 (R) : supp(fˆ) ⊆ (−∞, 0]}. 2 2 H+ (R) and H− (R) are known as the Hardy space and the conjugate Hardy space respectively. It is convenient at this point to mention that the results in this section are 2 stated in the context of the left Haar measure dµ and the Hardy space H+ (R) only. 2 2 Now, we let U (H+ (R)) be the set of all unitary operators on H+ (R). It is a 2 (R)) group with respect to the usual composition of mappings. Let π : U → U (H+ be the mapping deﬁned by x−b 1 (π(b, a)f )(x) = √ f , x ∈ R, a a 2 (R). Then it can be proved that π is an irreducible for all (b, a) in U and all f in H+ 2 2 (R). The raison d’ˆetre for H± (R) is that and unitary representation of U on H+ 2 the unitary representation π : U → U (L (R)) is not irreducible. 2 In fact, π is a square-integrable representation of U on H+ (R) in the sense 2 that there exists a nonzero function ϕ in H+ (R) such that ∞ ∞ db da |(ϕ, π(b, a)ϕ)L2 (R) |2 2 < ∞. (1.1) a 0 −∞ 2 We call any function ϕ in H+ (R) for which ϕL2 (R) = 1 and (1.1) is valid an 2 admissible wavelet for the square-integrable representation π of U on H+ (R). We have the following characterization of admissible wavelets.

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Theorem 1.1. The set AW (π) of all admissible wavelets for the square-integrable 2 representation π of U on H+ (R) is given by ∞ 2 |ϕ(ξ)| ˆ 2 AW (π) = ϕ ∈ H+ dξ < ∞ . (R) : ϕL2 (R) = 1, |ξ| 0 The proof of Theorem 1.1 follows from the fact that the integral in (1.1) is ∞ ˆ 2 equal to 2π 0 |ϕ(ξ)| |ξ| dξ. A proof can be found in Daubechies [3], Wong [8] and others. Let ϕ ∈ AW (π). Then we deﬁne the wavelet transform Wϕ f of a function f 2 in H+ (R) by (Wϕ f )(b, a) = (f, π(b, a)ϕ)L2 (R) , (b, a) ∈ U. Then we have the following resolution of the identity formula, which can also be found in Daubechies [3] and Wong [8]. Theorem 1.2. let ϕ ∈ AW (π) and let cϕ be the constant defined by ∞ 2 |ϕ(ξ)| ˆ dξ. cϕ = 2π |ξ| 0 2 Then for all f and g in H+ (R), ∞ ∞ db da 1 (f, π(b, a)ϕ)L2 (R) (π(b, a)ϕ, g)L2 (R) 2 . (f, g)L2 (R) = cϕ 0 a −∞

Let ϕ ∈ AW (π) and let F be a measurable function on U . Then we deﬁne 2 2 the localization operator LF,ϕ : H+ (R) → H+ (R) associated to the symbol F and the admissible wavelet ϕ by ∞ ∞ db da 1 F (b, a)(f, π(b, a)ϕ)L2 (R) (π(b, a)ϕ, g)L2 (R) 2 (LF,ϕ f, g)L2 (R) = cϕ 0 a −∞ for all f and g in L2 (R). It is shown in Wong [9] that if F (b, a) = β(b)α(a),

(b, a) ∈ U,

where α and β are suitable functions on (0, ∞) and (−∞, ∞) respectively, then the localization operator LF,ϕ is in fact a paracommutator studied in Janson and Peetre [4], Peng [5, 6] and Peng and Wong [7]. Furthermore, in Wong [9], it is proved that if F is a suitable function of b only, then LF,ϕ is given in terms of a paraproduct in the sense of Coifman and Meyer [2], and if F is a function of a only, then LF,ϕ is a Fourier multiplier. The aim of this paper is to extend the results hitherto described to the twodimensional wavelet transform in which a rotation is built in. The underlying Lie group, which we call the polar aﬃne group, is introduced in Section 2. The irreducible and unitary representations of the polar aﬃne group on L2 (R2 ) are analyzed in Section 3. This is in sharp contrast with the fact that the corresponding representations in the case of the ordinary wavelet transforms are not irreducible on L2 (R). Issues on square-integrability and admissible wavelets for the representations in Section 3 are investigated in Section 4. Polar wavelet transforms and

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localization operators are given, respectively, in Section 5 and Section 6. Under conditions similar to those in Wong [9], we show that the localization operators are given by paracommutators in Section 7, paraproducts in Section 8 and Fourier multipliers in Section 9. We end this introduction by mentioning that the polar aﬃne group and the polar wavelet transform are known, respectively, as the similitude group and the 2-D continuous wavelet transform in the book [1] by Antoine, Murenzi, Vandergheynst and Ali. The theme that underlies Sections 2–6 in this paper is the mathematical underpinnings of the most important results, which can be found in [1], on the polar aﬃne group and the polar wavelet transform from the stance of rigorous mathematical analysis. Sections 7–9 are devoted to localization operators on the polar aﬃne group.

2. The Polar Aﬃne Group Let U be the upper half space in R3 given by U = {(b, a) : b ∈ R2 , a > 0}. Then we let PU be the set deﬁned by PU = U × R/2πZ. It is well-known that the group R/2πZ can be identiﬁed with the closed interval [0, 2π]. Let θ ∈ [0, 2π]. Then we deﬁne the rotation Rθ on R2 by    cos θ sin θ x1   Rθ x =  x2 −sin θ cos θ   x1  in R2 . for all x =  x2 We can now deﬁne the binary operation · on PU by (b1 , a1 , θ1 ) · (b2 , a2 , θ2 ) = (b1 + a1 Rθ1 b2 , a1 a2 , θ1 + θ2 ) for all (b1 , a1 , θ1 ) and (b2 , a2 , θ2 ) in PU. Proposition 2.1. With respect to the multiplication ·, PU is a non-abelian group in (0, 1, 0) is the identity element and the inverse element of (b, a, θ) is which − a1 R−θ b, a1 , −θ for all (b, a, θ) in PU. The proof of Proposition 2.1 is straightforward and hence omitted. Proposition 2.2. The left and right Haar measures on PU are given, respectively, by db da dθ dµ = a3

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and dν =

db da dθ . a

Proof. Let f be an integrable function on PU with respect to dµ. Then for all (b , a , θ ) in PU, we get 2π ∞ db da dθ f ((b , a , θ ) · (b, a, θ)) dµ = f (b + a Rθ b, a a, θ + θ) . a3 0 0 PU R2 Let β = b + a Rθ b, α = a a and φ = θ + θ. Then 2π ∞ dβ dα dφ f ((b , a , θ ) · (b, a, θ)) dµ = f (β, α, φ) α3 2 0 PU R 0 = f (b, a, θ) dµ. PU

Therefore dµ is a left Haar measure. That dν is a right Haar measure can be proved similarly. Remark 2.3. With respect to the multiplication ·, PU is a Lie group on which the left Haar measure is diﬀerent from the right Haar measure. Thus, U is a nonunimodular group, which we call the polar aﬃne group. In contrast with the aﬃne group U in Section 1, let us note that the polar aﬃne group is two-dimensional and a rotation is built into it.

3. Representation Theory Let π : PU → U (L2 (R2 )) be the mapping of PU into the group U (L2 (R2 )) of all unitary operators on L2 (R2 ) given by R−θ (x − b) 1 (π(b, a, θ)f )(x) = f , x ∈ R2 , a a for all (b, a, θ) in PU and all f in L2 (R2 ). Proposition 3.1. π : PU → U (L2 (R2 )) is a representation of PU on L2 (R2 ). Proof. By straightforward computations, we get π(b1 , a1 , θ1 )π(b2 , a2 , θ2 ) = π((b1 , a1 , θ1 ) · (b2 , a2 , θ2 )) for all (b1 , a1 , θ1 ) and (b2 , a2 , θ2 ) in PU and π(b, a, θ)∗ = π((b, a, θ)−1 ) for all (b, a, θ) in PU. It remains to prove that π(b, a, θ)f → f in L2 (R2 ) as (b, a, θ) → (0, 1, 0) for all f in L2 (R2 ). But, by Plancherel’s theorem and the

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elementary properties of the Fourier transform, we get for all f in L2 (R2 ) such that fˆ ∈ C0∞ (R2 ), π(b, a, θ)f − f 2L2 (R2 ) 2 1 R−θ (x − b) f = − f (x) dx a 2 a R = |ae−ib·ξ fˆ(aR−θ ξ) − fˆ(ξ)|2 dξ R2 −ib·ξ ˆ 2 ˆ ≤ 2 |ae (f (aR−θ ξ) − f (ξ))| dξ + 2 R2

R2

|(ae−ib·ξ − 1)fˆ(ξ)|2 dξ.

It is clear from Lebesgue’s dominated convergence theorem that |(ae−ib·ξ − 1)fˆ(ξ)|2 dξ → 0 R2

as (b, a, θ) → (0, 1, 0). Now, for all ξ in R2 , |ae−ib·ξ (fˆ(aR−θ ξ) − fˆ(ξ))|2 → 0 as (b, a, θ) → (0, 1, 0). Moreover, for all a in 12 , 1 , |ae−ib·ξ (fˆ(aR−θ ξ) − fˆ(ξ))|2 ≤ 4χR (ξ) sup |fˆ(ξ)|, ξ∈R2

ξ ∈ R2 ,

where R is a ﬁxed positive number such that fˆ(ξ) = 0, |ξ| > R, and χR is the characteristic function on the ball {x ∈ R2 : |ξ| ≤ 2R}. Thus, by Lebesgue’s dominated convergence theorem, |ae−ib·ξ (fˆ(aR−θ ξ) − fˆ(ξ))|2 dξ → 0 R2

as (b, a, θ) → (0, 1, 0). Now, let f ∈ L2 (R2 ). Using the fact that the set W of all functions f in L2 (R2 ) such that fˆ ∈ C0∞ (R2 ) is dense in L2 (R2 ), we can ﬁnd a 2 2 sequence {fk }∞ k=0 of functions in W such that fk → f in L (R ) as k → ∞. So, for every positive number ε, let k0 be a positive integer such that 2ε fk0 − f L2 (R2 ) < . 3 Then there exists a positive number δ such that π(b, a, θ)f − f L2 (R2 ) ≤

π(b, a, θ)(f − fk0 )L2 (R2 ) + π(b, a, θ)fk0 − fk0 L2 (R2 )

<

+fk0 − f L2 (R2 ) ε

whenever (b, a, θ) is within δ-distance of (0, 1, 0). Therefore π(b, a, θ)f → f in L2 (R2 ) as (b, a, θ) → (0, 1, 0), and the proof is complete.

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Proposition 3.2. π : PU → U (L2 (R2 )) is an irreducible and unitary representation of PU on L2 (R2 ). Proof. Let M be a nonzero and closed subspace of L2 (R2 ) such that M is invariant with respect to π : PU → U (L2 (R2 )). Let g be a nonzero function in M . Then {π(b, a, θ)g : (b, a, θ) ∈ PU} ⊆ M. Let f ∈ L2 (R2 ) be such that f ∈ M ⊥ . Then for all (b, a, θ) in PU, R−θ (x − b) f (x)g dx = 0 a R2 and hence, by Plancherel’s theorem, eib·ξ fˆ(ξ)ˆ g (aR−θ ξ) dξ = 0. R2

So, fˆ(ξ)ˆ g (aR−θ ξ) = 0 for almost all ξ in R2 . Suppose that fˆ(ξ) = 0 for all ξ in a set S with positive measure. Then for all ξ in S, we get aR−θ ξ = 0 for all positive numbers a and all θ in [0, 2π]. Thus, gˆ = 0. This gives g = 0 and hence a contradiction.

4. Square-Integrability We show in this section that the representation π studied in Section 3 is squareintegrable in the sense that there exists a nonzero function ϕ in L2 (R2 ) with ϕL2 (R2 ) = 1 such that |(ϕ, π(b, a, θ)ϕ)L2 (R2 ) |2 dµ < ∞. (4.1) PU

Such a function ϕ is called an admissible wavelet and ϕ is said to satisfy the admissibility condition (4.1). The following result contains a characterization of all admissible wavelets. Theorem 4.1. For all functions ϕ and ψ in L2 (R2 ), PU

2

2

|(ϕ, π(b, a, θ)ψ)L2 (R2 ) | = (2π)

R2

2

|ϕ(ξ)| ˆ dξ

R2

2 ˆ |ψ(ξ)| dξ. |ξ|2

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Proof. First we begin with functions ϕ and ψ in W . We get 2π ∞ db da dθ |(ϕ, π(b, a, θ)ψ)L2 (R2 ) |2 a3 0 0 R2 2 2π ∞ R−θ (x − b) db da dθ = ϕ(x)ψ dx a a5 2 2 0 0 R R 2 2π ∞ db da dθ ib·ξ ˆ = e ϕ(ξ) ˆ ξ)dξ ψ(aR −θ a 0 0 R2 R2 2 2π ∞ db da dθ 2 −1 ib·ξ ˆ e ϕ(ξ) ˆ ψ(aR−θ ξ)dξ = (2π) (2π) a 0 0 R2 R2 2π ∞ 2 dξ da dθ ˆ = (2π)2 , ˆ ϕ(ξ)(D a R−θ ψ)(ξ) a 0 0 R2 where ξ ∈ R2 .

ˆ ˆ (Da R−θ ψ)(ξ) = ψ(aR −θ ξ), So,

|(ϕ, π(b, a, θ)ψ)L2 (R2 ) |2 dµ 2π ∞ 2 2 2 da dθ ˆ |ϕ(ξ)| ˆ |ψ(aR−θ ξ)| (2π) dξ. a 0 0 R2 PU

= To compute

2π ∞ 0

0

2 da dθ ˆ |ψ(aR −θ ξ)| a , let η = aR−θ ξ. Then

da dθ dη = 2. a |η| Therefore

0

So,

PU

2π

0

∞

2 da dθ ˆ = |ψ(aR −θ ξ)| a

|(ϕ, π(b, a, θ)ϕ)L2 (R2 ) |2 dµ = (2π)2

R2

R2

2 ˆ |ψ(η)| dη. |η|2

2 |ϕ(ξ)| ˆ dξ

R2

2 ˆ |ψ(ξ)| dξ. 2 |ξ|

Using the density of W in L2 (R2 ) and a density argument as in the proof of Theorem 18.8 in the book [8] by Wong, the proof is complete. Corollary 4.2. The set AW of all admissible wavelets is given by 2 |ϕ(ξ)| ˆ 2 2 dξ < ∞ . AW = ϕ ∈ L (R ) : ϕL2 (R2 ) = 1, |ξ|2 R2

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5. Polar Wavelet Transforms Let ϕ ∈ L2 (R2 ) be an admissible wavelet for the square-integrable representation π : PU → U (L2 (R2 )). For every function f in L2 (R2 ), we deﬁne the polar wavelet transform Wϕ f of f with respect to ϕ by (Wϕ f )(b, a, θ) = (f, π(b, a, θ)ϕ)L2 (R2 ) for all (b, a, θ) in PU. By Theorem 6.1 in the book [8] by Wong, we have the following resolution of the identity formula. Theorem 5.1. Let ϕ ∈ L2 (R2 ) be an admissible wavelet for the square-integrable representation π : PU → U (L2 (R2 )). Let cϕ be the constant defined by 2 |ϕ(ξ)| ˆ 2 dξ. cϕ = (2π) |ξ|2 R2 Then for all f and g in L2 (R2 ), 1 db da dθ (f, g)L2 (R2 ) = (Wϕ f )(b, a, θ)(Wϕ g)(b, a, θ) . cϕ PU a3

6. Localization Operators The following result is the PU-analog of Theorem 14.5 in the book [8] by Wong. Theorem 6.1. Let ϕ ∈ L2 (R2 ) be an admissible wavelet for the square-integrable representation π : PU → U (L2 (R2 )). Let F ∈ Lp (PU), 1 ≤ p ≤ ∞. Then the localization operator LF,ϕ : L2 (R2 ) → L2 (R2 ) associated to the symbol F and the admissible wavelet ϕ defined by 1 db da dθ F (b, a, θ)(Wϕ f )(b, a, θ)(Wϕ g)(b, a, θ) (LF,ϕ f, g)L2 (R2 ) = cϕ PU a3 for all f and g in L2 (R2 ) is in the Schatten–von Neumann class Sp and 1 LF,ϕ Sp ≤ F Lp (PU) , cϕ where Sp is the norm in Sp . We give in the following sections explicit formulas for the localization operators LF,ϕ : L2 (R2 ) → L2 (R2 ) under diﬀerent conditions imposed on the symbol F . These results are extensions of results obtained in the paper [9] for the standard wavelet transforms on the aﬃne group U.

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7. Paracommutators Let F be a function on PU given by (b, a, θ) ∈ PU,

F (b, a, θ) = β(b)α(a, θ),

where α and β are suitable functions on (0, ∞) × [0, 2π] and R2 respectively. Then using Plancherel’s theorem and Fubini’s theorem, we get for all u and v in L2 (R2 ), (LF,θ u, v)L2 (R2 ) 2π ∞ 1 db da dθ = F (b, a, θ)(Wϕ u)(b, a, θ)(Wϕ v)(b, a, θ) cϕ 0 a3 2 0 R 2π ∞ 1 da dθ = α(a, θ) β(b)(Wϕ u)(b, a, θ)(Wϕ v)(b, a, θ) db cϕ 0 a3 R2 0 2π ˆ − ξ)ˆ = A(ξ, η)β(η u(ξ)ˆ v (η) dξ dη, (7.1) cϕ R2 R2 where

2π

∞

A(ξ, η) = 0

0

da dθ a is a paracommutator

α(a, θ)ϕ(aR ˆ ˆ −θ ξ)ϕ(aR −θ η)

for all ξ and η in R2 . Thus, the localization operator LF,σ with Fourier kernel A and symbol β.

8. A Paraproduct Connection In this section we specialize to the case when the symbol F is a function of b only, i.e., F (b, a, θ) = β(b), (b, a, θ) ∈ PU, where β is a function on R2 . In order to simplify the computations that follow, let us introduce a notation. Let a be a positive number and let θ ∈ [0, 2π]. Then we deﬁne the polar Friedrich molliﬁer fa,θ of a function f in L2 (R2 ) by 1 R−θ x fa,θ (x) = 2 f , x ∈ R2 . a a An easy computation gives ˆ f a,θ (ξ) = f (aR−θ ξ),

ξ ∈ R2 .

Let ψ be the function on R2 deﬁned by ψ(x) = ϕ(−x),

x ∈ R2 .

Then, by (7.1), we get

=

(LF,ϕ u, v)L2 (R2 ) 2π ∞ 2π da ˆ dθ dξ vˆ(η) dη. u(ξ)β(η − ξ) ψa,θ (ξ)ϕ a,θ (ξ)ˆ cϕ R2 R2 a 0 0

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Since (f ∗ g)∧ = 2π fˆgˆ for suﬃciently nice functions f and g on R2 , it follows from Plancherel’s formula that for all u and v in L2 (R2 ), 2π ∞ 1 da (LF,ϕ u, v)L2 (R2 ) = ((β(ψa,θ ∗ u)) ∗ φ)(x) dθ v(x) dx (8.1) cϕ R2 a 0 0 and hence (LF,ϕ u)(x) =

1 cϕ

0

2π

0

∞

((β(ψa,θ ∗ u)) ∗ ϕa,θ )(x)

da dθ, a

x ∈ R2 .

Further manipulations of (8.1) using Fubini’s theorem give 1 (LF,ϕ u, v)L2 (R2 ) = β(y)pψ (u, v)(y) dy cϕ R2 for all u and v in L2 (R2 ), where pψ (u, v) is the paraproduct of u and v with respect to ψ given by 2π ∞ da dθ, y ∈ R2 . pψ (u, v)(y) = (ψa,θ ∗ u)(y)(ψa,θ ∗ v)(y) a 0 0

9. Fourier Multipliers Let F be a function on PU given by F (b, a, θ) = α(a, θ),

(b, a, θ) ∈ PU.

Then, by (7.1), we obtain for all u and v in L2 (R2 ),

=

(LF,ϕ u, v)L2 (R2 ) 2π ∞ 2π da α(a, θ)ϕ(aR ˆ ˆ u(ξ)ˆ v (η) dθ dξ dη −θ ξ)ϕ(aR −θ η)ˆ cϕ R2 R2 0 a 0 2

Let g(x) = e−|x| /2 , x ∈ R2 . For all positive numbers ε, let Iε be the number deﬁned by 2π ∞ 2π da Iε = α(a, θ)ϕ u(ξ)ˆ v (η)gε (ξ − η) dθ dξ dη, a,θ (ξ)ϕ a,θ (ξ)ˆ cϕ R2 R2 0 a 0 where gε is the Friedrich molliﬁer of g. Then 2π ∞ 2π da α(a, θ)ϕ u(ξ)((ϕ ˆ) ∗ gε )(ξ) dθ dξ. Iε = a,θ (ξ)ˆ a,θ v cϕ R2 a 0 0 Now, there exists a sequence {εj }∞ j=1 of positive numbers such that εj → 0 as j → ∞ and ˆ) ∗ gεj → 2π ϕ ˆ (ϕ a,θ v a,θ v

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in L2 (R2 ) and almost everywhere on R2 as j → ∞. Thus, 2π ∞ 4π 2 da Iεj → dθ dξ. α(a, θ)ϕ u ˆ (ξ) ϕ (ξ)ˆ v (ξ) a,θ a,θ cϕ R2 a 0 0 Obviously, using Lebesgue’s dominated convergence theorem, we see that Iεj → (LF,ϕ u, v)L2 (R2 ) as j → ∞. Therefore for all u and v in L2 (R2 ), (LF,ϕ u, v)L2 (R2 ) = (Tm u, v)L2 (R2 ) , where Tm is the Fourier multiplier with symbol m given by 4π 2 2π ∞ 2 da dθ, ξ ∈ R2 . m(ξ) = α(a, θ)|ϕ a,θ (ξ)| cϕ 0 a 0

References [1] J.-P. Antoine, R. Murenzi, P. Vandergheynst and S. T. Ali, Two-Dimensional Wavelets and their Relatives, Cambridge University Press, 2004. [2] R. R. Coifman and Y. Meyer, Au Del` a des Op´erateurs Pseudo-Diﬀ´erentiels, Ast´erisque 57, 1978. [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [4] S. Janson and J. Peetre, Paracommutators-boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305 (1988), 467–504. [5] L. Peng, On the compactness of commutators, Ark. Mat. 26 (1988), 315–325. [6] L. Peng, Wavelets and paracommutators, Ark. Mat. 31 (1993), 83–99. [7] L. Peng and M. W. Wong, Compensated compactness and paracommutators, J. London Math. Soc. 62(2) (2000), 505–520. [8] M. W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [9] M. W. Wong, Localization operators on the aﬃne group and paracommutators, in Progress in Analysis, World Scientiﬁc, 2003, 663–669. Yu Liu and M. W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada e-mail: [email protected] [email protected] Submitted: September 19, 2006 Revised: December 15, 2006

Integr. equ. oper. theory 58 (2007), 111–132 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010111-22, published online April 16, 2007 DOI 10.1007/s00020-007-1483-3

Integral Equations and Operator Theory

Test Function Criteria for Hankel Operators Sandra Pott, Martin Smith and David Walsh Abstract. It is well known that there are classes of test functions such that a Hankel operator is bounded if and only if it is bounded on those functions. Criteria are derived which determine whether a Hankel operator is compact or belongs to a particular Schatten class, in terms of its action on those test functions. Mathematics Subject Classification (2000). Primary 47B35, 47B10. Keywords. Hankel operators, test functions, reproducing kernels .

1. Introduction The characterisations of bounded, compact and Schatten class Hankel operators in terms of functional analytic properties of their symbols are due to Nehari [10], Hartman [5] and Peller [14] (see also Rochberg [16]) respectively (see e.g. [15] for details). Unfortunately, these conditions can prove rather diﬃcult to check in practice. In an attempt to ﬁnd criteria which are easier to verify, the notion of test functions has been introduced. By this we mean a small set of functions (of unit norm) with a simple form, such that operator theoretic properties of a Hankel operator may be determined solely by the operator’s action on these functions. In [1], Bonsall showed that a Hankel operator on the Hardy space of the disc is bounded precisely if it is bounded on the normalised reproducing kernels of the Hardy space. This is an important example of the reproducing kernel thesis; see [6], [11] and [17]. Reproducing kernels as test functions have also been used in control theory in order to ﬁnd accessible criteria for certain important properties of observation operators, see e.g. [13]. Other sets of test functions, which are “localised” in the frequency domain, are shown to characterise boundedness in [1], [2], [7] and [9]. It is well known that the compactness of a Hankel operator may also be characterised in terms of its behaviour on the reproducing kernels. In [8] and [18],

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characterisations of Schatten class membership for Hankel operators were found in terms of their actions upon reproducing kernels, at least for the Schatten Sp classes with p 2. For 1 p < 2, characterisation of Sp Hankel operators in terms of their actions on the derivatives of kernels was also found. In this paper, we characterise the compact and Schatten class Hankel operators in terms of various test functions. We ﬁnd a class of test functions which characterises boundedness, compactness and Sp membership, for all p > 1. We also consider Berezin transform- type expressions. Using these, we are able to test Sp membership, for all p 1, but not boundedness. We shall consider Hankel operators on the Lebesgue space L2 (R+ ); it is well known that these are unitarily equivalent to Hankel operators on the Hardy space of the disc, see e.g. [15] p. 46. Given a function (or more generally a distribution) k on R+ = (0, ∞), we formally deﬁne the Hankel operator with symbol k as follows: ∞ Γk f (x) = k(x + y)f (y)dy, Γk : L2 (R+ ) → L2 (R+ ), 0

Γk being densely deﬁned on smooth functions with compact support. For details on Hankel operators, we refer to [12] and [15]. For f ∈ L2 (R), let f ∧ denote the (unitary) Fourier transform of f , deﬁned by ∞ 1 f ∧ (x) = √ f (t)e−ixt dt, 2π −∞ and let f ∨ denote the inverse Fourier transform of f . Let B(H) and K(H) denote the collections of bounded operators and compact operators on a Hilbert space H, respectively. Throughout this paper, let Cp denote various absolute constants depending only upon p, whose values may change from line to line. 1.1. Classes of test functions We shall consider the following classes of test functions. These are considered in [9] and they are the analogues of the test functions on the unit circle which are employed in [1], [2], [7] and [8]. Let C+ = {z ∈ C : Imz > 0}. Given z ∈ C+ , we deﬁne the function Φz ∈ L2 (R+ ), √ Φz (x) = 2Imz e−izx χR+ (x), where χI denotes the characteristic function of a measurable subset I ⊆ R. Note that Φz ≡ 1 and that Φz is the Fourier transform of the normalised reproducing kernel of the Hardy space H 2 (C+ ), associated with z, i.e. √ i . (1) Φ∨ z = 2 πImz kz , where kz (t) = 2π(t − z)

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Given τ > 0 and u ∈ R we deﬁne the function Ψτ,u ∈ L2 (R+ ), Ψτ,u (x) = τ −1/2 e−iux χ(0,τ ) (x), so Ψτ,u ≡ 1. We note that, for z = u + iv, with v > 0, ∞ √ Φz = 2v 3/2 τ 1/2 e−vτ Ψτ,u dτ.

(2)

τ =0

This is easily proved; let x > 0, then we have ∞ √ √ 3/2 ∞ 1/2 −vτ 2v τ e Ψτ,u (x)dτ = 2v 3/2 e−iux e−vτ dτ, τ =0 τ =x √ = 2v 1/2 e−iux e−vx = Φz (x). Finally, given τ > 0 and u ∈ R we deﬁne the function Ωτ,u ∈ L2 (R+ ), Ωτ,u (x) = τ −1/2 e−iux χ(τ,2τ ) (x), so Ωτ,u ≡ 1. Note that the test functions Ψτ,u and Ωτ,u are closely related. It may be easily checked that ∞ √ 2−n/2 Ω2−n τ,u , (3) Ωτ,u = 2Ψ2τ,u − Ψτ,u and Ψτ,u = n=1

see [9]. We note that the test functions Ψτ,u and Ωτ,u are localised in the time domain.

2. Boundedness criteria Nehari’s theorem states that a Hankel operator Γk on L2 (R+ ) is bounded if and only if there exist a bounded function g ∈ L∞ (R) such that g ∧ χR+ = k. This in turn may be characterised in terms of functions of bounded mean oscillation, see e.g. [15] p. 52. We have the following equivalent criteria in terms of test functions. Theorem 2.1. Given a Hankel operator Γk , the following are equivalent. B1. Γk extends to a bounded operator on L2 (R+ ); B2. sup Γk Φz < ∞; z∈C+

B3. sup

Γk Ψτ,u < ∞;

sup

Γk Ωτ,u < ∞.

τ >0,u∈R

B4. τ >0,u∈R

Moreover, Γk is equivalent to each of the suprema given above, with absolute constants of equivalence.

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The fact that a Hankel operator Γk is bounded if and only if B2 holds is a consequence of Garsia’s equivalent norm on the functions of bounded mean oscillation, an observation ﬁrst made by Bonsall in [1]. The implication B1 ⇒ B4 is trivial and B4 ⇒ B3 ⇒ B2 may be deduced easily using (2) and (3). These results may be found in [9], see Theorem 2.3 and Corollary 2.10. See also [2] and [8] for analogous results for Hankel operators on the Hardy space of the disc.

3. Compactness criteria Hartman’s theorem states that a Hankel operator Γk on L2 (R+ ) is compact if and only if there exists a continuous function g on R, with equal limits at ±∞, such that g ∧ χR+ = k. This in turn may be characterised in terms of functions of vanishing mean oscillation, see e.g. [15] p. 53. We have the following equivalent criteria in terms of test functions. Theorem 3.1. Given a bounded Hankel operator Γk , the following are equivalent. K1. Γk ∈ K(L2 (R+ )); K2. lim sup Γk Φz = lim sup Γk Φz = 0; δ→0 Imz=δ

K3.

a.

R→∞ |z|=R

lim sup Γk Ψτ,u = 0,

τ →∞ u∈R

b.

lim sup Γk Ψτ,u = 0,

τ →0 u∈R

c. and, for any 0 < τ0 < τ1 < ∞, lim

sup

|u|→∞ τ ∈[τ0 ,τ1 ]

K4.

a.

Γk Ψτ,u = 0.

lim sup Γk Ωτ,u = 0,

τ →∞ u∈R

b.

lim sup Γk Ωτ,u = 0,

τ →0 u∈R

c. and, for any 0 < τ0 < τ1 < ∞, lim

sup

|u|→∞ τ ∈[τ0 ,τ1 ]

Γk Ωτ,u = 0.

Proof. K1 ⇔ K2. This is well known; see e.g. [11] pp. 220-221 or [15] p. 38 for the corresponding statement for Hankel operators on the Hardy space of the disc. This result may then be obtained by use of a standard unitary map taking the Hardy space of the disc to the Hardy space of the upper half plane (see [17]) and then taking Fourier transforms.

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K1 ⇒ K3. As compact operators convert weak convergence to norm convergence ([20] p. 10), it is suﬃcient to show that, for all f ∈ L2 (R+ ), lim sup | f, Ψτ,u | = lim sup | f, Ψτ,u | = lim

τ →∞ u∈R

τ →0 u∈R

sup

|u|→∞ τ ∈[τ0 ,τ1 ]

| f, Ψτ,u | = 0.

Also, as Ψτ,u ≡ 1, it is suﬃcient to test these assertions on dense subspaces of L2 (R+ ). The ﬁrst two assertions are easily proved by considering f ∈ L2 (R+ ) to be bounded and with compact support. To prove the third assertion, let f ∈ L2 (R+ ) be continuously diﬀerentiable and bounded. Using integration by parts, τ iu f, Ψτ,u = τ −1/2 f (x)iueiux dx, 0 τ f (x)eiux dx . = τ −1/2 f (τ )eiuτ − f (0) − 0

Thus, for τ ∈ [τ0 , τ1 ],

τ |u|| f, Ψτ,u | τ |f (x)|dx , and so |f (τ )| + |f (0)| + 0 −1/2 2χ[0,τ1 ] f ∞ + χ[0,τ1 ] f 1 τ0 → 0 as |u| → ∞. | f, Ψτ,u | |u| −1/2

K3 ⇒ K4. This is elementary as (3) implies that Γk Ωτ,u Γk Ψτ,u .

√ 2Γk Ψ2τ,u +

K4 ⇒ K3. Let us ﬁrst suppose that K4a holds and let ε > 0. Choose N ∈ N such that ∞ ε 2−n/2 Γk < . 2 n=N +1

By assumption, pick τ0 ∈ R+ such that τ > τ0 implies that for all u ∈ R, √ ( 2 − 1)ε . Γk Ωτ,u 2 By (3), if τ > 2N τ0 then Γk Ψτ,u

∞ n=1 N

2−n/2 Γk Ω2−n τ,u ,

ε 2−n/2 Γk Ω2−n τ,u + , 2 n=1 √ N ( 2 − 1)ε ε + , since 2−n τ > τ0 , n = 1, . . . N, 2−n/2 2 2 n=1 √ ∞ ( 2 − 1)ε ε + = ε, 2−n/2 2 2 n=1

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i.e. K3a holds. It is similarly shown that K4b implies K3b and K4c implies K3c. K3 ⇒ K2. By (2), for z = u + iv, with v > 0, √ 3/2 ∞ 1/2 −vτ √ τ e Γk Ψτ,u dτ = 2 Γk Φz 2v 0

0

∞

s1/2 e−s Γk Ψs/v,u ds, (4)

by a simple change of variable. Let ε > 0. By K3a and K3b, there exist M0 and M1 such that if s/v < M0 or s/v > M1 then Γk Ψs/v,u < ε. Choose δ and γ such that M1 δ ∞ 1/2 −s s e ds < ε and s1/2 e−s ds < ε. 0

M0 γ

So, if v < δ then by (4), M1 δ ∞ √ Γk Φz 2 Γk s1/2 e−s ds + ε s1/2 e−s ds , 0 M1 δ √ < 2ε (Γk + Γ(3/2)) , i.e. lim sup Γk Φz = 0.

(5)

Also, if v > γ then by (4), ∞ M0 γ √ Γk Φz 2 ε s1/2 e−s ds + Γk s1/2 e−s ds , 0 M0 γ √ < 2ε (Γk + Γ(3/2)) , i.e. lim sup Γk Φz = 0.

(6)

δ→0 Imz=δ

γ→∞ Imz=γ

Finally, we’ll show that there exists µ > 0 such that if δ v γ and |v| > µ then Γk Φz < ε, where z = u + iv. This, combined with (5) and (6), will show that limR→∞ sup|z|=R Γk Φz = 0. By (4), for such v we see that ∞ √ Γk Φz 2γ 3/2 τ 1/2 e−δτ Γk Ψτ,u dτ. τ =0

There exist τ0 , τ1 such that √ 3/2 τ0 1/2 −δτ 2γ τ e dτ < τ =0

∞ √ ε ε τ 1/2 e−δτ dτ < and 2γ 3/2 , 3Γk 3Γ k τ =τ1

(without loss of generality we may assume that τ1 > τ0 and Γk = 0). By K3c, there exists µ > 0 such that if τ ∈ [τ0 , τ1 ] and |u| > µ then √ Γk Ψτ,u < ε 3 2

τ1

τ 1/2 e−δτ dτ

−1 .

τ =τ0

Putting this together, we see that if δ v γ and |v| > µ then Γk Φz < ε.

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4. Sp criteria for arbitrary operators 4.1. Schatten Classes Any operator T ∈ K(H) has a Schmidt decomposition, so there exist orthonormal bases {en } and {σn } of H and a sequence {λn } with λn 0 and λn → 0 such that Tf =

∞

λn f, en σn ,

(7)

n=0

for all f ∈ H. Recall that, for 1 p < ∞, a compact operator T with such a decomposition belongs to the Schatten–von Neumann p-class, Sp (H), if and only if ∞ 1/p p λn < ∞. (8) T Sp = n=0

For T ∈ S1 (H), we deﬁne the trace of T by Tr(T ) =

∞

T en , en ,

n=0

for an arbitrary orthonormal basis {en }. Sp (L2 (R+ )) will be abbreviated to Sp and we shall henceforth refer to these classes as Schatten classes. Let Sp (Hank) denote the Banach space consisting of all Sp Hankel operators on L2 (R+ ). For further details about Schatten classes, see e.g. [12] Chapter 1, [15] Appendix 1 and [20] Chapter 1.4. When using test functions to determine boundedness of operators, there are obvious necessary conditions; e.g. if T ∈ B(L2 (R+ )) then supz∈C+ T Φz < ∞. Similarly, it is easy to describe necessary conditions for compactness, using weakconvergence arguments. The aim of this section is to derive necessary conditions for Sp membership, in terms of an operator’s action on the test functions. In fact, we shall ﬁnd conditions which are necessary for 2 p < ∞ and suﬃcient for 1 p 2. 4.2. Hilbert–Schmidt Operators Operators in S2 are known as Hilbert–Schmidt operators and they are particularly straightforward to characterise. Indeed, the following criteria apply to arbitary operators on L2 (R+ ), not just Hankel operators. Proposition 4.1. If T ∈ B(L2 (R+ )) then the following are equivalent: HS1. T ∈ S2 ; HS2.

sup v v>0

∞

u=−∞

T Φu+iv 2 du < ∞;

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HS3. sup τ τ >0

HS4.

∞

∞

u=−∞

τ =0

∞

u=−∞

Moreover, T 2S2 = sup 2v v>0

IEOT

T Ψτ,u2 du < ∞;

T Ωτ,u2 dudτ < ∞. ∞

u=−∞ ∞

T Φu+iv 2 du,

τ T Ψτ,u2 du 2π u=−∞ ∞ ∞ 1 T Ωτ,u 2 dudτ. = 2π log 2 τ =0 u=−∞ = sup τ >0

Proof. We shall only prove that HS1 is equivalent to HS3 and HS4; that HS1 is equivalent to HS2 is a continuous analogue of Lemma 1 from [18], which itself generalises [8] Theorem 1, and may anyway be proved in a similar manner to that below. To prove the equivalence of HS1 and HS3, let {en } be an arbitrary orthonormal basis of L2 (R+ ), so for τ > 0, ∞ ∞ ∞ 2 2 T Ψτ,u du = | T Ψτ,u , en | du = | T ∗ en , Ψτ,u |2 du. −∞

−∞ n

n

−∞

2

But, for all f ∈ L (R+ ), ∞ 2 ∨ 2π ∞ χ 2π χ 2 (0,τ ) f 2 , | f, Ψτ,u | du = (0,τ ) f (u) du = τ τ −∞ −∞ by the unitarity of the Fourier transform. So, ∞ τ T Ψτ,u2 du = sup χ(0,τ ) T ∗ en 2 , sup τ >0 2π −∞ τ >0 n T ∗en 2 = T ∗ 2S2 = T 2S2 , = n

see e.g. [20] pp. 17 and 21. The equivalence of HS1 and HS4 may be proved similarly; the key fact is, for f ∈ L2 (R+ ), ∞ ∞ ∞ ∞ 2 ∨ 1 χ 2 | f, Ωτ,u | dudτ = 2π (t,2τ ) f (u) dudτ, τ 0 −∞ 0 −∞ ∞ 1 χ (t,2τ ) f 2 dτ, = 2π τ 0 ∞ x dτ = 2π dx = 2π log 2f 2. (9) |f (x)|2 τ 0 x/2

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4.3. Sp Operators, with p = 2 We can use Proposition 4.1 to derive conditions which are necessary for Sp membership for p > 2 and suﬃcient for p < 2. Corollary 4.2. Let T ∈ B(L2 (R+ )) and consider the conditions

SP1.

∞

sup v v>0

u=−∞

SP2. sup τ τ >0

SP3.

∞

τ =0

T Φu+iv p du < ∞;

∞

u=−∞

∞

u=−∞

T Ψτ,up du < ∞;

T Ωτ,up dudτ < ∞.

1. Suppose that p > 2 and that T ∈ Sp . Then SP1, SP2 and SP3 hold and each expression is dominated by T pSp . 2. Suppose that 1 p < 2 and that any of SP1, SP2 and SP3 hold. Then T ∈ Sp , with an appropriate estimate on T pSp . Proof. Recall that, for a positive operator A on a Hilbert space, Tr(A) = A1/2 2S2 . We can therefore use Proposition 4.1 to derive the following trace formulae for positive operators on L2 (R+ ): ∞ Tr(A) = sup 2v

AΦu+iv , Φu+iv du, v>0 −∞ ∞ τ = sup

AΨτ,u , Ψτ,u du τ >0 2π −∞ ∞ ∞ 1

AΩτ,u , Ωτ,u dudτ. = 2π log 2 0 −∞ But, T pSp = Tr((T ∗ T )p/2 ), and if f is a unit vector in L2 (R+ ) then

(T ∗ T )p/2 f, f T ∗ T f, f p/2 , if p > 2

(T ∗ T )p/2 f, f T ∗ T f, f p/2 , if 1 p < 2, see e.g. [20] pp. 117. The results now follow easily.

We are interested in ﬁnding conditions which characterise precisely Schatten class membership for a Hankel operator in terms of its action on test functions, so it is appropriate to see whether the converses to the statements in Corollary 4.2 hold for Hankel operators. Unfortunately, this is not the case for SP1 and SP2; see [8] for analogous results on the disc. We will therefore seek new criteria in terms of these test functions. We shall be able to show that the converse to SP3 holds for Hankel operators, at least for p > 1.

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5. Sp Hankel operators, with p > 2 To help us characterise Sp (Hank) operators, with p > 2, we shall introduce some auxiliary test functions, which lie between the test functions Φz and Ψτ,u . For σ, τ > 0 and u ∈ R, let Ψσ (x) = τ −1/2 e−(σ/τ +iu)x χ[0,τ ] (x). τ,u

Also, let λ be the measure on C+ given by dλ(z) =

dA(z) , (Imz)2

where A denotes area measure on C+ . Theorem 5.1. If p > 2 then the following are equivalent: SPH1. Γk ∈ Sp (Hank); SPH2.

C+

SPH3.

SPH4. For all σ > 0,

∞ τ =0 ∞ τ =0

SPH5.

∞

τ =0

Γk Φz p dλ(z) < ∞;

∞

u=−∞

∞

u=−∞

∞

u=−∞

Γk Ψτ,u p dudτ < ∞;

Γk Ψστ,u p dudτ < ∞; Γk Ωτ,u p dudτ < ∞.

Moreover, Γk pSp is equivalent to the expressions above, with constants of equivalence depending only upon p (and σ in the case of SPH4). Proof. SPH1 ⇔ SPH2. This is a continuous analogue of [18] Theorem 3. The main tool is Peller’s characterisation of Schatten class Hankel operators in terms of the Besov space properties of their symbols (see [14] and [16]). SPH3 ⇒ SPH2. A simple change of variables shows that ∞ ∞ τ 1/2 e−vτ dτ = τ 1/2 e−τ dτ = Γ(3/2) v 3/2 0

0

for all v > 0. Consequently, Jensen’s inequality, applied to the convex function t → tp , implies that for some constant Cp , p ∞ ∞ 3/2 1/2 −vτ 3/2 τ e Γk Ψτ,u dτ Cp v τ 1/2 e−vτ Γk Ψτ,u p dτ, v 0 0 ∞ i.e. Γk Φz p Cp v 3/2 τ 1/2 e−vτ Γk Ψτ,u p dτ, 0

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by (4). Therefore, ∞ ∞ ∞ Γk Φz p dλ(z) Cp v −1/2 τ 1/2 e−vτ Γk Ψτ,u p dτ dudv, 0 −∞ 0 C+ ∞ ∞ ∞ 1/2 −1/2 −vτ τ v e dv Γk Ψτ,u p dudτ, = Cp 0

0

−∞

by Fubini and Tonelli’s Theorems. However, another easy change of variables shows that ∞ ∞ τ 1/2

v −1/2 e−vτ dv =

0

v −1/2 e−vτ dv = Γ(1/2)

0

for all τ > 0. Therefore, p Γk Φz dλ(z) Cp

0

C+

∞

∞ −∞

Γk Ψτ,u p dudτ,

for some constant Cp . SPH2 ⇒ SPH4. Let Tτ denote the right-shift by τ on L2 (R+ ), i.e.

f (x − τ ), x τ, Tτ f (x) = 0, x < τ. Note that Γk Tτ = Tτ∗ Γk , see e.g. [11] p. 273. If z = u + iσ/τ , then a simple veriﬁcation shows that Ψστ,u = (2σ)−1/2 Φz − e−(σ+iuτ ) Tτ Φz , which gives Γk Ψστ,u = (2σ)−1/2 Γk Φz − e−(σ+iuτ ) Tτ∗ Γk Φz , and soΓk Ψστ,u (2σ)−1/2 (1 + e−σ )Γk Φz , as Tτ∗ is a contraction. Consequently, ∞ ∞ (1 + e−σ )p ∞ ∞ σ p Γk Ψτ,u dudτ Γk Φu+iσ/τ p dudτ, (2σ)p/2 0 −∞ 0 −∞ σ(1 + e−σ )p Γk Φz p dλ(z), = (2σ)p/2 C+ letting z = u + iσ/τ = u + iv. SPH4 ⇒ SPH3. We’ll use a dyadic approach, introduced in [9]. For τ > 0, let Dτ be the collection of all dyadic subintervals of [0, τ ). Let Dτ,n = {I ∈ Dτ : |I| = 2−n τ√}. For an interval I, let lI denote the left endpoint of I. Fix some 1/2 < A < 1/ 2 and let σ = (2A − 1)/2 > 0. For any I ∈ Dτ , let hI (x) = |I|−1/2 e−σ/|I|(x−lI ) χI (x). Then (see [9] Lemma 2.4), for all τ > 0, there exist non-negative constants (αI )I∈Dτ such I∈Dτ αI hI converges uniformly on [0, τ ) to 1, i.e. αI hI = χ[0,τ ) . I∈Dτ

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Moreover, for n = 0, 1, 2, . . .,

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αI τ 1/2 2(n−2)/2 An .

(10)

I∈Dτ,n

Consequently,

Ψτ,u (x) = τ −1/2

αI e−iux hI (x),

I∈Dτ

= τ −1/2

αI |I|−1/2 e−iulI e−σ/|I|(x−lI )−iu(x−lI ) χ[0,|I|) (x − lI ),

I∈Dτ

i.e. Ψτ,u = τ

−1/2

αI e−iulI TlI Ψσ|I|,u ,

I∈Dτ

and thus Γk Ψτ,u = τ −1/2

αI e−iulI Tl∗I Γk Ψσ|I|,u ,

I∈Dτ

so Γk Ψτ,u τ −1/2

αI Γk Ψσ|I|,u .

(11)

I∈Dτ

Choose r, ε such that 1 1 r 1 1 < < ε < and > . p r 2 p 2(1 − ) Note that this is possible, e.g. let ε = 2/(p + 2) and choose any r such that 1/p < 1/r < ε. √ Let A = (1/2)r/p ; then 1/2 < A < 1/ 2, as 1/2 < r/p < 1. Note also that √ A1−ε = (1/2)r(1−ε)/p < 1/ 2, as r(1 − ε)/p > 1/2 and Aεp = (1/2)εr < 1/2 as εr > 1. By (11), we have p ∞ ∞ ∞ ∞ Γk Ψτ,u p dudτ τ −p/2 αI Γk Ψσ|I|,u dudτ 0

−∞

0

∞

=

τ −p/2

0

=2

∞

0 −p

τ −p/2

0

∞

∞ −∞ ∞

 

∞

n=0 I∈Dτ,n

−∞

I∈Dτ

p αI Γk Ψσ|I|,u  dudτ,

∞

τ 1/2 2(n−2)/2 An Γk Ψσ2−n τ,u dudτ, p ∞n=0 2n/2 An(1−ε) Aεn Γk Ψσ2−n τ,u dudτ,

−∞ ∞

−∞

p

n=0

√ using (10). However, since A1−ε < 1/ 2, ∞ n=0

2n/2 An(1−ε) < ∞.

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Therefore, applying Jensen’s inequality, we get ∞ ∞ ∞ ∞ ∞ Γk Ψτ,u p dudτ Cp 2n/2 An(1−ε) Aεpn Γk Ψσ2−n τ,u p dudτ 0

−∞

0

= Cp = Cp

∞ n=0 ∞

−∞ n=0

2

n/2

n(1−ε)

A

εpn

A

∞

−∞ ∞ ∞

0

2n/2 An(1−ε) Aεpn 2n

0

n=0

1−ε

As A

√ < 1/ 2 and Aεp < 1/2, ∞

∞

Γk Ψσ2−n τ,u p dudτ

−∞

Γk Ψστ,u p dudτ

2n/2 An(1−ε) Aεpn 2n < ∞,

n=0

which implies that ∞ ∞ 0

−∞

Γk Ψτ,u p dudτ Cp

∞

0

∞

−∞

Γk Ψστ,u p dudτ.

SPH5 ⇒ SPH3. By (3),

∞

∞

and so 0

−∞

∞

2−n/2 Γk Ω2−n τ,u n=1 p ∞ ∞ ∞ p −n/2 Γk Ψτ,u dudτ 2 Γk Ω2−n τ,u dudτ Γk Ψτ,u

0

−∞

n=1

As p > 2, choose ε > 0 such that 0 < ε < (p − 2)/(p − 1); it easily follows that ε + p(1 − ε) > 2. Then ∞ p ∞ p −n/2 −nε/2 −n(1−ε)/2 2 Γk Ω2−n τ,u = 2 2 Γk Ω2−n τ,u n=1

n=1 ∞

Cp

2−nε/2 2−np(1−ε)/2 Γk Ω2−n τ,u p ,

n=1

−nε/2 2 < ∞. Consequently, for some constant Cp by Jensen’s inequality, since ∞ ∞ ∞ ∞ ∞ Γk Ψτ,u p dudτ Cp 2−n(ε+p(1−ε))/2 Γk Ω2−n τ,u p dudτ 0

−∞

0

n=1 ∞

−∞ ∞ ∞

2n(1−(ε+p(1−ε))/2) 0 n=1 ∞ ∞ p = Cp Γk Ωτ,u dudτ,

= Cp

as

0

2

n(1−(ε+p(1−ε))/2)

−∞

Γk Ωτ,u p dudτ

−∞

< ∞ since ε + p(1 − ε) > 2.

SPH1 ⇒ SPH5. This follows directly from Corollary 4.2.

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6. Sp Hankel operators, with 1 p < 2 In [18] Theorem 4, a characterisation of Sp (Hank) for 1 p 2 is obtained in terms of a condition in which the reproducing kernels are replaced by their derivatives. In this context, for z ∈ C+ , let Λz (x) = 2(Imz)3/2 xe−izx χR (x), +

so that Λz is the normalised Fourier transform of the derivative of the reproducing kernel of H 2 (C+ ) associated with z. Then we have Theorem 6.1. For 1 p 2, Γk ∈ Sp (Hank) if and only if Γk Λz p dλ(z) < ∞. C+

Moreover, Γk pSp is equivalent to the above expression (with constants of equivalence dependent upon p). We would like to obtain characterisations of Sp membership in terms of the test functions Φz , Ψτ,u and Ωτ,u . However, we have not been able to do this for the ﬁrst two sets of test functions (recall that the converses to Corollary 4.2 SP1 and SP2 do not hold). However, we shall show that the converse to SP3 does hold, i.e. we have the following: Theorem 6.2. For 1 < p < ∞, Γk ∈ Sp (Hank) if and only if ∞ ∞ Γk Ωτ,u p dτ du < ∞. τ =0

(12)

u=−∞

Moreover, Γk pSp is equivalent to the above expression (with constants of equivalence dependent upon p). Note that, by Proposition 4.1 and Theorem 5.1, it only remains to show necessity and suﬃciency of (12) for 1 < p < 2. It may be easily shown that (12) fails when p = 1 even for a rank one Hankel operator Γk . We shall use Rochberg’s decomposition theorem for Sp Hankel operators in terms of p sums of rank one Hankel operators from [16]. Let d denote the hyperbolic metric on C+ . Given η > 0, a sequence {zn } of points in C+ is called an η-lattice if inf d(zn , zm ) > η/100 and for all z ∈ C+ , inf d(z, zn ) < η. n

n=m

2

Given g, h ∈ L (R+ ), let h ⊗ g denote the rank one operator on L2 (R+ ) deﬁned by (h ⊗ g)f = f, gh. For all z ∈ C+ , Φz ⊗ Φ−z is easily seen to be a rank one Hankel operator. Rochberg’s result states that, for 1 p < ∞, Γk ∈ Sp (Hank) if and only if there is an η-lattice {zn }, with η > 0 and a sequence {λn } ∈ p such that λn Φzn ⊗ Φ−zn . (13) Γk = n

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Test Function Criteria for Hankel Operators

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The following result implies that Theorem 6.2 holds for rank one Hankel operators. Lemma 6.3. For all 1 0, (2y)1/2 τ −1/2 e−i(z+u)τ e−i(z+u)τ − 1 , | Ωτ,u , Φ−z | = |z + u| (2y)1/2 τ −1/2 e−yτ (e−yτ + 1) (8y)1/2 τ −1/2 e−yτ . ((x + u)2 + y 2 )1/2 ((x + u)2 + y 2 )1/2 Therefore,

∞

∞

| Ωτ,u , Φ−z |p dudτ ∞ ∞ du p/2 −p/2 −pyτ (8y) τ e dτ , 2 2 p/2 ∞ −∞ ((x + u) + y ) ∞0 dw = 8p/2 s−p/2 e−ps ds = Cp , p/2 2 0 −∞ (w + 1) 0

−∞

using the substitutions s = yτ and w = (x + u)/y.

In order to extend this to arbitrary Sp (Hank) operators, we require the following technical result, which may be found in [19]. Proposition 6.4. For all η > 0 and 1 < p < 2, there exists a constant Cη,p such that for all η-lattices {zn } and all {αn } ∈ p , αn Φzn Cη,p {αn }p . 2 n L (R+ )

Proof. Recall that kz denotes the reproducing kernel of H 2 (C+ ), as deﬁned in (1). Since {zn } is an η-lattice, the points are uniformly discrete with respect to the pseudo-hyperbolic metric on C+ ; i.e. there exists a constant Cη > 0 depending upon η such that zm − zn Cη . inf m=n zm − zn It therefore follows that, for all 1 < p < 2, 1/2 αn (Imzn ) kzn Cη,p {αn }p , 2 n H (C+ )

p

for all {αn } ∈ , see [19]. The required result now follows from the unitarity of the Fourier transform.

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Proof of Theorem 6.2. By Proposition 4.2 we only need to show the necessity of (12). Suppose that Γk is decomposed as in (13). Then, by Proposition 6.4, p p p Γk Ωτ,u = λn Ωτ,u , Φ−zn Φzn Cη,p |λn |p | Ωτ,u , Φ−zn | . n n Therefore, by Lemma 6.3, ∞ ∞ τ =0

u=−∞

Γk Ωτ,u p dτ du Cη,p

|λn |p < ∞.

n

Let Lp (R × R+ , L2 (R+ )) denote the vectorial Banach space of functions taking values in L2 (R+ ). Deﬁne the operator A on Hankel operators by AΓk = Γk Ωτ,u . We have seen that A maps Sp (Hank) into Lp (R × (0, ∞), L2 (R+ )); we shall apply the Closed Graph Theorem to show that A is bounded and hence complete the proof. So, let {Γkn } and Γk be Hankel operators such that Γkn → Γk in Sp (Hank) and suppose that AΓkn → F in Lp (R × R+ , L2 (R+ )). Then Γkn − Γk → 0 and so Γkn Ωτ,u → Γk Ωτ,u for all (τ, u) ∈ (0, ∞) × R. Therefore, Γk Ωτ,u = F (τ, u) a.e., so A has a closed graph and thus is bounded. We conjecture that the operator A deﬁned above is of weak type (1, 1) i.e. there exists C1 such that for all Γk ∈ S1 (Hank) and t > 0, C1 Γk S1 . t We are able to show that this is true for rank one Hankel operators, using the calcuations in Lemma 6.3; the conjecture is open for general S1 (Hank) operators. |{(u, τ ) ∈ R × R+ : Γk Ωτ,u > t}|

7. Berezin Transform type criteria A Hankel operator Γk belongs to Sp (Hank) precisely when k ∨ belongs to the Besov space Bp . This characterisation can be seen in another way by means of a criteria involving an expression which is closely related to the Berezin transform (see e.g. [11] p.131). Theorem 7.1. For 1 < p < ∞, Γk ∈ Sp (Hank) if and only if p | Γk Φ−z , Φz | dλ(z) < ∞;

(14)

C+

moreover Γk pSp is equivalent to the above expression (with constants of equivalence dependent upon p). Proof. Elementary calculations show that, for z ∈ C+ , ∞ ∞

Γk Φ−z , Φz = 2Imz k(x + y)eiz(x+y) dydx, 0

0

Vol. 58 (2007)

Test Function Criteria for Hankel Operators

127

∞

xk(x)eizx dx, 0 ∞ d = −2iImz k(x)eizx dx, dz 0 √ = −2 2π iImz (k ∨ ) (z),

= 2Imz

(see also [1] Proposition 4). Consequently, (14) holds if and only if p ∨ p−2 dA(z) < ∞, (k ) (z) (Imz) C+

∨

i.e. precisely when k belongs to the analytic Besov space Bp of C+ , see [15] p. 239 and [16] Theorem 1. We will also characterise Sp (Hank) operators in terms of an analogous expression involving the test functions Ωτ,u . We show that: Theorem 7.2. If Γk ∈ B(L2 (R+ )) then, for 1 p < ∞, Γk ∈ Sp (Hank) if and only if ∞ ∞ | Γk Ωτ,u , Ωτ,−u |p dudτ < ∞. (15) τ =0

u=−∞

Moreover, Γk pSp is equivalent to the above expression (with constants of equivalence dependent upon p). In fact, when p = 2, the equivalence in Theorem 7.2 is (a constant multiple of) an isometry. We recall that (see e.g. [12] p. 67) ∞ Γk 2S2 = x|k(x)|2 dx. (16) 0

Lemma 7.3. There exists a constant c > 0 such that, for all Γk ∈ S2 (Hank) ∞ ∞ | Γk Ωτ,u , Ωτ,−u |2 dudτ = cΓk 2S2 . (17) τ =0

u=−∞

Proof. Elementary calculations show that 2τ 2τ

Γk Ωτ,u , Ωτ,−u = τ −1 k(x + y)e−iu(x+y) dydx, τ τ 4τ = τ −1 (τ − |x − 3τ |) k(x)e−iux dx, 2τ ∧

= (Στ k) (u), √ where Στ (x) = 2πτ −1 (τ − |x − 3τ |) χ(2τ,4τ ) (x). Therefore ∞ ∞ ∞ ∧ | Γk Ωτ,u , Ωτ,−u |2 dudτ = (Στ k) 2 dτ, 0 −∞ 0 ∞ Στ k2 dτ, = 0

128

Pott, Smith and Walsh

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(τ − |x − 3τ |)2 dτ dx τ2 0 x/4 ∞ 1/2 2 (τ − |1 − 3τ |) = 2π x|k(x)|2 dτ dx, τ2 0 1/4 = cΓk 2S2 , ∞

= 2π

|k(x)|2

x/2

where we have used the unitarity of the Fourier transform, Fubini and Tonelli’s Theorems, a simple change of variable and (16). Note that (17) may be polarised to give ∞ ∞

Γk Ωτ,u , Ωτ,−u Γj Ωτ,u , Ωτ,−u dτ du = cTr Γ∗j Γk , whenever

(18)

τ =0 u=−∞ Γ∗j Γk ∈ S1 .

We can show that condition (15) is necessary for arbitrary Sp operators. Lemma 7.4. For 1 p < ∞ there exists a constant Cp such that, if T ∈ Sp then ∞ ∞ | T Ωτ,u , Ωτ,−u |p dτ du Cp T Sp . (19) τ =0

u=−∞

Proof. Clearly, if T ∈ B(L2 (R+ )) then supτ >0,u∈R | T Ωτ,u , Ωτ,−u | T . Therefore, by interpolation of Schatten classes (see e.g. [20] p. 31) it suﬃces to prove (19) when p = 1. But for T decomposed as in (7), | T Ωτ,u , Ωτ,−u |

∞

|λn | | Ωτ,u , en σn , Ωτ,−u | ,

n=0

∞

∞

| T Ωτ,u , Ωτ,−u |dτ du

so 0

∞

n=0

|λn |

∞

0

∞

∞

∞

| Ωτ,u , en |2 dτ du

1/2

= 2π log 2

∞

0 ∞

∞

∞

| σn , Ωτ,−u |2 dτ du

1/2 ,

|λn |en σn = 2π log 2T S1 ,

n=0

where we have used the Cauchy-Schwarz inequality and (9). Let P be the averaging projection onto Hankel operators, so if ∞ Af (x) = K(x, y)f (y)dy, 0

for some kernel K then PA = ΓK , where 1 x K(x) = K(x − y, y)dy. x 0

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Test Function Criteria for Hankel Operators

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It is well known that P is the orthogonal projection from S2 onto S2 (Hank) and that P is bounded on Sp for 1 0, u ∈ R. Proof. It is easily veriﬁed that Ωτ,−u ⊗ Ωτ,u is the integral operator with kernel Kτ,u (x, y) = τ −1 χ(τ,2τ )(x)χ(τ,2τ ) (y)eiu(x+y) , , where and so P (Ωτ,−u ⊗ Ωτ,u ) = ΓK τ,u eiux (τ − |x − 3τ |) , K τ,u (x) = xτ = 0, otherwise. Therefore, Kτ,u

L1 (R

+)

4τ

= 2τ

2τ < x < 4τ,

τ − |x − 3τ | dx = xτ

4

2

1 − |y − 3| dy = M, y

by a simple change of variables. Since Γk S1 kL1 (R+ ) for any k ∈ L1 (R+ ) (see e.g. [12] p. 68), the result follows. Proof of Theorem 7.2. By Lemma 7.4, we only need to prove the suﬃciency of (15). First suppose that 1 < p < ∞ and (15) holds. Letting q = p/(p − 1) and using H¨ older’s inequality, we see that for all Γj ∈ Sq (Hank), ∞ ∞

Γj Ωτ,u , Ωτ,−u Γk Ωτ,u , Ωτ,−u dτ du

0

0

∞

−∞ ∞

−∞

1/p

p

| Γk Ωτ,u , Ωτ,−u | dτ du

i.e. c |Tr (Γ∗k Γj )| Cq

0

∞

∞ 0

∞

−∞

∞

−∞

q

1/q

| Γj Ωτ,u , Ωτ,−u | dτ du

| Γk Ωτ,u , Ωτ,−u |p dτ du

,

1/p Γj Sq ,

by (18) and Lemma 7.4. But the dual space of Sq (Hank) is isomorphic to Sp (Hank), under the pairing induced by the trace (this follows from the duality of Sp and Sq and the existence of the bounded projection P from Sp onto Sp (Hank)). It follows that Γk ∈ Sp , with the appropriate estimate on Γk Sp . Now suppose that p = 1 and (15) holds. It follows, using Lemma 7.5 that ∞ ∞

Γk Ωτ,u , Ωτ,−u P (Ωτ,−u ⊗ Ωτ,u ) dudτ 0 −∞ S ∞ 1∞ M | Γk Ωτ,u , Ωτ,−u |dudτ. 0

−∞

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Pott, Smith and Walsh

But, we claim that, if c is as in (17), then ∞ ∞

Γk Ωτ,u , Ωτ,−u P (Ωτ,−u ⊗ Ωτ,u ) dudτ. Γk = c−1 0

IEOT

(20)

−∞

Note that, as (τ, u) → Γk Ωτ,u , Ωτ,−u is a bounded function, (15) also holds for p = 2 and so Γk ∈ S2 (Hank), by Lemma 7.3. Thus, to show (20), it is suﬃcient to show that for all Γj ∈ S2 (Hank), ∞ ∞ ∗ −1 ∗ Tr(Γj Γk ) = c Tr Γj

Γk Ωτ,u , Ωτ,−u P (Ωτ,−u ⊗ Ωτ,u ) dudτ , 0

or equivalently since

−∞

Tr(Γ∗j (PT ))

Tr(Γ∗j Γk ) = c−1

∞

0

∞

−∞

= Tr(Γ∗j T ), for all T ∈ S2 ,

Γk Ωτ,u , Ωτ,−u Tr Γ∗j (Ωτ,−u ⊗ Ωτ,u ) dudτ

holds for all Γj ∈ S2 (Hank). Note that Tr(Γ∗j (Ωτ,−u ⊗Ωτ,u )) = Tr((Γ∗j Ωτ,−u )⊗Ωτ,u ) = Γ∗j Ωτ,−u , Ωτ,u = Γj Ωτ,u , Ωτ,−u . Therefore, it is suﬃcient to show that ∞ ∞ ∗ −1 Tr(Γj Γk ) = c

Γk Ωτ,u , Ωτ,−u Γj Ωτ,u , Ωτ,−u dτ du; 0

−∞

but this is just (18). Consequently, ∞ −1 Γk S1 c M 0

∞

−∞

| Γk Ωτ,u , Ωτ,−u |dudτ.

8. Conclusions and Extensions We note that there are analogous results for Hankel operators on the Hardy space of the disc, in terms of the test functions considered in [1], [2], [7] and [8]. We also note that the criteria which characterise bounded, compact and Schatten class Hankel operators in terms of their action on reproducing kernels may be shown to characterise the boundedness, compactness and Schatten class membership of other important classes of operators, including vectorial Hankel operators, Carleson embeddings and weighted composition operators, see [3], [4] and [18]. It would be interesting to see if the criteria in terms of the other test functions can be used. For instance, if p > 2 then Theorem 5.1 condition SPH2 is necessary for arbitrary Sp operators, see [18] Theorem 1. Is it true that, for all T ∈ Sp ∞ ∞ T Ψτ,up dτ du Cp T pSp , τ =0

u=−∞

i.e. is condition SPH3 necessary for arbitrary Sp operators?

Vol. 58 (2007)

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The ﬁrst author gratefully acknowledges ﬁnancial support provided by the Nuﬃeld Foundation and EPSRC. The second author gratefully acknowledges ﬁnancial support provided through EPSRC grant GR/R97610/01 and the European Community’s Human Potential Programme under contract HPRN-CT-2000-00116 (Analysis and Operators).

References [1] F. F. Bonsall, Boundedness of Hankel Matrices, J. London Math. Soc. (2) 29 (1984) 289–300. [2] F. F. Bonsall, Conditions for Boundedness of Hankel Matrices, Bull. London Math. Soc. 26 (1994) 171–176. [3] Z. Harper, Operator theory applications of the discrete Weiss conjecture, Integral Equations Operator Theory, to appear [4] Z. Harper and M. P. Smith Testing Schatten class Hankel operators, Carleson embeddings and weighted composition operators on reproducing kernels, preprint, (2004). [5] P. Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958) 862–866. [6] V. P. Havin and N. K. Nikolski, Stanislav Aleksandrovich Vinogradov, His Life and Mathematics, in Oper. Theory: Adv. Appl., Vol. 113, Birkh¨ auser, Basel-Boston, 2000, 1–18. [7] F. Holland and D. Walsh, Boundedness Criteria for Hankel Operators, Proc. R. Ir. Acad. 84A(2) (1984) 141–154. [8] F. Holland and D. Walsh, Hankel Operators in von–Neumann–Schatten Classes, Ill. J. Math. 32 (1988) 1–22. [9] B. Jacob, J. R. Partington and S. Pott, Conditions for admissibility of observation operators and boundedness of Hankel operators, Integral Equations Operator Theory 47 (2003) 315–338. [10] Z. Nehari, On bounded bilinear forms, Ann. Math., 64 (1957) 153–162. [11] N. K. Nikolski, “Operators, Functions and Systems : An Easy Reading Volume I : Hardy, Hankel and Toeplitz,” Mathematical Surveys and Monographs Volume 92, American Mathematical Society, 2002. [12] J. R. Partington, “An Introduction to Hankel Operators,” Cambridge University Press, Cambridge, 1988. [13] J. R. Partington, G.Weiss, Admissible observation operators for the right-shift semigroup, Math. Control Signals Systems 13 (2000), 179–192. [14] V. V. Peller, Hankel Operators of class Cp and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Math. USSRSbornik 41 (1982), 74–83. [15] V. V. Peller, “Hankel Operators and their Applications,” Springer monographs in mathematics, Springer-Verlag, New York, 2003. [16] R. Rochberg, Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J. 31 (1982) 913–925.

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[17] M. P. Smith, The Reproducing Kernel Thesis for Toeplitz Operators on the Paley– Wiener Space, Integral Equations Operator Theory 49 (2004) 111–122. [18] M. P. Smith, Testing Schatten class Hankel operators and Carleson embeddings via reproducing kernels, J. London Math. Soc.(2) 71 (2005), no. 1, 172–186. [19] M. P. Smith, Bounded evaluation operators from H p into q , Studia Mathematica, to appear [20] K. Zhu, “Operator Theory in Function Spaces,” Dekker, New York, 1990. Sandra Pott Department of Mathematics University of Glasgow Glasgow G12 8QW United Kingdom e-mail: [email protected] Martin Smith Holy Cross Sixth Form College Manchester Rd Bury BL9 9BB United Kingdom e-mail: [email protected] David Walsh Department of Mathematics NUI Maynooth Maynooth, Co. Kildare Ireland e-mail: [email protected] Submitted: September 16, 2005 Revised: January 19, 2007

Integr. equ. oper. theory 58 (2007), 133–152 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010133-20, published online December 26, 2006 DOI 10.1007/s00020-006-1478-5

Integral Equations and Operator Theory

Integral Equations on Function Spaces and Dichotomy on the Real Line Adina Luminit¸a Sasu Abstract. The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted T (R) and H(R) and we prove that if V, W are Banach function spaces with V ∈ T (R) and W ∈ H(R), then the admissibility of the pair (W (R, X), V (R, X)) for an evolution family U = {U (t, s)}t≥s implies the uniform dichotomy of U. In addition, we consider a subclass W(R) ⊂ H(R) and we prove that if W ∈ W(R), then the admissibility of the pair (W (R, X), V (R, X)) implies the uniform exponential dichotomy of the family U. This condition becomes necessary if V ⊂ W . Finally, we present some applications of the main results. Mathematics Subject Classification (2000). Primary 34D09; Secondary 34D05. Keywords. Integral equation, function spaces, dichotomy, evolution family.

1. Introduction In the study of dichotomy of evolution families an important tool is the solvability of an associated integral equation. Speciﬁcally, if J ∈ {R+ , R}, one associates with an evolution family U = {U (t, s)}t≥s the equation t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, t ≥ s, t, s ∈ J (EU ) s

with v ∈ I(J, X) called the input space and f ∈ O(J, X) called the output space and one studies the connections between the solvability of the equation (EU ) and the uniform (exponential) dichotomy of the family U. For the case J = R+ one considers the initial stable subspace Xs,O(R+ ,X) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)}. The work was supported by the CEEX Research Grant ET 4/2006.

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When studying the existence of exponential dichotomy on the half-line, a usual assumption is that the initial stable subspace is closed and complemented in X. The results obtained in recent years concerning the dichotomy on the half-line may be resumed in the following theorem: Theorem 1.1. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let p, q ∈ [1, ∞) with p ≥ q. Let n ∈ N∗ , q1 , . . . , qn ∈ [1, ∞) with min{q1 , . . . , qn } ≤ p and let W q1 ,...,qn (R+ , X) = Lq1 (R+ , X) ∩ . . . ∩ Lqn (R+ , X) ∩ C00 (R+ , X). Let m ∈ N∗ , p1 , . . . , pm ∈ (1, ∞) and let V p1 ,...,pm (R+ , X) = Lp1 (R+ , X) ∩ . . . ∩ Lpm (R+ , X) ∩ C00 (R+ , X). The following assertions are equivalent: (i) U is uniformly exponentially dichotomic; (ii) for every v ∈ C0 (R+ , X) the equation (EU ) has a solution f ∈ C0 (R+ , X) and the subspace Xs,C0 (R+ ,X) is closed and complemented in X; (iii) for every v ∈ C00 (R+ , X) the equation (EU ) has a solution f ∈ C0 (R+ , X) and the subspace Xs,C0 (R+ ,X) is closed and complemented in X; (iv) for every v ∈ Lp (R+ , X) the equation (EU ) has a solution f ∈ Lp (R+ , X) ∩ Cb (R+ , X) and the subspace Xs,Lp (R+ ,X) is closed and complemented in X; (v) for every v ∈ Lq (R+ , X) the equation (EU ) has a solution f ∈ Lp (R+ , X) and there exists a dichotomy projection family compatible with U; (vi) for every v ∈ W q1 ,...,qn (R+ , X) the equation (EU ) has a solution f ∈ Lp (R+ , X) and the subspace Xs,Lp (R+ ,X) is closed and complemented in X; (vii) for every v ∈ V p1 ,...,pm (R+ , X) the equation (EU ) has a solution f ∈ Cb (R+ , X) and the subspace Xs,Cb (R+ ,X) is closed and complemented in X. The equivalence (i)⇔(ii) was proved by Van Minh, R¨ abiger and Schnaubelt in [16], employing evolution semigroups arguments. Their result was extended in [13], for discrete evolution families as well as for evolution families, and consequently we proved the equivalence (i)⇔(iii). In 2001 Minh and Huy considered the integral equation (EU ) on Lp -spaces and obtained the equivalence (i)⇔(iv) in [17]. A diﬀerent dichotomy concept is characterized by Preda, Pogan and Preda in [19], under the assumption that there exists a dichotomy projection family compatible with U (see (i)⇔(v)). An approach which generalizes the above equivalences (see (i)⇔(vi)) was given in [25], treating both discrete and integral case. The investigation was completed in [26], where the author deduced the equivalence (i)⇔(vii), based on the homologous discrete-time result. For the case J = R, the pair (O(R, X), I(R, X)) is said to be admissible for the evolution family U = {U (t, s)}t≥s if for every v ∈ I(R, X) there exists a unique solution f ∈ O(R, X) of the equation (EU ). The characterizations for uniform exponential dichotomy published in the last few years may be stated in the following theorem.

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Theorem 1.2. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X and let p, q ∈ [1, ∞) with p ≥ q. Let n ∈ N∗ , p1 , . . . , pn , r ∈ (1, ∞) and V p1 ,...,pn (R, X) = Lp1 (R, X) ∩ . . . ∩ Lpn (R, X) ∩ C0 (R, X). The following assertions are equivalent: (i) the family U is uniformly exponentially dichotomic; (ii) one of the pairs (Cb (R, X), Cb (R, X)) or (C0 (R, X), C0 (R, X)) is admissible for U; (iii) the pair (Cb (R, X), C0 (R, X)) is admissible for U; (iv) the pair (Lp (R, X), Lq (R, X)) is admissible for U; (v) the pair (Cb (R, X), Lr (R, X)) is admissible for U; (vi) the pair (Cb (R, X), V p1 ,...,pn (R, X)) is admissible for U. The equivalence (i)⇔(ii) was ﬁrstly obtained by Latushkin, Randolph and Schnaubelt in [9], in the hypothesis that for every x ∈ X the mapping (t, s) → U (t, s)x is continuous. Their techniques were based on the use of certain properties of the evolution semigroup associated with U. Using the equivalence between the uniform exponential dichotomy of the evolution family U = {U (t, s)}t≥s and the uniform exponential dichotomy of the discrete evolution family ΦU = {U (m, n)}m≥n,m,n∈Z associated with U and discrete-time arguments, we proved the equivalences (i)⇔(ii)⇔(iii) in [22]. The equivalence (i)⇔(iv) was obtained in [21]. For p = q this equivalence was studied in [9] in the continuity hypothesis mentioned above. The equivalences (i)⇔(v) and (i)⇔(vi) were recently obtained in [20]. Speciﬁcally, these characterizations are consequences of the following theorem, proved in [20]: Theorem 1.3. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X and let B be a Banach function space such that B ∈ T (R). If B \ L1 (R, R) = ∅ and the pair (Cb (R, X), B(R, X)) is admissible for U, then U is uniformly exponentially dichotomic. An essential condition in [20] (see the example in Section 4) is that the input space contains a non-integrable function. Moreover the output space is the space of bounded continuous functions. In what follows we will remove the condition imposed in [20] on the input space and the class of output spaces will consist of spaces containing non-continuous functions. The purpose of this paper is to continue and complete the study described above in order to obtain general characterizations for uniform dichotomy as well as for uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted T (R) and H(R) and we prove that if V ∈ T (R) and W ∈ H(R), the admissibility of the pair (W (R, X), V (R, X)) implies the uniform dichotomy of the evolution family U. In what follows we study when these conditions are suﬃcient for uniform exponential dichotomy. To answer this question, we introduce a subclass of H(R), denoted W(R) and we prove that if W ∈ W(R) and V ∈ T (R), then the admissibility of the pair (W (R, X), V (R, X)) for an

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evolution family U = {U (t, s)}t≥s implies the uniform exponential dichotomy of U. Moreover, we will show that in the case when V ⊂ W , the family U is uniformly exponentially dichotomic if and only if the pair (W (R, X), V (R, X)) is admissible for U. We note that, in applications it is interesting to consider output spaces O(R, X) as general as possible, while the input space I(R, X) should be taken as small as possible. Taking into account that the class T (R) contains the Orlicz spaces, the space C0 (R, X), etc. and T (R) is closed to ﬁnite intersections, our main result leads to new and better applicability perspectives of the characterizations of uniform exponential dichotomy on the real line. We will focus on this problem in the last part of our paper, where we will give some applications of the central results.

2. Banach function spaces — notations and preliminary results Let M(R) be the linear space of all Lebesgue measurable functions u : R → R, identifying the functions equal almost everywhere. Definition 2.1. A linear subspace B of M(R) is called normed function space if there is a mapping | · |B : B → R+ such that: (i) (ii) (iii) (iv)

|u|B = 0 if and only if u = 0 a.e.; |αu|B = |α||u|B , for all (α, u) ∈ R × B; |u + v|B ≤ |u|B + |v|B , for all u, v ∈ B; if u, v ∈ B and |u| ≤ |v| a. e. then |u|B ≤ |v|B .

If (B, | · |B ) is complete, then B is called Banach function space. Definition 2.2. A Banach function space (B, | · |B ) is said to be invariant to translations if for every (u, s) ∈ B × R, the function us : R → R, us (t) = u(t − s) belongs to B and |us |B = |u|B . Let Cc (R, R) denote the linear space of all continuous functions v : R → R with compact support. Throughout this paper, we denote by T (R) the class of all Banach function spaces B, which are invariant to translations, Cc (R, R) ⊂ B and for every t > s there is α(t, s) > 0 such that t |u(τ )| dτ ≤ α(t, s) |u|B , ∀u ∈ B. s

For diverse examples of Banach function spaces from the class T (R) we refer to [20]. Remark 2.3. The following properties hold: (i) if B1 , B2 ∈ T (R), then B1 ∩ B2 ∈ T (R); (ii) If Oϕ is an Orlicz space with ϕ(1) < ∞, then Oϕ ∈ T (R) (see [20], Proposition 3.4).

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Let H(R) be the class of all Banach function spaces B ∈ T (R) with the property that if |u| ≤ |v| a.e. and v ∈ B, then u ∈ B. For every A ⊂ R denote by χA the characteristic function of the set A. Lemma 2.4. Let B ∈ H(R). The following assertions hold: (i) χ[a,b) ∈ B, for all a, b ∈ R with a < b; (ii) if un −→ u in B, then there is a subsequence (ukn ) such that ukn → u a.e. Proof. First assertion is immediate since Cc (R, R) ⊂ B and for the second we refer to [15]. Definition 2.5. Let B ∈ H(R). The mapping FB : (0, ∞) → R, FB (t) = |χ[0,t) |B is called the fundamental function of the space B. Remark 2.6. The fundamental function FB is non-decreasing. Lemma 2.7. Let B ∈ H(R) and g ∈ B. If f : R → R is a continuous function such that there are a, b ∈ R with a < b and f (t) = g(t), for all t ∈ R \ [a, b], then f ∈ B. Proof. Let K = supt∈[a,b] |f (t)|. Then |f (t)| ≤ |g(t)| + Kχ[a,b] (t), for all t ∈ R. Thus, it follows that f ∈ B. Proposition 2.8. Let B ∈ H(R) and ν > 0. If u : R → R+ is a function, which belongs to B and with the property that t+1 u(s) ds qu : R → R+ , qu (t) = t

belongs to B, then the functions f u : R → R+ ,

fu (t) =

t

−∞

g u : R → R+ ,

gu (t) =

e−ν(t−s) u(s) ds

∞

e−ν(s−t) u(s) ds

t

belong to B. Proof. We have that fu (t) =

∞ j=0

t−j

t−j−1

=

e−ν(t−s) u(s) ds ≤

∞

∞ j=0

e−νj qu (t − j − 1),

e−νj

t−j

u(s) ds = t−j−1

∀t ∈ R.

j=0

Since B is invariant to translations, the function ϕ : R → R+ ,

ϕ(t) =

∞ j=0

e−νj qu (t − j − 1)

(2.1)

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belongs to B and |ϕ|B ≤

∞

e−νj |qu (· − j − 1)|B =

j=0

|qu |B . 1 − e−ν

Then using (2.1) we deduce that fu ∈ B. Using similar arguments it follows that gu ∈ B. In what follows we denote by W(R) the class of all Banach function spaces B ∈ H(R) with the following properties: t+1 (i) for every u : R → R+ in B, the function qu : R → R+ , qu (t) = t u(s) ds belongs to B; (ii) sup FB (t) = ∞. t>0

Proposition 2.9. If 0 < ϕ(t) < ∞, then the Orlicz space Oϕ associated with ϕ, belongs to W(R). Proof. Orlicz spaces are rearrangement invariant (see e.g. [5]). Then, Orlicz spaces are interpolation spaces between L1 (R, R) and L∞ (R, R) (see e.g. [5], Theorem 2.2, p. 106). We consider the operator t+1 G : L∞ (R, R) → L∞ (R, R), (G(u))(t) = u(s) ds t

and we have that G is correctly deﬁned and bounded. Moreover, the restriction G| : L1 (R, R) → L1 (R, R) is correctly deﬁned and it is a bounded linear operator. Then G(Oϕ ) ⊂ Oϕ . The fact that sup FOϕ (t) = ∞ is trivial (see e.g. [11], Proposition t>0

2.1). In conclusion, we deduce that Oϕ ∈ W(R).

Let X be a real or complex Banach space. For every B ∈ T (R) we denote by B(R, X) the linear space of all Bochner measurable functions v : R → X with the property that the mapping Nv : R → R+ , Nv (t) = ||v(t)|| lies in B. With respect to the norm ||v||B(R,X) := |Nv |B , B(R, X) is a Banach space.

3. Evolution families on the real line Let X be a real or complex Banach space. The norm on X and on B(X)- the Banach algebra of all bounded linear operators on X, will be denoted by || · ||. Denote by I the identity operator on X. Definition 3.1. A family U = {U (t, s)}t≥s of bounded linear operators on X is called an evolution family if the following properties hold: (i) U (t0 , t0 ) = I and U (t, s)U (s, t0 ) = U (t, t0 ), for all t ≥ s ≥ t0 ; (ii) for every x ∈ X and every t0 ∈ R the mapping t → U (t, t0 )x is continuous on [t0 , ∞) and the mapping s → U (t0 , s)x is continuous on (−∞, t0 ]; (iii) there are M ≥ 1 and ω > 0 such that ||U (t, t0 )|| ≤ M eω(t−t0 ) , for all t ≥ t0 .

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Definition 3.2. An evolution family U = {U (t, s)}t≥s is said to be uniformly dichotomic if there are a family of projections {P (t)}t∈R and a constant K ≥ 1 such that: (i) U (t, t0 )P (t0 ) = P (t)U (t, t0 ), for all t ≥ t0 ; (ii) the restriction U (t, t0 )| : KerP (t0 ) → KerP (t) is an isomorphism, for all t ≥ t0 ; (iii) ||U (t, t0 )x|| ≤ K||x||, for all x ∈ ImP (t0 ) and all t ≥ t0 ; 1 (iv) ||U (t, t0 )y|| ≥ K ||y||, for all y ∈ KerP (t0 ) and all t ≥ t0 . Definition 3.3. An evolution family U = {U (t, s)}t≥s is said to be uniformly exponentially dichotomic if there exist a family of projections {P (t)}t∈R and two constants K ≥ 1 and ν > 0 such that: (i) U (t, t0 )P (t0 ) = P (t)U (t, t0 ), for all t ≥ t0 ; (ii) the restriction U (t, t0 )| : KerP (t0 ) → KerP (t) is an isomorphism, for all t ≥ t0 ; (iii) ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||, for all x ∈ ImP (t0 ) and all t ≥ t0 ; 1 (iv) ||U (t, t0 )y|| ≥ K eν(t−t0 ) ||y||, for all y ∈ KerP (t0 ) and all t ≥ t0 . Let U = {U (t, s)}t≥s be an evolution family on X and W ∈ H(R). For every t0 ∈ R, we consider the stable subspace Xs (t0 ) as the space of all x ∈ X with the property that the function U (t, t0 )x , t ≥ t0 δx : R → X, δx (t) = 0 , t < t0 belongs to W (R, X) and we deﬁne the unstable subspace Xu (t0 ) as the space of all x ∈ X with the property that there is a function ϕx ∈ W (R, X) such that ϕx (t0 ) = x and ϕx (t) = U (t, s)ϕx (s), for all s ≤ t ≤ t0 . Lemma 3.4. The following properties hold: (i) U (t, t0 )Xs (t0 ) ⊂ Xs (t), for all t ≥ t0 ; (ii) U (t, t0 )Xu (t0 ) = Xu (t), for all t ≥ t0 . Proof. The assertion (i) is immediate. To prove (ii), let t > t0 . Let x ∈ Xu (t0 ) and let ϕx ∈ W (R, X) be such that ϕx (t0 ) = x and ϕx (τ ) = U (τ, s)ϕx (s), for all s ≤ τ ≤ t0 . Setting y = U (t, t0 )x we have that  0 , τ >t  U (τ, t0 )x , τ ∈ [t0 , t] ϕy : R → X, ϕy (τ ) =  ϕx (τ ) , τ < t0 belongs to W (R, X) and ϕy (τ ) = U (τ, s)ϕy (s), for all s ≤ τ ≤ t. Since ϕy (t) = y, it follows that y ∈ Xu (t). Conversely, let z ∈ Xu (t) and ϕz ∈ W (R, X) with ϕz (t) = z and ϕz (τ ) = U (τ, s)ϕz (s), for all s ≤ τ ≤ t. Then, we have that ϕz (t0 ) ∈ Xu (t0 ) and z = U (t, t0 )ϕz (t0 ). This shows that Xu (t) ⊂ U (t, t0 )Xu (t0 ) and the proof is complete.

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4. Admissibility and uniform dichotomy Let X be a real or complex Banach space and let U = {U (t, s)}t≥s be an evolution family on X. Let V, W be two Banach function spaces such that V ∈ T (R) and W ∈ H(R). Definition 4.1. The pair (W (R, X), V (R, X)) is said to be admissible for U if for every v ∈ V (R, X) there exists a unique f ∈ W (R, X) such that the pair (f, v) satisﬁes the equation t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t ≥ s. (EU ) s

Remark 4.2. If the pair (f, v) satisﬁes the equation (EU ), then f is continuous. Remark 4.3. If the pair (W (R, X), V (R, X)) is admissible for U, then it makes sense to deﬁne the operator Q : V (R, X) → W (R, X),

Q(v) = f

where f ∈ W (R, X) is such that the pair (f, v) satisﬁes the equation (EU ). Proposition 4.4. If the pair (W (R, X), V (R, X)) is admissible for U, then Q is bounded. Proof. We prove that Q is a closed linear operator. Let (vn ) ⊂ V (R, X), v ∈ V (R, X) and f ∈ W (R, X) be such that vn −→ v in V (R, X) and Q(vn ) −→ f in n→∞

n→∞

W (R, X). Using Lemma 2.4 (ii) we have that there exists a subsequence (vkn ) ⊂ (vn ) and a set A ⊂ R of zero Lebesque measure such that (Q(vkn ))(t) −→ f (t), n→∞

∀t ∈ R \ A.

(4.1)

Using similar arguments to those used in the proof of Proposition 4.3 in [20], we have that t t U (t, τ )vkn (τ ) dτ −→ U (t, τ )v(τ ) dτ, ∀t > s. n→∞

s

s

Then, from (4.1) it follows that t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, s

∀t, s ∈ R \ A, t > s.

(4.2)

Let t ∈ A. Using (4.2) we deduce that for all s, r ∈ (R \ A) ∩ (−∞, t): t t U (t, s)f (s) + U (t, τ )v(τ ) dτ = U (t, r)f (r) + U (t, τ )v(τ ) dτ. s

r

Since in W (R, X) we identify the functions equal almost everywhere, we may consider that for t ∈ A t f (t) = U (t, r)f (r) + U (t, τ )v(τ ) dτ, r

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where r ∈ R \ A, r < t. Then, using the above representation and (4.2), an easy computation shows that t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t > s. s

It follows that f = Q(v) and the proof is complete.

Lemma 4.5. If the pair (W (R, X), V (R, X)) is admissible for U, then the following properties hold: (i) Xs (t0 ) ∩ Xu (t0 ) = {0}, for all t0 ∈ R; (ii) Xs (t0 ) + Xu (t0 ) = X, for all t0 ∈ R. Proof. (i) Let t0 ∈ R and x ∈ Xs (t0 ) ∩ Xu (t0 ). Since x ∈ Xu (t0 ) there is ϕx ∈ W (R, X) such that ϕx (t0 ) = x and ϕx (t) = U (t, s)ϕx (s), for all s ≤ t ≤ t0 . Taking into account that x ∈ Xs (t0 ) it follows that U (t, t0 )x , t ≥ t0 f : R → X, f (t) = ϕx (t) , t < t0 belongs to W (R, X). Moreover, f (t) = U (t, s)f (s), for all t ≥ s. This implies that f = Q(0) = 0. In particular, x = f (t0 ) = 0. (ii) This follows using similar arguments with those in the proof of Proposition 4.4 (ii) in [20]. Theorem 4.6. If the pair (W (R, X), V (R, X)) is admissible for the evolution family U, then the following assertions hold: (i) Xs (t0 ) is a closed linear subspace, for all t0 ∈ R; (ii) there is K ≥ 1 such that ||U (t, t0 )x|| ≤ K ||x||,

∀t ≥ t0 , ∀x ∈ Xs (t0 ).

Proof. Let M, ω > 0 be given by Deﬁnition 3.1. Let α : R → [0, 2] be a continuous 1 function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. (i) Let t0 ∈ R and (xn ) ⊂ Xs (t0 ) with xn −→ x. For every n ∈ N, let n→∞

vn : R → X, fn : R → X,

vn (t) = α(t − t0 )U (t, t0 )xn t fn (t) = α(τ − t0 ) dτ U (t, t0 )xn . −∞

We have that vn ∈ Cc (R, X), so vn ∈ V (R, X). Since xn ∈ Xs (t0 ), the function U (t, t0 )xn δxn : R → X, δxn (t) = 0

, t ≥ t0 , t < t0

belongs to W (R, X). Using Lemma 2.7 we obtain that fn ∈ W (R, X), for all n ∈ N. An easy computation shows that the pair (fn , vn ) satisﬁes the equation (EU ), so fn = Q(vn ), for all n ∈ N.

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Let v : R → X, v(t) = α(t − t0 )U (t, t0 )x. Then, v ∈ V (R, X). If f = Q(v), we have that From

||f − fn ||W (R,X) ≤ ||Q|| ||vn − v||V (R,X) ,

∀n ∈ N.

||vn (t) − v(t)|| ≤ α(t − t0 )M eω ||xn − x||,

∀t ∈ R,

(4.3)

we deduce that ||vn − v||V (R,X) ≤ |α|V M eω ||xn − x||,

∀n ∈ N.

(4.4)

Since xn −→ x, from (4.3) and (4.4) we have that fn −→ f in W (R, X). Taking n→∞

n→∞

into account that W ∈ H(R), from Lemma 2.4 (ii), we have that there exists a subsequence (fkn ) ⊂ (fn ) such that fkn −→ f a.e. In particular, there is s > t0 + 1 n→∞

such that fkn (s) −→ f (s). This implies that n→∞

f (s) = lim fkn (s) = lim U (s, t0 )xkn = U (s, t0 )x. n→∞

n→∞

Observing that f (t) = U (t, r)f (r), ∀t ≥ r ≥ t0 + 1 we obtain that f (t) = U (t, t0 )x, for all t ≥ s. Since f ∈ W (R, X) using similar arguments as in Lemma 2.7 we deduce that U (t, t0 )x , t ≥ t0 δx : R → X, δx (t) = 0 , t < t0 belongs to W (R, X). This shows that x ∈ Xs (t0 ), so Xs (t0 ) is a closed linear subspace. (ii) Let t0 ∈ R and x ∈ Xs (t0 ). We consider the functions v : R → X, f : R → X,

v(t) = α(t − t0 )U (t, t0 )x t α(τ − t0 ) dτ U (t, t0 )x. f (t) = −∞

Using similar arguments with those in the proof of (i), we deduce that (f, v) ∈ W (R, X) × V (R, X) and f = Q(v). From ||v(t)|| ≤ α(t − t0 ) M eω ||x||, we have that

∀t ∈ R

||v||V (R,X) ≤ |α|V M eω ||x||.

This implies that

||f ||W (R,X) ≤ ||Q|| |α|V M eω ||x||. (4.6) We observe that f (t) = U (t, t0 )x, for all t ≥ t0 + 1. Then, for t ≥ t0 + 2, we have that ||U (t, t0 )x|| χ[t−1,t) (s) ≤ M eω ||U (s, t0 )x|| χ[t−1,t) (s) ≤ M eω ||f (s)||,

∀s ∈ R.

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The above inequality implies that ||U (t, t0 )x|| FW (1) ≤ M eω ||f ||W (R,X) .

(4.7)

2 2ω

Setting λ = (||Q|| |α|V M e )/FW (1), from relations (4.6) and (4.7) it follows that ||U (t, t0 )x|| ≤ λ ||x||, ∀t ≥ t0 + 2. Then, for K = max{λ, M e2ω }, we deduce that ||U (t, t0 )x|| ≤ K ||x||,

∀t ≥ t0 .

Since K does not depend on t0 or x, we obtain the conclusion.

Theorem 4.7. If the pair (W (R, X), V (R, X)) is admissible for the evolution family U, then the following properties hold: (i) Xu (t0 ) is a closed linear subspace, for all t0 ∈ R; (ii) there exists K > 0 such that ||U (t, t0 )x|| ≥

1 ||x||, K

∀t ≥ t0 , ∀x ∈ Xu (t0 ).

Proof. Let M, ω > 0 be given by Deﬁnition 3.1. Let α : R → [0, 2] be a continuous 1 function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. (i) Let t0 ∈ R and (xn ) ⊂ Xu (t0 ) with xn −→ x. For every n ∈ N there is n→∞

ϕxn ∈ W (R, X) with ϕxn (t0 ) = xn and ϕxn (t) = U (t, s)ϕxn (s),

∀s ≤ t ≤ t0 .

For every n ∈ N, we consider the functions vn : R → X, vn (t) = −α(t − t0 )U (t, t0 )xn ∞ α(τ − t0 ) dτ U (t, t0 )xn , t ≥ t0 t fn : R → X, fn (t) = . ϕxn (t) , t < t0 Since vn ∈ Cc (R, X) we have that vn ∈ V (R, X), for all n ∈ N. Using the fact that ϕxn ∈ W (R, X) and Lemma 2.7 we deduce that fn ∈ W (R, X), for all n ∈ N. It is easy to verify that fn = Q(vn ), for all n ∈ N. Let v : R → X,

v(t) = −α(t − t0 )U (t, t0 )x.

Since xn −→ x, using analogous arguments as in Theorem 4.6 (i) we obtain that n→∞

||vn − v||V (R,X) −→ 0. Then, setting f = Q(v), from n→∞

||fn − f ||W (R,X) ≤ ||Q|| ||vn − v||V (R,X) ,

∀n ∈ N

we have that fn −→ f in W (R, X). Since W ∈ H(R) from Lemma 2.4 (ii) it n→∞

follows that there exists a subsequence (fkn ) ⊂ (fn ) such that fkn −→ f a.e. In particular, there is s < t0 such that fkn (s) −→ f (s). n→∞

n→∞

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From f = Q(v) we have that f (t) = U (t, r)f (r), for all r ≤ t ≤ t0 . Since f ∈ W (R, X), this implies that f (t0 ) ∈ Xu (t0 ). Taking into account that f (t0 ) = U (t0 , s)f (s) = lim U (t0 , s)fkn (s) = lim fkn (t0 ) = lim xn = x n→∞

n→∞

n→∞

we deduce that x ∈ Xu (t0 ), so Xu (t0 ) is closed. (ii) Let t0 ∈ R and x ∈ Xu (t0 ). Then, there is ϕx ∈ W (R, X) such that ϕx (t0 ) = x and ϕx (t) = U (t, s)ϕx (s), for all s ≤ t ≤ t0 . Let t > t0 . We consider the function v : R → X, v(s) = −α(s − t)U (s, t0 )x ∞ α(τ − t) dτ U (s, t0 )x , s ≥ t0 s f : R → X, f (s) = ϕx (s) , s < t0 . We have that v ∈ V (R, X) and f ∈ W (R, X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ), so f = Q(v). From ||v(τ )|| = α(τ − t) ||U (τ, t0 )x|| ≤ M eω ||U (t, t0 )x|| α(τ − t), it follows that

∀τ ∈ R

||v||V (R,X) ≤ M eω |α|V ||U (t, t0 )x||.

This implies that ||f ||W (R,X) ≤ ||Q|| M eω |α|V ||U (t, t0 )x||.

(4.8)

From x = U (t0 , s)f (s), for s ∈ [t0 − 1, t0 ), we have that ||x|| χ[t0 −1,t0 ) (s) ≤ M eω ||f (s)||,

∀s ∈ R.

From the above inequality we deduce that ||x|| FW (1) ≤ M eω ||f ||W (R,X) .

(4.9)

2 2ω

Setting K = (||Q|| |α|V M e )/FW (1), from relations (4.8) and (4.9) we obtain that 1 ||U (t, t0 )x|| ≥ ||x||. K Since K does not depend on t0 , t or x we have that 1 ||x||, ∀t ≥ t0 , ∀x ∈ Xu (t0 ). ||U (t, t0 )x|| ≥ K The main result of this section is: Theorem 4.8. If the pair (W (R, X), V (R, X)) is admissible for the evolution family U, then U is uniformly dichotomic. Proof. From Lemma 4.5, Theorem 4.6 (i) and Theorem 4.7 (i), it follows that Xs (t0 ) ⊕ Xu (t0 ) = X,

∀t0 ∈ R.

For every t0 ∈ R, let P (t0 ) be the projection with Im P (t0 ) = Xs (t0 ) and Ker P (t0 ) = Xu (t0 ). Then, from Lemma 3.4 we have that U (t, t0 )P (t0 ) = P (t)

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U (t, t0 ), for all t ≥ t0 . Moreover, from Lemma 3.4 (ii) and Theorem 4.7 (ii) we have that U (t, t0 )| : Ker P (t0 ) → Ker P (t) is an isomorphism, for all t ≥ t0 . Finally, from Theorem 4.6 (ii) and Theorem 4.7 (ii) it follows that the family U is uniformly dichotomic.

5. Uniform exponential dichotomy In this section we will obtain necessary and suﬃcient conditions for uniform exponential dichotomy of evolution families on the real line in terms of admissibility of pairs of function spaces. Let X be a real or complex Banach space and let U = {U (t, s)}t≥s be an evolution family on X. Let V, W be two Banach function spaces with V ∈ T (R) and W ∈ W(R). From the previous section we have that if the pair (W (R, X), V (R, X)) is admissible for U, then U is uniformly dichotomic with respect to the family of projections {P (t)}t∈R , where Im P (t) = Xs (t)

and

Ker P (t) = Xu (t),

∀t ∈ R.

In what follows we will prove that U is uniformly exponentially dichotomic with respect to this family of projections. Theorem 5.1. If the pair (W (R, X), V (R, X)) is admissible for the evolution family U, then there are K, ν > 0 such that ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||,

∀t ≥ t0 , ∀x ∈ Im P (t0 ).

Proof. Let λ > 0 be such that ||U (t, t0 )x|| ≤ λ ||x||,

∀t ≥ t0 , ∀x ∈ Im P (t0 ).

(5.1)

Let M, ω ∈ (0, ∞) be given by Deﬁnition 3.1 and let α : R → [0, 2] be a continuous 1 function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Since supt>0 FW (t) = ∞ there is h > 0 such that FW (h) ≥ e λ2 ||Q|| |α|V .

(5.2)

Let t0 ∈ R and let x ∈ Im P (t0 ) = Xs (t0 ). If U (t0 + 1, t0 )x = 0, we consider the functions U (t, t0 )x v : R → X, v(t) = α(t − t0 ) ||U (t, t0 )x|| t α(τ − t0 ) dτ U (t, t0 )x. f : R → X, f (t) = −∞ ||U (τ, t0 )x|| We have that v ∈ V (R, X). Setting t0 +1 α(τ − t0 ) a := dτ, ||U (τ, t0 )x|| t0

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we observe that f (t) = a U (t, t0 )x, for all t ≥ t0 + 1. Since x ∈ Xs (t0 ), the function U (t, t0 )x , t ≥ t0 δx : R → X, δx (t) = 0 , t < t0 belongs to W (R, X). Using Lemma 2.7 we deduce that f ∈ W (R, X). Taking into account that ||U (t0 + h + 1, t0 )x|| χ[t0 +1,t0 +h+1) (t) ≤ λ ||U (t, t0 )x|| χ[t0 +1,t0 +h+1) (t) ≤ ≤

λ ||f (t)||, a

∀t ∈ R

we obtain that ||U (t0 + h + 1, t0 )x||FW (h) ≤ Observing that 1 a≥ λ ||x|| using (5.3) we have that ||U (t0 + h + 1, t0 )x|| ≤

t0 +1

t0

λ ||f ||W (R,X) . a

α(τ − t0 ) dτ =

(5.3)

1 λ||x||

λ2 ||x|| ||x|| ||f ||W (R,X) ≤ ||f ||W (R,X) . (5.4) FW (h) e ||Q|| |α|V

An easy computation shows that f = Q(v). Since ||v(t)|| = α(t − t0 ), for all t ∈ R, using relation (5.4) we deduce that ||U (t0 + h + 1, t0 )x|| ≤

||x|| 1 ||v||V (R,X) = ||x||. e |α|V e

If U (t0 + 1, t0 )x = 0, then obviously ||U (t0 + h + 1, t0 )x|| ≤ (1/e) ||x||. Setting l = h + 1 and taking into account that l does not depend on t0 or x we obtain that 1 ||U (t0 + l, t0 )x|| ≤ ||x||, ∀t0 ∈ R, ∀x ∈ Im P (t0 ). (5.5) e Let t ≥ t0 and x ∈ Im P (t0 ). Then, there are k ∈ N and r ∈ [0, t0 ) such that t = t0 + kl + r. Using Lemma 3.4, relations (5.1) and (5.5) we have that ||U (t, t0 )x|| ≤ λ ||U (t0 + kl, t0 )x|| ≤ λe−k ||x|| ≤ Ke−ν(t−t0 ) ||x|| where ν = 1/l and K = λe, and the proof is complete.

Theorem 5.2. If the pair (W (R, X), V (R, X)) is admissible for the evolution family U, then there are K, ν > 0 such that 1 ν(t−t0 ) ||U (t, t0 )x|| ≥ e ||x||, ∀t ≥ t0 , ∀x ∈ Ker P (t0 ). K Proof. Let λ > 0 be such that 1 ||U (t, t0 )x|| ≥ ||x||, ∀t ≥ t0 , ∀x ∈ Ker P (t0 ). (5.6) λ Let M, ω ∈ (0, ∞) be given by Deﬁnition 3.1 and let α : R → [0, 2] be a continuous 1 function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1.

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Since supt>0 FW (t) = ∞, there is h > 0 such that FW (h) ≥ e λ2 ||Q|| |α|V .

(5.7)

Let t0 ∈ R and x ∈ Ker P (t0 ) = Xu (t0 ), x = 0. Then, U (t, t0 )x = 0, for all t ≥ t0 . Let ϕx ∈ W (R, X) be such that ϕx (t0 ) = x and ϕx (t) = U (t, s)ϕx (s), for all s ≤ t ≤ t0 . We consider the functions v(t) = −α(t − t0 − h)

v : R → X,

U (t, t0 )x ||U (t, t0 )x||

and f : R → X given by  ∞  t (α(τ − t0 − h)/||U (τ, t0 )x||) dτ U (t, t0 )x f (t) = a U (t, t0 )x  a ϕx (t) where

, t ≥ t0 + h , t ∈ [t0 , t0 + h) , t < t0

t0 +h+1

α(τ − t0 − h) dτ. ||U (τ, t0 )x|| t0 +h We have that v ∈ V (R, X) and using Lemma 2.7 we obtain that f ∈ W (R, X). Since ||v(t)|| = α(t − t0 − h), for all t ∈ R, we have that ||v||V (R,X) = |α|V . An easy computation shows that f = Q(v), so a=

||f ||W (R,X) ≤ ||Q|| ||v||V (R,X) = ||Q|| |α|V .

(5.8)

Using (5.6) we have that ||U (t, t0 )x|| ≥

1 ||x||, λ

∀t ∈ [t0 , t0 + h)

so

1 1 ||x|| χ[t0 ,t0 +h) (t) ≤ ||f (t)||, ∀t ∈ R. λ a From relations (5.8) and (5.9) we deduce that a FW (h) ||x|| ≤ ||Q|| |α|V . λ From relation (5.6) we have that ||U (t0 + h + 1, t0 )x|| ≥ Then

t0 +h+1

1 ||U (τ, t0 )x||, λ

(5.9)

(5.10)

∀τ ∈ [t0 + h, t0 + h + 1].

α(τ − t0 − h) dτ

1 = . (5.11) λ ||U (t0 + h + 1, t0 )x|| λ ||U (t0 + h + 1, t0 )x|| From relations (5.7), (5.10) and (5.11) it follows that ||U (t0 + h + 1, t0 )x|| ≥ e ||x||. Setting l = h + 1 and taking into account that l does not depend on t0 or x we obtain that a≥

t0 +h

||U (t0 + l, t0 )x|| ≥ e ||x||,

∀t0 ∈ R, ∀x ∈ Ker P (t0 ).

(5.12)

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Let t ≥ t0 and x ∈ Ker P (t0 ). Then, there are k ∈ N and r ∈ [0, t0 ) such that t = t0 + kl + r. Using Lemma 3.4, relation (5.6) and relation (5.12) we deduce that 1 1 1 ν(t−t0 ) ||U (t, t0 )x|| ≥ ||U (t0 + kl, t0 )x|| ≥ ek ||x|| ≥ e ||x|| λ λ K

where ν = 1/l and K = λe and the proof is complete. The main result of this section is:

Theorem 5.3. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X and let V, W be two Banach function spaces with V ∈ T (R) and W ∈ W(R). Then, the following assertions hold: (i) if the pair (W (R, X), V (R, X)) is admissible for U, then U is uniformly exponentially dichotomic; (ii) if V ⊂ W , then U is uniformly exponentially dichotomic if and only if the pair (W (R, X), V (R, X)) is admissible for U. Proof. (i) This follows from Theorem 4.8, Theorem 5.1 and Theorem 5.2. (ii) Necessity. Suppose that U is uniformly exponentially dichotomic with respect to the family of projections {P (t)}t∈R and the constants K, ν > 0. Let v ∈ V (R, X). Since V ⊂ W we have that v ∈ W (R, X). Using the fact that W ∈ W(R) and Proposition 2.8, we deduce that the functions t g : R → R+ , g(t) = e−ν(t−s) ||v(s)|| ds −∞

h : R → R+ ,

∞

h(t) = t

e−ν(s−t) ||v(s)|| ds

belong to W . We consider the function t U (t, s)P (s)v(s) ds − f : R → X, f (t) = −∞

∞

t

U (s, t)−1 | (I − P (s))v(s) ds

U (s, t)−1 |

denotes the inverse of the operator U (s, t)| : where for every s > t, Ker P (t) → Ker P (s). Since L := sup ||P (t)|| < ∞ (see e.g. [22]) and observing t∈R

that ||f (t)|| ≤ KLg(t) + K(L + 1)h(t),

∀t ∈ R

we obtain that f ∈ W (R, X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ). Since W ∈ W(R), there is α > 0 such that 1 ||u(s)|| ds ≤ α ||u||W (R,X) , ∀u ∈ W (R, X). 0

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Let ϕ ∈ W (R, X) be such that the pair (ϕ, v) satisﬁes the equation (EU ) and let δ = f − ϕ. Then δ ∈ W (R, X) and ∀t ≥ s.

δ(t) = U (t, s)δ(s),

Let δ1 (t) = P (t)δ(t) and δ2 (t) = (I − P (t))δ(t), for all t ∈ R. Let t0 ∈ R. For every n ∈ N, we have that ||δ1 (t0 )|| = ||U (t0 , s)δ1 (s)|| ≤ K e−ν(t0 −s) ||δ1 (s)|| ≤ ≤ Ke−νn ||δ1 (s)|| ≤ LKe−νn ||δ(s)||, ∀s ∈ [t0 − n − 1, t0 − n]. This inequality implies that t0 −n −νn ||δ1 (t0 )|| ≤ LKe ||δ(s)|| ds = = LKe−νn

t0 −n−1

1 0

||δ(s − (t0 − n − 1))|| ds ≤ α ||δ||W (R,X) LK e−νn ,

∀n ∈ N.

From the above inequality it follows that δ1 (t0 ) = 0. In addition ||δ2 (t0 )|| ≤ Ke−ν(s−t0 ) ||U (s, t0 )δ2 (t0 )|| ≤ ≤ Ke−νn ||δ2 (s)|| ≤ (L + 1)Ke−νn ||δ(s)||, which implies that −νn ||δ2 (t0 )|| ≤ (L + 1)Ke = (L + 1)K e

−νn

0

1

∀s ∈ [t0 + n, t0 + n + 1], ∀n ∈ N t0 +n+1 t0 +n

||δ(s)|| ds =

||δ(s − (t0 + n))|| ds ≤ α ||δ||W (R,X) (L + 1)Ke−νn ,

∀n ∈ N.

It follows that δ2 (t0 ) = 0, so δ(t0 ) = δ1 (t0 )+δ2 (t0 ) = 0. Since t0 ∈ R was arbitrary, we obtain that δ = 0, so f = ϕ. Thus, the pair (W (R, X), V (R, X)) is admissible for U. Suﬃciency. This follows from (i).

In what follows we present some applications of the main results. Theorem 5.4. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X. Let Oϕ be an Orlicz space with 0 < ϕ(t) < ∞, for all t > 0. Let n ∈ N∗ , let Oϕ1 , . . . Oϕn be Orlicz spaces such that ϕk (1) < ∞, for all k ∈ {1, . . . , n} and let V (R, X) := Oϕ1 (R, X) ∩ . . . ∩ Oϕn (R, X) ∩ C0 (R, X). The following assertions hold: (i) if the pair (Oϕ (R, X), V (R, X)) is admissible for U, then U is uniformly exponentially dichotomic; (ii) if V (R, X) ⊂ Oϕ (R, X), then U is uniformly exponentially dichotomic if and only if the pair (Oϕ (R, X), V (R, X)) is admissible for U. Proof. It follows from Theorem 5.3, Proposition 2.9 and Remark 2.3 (i).

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Lemma 5.5. Let p, q ∈ [1, ∞) with p ≥ q and let ν > 0. Then for every u ∈ Lq (R, R+ ) the functions t f : R → R+ , f (t) = e−ν(t−s) u(s) ds −∞ ∞

g : R → R+ ,

g(t) =

belong to Lp (R, R+ ).

e−ν(s−t) u(s) ds

t

Proof. This follows using H¨older’s inequality.

Theorem 5.6. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X and let p ∈ [1, ∞). Let n ∈ N∗ , q1 , . . . , qn ∈ [1, ∞) and V (R, X) = Lq1 (R, X) ∩ . . . ∩ Lqn (R, X) ∩ C0 (R, X). The following assertions hold: (i) if the pair (Lp (R, X), V (R, X)) is admissible for U, then U is uniformly exponentially dichotomic; (ii) if q = min{q1 , . . . , qn } ≤ p, then U is uniformly exponentially dichotomic if and only if the pair (Lp (R, X), V (R, X)) is admissible for U. Proof. (i) This is immediate from Theorem 5.4 (i). (ii) Necessity. Suppose that U is uniformly exponentially dichotomic with respect to the family of projections {P (t)}t∈R . For v ∈ V (R, X) we consider the function t ∞ f : R → X, f (t) = U (t, s)P (s)v(s) ds − U (s, t)−1 | (I − P (s))v(s) ds −∞

t

U (s, t)−1 |

denotes the inverse of the operator U (s, t)| : where for every s > t, Ker P (t) → Ker P (s). Using the fact that q ≤ p and Lemma 5.5, we deduce that f ∈ Lp (R, X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ). The uniqueness of f follows using similar arguments with those in the necessity of Theorem 5.3 (ii). Suﬃciency. This follows from (i).

Corollary 5.7. Let U = {U (t, s)}t≥s be an evolution family on the Banach space X and let p, q ∈ [1, ∞). The following assertions hold: (i) if the pair (Lp (R, X), Lq (R, X)) is admissible for U, then U is uniformly exponentially dichotomic; (ii) if p ≥ q, then U is uniformly exponentially dichotomic if and only if the pair (Lp (R, X), Lq (R, X)) is admissible for U. Remark 5.8. Using diﬀerent arguments and techniques, the above result was obtained in [21]. Remark 5.9. For the case p = q, the above corollary was proved in [9], using evolution semigroups techniques.

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References [1] A. Ben-Artzi, I. Gohberg, Dichotomies of systems and invertibility of linear ordinary diﬀerential operators. Oper. Theory Adv. Appl. 56 (1992), 90-119. [2] A. Ben-Artzi, I. Gohberg, Dichotomies of perturbed time-varying systems and the power method. Indiana Univ. Math. J. 42 (1993), 699-720. [3] A. Ben-Artzi, I. Gohberg, M. A. Kaashoek, Invertibility and dichotomy of diﬀerential operators on the half-line. J. Dynam. Diﬀerential Equations 5 (1993), 1–36. [4] J. A. Ball, I. Gohberg, L. Rodman, Interpolation of rational matrix functions. Operator Theory: Advances and Applications 45. Birkh¨ auser Verlag, Basel, 1990. [5] C. Bennett, R. Sharpley, Interpolation of Operators. Pure Appl. Math. 129, 1988. [6] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Diﬀerential Equations. Math. Surveys and Monographs 70 Amer. Math. Soc. 1999. [7] J. Daleckii, M. Krein, Stability of Diﬀerential Equations in Banach Space. Amer. Math. Soc., Providence, RI, 1974. [8] J. K. Hale, S. M. Verduyn-Lunel, Introduction to Functional Diﬀerential Equations. Applied Mathematical Sciences 99, New York, NY: Springer-Verlag, 1993. [9] Y. Latushkin, T. Randolph, R. Schnaubelt, Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces. J. Dynam. Diﬀerential Equations 10 (1998), 489-509. [10] J. J. Massera, J. L. Sch¨ aﬀer, Linear Diﬀerential Equations and Function Spaces. Academic Press, New-York, 1966. [11] M. Megan, B. Sasu, A. L. Sasu, On uniform exponential stability of evolution families. Riv. Mat. Univ. Parma 4 (2001), 27-43. [12] M. Megan, B. Sasu, A. L. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces. Integral Equations Operator Theory 44 (2002), 71-78. [13] M. Megan, A. L. Sasu, B. Sasu, Discrete admissibility and exponential dichotomy for evolution families. Discrete Contin. Dynam. Systems 9 (2003), 383-397. [14] M. Megan, A. L. Sasu, B. Sasu, Perron conditions for pointwise and global exponential dichotomy of linear skew-product ﬂows. Integral Equations Operator Theory 50 (2004), 489-504. [15] P. Meyer-Nieberg, Banach Lattices. Springer Verlag, Berlin, Heidelberg, New York, 1991. [16] N. Van Minh, F. R¨ abiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution families on the half-line. Integral Equations Operator Theory 32 (1998), 332-353. [17] N. Van Minh, N. Thieu Huy, Characterizations of exponential dichotomies of evolution equations on the half-line. J. Math. Anal. Appl. 261 (2001), 28–44. [18] V. A. Pliss, G. R. Sell, Robustness of the exponential dichotomy in inﬁnitedimensional dynamical systems. J. Dynam. Diﬀerential Equations 3 (1999), 471-513. [19] P. Preda, A. Pogan, C. Preda, (Lp , Lq )-admissibility and exponential dichotomy of evolutionary processes on the half-line. Integral Equations Operator Theory 49 (2004), 405-418. [20] A. L. Sasu, B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces. Integral Equations and Operator Theory 54 (2006), 113-130.

152

Sasu

IEOT

[21] A. L. Sasu, Exponential dichotomy for evolution families on the real line. Abstr. Appl. Anal. (2006), Article ID 31641. [22] A. L. Sasu, B. Sasu, Exponential dichotomy and admissibility for evolution families on the real line. Dynam. Contin. Discrete Impulsive Systems 13 (2006), 1-26. [23] A. L. Sasu, B. Sasu, Discrete admissibility, p -spaces and exponential dichotomy on the real line. Dynam. Contin.. Discrete Impulsive Systems 13 (2006). [24] B. Sasu, A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line. Math. Z. 253 (2006), 515-536. [25] B. Sasu, A. L. Sasu, Exponential dichotomy and (p , q )-admissibility on the half-line. J. Math. Anal. Appl. 316 (2006), 397-408. [26] B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line. J. Math. Anal. Appl. (2006), DOI 10.1016/j.jmaa.2005.12.002. [27] B. Sasu, A. L. Sasu, Input-output conditions for the asymptotic behavior of linear skew-product ﬂows and applications. Commun. Pure Appl. Anal. 5 (2006), 551-569. Adina Luminit¸a Sasu Department of Mathematics Faculty of Mathematics and Computer Science West University of Timi¸soara Bd. V. Pˆ arvan No. 4 300223-Timi¸soara Romania e-mail: [email protected] [email protected] Submitted: May 7, 2006 Revised: July 26, 2006

Integr. equ. oper. theory 58 (2007), 153–173 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020153-21, published online April 14, 2007 DOI 10.1007/s00020-007-1484-2

Integral Equations and Operator Theory

Hypercyclic Pairs of Coanalytic Toeplitz Operators Nathan S. Feldman Abstract. A pair of commuting operators, (A, B), on a Hilbert space H is said to be hypercyclic if there exists a vector x ∈ H such that {An B k x : n, k ≥ 0} is dense in H. If f, g ∈ H ∞ (G) where G is an open set with finitely many components in the complex plane, then we show that the pair (Mf∗ , Mg∗ ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈ H ∞ (G) such that the pair (Mf∗ , Mg∗ ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples. Mathematics Subject Classification (2000). Primary 47A16; Secondary 47B20. Keywords. Hypercyclic, supercyclic, semigroup.

1. Introduction Let H denote a separable complex Hilbert space and let A be a bounded linear operator on H. We say that A is hypercyclic if there exists a vector x ∈ H such that the orbit of x under A, Orb(A, x) := {An x : n ≥ 0} is dense in H. We say that A is supercyclic if if there exists a vector x ∈ H such that {αAn x : n ≥ 0, α ∈ C} is dense in H. There has been much work done on hypercyclic and supercyclic linear operators. The ﬁrst example of a hypercyclic operator constructed on a Banach space was by Rolewicz [16] in 1969. He showed that if B is the backward shift on p (N), then λB is hypercyclic if and only if |λ| > 1. Since that time, a “Hypercyclicity Criterion” has been developed independently by Kitai [15] and Gethner and Shapiro [12]. This criterion has been used to show that hypercyclic operators arise within the classes of composition operators [6], weighted shifts [17], adjoints of multiplication operators [13], and adjoints of subnormal and hyponormal operators [11].

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If A := (A1 , A2 , . . . , An ) is an n−tuple of commuting operators on H, then let F = FA = {Ak11 Ak22 · · · Aknn : ki ≥ 0} be the semigroup generated by A. Since A is a commuting tuple, then F is a ﬁnitely generated Abelian semigroup. If x ∈ H, then the orbit of x under the tuple A or under F is Orb(A, x) := Orb(F , x) := {Ax : A ∈ F}. We say that A (or F ) is hypercyclic on H if there exists an x ∈ H such that Orb(A, x) is dense in H. There is a growing literature on strongly continuous hypercyclic semigroups of linear operators, see for instance [4], [5], and [10]. However these are one-parameter families of operators and we are considering multi-parameter families of operators. Recently, K´erchy [14] has studied supercyclic properties of discrete abelian semigroups of operators. There are simple examples of hypercyclic semigroups, namely any semigroup that contains a hypercyclic operator. An easy example of this goes as follows: if B is the backward shift on 2 (N) and I denotes the identity operator on 2 (N), then the semigroup generated by the pair (B, 2I) will be hypercyclic because it will contain the hypercyclic operator 2B. In fact, if A is any supercyclic operator, then one can easily see that the semigroup generated by the tuple (A, 2I, 13 I, eiθ I) is hyperyclic whenever θ ∈ R is an irrational multiple of π; since in that case i { 32j eikθ : i, j, k ≥ 0} is dense in C. In this last example, if A is chosen to be a supercyclic operator such that no multiple of A is hypercyclic (see [17] or [11]), then the semigroup generated by (A, 2I, 13 I, eiθ I) will be hypercyclic yet contain no hypercyclic operator. We also see from above that the study of (discrete Abelian) hypercyclic semigroups includes the study of supercyclic operators. This paper mainly focuses on pairs or tuples of adjoints of multiplication operators on spaces of analytic functions (often called adjoint multiplication operators). If G is an open set in the complex plane, C, then let Hol(G) denote the space of all analytic functions on G. Also let H ∞ (G) denote the Banach space of all bounded analytic functions on G and we will use H(G) to denote a “Hilbert space of analytic functions on G” which will be carefully deﬁned below, but will include such spaces as the Hardy space and Bergman space over G. The following two results are samples of our main theorems. In what follows if f ∈ H ∞ (G), then Mf will denote the operator of multiplication by f on H(G). Theorem. Let f, g ∈ H ∞ (G) where G is an open set with finitely many components and let H(G) be a Hilbert space of analytic functions on G. If F = {Mf∗n Mg∗k : n, k ≥ 0}, then the following are equivalent: 1. The pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). 2. The semigroup F generated by (Mf∗ , Mg∗ ) contains a hypercyclic operator. 3. There exists integers n, k ≥ 0 such that f n g k is non-constant on every component of G and (f n g k )(Gi ) ∩ ∂D = ∅ for every i ∈ {1, . . . , N }.

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If G is connected and say |f (z)| > 1 and |g(z)| < 1 for all z ∈ G, then one may also add the following equivalent condition: 4. There does not exist a p > 0 such that |f (z)|p = 1/|g(z)| for all z ∈ G. If G has infinitely many components, then the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G) if and only if (Mf∗ , Mg∗ ) is hypercyclic on H(ΩN ) for each N ≥ 1, where ∞ ΩN = N i=1 Gi and {Gi }i=1 are the components of G. This latter characterization of when the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G) where G has inﬁnitely many components allows us to give an example of a pair of adjoint multiplication operators which is hypercyclic yet the semigroup they generate does not contain a hypercyclic operator. Theorem. If G is a bounded open set with infinitely many components and H(G) is a Hilbert space of analytic functions on G, then there exists f, g ∈ H ∞ (G) such that the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G), but the semigroup F generated by (Mf∗ , Mg∗ ) contains no hypercyclic operator.

2. Preliminaries If G is an open set and f ∈ H ∞ (G), then let f ∞ = sup{|f (z)| : z ∈ G} and let f inf = inf{|f (z)| : z ∈ G}. By a region in C we will mean an open connected set, however we are also interested in working on open sets that are not connected. This will correspond to working with direct sums of multiplication operators. Definition 2.1. If G is a open set (not necessarily connected) and H(G) ⊆ Hol(G), then H(G) is said to be a Hilbert space of analytic functions on G if the following conditions are satisfied: 1. H(G) is a vector subspace of Hol(G). 2. H(G) is complete with respect to an inner product on H(G). 3. For each point a ∈ G the point evaluation functional f → f (a) is continuous on H(G). 4. H ∞ (G) ⊆ H(G). 5. H(G) is invariant under multiplication by f for all f ∈ H ∞ (G). 6. Mf H(G) = f ∞ for all f ∈ H ∞ (G). Remark. If G is an open set and G1 is a component of G, and f is the characteristic function of G1 , then by property 6. multiplication by f has norm one, hence is a norm one idempotent, thus a (self-adjoint) projection. We will let H(G1 ) denote the range of this projection. It follows that if {Gi } are all the components of G, then H(G) is naturally isomorphic to ⊕i H(Gi ). Lemma 2.2 (Expansive Inequality). If H(G) is a Hilbert space of analytic functions on an open set G as in Definition 2.1, h ∈ H ∞ (G) and |h| ≥ 1 on G, then Mh∗ f ≥ f for all f ∈ H(G).

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Proof. To prove this fact, note that h is invertible and thus Mh∗ is invertible and the inequality above simply says that the inverse is a contraction. Since |h−1 | ≤ 1 on G, property 6. of Deﬁnition 2.1 implies that ((Mh )−1 )∗ = (Mh )−1 = h−1 ∞ ≤ 1. So the Expansive inequality is true. Examples of spaces that ﬁt into the above deﬁnition include the Hardy space H 2 (G), the Bergman space L2a (G), weighted Bergman spaces, pure P 2 (µ) spaces, representing the closure of the polynomials in L2 (µ) and certain (but not all) pure R2 (K, µ) spaces representing the closure of the rational functions with poles oﬀ K in L2 (µ). The Dirichlet space does not satisfy condition 6. above. The following result is a small variation of one due to Godefroy & Shapiro; see [13, Theorem 4.9]. Theorem 2.3 (Godefroy & Shapiro). If G is an open set in C with components {Gi } and H(G) is a Hilbert space of analytic functions on G as in Definition 2.1, and f ∈ H ∞ (G), then Mf∗ is hypercyclic on H(G) if and only if f |Gi is nonconstant for each i and f (Gi ) ∩ ∂D = ∅ for all i. We will need the following function theoretic result. Proposition 2.4. If G is a region in C, f, g ∈ Hol(G), f has no zeros in G, and p is an irrational real number such that |f (z)|p = |g(z)| for all z ∈ G, then f has an analytic logarithm on G; that is, there is an h ∈ Hol(G) such that f = eh . In particular, then f r is a well-defined analytic function on G for any r ∈ C, f r = erh . The author would like to thank Paul Bourdon for the following proof.

Proof. Recall that f has an analytic logarithm on G if and only if intγ ff dz = 0 for all rectiﬁable simple closed curves γ contained in G. Since we make take an exhaustion of G by a sequence of regions {Gn }∞ n=1 each of which is bounded by a ﬁnite number of disjoint smooth Jordan curves and each simple closed curve γ in G will be contained in some Gn . Thus it suﬃces to assume (which we will now do) that G itself is bounded by a ﬁnite number of disjoint smooth Jordan curves. Say C \ clG has n bounded components and choose a point ak from each of the bounded components such that Im(ak ) = Im(aj ) if k = j (where Im(z) denotes the imaginary part of the complex number z). By the Logarithmic Conjugation Theorem (see [2] or [3, p. 203]), there are real constants {bk }nk=1 and an analytic function h on G such that n bk ln |z − ak | for all z ∈ G. (1) ln |f (z)| = Re(h(z)) + k=1

It follows that |f (z)| = e

Re(h(z))

n k=1

ebk ln |z−ak | for all z ∈ G.

(2)

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n Let Γk = {z ∈ C : Im(z) = Im(ak ) and Re(z) ≤ Re(ak )} and Γ = k=1 Γk . If we let logπ (z) denote the principal branch of the logarithm on C \ (−∞, 0], then logπ (z − ak ) is analytic on G \ Γk and so logπ (z − ak ) is analytic on G \ Γ for all k. It follows from equation (2) that there exists a unimodular constant c such that f (z) = ceh(z)

n

ebk logπ (z−ak ) for all z ∈ G \ Γ.

(3)

k=1

Also taking pth powers of equation (2) we have |g(z)| = |f (z)|p = epRe(h(z))

n

epbk ln |z−ak | for all z ∈ G.

(4)

k=1

and thus there is a unimodular constant d such that n g(z) = deph(z) epbk logπ (z−ak ) for all z ∈ G \ Γ.

(5)

k=1

Now consider equation (3). The left hand side is continuous on G, whereas the right hand side is continuous on G if and only if bk is an integer for each k. Likewise, considering equation (5), since the left hand side is continuous on G, then the right hand side must also be, which happens if and only if pbk is an integer for all k. However, since p is irrational, the only way that bk and pbk can be an integer is is bk = 0 and this is true for all k. Thus equation (2) becomes |f (z)| = eRe(h(z)) for all z ∈ G.

(6)

Thus there is a unimodular constant α such that f = αeh . It follows that f has an analytic logarithm on G.

3. Pairs of multiplication operators on connected open sets In this section we consider the case where G is a region. We also begin by considering pairs of operators. We will see that we can easily derive an analogous result for n−tuples from the result for pairs. Theorem 3.1. Let H(G) be a Hilbert space of analytic functions as in Definition 2.1 on a region G. Also, let f, g ∈ H ∞ (G). If F = {Mf∗n Mg∗k : n, k ≥ 0}, then the following are equivalent: 1. The pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). 2. The semigroup F generated by (Mf∗ , Mg∗ ) contains a hypercyclic operator. 3. There exists integers n, k ≥ 0 such that f n g k is non-constant on G and (f n g k )(G) ∩ ∂D = ∅. 4. One of the following holds: (a) f is non-constant and f (G) ∩ ∂D = ∅. (b) g is non-constant and g(G) ∩ ∂D = ∅.

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(c) |f (z)| > 1 and |g(z)| < 1 for 1 for such that |f (z)|p = |g(z)| (d) |f (z)| < 1 and |g(z)| > 1 for 1 such that |f (z)|p = |g(z)| for

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all z ∈ G and there does not exist a p > 0 all z ∈ G. all z ∈ G and there does not exist a p > 0 all z ∈ G.

Proof of Theorem 3.1. It follows from Theorem 2.3 of Godefroy and Shapiro that 2. holds if and only if 3. holds. Also clearly 2. ⇒ 1. We will prove that 3. ⇔ 4. holds and that 1. ⇒ 4. This will prove the theorem. It is easy to see that if |f (z)| < 1 and |g(z)| < 1 for all z ∈ G or if |f (z)| > 1 and |g(z)| > 1 for all z ∈ G, then none of the conditions 1., 2., 3. or 4. hold. Also if either f or g maps G onto an open set that intersects the unit circle, then by Theorem 2.3 all four conditions are satisﬁed. Finally if, say, g is a unimodular constant, then 3. is satisﬁed if and only if f is nonconstant and f (G) ∩ ∂D = ∅ which happens if and only if 4. is satisﬁed. Also in this case where g is unimodular, conditions 1. and 2. hold if and only if f is nonconstant and f (G) ∩ ∂D = ∅. And ﬁnally, if f or g is identically zero, then all four conditions are equivalent, by Theorem 2.3. Let’s assume that |f (z)| > 1 for all z ∈ G and |g(z)| < 1 for all z ∈ G.

(*)

Assuming (*) we now show that 3. ⇔ 4.(c). There exists integers n, k ≥ 0 such that f n g k is non-constant and (f n g k )(G)∩ ∂D = ∅ if and only if there exist a, b ∈ G and integers n, k ≥ 0 such that |f (a)|n |g(a)|k < 1 and |f (b)|n |g(b)|k > 1.

(**)

Since g is not identically zero, we may assume that g(a) = 0, otherwise replace a by a where a ∈ G, g(a ) = 0 and a is suﬃciently close to a to preserve the above inequality. Taking logarithms of (**) gives n ln |f (a)| + k ln |g(a)| < 0 and n ln |f (b)| + k ln |g(b)| > 0 which (using (*)) may be rewritten as ln |f (a)| k k ln |f (b)| < and < . − ln |g(a)| n n − ln |g(b)| or equivalently k ln |f (b)| ln |f (a)| < < . − ln |g(a)| n − ln |g(b)| Thus there exists an a, b ∈ G and integers n, k ≥ 0 such that (**) holds if and only if the positive (extended real-valued) function w(z) :=

ln |f (z)| − ln |g(z)|

which is deﬁned on G is non-constant. Further, one easily checks that w is constant 1 if and only if there exists a p > 0 such that |f |p = |g| on G. Thus we have

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established 3. ⇔ 4.(c) assuming that (*) holds. The same argument will also show 3. ⇔ 4.(d) assuming |f (z)| < 1 for all z ∈ G and |g(z)| > 1 for all z ∈ G. It remains to show that 1. ⇒ 4.(c) (we are still assuming that (*) holds). So, assume that F is hypercyclic on H(G) and that 4.(c) does not hold. Then there exists a p > 0 such that |f (z)|p = 1/|g(z)| for all z ∈ G. There are now two cases, either p is rational or irrational. Case I: p is rational. Suppose that p = a/b where a, b ∈ N and gcd(a, b) = 1. Then since |f (z)|p = 1/|g(z)| for all z ∈ G, we have |f (z)|a |g(z)|b = 1 for all z ∈ G. Which implies that |f (z)a g(z)b | = 1 for all z ∈ G. Thus, there is a unimodular constant c such that f a g b = c on G or g b = c/f a . So, if n, k ≥ 0 and n = aq1 + r1 and k = bq2 + r2 where qi ≥ 0 and 0 ≤ r1 < a and 0 ≤ r2 < b, then f n g k = f r1 g r2 f aq1 g bq2 = f r1 g r2 f aq1 (c/f a )q2 = cq2 f r1 g r2 f a(q1 −q2 ) on G. Let φ ∈ H(G) be a vector with dense orbit under F . Also let F := {Mh∗ : h = αf n , α ∈ C, |α| = 1, n ∈ Z} and notice that Orb(F , φ) ⊆ {Mf∗r gs (Orb(F , φ)) : 0 ≤ r < a, 0 ≤ s < b}. Since |f (z)|p |g(z)| = 1 for all z ∈ G and since f is bounded, then g is bounded away from zero (and f is also, since |f | > 1 on G). So, f and g are invertible in H ∞ (G), thus Mf∗r gs is an invertible linear operator. So, if we can show that Orb(F , φ) is nowhere dense in H(G), then Orb(F , φ) will be contained in a ﬁnite union of nowhere dense sets, hence it will also be nowhere dense (meaning its closure has empty interior), contradicting the deﬁnition of φ. Claim: Orb(F , φ) is nowhere dense. Suppose that int[clOrb(F , φ)] = ∅. Then either (i) there exists z, w ∈ int[clOrb(F , φ)] \ Orb(F , φ) such that z > w > φ or (ii) there exists z, w ∈ int[clOrb(F , φ)] \ Orb(F , φ) such that z < w < φ. We will consider case (i); case (ii) is similar. Let > 0 be chosen such that < (1/3)(z − w). Since z ∈ clOrb(F , φ), then there exists an n ∈ N and an α ∈ C, |α| = 1 such that αMf∗n φ− z < . Hence, Mf∗n φ = αMf∗n φ ≥ z − . Now since w ∈ int[clOrb(F , φ)]\Orb(F , φ), then there exists a k > n and a β ∈ C, |β| = 1 such that βMf∗k φ − w < . It follows as above that Mf∗k φ < w + . Thus we have that Mf∗n φ > Mf∗k φ. However, since k > n this contradicts the ∗ ∞ fact that {Mf∗n φ}∞ n=0 is an increasing sequence. To see that {Mf n φ}n=0 is an ∗ increasing sequence notice that |f | > 1 on G that Mf is an expansive operator by Lemma 2.2. This contradiction implies that int[clOrb(F , φ)] = ∅ and so the claim follows. Thus we have that 1. ⇒ 4.(c) when p is rational. Case II: p is irrational.

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Let φ ∈ H(G) be a vector with dense orbit under F . Since f is never zero (recall |f | > 1 on G) and since |f (z)|p = 1/|g(z)| for all z ∈ G (which implies that g is never zero, hence 1/g is analytic on G) and since p is irrational, then Proposition 2.4 implies that f has an analytic logarithm on G. It follows that for every t ∈ R, f t is a well deﬁned bounded analytic function on G. Thus |f (z)|p = 1/|g(z)| may be written as |f (z)p g(z)| = 1 for all z ∈ G. Hence there is a unimodular constant c such that f (z)p g(z) = c for all z ∈ G. Thus we have that f n g k = ck f n−kp . Hence Orb(F , φ) ⊆ Orb(F , φ) where F = {Mh∗ : h = αf t , α ∈ C, |α| = 1, t ∈ R}. Notice that the map ϕ : [0, ∞) → H(G) deﬁned by ϕ(t) := Mf∗t φ is continuous and ϕ(0) = φ. Claim: The function t → ϕ(t) is continuous and increasing on [0, ∞). We will leave the continuity to the reader, for the increasing part we use the Expansive inequality in Lemma 2.2. To see that t → ϕ(t) is increasing, suppose that 0 < s < t. Since |f | > 1 on G, then |f |(t−s) > 1 on G. So, Mf∗t φ = Mf∗(t−s) Mf∗s φ ≥ Mf∗s φ. That establishes the claim. Since F is hypercyclic, then limt→∞ ϕ(t) = ∞. Thus, K := {t ∈ [0, ∞) : φ ≤ ϕ(t) ≤ 2φ} is a compact interval in [0, ∞). Hence it follows that Orb(F , φ) ∩ {h ∈ H(G) : φ ≤ h ≤ 2φ} = C := {αh : |α| = 1, h ∈ ϕ(K)}. But this latter set, C is compact since it is the continuous image of the compact set ∂D × K under the map (α, t) → αφ(t). Since compact sets in inﬁnite dimensions have empty interior, then C cannot be dense in {h ∈ H(G) : φ ≤ h ≤ 2φ}, thus Orb(F , φ) cannot be dense there either. But this contradicts the fact that Orb(F , φ) is dense. Thus it follows that 1. ⇒ 4.(c) when p is irrational. Thus we have proven that 1. ⇒ 4.(c) assuming that (∗) holds. A similar argument will show 1. ⇒ 4.(d) assuming |f (z)| < 1 for all z ∈ G and |g(z)| > 1 for all z ∈ G. The theorem now follows. Example 3.2. Let G be a region, H(G) a Hilbert space of analytic functions on G as in Definition 2.1, and f, g ∈ H ∞ (G). Also let F be the semigroup generated by Mf∗ and Mg∗ . 1. If f has a zero at some point in G, then the pair (Mf∗ , Mg∗ ) is hypercyclic if and only if f ∞ > 1 or g∞ > 1. 2. If G = D and f (z) = z, then the pair (Mf∗ , Mg∗ ) is hypercyclic if and only if g∞ > 1. 3. If G = D, f (z) = e(z+1) and g(z) = e−2(z+1) , then the pair (Mf∗ , Mg∗ ) is not hypercyclic.

4. Tuples of multiplication operators on connected open sets The case of hypercyclic n−tuples of adjoint multiplication operators will now follow easily from our result about hypercyclic pairs (Theorem 3.1).

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Theorem 4.1. Let H(G) be a Hilbert space of analytic functions as in Definition 2.1 on a region G. Let {f1 , , . . . , fn } ⊆ H ∞ (G). Also let F be the semigroup generated by the adjoints of the multiplication operators on H(G) with symbols f1 , . . . , fn . Then the following are equivalent: 1. The tuple (Mf∗1 , Mf∗2 , . . . , Mf∗n ) is hypercyclic on H(G). 2. The semigroup F generated by (Mf∗1 , Mf∗2 , . . . , Mf∗n ) contains a hypercyclic operator. 3. There exists integers k1 , k2 , . . . , kn ≥ 0 such that (f1k1 f2k2 · · · fnkn ) is nonconstant and (f1k1 f2k2 · · · fnkn )(G) ∩ ∂D = ∅. 4. There is a pair of indices i, j and integers k1 , k2 ≥ 0 such that fik1 fjk2 is non-constant and (fik1 fjk2 )(G) ∩ ∂D = ∅. 5. There is a pair of indices i, j such that one of the following holds: (a) fi is non-constant and fi (G) ∩ ∂D = ∅. (b) fj is non-constant and fj (G) ∩ ∂D = ∅. (c) |fi (z)| > 1 and |fj (z)| < 1 for all z ∈ G and there does not exist a p > 0 such that |fi (z)|p = |fj1(z)| for all z ∈ G. (d) |fi (z)| < 1 and |fj (z)| > 1 for all z ∈ G and there does not exist a p > 0 such that |fi (z)|p = |fj1(z)| for all z ∈ G. Proof. It follows from Theorem 3.1 that 4. ⇔ 5. holds. Also, by Theorem 2.3, 4. ⇒ 3. ⇒ 2. ⇒ 1. Hence it suﬃces to prove the implication 1 ⇒ 5. So assume that 1. holds and by way of contradiction assume that 5. does not hold. Since 4. and 5. are equivalent and we are assuming that 5. is not true, then there does not exist an index i such that fi is non-constant and fi (G) ∩ ∂D = ∅. Thus for every pair of indices i, j we have either min{fi inf , fj inf } ≥ 1 or max{fi ∞ , fj ∞ } ≤ 1 or the pair (fi , fj )

(*)

satisﬁes that there exits a p > 0 s.t. |fi | = 1/|fj | and 0 < |fj (z)| = 1, ∀z ∈ G. Let A = {i : fi ∞ ≤ 1 and fi is not a unimodular constant}, B = {i : fi inf ≥ 1 and fi is not a unimodular constant}, and C = {i : fi is a unimodular constant}. Then by the comments immediately preceding (*) we have that A ∪ B ∪ C = {1, 2, . . . , n}. Since we are assuming that 1. holds, then A = ∅ and B = ∅. Also since 1. holds fi must be nonconstant for some i ∈ A ∪ B, with out loss of generality assume that there is an i ∈ B so that fi is nonconstant (a similar argument would apply if i ∈ A). Now choose and ﬁx an i1 ∈ A and a j1 ∈ B and choose j1 such that fj1 is nonconstant. By (*) there is a p > 0 such that 1 on G. (**) |fi1 | = |fj1 |p p

Now if k ∈ B and k = j1 , then by applying (∗) to fi1 and fk we see that there exists a q > 0 such that 1 on G. |fi1 | = |fk |q

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It follows that

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1 on G. |fj1 |fk |q Hence |fk | = |fj1 |r where r = p/q > 0. Similarly, if k ∈ A, then by applying (∗) to fk and fj1 , we get that there exists an p > 0 such that |fk | = |fj1 |−p . It follows that for every i ∈ A ∪ B, there exists a unique pi ∈ R \ {0} such that (***) |fi | = |fj1 |pi on G. 1

|p

= |fi1 | =

There are two cases to consider now, either every pi is rational or there exists an i such that pi is irrational. Case I: For every i ∈ A ∪ B, pi is rational. For simplicity let f := fj1 . Say pi = ai /bi where ai ∈ Z \ {0} and bi ∈ N. Then from (***) we have that |fi | = |f |ai /bi on G. Hence there exists unimodular constants ci such that (†) fibi = ci f ai on G. Let k1 , k2 , . . . , kn ≥ 0 be integers. For each i ∈ A ∪ B, upon dividing ki by bi we see that there are integers qi , ri satisfying qi ≥ 0 and 0 ≤ ri < bi and ki = bi qi + ri . Thus, using (†) we have b q +r k1 k2 ki kn i i i fi fi · (f1 f2 · · · fn ) = i∈C

= α· = α·

i∈A∪B

i∈A∪B

·

firi

i∈A∪B

fibi qi

i∈A∪B

firi

·

ai qi

(ci f )

=β

i∈A∪B

firi

· fm

i∈A∪B

from some integer m(= i∈A∪B ai qi ) and for some unimodular constants α, β. Since f is invertible in H ∞ (G) we see from (†) that fi is also invertible for each i ∈ A ∪ B. Let firi : 0 ≤ ri < bi }. C={ i∈A∪B

Then C consists of a ﬁnite number of invertible functions in H ∞ (G). So, let F = {Mh∗ : h = cf m , c ∈ C, |c| = 1, m ∈ Z}. Since we are assuming that 1. holds, let φ ∈ H(G) be a function such that Orb(F , φ) is dense in H(G). It follows from the above equations that Orb(F , φ) is contained in {Mg∗ (Orb(F , φ)) : g ∈ C}. Since C is ﬁnite it suﬃces to show that Orb(F , φ) is nowhere dense in H(G); however this argument is identical to the one in Theorem 3.1. Thus Orb(F , φ) is nowhere dense, and so it follows that Orb(F , φ) is also nowhere dense, contradicting 1. Thus, 1. ⇒ 5. when all the pi ’s are rational. Case II: There exists an i ∈ A ∪ B such that pi is irrational.

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Again for simplicity let f := fj1 . If i0 ∈ A ∪ B and pi0 is irrational, then |f |pi0 = |fi0 | on G and |f | > 1 on G (because f = fj1 and j1 ∈ B) hence nonvanishing on G, thus by Proposition 2.4, f has an analytic logarithm on G. Then by (***) for each i ∈ A ∪ B, there exist unimodular constants ci such that fi = ci f pi on G. Thus for all integers k1 , k2 , . . . , kn ≥ 0, there exists a t ∈ R and an α ∈ ∂D such that (f1k1 f2k2 · · · fnkn ) = αf t . (Since f has a logarithm, f t is well deﬁned.) Hence if φ is a hypercyclic vector for F and F = {αMf∗t : t ∈ R, |α| = 1}, then Orb(F , φ) ⊆ Orb(F , φ). However, as in the proof of Theorem 3.1, we can show that Orb(F , φ) is not dense in H(G), contradicting the fact that F is hypercyclic. It now follows that 1. implies 5. when some pi is irrational. Hence the theorem follows.

5. Pairs of multiplication operators on disconnected open sets The following is a basic lemma showing that when |g| < 1 ( a similar result holds, when |g| > 1) which integers n, k have the property that f n g k is non-constant and has its range intersecting the unit circle. Proposition 5.2 is a more careful look at the same question when both f and g may have ranges that hit the circle. Lemma 5.1. If G is an open set in C and f, g ∈ Hol(G) \ {0} and |g(z)| < 1 for all z ∈ G, then for a pair of non-negative integers (n, k) there exists an a, b ∈ G such that |f (a)n g(a)k | < 1 and |f (b)n g(b)k | > 1 if and only if m < nk < M , where m = inf

z∈G

ln |f (z)| ln |f (z)| and M = sup . − ln |g(z)| − ln |g(z)| z∈G

Proof. The proof of this lemma is basically contained in the proof of Theorem 3.1. Notice that in the above lemma, n cannot equal zero. However, for the following proposition, if k > 0 and n = 0, then interpret nk as inﬁnity. However, notice that this will only occur in part 3. In parts 1. and 2. n cannot be zero. Proposition 5.2. Let Ω be a region in C and f, g ∈ H ∞ (Ω)\{0}, g not a unimodular constant. Define two subsets of Ω and four constants as follows: • Ω(1) = {z ∈ Ω : |g(z)| > 1}. • Ω(2) = {z ∈ Ω : |g(z)| < 1}. ln |f (z)| ln |f (z)| and M (1) = sup • m(1) = inf − ln |g(z)| z∈Ω(1) − ln |g(z)| (1) z∈Ω ln |f (z)| ln |f (z)| • m(2) = inf and M (2) = sup (2) − ln |g(z)| z∈Ω z∈Ω(2) − ln |g(z)| Then Ω(1) ∪ Ω(2) is a dense open subset of Ω and the following hold:

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1. Suppose Ω(1) = ∅. Then a pair of nonnegative integers (n, k), not both zero, satisfy that there exists an a, b ∈ Ω(1) such that |f (a)n g(a)k | < 1 and |f (b)n g(b)k | > 1 if and only if m(1) < M (1) and nk ∈ (m(1) , M (1) ). Furthermore, if nk ∈ / (m(1) , M (1) ), then either |f (z)n g(z)k | ≥ 1 for all z ∈ Ω(1) or n k |f (z) g(z) | ≤ 1 for all z ∈ Ω(1) . 2. Suppose Ω(2) = ∅. Then a pair of nonnegative integers (n, k), not both zero, satisfy that there exists an a, b ∈ Ω(2) such that |f (a)n g(a)k | < 1 and |f (b)n g(b)k | > 1 if and only if m(2) < M (2) and nk ∈ (m(2) , M (2) ). Furthermore, if nk ∈ / (m(2) , M (2) ), then either |f (z)n g(z)k | ≥ 1 for all z ∈ Ω(2) or n k |f (z) g(z) | ≤ 1 for all z ∈ Ω(2) . 3. Suppose Ω(1) = ∅, Ω(2) = ∅, and (m(1) , M (1) ) ∩ (m(2) , M (2) ) = ∅. Then a pair of nonnegative integers (n, k), not both zero, satisfy that there exists an a, b ∈ Ω such that |f (a)n g(a)k | < 1 and |f (b)n g(b)k | > 1 if and only if k (1) , M (2) } and β = max{m(1) , m(2) }. n ∈ (−∞, α)∪(β, ∞], where α = min{M k Furthermore, if n ∈ [α, β], then either |f (z)n g(z)k | ≥ 1 for all z ∈ Ω or |f (z)n g(z)k | ≤ 1 for all z ∈ Ω. 4. Suppose Ω(1) = ∅, Ω(2) = ∅, and (m(1) , M (1) ) ∩ (m(2) , M (2) ) = ∅. Then for every pair of nonnegative integers (n, k), not both zero, there exists an a, b ∈ Ω such that |f (a)n g(a)k | < 1 and |f (b)n g(b)k | > 1. Proof. For a point z ∈ G, |f (z)n g(z)k | < 1 if and only if n ln |f (z)|+k ln |f (z)| < 0. Thus, k (∗) |f (z)n g(z)k | < 1 if and only if ln |f (z)| < − ln |g(z)|. n if if

It then follows that if z ∈ Ω(1) , then − ln |g(z)| < 0, so (∗) holds if and only > nk . Similarly, if z ∈ Ω(2) , then − ln |g(z)| > 0, so (∗) holds if and only

ln |f (z)| − ln |g(z)| ln |f (z)| − ln |g(z)|

<

k n.

Similar statements hold describing when |f (z)n g(z)k | > 1.

If we deﬁne m(1) , m(2) , M (1) , M (2) as above, then the following statements

hold: (a) (b) (c) (d)

k n k n k n k n

> m(1) if and only if |f (z)n g(z)k | > 1 for some z ∈ Ω(1) . > m(2) if and only if |f (z)n g(z)k | < 1 for some z ∈ Ω(2) . < M (1) if and only if |f (z)n g(z)k | < 1 for some z ∈ Ω(1) . < M (2) if and only if |f (z)n g(z)k | > 1 for some z ∈ Ω(2) .

The above results follow from these facts by considering various cases. Example 5.3. Let f (z) = eaz+b   −∞ (1) m = −a   −(a + b)

and g(z) = ez , with a, b ∈ R. If   b>0 −(a + b) (1) M = −a b=0,   b<0 ∞

Ω = D, then b>0 b=0, b<0

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m

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(2)

  (b − a) = −a   −∞

b>0 b=0, b<0

M

(2)

  ∞ = −a   (b − a)

165

b>0 b=0. b<0

Thus, if α, β ∈ R with α < β, then there exists a, b ∈ R (with b > 0) such that (m(1) , M (1) ) = (−∞, α) and (m(2) , M (2) ) = (β, ∞). If α > β, then there exists a, b ∈ R (with b < 0) such that (m(1) , M (1) ) = (α, ∞) and (m(2) , M (2) ) = (−∞, β). Proof. For z ∈ D, let z = x+iy. Notice that since a, b ∈ R, then Re(az+b) = ax+b. Thus ln |f (z)| = (ax + b) and − ln |g(z)| = −x. Thus, ax + b b ln |f (z)| = = −a − . − ln |g(z)| −x x Also Ω(1) = {z ∈ D : Re(z) > 0} and Ω(2) = {z ∈ D : Re(z) < 0}. So the sup/inf of the above quantities simply amount to ﬁnding the sup/inf of the real function −a − b/x over the intervals (0, 1) and (−1, 0). Corollary 5.4. Keeping the same notation as in Proposition 5.2, let P(f, g) = {(n, k) ∈ N × N : f n g k is nonconstant on Ω and (f n g k )(Ω) ∩ ∂D = ∅}. If P(f, g) = ∅, then one of the following holds: (a) There is an open interval J = (a, b) ⊆ R such that (n, k) ∈ P(f, g) if and only if nk ∈ J, or (b) There is a compact interval K = [c, d] ⊆ R such that (n, k) ∈ P(f, g) if and / K. only if nk ∈ Furthermore, {a, b, c, d} ⊆ {m(1) , M (1) , m(2) , M (2) , ±∞}, c, d ∈ R. Proof. If P(f, g) = N × N, then (a) holds with J(f, g) = (0, ∞). If Ω = Ω(1) or Ω = Ω(2) , then by Proposition 5.2, (a) holds. The only cases when (a) is not satisﬁed is when condition 3. of Proposition 5.2 holds. In that case (b) holds. If condition (a) in Corollary 5.4 holds, then we will say that P(f, g) is a “sector” and if condition (b) in Corollary 5.4 holds, then we will say that P(f, g) is a “sector complement”. Notice that if J = (a, b) in Corollary 5.4, then (n, k) ∈ P(f, g) if and only if the point (n, k) lies in the sector or region strictly between the two lines y = ax and y = bx. Similarly, if P(f, g) is a sector complement, then there is a sector S (the region between two lines through the origin) such that P(f, g) = (N × N) \ S. Theorem 5.5 (Finitely many Components). Let G be an open set in C with finitely many components. Suppose that {Gi }N i=1 are the components of G. Also let H(G) be a Hilbert space of analytic functions as in Definition 2.1 on G. If f, g ∈ H ∞ (G) and F = {Mf∗n Mg∗k : n, k ≥ 0}, then the following are equivalent: 1. The pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). 2. The semigroup F generated by (Mf∗ , Mg∗ ) contains a hypercyclic operator.

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3. There exists integers n, k ≥ 0 such that f n g k is non-constant on every component of G and (f n g k )(Gi ) ∩ ∂D = ∅ for every i ∈ {1, . . . , N }. Proof. We will only prove the Theorem for N = 2, the general case is similar. By Theorem 2.3 one easily sees that 3. implies 2. implies 1. We must show that 1. implies 3. So assume that 1. holds and by way of contradiction assume that 3. does not hold. For i ∈ {1, 2}, let fi = f |Gi and let gi = g|Gi . Since for each i ∈ {1, 2}, Fi := F |H(Gi ) is hypercyclic on H(Gi ), then by Theorem 3.1, (∗)∀i ∈ {1, 2}, ∃n, k ≥ 0 such that fin gik is non-constant and (fin gik )(Gi ) ∩ ∂D = ∅. But the n and k may depend on i. Furthermore by Theorem 3.1, if n and k are any non-negative integers, not both zero, and fin gik is constant on Gi for some i, then it cannot be a unimodular constant. It follows from (∗) that P(f1 , g1 ) = ∅ and P(f2 , g2 ) = ∅. Since we are assuming that (3) does not hold, then (∗∗) P(f1 , g1 ) ∩ P(f2 , g2 ) = ∅. Hence by Corollary 5.4 it follows that there are two cases, either P(f1 , g1 ) and P(f2 , g2 ) are disjoint sectors or one is a sector which is disjoint from the other one which is a sector complement. Case 1: P(f1 , g1 ) and P(f2 , g2 ) are disjoint sectors. We will use the notation from Proposition 5.2 and Corollary 5.4. For each (1) (1) (2) (2) i ∈ {1, 2}, let Ji = (ai , bi ) with ai , bi ∈ {mi , Mi , mi , Mi , ±∞} be as in Corollary 5.4 (also see below), so that (n, k) ∈ P(fi , gi ) if and only if nk ∈ Ji . Since P(f1 , g1 ) and P(f2 , g2 ) are disjoint, it follows that J1 ∩ J2 = ∅. ln |fi (z)| . Then by (∗) above and PropoFor each i ∈ {1, 2}, let wi (z) = − ln |gi (z)| (j) (j) (j) sition 5.2, wi is nonconstant. Let mi = inf{wi (z) : z ∈ Gi } and Mi = (j) sup{wi (z) : z ∈ Gi }. Since J1 ∩ J2 = ∅ either b1 ≤ a2 or b2 ≤ a1 . We will suppose that b1 ≤ a2 , the other case is similar. Thus, b1 ∈ R and so by Proposition 5.2 we (1) (2) have that either G1 = G1 (|g1 | > 1 on G1 ) or G1 = G1 (|g1 | < 1 on G1 ). We’ll (2) suppose that G1 = G1 . (2) (2) (j) Thus a1 = m1 , b1 = M1 and a2 = m2 for some j ∈ {1, 2}. Let (2) (j) (2) m ∈ [b1 , a2 ] = [M1 , m2 ]. If nk ≥ m(≥ M1 ), then item (d) in the proof of (2) (j) Proposition 5.2 implies that |f1n g1k | ≤ 1 on G1 = G1 . However, if nk < m(≤ m2 ), (1) then item (a) or (b) in the proof of Proposition 5.2 implies that |f2n g2k | ≤ 1 on G2 (2) or |f2n g2k | ≥ 1 on G2 . Since nk < m ≤ a2 , then nk ∈ / J2 , so |f2 (z)n g2 (z)k | = 1 for n k n k any z ∈ G2 , thus either |f2 g2 | ≤ 1 on G2 or |f2 g2 | ≥ 1 on G2 . Since both cases are similar, let’s assume that j = 2 and so |f2n g2k | ≥ 1 on G2 . Now let φ ∈ H(G) be a hypercyclic vector for F and let φi := φ|Gi . Now suppose that h is in the closure of the orbit of φ under F . Then there exists integers nj , kj such that Mf∗nj gkj φ → h. Now either there exist inﬁnitely many j’s such

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that

kj nj

Hypercyclic Pairs of Coanalytic Toeplitz Operators

≥ m or inﬁnitely many j’s such that

kj nj

167 n

k

< m. In the ﬁrst case |f1 j g1 j | ≤ 1

on G1 and hence Mf∗nj gkj |H(G1 ) ≤ 1. In the second case, |f2 j g2 j | ≥ 1 on G2 and hence M ∗nj kj |H(G2 ) is an expansive operator (meaning M ∗nj kj x ≥ x n

f2 g2

k

f2 g2

for all x ∈ H(G2 )). Thus it follows that either h|G1 ≤ φ1 or h|G2 ≥ φ2 . This restriction on h contradicts the fact that F is hypercyclic. Thus in this case it follows that 1. implies 3. Case 2: P(f1 , g1 ) is a sector that is disjoint from P(f2 , g2 ) which is a sector complement. Since P(f1 , g1 ) is a sector let J1 = (a, b) be as in Corollary 5.4 and let J2 = R \ K = (−∞, c) ∪ (d, ∞) where K = [c, d] is the compact interval guaranteed by Corollary 5.4. Since P(f1 , g1 ) ∩ P(f2 , g2 ) = ∅, then J1 ∩ J2 = ∅, thus c ≤ a < b ≤ d. (1)

(1)

(2)

(2)

By Corollary 5.4, either (a, b) = (m1 , M1 ) or (a, b) = (m1 , M1 ). With (1) (1) out loss of generality we’ll suppose that (a, b) = (m1 , M1 ). Similarly without (2) (1) loss of generality, we’ll suppose that [c, d] = [M2 , m2 ]. (2)

(1)

(2)

Now if nk ∈ [c, d] = [M2 , m2 ], then nk ≥ c = M2 , so by part (d) of (2) Proposition 5.2 it follows that |f2n g2k | ≤ 1 for all z ∈ G2 . Now since nk ≤ d, / J2 , so the range of f2n g2k cannot hit the unit circle on G2 , thus we must then nk ∈ n k have |f2 g2 | < 1 for all z ∈ G2 . Which implies that Mf∗n gk |H(G2 ) is a contraction, 2

2

whenever nk ∈ [c, d]. Now if nk < c, then it follows that Mf∗n gk |H(G1 ) is a contraction and if 1 1 then it follows that Mf∗n gk |H(G1 ) is expansive. 1

k n

> d,

1

Now if φ ∈ H(G) is a hypercyclic vector for F and h is in the closure of the orbit of φ under F . Then there exists integers nj , kj such that Mf∗nj gkj φ → h. Now either there exist inﬁnitely many j’s such that kj nj

kj nj

∈ [c, d] or there exist inﬁnitely k

many j’s such that < c or there exist inﬁnitely many j’s such that njj > d. In either of these cases, because of the contractive or expansive properties we’ve established, there will be restrictions on the norm of h|G1 and/or the norm of h|G2 . Thus h cannot be arbitrary, so F is not hypercyclic. This contradiction implies that 1. implies 3. in case 2. It now follows that 1. always implies 3., hence the theorem follows. We now give an example showing how one might ﬁnd an f and g on a disconnected open set so that Mf∗ and Mg∗ are not hypercyclic, but the semigroup, F , that they generate is hypercyclic. ln(d) Lemma 5.6. If 0 < a 1 if and only if ln(c) ln(d) k − ln(a) < n < − ln(b) .

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Proof. Since ln(x) is increasing and − ln(x) is decreasing it follows that ln(d) − ln(b) <1< . − ln(a) ln(c) From this the ﬁrst inequality holds. For the second, simply take the logarithm of both inequalities and solve for nk . Example 5.7. Keep the same notation as in Theorem 5.5 and assume that in addition |f (z)| > 1 for all z ∈ G and |g(z)| < 1 for all z ∈ G. Also suppose that there exists an r, s such that 0 < r < 1 < s and for each i ∈ {1, . . . , N }, there exists zi , wi ∈ Gi such that the following hold: 1. for each i ∈ {1, . . . , N }, |f (zi )| > s, 2. for each i ∈ {1, . . . , N }, |f (wi )| < s, 3. for each i ∈ {1, . . . , N }, |g(zi )| > r, 4. for each i ∈ {1, . . . , N }, |g(wi )| < r, then the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). Proof. Choose a, b, c, d ∈ R such that max |g(wi )| < a < r 1. Thus for each i, |f n (zi )g k (zi )| > dn bk > 1 and |f n (wi )g k (wi )| < cn ak < 1. It follows that 3. holds from Theorem 5.5, hence the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). Next we give an example of f, g ∈ H ∞ (G) such that the pair (Mf∗ , Mg∗ ) is not hypercyclic on H(G), however, (Mf∗ , Mg∗ ) is hypercyclic on H(Gi ) for each i. If r, s ≥ 0, then let A(r, s) = {z ∈ C : r < |z| < s}. Example 5.8. (a) Let G1 , G2 be two disjoint open sets in C. If 0 < a1 < b1 < a2 < b2 < 1 < c1 < d1 < c2 < d2

(*)

∞

and fi , gi ∈ H (Gi ) for i ∈ {1, 2}, and fi (Gi ) ⊆ A(ci , di ) and gi (Gi ) ⊆ A(ai , bi ), then there is no pair of integers, n, k ≥ 0 such that for all i ∈ {1, 2}, fin gik is nonconstant on Gi and (fin gik )(Gi ) ∩ ∂D = ∅. That is, the pair (Mf∗ , Mg∗ ) is not hypercyclic on H(G), where f |Gi = fi and g|Gi = gi for i ∈ {1, 2}. (b) Keeping the notation from part (a), clearly fi and gi may be chosen to satisfy the conditions of part (a) and yet also satisfy condition 4. of Theorem 3.1 (e.g. if fi and gi are all linear polynomials), then the pair (Mf∗ , Mg∗ ) is hypercyclic on H(Gi ) for each i, but the pair (Mf∗ , Mg∗ ) is not hypercyclic on H(G). Proof. (a) By way of contradiction, suppose that there are integer n, k ≥ 1 such that for i ∈ {1, 2}, fin gik is nonconstant on Gi and (fin gik )(Gi ) ∩ ∂D = ∅, then ln(d1 ) ln(c2 ) ln(d2 ) 1) k k by Lemma 5.6 it follows that −ln(c ln(a1 ) < n < − ln(b1 ) and − ln(a2 ) < n < − ln(b2 ) . However, by (*) and Lemma 5.6 we have that contradiction, and no such n, k exist.

ln(c2 ) − ln(a2 )

>

ln(d1 ) − ln(b1 ) ,

hence we have a

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Remark. The previous example shows that if (A1 , A2 ) and (B1 , B2 ) are each hypercyclic pairs, then the pair (A1 ⊕ B1 , A2 ⊕ B2 ) need not be hypercyclic.

6. Some General Observations If F is any collection of operators, then let Orb(F , x) = {T x : T ∈ F }. Also let HC(F ) = {x ∈ X : Orb(F , x) is dense in X} be the set of hypercyclic vectors for the collection F . Theorem 6.1. Suppose that F is a collection of commuting operators on a separable Banach space X. Then the following are equivalent. 1. HC(F ) is dense in X. 2. HC(F ) is a dense Gδ in X. 3. For any two nonempty open sets U, V in X, there exists a T ∈ F such that T (U ) ∩ V = ∅. If every operator in F has dense range, then the above conditions are also equivalent to: 4. HC(F ) is nonempty. Proof. If {U }n is a countable basis for the space X, then one easily sees that n HC(F ) = n T ∈F T −1 (Un ). If condition 3. is satisﬁed, then T ∈F T −1 (Un ) is a dense open set, hence by the Baire Category Theorem, HC(F ) is a dense Gδ in X, so condition 2. holds. Clearly 2. implies 1. To see that 1. implies 3., assume that HC(F ) is dense in X and let U, V be two nonempty open sets. Then there exists a x ∈ HC(F ) ∩ U and a T ∈ F such that T x ∈ V . Thus, T (U ) ∩ V = ∅. So 1. implies 3. Thus we have 1. ⇒ 3. ⇒ 2. ⇒ 1. Clearly, 1. ⇒ 4. Now assume that 4. holds and that every operator in F has dense range and we will prove that (1) holds. Let x ∈ X be such that Orb(F , x) is dense in X. If T ∈ F, then since F is commutative, Orb(F , T x) = T (Orb(F , x)) and since T has dense range and Orb(F , x) is dense we get that Orb(F , T x) is dense in X. Thus T x ∈ HC(F ). Since this holds for each T ∈ F, then we have Orb(F , x) ⊆ HC(F ). Thus HC(F ) is dense and 1. holds. ∞ ∞ Theorem i=1 Ai and B = i=1 Bi be commuting operators on 6.2. Let A = H= ∞ H . Suppose that for each n ≥ 1, that the semigroup Fn generated by i i=1 A1 ⊕ · · ·⊕An and B1 ⊕ · · ·⊕Bn on H1 ⊕ · · ·⊕ Hn is hypercyclic and has a dense set of hypercyclic vectors, then the semigroup F generated by A and B is hypercyclic on H and has a dense set of hypercyclic vectors. Proof. Let F be the semigroup group generated by A and B on H. We will use Theorem 6.1 to show that F is hypercyclic on H. Let U, V be two nonempty open sets. Since the set of vectors in H that have only ﬁnitely many nonzero coordinates is dense in H, then there exists vectors x, y ∈ H with only ﬁnitely many nonzero coordinates such that x ∈ U and y ∈ V . Also choose an > 0 such that B(x, ) ⊆ U and B(y, ) ⊆ V . Let x = (x1 , x2 , . . .) and y = (y1 , y2 , . . .). Let n ≥ 1 be large

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enough such that xk = 0 and yk = 0 if k ≥ n. Let x[n] := (x1 , x2 , . . . , xn ) and y[n] = (y1 , y2 , . . . , yn ). Then by assumption the semigroup Fn is hypercyclic on H1 ⊕ · · · ⊕ Hn and has a dense set of hypercyclic vectors, thus there exists a T ∈ Fn such that T (B(x[n], )) ∩ B(y[n], ) = ∅. Let z[n] = (z1 , z2 , . . . , zn ) ∈ B(x[n], ) such that T z[n] ∈ B(y[n], ). If p and q are nonnegative integers such that T = (A1 ⊕ · · · ⊕ An )p (B1 ⊕ · · · ⊕ Bn )q and we let z = (z1 , z2 , . . . , zn , 0, 0, . . .), then one checks that z ∈ B(x, ) and T z ∈ B(y, ) where T = Ap B q . It follows that T (U ) ∩ V = ∅. Hence by Theorem 6.1, F is hypercyclic on H and has a dense set of hypercyclic vectors.

7. Open Sets with Infinitely Many Components Theorem 7.1. Let G be an open set in C with infinitely many components. Suppose that {Gi }∞ i=1 are the components of G. Also let H(G) be a Hilbert space of analytic functions as in Definition 2.1 on G. If f, g ∈ H ∞ (G) and F = {Mf∗n Mg∗k : n, k ≥ 0}, then the following are equivalent: 1. The pair (Mf∗ , Mg∗ ) is hypercyclic on H(G). N 2. For every N ≥ 1, the pair (Mf∗ , Mg∗ ) is hypercyclic on H( i=1 Gi ). N 3. For every N ≥ 1, F |H( i=1 Gi ) contains a hypercyclic operator. 4. For every N ≥ 1, there exists integers n, k ≥ 0 such that for every i ∈ {1, . . . , N }, f n g k is non-constant on Gi and (f n g k )(Gi ) ∩ ∂D = ∅. Proof. It follows from Theorem 5.5 that 2., 3., and 4. are all equivalent. Clearly, 1. implies 2. and it follows from Theorem 6.2 that 4. implies if fi = f |Gi ∞ 1. Because ∞ for all i, then Mf on H(G) is unitarily equivalent to i=1 Mfi on i=1 H(Gi ) = H(G), similarly for Mg . Thus we may apply Theorem 6.2.

8. A hypercyclic semigroup containing no hypercyclic operators In this section we give an example of a hypercyclic commutative semigroup generated by a pair of pure cosubnormal operators, yet the semigroup does not contain a hypercyclic operator. The cosubnormal operators are adjoints of multiplication operators on a Hilbert space of analytic functions on an open set with inﬁnitely many components. As mentioned in the introduction, if A is any supercyclic operator, then one can easily see that the semigroup generated by the tuple (A, 2I, 13 I, eiθ I) is i hyperyclic if θ ∈ R is an irrational multiple of π. This follows because { 32j eikθ : i, j, k ≥ 0} is dense in C. Now there exists a bounded open set G with inﬁnitely many components such that if A is the adjoint of multiplication by z on the Bergman space of G, then A is supercyclic, but no multiple of A is hypercyclic (see [11]). With that operator A, then the semigroup generated by (A, 2I, 13 I, eiθ I) will be hypercyclic (consist entirely of cosubnormal operators and) yet contain no

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hypercyclic operator. However, this semigroup has four generators. We now show how to give an example of such a semigroup generated by two operators. Theorem 8.1. If G is a bounded open set with infinitely many components and H(G) is a Hilbert space of analytic functions on G as in Definition 2.1, then there exists f, g ∈ H ∞ (G) such that the pair (Mf∗ , Mg∗ ) is hypercyclic on H(G), but F = {Mf∗n Mg∗k : n, k ≥ 0} contains no hypercyclic operator. First two simple lemmas which we leave to the reader. Recall that A(r, s) = {z ∈ C : r < |z| < s}, f inf,G = inf{|f (z)| : z ∈ G}, and f ∞,G = sup{|f (z)| : z ∈ G}. Lemma 8.2. If G is any bounded open set in C, then given any s > r > 0, there exists a nonconstant linear polynomial f such that f (G) ⊆ A(r, s) and f inf,G = r. There also exists a nonconstant linear polynomial g such that g(G) ⊆ A(r, s) and f ∞,G = s. Lemma 8.3. Let f (z) = az + b and g(z) = cz + d where ac = 0. If G is an open 1 on G. set where f and g are non-zero, then there is no p > 0 such that |f |p = |g| Proof of Theorem 8.1. Let {Gi }∞ i=1 be the components of G. Using Lemma 8.2 we can choose inductively constants ai , bi and nonconstant linear polynomials fi , gi such that the following hold: 1. 12 < ai+1 < ai < 1 and 2 < bi+1 < bi < 3 for all i ≥ 1. 2. limi→∞ ai = 12 and limi→∞ bi = 2. 3. fi (Gi ) ⊆ A( 12 , ai ) and gi (Gi ) ⊆ A(2, bi ). 4. For each i, fi inf,Gi = 12 and fi ∞,Gi = ai . 5. For each i, gi inf,Gi = 2 and fi ∞,Gi = bi . Now let f, g be deﬁned on G as f |Gi = fi and g|Gi = gi . Then f, g ∈ H ∞ (G). Let ln |fi (z)| . wi (z) = − ln |gi (z)| Then since fi and gi are nonconstant linear polynomials on Gi , then by Lemma 8.3 and the proof of Theorem 3.1, wi is non-constant on Gi . Thus Ji := wi (Gi ) is an open interval. By Lemma 5.6 we have that

ln(bi ) Ji := wi (D) = 1, . − ln(ai ) N k ∗ ∗ It follows that for each N > 1, N i=1 Ji = ∅ and if n ∈ i=1 Ji , then Mf n Mgk N is hypercyclic on H( i=1 Gi ). In particular, for each N > 1, the pair (Mf∗ , Mg∗ ) ∗ ∗ is hypercyclic on H( N i=1 Gi ). It now follows from Theorem 7.1 that (Mf , Mg ) is hypercyclic on H(G). However since ∞ i=1 Ji = ∅, then F contains no hypercyclic operator. Because for any nonnegative integers n, k, there exists an i such that k/n ∈ / Ji , thus by Lemma 5.1 (fin gik )(Gi ) ∩ ∂D = ∅. So, Mf∗n Mg∗k is not hypercyclic.

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9. Final Remarks & Questions There are a lot of open questions about the hypercyclicity of pairs or tuples of operators, or equivalently of ﬁnitely generated commutative hypercyclic semigroups. Here are a few, there are many others. Note that K´erchy [14] has some results about supercyclic semigroups in ﬁnite dimension, some results involving weighted shifts and supercyclic semigroups, and a “supercyclicity criterion” for a semigroup. Question 9.1. Can one characterize the finitely generated commutative hypercyclic semigroups in finite dimensions? There are non-trivial examples of such in every dimension, see K´erchy [14]. Question 9.2. Can one characterize the pairs (tuples) of cosubnormal (cohyponormal) operators that are hypercyclic? Question 9.3. Can one characterize the pairs (tuples) of weighted shifts that are hypercyclic? Are there non-trivial examples in this case? Question 9.4. Is there a “hypercyclicity criterion” for pairs or tuples of operators? Question 9.5. If F is a finitely generated commutative hypercyclic semigroup, then must F contain a cyclic operator? Question 9.6. If (T1 , T2 ) is a hypercyclic pair, then is (T1 ⊕ T1 , T2 ⊕ T2 ) also a hypercyclic pair? Notice that this reduces to Herrero’s question when T2 is the identity operator.

References [1] S.I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), 374–383. [2] S. Axler, Harmonic Functions from a Complex Analysis Viewpoint, Amer. Math. Monthly, 93, No. 4, (1986), 246–258. [3] S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory Second edition. Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 2001. [4] T. Berm´ udez, A. Bonilla, A. Martin´ on On the existence of chaotic and hypercyclic semigroups on Banach spaces Proc. Amer. Math. Soc. 131 (2003), no. 8, 2435–2441 [5] T. Berm´ udez, A. Bonilla, J.A. Conejero, A. Peris Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces Studia Math. 170 (2005), no. 1, 57–75. [6] P.S. Bourdon and J.H. Shapiro, Cyclic Phenomena for Composition Operators, Memoirs of the AMS, 125, AMS, Providence, RI, 1997. [7] P.S. Bourdon and N.S. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J. 52 (2003), No. 3, 811-819. [8] J.B. Conway, Spectral Properties of Certain Operators on Hardy spaces of Planar Regions, Int. Eqns. Oper. Th. 10 (1987), 659–706. [9] J.B. Conway, The Theory of Subnormal Operators, Amer. Math. Soc., Providence, RI, 1991.

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[10] G. Costakis and A. Peris Hypercyclic semigroups and somewhere dense orbits C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 895–898. [11] N.S. Feldman, T.L. Miller, and V.G. Miller, Hypercyclic and Supercyclic Cohyponormal Operators, Acta Sci. Math. (Szeged) 68 (2002), 303–328. [12] R.M. Gethner and J.H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. AMS, 100 (1987), 281–288. [13] G. Godefroy and J.H. Shapiro, Operators with dense invariant cyclic manifolds, J. Func. Anal., 98 (1991), 229–269. [14] L. K´erchy Cyclic properties and stability of commuting power bounded operators, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 299–312. [15] C. Kitai, Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982. [16] S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17–22. [17] H.N. Salas, Hypercyclic weighted shifts, Trans AMS, 347 (1995), 993–1004. [18] H.N. Salas, Supercyclicity and weighted shifts, Studia Math. 135 (1999), no. 1, 55–74. Nathan S. Feldman Mathematics Department Washington & Lee University Lexington, VA 24450 USA e-mail: [email protected] Submitted: October 3, 2005 Revised: October 30, 2006

Integr. equ. oper. theory 58 (2007), 175–204 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020175–174, published online April 14, 2007 DOI 10.1007/s00020-007-1482-4

Integral Equations and Operator Theory

On Weakly Formulated Sylvester Equations and Applications Luka Grubiˇsi´c and Kreˇsimir Veseli´c Abstract. We use a “weakly formulated” Sylvester equation H1/2 T M−1/2 − H−1/2 T M1/2 = F to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. Another application is a bound for the square root of a positive selfadjoint operator which extends the known rule: “The relative error in the square root is bounded by the one half of the relative error in the radicand”. Both bounds are illustrated on diﬀerential operators which are deﬁned via quadratic forms. Mathematics Subject Classification (2000). 65F15, 49R50, 47A55, 35Pxx. Keywords. Eigenvalues, eigenvectors, variational methods for eigenvalues of operators, perturbation theory.

1. Preliminaries In this work we will study properties of nonnegative selfadjoint operators in a Hilbert space which are close in the sense of the inequality (1.1) |h(φ, ψ) − m(φ, ψ)| ≤ η h[φ]m[ψ] where the sesquilinear forms h, m belong to the operators H, M respectively and m[ψ] = m(ψ, ψ), h[φ] = h(φ, φ). By Q(h) we denote the domain space of a sesquilinear form h and in (1.1) we assume that Q(h) = Q(m). In the ﬁrst part of the paper we show that (1.1) implies an estimate of the separation between “matching” eigensubspaces of H and A. To be more precise one of the typical situations is: Let 0 ≤ λ1 (H) ≤ λ2 (H) ≤ · · · ≤ λn (H) < D < λn+1 (H) ≤ · · ·

(1.2)

0 ≤ λ1 (M) ≤ λ2 (M) ≤ · · · ≤ λn (M) < D < λn+1 (M) ≤ · · ·

(1.3)

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be the eigenvalues of the operators H and M which satisfy (1.1), then Dλ (H) Dλ (M) n n , η. EH (D) − EM (D) ≤ min D − λn (H) D − λn (M) Such an estimate was implicit in [7]. We then generalize this inequality to hold both for the operator norm · and the Hilbert–Schmidt norm ||| · |||HS . We also allow that EH (D) and EM (D) be possibly inﬁnite dimensional. For recent estimates of the separation between eigensubspaces see [10]. In the second part of the paper we establish estimates for a perturbation of the square root of a positive operator. It will be shown that the inequality (1.1) implies η h2 [φ]m2 [ψ], |h2 (φ, ψ) − m2 (φ, ψ)| ≤ 2 where the sesquilinear forms h2 , m2 belong to the operators H1/2 , M1/2 , respectively. This will show that it is meaningful to consider weakly formulated Sylvester equations where all the coeﬃcient operators are unbounded, cf. (1.4). Both of this problems will be solved through a study of the weak Sylvester equation, which reads formally HT − T M = H1/2 F M1/2 .

(1.4)

These two case studies represent two diﬀerent classes of additional assumptions which have to be imposed on the coeﬃcient operators H, M and F in order that (1.4) deﬁnes a meaningful operator T . The main novelty (and contribution) of this work is that we present an abstract study of the operator equation (1.4) in the case when only F is a bona ﬁde operator. The expression H1/2 F M1/2 need not possess an operator representation. In comparison, H1/2 F M1/2 was always a bounded operator for the Sylvester equations which were studied in [1, 2, 12]. Most recent and most general result of this type in the case of matrix coeﬃcients is [12, Theorem 1] which reads Let M and H be positive semi deﬁnite (ﬁnite) matrices such that the intersection of their spectra is empty. Then the solution T of (1.4) satisﬁes ||| F ||| π . ||| T |||≤ 2 min{| ln λ/µ| : λ ∈ σ(H), µ ∈ σ(M )} Here σ(H) and σ(M ) denote the spectra of H and M and ||| . ||| is any unitary invariant matrix norm. We consider a very general class of (unbounded) operator coeﬃcients for the weak Sylvester equation. In order to regularize the problem we need to impose more stringent conditions (as compared with those in the result we have just stated) on the location of σ(H) and σ(M) or on the unitary invariant norm ||| · |||, see Theorems 2.1, 2.4, 2.7, 2.8 and 5.1 below. It should be noted that in the matrix case and in the situation in which all of these results apply their numerical performance is comparable, cf. [12].

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A main technique which led to [12, Theorem 1] is the inequality ||| ln(H)T − T ln(M) |||≤||| H1/2 T M−1/2 − H−1/2 T M1/2 |||=||| F ||| .

(1.5)

This deep result from [9] can unfortunately only be assumed as formally correct in our setting since the products ln(H)T and T ln(M) do not have to be bona ﬁde operators. To some extent it could be said that the main novelty in this work is a form theoretic approach to the problem of regularizing the equation (1.4). More speciﬁcally, our ﬁrst main result—contained in Theorem 2.1 below— extends our previous result from [7] in various ways. In particular, we allow the perturbed projection to be inﬁnite dimensional. In the proof we also overcome a technical error contained in [7]. We then extend this result to the case of other unitary invariant operator norms1 . Particular attention is paid to the Hilbert– Schmidt norm because of its possible importance in applications. This special case is handled by another technique which allows an arbitrary interlacing of the involved spectra. 1.1. Notation and Lemmata The main object in this work shall be a closed nonnegative symmetric form in a Hilbert space. When dealing with symmetric forms in a Hilbert space, we shall follow the terminology of Kato, cf. [8]. For reader’s convenience we now give deﬁnitions of some terms that will frequently be used, cf. [3, 8]. Definition 1.1. Let h be a positive deﬁnite symmetric form in H. A sesquilinear form a, which need not be closed, is said to be h-bounded, if Q(h) ⊂ Q(a) and there exists η ≥ 0 |a[u]| ≤ ηh[u] u ∈ Q(h). If h is positive deﬁnite the space (Q(h), h) can be considered as a Hilbert space and h-bounded form a deﬁnes a bounded operator on (Q(h), h). Definition 1.2. A bounded operator A : H → U is called degenerate if its range space R(A) := {Au : u ∈ H} is ﬁnite dimensional. Definition 1.3. If H is a selfadjoint operator and P a projection, to say that P commutes with H means that u ∈ D(H) implies P u ∈ D(H) and HP u = P Hu,

u ∈ D(H).

Definition 1.4. Let H and M be nonnegative operators. We deﬁne the order relation ≤ between the nonnegative operators by saying that M ≤ H if and only if D(H1/2 ) ⊂ D(M1/2 ) and M1/2 u ≤ H1/2 u,

u ∈ D(H1/2 ),

or equivalently m[u] ≤ h[u], u ∈ Q(h) := D(H1/2 ), when m and h are nonnegative forms deﬁned by the operators M and H and M ≤ H. 1 Also called “cross-norms” in the terminology of [8] or “symmetric norms” in the terminology of [4, 17].

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As a notational convention we use normal math-script letters (e.g. M ) to denote bounded operators and matrices and boldface math-script letters (e.g. H) to denote unbounded operators. A main principle we shall use to develop the perturbation theory will be the monotonicity of the spectrum with regard to the order relation between nonnegative operators. This principle can be expressed in many ways. The relevant results, which are scattered over the monographs [3, 8], are summed up in the following theorem, see also [11, Corollary A.1]. Theorem 1.5. Let M = λ dEM (λ) and H = λ dEH (λ) be nonnegative operators in H and let M ≤ H. Let the eigenvalues of H and M be as in (1.2) and (1.3), then 1. λe (M) ≤ λe (H) 2. dim EH (γ) ≤ dim EM (γ), for every γ ∈ R 3. λk (M) ≤ λk (H), k = 1, 2, . . . . The inﬁmum of the essential spectrum of some operator H is denoted by λe (H). With this theorem in hand we review spectral properties of operators H and M, for which there exists 0 ≤ ε < 1 such that (1 − ε)m[u] ≤ h[u] ≤ (1 + ε)m[u],

u ∈ Q := Q(h) = Q(m).

(1.6)

Let us assume h[u] > 0, then m[u] > 0 and ε ε )h[u] ≤ m[u] ≤ (1 + )h[u]. (1.7) (1 − 1−ε 1−ε Inequality (1.6) implies that N(H) = N(M), so (1.7) holds for all u ∈ Q. By N(H) we denote the null space of some operator H. Lemma 1.6. Let m and h be nonnegative forms such that λe (M) > 0 and λe (H) > 0 and let (1.6) hold. Then |λi (H) − λi (M)| ≤ ελi (M) (1.8) ε λi (H) (1.9) |λi (H) − λi (M)| ≤ 1−ε λi (H) and λi (M) are as in (1.2) and (1.3). Assume that λi−1 (H) < λi (H) < λi+1 (H) and λ (H) − λ (H) λ (H) − λ (H) ε i+1 i i i−1 < max , ,1 , (1.10) 1−ε λi+1 (H) + λi (H) λi (H) + λi−1 (H) then min

λj (M)

|λi (H) − λi (M)| |λi (H) − λj (M)| = < 1. λi (H) λi (H)

If λi−1 (H) < λi (H) = · · · = λi+n−1 (H) < λi+n (H) and λ (H) − λ (H) λ (H) − λ (H) ε i+n i i i−1 < max , ,1 , 1−ε λi+n (H) + λi (H) λi (H) + λi−1 (H)

(1.11)

(1.12)

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then argmin

|λi−1 (H) − λj (M)| ≤i−1 λi−1 (H)

(1.13)

argmin

|λi+n (H) − λj (M)| ≥ i + n. λi+n (H)

(1.14)

j∈N

j∈N

Proof. Estimates (1.8)–(1.9) are a consequence of (1.6)–(1.7) and Theorem 1.5. The rest of the theorem follows from a proof which analogous to the proof of [5, Theorem 4.16]. We repeat the argument in this new setting. Let i = j, then |λi (H) − λj (M)| |λi (H) − λj (H)| λi (H) + λj (H) |λj (H) − λj (M)| λj (H) ≥ − λi (H) λi (H) + λj (H) λi (H) λj (H) λi (H) ε λj (H) λj (H) − >γ ≥γ 1+ λi (H) 1 − ε λi (H) |λi (H) − λi (M)| > . λi (H) With this we have established (1.11). (1.13)–(1.14) are a way to state (1.11) in a presence of a multiple eigenvalue λi (H). The proof follows by a repetition of the previous argument for j ≥ i and j ≤ i + n − 1. For instance, we establish (1.13) by proving |λi−1 (H) − λi−1 (M)| |λi−1 (H) − λj (M)| > λi−1 (H) λi−1 (H) for all j ≥ i.

Remark 1.7. The signiﬁcance of this lemma is that it detects which spectral subspaces should be compared. When we were comparing discrete eigenvalues, the order relation between the real numbers (eigenvalues) solved this problem automatically. For spectral subspaces we need to assume more than (1.6) in order to be able to construct meaningful estimates. Assumptions (1.10) and (1.12) show how much more we (will) assume. Next we show that (1.6) implies (1.1) with η = ε(1 − ε)−1/2 . To establish this claim we need a notion of a pseudo inverse of a closed operator. A deﬁnition from [18] will be used. The pseudo inverse of a selfadjoint operator H is the selfadjoint operator H† deﬁned by D(H† ) = R(H) ⊕ D(H)⊥ , H† (u + v) = H−1 u,

u ∈ R(H), g ∈ D(H)⊥ .

It follows that H† = H−1 in R(H). Note that we did not assume H† to be bounded or densely deﬁned. The operator H† will be bounded if and only if R(H) is closed in

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H, see [15]. The operator H† could have also been deﬁned by the spectral calculus, since 0, λ = 0, † f (λ) = 1 H = f (H), λ , λ = 0. In [18] Weidmann has given a short survey of the properties of the pseudo inverse of a nondensely deﬁned operator H. In particular, let H1 and H2 be two nonnegative operators in D(H1 ) and D(H2 ) respectively, then 1/2

1/2

1/2†

H1 u ≤ H2 u ⇐⇒ H2

1/2†

u ≤ H1

u.

(1.15)

Analogously, let h1 and h2 be two closed, not necessarily densely deﬁned, positive deﬁnite forms and let H1 and H2 be the selfadjoint operators deﬁned by h1 and h2 in Q(h1 ) and Q(h2 ). We say h1 ≤ h2 when Q(h2 ) ⊂ Q(h1 ) and 1/2

1/2

h1 [u] = H1 u2 ≤ h2 [u] = H2 u2 ,

u ∈ Q(h2 ).

(1.16)

Equivalently, we write H1 ≤ H2 when h1 ≤ h2 . Now, we can write the fact (1.15) as H1 ≤ H2 ⇐⇒ H†2 ≤ H†1 . (1.17) In one point we will depart from the conventions in [8]. Definition 1.8. A nonnegative form h(u, v) = (H1/2 u, H1/2 v) will be called nonnegative deﬁnite when H† is bounded. Analogously, the nonnegative operator H such that H† is bounded will also be called nonnegative deﬁnite. In the sequel we establish a connection between (1.6) and (1.1) when h and m are nonnegative deﬁnite forms. Lemma 1.9. Let H and M be nonnegative deﬁnite operators in a Hilbert space H such that (1.6) holds for 0 ≤ ε < 1. Let S = H1/2 M†1/2 − H†1/2 M1/2 ,

(1.18)

then S is bounded and

ε ψφ. 1−ε Proof. The closed graph theorem implies that the operator |(ψ, Sφ)| ≤ √

(1.19)

S = H1/2 M†1/2 − H†1/2 M1/2 is bounded. Also, N(H) = N(M) = N(S) and PN(S) commutes with S. It is suﬃcient to prove the estimate for x, y ∈ R(H). The assumption (1.6) gives | h − m (H†1/2 x, M†1/2 y)| ≤ εy m[H†1/2 x]1/2 . Analogously, (1.6) implies 1 M1/2 H†1/2 ≤ √ . 1−ε Altogether, the estimate (1.19) follows.

(1.20)

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Now, we rewrite the conclusion of this lemma in the symmetric form setting. The result is given in the form of a proposition which we present without proof. Proposition 1.10. Let m and h be nonnegative deﬁnite forms and let there exist 0 ≤ ε < 1 such that (1.6) holds, then N(H) = N(H) and ε h[u]m[v]. |h(u, v) − m(u, v)| ≤ √ 1−ε When we only know that h and m satisfy (1.1), then we can establish a similar result about N(H) and N(M). Proposition 1.11. Let m and h be nonnegative deﬁnite forms such that (1.1) holds, then S = H1/2 M†1/2 − H†1/2 M1/2 S ∗ = M†1/2 H1/2 − M1/2 H†1/2 are bounded operators and S ∗ = S ≤ η. Furthermore, N(H) = N(M) and a fortiori R(H) = R(M). The operator S has a special structure. Assume Mu = µu and Hv = λv, then (v, Su) = λ1/2 (v, u)µ1/2 − λ−1/2 (v, u)µ1/2 λ−µ = √ (v, u) . λµ

(1.21)

The equation (1.21) suggests the distance function |λ − µ| √ λµ which measures the distance between the eigenvalues of operators H and M. We state this result as the following corollary.. Corollary 1.12. Let Mu = µu, u = 1 and Hv = λv, v = 1 and let S be as in Proposition 1.11, then η |λ − µ| √ . ≤ |(u, v)| λµ Our theory is designed to be directly applicable to diﬀerential operators given in a weak form. This will enable us to obtain estimates for the diﬀerence between the spectral projections of the operators to which the theory of [1, 2] does not apply, see Example 3.4 below.

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2. Weak Sylvester equation Let us outline the general picture. We have an unbounded positive deﬁnite operator A and a bounded positive deﬁnite operator M . They are deﬁned in, possibly, diﬀerent subspaces of the environment Hilbert space H. Thus, HM = R(M ) is (of necessity) a closed subspace of H and likewise D(A1/2 )

H

= R(A1/2 ) = HA .

Let the bounded operator F : HM → HA be given, then we are looking for the bounded operator T : HM → HA such that (A1/2 v, T M −1/2 u) − (A−1/2 v, T M 1/2 u) = (v, F u) ,

v ∈ D(A1/2 ), u ∈ HM . (2.1)

Formally, we say that T solves the equation AT − T M = A1/2 F M 1/2 .

(2.2)

Here G = A1/2 F M 1/2 is naturally only a formal expression and does not represent a bona ﬁde operator. In the case in which G be a bona ﬁde operator equation (2.2) becomes the rigorous equation AT − T M = G, called the (standard) Sylvester equation, cf. [1, 2]. The case when A and M are ﬁnite matrices has been considered in [12] where (2.2) was called the structured Sylvester equation. We call the relation (2.1) the weak Sylvester equation. This equation has the same form as (1.4), but its coeﬃcients are less general since we assume M to be a bounded operator. On the other hand, this “special” Sylvester equation allows us to tackle the perturbation problem for EH (D) and EM (D) in full generality (e.g. take A : HA → HA as the compression of H on the subspace HA := R(EH (D))⊥ and M : HM → HM as the compression of M on the subspace HM := R(EM (D)), for details see Section 3). We have adapted the notation to reﬂect this structural fact. The weak Sylvester equation represents a generalization of the concept of the structured Sylvester equation (2.2) from ﬁnite matrix setting to unbounded operator setting. The following theorem slightly generalizes the corresponding result from the joint paper [7] and corrects a technical glitch in one of the proofs. Theorem 2.1. Let A and M be positive deﬁnite operators in HA and HM , respectively and let F be a bounded operator from HM into R(A1/2 ) = HA . If M is bounded and 1 , (2.3) M < A−1 then the weakly formulated Sylvester equation A1/2 v, T M −1/2 u − v, A−1/2 T M 1/2 u = (v, F u) (2.4)

Vol. 58 (2007)

On Weakly Formulated Sylvester Equations

has a unique solution T , given by τ (v, u) = (v, T u) and

∞ 1 (A1/2 v, (A − iζ − d)−1 F (M − iζ − d)−1 M 1/2 u)dζ, τ (v, u) = − 2π −∞

183

(2.5)

where d is any number satisfying M < d <

1 . A−1

Proof. The uniqueness means that A1/2 v, W M −1/2 u − v, A−1/2 W M 1/2 u = 0,

(2.6)

(2.7)

for u ∈ HM , v ∈ D(A1/2 ), has the only bounded solution W = 0. Let

n d EA1/2 (λ), En = 0

then in particular A1/2 v, En W M −1/2 u − v, A−1/2 En W M 1/2 u = 0, for u ∈ HM , v ∈ D(A1/2 ) ∩ En H. Deﬁne the cut–oﬀ function x, D ≤ x ≤ n fn (x) = n, n ≤ x with D = 1/A−1 . The operator fn (A1/2 ) is bicontinuous and fn (A1/2 )En W M −1/2 − fn (A1/2 )−1 En W M 1/2 = 0.

(2.8)

Since fn (A1/2 ) and M 1/2 are bounded and positive deﬁnite operators, the standard Sylvester equation (2.8) has the unique solution n∈N.

En W = 0,

(2.9)

This is a consequence of the standard theory of the Sylvester equation with bounded coeﬃcients, see [1, 2]. The statement (2.9) implies W = 0. Now for the existence. We use the spectral integral A = λ dE(λ) to compute

∞

∞ −2 −1 1/2 2 (A + iζ − d) A v dζ = (A1/2 v, A − iζ − d Av) dζ −∞

−∞ ∞

∞

λ d(E(λ)A1/2 v, A1/2 v) (λ − d)2 + ζ 2 −∞ D

∞

∞ dζ = λ d(E(λ)A1/2 v, A1/2 v) 2 2 D −∞ (λ − d) + ζ

∞ πλ d(E(λ)A1/2 v, A1/2 v) = λ−d D −1 (2.10) = π(A(A − d) v, v). =

dζ

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Analogously, one establishes

∞ (M − iζ − d)−1 M 1/2 u2 dζ = π(M (d − M )−1 u, u).

(2.11)

−∞

The convergence of these integrals justiﬁes the following computation. Set

∞ 1 τ (v, u) = − (A1/2 v, (A − iζ − d)−1 F (M − iζ − d)−1 M 1/2 u)dζ 2π −∞ and then compute using (2.10) and (2.11)

2 1 ∞ −1 1/2 −1 1/2 |τ (v, u)|2 = ((A + iζ − d) A v, F (M − iζ − d) M u)dζ (2π)2 −∞

2 F 2 ∞ −1 1/2 −1 1/2 ≤ (A + iζ − d) A v (M − iζ − d) M udζ (2π)2 −∞ ≤

F 2 (A(A − d)−1 v, v)(M (d − M )−1 u, u). 4

(2.12)

This in turn implies that the operator τ (v, u) = (v, T u) is a bounded operator and also gives the meaning to the formula (2.5). Now we will prove that this T satisﬁes the equation (2.4). Note that A(A − ρ − d)−1 = I + (ρ + d)(A − ρ − d)−1 ,

ρ ∈ σ(A)

and then take v ∈ D(A) to compute (A1/2 v, T M −1/2 u) − (A−1/2 v, T M 1/2 u) =

1 ∞ =− (Av, (A − iζ − d)−1 F (M − iζ − d)−1 u) dζ 2π −∞

∞ (v, (A − iζ − d)−1 F (M − iζ − d)−1 M u) dζ − −∞

∞ 1 v.p. (v, F (M − iζ − d)−1 u) dζ =− 2π −∞

∞ + (iζ + d)((A − iζ − d)−1 v, F (M − iζ − d)−1 u) dζ −∞

∞ (iζ + d)((A − iζ − d)−1 v, F (M − iζ − d)−1 u) dζ − −∞

∞ ((A − iζ − d)−1 v, F u) dζ − v.p. −∞

= (v, F u). By a usual density argument we conclude that the operator T satisﬁes (2.4).

Vol. 58 (2007)

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Theorem 2.2. Let A, M and F be as in Theorem 2.1, then F DM T ≤ (D − d)(d − M ) 2 for any M < d < D. The optimal d is d = (M + D)/2 and then we obtain DM F (2.13) T ≤ (D − M ) Proof. Estimate (2.12) yields F F T ≤ A(A − d)−1 M (d − M )−1 ≤ 2 2

DM (D − d)(d − M )

This in turn implies the desired estimate. The optimality of the d = (M + D)/2 can now be checked by a direct computation. Remark 2.3. In fact, we will see that the estimate of Theorem 2.2 is optimal in the following sense. Let us consider the equation (2.4) in another light. Theorem 2.1 gives a set of conditions when the equation (2.1) has a unique solution. Theorem 2.2, then provides us with an estimate of this solution. Since for given F , under the conditions of Theorem 2.1, there exists the unique T such that (2.4) holds, we can deﬁne the so called “Sylvester operator” which associates the solution T to every operator F . The estimate (2.13) is then an estimate of the norm of the inverse of such an operator. The bound (2.13) is sharp in this sense as shows the following example. Let M and A be such that M q = M q,

Ap = Dp,

for p and q one dimensional projections and let F = pq. Then (2.4) is obviously satisﬁed by DM pq. T = D − M 2.1. Allowing for a more general relation between σ(M ) and σ(A). An analogue of Theorem 2.1 holds, if the assumption (2.6) is replaced by a more general one, namely that the interval M −1 −1 , M be contained in the resolvent set of the operator A. We omit the proof of the following result.

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11111 00000 11111 00000

λ ≤ λm−1

0

D−

g

M −1 −1

IEOT λm+n ≤ λ

M

d

D+

Figure 1. The spectral gaps Theorem 2.4. Let the operators A, M and F be as in Theorem 2.1, and let their spectra be arranged as on Figure 1, then (in the sense of (2.5))

∞ 1 T = − A1/2 (A − iζ − d)−1 F (M − iζ − d)−1 M 1/2 dζ 2π −∞

∞ 1 A1/2 (A − iζ − g)−1 F (M − iζ − g)−1 M 1/2 dζ, + 2π −∞ where d, g are chosen from the right and left spectral gap, see Figure 1, is the solution of the weak Sylvester equation (2.4). We also have the estimate M −1 −1 D D+ M − + F . T ≤ M −1 −1 − D− D+ − M 2.2. Estimates in the Hilbert–Schmidt norm A bounded operator H : H → H is a Hilbert–Schmidt operator if H ∗ H is trace class and then, cf. [8, Ch. X.1.3], √ ||| H |||HS := Tr H ∗ H. (2.14) Let A and M be positive deﬁnite operators in HA ⊂ H and HM ⊂ H, respectively. We will analyze the weakly formulated Sylvester equation under the assumption that ||| F |||HS < ∞ and gap(σ(M), σ(H)) :=

|µ − λ| √ > 0. µ∈σ(M), µλ inf

(2.15)

λ∈σ(A)

To prove our result, we will need a basic result on the spectral representation of selfadjoint operators, see [19, Satz 8.17]. Theorem 2.5 (Spectral representation). For every selfadjoint operator H in a sparable Hilbert space H there exists a σ-ﬁnite measure space (M, µ), a µ-measurable function h : M → R and a unitary operator V : H → L2 (M, µ) such that H = V −1 HV. 2 2 : L (M, µ) → L (M, µ) is the multiplication operator which is deﬁned by Here H the function h. We will also need the following theorem on the integral representation of Hilbert–Schmidt operators. For the proof see [19, Satz 3.19].

Vol. 58 (2007)

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Theorem 2.6. A bounded operator T : L2 (M1 , µ) → L2 (M2 , ν) is a Hilbert– Schmidt operator if and only if there exists a function t ∈ L2 (M1 × M2 , µ × ν) such that

(T g)(y) = t(x, y)g(x)dµ ν-almost everywhere, g ∈ L2 (M1 , µ). M1

Furthermore, we have ||| T |||HS = tL2 (M1 ×M2 ,µ×ν) . We now prove a “Hilbert–Schmidt” version of Theorem 2.1. We will assume that ||| F |||HS < ∞ and that H be separable. On the other hand, the spectra of A and M may be arbitrarily interlaced. Theorem 2.7. Let A and M be positive deﬁnite operators in HA and HM , respectively and let F : HM → HA be a bounded operator. Assume further that ||| F |||HS < ∞ and gap(σ(M), σ(H)) > 0, then there exists a unique Hilbert– Schmidt operator T such that A1/2 v, T M−1/2 u − v, A−1/2 T M1/2 u = (v, F u) (2.16) and ||| T |||HS ≤

||| F |||HS . gap(σ(M), σ(H))

(2.17)

Proof. The uniqueness of the bounded solution of the equation (2.16) follows by a double cut-oﬀ argument analogous to the one used in (2.8)–(2.9). We leave out the details. By Theorem 2.6 there exist measure spaces (MM , µ) and (MA , µ), measurable functions m : MM → R and a : MA → R and unitary operators U : H → L2 (MM , µ) and V : H → L2 (MA , µ) such that A = V −1 AV M = U −1 MU. and M to be the multiplication operators which were Here we have taken A deﬁned by the functions a and m respectively. Since ||| F |||HS < ∞, the operator V F U : L2 (MM , µ) → L2 (MA , µ) is obviously a Hilbert–Schmidt operator and ||| V F U |||HS =||| F |||HS . We can therefore assume, without loosing generality, that M=M we work with HM = L2 (MM , µ), HA = L2 (MH , ν) and that A = A, and F = V F U . Theorem 2.6 implies that there exists a function f ∈ L2 (MM × MA , µ × ν) such that

(F g)(y) = f (x, y)g(x)dµ ν-almost everywhere, g ∈ L2 (MM , µ). MM

Set t(x, y) =

f (x, y) a(y)1/2 m(x)1/2

−

m(x)1/2 a(y)1/2

,

µ × ν-almost everywhere.

(2.18)

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Relation (2.15) and the positive deﬁniteness of A and M imply that

1 a(·)1/2 m(··)1/2 L∞ (MM ×MA ,µ×ν) ≤ a(·) − m(··) gap(σ(M), σ(A))

thus t ∈ L2 (MM × MA , µ × ν) and tL2 (MM ×MA ,µ×ν) ≤

1 f L2(MM ×MA ,µ×ν) . gap(σ(M), σ(A))

(2.19)

Now (2.18) can be rewritten as a(y)1/2 t(x, y)m(x)−1/2 − a(y)−1/2 t(x, y)m(x)1/2 = f (x, y)

(2.20)

The kernel t deﬁnes a Hilbert–Schmidt operator T with

(v, T u) = v(y)t(x, y)u(x)dµ dν. By taking integrals for v ∈ D(A1/2 ) and u ∈ D(M1/2 ) we establish that the equation (2.20) is equivalent to (2.16) and the estimate (2.19) implies (2.17). 2.3. Estimates by other unitary invariant operator norms Let L(H) be the algebra of all bounded operators on the Hilbert space H. We will consider symmetric norms ||| · ||| on a subspace S of L(H). To say that the norm is symmetric on S ⊂ L(H) means that, beside the usual properties of any norm, it additionally satisﬁes: (i). If B ∈ S, A, C ∈ L(H), then ABC ∈ S and ||| ABC |||≤ A ||| B ||| C. (ii). If A has rank 1, then ||| A |||= A, where · always denotes the standard operator norm on L(H). (iii). If A ∈ S and U, V are unitary on H, then U AV ∈ S and ||| U AV |||=||| A |||. (iv). S is complete under the norm ||| · |||. The subspace S is deﬁned as a ||| · |||–closure of the set of all degenerate operators in L(H). Such S is an ideal in the algebra L(H), cf. [4, 17]. Symmetric norms were used in [1] in the context of subspace estimates. If we assume, additionally to the assumptions of Theorem 3.2 that ||| F |||< ∞, then there exists a unique bounded solution T of the weak Sylvester equation and DM ||| T |||≤ ||| F ||| . D − M We now prove this fact. Theorem 2.8. Let A and M be the selfadjoint operators which satisfy the assumptions of Theorem 2.1 and let the symmetric norm ||| · ||| have the property (P) If sup ||| An |||< ∞ and A = w-limn An , then A ∈ S and ||| A |||≤ sup ||| An ||| .

Vol. 58 (2007)

On Weakly Formulated Sylvester Equations

189

If ||| F |||< ∞, then there exists a unique bounded operator T such that A1/2 v, T M −1/2 u − v, A−1/2 T M 1/2 u = (v, F u) and ||| T |||≤

DM ||| F ||| . D − M

Proof. The proof follows by a cut-oﬀ argument. We (re)use the construction which was used in (2.8). Let fn (A1/2 ) and En be as in (2.8). The equation (2.21) fn (A1/2 )v, Tn M −1/2 u − fn (A1/2 )−1 v, Tn M 1/2 u = (v, En F u) can now be written as the standard Sylvester equation fn (A1/2 )2 Tn − Tn M = fn (A1/2 )En F M 1/2 which has the unique bounded solution Tn : HM → R(En ) and ||| Tn |||< ∞ (this follows from [2, Theorem 5.2]). The operator En T is bounded and satisﬁes the equation (2.21) therefore Tn = En T . Here we have tacitly assumed L(HA ) ⊂ L(H). Furthermore, (2.22) A1/2 En T M −1/2 − A−1/2 En T M 1/2 = En F. We compute, using the property (i), M 1/2 ||| A−1/2 En T M 1/2 ||| ≤ ||| En T ||| A−1 −1 A−1 −1 1/2 −1/2 ||| En T ||| . ||| A En T M ||| ≥ M 1/2 From these estimates and (2.22) we obtain the uniform upper bound DM DM ||| En F |||≤ ||| F ||| . ||| En T |||≤ D − M D − M

(2.23)

Since En T → T in the strong operator topology, Property (P) and the uniform bound (2.23) imply ||| T |||< ∞ and the desired estimate follows.

3. Perturbations of spectral subspaces When comparing two spectral subspaces of operators H and M, which satisfy (1.1), we have to make an additional assumption on the location of the spectra. Namely we assume that there exist D1 < D2 such that the interval [D1 , D2 ] ⊂ R is contained in the resolvent sets of both H and M. Let Q = EH (D1 ) and P = EM (D1 ). We want to estimate the norm of P − Q. The following description of a relation between a pair of orthogonal projections in a Hilbert space will be suﬃcient for our considerations. For the proof see [8].

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Theorem 3.1 (Kato). Let P and Q be two orthogonal projections such that P (I − Q) < 1. Then we have the following alternative. Either 1. R(P ) and R(Q) are isomorphic and P (I − Q) = Q(I − P ) = P − Q

or

2. R(P ) is isomorphic to true subspace of R(Q) and Q(I − P ) = P − Q = 1. To ease the presentation set P⊥ = I−P and Q⊥ = I−Q. First, let us consider the case when h and m are positive deﬁnite. With the help of Proposition 1.11 we shall later reduce the nonnegative deﬁnite case to the positive deﬁnite one. We deﬁne the operators A = Q⊥ HQ⊥ ,

M = P MP

H = QHQ,

and W = P⊥ MP⊥ .

(3.1)

We shall not notationally distinguish the operators A, M , W and H from their restrictions to the complement of their respective null spaces. Obviously, H = H + A,

M=M +W

and we compute, for S from (1.18), Q⊥ SP = (H1/2 Q⊥ P M−1/2 − H−1/2 Q⊥ P M1/2 )P = A1/2 Q⊥ P M −1/2 − A−1/2 Q⊥ P M 1/2 = A1/2 T M −1/2 − A−1/2 T M 1/2 .

(3.2)

Here we have deﬁned T = Q⊥ P . If we assume that dim(Q) = dim(P ) < ∞, then Theorem 3.1 yields P − Q = T . The case when dim(Q) = dim(P ) = ∞ will follow in a similar fashion. The operator equation can be written in the following variational form (A1/2 v, T M −1/2 u) − (A−1/2 v, T M 1/2 u) = (v, Su), 1/2

v ∈ D(A

),

(3.3)

u ∈ R(P ),

which we have called the weakly formulated Sylvester equation. Theorem 3.2. Let the positive deﬁnite forms m and h be given such that (1.1) holds. Let there exist D1 < D2 such that the interval [D1 , D2 ] ⊂ R be contained in the resolvent sets of both H and M. Set Q = EH (D1 ), P = EM (D1 ) and assume η < (D2 − D1 )(D2 D1 )−1/2 , then √ D2 D1 η. (3.4) P − Q ≤ D2 − D1

Vol. 58 (2007)

On Weakly Formulated Sylvester Equations

191

Proof. T = Q⊥ P is the unique solution of the equation (3.3). Theorem 2.2 implies D2 λn (M) η . T ≤ 2 (D2 − d)(d − λn (M)) for any λn (M) < d < D2 . The optimal d equals (D2 +λ2n (M)) and since M < D1 we conclude √ D2 λn (M) D2 D1 ≤η Q⊥ P ≤ η < 1. D2 − λn (M) D2 − D1 Analogous argumentation for T = P⊥ Q, with the roles of H and M in (3.3) being interchanged, yields the inequality √ D2 D1 P⊥ Q ≤ η < 1. D2 − D1 Theorem (3.1) now implies that Q⊥ P = P⊥ Q = Q − P .

This in turn establishes (3.4).

In the case in which h is only nonnegative deﬁnite, assumption (1.1) implies that N(M) = N(H) and R(H) = R(M), since H and M are selfadjoint. This in turn allows us to conclude that N := R(P ) ∩ N(H) ⊂ R(Q), so = Q − PN , Q

P = P − PN

are orthogonal projections and − P . Q − P = Q ⊂ R(H) we can reduce the problem to the positive Since R(P ) ⊂ R(H) and R(Q) deﬁnite case. Theorem 3.3. Let the positive deﬁnite forms m and h be given such that (1.1) holds. Let there exist 0 < L1 < L2 < D1 < D2 such that the intervals [L1 , L2 ] ⊂ R and [D1 , D2 ] ⊂ R be contained in the resolvent sets of both H and M. Set Q = EH [L1 , L2 ], P = EM [D1 , D2 ] and assume √

√D D L2 L1 2 1 η < 1, + D2 − D1 L2 − L1 then

√

√D D L2 L1 2 1 P − Q ≤ η. + D2 − D1 L2 − L1

(3.5)

Proof. The assumption L1 > 0 implies that we may assume, without losing any generality, that we have the positive deﬁnite forms m and h. Theorem 2.7 and the same argument as in Theorem 3.2 implies P⊥ Q = Q⊥ P = P − Q.

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This in turn allows us to conclude that √

√D D L2 L1 2 1 P − Q ≤ + η. D2 − D1 L2 − L1

(3.6)

Numerical experiments with the Sturm–Liouville eigenvalue problem, which were performed in [7], illustrated that in some situations the results of Theorems 3.2 and 3.3 yield considerably sharper estimates of the perturbations of the spectral subspaces than the results of [1, 2]. We now show that our theorems also apply in situations in which the theory from [1, 2] does not. Example 3.4. Take H, M as selfadjoint realizations of the diﬀerential operators ∂ ∂ ∂ ∂ − α(x) , − β(x) , ∂x ∂x ∂x ∂x respectively, in the Hilbert space H = L2 (I), I a (ﬁnite or inﬁnite) interval with, say, Dirichlet boundary conditions and non-negative bounded measurable functions α(x), β(x) which satisfy |β(x) − α(x)| ≤ η β(x)α(x). Now, the form δ(u, v) = h(u, v) − m(u, v) is not—in general—representable by a bounded operator. This rules out an application of the subspace perturbation theorems from [1, 2]. On the other hand both of our Theorems 3.2 and 3.3 apply and yield the corresponding estimates, e.g. √ D2 D1 η, Eα (D1 ) − Eβ (D1 ) ≤ D2 − D1 when we know that [D1 , D2 ] is contained in the resolvent sets of both H and M. Theorem 2.7 can also be applied to yield a Hilbert–Schmidt version of Theorems 3.2 and 3.3. Theorem 3.5. Let the positive deﬁnite forms m and h be given such that (1.1) holds. Assume P and Q are projections which commute with the operators H and M respectively and let A, M , W, H as in (3.1). If both ||| Q⊥ SP |||HS < ∞, ||| P⊥ S ∗ Q |||HS < ∞ and both gap(σ(A), σ(M )),

gap(σ(W), σ(H)),

are positive, then Q⊥ P , P⊥ Q and P − Q are Hilbert–Schmidt operators and ||| Q⊥ SP |||2HS gap(σ(A), σ(M ))2 ||| QSP⊥ |||2HS ≤ gap(σ(W), σ(H))2 ||| Q⊥ SP |||2HS ||| QSP⊥ |||2HS ≤ + . 2 gap(σ(A), σ(M )) gap(σ(W), σ(H))2

||| Q⊥ P |||2HS ≤

(3.7)

||| P⊥ Q |||2HS

(3.8)

||| P − Q |||2HS

(3.9)

Vol. 58 (2007)

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Proof. By construction (3.1) the operator T = T1 = Q⊥ satisﬁes the Sylvester equation (3.3), which in this setting has the form, cf. (2.16), (A1/2 v, T M −1/2 u) − (A−1/2 v, T M 1/2 u) = (v, Q⊥ SP u), 1/2

v ∈ D(A

),

u ∈ D(M

1/2

(3.10)

).

On the other hand, the operator T = T2 = P⊥ Q satisﬁes the “dual” equation, cf. Proposition 1.11, (W1/2 v, T H −1/2 u) − (W−1/2 v, T H 1/2 u) = (v, P⊥ S ∗ Qu), v ∈ D(W

1/2

),

u ∈ D(H

1/2

(3.11)

).

Now, (P − Q)2 = Q⊥ P + P⊥ Q, where by Theorem 2.7 both Q⊥ P and P⊥ Q are Hilbert–Schmidt2 and ||| P − Q |||2HS = Tr(Q⊥ P + P⊥ Q) = Tr(P Q⊥ P ) + Tr(QP⊥ Q) =||| Q⊥ P |||2HS + ||| P⊥ Q |||2HS

Using (2.17), we see that estimates (3.7)–(3.9) hold.

Corollary 3.6. Let the positive deﬁnite forms m and h be given such that (1.1) holds. If ||| S |||HS < ∞ and the other conditions of Theorem 3.5 hold. Then ||| P − Q |||2HS ≤

||| S |||2HS . min gap(σ(A), σ(M ))2 , gap(σ(W), σ(H))2

(3.12)

Proof. Just note that ||| Q⊥ SP |||2HS + ||| QSP⊥ |||2HS ≤||| S |||2HS .

4. Further properties of the operator S — an application in the numerical analysis We will now present an application of Theorem 2.2 in numerical analysis. This will also demonstrate a role played by the new Hilbert–Schmidt norm estimates. Assume now that we are given a positive deﬁnite operator H such that (1.2) holds. Let P be an orthogonal projection such that R(P ) ⊂ Q(h) and dim R(P ) = n. We aim to obtain estimates of ||| EH (D) − P |||

(4.1)

for ||| · |||= · and ||| · |||=||| · |||HS . 2 To prove this equality one can use the singular value analysis from [2]. Alternatively, one could use the property (P) from Theorem 2.8.

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We estimate (4.1) by an application of Theorem 2.2 (equivalently Theorem 3.5). Theorem 2.2 will allow us to improve [7, Theorem 3.2] inasmuch as that we establish estimates for the Hilbert–Schmidt norm and not just the spectral norm. The properties of the main perturbation construction from [7], cf. [5, 6], will be summarized for reader’s convenience. We start by deﬁning the positive deﬁnite form hP (u, v) = h(P u, P v) + h(P⊥ u, P⊥ v)

(4.2)

and a selfadjoint operator HP which represents the form hP in the sense of Kato. It was shown, see [5, 6, 7] that 1. the form hP is positive deﬁnite , hence there exists the positive deﬁnite operator HP which represents hP in the sense of Kato. 2. Q(h) = Q(hP ) 3. R(P ) reduces HP . 4. H−1 − H−1 P is a degenerate selfadjoint operator. 5. Let δhP := h − hP and let δHsP be the bounded selfadjoint operator such that −1/2 −1/2 (u, δHsP v) = δhP (HP u, HP v), (4.3) then δHsP is a degenerate operator and dimR(δHsP ) = 2n. 6. The values 1/2 (ψ, H−1 ψ) − (ψ, H−1 ψ) P : ψ ∈ S max min ηi = S⊂R(P ), dimS=n−i (ψ, H−1 ψ)

(4.4)

together with their negatives are all non-zero eigenvalues of δHsP . Furthermore, ηi are all the singular values of the operator δHsP P . 7. |δhP (φ, ψ)| ≤ ηn

hP [ψ]hP [φ].

(4.5)

The estimates from [7, Theorem 3.2] only use information which is contained in ηn . New theory allows us to take advantage of other ηi . Proposition 4.1. Let P and hP be as in (4.2) and let −1/2

S = H1/2 HP

1/2

− H−1/2 HP ,

then ||| δH P Q ||| ||| SQ ||| ≤ √ s 1 − ηn ||| δH P ||| ||| S ||| ≤ √ s . 1 − ηn Here δHsP is the degenerate operator from (4.3), Q is any projection, ηn is given by (4.4) and ||| · ||| is any unitary invariant norm.

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Proof. −1/2 −1/2 1/2 −1/2 −1/2 (ψ, SQφ) = δhP (H−1/2 ψ, HP Qφ) = δhP (HP HP H ψ, HP Qφ) ∗ 1/2 = (ψ, HP H−1/2 δHsP Qφ), φ, ψ ∈ H

√ 1/2 (4.5) and (1.20) imply HP H−1/2 ≤ 1/ 1 − ηn . Property (i) of the symmetric norm ||| · ||| and the fact that δHsP Q is a degenerate operator allow us to complete the proof. This proposition leads to an improved version of [7, Theorem 3.2]. Observe that ||| δHsP ||| depends only on ηi from (4.4). Theorem 4.2. Let h be as in (1.2) and let P and hP be as in (4.2) and ηi as in (4.4). Set h[ψ] DP := max ψ∈R(P ) ψ2 and assume ηn (1 − ηn )−1 < (D − DP )(D + DP ), then √ DDP ||| δHsP P ||| √ ||| EH (D)P⊥ |||≤ . D − DP 1 − ηn

(4.6)

Here ||| · ||| is any unitary invariant norm which has Property (P). Proof. Set T = (EH (D))⊥ P and apply Theorem 2.8 to estimate the norm ||| (EH (D))⊥ P |||. Proposition 4.1 now implies (4.6), cf. Corollary 3.6, [2, Corollary 3.1] and [2, Proposition 6.1]. Assume ||| · |||=||| · |||HS , then Theorem 4.2 yields the estimate √ DDP η12 + · · · + ηn2 √ . ||| (EH (D))⊥ P |||HS ≤ D − DP 1 − ηn

(4.7)

Remark 4.3. If ||| · |||= · , then under the conditions of Theorem 4.2 the identity (EH (D))⊥ P = EH (D) − P holds, cf. Theorem 3.1. A similar relation holds for a general unitary invariant norm since according to [2, Corollary 3.1] and [2, Section 2] we have ||| (EH (D))⊥ P |||=||| P⊥ EH (D) ||| and ||| EH (D) − P |||=||| (EH (D))⊥ P + P⊥ EH (D) ||| .

(4.8)

Theorem 4.2 is therefore our version (generalization) of the sin Θ theorem from [2, Appendiy 6.]. Same as in [2, Proposition 6.1], an estimate of (4.8) is obtained by a combination of Proposition 4.1 and available (depending on an application) information on the separation of the involved spectra, cf. Corollary 3.6. We have not speciﬁed a general estimate on ||| EH (D) − P ||| since we consider such an estimate to be highly application dependent and we would not like to prejudice its form.

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N

5

6

7

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8

9

10

||| (EH (D))⊥ P |||HS

4.4e-3 2.0e-3 1.1e-3 6.0e-4 3.7e-4 2.4e-4

λ3 DPN η 2 + η22 √1 λ3 − DPN 1 − η2

2.2e-2 1.0e-2 5.3e-3 3.3e-3 2.2e-3 1.5e-3

s1 (R2N ) + s2 (R2N ) λ3 − DPN

2.0e-2 1.4e-2 9.6e-3 7.2e-3 5.5e-3 4.4e-3

Table 1. Error estimate from Theorem 4.2 and the true error

We will now evaluate (4.7) on the example from [7, Section 4]. There we have considered the positive deﬁnite operator H which is deﬁned by the symmetric form

2π u v − αuv dt h(u, v) = 0

u, v ∈ {f : f, f ∈ L2 [0, 2π], eiθ f (0) = f (2π)} = D(h). The eigenvalues and eigenvectors of the operator H are 2 θ θ ±k + − α, z±k (t) = e−i(±k+ 2π )t , ω±k = 2π 2 θ θ ω0 = − α, z0 (t) = e−i 2π t . 2π

k∈N

In standard notation we have λ1 (H) = ω0 , λ2 (H) = ω−1 , λ3 (H) = ω1 , u1 = z0 , u2 = z−1 , u3 = z1 . For numerical experiments we chose θ = π − 10−4 and α = 0.2499 so that the eigenvaluesλ1 and λ2 are “small” and tightly clustered. As a test space we chose 3 = span w1N , w2N , where w1N and w2N are generated by the smooth N point YN equidistant cubic interpolation of the known eigenfunctions u1 and u2 . Take PN 3 3 such that R(PN ) = YN . Since YN ⊂ D(H) both Theorem 4.2 and the bounds from [2] apply. Set rφ = Hφ + (φ, Hφ)φ. Since w1N , w2N ∈ D(H) we conclude that rw1N and rw2N are bona ﬁde vectors. Set (rw1N , rw1N ) (rw1N , rw2N ) R2N = . (rw2N , rw1N ) (rw2N , rw2N )

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The competing bound from [2] is s1 (R2 ) + s2 (R2 ) |||HS ≤ . λ3 − DPN

||| (EH (D))⊥ PN

(4.9)

We see that with the improvement of the approximation the advantage of the bound from Theorem 4.2 over (4.9) grows, see Table 1. On Table 1 we have displayed the actual measured error in the ﬁrst line, in the second line we display the bound from (4.7) and in the third line Davis–Kahan bound (4.9). Further examples, where a numerical advantage of (4.6) over (4.9) is more stunning, are given in [7]. We repeat the results of the numerical experiments from [7] on Table 2. There we try to estimate the approximation error in the vector w1N in the · -norm by an application of Theorem 3.2. Otherwise the makeup of Table 2 is the same as the makeup of Table 1.

N

6

7

8

9

10

EH (D) − PN

2.0e-3

1.1e-3

6.0e-4

3.7e-4

2.4e-4

λ2 dPN η2 √ λ2 − dPN 1 − η2

1.5e0

6.2e-1

3.5e-1

2.2e-1

1.5e-1

s1 (R2N ) λ2 − dPN

3.6e+2 2.1e+2 1.5e+2 1.1e+2 8.9e+1

Table 2. Approximations for u1 (here we use dPN := minψ∈R(PN )

h[ψ] ψ2 )

We now present a variation on this example where (4.9) does not apply 1 = span l1N , l2N , where whereas (4.6) still gives useful information. We chose YN l1N and l2N are generated by the N point equidistant continuous linear interpolation of u1 and u2 , then rlN and rlN are no longer bona ﬁde vectors. Subsequently, (4.9) 1 2 does not apply any more but Theorem 4.2 is still applicable. Take now QN such 1 . The results are presented on Table 3. that R(QN ) = YN The performance of the bound (4.6) is inﬂuenced by the quotient |DPN − λ2 | . DPN

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100

IEOT

120

140

||| (EH (D))⊥ QN |||HS

5.2024e-005 3.6126e-005

λ3 DQN ||| δHsQN |||HS √ λ3 − DQN 1 − ηn

8.7374e-003 6.9293e-003 5.7302e-003

2.6541e-5

Table 3

DPN is an approximation3 of λ2 and in this example we have measured |DPN − λ2 | > 0.17, DPN

N = 100, 120, 140.

The (under)performance of the bound (4.6) correctly detects this approximation feature of R(QN ), cf. Table 3.

5. Estimates for perturbations of the square root of a nonnegative operator In this section we will show that there are interesting applications of the equation (2.1) even when all of the coeﬃcients A, M and F are unbounded. To demonstrate this we will generalize the known scalar inequality 4 √ √ | m − h| |m − h| √ ≤ √ , h, m > 0. (5.1) 4 mh 2 mh to positive deﬁnite Hermitian matrices or, more generally, to positive, possibly unbounded, operators in an arbitrary Hilbert space. One of the obtained bounds is 1 M −1/4 (M −1/2 − H −1/2 )H −1/4 ≤ M −1/2 (M − H)H −1/2 . (5.2) 2 In [13] a related bound for ﬁnite matrices was obtained. It reads η (5.3) H −1/4 (M −1/2 − H −1/2 )H −1/4 ≤ + O(η 2 ), 2 η = H −1/2 (M − H)H −1/2 . This is a more common type of estimate — the error is measured by the “unperturbed operator” only — while in our estimate the error is measured by H and M in a symmetric way. The latter type of estimate is convenient, if both operators H 3 To

be more precise DPN is Rayleigh–Ritz approximation to λ2 (H) from the subspace R(PN ). For more on the Rayleigh–Ritz eigenvalue approximations see [7]. 4 “The relative error in the square root is bounded by the half relative error in the radicand”.

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and M are known equally well and we are interested in a possibly sharp bound. Our bound is obviously as sharp as its scalar pendant. It is also rigorous, in contrast to (5.3) which is only asymptotic. Moreover, (5.2) will retain its validity for fairly general positive selfadjoint operators in a Hilbert space. The bound (5.2) is a “relative bound” which may be convenient in computing or measuring environments (cf. related bounds obtained for the eigenvalues and eigenvectors of the Hermitian matrices in [14] and the literature cited there). Also, this bound is readily expressed in terms of quadratic forms, which will be convenient for application with elliptic diﬀerential operators as will be shown below. The idea of the proof is very simple, especially in the ﬁnite dimensional case which we present ﬁrst, also in order to accommodate readers not interested in inﬁnite dimension technicalities. The basis of our proof is the obvious Sylvester equation (cf. [16]) M 1/2 (M 1/2 − H 1/2 ) + (M 1/2 − H 1/2 )H 1/2 = M − H,

(5.4)

valid for any Hermitian, positive deﬁnite matrices H and M . We rewrite this equation in the equivalent form M 1/4 T H −1/4 + M −1/4 T H 1/4 = F

(5.5)

with F = M −1/2 (M − H)H −1/2 ,

T = M −1/4 (M −1/2 − H −1/2 )H −1/4 ,

which is immediately veriﬁed. This equation has a unique solution

∞ −1/2 −1/2 t t e−M M −1/4 F H −1/4 e−H dt. T =

(5.6)

(5.7)

0 −1/2

−1/2

t t −1/4 (just premultiply (5.5) by e−M M −1/4 , postmultiply by e−H H , integrate from 0 to ∞ and perform partial integration on its left hand side). Hence for arbitrary vectors φ, ψ we have 2 ∞ −1/2 −1/2 t t −1/4 e−M M −1/4 φe−H H ψdt |(T ψ, φ)|2 ≤ F 2

≤ F

2

0

0

∞

e

−M −1/2 t

M

−1/4

2

φ dt 0

∞

e−H

F 2 ψ2 φ2 , 4 where we have used the obvious identity

∞ 1 e−2Ct Cdt = I 2 0 =

−1/2

t

H −1/4 ψ2 dt (5.8)

(5.9)

for C = H −1/2 , M −1/2 . Thus, (5.2) holds true. We now turn to the Hilbert space H of arbitrary dimension. We assume that H and M are positive selfadjoint operators. This implies that all fractional powers

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of H and M are also positive. Neither of these operators need be bounded (or have bounded inverse). Theorem 5.1. Let H and M be positive selfadjoint operators in a Hilbert space X having the following property (A): D(H1/2 ) = D(M1/2 ) and the norms H1/2 · and M1/2 · are topologically equivalent. Then the same property is shared by H1/2 and M1/2 . The operators M−1/2 H1/2 ,

M1/2 H−1/2 ,

H−1/2 M1/2 ,

H1/2 M−1/2 ,

M−1/4 H1/4 ,

M1/4 H−1/4 ,

H−1/4 M1/4 ,

H1/4 M−1/4

(5.10)

are well deﬁned and bounded. Let F = M1/2 H−1/2 − M−1/2 H1/2

(5.11)

T = M1/4 H−1/4 − M−1/4 H1/4 ,

(5.12)

and

then T ≤

1 F . 2

(5.13)

Proof. The fact that the square roots inherit the property (A) is a consequence of L¨ owner type theorems (see e.g. [8], Ch.V, Th. 4.12). The corresponding pairs ∗ of operators in (5.10) are mutually adjoint e.g. M−1/2 H1/2 = H1/2 M−1/2 etc. Obviously, (5.11) and (5.12) reduce to F, T from (5.11), if the space is ﬁnite dimensional. The equation (5.5) becomes here (T H−1/4 u, M1/4 v) + (T H1/4 u, M−1/4 v) = (F u, v)

(5.14)

for u ∈ DA = D(H1/4 ) ∩ D(H−1/4 ) and similarly for v and M. We will now prove this. The left-hand side of (5.14) equals (M1/4 H−1/4 H−1/4 u, M1/4 v) − (H−1/4 u, H1/4 M−1/4 M1/4 v) +(H1/4 u, H−1/4 M1/4 M−1/4 v) − (M−1/4 H1/4 H1/4 u, M−1/4 v) = (H−1/2 u, M1/2 v) − (u, v) + (u, v) − (H1/2 u, M−1/2 v) = (M1/2 H−1/2 u, v) − (M−1/2 H1/2 u, v) = (F u, v). Now, substitute in (5.14) v = e−M

−1/2

t

M−1/4 φ,

−1/2

u = e−H

t

H−1/4 ψ

(5.15)

for any φ ∈ D(M−1/2 ), ψ ∈ D(H−1/2 ). Note that subspaces M−1/4 D(M−1/2 ) and −1/2 −1/2 t t H−1/4 D(H−1/2 ) are invariant under e−M , e−H , respectively so, in (5.15)

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we have u ∈ DA and v ∈ DM . Then integrate (5.15) and use partial integration:

s

s −1/2 −1/2 −1/2 −1/2 t −1/2 t t t (T e−H H ψ, e−M φ)dt + (T e−H ψ, e−M M−1/2 φ)dt 0

s

0 s −1/2 −1/2 d −H−1/2 t −M−1/2 t t t =− (T e ψ, e φ)dt + (T e−H ψ, e−M M−1/2 φ)dt dt 0 0 −1/2

−1/2

−1/2

−1/2

s s = (T ψ, φ) − (T e−H ψ, e−M φ)

s −1/2 −1/2 t t + (T e−H ψ, (−e−M M−1/2 )φ)dt 0

s −1/2 −1/2 t t + (T e−H ψ, e−M M−1/2 φ)dt 0

s s = (T ψ, φ) − (T e−H ψ, e−M φ)

s −1/2 −1/2 t −1/4 t = (F e−H H ψ, e−M M−1/4 φ)dt. 0

In the limit s → ∞ by using the functional calculus for H, M, respectively and monotone convergence for spectral integrals we obtain −1/2

e−H in the norm. Hence

(T ψ, φ) =

∞

s

ψ → 0,

−1/2

(F e−H

0

t

e−M

−1/2

s

φ→0

H−1/4 ψ, e−M

−1/2

t

M−1/4 φ)dt

(5.16)

where the integral on the right hand side is, in fact, Lebesgue as shows the chain of inequalities in (5.8) which are valid in this general case as well. Here the identity (5.9) is used in the weak sense:

∞ (e−2Ct Cφ, φ)dt = (φ, φ)/2, φ ∈ D(C) 0

for any positive selfadjoint C. Thus, |(T ψ, φ)|2 ≤ F 2 (ψ, ψ)(φ, φ)/4.

Remark 5.2. The main assertion (5.13) of Theorem 5.1 is obviously equivalent to the following statement: the inequality |m(φ, ψ) − h(φ, ψ)| ≤ ε h(φ, φ)m(φ, ψ) implies |m2 (φ, ψ) − h2 (φ, ψ)| ≤

ε h2 (φ, φ)m2 (ψ, ψ) 2

where the sesquilinear forms h, m, h2 , m2 belong to the operators H, M, H1/2 , M1/2 , respectively. Thus, our theorem will be directly applicable to diﬀerential operators given in weak form.

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Example 5.3. Let again H and M be as in Example 3.4. That is to say take H, M as selfadjoint realizations of the diﬀerential operators ∂ ∂ ∂ ∂ − α(x) , − β(x) , ∂x ∂x ∂x ∂x in the Hilbert space H = L2 (I) (again I can be a ﬁnite or inﬁnite interval) with the Dirichlet boundary conditions and non-negative bounded measurable functions α(x), β(x) which satisfy |β(x) − α(x)| ≤ ε β(x)α(x) Now

2 |(M1/2 φ, M1/2 ψ) − (H1/2 φ, H1/2 ψ)|2 ≤ |β(x) − α(x)||ψ (x)φ (x)|dx I

2 2 2 2 ≤ε α(x)|ψ (x)| dt β(x)|φ (x)| dt = ε H1/2 φ2 M1/2 ψ2 I

I

hence F ≤ ε and Theorem 5.1 applies yielding |(M1/4 φ, M1/4 ψ) − (H1/4 φ, H1/4 ψ)| ≤

ε H1/4 φM1/4 ψ 2

or, equivalently, in the terms as in Remark 5.2 ε |m2 (φ, ψ) − h2 (φ, ψ)| ≤ h2 (φ, φ)m2 (ψ, ψ). 2

6. Conclusion With this work we complete our study of the weak Sylvester equation which started in [7]. A notion of a weak Sylvester equation was introduced in [7] as a tool on a way to obtain invariant subspace estimates for unbounded positive deﬁnite operators. With this paper we show that there are applications of the concept of a weak Sylvester equation outside the theory of Rayleigh–Ritz spectral approximations. We have extended out theory to inﬁnite dimensional invariant subspaces and have obtained estimates of the diﬀerence between the corresponding spectral projections in all unitary invariant norms. With this results we have developed a counterpart of the sin Θ theorems from [1] for perturbations of operators which are only deﬁned as quadratic forms. Due to the very singular nature of integral representations (which can not be avoided by reformulation of the integrals) of the solution to the equation (2.4), cf. formula (2.5), we were not able to extend the technique from [1] to prove that in the setting of Theorem 2.7 assumption ||| F |||< ∞ also implies that there exists a bounded solution T such that ||| T |||< ∞. We believe that this statement is true, but the proof will have to remain a task for the future and will most likely require another technique. The technique behind [12, Theorem 1.] could be a way to overcome this diﬃculty since the inequality (1.5) holds for bounded and invertible operators H and M . In order to complete this agenda a new way to regularize the weak Sylvester equation has to be found. We believe that the results of this article

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well illustrate the advantages and limitations of our form theoretic approach to weak Sylvester equation. An application of the concept to a perturbation of the square root of a positive deﬁnite operator shows that there are other application areas for weakly formulated operator equations and that the developed techniques are (and hopefully will be) easily adaptable to new situations. The applications which we have reported in this paper are presented as an illustration only. Further applications will be the subject of the future work, cf. [6].

References [1] R. Bhatia, C. Davis, and A. McIntosh. Perturbation of spectral subspaces and solution of linear operator equations. Linear Algebra Appl., 52/53:45–67, 1983. [2] C. Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal., 7:1–46, 1970. [3] W. G. Faris. Self-adjoint operators. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 433. [4] I. C. Gohberg and M. G. Kre˘in. Introduction to the theory of linear nonselfadjoint operators. Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969. [5] L. Grubiˇsi´c. On eigenvalue estimates for nonnegative operators. to appear in SIAM J. Matrix Anal. Appl. [6] L. Grubiˇsi´c. Ritz value estimates and applications in Mathematical Physics. PhD thesis, Fernuniversit¨ at in Hagen, dissertation.de Verlag im Internet, ISBN: 3-89825998-6, 2005. [7] L. Grubiˇsi´c and K. Veseli´c. On Ritz approximations for positive deﬁnite operators I (theory). Linear Algebra Appl., 417(2-3):397–422, 2006. [8] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin, second edition, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. [9] H. Kosaki. Arithmetic-geometric mean and related inequalities for operators. J. Funct. Anal., 156(2):429–451, 1998. [10] V. Kostrykin, K. A. Makarov, and A. K. Motovilov. On the existence of solutions to the operator Riccati equation and the tan Θ theorem. Integral Equations and Operator Theory, 51(1):121–140, 2005. [11] S. Levendorskiˇı. Asymptotic distribution of eigenvalues of diﬀerential operators, volume 53 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian. [12] R.-C. Li. A bound on the solution to a structured Sylvester equation with an application to relative perturbation theory. SIAM J. Matrix Anal. Appl., 21(2):440–445 (electronic), 1999. [13] R. Mathias. A bound for the matrix square root with application to eigenvector perturbation. SIAM J. Matrix Anal. Appl., 18(4):861–867, 1997. [14] R. Mathias and K. Veseli´c. A relative perturbation bound for positive deﬁnite matrices. Linear Algebra Appl., 270:315–321, 1998.

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Grubiˇsi´c and Veseli´c

IEOT

[15] Z. M. Nashed. Perturbations and approximations for generalized inverses and linear operator equations. In Generalized inverses and applications (Proc. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973), pages 325–396. Publ. Math. Res. Center Univ. Wisconsin, No. 32. Academic Press, New York, 1976. [16] B. A. Schmitt. Perturbation bounds for matrix square roots and Pythagorean sums. Linear Algebra Appl., 174:215–227, 1992. [17] B. Simon. Trace ideals and their applications, volume 35 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1979. [18] J. Weidmann. Stetige Abh¨ angigkeit der Eigenwerte und Eigenfunktionen elliptischer Diﬀerentialoperatoren vom Gebiet. Math. Scand., 54(1):51–69, 1984. [19] J. Weidmann. Lineare Operatoren in Hilbertr¨ aumen. Teil 1. Mathematische Leitf¨ aden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart, 2000. Grundlagen. [Foundations]. Luka Grubiˇsi´c LG Mathematische Physik FernUniversit¨ at in Hagen Feithstr. 140 D-58084 Hagen Germany Current address: Institut fuer Reine und Angewandte Mathematik RWTH Aachen University Templergraben 55 D-52056 Aachen Germany (On leave from Department of Mathematics, University of Zagreb, Croatia) e-mail: [email protected] Kreˇsimir Veseli´c LG Mathematische Physik FernUniversit¨ at in Hagen Feithstr. 140 D-58084 Hagen Germany e-mail: [email protected] Submitted: July 28, 2005 Revised: June 14, 2006

Integr. equ. oper. theory 58 (2007), 205–238 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020205-34, published online April 14, 2007 DOI 10.1007/s00020-007-1481-5

Integral Equations and Operator Theory

Marcinkiewicz Integrals with Non-Doubling Measures Guoen Hu, Haibo Lin and Dachun Yang Abstract. Let µ be a positive Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr n for all x ∈ Rd , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some H¨ ormander-type condition, and assume that it is bounded on L2 (µ). We then establish its boundedness, respectively, from the Lebesgue space L1 (µ) to the weak Lebesgue space L1,∞ (µ), from the Hardy space H 1 (µ) to L1 (µ) and from the Lebesgue space L∞ (µ) to the space RBLO(µ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space Lp (µ) with p ∈ (1, ∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(µ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger H¨ ormander-type condition, respectively, from Lp (µ) with p ∈ (1, ∞) to itself, from the space L log L(µ) to L1,∞ (µ) and from H 1 (µ) to L1,∞ (µ). Some of the results are also new even for the classical Marcinkiewicz integral. Mathematics Subject Classification (2000). Primary 42B25; Secondary 47B47, 42B20, 47A30. Keywords. Non-doubling measure, Marcinkiewicz integral, commutator, Hardy space, Lebesgue space, RBMO(µ), RBLO(µ), L log L(µ).

1. Introduction As an analogy of the classical Littlewood-Paley g function, Marcinkiewicz in [13] introduced the operator π 1/2 |F (x + t) + F (x − t) − 2F (x)|2 M(f )(x) = dt , x ∈ [0, 2π], t3 0 The third (corresponding) author was supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of China.

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x where F (x) = 0 f (t)dt. This operator is now called the Marcinkiewicz integral. Zygmund in [28] proved that the operator M is bounded on the Lebesgue space Lp ([0, 2π]) for p ∈ (1, ∞). Stein in [20] generalized the above Marcinkiewicz integral to the following higher-dimensional case. Let Ω be homogeneous of degree zero in Rd for d ≥ 2, integrable and have mean value zero on the unit sphere S d−1 . The higher-dimensional Marcinkiewicz integral is then deﬁned by ∞ 2 dt 1/2 Ω(x − y) f (y) dy , x ∈ Rd . MΩ (f )(x) = 3 d−1 |x − y| t 0 |x−y|≤t Stein in [20] proved that if Ω ∈ Lipα (S d−1 ) for some α ∈ (0, 1], then MΩ is bounded on Lp (Rd ) for any p ∈ (1, 2], and is also bounded from L1 (Rd ) to L1, ∞ (Rd ). Since then, a lot of papers focus on this operator. For some recent development, we mention that Al-Salman et al in [2] obtained the Lp (Rd )-boundedness for p ∈ (1, ∞) of MΩ if Ω ∈ L(logL)1/2 (S d−1 ); Fan and Sato in [6] proved that MΩ is bounded from the Lebesgue space L1 (Rd ) to the weak Lebesgue space L1, ∞ (Rd ) if Ω ∈ LlogL(S d−1 ); and Ding et al in [4] established its boundedness from the classical Hardy space H 1 (Rd ) to the Lebesgue space L1 (Rd ) if Ω satisﬁes the L1 Dini condition. There are many other interesting works on this operator, among them we refer to [19, 27] and their references. On another hand, Torchinsky and Wang in [26] ﬁrst introduced the commutator generated by the Marcinkiewicz integral MΩ and the classical BMO(Rd ) function, and established its Lp (Rd )boundedness for p ∈ (1, ∞) when Ω ∈ Lipα (S d−1 ) for some α ∈ (0, 1]. Such boundedness of this commutator was further discussed in [8, 11] when Ω only satisﬁes certain size conditions. Moreover, its weak type endpoint estimate was obtained in [5] when Ω ∈ Lipα (S d−1 ) for some α ∈ (0, 1]. The main purpose of this paper is to establish a similar theory for the Marcinkiewicz integral and the associated commutator on Rd with a positive Radon measure which may be non doubling. To be precise, let µ be a positive Radon measure on Rd which only satisﬁes the following growth condition that for all x ∈ Rd and all r > 0, µ(B(x, r)) ≤ C0 rn ,

(1.1)

where C0 and n are some positive constants and 0 < n ≤ d, and B(x, r) is the open ball centered at x and having radius r. We recall that µ is said to be a doubling measure, if there is a positive constant C such that for any x ∈ supp (µ) and r > 0, µ(B(x, 2r)) ≤ Cµ(B(x, r)), and that the doubling condition is a key assumption in the classical theory of harmonic analysis. In recent years, many classical results concerning the theory of Calder´ on-Zygmund operators and function spaces have been proved still valid if the Lebesgue measure is substituted by a measure µ as in (1.1); see [14, 15, 21, 22, 23, 16, 17, 24, 9, 10, 7, 12, 3]. We mention that the analysis on non-homogeneous spaces played an essential role in solving the long-standing open Painlev´e’s problem by Tolsa in [25]. To outline the structure of this paper, we ﬁrst recall some notation and deﬁnitions. For a cube Q ⊂ Rd we mean a closed cube whose sides parallel to

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the coordinate axes and we denote its side length by l(Q) and its center by xQ . Let α > 1 and β > αn . We say that a cube Q is an (α, β)-doubling cube if µ(αQ) ≤ βµ(Q), where αQ denotes the cube with the same center as Q and l(αQ) = αl(Q). For deﬁniteness, if α and β are not speciﬁed, by a doubling cube we mean a (2, 2d+1 )-doubling cube. Especially, for any given cube Q, we denote the smallest doubling cube which contains Q and has the same center as Q. by Q Given two cubes Q ⊂ R in Rd , set

NQ, R

SQ, R = 1 +

k=1

µ(2k Q) n , l(2k Q)

where NQ, R is the smallest positive integer k such that l(2k Q) ≥ l(R). The concept of SQ, R ﬁrst appeared in [21], where some useful properties of SQ, R can be found. 1, ∞ The following atomic Hardy space Hatb (µ) was introduced by Tolsa in [21]. Definition 1.1. For a ﬁxed ρ > 1, a function b ∈ L1loc (µ) is called an atomic block if (i) there exists some cube R such that supp (b) ⊂ R; (ii) Rd b dµ = 0; (iii) there are functions

aj with supports in cubes Qj ⊂ R and numbers λj ∈ R such that b = j λj aj , and −1 aj L∞ (µ) ≤ µ(ρQj )SQj , R . Deﬁne |b|H 1, ∞ (µ) =

atb

|λj |.

j

1, ∞ (µ) if there are atomic blocks bi such that We say that f ∈ Hatb

f= with

∞

bi ,

i=1

i

1, ∞ |bi |H 1, ∞ (µ) < ∞. The Hatb (µ) norm of f is deﬁned by atb |bi |H 1, ∞ (µ) , f H 1, ∞ (µ) = inf atb

i

atb

where the inﬁmum is taken over all the possible decompositions of f in atomic blocks. 1, ∞ (µ) is It was proved by Tolsa in [21] that the deﬁnition of the space Hatb 1, ∞ independent of the choice of the constant ρ > 1. Moreover, the space Hatb (µ) was proved to be the Hardy space H 1 (µ) in [24] with equivalent norms. For conve1, ∞ nience, in what follows, we denote the space Hatb (µ) and the norm · H 1, ∞ (µ) , atb 1 respectively, by H (µ) and · H 1 (µ) . Tolsa in [21] proved the dual space of H 1 (µ) is the following space RMBO(µ).

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Definition 1.2. Let ρ > 1 be a ﬁxed constant. A function f ∈ L1loc (µ) is said to be in the space RMBO(µ) if there exists some constant C > 0 such that for any cube Q centered at some point of supp (µ), 1 |f (y) − mQ (f )| dµ(y) ≤ C µ(ρQ) Q and |mQ (f ) − mR (f )| ≤ CSQ,R for any two doubling cubes Q ⊂ R, where mQ (f ) denotes the mean of f over cube Q. The minimal constant C as above is deﬁned to be the norm of f in the space RMBO(µ) and denoted by f RBMO(µ) . Tolsa in [21] proved that the deﬁnition of the space RBMO(µ) is independent of the choice of ρ. The following space RBLO(µ) was introduced in [12]. Obviously, RBLO(µ) ⊂ RBMO(µ). Definition 1.3. An f ∈ L1loc (µ) is said to be in√the space √ RBLO(µ) if there exists some positive constant C such that for any (4 d, (4 d)n+1 )-doubling cube Q, mQ (f ) − essinf x∈Q f (x) ≤ C

(1.2)

and (1.3) mQ (f ) − mR (f ) ≤ CSQ, R √ n+1 √ )-doubling cubes Q ⊂ R. The minimal constant C for any two (4 d, (4 d) as above is deﬁned to be the norm of f in the space RBLO(µ) and denoted by f RBLO(µ) . We now introduce the Marcinkiewicz integral related to the measure µ as in (1.1). Let K be a locally integrable function on Rd × Rd \ {(x, y) : x = y}. Assume that there exists a constant C > 0 such that for all x, y ∈ Rd with x = y, |K(x, y)| ≤ C|x − y|−(n−1) , and |x−y|≥2|y−y |

|K(x, y)−K(x, y )|+|K(y, x)−K(y , x)|

(1.4) 1 dµ(x) ≤ C (1.5) |x − y|

for any y, y ∈ Rd . The Marcinkiewicz integral M(f ) associated to the above kernel K and the measure µ as in (1.1) is deﬁned by ∞ 2 dt 1/2 K(x, y)f (y) dµ(y) 3 , x ∈ Rd . (1.6) M(f )(x) = t 0 |x−y|≤t Throughout this paper, we always assume that M is bounded on L2 (µ). Obviously, if µ is the d-dimensional Lebesgue measure in Rd , and K(x, y) =

1 Ω(x − y) |x − y|d−1

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with Ω homogeneous of degree zero and Ω ∈ Lipα (S d−1 ) for some α ∈ (0, 1], then it is easy to verify that K satisﬁes (1.4) and (1.5), and M in (1.6) is just the Marcinkiewicz integral MΩ introduced by Stein in [20]. Thus, M in (1.6) is a natural generalization of the classical Marcinkiewicz integral in the current setting. For m ∈ N and a function b ∈ RBMO(µ), the mth order commutator generated by the Marcinkiewicz integral M and the function b is deﬁned by ∞ 2 dt 1/2 m Mb, m (f )(x) = [b(x) − b(y)] K(x, y)f (y) dµ(y) 3 , (1.7) t 0 |x−y|≤t where x ∈ Rd . The organization of this paper is as follows. In Section 2, we establish the boundedness of M in (1.6) with kernel K satisfying (1.4) and (1.5), respectively, from the Lebesgue space L1 (µ) to the weak Lebesgue space L1,∞ (µ), from the Hardy space H 1 (µ) to L1 (µ) and from the Lebesgue space L∞ (µ) to the space RBLO(µ). The last result is also new even for the classical Marcinkiewicz integral; see Theorem 2.4 below. As a corollary, in this section, we obtain the boundedness of the Marcinkiewicz integral M in the Lebesgue space Lp (µ) with p ∈ (1, ∞). In Section 3, we obtain the boundedness of the commutator Mb,m in (1.7), respectively, from Lp (µ) to itself for p ∈ (1, ∞), and from the space L log L(µ) to L1,∞ (µ), where the kernel K satisﬁes a H¨ormander-type condition which is slightly stronger than (1.5); see (3.1) below. The latter result improves the corresponding result in [5] even for the classical Marcinkiewicz integral. Moreover, we establish the boundedness of Mb,1 from H 1 (µ) to L1,∞ (µ), which is also new even for the classical Marcinkiewicz integral. Throughout this paper, C denotes a constant that is independent of the main parameters involved but whose value may diﬀer from line to line. We use the constant with subscripts to indicate its dependence on the parameters in the subscripts. We denote simply by A B if there exists a constant C > 0 such that A ≤ CB; and A ∼ B means that A B and B A. For a µ-measurable set E, χE denotes its characteristic function. For any p ∈ [1, ∞], we denote by p its conjugate index, namely, 1/p + 1/p = 1.

2. The Marcinkiewicz integral We ﬁrst establish the following boundedness from L1 (µ) to L1, ∞ (µ) of M in (1.6). Theorem 2.1. Let K satisfy (1.4) and (1.5), and M be as in (1.6). If M is bounded on L2 (µ), then it is also bounded from L1 (µ) into L1, ∞ (µ), namely, there is a positive constant C such that for all λ > 0 and all f ∈ L1 (µ), f L1(µ) . (2.1) x ∈ Rd : M(f )(x) > λ ≤ C λ To prove Theorem 2.1, we need the following Calder´ on-Zygmund decomposition with non doubling measures in [21] and [23]. µ

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Lemma 2.2. Let 1 ≤ p < ∞. For any f ∈ Lp (µ) and λ > 0 (λ > 2d+1 f L1 (µ) /µ if µ < ∞), we have

(a) There exists a family of almost disjoint cubes {Qj }j (that is, j χQj ≤ C) such that λp 1 |f |p dµ > d+1 , µ(2Qj ) Qj 2 p 1 λ |f |p dµ ≤ d+1 f or all η > 2, µ(2ηQj ) ηQj 2 |f | ≤ λ µ − a. e. on Rd \ ∪j Qj . (b) For each j, let Rj be the smallest (6, 6n+1 )-doubling cube of the form 6k Qj , χQ k ∈ N, and let ωj = χjQ . Then, there exists a family of functions ϕj with k k supp (ϕj ) ⊂ Rj and with constant sign satisfying ϕj dµ = f ωj dµ, Rd

Qj

|ϕj | ≤ Bλ

j

(where B is some constant), and when p = 1, |f | dµ; ϕj L∞ (µ) µ(Rj ) ≤ C Qj

when 1 2d+1 f L1 (µ) /µ (note that if 0 < λ ≤ 2d+1 f L1(µ) /µ, the estimate (2.1) obviously holds ). Applying Lemma 2.2 to f and λ, we obtain a family of almost disjoint cubes {Qj }j . With the notation wj , ϕj and Rj the same as in Lemma 2.2, we can write f = g + b, with g = f χRd \ j Qj + ϕj j

and b=

j

(ωj f − ϕj ) =

bj .

j

It is easy to see that gL∞ (µ) λ and gL1(µ) f L1 (µ) . Thus, by the boundedness of M in L2 (µ),

µ x ∈ Rd : M(g)(x) > λ ≤ λ−2 M(g)L2 (µ) λ−1 f L1 (µ) . From Lemma 2.2 (a), it follows that 1

1 |f (x)| dµ(x) |f (x)| dµ(x), µ ∪j 2Qj λ j Qj λ Rd

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and therefore, the proof of Theorem 2.1 can be deduced to proving that

1 d µ x ∈ R \ ∪j 2Qj : M(b)(x) > λ |f (x)| dµ(x). λ Rd Note that

µ x ∈ Rd \ ∪j 2Qj : M(b)(x) > λ ≤ λ−1 M(b)(x) dµ(x) Rd \∪j 2Qj

−1

≤λ

Rd \2Rj

j

M(bj )(x) dµ(x) +

j

2Rj \2Qj

M(bj )(x) dµ(x) .

Thus, it suﬃces to prove that for each ﬁxed j, M(bj )(x) dµ(x) bj L1 (µ)

(2.2)

Rd \2Rj

and

2Rj \2Qj

M(bj )(x) dµ(x)

|f (x)| dµ(x).

(2.3)

Qj

To verify (2.2), for each ﬁxed j, write M(bj )(x) dµ(x) |x−xR |+√dl(Rj ) 2 dt 1/2 j K(x, y)b (y) dµ(y) dµ(x) ≤ 3 j t d 0 R \2Rj |x−y|≤t ∞ 2 dt 1/2 + K(x, y)bj (y) dµ(y) 3 dµ(x) √ t d |x−xRj |+ dl(Rj ) R \2Rj |x−y|≤t = I1 + I 2 .

Rd \2Rj

The Minkowski inequality along with (1.1) and (1.4) leads to that 1/2 |x−xRj |+√dl(Rj ) dt |bj (y)| I1 dµ(y) dµ(x) 3 |x − y|n−1 d |x−y| Rd \2Rj R t 1/2 1 l(Rj ) |bj (y)| dµ(y) dµ(x) n+1/2 |x − x d d Rj | R R \2Rj bj L1 (µ) . On the other hand, by the vanishing moment of bj and the smoothness condition (1.5) we obtain that ∞ 2 dt 1/2 I2 = K(x, y)b (y) dµ(y) dµ(x) 3 j √ t d d |x−xRj |+ dl(Rj ) R \2Rj R 1 √ dµ(x) K(x, y) − K(x, xRj ) bj (y) dµ(y) |x − xRj | + dl(Rj ) Rd \2Rj Rd

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Rd

|bj (y)|

Rd \2Rj

|K(x, y) − K(x, xRj )|

IEOT

1 dµ(x) dµ(y) |x − xRj |

bj L1 (µ) . Combining the estimates for I1 and I2 yields the estimate (2.2). To prove the estimate (2.3), we ﬁrst observe that if supp(h) ⊂ I for some cube I then for any s > 1 and x ∈ sI, by (1.4), ∞ 1/2 |h(y)| dt dµ(y) M(h)(x) ≤ C n−1 3 |x−y| t I |x − y| 1 ≤ Cs |h(y)| dµ(y). (2.4) |x − xI |n I On another hand, by Lemma 2.3 in [23] (see also Lemma 2.1 in [21]), we have 1 n dµ(s) 1. 2Rj \2Qj x − xQj Thus, from this and (2.4) together with supp (ωj f ) ⊂ Qj and |ωj | ≤ 1, it follows that 1 M(ωj f )(x) dµ(x) dµ(x) |f (y)| dµ(y) n 2Rj \2Qj Qj 2Rj \2Qj |x − xQj | |f (y)| dµ(y). Qj

The last estimate and the following trivial estimate that 1/2 2 M(ϕj )(x) dµ(x) ≤ |M(ϕj )(x)| dµ(x) µ(2Rj )1/2 2Rj 2Rj 1/2 2 |ϕj (x)| dµ(x) µ(Rj )1/2 2Rj |f (x)| dµ(x), Qj

which is obtained by the H¨ older inequality and the L2 (µ)-boundedness of M, imply the inequality (2.3). This ﬁnishes the proof of Theorem 2.1. Applying Theorem 2.1, we can obtain the following boundedness of M in (1.6) from H 1 (µ) into L1 (µ). Theorem 2.3. Let K satisfy (1.4) and (1.5), and M be as in (1.6). If M is bounded on L2 (µ), then it is also bounded from H 1 (µ) into L1 (µ). Proof. Recalling that the deﬁnition of H 1 (µ) is independent of the choice of the constant ρ > 1, we may assume that ρ = 2 in Deﬁnition 1.1. By Theorem 2.1, the operator M is bounded from L1 (µ) to L1, ∞ (µ). By a standard argument, it is enough to prove that M(b)L1 (µ) |b|H 1 (µ) for any atomic block b with supp (b) ⊂ R. To this end, write

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Rd

M(b)(x) dµ(x) =

213

Rd \2R

M(b)(x) dµ(x) +

M(b)(x) dµ(x) = J1 + J2 . 2R

By (2.2) and Deﬁnition 1.1, we have

write

J1 bL1(µ) |b|H 1 (µ) . (2.5)

To estimate the term J2 , let b = j λj aj be as in Deﬁnition 1.1 (iii), and M(b)(x) dµ(x) ≤ 2R

j

+

|λj |

M(aj )(x) dµ(x) 2Qj

|λj |

j

M(aj )(x) dµ(x). 2R\2Qj

The L2 (µ) boundedness of M via the H¨older inequality states that for each ﬁxed j, 1/2 M(aj )(x) dµ(x) ≤ M(aj )L2 (µ) µ(2Qj ) aj L∞ (µ) µ(2Qj ) 1. 2Qj

On the other hand, by the inequality (2.4), M(aj )(x) dµ(x)

1 dµ(x)aj L1 (µ) |x − xQj |n 2R\2Qj SQj , R aj L∞ (µ) µ(Qj ) 1.

2R\2Qj

This in turn leads to that

M(b)(x) dµ(x) 2R

From this, it follows that

|λj |.

j

M(b)(x) dµ(x) |b|H 1 (µ) .

(2.6)

2R

Combining the estimates (2.5) and (2.6) then completes the proof of Theorem 2.3. We also obtain the following boundedness of M in (1.6) from L∞ (µ) to RBLO(µ). Theorem 2.4. Let K satisfy (1.4) and (1.5), and M be as in (1.6). If M is bounded on L2 (µ), then for f ∈ L∞ (µ), M(f ) is either infinite everywhere or finite almost everywhere. More precisely, if M(f ) is finite at some point x0 ∈ Rd , then M(f ) is µ-finite almost everywhere and M(f )RBLO(µ) ≤ Cf L∞ (µ) , where the constant C > 0 is independent of f .

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Proof. First, we √ claim √ that there is a positive constant C such that for any f ∈ L∞ (µ) and (4 d, (4 d)n+1 )-doubling cube Q, 1 M(f )(y) dµ(y) ≤ Cf L∞ (µ) + inf M(f )(y). (2.7) y∈Q µ(Q) Q To prove this, for each ﬁxed cube Q, let B be √ the smallest ball which contains Q and has the same center as Q. Then 2B ⊂ 4 dQ. We then decompose f as f (x) = f (x)χ2B (x) + f (x)χRd \2B (x) = f1 (x) + f2 (x). By the H¨older inequality and L2 (µ)-boundedness of M, we have 1/2 1 1 2 M(f1 )(x) dµ(x) ≤ [M(f1 )(x)] dµ(x) 1/2 µ(Q) Q Rd µ(Q) √ 1/2 µ(4 dQ) 1/2 f L∞ (µ) µ(Q) f L∞ (µ) .

(2.8)

Denote by r the radius of B. Noting that |y −z| ≥ r for any y ∈ Q and z ∈ Rd \2B, by the Minkowski inequality and (1.4), we have that for any y ∈ Q, 2 1/2 ∞ dt K(y, z)f (z) dµ(z) M(f2 )(y) = 2 t3 r |y−z|≤t 2 1/2 ∞ dt ≤ K(y, z)f (z) dµ(z) 3 t 2 1/2 r ∞ |y−z|≤t dt + K(y, z)f1 (z) dµ(z) 3 t r |y−z|≤t ∞ 2 1/2 dt ≤ M(f )(y) + |K(y, z)f1 (z)| dµ(z) t3 r |y−z|≤4r 1 M(f )(y) + f L∞ (µ) r−1 dµ(z) |y − z|n−1 |y−z|≤4r M(f )(y) + f L∞ (µ) . (2.9) Thus, the proof of the estimate (2.7) can be reduced to proving that for any x, y ∈ Q, |M(f2 )(x) − M(f2 )(y)| f L∞(µ) . (2.10) To prove (2.10), set ∞ A1 =

2 1/2 dt |K(y, z)||f2 (z)| dµ(z) , t3 |y−z|≤t<|x−z|

0

A2 = 0

∞

2 1/2 dt |K(x, z)||f2 (z)| dµ(z) t3 |x−z|≤t<|y−z|

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and

∞

A3 = 0

2 1/2 dt |K(y, z) − K(x, z)||f2 (z)| dµ(z) , t3 (|y−z|∨|x−z|)≤t

where and in what follows, we denote max{a, b} for any a, b ∈ R simply by a ∨ b. By the Minkowski inequality, we have M(f2 )(x) ∞ ≤ 0

2 1/2 dt K(x, z)f2 (z) dµ(z) − K(y, z)f2 (z) dµ(z) 3 t |x−z|≤t |y−z|≤t

+M(f2 )(y) ≤ A1 + A2 + A3 + M(f2 )(y), which together with symmetry gives |M(f2 )(y) − M(f2 )(x)| ≤ A1 + A2 + A3 .

(2.11)

Applying the Minkowski inequality and (1.4) yields that for x, y ∈ Q, 1/2 |f2 (z)| dt A1 dµ(z) n−1 3 |y−z|<|x−z| |y−z|≤t<|x−z| t |y − z| 1/2 1 |f (z)| dµ(z) l(Q) |x − z|n+1/2 d Q R \2B f L∞ (µ) , and by (1.5), A3

Rd

|K(y, z) − K(x, z)||f2 (z)|

Rd \2B

(|y−z|∨|x−z|)≤t

|K(y, z) − K(x, z)||f (z)|

f L∞(µ) .

dt t3

1/2 dµ(z)

1 dµ(z) |xQ − z|

By symmetry, we have A2 f L∞ (µ) . Combining the estimates for A1 , A2 and A3 gives us the estimate (2.10). Thus, (2.7) holds. From (2.7), it follows that for f ∈ L∞ (µ), if M(f )(x0 ) < ∞ for some point x0 ∈ Rd , then M(f ) is µ-ﬁnite almost everywhere and in this case, mQ (M(f )) − essinf x∈Q M(f )(x) f L∞ (µ) , √ √ provided that Q is a (4 d, (4 d)n+1 )-doubling cube. To prove that M(f ) ∈ RBLO(µ), by Deﬁnition 1.3, we still need to prove that M(f ) satisﬁes (1.3). Let

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√ √ Q ⊂ R be any two (4 d, (4 d)n+1 )-doubling cubes. Set N = NQ, R + 1. Write NQ, R M(f )(x) ≤ M(f χ2Q )(x) + M(f χ2k+1 Q\2k Q )(x) k=1 + M(f χRd \2N Q )(x) − M(f χRd \2N Q )(y) + M(f χRd \2N Q )(y).

By an estimate similar to that for (2.9), we have that for y ∈ R, M(f χRd \2N Q )(y) M(f )(y) + f L∞ (µ) . On the other hand, by the estimate same as that for (2.10), we have that for x, y ∈ R, M(f χRd \2N Q )(x) − M(f χRd \2N Q )(y) f L∞(µ) . For x ∈ Q and each ﬁxed k ∈ N, an application of the Minkowski inequality together with (1.4) says that 1 M(f χ2k+1 Q\2k Q )(x) f L∞ (µ) dµ(y). n 2k+1 Q\2k Q |y − xQ | Therefore, for any x ∈ Q and y ∈ R, M(f )(x) M(f χ2Q )(x) + M(f )(y) + SQ, R f L∞ (µ) . Taking mean value over Q for x, and over R for y, then yields mQ (M(f )) − mR (M(f )) SQ, R f L∞ (µ) + mQ (M(f χ2Q )) SQ, R f L∞ (µ) , where we used the fact that mQ M(f χ2Q ) f L∞ (µ) , which can be proved by a way similar to that for the estimate (2.8). This ﬁnishes the proof of Theorem 2.4. As a simple corollary of Theorem 2.1 and Theorem 2.4 together with Theorem 3.1 in [10], we can obtain that Corollary 2.5. Let K satisfy (1.4) and (1.5), and M be as in (1.6). If M is bounded on L2 (µ), then M is also bounded on Lp (µ) for any 1 0, y, y ∈Rd |y−y |≤r

l=1

2l r<|x−y|≤2l+1r

+|K(y, x) − K(y , x)|

1 dµ(x) ≤ C, |x − y|

(3.1)

which is slightly stronger than (1.5). We ﬁrst have the following Lp (µ)-boundedness of Mb,m in (1.7) for p ∈ (1, ∞).

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Theorem 3.1. Let K satisfy (1.4) and (3.1), and p ∈ (1, ∞). If M is bounded on L2 (µ), then for any b ∈ RBMO(µ) and positive integer m, the commutator Mb, m in (1.7) is bounded on Lp (µ) with bound no more than Cm,p bm RBMO(µ) . To prove Theorem 3.1, we ﬁrst recall some basic facts and establish a technical lemma. The following John-Nirenberg inequality was established by Tolsa in [21]. Lemma 3.2. Let b ∈ RBMO(µ). Then there exist constants C1 > 0 and C2 > 0 such that for any cube Q and any λ > 0, −C2 λ ≤ C1 µ(ρQ) exp µ x ∈ Q : b(x) − mQ (b) > λ bRBMO(µ) with C1 and C2 depending on the constant ρ > 1 in Definition 1.2, but not on b, Q and λ > 0. From Lemma 3.2, it is easy to deduce that for a ﬁxed ρ > 1, there are two positive constants B1 and B2 such that for any cube Q and b ∈ RBMO(µ), |b(x) − mQ (b)| 1 exp (3.2) dµ(x) ≤ B2 . µ(ρQ) Q B1 bRBMO(µ) Lemma 3.3. Let m be a positive integer. Then there is a positive constant Cm such that for any a > 0 and t1 , t2 ≥ 0, m −1 exp t2 . t1 tm 2 ≤ Cm t1 log (2 + at1 ) + a Proof. Note that for any t1 , t2 ≥ 0, m t1 tm 2 ≤ Cm [t1 log (2 + t1 ) + exp t2 ] ;

see [1, Chap. 8]. Lemma 3.3 follows from the last inequality immediately by chang ing t1 into at1 . For a µ-locally integrable function f , let M f be the sharp maximal function of f , namely, for x ∈ Rd , 1 M f (x) = sup |f (y) − mQ (f )| dµ(y) µ( 32 Q) Q Qx Q cube

+

sup x∈Q⊂R

Q, R doubling cubes

|mQ (f ) − mR (f )| . SQ, R

1/r For 0 < r < ∞, let Mr f (x) = M (|f |r )(x) for x ∈ Rd . A straightforward computation proves that if 0 < r < 1, Mr f (x) ≤ Cr M f (x),

x ∈ Rd ,

(3.3)

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where Cr > 0 is independent of f and x; see [9]. For 0 < s < ∞ and η > 1, let Ms, (η) be the non centered maximal operator deﬁned by Ms, (η) f (x) = sup Qx

1 µ(ηQ)

1/s |f (y)|s dµ(y) ,

Q cube

x ∈ Rd .

Q

It is well known that Ms, (η) is bounded on Lp (µ) provided that s < p < ∞; see [21]. The following technical lemma is of independent interest. In what follows, for convenience, we denote Mb, 0 simply by M. Lemma 3.4. Let K satisfy (1.4) and (3.1), s ∈ (1, ∞), p0 ∈ (1, ∞) and b ∈ L∞ (µ). If M is bounded on L2 (µ), then there is a positive constant C such that for all f ∈ L∞ (µ) ∩ Lp0 (µ) and all x ∈ Rd , M [Mb, m (f )] (x) m−1 m−k m ≤C bRBMO(µ) Ms, (3/2) Mb, k (f ) (x) + bRBMO(µ) f L∞ (µ) . k=0

Proof. Without loss of generality, we may assume that ρ = 9/8 in Deﬁnition 1.2 and bRBMO(µ) = 1. As in the proof of Theorem 9.1 in [21], it suﬃces to prove that 1 µ( 32 Q)

|Mb, m (f )(y) − hQ | dµ(y) Q

m−1

Ms, (3/2) Mb, k (f ) (x) + f L∞ (µ)

k=0

(3.4) for all x and Q with x ∈ Q, and m+1 m−1 |hQ − hR | SQ, R Ms, (3/2) Mb, k (f ) (x) + f L∞(µ)

(3.5)

k=0

for all cubes Q ⊂ R with x ∈ Q, where Q is an arbitrary cube and R is a doubling cube, m hQ = mQ M mQ (b) − b f χRd \ 43 Q and

m hR = mR M [mR (b) − b] f χRd \ 43 R .

Recall that M is bounded on Lp0 (µ) by Corollary 2.5. This fact and the assumptions that b ∈ L∞ (µ) and f ∈ Lp0 (µ) together with the H¨ older inequality imply that both hQ and hR are ﬁnite.

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We ﬁrst establish the estimate (3.4). For a ﬁxed cube Q, x ∈ Q and f ∈ L∞ (µ), decompose f as f (y) = f (y)χ 43 Q (y) + f (y)χRd \ 43 Q (y) = f1 (y) + f2 (y). With the aid of the formula that for y, z ∈ Rd , m mQ (b) − b(z) = [b(y) − b(z)]m m−1 m−k k + , [b(y) − b(z)]k mQ (b) − b(y) m

(3.6)

k=0

we can write

1 |Mb, m (f )(y) − hQ | dµ(y) µ( 32 Q) Q m−1 m−k 1 m (b) − b(y) Mb, k (f )(y) dµ(y) Q 3 µ( Q) Q 2 k=0 m 1 + 3 M mQ (b) − b(y) f1 (y) dµ(y) µ( 2 Q) Q m 1 + 3 M mQ (b) − b(y) f2 (y) − hQ dµ(y) µ( 2 Q) Q = D1 + D2 + D3 .

It follows from the H¨ older inequality and (3.10) in [21] that 1/s m−1 1 m (b) − b(y)(m−k)s dµ(y) D1 ≤ µ( 32 Q) Q Q k=0 1/s 1 s × [Mb, k (f )(y)] dµ(y) µ( 32 Q) Q m−1 Ms, (3/2) [Mb, k (f )] (x). k=0

Similarly, by the H¨ older inequality, the L2 (µ)-boundedness of M, and (3.10) in [21], 1/2 2 1 m D2 ≤ M [mQ (b) − b] f1 (y) dµ(y) µ( 3 Q) Q 2 1/2 2m 1 2 m 4 Q (b) − b(y) |f (y)| dµ(y) 3 µ( 32 Q) 43 Q m +f L∞(µ) m 4 Q (b) − mQ (b) f L∞ (µ) ,

3

where the last inequality follows from the fact that m 4 Q (b) − mQ (b) SQ, 43 Q 1; 3

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see the estimate (2.10) and Lemma 2.1 in [21]. To estimate the term D3 , set ∞ 2 1/2 m dt I1 (x, y) = |K(y, z)| mQ (b) − b(z) |f2 (z)| dµ(z) 3 , t 0 |y−z|≤t<|x−z| I2 (x, y) = 0

∞

2 1/2 m dt |K(x, z)| mQ (b) − b(z) |f2 (z)| dµ(z) 3 t |x−z|≤t<|y−z|

and

∞

|K(y, z) − K(x, z)|

I3 (x, y) = 0

(|y−z|∨|x−z|)≤t

2 1/2 m dt . × mQ (b) − b(z) |f2 (z)| dµ(z) t3 It is easy that for any x, y ∈ Q, 3 m m Ij (x, y); M mQ (b) − b f2 (y) − M mQ (b) − b f2 (x) ≤ j=1

see (2.11). A trivial computation involving the Minkowski inequality and (1.4) proves that for x, y ∈ Q, 1/2 |mQ (b) − b(z)|m |f2 (z)| dt I1 (x, y) dµ(z) 3 |y − z|n−1 |y−z|<|x−z| |y−z|≤t<|x−z| t m 1/2 1 l(Q) mQ (b) − b(z) |f (z)| dµ(z) n+1/2 4 |x − z| d Q R \3Q ∞ m 1 n 2−k/2 (b) − b(z) |f (z)| dµ(z) m 4 k k Q) 2 Q 4 k 3 l(2 32 Q k=1 ∞ 1 m −k/2 n + k 2 |f (z)| dµ(z) 4 k l(2k Q) 32 Q k=1 f L∞ (µ) , where we used the facts that by (1.1) together with (3.10) in [21], m 1 n (b) − b(z) dµ(z) 1, m 4 k 4 k l(2k Q) 32 Q 2 Q 3 and that by (2.10) and Lemma 2.1 in [21], (b) mQ (b) − m SQ, 43 2k Q k. 4 k 32

Q

Similarly, by symmetry, we have that for x, y ∈ Q, I2 (x, y) f L∞ (µ) .

(3.7)

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Note that for x, y ∈ Q, by the Minkowski inequality, I3 (x, y)

m |K(y, z) − K(x, z)| mQ (b) − b(z) |f2 (z)|

Rd

× (|y−z|∨|x−z|)≤t ∞

f L∞ (µ)

dt t3

1/2 dµ(z)

4 k 4 k−1 Q 3 2 Q\ 3 2

k=1

|K(y, z) − K(x, z)|

m ×m (b) − b(z) 4 k

1 dµ(z) |y − z| ∞ m +f L∞ (µ) (b) mQ (b) − m 4 k 32 Q

×

32

k=1

4 k 4 k−1 Q 3 2 Q\ 3 2

Q

|K(y, z) − K(x, z)|

1 dµ(z) |y − z|

= I13 + I23 . By (3.7), (1.4) and (3.1), we obtain I23

f L∞ (µ)

∞

k

m 4 k 4 k−1 Q 3 2 Q\ 3 2

k=1

|K(y, z) − K(x, z)|

1 dµ(z) f L∞ (µ) . |y − z|

Similarly, applying Lemma 3.3, (3.2), (1.4) and (3.1) leads to that ∞

2−k

4 I13 f L∞(µ) µ 3 2k+1 Q k=1 +f L∞(µ)

∞ k=1



4 k 32 Q

4 k 4 k−1 Q 3 2 Q\ 3 2

exp 

(b) − b(z)| |m 4 k 32 Q

B1

  dµ(z)

|K(y, z) − K(x, z)|

1 4 k+1 1 2 Q |K(y, z) − K(x, z)| dµ(z) × log 2 + 2 µ 3 |y − z| |y − z| ∞ f L∞(µ) + f L∞ (µ) km |K(y, z) − K(x, z)|

m

k

× logm 2 + µ f L∞(µ) .

k=1

4 k+1 2 Q 3

4 k 4 k−1 Q 3 2 Q\ 3 2

1 1 1 dµ(z) n−1 |y − z| |xQ − z| |y − z|

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Combining the estimates above then gives that 1 1 D3 ≤ 3 M [mQ (b) − b]m f2 (y) µ( 2 Q) µ(Q) Q Q m −M [mQ (b) − b] f2 (x) dµ(y) dµ(x) f L∞ (µ) , and the estimate (3.4) is proved. We now verify (3.5). For any cubes Q ⊂ R with x ∈ Q, where Q is an arbitrary and R is a doubling cube, denote NQ, R + 1 simply by N . Write |hQ − hR | ≤ mR M [mQ (b) − b]m f χRd \2N Q − mQ M [mQ (b) − b]m f χRd \2N Q + mR M [mR (b) − b]m f χRd \2N Q − mR M [mQ (b) − b]m f χRd \2N Q + mQ M [mQ (b) − b]m f χ2N Q\ 43 Q + mR M [mR (b) − b]m f χ2N Q\ 43 R = E1 + E2 + E3 + E4 . As in the estimate for the term D3 , we have E1 f L∞ (µ) . On the other hand, via the Minkowski inequality and (1.4), it is easy to see that for y ∈ R, M [mR (b) − b]m f χ2N Q\ 43 R (y) ∞ dt 1/2 m |K(y, z)| |mR (b) − b(z)| f χ2N Q\ 43 R (z) dµ(z) 3 |y−z| t Rd 1 m |mR (b) − b(z)| dµ(z) f L∞ (µ) [l(R)]n 2N Q f L∞ (µ) , where we used the fact that 1 |mR (b) − b(z)|m dµ(z) [l(R)]n 2N Q 1 m % (b) − b(z)m dµ(z) + m % (b) − mR (b)m 4R 4R µ(6R) 4R 1 + SR, 4R 1,

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by (3.10), (2.10) and Lemma 2.1 in [21]. Hence, E4 f L∞ (µ) . To estimate E2 , for y ∈ R, by an argument similar to that for the estimate (2.11), (3.6) and (2.10) in [21], we have m M [mR (b) − b]m f χRd \2N Q (y) − M mQ (b) − b f χRd \2N Q (y) m m ≤ M fχRd \2N Q (y) mR (b) − b − mQ (b) − b m−1 m−k k

M [mR (b) − b]k f χRd \2N Q (y) ≤ mQ (b) − mR (b) m k=0

m−1

k m−k k−i SQ, R |mR (b) − b(y)| Mb, i (f )(y) i=0

k=0 m−1

k=0 m−1

+

+

SQ, R SQ, R

m−k

M [mR (b) − b]k f χ2N Q\ 43 R (y)

m−k

M [mR (b) − b]k f χ 34 R (y).

k=0

By the fact that µ( 32 R) ≤ µ(2R) µ(4R), since R is doubling, and an estimate similar to that for D1 , we obtain that for each k with 0 ≤ k ≤ m − 1, i with 0 ≤ i ≤ k and x ∈ Q, k−i Mb, i (f ) Ms, (3/2) [Mb, i (f )] (x); mR |mR (b) − b| and as in the estimate for E4 , for each y ∈ R M [mR (b) − b]k f χ2N Q\ 43 R (y) f L∞(µ) . An argument similar to the estimate for D2 gives that mR M [mR (b) − b]k f χ 34 R f L∞(µ) . We thus obtain that m E2 SQ, R

m−1

Ms, (3/2) [Mb, k (f )](x) + f L∞(µ)

.

k=0

Finally, by the Minkowski inequality and (1.4), for each y ∈ Q, M [mQ (b) − b]m f χ2N Q\ 43 Q (y) ≤ M [mQ (b) − b]m f χ2N Q\2Q (y) + M [mQ (b) − b]m f χ2Q\ 43 Q (y) 1/2 m ∞ 1 dt |f (z)| dµ(z) mQ (b) − b(z) 3 |y − z|n−1 |y−z| t 2N Q\2Q

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+

2Q\ 43 Q

m mQ (b) − b(z)

1 |f (z)| |y − z|n−1

IEOT

∞

|y−z|

dt t3

1/2 dµ(z)

m 1 n (b) − b(z) dµ(z) m Q k Q) k+1 k l(2 2 Q\2 Q k=1 m 1 n +f L∞(µ) mQ (b) − b(z) dµ(z) l(Q) 2Q\ 43 Q N −1 m µ(2k+2 Q) n SQ, 2k+1 Q ∞ f L (µ) + 1 + f L∞ (µ) l(2k+1 Q) k=1 SQ, R ]m+1 f L∞ (µ) , f L∞ (µ)

N −1

where in the second-to-last inequality, we used the estimates (2.10), (3.10) and Lemma 2.1 in [21]; see the estimate for D2 . From this estimate, it follows that m+1 f L∞ (µ) . E3 SQ, R Combining the estimates through E1 to E4 establishes (3.5), which completes the proof of Lemma 3.4. Proof of Theorem 3.1. We prove the theorem by induction on m. If m = 0, then Mb, m is just the Marcinkiewicz integral, and the conclusion of Theorem 3.1 is true in this case. Now let m be a positive integer, and we assume that for any integer k with 0 ≤ k ≤ m − 1, Mb, k is bounded on Lp (µ) with bound no more than CbkRBMO(µ) for any p ∈ (1, ∞). Our goal is now to prove that Mb, m (f )Lp (µ) bm RBMO(µ) f Lp (µ) .

(3.8)

Let 0 < r < 1. We claim that for any p ∈ (1, ∞), b ∈ L∞ (µ) and all bounded functions f with compact support, p d λ−p bmp (3.9) µ x ∈ R : Mr [Mb, m (f )] (x) > λ RBMO(µ) f Lp (µ) . Once this estimate is established, it follows form the Marcinkiewicz interpolation theorem that for any p ∈ (1, ∞), b ∈ L∞ (µ) and all bounded functions f with compact support, Mr [Mb, m (f )]Lp (µ) bm RBMO(µ) f Lp (µ) . This via Theorem 6.2 in [21] states that for any p ∈ (1, ∞), b ∈ L∞ (µ) and all bounded functions f with compact support and integral zero, Mb, m (f )Lp (µ) bm RBMO(µ) f Lp (µ) . A routine argument involving the density in RBMO(µ) of L∞ (µ) (see Lemma 3.3 in [21]) and the density in Lp (µ) of bounded functions with compact support and integral zero (see [21, p. 135]) then leads to the inequality (3.8).

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Now we prove (3.9). Without loss of generality, we may assume that ρ = 3/2 in Deﬁnition 1.2 and bRBMO(µ) = 1. For each ﬁxed λ > 0 and each bounded function f with compact support, applying the Calder´on-Zygmund decomposition to |f |p at level λp as Lemma 2.2, we can decompose f as f (x) = g(x)+ h(x), where ϕj (x) g(x) = f (x)χRd \∪j Qj (x) + j

and h(x) = f (x) − g(x) =

[ωj (x)f (x) − ϕj (x)] =

j

Lp (µ)

λ

& & & & & ϕj & & &

L1 (µ)

j

1/p 1/p µ(Rj ) |ϕj (x)| dµ(x)

p−1

other hand, by Lemma 2.2 (b) together

L∞ (µ)

j

hj (x).

j

It is obvious that gL∞ (µ) λ. On the with the H¨older inequality, & &p−1 & &p & & & & & & & ϕj & ≤& |ϕj |& & & j

p

Rj

j

|f (x)|p dµ(x)

Qj

j

f pLp(µ) , which further tells us that gLp(µ) f Lp(µ) . This, together with estimate (3.3), Lemma 3.4 and the inductive assumption, says that d µ x ∈ R : Mr (Mb, m (g))(x) > 2Cλ m−1 Ms, (3/2) [Mb, k (g)](x) > λ ≤ µ x ∈ Rd : λ−p

m−1

k=0

Ms, (3/2) [Mb, k (g)]pLp (µ)

k=0

λ−p f pLp(µ) , where 1 < s < p. Now we turn our attention to Mr (Mb, m (h)). Using (3.6) and the fact that Mr [Mb, m (h)](x) Mr, (3/2) [Mb, m (h)](x), which can be deduced from the H¨ older inequality, we can write

µ x ∈ Rd : Mr [Mb, m (h)](x) > λ m (b) M(h ) (x) > λ/2 ≤ µ x ∈ Rd : Mr, (3/2) b − mQ j %j j

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m−1 m−k b − mQ +µ x ∈ Rd : Mr, (3/2) Mb, k (b) h (x) > Cλ j %j j

k=0

=

I14

+

I24 .

Choose p1 with 1 < p1 < p. Repeating the argument used in [15, p. 471], we see that for any σ > 0,

σµ x ∈ Rd : Mr, (3/2) u(x) > σ sup τ µ({x ∈ Rd : |u(x)| > Cτ }).. (3.10) τ >Cσ

Thus, from this and the inductive assumption, it follows that m−1 m−k 2 −1 d I4 λ b − mQ sup τ µ x ∈ R : Mb, k hj (x) > Cτ %j (b) τ >Cλ

λ−1 λ−p1 +1

k=0

& m−1

j

m−k & &p1 & & &Mb, k b − m (b) h j % & & Qj

Lp1 (µ)

j

k=0

m−1 &p1 m−k & & & −p1 & b − m λ (b) f ω & p j % & Qj L 1 (µ) j

k=0

m−1 &

& & &

−p1

+λ

k=0

m−k & &p1 b − mQ ϕj & %j (b) &

Lp1 (µ)

j

= F1 + F2 . Another application of the H¨ older inequality and Lemma 2.2 (a) gives that p1 /p m−1 |f (x)|p dµ(x) F1 λ−p1 × λ−p

k=0

Qj Rd

j

Qj

1−p1 /p (m−k)p1 (p/p1 ) (b) dµ(x) b(x) − mQ %j |f (x)|p dµ(x).

To estimate F2 , by H¨ older’s inequality for the series and the fact that Bλ, we have  p1  m−k |ϕ (x)|  j b(x) − mQ %j (b)  λ  j

≤

j

j

−1

λ

j

|ϕj (x)| ≤

p1 /p1 (m−k)p1 −1 |ϕj (x)|b(x) − mQ λ |ϕj (x)| %j (b)

(m−k)p1 λ−1 |ϕj (x)|b(x) − mQ (b) . %j

j

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Recall that Rj is (6, 6n+1 )-doubling, and by (2.10) and Lemma 2.1 in [21], (b) − m (b) (3.11) SQj , Rj 1. mQ %j %j R Therefore, from these estimates, (3.10) in [21] and Lemma 2.2 (b) together with the H¨older inequality, it follows that m−1 (m−k)p1 F2 λ−1 |ϕj (x)|b(x) − mQ (b) dµ(x) %j λ−1

k=0

j

k=0

j

1/p |ϕj (x)|p dµ(x)

m−1

×

Rj

+λ−1

Rj

1/p (m−k)p1 p dµ(x) b(x) − mR %j (b) (m−k)p1 m (b) − m (b) Q %j %j R

m−1 k=0

λ−p

Rj

|f (x)|p dµ(x).

Qj

j

It remains to estimate I14 . I14 λ−1

+λ−1 +λ−1

Employing the estimate (3.10) again, we have m b(x) − mQ %j (b) M(hj )(x) dµ(x)

Rd \2Rj

j

+λ−1

|ϕj (x)| dµ(x)

Rj

j

m (b) b(x) − mQ M(ϕj )(x) dµ(x) %j

j

2Rj

j

4 3 Qj

j

2Rj \ 43 Qj

m (b) b(x) − mQ M(ωj f )(x) dµ(x) %j m (b) M(ωj f )(x) dµ(x) b(x) − mQ %j

= G + H + J + L. For each ﬁxed j, write m b(x) − mQ %j (b) M(hj )(x) dµ(x) d R \2Rj m ≤ b(x) − mQ %j (b) Rd \2Rj

×

0

+ Rd \2Rj

2 1/2 dt K(x, y)hj (y) dµ(y) 3 dµ(x) t |x−y|≤t m (b) b(x) − mQ %j

√ |x−xRj |+ dl(Rj )

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×

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2 1/2 dt K(x, y)hj (y) dµ(y) 3 dµ(x) t |x−y|≤t

∞

√ |x−xRj |+ dl(Rj )

= G1 + G2 . Observe that for each ﬁxed j, by (2.10) and Lemma 2.1 in [21], (b) − mQ (b) m2k+1 SQj , 2k+1 Rj (SQj , Rj + SRj , 2k+1 Rj ) k. % j R j

It follows from this estimate, the Minkowski inequality, (1.4), (3.10) in [21] and (1.1) that m (b) G1 ≤ b(x) − mQ |K(x, y)||hj (y)| %j Rd \2Rj

×

Rd

√ |x−xRj |+ dl(Rj )

|x−y|

|hj (y)| dµ(y) ∞

Rd

dt t3

1/2 dµ(y) dµ(x)

1/2 l(Rj ) dµ(x) × j k+1 R \2k R |x − xRj |n+1/2 j j k=1 2 ∞ m l(R ) 1/2 j (b) − mQ dµ(x) + m2k+1 %j (b) Rj |x − xRj |n+1/2 k+1 R \2k R j j k=1 2 1/2 ∞ k m µ(2k+1 Rj ) l(Rj ) |hj (y)| dµ(y) n+1/2 Rd l(2k Rj ) k=1 hj (y) dµ(y). m (b) b(x) − m2k+1 R

Rd

Similar to the estimate for terms I13 and I23 , by the vanishing moment of hj , we can write 2 m |b(x) − mR K(x, y)hj (y) dµ(y) G = %j (b)| Rd \2Rj Rd ∞ 1/2 dt × dµ(x) √ 3 |x−xRj |+ dl(Rj ) t ∞ m m |hj (y)| (b) + (b) − m m b(x) − m2k+1 %j R 2k+1 R R Rd

k=1

2k+1 Rj \2k Rj

j

1 dµ(x) dµ(y) ×|K(x, y) − K(x, xRj )| |x − xRj | |hj (y)| dµ(y). Rd

j

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The desired estimate for G follows from the above estimates and the fact that 1/p 1/p 1/p p µ(Qj ) |hj (y)| dµ(y) |f (y)| dµ(y) + ϕj Lp (µ) µ(Rj ) Rd

Qj

λ−p+1 f pLp(µ) ,

which follows from the H¨older inequality and Lemma 2.2. For the term H, it follows from the H¨ older inequality, (3.10) in [21] and the Lp (µ)-boundedness of M together with Lemma 2.2 that m −1 H λ M(ϕj )(x) dµ(x) mQ %j (b) − m2Rj (b) j

+λ−1

2Rj

j

λ−1 ×

λ−1

2Rj

m b(x) − m2Rj (b) M(ϕj )(x) dµ(x)

M(ϕj )Lp (µ)

j

1/p µ(2Rj ) +

2Rj

1/p p m dµ(x) b(x) − m2Rj (b)

1/p ϕj Lp (µ) µ(6Rj )

j

λ−p f pLp(µ) . Similarly, −1

J λ

j

λ−1

f ωj Lp (µ)

4 3 Qj

1/p f ωj Lp (µ) µ(2Qj )

j −1

≤λ

−p/p

λ

f ωj Lp (µ) λ

1/p |f (x)| dµ(x) p

Qj

j −p

1/p p m dµ(x) b(x) − mQ %j (b)

f pLp(µ) .

Observe that for each ﬁxed j, by (3.10) in [21] and (3.11),

m |b(x) − mQ %j (b)| 2Rj \Rj

|x − xQj |n

m 1 n µ(2Rj )mQ %j (b) − m2Rj (b) l(Rj ) m 1 n + b(x) − m2Rj (b) dµ(x) l(Rj ) 2Rj 1,

dµ(x)

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and further by Lemma 2.1 in [21] and the proof of Lemma 2.3 in [23], m |b(x) − mQ %j (b)| dµ(x) |x − xQj |n Rj \Qj Nj −1 m 1 n − m k+1 (b) b(x) dµ(x) 6 kQ ) Qj k+1 Q \6k Q l(6 6 j j j k=0 Nj −1

+

k=0

m 1 n µ(6k+1 Qj )mQ (b) − m (b) %j 6k+1 Qj l(6k Qj )

Nj −1

µ(6k+2 Qj ) n SQj , Rj l(6k Qj ) k=0

1, where Nj is a positive constant such that Rj = 6Nj Qj . The above estimates via the inequality (2.4) and the H¨older inequality tell us that m |b(x) − mQ %j (b)| −1 Lλ |f (y)| dµ(y) dµ(x) |x − xQj |n Qj 2Rj \Rj j m |b(x) − mQ %j (b)| −1 +λ |f (y)| dµ(y) dµ(x) |x − xQj |n Qj Rj \Qj j 1/p 1/p λ−1 µ(Qj ) |f (y)|p dµ(y) j −p

λ

Qj

f pLp(µ) .

Combining the estimate for the terms G, H, J and L yields the desired estimate for I14 and then establishes the inequality (3.9), which completes the proof of Theorem 3.1. In the sequel, for α ≥ 1, let Φα (t) = t logα (2 + t) and Ψ1/α (t) = exp t1/α . Using Theorem 2.1 and Theorem 3.1, we now establish the boundedness of Mb, m from L(log L)m (µ) to L1, ∞ (µ). Theorem 3.5. Let K satisfy (1.4) and (3.1). If M is bounded on L2 (µ), then for any b ∈ RBMO(µ) and positive integer m, there is a positive constant C such that for all λ > 0 and all bounded functions f with compact support, |f (y)| µ({x ∈ Rd : Mb, m (f )(x) > λ}) ≤ CΦm (bm ) Φ dµ(y). m RBMO(µ) λ Rd

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Proof. Again we prove this theorem by induction on m. Recall that M is bounded from L1 (µ) to L1, ∞ (µ) by Theorem 2.1. Let m be a positive integer, we assume that for any integer k with 0 ≤ k ≤ m − 1, the estimate |f (y)| d k Φk µ({x ∈ R : Mb, k (f )(x) > λ}) Φk (bRBMO(µ) ) dµ(y) λ Rd holds. By the fact that for s > 0 and t1 , t2 > 0, Φs (t1 t2 ) Φs (t1 )Φs (t2 ),

(3.12)

without loss of generality, we may assume that bRBMO(µ) = 1; see also the proof of Theorem 4 in [10]. For each ﬁxed λ > 0 and bounded function f with compact support, we apply Lemma 2.2 for |f | at the level λ. With the same notation Qj , Rj , ϕj , ωj as in Lemma 2.2, we again decompose f as f = g + h, where ϕj (x) g(x) = f χRd \∪j Qj (x) + j

and h(x) = f (x) − g(x) =

[ωj (x)f (x) − ϕj (x)] .

j

Note that gL1 (µ) f L1(µ) . The L2 (µ)-boundedness of Mb, m (f ) from Theorem 3.1 and the fact that |g(x)| λ µ − a. e. proves that µ({x ∈ Rd : Mb, m (g)(x) > λ}) λ−1 |f (y)| dµ(y). Rd

Taking into account the fact from Lemma 2.2 that

−1 µ ∪j 2Qj λ |f (y)| dµ(y), Rd

we see that the proof of Theorem 3.5 can be reduced to proving that |f (y)| |f (y)| m d log µ x ∈ R \ ∪j 2Qj : Mb, m (h)(x) > λ 2+ dµ(y). λ λ Rd For each ﬁxed j, set bj (x) = b(x) − mQ %j (b) and hj (x) = ωj (x)f (x) − ϕj (x). By (3.6), we can split Mb, m (h)(x) as m−1 k m m−k Mb, m (h)(x) ≤ |bj (x)| M(hj )(x) + (bj ) hj (x) Mb, k m j

k=0

j

= I(x) + II(x). The estimate for I(x) is easy. In fact, as in the estimates for the terms G, H and L in the proof of Theorem 3.1, we have d µ x ∈ R \ ∪j 2Qj : I(x) > λ/2 |I(x)| dµ(x) Rd \

j

2Qj

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j

+ +

Rd \2Rj

|bj (x)|m M(hj )(x) dµ(x)

j

2Rj \2Qj

j

2Rj

Rd

IEOT

1 dµ(x) |x − xQj |n

|f (y)| dµ(y) Qj

|bj (x)|m M(ϕj )(x) dµ(x)

|f (y)| dµ(y).

To estimate II(x), from the inductive assumption, it follows that d µ x ∈ R \ ∪j 2Qj : II(x) > λ m−1 k d m−k (bj ) ωj f (x) > λ/2 Mb, k ≤ µ x ∈ R \ ∪j 2Qj : m k=0

j

m−1 k d m−k (bj ) ϕj (x) > λ/2 Mb, k +µ x ∈ R \ ∪j 2Qj : m

m−1 k=0

+

m−1 k=0

Φk

Qj

j

Rd

Φk

k=0

j

m−k |f (y)| dµ(y) b(y) − mQ %j (b) λ

j

m−k |ϕj (y)| dµ(y) b(y) − mQ %j (b) λ

= II1 + II2 , where in the ﬁrst term of the second-to-last inequality, we used the almost disjoint property of cubes {Qj }j ; see Lemma 2.2. By the fact that for s, t > 0, Φk (st) Φm (s) + Ψ1/(m−k) (t) (see [18, p. 26]), (3.12), (3.2) and Lemma 2.2, we then obtain |b(y) − m (b)|m−k %j Q Ψ1/(m−k) m−k B1 Qj k=0 j |f (y)| m−k B1 +Φm dµ(y) λ |f (y)| Φm µ(2Qj ) + dµ(y) λ Qj j |f (y)| Φm dµ(y). λ d R

II1

m−1

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To estimate II2 , let Λ ⊂ N be a ﬁnite index set and rj (y) = λ−1 |ϕj (y)|. Using the convexity of Φk and (3.12), we have m−k |ϕj (y)| Φk b(y) − mQ %j (b) λ j∈Λ m−k rj (y) rj (y) rl (y) Φk b(y) − mQ %j (b) rl (y) j∈Λ l∈Λ

l∈Λ

m−k rj (y)Φk b(y) − mQ (b) . %j

j∈Λ

By the continuity of Φk , we further know the above estimate also holds if Λ = N. This together with some estimates similar to that for F2 in turn gives m−k II2 λ−1 ϕj L∞ (µ) Φk b(y) − mQ (b) dµ(y) %j Rj

j

λ−1

j

−1

λ

ϕj L∞ (µ)

Rj

m−k m−k − m (b) (b) 1 + dµ(y) b(y) − mQ b(y) %j %j Q

ϕj L∞ (µ) µ(Rj )

j

λ−1

Rd

|f (y)| dµ(y),

which completes the proof of Theorem 3.5.

When m = 1, we can further prove that Mb, 1 is bounded from H (µ) to L1, ∞ (µ) by using Theorem 2.1. Moreover, we can prove this without using the Lp (µ)-boundedness condition of Mb, 1 for 1 < p < ∞. In the sequel, we denote Mb, 1 simply by Mb . 1

Theorem 3.6. Let K satisfy (1.4), (3.1) with m = 1, and b ∈ RBMO(µ). If M is bounded on L2 (µ), then the commutator Mb is also bounded from H 1 (µ) to L1, ∞ (µ), namely, there exists a constant C > 0 such that for all λ > 0 and for all functions f ∈ H 1 (µ), µ({x ∈ Rd : Mb (f )(x) > λ}) ≤ CbRBMO(µ) λ−1 f H 1 (µ) . In order to prove Theorem 3.6, let us ﬁrst recall the following characterization of the Hardy space H 1 (µ) established in [9]. Lemma 3.7. Let ρ > 1 and γ ∈ N. A function f ∈ H 1 (µ) if and only if ∞ f= bi ,

(3.13)

i=1

where bi is a (∞, γ)-atomic block, which means that b ∈ L1loc (µ) and satisfies (i) and (ii) of Definition 1.1, and

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(iii) for j = 1, 2, there are functions aj supported on cubes Qj ⊂ R and numbers λj ∈ R such that b = λ1 a1 + λ2 a2 , and −γ , aj L∞ (µ) ≤ [µ(ρQj )]−1 SQj , R and if we let then

i

|b|H 1, ∞

atb, γ (µ)

|bi |H 1, ∞

atb, γ (µ)

< ∞. Moreover, f H 1 (µ) ∼ inf

= |λ1 | + |λ2 |,

i

|bi |H 1, p

atb, γ (µ)

,

where the infimum is taken over all the possible decompositions of f in (∞, γ)atomic blocks as in (3.13). Proof of Theorem 3.6. Again we may assume that bRBMO(µ) = 1. By Lemma 3.7 with γ = 2 and ρ = 4, we have the decomposition hj , f= j

where hj ’s are (∞, 2)-atomic blocks, supp (hj ) ⊂ Rj , and |hj |H 1, ∞ (µ) ≤ 2f H 1 (µ) . j

atb, 2

Moreover, by Lemma 3.7, we can further write hj as hj (x) = rj1 a1j (x) + rj2 a2j (x), where rji ∈ R for i = 1, 2, |hj |H 1, ∞ (µ) = |rj1 | + |rj2 |, aij for i = 1, 2 is a bounded atb, 2

function supported on some cube Qij ⊂ Rj and satisﬁes 2 −1 . aij L∞ (µ) ≤ µ(4Qij ) SQij , Rj Write

Mb (f )(x) ≤ mR b(x) − mR %j (b)M(hj )(x) + M %j (b) − b hj (x) j

j

= I15 (x) + I25 (x).

The (L1 (µ), L1, ∞ (µ))-boundedness of M from Theorem 2.1 states that ∞ 1

d 2 µ x ∈ R : I5 (x) > λ b(x) − mR %j (b) |hj (x)| dµ(x) λ j=1 Rj 1 1 1 |rj | b(x) − mR %j (b) |aj (x)| dµ(x) λ j Rj 2 |a +|rj2 | (b) (x)| dµ(x) b(x) − mR %j j Rj

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Marcinkiewicz Integrals

=

235

1 1 Uj + U2j . λ j

The inequality (2.10) in [21] via a trivial computation leads to 1 dµ(x) + U1j ≤ |rj1 |a1j L∞ (µ) (b) (b) − m (b) ) m b(x) − mQ µ(Q %1 %j %1 j R Q j j Q1j 2 −1 µ(2Q1j ) + SQ1j , Rj µ(Q1j ) |rj1 | µ(4Qij ) SQij , Rj 1 |rj |. Similarly, U2j |rj2 |. We thus have that µ

x ∈ Rd : I25 (x) > λ λ−1 f H 1 (µ) .

Now we deal with the estimate for I15 (x). Write

µ x ∈ Rd : I15 (x) > λ ≤ λ−1 (b) M(hj )(x) dµ(x) b(x) − mR %j 2Rj j +λ−1 (b) b(x) − mR M(hj )(x) dµ(x) %j j

Rd \2Rj

= V1 + V2 . As in the estimate for the term G in the proof of Theorem 3.1, we obtain V2 λ−1 hj L1 (µ) λ−1 hj H 1 (µ) . It remains to estimate the term V1 . For each ﬁxed j, write b(x) − mR %j (b) M(hj )(x)| dµ(x) 2Rj ≤ |rj1 | (b) M(a1j )(x) dµ(x) b(x) − mR %j 2Rj

+|rj2 |

2Rj

(b) M(a2j )(x) dµ(x). b(x) − mR %j

Since the two terms in the right hand of the last inequality can be estimated in the same way, we only deal with the ﬁrst one. Write (b) M(a1j )(x) dµ(x) b(x) − mR %j 2Rj ≤ (b) M(a1j )(x) dµ(x) b(x) − mR %j 2Rj \2Q1j

+

2Q1j

b(x) − m2Q1 (b)M(a1j )(x) dµ(x) j

236

Hu, Lin and Yang

+m2Q1 (b) − mR (b) %j

2Q1j

j

IEOT

M(a1j )(x) dµ(x)

= Wj1 + Wj2 + Wj3 . The H¨older inequality together with (2.10) in [21] and the L2 (µ)-boundedness of M now gives us that 1/2 1/2 SQ1j , Rj a1j L2 (µ) µ(2Q1j ) 1. Wj3 SQ1j , Rj M(a1j )L2 (µ) µ(2Q1j ) Employing the L2 (µ)-boundedness of M, the H¨older inequality and (3.10) in [21], we have 1/2 2 M(a1j )L2 (µ) Wj2 b(x) − m2Q1 (b) dµ(x) 2Q1

j

j 1/2 1/2 µ(4Q1j ) a1j L∞ (µ) µ(Q1j ) 1.

Set N1 = N2Q1j , 2Rj . A straightforward computation via the estimate (2.4), and (2.10) and Lemma 2.1 in [21] proves that |b(x) − mR %j (b)| dµ(x) Wj1 a1j L∞ (µ) µ(Q1j ) |x − xQ1j |n 2Rj \2Q1j N |b(x) − m k+1 (b)| 1 2 Q1j 1 1 aj L∞ (µ) µ(Qj ) dµ(x) |x − xQ1j |n 2k+1 Q1j \2k Q1j k=1 1 +m k+1 (b) − mR dµ(x) %j (b) n 2 Q1j 2k+1 Q1j \2k Q1j |x − xQ1j | N1 k+2 1 k+1 1 Q ) µ(2 Q ) µ(2 j j n + S2Q1j , 2Rj n a1j L∞ (µ) µ(Q1j ) l(2k+2 Q1j ) l(2k+1 Q1j ) k=1 2 a1j L∞ (µ) µ(Q1j ) SQ1j , Rj 1. We ﬁnally obtain

µ({x ∈ Rd : I15 (x) > λ}) λ−1 f H 1 (µ) ,

and then conclude the the proof of Theorem 3.6.

References [1] R. A. Admas, Sobolev spaces, Academic Press, New York, 1975. [2] A. Al-Salman, H. Al-Qassem, L. C. Cheng and Y. Pan, Lp bounds for the function of Marcinkiewicz, Math. Res. Lett. 9 (2002), 697–700.

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[3] D. Deng, Y. Han and D. Yang, Besov spaces with non-doubling measures, Trans. Amer. Math. Soc. 358 (2006), 2965–3001. [4] Y. Ding, S. Lu and Q. Xue, Marcinkiewicz integral on Hardy spaces, Integral Equations Operator Theory 42 (2002), 174–182. [5] Y. Ding, S. Lu and P. Zhang, Weighted weak type estimates for commutators of the Marcinkiewicz integrals, Sci. China Ser. A 47 (2004), 83–95, [6] D. Fan and S. Sato, Weak type (1, 1) estimates for Marcinkiewicz integrals with rough kernels, Tˆ ohoku Math. J. (2) 53 (2001), 265–284. [7] Y. Han and D. Yang, Triebel-Lizorkin spaces with non-doubling measures, Studia Math. 162 (2004), 105–140. [8] G. Hu, Lp (Rn ) boundedness for a class of g-functions and applications, Hokkaido Math. J. 32 (2003), 497–521. [9] G. Hu, Y. Meng and D. Yang, New atomic characterization of H 1 space with nondoubling measures and its applications, Math. Proc. Camb. Phil. Soc. 138 (2005), 151–171. [10] G. Hu, Y. Meng and D. Yang, Multilinear commutators of singular integrals with non doubling measures, Integral Equations Operator Theory 51 (2005), 235–255. [11] G. Hu and D. Yan, On the commutator of the Marcinkiewicz integral, J. Math. Anal. Appl. 283 (2003), 351–361. [12] Y. Jiang, Spaces of type BLO for non-doubling measures, Proc. Amer. Math. Soc. 133 (2005), 2101–2107. [13] J. Marcinkiewicz, Sur quelques int´ egrales du type de Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50. [14] F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calder´ on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1997, no. 15, 703–726. [15] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calder´ on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, no. 9, 463–487. [16] F. Nazarov, S. Treil and A. Volberg, Accretive system T b-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2002), 259–312. [17] F. Nazarov, S. Treil and A. Volberg, The T b-theorem on non-homogeneous spaces, Acta Math. 190 (2003), 151–239. [18] C. P´erez and G. Pradolini, Sharp weight endpoint estimates for commutators of singular integrals, Michigan Math. J. 49 (2001), 23–37. [19] N. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions, Studia. Math. 135 (1999), 103–142. [20] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466. on-Zygmund operators for non doubling measures, [21] X. Tolsa, BMO, H 1 and Calder´ Math. Ann. 319 (2001), 89–149. [22] X. Tolsa, Littlewood-Paley theory and the T (1) theorem with non-doubling measures, Adv. Math. 164 (2001), 57–116.

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[23] X. Tolsa, A proof of weak (1,1) inequality for singular integrals with non doubling measures based on a Calder´ on-Zygmund decomposition, Publ. Mat. 45 (2001), 163– 174. [24] X. Tolsa, The space H 1 for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc. 355 (2003), 315–348. [25] X. Tolsa, Painlev´e’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105–149. [26] A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235–243. [27] H. Wu, On Marcinkiewicz integral operators with rough kernels, Integral Equations Operator Theory 52 (2005), 285–298. [28] A. Zygmund, Trigonometric series, 3rd Edition, Cambridge University Press, Cambridge, 2002. Guoen Hu Department of Applied Mathematics University of Information Engineering Zhengzhou 450002 People’s Republic of China e-mail: [email protected] Haibo Lin and Dachun Yang School of Mathematical Sciences Beijing Normal University Beijing 100875 People’s Republic of China e-mail: [email protected] [email protected] Submitted: July 20, 2005 Revised: February 16, 2007

Integr. equ. oper. theory 58 (2007), 239–253 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020239-15, published online April 14, 2007 DOI 10.1007/s00020-007-1485-1

Integral Equations and Operator Theory

Elements of the Theory of Linear Volterra Operators in Banach Spaces Elena Litsyn Abstract. The new deﬁnition of Volterra operator introduced in [5] allows speciﬁcation of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations. Mathematics Subject Classiﬁcation (2000). Primary 47A05; Secondary 47B38. Keywords. Volterra operators, memory of operators.

1. Introduction We attempt considering some divisions in the theory of Linear Equations in Banach Spaces from the view point of Volterra operators. As far as we know, such an attempt has not been made before. The reason might be that the earlier existed deﬁnitions of Volterra type operator did not allow extracting the speciﬁcs of such operators in the frames of the mentioned theory. The history of Volterra operator traces back to a Volterra’s paper of 1913, where he studied an integro-diﬀerential equation with the integral operator t K(t, s)x(s)ds. (Kx)(t) = a

Afterwards operators of such type appeared in a more general form in works by Tonelli (1929) and Tikhonov (1938). The deﬁnition of Volterra operator introduced by Tikhonov is very easy to grasp: An operator is Volterra if any two functions coinciding on an interval [a, t] have equal images on [a, t], t ∈ [a, b]. Inspired by the success of the notion of Volterra operator in study of equations e.g. in the space of continuous functions, researchers dealing with equations in

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abstract spaces were tempted to introduce some equivalent of this deﬁnition into their considerations. This led to a series of works providing a notion reminiscent of the Volterra operator in each particular situation. Since the notion of Volterra operator appeared simultaneously in several areas of mathematics there is no common accepted body of deﬁnitions and terminology. Even the term itself - Volterra operator - is not always used, other ones are: Volterra type, delay, hereditary, causal, non-anticipative operator, etc. Usually the deﬁnitions are based on such important properties of the Volterra operator as evolutionarity, compactness and quasi-nilpotence. The singled out classes of Volterra type operators were based on one of the mentioned above properties, or on their combination. Mainly, some of the authors addressed the compactness and quasi-nilpotence properties of this operator, while others concentrated on its evolutionary side. However, all these classes preserved the name ”Volterra” operator (or one of the equivalent terms as it has been mentioned above). A brief review of results concerning the considered operators can be found, for example, in [5] (see also [1], §2.4). In our opinion, all the previous approaches are deﬁcient in the following sense: • They use essentially the topology of the underlying space, e.g. employ completeness, convergence, etc. • Being formulated for a particular space, they do not cover all the situations described by the original deﬁnition (for example, operator (F x)(t) = x(t/2) is not Volterra according to [6], but is Volterra in the original sense for the interval [0, a], a ∈ R+ ). • Apparently, there is no clear way of extending the introduced notions to spaces other than the ones under the particular consideration. In [5], basing on the notions of operator’s memory and chain we singled out a class of operators possessing the evolutionary property, which we call Volterra. The deﬁnition we gave requires only existence of a σ-algebra on a metric space and it avoids the aforementioned shortcomings. Our approach stems mainly from the initial considerations of Volterra-Tonelli-Tikhonov, i.e. uses mainly the evolutionary nature of the Volterra operator. The present paper is organized as follows. In Section 2 we recall the deﬁnitions related to the notion of memory [4]. In Section 3, following [5], we single out a class of Volterra operators, basing on the notions of operator’s memory and chain. We also provide some useful illustrative examples. In Section 4 the correct and normal solvability of linear equations with Volterra operators is under consideration. Section 5 is devoted to the conjugate equation to a linear equation with Volterra operator, and some results connecting the properties of the initial and the conjugate equations are established. Finally, in Section 6, we derive some conditions under which a linear equation with Volterra operator is Fredholm.

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2. Notation and preliminaries Let (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) be two measure spaces, and Σ01 ⊂ Σ1 , Σ02 ⊂ Σ2 ˜ i := Σi /Σ0 , be the σ-ideals of µ1 - and µ2 -nullsets respectively. We denote by Σ i ˜ i (i.e. i = 1, 2, the respective measure algebras (see § 42 of [8]). The elements of Σ the equivalence classes of sets) will be denoted e˜i or [ei ], i = 1, 2. Further on we will however frequently abuse the notation and identify the elements of the measure ˜ i with the elements of the respective original σ-algebras of sets Σi . algebras Σ A measure space (Ω, Σ, µ) is called standard, if Ω is a Polish space, Σ is either the Borel σ-algebra or its completion with respect to ﬁnite or σ-ﬁnite Borel measure µ. By X(Ω, Σ, µ; Y) we will understand a linear space of measurable functions, deﬁned on Ω and taking values in linear space Y. A topology in X will be deﬁned explicitly depending on the particular problem under consideration. Further, the notation Lp (Ω, Σ, µ; Y), where Y is a separable Banach space, will stand, as usual, for the classical Lebesgue space of Y-valued functions measurable with respect to Σ and µ-summable with power p (if p ∈ [1, +∞)) or µessentially bounded (if p = +∞). These spaces are silently assumed to be equipped with their strong topologies. Whenever there is no possibility of confusion, the references to Y, Ω, Σ and/or µ will be omitted. We will also omit in sequel the sign (˜·) , assuming that all the considerations are modulo the equivalence classes of sets. Let Xi := X(Ωi , Σi , µi ; Yi ), i = 1, 2. Consider an operator T : X1 → X2 . Following [4] (see also [2], [3]) we introduce now the concept of memory and the related concept of comemory. Deﬁnition 2.1. We call the memory of an operator T : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) on a set e2 ∈ Σ2 , the family of all possible e1 ∈ Σ1 such that for any x, y ∈ X1 satisfying x |e1 = y |e1 it follows that T (x) |e2 = T (y) |e2 . In other words, MemT (e2 ) := {e1 ∈ Σ1 : x |e1 = y |e1 ⇒ T (x) |e2 = T (y) |e2 } . Similarly, the comemory of operator T on a set e1 ∈ Σ1 is the family ComemT (e1 ) := {e2 ∈ Σ2 : x |e1 = y |e1 ⇒ T (x) |e2 = T (y) |e2 } . Recall that according to our convention all the equalities in the above deﬁnition should be understood in almost everywhere sense. It is clear from the deﬁnitions that e1 ∈ MemT (e2 ) ⇐⇒ e2 ∈ ComemT (e1 ).

(2.1)

Properties of memory and comemory along with some examples helping to understand deeper the deﬁnitions given above, could be found in [4].

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3. Volterra Operator In this section, basing on the notions of chain and memory, we single out a class of operators possessing the evolutionary property. Deﬁnition 3.1. A collection of subsets {eν }, eν ∈ Σ, ν ∈ [0, ∞], in a measure space (Ω, Σ, µ) is said to be chain if the following conditions are satisﬁed: 1. µ(e0 ) = 0; 2. eν1 ⊂ eν2 if ν1 ≤ ν2 ; 3. for every α ∈ (0, µ(Ω)) there exists a set eβ ∈ {eν } such that µ(eβ ) = α. Remark 3.1. Let {e1ν } and {e2ν }, ν ∈ [0, ∞], be two chains in the space (Ω, Σ, µ). Then the collection of subsets {e1ν ∪ e2ν } is also a chain in the same space. It is clear, that the union of a countable number of chains is not, in general, a chain. Example 3.1. 1. Ω = [0, 1], {eν } = {[0, ν]}, ν ∈ [0, 1]. 2. Ω = [0, 1], {eν } = {[1 − ν, 1]}, ν ∈ [0, 1]. 3. Ω = [0, 1], {eν } = {[ 12 − ν, 12 + ν]}, ν ∈ [0, 12 ]. 4. Ω = [0, 1], {eν } = {[0, ν] ∪ [1 − ν, 1]}, ν ∈ [0, 12 ]. Deﬁnition 3.2. An operator T : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ), is called Volterra (this will be denoted T ∈ V ), if there exists a pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 , such that for every member e2α of the chain {e2λ }, the corresponding element e1α of the chain {e1ν } satisﬁes e1α ∈ MemT (e2α )

(3.1)

The correspondence between the pair of chains here is provided by the same lower index α. Remark 3.2. Taking into account (2.1), the inclusion e1α ∈ MemT (e2α ) in the above deﬁnition can be replaced by an equivalent one: e2α ∈ ComemT (e1α )

(3.2)

Remark 3.3. Let operator T1 : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e11ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 ,, and operator T2 : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e12ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . Then, by Remark 3.1, operator T = (T1 + T2 ) : X1 → X2 is Volterra with respect to the pair of chains {e11ν ∪ e12ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . Remark 3.4. Let operator T1 : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 , and operator T2 : X2 (Ω2 , Σ2 , µ2 ; Y2 ) → X3 (Ω3 , Σ3 , µ3 ; Y3 ) be Volterra with respect to the pair of chains {e2λ } ⊂ Σ2 , {e3δ } ⊂ Σ3 . Then evidently operator T2 T1 : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X3 (Ω3 , Σ3 , µ3 ; Y3 )

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{e3δ } ⊂ Σ3 .

Example 3.2. 1. Ti : X1 ([0, 1], Σ, m; Y1 ) → X2 ([0, 1], Σ, m; Y2 ), i = 1, 2, 3, 4, 5, 6, where t (T1 x)(t) = K(t, s)x(s)ds, t ∈ [0, 1]; (T2 x)(t) =

0

1

K(t, s)x(s)ds, 1−t

t ∈ [0, 1];

(T3 x)(t) = B(t)x(g(t)),

t ∈ [0, 1], g(t) ≤ t, x(ζ) = 0 if ζ < 0;

(T4 x)(t) = B(t)x(τ (t)),

t ∈ [0, 1], τ (t) ≥ t, x(ζ) = 0 if ζ > 1;

(T5 x)(t) = B(t)x(1 − t), t ∈ [0, 1]; t (T6 x)(t) = K(t, s)x(1 − s)ds, t ∈ [0, 1]. 0

Here the operators T1 and T3 are Volterra with respect to the chain pair {[0, t]} and {[0, t]}; T2 and T4 are Volterra with respect to {[1 − t, 1]} and {[1 − t, 1]}, and, ﬁnally, T5 and T6 are Volterra with respect to {[1 − t, 1]} and {[0, t]}. 2. Si : X1 ([0, 1], Σ1 , m; Y1 ) → X2 ([0, 12 ], Σ2 , m; Y2 ), i = 1, 2; 12 +t 1 (S1 x)(t) = K(t, s)x(s)ds, t ∈ [0, ]; 1 2 −t 2

(S2 x)(t) =

0

t

K1 (t, s)x(s)ds +

1 2 +t 1 2

K2 (t, s)x(s)ds,

1 t ∈ [0, ]. 2

Here S1 is Volterra with respect to the pair of chains {[ 12 − t, 12 + t]} ∈ Σ1 and {[0, t]} ∈ Σ2 , and S2 is Volterra with respect to the pair of chains {[0, t] ∪ [ 12 , 12 + t]} ⊂ Σ1 and {[0, t]} ⊂ Σ2 . Examples of nonlinear Volterra operators can be found in [5].

4. Correct and Normal Solvability Let X1 (Ω1 , Σ1 , µ1 ; Y1 ) be a Banach space and assume that a linear operator L is deﬁned on some linear manifold D(L) in X1 and that L takes D(L) into a Banach space X2 (Ω2 , Σ2 , µ2 ; Y2 ). The set D(L) is called the domain of the operator L. Consider the equation Lx = f, (4.1) where f is a given element of X2 , and x is the unknown elemant in D(L). The collection of all y ∈ X2 such that equation (4.1) is solvable is a linear manifold in X2 , called the range R(L) of the operator L. The collection of all the solutions to the corresponding homogeneous equation Lx = 0 is a linear manifold in X1 called

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the null-space or the kernel N (L) of the operator L. [Sometimes the range of the operator L is denoted by ImL (image of L), and its kernel - by KerL.] Equation (4.1) is said to be uniquely solvable on R(L) if the homogeneous equation Lx = 0 has only the null solution, i.e., if N (L) = 0. In this case , for each f ∈ R(L), there is only one solution of the equation Lx = f , and so the operator L has an inverse L−1 on R(L) : L−1 f = x (f ∈ R(L)). Equation (4.1) is said to be correctly solvable on R(L) if the inequality

x X1 ≤ k Lx X2 holds for all x ∈ D(L), where k > 0 and does not depend on x. Correct solvability implies unique solvability. If equation (4.1) is correctly solvable, then the operator L has a bounded inverse on R(L). Equation (4.1) is normally solvable if R(L) is a (closed!) subspace of X2 : R(L) = R(L). Equation (4.1) is densely solvable if R(L) is dense in X2 : R(L) = X2 . Equation (4.1) is everywhere solvable if R(L) = X2 . A linear operator L is called closed if whenever xn → x and Lxn → f we have x ∈ D(L) and Lx = f. Deﬁnition 4.1. We say that a space X possesses property X (and write X ∈ X ) if (∀e ∈ Σ)(∀x ∈ X), the function xe deﬁned by xe (t) = χe (t)x(t),

t ∈ Ω,

(4.2)

also belongs to X. Here χe is the characteristic function of e. Let X ∈ X , e ∈ Σ. Choose a subspace Xe of X as follows: to every function x ∈ X we correspond the function xe ∈ Xe deﬁned by (4.2). Let us deﬁne by Le the reduction of L : X1 → X2 to the subspace X1e . Everywhere below we assume that the space X1 , where the operator L is deﬁned, possesses the property X . Theorem 4.1. Let the closed operator L : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . Then the following statements are equivalent: 1. there exist integers r > 0, δ > 0 such that for any e ∈ Σ1 , µ1 (e) ≤ δ the following inequality holds: (∀x ∈ D(L))

xe X1 ≤ r Lxe X2 ;

(4.3)

2. the operator L is correctly solvable; 3. the operator L is uniquely and normally solvable. Proof. The validity of the implications 2. ⇔ 3. is a well-known fact in the theory of linear operators in Banach spaces (see, for example, [7]). In virtue of the deﬁnitions of the solvability, the implication 2. ⇒ 1. is also true. Thus, to complete the proof it is enough to show that 1. implies 3. (1. ⇒ 3.).

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Let us choose such an element e1α of the chain {e1ν } that µ1 (e1α ) ≤ δ, then for any x ∈ D(L) ⊂ X1 condition 1. implies that

xe1α X1 ≤ r Lxe1α X2 . This means that operator Le1α : X1eα → X2 is correctly solvable. Since, in virtue of the Volterra property, (∀x ∈ D(L)) (Lx)(t) = (Le1α xe1α )(t), then is uniquely solvable on

e2α

(Lx)(t) = f (t), and

t ∈ e2α ,

t ∈ e2α ,

[R(L)]e2α = [R(Le1α )]e2α = [R(L)]e2α . Now let us take an element e1β ∈ {e1ν }, satisfying the ineualities 0 < µ1 (e1β \ e1α ) ≤ δ. The correct solvability, in virtue of 1., of the operator Le1β \e1α : X1e1β \e1α → X2 together with the unique solvability of the operator Le1α allows to conclude on the unique solvability of the operator Le1β : X1e1β → X2e2β . Indeed, let x1e1β and x2e1β are two solutions. Then

x1e1β − x2e1β X1 = x1e1α + x1e1β \e1α − x2e1α − x2e1β \e1α X1 = x1e1β \e1α − x2e1β \e1α X1 ≤ r L(x1e1β \e1α − x2e1β \e1α ) X2 = 0. The normal solvability of the operator Le1β : X1e1β → X2e2β follows from the sequence of the equalities [R(L)]e2β = [R(Le1β )]e2β = [R(Le1α )]e2β ∪ [R(Le1β \e1α )]e2β = [R(Le1α )]e2 ∪ [R(Le1β \e1α )] β

= [R(Le1β )]

e2β

e2β

= [R(L)]e2 . β

Continuing this process we will obtain the validity of the implication 1. ⇒ 3.

Corollary 4.1. Let the closed operator L : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ∈ Σ1 , {e2λ } ∈ Σ2 . Then for the normal solvability of (4.1) it is necessary and suﬃcient that the following condition holds: there exist integers r > 0 and δ > 0 such that for every y ∈ R(L) there exist and element x ∈ D(L) such that y = Lx and for any set e ∈ Σ1 µ1 (e) ≤ δ,

xe X1 ≤ r Lxe X2 . Moreover r, δ do not depend on y ∈ R(L). Proof. To prove the statement it is convenient to call on the factored equation. Since the kernel N (L) of the operator L is closed, one can take the quotient space X1 /N (L) whose elements are cosets of the elements x ∈ X1 relative to the subspace N (L) : Ξ = {x + z} (z ∈ N (L)). The quotient space is a Banach space relative to the norm

Ξ = inf x − z . z∈N (L)

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In a natural way, the operator L induces a linear operator L˜ deﬁned on D(L)/N (L) by the formula: ˜ = Lx (x ∈ D(L)). LΞ The kernel of L˜ reduces to zero, i.e., the factored or quotient equation ˜ =f LΞ

(4.4)

is uniquely solvable. On the other hand, equations (4.1) and (4.4) are simultane˜ Reference to Theorem 4.1 completes ously normally solvable or not (R(L) = R(L).) the proof. Theorem 4.2. Let the closed operator L : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . If there exists an integer δ > 0 such that for any set e ∈ Σ1 , µ1 (e) ≤ δ the space X2 decomposes into a direct sum X2 = R(Le ) ⊕ Z e , where Z e is closed linear manifold of X2 , then (4.1) is normally solvable. Proof. Choose an element e1α of the chain {e1ν } from the condition µ1 (e1α ) ≤ δ. The operator Le1α : X1 → X2 satisﬁes all the conditions of the well-known in the theory of linear equations in Banach spaces (see, for example Theorem 2.4, p. 11, [7]). This theorem allows to conclude that R(Le1α ) = R(Le1α ). Further, choose e1β ∈ {e1ν }, X1 → X2 the equalities

0 < µ1 (e1β \ e1α ) ≤ δ. Again, for the operator Le1β \e1α : R(Le1β \e1α ) = R(Le1β \e1α )

are true. In virtue of the Volterra property R(Le1β ) = R(Le1α ) ∪ R(Le1β \e1α ). Thus, R(Le1β ) = R(Le1β ). Continuing this process we obtain that R(L) = R(L), i.e. equation (4.1) is normally solvable.

5. Dense and Everywhere Solvability Let a collection of subsets {eν }, eν ∈ Σ, ν ∈ [0, ∞], be a chain in a measure space (Ω, Σ, µ), µ(Ω) < ∞. Let us deﬁne by ν¯ the exact upper bound of the values of ν and assume that ν¯ < ∞.

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Deﬁnition 5.1. A chain {e∗ν ∗ }, e∗ν ∗ ∈ Σ, ν ∗ ∈ [0, ∞] in a measure space (Ω, Σ, µ) is called the dual chain to the chain {eν } if the following equality e∗ν ∗ = Ω\eν ,

ν ∗ = ν¯ − ν.

(5.1)

holds. The dual chain will be denoted {e∗ν ∗ } := {eν }∗ . Assumption 5.1. Everywhere below we assume that the domain of the operator L : X1 → X2 coincides with the whole space (D(L) = X1 ). Theorem 5.1. Let Y1 and Y2 be reﬂexive Banach spaces. If a linear bounded operator L : Lp (Ω1 , Σ1 , µ1 ; Y1 ) → Lq (Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p, q < ∞, is Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 , then the conjugate operator ∗ ∗ L∗ : Lq (Ω2 , Σ2 , µ2 ; Y2∗ ) → Lp (Ω1 , Σ1 , µ1 ; Y1∗ ) is Volterra with respect to the pair of chains {e2λ }∗ ⊂ Σ2 , {e1ν }∗ ⊂ Σ1 . Here 1p + p1∗ = 1, 1q + q1∗ = 1. Proof. Consider the identity < (Lx)(s2 ), y ∗ (s2 ) >2 dµ2 (s2 ) = Ω2

Ω1

< x(s1 ), (L∗ y ∗ )(s1 ) >1 dµ1 (s1 ), (5.2)

∗

where x ∈ Lp (Ω1 , Σ1 , µ1 ; Y1 ), y ∗ ∈ Lq (Ω2 , Σ2 , µ2 ; Y2∗ ) and < ·, · >i , i = 1, 2, is the natural duality between the spaces Yi and Yi∗ . Let x ∈ Lp (Ω1 , Σ1 , µ1 ; Y1 ), x(s1 ) = 0, s1 ∈ e1α , e1α ∈ {e1ν }, α ∈ (0, ν¯); ∗ ∗ y ∈ Lq (Ω2 , Σ2 , µ2 ; Y2∗ ), y ∗ (s2 ) = 0, s2 ∈ Ω2 \e2α . The fact that L is Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 implies that (Lx)(s2 ) = 0, s2 ∈ e2α . Then the integral in the l.h.s of the identity (5.2) equals zero for any ∗ 2 ∗ function y ∗ ∈ Lq (Ω2 , Σ2 , µ2 ; Y2∗ ) and any element of the chain e2∗ α∗ ∈ {eλ } . This ∗ 2 2∗ ∗ ∗ means that the equality y (s2 ) = 0, s2 ∈ Ω2 \eα = eα∗ implies (L y )(s1 ) = 0, s1 ∈ Ω1 \e1α = e1∗ α∗ . The theorem is proved. Taking into account the symmetry of (5.2) and the equalities {{e1ν }∗ }∗ = {e1ν },

{{e2λ }∗ }∗ = {e2λ },

we can formulate the following criterion. Theorem 5.2. Let Y1 and Y2 be reﬂexive Banach spaces. a linear bounded operator L : Lp (Ω1 , Σ1 , µ1 ; Y1 ) → Lq (Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p, q < ∞, is Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 iﬀ the conjugate operator ∗ ∗ L∗ : Lq (Ω2 , Σ2 , µ2 ; Y2∗ ) → Lp (Ω1 , Σ1 , µ1 ; Y1∗ ) is Volterra with respect to the pair 2 ∗ 1 ∗ of chains {eλ } ⊂ Σ2 , {eν } ⊂ Σ1 . Here 1p + p1∗ = 1, 1q + q1∗ = 1. Lemma 5.1. Let operator L : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . If there exists δ > 0 such that for any e ∈ Σ1 µ(e) < δ implies N (Le ) = 0, then N (L) = 0. Proof. To prove the lemma we will apply the scheme exploited in the proof of Theorem 2.1. Choose an element e1α ∈ {e1ν } ⊂ Σ1 from the condition 0 < µ1 (e1α ) ≤ δ.

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Then by the conditions of the lemma, the following equalities are valid: [N (L)]e1α = N (Le1α ) = 0. Now, choose an element

e1β

∈ {e1ν } ⊂ Σ1 from the condition 0 < µ1 (e1β \ e1α ) ≤ δ.

Then [N (L)]e1β = [N (L)]e1α ∪ N (Le1β \e1α ) = 0. Continuing the process we get the statement of the lemma.

Theorem 5.3. Let Y1 and Y2 be reﬂexive Banach spaces and a linear bounded operator L : Lp (Ω1 , Σ1 , µ1 ; Y1 ) → Lq (Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p, q < ∞, be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e1λ } ⊂ Σ2 . Then for equation (4.1) to be densely solvable (R(L) = Lq (Ω2 , Σ2 , µ2 ; Y2 )), it is necessary and suﬃcient that there exists δ > 0 such that (∀e ∈ Σ2 ) µ2 (e) < δ

⇒ N (L∗e ) = 0.

(5.3)

Proof. The necessity is obvious. Suﬃciency. Theorem 5.1 implies that the conjugate operator L∗ : q∗ L (Ω2 , Σ2 , µ2 ; Y2∗ ) → Lp∗ (Ω1 , Σ1 , µ1 ; Y1∗ ) is Volterra with respect to the pair of chains {e2λ }∗ ⊂ Σ2 , {e1ν }∗ ⊂ Σ1 . In virtue of Lemma 5.1, condition (5.3) implies that N (L∗ ) = 0. To complete the proof it is enough to use the following well-known criterion (see, for example, [7]): equation (4.1) is densely solvable iﬀ the conjugate equation is uniquely solvable. Theorem 5.4. Let Y1 and Y2 be reﬂexive Banach spaces and a linear bounded operator L : Lp (Ω1 , Σ1 , µ1 ; Y1 ) → Lq (Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p, q < ∞, be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e1λ } ⊂ Σ2 . Then for equation (4.1) to be everywhere solvable (R(L) = Lq (Ω2 , Σ2 , µ2 ; Y2 )), the validity of the following condition is necessary and suﬃcient: there exist r, δ > 0 such that (∀y ∗ ∈ Lq∗ (Ω2 , Σ2 , µ2 ; Y2∗ )) µ2 (e) < δ

⇒ ye∗ Lq∗ ≤ r L∗ ye∗ Lp∗ .

(5.4)

Proof. In virtue of Theorem 5.1 the conjugate operator L∗ : Lq∗ (Ω2 , Σ2 , µ2 ; Y2∗ ) → Lp∗ (Ω1 , Σ1 , µ1 ; Y1∗ ) is Volterra with respect to the pair of chains {e2λ }∗ ⊂ Σ2 , {e1ν }∗ ⊂ Σ1 . Note that the conjugate operator to the closed operator is also closed. Then, in virtue of Theorem 4.1, condition (5.4) implies that operator L∗ is uniquely and closely solvable (N (L∗ ) = 0, R(L∗ ) = R(L∗ )). The unique solvability implies, in virtue of Theorem 5.3, the dense solvability of (4.1), while the close solvability of the conjugate equation implies the normal solvability of (4.1). Thus, R(L) = Lq (Ω2 , Σ2 , µ2 ; Y2 ), i.e. equation (4.1) is everywhere solvable.

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6. Fredholm Equations Let us cite some well-known deﬁnitions and statements of the theory of linear equations in Banach spaces (see, for example, [7]), which will enable formulating the main results of this section. A normally solved equation (4.1) given by a closed operator L with the ﬁnite dimensional null-space N (L) is called n-normal. By n(L) we will denote the dimension of N (L). The defect d(L) of the operator L : X1 → X2 is the defect of the subspace R(L) (defR(L)) in X2 , i.e. the dimension of the orthogonal compliment to the subspace R(L). Having in mind Assumption 5.1, the following equalities hold: d(L) = defR(L) = dimR(L)⊥ = dimN (L∗ ) = n(L∗ ).

(6.1)

If L is closed, we say that equation (4.1) is d-normal if it is normally solvable and has a ﬁnite defect (d(L) < ∞). Note, that in virtue of Assumption 5.1, equation (4.1) is d-normal iﬀ equation L∗ y = g,

(y ∈ X∗2 ,

g ∈ X∗1 )

(6.2)

is n-normal. Equation (4.1) is called Noetherian if it is both n-normal and d-normal; the corresponding operator L is called a Noetherian operator. [In literature, these operators are sometimes called Fredholm operators. The operators L appearing in n-normal or d-normal equations (4.1) are called semi-Fredholm.] The number ind(L) = n(L) − d(L) (6.3) is called the index of the equation (4.1) or the index of the operator L. Equation (4.1) is called a Fredholm equation if it is Noetherian and has index zero. The corresponding operator L is called a Fredholm operator. Each operator L : X1 → X2 having a bounded inverse U = L−1 : X2 → X1 deﬁned on the entire space X2 is obviously Fredholm (n(t) = d(L) = ind(L) = 0). Let us now consider an operator T : X(Ω1 , Σ1 , µ1 ; Y1 ) → Lp ((Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p ≤ ∞, of the form (6.4) T = U −1 + L, where U - is a bounded operator from Lp into X, having a bounded inverse. Lemma 6.1. Let for a linear continuous operator L : X(Ω1 , Σ1 , µ1 ; Y1 ) → Lp (Ω2 , Σ2 , µ2 ; Y2 ),

µ2 (Ω2 ) < ∞,

1 ≤ p ≤ ∞,

where X is a Banach space satisfying the property X the following conditions hold: 1. There exists a pair of chains {e1ν } ⊂ Σ1 , {e2ν } ⊂ Σ2 , such that L is Volterra with respect to it; 2. (∀ε > 0)(∃δ > 0)(∀e ∈ Σ1 ) : µ1 (e) < δ ⇒ Le Xe →Lp < ε; 3. (∀e1α ∈ {e1ν })(∀t1 , t2 ∈ Ω2 \ e2α )(∀x ∈ X) : (Le1α xe1α )(t1 ) = (Le1α xe1α )(t2 ). Then operator T , deﬁned by (6.4), is Fredholm.

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Proof. As it is known (see, for example, [7]) every operator T : X1 → X2 of the form (6.4), where U - is a bounded operator from X2 into X1 , and L is completely continuous operator from X1 into X2 , is Fredholm. Thus, to prove the theorem we have to show that its’ conditions garantee the complete continuity of the operator L : X → Lp , 1 ≤ p ≤ ∞. 1. First let us consider the case 1 ≤ p < ∞. To prove the theorem it is enough to establish integral equicontinuity of the image of any bounded set from X. Fix ε, ε > 0, and using Condition 2 pick sets e1αi , i = 1, 2, ..., k, belonging to the chain {e1ν } ⊂ Σ1 , in such a way that

Le1α

i

\e1αi−1 Xe1

→L 1 αi \eαi−1

p

< ε,

i = 1, 2, ..., k.

(6.5)

Here we denote: e1α0 = e10 , e1αk = Ω1 , and it is assumed that e1αi ⊂ e1αj if 0 ≤ i ≤ j ≤ k. Let us correspond to each point t ∈ Ω2 a point t ∈ Ω2 satisfying the following condition: t ∈ e2αi \ e2αi−1 implies t ∈ e2αi \ e2αi−1 , i = 1, 2, ..., k. Let us estimate

(Lx)(t ) − (Lx)(t) Y2 , t ∈ Ω2 , for an arbitrary x ∈ X. Let t ∈ e2αi \ e2αi−1 . Since X ∈ X , (Lx)(t) = [L(xe1α

+ xe1α

i−1

i

\e1αi−1

+ xe1α

k

\e1αi )](t).

By the conditions of the theorem: (Lxe1α

i−1

(Lxe1α

k

)(t) = (Lxe1α

\e1αi )(t)

i−1

= (Lxe1α

)(t ), k

t ∈ e2αi \ e2αi−1 ,

\e1αi )(t

) t, t ∈ e2αi .

Therefore,

(Lx)(t ) − (Lx)(t) Y2 = (Lxe1α = (Le1α

i

\e1αi−1 xe1αi \e1αi−1 )(t

Thus

Ω2

=

i

\e1α

) − (Le1α

i−1

i

)(t ) − (Lxe1α

i

\e1α

i−1

)(t) Y2

\e1αi−1 xe1αi \e1αi−1 )(t) Y2

≤ 2ε x X .

(6.6)

(Lx)(t ) − (Lx)(t) pY2 dµ2 (t)

k i=1

e2α \e2α i

(Lx)(t ) − (Lx)(t) pY2 dµ2 (t) i−1

≤ (2ε x X )p µ2 (Ω2 ). To accomplish the proof we use the compactness conditions in Lp ,

1 ≤ p < ∞.

2. Now let us consider the case p = ∞. Taking into account (6.6), we obtain essup

(Lx)(t ) − (Lx)(t) Y2 t,t ∈e2α \e2α i i−1 i=1,2,...,k

=

max

(Le1α

essup

i∈(1,...,k) t,t ∈e2 \e2 α α i

i

\e1α

i−1

xe1α

i

\e1α

i−1

i−1

≤ 2ε x X .

)(t ) − (Le1α

i

\e1α

i−1

xe1α

i

\e1α

i−1

)(t) Y2

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Last estimate means that every equivalence class Lx contains the continuous function. Since for a bounded set from X this estimate holds equicontinuously, then for completeness of the proof it is enough to refer to the compactness conditions in the space of continuous functions. Remark 6.1. If we transform the equation (U −1 + L)x = f

(6.7)

using the operator U , then it takes the form (I + U L)x = U f.

(6.8)

(Here I : X → X is the identity operator.) Equation (6.8) is deﬁned in X and is equivalent to the equation (6.7), since U is bounded and invertible. Traditionally, equation of type (6.8) is called a canonical Fredholm, and the operator U , which makes the equivalent transformation of (6.7) into (6.8) is called a left equivalenceregularizer. Remark 6.2. Let operator T : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → X2 (Ω2 , Σ2 , µ2 ; Y2 ) be Volterra with respect to the pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 . Then for any α ∈ (0, µ2 (Ω2 )) one can deﬁne an operator Tα : X1α → X2α , where X1α = X1 (e1α , Σ1α , µ1 ; Y1 ), X2α = X2 (e2α , Σ2α , µ2 ; Y2 ),

(here Σiα is a restriction of the σ-algebra Σi on the set eiα , i = 1, 2) as follows: (∀x ∈ X1α , y ∈ X1 ) y(t) = x(t), t ∈ e1α ⇒ (Tα x)(t) = (T y)(t), t ∈ e2α . Everywhere below the notation {e1ν }α , {e2λ }α will stand for the parts of the chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 , belonging to the σ -algebras Σ1α , Σ2α respectively. Notice also that (∀α ∈ µ2 (Ω2 )) operator Tα : X1α → X2α is Volterra with respect to the pair of chains {e1ν }α ⊂ Σ1α , {e2λ }α ⊂ Σ2α . Let us now consider an operator T : X1 (Ω1 , Σ1 , µ1 ; Y1 ) → Lp (Ω2 , Σ2 , µ2 ; Y2 ), 1 ≤ p ≤ ∞, of the form T = P + L. (6.9) Theorem 6.1. Let for a linear continuous operator L : X(Ω1 , Σ1 , µ1 ; Y1 ) → Lp (Ω2 , Σ2 , µ2 ; Y2 ), µ2 (Ω2 ) < ∞, 1 ≤ p ≤ ∞, where X is a Banach space satisfying the property X , the conditions 1., 2. and 3. of the Lemma 6.1 are fulﬁlled. Let for a linear continuous operator P : X(Ω1 , Σ1 , µ1 ; Y1 ) → Lp (Ω2 , Σ2 , µ2 ; Y2 ) the following conditions hold: 1) P is everywhere solvable, i.e. R(P ) = Lp (Ω2 , Σ2 , µ2 ; Y2 ); 2) P is Volterra with respect to the same pair of chains {e1ν } ⊂ Σ1 , {e2λ } ⊂ Σ2 as L; 3) there exists δ > 0 such that for any e ∈ Σ1 µ1 (e) < δ implies N (P e) = 0.

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Then for any α ∈ (0, µ2 (Ω2 )] an operator Tα : X(e1α , Σ1α , µ1 ; Y1 ) → Lp (e2α , Σ2α , µ2 ; Y2 ), 1 ≤ p ≤ ∞ such that

Tα = Pα + Lα

(6.10)

is Fredholm. Proof. In virtue of Lemma 5.1 the operator Pα : X(e1α , Σ1α , µ1 ; Y1 ) → Lp (e2α , Σ2α , µ2 ; Y2 ) is invertible for every α ∈ (0, µ2 (Ω2 )]. According to the Banach theorem, the inverse operator Pα−1 : Lp (e2α , Σ2α , µ2 ; Y2 ) → X(e1α , Σ1α , µ1 ; Y1 ) is bounded. To complete the proof one just has to refer to Lemma 6.1, since for every α ∈ (0, µ2 (Ω2 )] the operator Lα : X(e1α , Σ1α , µ1 ; Y1 ) → Lp (e2α , Σ2α , µ2 ; Y2 ), 1 ≤ p ≤ ∞, satisﬁes all the conditions of the lemma. Remark 6.3. Under the conditions of Theorem 6.1 P −1 : Lp (Ω2 , Σ2 , µ2 ; Y2 ) → X(Ω1 , Σ1 , µ1 ; Y1 ) (P −1 = Pα−1 , α = µ2 (Ω2 )) is Volterra with respect to the pair of chains {e2λ } ⊂ Σ2 , {e1ν } ⊂ Σ1 . Proof. Indeed, the invertibility of the operator Pα : X(e1α , Σ1α , µ1 ; Y1 ) → Lp (e2α , Σ2α , µ2 ; Y2 ) implies that

e2α ∈ MemPα−1 (e1α ). Taking into account the deﬁnition of Pα , the last means that (∀α ∈ (0, µ1 (Ω1 )) e2α ∈ MemP −1 (e1α ). To complete the proof one has to refer to Deﬁnition 3.2.

References [1] N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, Elements of Modern Theory of functional Diﬀerential Equations. Methods and Applications. Moscow, Institute for computer studies, 2002, (in Russian) [2] M.E. Drakhlin, A. Ponosov and E. Stepanov. On Some Classes of Operators Determined by the Structure of their Memory. Proc. Edinburgh. Math. Soc., 45(2):467–490, 2002. [3] M.E. Drakhlin, E. Litsyn, A. Ponosov and E. Stepanov. Generalizing the Property of Locality: Atomic/Coatomic Operators and Applications. Journal of Nonlinear and Convex Analysis, 7(2):139–162, 2006. [4] M.E. Drakhlin and E. Litsyn. On the Memory of Atomic Operators. Journal of Nonlinear and Convex Analysis, 6(2):235–249, 2005. [5] M.E. Drakhlin and E. Litsyn. Volterra Operator: Back to the Future. Journal of Nonlinear and Convex Analysis, 6(3):375–391, 2005.

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[6] I.Z. Gohberg, M.G. Krein. Theory of Volterra Operator in Hilbert Space and its’ Applications. Moscow, Nauka, 1967. (in Russian). [7] S.G. Krein, Linear Equations in Banach Spaces, Birkh¨ auser, 1982, (Translated from Russian). [8] R. Sikorski, Boolean Algebras. Springer-Verlag, Berlin, 1960. [9] A.N. Tikhonov, On Functional Equations of Volterra Type and their Applications to Some Problems of Mathematical Physics. Bul. Of Moscow University, section A, 1(8):1–25, 1938 (in Russian). [10] L. Tonelli, Sulle Equazioni Funzionali del Tipo di Volterra, Bull. Calcula, Math. Soc., 20:31–48, 1929 (Opere scelte 4, pp. 198-212). ´ ´ [11] V. Volterra, Le¸cons sur les Equations Integrales et les Equations IntegroDiﬀerentielles, Paris, 1913. Elena Litsyn Department of Mathematics Ben Gurion University of the Negev Beer Sheva, 84105 Israel e-mail: [email protected] Submitted: December 15, 2005 Revised: December 7, 2006

Integr. equ. oper. theory 58 (2007), 255–272 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020255-18, published online April 14, 2007 DOI 10.1007/s00020-007-1492-2

Integral Equations and Operator Theory

Characterizations of Positive Linear Volterra Integro-diﬀerential Systems Toshiki Naito, Satoru Murakami, Jong Son Shin and Pham Huu Anh Ngoc Abstract. We ﬁrst give a criterion for positivity of the solution semigroup of linear Volterra integro-diﬀerential systems. Then, we oﬀer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L2 [0, +∞). Mathematics Subject Classiﬁcation (2000). Primary 34A30; Secondary 34K20. Keywords. Volterra integro-diﬀerential system, positive system, asymptotic behavior.

1. Introduction Generally speaking, a dynamical system is called positive if for any nonnegative initial condition, the corresponding solution of the system is also nonnegative. In particular, a dynamical system with state space Rn is positive if any trajectory of the system starting at an initial state in the positive orthant Rn+ remains forever in Rn+ . Positive dynamical systems play an important role in the modelling of dynamical phenomena whose variables are restricted to be nonnegative. They are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modelling transport and accumulation phenomena of substances, see e.g. [3], [7], [22]. Concrete examples of positive systems are such as an electrical circuit consisting of resistors, capacitors and voltage sources or an electrically heated oven. The mathematical theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius. As references we mention [3], [7]. The first and last author are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049. Corresponding author: Pham Huu Anh Ngoc, Email: [email protected].

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Positive systems are objects for many interesting problems in Mathematics, Physics, Economics, Biology, etc. Moreover, obtained results of problems for a class of positive systems are often very interesting, see e.g. [3], [7], [10]-[15], [22], [30]-[32], [36], [37]. In recent time, problems of positive systems have attracted a lot of attention of many researchers, see e.g. [2], [8], [10]-[13], [15], [17]-[18], [30]-[32], [35]-[37]. In the literature, there are some criteria for familiar positive linear systems such as positive linear invariant-time diﬀerential (diﬀerence) system, positive linear invariant time-delay system of retarded type. For example, it is well-known that a linear time-delay system of the form x(t) ˙ = A0 x(t) + A1 x(t − h), t ≥ 0, is positive if and only if A0 is a Metzler matrix and A1 is a nonnegative matrix and a linear discrete system of the form x(k + 1) = A0 x(k) + A1 x(k − h), k ∈ N, k ≥ h is positive if and only if A0 , A1 are nonnegative matrices, see e.g. [28], [30], [36]. Essentially, it is worth noticing that each of the above systems is positive if and only if its solution semigroup is positive. In this paper, we ﬁrst prove that the solution semigroup of a linear Volterra integro-diﬀerential system of the form t d x(t) = Ax(t) + B(t − s)x(s)ds, t ≥ 0, (1) dt −∞ x(t) = φ(t),

t ≤ 0,

φ ∈ Cl (−∞, 0],

(2)

is a Metzler matrix and B(t) ∈ R is a is positive if and only if A ∈ R nonnegative matrix for every t ≥ 0. Then, we explore asymptotic behavior of solutions of positive Volterra systems of the form (1). It is important to note that Volterra equations are studied extensively in many various areas such as : Control Theory, Optimization, Probability and Statistics, Economics... In particular, problems of stability, robust stability of Volterra equations have been studied in along time, see e.g. [1], [4]-[5], [9], [19]-[20], [23]-[27], [39]-[40]. However, to the best of our knowledge, aspects of positivity of problems of Volterra equations have not been exploited yet in the literature and the main purpose of this paper is to ﬁll this gap. The organization of the paper is as follows. In the next section, we give some notations and preliminary results which will be used in the sequel. In Section 3, we oﬀer an explicit criterion for positivity of solution semigroups of linear Volterra integro-diﬀerential systems of the form (1). Finally, in the last section, we give some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L2 [0, +∞). n×n

n×n

2. Preliminaries Let K = C or R and n, l, q be positive integers. For a complex number s, denote by s the real part of s. Inequalities between real matrices or vectors will be understood componentwise, i.e. for two real l×q-matrices A = (aij ) and B = (bij ),

Vol. 58 (2007)

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the inequality A ≥ B means aij ≥ bij for i = 1, · · · , l, j = 1, · · · , q. The set of n all nonnegative l × q-matrices is denoted by Rl×q + . If x = (x1 , x2 , ..., xn ) ∈ K and l×q we deﬁne |x| = (|xi |) and |P | = (|pij |) . It is easy to see that P = (pij ) ∈ K |CD| ≤ |C||D|. For any matrix A ∈ Kn×n the spectral radius, spectral abscissa of A is denoted respectively, by ρ(A) = max{|λ| : λ ∈ σ(A)}, µ(A) = max{Re λ : λ ∈ σ(A)}, where σ(A) := {z ∈ C : det(zIn − A) = 0} is the set of all eigenvalues of A. A norm · on Kn is said to be monotonic if |x| ≤ |y| implies x ≤ y for all x, y ∈ Kn . Every p-norm on Kn , 1 ≤ p ≤ ∞, is monotonic. In this paper, the norm M of a matrix M ∈ Kl×q is always understood as the operator norm deﬁned by M = maxy=1 M y where Kq and Kl are provided with some monotonic vector norms. Then, the operator norm · has the following monotonicity property, see e.g. [14], P ∈ Kl×q , Q ∈ Rl×q + , |P | ≤ Q ⇒ P ≤ |P | ≤ Q.

(3)

A matrix A ∈ Rn×n is called a Metzler matrix if all the oﬀ-diagonal entries of A are nonnegative. It is obvious that A ∈ Rn×n is a Metzler matrix if and only if tIn + A ≥ 0, for some t ≥ 0. The next theorem summarizes some basic properties of Metzler matrices which will be used in the next section. Theorem 2.1. [34]. Let A ∈ Rn×n be a Metzler matrix. Then (i) (Perron-Frobenius Theorem) µ(A) is an eigenvalue of A and there exists a nonnegative eigenvector x ≥ 0, x = 0 such that Ax = µ(A)x. (ii) Given α ∈ R, there exists a nonzero vector x ≥ 0 such that Ax ≥ αx if and only if µ(A) ≥ α. (iii) (tIn − A)−1 exists and is nonnegative if and only if t > µ(A). n×n (iv) Given B ∈ Rn×n . Then + ,C ∈ C |C| ≤ B

=⇒

µ(A + C) ≤ µ(A + B).

Let J be an interval of R. For a matrix function φ : J → Rm×n , the notation φ ≥ 0 means that φ(θ) ≥ 0 for every θ ∈ J. Denote by C(J, Kn ), the space of all continuous functions on J with values in Kn . To make the presentation selfcontained we present here some basic facts on vector functions of bounded variation and relative knowledge. A matrix function η(.) : [α, β] → Rl×q is called a increasing matrix function, if η(θ2 ) ≥ η(θ1 ) A matrix function η(·) : [α, β] → K

for m×n

Var(η; α, β) := sup

P [α,β] k

α ≤ θ1 ≤ θ2 ≤ β.

is said to be of bounded variation if η(θk ) − η(θk−1 ) < +∞,

(4)

where the supremum is taken over the set of all ﬁnite partitions of the interval [α, β]. The set BV([α, β], Km×n ) of all matrix functions η(·) of bounded variation on [α, β] satisfying η(α) = 0 is a Banach space endowed with the norm η = Var(η; α, β). Since all matrix norms on Km×n are equivalent, it follows that the

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matrix function η(·) = (ηij (·)) ∈ Km×n is of bounded variation if and only if each ηij (·) is of bounded variation. Given η(·) ∈ BV([α, β], Km×n ) then for any continuous functions γ ∈ C([α, β], K) and φ ∈ C([α, β], Kn ), the integrals β β γ(θ)d[η(θ)] and d[η(θ)]φ(θ) α α p exist and are deﬁned respectively as the limits of S1 (P ) := k=1 γ(ζk )(η(θk ) − p η(θk−1 )) and S2 (P ) := k=1 (η(θk )−η(θk−1 ))φ(ζk ) as d(P ) := maxk |θk −θk−1 | → 0, where P = {θ1 = α ≤ θ2 ≤ · · · ≤ θp = β} is any ﬁnite partition of the interval [α, β] and ζk ∈ [θk−1 , θk ]. It is immediate from the deﬁnition that β α γ(θ)d[η(θ)] ≤ maxθ∈[α,β] |γ(θ)| η, (5) β α d[η(θ)]φ(θ) ≤ maxθ∈[α,β] φ(θ) η. Let Kn be endowed with a vector norm · and C([α, β], Kn ) be a Banach space of all continuous functions on [α, β] with values in Kn normed by the maximum norm φ = maxθ∈[α,β] φ(θ). Let L : C([α, β], Kn ) → Kn be a linear bounded operator. Then, by the Riesz representation theorem, there exists a unique matrix function η = (ηij (·)) ∈ BV([α, β], Kn×n ) which is continuous from the left (or brieﬂy c.f.l.) on (α, β) such that β Lφ = d[η(θ)]φ(θ), ∀φ ∈ C([α, β], Kn ). (6) α

For any vector norm on Kn , we have by (5), L ≤ η. Let X be a subspace of C([α, β], Rn ). Then the operator L is called positive on X if Lφ ≥ 0, for every φ ∈ X, φ ≥ 0. In the subsequent sections the following subspace of BV([α, β], Km×n ) will be used: NBV([α, β], Km×n ) := {η ∈ BV([α, β],Km×n ); η(α) = 0, η is c.f.l. on [α, β]}. ) is closed in BV([α, β], K It is clear that NBV([α, β], K Banach space with the norm δ = Var(δ; α, β). m×n

m×n

(7)

) and thus it is a

3. An explicit criterion for positive linear Volterra integro-diﬀerential systems Consider a linear Volterra integro-diﬀerential system of the form (1), where A ∈ Rn×n is a given matrix and B : [0, +∞) → Rn×n is a given continuous matrix function. Throughout this section, we assume that B(·) ∈ L1 ([0, +∞), Rn×n ). In what follows, we will write L [0, +∞) instead of L ([0, +∞), R p

p

(8) n×n

), p ≥ 1.

Deﬁnition 3.1. A continuous function x : R → Rn is called a solution of (1) with the initial condition (2) if

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(i) x is continuously diﬀerentiable on [0, +∞) and (1) is satisﬁed for t ≥ 0, (ii) x(s) = φ(s), s ∈ (−∞, 0]. Let us deﬁne Cl (−∞, 0] := {φ ∈ C((−∞, 0], Rn ) :

lim φ(s) ∈ Rn }.

s→−∞

Then, Cl (−∞, 0] is a Banach space endowed with the supremum norm. It is important to note that under the condition (8), the initial value problem (1)-(2) has a unique solution x(·, φ) on R, for every φ ∈ Cl (−∞, 0], see e.g. [6], [21]. We associate the system (1)-(2) with a semigroup of solution operator on Cl (−∞, 0]. The semigroup is strongly continuous and is given by translation along the solution of (1)-(2): T (t)φ := xt (·; φ), t ≥ 0, where xt (s; φ) := x(t + s; φ), s ∈ (−∞, 0]. For further information and details, see [1]. Let us denote +∞ e−λt B(t)dt), (9) ∆(λ) := (λIn − A − 0

for appropriate λ ∈ C. The following was found in [1]. Theorem 3.2. (T (t))t≥0 is a C0 −semigroup on Cl (−∞, 0]. The inﬁnitesimal generator A of the semigroup is given by dφ , s ∈ (−∞, 0], Aφ = ds where 0 dφ dφ D(A) = φ ∈ Cl (−∞, 0] : ∈ Cl (−∞, 0], (0) = Aφ(0)+ B(−s)φ(s)ds . ds ds −∞ Moreover, the resolvent of A is given by 0

B(−σ) R(λ, A)g (s) = eλs ∆−1 (λ) g(0) + +e

λs

−∞ 0

0

e−λ(σ−τ ) g(τ )dτ ds

(10)

σ

e−λτ g(τ )dτ,

s ∈ (−∞, 0],

s

where ∆(·) is given by (9) and λ ∈ C with λ large enough. The semigroup (T (t))t≥0 is called the solution semigroup of the linear Volterra integro-diﬀerential system (1). Deﬁnition 3.3. A linear Volterra integro-diﬀerential system of the form (1) is said to be positive if its solution semigroup is a positive semigroup. Remark 3.4. Recall that the semigroup (T (t))t≥0 is called positive if, by deﬁnition, T (t)φ ≥ 0, for every φ ∈ Cl (−∞, 0], φ ≥ 0. Moreover, by the deﬁnition, it is obvious that the system (1) is positive if and only if for any initial function φ ∈ Cl (−∞, 0], φ ≥ 0 the corresponding solution x(·, φ) of (1)-(2) satisﬁes x(t, φ) ∈ Rn+ for every t ≥ 0.

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To prove a criterion for positive linear Volterra equations, we need some technical lemmas. Lemma 3.5. [32] Let h > 0 and C0 ([−h, 0], Rn ) := {φ ∈ C([−h, 0], Rn ) : φ(0) = 0}. Suppose that the linear operator L is deﬁned by 0 n n L : C0 ([−h, 0], R ) → R , φ → Lφ = d[η(θ)]φ(θ), −h

where η ∈ N BV ([−h, 0], R ) is given. Then L is a positive operator if and only if η is an increasing matrix function. n×n

Let h : [0, +∞) → R. Then the Laplace transform of h is formally deﬁned to be +∞ ˆ h(λ) := e−λt h(t)dt. +∞

0 −βt

ˆ If β ∈ R and 0 e |h(t)|dt < +∞, then h(λ) exists for λ ∈ C, λ ≥ β. ˆ Furthermore, h(λ) is an analytic function in the domain {λ ∈ C : λ > β}. If D(t) = (dij (t)) is a matrix function then we deﬁne ˆ := (dˆij ). D We now rewrite the system (1)-(2) as t 0 d x(t) = Ax(t) + B(t − s)x(s)ds + B(t − s)φ(s)ds, dt 0 −∞

t ≥ 0,

(11)

where φ ∈ Cl (−∞, 0]. Then, we associate the system (11) with the following system t d x(t) = Ax(t) + B(t − s)x(s)ds, t ≥ 0. (12) dt 0 It is well-known that there always exists a unique solution x(t, x0 ), t ≥ 0, of (12) satisfying the initial condition x(0) = x0 , for every given x0 ∈ Rn . Let Z(t) be the matrix whose columns are solutions of (12) with Z(0) = In . Then, Z(t) satisﬁes the resolvent equation t d Z(t) = AZ(t) + B(t − s)Z(s)ds, Z(0) = In , (13) dt 0 and is called the fundamental solution of the system (12), see e.g. [4], [5]. Lemma 3.6. Suppose that for every x0 ∈ Rn+ , the corresponding solution x(t, x0 ), t ≥ 0 of (12) satisﬁes x(t, x0 ) ≥ 0, ∀t ≥ 0. Then, A ∈ Rn×n is a Metzler matrix −1 ˆ and (sIn − A − B(s)) ≥ 0, for s ∈ R large enough. Proof. By the assumption, Z(t) ≥ 0, ∀t ≥ 0. It is well-known that Z is of exponential order, see [4], page 29. Therefore, taking the Laplace transforms to two sides of the equation (13), we get ˆ ˆ Z(s) = Z(0) = In , [sIn − A − B(s)]

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ˆ for s ∈ R large enough. From Z(t) ≥ 0, ∀t ≥ 0, it follows that Z(s) = (sIn − −1 ˆ A − B(s)) ≥ 0, for s ∈ R large enough. It is only to show that A is a Metzler matrix. Let A = (aij ) and assume contrary that ai0 j0 < 0 for some i0 = j0 . ˆ Since B(·) ∈ L1 [0, +∞), it follows that B(s) → 0, as s → +∞. Therefore, we can represent the following

−1 −1 ˆ ˆ = s−1 In − s−1 (A + B(s)) (sIn − A − B(s)) ˆ = s−1 In + s−2 (A + B(s)) +

+∞

k ˆ s−(k+1) (A + B(s)) ,

k=2

for s > 0 large enough. We thus get, ˆ sIn + (A + B(s)) +

+∞

k ˆ s−(k−1) (A + B(s)) ≥ 0,

(14)

k=2

for s > 0 large enough. It is important to note that lim

s→+∞

+∞

k ˆ s−(k−1) (A + B(s)) =0

k=2

Then, from (14) it follows that the entry bi0 j0 of the matrix on the left-hand side of (14) is negative for s > 0 large enough. It is a contradiction. Hence, A must be a Metzler matrix. This completes our proof. We are now in the position to prove the main result of this paper. Theorem 3.7. The system (1) is positive if and only if A is a Metzler matrix and B(t) ∈ Rn×n + , for all t ≥ 0. Proof. (⇒) Let the system (1) be positive. For a ﬁxed k ∈ N and a ﬁxed x0 ∈ Rn+ , we consider the function φk ∈ Cl (−∞, 0] deﬁned by 0 if s ∈ (−∞, − k1 ] φk (s) := (ks + 1)x0 if s ∈ [− k1 , 0]. Denote by xk (t) := x(t, φk ), t ∈ R, the solution of (1)-(2) with the initial function φk . By (11), it is easy to see that xk (t) satisﬁes the following t t xk (t) = eAt x0 + eA(t−s) gk (s)ds + eA(t−s) fk (s)ds t ≥ 0, (15) 0

where gk (s) :=

0

s

0

B(s − τ )xk (τ )dτ, s ≥ 0

and fk (s) :=

0

−∞

B(s − τ )φk (τ )dτ, s ≥ 0.

Since the system (1) is positive, it follows that xk (t) ≥ 0, ∀t ≥ 0. Let x(t) := x(t, x0 ), t ≥ 0 be the solution of (12) with the initial state x0 . Then, x(t) satisﬁes

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the following equation At

x(t) = e x0 + s

t

eA(t−s) g(s)ds,

0

t ≥ 0,

(16)

B(s − τ )x(τ )dτ, s ≥ 0. From (15)-(16), it follows that t A(t−s) s xk (t) − x(t) = e B(s − τ )(xk (τ ) − x(τ ))dτ ds

where g(s) :=

0

0

0

t

+ 0

eA(t−s) fk (s)ds

t ≥ 0,

Interchanging the order of integration in the ﬁrst integral, we get t t xk (t) − x(t) = eA(t−τ ) B(τ − s)dτ (xk (s) − x(s))ds 0

s

t

t ≥ 0.

eA(t−s) fk (s)ds,

+ 0

This implies that xk (t) − x(t) ≤

t

0

t

eA(t−τ )B(τ − s)dτ xk (s) − x(s)ds

s

+ 0

t

eA(t−s) fk (s)ds,

for every t ≥ 0. For a ﬁxed T > 0, by the continuity, there exists constants M1 , M2 > 0 such that t A(t−s) e ≤ M1 , eA(t−τ ) B(τ − s)dτ ≤ M2 , 0 ≤ s ≤ t ≤ T. s

We thus get, xk (t) − x(t) ≤

0

t

M2 xk (s) − x(s)ds +

0

t

M1 fk (s)ds,

Using Gronwall’s inequality, we derive that t M2 T xk (t) − x(t) ≤ M1 e fk (s)ds, 0

t ∈ [0, T ].

t ∈ [0, T ].

(17)

On the other hand, it is easy to see that the function sequence (fk )k uniformly converges to 0 on [0, t], as k → +∞, for every 0 < t ≤ T . Therefore, it follows from (17) that xk (t) → x(t) as k → +∞, for every t ∈ [0, T ]. Hence, x(t) = x(t, x0 ) ≥ 0, for t ∈ [0, T ]. Since T > 0 is arbitrary, we have x(t, x0 ) ≥ 0, ∀t ≥ 0. By Lemma 3.6, A ∈ Rn×n is a Metzler matrix.

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We now show that B(t) ≥ 0, ∀t ≥ 0. To do so, we ﬁx h > 0, Ψ ∈ C0 ([−h, 0], Rn ), Ψ ≥ 0, k ∈ N and then consider the function ψk ∈ Cl (−∞, 0] deﬁned by   if s ∈ (−∞, −h − 1/k) 0 ψk (s) := ksΨ(−h) + Ψ(−h)(kh + 1) if s ∈ [−h − 1/k, −h).   Ψ(s) if s ∈ [−h, 0]. It is clear that ψk ≥ 0, ∀k ∈ N. Let yk (t) := x(t, ψk (s)), t ∈ R be the solution of the solution of (1)-(2) with the initial function ψk . Then yk (t) satisﬁes the following 0 −h 0 dyk B(−s)ψk (s)ds = B(−s)Ψ(s)ds + B(−s)ψk (s)ds. (0) = dt −∞ −h −h−1/k Note that −h 1 max B(−s)ψk (s)ds ≤ Ψ(−h) B(s) → 0 as k → +∞. k s∈[−h−1,−h] −h−1/k Since the system (1) is positive, it follows that 0 −h yk (t) dyk (0) = lim = 0≤ B(−s)Ψ(s)ds + B(−s)ψk (s)ds. dt t t→0+ −h −h−1/k 0 This implies that −h B(−s)Ψ(s)ds ≥ 0, for every Ψ ∈ C0 ([−h, 0], Rn ), Ψ ≥ 0. Thus, the linear operator deﬁned by 0 n n L : C0 ([−h, 0], R ) → R , Ψ → LΨ := B(−s)Ψ(s)ds, −h

is a positive operator. Applying Lemma 3.5 to the positive operator L, we conclude that the function s

B(−τ )dτ,

η(s) = −h

s ∈ [−h, 0],

is an increasing matrix function. This gives B(t) ≥ 0, for every t ∈ [0, h]. Since h > 0 is arbitrary, it follows that B(t) ≥ 0, for every t ≥ 0. (⇐) By the standard property of a C0 −semigroup, k k k T (t)φ = lim R ,A φ, t > 0, k→∞ t t for every φ ∈ Cl (−∞, 0]. So, we only have to show that R(s, A) ≥ 0 for s ∈ R, s > 0 large enough. In view of (10), it is suﬃcient to show that ∆−1 (s) ≥ 0, for s > 0 large enough. Since A is a Metzler matrix and B(t) ≥ 0, for all t ≥ 0, it implies that +∞ A + 0 e−st B(t)dt is also a Metzler matrix for every s > 0. Taking into account +∞ the fact that lims→+∞ 0 e−st B(t)dt = 0, by the continuity of the spectral abscissa µ(X) in X, we have +∞ µ(A + e−st B(t)dt) < µ(A) + 1, for every s ≥ s1 , 0

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for some s1 > 0. Finally, it follows from Theorem 2.1(iii) that +∞ −1 −1 −st ∆(s) = sIn − A + e B(t)dt ≥ 0, 0

for every s > max{s1 , µ(A) + 1}. This completes our proof.

4. Asymptotic behavior of positive linear Volterra integro-diﬀerential systems In this section, we explore asymptotic behavior of solution of positive linear Volterra integro-diﬀerential systems of the form (1). It is important to note that, asymptotic behavior of solution of linear Volterra systems of the form (1) has been studied in [1], [9], [20], [23]. However, most of the results of these papers are still in primary forms that are not easy to verify. The main purpose of our work here is to exploit positivity of the solution semigroup of positive systems in order to simplify the main results of [1] and give an extension of one of them to perturbed systems. Consider again the linear Volterra integro-diﬀerential system of the form (1), where A ∈ Rn×n is a given matrix and B : [0, +∞) → Rn×n is a given continuous matrix function. Deﬁnition 4.1. The system (1) is said to be exponentially stable in Cl (−∞, 0] if there exist the positive numbers M, β such that x(t, φ) ≤ M e−βt φ,

t ≥ 0,

for all φ ∈ Cl (−∞, 0]. The following theorem summarizes the main results of [1]. Theorem 4.2. (i) Assume that B(·) ∈ L1 [0, +∞). Then following statements are equivalent. a) For every > 0, there is a positive number M such that T (t) ≤ M et , t ≥ 0, ˆ b) det(λIn − A − B(λ)) = 0, λ ∈ C, λ > 0. α· 1 ˆ = 0, λ ∈ (ii) If e B(·) ∈ L [0, +∞) for some α > 0 and det(λIn − A − B(λ)) C, λ > −α, then the system (1) is exponentially stable in Cl (−∞, 0]. ˆ = 0, λ ∈ C, λ ≥ 0. (iii) Suppose that B(·) ∈ L1 [0, +∞) and det(λIn − A − B(λ)) Then, for every φ ∈ Cl (−∞, 0]∩L1 (−∞, 0], the solution x(·, φ) of (1) belongs to L2 [0, +∞) and x(·, φ)L2 ≤ M (φCl (−∞,0] + φL1 ), for some M > 0. For positive systems, the above theorem can be reﬁned as follows. for every t ≥ 0. Theorem 4.3. Let A ∈ Rn×n be a Metzler matrix and B(t) ∈ Rn×n + 1 (i) Assume that B(·) ∈ L [0, +∞). Then following statements are equivalent. a) For every > 0, there is a positive number M such that T (t) ≤ M et , t ≥ 0,

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+∞ b) µ(A + 0 B(t)dt) ≤ 0. +∞ (ii) If eα· B(·) ∈ L1 [0, +∞) for some α > 0 and µ(A + 0 eαt B(t)dt) ≤ −α, then the system (1) is exponentially stable in Cl (−∞, 0]. +∞ (iii) Suppose that B(·) ∈ L1 [0, +∞) and µ(A + 0 B(t)dt) < 0. Then, for every φ ∈ Cl (−∞, 0] ∩ L1 (−∞, 0], the solution x(·, φ) of (1) belongs to L2 [0, +∞) and x(·, φ)L2 ≤ M (φCl (−∞,0] + φL1 ), for some M > 0. Proof. (i) It is only to show that ˆ = 0, λ ∈ C, λ > 0 ⇔ µ(A + det(λIn − A − B(λ))

+∞ 0

B(t)dt) ≤ 0.

ˆ In fact, if det(λIn − A − B(λ)) = 0, for some λ ∈ C, λ > 0, then taking into account the fact that A ∈ Rn×n is a Metzler matrix and B(t) ∈ Rn×n + , t ≥ 0, by Theorem 2.1(iv), we have λ > 0 and +∞ +∞ +∞ e−λt B(t)dt) ≤ µ(A+ e−λt B(t)dt) ≤ µ(A+ B(t)dt). λ ≤ µ(A+ 0

0

0

ˆ Conversely, let det(λIn − A − B(λ)) = 0, λ ∈ C, λ > 0, we show that µ(A + +∞ +∞ B(t)dt) ≤ 0. Assume contrary that µ(A + 0 B(t)dt) > 0. Consider the 0 following continuous real function +∞ f (θ) := θ − µ(A + e−θt B(t)dt), θ ≥ 0. (18) 0

From f (0) < 0 and limθ→+∞ f (θ) = +∞, it follows that f (θ0 ) = 0, for some +∞ θ0 > 0. That is, θ0 = µ(A+ 0 e−θ0 t B(t)dt). Then, by Theorem 2.1(i), det(θ0 In − +∞ A − 0 e−θ0 t B(t)dt) = 0. However, this conﬂicts with our assumption. This completes the proof of (i). The proof of (ii), (iii) can be done by the same way. Furthermore, we now show that the converse of Theorem 4.3 (ii) also holds true. Theorem 4.4. Let A be a Metzler matrix and B(t) ≥ 0, ∀t ≥ 0. Assume that B(·) ∈ L1 [0, +∞) and the system (1) is exponentially stable in Cl (−∞, 0]. Then +∞ eα· B(·) ∈ L1 [0, +∞) and µ(A + eαt B(t)dt) ≤ −α, 0

for some α > 0. Proof. Let Z(·) be the fundamental solution of the system (12). Since A is a Metzler matrix and B(t) ≥ 0, ∀t ≥ 0, it is easy to see that Z(t) ≥ 0, t ≥ 0. Therefore, Z(t) =

max

x∈Rn + ,x=1

Z(t)x,

(19)

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for every t ≥ 0, see e.g. [3]. For a ﬁxed x0 ∈ Rn+ , consider the function deﬁned by 0 if s ∈ (−∞, − k1 ] φk (s) := (ks + 1)x0 if s ∈ (− k1 , 0]. It is obvious that φk ∈ Cl (−∞, 0], for every k ∈ N. Since the system (1) is exponentially stable in Cl (−∞, 0], we have x(t, φk ) ≤ M e−βt φk ≤ M e−βt x0 ,

t ≥ 0,

(20)

for some positive numbers M, β. Let x(t, x0 ), t ≥ 0 be the solution of the system (12) with the initial state x0 . It is important to note that in the proof of Theorem 3.7, we showed that x(t, φk ) → x(t, x0 ) as k → +∞, for every t ≥ 0. Letting k → +∞ in (20), we get x(t, x0 ) = Z(t)x0 ≤ M e−βt x0 ,

t ≥ 0.

Taking (19) into account, we get Z(t) ≤ M e−βt ,

t ≥ 0.

(21)

1

By B(t) ≥ 0, ∀t ≥ 0, B(·) ∈ L [0, +∞) and (21), it follows from Theorem 2 of [25] (also, see [26]) that eγ· B(·) ∈ L1 [0, +∞), for some positive number γ > 0. On the other hand, (21) implies that the system (12) is uniformly asymptotically stable, see e.g. [21]. This is equivalent to det(zIn − ˆ A − B(z)) = 0, for all z ∈ C, s ≥ 0, see e.g. [23]. Since A is a Metzler matrix and B(t) ≥ 0, ∀t ≥ 0, by a similar argument as in the proof of Theorem 4.3, +∞ we get µ(A + 0 B(t)dt) < 0. Consider the continuous real function deﬁned +∞ by g(θ) := θ + µ(A + 0 eθt B(t)dt), θ ∈ [0, γ]. From g(0) < 0, it follows that +∞ g(α) ≤ 0, for α > 0, small enough. This means that µ(A + 0 eαt B(t)dt) ≤ −α, for α > 0, small enough which completes our proof. Consider a linear Volterra integro-diﬀerential system deﬁned by t d x(t) = Ax(t) + C(t − s)x(s)ds, t≥0 dt −∞

(22)

where A ∈ Rn×n and C(t) ∈ C([0, +∞), Rn×n ) are given. Corollary 4.5. Let A ∈ Rn×n be a Metzler matrix and B(t) ∈ Rn×n for every + t ≥ 0. Suppose that |C(t)| ≤ B(t), ∀t ≥ 0 and B(·) ∈ L1 [0, +∞). Let (TC (t))t≥0 be the solution semigroup of the system (22). Then, +∞ (i) If µ(A + 0 B(t)dt) ≤ 0 then for every > 0, there is a positive number M such that TC (t) ≤ M et , t ≥ 0. +∞ (ii) If eα· B(·) ∈ L1 [0, +∞) for some α > 0 and µ(A + 0 eαt B(t)dt) ≤ −α, then the system (22) is exponentially stable in Cl (−∞, 0].

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+∞ (iii) If µ(A + 0 B(t)dt) < 0 then, for every φ ∈ Cl (−∞, 0] ∩ L1 (−∞, 0], the solution xC (·, φ) of (22) belongs to L2 [0, +∞) and xC (·, φ)L2 ≤ M (φCl (−∞,0] + φL1 ), for some M > 0. Proof. The proof is immediate from Theorem 4.2 and Theorem 2.1 (iv).

Remark 4.6. It is important to note that in the above corollary, the condition that A is a Metzler matrix cannot be omitted. To see this, we consider the system (22) where √ − 2 0 0 −e−t √ A= , t ≥ 0. , C(t) = 0 0 −2 − 2 Then, it is easy to see that the characteristic equation of the system +∞ √ ˆ = ((z + 2)2 − 2 e−(z+1)t dt) = 0 det(zI2 − A − C(z)) 0

has the root z0 = 0. Therefore, there exists a nonzero vector x0 ∈ R2 such ˆ that (0I2 − A − C(0))x 0 = 0. Then, the system (22) admits a constant solution x(t) = x0 , t ≥ 0, with the initial function φ0 (t) = x0 , t ∈ (−∞, 0]. Therefore, the system (22) cannot be exponentially stable in Cl (−∞, 0], although we have √ +∞ 1 µ(A + 0 e 2 t B(t)dt) = − 2 < − 12 , where B(t) := |C(t)|, t ≥ 0. We conclude the paper by an extension of Theorem 4.3 (iii) to perturbed systems. To do so, we consider perturbed systems of the form t d x(t) = (A + D0 ∆E0 )x(t) + (B(t − s) + D1 δ(t − s)E1 )x(s)ds, t ≥ 0. (23) dt −∞ Here Di ∈ Rn×li , Ei ∈ Rqi ×n , i ∈ I := {0, 1}, are given matrices deﬁning the structure of perturbations and ∆ ∈ Rl0 ×q0 , δ ∈ C([0, +∞), Rl1 ×q1 )∩L1 ([0, +∞), Rl1 ×q1 ) are unknown perturbations. We shall measure the size of each perturbation (∆, δ(·)) by the norm +∞ δ(t)dt. (∆, δ(·)) := ∆ + 0

1

Theorem 4.7. Let B(·) ∈ L [0, +∞) and the system (1) be positive. Suppose that +∞ i µ(A + 0 B(t)dt) < 0 and Di ∈ Rn×l , Ei ∈ Rq+i ×n , i ∈ I. Then, for every + perturbation (∆, δ(·)) satisfying (∆, δ(·)) <

1 +∞ maxi,j∈I Ei (−A − 0 B(t)dt)−1 Dj

the solution x(·, φ) of the perturbed system (23) belongs to L2 [0, +∞), for every φ ∈ Cl (−∞, 0] ∩ L1 (∞, 0] and x(·, φ)L2 ≤ M (φCl (−∞,0] + φL1 ), for some M > 0. To prove this theorem, we need the following technical lemma.

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+∞ Lemma 4.8. Suppose that the system (1) is positive, µ(A + 0 B(t)dt) < 0 and q×n D ∈ Rn×l + , E ∈ R+ . Then, +∞ +∞ max E(λIn − A − e−λt B(t)dt)−1 D = E(−A − B(t)dt)−1 D. λ∈C,λ≥0

0

0

Proof. Since the system (1) is positive, it follows that A is a Metzler matrix and B(t) ≥ 0, ∀t ≥ 0. For every λ ∈ C, λ ≥ 0, by Theorem 2.1 (iv), we get +∞ +∞ +∞ e−λt B(t)dt) ≤ µ(A + e−λt B(t)dt) ≤ µ(A + B(t)dt). µ(A + 0

+∞

0

0

+∞

−λt

Therefore, µ(A + 0 e B(t)dt) ≤ µ(A + 0 B(t)dt) < 0, for every λ ∈ C, λ ≥ 0. For a ﬁxed λ ∈ C, λ ≥ 0, we can represent the following +∞ +∞ +∞ −λt −1 λIn − (A + e−λt B(t)dt) x= e−λθ eθ(A+ 0 e B(t)dt) xdθ, x ∈ Cn , 0

0

(24) for every λ ∈ C, λ ≥ 0, see [28], [33]. Since A is a Metzler matrix, there exists a real number α0 > 0 such that (A + α0 In ) ≥ 0. From (A + α0 In ) ≥ 0 and B(t) ≥ 0, ∀t ≥ 0, it follows that eα0 θ |eθ(A+

+∞ 0

e−λt B(t)dt)

= |eθ((A+α0 In )+ = eα0 θ eθ(A+

+∞ 0

+∞ 0

| = |eα0 θIn eθ(A+

e−λt B(t)dt)

B(t)dt)

,

+∞ 0

e−λt B(t)dt)

| ≤ eθ((α0 In +A)+

+∞ 0

|

B(t)dt)

θ ≥ 0.

This implies that |eθ(A+

+∞ 0

e−λt B(t)dt)

| ≤ eθ(A+

+∞ 0

B(t)dt)

Taking (24), (25) into account, we get +∞ e−λt B(t)dt)−1 x| ≤ |(λIn − A − 0

=

θ ≥ 0, λ ∈ C, λ ≥ 0.

,

+∞

(25)

+∞

eθ(A+ 0 B(t)dt) dθ|x| 0 +∞ B(t)dt)−1 |x|, (−A − 0

for every λ ∈ C, λ ≥ 0. Furthermore, since D, E are the nonnegative matrices, it follows that +∞ +∞ |E(λIn − A − e−λt B(t)dt)−1 Dx| ≤ E(−A − B(t)dt)−1 D|x|, x ∈ Cn , 0

0

for every λ ∈ C, λ ≥ 0. By monotonicity property of the vector norm and the deﬁnition of operator norm, we get +∞ +∞ −λt −1 e B(t)dt) D ≤ E(−A − B(t)dt)−1 D, E(λIn − A − 0

for every λ ∈ C, λ ≥ 0. This completes our proof.

0

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Proof of Theorem 4.7. Suppose that ∆ ∈ Rl0 ×q0 , δ ∈ C([0, +∞), Rl1 ×q1 )∩L1 ([0, +∞), Rl1 ×q1 ) is a perturbation such that the solution x(·, φ) of the perturbed system (23) does not belong to L2 [0, +∞). By Theorem 4.2 (iii), there exist λ ∈ C, λ ≥ 0 and a non-zero vector x ∈ Cn such that +∞ −λt A + D0 ∆E0 + e (B(t) + D1 δ(t)E1 )dt x = λx. 0

+∞

+∞ By µ(A+ 0 B(t)dt) < 0 and Theorem 2.1(iv), det(λIn −A− 0 e−λt B(t)dt) = 0. It follows that −1 +∞ +∞ e−λt B(t)dt e−λt D1 δ(t)dt E1 x = x. D0 ∆E0 + D1 λIn − A − 0

0

(26) Let i0 ∈ I be an index such that Ei0 x = max(E0 x, E1 x). Then, it follows from (26) that Ei0 x = 0. Then, we thus get −1 +∞ −λt e B(t)dt D0 ∆E0x+ Ei0 λIn − A − Ei0 λIn − A −

0

+∞

e

−λt

−1

B(t)dt

0

This gives, max Ei λIn − A − i,j∈I

+∞

e

−λt

D1 −1

B(t)dt

0

0

e−λt δ(t)dtE1 x ≥ Ei0 x.

Dj ∆ +

+∞

0

Using Lemma 4.8, we have max Ei − A − i,j∈I

+∞

+∞

B(t)dt

0

−1

δ(t)dt

≥ 1.

Dj (∆, δ(·)) ≥ 1.

This is equivalent to (∆, δ(·)) ≥

1

maxi,j∈I Ei − A −

which completes the proof.

+∞ 0

B(t)dt

−1

Dj

,

References [1] V. Barbu, S.I. Grossman, Asymptotic behavior of linear integro-diﬀerential systems. Transactions of the American Mathematical Society 173 (1972), 277–288. [2] L. Benvenuti, L. Farina, Eigenvalue regions for positive systems. Systems Control Lett. 51 (2004), 325–330. [3] A. Berman, R.J. Plemmons, Nonnegative Matrices in Mathematical Sciences. Acad. Press, New York, 1979. [4] T.A. Burton, Volterra Integral and Diﬀerential Equations. Mathematics in Science and Engineering 167, Acad. Press, New York, 1983.

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[5] T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Diﬀerential Equations. Mathematics in Science and Engineering 178, Acad. Press, New York, 1985. [6] R.D. Driver, Existence and continuous dependence of solutions of a neutral functional-diﬀerential equation. Arch. Rational Mech. Anal. 19 (1965), 149–166. [7] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications. John Wiley and Sons, New York, 2000. [8] E. Fornasini, M.E. Valcher, Controllability and reachability of 2-D positive systems: a graph theoretic approach. IEEE Trans. Circuits Syst. I Regul. Pap. 52 (2005), 576– 585. [9] G.S. Grossman, R.K. Miller, Nonlinear Volterra integro-diﬀerential systems with L1 −kernels. Journal of Diﬀerential Equations 13 (1973), 551–566. [10] W.M. Haddad, V. Chellaboina, Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay, Advances in time-delay systems, pp. 421–435, Lect. Notes Comput. Sci. Eng., 38 (2004), Springer, Berlin. [11] W.M. Haddad, V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with delay. Systems & Control Letters 51 (2004), 355–361. [12] W.M. Haddad, V. Chellaboina, Stability and dissipativity theory for nonnegative dynamical systems: a uniﬁed analysis framework for biological and physiological systems. Nonlinear Anal. Real World Appl. 6 (2005), 35–65. [13] W.M. Haddad, T. Hayakawa, Adaptive control for nonlinear nonnegative dynamical systems. Automatica 40 (2004), 1637–1642. [14] D. Hinrichsen, N.K. Son, µ-analysis and robust stability of positive linear systems. Appl. Math. and Comp. Sci. 8 (1998), 253–268. [15] D. Hinrichsen, N.K Son, P.H.A. Ngoc, Stability radii of positive higher order diﬀerence systems. Systems & Control Letters 49 (2003), 377–388. [16] R.A. Horn, C.R. Johnson, Matrix Analysis. Cambridge University Press, Cambridge, 1993. [17] G. James, S.P. Kostova, V.G. Rumchev, Pole-assignment for a class of positive linear systems. Internat. J. Systems Sci. 32 (2001), 1377–1388. [18] G. James, V.G. Rumchev, Stability of positive linear discrete-time systems. Systems Sci. 30 (2004), 51–67. [19] G.S. Jordan, R.L. Wheeler, Structure of resolvents of Volterra integral and integrodiﬀerential systems. SIAM J. Math. Anal. 11 (1980), 119–132. [20] G.S. Jordan, O.J. Staﬀans, R.L. Wheeler, Local analyticity in weighted L1 -spaces and applications to stability problems for Volterra equations. Trans. Amer. Math. Soc. 274 (1982), 749–782. [21] V. Lakshmiskantham, M. Rama Mohana Rao, Theory of Integro-Diﬀerential Equations. Stability and Control: Theory Methods and Applications Volume 1, Gordon and Breach Science Plublisher, 1995. [22] D.G. Luenberger, Introduction to Dynamic Systems, Theory, Models and Applications. J. Wiley, New York, 1979. [23] R.K. Miller, Asymptotic stability properties of Volterra integro-diﬀerential systems. Journal of Diﬀerential Equations 10 (1971), 485–506.

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[24] R.K. Miller, Structure of solutions of unstable linear Volterra integro-diﬀerential equations. Journal of Diﬀerential Equations 15 (1974), 129–157. [25] S. Murakami, Exponential stability for fundamental solutions of some linear functional diﬀerential equations. In T. Yoshizawa and J. Kato, editors, Proceedings of the International Symposium: Functional Diﬀerential Equations, pp. 259–263, Singapore, 1990, World Scientiﬁc. [26] S. Murakami, Exponential asymptotic stability for scalar linear Volterra equations. Diﬀerential Integral Equations 4 (1991), 519–525. [27] S. Murakami, Y. Nagabuchi, Stability properties and asymptotic almost periodicity for linear Volterra diﬀerence equations in a Banach space Japan. J. Math. (N.S.) 31 (2005), 193–223. [28] R. Nagel (Ed.), One-Parameter Semigroups of Positive Operators. Springer-Verlag, Berlin, 1986. [29] P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems. International Journal of Robust and Nonlinear Control 15 (2005), 459–472. [30] P.H.A. Ngoc, N.K. Son, Stability radii of positive linear diﬀerence equations under aﬃne parameter perturbations. Applied Mathematics and Computation 134 (2003), 577–594. [31] P.H.A. Ngoc, N.K. Son, Stability radii of linear systems under multi-perturbations. Numer. Funct. Anal. Optim. 25 (2004), 221–238. [32] P.H.A. Ngoc, T. Naito, J.S. Shin, Characterizations of postive linear functional differential equations. To appear in Funkcialaj Ekvacioj (2006). [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Diﬀerential Equations. Springer-Verlag, Berlin, 1983. [34] N.K. Son, D. Hinrichsen, Robust stability of positive continuous time systems. Numer. Funct. Anal. Optim. 17 (1996), 649–659. [35] N.K. Son, P.H.A. Ngoc, Stability radius of linear delay systems, in Proceedings of the American Control Conference, San Diego, California, June 1999, pp. 815–816. [36] N.K. Son, P.H.A. Ngoc, Robust stability of positive linear time delay systems under aﬃne parameter perturbations. Acta Mathematica Vietnamica 24 (1999), 353–372. [37] N.K. Son, P.H.A. Ngoc, Robust stability of linear functional diﬀerential equations. Advanced Studies in Contemporary Mathematics 3 (2001), 43–59. [38] W. Rudin, Real and Complex Analysis. McGraw-Hill, New York, 1987. [39] B. Zhang, Asymptotic stability criteria and integrability properties of the resolvent of Volterra and functional equations. Funkcial. Ekvac. 40 (1997), 335–351. [40] B. Zhang, Necessary and suﬃcient conditions for stability in Volterra equations of nonconvolution type. Dynam. Systems Appl. 14 (2005), 525–549. Toshiki Naito Department of Mathematics, The University of Electro-Communications, P.O. Box 182-8585 Chofu, Tokyo Japan e-mail: [email protected]

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Satoru Murakami Department of Applied Mathematics, Okayama University of Science Ridai, Okayama, Okaya 700 Japan e-mail: [email protected] Jong Son Shin Department of Mathematics, The University of Electro-Communications, P.O. Box 182-8585 Chofu, Tokyo Japan e-mail: [email protected] Pham Huu Anh Ngoc Department of Mathematics, The University of Electro-Communications, P.O. Box 182-8585 Chofu, Tokyo Japan e-mail: [email protected] Submitted: June 12, 2006 Revised: October 23, 2006

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Integr. equ. oper. theory 58 (2007), 273–299 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020273-27, published online April 14, 2007 DOI 10.1007/s00020-007-1486-0

Integral Equations and Operator Theory

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators M. A. Nudelman Abstract. Let jmm =

Im 0

0 −Im

,

Jm =

0 iIm

−iIm 0

,

Im is the identity matrix of order m. Let W (λ) be an entire matrix valued function of order 2m, W (0) = I2m , the values of W (λ) are jmm -unitary at the imaginary axis and strictly jmm -expansive in the open right half-plane. The blocks of order m of the matrix W (λ) with appropriate signs are treated as coeﬃcients of algebraic Riccati equation. It is proved that for any λ with positive real part this equation has a unique contractive solution θ(λ). The matrix valued function θ(λ) can be represented in a form θ(λ) = θA (iλ) where θA (µ) is the characteristic function of some maximal dissipative operator A. This operator is in a natural way constructed starting from the Hamiltonian system of the form dx(τ ) = iJm K(τ )x(τ ), τ ∈ [0; +∞) dτ with periodic coeﬃcients. Mathematics Subject Classiﬁcation (2000). Primary 34L05; Secondary 15A24. Keywords. Algebraic Riccati equations, Hamiltonian systems, maximal dissipative operators, characteristic functions.

1. Introduction There exists a vast literature devoted to investigation of algebraic Riccati equations of the form XAX + BX + XC + D = 0 (1.1)

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(here A, B, C, D are given square matrices of some order n with complex entries, X is the unknown complex square matrix of the order n). Investigation of the matrix equation of the form (1.1) constitutes the important self-dependent algebraic problem which has numerous applications in the framework of linear algebra (such as investigation of invariant subspaces of ﬁnite dimensional linear operators, factorization of rational matrix valued functions and others) as well as in the control theory (in the questions connected with the Kalman-Yakubovich lemma, in the theory of Kalman ﬁlters, in the theory of H ∞ -optimization and others); the inﬁnite dimensional generalizations of the equation (1.1) have found the applications in the theory of block self-adjoint operators and in the scattering theory. The most complete exposition of the algebraic theory of the equations of the form (1.1) and also of some applications of this theory to the problems of control theory see in the book [7]: see also the bibiliography in this book. On the other hand, in the operator theory the concept of the characteristic function is well known. The most completely the theory of characteristic functions is elaborated in the works by B. Sz¨okefalvi-Nagy and C. Foias for the case of the contractive linear operators acting in a separable Hilbert space (recall that an operator is said to be contractive if its norm not exceeds 1); see [15]. Using the concept of unitary node (see [4]), one can expose the essence of the Sz¨okefalvyNagy and Foias theory in the following way. Let T be a contractive operator acting in some separable Hilbert space H. Then there exist such auxiliary separable Hilbert (may be, ﬁnite dimensional) spaces U , V and bounded operators F , G, H acting respectively from U to H, from H to V and from U to V that the block operator T F Λ= G H accomplishes the isometric mapping of the space H ⊕ U onto the space H ⊕ V . The aggregate of the spaces H, U , V and the operators T , F , G, H is called the unitary node (we shall denote it by the letter α). The contractive operator T is said to be included into the unitary node Λ. The characteristic function of the contractive operator T is the operator valued function −1 ∗ θT (z) = H ∗ + zF ∗ IH − zT ∗ G (1.2) (here IH is the identity operator acting in the space H) the values of which are the bounded operators acting from the space V to the space U . In the Sz¨okefalvi-Nagy and Foias theory it is proved that the function θT (z) is deﬁned and holomorphic in the open unit disk and moreover when |z| < 1 the inequality θT (z) 1 holds. Under some natural restrictions which are put on the unitary node α the function θT (z) is deﬁned unambiguously up to isomorphisms acting in the spaces U and V .

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The apparatus of characteristic functions is the powerful means for investigation of the spectral properties of contractive operators (such as spectrum, comleteness, invariant subspaces and others). Recall that closed (may be, unbounded) linear operator A acting in some Hilbert space H, is said to be maximal dissipative if ∀h ∈ DA and

Im Ah, h 0

R iIH + A = H

(1.3)

(1.4)

(here DA i s the domain of operator A, the symbol R denotes the image of operator, the angle brackets denote the inner product). There are known the numerous approaches to the deﬁnition of the concept of characteristic function of maximal dissipative operator acting in a separable Hilbert space. The most general approach is developed in the paper by D. Z. Arov and the author [2] in which the concept of conservative scattering system with continuous time is deﬁned and investigated. This concept is completely analogous to the concept of unitary node. The design of conservative scattering system with continuos time is based on the constructions due to Yu. L. Shmulyan [14] (see also [13]) who pointed at the connection of the concerned theory with the P. Lax and R. Phillips scattering theory. The brief information on the conservative systems with continuous time and the brief description of the Lax-Phillips scattering scheme see in the § 3 of the present paper. The approach to the concept of characteristic function of maximal dissipative operator which is developed in the paper [2] opens up possibilities of investigation of the spectral properties of ordinary diﬀerential operators with dissipative boundary conditions. The example of such investigation is contained in the paper [10] by the author in which the question about completeness of the operator of nonhomogeneous string with friction at the left end was considered. Analogous investigation based on Theorem 6.3 of the present work can be staged for the maximal dissipative operator which is generated by the Hamiltonian system with periodic coeﬃcients at the semi-axis (see construction of this operator in the § 6 of the present paper, in particular, Theorem 6.1). The author plans to stay on this point in detail in one of his subsequent publications. The main result of the present paper is based on the well known V. P. Potapov theorem [11] which consisits in the following. Let the square matrix J with complex entries satisﬁes the equalities J = J ∗ = J −1 .

(1.5)

The examples are the identical matrix In of an arbitrary order n and also the matrices Im 0 jmm = (1.6) 0 −Im

276

Nudelman

and

Jm =

0 iIm

−iIm 0

IEOT

,

(1.7)

in the both formulas (1.6) and (1.7) m is an arbitrary natural number. As is well known, the square matrix valued function G(λ) (we assume that its order equals to the order of the matrix J) which is deﬁned when Re λ > 0 is said to be J-inner if for all λ from open right half-plane G(λ)∗ JG(λ) − J 0

(1.8)

holds in the Hermitian sense and for the boundary values the equality G(iα)∗ JG(iα) − J = 0

(1.9)

holds for almost all real α. In particular, when J = −I (I is the identical matrix of arbitrary order) we obtain the deﬁnition of square inner function. Recall that the monodromy matrix of the vector diﬀerential equation dx(τ ) = A(τ )x(τ ), dτ

τ ∈ 0, T ,

T >0

(here A(τ ) is the square matrix valued function of some order n the entries of which are summable on 0, T , x(τ ) ∈ Cn is the value X(T ) of the solution of the matrix Cauchy problem dx(τ ) = dτ X(0) =

A(τ )x(τ ),

τ ∈ 0, T ,

T >0

In

for τ = T . The V. P. Potapov theorem claims that the matrix-valued function F (λ) is the monodromy matrix of some system of diﬀerential equations of the form dx(τ ) = λJH(τ )x(τ ), dτ

τ ∈ 0, T ,

T >0

(1.10)

(here J is the matrix satisfying the equalities (1.5), H(τ ) is positive semideﬁnite Hermitian matrix for all τ ∈ 0, T , the matrix valued function H(τ ) is summable on [0, T ], x(τ ) ∈ Cn where n is the order of the matrices J, H(τ )) iﬀ it is the entire J-inner in the right half-plane matrix valued function which is normed by the condition F (0) = In . In this paper the algebraic Riccati equations of the form θ(λ)β(λ)θ(λ) + θ(λ)α(λ) − δ(λ)θ(λ) − γ(λ) = 0

(1.11)

are studied where the square matrix valued functions of order m are deﬁned by the equality α(λ) β(λ) G(λ) = , γ(λ) δ(λ)

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where G(λ) is some entire jmm -inner (see (1.6)) in the right half-plane matrix valued function normed by the condition G(0) = I2m and moreover the inequality (1.8) is fulﬁlled for it in the strengthened form: G(λ)∗ jmm G(λ) − jmm > 0,

Re λ > 0,

(1.12)

θ(λ) is the unknown contractive (that is satisfying the inequality θ(λ)∗ θ(λ) Im ) matrix valued function which is deﬁned in the inner right half-plane. We shall show that the condition (1.12) implies existence and uniqueness of the contractive solution of the equation (1.11) for any ﬁxed value of λ (Re λ > 0). The main result of this paper is that this solution can be naturally interpreted as the characteristic function of some maximal dissipative operator which is in some standard way generated by some Hamiltonian (that is corresponding to J = Jm , see (1.7)) system of the form (1.10). The author expresses his gratitude to L. A. Sakhnovich for useful discussions and to S. A. Kupin for the stimulating interest.

2. The existence and uniqueness of the contractive solution of the equation (1.11) where the parameter λ is ﬁxed Lemma 2.1. Let the matrix jmm be deﬁned by the equality (1.6) and let G be some non-degenerate strictly jmm -expansive (that is satisfying the inequality G∗ jmm G − jmm > 0,

(2.1)

compare with (1.12)) matrix of order 2m. Then G can be represented in a form K1 0 G=V (2.2) V −1 0 K2 where K1 and K2 are the non-degenerate matrices of order m, K1−1 < 1 and K2 < 1 (compare with [12], formula (2.14) of the chapter 6). Proof. The proof of this lemma is known (this is a folklore of V. P. Potapov school); we shall expose it for the sake of completeness. First of all, note that the matrix G has not eigenvalues with modulus equal to one. Indeed, let λ be the eigenvalue of the matrix G and x be the corresponding to it eigenvector. In view of the inequality (2.1), we have: 0 < x∗ G∗ jmm G − jmm x = (Gx)∗ jmm (Gx) − x∗ jmm x = = and therefore

(λx)∗ jmm (λx) − x∗ jmm x = (|λ|2 − 1)(x∗ jmm x)

|λ|2 − 1 = 0. Let us reduce the matrix G to the Jordan form. Then it can be written in a form D1 0 (2.3) U −1 G=U 0 D2

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where U is some non-degenerate matrix of order 2m, D1 is the aggregate of the Jordan blocks corresponding to the eigenvalues the modulus of which more then 1 and D2 is the aggregate of the Jordan blocks corresponding to the eigenvalues the modulus of which less then 1. Let us show that D1 and D2 are the matrices of order m. If not, let, for example, the order p of the matrix D1 less then m; in this case the order q of the matrix D2 more then m. Consider the expression ∗ ∗k ∗ 0 0 k ∗ ∗ 0 h U G jmm G U U jmm U − 0 h (2.4) h h where h is an arbitrary column of the height q. In view of the formula (2.3) we have: 0 0 k =U −→ 0; G U h D2k h k→+∞ so we see that the limit of the expression (2.4) as k → +∞ equals to 0 . − 0 h∗ U ∗ jmm U h At the same time the inequality (2.1) implies the strict positiviness of the expression (2.4) for all h = 0 and so for all columns h of the height q the inequality 0 0 h∗ U ∗ jmm U 0 (2.5) h holds. But in view of the assumtion q > m, in the 2m-dimensional space of the columns of the height 2m the q-dimensional lineal of the columns of the form 0 U has non-zero intersection with the m-dimensional lineal of the vectors of h g the form where g is an arbitrary column of the height m. Since for g = 0 0 the relation ∗ g g 0 jmm = g∗g > 0 0 is valid, we have obtained the contradiction with the inequality (2.5) which proves the inequalities q m, p m. Using the relation G∗ −1 jmm G−1 − jmm < 0 which is the consequence of the inequality (2.1), one can in the same way prove the inequalities p m, q m. So p = q = m. It remains to show that the matrix D1 is similar to some non-degenerate matrix K1 satisfying the inequality K1−1 < 1 and the matrix D2 is similar to some non-degenerate matrix K2 satisfying the inequality K2−1 < 1. By the

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construction of matrices D1 and D2 and taking into account the non-singularity of the matrix G, it is suﬃcient to prove that the Jordan block   λ 1 0 ... 0 0  0 λ 1 ... 0 0     T =  .....................   0 0 0 ... λ 1  0 0 0 ... 0 λ for which λ < 1 is similar to some S such that S < 1. Indeed, let r be the order of matrix T . Then T = λIr + T0 where

   T0 =   

Let

    A=   

0 1 0 ... 0 0 0 0 1 ... 0 0 .................... 0 0 0 ... 0 1 0 0 0 ... 0 0

     

0 0 a1 0 0 . . . 0 a2 0 . . . 0 0 0 0 a3 . . . 0 0 ........................... 0 0 0 . . . ar−1 0 0 0 0 ... 0 ar

       

be some non-degenerate diagonal matrix of order r. We have: A−1 T A = λIr + A−1 T0 A. A direct computation shows that  0 0 aa21 0 . . . 0  0 0 a3 . . . 0 0 a2  . . . . . . . . . ................. A−1 T0 A =   ar  0 0 0 . . . 0 ar−1 0 0 0 ... 0 0

(2.6)    ;  

therefore in the sum in the right side of the equality (2.6) the norm of the ﬁrst item, which is equal to |λ|, strictly less than 1 and the norm of the second item can be done arbitrary small due to the choice of diagonal matrix A. So for some choice of A one can take S = A−1 T A and at that the inequality S < 1 holds.

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Theorem 2.2. Let the matrix jmm be deﬁned by the equality (1.6) and let α β G= γ δ be some non-degenerate strictly jmm -expansive (that is satisfying the inequality (2.1)) matrix of order 2m the blocks α, β, γ, δ are assumed to be the matrices of order m. Then the Riccati equation θβθ +θα−δθ−γ = 0

(2.7)

(compare with (1.11)) has the unique solution θ in the class of contractive matrices of order m. This solution is strictly contractive that is satisﬁes the inequality ∗

θ θ < Im . Proof. Consider the matrix linear-fractional transform −1 Γ : θ → γ +δθ α+βθ

(2.8)

where θ is a square matrix of order m. From now on we shall mean linear-fractional transform of matrices speaking about linear-fractional transforms. Im Multiplying the inequality (2.1) by the matrix of size 2m × m from θ the right and by the conjugate to it matrix Im θ∗ from the left we ﬁnd: ∗ ∗ α + β θ α + β θ − γ + δ θ γ + δ θ − Im − θ∗ θ > 0. (2.9) From this it is clear that if the matrix θ is contractive then the matrix α + β θ is non-degenerate and so for contractive θ the linear-fractional transform (2.8) is well deﬁned. Consider the ﬁxed points of this linear-fractional transform. Multiplying the equation −1 =θ (2.10) γ +δθ α+βθ by α + β θ from the right, we come to the Riccati equation (2.7). If the matrix α+β θ is non-degenerate then the inverse passage is correct also. So for contractive matrices θ the equations (2.7) and (2.10) are equivalent. Fix the matrix θ in such a way that Im − θ∗ θ > 0. Apply Lemma 2.1. Let the matrix V which ﬁgures in the formula (2.2) has the block fragmentation a b V = (2.11) c d Im −1 has the block fragmentation and the matrix V θ p Im V −1 = (2.12) θ q where a, b, c, d, p, q are the square matrices of order m.

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Consider the matrix linear-fractional transform Γk (k 1) which is deﬁned by the matrix Gk in the same way as the linear-fractional transform (2.8) is deﬁned by the matrix G. In view of the formula (2.2), the equality k 0 K1 Gk = V (2.13) V −1 0 K2k takes place and so, in view of the formulas (2.11), (2.12) the result of application of the transformation Γk to the choosen by us strictly contractive matrix θ can be written in a form Γk θ = (cK1k p + dK2k q)(aK1k p + bK2k q)−1 . It is easy to see that the matrix Gk is strictly jmm -expansive so as the matrix G and therefore the inequality (a1 K1k p + bK2k q)∗ (aK1k p + bK2k q) > Im − θ∗ θ > 0 holds (compare with (2.9)). Since K2 < 1, the norm of the second item in each of the pairs of brackets tends to zero while the power k tends to inﬁnity and so for suﬃciently large k the inequality (aK1k p)∗ (aK1k p) > 0 takes place. From this we see that the both matrices a and p are non-degenerate. Let us show that the matrix ca−1 is strictly contractive. Really, as in the previous paragraph, let arbitrarily some strictly contractive matrix θ and us ﬁx ∞ consider the sequence Γk θ k=1 taking into account that, as it is well known, the product of matrices corresponds to the composition of linear-fractional transforms which are deﬁned by them. Let us use the formula (2.13). The result of application of the linear-fractional transform corresponding to the matrix V −1 to the matrix θ is, in the notations of the previous paragraph, the matrix qp−1 which is well deﬁned. Then applying matrix to this the linear-fractional transform corresponding to the 0 K1k block matrix we obtain the matrix K2k qp−1 K1−k which tends to zero 0 K2k as k → ∞. in view of the inequalities K1−1 < 1, K2 < 1. From this it is clear that Γk θ tends to the image of the zero matrixunder the linear-fractional transform a b corresponding to the matrix V = that is to ca−1 . c d Multiplying the inequality (2.9) by (α + β θ)∗−1 from the left and by (α + −1 β θ) from the right and taking into account the obvious fact that the matrix linear-fractional transform Γ is continuous at the set of contractive matrices B = {θ | θ 1} we obtain that the image ΓB is some compact set consisting of strictly contractive matrices. From this it is clear that ca−1 = lim Γk θ is the k→∞

strictly contractive matrix. Let us denote it by θ and prove that it is the unique contractive solution of the equation (2.7) (or, equivalently, of the equation (2.10)).

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Really, the equation Γθ = θ which is equivalent to the equation (2.10), immediately follows from the formula (2.2) and the relation θ = F 0 where F is the linear-fractional transform corresponding to the matrix V . Let us prove that θ is unique contractive solution of the equation (2.10). Let θ be an arbitrary contractive solution of this equation. The matrix θ belongs to < 1. the image ΓB of the set of contractive matrices because θ = Γθ and so θ k −1 From this, as it has been shown above, follows that lim Γ θ = ca = θ. Taking k→∞ ∞ into account that all members of the sequence Γk θ equal to θ we obtain that k=1

θ = θ.

By the same method it can be proved the following Theorem 2.3. Let

G=

α γ

β δ

be the strictly jmm -contractive (that is satisfying the inequality G∗ jmm G − jmm < 0)

(2.14)

matrix of order 2m, the blocks α, β, γ, δ are assumed to be the matrices of order m. Then the Riccati equation θγθ + θδ − αθ − β = 0

(2.15)

has the unique solution θ in the class of contractive matrices of order m. This solution is strictly contractive that is it satisﬁes the inequality ∗

θ θ < Im . Remark 2.4. Note that under the condition (2.14) for contractive θ the equation (2.15) is equivalent to the equation (αθ + β)(γθ + δ)−1 = θ (Compare with (2.10)).

3. Basic information about the conservative scattering systems with continuous time Let B be a closed (may be, unbounded) linear operator acting in some ﬁnite dimensional or separable Hilbert space H. Consider the conjugate to it linear operator B ∗ . The domain DB ∗ of this operator can be naturally endowed by the graph norm: h ∈ DB ∗ h 2+ = h 2 + B ∗ h 2 , and, in view of closedness of the operator B ∗ , it is the Hilbert space respectively to this norm. Let us denote this Hilbert space by H+ (B ∗ )∗ .

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Let us introduce the following notation: H− (B ∗ ) = (H+ (B ∗ ))∗ . Since for any vector h ∈ H the functional fh (ξ) = ξ, h H , ξ ∈ DB ∗ where ( · , · )H is the inner product in the space H+ (B ∗ ), then the inclusion

H is continuous in the metric of

H+ (B ∗ ) ⊂ H ⊂ H− (B ∗ ) takes place. As it is well known, such construction is called the rigged Hilbert space (see, for example, [3]). The operator B ∗ can be considered as the bounded operator acting from H+ (B ∗ ) to H. The conjugate operator B which act from H to H− (B ∗ ) is called the natural extension of the operator B (see, for example, [16]). Recall that the C0 -semigroup is the family V (t) (t 0) of bounded linear operators acting in a Banach (in particular, may be, Hilbert) space R having the following properties: (i) (ii) (iii)

V (0) = IR V (t + s) = V (t)V (s), t 0, s 0 s − lim V (t) = V (t0 ), t0 0 t→t0

(here IR is the identity operator acting in the space R; s − lim is the limit in the strong operator topology). It is well known (see, for example, [5], chapter VIII, § 1) that for any such semigroup the set D of such k ∈ R for which the limit V (t)k − k t exists constitutes the lineal which is dense in R and the operator C which is set by the formula V (t)k − k , k ∈ D, Ck = lim t→+0 t is a closed densely deﬁned (may be, unbounded) operator acting in the space R. Let the operator B which takes part in the construction of rigged Hilbert space be the generator of some C0 -semigroup in the space H. A conservative scattering system µ with continuous time has a form [2] lim

t→+0

d h(t) = dt ϕ(+) (t) =

Bh(t) + Lϕ(−) (t), N (h(t), ϕ(−) (t)); −

(3.1)

here h(t) ∈ H is the state of the system, ϕ(−) (t) ∈ N is its input data, ϕ(+) (t) ∈ N+ is its output data; N− , N+ are the ﬁnite dimensional or separable inﬁnite − dimensional Hilbert spaces; L is the bounded linear operator acting from N to ∗ H− (B ). It is assumed that the set − + Lϕ(−) ∈ H} DB,L = {(h, ϕ(−) ) ∈ H ⊕ N | Bh

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−

is a dense lineal in the space H ⊕ N and a Hilbert space relatively to the norm 2 + Lϕ(−) 2 , = h 2 + ϕ(−) 2 + Bh (h, ϕ(−) ) ∈ DB,L . (h, ϕ(−) ) DB,L

The operator N is assumed to be a bounded linear operator acting from the space + DB,L to the space N . ∈ H is equivalent to the inclusion h ∈ It is known [16] that the inclusion Bh DB or, in another notation, h ∈ H+ (B). Cosequently, the operator N generates the bounded linear operator M :

H+ (B) → N+

by the formula M h = N (h, 0). It can be shown [4] that the inclusion (h, ϕ(−) ) ∈ DB,L implies the inclusion −1 Lϕ(−) ∈ DB ; h − (λI − B) I is the identical operator acting in here λ is the regular point of the operator B, ∗ the space H− (B ). The obvious identity −1 Lϕ(−) ) + N ((λI − B) −1 Lϕ(−) , ϕ(−) ) N (h, ϕ(−) ) = M (h − (λI − B) shows that the coeﬃcients of the system (3.1) are uniquely deﬁned by the four objects: B, L, M and θµ (λ) where θµ (λ) is the transfer function of the system (3.1) which is deﬁned by the formula −1 ϕ(−) , ϕ(−) ). θµ (λ)ϕ(−) = N ((λI − B) The system (3.1) is said to be conservative if the equality d h(t) 2 = ϕ(−) (t) 2 − ϕ(+) (t) 2 dt or, more exactly, + Lϕ(−) , h 2Re Bh

H

= ϕ(−) 2 − N (h, ϕ(−) ) 2 ,

(h, ϕ(−) ) ∈ DB,L

holds for it and the analogous equality holds for the conjugate system which is deﬁned by the four (B ∗ , M ∗ , L∗ , (θµ (λ))∗ ). Now recall the connection between conservative scattering systems with continuous time and unitary Lax-Phillips semigroups. Accordingly to [14], the unitary Lax-Phillips semigroup is the unitary C0 + semigroup W (t) acting in the orthogonal sum L2 ( 0, +∞ , N )⊕ H ⊕L2 ( 0, +∞ ) and having the following properties: 1) the ﬁrst item in the orthogonal sum is invariant relatively to the semigroup W (t) (t 0) and moreover this semigroup acts at this subspace as the semigroup of right shifts: f (s − t), s t, + (3.2) (W (t) f )(s) = f ∈ L2 ([0, +∞), N ) 0, 0 s < t,

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2) the third item in the orthogonal sum is invariant relatively to the semigroup W (t)∗ (t 0) and moreover the semigroup acts at this subspace as the semigroup of right shifts (see the formula (3.2)). When ommitting the condition of unitarity of the semigroup W (t) this deﬁnition is somewhat more complicated [14]. According to [2] (see also [14]), there exists the canonical one-to-one correspondence between the unitary Lax-Phillips semigroups and the conservative scattering systems of the form (3.1). The evolution of the unitary Lax-Phillips semigroup W (t) reproduces the dynamics of the corresponding conservative scattering system in the following sense. Let the vector valued function ϕ− (t) belong to the Sobolev space − + Lϕ− (0) ∈ H is valid. W21 ([0, +∞), N ) and for some a ∈ H the inclusion Ba Then there exists the diﬀeretiable vector valued function h(t) (t 0) taking its values in the space H that for all t 0 the equality d h(t) = Bh(t) + Lϕ− (t) dt is valid (see [14] and also [2]). Let ϕ+ (t) = N (h(t), ϕ− (t)),

t 0. +

Let λ(t) be some element of the space L2 ([0, +∞), N ). Consider the el+ ement x(s) = (λ(s), a, ϕ− (s)) of the orthogonal sum L2 ([0, +∞), N ) ⊕ H ⊕ − L2 ([0, +∞), N ). It occurs that for any t 0 the element W (t)x (s) of this orthogonal sum is W (t)x (s) = (µt (s), h(t), νt (s)) where λ(s − t), st µt (s) = ϕ+ (t − s), 0 s < t, νt (s) = ϕ− (t + s) (see [14]).

4. Auxiliary constructions Consider the system of diﬀerential equations (canonical system) of the form dx(τ ) = λjmm H(τ )x(τ ), τ ∈ [0, +∞), (4.1) dτ where x(τ ) ∈ C2m (m 1), λ ∈ C, the matrix jmm is deﬁned by the formula (1.6), H(τ ) is the Hermitian nonnegative locally summable matrix valued function of order 2m satisfying the condition of periodicity H(τ + T ) = H(τ ), T > 0. Recall (see, for example, [4, chapter XIII, § 5] that any Hermitian nonnegative locally summable matrix valued function H(τ ) of order n 1 which is deﬁned on some open interval ∆ ⊂ R deﬁnes the Hilbert space L2Hdτ (∆) according to the

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following rule: this space is the factor-space of the linear space of measurable column functions f (τ ) of the height n satisfying the condition f (τ )∗ H(τ )f (τ )dτ < +∞ (4.2) ∆

relatively to its subspace consisting of such measurable column functions g(τ ) of the height n for which g(τ )∗ H(τ )g(τ )dτ = 0. ∆

The left part of the inequality (4.2) gives the square of the norm in the space L2Hdτ . Now recall that the product of matrices of order 2m corresponds to the composition of linear-fractional transforms which are deﬁned by the formula (2.8) (variant: deﬁned as in the remark to Theorem 2.3). Accordingly, the inverse matrix corresponds to the inverse linear-fractional transform. Assume that the monodromy matrix of the periodic canonical system (4.1) (let us denote it by G(λ)) satisﬁes the condition (1.12). Let Re λ > 0. Since the matrix G(λ) is non-degenerate and satisﬁes the inequality (1.12) the inverse to it G1 (λ) = G(λ)−1 exists and satisﬁes the inequality (2.14). So in view of Theorem 2.3 and the remark to it the linear-fractional transform which is built by the matrix G1 (λ) as in the remark to Theorem 2.3 has the unique ﬁxed point θr (λ) in the class of contractive matrices of order m. Consequently, the inverse to it linear-fractional transform which is built by the same way by the matrix G(λ) has the unique (the same) ﬁxed point in this class. Analogously, let Re λ < 0. Then the matrix G(λ) is non-degenerate and satisﬁes the inequality (2.14) (the last assertion is the consequence of the identity G(−λ)∗ jmm G(λ) = jmn which is, in turn, the consequence of the equality (1.9) and analyticity of the matrix valued function G(λ)). The inverse to it matrix G1 (λ) = G(λ)−1 exists and satisﬁes the inequality (1.12). Therefore, in view of Theorem 2.2, the linear-fractional transform (2.8) corresponding to the matrix G1 (λ) has the unique ﬁxed point θl (λ) in the class of contractive matrices of order m. Consequently, the inverse to it linear-fractional transform corresponding to G(λ) has the unique (the same) ﬁxed point in this class. Lemma 4.1. Assume that the matrix valued function G(λ) satisﬁes the condition (1.12). a) Let Re λ > 0, ∆ = (0, +∞), H(τ ) = H(τ ). Then the set M of initial data of the canonical system with periodic coeﬃcients (4.1) corresponding to the solutions x(τ ) satisfying the condition (4.2) is θr (λ) a | a ∈ Cm . Im

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b) Let Re λ < 0, ∆ = (0, +∞), H(τ ) = H(τ ). Then the set M of initial data of canonical system with periodic coeﬃcients (4.1) satisfying the condition (4.2) is Im m a|a ∈ C . θl (λ) Proof. We shall prove only the item a); the item b) can be proved analogously. So let Re λ > 0. Let us start from the inclusion θr (λ) m a|a ∈ C ⊂ M. (4.3) Im Consider the sequence {Un }∞ n=1 which is given by the formula nT Un =

x(τ )∗ H(τ )x(τ )dτ,

0

where x(τ ) is some solution of the system (4.1). It is clear that as applied to the function x(τ ) the condition (4.2) is equivalent to the boundedness of the sequence {Un }. The following formula is well known (it is the consequence of the NewtonLeibniz formula): if x(τ ) is the solution of the system (4.1) then for any E > 0 E

x(τ )∗ H(τ )x(τ )dτ =

0

1 (x(E)∗ jmm x(E) − x(0)∗ jmm x(0)). 2Re λ

somenatural n then x(E) = G(λ)n x(0). θr a for some a ∈ Cm . By construction of the matrix θr , Im θr θr of sizes 2m × m has a form s for some nonIm Im n degenerate matrix s of order m. According to this, the vector x(nT ) = G(λ) x(0) θr appears to be equal sn a. Im Since, by assumption, thematrix G(λ) satisﬁes the inequality (1.12), so, using θr θr the equality G(λ) = s, we obtain: Im Im θr θr − (θr∗ Im )jmm 0 < (θr∗ Im )G(λ)∗ jmm G(λ) Im Im θr θr = s∗ (θr∗ Im )jmm s − (θr∗ Im )jmm Im Im If E = nT for Let x(0) = the matrix G(λ)

= s∗ (θr∗ θr − Im )s − (θr∗ θr − Im ). Let κ = (Im − θr∗ θr )1/2 . We have s∗ κ 2 s < κ 2 .

(4.4)

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Multiplying the both parts of the last inequality from the left and from the right by κ −1 (this is legitimate because, in view of Theorem 2.3, the matrix Im − θr∗ θr is non-degenerate), we obtain: (κsκ −1 )∗ (κsκ −1 ) < Im and so the spectrum of the matrix s is situated inside of unit disk. Therefore the limit nT lim Un = lim x(τ )∗ H(τ )x(τ )dτ n→∞

n→∞

0

1 (x(nT )∗ jmm x(nT ) − x(0)∗ jmm x(0)) 2Re λ θr θr ∗ ∗n ∗ n ∗ ∗ lim a s (θr Im )jmm s a − a (θr Im )jmm a Im Im n→∞ θr a −a∗ (θr∗ Im )jmm Im

=

lim

n→∞

= =

exists and consequently the sequence {Un }∞ n=1 is bounded which proves the inclusion (4.3). Thus the dimension of the lineal M not less then m. To complete the proof of the item a) of our lemma let us show that this dimension not more then m. For this let us build such a lineal N in the space C2m that M ∩ N = {0} and the dimension of N equals to m. Since the matrix G(λ) satisﬁes the inequality (1.12) then, in view of Theorem 2.2, the linear-fractional transform of the form (2.8) which is built by this matrix has the unique ﬁxed point ϕ(λ) in the class of contractive matrices of order m and moreover ϕ(λ) < 1. We have: Im Im G(λ) = l, ϕ(λ) ϕ(λ) where l is some non-degenerate square matrix of order m. So repeating the previous considerations we obtain that for any b ∈ Cm the equality nT

∗

x(τ ) H(τ )x(τ )dτ 0

=

1 Im ∗ ∗ ∗ (b l (Im ϕ(λ) )jmm ln b ϕ(λ) 2Re λ Im −b∗ (Im ϕ(λ)∗ )jmm b) ϕ(λ)

(4.5)

holds where x(τ ) is the solution of the system (4.1) corresponding to the initial Im b, n ∈ N. condition ϕ(λ) By the same consideration as above it can be shown that the spectrum of the matrix l lies strictly out of the unit disk (the inequality (4.4) will be placed by the analogous inequality in which the sign ”<” will be placed by the sign ”>”). From

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this and from the formula (4.5) it follows that if b = 0 then for the considered by us solution the inequality (4.2) fails. Im m b|b∈C . It is clear that the dimenDenote by N the lineal ϕ(λ) sion of N equals to m. From the considerations of the previous paragraph it is clear also that M ∩ N = {0}. Consequently, the dimension of M not more then m and so θr (λ) a | a ∈ Cm . M= Im The item a) is proved. The item b) can be proved analogously.

Recall some known deﬁnitions. Deﬁnition 4.2. A linear relation in the Hilbert space

H ⊕ H.

H is a lineal U in the space

It is well known that if the set of elements of the form 0 ⊕ h belonging to some linear relation L in the space H consists of the unique element 0 ⊕ 0 then L is identiﬁed with some (may be, not closely deﬁned) linear operator acting in the space H for which U serves as the graph. Deﬁnition 4.3. Let L be a linear relation in the Hilbert space linear relation (notation: L∗ ) is the linear relation

H. The conjugate

U ∗ = {f ⊕ g ∈ H ⊕ H | ∀ϕ ⊕ ψ ∈ L (f, ψ) = (g, ϕ)} (here and from now on the round brackets denote the scalar product). Deﬁnition 4.4. A linear relation L in the Hilbert space if L ⊂ L∗ .

H is said to be symmetric

Deﬁnition 4.5. A linear relation L in the Hilbert space H is said to be self-adjoint if L = L∗ . Assume again that the monodromy matrix G(λ) of periodic canonical system (4.1) satisﬁes the condition (1.12). Let us deﬁne the matrix valued function HR (τ ) by the formula: H(τ ), τ 0 (4.6) HR (τ ) = I2m , τ < 0. Our ﬁrst goal is to build the self-adjoint linear relation S in the space L2HR dτ . Consider the system of diﬀerential equations dx(τ ) = ijmm HR (τ )x(τ ), τ ∈ (−∞, +∞). (4.7) dτ Following the paper [9], consider the symmetric linear relation Smin in the space L2HR dτ which is deﬁned as follows: the element f ⊕ g ∈ L2HR dτ ⊕ L2HR dτ belongs to Smin if and only if the measurable functions f ∈ f, g ∈ g with the

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compact supports exist such that the function f is absolutely continuous and for any real τ the equality df (τ ) = ijmm HR (τ )g(τ ) dτ takes place (recall that any element of the space L2HR dτ represents some lineal in the set of measurable vector valued functions because it is deﬁned as a factorspace). Lemma 4.6. The linear relation S = S min (the bar denotes closure) is self-adjoint. Proof. As is well known, in the theory of extensions of symmetric operators (see [1]) the defect indices of an operator D are deﬁned as dimensions of subspaces Ker (D∗ − λI), Im λ = 0. If the operator D is symmetric then these numbers depend only on the sign of the imaginary part Im λ and so any symmetric operator generates the pair of defect numbers (indices of defect) (m, n) where the index m corresponds to the values of λ which are chosen in the open lower half-plane and the index n – to the values of λ which are chosen from the open upper half-plane. It is well known (see Theorem 3 from the Section 101 of the book [1]) that a closed symmetric operator is self-adjoint if and only if the both of its defect numbers equal to zero. For an arbitrary linear relation U in the Hilbert space H and an arbitrary complex number α, as usually, we deﬁne the lineal Ker (U − αIH ) as the set of such h ∈ H that h ⊕ αh ∈ U . After this all that was said in the previous paragraph without any changes can be extended from the case of symmetric linear operators to the case of symmetric linear relations. Thus for the proof of the lemma it is suﬃcient to verify the equalities Ker (S ∗ − λIL2H

R dτ

) = {0},

Im λ = 0.

(4.8)

It is necessary to consider separately the cases Im λ > 0 and Im λ < 0; we shall consider the case Im λ < 0 in detail (the case Im λ > 0 can be investigated analogously). ∗ consists of such According to the paper [13], the linear relation S ∗ = Smin 2 2 elements f ⊕ g ∈ LHR dτ ⊕ LHR dτ for which the measurable functions f ∈ f, g ∈ g exist such that the function f is absolutely continuous and for all real τ the equality df (τ ) = ijmm HR (τ )g(τ ) dτ holds (note that, unlike to the cited above construction of the linear relation Smin , we don’t demand now the compactness of the supports of the functions f and g). Let f ∈ Ker (S ∗ − λIL2H dτ ) ⊂ L2HR dτ . Then the absolutely continuous funcR tion f ∈ f exists such that df (τ ) = iλjmm HR (τ )f (τ ). dτ

(4.9)

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f1 (τ ) where f1 (τ ) and f2 (τ ) are column valued functions of f2 (τ ) the dimension m. Then for τ < 0 we obtain the system of the vector diﬀerential equations   df1 (τ )   = iλf1 (τ ) dτ τ < 0.    df2 (τ ) = −iλf2 (τ ), dτ f1 (0) a1 If = then the solution of this system is the vector-function f2 (0) a2 a1 eiλτ . Thus the inclusion f ∈ L2HR dτ implies the equality a2 = 0 (recall a2 e−iλτ that we consider the case Im λ < 0 and, consequently, Re (−iλ) < 0).But, in view f1 (0) θr (iλ) of Lemma 4.1, this implies also that the vector has a form a f2 (0) Im and so f (0) = 0, f (τ ) ≡ 0. Let f (τ ) =

5. Hamiltonian systems with periodic coeﬃcients and their Weyl functions Consider the Hamiltonian system dx(τ ) = λJm K(τ )x(τ ), τ 0, (5.1) dτ where K(τ ) is a Hermitian positive semideﬁnite locally summable matrix valued function of order 2m satisfying the periodicity condition K(τ + T ) = K(τ ), T > 0 0 −iIm (recall that Jm = , (see (1.7)). iIm 0 Then let the monodromy matrix Q(λ) of the periodic system (5.1) be strictly Jm -expansive in the open right half-plane that is Q(λ)∗ Jm Q(λ) − Jm > 0,

Re λ > 0.

(5.2)

The inequality (5.2) is equivalent to the to the inequality T H(τ ) dτ > 0, 0

see [6], chapter VI, formula (1.16). For any ﬁxed λ from open right half-plane let R be the lineal of initial data of the system (5.1) corresponding to such solutions x(τ ) for which the inequality (4.2) with ∆ = (0, +∞), H(τ ) = K(τ ) is fulﬁlled. Lemma 5.1. For any λ for which Re λ > 0 the dimension of lineal R(λ) equals to m.

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Proof. Let 1 H(τ ) = √ 2

iIm −iIm

Im Im

1 K(τ ) √ 2

IEOT

−iIm Im

iIm Im

Then the canonical system (4.1), as it can be easily seen, has the monodromy matrix 1 1 iIm Im −iIm iIm G(λ) = √ Q(λ) √ , −iIm Im Im Im 2 2 satisfying the condition (1.12). It is clear also that the set of solutions of the system (5.1) is the set of functions of the form 1 −iIm iIm √ x(τ ), τ 0, (5.3) Im Im 2 where x(τ ) is a solution of the system (4.1). Thus in view of the part a) of Lemma 4.1 1 θr (λ) iIm Im m a|a ∈ C (5.4) R(λ) = √ Im −iIm Im 2 Now the assertion of the lemma is clear.

Lemma 5.2. For any λ for which Re λ > 0 there exists the unique basis of the lineal R(λ) the elements of which represent the columns of the matrix of the form Im (5.5) W (λ) where W (λ) is some square matrix of order m and moreover the equality Im Im = d(λ) Q(λ) W (λ) W (λ)

(5.6)

holds for some non-degenerate matrix d(λ) of order m. Proof. As it is seen from the formula (5.4), the set of all matrices with the columns serving as a bases of the lineal R(λ) is 1 1 θr (λ) −iIm iIm i(Im − θr (λ)) √ k (5.7) k = √ Im Im + θr (λ) Im Im 2 2 where k runs the set of all non-degenerate matrices of order m. The matrix Im − θr (λ) is not degenerate because, in view of Lemma 4.1, the norm of the matrix θr (λ) is strictly less than 1. So the elements of the basis of the lineal R(λ) represent the columns of the matrix of the form (5.4) when √ −1 k = −i 2 Im − θr (λ) and only for this value of k. Thus −1 . W (λ) = −i Im + θr (λ) I − θr (λ)

(5.8)

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To prove the relation (5.6), consider the values of the solutions of the system (5.1) corresponding to the initial data which are written in the columns of the matrix (5.5), for τ = T . These values form the columns of the matrix Im Q(λ) . W (λ) On the other hand, in view of periodicity, they belong to the lineal R(λ) and so for some non-degenerate matrix k the equality 1 Im i(Im − θr (λ)) = √ k Q(λ) Im + θr (λ) W (λ) 2 is valid (see (5.7)). Now putting 1 d(λ) = √ i Im − θr (λ) k, 2 we obtain the equality (5.5).

It is natural to name the matrix valued function W (λ) the Weyl function of the system (5.1).

6. The main result Consider the Hamiltonian system (5.1) with λ = i and, analogously to (4.7), let us extend it to the whole real line, that is consider the Hamiltonian system dx(τ ) = iJm KR (τ )x(τ ); dτ

where KR (τ ) =

K(τ ), I2m ,

τ ∈R τ 0 τ < 0.

(6.1)

(6.2)

Let SK be the linear relation in the space L2KR dτ which is generated by the system (6.1) in the same way as the lineal S is generated by the system (4.7). From the relation (5.3) and Lemma 4.6 we have that the relation SK is self-adjoint. Therefore the relation iSK is skew-self-adjoint. Let us pick out from it the operator part. The standard procedure is such: we consider the orthogonal complement Y = iSK Z where Z = {(0, g) | (0, g) ∈ iSK } . Y . This relation is the graph of some operator which we denote by SK 2 It can be easily seen that the subspaces Z and L4m (−∞, 0) of the space L2KR dτ are orthogonal so that

Y ⊃ L24m (−∞, 0) ∩ iSK .

(6.3)

This circumstance allows to construct the unitary Lax-Phillips semigroup starting from the Hamiltonian system (6.1).

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Y Let SK be the operator part of the linear relation SK . It is easy to show (for example, passing to the unitary operators via Cayley transform) that the operator Y is self-adjoint (may be, unbounded). SK Consider the evolutionary equation d f f f Y Y ⊂ L2KR dτ ; dim f = dim g = m, , ∈ Dom SK = −iSK g g g dt (6.4) Y Y where Dom SK is the domain of the operator SK . Y is skew-self-adjoint, it generates some strongly conSince the operator −iSK tinuous unitary semigroup V (t) (see [8], Addendum 1, Theorem 2) which deﬁnes the evolution of the equation (6.4). For this equation when τ < 0 we have:

∂g(t, τ ) ∂f (t, τ ) = −i ∂t ∂τ ∂g(t, τ ) ∂g(t, τ ) = i . ∂t ∂τ Performing the change of variables α = t + τ, β = t − τ, we ﬁnd:

    

∂ (f (t, τ ) + ig(t, τ )) = 0, ∂α  ∂    (f (t, τ ) − ig(t, τ )) = 0. ∂β Let us introduce the following notation: k+ (t, τ )

=

k− (t, τ )

=

(6.5)

1 √ (f (t, τ ) + ig(t, τ )) 2 1 √ (f (t, τ ) − ig(t, τ )) 2

The linear transformation 1 f Im → √ g I m 2

iIm −iIm

f g

(6.6)

deﬁnes the splitting of the Hilbert space L22m (−∞, 0) into the orthogonal sum of two Hilbert spaces L22m (−∞, 0) = K+ ⊕ K− according to the following rule: the element f (·) r= ∈ L22m (−∞, 0) g(·)

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belongs to the summand K+ if the lower subvector of the result of the transformation (6.6) equals to zero almost everywhere, that is 1 √ (f (τ ) − ig(τ ) = 0 a.e. at (−∞, 0) 2 and belongs to K− if its upper subvector equals to zero almost everywhere that is 1 √ (f (τ ) + ig(τ ) = 0 a.e. at (−∞, 0). 2 Since the matrix Im iIm ∆= , Im −iIm which deﬁnes the transformation (6.6), is unitary, the natural splitting of the space L22m into orthogonal sum L22m = L2m ⊕ L2m (the ﬁrst item consists of the vectors with zero lower subvector and the second – of the vectors with zero upper subvector) deﬁnes the natural isomorphisms of each of the spaces K+ , K− to the Hilbert space L2m (−∞, 0). Now, recalling the concept of the Lax-Phillips semigroup (see § 3), we see that, in view of the relations (6.5), the combination 1 (6.7) k− (t, τ ) = √ f (t, τ ) − ig(t, τ ) 2 can be treated as the output wave in the Lax-Phillips scattering scheme (that is − the space K− plays the role of the space L2 ([0, +∞), N ) in the notation of § 3) and, analogously, the combination 1 k+ (t, τ ) = √ f (t, τ ) + ig(t, τ ) (6.8) 2 can be treated as the input wave (that is the space K+ plays the role of the space + L2 ([0, +∞), N ). So we now can treat the C0 -semigroup V (t) which is generated Y as the Lax-Phillips semigroup in terms of § 3. by the operator −iSK Let BK be the main operator of the conservative scattering system H which is canonically connected with the Lax-Phillips semigroup V (t)∗ (note that for this latter semigroup k− (t, τ ) is the input wave and k+ (t, τ ) is the output wave). Let AK = −iBK . Theorem 6.1. The operator AK is the maximal dissipative operator which acts in the space Y L2 (−∞, 0). H = Dom SK 2m Proof. This assertion is trivial because the main operator B of a conservative scattering system with continuous time always has a form B = iA where A is some maximal dissipative operator (see [2]). We shall say that the operator AK is associated with the Hamiltonian system (5.1).

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Theorem 6.2. The characteristic function θAK (µ) of the operator AK can be calculated by the formula −1 (6.9) θAK (µ) = Im − iW (−iµ) Im + iW (−iµ) where W (λ) is the Weyl matrix valued function of the Hamiltonian system (5.1). Proof. As it is follows from the considerations of the § 3, the characteristic function θAK (µ) equals to θρ (−iµ) where θρ (λ) is the transfer function of the conservative scattering system ρ which is connected with the Lax-Phillips semigroup V (t). −1 Let ϕ(−) ∈ Cm , Re λ > 0, ϕ(−) = 0. Let y = λIH − βρ Lρ ϕ(−) (here βρ ,

(−) Lρ are the coeﬃcients of the system ρ). Then (y, ϕ ) ∈ Dβρ ,Lρ [2]. f (τ ) Let y = . Then, by the formula (6.8), g(τ ) 1 ϕ(−) = √ f (0) + ig(0) . 2 On the other hand, the relation βρ y + Lρ ϕ(−) = λy

f (τ ) g(τ ) satisﬁes the equation (5.1) (see [14]) and belongs to the space L2Kdτ (0; +∞). So, by Lemma 5.2, we have the equality f (0) Im = c g(0) W (λ) takes place and hence, by the construction of the semigroup v(t), the pair

for some c ∈ C2m . Now let ϕ(+) = θρ (λ)ϕ(−) that is

ϕ(+) = Nρ y, ϕ(−) .

Then, by the formula (6.7), 1 1 ϕ(+) = √ f (0) − ig(0) = √ Im − iW (λ) C. 2 2 Note that, in view of the formula (5.7), the relation −1 iW (λ) = Im + θr (λ) Im − θr (λ) holds where the matrix θr (λ) is strictly constractive and so the matrix In + iW (λ) is non-generate. So we can write: √ −1 (−) c = 2 Im + iW (λ) ϕ . Thus for any ϕ(−) ∈ Cm we have: −1 (−) ϕ(+) = Im − iW (λ) Im + iW (λ) ϕ and so

−1 θρ (λ) = Im − iW (λ) Im + iW (λ) .

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It remains to recall that θAK (µ) = θρ (−iµ). Theorem 6.3. The characteristic function θAK (µ) is the (unique by Theorem 2.2) contracrive ﬁxed point of the linear-fractional transform which is deﬁned by the matrix 1 Im Im Im iIm G(−iµ) = √ Q(−iµ) Im −iIm −iIm iIm 2 as in the formula (2.8) (here Q(λ) is the monodromy matrix of the system (5.1)). Proof. This theorem immediately follows from the formula (6.9) and the formula (5.6) which shows that the Weyl function W (λ) is the ﬁxed point of the linearfractional transform which is deﬁned by the monodromy matrix Q(λ) as in the formula (2.8). By assumption (see § 5), the monodromy matrix Q(λ) is strictly Jm -expansive in the open right half-plane. It is easy to see that 1 1 Im iIm Im Im √ Jm √ = −jmm Im −iIm −iIm iIm 2 2 and so the matrix G1 (λ) is strictly jmm -contractive and inverse to it matrix G(λ) is strictly jmm -contractive in the open right half-plane. Now note (see [2]) that the matrix valued function θAK (µ) = θρ (−iµ) belongs to the Schur class in the open upper half-plane that is when Im µ > 0 it satisﬁes the inequality θA (µ) 1 k and analytically depends from the parameter µ. Now we are in a position to formulate and prove our main result:

α(λ) β(λ) G(λ) = γ(λ) δ(λ) be an entire matrix valued function of order 2m satisfying the following three conditions: Theorem 6.4. Let

(i) (ii) (iii)

G(λ)∗ jmm G(λ) − jmm > 0, ∗

G(λ) jmm G(λ) − jmm = 0, G(0) = I2m .

Re λ > 0, Re λ = 0,

Then the contractive solution θ(λ) of the algebraic equation θ(λ)β(λ)θ(λ) + θ(λ)α(λ) − δ(λ)θ(λ) − γ(λ) = 0,

Re λ > 0

(6.10)

(such solution exists and unique in view of Theorem 2.2) can be represented as a characteristic function θ(λ) = θAK (iλ)

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where AK is the maximal dissipative operator which is associated with some periodic Hamiltonian system of the form (5.1). In particular, the matrix valued function θ(λ) is analytic in the open right half-plane. Proof. As it was noted above (see the proof of Theorem 2.2), for contractive θ(λ) the equality (6.10) is valid if and only if the matrix θ(λ) is the ﬁxed point of the matrix linear-fractional transform corresponding to the matrix G(λ) as in the formula (2.8). At that, in view of Theorem 2.2, the above mentioned linearfractional transform has the unique ﬁxed point in the class of the contractive matrices and θ(λ) < 1. Let G1 (λ) = G(λ)−1 (as it was noted above, see the paragraph before the formulation of Lemma 4.1 in § 4, the matrix G(λ) is non-degenerate for any complex λ). Then the entire matrix-valued function G1 (λ) will satisfy the conditions (i)-(iii) of our theorem if we replace the matrix jmm by −jmm . It generates by the formula (2.8) the linear-fractional transform for which the matrix θ(λ) serves as the contractive ﬁxed point. Let 1 1 Im Im Im iIm Q(λ) = √ G1 (λ) √ . −iIm iIm Im −iIm 2 2 Then the matrix valued function Q(λ) satisﬁes the condition of the V. P. Potapov theorem (see its formulation in the introduction to this paper) for J = Jm and moreover it is strictly Jm -expansive when Re λ > 0. Thus, it is the monodromy matrix of some Hamiltonian system of the form (5.1) and, in view of the inequality (5.2), all above constructions of §§ 5,6 are valid. So one may construct the maximal dissipative operator AK associated with the system (5.1). Then, repeating the considerations, preceding the formulation of this theorem, we ﬁnd that the characteristic function θAK (iλ) is the ﬁxed point of the linear-fractional transform generated by the matrix G1 (λ) and therefore it is the ﬁxed point of the linearfractional transform generated by the matrix G(λ) = G1 (λ)−1 . It is left to note that θAK (iλ) 1 and, since, by Theorem 2.2, the contractive ﬁxed point of the last transform is unique, we have: θ(λ) = θAK (iλ).

References [1] N. Akhiezer and I. Glazman, Theory of Linear Operators on Hilbert Space I/II, Frederick Ungar Publ., New York, 1961/63. [2] D. Z. Arov and M. A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations and Operator Theory 24 (1) (1996), 1–45. [3] Yu. M. Berezanskiy, Decomposititon Respectively to the Eigen Functions of the Selfadjoint Operators, Naukova Dumka, Kiev, 1965, in Russian.

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[4] M. S. Brodskiy, Unitary operator colligations and their characteristic functions, Uspekhi Mat. Nauk 33 (4) (1978), 141–168, in Russian. [5] N. Dunford and J. T. Schwartz, Linear Operators, part I,II Interscience Publishers, 1963. [6] I. C. Gohberg and M. G. Krein, Theory of Volterra Operators in Hilbert Space and its Applications, Nauka, Moscow, 1967, in Russian. [7] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford University Press, 1995. [8] P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, 1967. [9] M. Lesch and M. Malamud, On the deﬁciency indices and self-adjointness of symmetric Hamiltonian systems, Journal of Diﬀerential Equations, 189 (2003), 556–615. [10] M. A. Nudelman, The Krein string and characteristic functions of maximal dissipative operators, Zapiski nauch. sem. POMI, 290 (2002), 138–167, in Russian. [11] V. P. Potapov, The multiplicative structure of J-contractive matrix functions, Amer. Math. Soc. Transl. 15 (2) (1960), 131–224. [12] L. A. Sakhnovich, The Spectral Theory of Canonical Diﬀerential Systems. Method of Operator Identities, Operator Theory: Advances and Applications, 107, Birkh¨ auser Verlag, Basel, 1999. [13] D. Salamon, Realization theory in Hilbert space, Math. Systems Theory 21 (1989), 147–164. [14] Yu. L. Shmuljan, Invariant subspaces of semigroups and Lax-Phillips scheme, dep. in VINITI, no. 8009–B86, Odessa, 1986. [15] B. Sz¨ okefalvi-Nagy and C. Foia¸s, Harmonic Analysis of Operators on Hilbert Space, Acad´emiai Kiad´ o, Budapest, 1970. [16] E. R. Tsekanovskiy and Yu. L. Shmuljan, The Method of Distributions in the Theory of Extensions of Unbounded Linear Operators, DonSU, Donetsk, 1973, in Russian. M. A. Nudelman Integrated Banking Information Systems P.O. Box 4 Uspenskaya 22 Odessa 65014 Ukraine e-mail: [email protected] Submitted: March 6, 2006 Revised: February 14, 2007

Integr. equ. oper. theory 58 (2007), 301–314 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030301-14, published online April 14, 2007 DOI 10.1007/s00020-007-1488-y

Integral Equations and Operator Theory

Quasi-homogeneous Hilbert Modules Yongjiang Duan Abstract. This paper is to study the quasihomogeneous Hilbert modules and generalize a result of Arveson [3] which relates the curvature invariant to the index of the Dirac operator. Mathematics Subject Classification (2000). 47A13; 47A20; 46H25; 46C99. Keywords. Quasi-homogenous Hilbert modules, curvature invariant, Dirac operator.

1. Introduction In the study of multivariable operator theory, there is a natural approach via Hilbert modules [7, 10]. First we will recall it brieﬂy. Let T = (T1 , · · · , Td ) be a tuple of commuting operators acting on a common Hilbert space H and A = C[z1 , · · · , zd ] be the polynomial ring of d complex variables. One can naturally view H as a Hilbert module over A as follows: P · ξ = P (T1 , · · · , Td )ξ, P ∈ A, ξ ∈ H. Following Arveson’s language [1, 2], a Hilbert module is said to be row contractive if T1 T1∗ + · · · + Td Td∗ ≤ I, equivalently, T1 ξ1 + · · · + Td ξd 2 ≤ ξ1 2 + · · · + ξd 2 , ξ1 , · · · , ξd ∈ H. Such an d-tuple is called a d -contraction, and its defect operator is deﬁned to be ∆ = (I − T1 T1∗ − · · · Td Td∗ )1/2 . The defect operator theory carries key information about the operator theory and structure of submodules, as is shown in [2, 16, 17]. The dimension of the range of the defect operator is called the rank of H, and is denoted by rank(H ). Among all the d -contractions, there is a distinguished one called the d -shift, which has been studied comprehensively by Arveson [1, 2, 3]. To see the d -shift, let us recall a function space called the symmetric Fock space. It is a reproducing This work was partially supported by NKBRPC (2006CB805905) and SRFDP.

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kernel function space, denoted by Hd2 , deﬁned on the unit ball Bd and derived by the reproducing kernel 1 , Kλ (z) = 1 − z, λ d where z, λ = j=1 zj λj . The d -shift is the tuple of the multiplication operators {Mz1 , · · · , Mzd } acting on Hd2 by the coordinate functions. Given a d -contraction T = (T1 , · · · , Td ), we deﬁne an operator transform: Ψ(X) =

d

Ti XTi∗ , X ∈ B(H).

i=1

It is a completely positive map of B(H). If Ψn (I) → 0 (SOT ), as n → ∞, then the associated Hilbert module H is said to be pure. Arveson’s dilation theorem [1, 4] shows that every pure contractive Hilbert module H of ﬁnite or inﬁnite rank r is unitarily equivalent to a quotient F/M , where F = r · Hd2 is the free Hilbert module of rank r and M is a closed submodule of F, so the symmetric Fock space and the d -shift play important roles in the study of a d -contraction. Arveson [2] introduced a notion of curvature invariant of a contractive, ﬁnite rank Hilbert module H. For any z ∈ Bd , deﬁne T (z) = z¯1 T1 + · · · + z¯d Td , and F (z) = ∆(1 − T (z)∗ )−1 (1 − T (z))−1 ∆. Then the curvature invariant K(H) is deﬁned as follows: K(H) = lim (1 − r2 )tr(F (rz))dσ(z), r→1

Sd

where dσ(z) is the normalized Lebesgue measure on the unit sphere Sd in Cd . It is easy to see that K(H) takes real values in the interval [0, r] where r = rank(H ). The curvature invariant is an important invariant, as shown in [2], there are signiﬁcant operator-theoretic implications when the curvature invariant takes values as 0 or rank(H ). Many people studied the curvature invariant from diﬀerent aspects. When H is pure, ﬁnite rank and graded, Arveson [2] has shown that K(H) coincides with the Euler characteristic of H. Recall that the Euler characteristic of H is deﬁned as follows. Let ζ1 , . . . , ζr be a linear basis for ∆H, MH be the set of “linear combinations” MH = span{f1 · ζ1 + · · · + fr · ζr , f1 , . . . , fr ∈ A}. In particular, MH is a ﬁnitely generated algebraic module. By Hilbert syzygy theorem, MH has a ﬁnite free resolution, i.e., an exact sequence of A-modules: 0 → Fn → Fn−1 → · · · → F2 → F1 → F0 → MH → 0,

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where Fj is a free module of ﬁnite rank βj , Fj = A · · ⊕ A . ⊕ · βj times

The alternating sum χ(H) =

n

(−1)j βj does not depend on the particular ﬁnite

j=0

free resolution of MH and is called the Euler characteristic of H. Fang generalized Arveson’s result to polynomially generated modules [12]. Furthermore, Arveson conjectured that the curvature invariant of a pure ﬁnite rank Hilbert module is always an integer. Indeed, Greene, Richter and Sundberg showed that this is true [19]. Moreover, they showed that the curvature invariant can be expressed in terms of the (almost everywhere constant) rank of the boundary values of a certain operator-valued “inner” function that is naturally associated with T via dilation theory. However, it is not clear how it can be computed in terms of the actions of the operator tuple T. Furthermore, to detect whether the curvature invariant is stable under compact perturbation, Arveson introduced a Dirac operator [3], which we will recall below. Let {ei | 1 ≤ i ≤ d} be an orthonormal basis for Cd , the complex Hilbert space of dimension d. Let {ei1 ∧ · · · ∧ eik | 1 ≤ i1 < · · · < ik ≤ d} be an orthonormal basis for the k th exterior power of Cd (denoted by Λk Cd , for 1 ≤ k ≤ d). Set d ΛCd = Λk Cd , where Λ0 Cd = C, then ΛCd forms a Hilbert space spanned by k=0

the orthonormal basis {1} ∪ {ei1 ∧ · · · ∧ eik | 1 ≤ i1 < · · · < ik ≤ d, 1 ≤ k ≤ d}. Deﬁne the canonical creation operators C1 , · · · , Cd on ΛCd as follows: Ci : ξ → ei ∧ ξ, ξ ∈ ΛCd . The operators C1 , · · · , Cd satisfy the canonical anticommutation relations Ci Cj + Cj Ci = 0, Ci∗ Cj + Cj Ci∗ = δij 1. Given a d -contraction T = (T1 , · · · , Td ) acting on H, one deﬁnes the Koszul complex of H as follows: 0 → Ω0 → Ω1 → . . . → Ωd → 0, where Ωk = H ⊗ Λk Cd with cohomological boundary operator B = T1 ⊗ C1 + · · · + Td ⊗ Cd . It is easy to see B 2 = 0. We denote the restriction of B to Ωk by Bk , and hence ranBk−1 ⊆ kerBk . Deﬁne the k -th cohomology space as: H k (H) = kerBk /ranBk−1 , k = 1, . . . , d, H 0 (H) = ker B0 . Definition 1.1. The Dirac operator is a self-adjoint operator D acting on the Hilbert ˜ = H ⊗ ΛCd , D = B + B ∗ . space H

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Let ˜+ = H

˜− = H ⊗ Λ k Cd , H

k even

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H ⊗ Λ k Cd ,

k odd

˜ − , DH ˜− ⊆ H ˜ + . Let D+ be the restriction of D to H ˜ + . Arveson [3] ˜+ ⊆ H then DH proved that when the Hilbert module is pure, ﬁnite rank and graded, the curvature invariant is connected with the index of the Dirac operator as follows: ∗ . (−1)d K(H) = dim ker D+ − dim ker D+

And he conjectured that it is also true in general. The general case remains unknown. In this paper we will give this question an aﬃrmative answer in the case of quasi-homogenous Hilbert modules. Recently, Gleason, Richter and Sundberg showed in [13] that if T is a pure d -contraction with ﬁnite rank acting on a Hilbert space H, then σe (T ) ∩ Bd is contained in the zero set of a nonzero bounded analytic function and (−1)d ind(T − λ) = K(H), λ ∈ Bd \σe (T ),

(1.1)

where σe (T ) is the essential spectrum of T, and Bd is the unit ball of C . If we assume further that H is quasihomogeneous and its associated contraction T is Fredholm, then the above formula coincides with ours. However, we don’t know whether the contraction associated with a quasihomogenenous module is Fredholm until now, which is not needed to be assumed in our theorem. d

2. Quasi-homogeneous Hilbert Modules We now introduce the notion of quasi-homogeneous Hilbert modules. Throughout this paper, we ﬁx K = (K1 , K2 , . . . , Kd ), here each Ki is a positive integer. We will give an example ﬁrst. Example. For K = (K1 , K2 , . . . , Kd ), a polynomial P is said to be K- quasihomogeneous of degree m ≥ 0 if a α z α , z ∈ Cd , (2.1) P (z) = K,α=m

where K, α =

d

Kj αj and where not all aα vanish.

j=1

More generally, for a ﬁnite dimensional Hilbert space L, if in (2.1), each aα belongs to L, then the above deﬁned polynomial P is said to be L-valued K quasihomogeneous of degree m ≥ 0. The zero polynomial is considered to be K quasihomogeneous of degree −∞. The theory of quasi-homogenous polynomials is widely used in the ﬁelds of partial diﬀerential equations and algebraic geometry [6, 20]. An interesting example is that for positive integers k1 , . . . , kd , the polynomial p(z) = z k1 + · · · + zdkd is K-quasihomogeneous, where Ki = k1 · · · ki−1 ki+1 · · · kd for i = 1, . . . , d.

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For K = (K1 , K2 , . . . , Kd ), let Ln denote the space consisting of K- quasihomogeneous polynomials of degree n (n ≥ 0). (Note Ln may be {0} for some n, for example, K = (3, 5), then L7 = {0}.) Then it is easy to see that the symmetric Fock space can be decomposed into sum of orthogonal subspaces as follows: ∞ Ln . (2.2) Hd2 = n=0

Now one can see from this decomposition that there is an positive integer n0 such that d zi Ln−Ki , n ≥ n0 . Ln = i=1

Consequently, the symmetric Fock space can be generated by ﬁnitely many K quasihomogeneous polynomials. We call a submodule of the symmetric Fock space K -quasihomogeneous if it is generated by ﬁnitely many K-quasihomogeneous polynomials. The vectorvalued K -quasihomogeneous submodule is deﬁned analogously. Similarly, a K -quasihomogeneous submodule M of the symmetric Fock space have the decomposition: ∞ Mn , (2.3) M= n=0

where Mn = M ∩ Ln . Indeed, if M is generated by the K -quasihomogeneous polynomials q1 , . . . , ql , with degree d1 , . . . , dl respectively, then Mn =

l

qk Ln−dk .

k=1

From this the decomposition (2.3) follows directly. Now we introduce: Definition 2.1. A pure contractive Hilbert module H of ﬁnite rank r is said to be K -quasihomogeneous if H is unitarily equivalent to Hd2 ⊗ Cr /M , where M is a K-quasihomogeneous submodule of Hd2 ⊗ Cr , generated by ﬁnitely many Kquasihomogeneous polynomials with values in Cr , and r is a positive integer. Note H can be seen as H = Hd2 ⊗Cr M, let PH be the orthogonal projection from Hd2 ⊗ Cr onto H, ICr be the identity operator on Cr , then the canonical contraction T associated with H is: T = (T1 , . . . , Td ), Ti = PH Mzi ⊗ ICr |H , 1 ≤ i ≤ d. Now with Hd2 and M decomposed into (2.2) and (2.3) respectively, we can decompose H into: ∞ Hn , (2.4) H= n=0

where Hn = (Ln ⊗ Cr ) Mn .

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For our main result, the following proposition is needed. This generalizes Proposition 5.4 in [2]. Proposition 2.2. Given a K-quasihomogeneous finite rank Hilbert module H, the following statements are equivalent: (1) H is pure in the sense that its associated completely positive map of B(H), Ψ(B) = T1 BT1∗ + . . . + Td BTd∗ satisfies Ψn (1) ↓ 0 (SOT ) as n → ∞. (2) The algebraic submodule MH = span{f · ∆ζ : f ∈ A, ζ ∈ ∆H} is dense in H. (3) With H decomposed into (2.4), there is an integer n0 such that En = 0 for each n < n0 , where En is the projection from H onto Hn . Moreover, if these conditions are satisfied, then Hn are all finite dimensional for n ∈ Z. Proof. Proofs of (1) ⇒ (2) and (2) ⇒ (3) are analogous to those in [2], so we only prove (3) ⇒ (1). Without loss of generality, assume K1 = min{K1 , . . . , Kd }. Then we have Ψ(1) = Ψ(

∞

∞

Ep ) =

p=n0

Ψ(Ep ) =

p=n0

∞ d

Ti Ep Ti∗ =

p=n0 i=1

d ∞ i=1 p=n0

For 1 ≤ i ≤ d, using Ti En = EKi +n Ti , we have ∞

Ti Ep Ti∗ =

p=n0

where Fj =

∞

∞

Ep+Ki Ti Ti∗ = Fn0 +Ki Ti Ti∗ ,

p=n0

Ek . By

k=j

En = 0 if n < n0 , and Ti En ⊆ En+Ki . It is easy to show that Ti∗ En = 0, for n < n0 + Ki . Consequently, Ψ(1) =

d

Fn0 +Ki Ti Ti∗ Fn0 +Ki

i=1

=

d

Fn0 +K1 Ti Ti∗ Fn0 +K1

i=1

=

Fn0 +K1 Ψ(1)Fn0 +K1

≤

Fn0 +K1 .

By reduction of n, we have Ψn (1) ≤ Fn0 +nK1 =

∞ p=n0 +nK1

Ep .

Ti Ep Ti∗ .

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This gives the conclusion that limn Ψn (1) = 0 immediately.

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3. Curvature Invariant and Dirac Operator associated with Quasi-homogeneous Hilbert Modules Now we reach at the main result in this paper: Theorem 3.1. Given K = (K1 , K2 , . . . , Kd ), let H be a pure K-quasihomogeneous Hilbert module of finite rank and let D be its Dirac operator, then both ker D+ and ∗ ker D+ are finite dimensional and ∗ (−1)d K(H) = dim ker D+ − dim ker D+ .

(3.1)

Remark 3.2. We have not assumed that D is a Fredholm operator, i.e., the canonical d -contraction T is Fredholm. If the Fredholmness of T is assumed, then (1.1) coincides with (3.1). To prove the theorem, we need two lemmas. Lemma 3.3. [12, Theorem 18] If M is a polynomially generated submodule of Hd2 ⊗ CN (N ∈ N) and H = (Hd2 ⊗ CN )/M , then we have χ(H) = K(H).

(3.2)

Thus we immediately know that for every pure K-quasihomogeneous Hilbert module H of ﬁnite rank, its canonical algebraic module MH satisﬁes (3.2). Lemma 3.4. [3, Lemma 2] Every finitely generated A-module M is of finite type, i.e., H k (M ) is finite dimensional, for 0 ≤ k ≤ d. Moreover, e(M ) = (−1)d χ(M ), where e(M ) is defined to be

d

(−1)k dim H k (M ), and H k (M ) is the k-th coho-

k=0

mology space of the Koszul complex of M. Proof of Theorem 3.1. Since H has ﬁnite rank, it means that the range of the defect operator ∆ = (1 − T1 T1∗ − . . . − Td Td∗ )1/2 is ﬁnite dimensional. So the canonical algebraic module MH = span{f (T1 , . . . , Td )ξ : f ∈ A, ξ ∈ ∆H} is a ﬁnitely generated A-module. By Proposition 2.2 (2), MH is dense in H, and ˜ = H ⊗ ΛCd . Let D ∈ B(H) ˜ be the Dirac it follows that MH ⊗ ΛCd is dense in H ∗ are ﬁnite dimensional, and operator. We will show that both kerD+ and kerD+ ∗ dim ker D+ = dim H k (MH ), dim ker D+ = dim H k (MH ). k even

k odd

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From Lemma 3.3, K(H) = χ(H). And χ(H) = χ(MH ) holds by deﬁnition. Using Lemma 3.4, MH is of ﬁnite type, and χ(MH ) = (−1)d e(MH ). This implies that (−1)d K(H) = e(MH ) =

d

(−1)k dim H k (MH ).

k=0

Let Bk be the restriction of B to MH ⊗ Λk Cd , then

H k (MH ) = kerBk /ranBk−1 , k = 1, . . . , d, H 0 (MH ) = ker B0 , thus (−1)d K(H) =

d

(−1)k dim(ker Bk /ranBk−1 ) + dim ker B0 .

(3.3)

k=1

Now we turn to the Dirac operator D. Write the Koszul complex of H as follows: 0 → Ω0 → Ω1 → . . . → Ωd → 0, where Ωk = H ⊗ Λk Cd with cohomological boundary operator B = T1 ⊗ C1 + . . . + Td ⊗ Cd . Since the Dirac operator D = B + B ∗ , and ˜+ = ˜− = H Ωk , H Ωk , k even

k odd

we have ∗ ∗ ˜ ˜+ ⊆ H ˜ − , D+ H ˜ − , D+ ˜ +. H− ⊆ H D+ =D|H

Below we compute dimensions of the kernels of the two operators. Note that B 2 = 0, which implies that D2 = B ∗ B + BB ∗ , thus ker D = ker B ∩ ker B ∗ . Therefore, dim ker D+

=

dim(kerD ∩ Ωk )

k even

=

dim(kerB ∩ kerB ∗ ∩ Ωk )

k even

=

dim(kerB ∩ ranB ⊥ ∩ Ωk )

k even

=

dim(kerBk /ranB k−1 ),

k even

where Bk is the restriction of B to Ωk = H ⊗ Λk Cd . Similarly, ∗ dim ker D+ = dim(kerBk /ranB k−1 ). k odd

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It follows that ∗ dim ker D+ − dim ker D+ d

=

(−1)k dim(kerBk /ranB k−1 ) + dim ker B0 .

(3.4)

k=1

By Lemma 3.4, for 1 ≤ k ≤ d,

dimH k (MH ) = kerBk /ranBk−1 < ∞, dim ker B0 < ∞.

(3.5)

From (3.3) (3.4) and (3.5), it is enough to prove that

dim ker B0 = dim ker B0 , and

dim(kerBk /ranBk−1 ) = dim(kerBk /ranB k−1 ), for 1 ≤ k ≤ d,

(3.6)

where Bk is the restriction of B to MH ⊗ Λk Cd . By the decomposition of H, it is easy to show that ker B0 is dense in ker B0 . Since dim ker B0 < ∞, it follows that

dim ker B0 = dim ker B0 . So it remains to prove (3.6). For (3.6), we claim:

Claim 1. For 1 ≤ k ≤ d, ker Bk is dense in ker Bk . Proof of Claim 1. For 1 ≤ k ≤ d, let Ik = {(i1 , · · · , ik ) | 1 ≤ i1 < · · · < ik ≤ d}. Let

ξ=

ξi1 ...ik ei1 ∧ ei2 ∧ . . . ∧ eik ∈ ker Bk ,

1≤i1
here ξi1 ...ik ∈ H for each (i1 , · · · , ik ) ∈ Ik . From Bk ξ = 0, we have Bξ

=

d

Tp ξi1 ...ik ep ∧ ei1 ∧ . . . ∧ eik

1≤i1

=

k+1

1≤j1 <···<jk+1 ≤d i=1

(−1)i−1 Tji ξj1 ···j i ···jk+1 ej1 ∧ . . . ∧ eji ∧ . . . ∧ ejk+1

= 0. Then the above equality is equivalent to k+1 i=1

(−1)i−1 Tji ξj1 ...j i ...jk+1 = 0,

(3.7)

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for each (j1 , . . . , jk+1 ) ∈ Ik+1 (the term j1 . . . j i · · · jk+1 means that ji is omitted in the expression). Now by Proposition 2.2(3), there is n0 such that for any n < n0 , Hn is {0}. So for any (i1 , · · · , ik ) ∈ Ik , we can write ξi1 ···ik =

∞

(n)

(n)

ξi1 ···ik , ξi1 ···ik ∈ Hn .

n=n0

Since the subspaces Hn are mutually orthogonal, from (3.7) and the fact that Ti Hn ⊆ Hn+Ki , we conclude that ξ ∈ ker Bk if and only if ξ satisﬁes k+1 i=1

(n−Kji )

1 ...ji ...jk+1

(−1)i−1 Tji ξj

= 0,

(3.8)

for each n and (j1 , . . . , jk+1 ) ∈ Ik+1 . Choose a suﬃciently large N ∈ N, and N > n0 . For each (i1 · · · ik ) ∈ Ik , let Ni1 ···ik

ξiN1 ···ik

=

(n)

ξi1 ···ik , where Ni1 ···ik = N +

n=n0

k

Kiq ,

q=1

and

ξN =

ξiN1 ···ik ei1 ∧ · · · ∧ eik .

1≤i1 <···
By Proposition 2.2, Hn is a ﬁnite dimensional subspace of MH for each n, so we have ξN ∈ MH ⊗Λk Cd . Moreover, using (3.8) we can easily check that ξN ∈ ker Bk . Indeed, given (j1 , . . . , jk+1 ) ∈ Ik+1 , when n>N+

k+1

Kji ,

i=1

then all the terms of the left of (3.8) vanish; when n≤N+

k+1

Kji ,

i=1

(3.8) is obviously satisﬁed. Now let N → ∞, then ξN → ξ. This proves Claim 1. Now we come back to prove (3.6). Deﬁne a map

Q : X = ker Bk /ranBk−1 → Y = ker Bk /ranB k−1 as follows:

Q([ξ]) = [ξ], = ξ + ranB k−1 ∈ Y. Since ker B where ξ ∈ ker Bk , [ξ] = ξ + ranBk−1 ∈ X and [ξ] k is dense in ker Bk , it follows that

ranQ = (ker Bk + ranB k−1 )/ranB k−1

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is dense in Y. Now assume that {[ξi ] | i ∈ Λ} is an basis for X, here Λ is a ﬁnite m m It = αi [ξ]. αi [ξi ], and thus [ξ] set. Then for each [ξ] ∈ X, we have [ξ] = i=1

follows that

k=1

i ] | i ∈ Λ} = ranQ. span{[ξ Combined with (3.5) it yields that

dim ranQ ≤ dim(kerBk /ranBk−1 ) < ∞. Now let l = dim(kerBk /ranB k−1 ), since ranQ is dense in Y, we conclude that l < ∞. Thus to prove (3.6), it suﬃces to show that Q is injective. For this purpose, we claim:

Claim 2. ranB k−1 ∩ ker Bk = ranBk−1 , 1 ≤ k ≤ d. Assume the claim has been proved, we will easily get the conclusion that Q is injective. Indeed, let

ξ ∈ ker Bk , [ξ] ∈ ker Bk /ranBk−1 and Q[ξ] = 0,

then we have ξ ∈ ranB k−1 , hence ξ ∈ ranB k−1 ∩ ker Bk . Then by claim 2, we have ξ ∈ ranBk−1 , thus [ξ] = 0. The desired conclusion follows immediately. Hence it suﬃces to prove Claim 2.

Proof of Claim 2. Since ranBk−1 is dense in ranBk−1 , it follows that ranB k−1 = ranB k−1 . So we only prove that

ranB k−1 ∩ ker Bk = ranBk−1 .

(3.9)

One readily see that ranBk−1 ⊆ ranB k−1 ∩ ker Bk . Now for each ξ ∈ ranB k−1 ∩ ker Bk , there exists a sequence {ξn } ⊆ ranBk−1 such that ξ = limn→∞ ξn . Since ξ ∈ ker Bk ⊆ MH ⊗ Λk Cd , it has the form as follows:

ξ=

Ni1 ···ik

1≤i1
where

(m)

ξi1 ···ik ei1 ∧ · · · ∧ eik ,

(m)

(i1 , · · · , ik ) ∈ Ik , ξi1 ···ik ∈ Hm , Ni1 ···ik ∈ N.

Since Ik is a ﬁnite set, there exists a ﬁxed N ∈ N such that N ≥ Ni1 ···ik for each (i1 , · · · , ik ) ∈ Ik . Since ξn ∈ ranBk−1 , there exists ηn ∈ MH ⊗ Λk−1 Cd such that ξn = Bk ηn . Write ξn and ηn as follows: ξn = ξn,i1 ···ik ei1 ∧ · · · ∧ eik 1≤i1
and ηn =

1≤j1 <···<jk−1 ≤d

ηn,j1 ...jk−1 ej1 ∧ · · · ∧ ejk−1

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where

∞

ξn,i1 ···ik =

IEOT

(m)

ξn,i1 ···ik , ηn,j1 ...jk−1 =

m=n0

with

(m) ξn,i1 ···ik ,

ξn

(m) ηn,j1 ...jk−1

∞

(m)

ηn,j1 ...jk−1 , ,

m=n0

∈ Hm . Then we have

= Bk ηn

= Bk (

ηn,j1 ...jk−1 ej1 ∧ · · · ∧ ejk−1 )

1≤j1 <···<jk−1 ≤d

=

k

1≤i1
(−1)l−1 Til ηn,i1 ...i l ...ik ei1 ∧ · · · ∧ eik ,

therefore, ∞

(m) ξn,i1 ···ik

=

m=n0

∞ k m=n0 l=1

(m)

(−1)l−1 Til ηn,i

1 ...il ...ik

.

From the decomposition of H, we have (m)

ξn,i1 ···ik =

k l=1

(m−Kil ) .

1 ...il ...ik

(−1)l−1 Til ηn,i

Let

Nj1 ···jk−1 = N +

k−1

Nj1 ···j

Kjq , ηn,j1 ...jk−1 =

ηn =

(p)

ηn,j1 ...jk−1 ,

p=n0

q=1

and

k−1

ηn,j1 ...jk−1 ej1 ∧ · · · ∧ ejk−1 , and ξn = Bk ηn .

1≤j1 <j2 <...<jk−1 ≤d

By a careful observation, we see that ξn is in fact a truncation from ξn . Since ξ = limn→∞ ξn , and Hn ⊥ Hm if n = m, it follows that ξ = limn→∞ ξn . d Let q = N + Ki , then ηn belongs to the ﬁnite dimensional subspace S = i=1

(⊕qm=n0 Hm ) ⊗ Λk−1 Cd , for each n. We conclude that Bk S is closed, because it is ﬁnite dimensional. Note ξn ∈ Bk S, it follows that ξ ∈ ranBk−1 , which shows that ranB k−1 ∩ ker Bk ⊆ ranBk−1 . We thus completed the proof of Theorem 3.1.

Acknowledgments. This paper was written under the direction of Prof. Kunyu Guo. The author would like to thank Prof. Kunyu Guo for many valuable discussions and suggestions. The author also wish to express his gratitude to the referee for making him aware of the related result of Gleason, Richter and Sundberg, and for suggestions which helped to make this paper more readable.

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References [1] W. Arveson, Subalgebras of C ∗ -algebras III: Multivariable operator theory, Acta Math. 181(1998), 159–228. [2] W. Arveson, The curvature invariant of a Hilbert module over C[z1 , . . . , zn ], J. Reine. Angew. Math. 522 (2000), 173–236. [3] W. Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189 (2002), 53–79. [4] W. Arveson, The curvature invariant of a Hilbert module over C[z1 , · · · , zn ], Proc. Nat. Acad. Sci. (USA) 96 (1999), 11096–11099. [5] W. Arveson, p-summable commutators in dimension d, J. Oper. Theory. 54 (2005), 101–117. [6] R. Braun, R. Meise and B. Taylor, characterization of the the homogenous polynomails P for which (P + Q)(d) admits a continous linear right inverse for all lower order perturbations Q, Pacfic. J. Math. 192 (2000), 201–218. [7] X. Chen and K. Guo, Analytic Hilbert modules, π-Chapman & Hall/CRC Research Notes in Math. 433, 2003. [8] R. Curto, Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129–159. [9] R. Curto, Applications of Several Complex Variables to Multiparameter Spectral Theory, Surveys of Some Results in Operator theory (J. B. Conway and B. B. Morrel,eds), Pitman Research Notes in Math. 192, 1988. [10] R. Douglas and V. Paulsen, Hilbert modules over function algebras, Pitman Research Notes in Math. 217, 1989. [11] D. Eisenbud, Commutative Algebra. With a View toward Algebaic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995. [12] X. Fang, Hilbert polynomials and Arveson’s curvature invariant, J. Funct. Anal. 198 (2003), 445–464. [13] J. Gleason, S. Richter and 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. [14] K. Guo, Characteristic spaces and rigidity for Hilbert modules, J. Funct. Anal. 163 (1999), 133–151. [15] K. Guo, Equivalence of Hardy submodules generated by polynomials, J. Funct. Anal. 178 (2000), 343–371. [16] K. Guo, Defect operators, defect functions and defect indices for analytic submodules, J. Funct. Anal. 213 (2004), 380–411. [17] K. Guo, Defect operators for submodules of Hd2 , J. Reine. Angew. Math. 573 (2004), 181–209. [18] K. Guo and D. Zheng, Invariant subspaces, quasi-invariant subspaces and Hankel operators, J. Funct. Anal. 187 (2001), 308–342. [19] D. Greene, S. Richter and C. Sundberg, The structure of inner multiplications on spaces with complete Nevanlinna Pick kernels, J. Funct. Anal. 194 (2002), 311–331. [20] J. Milnor and P. Orlik, Isolated singularities defined by weighted homogenous polynomials, Topology. 12 (1973), 19–32.

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Duan

IEOT

[21] J. Taylor, A joint Spectrum for Several Commuting Operators, J. Funct. Anal. 6 (1970), 172–191. Yongjiang Duan School of Mathematics Fudan University Shanghai, 200433 People’s Republic of China e-mail: [email protected] Submitted: March 21, 2006 Revised: November 15, 2006

Integr. equ. oper. theory 58 (2007), 315–339 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030315-25, published online April 14, 2007 DOI 10.1007/s00020-007-1489-x

Integral Equations and Operator Theory

Collinear Systems and Normal Contractive Projections on JBW∗ -Triples Remo V. H¨ ugli Abstract. Given a family {xk }k∈K of elements xk in the predual A∗ of a JBW∗ -triple A, such that the support tripotents ek of xk form a collinear system in the sense of [31], necessary and suﬃcient criteria for the existence n of a contractive projection from A∗ onto the subspace lin{xk : k ∈ K} are provided. Preparatory to these results, and interesting in itself, is a set of necessary and suﬃcient algebraic conditions upon a contractive projection P on A for its range P A to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space. Mathematics Subject Classification (2000). Primary 17C65; Secondary 47D27. Keywords. JBW∗ -triple, collinearity, contractive projection.

1. Introduction In recent years, considerable eﬀort has been devoted to the investigations of contractive projections on an operator algebra and its related Banach spaces. The question of whether or not there exists a contractive projection onto a given subspace of a Banach space is of fundamental signiﬁcance. In the presence of algebraic structure on the Banach space in consideration, it is natural to tie this question to algebraic conditions upon the given subspace. Among the extensive literature on these topics we refer the reader to [4] [6] [7] [8] [22] [30] [33] [38]. The aim of this article is to investigate normal contractive projections in connection with certain algebraic conditions on generalized operator algebras. It has been observed earlier that the structures known as JB∗ -triples and their weak∗ closed analogues, JBW∗ -triples, provide a natural setting for studying contractive projections or normal contractive projections. As shown by Kaup in [27] and independently by Stach` o in [34], the range P A of a contractive projection P on a Supported by the Irish Research Council for Science, Engineering and Technology, Grant No. R 9854.

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JB∗ -triple A is itself a JB∗ -triple in a canonical way. The algebraic structure on A is given by a ternary product {. . .} : A × A × A → A, that on P A by P {. . .}. This generalizes earlier results by Choi and Eﬀros [5], Eﬀros and Størmer [18], Friedman and Russo [21] and others. For further reading on the subject and its connections to holomorphy see for example [23] [26] [28] [31] [35] [36]. In joint work with Edwards and R¨ uttimann [11] [12] contractive projections have been studied by the author for the case in which the global vector space is the Banach predual A∗ of a JBW∗ -triple A. By standard Banach space theory, the adjoints of the contractive projections on A∗ are precisely the normal contractive projections on A. A main result in [12] states that, for any family {Pk }k∈K of contractive projections on A∗ , such that, for k = j, all elements x in Pj (A∗ ) and y in Pk (A∗ ) are L-orthogonal in that x ± y = x + y, there exists a contractive n projection onto the norm closed subspace k∈K Pk (A∗ ) of A∗ . In particular, there is a contractive projection from A∗ onto the norm-closed subspace spanned by any family {xk }k∈K of pairwise L-orthogonal elements in A∗ . Moreover, such a projection is explicitly given in terms of the support tripotents ek in A of the elements xk . By [17] and [19], the L-orthogonality of the elements xj and xk in A∗ is equivalent to the algebraic orthogonality of their respective support tripotents ej and ek in A, or in the terminology of [31], to {ek }k∈K being an orthogonal system. For details on these and related results, see also [25]. In this article we investigate the similar situation in which the support tripotents ek of the elements xk form a collinear system in the sense of [31]. Under this global assumption, we obtain criteria upon {ek }k∈K and {xk }k∈K which are equivalent to the existence of n a contractive projection on A∗ with range G = lin{xk : k ∈ K} . One set of these conditions involves only the triple product on A and its duality with A∗ . Moreover, w∗

a necessary condition is that G is a Hilbert space with dual H = {ek : k ∈ K} and H is a subtriple of A. Further conditions are formulated in terms of GLprojections on A∗ , which were introduced in [11]. Neal and Russo classiﬁed the atomic contractively complemented subspaces of B(H) up to complete isometry using the theory of operator spaces [30]. The subtriple H given above is an example of an atomic JBW∗ -triple. Our considerations proceed within the theory of of JBW∗ -triples and their preduals, and hence provide detailed information related to the general triple structure of these spaces. The article is organized as follows. In Section 2 some well known facts about JB∗ -triples and JBW∗ -triples are reviewed. Section 3 is devoted to the triple structure of a Hilbert space H. It is shown that a certain set of isometries, obtained from the triple product on H, acts transitively on the unit-sphere of H. In Theorem 3.3, pivotal in proving the main theorems, that observation is generalized to JB∗ -triples which are generated by a collinear system of tripotents. Of similar signiﬁcance is Theorem 4.1 in Section 4. It provides algebraic conditions on a contractive projection P on a JB∗ -triple A which are necessary and suﬃcient for P A to be a subtriple. This result draws heavily on the conditional expectation formulas in [20], and [27]. In Section 5, the principal question as outlined above is

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Normal Contractive Projections on JBW*-Triples

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answered in Theorem 5.3, Theorem 5.2 and Theorem 5.4. Further connections between contractive projections and Hilbert spaces are explored in Theorem 5.7 and Theorem 5.8. When dealing with collinearity, the technical diﬃculties are considerably greater than those arising from orthogonality. In particular, the existence of the sought-after projections is automatically ensured by orthogonality, but not by collinearity of {ek }k∈K . Therefore, our main theorems provide non-trivial information, even in the case of ﬁnite dimensional spaces, as can be seen from Example 5. The relation of collinearity was considered for the purpose of a classiﬁcation of Jordan-triples by Dang and Friedman [9], Horn [24] McCrimmon and Meyberg [29], and by Neher [31]. Moreover, Edwards and R¨ uttimann [16] and Wright [37] used collinearity to represent the relation of quantum-decoherence of states, and JBW∗ -triples are used to describe non-classical statistical systems.

2. Preliminaries A JB∗ -triple is a complex Banach space A endowed with a triple product {. , . , .} : A × A × A → A, which has the following properties, the axioms of the theory: (A1) The expression {a, b, c} is symmetric and linear in the variables a and c and is conjugate linear in b. (A2) For all elements a and b of A, the linear maping c → {a, b, c} on A denoted by D(a, b), satisﬁes the Jordan triple identity, D(a, b){c, d, e} = {D(a, b)c, d, e} + {c, b, D(a, b)e} − {c, D(b, a)d, e}, (A3) For all elements a of A, the linear operator D(a, a) on A has nonnegative spectrum (cf. [2]), and norm equal to a2 . (A4) For all real numbers t, the linear operator exp(itD(a, a)) is an isometry of A. By [26] Proposition 5.5 and [24] Proposition 2.4, two JB∗ -triples are isometrically isomorphic as Banach spaces if and only if they are triple isomorphic. Hence the group of all triple automorphisms, denoted Aut(A), coincides with the group of all bijective linear isometries of the JB∗ -triple A. If A is also the dual of a Banach space A∗ , then A is said to be a JBW ∗ -triple. To enhance the clarity of notation in later calculations, we will henceforth write (x · a) for the dual pairing of the elements x in A∗ and a in A. A subspace B of A is a subtriple if {B, B, B} is a subset of B, which is the case if and only if for all b ∈ B the element {b, b, b} lies in B. For an element b of A, the conjugate linear operator Q(b) : A → A is deﬁned by Q(b)a = {b, a, b}. The operators D(a, b) and Q(b) are norm-continuous on the JB∗ -triples and weak∗ continuous on JBW∗ -triples. On the latter, the triple product is separately weak∗ continuous [1] [28]. An element u of A is said to be a tripotent if {u, u, u} = u. The set of tripotents of A is denoted by U(A). Let j, k and l be equal to 0, 1 or 2. For each

318

H¨ ugli

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tripotent u of A, the norm- and weak∗ -continuous projections P2 (u) = Q(u)2 , P1 (u) = 2(D(u, u) − Q(u)2 ), P0 (u) = idA − 2D(u, u) + Q(u)2 are referred to as the Peirce projections corresponding to u. It can be seen that P0 (u) + P1 (u) + P2 (u) equals the identity idA on A and that if j = k, then Pj (u)Pk (u) equals zero. The ranges, Ak (u) of Pk (u) are weak∗ -closed subtriples of A, referred to as the Peirce spaces of u. Moreover, for all elements a of A, k (2.1) if and only if D(u, u)a = a. a ∈ Ak (u) 2 Extensive use will be made of the Peirce rules, Aj−k+l (u) if j − k + l ∈ {0, 1, 2} {Aj (u), Ak (u), Al (u)} ⊆ (2.2) {0} else {A2 (u), A0 (u), A}

= {A0 (u), A2 (u), A} = {0}.

(2.3)

A pair u, v of tripotents in A is said to be orthogonal, denoted u⊥v, if u ∈ A0 (v) and v ∈ A0 (u). The relation ⊥ is symmetric, and u⊥v is equivalent to D(u, u)v = 0, to D(v, v)u = 0 and to D(u, v) being identically zero on A. Moreover v is said to be less than or equal to u, denoted v ≤ u, if (u − v) ∈ U(A) and (u − v)⊥v. The relation ≤ is a partial order on the set U(A) [28]. If A is a JBW∗ -triple, then, for each element x in the predual A∗ of A, there exists the support tripotent ex of x which is the smallest of all tripotents u in A with the property that (u · x) = x. Also, x lies in A2 (ex )∗ [19]. For a non-empty subset G of A∗ , the support space s(G) of G is deﬁned to be the weak∗ -closed subspace lin{ex : x ∈ G}

w∗

of A [11].

A pair u, v of tripotents is said to be collinear, denoted u v, if u ∈ A1 (v) and v ∈ A1 (u). By (2.1) the relation u v holds if and only if 1 1 (2.4) D(u, u)v = v and D(v, v)u = u. 2 2 A family {uk }k∈K of tripotents is said to be a collinear system if the relation uk ul holds whenever k = l. Observe that a collinear system not equal to {0} is linearly independent. When G and H are Hilbert spaces, the set B(G, H) of bounded linear operators from G to H is a JBW∗ -triple with triple product, deﬁned for elements a, b, c in B(G, H) by 1 {a, b, c} = (ab∗ c + cb∗ a). (2.5) 2 A JBW∗ -triple isomorphic to B(G, H) is said to be rectangular. When G and H have ﬁnite dimension, i.e. A is represented as a matrix algebra, examples of collinear tripotents are provided by distinct matrix units ui,j and uk,l , i.e. matrices

Vol. 58 (2007)

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with entry 1 at position (i, j) and (k, l), respectively, and zero elsewhere. Then ui,j uk,l if either i = k or j = l. Further examples of JBW∗ -triples are provided by W∗ -algebras and JBW∗ -algebras, the spin factors, obtained from Hilbert spaces 8 . The latter are the equipped with a conjugation, or by the bi-Cayley numbers C1,2 8 1 by 2 matrices with entries in the Cayley numbers C over the ﬁeld of complex numbers. For more details on these structures, see for example [9] [24] [29] [31]. We conclude this section with a result which is easily obtained from the aforesaid. Proposition 2.1. Let u be a tripotent element in the JBW∗ -triple A. For k = 0, 1, 2, denote by Pk the pre-adjoint Pk (u)∗ of the Peirce projection Pk (u). Then the range Pk A∗ of Pk is the eigenspace of the pre-adjoint D(u, u)∗ of D(u, u), with corresponding eigenvalue k/2. Proof. Using the Peirce-rules it can be seen that D(u, u) commutes with Pk (u) = Pk∗ . Let x be an element of Pk A∗ . By (2.1), for all a in A, (D(u, u)∗ x · a) = (Pk∗ x · D(u, u)a) = (x · Pk∗ D(u, u)a) = (x · D(u, u)Pk∗ a) k (x · a). = 2 Therefore, x lies in the eigenspace of D(u, u)∗ with eigenvalue k/2. Conversely, suppose that D(u, u)∗ x equals (k/2)x. Notice that D(u, u)∗ = P2 + 21 P1 and that P2 + P1 + P0 is the identity on A∗ . Hence, 1 k P2 x + P1 x = (P2 x + P1 x + P0 x). 2 2 Since Pi Pj = 0, setting k equal to 0, 1 or 2 implies, in each case, that x = Pk x, as required.

3. Hilbert spaces Since the triple structure of Hilbert spaces plays a signiﬁcant part in subsequent considerations, it is necessary to establish some preliminary results on Hilbert spaces. Let H be a complex Hilbert space with scalar product . , . : H ×H → H which is linear in the ﬁrst and conjugate linear in the second variable. For elements a, b and c in H, let the triple product {a, b, c} be deﬁned by 1 (3.1) {a, b, c} = (a, b c + c, b a) . 2 In this way, the Hilbert space H is a JBW∗ -triple. We remark that another triple product can be deﬁned on H to obtain what is known as a spin triple. This, however, requires the presence of additional structure, such as a conjugation, as well as a new norm on H. We will only be working with the triple product described above. Notice that (3.1) concurs with (2.5) for the case in which G equals C, i.e. H is identyﬁed with the rectangular triple B(C, H).

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Denote the unit sphere of H by S1 (H). It is straightforward from Equation (3.1) that S1 (H) ∪ {0} coincides with the set U(H) of tripotents of H, and that the relation of collinearity on U(H) is the usual Hilbert-orthogonality restricted to U(A). For an non-empty set S, let l2 (S) be the Hilbert space of all functions f : S → C, such that s∈S |f (s)|2 < ∞, equipped with the inner product f, g = 2 s∈S f (s)g(s), for f, g ∈ l (S). The following lemma provides an algebraic characterization of Hilbert spaces among the JB∗ -triples and JBW∗ -triples. Similar results may be obtained from classiﬁcation theory, as carried out e.g. in [31] and [9]. Lemma 3.1. Let C be a collinear system in the JB∗ -triple A. Denote by H the n closed subspace linC spanned by C. Then the following conditions are equivalent: (1.) The subspace H is a subtriple of A. (2.) Either |C| ≤ 2, or, for any three distinct elements u, v and w of C, {u, v, w} = 0. (3.) The subspace H is a Hilbert space with orthonormal basis C, i.e. H ∼ = l2 (C), and the restriction of the triple product of A to H coincides with the triple product given by (3.1) on H. If these hold, and if in addition A is a JBW∗ -triple, then H is also weak∗ -closed, hence a JBW∗ -subtriple of A. Proof. (1.) ⇒ (2.): Suppose that C has at least three elements, and choose u, v and w in C to be distinct. The assumption of collinearity of C implies that, H ⊆ Cv ⊕ A1 (v) ⊆ A2 (v) ⊕ A1 (v). Since H is a subtriple, {u, v, w} ∈ A2 (v) ⊕ A1 (v). The Peirce rules (2.2), (2.3) imply that {u, v, w} ∈ A0 (v). It follows that {u, v, w} = {0}. (2.) ⇒ (1.): It is easily seen that the arguments can be simpliﬁed when |C| ≤ 2. Therefore, assume again that |C|≥ 3. An element a in H can be written as the norm convergent sum a = k∈N αk uk , for uk ∈ C. The assumptions and (2.2),(2.3) imply that, for distinct indices j, k and l in N, {uj , uk , ul } = 2 {uj , ul , ul } =

{uk , uj , uk } = 0, 2 {ul , ul , uj } = {uj , uj , uj } = uj .

(3.2) (3.3)

Since {a, a, a} is a (possibly inﬁnite) linear combination of these expressions, and is well deﬁned in A, it follows that {a, a, a} lies in H. Hence H is a subtriple of A, which proves (1.). (2.) ⇒ (3.): The equalities (3.2) and (3.3) determine the triple product on H completely. In particular, for any ﬁnite subset F ⊆ C, linF is a JB∗ -triple, and as such is isomorphic to l2 (F ). Isomorphic JB∗ -triples are also isometrically

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isomorphic as Banach spaces. Hence, for any ﬁnite subset F of C, linF is isometric to the Hilbert space l2 (F ). This entails the norm of an arbitrary a ∈ H to be 1 αk uk = ( |αk |2 ) 2 . a = k∈N

k∈N

It follows that H ∼ = l (C), proving (3.). 2

(3.) ⇒ (2.): This is obvious from (3.1). Suppose that A is a JBW∗ -triple and that (1.), (2.) and (3.) hold. Then, for all elements a and b in H, Q(a)(b) = {a, b, a} = a, b a ∈ Ca.

(3.4)

∗

By separate weak -continuity of the triple product, (3.4) is preserved when a w∗ and b are chosen in the weak∗ -closure H of H. When a and b are linearly w∗ independent, (3.1) and (3.4) also imply that Ca ⊕ Cb is a subtriple of H . Up to triple-isomorphism, there are only two diﬀerent JBW∗ -triples of dimension 2, namely C2 with the componentwise triple product, and C2 as a Hilbert space. The former contains elements for which (3.4) does not hold. Therefore Ca ⊕ Cb is (isometrically) equal to the Hilbert space C2 . It follows that every two dimensional w∗ w∗ subspace of H is a Hilbert space. Clearly H is complete, hence is itself a w∗ Hilbert space. To see that H = H , extend C to an orthonormal basis C ∪ C w∗ w∗ of H . For every u ∈ C ∪ C , the linear functional xu , deﬁned for a ∈ H by w∗ ∗ ∗ (xu ·a) = a, u is weak -continuous on H . Therefore, xu has a weak -continuous ˜u can even be chosen to have the same extension x ˜u on A. We remark that by [3], x w∗ norm as xu . However, if u is chosen in C , then x˜u annihilates H, hence also H . This contradicts the obvious equality (˜ xu · u) = 1. Consequently, C is empty and w∗ H coincides with H . The inner derivations of a JB∗ -triple A, that is the mappings of the form D(a, b) − D(b, a) and their exponentials, the inner automorphisms, have been investigated in numerous works. See e.g. [28], [35]. In the following lemma, a symmetry property of the unit sphere S1 (H) of the Hilbert space H with respect to inner automorphisms of the form exp itD(a, a) is established. The result may be deduced from the general theory of inner automorphisms. Since we need this result only for the special case of Hilbert spaces, we state it separately and give an independent, more elementary proof. Lemma 3.2. Let H be a Hilbert space, equipped with the triple product, given by (3.1). Let S1 (H) and S1 (C) be the unit spheres of H and of the complex plane C respectively. Then, the set E = {σ exp itD(a, a) : σ ∈ S1 (C), a ∈ H, t ∈ R} consists of linear isometries of H and acts transitively on S1 (H).

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Proof. Suppose ﬁrst that H is two-dimensional. Let Bs = {b1 , b2 } be the standard basis of H, coordinatized in the usual way. For any complex number λ of modulus one, let a = a(λ) be the element of H, deﬁned by 1 1 a = √ . λ 2 Denote the operator D(a, a) by Dλ , to indicate its dependence of λ. An elementary caclulation shows that, with respect to Bs , the operator exp itDλ , is given by ¯ it − eit/2 ) 1 λ(e eit + eit/2 . exp itDλ = λ(eit − eit/2 ) eit + eit/2 2 In particular, the unit vector b1 is mapped to 1 (exp itDλ )(b1 ) = (exp itDλ ) = 0

1 2

eit + eit/2 λ(eit − eit/2 )

.

(3.5)

This vector is of norm one, for all reals t. The modulus of its ﬁrst component is i |eit + e 2 t |/2 and attains all values between 0 and 1 when t runs through R. If c is any vector with components γ1 and γ2 , and if c has norm one in H, then |γ2 | equals 1 − |γ1 |2 and t can be chosen such that 1 it 1 it |e + eit/2 | = |γ1 |, |e − eit/2 | = |γ2 |. and (3.6) 2 2 If γ1 = 0, then σ(exp 2πiDλ )b1 equals c, for some σ ∈ S1 (C). Otherwise set λ = γ2 (eit + eit/2 )/γ1 (eit − eit/2 ). The equations (3.6) and (3.5) imply that 2γ1 2γ1 1 , and |λ| = 1 = it (exp itDλ ) = c. 0 e + eit/2 eit + eit/2 This shows that, exp itDλ acts transitively on S1 (H), up to multiplication by a complex number of modulus one. The operators D(a, a) and exp itD(a, a) are deﬁned on any Hilbert space, in fact, on any JB∗ -triple containing the element a. In the case when H is of arbitrary dimension, and b and c are any two elements of H such that b = c, the above arguments can applied to the subspace lin{b, c} which is also a subtriple of H. Hence, there exist elements t of R, σ of S1 (C), and a of lin{b, c}, such that σ(exp itD(a, a))b = c, and σexp itD(a, a) is an isometry of the whole space H. The two foregoing lemmas can be combined as follows. Theorem 3.3. Let C be a collinear system in a JB∗ -triple A, such that the subspace n H = linC is a subtriple of A. Then, (1.) The subtriple H is a Hilbert space with orthonormal basis C, and the triple product on H given by (3.1) conicides with the restriction of the triple product of A to H.

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(2.) The set E = {σ exp itD(a, a) : σ ∈ S1 (C), a ∈ H} consists of linear isometric triple isomorphisms on the whole space A and acts transitively on the unit sphere S1 (H) of H. (3.) Either |C| ≤ 2, or, for any three distinct elements u, v, w of C, the product {u, v, w} vanishes. If A is a JBW∗ -triple, then H is also weak∗ -closed in A, and therefore, is a JBW∗ subtriple of A. Proof. For any element a of H, the linear mapping exp(itD(a, a)) is deﬁned on the whole space A, and, by (A4), is an isometry on A. The remaining assertions follow from Lemma 3.1 and Lemma 3.2.

4. Contractive projections By a projection we always mean a linear mapping P on a vector space E which is such that P 2 = P . When E is endowed with a norm ., and if P is such that, for all elements x of E, P x ≤ x, then P is said to be contractive. In this case, the range P ∗ E ∗ of the adjoint projection P ∗ is isometrically isomorphic to the dual (P E)∗ of the range P E of P . Observe also that, if P and S are continuous projections on E, such that P E = SE and P ∗ E ∗ = S ∗ E ∗ , then P and S coincide. The set of all bounded linear mappings on E is denoted by B(E). For an arbitrary set K, let K f denote the set of all ﬁnite subsets of K, partially ordered by setinclusion. Given a family {Pk }k∈K of projections, or of general elements in B(E), the formal sum k∈K Pk is said to be convergent in the strong operator topology or SOT-convergent in B(E) if there exists an element P of B(E) such that, for each x of E, the net { k∈F Pk x : F ∈ K f } converges to P x in norm. A pair of elements x, y in E is said to be L-orthogonal, denoted xy, if x±y equals x + y. The L-complement F of a non-empty subset F of E is deﬁned to be the set F = {y ∈ E : x y ∀x ∈ F }. A contractive projection P on E is said to be a GL-projection if the L-complement (P E) of its range P E is a subset of its kernel ker P . The set of all GL-projections on E is denoted by GL(E). The results of [27] and [34] show that the range P A of a contractive projection P on a JB∗ -triple A is itself a JB∗ -triple when equipped with the restricted triple product {. . .}P A , deﬁned for elements a, b and c in P A by {a, b, c}P A = P {a, b, c}.

(4.1)

When A is a JBW∗ -triple with predual A∗ , and if P is the adjoint P = R∗ of a contractive projection R on A∗ , then P A is a JBW∗ -triple with product given by

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(4.1). Further algebraic relations, referred to as conditional expectation formulas hold. P {P a, P b, P c} =

P {P a, b, P c},

(4.2)

P {P a, P b, P c} =

P {a, P b, P c}.

(4.3)

The equality (4.2) was proved in [27], and (4.3) was proved in [20]. As shown next, the conditional expectation formulas provide some interesting connections between contractive projections and norm-closed subtriples of JB∗ -triples. Theorem 4.1. Let A be a JB∗ -triple, and let P be a contractive projection on A. Then, the following conditions are equivalent. (1.) The range P A of P is a subtriple of A. (2.) For each element b in P A, the operator Q(b) on A commutes with P . (3.) For each element b in P A, the operator D(b, b) on A commutes with P . (4.) For elements a and b in P A, the operator D(a, b) on A commutes with P . Proof. Suppose that P A is a subtriple of A. Consider elements b of P A and a of A. Then, by (4.2), Q(b)P a

= {b, P a, b} = {P b, P a, P b} = P {P b, P a, P b} = P {P b, a, P b} = P {b, a, b} = P Q(b)a,

which proves (2.). Conversely, if b lies in P A and P commutes with Q(b) then P {b, b, b} = P Q(b)b = Q(b)P b = Q(b)b = {b, b, b}. It follows that, for all elements b of P A, the product {b, b, b} lies in P A which is therefore a subtriple of A. The equivalence of (1.), (3.) and (4.) can be derived in a similar way using (4.3). The following results relate collinear systems in a JB∗ -triple A with contractive projections on A. Lemma 4.2. Let {u1 , ..., un } be a collinear system in a JB∗ -triple A, For k = 1, ..., n, let Pk be a contractive projection onto n the one-dimensional subspace Cuk of A, and deﬁne the linear map P by P = k=1 Pk . Then, for j = k, the product Pj Pk vanishes, and P is a projection. n Proof. Clearly, k=1 Pk is a continuous linear mapping on A. Consider two distinct indices k and j in {1, ...., n}. By Lemma 3.1, H = Cuk ⊕ Cuj is a Hilbert space. The restrictions Pk |H and Pj |H of Pk and Pj to H are orthogonal orthoprojections non H. Therefore, the products Pk Pj and Pj Pk vanish on A. This implies that k=1 Pk is a projection.

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Theorem 3.3 and Theorem 4.1 can be combined to obtain an algebraic criterion for contractivity of the continuous projection described in Lemma 4.2. The corollary below serves as a key ingredient in the proof of the main theorems in the next section. Corollary 4.3. Under the conditions of Lemma 4.2, suppose that lin{u1 , ..., un } is a subtriple of A. Then the projection P is contractive if and only if it commutes with D(uj , uk ), for all j, k = 1, ...., n. n Proof. If lin{u1 , ..., un } is a subtriple and P = k=1 Pk is contractive, then by Theorem 4.1 P commutes with D(uj , uk ). Conversely, suppose that P and D(uj , uk ) commute. By linearity and continuity, P commutes also with D(b, b) and with exp itD(b, b), for all elements b of H. From Theorem 3.3 it follows that, for all a ∈ A and a ﬁxed j ∈ {1, ..., n}, there exist elements b of H and t of R such that the isometry φ = exp itD(b, b) of A satisﬁes, φP (a) = P φ(a) ∈ Pj A = Cuj . Therefore, Pj φP (a) = φP (a). Moreover, from Lemma 4.2 it can be seen that Pj P = Pj . It follows that for all a ∈ A, P (a) = ≤

φP (a) = Pj φP (a) = Pj P φ(a) = Pj φ(a) φ(a) = a.

Hence, P is contractive.

5. Normal contractive projections Attention is now turned to the case when A is a JBW∗ -triple. The set GL(A∗ ) of GL-projections on the predual A∗ of A will be a valuable tool. Several characterizations of GL(A∗ ) can be given in terms of the support space s(P A∗ ) of the range P A∗ of P . Theorem 4.6 in [11] states that, for a contractive projection P on A∗ , the following conditions are equivalent P ∈ s(P A∗ ) =

GL(A∗ ), P ∗ A,

(5.1) (5.2)

s(P A∗ ) ⊇ s(P A∗ ) ⊆

P ∗ A, P ∗ A.

(5.3) (5.4)

In fact, the condition s(P A∗ ) ⊆ P ∗ A forces P to be in GL(A∗ ), even when P is any projection deﬁned on A∗ [12]. Moreover, for any contractive projection Q on A∗ , the subspace s(QA∗ ) is a subtriple of A, and there exists a unique element P in GL(A∗ ) such that P A∗ = QA∗ [11]. The Hahn-Banach theorem implies the existence of a contractive projection onto any one-dimensional subspace of A∗ . The

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corresponding unique GL-projection can be explicitly described as follows [12] [25]. Given an element x = 0 of A∗ , deﬁne the mapping Px : A∗ → A∗ by (ex · z) x. (5.5) z → Px (z) = x Then, Px is an element of GL(A∗ ), and it is easy to see that the adjoint Px∗ of Px on A is given by Px∗ a = (a · x)ex . For x = 0, the mapping Px is deﬁned to be the zero-projection on A∗ . Corollary 5.1. Let (xk )k∈K a family of elements of the predual A∗ of a JBW∗ -triple A, such that the corresponding support tripotents ek form a collinear system. Then, for distinct indices j and k in K, (ej · xk ) = δj,k xk . Proof. Let Pk be the unique element of GL(A∗ ) with range Cxk . Observe that (ej · xj ) = xj , by deﬁnition of ej . For j = k the desired equality follows either by combining Lemma 4.2 with Equation (5.5), or from the Peirce rules which imply that (ej · xj ) = (P1 (ek )ej · P2 (ek )∗ xk ) = 0. It is now possible to address the main problem of this article. First, in Theorem 5.2, we focus on the properties and the explicit description of the various projections involved. The properties (2.) and (4.) given therein, are obtained from the connection between GL-projections, support spaces and support tripotents. Theorem 5.2. Let A be a JBW∗ -triple with predual A∗ , and let {xk }k∈K be a family of non-zero elements of A∗ , such that the corresponding support tripotents form a collinear system {ek }k∈K . Let the subspaces H and G of A and A∗ be w∗ n deﬁned by H = lin{ek : k ∈ K} and G = lin{xk : k ∈ K} respectively. For k in K, let Pk be the GL-projection onto Cxk , described by (5.5). Then, the following conditions are equivalent. (1.) There exists a contractive projection P on A∗ with range G. (2.) For any ﬁnite subset F of K, the projection PF = k∈F Pk on A∗ is contractive. In this case, PF is also the unique element of GL(A∗ ) with range lin{xk : k ∈ F }. (3.) The formal sum k∈K Pk is SOT-convergent in B(A∗ ) and is a contractive projection on A∗ . (4.) The formal sum k∈K Pk is SOT-convergent in B(A∗ ) and is the unique GL-projection on A∗ with range G. (5.) When x is a nonzero element of G such that x = k∈K αk xk , and 2 2 k∈K |αk | = ||x|| , then the support tripotent ex of x is αk ek ex = nk∈K 1 . ( k=1 |αk |2 ) 2

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All the sums converge in norm in the corresponding spaces. (6.) For all ﬁnite subsets F of K, and any element x ∈ lin{xj : j ∈ F } the support tripotent ex of x lies in lin{ej : j ∈ F }. If these conditions hold, then H is a subtriple of A, isometrically isomorphic to a Hilbert space, and is equal to the dual G∗ and to the support space s(G) of G. Moreover, H and G have orthonormal basis {ek }k∈K and {xk /xk : k ∈ K} respectively. Proof. (1.) ⇒ (2.): Let P be a contractive projection with range P A∗ = G. By [12] Theorem 3.1, it can be assumed that P lies in GL(A∗ ). Then, by (5.1) the subspaces s(P A∗ ) and P ∗ A of A coincide. Since P is contractive, P ∗ A and P A∗ form a dual pair. Denote by H◦ and (H◦ )◦ the annihilator and the bi-annihilator of H in P A∗ and P ∗ A respectively. Corollary 5.1 shows that if j = k then xj ∈ {xk }◦ . Hence, H◦ ⊆ ({xk }k∈K )◦ = {xk }◦ = {0}. k∈K

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

(5.6)

By [14] Lemma 5.1, H is a subtriple of A, and by Theorem 3.3, H is a Hilbert space. Therefore, also G is a Hilbert space of the same dimension as H. Let K f denote the of all ﬁnite subsets of K. For every set F ∈ K f , let the projection PF on A∗ , be deﬁned by Pk . PF = k∈F

Its range PF A∗ is a subspace of the Hilbert space G = P A∗ , with ﬁnite dimension |F |. Hence, PF A∗ is the range of a contractive projection on A∗ . As argued before, there is a unique element QF of GL(A∗ ) such that QF A∗ = PF A∗ ,

and

Q∗F A = s(QF A∗ ) = lin{ek : k ∈ F } = PF∗ A.

It follows that PF and QF coincide. (2.) ⇒ (3.): Repeating the above argument with K replaced by F shows that if PF is contractive, then PF A∗ and PF∗ A are Hilbert spaces, both having dimension |F | < ∞. Moreover, PF ∈ GL(A∗ ), which implies that s(PF A∗ ) = PF∗ A. The restriction Pk |PF A∗ of Pk to PF A∗ is an orthoprojecion on PF A∗ . By Lemma 4.2, Pk annihilates xj , if j = k. It follows that the elements xj and xk are Hilbertorthogonal in PF A∗ and that, for all elements x of A∗ , Pi x2 = Pi x2 = PF x2 ≤ x2 . (5.7) i∈F

i∈F

Therefore, {PF x}F ∈K f is a Cauchy net with respect to the norm topology on A∗ , its norm limit exists for every x in A∗ , and k∈K Pk converges in the strong

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operator topology to a linear projection P which, by (5.7), is also contractive. This also shows that {xk /xk }k∈K is an orthonormal basis of G. (3.) ⇒ (4.): Set P = k∈K Pk . It is clear that P has range G. That P is a GL-projection is veriﬁed as follows. For each k in K, the L-complement (Pk A∗ )♦ of Pk A∗ is a subset of kerPk . It follows that n

(P A∗ )♦ = (lin Pk A∗ )♦ ⊆ ( Pk A∗ )♦ = (Pk A∗ )♦

⊆

k∈K

k∈K

k∈K

kerPk = {x ∈ A∗ : Pk (x) = 0, ∀k ∈ K}

k∈K

⊆

kerP.

Since P is also assumed to be contractive, it lies in GL(A∗ ). The uniqueness of P among the GL-projections with given range is obtained from Corollary 4.7 in [11]. Hence, (3.) implies (4.). (4.) ⇒ (1.): Since GL-projections are contractive by deﬁnition, this implication is obvious. (4.) ⇒ (5.): If (4.) holds then, as it has already been shown, {xk /xk }k∈K is an orthonormal basis of G. For any x ∈ G there are coeﬃcients αk in C, such that xk x = αk |αk |2 , , and x2 = xk k∈K

k∈K

Since by (5.1) the support tripotent ex of x lies in the subtriple H = s(P A∗ ), and since {ek }k∈K is an orthonormal basis of H, it follows that there are coeﬃcients βk , (k ∈ K) with ex = βk e k . and |βk |2 = ex 2 = 1. k∈K

k∈K

Here it is assumed that x is not zero, and hence that ex is not zero. We identify H and G with l2 (K)∗ andl2 (K) respectively, with the conventional inner product (αk )k∈K , (βk )k∈K = k∈K ak βk . Using Corollary 5.1 and the CauchyBunyakowsky-Schwarz (CBS) inequality, ak βk )2 = (αk )k∈K , (βk )k∈K 2 x2 = (ex · x)2 = ( ≤

(

k∈K

2

|αk | )(

k∈K

|βk |2 ) ≤ x2 .

k∈K

This shows that equality holds in (CBS), and it follows that there exists a complex number z such that αk = zβk , for all k ∈ K. Hence, αk βk )2 = ( zβk βk )2 = |z|2 , x2 = ( k∈K

k∈K

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αk αk = . 2 |z| x2

βk = This proves (5.). (5.) ⇒ (6.) This is trivial.

(6.) ⇒ (2.) When F is a ﬁnite subset of K, then by Lemma 4.2, j∈F Pj is a projection on A∗ with adjoint j∈F Pj∗ on A. The assumption implies that s( j∈F Pj A∗ ) is a subset of j∈F Pj∗ A. Theorem 3.18 in [12] shows that j∈F Pj lies in GL(A∗ ), and is therefore contractive. This completes the proof. Having explicitly described the GL-projections involved, it is now possible to turn the attention to algebraic conditions. The case in which the index set K consists of two elements diﬀeres in some details from all other cases and is therefore treated separately. Theorem 5.3. Let A be a JBW∗ -triple with predual A∗ , and let x and y be elements of A∗ the support tripotents ex and ey of which are collinear. Let Px and Py the GL-projections onto Cx and Cy deﬁned by (5.5). Then the following conditions are equivalent. (1.) There exists a contractive projection P from A∗ onto Cx ⊕ Cy. (2.) The projection Px∗ + Py∗ commutes with D(ex , ey ), D(ey , ex ), D(ex , ex ) and with D(ey , ey ). (3.) The elements x and y satisfy 2D(ex , ey )∗ x =

2D(ex , ex )∗ y = y,

2D(ey , ex )∗ y

2D(ey , ey )∗ x = x.

=

In particular, x lies in A1 (ey )∗ and y lies in A1 (ex )∗ . Proof. (1.) ⇔ (2.): By Lemma 4.2, Px +Py and Px∗ +Py∗ are projections. The range Cex ⊕ Cey of Px∗ + Py∗ is a subtriple of A. Combining Theorem 5.2 with Corollary 4.3 gives the equivalence of (1.) and (2.), as required. (2.) ⇔ (3.): To obtain this equivalence, the commutators in (2.) are to be calculated explicitely. For all elements a of A, (Px∗ + Py∗ )D(ex , ex )a

= (x · {ex ex a})ex + (y · {ex ex a})ey = (D(ex , ex )∗ x · a)ex + (D(ex , ex )∗ y · a)ey = (x · a)ex + (D(ex , ex )∗ y · a)ey

and D(ex , ex )(Px∗ + Py∗ )a

=

D(ex , ex )((x · a)ex + (y · a)ey )

=

(x · a)ex + (y · a){ex ex ey } 1 (x · a)ex + (y · a)ey . 2

=

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Hence Px∗ + Py∗ commutes with D(ex , ex ), if and only if 1 y. (5.8) 2 By Proposition 2.1, the equality (5.8) is equivalent to y lying in the predual A1 (ex )∗ of the Peirce space A1 (ex ) of ex . Exchanging x and y in these calculations shows that (Px + Py )∗ commutes with D(ey , ey ) if and only if x lies in A1 (ey )∗ , or, equivalently, D(ex , ex )∗ y =

1 x. (5.9) 2 To obtain the required expressions for D(ex , ey )∗ x and D(ey , ex )∗ y, observe that, by the relation ex ey and the Peirce rules, D(ey , ey )∗ x =

D(ey , ex )∗ x =

D(ex , ey )∗ y = 0,

(5.10)

{ex , ey , ex } =

{ey , ex , ey } = 0.

(5.11)

From Equation (5.10) it follows that, for any element a in A, (Px + Py )∗ D(ex , ey )(a)

=

(Px + Py )∗ {ex , ey , a}

= =

(x · {ex , ey , a})ex + (y · {ex , ey , a})ey (x · {ex , ey , a})ex

=

(D(ex , ey )∗ x · a)ex ,

and from (2.1) and (5.11) that D(ex , ey )(Px + Py )∗ (a) =

{ex , ey , (Px + Py )∗ a}

(x · a){ex , ey , ex } + (y · a){ex , ey , ey } 1 (y · a)ex . = 2 This shows that Px∗ + Py∗ and D(ex , ey ) commute if and only if =

1 y. 2 Similarly, D(ey , ex ) and (Px + Py )∗ commute if and only if D(ex , ey )∗ x =

(5.12)

1 x. (5.13) 2 The relations (5.8), (5.9), (5.12) and (5.13) provide the equivalence of (2.) and (3.). This completes the proof. D(ey , ex )∗ y =

In order to pass to arbitrary families of elements in A∗ , a careful re-examination of the above proof is required. In particular, further conditions need to be added to those in (3.) of Theorem 5.3, and it must be proved that the stated algebraic criterias imply the convergence of the formal sums as described in Theorem 5.2.

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Theorem 5.4. Let A be a JBW∗ -triple with predual A∗ , and let {xk }k∈K with |K| ≥ 3 be a family of elements of A∗ , such that the corresponding support tripotents form a collinear system {ek }k∈K . Let H, G and Pk be deﬁned as in Theorem 5.2. Then, the following conditions are equivalent. (1.) There exists a contractive projection on A∗ with range G. (2.) The formal sum k∈K Pk∗ is SOT-convergent in B(A) and commutes with D(ej , ek ), for all elements j, k in K. (3.) For distinct indices j, k, l in K, the following relations hold; (i)

2D(ej , ej )∗ xk = xk ,

(ii)

2D(ek , ej )∗ xk = xk ,

(iv) {ej , ek , el } = 0.

(iii) D(ej , ek )∗ xl = 0,

In particular, xk lies in A1 (ej )∗ and H is a subtriple of A that is isometrically isomorphic to a Hilbert space with orthonormal basis {ek }k∈K . (4.) For any three distinct indices j, k and l in K, there exists a contractive projection on A∗ with range lin{xj , xk , xl }. Proof. (1.) ⇒ (2.): Theorem 5.2 (4.) shows that P = k∈K Pk lies in GL(A∗ ). By (5.1) s(P A∗ ) and P ∗ A coincide, and hence P ∗ A is a weak∗ -closed subtriple of A. Theorem 4.1 shows that, for a, b in H, P ∗ commutes with D(a, b). (2.) ⇒ (3.): Following the strategy used in the proof of Theorem 5.3 we calculate the commutators explicitly. Recall that xj lies in A2 (ej )∗ and that D(ej , ej )∗ xj equals xj . Hence, for any element a in A, Pk∗ )D(ej , ej )a = ( Pk∗ ){ej , ej , a} = (xk · {ej , ej , a})ek ( k∈K

k∈K

k∈K

= (xj · {ej , ej , a})ej +

(xk · {ej , ej , a})ek

k∈K\{j}

= (xj · a)ej +

(xk · {ej , ej , a})ek .

k∈K\{j}

The relation ej ek implies that Pk∗ )a = D(ej , ej ) (xk · a)ek = (xk · a){ej , ej , ek } D(ej , ej )( k∈K

k∈K

= =

k∈K

(xj · a){ej , ej , ej } + (xj · a)ej +

k∈K\{j}

(xk · a){ej , ej , ek })

k∈K\{j}

1 (xk · a)ek . 2

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Therefore, the operators D(ej , ej ) and a ∈ A, 1 (xk · a)ek = 2

k∈K

Pk∗ commute if and only if, for all

k∈K\{j}

(xk · {ej , ej , a})ek .

(5.14)

k∈K\{j}

From the linear independence of the set {ek }k∈K and Equation (5.14) we conclude that (xk · {ej , ej , a}) equals (xk · a)/2, i.e. D(ej , ej )∗ xk =

1 xk . 2

(5.15)

Therefore, (3.)(i) holds. Now assume j and l to be distinct indices in K. Equation (5.10) implies that Pk∗ )D(ej , el )a = ( Pk∗ ){ej , el , a} = (xk · {ej , el , a})ek ( k∈K

k∈K

k∈K

= (xj · {ej , el , a})ej + (xl · {ej , el , a})el + = (xj · {ej , el , a})ej +

(xk · {ej , el , a})ek

k∈K\{j,l}

(xk · {ej , el , a})ek ,

k∈K\{j,l}

and Equation (5.11) implies that D(ej , el )( Pk∗ )a = D(ej , el ) (xk · a)ek k∈K

k∈K

= (xj · a){ej , el , ej } + (xl · a){ej , el , el } +

(xk · a){ej , el , ek }

k∈K\{j,l}

1 (xl · a)ej + (xk · a){ej , el , ek }. 2 k∈K\{j,l} Hence, D(ej , el ) and k∈K Pk∗ commute if and only if, for all a ∈ A, (xj · {ej , el , a})ej + (xk · {ej , el , a})ek =

(5.16)

k∈K\{j,l}

=

1 (xl · a)ej + 2

(xk · a){ej , el , ek }.

(5.17)

k∈K\{j,l}

Choose an index m ∈ K, distinct from j and l, and set a = em . Using again Corollary 5.1, the right hand side (5.17) of the above equation becomes {ej , el , em }. The Peirce rules show that, {ej , el , em } ∈ A2 (ej ) ∩ A0 (el ) ∩ A2 (em ).

(5.18)

By Equation (5.15), xk lies in A1 (ej )∗ ∩ A1 (em )∗ when k = j, m, and xj lies in A1 (em )∗ . Therefore, the left hand side (5.16) vanishes, and we obtain, {ej , el , em } = 0.

(5.19)

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This proves (3.)(iv). By inserting (5.19) back into (5.17), the equality (5.17) = (5.16) simpliﬁes as 1 (xk · {ej , el , a})ek = (xl · a)ej . (xj · {ej , el , a})ej + 2 k∈K\{j,l}

The linear independence of {ek }k∈K shows that (xj · {ej , el , a}) = (xl · a)/2, and (xk · {ej , el , a}) = 0, which implies that 1 xl , and D(ej , el )∗ xk = 0. 2 This proves (3.)(ii) and (3.)(iii). The condition (3.)(iv) implies also that H is a Hilbert space and a subtriple of A, and that {ek }k∈K is and orthonormal basis of H. D(ej , el )∗ xj =

(3.) ⇒ (1.): It is enough to prove that the condition (2.) of Theorem 5.2 holds. For any ﬁnite subset F of K, set HF = lin{ek : k ∈ F }. By Lemma 4.2, PF = k∈F Pk is a projection on A∗ . Theorem 3.3 and (3.iv) show HF to be a subtriple of A isomorphic to the Hilbert space with triple product (2.5). Reinspecting the relations (5.14), (5.16) and (5.17) shows that, for j, l in F , the ∗ P projection PF∗ = k∈F k commutes with D(ej , el ) and with D(ej , ej ) on A. Hence, by Corollary 4.3, PF is contractive, as required. (3.) ⇔ (4.): This follows from the equivalence of (1.) and (3.) and the fact that (3.) is formulated in terms of all possible triples (j, k, l) ∈ K × K × K. The proof is now complete. Let {xk }k∈K a family of elements in A∗ such that the corresponding support tripotents {ek }k∈K form a collinear system. Then {xk }k∈K is said to be projectively n collinear if there exists a contractive projection onto lin{xk : k ∈ K} . Corollary 5.5. Let {xk }k∈K be an arbitrary family of elements of A∗ . Then, {xk }k∈K is projectively collinear if and only if the conditions (3.i)–(3.iv) in Theorem 5.4 hold for {xk }k∈K and {ek }k∈K . In particular, these conditions are suﬃcient for {ek }k∈K to be a collinear system. Proof. The condition (3.i) and Proposition 2.1 imply that xk ∈ A1 (el )∗ (k = l). By Proposition 3 in [19], ek lies in A1 (el ) and el lies in A1 (ek ), i.e. {ek }k∈K is a collinear system. Theorem 5.4 completes the proof. At this stage, two questions have to be clariﬁed. First, is there a non-trivial example of a projectively collinear system, and, secondly, is there an example of a family of elements in A∗ with pairwise collinear support tripotent which is not projectively collinear. A negative answer to either of the questions would render our main results redundant. The following example answers both questions aﬃrmatively.

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Example. Let A be the W∗ -algebra B(C4 ), represented by 4 × 4-matrices. be the normalized trace on A, and let mx , my and mz be the matrices      2 0 0 0 0 0 0 0 2 1 0  0 2 0 0   0 0 0 0   1 2 0     mx =   0 0 0 0  , my =  2 0 0 0  , mz =  0 0 0 0 0 0 0 0 2 0 0 0 0 0

Let τ  0 0   0  0

Deﬁne the linear functionals x, y and z, for any element a of A with entries aij (i, j = 1, 2, 3, 4) by 1 (a11 + a22 ), 2 1 (y · a) = τ (my a) = (a13 + a24 ), 2 1 (z · a) = τ (mz a) = (2a11 + a21 + a12 + 2a22 ). 4 Then the elements u and v of A, given by     1 0 0 0 0 0 1 0  0 1 0 0   0 0 0 1  ,   u =  v =  0 0 0 0   0 0 0 0 . 0 0 0 0 0 0 0 0 (x · a) =

τ (mx a) =

form a collinear pair of tripotents of A and are such that u = ex = ez and v = ey . The mapping P deﬁned for a linear functional s on A, by s → (u · s)x + (v · s)y is a contractive projection and is also the unique GL-projection from A∗ onto Cx⊕ Cy, whereas there is no contractive projection from A∗ onto Cz ⊕ Cy. Proof. It is obvious that (x · u) = (z · u) = (y · v) = 1, and the norms of mx , my and mz are, respectively, mx = 2, my = 2, and mz = 3. The following trace inequality, obtained in [32], is useful in subsequent calculations. For all elements a, b in a C∗ -algebra A and a continuous linear functional τ on A, the condition |τ (ab)| ≤ b τ (|a|)

(5.20)

holds if and only if τ is a positive normalized trace on A. From (5.20) it follows that, for all b in A, and m equal to mx , my , or to mz , |τ (mb)|

≤ b τ (|m|) = b,

hence, that x, y and z are of norm one in A∗ . Since x attains its norm at u, it follows that ex ≤ u. In particular, ex lies in A2 (u). Clearly, ex is not zero. Consider any tripotent w of A with w ≤ u. Observe that u is a partial isometry of rank two. If w is not equal to zero or to u, then w is a partial isometry of rank one, that is trace(|w|) = 1. Then, by (5.20), |(x · w)| = |τ (mx w)| ≤ mx τ (|w|) =

1 . 2

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It follows that ex has rank two. This and the condition that ex ≤ u, implies that ex = u. Similarly, (z · u) = 1 = z implies that ez ≤ u, and if w is as above, |(z · w)| = |τ (mz w)| ≤ mz τ (|w|) =

3 . 4

Hence ez = ex = u. Also v has rank two. The same arguments as those used above show that ey ≤ y. And, for any tripotent w with w ≤ v and trace(|w|) = 1, |(y · w)| = |τ (my w)| ≤ my τ (|w|) =

1 . 2

Hence ey = v. We have shown that the subsets {x, y} and {z, y} satisify the assumptions of Theorem 5.3. It can be seen from elementary calculations that for all a ∈ A, (D(u, u)∗ y · a) = (D(u, v)∗ x · a) = (D(v, u)∗ y · a) = (D(v, v)∗ x · a) =

1 (a13 + a24 ) 4 1 (a13 + a24 ) 4 1 (a11 + a22 ) 4 1 (a11 + a22 ) 4

1 (y · a), 2 1 = (y · a), 2 1 = (x · a), 2 1 = (x · a). 2 =

By Theorem 5.3, the mapping P deﬁned for a linear functional s on A, by s → (u · s)x + (v · s)y is a contractive projection and is also the unique GL-projection from A∗ onto Cx ⊕ Cy. On the other hand, (D(u, v)∗ z · a) = (D(v, v)∗ z · a) =

1 (2a13 + a23 + a14 + 2a24 ), 8 1 1 (2a11 + a12 + a21 + 2a22 ) = (z · a). 8 2

(5.21) (5.22)

Equation (5.22) shows that z lies in A1 (v), but (5.21) condradicts (3.) in Theorem 5.3. Hence, there is no contractive projection from A∗ onto Cz ⊕ Cy. The pair {y, z} in Example 5 shows that in general the condition that {xk }k∈K be projectively collinear is not necessary for the existence of a normal w∗ contractive projection onto H = lin{ek : k ∈ K} . Given a normal contractive projecton R from A onto H, the range R∗ A∗ of its pre-adjoint R∗ may be difn ferent from G = lin{xk : k ∈ K} . In the remainder of this section, we assume the existence of a normal contractive projection R on A with range RA equal to the weak∗ -closed span of an arbitrary collinear system. We also investigate the properties of R∗ . A tripotent u in a JBW∗ -triple is said to be σ-ﬁnite if the following implication holds. Whenever B is an orthogonal system in U(A) with the property that

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B ≤ u then B is countable. The set of all σ-ﬁnite tripotents is denoted by Uσ (A). The results of [15] and [19], show that Uσ (A) = {ex : x ∈ A∗ }.

(5.23)

This characterizes the support tripotents in terms of algebraic criteria, and the next lemma can established from earlier results. By [11] Corollary 4.8 (iii), a contractive projection P on A∗ lies in GL(A∗ ) if and only if Uσ (P ∗ A) = Uσ (A) ∩ s(P A∗ ).

(5.24)

Lemma 5.6. Let A be a JBW∗ -triple, and let H be a subtriple of A isometrically isomorphic as subtriple to a Hilbert space. Denote by S1 (H) the unit sphere of H, a subset of U(A). If S1 (H) ∩ Uσ (A) is not empty, then S1 (H) ⊆ Uσ (A). Proof. Since σ-ﬁniteness is an algebraic property, it is preserved under triple automorphisms of A. By Theorem 3.3, the set of triple automorphisms of A acts transitively on S1 (H). This completes the proof. When R is any normal contractive projection on A, denote by R∗ its preadjoint, a contractive projection on A∗ . Recall that the range RA of R is a JBW∗ triple when equipped with the restricted triple product {. . .}RA deﬁned in (4.1). Theorem 5.7. Let C be a collinear system in the JBW∗ -triple A. Denote by HC the w∗ subspace linC of A. Let R be a normal contractive projection on A with range HC and with pre-adjoint R∗ . Let P be the unique GL-projection on A∗ such that P A∗ = R∗ A∗ . Then, the following results hold, (1.) There exists a projectively collinear system {xk }k∈K in A∗ such that the range R∗ A∗ of R∗ is the norm closed span of {xk }k∈K . (2.) The JBW∗ -triple HC with the restricted triple product {. . .}RA is isomorphic w∗ to the subtriple H = lin{ek : k ∈ K} spanned by the support tripotents ek of xk . In particular, HC is a Hilbert space, isomorphic to H. (3.) The restrictions R|H and P ∗ |HC of R and P ∗ to H and HC are triple isomorphisms and inverse of each other. (4.) When u, v w are distinct elements of C (provided that |C| ≥ 3), then {u, v, w} lies in kerR. Proof. By the conditional expectation formulas (4.2) and (4.3), it can be seen that, for two diﬀerent elements u and v of C. 1 1 {u, u, v}RA = R{u, u, v} = Rv = v. 2 2 Therefore, the relation of collinearity holds also with respect to the restricted triple product. By [14] Lemma 5.1 and [19] Proposition 2.2, it can be seen that the triples (RA, {. . .}RA ) and (s(R∗ A∗ ), {. . }) are isomorphic. When P is the unique element of GL(A∗ ) with P A∗ = R∗ A∗ , then by [11] Lemma 3.2 (iv), P |RA is a

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triple isomorphism and inverse to R|R∗ A . It follows that P ∗ C = {P ∗ u : u ∈ C} is a collinear system in the Hilbert space H. Consequently, by Lemma 5.6, P ∗ C is a subset of Uσ (A). From (5.23) and (5.24), it can be seen that, for each u in C the element P ∗ u is the support tripotent of some xu in R∗ A∗ . This proves (1.), (2.) and (3.). For distinct elements u, v and w in C, the above arguments and Theorem 5.4 (3.iv) show that {P ∗ u, P ∗ v, P ∗ w} = 0. Since R|P ∗ A is a triple isomorphism with inverse of P |RA , it follows that R{u, v, w} is zero, proving (4.). The above Theorem can be generalized as follows. Theorem 5.8. Let Q be a normal contractive projection on a JBW∗ -triple A, with preadjoint Q∗ on A∗ . Then QA is isometrically a Hilbert space if and only if Q∗ A∗ is the norm-closed span of a projectively collinear system. Proof. Let P be the uniqe element of GL(A∗ ) with P A∗ = Q∗ A∗ . Then, as argued before, P ∗ |QA and Q|P ∗ A are isometries and inverse of each other, and P ∗ A is a subtriple of A. Applying Theorem 5.7 gives the proof. The results presented in this article apply to the more restricted relation of rigid collinearity [13]. In that case, the exchange automorphisms introduced in [29] can be used to answer our main question, whether or not a contractive projection with the prescribed range exists. Details and proofs are part of current investigations, and will be presented in forthcoming publications.

References [1] T. J. Barton and R. M. Timoney: Weak∗ -continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), 177-191. [2] F. F. Bonsall and J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Notes Series 2, Cambridge University Press 1971. [3] L. J. Bunce: Norm preserving extensions in JBW∗ -triple preduals, Quart. J. Math. 52 (2000), 133-136. [4] L. J. Bunce and A. Peralta: Images of contractive projections on operator algebras, J. Math. Anal. Appl. 272 no. 1 (2002), 55-66. [5] M. D. Choi and E. G. Eﬀros: Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156-209. [6] E. Christensen and A. Sinclair: On von Neumann algebras which are complemented subspaces of B(H), J. Funct. Anal. 122 No.1 (1994), 91-102. [7] C.-H. Chu and B. Iochum: Complementation of Jordan-triples in von Neumannalgebras, Proc. Amer. Math. Soc. 108 no.1 (1990), 19-24.

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[8] C.-H. Chu, M. Neal and B. Russo: Normal contractive projections preserve the type, J. Operator Theory 51 no.2 (2004), 281-301. [9] T. Dang, Y. Friedman Classiﬁaction of JBW∗ -triple factors and applications, Math. Scand. 61 (1987), 292-330. [10] S. Dineen: The second dual of a JB∗ -triple system, in: Complex Analysis, Functional Analysis and Approximation Theory ( J. Mujica ed.), North-Holland, Amsterdam, 1986, 67-69. [11] C. M. Edwards, R. V. H¨ ugli and G. T. R¨ uttimann: A geometric characterization of structural projections on a JBW∗ -triple, J. Funct. Anal., 202 (2003), 174-194. [12] C. M. Edwards, R. V. H¨ ugli: Order structure of GL-projections on a complex Banach space, Atti Sem. Mat. Fis. Univ. Modena to appear. [13] C. M. Edwards, R. V. H¨ ugli: Decoherence in pre-symmetric spaces. Preprint 2007. [14] C. M. Edwards and G. T. R¨ uttimann: Structural projections on JBW∗ -triples, J. London Math. Soc. 53 (1996), 354-368. [15] C. M. Edwards and G. T. R¨ uttimann: Exposed faces of the unit ball in a JBW∗ triple, Math. Scand. 82 (1998), 287-304. [16] C. M. Edwards and G. T. R¨ uttimann: Gleason’s Theorem for rectangular JBW∗ triples, Commun. Math. Phys. 203 (1999), 269-295. [17] C. M. Edwards and G. T. R¨ uttimann: Orthogonal faces of the unit ball in a Banach space. Atti Sem. Mat. Fis. Univ. Modena 49 (2001), 473-493. [18] E. G. Eﬀros and E. Størmer: Positive projections and Jordan structure in operator algebras, Math. Scand. 45 (1979), 127-138. [19] Y. Friedman and B. Russo: Structure of the predual of a JBW∗ -triple, J. Reine Angew. Math. 356 (1985), 67-89. [20] Y. Friedman and B. Russo: Conditional expectation and bicontractive projections on Jordan C ∗ -algebras and their generalisations, Math. Z. 194 (1987), 227-236 . [21] Y. Friedman and B. Russo: Solution of the contractive projection problem, J. Funct. Anal. 60 (1985), 56-79. [22] U. V. Haagerup, E. Størmer: Positive Projections of von Neumann algebras onto JWalgebras, Proceedings of the XXVII Symposium on Mathematical Physics (Toru´ n, 1994). Rep. Math. Phys. 36 no. 2-3 (1995), 317-330. [23] L. A. Harris: Bounded symmetric domains in inﬁnite dimensional spaces, in T. L. Hayden and T. J. Suﬀridge (Eds.): Proceedings on Infinite Dimensional Holomorphy, Lecture Notes in Mathematics 364 (13-40), Springer Berlin/Heidelberg/New York, 1974. [24] G. Horn: Characterisation of the predual and ideal structure of a JBW∗ -triple, Math. Scand. 61 (1983), 117-133. [25] R. V. H¨ ugli: Structural Projections on a JBW∗ -Triple and GL-Projections on its Predual, J. Korean Math. Soc. 41, No. 1 (2004), 107-130. [26] W. Kaup: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503-529. [27] W. Kaup: Contractive projections on Jordan C∗ -algebras and generalisations. Math. Scand. 54 (1984), 95-100.

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[28] O. Loos: Bounded Symmetric Domains and Jordan Pairs Mathematical Lectures, University of California, Irvine, 1977. [29] K. McCrimmon and K. Meyberg: Coordinatization of Jordan triple systems, Communications in Algebra, 9(14) (1981), 1495-1542. [30] M. Neal and B. Russo: Contractive projections and operator spaces Trans. American Math. Soc. 355 No. 6 (2003), 2223-2262 [31] E. Neher: Jordan Triple Systems by the Grid Approach, Lecture Notes in Mathematics 1280, Springer, Berlin/Heidelberg/New York, 1987. [32] D. Petz and J. Zemanek: Characterizations of the trace, Lin. Alg. Appl. 111 (1988), 43-52. [33] G. Pisier: Projections from a von Neumann algebra onto a subalgebra, Bull. Soc. Math. France, 123 (1995), 139-153. [34] L. L. Stach` o: A projection principle concerning biholomorphic automorphisms, Acta. Sci. Math. 44 (1982), 99-124. [35] H. Upmeier: Symmetric Banach Manifolds and Jordan∗ -Algebras. North-Holland Math. Studies 104 1985. [36] J. P. Vigu´e: Les automorphismes analytiques isom´etriques d’un vari´et´e complexe norm´ee, Bull. Soc. Math. France 110 (1982), 49-73. [37] J. D. M. Wright: The structure of decoherence functionals for von Neumann quantum histories. J. Math. Phys. 36 (1995), 5409-5413. [38] M. A. Youngson: Completely contractive projections on C∗ -algebras, Quart. J. Math. Oxford Ser. (2) 34 (1983), 507-511. Remo V. H¨ ugli School of Mathematical Sciences University College Dublin Belﬁeld D-4 Ireland e-mail: [email protected] Submitted: March 27, 2006 Revised: February 27, 2007

Integr. equ. oper. theory 58 (2007), 341–362 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030341-22, published online April 14, 2007 DOI 10.1007/s00020-007-1494-0

Integral Equations and Operator Theory

The Fredholm Index for Elements of Toeplitz-Composition C*-Algebras Michael T. Jury Abstract. We analyze the essential sectrum and index theory of elements of Toeplitz-composition C*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C*-algebra for the action of the symbol on the boundary circle. Using this result we obtain suﬃcient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse. Mathematics Subject Classification (2000). Primary 47B33; Secondary 47A53, 47L80. Keywords. Composition operator, Toeplitz operator, Fredholm operator, Fredholm index.

1. Introduction The C*-algebras generated by Toeplitz operators have been much studied, and have been shown to have a rich and interesting structure. For example, the wellknown theorem of Coburn shows that the C*-algebra T generated by the Toeplitz operators with continuous symbol, acting on the Hardy space of the unit disk H 2 (D), contains the C*-algebra of compact operators K and there is an exact sequence of C*-algebras 0 → K → T → C(T) → 0. The quotient map takes a Toeplitz operator Tf to its symbol f . From this can be deduced the Toeplitz index theorem, which says that Tf is Fredholm if and only

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if f is nonvanishing on the unit circle T, in which case the Fredholm index of Tf is equal to minus the winding number of f about the origin. For these and other basic results concerning Toeplitz operators we refer to the book of R. Douglas [6, Chapter 7]. In this paper we study the C*-algebras generated by T and a linear-fractional composition operator on the Hardy space H 2 of the unit disk. In particular we are interested in the C*-algebras T Cϕ = C ∗ (S, Cϕ ) where S denotes multiplication by z on H 2 (the unilateral shift) and Cϕ is a composition operator with ϕ : D → D a linear fractional map, either an automorphism or non-parabolic non-automorphism. In all of these cases the quotient C*-algebra T Cϕ /K has a discernible structure, and this structure can be used to attack the problem of deciding when an element of T Cϕ is Fredholm, and in such cases computing its index. This study is motivated by several results: ﬁrst, earlier work on algebras related to T Cϕ by the author in [7] (which considers the C*-algebras generated by composition operators with symbols in a Fuchsian group) and by Kriete, MacCluer and Moorhouse [8] on the algebra T Cϕ for certain non-automorphic ϕ. Second, the work of M.D. Choi and F. Latr´emoli`ere [3] on the C*-algebras C(D) ϕ Z (for disk automorphisms ϕ) describes the representation theory of these algebras and is closely related to the C*-algebras C(T)ϕ Z which we obtain as quotients of T Cϕ . Finally, a theorem of E. Park [9] describes the Fredholm index of operators in a Toeplitz-like extension of irrational rotation C*-algebras. This extension turns out to be a special case of the extensions of C(T)ϕ Z given by T Cϕ , and this index result generalizes readily to our situation. We thus have tools available to study the Fredholm theory in T Cϕ . The paper is divided as follows: Section 2 treats results concerning T Cϕ common to all automorphisms ϕ. We prove that for any automorphism ϕ there is an exact sequence of C*-algebras 0 → K → T Cϕ → C(T)ϕ Z → 0. Using this exact sequence and a computation of the K-theory of C(T)ϕ Z, we obtain an integral formula for the Fredholm index in T Cϕ which generalizes a result of E. Park [9] for the irrational rotation algebras. We also prove that for many automorphisms ϕ, the inclusion of the shift S as a generator of T Cϕ is unnecessary, that is, we give a suﬃcient condition on ϕ for S to be contained in C ∗ (Cϕ ). Finally we review some basic facts about analytic automorphisms of D needed in the subsequent sections. In Sections 3 through 5 we ﬁnd conditions under which elements of T Cϕ are Fredholm, when ϕ is elliptic (of ﬁnite order), hyperbolic, or parabolic respectively. By virtue of the exact sequence obtained in Section 2, this amounts to a study of invertibility in C(T)ϕ Z. The ﬁnite-order elliptic case, Section 3, can be handled by fairly elementary methods, and we also obtain here a more topological form of the index formula of Section 2. Section 4, the hyperbolic case, is the longest of the three sections, and requires an analysis of representations of C(T)ϕ Z which draws on recent related work of M.D. Choi and F. Latr´emoli`ere [3]. Since parabolic

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automorphisms can be viewed as degenerate hyperbolics, the results of Section 5 are obtained by slight modiﬁcation of the arguments of Section 4. It should be noted here that we do not consider invertibility in C(T) ϕ Z when ϕ is elliptic of inﬁnite order. This is the case of the irrational rotation algebras Aθ , and because these are known to be simple C*-algebras, there are no accessible “local” criteria for invertibility as in the other cases. Indeed the problem of determining spectra of elements of Aθ is extremely diﬃcult even for very simple symbols. For example the elements of the form z + z + uϕ + u∗ϕ are the so-called “almost Mathieu operators,” and the spectral theory even for these self-adjoint operators is quite diﬃcult and is the subject of a recent book by F.-P. Boca [1]. Finally, Section 6 treats the non-automorphic linear fractional maps considered by Kriete, MacCluer, and Moorhouse [8]. They obtain an exact sequence 0 → K → T Cϕ → D → 0, where ϕ is a non-automorphic linear fractional map of D into itself without a boundary ﬁxed point (so that Cϕ2 is compact) and D is an explicitly described type I C*-algebra. They also provide an explicit expression for the essential spectrum of elements of T Cϕ (which provides a characterization of the Fredholm operators in T Cϕ ) and from that expression we prove an index theorem. It turns out that the situation here is rather simpler than in the case of automorphisms; in fact we prove that the C*-algebra D is homotopy equivalent to C(T), from which the index results follow readily.

2. General considerations The Hardy space H 2 is deﬁned to be the Hilbert space of functions analytic in the open unit disk D such that 2π 1 2 f = sup |f (reiθ )|2 dθ 0
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shift S = Tz and Cϕ , for ϕ either an automorphism of D or a non-parabolic nonautomorphism. We let T Cϕ denote this C*-algebra. We observe that since Tz is included as a generator, T Cϕ always contains the C*-algebra of compact operators K. In the remainder of this section we assume ϕ is an automorphism of D. We will write ϕn for the nth iterate of ϕ and ϕ−n for the nth iterate of the inverse automorphism ϕ−1 . The basic result concerning the structure of the C*-algebras T Cϕ is the following: Theorem 2.1. There is an exact sequence of C*-algebras 0 → K → T Cϕ → C(T)ϕ Z → 0.

(2.1)

Proof. The proof is essentially the same as the proof for discrete groups of automorphisms given in [7]. Let Uϕ denote the unitary operator (Cϕ Cϕ∗ )−1/2 Cϕ . The calculations of [7] show that the following relations hold: • For all f ∈ C(T), Uϕ∗ Tf Uϕ − Tf ◦ϕ−1 is compact. • For all integers m, n, Uϕm Uϕn − Uϕm+n is compact. It follows that, letting f and uϕ denote the images in the Calkin algebra of Tf and Uϕ respectively, the quotient T Cϕ /K is generated by a copy of C(T) and a unitary representation of Z satisfying the covariance relation u∗ϕ f uϕ = f ◦ ϕ−1 . Since the crossed product is the universal C*-algebra generated by these relations, the quotient T Cϕ /K is therefore isomorphic to a quotient of the crossed product C(T)ϕ Z. By the same argument as in [7], since the action of ϕ on T is amenable and topologically free, in fact T Cϕ /K ∼ = C(T)ϕ Z. Lemma 2.2. The group K1 (C(T)ϕ Z) is isomorphic to Z ⊕ Z, generated by [z]1 and [uφ ]1 . Proof. This result is proved using the Pimsner-Voiculescu six-term exact sequence, and is a simple generalization of the well-known result for the irrational rotation algebras Aθ [5, Example VII.5.2]. The irrational rotation algebras appear in our setting when ϕ(z) = λz with λ = e2πiθ . The Pimsner-Voiculescu exact sequence is K1 (C(T))  δ0 

1−α

i

i

1−α

∗ −−−−→ K1 (C(T)) −−−∗−→ K1 (C(T)ϕ Z)  δ 1

∗ K0 (C(T)ϕ Z) ←−− −− K0 (C(T)) ←−−−∗−

K0 (C(T))

The group K1 (C(T)) is generated by [z]1 , and by deﬁnition α∗ ([z]1 ) = [ϕ]1 . Since ϕ has winding number 1 about the origin, we have [ϕ]1 = [z]1 . Moreover, ϕ ﬁxes the unit which generates K0 (C(T)) and hence α∗ induces the identity map on K∗ (C(T)). Thus, the maps 1 − α∗ are 0 and K1 (C(T) ϕ Z) ﬁts into the short exact sequence i

δ

0 −−−−→ K1 (C(T)) −−−∗−→ K1 (C(T)ϕ Z) −−−1−→ K0 (C(T)) −−−−→ 0.

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Since the K-groups of C(T) are both isomorphic to Z, the above sequence splits and we obtain K1 (C(T)ϕ Z) ∼ = Z ⊕ Z. To ﬁnd the generators, we observe that the ﬁrst map is induced by inclusion, so [z]1 is a generator. The second map is the connecting map in the P-V exact sequence, and by construction this map takes the class [uϕ ]1 to the class of the unit in K0 (C(T)), which is its generator. With the exact sequence of Theorem 2.1 and the above description of the K1 group of C(T) ϕ Z, we can now obtain an integral formula for the index of Fredholm operators in T Cϕ which is a straightforward generalization of the formula obtained by E. Park [9] in the case of the irrational rotation algebras. In fact given Theorem 2.1 and Lemma 2.2 the arguments used in [9] go through almost verbatim, so we only sketch the details. To begin with, we deﬁne a “smooth subalgebra” A∞ ϕ of C(T)ϕ Z by ∞ k ∞ Aϕ = fk uϕ : fk ∈ C (T), {fk }k∈Z rapidly decreasing k∈Z

and a space of “smooth 1-forms” 1 ∞ k 1 Ω (Aϕ ) = ωk uϕ : ωk ∈ Ω (T), {ωk }k∈Z rapidly decreasing . k∈Z 1 ∞ ∞ The left module action of A∞ ϕ on Ω (Aϕ ) and the exterior derivative d : Aϕ → 1 ∞ Ω (Aϕ ) are deﬁned exactly as in [9], using the diﬀeomorphism ϕ. We then obtain 1 a map ν : Mn (Ω1 (A∞ ϕ )) → Ω (T) k ν ωk uϕ = Tr ω0 k∈Z

: GL(n, A∞ ) → Ω1 (T) by where Tr is the usual trace on Mn . Finally we deﬁne Ch ϕ 1

ν(X −1 dX). Ch(X) =− 2πi With these deﬁnitions, the proofs of Lemma 6 and Proposition 7 of [9] go through

induces a group homomorphism Ch : K1 (A∞ ) → unchanged, so that the map Ch ϕ 1 ) given by the series HdR (T). Finally, for X in GL(n, A∞ ϕ X= fk ukϕ k∈Z ∞

(note that fk ∈ Mn (C (T)) here), deﬁne TX ∈ T Cϕ by Tfk Uϕk . TX = k∈Z

Using Theorem 2.1 and Lemma 2.2 we obtain the following generalization of Theorem 8 of [9], with the same proof:

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Theorem 2.3. For every X in GL(n, A∞ ϕ ), ind TX = Ch(X). T

Proof. We prove that the two homomorphisms ind, T Ch(·) agree on the generators of the group K1 (C(T)ϕ Z). The map ind is the index map associated to the extension (2.1) and is computed on the generators easily: ind([uϕ ]1 ) = 0, since by construction uϕ lifts to the unitary Uϕ ∈ T Cϕ , and ind([z]1 ) = −1 since z lifts to the unilateral shift, which has Fredholm index −1. On the other hand, using the deﬁnition of Ch we ﬁnd 1 dz 1 1 · d1 and Ch(z) = − Ch(uϕ ) = − 2πi 2πi z so that T Ch(uϕ ) = 0 and T Ch(z) = −1. For the operators appearing in the later sections of the paper a more concrete expression of the above integral formula can be obtained. If we consider an element of T Cϕ of the form N TX = Tfk Uϕk k=0

N −1 with each fk ∈ C (T), then we have X = k=0 fk ukϕ ∈ A∞ ∈ A∞ ϕ and X ϕ is expressible as the norm convergent series X −1 = gk ukϕ ∞

k∈Z ∞

with each gk ∈ C (T). The index formula then takes the form N 1 ind TX = − g−k (ϕk (z))dfk (z). 2πi T

(2.2)

k=0

From this description it is also apparent that the index homomorphism from ∼ K1 (A∞ ϕ ) = K1 (C(T) ϕ Z) to C given by [X]1 → T Ch(X) coincides with the homomorphism given by the character of the 1-trace constructed by Connes [4, Theorem 1.5] for (reduced) crossed products of actions of countable groups on T. In particular, the validity of the index formula of Theorem 2.3 extends beyond the smooth subalgebra A∞ ϕ to any symbol X ∈ C(T)ϕ Z with the property that, in the natural representation π of C(T)ϕ Z on L2 (T), the commutator [P, π(X)] is trace class (here P : L2 → H 2 is the Riesz projection). While we will work throughout the next three sections with the C*-algebra T Cϕ = C ∗ (S, Cϕ ), it is worth noting that the inclusion of the unilateral shift S as a generator is often unnecessary; that is, it is often the case that S ∈ C ∗ (Cϕ ). Theorem 2.6 below gives a suﬃcient condition on the dynamics of ϕ (in fact valid for any inner function ϕ) for this to occur. This condition is satisﬁed, for example, by all parabolic automorphisms and by all hyperbolic automorphisms for which the ﬁxed points are not the endpoints of a diameter.

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Lemma 2.4. Let S denote the unilateral shift on H 2 . For any x ∈ C, let A = S + xSS ∗ . Then S ∈ C ∗ (A, I). Proof. We may obviously assume x = 0. We calculate |x|2 SS ∗ = I + xA∗ + xA − A∗ A ∈ C ∗ (A, I), and hence S = A − xSS ∗ ∈ C ∗ (A, I).

The forward direction of the following lemma was ﬁrst proved by Bourdon and MacCluer [2, Proposition 3]; we give here a diﬀerent proof which also suggests the proof of the converse. Lemma 2.5. If ϕ is any inner function then Cϕ∗ Cϕ = Tf∗ Tf = T|f |2 , where

(1 − |ϕ(0)|2 )1/2

. 1 − ϕ(0)z Conversely, if Cϕ∗ Cϕ is a Toeplitz operator, then ϕ is inner. f (z) =

Proof. As before let S denote the unilateral shift, and recall that an operator T ∈ B(H 2 ) is a Toeplitz operator if and only if S ∗ T S = T . Since ϕ is inner, Tϕ is an isometry and thus S ∗ Cϕ∗ Cϕ S = Cϕ∗ Tϕ∗ Tϕ Cϕ = Cϕ∗ Cϕ , so Cϕ∗ Cϕ is Toeplitz. Let g be its symbol. To ﬁnd g, we ﬁrst observe that since Cϕ∗ Cϕ is positive, g must be positive, and the projection of g into H 2 is given by h = Tg 1 = Cϕ∗ Cϕ 1 = Cϕ∗ 1 = kϕ(0) (z) = (1 − ϕ(0)z)−1 . Since g must be real-valued, it follows that g(z) = h(z) + h(z) − h(0) = |f (z)|2 . The factorization T|f |2 = Tf∗ Tf is valid because f is analytic. To prove the converse, the assumption that Cϕ∗ Cϕ is Toeplitz implies that ∗ Cϕ Cϕ = S ∗ Cϕ∗ Cϕ S = Cϕ∗ Tϕ∗ Tϕ Cϕ , or Cϕ∗ (Tϕ∗ Tϕ − I)Cϕ = 0. We therefore have 0 = Cϕ∗ (Tϕ∗ Tϕ − I)Cϕ 1, 1

= (Tϕ∗ Tϕ − I)1, 1

= ϕ2H 2 − 1. Thus, the L2 norm (using normalized arc length measure) of |ϕ| on the circle is 1, but since ϕ is a self-map of D its L∞ norm on the circle is at most 1, which implies that |ϕ(z)| = 1 almost everywhere on T; that is, ϕ is inner. Theorem 2.6. Let ϕ be an inner function. If the orbit (ϕn (0))∞ n=1 does not lie on a diameter of D, then S ∈ C ∗ (Cϕ ).

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Proof. By Lemma 2.5 we have, for each n, Cϕ∗ n Cϕn = Tf∗n Tfn where fn (z) = −1

(1 − |ϕn (0)|2 )1/2 (1 − ϕn (0)z) C ∗ (Cϕ ) contains

. Now Tf∗n Tfn is invertible, with inverse T f1 T ∗1 So n

fn

(1 − |ϕn (0)|2 )T f1 T ∗1 = 1 − ϕn (0)S − ϕn (0)S + |ϕn (0)|2 SS ∗ n

fn

for each n. By the hypothesis on the orbit, there exist n, m such that a = ϕn (0) and b = ϕm (0) are linearly independent over R. Since C ∗ (Cϕ ) contains an invertible operator, it contains I, so we have the operators aS + aS ∗ − |a|2 SS ∗ and bS + bS ∗ − |b|2 SS ∗ in C ∗ (Cϕ ). Since a and b are linearly independent over R, we may ﬁnd a linear combination of these operators of the form S + xSS ∗ for some scalar x. Applying the lemma, we ﬁnd S ∈ C ∗ (Cϕ ). The remainder of the paper concerns the Fredholm theory in T Cϕ for various linear fractional maps ϕ. When ϕ is an automorphism, the structure of the algebras T Cϕ and of the quotients C(T)ϕ Z is dependent upon the ﬁxed point behavior of ϕ. We therefore divide our analysis into three sections, for ϕ elliptic, hyperbolic, and parabolic respectively. This classiﬁcation of automorphisms and the properties of each class are well known, we brieﬂy recall the facts we require. Every (nontrivial) M¨ obius transformation of the Riemann sphere C ∪ {∞} ﬁxes exactly two points (counting multiplicity). A M¨obius transformation of the disk is called hyperbolic if it has two distinct ﬁxed points in the closed disk D, these necessarily lie on the boundary ∂D. If ϕ has only one ﬁxed point in D, then ϕ is called elliptic if the ﬁxed point lies in the interior D, and parabolic if the ﬁxed point lies on ∂D (in which case this point is a ﬁxed point of multiplicity two for the automorphism of the Riemann sphere given by ϕ). Obviously, the automorphism ϕ−1 is of the same type as ϕ, as are the iterates ϕn of ϕ. Every elliptic automorphism is conjugate to a rotation of D about the origin. If we replace the disk by the upper half-plane, every hyperbolic automorphism is conjugate to a dilation z → rz of the upper half plane (for some nonzero real r) and every parabolic automorphism is conjugate to a translation z → z + r for some real r. When ϕ is hyperbolic, the ﬁxed points can be distinguished by the modulus of the derivative: at one ﬁxed point λ+ , called the attracting ﬁxed point, we have |ϕ (λ+ )| < 1 and at the repelling ﬁxed point λ− we have |ϕ (λ− )| > 1. Moreover for each w ∈ D \ {λ− }, we have lim ϕn (w) = λ+ .

n→∞

When ϕ is replaced by ϕ−1 , the roles of the attracting and repelling ﬁxed points are reversed; and in particular we have for all w ∈ D \ {λ+ } lim ϕ−n (w) = λ− .

n→∞

When ϕ is parabolic the single ﬁxed point λ is called indiﬀerent, since it is neither attracting nor repelling in the topological sense, however we do have for

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all w ∈ D lim ϕn (w) = λ.

n→±∞

its Finally, we recall that for any automorphism ϕ of the Riemann sphere C,

on which the iterates of ϕ are a normal Fatou set (that is, the largest subset of C family) is the complement of its ﬁxed point set. In particular the set of iterates of ϕ is equicontinuous on T \ {λ+ , λ− } (in the hyperbolic case) and T \ {λ} (in the parabolic case).

3. Elliptic automorphisms of finite order In this section we assume ϕ is an elliptic automorphism of ﬁnite order, that is, there exists a nonnegative integer q such that ϕq (z) = z. Conjugating by an automorphism if necessary, we may assume that the ﬁxed point of ϕ is the origin, so that ϕ(z) = λz where λ = e2πi(p/q) with p/q in lowest terms. It is well-known that the crossed product C(T)ϕ Z is isomorphic to the subalgebra of Mq ⊗ C(T) consisting of elements of the form   f0 f1 ··· fq−1      fq−1 ◦ ϕ  f ◦ ϕ · · · f ◦ ϕ 0 q−2    .     · · · · · · · · · · · ·     f1 ◦ ϕ(q−1) ··· · · · f0 ◦ ϕ(q−1) In particular, in the regular representation π of C(T)ϕ Z on L2 (T)(q) , we have π(f ) = diag(Mf ◦ϕj ) for all f ∈ C(T) and π(uϕ ) is the permutation matrix with a 1 in the (i, j) entry if j − i ≡ 1 mod q and zeroes elsewhere. With this description of C(T)ϕ Z the Fredholm theory in T Cϕ is easily worked out, and we also obtain a more topological form of the index formula of Section 2. q Theorem 3.1. Let T = j=1 Tfj Cϕj + K and suppose each fj ∈ C 1 (T). Then T is Fredholm if and only if the C(T)-valued determinant f0 f1 ··· fq−1 fq−1 ◦ ϕ f ◦ ϕ · · · f ◦ ϕ 0 q−2 hT = · · · · · · · · · · · · (q−1) f1 ◦ ϕ(q−1) ··· · · · f0 ◦ ϕ is nonvanishing on T, in which case −1 ind(T ) = 2πiq

T

dhT , hT

(3.1)

350

that is,

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times the winding number of hT about the origin.

Proof. By the exact sequence 2.1 the operator T is Fredholm if and only if the q element f = j=0 fj ujϕ is invertible in C(T)ϕ Z, and by the above description of the regular representation this is the case if and only if hT is nonvanishing. To prove the index formula, we observe that it follows from the matrix-valued Toeplitz index theorem that if H(z) is invertible in Mq ⊗ C(T) and h(z) = det H(z), the (integer) quantity 1 dh χ(h) = (3.2) 2πi T h is homotopy invariant. By restricting to the subalgebra C(T)ϕ Z ⊂ Mq ⊗ C(T) (identiﬁed with its image in the regular representation), we obtain a homomorphism from K1 (C(T)ϕ Z) to Z. To prove the index formula we must show that this map agrees with the index map of the extension 2.1. Since this K1 group is generated by the elements w(z) = z and uϕ , it suﬃces to check that the two maps agree on these generators. On w(z) = z, we have det π(w)(z) = qj=1 ϕj (z) so the formula (3.1) gives (−1/q) · q = −1, which is equal to the index of the unilateral shift Tz , which is the lifting of w(z) = z to T Cϕ . On the other hand, applied to uϕ the formula (3.1) gives 0, which agrees with the index map since uϕ lifts to the unitary Uϕ .

4. Hyperbolic automorphisms In this section, ϕ is a hyperbolic automorphism with attracting ﬁxed point λ+ and repelling ﬁxed point λ− . While the problem of determining the spectrum of an arbitrary element f ∈ C(T) ϕ Z is likely intractable, we can obtain some suﬃcient conditions for invertibility (and hence for Fredholmness in T Cϕ ). 4.1. Localization in C(T)ϕ Z Fix an orthonormal basis {ξn }n∈Z for 2 (Z). We let u denote the bilateral shift on this basis, that is, uξn = ξn+1 . For each x ∈ T we deﬁne a representation πx : C(T)ϕ Z → B(2 (Z)) as follows: let πx (uϕ ) = u and for g ∈ C(T) let πx (g)(ξn ) = g(ϕn (x))ξn . In this subsection we collect some results about the representations πx which we will require for our main theorems. The main result of this section is Theorem 4.4, which reduces the question of invertibility in C(T)ϕ Z to invertibility in the “local” representations πx (by [3, Lemma 3.9], πx is irreducible for x = λ± ). Lemma 4.1. For each f ∈ C(T)ϕ Z, the function x → πx (f ) is continuous in the point-norm topology from T \ {λ+ , λ− } to B(2 (Z)). Proof. We prove the lemma for the “polynomial” expressions n fk ukϕ . f= k=m

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Since these are norm dense in C(T) ϕ Z, the general case follows by an “ /3” argument. Let f ∈ C(T)ϕ Z be as above, ﬁx x ∈ T \ {λ+ , λ− }, and let > 0 be given. Since each of the ﬁnitely many functions fk is uniformly continuous, there exists δ1 > 0 such that for all k = m, . . . n and all y, z ∈ T such that |y − z| < δ1 , . |fk (y) − fk (z)| < n−m+1 Since the iterates of ϕ are equicontinuous on T \ {λ+ , λ− }, there exists δ > 0 such that for all |y − x| < δ, we have supj |ϕj (y) − ϕj (x)| < δ1 . Using the fact that πx (uϕ ) = πy (uϕ ) = u, we then have for all |y − x| < δ πy (f ) − πx (f ) ≤ = =

n k=m n k=m n k=m

πy (fk ) − πx (fk )u diag(fk (ϕj (y)) − fk (ϕj (x))) sup |fk (ϕj (y)) − fk (ϕj (x))| j∈Z

< where the last inequality holds because of the choices of δ and δ1 .

(4.1)

Lemma 4.2. Let x1 and x2 be points in T \ {λ+ , λ− }. The representations πx1 and πx2 are equivalent if and only if x1 and x2 lie on the same orbit of ϕ. Proof. This is a special case of [3, Lemma 3.9].

Lemma 4.3. There exist closed arcs I1 , I2 ⊂ T such that for each x ∈ T \ {λ+ , λ− } there is a y ∈ I1 ∪ I2 such that πx is equivalent to πy . Proof. Choose one point from each of the two open arcs of T \ {λ+ , λ− }, call these points x1 and x2 . Let Ii be the closed arc with endpoints xi and ϕ(xi ). Since the translates of I1 and I2 cover T \ {λ+ , λ− }, each point of the latter set lies on the orbit of some point of I1 ∪ I2 . The lemma now follows from Lemma 4.2. Theorem 4.4. Let f ∈ C(T) ϕ Z. Then f is invertible if and only if πx (f ) is invertible for each x ∈ T. Proof. The “only if” statement is immediate, so we must prove the “if” statement. Let I1 and I2 be the closed arcs provided by Lemma 4.3 and let {xn } be a countable dense subset of I = I1 ∪ I2 which includes the endpoints of the intervals. We ﬁrst claim that the representation π := πλ+ ⊕ πλ− ⊕ {⊕n πxn } of C(T)ϕ Z is faithful. Since the action of ϕ on T is topologically free, a nontrivial ideal in C(T)ϕ Z must have nontrivial intersection with C(T). It therefore suﬃces to show that the restriction of πx to C(T) is faithful. Let g ∈ C(T) be a nonzero function. If πλ± (g) is nonzero, we are done. Otherwise, let x ∈ T \ {λ+ , λ− } with

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g(x) nonzero. Then g is nonvanishing in a neighborhood U of x, and by Lemma 4.3 there exists an integer k such that ϕk (U ) ∩ I = ∅. For any x in this intersection πx (g) = 0, and by Lemma 4.1 πxn (g) = 0 for some xn . Now let f ∈ C(T)ϕ Z and suppose πx (f ) is invertible for each x ∈ T. Since the representation π is faithful, it suﬃces to prove that π(f ) is invertible, and to prove this it suﬃces to show that the sequence πxn (f )−1 is bounded. By Lemma 4.1 and the continuity of the holomorphic functional calculus, for each n there is a neighborhood Un of xn such that πx (f )−1 − πxn (f )−1 < 1 for all x ∈ Un . By compactness, I is covered by ﬁnitely many of these Un , say Un1 , . . . Unk . Then for all x ∈ I (and in particular for all xn ) πx (f )−1 ≤ max{πxnk (f )−1 + 1}, k

−1

so the sequence πxn (f )

is bounded.

4.2. Fredholm criteria From the exact sequence 2.1 we know that an element of T Cϕ of the form T =

N

Tfn Uϕn

n=0

is Fredholm if and only if f=

N

fn unϕ

n=0

is invertible in C(T) ϕ Z. Under the simplifying assumption that the leading coeﬃcient f0 is nonvanishing on T (in other words, invertible in C(T)), we obtain a necessary and suﬃcient condition for the invertibility of f . The ﬁxed point polynomials of f , deﬁned below, play a central role in this analysis. Definition 4.5. For a ﬁnite sum f=

N

fn unϕ ,

n=0 − we deﬁne the ﬁxed point polynomials ρ+ f and ρf by

ρ+ f (z) =

N n=0

fn (λ+ )z n ,

ρ− f (z) =

N

fn (λ− )z n .

n=0

Before stating and proving the next theorem we introduce some notation. + For each n ∈ Z we let H+ n be the closed span of {ξk : k ≥ n}, and Qn the 2 − orthogonal projection from (Z) onto Hn . Similarly, we let Hn be the closed span of {ξk : k ≤ n} and Q− n the corresponding orthogonal projection. (Note that + 2 Q− = I − Q .) We deﬁne a unitary operator Un : H+ n n → H (T) by sending n+1 the orthonormal basis {ξn , ξn+1 , . . . , ξn+k , . . . } for Hn to the orthonormal basis

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2 {1, z, . . . z k . . . } for H 2 . Similarly, Vn is the unitary operator taking H− n to H via n−k 2 ξk → z . Finally, for an operator A ∈ B( (Z)), we write

[A]ij = Aξi , ξj

for the matrix elements of A. N n Throughout this section we ﬁx an element f = n=0 fn uϕ ∈ C(T) ϕ Z. We ﬁrst examine matricial structure of the operator πx (f ) with respect to the orthonormal basis {ξn }. From the deﬁnition of the representation πx , the matrix elements of πx (f ) are fi−j (ϕj (x)) 0 ≤ i − j ≤ N, [πx (f )]ij = 0 otherwise. Note that if x is a ﬁxed point of ϕ then the matrix of πx (f ) is a bi-inﬁnite Toeplitz matrix, with fk (x) on the k th subdiagonal. In particular, this shows that for x = λ± 2 the operator πλ± (f ) is unitarily equivalent to multiplication by ρ± f on L (T), since the latter operators are represented, with respect to the basis {e2πikθ } of L2 (T), by the same Toeplitz matrix as πλ± (f ). The following lemma tells us that for x not a ﬁxed point, the matrix of πx (f ) has an “asymptotic” Toeplitz structure. ± n Lemma 4.6. Let f = N n=0 fn uϕ ∈ C(T)ϕ Z with ﬁxed point polynomials ρf . For + each > 0 and each x ∈ T, there exists an integer Nx such that + ∗ Q+ n πx (f )Qn − Un Tρ+ Un < f

Nx+ .

for all n ≥ Similarly, for each > 0 and each x ∈ T there is an integer Nx− such that − ∗ ∗ Q− n πx (f )Qn − Vn Tρ− Vn < f

for all n ≤ Nx− . Moreover, each of these diﬀerences is compact. Proof. Since lim fk (ϕn (x)) = fk (λ+ )

n→∞

for all x ∈ T \ {λ+ , λ− } and all 0 ≤ k ≤ N , it follows that given any > 0 there exists an integer Nx+ such that . (4.2) max sup |fk (ϕn (x)) − fk (λ+ )| < 0≤k≤N N +1 n≥Nx+ + ∗ Now, ﬁx n ≥ Nx+ . Each of the operators Q+ Un has its matrix n πx (f )Qn , Un Tρ+ f elements supported in the band 0 ≤ i − j ≤ N , and within this band we have + j [Q+ n πx (f )Qn ]ij = fi−j (ϕ (x));

[Un∗ Tρ+ Un ]ij = fi−j (λ+ ). f

It follows that the diﬀerence of these operators is also supported in the band 0 ≤ i − j ≤ N , and by (4.2) each of its nonzero matrix elements is less that /(N + 1) in absolute value. It follows that the norm of this diﬀerence is less

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than . Moreover since the diﬀerence is supported in a ﬁnite band and its matrix elements tend to 0 on each subdiagonal, it is compact. − The proof for Q− n πx (f )Qn is entirely analogous. Lemma 4.7. If the ﬁxed point polynomials ρ± f do not vanish in D, then for each x ∈ T there exists integers Nx± such that for all n ≥ Nx+ , the operator + Q+ n πx (f )Qn

is invertible, and for all n ≤ Nx− the operator − Q− n πx (f )Qn

is invertible. Proof. Since the ﬁxed point polynomial ρ+ f does not vanish in D, the Toeplitz operator Tρ+ is invertible. It follows that for any n, any suﬃciently small perturf bation of Un∗ Tρ+ Un will also be invertible. By the previous lemma we may choose f

+ + ∗ the integer Nx+ to make Q+ Un n πx (f )Qn , for all n ≥ Nx , as close in norm to Un Tρ+ f as we like. The claimed invertibility follows, and an identical argument works for − Q− n πx (f )Qn .

Theorem 4.8. Let f =

N

fn unϕ ∈ C(T)ϕ Z, and suppose the ﬁxed point polyno-

n=0

− mials ρ+ f (z) and ρf (z) have no zeroes in the closed unit disk. Then for each x ∈ T, πx (f ) is invertible if and only if f0 does not vanish on the orbit Ox .

Proof. Since the polynomials ρ± f do not vanish in D, we have in particular ρ± f (0) = f0 (λ± ) = 0, that is, f0 does not vanish at the ﬁxed points of ϕ, so we assume x ∈ T \ {λ+ , λ− } and that f0 (y) = 0 for some y ∈ Ox ; that is, f0 (ϕn (x)) = 0 for some n ∈ Z. Since f0 is continuous, it is nonvanishing in neighborhoods of λ+ and λ− . Hence for all x ∈ T \ {λ+ , λ− }, lim [πx (f )]nn = lim f0 (ϕn (x)) = f0 (λ± ) = 0

n→±∞

n→±∞

and thus [πx (f )]nn = f0 (ϕn (x)) is nonzero for all n suﬃciently large in absolute value. It follows that πx (f ) has only ﬁnitely many zeroes on the diagonal, and hence there is some largest m such that [πx (f )]mm = 0. By Lemma 4.7, the compression + Q+ M πx (f )QM is invertible for all suﬃciently large M > m. We may now write + Q+ m πx (f )Qm in block diagonal form as A 0 + + Qm πx (f )Qm = K B + where B = Q+ M πx (f )QM is invertible and A is an (M − m) × (M − m) lower + triangular matrix whose ﬁrst row is 0. Thus we can ﬁnd v ∈ H+ m HM such that −1 + Av = 0. It follows that the vector v ⊕ (−B Kv) lies in the kernel of Q+ m πx (f )Qm ,

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and hence 0 ⊕ v ⊕ (−B −1 Kv) lies in the kernel of πx (f ) and hence πx (f ) is not invertible. Conversely, if f does not vanish on Ox then we may decompose πx (f ) into a block lower triangular form with each diagonal block invertible. Indeed choose + M according to Lemma 4.7 so that Q+ M πx (f )QM is invertible, and similarly we − − choose L ∈ Z, L < M so that QL πx (f )QL is invertible. The operator πx (f ) thus admits a block triangular decomposition   − 0 0 QL πx (f )Q− L , ∗ C 0 πx (f ) =  + π (f )Q ∗ ∗ Q+ M x M where the matrix C is an (M − L) × (M − L) lower triangular matrix which does not vanish on the diagonal, and is hence invertible. Thus πx (f ) admits a block triangular form with invertible diagonal blocks, and is hence invertible. Theorem 4.9. Let T =

N

− Tfn Uϕn + K. If the ﬁxed point polynomials ρ+ f and ρf

n=0

have no zeroes in the closed unit disk, then T is Fredholm if and only if Tf0 is Fredholm. Furthermore, in this case ind(T ) = ind(Tf0 ). Proof. T is Fredholm if and only if its symbol χ(T ) =

N

fn unϕ

n=0

is invertible. By Theorems 4.8 and 4.4, χ(T ) is invertible if and only if f0 does not vanish on T. This is the case if and only if the Toeplitz operator Tf0 is Fredholm. To prove the statement about the indices, we will construct an explicit homotopy between the symbols f and f0 through invertible elements of C(T)ϕ Z. Since T and Tf0 are liftings of f and f0 respectively, the equality of the indices follows. To construct the homotopy, we ﬁrst observe that since the ﬁxed point polynomials have no zeroes in D, each is homotopic to a nonzero constant polynomial, and the homotopy may be taken through polynomials without zeroes in D. Indeed, for 0 ≤ t ≤ 1 put ± p± t (z) = ρf (tz) and observe that if the zeroes of ρ± f have modulus greater than one then the same is true for each p± . We now deﬁne the homotopy between f and f0 by t g(t) =

N

fn tn unϕ .

n=0 − For each t, the ﬁxed point polynomials of g(t) are p+ t and pt respectively. Thus, by Theorem 4.8, g(t) is invertible for all t. By construction g(0) = f0 and g(1) = f .

This result can be restated as a spectral inclusion theorem:

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Corollary 4.10. Let f =

N n=0

IEOT

fn unϕ ∈ C(T)ϕ Z, with ϕ hyperbolic. Then

− σ(f ) ⊆ f0 (T) ∪ ρ+ f (D) ∪ ρf (D).

Proof. Suppose λ ∈ C does not belong to the above union of sets. Then the leading coeﬃcient f0 − λ of f − λ has no zeroes on the circle, and since the ﬁxed point polynomials of f − λ are ρ± f − λ, these have no zeroes in the closed disk. Therefore f − λ is invertible. In general this inclusion is strict (in the example following Theorem 4.12, we − have 0 ∈ ρ+ f (D) ∪ ρf (D) but f is invertible). A more reﬁned spectral inclusion will follow from that theorem. Under the assumption that f0 is invertible, but now allowing zeroes ofnthe ﬁxed point polynomials in D, we characterize the invertibility of f = fn uϕ in terms of the invertibility of an additional matrix-valued function on T. Again, we ﬁx x ∈ T and examine invertibility in the representation πx . First, we observe that since ρ+ f is an analytic polynomial with d zeroes in D, the Toeplitz operator Tρ+ has a d-dimensional cokernel and trivial kernel (and f

+ hence Fredholm index −d). By Lemma 4.6 the same will be true of Q+ M πx (f )QM for all suﬃciently large M . We now ﬁx some notation: for a ﬁxed M as above, put + F+ (x) = Q+ M πx (f )QM ,

− F− (x) = Q− M πx (f )QM ,

− K(x) = Q+ M πx (f )QM

and note that K(x) is ﬁnite rank and hence compact. With this notation πx (f ) has the block triangular form F− (x) 0 πx (f ) = . K(x) F+ (x) Observe also that by compactness, we may choose a single M so that this decomposition is valid for all x ∈ I1 ∪ I2 (see subsection 4.1). − Lemma 4.11. Suppose the ﬁxed point polynomials ρ+ f , ρf are nonvanishing on ∂D and have d+ and d− zeroes in D, respectively. Then for each x ∈ ∂D \ {λ+ , λ− }, the operator πx (f ) is Fredholm and ind(πx (f )) = d− − d+ . Furthermore, if f0 does not vanish on Ox then ker F− (x)∗ = {0}.

Proof. By Lemma 4.6 and the above triangular form of πx (f ), we see that πx (f ) is unitarily equivalent to a compact perturbation of the operator ∗ 0 Tρ− f

0

Tρ+ f

on H ⊕ H . The assumption on the zeroes of ρ± f then implies that the operators Tρ∗− and Tρ+ are Fredholm of index d− and −d+ , respectively. This proves the ﬁrst 2

f

statement.

2

f

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For the second statement, we consider a further block decomposition of − F− (x). By Lemma 4.6 we can choose L < M so that Q− L πx (f )QL is unitarily ∗ equivalent to an arbitrarily small perturbation of Tρ− . Since this latter operator f

− has trivial cokernel, we can choose L so that Q− L πx (f )QL also has trivial cokernel. We then obtain a block triangular form for F− (x): − QL πx (f )Q− 0 L F− (x) = X C

where X is ﬁnite rank and C is an (M − L) × (M − L) lower triangular matrix which does not vanish on the diagonal, and is hence invertible. We now consider ker F− (x)∗ : suppose v, w are vectors such that − ∗ v 0 (QL πx (f )Q− X∗ L) = . w 0 0 C∗ − ∗ Then w = 0 since C ∗ is invertible, and therefore v = 0 since (Q− L πx (f )QL ) has ∗ trivial kernel. Thus ker F− (x) is trivial.

We now return to the triangular form of πx (f ). Assume now that the ﬁxed point polynomials each have d zeroes in D (and are nonvanishing on ∂D) and that f0 does not vanish on Ox . In the block decomposition of πx (f ), the operator F+ (x) has trivial kernel and, by the previous lemma, F− (x) has trivial cokernel. Since these operators are also Fredholm, they have closed range and we can deﬁne two projection-valued functions on I1 ∪ I2 by P+ (x) ≡ 1 − F+ (F+∗ F+ )−1 F+∗ ;

P− (x) ≡ 1 − F−∗ (F− F−∗ )−1 F−

(here we have suppressed the x-dependence on the right-hand side). In other words, P+ (x) is the projection onto the cokernel of F+ (x), and P− (x) is the projection onto the kernel of F− (x). We are now ready to state and prove our main result about the invertibility of πx (f ). Theorem 4.12. Let f = fn unϕ with f0 nonvanishing and suppose the ﬁxed point polynomials ρ± f each have d zeroes in D (and are nonvanishing on the boundary). Then f is invertible if and only if for all x ∈ ∂D \ {λ+ , λ− }, the matrix-valued function D(x) ≡ P+ (x)K(x)P− (x) is invertible from ker F− (x) to ker F+ (x)∗ . Proof. By the assumptions on ρ± f , Lemma 4.11 shows that πx (f ) is Fredholm of index 0. It is therefore invertible if and only if it has trivial kernel. Considering the block triangular form of πx (f ), suppose a nonzero vector v ⊕ w lies in the kernel of πx (f ). Then, since kerF+ is trivial, v ∈ kerF− must be nonzero, and we must have Kv ∈ ranF+ . In other words, v must be a nonzero vector in the kernel of P+ (x)K(x)P− (x). Conversely, given any such v we may solve for w such that Kv + F+ (x)w = 0, whence v ⊕ w lies in kerπx (f ). Thus πx (f ) is invertible if and only if D(x) is invertible. The theorem then follows from Theorem 4.4.

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Exactly as in the case of Theorem 4.9, this result can be stated as a spectral inclusion. To account for the necessary condition that the ﬁxed point polynomials have the same number of zeroes in D, let Wρ ⊂ C denote the set of all points − w such that either 1) ρ+ f and ρf have diﬀerent winding numbers about w, or 2) − w ∈ ρ+ f (T) ∪ ρf (T) (that is, at least one of the winding numbers is undeﬁned). It is not hard to see that Wρ is compact. Finally, let WD denote the subset of the complement of f0 (T) ∪ Wρ such that the matrix-valued function D constructed fromf − w in the manner above is not invertible. N Corollary 4.13. Let f = n=0 fn unϕ ∈ C(T)ϕ Z. Then σ(f ) ⊆ f0 (T) ∪ Wρ ∪ WD . Unlike in Theorem 4.8, it is possible to have the symbol of a Fredholm operator satisfy the hypotheses of Theorem 4.12 but ind(T ) = ind(Tf0 ): Example. Let 0 < r < 1 and put T = rTz + Uϕ . Then f = χ(T ) = rz + uϕ is invertible, since uϕ is unitary and f − uϕ = rz∞ < 1. We have f0 (z) = rz nonvanishing on the unit circle, and the ﬁxed point polynomials are ρ+ f (w) = rλ+ + w,

ρ− f (w) = rλ− + w;

each of which has one zero in D and is nonvanishing on the unit circle. Thus f satisﬁes the hypotheses of Theorem 4.12. However, since f − uϕ < 1, the symbol f is homotopic to uϕ . It follows that ind(T ) = ind(Uϕ ) = 0 while ind(Tf0 ) = ind(Trz ) = −1. If a Toeplitz operator Tg is Fredholm and ind(Tg ) = 0, then Tg is invertible. We do not know if the corresponding statement is true in T Cϕ , even for the Fredholm operators described by the theorems in this section: Question. If T ∈ T Cϕ is Fredholm and ind(T ) = 0, is T invertible?

5. Parabolic automorphisms For the present purposes, it is best to view a parabolic automorphism as a degenerate hyperbolic automorphism, for which the attracting and repelling ﬁxed points coalesce. Taking this point of view, the main theorems of the previous section are easily modiﬁed to apply to the parabolic case, with essentially the same proofs. We will therefore only sketch the proofs in this section, taking care to highlight the necessary changes.

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We let ϕ now denote a parabolic automorphism of D, with ﬁxed point λ ∈ T. As before we consider operators of the form f=

N

fn unϕ

n=0

and deﬁne the ﬁxed point polynomial ρf by ρf (z) =

N

fn (λ)z n .

n=0

In the hyperbolic case, the ﬁxed points λ+ and λ− are attracting for ϕ and ϕ−1 , respectively. Now that ϕ is parabolic, the point λ is attracting for both ϕ and ϕ−1 , in the sense that for any x ∈ T, x = λ, we have lim ϕn (x) = lim ϕ−n (x) = λ.

n→∞

n→∞

With this fact in mind, the obvious modiﬁcations to the proof of Lemma 4.6 show that the conclusion of that lemma holds for parabolic ϕ if we replace both ρ+ f and ρ− f with ρf . The statement of Lemma 4.7 is then also valid in the parabolic case. With these substitute lemmas in hand, the following theorem is proved in the same way as Theorem 4.8, mutatis mutandis. Theorem 5.1. For a parabolic automorphism ϕ, let f =

N

fn unϕ ∈ C(T)ϕ Z, and

n=0

suppose the ﬁxed point polynomial ρf (z) has no zeroes in the closed unit disk. Then for each x ∈ T, πx (f ) is invertible if and only if f0 does not vanish on the orbit Ox . In turn, using this result, the following analog of Theorem 4.9 is seen to be − true, by letting ρ+ f = ρf = ρf in its proof: Theorem 5.2. Let T =

N

Tfn Uϕn + K. If the ﬁxed point polynomial ρf has no

n=0

zeroes in the closed unit disk, then T is Fredholm if and only if Tf0 is Fredholm. Furthermore, in this case ind(T ) = ind(Tf0 ). Continuing as in the discussion following Theorem 4.9 for each x ∈ T, x = λ, we have the block triangular form for the matrix F− (x) 0 πx (f ) = . K(x) F+ (x) From this follows a simpler version of Lemma 4.11: Lemma 5.3. Suppose the ﬁxed point polynomial ρf is nonvanishing on ∂D. Then for each x ∈ ∂D \ {λ}, the operator πx (f ) is Fredholm and ind(πx (f )) = 0. Furthermore, if f0 does not vanish on Ox then ker F− (x)∗ = {0}.

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Proof. Using the block triangular form of πx (f ) and the parabolic version of Lemma 4.6, we ﬁnd that πx (f ) is unitarily equivalent to a compact perturbation of ∗ 0 Tρf 0 Tρf on H 2 ⊕ H 2 . This operator is Fredholm of index 0. The second part of the lemma is proved as in the hyperbolic case. Finally, since we have now shown that πx (f ) is always Fredholm of index 0 when ϕ is parabolic, using the same notation as in Theorem 4.12 we have a parabolic version of that theorem and its corresponding spectral inclusion: Theorem 5.4. Let f = fn unϕ with f0 nonvanishing and suppose the ﬁxed point polynomial ρf is nonvanishing on T. Then f is invertible if and only if for all x ∈ T, x = λ, the matrix-valued function D(x) ≡ P+ (x)K(x)P− (x) is invertible from ker F− (x) to ker F+ (x)∗ . Since we have only a single ﬁxed point polynomial, the corresponding spectral inclusion theorem is somewhat simpler in the parabolic case. We here let WD denote those points w of the complement of f0 (T) ∪ ρf (T) for which the function D associated to f − w is not invertible. Corollary 5.5. Let f = fn unϕ ∈ C(T)ϕ Z. Then σ(f ) ⊆ f0 (T) ∪ ρf (T) ∪ WD .

6. Non-automorphic linear fractional maps In this section we consider the extension 0 → K → T Cϕ → D → 0

(6.1)

constructed by Kriete, MacCluer and Moorhouse [8]. We ﬁrst recall the notation and results of [8]. Throughout this section, ϕ is a non-automorphic linear fractional map ϕ : D → D such that ϕ(ζ) = η for some ζ = η ∈ ∂D. The compact Hausdorﬀ space Λ consists of the disjoint union of ∂D and the closed interval [0, 1] with the points ζ, η and 0 identiﬁed, so that Λ is homeomorphic to a ﬁgure eight with a closed interval attached by one endpoint to the vertex, which we denote p. The C*-algebra D consists of all functions b : Λ → M2 (C) satisfying the following conditions: there exist w ∈ C(∂D) and G ∈ M2 (C([0, 1])) such that • For λ ∈ C(∂D) \ {ζ, η}, w(λ) 0 b(λ) = . 0 w(λ) • For λ ∈ [0, 1], b(λ) = G(λ).

Vol. 58 (2007) The Fredholm Index in Toeplitz-Composition C*-Algebras

• At the vertex p,

w(ζ) b(p) = G(0) = 0

361

0 . w(η)

The set of such functions is a C*-algebra when equipped with the supremum norm, pointwise operations, and the obvious involution. We will denote elements of D by (w, G) where w and G are as above. Note that in this description products and adjoints are taken entrywise: (w1 , G1 ) · (w2 , G2 ) = (w1 w2 , G1 G2 ), and (w, G)∗ = (w, G∗ ). For the next theorem, we recall two deﬁnitions: ﬁrst, if A and B are C*algebras and ρ, σ : A → B are *-homomorphisms, we say that ρ and σ are homotopic if there exists a path of *-homomorphisms ρt : A → B, 0 ≤ t ≤ 1, with ρ = ρ0 and σ = ρ1 . The path is required to be continuous in the sense that for each a ∈ A, the map t → ρt (a) is continuous from [0, 1] to B (equipped with the norm topology). Secondly, a pair of C*-algebras A and B are called homotopy equivalent if there exist *-homomorphisms θ : A → B and ψ : B → A such that the compositions θ ◦ ψ and ψ ◦ θ are homotopic to idA and idB respectively. Theorem 6.1. The C*-algebras D and C(T) are homotopy equivalent. Proof. For w ∈ C(T), let W be the M2 (C)-valued function on [0, 1] which is identically equal to diag(w(ζ), w(η)). We deﬁne *-homomorphisms θ : C(T) → D and ψ : D → C(T) by θ(w) = (w, W ),

ψ(w, G) = w.

Obviously ψ ◦ θ = idC(T) . We have θ ◦ ψ(w, G) = (w, W ). Note that W (λ) = G(0) for all λ ∈ [0, 1]. For t ∈ [0, 1] deﬁne the *-homomorphism ρt : D → D by ρt (w, G) = (w, Gt ) where Gt (λ) = G(tλ). Thus ρt is a homotopy between ρ1 = idD and ρ0 = θ ◦ψ. Theorem 6.2. Let T = Tw + C + K ∈ T Cϕ . If T is Fredholm then Tw is Fredholm and ind(T ) = ind(Tw ). Proof. Since the image of T in D is of the form (w, G), T is Fredholm if and only if w and G are pointwise invertible, so if T is Fredholm then w is invertible and the Toeplitz operator Tw is Fredholm. To prove the index statement, it suﬃces to prove that the images of T and Tw are homotopic through invertibles in D. Using the homotopy (w, Gt ) of the previous theorem, we see that (w, G) is homotopic to (w, W ), and since G is assumed invertible each Gt is invertible, and hence (w, Gt ) is invertible in D for all t ∈ [0, 1]. Finally, the element (w, W ) is by construction the image of Tw in D. Combining the previous two theorems we get the following corollary:

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Corollary 6.3. The group Ext(D) is isomorphic to Z and is generated by the class of the extension (6.1). Proof. The ﬁrst statement follows from Theorem 6.1 and the homotopy invariance of the Ext functor. Since Ext(T) ∼ = Z is generated by the class of an extension for which the function w(z) = z lifts to an operator of index ±1, the group Ext(D) will be generated by an extension for which the element θ(w) = (w, W ) lifts to an operator of index ±1. By Theorem 6.2 the extension (6.1) has this property ((w, W ) lifts to the unilateral shift).

References [1] Florin-Petre Boca. Rotation C ∗ -algebras and almost Mathieu operators, Theta Series in Advanced Mathematics. The Theta Foundation, Bucharest, 2001. [2] Paul S. Bourdon and Barbara MacCluer. Selfcommutators of automorphic composition operators. Indiana Univ. Math. J., to appear. [3] Man-Duen Choi and Fr´ed´eri Latr´emoli`ere. Crossed-product C*-algebras for conformal automorphisms of the disk, arXiv:math.OA/0511331 [4] Alain Connes. Cyclic cohomology and the transverse fundamental class of a foliation. In Geometric methods in operator algebras (Kyoto, 1983), volume 123 of Pitman Res. Notes Math. Ser., pages 52–144. Longman Sci. Tech., Harlow, 1986. [5] Kenneth R. Davidson. C ∗ -algebras by example, Fields Institute Monographs, volume 6. American Mathematical Society, Providence, 1996. [6] Ronald G. Douglas. Banach algebra techniques in operator theory, volume 179 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. [7] Michael T. Jury. C*-algebras generated by groups of composition operators, arXiv:math.OA/0509614. [8] Thomas Kriete, Barbara MacCluer, and Jennifer Moorhouse. Toeplitz-composition C*-algebras. Journal of Operator Theory, to appear, arXiv:math.OA/0608445. [9] Efton Park. Toeplitz algebras and extensions of irrational rotation algebras. Canad. Math. Bull., 48(4):607–613, 2005. Michael T. Jury Department of Mathematics University of Florida Gainesville, Florida 32603 USA e-mail: [email protected] Submitted: October 26, 2006 Revised: November 29, 2006

Integr. equ. oper. theory 58 (2007), 363–405 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030363-43, published online April 14, 2007 DOI 10.1007/s00020-007-1498-9

Integral Equations and Operator Theory

Soft Ideals and Arithmetic Mean Ideals Victor Kaftal and Gary Weiss Abstract. This article investigates the soft-interior (se) and the soft-cover (sc) of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study in [10] of the multiplicity of traces. Many classical ideals are “soft”, i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals in [6, 7] and arithmetic mean operations at infinity (am-∞) were studied in [10]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-∞ interior of an ideal. Mathematics Subject Classification (2000). Primary 47B47, 47B10, 47L20; Secondary 46A45, 46B45. Keywords. Arithmetic means, operator ideals, countably generated ideals, Lorentz ideals, Orlicz ideals, Marcinkiewicz ideals, Banach ideals.

1. Introduction Central to the theory of operator ideals (the two-sided ideals of the algebra B(H) of bounded operators on a separable Hilbert space H) are the notions of the commutator space of an ideal I (the linear span of the commutators T A − AT , A ∈ I, T ∈ B(H)) and of a trace supported by the ideal. In this context, the arithmetic (Cesaro) mean of monotone sequences ﬁrst appeared implicitly in [21], then played in [15] an explicit and key role for determining the commutator space of the trace class, and more recently entered center stage in [6, 7] by providing the framework for the characterization of the commutator space of arbitrary ideals. This prompted [7] to associate more formally to a given ideal I the arithmetic mean ideals Ia , a I, I o = (a I)a (the am-interior of I) and I − = a (Ia ) (the am-closure Both authors were partially supported by grants of the Charles Phelps Taft Research Center; the second named author was partially supported by NSF Grants DMS 95-03062 and DMS 97-06911.

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of I). (See Section 2 for deﬁnitions.) In particular, the arithmetically mean closed ideals (those equal to their am-closure) played an important role in the study of single commutators in [7]. This paper and [10]-[13] are part of an ongoing program announced in [9] dedicated to the study of arithmetic mean ideals and their applications. In [10] we investigated the question: “How many traces (up to scalar multiples) can an ideal support?” We found that for the following two classes of ideals which we call “soft” the answer is always zero, one or uncountably many: the soft-edged ideals that coincide with their soft-interior se I := IK(H) and the softcomplemented ideals that coincide with their soft complement sc I := I : K(H) (K(H) is the ideal of compact operators on H and for quotients of ideals see Section 3). Softness properties have often played a role in the theory of operator ideals, albeit that role was mainly implicit and sometimes hidden. Taking the product of I by K(H) corresponds at the sequence level to the “little o” operation, which ﬁgures so frequently in operator ideal techniques. M. Wodzicki employs explicitly the notion of soft interior of an ideal (although he does not use this terminology) to investigate obstructions to the existence of positive traces on an ideal (see [22, Lemma 2.15, Corollary 2.17]. A special but important case of quotient is the celebrated K¨othe dual of an ideal and general quotients have been studied among others by Salinas [18]. But to the best of our knowledge the power of combining these two soft operations has gone unnoticed along with their investigation and a systematic use of their properties. Doing just that permitted us in [10] to considerably extend and simplify our study of the codimension of commutator spaces. In particular, we depended in a crucial way on the interplay between soft operations and arithmetic mean operations. Arithmetic mean operations on ideals were ﬁrst introduced in [7] and further studied in [10]. For summable sequences, the arithmetic mean must be replaced by the arithmetic mean at inﬁnity (am-∞ for short), see for instance [1, 7, 14, 22]. In [10] we deﬁned am-∞ ideals and found that their theory is in a sense dual to the theory of am-ideals, including the role of ∞-regular sequences studied in [10, Theorem 4.12]). In [10] we considered only the ideals a I, Ia , a∞ I, and Ia∞ , and so in this paper we focus mostly on the other am and am-∞ ideals. In Section 2 we prove that the sum of two am-closed ideals is am-closed (Theorem 2.5) by using the connection between majorization of inﬁnite sequences and inﬁnite substochastic matrices due to Markus [16]. (Recent outgrowths from [ibid] from the classical theory for ﬁnite sequences and stochastic matrices to the inﬁnite is one focus of [11].) This leads naturally to deﬁning a largest amclosed ideal I− ⊂ I. We prove that I− = a I for countably generated ideals (Theorem 2.9) while in general the inclusion is proper. An immediate consequence is that a countably generated ideal is am-closed (I = I − ) if and only if it is amstable (I = Ia ) (Theorem 2.11). This generalizes a result from [2, Theorem 3.11]. Then we prove that arbitrary intersections of am-open ideals must be am-open

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(Theorem 2.17) by ﬁrst obtaining a characterization of the am-interior of a principal ideal (Lemma 2.14) and then of an arbitrary ideal (Corollary 2.16). This leads naturally to deﬁning the smallest am-open ideal I oo ⊃ I. In Section 3 we obtain analogous results for the am-∞ case. But while the statement are similar, the techniques employed in proving them are often substantially diﬀerent. For instance, the proof that the sum of two am-∞ closed ideals is am-∞ closed (Theorem 3.2) depends on a w∗ -compactness argument rather than a matricial one. In Section 4 we study soft ideals. The soft-interior se I and the soft-cover sc I are, respectively, the largest soft-edged ideal contained in I and the smallest soft-complemented ideal containing I. The pair se I ⊂ sc I is the generic example of what we call a soft pair. Many classical ideals, i.e., ideals whose characteristic set is a classical sequence space, turn out to be soft. Among soft-edged ideals are (o) minimal Banach ideals Sφ for a symmetric norming function φ, Lorentz ideals (o)

L(φ), small Orlicz ideals LM , and idempotent ideals. To prove soft-complementedness of an ideal we often ﬁnd it convenient to prove instead a stronger property which we call strong soft-complementedness (Deﬁnition 4.4, Proposition 4.5). Among strongly soft complemented ideals are principal and more generally countably generated ideals, maximal Banach ideals (o) ideals Sφ , Lorentz ideals L(φ), Marcinkiewicz ideals a (ξ), and Orlicz ideals LM . K¨ othe duals and idempotent ideals are always soft-complemented but can fail to be strongly soft-complemented. (o)

Employing the properties of soft pairs for the embedding Sφ ⊂ Sφ in the principal ideal case, we present a simple proof of the fact that if a principal ideal is a Banach ideal then its generator must be regular, which is due to Allen and Shen [2, Theorem 3.23] and was also obtained by Varga [20] (see Remark 4.8(iv) and [7, Theorem 5.20]). The same property of the embedding yields a simpler proof of part of a result by Salinas in [18, Theorem 2.3]. Several results relating small Orlicz and Orlicz ideals given in theorems in [7] follow immediately from the fact (o) that LM ⊂ LM are also soft pairs (see remarks after Proposition 4.11). Various operations on ideals produce additional soft ideals. Powers of soft-edged ideals, directed unions (by inclusion) of soft-edged ideals, ﬁnite intersections and ﬁnite sums of soft-edged ideals are all soft-edged. Powers of softcomplemented ideals and arbitrary intersections of soft-complemented ideals are also soft-complemented (Section 4). As consequences follow the softness properties of the am and am-∞ stabilizers of the trace-class L1 (see Sections 2 and 3 for the deﬁnitions) which play an important role in [9]-[10]. However, whether the sum of two soft-complemented ideals or even two strongly soft-complemented ideals is always soft-complemented remains unknown. We prove that it is under the additional hypothesis that one of the ideals is countably generated and the other is strongly soft-complemented (Theorem 5.7).

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Some of the commutation relations between the soft-interior and soft-cover operations and the am and am-∞ operations played a key role in [10]. We investigate the commutation relations with the remaining operations in Section 6 (Theorems 6.1, 6.4, 6.9, and 6.10). As a consequence we obtain which operations preserve soft-complementedness and soft-edgedness. Some of the relations remain open, e.g., we do not know if sc Ia = (sc I)a (see Proposition 6.8). Following this paper in the program outlined in [9] will be [11] where we clarify the interplay between arithmetic mean operations, inﬁnite convexity, and diagonal invariance and [12] where we investigate the lattice properties of several classes of operator ideals proving results of the kind: between two proper ideals, at least one of which is am-stable (resp., am-∞ stable) lies a third am-stable (resp., am-∞ stable) principal ideal and applying them to various arithmetic mean cancellation and inclusion properties (see [9, Theorem 11 and Propositions 12–14]. Example, for which ideals I does the Ia = Ja (resp., Ia ⊂ Ja , Ia ⊃ Ja ) imply I = J (resp., I ⊂ J, I ⊃ J) and in the latter cases, is there an “optimal” J?

2. Preliminaries and Arithmetic Mean Ideals Calkin [5] established a correspondence between two-sided ideals of bounded operators on a complex separable inﬁnite dimensional Hilbert space and characteristic sets, i.e., hereditary (i.e., solid) cones Σ ⊂ c∗o (the collection of sequences decreasing to 0), that are invariant under ampliations. For each m ∈ N , the m-fold ampliation Dm is deﬁned by: c∗o ξ −→ Dm ξ := ξ1 , . . . , ξ1 , ξ2 , . . . , ξ2 , ξ3 , . . . , ξ3 , . . . with each entry ξi repeated m times. The Calkin correspondence I → Σ(I) induced by I X → s(X) ∈ Σ(I), where s(X) denotes the sequence of the s-numbers of X, is a lattice isomorphism between ideals and characteristic sets and its inverse is the map from a characteristic set Σ to the ideal generated by the collection of the diagonal operators {diag ξ | ξ ∈ Σ}. For a sequence 0 ≤ ξ ∈ co , denote by ξ ∗ ∈ c∗o the decreasing rearrangement of ξ, and for each ξ ∈ c∗o denote by (ξ) the principal ideal generated by diag ξ, so that (s(X)) denotes the principal ideal generated by the operator X ∈ K(H) (the ideal of compact operators on the Hilbert space H). Recall that η ∈ Σ((ξ)) precisely when η = O(Dm ξ) for some m. Thus the equivalence between ξ and η (ξ η if ξ = O(η) and η = O(ξ)) is only suﬃcient for (ξ) = (η). It is also necessary if one of the two sequences (and hence both) satisfy the ∆1/2 -condition. Following the notations of [22], we say that ξ satisﬁes n < ∞, i.e., D2 ξ = O(ξ), which holds if and only if the ∆1/2 -condition if sup ξξ2n Dm ξ = O(ξ) for all m ∈ N. Dykema, Figiel, Weiss and Wodzicki [6, 7] showed that the (Cesaro) arithmetic mean plays an essential role in the theory of operator ideals by using it to characterize the normal operators in the commutator space of an ideal. (The commutator space [I, B(H)] of an ideal I, also called the commutator ideal

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of I, is the span of the commutators of elements of I with elements of B(H)). This led them to introduce and study the arithmetic mean and pre-arithmetic mean of an ideal and the consequent notions of am-interior and am-closure of anideal. n The arithmetic mean of any sequence η is the sequence ηa := n1 i=1 ηi . For every ideal I, the pre-arithmetic mean ideal a I and the arithmetic mean ideal Ia are the ideals with characteristic sets Σ(a I) = {ξ ∈ c∗o | ξa ∈ Σ(I)}

Σ(Ia ) = {ξ ∈ c∗o | ξ = O(η) for some η ∈ Σ(I)}.

A consequence of one of the main results in [7, Theorem 5.6] is that the positive part of the commutator space [I, B(H)] coincides with the positive part of the pre-arithmetic mean ideal a I, that is: [I, B(H)]+ = (a I)+ In particular, ideals that fail to support any nonzero trace, i.e., ideals for which I = [I, B(H)], are precisely those for which I = a I (or, equivalently, I = Ia ) and are called arithmetically mean stable (am-stable for short). The smallest nonzero am-stable ideal is the upper stabilizer of the trace-class ideal L1 (in the notations of [7]) ∞ ∞ sta (L1 ) := (ω)am = (ω logm ) m=0

m=0

where ω = 1/n denotes the harmonic sequence (see [10, Proposition 4.18]). There is no largest proper am-stable ideal. Am-stability for many classical ideals was studied in [7, Sections 5.11–5.27]. Arithmetic mean operations on ideals were introduced in [7, Sections 2.8 and 4.3] and employed, in particular, in the study of single commutators [7, Section 7]: the arithmetic mean closure I − and the arithmetic mean interior I o of an ideal I are deﬁned respectively as I − := a (Ia ) and I o := (a I)a . The following 5-chain inclusion holds: o − a I ⊂ I ⊂ I ⊂ I ⊂ Ia Ideals that coincide with their am-closure (resp., am-interior) are called am-closed (resp., am-open), and I − is the smallest am-closed ideal containing I (resp., I o is the largest am-open ideal contained in I). We list here some of the elementary properties of am-closed and am-open ideals, and since there is a certain symmetry between them, we shall consider both in parallel. An ideal I is am-closed (resp., am-open) if and only if I = a J (resp., I = Ja ) for some ideal J. The necessity follows from the deﬁnition of I − (resp., I o ) and the suﬃciency follows from the identities Ia = (a (Ia ))a and a I = a ((a I)a ) that are simple consequences of the 5-chain of inclusions listed above. The characteristic set Σ(L1 ) of the trace-class ideal is ∗1 , the collection of monotone nonincreasing nonnegative summable sequences. It is elementary to show L1 = a (ω), L1 is the smallest nonzero am-closed ideal, (ω) = Fa = (L1 )a , and so (ω) is the smallest nonzero am-open ideal (F denotes the ﬁnite rank ideal.)

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In terms of characteristic sets: Σ(I − ) = {ξ ∈ c∗o | ξa ≤ ηa for some η ∈ Σ(I)} Σ(I o ) = {ξ ∈ c∗o | ξ ≤ ηa ∈ Σ(I) for some η ∈ c∗o } Here and throughout, the relation between sequences “≤” denotes pointwise, i.e., for all n. The relation ξ ≺ η deﬁned by ξa ≤ ηa is called majorization and plays an important role in convexity theory (e.g., see [16, 17]). We will investigate it further in this context in [11] (see also [9]). But for now, notice that I is am-closed if and only if Σ(I) is hereditary (i.e., solid) under majorization. The two main results in this section are that the (ﬁnite) sum of am-closed ideals is am-closed and that intersections of am-open ideals are am-open. These will lead to two additional natural arithmetic mean ideal operations, I− and I oo , see Corollary 2.6 and Deﬁnition 2.18. We start by determining how the arithmetic mean operations distribute with respect to direct unions and intersections of ideals and with respect to ﬁnite sums. Recall that the union of a collection of ideals that is directed by inclusion and the intersection of an arbitrary collection of ideals are ideals. The proofs of the following three lemmas are elementary, with the exception of one of the inclusions in Lemma 2.2(iii) which is a simple consequence of Theorem 2.17 below. Lemma 2.1. For {Iγ , γ ∈ Γ} a collection of ideals directed by inclusion: (i) a ( γ Iγ ) = γ a (Iγ ) (ii) ( γ Iγ )a = γ (Iγ )a (iii) ( γ Iγ )o = γ (Iγ )o (iv) ( γ Iγ )− = γ (Iγ )− (v) If all Iγ are am-stable, (resp., am-open, am-closed) then γ Iγ is am-stable, (resp., am-open, am-closed). Lemma 2.2. For {Iγ , γ ∈ Γ} a collection of ideals: (i) a ( γ Iγ ) = γ a (Iγ ) (ii) ( γ Iγ )a ⊂ γ (Iγ )a (inclusion can be proper by Example 2.4(i)) (iii) ( γ Iγ )o = γ (Iγ )o (equality holds by Theorem 2.17) (iv) ( γ Iγ )− ⊂ γ (Iγ )− (inclusion can be proper by Example 2.4(i)) (v) If all Iγ are am-stable, (resp., am-open, am-closed) then γ Iγ is am-stable, (resp., am-open, am-closed). Lemma 2.3. For all ideals I, J: (i) (ii) (iii) (iv) (v)

Ia + Ja = (I + J)a (the inclusion can be proper by Example 2.4(ii)) a I + a J ⊂ a (I + J) I o + J o ⊂ (I + J)o (the inclusion can be proper by Example 2.4(ii)) I − + J − ⊂ (I + J)− (equality is Theorem 2.5) If I and J are am-open, so is I + J.

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Example 2.4. (i) In general, equality does not hold in Lemma 2.2(ii) or, equivalently, in (iv) even when Γ is ﬁnite. Indeed it is easy to construct two nonsummable sequences ξ and η in c∗o such that min(ξ, η) is summable. But then, as it is elementary to show, (ξ) ∩ (η) = (min(ξ, η)) and hence ((ξ) ∩ (η))a = (ω) while (ξ)a ∩ (η)a = (ξa ) ∩ (ηa ) = (min(ξa , ηa )) (ω), the inclusion since ω = o(ξa ), ω = o(ηa ), hence ω = o(min(ξa , ηa )), and the inequality since ω satisﬁes the ∆1/2 -condition and then equality leads to a contradiction. (ii) In general, equality does not hold in Lemma 2.3(ii) or (iii). Indeed take the principal ideals generated by two sequences ξ and η in c∗o such that ξ + η = ω but ω = O(ξ) and ω = O(η), which implies that a (ξ)

= a (η) = {0} = L1 = a (ω) = a ((ξ) + (η)).

The same example shows that (ξ)o + (η)o = {0} = (ω) = ((ξ) + (η))o . That the sum of ﬁnitely many am-open ideals is am-open (Lemma 2.3(v)), is an immediate consequence of Lemma 2.3(iii). Less trivial is the fact that the sum of ﬁnitely many am-closed ideals is am-closed, or, equivalently, that equality holds in Lemma 2.3(iv). This result was announced in [9]. The proof we present here exploits the role of substochastic matrices in majorization theory ∞([16], see ≥ 0, also [11]). Recall that a matrix P is called substochastic if P ij i=1 Pij ≤ 1 ∞ for all j and j=1 Pij ≤ 1 for all i. By extending the well-known result for ﬁnite sequence majorization (e.g., see [17]), Markus showed in [16, Lemma 3.1] that if η, ξ ∈ c∗o , then ηa ≤ ξa if and only if there is a substochastic matrix P such that η = P ξ. Finally, recall also the Calkin [5] isomorphism between proper two sided ideals of B(H) and ideals of ∞ that associates to an ideal J the symmetric sequence space S(J) deﬁned by S(J) := {η ∈ co | diag η ∈ J} (e.g., see [5] or [7, Introduction]). It is immediate to see that S(J) = {η ∈ co | |η|∗ ∈ Σ(J)} and that for any two ideals, S(I + J) = S(I) + S(J). Theorem 2.5. (I + J)− = I − + J − for all ideals I, J. In particular, the sum of two am-closed ideals is am-closed. Proof. The inclusion I − + J − ⊂ (I + J)− is elementary and was stated in Lemma 2.3(iv). Let ξ ∈ Σ((I + J)− ), then ξa ∈ Σ((I + J)a ) so that ξa ≤ (ρ + η)a for some ρ ∈ Σ(I) and η ∈ Σ(J). Then by Markus’ lemma [16, Lemma 3.1], there is a substochastic matrix P such that ξ = P (ρ+η). Let Π be a permutation matrix monotonizing P ρ, i.e., (P ρ)∗ = ΠP ρ, then ΠP too is substochastic and hence by the same result, ((P ρ)∗ )a ≤ ρa , i.e., (P ρ)∗ ∈ Σ(I − ), or equivalently, P ρ ∈ S(I − ). Likewise, P η ∈ S(J − ), whence ξ ∈ S(I − ) + S(J − ) = S(I − + J − ) and hence ξ ∈ Σ(I − + J − ). Thus (I + J)− ⊂ I − + J − , concluding the proof. As a consequence, the collection of all the am-closed ideals contained in an ideal I is directed and hence its union is an am-closed ideal by Lemma 2.1(v).

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Corollary 2.6. Every ideal I contains a largest am-closed ideal denoted by I− , which is given by I− := {J | J ⊂ I and J is am-closed}. Thus I− ⊂ I ⊂ I − and I is am-closed if and only if one of the inclusions and hence both of them are equalities. Since a I ⊂ I and a I is am closed, a I ⊂ I− . The inclusion can be proper as seen by considering any am-closed but not am-stable ideal I, e.g, I = L1 where a (L1 ) = {0}. If equality holds, we have the following equivalences: Lemma 2.7. For every ideal I, the following conditions are equivalent. (i) I− = a I (ii) If J − ⊂ I for some ideal J, then J − ⊂ a I. (iii) If J − ⊂ I for some ideal J, then Ja ⊂ I. (iv) If a J ⊂ I for some ideal J, then J o ⊂ I. We leave the proof to the reader. Notice that the converses (ii)–(iv) hold trivially for any pair of ideals I and J. Theorem 2.9 below will show that for countably generated ideals the equality a I = I− always holds, i.e., a I is the largest am-closed ideal contained in I. We ﬁrst need the following lemma. Lemma 2.8. If I is a countably generated ideal and L1 ⊂ I, then (ω) ⊂ I. In particular, (ω) is the smallest principal ideal containing L1 . Proof. Let ρ(k) be a sequence of generators for the characteristic set Σ(I), i.e., for every ξ ∈ Σ(I) there are m, k ∈ N for which ξ = O(Dm ρ(k) ). By adding if necessary to this sequence of generators all their ampliations and then by passing to the sequence ρ(1) + ρ(2) + · · · + ρ(k) , we can assume that ρ(k) ≤ ρ(k+1) and that then ξ ∈ Σ(I) if and only if ξ = O(ρ(m) ) for some m ∈ N. Thus if ω ∈ / Σ(I) there ω )nk ≥ k 3 for all k ≥ 1. Set is an increasing sequence of indices nk such that ( ρ(k) no := 0 and deﬁne ξj := k21nk for nk−1 < j ≤ nk and k ≥ 1. Then it is immediate that ξ ∈ ∗1 . On the other hand, ξ = O(ρ(m) ) for any m ∈ N since for every k ≥ m, ξ ξ 1 ≥ = ≥ k. (k) 2 ρ(m) nk ρ(k) nk k nk ρnk and hence ξ ∈ / Σ(I), against the hypothesis L1 ⊂ I. Theorem 2.9. If I is a countably generated ideal, then I− =

a I.

Proof. Let η ∈ Σ(I− ). Then (η)− ⊂ I− ⊂ I. We claim that ηa ∈ Σ(I), i.e., I− ⊂ a I and hence equality holds from the maximality of I− . If 0 = η ∈ ∗1 , then / ∗1 , (η)− = L1 , hence by Lemma 2.8, (ω) ⊂ I and thus ηa ω ∈ Σ(I). If η ∈ assume by contradiction that ηa ∈ / Σ(I). As in the proof of Lemma 2.8, choose a sequence of generators ρ(k) for Σ(I) with ρ(k) ≤ ρ(k+1) and such that for every ξ ∈ Σ(I) there is an m ∈ N for which ξ = O(ρ(m) ). Then there is an increasing

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a sequence of indices nk such that ( ρη(k) )nk ≥ k for every k. Exploiting the nonnk 1 1 summability of η we can further require that nk −n i=nk−1 +1 ηi ≥ 2 (ηa )nk k−1 for every k. Set no := 0 and deﬁne ξj = (ηa )nk for nk−1 < j ≤ nk . We prove by induction that (ξa )j ≤ (2ηa )j . The inequality holds trivially for j ≤ n1 and assume it holds also for all j ≤ nk−1 . If nk−1 < j ≤ nk , it follows that

j

ξi = nk−1 (ξa )nk−1 + (j − nk−1 )(ηa )nk

i=1

≤ 2nk−1 (ηa )nk−1 + (j − nk−1 )(ηa )nk nk−1

≤2

ηi + 2

i=1 nk−1

≤2

i=1

ηi + 2

j − nk−1 nk − nk−1 j

nk

ηi

i=nk−1 +1

ηi = 2j(ηa )j

i=nk−1 +1

1 j where the last inequality follows because j−n i=n+1 ηi is monotone nonincreasing in j for j > n. Thus ξ ∈ Σ((η)− ) ⊂ Σ(I). On the other hand, for every m ∈ N and ξ ξ a k ≥ m, ( ρ(m) )nk ≥ ( ρ(k) )nk = ( ρη(k) )nk ≥ k and thus ξ ∈ / Σ(I), a contradiction.

By Theorem 2.5, I− + J− is am-closed for any pair of ideal I and J and it is contained in I + J. Hence I− + J− ⊂ (I + J)− and this inclusion can be proper by Theorem 2.9 and Example 2.4(ii). Corollary 2.10. If I is a countably generated ideal, then Ia is the smallest countably generated ideal containing I − . Proof. By the ﬁve chain inclusion, I − ⊂ Ia and if I − ⊂ J for some countably generated ideal J, then I − ⊂ J− = a J and hence Ia = (I − )a ⊂ J o ⊂ J. As a consequence of Theorem 2.9 we obtain also an elementary proof of the following, which was obtained for the principal ideal case by [2, Theorem 3.11]. Theorem 2.11. A countably generated ideal is am-closed if and only if it is amstable. Proof. If I is a countably generated am-closed ideal, then I = I− and hence I = a I by Theorem 2.9, i.e., I is am-stable. On the other hand, every am-stable ideal is am-closed by the ﬁve chain inclusion. L1 is an example of a non countably generated ideal which is am-closed (and also am-∞ closed) but is neither am-stable nor am-∞ stable. Now we pass to the second main result of this section, namely that the intersection of am-open ideals is am-open (Theorem 2.17). To prove it and to provide tools for our study in Section 6 of the commutation relations between the se and sc operations and the am-interior operation, we need the characterization

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of the am-interior I o of an ideal I given in Corollary 2.16 below. This in turn will lead naturally to a characterization of the smallest am-open ideal I oo containing I (Deﬁnition 2.18 and Proposition 2.21). Both characterizations depend on the principal ideal case. As recalled earlier, an ideal I is am-open if I = Ja for some ideal J (e.g., J = I − = a (Ia )). In terms of sequences, I is am-open if and only if for every ξ ∈ Σ(I), one has ξ ≤ ηa ∈ Σ(I) for some η ∈ c∗o . Remark 2.15(iii) show that there is a minimal ηa ≥ ξ. First we note when a sequence is equal to the arithmetic mean of a c∗o -sequence. The proof is elementary and is left to the reader. Lemma 2.12. A sequence ξ is the arithmetic mean ηa of some sequence η ∈ c∗o if and only if 0 ≤ ξ → 0 and ωξ is monotone nondecreasing and concave, i.e., ( ωξ )n+1 ≥ 12 (( ωξ )n + ( ωξ )n+2 ) for all n ∈ N and ξ1 = ( ωξ )1 ≥ 12 ( ωξ )2 . It is elementary to see that for every η ∈ c∗o , (η)a = (ηa ) and that ηa satisﬁes the ∆1/2 -condition because ηa ≤ Dm ηa ≤ mηa for every m ∈ N. In particular, all the generators of the principal ideal (ηa ) are equivalent. Lemma 2.13. If I is a principal ideal, then the following are equivalent. (i) I is am-open (ii) I = (ηa ) for some η ∈ c∗o (iii) I = (ξ) for some ξ ∈ c∗o for which ωξ is monotone nondecreasing. Proof. (i) ⇔ (ii). Assume that I = (ξ) for some ξ ∈ c∗o and that I is am-open and hence I = Ja for some ideal J. Then ξ ≤ ηa for some ηa ∈ Σ(I) and hence ηa ≤ M Dm ξ for some M > 0 and m ∈ N. Since ηa Dm ηa , it follows that ξ ηa and hence (ii) holds. The converse holds since (ηa ) = (η)a . (ii) ⇒ (iii) is obvious. (iii) ⇒ (ii). ωξ is quasiconcave, i.e., ωξ is monotone nondecreasing and ω ωξ is monotone nonincreasing. Adapting to sequences the proof of Proposition 5.10 in Chapter 2 of [3] (see also [7, Section 2.18]) shows that if ψ is the smallest concave sequence that majorizes ωξ , then ωξ ≤ ψ ≤ 2 ωξ and hence ψ ωξ . Moreover, ψ1 = ξ1 = ( ωξ )1 since otherwise we could lower ψ1 and still maintain the concavity of ψ. And since the sequence ξω1 is concave and ξω1 ≥ ωξ , it follows by the minimality of ψ that ψ ≤ ξω1 and so, in particular, ψ1 = ξ1 ≥ 12 ψ2 . Since ψ is concave and nonnegative, it follows that it is monotone nondecreasing. But then, by Lemma 2.12 applied to ωψ, one has ωψ = ηa for some sequence η ∈ c∗o and thus (ξ) = (ωψ) = (ηa ). We need now the following notations from [7, Section 2.3]. The upper and lower monotone nondecreasing and monotone nonincreasing envelopes of a realvalued sequence φ are: und φ := max φi , lnd φ := inf φi , uni φ := sup φi , lni φ := min φi . i≤n

i≥n

i≥n

i≤n

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Lemma 2.14. For every ξ ∈ c∗o :

(i) (ξ)o = (ω lnd ωξ ) (ii) (ω und ωξ ) is the smallest am-open ideal containing (ξ).

Proof. (i) We ﬁrst prove that ω lnd ωξ is monotone nonincreasing. Indeed, in the case when (lnd ωξ )n = (lnd ωξ )n+1 , then (ω lnd ωξ )n+1 ≤ (ω lnd ωξ )n , but if on the other hand (lnd ωξ )n = (lnd ωξ )n+1 , then (lnd ωξ )n = ( ωξ )n and hence also (ω lnd ωξ )n+1 ≤ ξn+1 ≤ ξn = (ω lnd ωξ )n . Moreover, ω lnd ωξ → 0 since ω lnd ωξ ≤ ξ. Thus (ω lnd ωξ ) ⊂ (ξ). By Lemma 2.13(i) and (iii), (ω lnd ωξ ) is am-open and hence (ω lnd ωξ ) ⊂ (ξ)o . For the reverse inclusion, if µ ∈ Σ((ξ)o ), then µ ≤ ζa for some ζa ∈ Σ(ξ), i.e., ζa ≤ M Dm ξ for some M > 0 and m ∈ N. Then Dm ζa ≤ mζa ≤ mM Dm ξ, whence ζωa ≤ mM ωξ . As ζωa is monotone nondecreasing, also ζωa ≤ mM lnd ωξ so that µ ≤ mM ω lnd ωξ . Thus (ξ)o ⊂ (ω lnd ωξ ). (ii) A similar proof as in (i) shows that ω und ωξ ∈ c∗o . Since by deﬁnition ξ ≤ ω und ωξ , we have that (ξ) ⊂ (ω und ωξ ), and the latter ideal is am-open by Lemma 2.13. If I is any am-open ideal containing (ξ), then ξ ≤ ζa for some ζa ∈ Σ(I) and again, since ζωa is monotone nondecreasing, ω und ωξ ≤ ζa , hence (ω und ωξ ) ⊂ I. Remark 2.15. (i) Lemma 2.14(i) shows that the am-interior (ξ)o of a principal ideal (ξ) is always principal and its generator ω lnd ωξ is unique up to equivalence by Lemma 2.13 and preceding remarks. Notice that (ω) being the smallest nonzero am-open ideal, (ξ)o = {0} if and only if (ω) ⊂ (ξ). In terms of sequences, this corresponds to the fact that that lnd ωξ = 0 if and only if (ω) ⊂ (ξ). (ii) While (ω lnd ωξ ) is the largest am-open ideal contained in (ξ) by Lemma 2.14(i), it is easy to see that there is no (pointwise) nonzero largest arithmetic mean sequence majorized by ξ unless ξ is itself an arithmetic mean. However, there is an arithmetic mean sequence ηa majorized by ξ which is the largest in the O-sense (actually up to a factor of 2). Indeed, let ψ be the smallest concave sequence that majorizes the quasiconcave sequence 12 lnd ωξ . Then, as in the proof of Lemma 2.13(iii) ⇒ (ii), ψ = ηωa for some η ∈ c∗o and ψ ≤ lnd ωξ and hence ηa ≤ ξ. Moreover, for every ρ ∈ c∗o with ρa ≤ ξ, since ρωa is monotone nondecreasing, it follows that ρωa ≤ lnd ωξ ≤ 2 ηωa and hence ρa ≤ 2ηa . (iii) Lemma 2.14(ii) shows that (ω und ωξ ) is the smallest am-open ideal containing (ξ), and moreover, from the proof of Lemma 2.13(iii) we see that (ω und ωξ ) = (ηa ) where ηωa is the smallest concave sequence that majorizes the quasiconcave sequence und ωξ . In contrast to (ii), ηa is also the (pointwise) smallest arithmetic mean that majorizes ξ. Indeed, if ρa ≥ ξ then ρωa ≥ und ωξ because ρa ρa ηa ρa ω is monotone nondecreasing and moreover ω ≥ ω because ω is concave. ξ (iv) By [7, Section 2.33], ω lnd ω is the reciprocal of the fundamental sequence of the Marcinkiewicz norm for a (ξ).

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Corollary 2.16. For every ideal I: (i) Σ(I o ) = {ξ ∈ c∗o | ω und ωξ ∈ Σ(I)} = {ξ ∈ c∗o | ξ ≤ ω lnd ωη for some η ∈ Σ(I)}. (ii) If I is an am-open ideal, then ξ ∈ Σ(I) if and only if ω und ωξ ∈ Σ(I). Proof. If ξ ∈ Σ(I o ), then (ξ) ⊂ (ω und ωξ ) ⊂ I o by Lemma 2.14(ii), whence ω und

ξ

ω und ωξ ∈ Σ(I). If ω und ωξ ∈ Σ(I), then ξ ≤ ω und ωξ = ω lnd( ω ω ). Finally, if ξ ≤ ω lnd ωη for some η ∈ Σ(I), then ω lnd ωη ∈ Σ((η)o ) ⊂ Σ(I o ) by Lemma 2.14(i) and hence ξ ∈ Σ(I o ). Thus all three sets are equal. This proves (i) and (ii) is a particular case. An immediate consequence of Corollary 2.16(ii) is the following result. Theorem 2.17. Intersections of am-open ideals are am-open. Since I ⊂ Ia , the collection of all am-open ideals containing I is always nonempty. By Theorem 2.17 its intersection is am-open, hence it is the smallest am-open ideal containing I. Definition 2.18. For each ideal I, denote I oo := {J | J ⊃ I and J is am-open}. Remark 2.19. Lemma 2.14 aﬃrms that if I is principal, so are I o and I oo . Notice that I o ⊂ I ⊂ I oo and I is am-open if and only if one of the inclusions and hence both of them are equalities. Since I ⊂ Ia and Ia is am-open, I oo ⊂ Ia . The inclusion can be proper even for principal ideals. Indeed if ξ ∈ c∗o and ξa is irregular, i.e., ξa2 = O(ξa ), then I = (ξa ) is am-open and hence I = I oo , but Ia = (ξa2 ) = (ξa ) = I oo . Of course, if I is am-stable then I = I oo = Ia , and if {0} = I ⊂ L1 then (ω) = I oo = Ia , but as the following example shows, the equality I oo = Ia can hold also in other cases. 1 for ((k − 1)!)2 < j ≤ (k!)2 . Then direct computations Example 2.20. Let ξj = k! show that ξ is irregular, indeed does not even satisfy the ∆1/2 -condition, is not summable, but ξa = O(ω und ωξ ) and hence by Lemma 2.14 (ii), (ξ)oo = (ξ)a .

The characterization of I oo = (ω und ωη ) provided by Lemma 2.14(ii) for principal ideals I = (ξ) extends to general ideals. Proposition 2.21. For every ideal I, the characteristic set of I oo is given by

η Σ(I oo ) = ξ ∈ c∗o | ξ ≤ ω und for some η ∈ Σ(I) . ω Proof. Let Σ = {ξ ∈ c∗o | ξ ≤ ω und ωη for some η ∈ Σ(I)}. First we show that Σ is a characteristic set. Let ξ, ρ ∈ Σ, i.e., ξ ≤ ω und ωη and ρ ≤ ω und ωµ for some η, µ ∈ Σ(I). Since ω und ωη + ω und ωµ ≤ 2ω η+µ ω and η + µ ∈ Σ(I), it follows that ξ + ρ ∈ Σ. Moreover, if ξ ≤ ω und ωη , then for all m, Dm ξ ≤ Dm ωDm und

Dm η η η = Dm ω und Dm ≤ mω und ω ω ω

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and hence Dm ξ ∈ Σ, i.e., Σ is closed under ampliations. Clearly, Σ is also closed under multiplication by positive scalars and it is hereditary. Thus Σ is a characteristic set and hence Σ = Σ(J) for some ideal J. Then J ⊃ I follows from the inequality ξ ≤ ω und ωξ . If η ∈ Σ(J), i.e., η ≤ ω und ωξ for some ξ ∈ Σ(I), then also ω und ωη ≤ ω und ωξ and hence ω und ωη ∈ Σ(J). By Corollary 2.16, this implies that J is am-open and hence J ⊃ I oo . For the reverse inclusion, if η ∈ Σ(J), i.e., η ≤ ω und ωξ for some ξ ∈ Σ(I), then ω und ωξ ∈ Σ((ξ)oo ) ⊂ Σ(I oo ) by Lemma 2.14(ii). Thus η ∈ Σ(I oo ), hence J ⊂ I oo , and we have equality. As a consequence of this proposition and by the subadditivity of “und”, we see that (I + J)oo = I oo + J oo for any two ideals I and J. For completeness’ sake we collect in the following lemma the distributivity properties of the I oo and I− operations. Lemma 2.22. For all ideals I, J: (i) I oo + J oo = (I + J)oo (paragraph after Proposition 2.21) (ii) I− + J− ⊂ (I + J)− and the inclusion can be proper (remarks after Theorem 2.9). Let {Iγ , γ ∈ Γ} be a collection of ideals. Then (iii) ( γ Iγ )oo ⊂ γ (Iγ )oo (the inclusion can be proper by Example 2.23(i)) (iv) ( γ Iγ )− = γ (Iγ )− (by Lemma 2.2(v)) If {Iγ , γ∈ Γ} is directed by inclusion, then (v) ( γ Iγ )oo = γ (Iγ )oo (by Lemma 2.1(v)) (vi) ( γ Iγ )− ⊃ γ (Iγ )− (the inclusion can be proper by Example 2.23(ii)) Example 2.23. (i) The inclusion in (iii) can be proper even if Γ is ﬁnite. Indeed for the same construction as in Example 2.4(i), ((ξ) ∩ (η))oo = (min(ξ, η))oo = (ω) since min(ξ, η) is summable, while ω = o(ω und ωξ ) and ω = o(ω und ωη ) since ξ and η are not summable. Thus ω = o(min(ω und ωξ , ω und ωη )) and hence ξ η (ω) ⊂ ω und ∩ ω und = (ξ)oo ∩ (η)oo ω ω (ii) The inclusion in (vi) can be proper. L1 as every ideal with the exception the smallest amof {0} and F , is the directed union of distinct ideals Iγ . Since L1 is closed ideal, (Iγ )− = {0} for every γ. Thus L1 = ( γ Iγ )− while γ (Iγ )− = {0}.

3. Arithmetic Mean Ideals at Infinity The arithmetic mean is not adequate for distinguishing between nonzero ideals contained in the trace-class since they all have the same arithmetic mean (ω) and the same pre-arithmetic mean {0}. The “right” tool for ideals in the trace-class is

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the arithmetic mean at inﬁnity which was employed for sequences in [1, 7, 14, 22] among others. For every summable sequence η, ∞ 1 ηa∞ := ηj . n n+1 Many of the properties of the arithmetic mean and of the am-ideals have a dual form for the arithmetic mean at inﬁnity but there are also substantial diﬀerences often linked to the fact that contrary to the am-case, the arithmetic mean at inﬁnity ξa∞ of a sequence ξ ∈ ∗1 may fail to satisfy the ∆1/2 condition and also may fail to majorize ξ (in fact, ξa∞ satisﬁes the ∆1/2 condition if and only if ξ = O(ξa∞ ), see [10, Corollary 4.4]). Consequently the results and proofs tend to be harder. In [10] we deﬁned for every ideal I = {0} the am-∞ ideals a∞ I (pre-arithmetic mean at inﬁnity) and Ia∞ (arithmetic mean at inﬁnity) with characteristic sets: Σ(a∞ I) = {ξ ∈ ∗1 | ξa∞ ∈ Σ(I)}

Σ(Ia∞ ) = {ξ ∈ c∗o | ξ = O(ηa∞ ) for some η ∈ Σ(I ∩ L1 )}

Notice that ξa∞ = o(ω) for all ξ ∈ ∗1 . Let se(ω) denote the ideal with characteristic set {ξ ∈ c∗o | ξ = o(ω)} (see Deﬁnition 4.1 for the soft-interior se I of a general ideal I). Thus a∞ I

=

a∞ (I

∩ se(ω)) ⊂ L1

and

Ia∞ = (I ∩ L1 )a∞ ⊂ se(ω).

In [10, Corollary 4.10] we deﬁned an ideal I to be am-∞ stable if I = a∞ I (or, equivalently, if I ⊂ L1 and I = Ia∞ ). There is a largestam-∞ stable ideal, (L1 ), which namely the lower stabilizer at inﬁnity of L1 , sta∞ (L1 ) = ∞ n=0 an ∞ together with the smallest nonzero am-stable ideal sta (L1 ) deﬁned earlier plays an important role in [10]. Natural analogs to the am-interior and am-closure are the am-∞ interior of an ideal I I o∞ := (a∞ I)a∞ = (I ∩ se(ω))o∞ and the am-∞ closure of an ideal I I −∞ :=

a∞ (Ia∞ )

= (I ∩ L1 )−∞ .

We call an ideal I am-∞ open (resp., am-∞ closed) if I = I o∞ (resp., I = I −∞ ). In [10, Proposition 4.8] we proved the analogs of the 5-chain of inclusions for am-ideals (see Section 2 paragraph 5 and [10, Section 2]): a∞ I

⊂ I o∞ ⊂ I ∩ se(ω)

and

I ∩ L1 ⊂ I −∞ ⊂ Ia∞ ∩ L1 and the idempotence of the maps I → I o∞ and I → I −∞ , a consequence of the more general identities a∞ I

=

a∞ ((a∞ I)a∞ )

and

Ia∞ = (a∞ (Ia∞ ))a∞ .

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Thus, like in the am-case, an ideal I is am-open (resp., am-∞ closed) if and only if there is an ideal J such that I = Ja∞ (resp., I = a∞ J). As (L1 )a∞ = se(ω) and a∞ se(ω) = L1 (see [10, Lemma 4.7, Corollary 4.9]), se(ω) and L1 are, respectively, the largest am-∞ open and the largest am-∞ closed ideals. The ﬁnite rank ideal F is am-∞ stable and hence it is the smallest nonzero am-∞ open ideal and the smallest nonzero am-∞ closed ideal. Moreover, every nonzero ideal with the exception of F contains a nonzero principal am-∞ stable ideal (hence both am-∞ open and am-∞ closed) distinct from F [12]. Contrasting these properties for the am-∞ case with the properties for the am case, (ω) is the smallest nonzero am-open ideal, while L1 is the smallest nonzero am-closed ideal, and every principal ideal is contained in an am-stable principal ideal (hence both am-open and am-closed) and so there are no proper largest am-closed or am-open ideals. We leave to the reader to verify that the exact analogs of Lemmas 2.1, 2.2 and 2.3 hold for the am-∞ case. Here Theorem 3.2 plays the role of Theorem 2.5 for the equality in Lemma 2.3(iv) and Theorem 3.11 plays the role of Theorem 2.17 for the equality in Lemma 2.2(iii). The same counterexample to equality in Lemma 2.2. (ii) given in Example 2.4(i) provides a counterexample to the equality in the analog am-∞ case: by [10, Lemma 4.7], ((ξ) ∩ (η))a∞ = (min(ξ, η))a∞ = ((min(ξ, η))a∞ ) while (ξ)a∞ = (η)a∞ = se(ω). The counterexample to the equality in Lemma 2.3(iii) and hence (ii) given in Example 2.4(ii) provides also a counterexample to the same equalities in the am-∞ analogs, but we postpone verifying that until after Lemma 3.9. The distributivity of the am-∞ closure over ﬁnite sums, i.e., the am-∞ analog of Theorem 2.5, also holds, but for its proof we no longer can depend on the theory of substochastic matrices. Instead we will use the following ﬁnite dimensional lemma and then we will extend it to the inﬁnite dimensional case via the w∗ compactness of the unit ball of 1 . Lemma 3.1. Let ξ, η, and µ ∈ [0, ∞)n for some n ∈ N. If for all 1 ≤ k ≤ n, k k k ξj , then there exist η˜ and µ ˜ ∈ [0, ∞)n for which j=1 ηj + j=1 µj ≤ k j=1 k k k ˜j for all 1 ≤ k ≤ n. ξ = η˜ + µ ˜, j=1 ηj ≤ j=1 η˜j , and j=1 µj ≤ j=1 µ Proof. The proof is by induction on n. The case n = 1 is trivial, so assume the property is true for all integers less than equal to n − 1. Assume without loss of k generality that j=1 ξj > 0 for all 1 ≤ k ≤ n and let k γ = max

j=1

1≤k≤n

ηj + k

k

j=1

j=1 ξj

µj

,

which maximum γ ≤ 1 is achieved for some k. Then m j=1

ηj +

m j=1

µj ≤ γ

m j=1

ξj

for all 1 ≤ m ≤ k,

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with equality holding for m = k, so also m

m

ηj +

j=k+1

µj ≤ γ

j=k+1

m

ξj

for all k + 1 ≤ m ≤ n.

j=k+1

Thus if we apply the induction hypothesis separately to the truncated sequences γξχ[1,k] , ηχ[1,k] and µχ[1,k] and to γξχ[k+1,n] , ηχ[k+1,n] , and µχ[k+1,n] we obtain ηj ≤ m that γξχ[1,k] = ρ+σ for two sequences ρ, σ ∈ [0, ∞)k for which m j=1 j=1 ρj m m and j=1 µj ≤ j=1 σj for all 1 ≤ m ≤ k. Similarly γξχ[k+1,n] = ρ + σ for m m m m ρ , σ ∈ [0, ∞)n−k and j=k+1 ηj ≤ j=k+1 ρj , j=k+1 µj ≤ j=k+1 σj for all k + 1 ≤ m ≤ n. But then it is enough to deﬁne η˜ = γ1 ρ, ρ and µ ˜ = γ1 ρ, ρ and verify that it satisﬁes the required condition. Theorem 3.2. (I + J)−∞ = I −∞ + J −∞ for all ideals I, J. In particular, the sum of two am-∞ closed ideals is am-∞ closed. Proof. Let ξ ∈ Σ((I + J)−∞ ), i.e., ξa∞ ≤ (η + µ)a∞ = ηa∞ + µa∞ for some η ∈ Σ(I ∩ L1 ) and µ ∈ Σ(J ∩ L1 ). By if necessary the values of ξ1 or η1 , ∞increasing ∞ ∞ ξ = η + we can assume that j=1 j j=1 j j=1 µj and hence ηa + µa ≤ ξa . By applying Lemma 3.1 to the truncated sequences ξχ[1,n] , ηχ[1,n] , and µχ[1,n] , we obtain two sequences (n)

(n)

η (n) := η1 , η2 , . . . , ηn(n) , 0, 0, . . . (n)

for which ξj = ηj m j=1

(n)

+ µj

ηj ≤

m j=1

(n)

for all 1 ≤ j ≤ n and

(n)

ηj

(n)

and µ(n) := µ1 , µ2 , . . . , µ(n) n , 0, 0, . . .

and

m j=1

µj ≤

m j=1

(n)

µj

for all m ≤ n.

Since 0 ≤ η (n) and µ(n) ≤ ξ, by the sequential compacteness of the unit ball of 1 in the w∗ -topology (as dual of co ), we can ﬁnd converging subsequences η (nk ) →∗ η˜, w

µ(nk ) →∗ µ ˜. It is now easy to verify that ξ = η˜ + µ ˜, that η˜ ≥ 0, µ ˜ ≥ 0, and w n n n n ηj ≤ η˜ and µ ≤ ˜j for all n. It follows from that j=1 j=1 j=1 j=1 µ ∞ ∞j ∞j ∞ ∞ ∞ ∞ ˜j = j=1 ηj and j=1 µ ˜j = j=1 µj , j=1 ξj = j=1 ηj + j=1 µj that j=1 η ∞ and hence ∞ ˜j ≤ ∞ ˜j ≤ ∞ µj for all n. Let η˜∗ , µ ˜∗ be j=n η j=n ηj and j=n µ j=n ∞ ∞ ∗ the decreasing rearrangement of η˜ and µ ˜. Since j=n η˜j ≤ j=n η˜j for every n, ∗ ∗ it follows that (˜ η )a∞ ≤ ηa∞ , i.e., η˜ ∈ Σ(I −∞ ). Thus η˜ ∈ S(I −∞ ). Similarly, −∞ µ ˜ ∈ S(J ). But then ξ ∈ S(I −∞ )+S(J −∞ ) = S(I −∞ +J −∞ ), which proves that −∞ + J −∞ ) and hence (I + J)−∞ ⊂ I −∞ + J −∞ . Since the am-∞ closure ξ ∈ Σ(I operation preserves inclusions, I −∞ +J −∞ ⊂ (I +J)−∞ , concluding the proof. As a consequence, as in the am-case the collection of all the am-∞ closed ideals contained in an ideal I is directed and hence its union is an am-∞ closed ideal by the am-∞ analog of Lemma 2.1(v).

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Corollary 3.3. For every ideal I, I−∞ := {J | J ⊂ I and J is am-∞ closed} is the largest am-closed ideal contained in I. Notice that I−∞ ⊂ I ∩ L1 ⊂ I −∞ and I is am-∞ closed if and only if I−∞ = I if and only if I = I −∞ . Moreover, a∞ I is am-∞ closed, so a∞ I ⊂ I−∞ . The inclusion can be proper: consider any ideal I that is am-∞ closed but not am-∞ stable, e.g., L1 . Analogously to the am-case, we can identify I−∞ for I countably generated. Theorem 3.4. If I is a countably generated ideal, then I−∞ = a∞ I. Proof. Let η ∈ Σ(I−∞ ). Since I−∞ ⊂ L1 , the largest am-∞ closed ideal, η ∈ ∗1 . We claim that ηa∞ ∈ Σ(I), i.e., η ∈ Σ(a∞ I). This will prove that I−∞ ⊂ a∞ I and / Σ(I) and as in the proof hence the equality. Assume by contradiction that ηa∞ ∈ of Lemma 2.8, choose a sequence of generators ρ(k) for Σ(I) with ρ(k) ≤ ρ(k+1) and so that for every ξ ∈ Σ(I), ξ = O(ρ(m) ) for some m ∈ N. Then there is an η ∞ )n ≥ k for every k ∈ N. By the increasing sequence of indices nk such that ( ρa(k) nkk ∞ ηj ≥ 12 j=nk−1 +1 ηj . summability of η, we can further request that j=n k−1 +1 Set no := 0 and deﬁne ξj = (ηa∞ )nk for nk−1 < j ≤ nk . Then ∞

∞

ξi = (nk − j)ξnk +

i=j+1

(ni − ni−1 )ξni

i=k+1

=

nk − j nk ∞

≤

∞

ηi +

i=nk +1

ηi + 2

i=nk +1 ∞

≤3

i=nk +1

∞

∞ ni − ni−1 ni

i=k+1 ni+1

∞

ηm

m=ni +1

ηm

i=k+1 m=ni +1 ∞

ηi ≤ 3

ηi .

i=j+1

Thus ξ ∈ Σ((η)−∞ ) ⊂ Σ(I−∞ ) ⊂ Σ(I). On the other hand, for every m ∈ N η ∞ ξ ξ )nk ≥ ( ρ(k) )nk = ( ρa(k) )nk ≥ k, whence ξ ∈ Σ(I), and for every k ≥ m, ( ρ(m) a contradiction. Precisely as for the am-case we have: Theorem 3.5. A countably generated ideal is am-∞ closed if and only if it is am-∞ stable. Now we investigate the operations I → I o∞ and I → I oo∞ , where I oo∞ is the am-∞ analog of I oo and will be deﬁned in Deﬁnition 3.12. While the statements are analogous to the statements in Section 2, the proofs are sometimes substantially diﬀerent. The analog of Lemma 2.12 is given by: Lemma 3.6. A sequence ξ is the arithmetic mean at infinity ηa∞ of some sequence η ∈ ∗1 if and only if ωξ ∈ c∗o and is convex, i.e., ( ωξ )n+1 ≤ 12 (( ωξ )n + ( ωξ )n+2 ) for all n.

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The analog of Lemma 2.13 is given by: Lemma 3.7. For every principal ideal I, the following are equivalent. (i) I is am-∞ open. (ii) I = (ηa∞ ) for some η ∈ ∗1 . (iii) I = (ξ) for some ξ for which

ξ ω

∈ c∗o .

Proof. (i) ⇔ (ii). Assume I is am-∞ open and that ξ ∈ c∗o is a generator of I. Then I = Ja∞ for some ideal J, i.e., ξ ≤ ηa∞ for some η ∈ ∗1 such that ηa∞ ∈ Σ(I) and thus (ξ) = (ηa∞ ). The other implication is a direct consequence of the equality (ηa∞ ) = (η)a∞ obtained in [10, Lemma 4.7]. η (ii) ⇒ (iii). Obvious as aω∞ ↓ 0. (iii) ⇒ (ii). Since F = (1, 1, 0, 0, . . .)a∞ , we can assume without loss of generality that ξj > 0 for all j. Let ψ be the largest (pointwise) convex sequence majorized by ωξ . It is easy to see that such a sequence ψ exists, that ψ > 0, and that being convex, ψ is decreasing, hence ψ ∈ c∗o and by Lemma 3.6 , ωψ = ηa∞ for some η ∈ ∗1 . By deﬁnition, ξ ≥ ηa∞ and hence (ηa∞ ) ⊂ (ξ) = I. To prove the reverse inclusion, ﬁrst notice that the graph of ψ (viewed as the polygonal curve through the points {(n, ψn ) | n ∈ N}) must have inﬁnitely many corners since ψn > 0 for all n. Let {kp } be the strictly increasing sequence of all the integers where the corners occur, starting with k1 = 1, i.e., for all p > 1, ψkp −1 − ψkp > ψkp − ψkp +1 . By the pointwise maximality of the convex sequence ψ, ψkp = ( ωξ )kp for every p ∈ N (including p = 1) since otherwise we could contradict maximality by increasing ψkp and still maintain convexity and majorization by ωξ . Denote by D 12 the operator (D 12 ζ)j = ζ2j for ζ ∈ c∗o . We claim that for every j, (D 12 ωξ )j < 2ψj . Assume

otherwise that there is a j ≥ 1 such that ( ωξ )2j ≥ 2ψj and let p be the integer for which kp ≤ j < kp+1 . Then kp < 2j and also 2j < kp+1 because otherwise we would have the contradiction 2ψj ≤ ( ωξ )2j ≤ ( ωξ )kp+1 = ψkp+1 ≤ ψj . Moreover, since kp and kp+1 are consecutive corners, between them ψ is linear, i.e., ψj = ψkp +

ψkp+1 − ψkp kp+1 − j j − kp (j − kp ) = ψk + ψk kp+1 − kp kp+1 − kp p kp+1 − kp p+1

and hence kp+1 − j ψj ≥ ψk > kp+1 − kp p ( ωξ )j

1−

1 ξ 1 ξ ≥ ≥ ψj . ψkp > kp+1 2 ω kp 2 ω 2j j

This contradiction proves that D 21 ωξ < 2ψ. It is now easy to verify that

≤ (D3 D 12 ωξ )j < 2(D3 ψ)j for j > 1, and hence I = (ξ) ⊂ (ηa∞ ) because ξj < 2ωj (D3 ψ)j ≤ 2(D3 ω)j (D3 ψ)j = 2(D3 (ωψ))j = 2D3 (ηa∞ )j . Example 3.8. In the proof of the implication (iii) ⇒ (ii), one cannot conclude that 1 for k! ≤ j < (k + 1)! where it is elementary ξ = O(ηa∞ ). Indeed consider ξj = jk! to compute ψj =

j−k! 1 k! (1 − (k+1)! )

for k! ≤ j < (k + 1)!. Also, this example shows that

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while in the am-case the smallest concave sequence ηωa that majorizes ωξ (when ωξ is monotone nondecreasing) provides also the smallest arithmetic mean ηa that majorizes ξ (see Remark 2.15(iii)), this is no longer true for the am-∞ case. We have seen in Lemma 2.14 that the am-interior of a nonzero principal ideal is always principal and it is nonzero if and only if the ideal is large enough (that is, it contains (ω)). Furthermore, there is always a smallest am-open ideal containing it and it too is principal. The next lemma shows that the am-∞ interior of a nonzero principal ideal is principal if only if the ideal is small enough (that is, it does not contain (ω)). Furthermore, if the principal ideal is contained in se(ω), which is the largest am-∞ open ideal, then there is a smallest am-∞ open ideal containing it and it is principal. Lemma 3.9. For every ξ ∈ c∗o : (ω lni ωξ ) if ω ⊂

(ξ) o∞ = (i) (ξ) se(ω) if ω ⊂ (ξ) (ii) If (ξ) ⊂ se(ω), then (ω uni ωξ ) is the smallest am-∞ open ideal containing (ξ). Proof. (i) If (ξ) = F , then also (ω lni ωξ ) = (ω uni ωξ ) = F , so assume that ξ ∈ Σ(F ). If (ω) ⊂ (ξ), then se(ω) = (ξ)o∞ because se(ω) is the largest am-∞ open ideal. If (ω) ⊂ (ξ), in particular ω = O(ξ) and hence lni ωξ ∈ c∗o . But then by Lemma 3.7, (ω lni ωξ ) = (ηa∞ ) for some η ∈ ∗1 , and since (ηa∞ ) = (η)a∞ by [10, Lemma 4.7], it follows that (ω lni ωξ ) is am-∞ open. Since ω lni ωξ ≤ ξ and hence (ω lni ωξ ) ⊂ (ξ), it follows that (ω lni ωξ ) ⊂ (ξ)o∞ . For the reverse inclusion, if ζ ∈ Σ((ξ)o∞ ), then ρ ζ ≤ ρa∞ ≤ M Dm ξ for some ρ ∈ ∗1 , M > 0 and m ∈ N. But then aω∞ ≤ lni M Dωm ξ ρa∞ because ω is monotone nonincreasing, and from this and ω ≤ Dm ω ≤ mω, it follows that ξ Dm ξ ξ ≤ mM ω lni Dm ρa∞ ≤ M ω lni = mM ωDm lni ω ω ω ξ ξ ≤ mM (Dm ω) Dm lni = mM Dm ω lni ω ω where the ﬁrst equality follows by an elementary computation. Thus ζ ∈ Σ(ω lni ωξ ), i.e., (ξ)o∞ ⊂ (ω lni ωξ ) and the equality of these ideals is established. (ii) If (ξ) ⊂ se(ω), then uni ωξ ∈ c∗o , hence (ω uni ωξ ) is am-∞ open by Lemma 3.7. Clearly, (ξ) ⊂ (ω uni ωξ ) and if (ξ) ⊂ I for an am-∞ open ideal I, ρ then ξ ≤ ρa∞ for some ρa∞ ∈ Σ(I). Since aω∞ is monotone nonincreasing, by the minimality of “uni”, ω uni ωξ ≤ ρa∞ and hence (ω uni ωξ ) ⊂ I. As a consequence of this lemma we see that if (ω) = (ξ) + (η) but (ω) ⊂ (ξ) and (ω) ⊂ (η) as in Example 2.4(ii), then (ω)o∞ = se(ω) is not principal but η ξ η ξ o∞ o∞ (ξ) + (η) = ω lni + ω lni = ω lni + ω lni ω ω ω ω

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which is principal. By the same token, a∞ (ξ) + a∞ (η) = a∞ ((ξ) + (η)) and in view of Theorem 3.4, (ξ)−∞ + (η)−∞ = ((ξ) + (η))−∞ . ¿From this lemma we obtain an analog of Corollary 2.16. Corollary 3.10. Let I be an ideal. Then (i) Σ(I o∞ ) = {ξ ∈ Σ(se(ω)) | ω uni

ξ ∈ Σ(I)} ω

η for some η ∈ Σ(I ∩ se(ω))}. ω If I is am-∞ open and ξ ∈ c∗o , then = {ξ ∈ c∗o | ξ ≤ ω lni

(ii) ξ ∈ Σ(I) if and only if ω uni ωξ ∈ Σ(I). Proof. (i) If ξ ∈ Σ(I o∞ ), then ξ ∈ Σ(se(ω)) and hence ω uni ωξ ∈ Σ(I o∞ ) ⊂ Σ(I) by Lemma 3.9(ii). If ξ ∈ Σ(se(ω)) and ω uni ωξ ∈ Σ(I), then ω uni ωξ ∈ Σ(I ∩ se(ω)) and ξ ≤ ω uni ωξ = ω lni

ω uni ω

ξ ω

. Thus

Σ(I o∞ ) ⊂ {ξ ∈ Σ(se(ω)) | ω uni

ξ ∈ Σ(I)} ω

η for some η ∈ Σ(I ∩ se(ω))}. ω Finally, let ξ ∈ c∗o , ξ ≤ ω lni ωη for some η ∈ Σ(I ∩ se(ω)). From the inequality ξ ≤ ω uni ωξ ≤ ω lni ωη , it follows by by Lemma 3.9(i) that ξ ∈ Σ((η)o∞ ) ⊂ Σ(I o∞ ), which concludes the proof. (ii) Just notice that ξ ≤ ω uni ωξ ∈ Σ(I) ⊂ Σ(se(ω)). ⊂ {ξ ∈ c∗o | ξ ≤ ω lni

Now Theorem 2.17, Deﬁnition 2.18 and Proposition 2.21 extend to the am-∞ case with proofs similar to the am-case. Theorem 3.11. The intersection of am-∞ open ideals is am-∞ open. Definition 3.12. For every ideal I, deﬁne

I oo∞ := {J | I ∩ se(ω) ⊂ J and J is am-∞ open}. Remark 3.13. Lemma 3.9 aﬃrms that if I is principal then I o∞ is principal if and only if (ω) ⊂ (ξ) and I oo∞ is principal if and only if (ξ) ⊂ se(ω). The next proposition generalizes to general ideals the characterization of I oo∞ given by Lemma 3.9 in the case of principal ideals. Proposition 3.14. For every ideal I, the characteristic set of I oo∞ is given by:

η oo∞ ∗ ) = ξ ∈ co | η ≤ ω uni for some η ∈ Σ(I ∩ se(ω)) . Σ(I ω

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Notice that I o∞ ⊂ I ∩ se(ω) ⊂ I oo∞ and I is am-∞ open if and only if one of the inclusions and hence both of them are equalities. Also, I ∩ se(ω) ⊂ Ia∞ and Ia∞ is am-∞ open so I oo∞ ⊂ Ia∞ . As for the am-case, we see by considering an am-∞ open principal ideal that is not am-∞ stable that the inclusion may be proper, and by considering am-∞ stable ideals that it may become an equality. Example 3.15. Let ξj = 2k1k! for (k − 1)! < j ≤ k! for k > 1. Then a direct ξ computation shows that (uni ωξ )j = 21k for (k − 1)! < j ≤ k! and that uni ωξ aω∞ . ω uni

ξ

Thus by Lemma 3.9, (ξ)oo∞ = (ξ)a∞ . On the other hand, ( ξ ω )(k−1)! = k and hence ξa∞ = O(ξ). By [10, Theorem 4.12], ξ is ∞-irregular, i.e., (ξ) = (ξ)a∞ . A consequence of Proposition 3.14 and the subadditivity of “uni” is that I oo∞ + J oo∞ = (I + J)oo∞ for any two ideals I and J. Proposition 3.14 also permits us to determine simple suﬃcient conditions on I under which I −∞ (resp., I oo∞ ) is the largest am-∞ closed ideal L1 (resp., the largest am-∞ open ideal se(ω)). Lemma 3.16. Let I be an ideal. (i) If I ⊂ L1 , then I −∞ = L1 . (ii) If I ⊂ se(ω), then I oo∞ = se(ω). Proof. (i) Let ξ ∈ Σ(I) \ ∗1 . Then se(ω) = (ξ)a∞ ⊂ Ia∞ by [10, Lemma 4.7]. Since Ia∞ ⊂ se(ω) holds generally, Ia∞ = se(ω) and thus I −∞ = a∞ se(ω) = L1 . (ii) Let η ∈ Σ(se(ω)), set α := uni ωη , αo = α1 , and choose an arbitrary ξ ∈ Σ(I)\Σ(se(ω)). Then there is an increasing sequence of integers nk with n0 = 0 αn and an ε > 0 such that ξnk ≥ εωnk for all k ≥ 1. Set µj = n1k and ρj = nk−1 for k 1 ∗ nk−1 < j ≤ nk and k ≥ 1. Then µ, ρ ∈ co , µ ≤ ε ξ, hence µ ∈ Σ(I) and ρ = o(ω), ρ ≤ α1 µ, hence ρ ∈ Σ(I ∩ se(ω)). Moreover, max{( ωρ )j | nk−1 < j ≤ nk } = αnk−1 and thus (uni ωρ )j = αnk−1 for nk−1 < j ≤ nk . But then, α ≤ uni ωρ and hence η ≤ αω ≤ ω uni ωρ . By Proposition 3.14, η ∈ Σ(I oo∞ ), which proves the claim. Finally, it is easy to see that the exact analog of Lemma 2.22 holds.

4. Soft Ideals It is well-known that the product IJ = JI of two ideals I and J is the ideal with characteristic set Σ(IJ) = {ξ ∈ c∗o | ξ ≤ ηρ for some η ∈ Σ(I) and ρ ∈ Σ(J)} and that for all p > 0, the ideal I p is the ideal with characteristic set Σ(I p ) = {ξ ∈ c∗o | ξ 1/p ∈ Σ(I)} (see [7, Section 2.8] as but one convenient reference). Recall also from [7, Sections 2.8 and 4.3] that the quotient Σ(I) : X of a characteristic set Σ(I) by a nonempty subset X ⊂ [0, ∞)N is deﬁned to be the characteristic set ∗ ξ ∈ c∗o | (Dm ξ)x ∈ Σ(I) for all x ∈ X and m ∈ N .

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Whenever X = Σ(J), denote the associated ideal by I : J. A special important case is the K¨ othe dual X × of a set X, which is the ideal with characteristic ∗ set 1 : X. In [9] and [10] we introduced the following deﬁnitions of soft ideals. Definition 4.1. The soft interior of an ideal I is the product se I := IK(H), i.e., the ideal with characteristic set Σ(se I) = {ξ ∈ c∗o | ξ ≤ αη for some α ∈ c∗o , η ∈ Σ(I)}. The soft cover of an ideal I is the quotient sc I := I : K(H), i.e., the ideal with characteristic set Σ(sc I) = {ξ ∈ c∗o | αξ ∈ Σ(I) for all α ∈ c∗o }. An ideal is called soft-edged if se I = I and soft-complemented if sc I = I. A pair of ideals I ⊂ J is called a soft pair if se J = I and sc I = J. This terminology is motivated by the fact that I is soft-edged if and only if, for every ξ ∈ Σ(I), one has ξ = o(η) for some η ∈ Σ(I). Analogously, an ideal I is soft-complemented if and only if, for every ξ ∈ c∗o \ Σ(I), one has η = o(ξ) for some η ∈ c∗o \ Σ(I). Below are some simple properties of the soft interior and soft cover operations that we shall use frequently throughout this paper. Lemma 4.2. For all ideals I, J: (i) se and sc are inclusion preserving, i.e., se I ⊂ se J and sc I ⊂ sc J whenever I ⊂ J. (ii) se and sc are idempotent, i.e., se(se I) = se I and sc(sc I) = sc I and so se I and sc I are, respectively, soft-edged and soft-complemented. (iii) se I ⊂ I ⊂ sc I (iv) se(sc I) = se I and sc(se I) = sc I (v) se I and sc I form a soft pair. (vi) If I ⊂ J form a soft pair and L is an intermediate ideal, I ⊂ L ⊂ J, then I = se L and J = sc L. (vii) If I ⊂ J, I = se I, and J = sc J, then I and J form a soft pair if and only if sc I = J if and only if se J = I. Proof. (i) and (iii) follow easily from the deﬁnitions. From K(H) = K(H)2 follows the idempotence of se in the ﬁrst part of (ii) and the inclusion sc(sc I) ⊂ sc I, while the equality here follows from (iii) and (i). That se(sc I) ⊂ I ⊂ sc(se I) is immediate by Deﬁnition 4.1. Applying se to the ﬁrst inclusion, by (i)–(iii) follows the ﬁrst equality in (iv) and the second equality follows similarly. (v), (vi) and (vii) are now immediate. An easy consequence of this proposition and of Deﬁnition 4.1 is:

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Corollary 4.3. For every ideal I, (i) se I is the largest soft-edged ideal contained in I and it is the smallest ideal whose soft cover contains I (ii) sc I is the smallest soft-complemented ideal containing I and it is the largest ideal whose soft interior is contained in I. The rest of this section is devoted to showing that many ideals in the literature are soft-edged or soft-complemented (or both) and that soft pairs occur naturally. Rather than proving directly soft-complementedness, it is sometimes easier to prove a stronger property: Definition 4.4. An ideal I is said to be strongly soft-complemented (ssc for short) if for every countable collection of sequences {η (k) } ⊂ c∗o \ Σ(I) there is a sequence (k) of indices nk ∈ N such that ξ ∈ Σ(I) whenever ξ ∈ c∗o and ξi ≥ ηi for all k and for all 1 ≤ i ≤ nk . Proposition 4.5. Strongly soft-complemented ideals are soft-complemented. Proof. Let I be an ssc ideal, let η ∈ Σ(I), and for each k ∈ N, set η (k) := k1 η. Since {η (k) } ⊂ c∗o \ Σ(I), there is an associated sequence of indices nk which, without loss of generality, can be taken to be strictly increasing. Set no = 0 and (k) deﬁne αi := k1 for nk−1 < i ≤ nk . Then α ∈ c∗o and (αη)i ≥ ηi for all 1 ≤ i ≤ nk and all k. Therefore αη ∈ Σ(I) and hence, by the remark following Deﬁnition 4.1, I is soft-complemented. Example 4.15 and Proposition 5.3 provide soft-complemented ideals that are not strongly soft-complemented. Proposition 4.6. (i) Countably generated ideals are strongly soft-complemented and hence soft-complemented. (ii) If I is a countably generated ideal and if {ρ(k) } is a sequence of generators for its characteristic set Σ(I), then I is soft-edged if and only if for every k ∈ N there are m, k ∈ N for which ρ(k) = o(Dm ρ(k ) ). In particular, a principal ideal (ρ) is soft-edged if and only if ρ = o(Dm ρ) for some m ∈ N. If a principal ideal (ρ) is soft-edged, then (ρ) ⊂ L1 . Proof. (i) As in the proof of Lemma 2.8, choose a sequence of generators ρ(k) for Σ(I) with ρ(k) ≤ ρ(k+1) and such that ξ ∈ Σ(I) if and only if ξ = O(ρ(m) ) for some m ∈ N. Let {η (k) } ⊂ c∗o \Σ(I). Then, in particular, η (k) = O(ρ(k) ) for every k. Thus (k) (k) there is a strictly increasing sequence of indices nk ∈ N such that ηnk ≥ kρnk for (k) all k. If ξ ∈ c∗o and for each k, ξi ≥ ηi for all 1 ≤ i ≤ nk , then for all k ≥ m, (k) (k) (m) ξnk ≥ ηnk ≥ kρnk ≥ kρnk . Hence ξ = O(ρ(m) ) for each m and thus ξ ∈ Σ(I), establishing that I is ssc. (ii) Assume that I is soft-edged and let k ∈ N. By the remarks following Deﬁnition 4.1, ρ(k) = o(ξ) for some ξ ∈ Σ(I). But also ξ = O(Dm ρ(k ) ) for some m and k and hence ρ(k) = o(Dm ρ(k ) ). Conversely, assume that the condition holds

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and let ξ ∈ Σ(I). Then ξ = O(Dm ρ(k) ) for some m and k and ρ(k) = o(Dp ρ(k ) ) for some p and k . Since Dm Dp = Dmp , one has (k) (k) Dm ρ(k) n ρ ρ = lim Dm lim = lim = 0, ) ) (k (k ) (k n n j D ρ D Dmp ρ p pρ n j n

i.e., Dm ρ(k) = o(Dmp ρ(k ) ), whence ξ ∈ Σ(se I) and I is soft-edged. Thus, if I is a soft-edged principal ideal with a generator ρ, then ρ = o(Dm ρ) for some m ∈ N. As a consequence, ρmk ≤ m12 ρmk−1 for k large enough, from which it follows that ρ is summable. Next we consider Banach ideals. These are ideals that are complete with respect to a symmetric norm (see for instance [7, Section 4.5]) and were called uniform-cross-norm ideals by Schatten [19], symmetrically normed ideals by Gohberg and Krein [8], and symmetric norm ideals by other authors. Recall that the norm of I induces on the ﬁnite rank ideal F (or, more precisely, on S(F ), the associated space of sequences of co with ﬁnite support) a symmetric norming function φ, and the latter permits one to construct the so-called minimal and maximal (o) Banach ideals Sφ = cl(F ) contained in I (the closure taken in the norm of I) and Sφ containing I where Σ Sφ = ξ ∈ c∗o | φ(ξ) := sup φ ξ1 , ξ2 , . . . , ξn , 0, 0, . . . < ∞ (o) Σ Sφ = ξ ∈ Σ Sφ | φ ξn , ξn+1 , . . . −→ 0 . (o)

As the following proposition implies, the ideals Sφ and Sφ can be obtained (o)

from I through a “soft” operation, i.e., Sφ = se I and Sφ = sc I, and the embed(o)

ding Sφ ⊂ Sφ is a natural example of a soft pair. In particular, if I is a Banach ideal, then so also are se I and sc I. (o)

Proposition 4.7. For every symmetric norming function φ, Sφ Sφ is ssc, and

(o) Sφ

is soft-edged,

⊂ Sφ is a soft pair. (o)

(o)

Proof. We ﬁrst prove that Sφ is soft-edged. For every ξ ∈ Σ(Sφ ), that is, φ(ξn , ξn+1 , . . .) → 0, choose a strictly increasing sequence of indices nk with no = 0 for which φ(ξnk +1 , ξnk +2 , . . .) ≤ 2−k and kξnk ↓ 0. Set βi := k for all nk−1 < i ≤ nk and η := lni βξ. Then η ∈ c∗o since ηnk ≤ βnk ξnk = kξnk → 0 and ξ = o(η) because for every k and nk−1 < n ≤ nk , ηn = min{βi ξi | i ≤ n} = min min jξi | nj−1 < i ≤ nj | 1 ≤ j ≤ k − 1 , min kξi | nk−1 < i ≤ n = min jξnj | 1 ≤ j ≤ k − 1 , kξn = min (k − 1)ξnk−1 , kξn ≥ (k − 1)ξn .

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(o)

Furthermore, η ∈ Σ(Sφ ) which establishes that Sφ is soft-edged. Indeed, for all h > k > 1, h−1 φ ηnj +1 , . . . , ηnj+1 , 0, 0, . . . φ ηnk +1 , . . . , ηnh , 0, 0, . . . ≤ j=k

≤

h−1

(j + 1)φ ξnj +1 , . . . , ξnj+1 , 0, 0, . . .

j=k

≤

h−1

(j + 1)φ ξnj +1 , ξnj +2 , . . .

j=k

≤

h−1 j=k

Thus

j+1 . 2j

φ ηnk +1 , ηnk +2 , . . . = sup φ ηnk +1 , . . . , ηn , 0, 0, . . . n

≤

∞ j+1 j=k

2j

−→ 0 as k −→ ∞.

from which it follows that φ(ηn , ηn+1 , . . .) → 0 as n → ∞. Next we prove that Sφ is ssc. For every {η (k) } ⊂ c∗o \ Σ(Sφ ), that is, (k) (k) (k) supn φ(η1 , η2 , . . . , ηn , 0, 0, . . .) = ∞ for each k, choose a strictly increasing (k) (k) sequence of indices nk ∈ N for which φ(η1 , . . . , ηnk , 0, 0, . . .) ≥ k. Thus, if ξ ∈ c∗o (k) and for each k, ξi ≥ ηi for all 1 ≤ i ≤ nk , then φ(ξ1 , ξ2 , . . . , ξnk , 0, 0, . . .) ≥ k and hence ξ ∈ Σ(Sφ ), which shows that Sφ is ssc. (o)

Finally, to prove that Sφ ⊂ Sφ form a soft pair, in view of Lemma 4.2(vii), Corollary 4.3(i) and the ﬁrst two results in this proposition, it suﬃces to show that (o) se(Sφ ) ⊂ Sφ . Let ξ ∈ Σ(se(Sφ )), i.e., ξ ≤ αη for some α ∈ c∗o and η ∈ Σ(Sφ ). (o)

Then φ(ξn , ξn+1 , . . .) ≤ αn φ(ηn , ηn+1 , . . .) ≤ αn φ(η) → 0, i.e., ξ ∈ Σ(Sφ ). Remark 4.8. (i) In the notations of [7] and of this paper, Gohberg and Krein [8] showed that the symmetric norming function φ(η) := sup ηξaa induces a complete norm on the am-closure (ξ)− of the principal ideal (ξ) and for this norm cl(F ) = Sφ ⊂ cl(ξ) ⊂ Sφ = (ξ)− . (o)

(ii) The fact that Sφ is soft-complemented was obtained in [18, Theorem (o) 3.8], but Salinas proved only that (in our notations) se Sφ ⊂ Sφ [18, Remark 3.9]. Varga reached the same conclusion in the case of the am-closure of a principal ideal with a non-trace class generator [20, Remark 3].

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(iii) By Lemma 4.2(vi), if I is a Banach ideal such that Sψ ⊂ I ⊂ Sψ for some symmetric norming function Ψ and if φ is the symmetric norming function (o) (o) induced by the norm of I on Σ(F ), then Sφ = Sψ and Sφ = Sψ and hence φ and ψ are equivalent (cf. [8, Chapter 3, Theorem 2.1]). (o) (iv) The fact that Sφ ⊂ Sφ is always a soft pair yields immediately the equivalence of parts (a)–(c) in [18, Theorem 2.3] without the need to consider norms and hence establish (d) and (e). (o)

That Sφ ⊂ Sφ is a soft pair can help simplify the classical analysis of principal ideals. In [2, Theorem 3.23] Allen and Shen used Salinas’ results [18] on (second) K¨ othe duals to prove that (ξ) = cl(ξ) if and only if ξ is regular (i.e., ξ ξa , or in terms of ideals, if and only if (ξ) is am-stable). In [20, Theorem 3] Varga gave an independent proof of the same result. This result is also a special case of [7, Theorem 2.36], obtained for countably generated ideals by yet independent methods. A still diﬀerent and perhaps simpler proof of the same result follows (o) immediately from Theorem 2.11 and the fact that Sφ ⊂ Sφ form a soft pair. Proposition 4.9. (ξ) = cl(ξ) if and only if ξ is regular. Proof. The inclusion Sφ ⊂ (ξ) = cl(ξ) ⊂ Sφ = (ξ)− and the fact that (ξ) is soft complemented by Proposition 4.6(i), Sφ is soft complemented by Proposition 4.7, (o) and Sφ ⊂ Sφ is a soft pair (ibid), proves by applying the sc operation to the above inclusion that (ξ) = (ξ)− . The conclusion now follows from Theorem 2.11. (o)

Remark 4.10. If (ξ)− is countably generated, so in particular if it is principal, by Theorem 2.11 it is am-stable and hence (ξ)− = ((ξ)− )a = (ξ)a = (ξa ), so that ξa is regular. This implies that ξ itself is regular, as was proven in [7, Theorem 3.10] and as is implicit in [20, Theorem IRR]. This conclusion fails for general ideals: we construct in [12] a non am-stable ideal with an am-closure that is countably generated and hence am-stable by Theorem 2.11. Next we consider Orlicz ideals which provide another natural example of soft pairs. Recall from [7, Sections 2.37 and 4.7] that if M is a monotone nondecreasing (o) function on [0, ∞) with M (0) = 0, then the small Orlicz ideal LM is the ideal with ∗ characteristic set {ξ ∈ co | n M (tξn ) < ∞ for all t > 0} and the Orlicz ideal LM is the ideal with characteristic set {ξ ∈ c∗o | n M (tξn ) < ∞ for some t > 0}. (o) (o) If the function M is convex, then LM and LM are respectively the ideals Sφ and Sφ for the symmetric norming function deﬁned by

n 1 φ ξ1 , ξ2 , . . . , ξn , 0, 0, . . . := inf | M (tξi ) ≤ 1 . t>0 t i=1 (o)

Thus, when M is convex, LM ⊂ LM form a soft pair by Proposition 4.7. In fact, the same can be proven directly without assuming convexity for M .

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Proposition 4.11. Let M be a monotone nondecreasing function on [0, ∞) with (o) (o) M (0) = 0. Then LM is soft-edged, LM is ssc, and LM ⊂ LM is a soft pair. (o)

Proof. Take ξ∞∈ Σ(LM ) and choose a strictly increasing sequence of indices nk ∈ N such that i=nk−1 +1 M (k 2 ξi ) ≤ 2−k and kξnk ↓ 0. As in the proof of Proposition 4.7, set n0 = 0 and βi := k for all nk−1 0 be arbitrary and ﬁx an integer k ≥ t. Then since η ≤ βξ and M is monotone nondecreasing, it follows that ∞

M (tηi ) ≤

i=nk +1

∞

M (kβi ξi ) =

i=nk +1

≤

M (j 2 ξi ) ≤

j=k+1 i=nj−1 +1 (o) Σ(LM ),

nj

M (kjξi )

j=k+1 i=nj−1 +1 ∞

∞

∞

∞

2−j < ∞.

j=k+1

(o) LM

Therefore η ∈ which proves that is soft-edged. Next we prove that LM is ssc. For every countable collection of sequences (k) η (k) ∈ c∗o \ Σ(LM ), since i M ( k1 ηi ) = ∞ for all k, we can choose a strictly nk (k) M ( k1 ηi ) ≥ k. If ξ ∈ c∗o and increasing sequence of indices nk ∈ N such that i=1 (k) ξi ≥ ηi for all 1 ≤ i ≤ nk , then for all m and all k ≥ m it follows that nk

n

n

k k 1 1 1 (k) M ( ξi ) ≥ M ( ξi ) ≥ M ( ηi ) ≥ k m k k i=1 i=1 i=1

and hence i M (tξi ) = ∞ for all t > 0. Thus ξ ∈ Σ(LM ), which proves that LM is ssc. (o) (o) To prove that LM ⊂ LM is a soft pair, since LM is soft-edged and LM (o) is soft-complemented, by Lemma 4.2(vii) it suﬃces to prove that se LM ⊂ LM . Let ξ ∈ Σ(LM ), let to > 0 be such that n M (to ξn ) < ∞, and let α ∈ c∗o . For each t > 0 choose N so that tαn ≤ to for n ≥ N . By the monotonicity of M , ∞ (o) n=N M (tαn ξn ) < ∞ and hence αξ ∈ Σ(LM ). (o)

The fact that LM ⊂ LM forms a soft pair can simplify proofs of some properties of Orlicz ideals. Indeed, together with [10, Proposition 3.4] that states that for an ideal I, se I is am-stable if and only if sc I is am-stable if and only if Ia ⊂ sc I, and combined with Lemma 4.16 below it yields an immediate proof of the following results in [7]: the equivalence of (a), (b), (c) in Theorem 4.21 and hence the equivalence of (a), (b), (c) in Theorem 6.25, the equivalence of (b), (c), and (d) in Corollary 2.39, the equivalence of (b) and (c) in Corollary 2.40, and the equivalence of (a), (b), and (c) in Theorem 3.21.

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Next we consider Lorentz ideals. If φ is a monotone nondecreasing nonnega< ∞, then in the notations tive sequence satisfying the ∆2 -condition, i.e., sup φφ2n n of [7, Sections 2.25 and 4.7] the Lorentz ideal L(φ) corresponding to the sequence space (φ) is the ideal with characteristic set

Σ(L(φ)) := ξ ∈ c∗o | ξ(φ) := ξn (φn+1 − φn ) < ∞ . n

A special case of Lorentz ideal is the trace class L1 which corresponds to the sequence φ = n and the sequence space (φ) = 1 . Notice that L(φ) is also the K¨othe dual {φn+1 − φn }× = ∗1 : {φn+1 − φn } of the singleton set consisting of the sequence φn+1 − φn (cf. [7, Section 2.8(iv)]). L(φ) is a Banach ideal with norm induced by the cone norm · (φ) on (φ)∗ if and only if the sequence φ is concave (cf. [7, Lemma 2.29 and Section 4.7]), and it (o) is easy to verify that in this case (φ)∗ = Sψ = Sψ where ψ is the restriction of ·(φ) to Σ(F ). Thus by Proposition 4.7, L(φ) is both strongly soft-complemented and soft-edged. In fact, the same holds without the concavity assumption for φ as we see in the next proposition. Proposition 4.12. If φ be a monotone nondecreasing nonnegative sequence satisfying the ∆2 -condition, then L(φ) is both soft-edged and strongly soft-complemented. Proof. For ξ ∈ Σ(L(φ)), ∞ choose a strictly increasing sequence of indices nk ∈ N with kξnk ↓ 0 and i=nk ξi (φi+1 − φi ) ≤ 2−k . As in Proposition 4.7(proof), set no = 0, βi := k for all nk−1 < i ≤ nk , hence η = lni βξ ∈ c∗o and ξ = o(η). Then ∞ i

ηi (φi+1 −φi ) ≤

∞ i

βi ξi (φi+1 −φi ) =

∞

nk

kξi (φi+1 −φi ) ≤

k=1 i=nk−1 +1

∞

k2−k+1 < ∞,

k=1

whence η ∈ Σ(L(φ)). Thus ξ ∈ Σ(se L(φ)) and hence L(φ) is soft-edged. Finally, for every sequence of sequences {η (k) } ⊂ c∗o \ Σ(L(φ)), choose a nk (k) strictly increasing sequence nk ∈ N such that for all k, i=1 ηi (φi+1 − φi ) ≥ k. (k) nk ∗ Thus if ξ ∈ co and ξi ≥ ηi for all 1 ≤ i ≤ nk , then i=1 ξi (φi+1 − φi ) ≥ k and hence ξ ∈ Σ(L(φ)), thus proving that L(φ) is ssc. In particular, we use frequently that L1 is both soft-edged and soft-complemented. As the next proposition shows, any quotient with a soft-complemented ideal as numerator is always soft-complemented (cf. ﬁrst paragraph of this section for the deﬁnition of quotient), but as Example 4.15 shows, even a K¨ othe dual of a singleton can fail to be strongly soft-complemented. Proposition 4.13. Let I be a soft-complemented ideal and let X be a nonempty subset of [0,∞)N . Then the ideal with characteristic set Σ(I) : X is soft-complemented. Proof. Let ξ ∈ c∗o \ (Σ(I) : X), i.e., ((Dm ξ)x)∗ ∈

Σ(I) for some m ∈ N and x ∈ X. As I is soft-complemented, there exists α ∈ c∗o such that α((Dm ξ)x)∗ ∈ Σ(I).

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Let π be an injection that monotonizes (Dm ξ)x, i.e., (((Dm ξ)x)∗ )i = ((Dm ξ)x)π(i) for all i. Deﬁne απ−1 (j) if j ∈ π(N ) γj := . 0 if j ∈ π(N ) Then γ → 0 and hence uni γ ∈ c∗o . Thus for all i, (α((Dm ξ)x)∗ )i = γπ(i) (Dm ξ)π(i) xπ(i) ≤ (uni γ)π(i) (Dm ξ)π(i) xπ(i) ≤ (Dm ((uni γ)ξ))π(i) xπ(i) . From this inequality, and from the elementary fact that for two sequence ρ and µ, 0 ≤ ρ ≤ µ implies ρ∗ ≤ µ∗ , it follows that α((Dm ξ)x)∗ ≤ ((Dm ((uni γ)ξ))x)∗ . Thus ((Dm ((uni γ)ξ))x)∗ ∈ Σ(I), i.e., (uni γ)ξ ∈ Σ(I) : X, proving the claim. Remark 4.14. If X is itself a characteristic set, the above result follows by the simple identities for ideals I, J, L analogous to the numerical quotient operation “÷”: (I : J) : L = I : (JL) = (I : L) : J Indeed if in these identities we set L = K(H) (the ideal of compact operators), we obtain sc(I : J) = I : se J = sc I : J. Thus if I is soft-complemented or J is soft-edged it follows that I : J is soft-complemented. As an aside: (I : J)J ⊂ I ⊂ (IJ : J) ⊂ I : J and each of the embeddings can be proper (see also [18]). Example 4.15. The K¨ othe dual I := {en }× of the singleton {en } is softcomplemented by Proposition 4.13 but it is not strongly soft-complemented. ∗ Indeed, by deﬁnition, ξ ∈ Σ(I) if and only if ((Dm ξ)en )∗ ∈ 1 (or, equivalently, n mn to < ∞ for (Dm ξ)e ∈ 1 ) for every m, whichin turns is equivalent n ξn e ∗ n 2n every m. Choose η ∈ co such that n ηn e < ∞ but n ηn e = ∞ and hence (k) η ∈ Σ(I), and set η (k) := D1/k η, i.e., ηi = ηki for all i. As (D2k η (k) )i ≥ ηi for i ≥ k, it follows that for every k, D2k η (k) and hence η (k) are not in Σ(I). Let nk ∈ N be an arbitrary strictly increasing sequence of indices, set no = 0 and (k) deﬁne ξi := ηi for nk−1 < i ≤ nk . As η (k+1) ≤ η (k) , it follows that ξ is monotone (k) nonincreasing and for all k, ξi ≥ ηi for 1 ≤ i ≤ nk . On the other hand, for all m and for all k ≥ m, nk

ξi emi ≤

i=nk−1 +1

and thus

nk

ηki eki ≤

i=nk−1 +1 ∞

i=nm−1 +1

ξi emi ≤

knk i=knk−1 +1

∞

ηi ei < ∞,

i=nm−1 +1

which proves that ξ ∈ Σ(I) and hence that I is not ssc.

ηi ei

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Next we consider idempotent ideals, i.e., ideals for which I = I 2 . Notice that an ideal is idempotent if and only if I = I p for some p = 0, 1, if and only if I = I p for all p = 0. The following lemma is an immediate consequence of Deﬁnition 4.1, the remarks following it, and of Deﬁnition 4.4. Lemma 4.16. For every ideal I and p > 0: (i) se(I p ) = (se I)p and sc(I p ) = (sc I)p In particular, if I is soft-edged or soft-complemented, then so respectively is I p. (ii) If I ⊂ J is a soft pair, then so is I p ⊂ J p . (iii) If I is ssc, then so is I p . Proposition 4.17. Idempotent ideals are both soft-edged and soft-complemented. Proof. Let I be an idempotent ideal. That I is soft-edged follows from the inclusions I = I 2 ⊂ K(H)I = se I ⊂ I. That I is soft-complemented follows from the inclusions sc I = sc(I 2 ) = (sc I)2 ⊂ K(H) sc I = se(sc I) = se I ⊂ I ⊂ sc I which follows from Lemmas 4.16 and 4.2(iii),(iv).

The remarks following Proposition 5.3 show that idempotent ideals may fail to be strongly soft-complemented. Finally, we consider the Marcinkiewicz ideals namely, the pre-arithmetic means of principal ideals, and we consider also their am-∞ analogs. That these ideals are strongly soft-complemented follows from the following proposition combined with Proposition 4.6(i). Proposition 4.18. The pre-arithmetic mean and the pre-arithmetic mean at infinity of a strongly soft-complemented ideal is strongly soft-complemented. In particular, Marcinkiewicz ideals are strongly soft-complemented. Proof. Let I be an ssc ideal. We ﬁrst prove that a I is ssc. Let {η (k) } ⊂ c∗o \ Σ(a I), (k) i.e., {ηa } ⊂ c∗o \ Σ(I), and let nk ∈ N be a strictly increasing sequence of indices (k) for which if ζ ∈ c∗o and ζi ≥ (ηa )i for all 1 ≤ i ≤ nk and all k, then ζ ∈ Σ(I). (k) Let ξ ∈ c∗o and ξi ≥ (η (k) )i for all 1 ≤ i ≤ nk and all k. But then (ξa )i ≥ (ηa )i for all 1 ≤ i ≤ nk and all k and hence ξa ∈ Σ(I), i.e., ξ ∈ Σ(a I). We now prove that a∞ I is ssc. Let {η (k) } ⊂ c∗o \ Σ(a∞ I). Assume ﬁrst that inﬁnitely many of the sequences η (k) are not summable. Since the trace class L1 is ssc by Proposition 4.12, there is an associated increasing sequence of indices (k) nk ∈ N so that if ξ ∈ c∗o and ξi ≥ ηi for all 1 ≤ i ≤ nk , then ξ ∈ Σ(L1 ) and hence ξ ∈ Σ(a∞ I) since a∞ I ⊂ L1 . Thus assume without loss of generality that (k) all η (k) are summable and hence ηa∞ ∈ Σ(I). Let nk ∈ N be a strictly increasing (k) sequence of indices for which ζ ∈ Σ(I) whenever ζ ∈ c∗o and ζi ≥ (ηa∞ )i for all 1 ≤ i ≤ nk and all k. For every k and n choose an integer p(k, n) ≥ n for which

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p(k,n)

∞ (k) (k) ηi ≥ 12 i=n ηi . Set Nk := max{p(k, n) | 1 ≤ n ≤ nk + 1}. For any (k) ξ ∈ c∗o such that ξi ≥ ηi for all 1 ≤ i ≤ Nk consider two cases. If ξ is not summable then ξ ∈ Σ(a∞ I) trivially. If ξ is summable, then for all 1 ≤ n ≤ nk and for all k Nk Nk ∞ 1 1 1 (k) ξi ≥ ξi ≥ η ξa∞ n = n i=n+1 n i=n+1 n i=n+1 i i=n

≥

1 n

p(k,n+1)

i=n+1

(k)

ηi

≥

∞ 1 (k) 1 ηi = ηa(k) ∞ n 2n i=n+1 2

and hence ξa∞ ∈ Σ(I), i.e., ξ ∈ Σ(a∞ I).

That Marcinkiewicz ideals are ssc can be seen also by the following consequence of Proposition 4.7. If I is a Marcinkiewicz ideal, then I = a (ξ) = a ((ξ)o ) for some ξ ∈ c∗o . By Lemma 2.13, (ξ)o = (ηa ) = (η)a for some η ∈ c∗o . Thus I = a ((η)a ) = (η)− and (η)− is ssc by Proposition 4.7 and Remark 4.8(i). Corollary 6.7 and Proposition 6.11 below show that the pre-arithmetic mean (resp., the pre-arithmetic mean at inﬁnity) also preserve soft-complementedness. They also show that the am-interior and the am-closure of a soft-edged ideal are soft-edged, that the am-interior of a soft-complemented ideal is soft-complemented by Proposition 6.11, and that the same holds for the corresponding am-∞ operations. However, as mentioned prior to Proposition 6.8, (resp., Proposition 6.11) we do not know whether the am-closure (resp., the am-∞ closure) of a softcomplemented ideal is soft-complemented. Likewise, we do not know whether the am-closure (resp., am-∞ closure) of an ssc ideal is ssc. One non-trivial case in which we can prove directly that the am-closure of an ssc ideal is scc is the following. If I is countably generated, then Ia too is countably generated and hence, by Propositions 4.6(i) and 4.18(i), its am-closure p I − is also ssc, and then by Lemma 4.16 so is (I − ) for any p > 0. The latter ideal is in general not countably generated (e.g., if 0 = ξ ∈ ∗1 , then (ξ)− = L1 is not countably generated) but Lemma 4.19 below shows that nevertheless its am-closure is ssc. Lemma 4.19. For every ideal I, − p −

((I ) ) =

p

(I − ) − (I p )

for 0 < p ≤ 1 for p ≥ 1.

Proof. Let ξ ∈ Σ(((I − )p )− ). By deﬁnition, ξa ≤ ηa for some η ∈ Σ((I − )p ), i.e., η 1/p ∈ Σ(I − ), which in turns holds if and only if (η 1/p )a ≤ ρa for some ρ ∈ Σ(I). Recall from [17, 3.C.1.b] that if µ and ν are monotone sequences and µa ≤ νa , then (µq )a ≤ (ν q )a for q ≥ 1. Thus, if p ≤ 1, (ξ 1/p )a ≤ (η 1/p )a ≤ ρa and consequently ξ 1/p ∈ Σ(I − ), i.e., ξ ∈ Σ((I − )p ). Thus ((I − )p )− ⊂ (I − )p , which then implies equality since the reverse inclusion is automatic. If p > 1, the inequality (η 1/p )a ≤ ρa implies for the same reason that ηa ≤ (ρp )a . Hence ξa ≤ (ρp )a , i.e., ξ ∈ Σ((I p )− ).

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Thus ((I − )p )− ⊂ (I p )− , which then implies equality since the reverse inclusion is again automatic. Proposition 4.20. If I is countably generated and 0 < p < ∞, then ((I − )p )− is strongly soft-complemented.

5. Operations on Soft Ideals In this section we investigate the soft interior and soft cover of arbitrary intersections of ideals, unions of collections of ideals directed by inclusion, and ﬁnite sums of ideals. Proposition 5.1. For every collection of ideals {Iγ , γ ∈ Γ}: (i) γ se Iγ ⊃ se( γ Iγ ) (ii) γ sc Iγ = sc( γ Iγ ) In particular, the intersection of soft-complemented ideals is soft-complemented. Proof. (i) and the inclusion γ sc Iγ ⊃ sc( γ Iγ ) are immediate consequences of Lemma 4.2(i). For the reverse inclusion in (ii), by (i) and Lemma 4.2 (i)–(iv) we have:

sc Iγ ⊃ Iγ ⊃ se Iγ = se(sc Iγ ) ⊃ se sc Iγ γ

and hence sc

γ

γ

Iγ

γ

⊃ sc se

γ

γ

sc Iγ

= sc

γ

sc Iγ

γ

⊃

sc Iγ .

γ

It follows directly from Deﬁnition 4.1 that if Γ is ﬁnite, then equality holds in (i). In general, equality in (i) does not hold, as seen in Example 5.2 below, where the intersection of soft-edged ideals fails to be soft-edged, thus showing that the inclusion in (i) is proper. Example 5.2. Let ξ ∈ c∗o be a sequence that satisﬁes the ∆1/2 -condition, i.e., n sup ξξ2n < ∞, and let {Iγ }γ∈Γ be the collection of all soft-edged ideals containing the principal ideal (ξ). Then I := γ Iγ is not soft-edged. Indeed, assume that it is and hence ξ = o(η) for some η ∈ Σ(I). By Lemma 6.3 of the next section, there is a sequence γ ↑ ∞ for which γ ≤ ηξ and µ := γξ ∈ c∗o . Then (ξ) ⊂ se(µ) ⊂ (µ) ⊂ (η) ⊂ I. Then se(µ) ∈ {Iγ }γ∈Γ , hence I ⊂ se(µ), and thus se(µ) = (µ). By Proposition 4.6(ii), this implies that µ = o(Dm µ) for some integer m. This is impossible ξn n n ξn = γγ2n since µµ2n ξ2n ≤ ξ2n which implies that µ too satisﬁes the ∆1/2 -condition and hence Dm µ = O(µ), a contradiction.

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Notice that the conclusion that γ Iγ is not soft-edged follows likewise if {Iγ } is a maximal chain of soft-edged ideals that contain the principal ideal (ξ). Moreover, Example 5.2 shows that in general there is no smallest soft-edged cover of an ideal. The next proposition shows that an intersection of strongly soft-complemented ideals, which is soft-complemented by Proposition 5.1(ii), can yet fail to be strongly soft-complemented. Proposition 5.3. The intersection of an infinite countable strictly decreasing chain of principal ideals is never strongly soft-complemented. Proof. Let {Ik } be the chain of principal ideals with Ik Ik+1 and set (k) I = ∈ c∗o for the ideals Ik such that k Ik . First we ﬁnd generators η (k) (k+1) η ≥ η . Assuming the construction up to η (k) , if ξ is a generator of Ik+1 1 then ξ ≤ M Dm η (k) for some M > 0 and m ∈ N. Set η (k+1) := M D1/m ξ, where (k+1) ∗ (k+1) (k) (D1/m ξ)i = ξmi . Then η ∈ co and η ≤η since D1/m Dm = id. More1 over, η (k+1) ≤ M ξ and by an elementary computation, ξi ≤ (D2m D1/m ξ)i for i ≥ m so that (ξ) ⊂ (η (k+1) ) and hence Ik+1 = (ξ) = (η (k+1) ). By assumption, η (k) ∈ Σ(I) for all k. For any given strictly increasing sequence of indices nk ∈ N, (k) set no = 0 and ξi := ηi for nk−1 < i ≤ nk . Since η (k) ≥ η (k+1) for all k, it (k) (k) follows that ξ ∈ c∗o and ξi ≥ ηi for 1 ≤ i ≤ nk . Yet, since ξi ≤ ηi for all (k) i ≥ nk , one has ξ ∈ Σ(η ) for all k and hence ξ ∈ Σ(I). Thus I is not strongly soft-complemented. Notice that if in the above construction η (k) = ρk for some ρ ∈ c∗o that k satisﬁes the ∆1/2 -condition, then I = k (ρ ) is also idempotent. This shows that while idempotent ideals are soft-complemented by Proposition 4.17, they can fail to be strongly soft-complemented. Proposition 5.4. For {Iγ }γ∈Γ a collection of ideals directed by inclusion: (i) γ se Iγ = se( γ Iγ ) In particular, the directed union of soft edged ideals is soft-edged. (ii) γ sc Iγ ⊂ sc( γ Iγ ) Proof. As in Proposition 5.1, (ii) and the inclusion γ se Iγ ⊂ se( γ Iγ ) in (i) are immediate. For the reverse inclusion in (i), from (ii) and Lemma 4.2(iii) and (iv) we have se Iγ ⊂ se sc(se Iγ ) ⊂ se sc se Iγ se Iγ ⊂ se Iγ . = se γ

γ

γ

γ

γ

It follows directly from Deﬁnition 4.1 that if Γ is ﬁnite, then equality holds in (ii), but in general, it does not. Indeed, any ideal I is the union of the collection of all the principal ideals contained in I and this collection is directed by inclusion

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since (η) ⊂ I and (µ) ⊂ I imply that (η), (µ) ⊂ (η + µ) ⊂ I. By Proposition 4.6(i), principal ideals are ssc, hence soft-complemented. Notice that by assuming the continuum hypothesis, every ideal I is the union of an increasing nest of countably generated ideals [4], so then even nested unions of ssc ideals can fail to be softcomplemented. ∞ a m The smallest nonzero am-stable ideal ∞ st (L1 ) = m=0 = (ω)a and the largest am-∞ stable ideal sta∞ (L1 ) = m=0 am (L ) (see Section 2) play an im1 ∞ portant role in [9, 10]. Proposition 5.5. The ideals sta (L1 ) and sta∞ (L1 ) are both soft-edged and softcomplemented, sta (L1 ) is ssc, but sta∞ (L1 ) is not ssc. Proof. For every natural number m, (ω)am = (ωam ) = (ω logm ) is principal, hence Σ(sta (L1 )) is generated by the collection{ω logm }m . Since ω logm = o(ω logm+1 ) for all m, by Proposition 4.6(i) and (ii), sta (L1 ) is both soft-edged and ssc. From ∞ [10, Proposition 4.17 (ii)], sta∞ (L1 ) = m=0 L(σ(logm )), where using the notaideal with chartions of [7, Sections 2.1, 2.25, 4.7], L(σ(logm )) is the Lorentz ∞ acteristic set {ξ∈ c∗o | ξ(log)m ∈ 1 }. Thus if ξ ∈ Σ( m=0 L(σ(logm ))), then ∞ also ξ log ∈ Σ( m=0 L(σ(logm ))) and hence sta∞ (L1 ) is soft-edged. By Propositions 4.12 and 5.1(ii), sta∞ (L1 ) is soft-complemented. However, sta∞ (L1 ) is not ssc. Indeed, set η (k) := ω(log)−k . Then η (k) ∈ Σ(sta∞ (L1 )) for all k, but η (k) ∈ Σ(L(σ(logk−2 ))) for each k ≥ 2. For any arbitrary sequence of increasing indices nk , set no = 0 and ξj := (η (k) )j for nk−1 < j ≤ nk . Then ξj ≥ (η (k) )j for 1 ≤ j ≤ nk but also ξj ≤ (η (k) )j for j ≥ nk . Thus ξ ∈ Σ(L(σ(logk−2 ))) for all k ≥ 2, hence ξ ∈ Σ(sta∞ (L1 )) which shows that sta∞ (L1 ) is not ssc. Now consider ﬁnite sums of ideals. Clearly, K(H)(I + J) = K(H)I + K(H)J, i.e., se(I +J) = se I +se J and hence ﬁnite sums of soft-edged ideals are soft-edged. The situation is far less simple for the soft-cover of a ﬁnite sum of ideals. The inclusion sc(I +J) ⊃ sc I +sc J is trivial, but so far we are unable to determine whether or not equality holds in general or, equivalently, whether or not the sum of two soft-complemented ideals is always soft-complemented. We also do not know if the sum of two ssc ideals is always soft-complemented. However, the following lemma permits us to settle the latter question in the aﬃrmative when one of the ideals is countably generated. Recall that if 0 ≤ λ ∈ co , then λ∗ denotes the decreasing rearrangement of λ. Lemma 5.6. For all ideals I, J and sequences ξ ∈ c∗o : ξ ∈ Σ(I + J) if and only if (max((ξ − ρ), 0))∗ ∈ Σ(I) for some ρ ∈ Σ(J). Proof. If ξ ∈ Σ(I + J), then ξ ≤ ζ + ρ for some ζ ∈ Σ(I) and ρ ∈ Σ(J). (Actually, one can choose ζ and ρ so that ξ = ζ + ρ but equality is not needed here.) Thus ξ − ρ ≤ ζ, and so max((ξ − ρ), 0) ≤ ζ. But then, by the elementary fact that if for two sequence 0 ≤ ν ≤ µ, then ν ∗ ≤ µ∗ , it follows that max((ξ−ρ), 0)∗ ≤ ζ ∗ = ζ and hence (max((ξ − ρ), 0))∗ ∈ Σ(I). Conversely, assume that (max((ξ − ρ), 0))∗ ∈ Σ(I)

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for some ρ ∈ Σ(J). Since 0 ≤ ξ ≤ max((ξ − ρ), 0) + ρ, ξ = ξ ∗ ≤ (max((ξ − ρ), 0) + ρ)∗ ≤ D2 (max((ξ − ρ), 0)∗ ) + D2 ρ ∈ Σ(I + J), where the second inequality, follows from the fact that (ρ + µ)∗ ≤ D2 ρ∗ + D2 µ∗ for any two non-negative sequences ρ and µ, which fact is likely to be previously known but is also the commutative case of a theorem of K. Fan [8, II Corollary 2.2, Equation (2.12)]. Theorem 5.7. The sum I + J of an ssc ideal I and a countably generated ideal J is ssc and hence soft-complemented. Proof. As in the proof of Lemma 2.8 there is an increasing sequence of generators ρ(k) ≤ ρ(k+1) for the characteristic set Σ(J) such that µ ∈ Σ(J) if and only if µ = O(ρ(m) ) for some integer m. By passing if necessary to the sequences kρ(k) , we can further assume that µ ∈ Σ(J) if and only if µ ≤ ρ(m) for some integer m. Let {η (k) } ⊂ c∗o \Σ(I +J). By Lemma 5.6, for each k, (max((η (k) −ρ(k) ), 0))∗ ∈ Σ(I) (k) (k) so, in particular, ηi > ρi for inﬁnitely many indices i. Let πk : N → N be a monotonizing injection for max((η (k) − ρ(k) ), 0), i.e., for all i ∈ N, ∗ max η (k) − ρ(k) , 0 = max η (k) − ρ(k) , 0 = η (k) − ρ(k) > 0. πk (i)

i

πk (i)

Since I is ssc, there is a strictly increasing sequence of indices nk ∈ N such that if ζ ∈ c∗o and ζi ≥ (max((η (k) − ρ(k) ), 0))∗i for all 1 ≤ i ≤ nk , then ζ ∈ Σ(I). Choose integers Nk ≥ max{πk (i) | 1 ≤ i ≤ nk } so that Nk is increasing. We claim (k) that if ξ ∈ c∗o and ξi ≥ ηi for all 1 ≤ i ≤ Nk and all k, then ξ ∈ Σ(I + J), which would conclude the proof. Indeed, for any given m ∈ N and for each k ≥ m, 1 ≤ j ≤ nk and 1 ≤ i ≤ j, it follows that πk (i) ≤ Nk and hence ∗ ξ − ρ(m) ≥ η (k) − ρ(k) = max η (k) − ρ(k) , 0 πk (i) πk (i) i ∗ ≥ max η (k) − ρ(k) , 0 . j

Thus there are at least j values of (ξ −ρ(m) )n that are greater than or equal to (max((η (k) − ρ(k) ), 0))∗j and hence (max((ξ − ρ(m) ), 0))∗j ≥ (max((η (k) − ρ(k) ), 0))∗j . By the deﬁning property of the sequence {nk }, (max((ξ − ρ(m) ), 0))∗ ∈ Σ(I) for every m. But then, for any µ ∈ Σ(J) there is an m such that µ ≤ ρ(m) so that (max((ξ − µ), 0))∗ ≥ (max((ξ − ρ(m) ), 0))∗ and hence (max((ξ − µ), 0))∗ ∈ Σ(I). By Lemma 5.6, it follows that ξ ∈ Σ(I + J), which concludes the proof of the claim and thus of the theorem.

6. Arithmetic Means and Soft Ideals The proofs of the main results in [10, Theorems 7.1 and 7.2] depend in a crucial way on some of the commutation relations between the se and sc operations and the pre and post-arithmetic means and pre and post arithmetic means at inﬁnity

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operations. In this section we shall investigate these relations. We start with the arithmetic mean and for completeness, we list the relations already obtained in [10, Lemma 3.3] as parts (i)–(ii ) of the next theorem. Theorem 6.1. Let I be an ideal. (i) (i ) (ii) (ii ) (iii) (iv) (iv )

sc a I ⊂ a (sc I) sc a I = a (sc I) if and only if ω ∈ Σ(sc I) \ Σ(I) se Ia ⊂ (se I)a se Ia = (se I)a if and only if I = {0} or I ⊂ L1 sc Ia ⊃ (sc I)a se a I ⊃ a (se I) se a I = a (se I) if and only if ω ∈ Σ(I) \ Σ(se I).

The “missing” reverse inclusion of (iii) will be explored in Proposition 6.8. The proof of parts (iii)–(iv ) of Theorem 6.1 depend on the following two lemmas. Lemma 6.2. (i) Fa = (L1 )a = (ω) and a (ω) = L1 Consequently (ω) and L1 are, respectively, the smallest nonzero am-open ideal and the smallest nonzero am-closed ideal. (ii) {0} = a I if and only if L1 ⊂ a I if and only if ω ∈ Σ(I) (iii) L1 = a I if and only if ω ∈ Σ(I) \ Σ(se I) (iv) L1 a I if and only if ω ∈ Σ(se I) Proof. Notice that ηa ω for every 0 = η ∈ ∗1 and that ω = o(ηa ) for every η ∈ ∗1 . Thus (ii) and the equalities in (i) follow directly from the deﬁnitions. Recall from the paragraphs preceding Lemma 2.1 that an ideal is am-open (resp., am-closed) if and only if it is the arithmetic mean of an ideal, in which case if it is nonzero, it contains Fa = (ω) (resp., if and only if it is the prearithmetic mean of an ideal, in which case by (ii), it contains L1 ). Thus the minimality of (ω) (resp., L1 ) are established. (iii) follows immediately from (ii) and (iv). (iv) Assume ﬁrst that L1 a I. Then L1 ⊂ se a I since L1 is soft-edged (Proposition 4.12) and hence by (i), (ω) = (L1 )a ⊂ (se a I)a = se((a I)a ) = se I o ⊂ se I where the second equality follows from Theorem 6.1(ii ) applied to a I which is not contained in L1 . Conversely, assume that ω ∈ Σ(se I), i.e., ω = o(η) for some η ∈ Σ(I). Then L1 ⊂ a I by (ii). It follows directly from the deﬁnition of lnd (see paragraph preceding Lemma 2.14) that ω = o(ω lnd ωη ). By Lemma 2.14(i), ω lnd ωη ∈ Σ(I o ), i.e., ω lnd ωη ≤ ρa ∈ Σ(I) for some ρ ∈ Σ(a I). But ρ ∈ ∗1 since ω = o(ρa ) and hence L1 = a I. Lemma 6.3. For η ∈ c∗o and 0 < β → ∞, there is a sequence 0 < γ ≤ β with γ ↑ ∞ for which γη is monotone nonincreasing.

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Proof. The case where η has ﬁnite support is elementary, so assume that for all i, ηi > 0. By replacing if necessary β with lnd β we can assume also that β is monotone nondecreasing. Starting with γ1 := β1 , deﬁne recursively 1 γn := min(γn−1 ηn−1 , βn ηn ). ηn It follows immediately that γ ≤ β and that γη is monotone nonincreasing. Moreover, γn ≥ γn−1 for all n since both βn ≥ βn−1 ≥ γn−1 and γn−1 ηn−1 ηn ≥ γn−1 . In the case that γn = βn inﬁnitely often, then γ → ∞. In the case that γn = βn for γm → ∞ since ηn → 0 and all n > m, then γn ηn = γn−1 ηn−1 and so also γn = ηηm n ηm γm = 0. Proof of Theorem 6.1. (i)–(ii ) See [10, Lemma 3.3]. (iii) If ξ ∈ Σ((sc I)a ), then ξ ≤ ηa for some η ∈ Σ(sc I). So for every α ∈ c∗o , αη ∈ Σ(I) and αξ ≤ αηa ≤ (αη)a ∈ Σ(Ia ), where the last inequality follows from the monotonicity of α. Thus ξ ∈ Σ(sc Ia ). (iv) Let ξ ∈ Σ(a (se I)), i.e., ξa ≤ αη for some α ∈ c∗o and η ∈ Σ(I). Since 1 ( α ξ)a ≤ α1 ξa ≤ η ∈ c∗o where the ﬁrst inequality follows from the monotonicity of α, by Lemma 6.3 there is a sequence γ ↑ ∞ such that γ ≤ α1 and γξ is monotone nonincreasing. Thus (γξ)a ≤ η ∈ Σ(I), i.e., γξ ∈ Σ(a I), and hence ξ ∈ Σ(se a I). (iv ) There are three cases. If ω ∈ Σ(I), then by Lemma 6.2(ii), both a I = {0} and a (se I) = {0} and hence the equality holds. If ω ∈ Σ(I) \ Σ(se I), then L1 = a I by Lemma 6.2(iii) and hence se a I = L1 since L1 is soft-edged by Proposition 4.12. But a (se I) = {0) by Lemma 6.2(ii), so the inclusion in (iv) fails. For the ﬁnal case, if ω ∈ Σ(se I), then by Lemma 6.2(iv), L1 a I. Let ξ ∈ Σ(se a I), i.e., ξ = o(η) for some η ∈ Σ(a I). By adding to η, if necessary, a nonsummable sequence in Σ(a I), we can assume that η is itself not summable. But then it is easy to verify that ξa = o(ηa ), i.e., ξa ∈ Σ(se I) and hence ξ ∈ Σ(a (se I)). Now we examine how the operations sc and se commute with the arithmetic mean operations of am-interior I o := (a I)a and am-closure I − := a (Ia ). Theorem 6.4. Let I be an ideal. (i) sc I − ⊃ (sc I)− (ii) se I − = (se I)− (iii) sc I o ⊂ (sc I)o (iii ) sc I o = (sc I)o if and only if ω ∈ Σ(sc I) \ Σ(I) (iv) se I o ⊃ (se I)o (iv ) se I o = (se I)o if and only if ω ∈

Σ(I) \ Σ(se I) Proof. (i) The case I = {0} is obvious. If I = {0}, then ω ∈ Σ(Ia ) and hence, by Theorem 6.1(i ) and (iii), it follows that sc I − = sc a (Ia ) = a (sc Ia ) ⊃ a ((sc I)a ) = (sc I)− . (ii) There are three possible cases. The case when I = {0} is again obvious. In the second case when {0} = I ⊂ L1 , then I − = L1 and (se I)− = L1 since L1 is

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the smallest nonzero am-closed ideal by Lemma 6.2(i). Since L1 is soft-edged by Proposition 4.12, se I − = L1 , so equality in (ii) holds. In the third case, I ⊂ L1 . Then L1 I − and ω ∈ Σ(se Ia ) by Lemma 6.2(iv). Then se I − = se a (Ia ) = a (se(Ia )) = a ((se I)a ) = (se I)− where the second and third equalities follow from Theorem 6.1(iv ) and (ii ). (iii) Let ξ ∈ Σ(sc I o ) and let α ∈ c∗o . By the deﬁnition of “und” (see the paragraph preceding Lemma 2.14) it follows easily that αω und ωξ ≤ ω und αξ ω and ξ o ∈ Σ(I) since αξ ∈ Σ(I ). Thus αω und ∈ Σ(I) by Corollary 2.16, that ω und αξ ω ω ξ o and hence ω und ω ∈ Σ(sc I). But then, again by Corollary 2.16, ξ ∈ Σ((sc I) ). (iii ) If ω ∈ Σ(sc I) \ Σ(I), then sc I o = sc(a I)a ⊃ (sc(a I))a = (a (sc I))a = (se I)o by Theorem 6.1(iii) and (i ). If on the other hand ω ∈ Σ(sc I) \ Σ(I), then by Lemma 6.2(ii) a (sc I) = {0} and hence (sc I)o = {0), while a (I) = {0} and hence sc(I)o = {0}. (iv) and (iv ). There are three possible cases. If ω ∈ Σ(I), then I o = {0} by Lemma 6.2(ii) and so se I o = {0} and (se I)o = {0}, i.e., (iv ) holds trivially. If ω ∈ Σ(I) \ Σ(se I), then I o = {0} and (se I)o = {0} again by Lemma 6.2(ii). But then se I o = {0}, so (iv) holds but (iv ) does not. Finally, when ω ∈ Σ(se I), then L1 a I by Lemma 6.2 (iv) and hence se I o = se(a I)a = (se(a I))a = (a (se I))a = (se I)o by Theorem 6.1(ii ) and (iv ).

We were unable to ﬁnd natural conditions under which the reverse inclusion of Theorem 6.4(i) holds (see also Proposition 6.8), nor examples where it fails. Corollary 6.5. (i) If I is an am-open ideal, then sc I is am-open while se I is am-open if and only if I = (ω). (ii) If I is an am-closed ideal, then sc I and se I are am-closed. Proof. (ii) and the ﬁrst implication in (i) are immediate from Theorem 6.4. For the second implication of (i), assume that I is am-open and that 0 = I = (ω). Then by Lemma 6.2(i), (ω) I and L1 = a (ω) ⊂ a I. But L1 = a I follows from (L1 )a = (ω) = I = (a I)a . Then ω ∈ Σ(se I) by Lemma 6.2(iv), hence se I = se I o = (se I)o by Theorem 6.4(iv ) and thus se I is am-open. If I = {0}, then se I = {0} too is am-open. If I = (ω), then se I (ω) cannot be am-open, again by Lemma 6.2(i). For completeness’ sake we list also some se and sc commutation properties for the largest am-closed ideal I− contained in I and the smallest am-open ideal I oo containing I (see Corollary 2.6 and Deﬁnition 2.18). Proposition 6.6. For every ideal I: (i) sc I− = (sc I)− (ii) se I− ⊂ (se I)−

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(iii) sc I oo ⊃ (sc I)oo (iv) se I oo ⊂ (se I)oo (iv ) se I oo = (se I)oo if and only if either I = {0} or I ⊂ (ω) Proof. (i)–(iii) Corollary 6.5 and the maximality (resp., minimality) of I− (resp., I oo ) yield the inclusions sc I− ⊂ (sc I)− , se I− ⊂ (se I)− , and sc I oo ⊃ (sc I)oo . From the second inclusion it follows that se((sc I)− ) ⊂ (se(sc I))− = (se I)− ⊂ I− and hence (sc I)− ⊂ sc(sc I)− = sc(se((sc I)− )) ⊂ sc I− so that equality holds in (i). (iv) If η ∈ Σ((se I oo ), then by Proposition 2.21, η ≤ αω und ωξ for some ξ ∈ Σ(I) and α ∈ c∗o . As remarked in the proof of Theorem 6.4(iii), it follows that oo η ≤ ω und αξ ω and hence η ∈ Σ((se I) ), again by Proposition 2.21. (iv ) There are three cases. If I = {0}, (iv ) holds trivially. If {0} = I ⊂ (ω), then by the minimality of (ω) among nonzero am-open ideals, I oo = (ω) and (se I)oo = (ω), so the inclusion in (iv ) fails. If I ⊂ (ω), then I oo = (ω) and hence by Corollary 6.5(i), se I oo is am-open and by minimality of (se I)oo , (iv ) holds. It is now an easy application of the above results to verify that the following am-operations preserve softness. Corollary 6.7. (i) If I is soft-complemented, then so are a I, I o , and I− . (ii) If I is soft-edged, then so are a I, I o , and I − . (iii) If I is soft-edged, then Ia is soft-edged if and only if either I = {0} or I ⊂ L1 . (iv) If I is soft-edged, then I oo is soft-edged if and only if either I = {0} or I ⊂ (ω). Several of the “missing” statements that remain open are equivalent as shown in the next proposition. Proposition 6.8. For every ideal I, the following conditions are equivalent. (i) (ii) (iii) (iv)

sc Ia ⊂ (sc I)a (sc I)a is soft-complemented (sc I)− is soft-complemented sc I − ⊂ (sc I)−

Proof. Implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) are easy consequences of Theorem 6.1 and Corollary 6.7. We prove that (iv) ⇒ (i). The case I = {0} being trivial, assume I = {0}. Then ω ∈ Σ(Ia ), hence sc I − ⊃ a sc(Ia ) by Theorem 6.1(i ). Moreover, since Ia is am-open, then so is sc Ia by Corollary 6.5, i.e., sc Ia = (sc Ia )o . Then sc Ia = (a (sc Ia ))a ⊂ (sc I − )a ⊂ ((sc I)− )a = (sc I)a , the latter equality following from the general identity (a (Ja ))a = Ja .

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Now we investigate the relations between arithmetic means at inﬁnity and the se and sc operations and we list some results already obtained in [10, Lemma 4.19] as parts (i) and (ii) of the next theorem. Theorem 6.9. For every ideal I = {0}: (i) sc a∞ I = a∞ (sc I) (ii) se Ia∞ = (se I)a∞ (iii) sc Ia∞ ⊃ (sc I)a∞ (iv) se a∞ I = a∞ (se I) Proof. (i)–(ii) See [10, Lemma 4.19]. (iii) If ξ ∈ Σ((sc I)a∞ ), ξ ≤ ηa∞ for some η ∈ Σ(sc I ∩ L1 ). In [10, Lemma 4.19 (i)](proof) we showed that for every α ∈ c∗o , αηa∞ ≤ (α η)a∞ for some α ∈ c∗o . But then α η ∈ Σ(I ∩ L1 ) and so αξ ≤ (α η)a∞ ∈ Σ(Ia∞ ), i.e., ξ ∈ Σ(sc Ia∞ ). (iv) Let ξ ∈ Σ(se a∞ I), then ξ ≤ αη for some α ∈ c∗o and η ∈ Σ(a∞ I). But then by the monotonicity of α, ξa∞ ≤ (αη)a∞ ≤ αηa∞ ∈ Σ(se I). Thus ξ ∈ Σ(a∞ (se I)) which proves the inclusion se a∞ I ⊂ a∞ (se I). Now let ξ ∈ Σ(a∞ (se I)), i.e., ξa∞ ≤ αη for some α ∈ c∗o and η ∈ Σ(I). We construct a sequence γ ↑ ∞ such that γξ is monotone nonincreasing and (γξ)a∞ ≤ η. Without loss generality assume that ξn = 0 and hence αn = 0 for all n. We choose a strictly increasing indices nk (with no = 0) such that ∞ sequence of ∞ for k ≥ 1, αnk ≤ 2−k−2 and nk+1 +1 ξi ≤ 14 nk +1 ξi for all k. Set βn = 2k for nk < n ≤ nk+1 . Then for all k ≥ 0 and nk < n + 1 ≤ nk+1 we have ∞

βi ξi = 2k

n+1

≤ 2k ≤ 2k

nk+1

nk+2

ξi + 2k+2

n+1

nk+1 +1

nk+1

∞

 ξi + 2k+1 

n+1

nk+1 +1 ∞

ξi + 2k+2

nk+1 +1

nk+3

ξi + · · ·

nk+2 +1

nk+1 n+1

≤

ξi + 2k+1

∞

ξi + 2

ξi + 22

nk+2 +1

ξi ≤ 2k+2

∞

∞

 ξi + · · · 

nk+3 +1

ξi

n+1

∞ ∞ 1 1 n ξi ≤ ξi = (ξa )n ≤ nηn . αnk n+1 αn n+1 αn ∞

This proves that (βξ)a∞ ≤ η. Now Lemma 6.3 provides a sequence γ ≤ β, with γ ↑ ∞ and γξ monotone nonincreasing, and hence (γξ)a∞ ≤ (βξ)a∞ ≤ η. Thus γξ ∈ Σ(a∞ I) and hence ξ = γ1 (γξ) ∈ Σ(se a∞ I). The reverse inclusion in Theorem 6.9(iii) does not hold in general. Indeed, whenever Ia∞ = se(ω) (which condition by [10, Corollary 4.9 (ii)] is equivalent to I −∞ = a∞ (Ia∞ ) = L1 and in particular is satisﬁed by I = L1 ), it follows that sc Ia∞ = (ω) while (sc I)a∞ ⊂ se(ω). We do not know of any natural suﬃcient condition for the reverse inclusion in Theorem 6.9(iii) to hold.

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Many of the other results obtained for the arithmetic mean case have an analog for the am-∞ case: Theorem 6.10. For every ideal I: (i) (ii) (iii) (iii ) (iv)

sc I −∞ ⊃ (sc I)−∞ se I −∞ = (se I)−∞ sc I o∞ ⊃ (sc I)o∞ sc I o∞ = (sc I)o∞ if and only if sc I o∞ ⊂ se(ω) se I o∞ = (se I)o∞

Proof. (i), (ii), (iii), and (iv) follow immediately from Theorem 6.9. (iii ) Since every am-∞ open ideal is contained in se(ω), it follows that o∞ sc I = (sc I)o∞ ⊂ se(ω). Assume now that sc I o∞ ⊂ se(ω), let ξ ∈ Σ(sc I o∞ ), and let α ∈ c∗o . Since ξ = o(ω), there is an increasing sequence of integers nk with no = 0 for which (uni ωξ )j = ( ωξ )nk for nk−1 < j ≤ nk . Deﬁne α ˜ j = α1 for ˜ j = αnk for nk < j ≤ nk+1 for k ≥ 1. Then α ˜ ∈ c∗o and for all 1 < j ≤ n1 and α k ≥ 1 and nk−1 < j ≤ nk α ˜ξ ξ ξ α ˜ξ α ˜ξ ξ = αj ≤ αnk−1 = ≤ uni ≤ uni . α uni ω j ω nk ω nk ω nk ω nk ω j ˜ Since αξ ˜ ∈ Σ(I o∞ ) by hypothesis, it follows that ω uni αξ ω ∈ Σ(I) by Corollary 3.10. ξ ξ ∗ But then αω uni ω ∈ Σ(I) for all α ∈ co , i.e., ω uni ω ∈ Σ(sc I). Hence, again by Corollary 3.10, ξ ∈ Σ((sc I)o∞ ) and hence sc I o∞ ⊂ (sc I)o∞ . By (iii) we have equality.

The necessary and suﬃcient condition in Theorem 6.10 (iii ) is satisﬁed in the case of most interest, namely when I ⊂ L1 . As in the am-case, we know of no natural conditions under which the reverse inclusion of (i) holds nor examples where it fails (see also Proposition 6.8). In the following proposition we collect the am-∞ analogs of Corollary 6.5, Proposition 6.6, and Corollary 6.7. Recall by Lemma 3.16 that I oo∞ = se(ω) for any ideal I ⊂ se(ω). Proposition 6.11. Let I = {0} be an ideal. (i) If I is am-∞ open, then so is se I. (i ) If I is am-∞ open, then sc I is am-open if and only if sc I ⊂ se(ω). (ii) If I is am-∞ closed, then so are se I and sc I. (iii) se I oo∞ = (se I)oo∞ (iv) sc I oo∞ ⊃ (sc I)oo∞ (v) se I−∞ ⊂ (se I)−∞ (vi) sc I−∞ = (sc I)−∞ (vii) If I is soft-edged, then so are a∞ I, Ia∞ , I −∞ , I o∞ , and I oo∞ . (viii) If I is soft-complemented, then so is a∞ I and I−∞ . (viii ) If I is soft-complemented, then I o∞ is soft-complemented if and only if sc I o∞ ⊂ se(ω).

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Proof. (i) Immediate from Theorem 6.10(iv). (i ) If sc I ⊂ se(ω) then sc I = (sc I)o∞ by Theorem 6.10(iii ) and hence sc I is am-∞ open. The necessity is clear since se(ω) is the largest am-∞ open ideal. (ii) se I is am-∞ closed by Theorem 6.10(ii). By Theorem 6.10(i) and the am-∞ analog of the 5-chain of inclusions given in Section 2, sc I = sc I −∞ ⊃ (sc I)−∞ ⊃ sc I ∩ L1 = sc I, where the last equality holds because L1 is the largest am-∞ closed ideal so contains I, and being soft-complemented it contains sc I. (iii) By (i), se I oo∞ is am-∞ open and by Deﬁnition 3.12 and Proposition 5.1 and following remark, it contains se(I ∩ se(ω)) = se I ∩ se(ω), hence it must contain (se I)oo∞ . On the other hand, if ξ ∈ Σ(se I oo∞ ), then by Proposition 3.14 there is an α ∈ c∗o and η ∈ Σ(I ∩ se(ω)) such that ξ ≤ αω uni ωη . Then, by the ˜ ˜ ∈ c∗o such that αω uni ωη ≤ ω uni αη proof in Theorem 6.10(iii ), there is an α ω . Since oo∞ α ˜ η ∈ Σ(se I ∩ se(ω)), Proposition 3.14 yields again ξ ∈ Σ((se I) ) which proves the equality in (iii). (iv) Let ξ ∈ Σ((sc I)oo∞ ). By Proposition 3.14 there is an η ∈ Σ((sc I)∩se(ω)) such that ξ ≤ ω uni ωη . Then, by the proof in Theorem 6.10(iii ), for every α ∈ c∗o ˜ there is an α ˜ ∈ c∗o such that αξ ≤ αω uni ωη ≤ ω uni αη ˜ η ∈ Σ(I ∩ se(ω)), ω . As α oo∞ again by Proposition 3.14, αξ ∈ Σ(I ) and hence ξ ∈ Σ(sc(I oo∞ )). (v) This is an immediate consequence of (ii). (vi) The inclusion sc I−∞ ⊂ (sc I)−∞ is similarly an immediate consequences of (ii). The reverse inclusion follows from (v) applied to the ideal sc I: se(sc I)−∞ ⊂ (se sc I)−∞ = (se I)−∞ ⊂ I−∞ hence (sc I)−∞ ⊂ sc(sc I)−∞ = sc(se(sc I)−∞ ) ⊂ sc I−∞ . (vii) The ﬁrst two statements follow from Theorem 6.9 (iv) and (ii), the next two from Theorem 6.10(ii) and (iv), and the last one from (iii). (viii), (viii ) follow respectively from Theorem 6.9(i) and Theorem 6.10(iii ). Acknowledgments We wish to thank Ken Davidson for his input on the initial phase of the research and Daniel Beltita for valuable suggestions on this paper.

References [1] S. Albeverio, D. Guido, A. Posonov, and S. Scarlatti, Singular traces and compact operators. J. Funct. Anal. 137 (1996), 281–302. [2] G. D. Allen and L. C. Shen, On the structure of principal ideals of operators. Trans. Amer. Math. Soc. 238 (1978), 253–270. [3] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, vol. 129, Academic Press, 1988.

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[4] A. Blass and G. Weiss, A characterization and sum decomposition for operator ideals. Trans. Amer. Math. Soc. 246 (1978), 407–417. [5] J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. of Math. (2) 42 (1941), 839–873. [6] K. Dykema, G. Weiss, and M. Wodzicki, Unitarily invariant trace extensions beyond the trace class. Complex analysis and related topics (Cuernavaca, 1996), Oper. Theory Adv. Appl. 114 (2000), 59–65. [7] K. Dykema, T. Figiel, G. Weiss, and M. Wodzicki, The commutator structure of operator ideals. Adv. Math. 185/1 (2004), 1–79. [8] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, 1969. [9] V. Kaftal and G. Weiss, Traces, ideals, and arithmetic means. Proc. Natl. Acad. Sci. USA 99 (2002), 7356–7360. , Traces on operator ideals and arithmetic means, preprint. [10] [11] , Majorization for infinite sequences and operator ideals, in preparation. [12] , B(H) Lattices, density, and arithmetic mean ideals, preprint. , Second order arithmetic means in operator ideals, J. Operators and Matrices, [13] to appear. [14] N. J. Kalton, Unusual traces on operator ideals. Math. Nachr. 136 (1987), 119–130. [15] , Trace-class operators and commutators. J. Funct. Anal. 86 (1989), 41–74. [16] A. S. Markus, The eigen- and singular values of the sum and product of linear operators. Uspekhi Mat. Nauk 4 (1964), 93–123. [17] A. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, Mathematics in Science and Engineering, vol. 143, Academic Press, New York, 1979. [18] N. Salinas, Symmetric norm ideals and relative conjugate ideals. Trans. Amer. Math. Soc. 138 (1974), 213–240. [19] R. Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und irher Grenzgebiete, Neue Folge, Heft 27, Springer Verlag, Berlin, 1960. [20] J. Varga, Traces on irregular ideals. Proc. Amer. Math. Soc. 107 (1989), 715–723. [21] G. Weiss, Commutators and Operator ideals, dissertation (1975), University of Michigan microfilm. [22] M. Wodzicki, Vestigia investiganda. Mosc. Math. J. 4 (2002), 769–798, 806. Victor Kaftal and Gary Weiss University of Cincinnati Department of Mathematical Sciences Cincinnati, OH 45221-0025 USA e-mail: [email protected] [email protected] Submitted: April 15, 2006 Revised: January 23, 2007

Integr. equ. oper. theory 58 (2007), 407–431 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030407-25, published online April 14, 2007 DOI 10.1007/s00020-007-1487-z

Integral Equations and Operator Theory

Scattering Matrix, Phase Shift, Spectral Shift and Trace Formula for One-dimensional Dissipative Schr¨ odinger-type Operators Hagen Neidhardt and Joachim Rehberg In friendship dedicated to P. Exner on the occasion of his 60th birthday

Abstract. The paper is devoted to Schr¨ odinger operators with dissipative boundary conditions on bounded intervals. In the framework of the LaxPhillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are veriﬁed directly. The results are exploited for dissipative Schr¨ odinger-Poisson systems. Mathematics Subject Classification (2000). Primary 47A20; Secondary 47B44, 47A40. Keywords. Dissipative Schr¨ odinger-type operators, Sturm-Liouville operators, self-adjoint dilation, characteristic function, Lax-Phillips scattering theory, scattering matrix, phase shift, spectral shift, trace formula, Birman-Krein formula.

1. Introduction Dissipative Schr¨ odinger operators are important examples of non-selfadjoint operators which admit a detailed investigation. The powerful tool for this is the dilation and model theory for dissipative operators, cf. [14]. There is a rich literature on dissipative Schr¨odinger operators, their dilations and eigenfunction expansions mainly for Sturm-Liouville operators [2, 4, 6, 7, 19], [25]–[28] but also for Schr¨ odinger operator in higher dimensions, cf. [24]. The investigations are extended to matrix-valued dissipative Sturm-Liouville operators also, see [3, 5, 8]. In This work was supported by DFG, Grant 1480/2.

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the following we consider one-dimensional Schr¨ odinger-type operators H[κa , κb , V ] deﬁned by (H[κa , κb , V ]g)(x)

=

dom(H[κa , κb , V ]) =

(l[V ]g)(x), g ∈ dom(H[κa , κb , V ]),  1 1,2 ,   m(x) f (x) ∈ W 1 f ∈ W 1,2 (Ω) : 2m(a) f (a) = −κa f (a),  1  f (b) = κb f (b) 2m(b)

where

    

1 d 1 d g(x) + V (x)g(x), 2 dx m(x) dx such that the boundary coeﬃcients obey κa , κb ∈ C+ := {z ∈ C : m(z) ≥ 0} and the potential V ∈ L∞ (Ω) is real. Throughout the paper we always assume that m is a real function satisfying 1 ∈ L∞ (Ω) 0≤m+ m without mentioning this explicitly in the following. Such dissipative Schr¨ odingertype operators naturally appear in the theory of dissipative Schr¨odinger-Poisson system which is used to describe quantum transport in semi-conductors, see [9, 17, 23]. From [14] it s known that purely dissipative operators are completely described by the characteristic function which is an analytic contraction-valued operator function deﬁned in the lower half-plane. It turns out that the characteristic function of a dissipative operator can be regarded as the scattering matrix of a suitable posed Lax-Phillips scattering theory, cf. [22]. In [19] we have analyzed the characteristic function Θ[κa , κb , V ], the self-adjoint dilation K[κa , κb , V ] of H[κa , κb , V ] as well as the generalized eigenfunctions of K[κa , κb , V ] for the case κa , κb ∈ C+ := {z ∈ C+ : m(z) > 0} which was necessary to deﬁne carrier and current densities for dissipative Schr¨odinger-Poisson systems, see [17]. Now we are interested in the associated Lax-Phillips scattering theory, in particular, in the phase shift and its high-energy asymptotic behaviour. In Theorem 4.8 and Corollary 4.9 it is shown that the phase shift of the Lax-Phillips scattering matrix possesses the same high-energy asymptotic behaviour as the eigenvalue distribution function of the Dirichlet boundary problem. Moreover, Theorem 5.4 establishes an intimate connection between the spectral shift, phase shift and eigenvalue distribution function of the Dirichlet boundary value problem. Using this connection one gets a simple proof of the so-called Birman-Krein formula, cf. [11], for the the Lax-Phillips scattering theory under consideration. The results are interesting from a pure operator theoretic view point but, additionally, provide optimal estimates for the carrier density operator of a dissipative Schr¨ odinger-Poisson system in [23]. In The paper is organized as follows. In Section 2 we introduce a boundary triplet which allows us to describe appropriately self-adjoint and maximal dissipative Schr¨ odinger-type operators used in the following. In particular, we verify in this (l[V ]g)(x) := −

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way some properties of Schr¨odinger-type operators not proven in [19] and introduce the characteristic function quite diﬀerently from [19] in terms of that boundary triplet. In Section 3 we give a short introduction to the Lax-Phillips scattering theory for Schr¨ odinger-type operators. Section 4 is devoted to the phase shift of the Lax-Phillips scattering theory; in particular, high-energy estimates from above of the phase shift are are found. Finally, in Section 5 we introduce the spectral shift for the pair {H[κa , κb , V ], HD [V ]} where HD [V ] is the self-adjoint operator generated by l[V ] with Dirichlet boundary conditions and verify the trace formula and the Birman-Krein formula directly. Notice that the existence of the spectral shift follows already from an abstract result proven in [1]. Notation: Hilbert spaces are denoted by Gothic letters, for instance H = L2 (Ω), the dilation space K, etc, where Lp (Ω), 1 ≤ p ≤ ∞, denotes the usual Banach spaces of p-summable functions on Ω ⊆ R. If we have in mind real functions, we write LpR (Ω). By W l,p (Ω), p ≥ 1, l ≥ 1, we denote the standard Sobolev spaces. The norm of a Banach space X is denoted by · X or simply by · . The scalar product of a Hilbert space H is denoted by (·, ·)H or simply by (·, ·). In the special case of the Hilbert space C2 we use the notation ·, · for the scalar product. The set of bounded operators on some Banach space X is denoted by B(X). For a densely deﬁned linear operator A : X −→ X we denote by A∗ , spec(A) and res(A) its adjoint operator, the spectrum and resolvent set, respectively.

2. Dissipative Schr¨ odinger-type operators 2.1. Boundary triplets, Weyl function and γ-field We note that the operators H[κa , κb , V ], κa , κb ∈ C+ , and HD [V ] can be regarded as dissipative or self-adjoint extensions of one and the same closed symmetric operator S[V ], (S[V ]g)(x)

:=

dom(S[V ])

=

(l[V ]g)(x), g ∈ dom(S[V ]),   1 1,2   mg ∈ W 1 g ∈ W 1,2 (Ω) : g(b) = 2m(b) g (b) = 0   1 g(a) = 2m(a) g (a) = 0

(2.1)

which has the deﬁciency indices (2, 2). The adjoint operator S[V ]∗ is given by (S[V ]∗ g)(x) dom(S[V ]∗ )

:= (l[V ]g)(x), g ∈ dom(S[V ]∗ ), 1 = g ∈ W 1,2 (Ω) : m g ∈ W 1,2 .

It is straightforward to verify that (C2 , Γ0 , Γ1 ) performs a boundary triplet for S[V ]∗ , for deﬁnition see [16] and references therein, where Γ0 , Γ1 : dom(S[V ]∗ ) → C2 are linear operators, given by

1 1 g(b) m(b) g (b) Γ0 g := and Γ1 g := − . (2.2) 1 −g(a) 2 m(a) g (a)

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That is, one has to show that Green’s identity (S[V ]∗ f, g) − (f, S[V ]∗ g) = Γ1 f, Γ0 g − Γ0 f, Γ1 g ,

f, g ∈ dom(S[V ]∗ ),

is satisﬁed and the operator Γ : H −→ C2 ⊕ C2 , Γf := Γ0 f ⊕ Γ1 f,

f ∈ dom(Γ) := dom(S[V ]∗ ),

is surjective, which can be easily seen. We note that the selfadjoint extension HD [V ] := S[V ]∗ ker(Γ0 ) corresponds to the Dirichlet boundary conditions, that is, 1 dom(HD [V ]) = g ∈ W 1,2 (Ω) : g ∈ W 1,2 (Ω), f (a) = f (b) = 0 . m Let B a dissipative or self-adjoint operator on the Hilbert space C2 . By HB [V ] := S[V ]∗ ker(Γ1 − BΓ0 ) one deﬁnes a maximal dissipative or self-adjoint extension of the symmetric operator S[V ]. Setting

κb 0 , κ a , κ b ∈ C+ , κ := 0 κa we ﬁnd that H−κ [V ] = H[κa , κb , V ]. The defect subspace of S[V ] at the point z ∈ C is denoted by Nz [V ], i.e., Nz [V ] := ker(S[V ]∗ − z), z ∈ C+ . For every z ∈ res(HD [V ]) we set γ[V ](z) := (Γ0 Nz [V ])−1

and M [V ](z) := Γ1 γ[V ](z).

The functions res(HD [V ]) z −→ γ[V ](z) and res(HD [V ]) z −→ M [V ](z) are called the γ-ﬁeld and the Weyl function corresponding to S[V ] and the boundary triplet {C2 , Γ0 , Γ1 }. We note that the Weyl function is a Nevanlinna function, that is, a holomorphic operator-valued function in C+ and C− such that m(M [V ](z)) ≥ 0 for z ∈ C+ , and M [V ](z)∗ = M [V ](z),

z ∈ res(HD [V ]).

In the present case the Weyl function is meromorphic in C with poles on R which coincide with the eigenvalues of HD [V ]. For any dissipative or self-adjoint operator B on C2 the so-called Krein’s formula (HB [V ] − z)−1 = (HD [V ] − z)−1 + γ(z)(B − M [V ](z))−1 γ(z)∗ ,

z ∈ C+ ,

holds, cf. [12]. In particular, one has the Krein’s formula (H[κa , κb , V ]−z)−1 = (HD [V ]−z)−1 −γ(z)(κ+M [V ](z))−1 γ(z)∗ ,

z ∈ C+ . (2.3)

The Schr¨ odinger-type operator H[κa , κb , V ] is maximal dissipative if either κa ∈ C+ or κb ∈ C+ . In both cases the operator is completely non-selfadjoint, see [18]. In corresponding to [19] we consider only the case κa , κb ∈ C+ in the following. The spectrum of H[κa , κb , V ] consists of isolated eigenvalues in the lower half-plane with the only accumulation point at inﬁnity. Since the operator H[κa , κb , V ] is

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completely non-selfadjoint, its eigenvalues are non-real. The extension H[qa , qb , V ], qa , qb ∈ R, of S is self-adjoint and semi-bounded from below. Lemma 2.1. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 lim =0 B(H) |κa | → ∞ |κb | → ∞

(2.4)

for z ∈ C+ . Proof. We note that the γ-ﬁeld γ[V ](z) as well as the Weyl function M [V ](z) are independent from κa , κb ∈ C+ . Using Krein’s formula (2.3) we immediately verify the relation (2.4). 2.2. Characteristic function If B is dissipative operator, then in accordance with [13] the characteristic function ΘHB [V ] (z), z ∈ C− , of the maximal dissipative operator HB [V ] is given by ΘHB [V ] (z) = IC2 − 2i −m(B)(B ∗ − M [V ](z))−1 −m(B) ran(m(B)), 1 (B−B ∗ ). The characteristic function is analytic for z ∈ C− , where m(B) := 2i and its values are contractions, if z ∈ C− . In the present case the characteristic function admits a meromorphic continuation to C+ for any dissipative operator B. The characteristic function entirely characterizes the non-selfadjoint part of the maximal dissipative operator HB [V ], cf. [14]. In the following we use the representations

α2a α2 and κb = qb + i b , 2 2 where qa , qb ∈ R and αa , αb > 0. If B = −κ, then

1 1 α2b 0 ∗ −m(B) = (κ − κ ) = . 2i 2 0 α2a κa = qa + i

Hence we obtain 1 −m(B) = √ α, 2

α :=

αb 0

0 . αa

Setting Θ[κa , κb , V ](z) := ΘH−κ [V ](z), z ∈ C− , and using the deﬁnition (2.2) we get Θ[κa , κb , V ](z) = IC2 + iα(κ∗ + M [V ](z))−1 α, z ∈ C− . (2.5) Since the spectrum of H[κa , κb , V ] is non-real the characteristic function Θ[κa , κb , V ](·) is well-deﬁned on R and, moreover, holomorphic in a neighborhood of R. Furthermore, a straightforward computation shows that Θ[κa , κb , V ](λ) is unitary for of λ ∈ R. Since the maximal dissipative operator H[κa , κb , V ] is completely non-selfadjoint for κa , κb ∈ C+ , the characteristic function Θ[κa , κb , V ](·) completely characterizes H[κa , κb , V ].

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The characteristic function of the operator H[κa , κb , V ] can be represented by the operator H[κa , κb , V ] itself and αa , αb . Indeed, multiplying Krein’s formula on the left by Γ0 we obtain G[κa , κb , V ](z) := Γ0 (H[κa , κb , V ] − z)−1 = −(κ + M [V ](z))−1 γ(z)∗ ,

z ∈ C+ .

Taking the adjoint we get G[κa , κb , V ](z)∗ = −γ(z)(κ∗ + M [V ](z)∗ )−1 ,

z ∈ C+ .

(2.6)

Multiplying again this equation on the left by Γ0 we ﬁnd Γ0 G[κa , κb , V ](z)∗ = −(κ∗ + M [V ](z)∗ )−1 ,

z ∈ C+ .

∗

Since M [V ](z) = M [V ](z), z ∈ res(HD [V ]), we ﬁnally get Γ0 G[κa , κb , V ](z)∗ = −(κ∗ + M [V ](z))−1 ,

z ∈ C− .

Inserting this expression into (2.5) one obtains Θ[κa , κb , V ](z) = IC2 − iαΓ0 G[κa , κb , V ](z)∗ α,

z ∈ C− . 2

In [19] the operator-valued function T [κa , κb , V ](z) : H −→ C ,

αb ((H[κa , κb , V ] − z)−1 f )(b) , T [κa , κb , V ](z)f := −αa ((H[κa , κb , V ] − z)−1 )f (a)

f ∈ H,

was introduced for z ∈ res(H[κa , κb , V ]). We note that T [κa , κb , V ](z) = αΓ0 (H[κa , κb , V ] − z)−1 = αG[κa , κb , V ](z), ∗

2

z ∈ C+ .

2

Hence the adjoint operator T [κa , κb , V ](z) : C −→ L (Ω) exists and admits the representation T [κa , κb , V ](z)∗ = G[κa , κb , V ](z)∗ α,

z ∈ C+ .

Taking into account (2.6) we ﬁnd ran(T [κa , κb , V ](z)∗ ) ⊆ Nz [V ] ⊆ W 1,2 (Ω),

z ∈ C+ .

2

In [19] the operator α : L (Ω) −→ C,

αb f (b) α f = , −αa f (a)

¯ f ∈ dom( α) := C(Ω),

(2.7)

was introduced. Since α f = αΓ0 f,

f ∈ dom(S[V ]∗ ) ⊆ W 1,2 (Ω),

the characteristic function Θ[κa , κb , V ](·) admits the representation Θ[κa , κb , V ](z) = IC2 − i αT [κa , κb , V ](z)∗ ,

z ∈ C− ,

(2.8)

which coincides with the representation of the characteristic function of [19]. Using the representation (2.8) we prove the following lemma. Lemma 2.2. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then the characteristic function Θ[κa , κb , V ](·) is holomorphic in a neighborhood of R and obeys lim Θ[κa , κb , V ](λ) − IC2 B(C2 ) = 0.

λ→−∞

(2.9)

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Proof. For simplicity we set H[V ] := H[qa , qb , V ]. Obviously, we have H[V ] := H[0] + V,

V ∈ L∞ R (Ω).

We note that inf spec(H[V ]) =: ς[V ] is ﬁnite. Let us introduce the operator U [V ](λ) := α (H[V ] − λ)−1/2 ,

λ < ς[V ],

where α is deﬁned by (2.7). A straightforward computation shows that the representation

−1 i T [V ](λ) = U [V ](λ) I − U [V ](λ)∗ U [V ](λ) (H[V ] − λ)−1/2 2 is valid for λ < ς[V ]. Hence the characteristic function admits the representation

−1 i ∗ Θ[κa , κb , V ](λ) = I − iU [V ](λ) I + U [V ](λ) U [V ](λ) U [V ](λ)∗ 2 for λ < ς[V ]. Using the representation U [V ](λ) = U [V ](λ0 )D[V ](λ),

D[V ](λ) := (H[V ] − λ0 )1/2 (H[V ] − λ)−1/2 ,

λ0 , λ < ς[V ], we have Θ[κa , κb , V ](λ) = IC2

−1 i ∗ − iU [V ](λ0 )D[V ](λ) I + U [V ](λ) U [V ](λ) D[V ](λ)U [V ](λ0 )∗ 2

for λ0 , λ < ς[V ]. Since s−limλ→−∞ D[V ](λ) = 0 we obtain s−limλ→−∞ Θ[V ](λ) = IC2 which yields immediately the operator-norm convergence of (2.9).

3. Dilation and Lax-Phillips scattering Since H[κa , κb , V ] is a maximal dissipative operator there is a larger Hilbert space K ⊇ H and a self-adjoint operator K[κa , κb , V ] on K such that PHK (K[κa , κb , V ] − z)−1 H = (H[κa , κb , V ] − z)−1 ,

m(z) > 0,

(3.1)

see [14]. The operator K[κa , κb , V ] is called a self-adjoint dilation of the maximal dissipative operator H[κa , κb , V ]. Obviously, from the condition (3.1) one gets PHK (K[κa , κb , V ] − z)−1 H = (H[κa , κb , V ]∗ − z)−1 ,

m(z) < 0.

If the condition clospanz∈C\R (K[κa , κb , V ] − z)−1 H = K is satisﬁed, then K[κa , κb , V ] is called a minimal self-adjoint dilation of H[κa , κb , V ]. Minimal self-adjoint dilations of maximal dissipative operators are determined up to an isomorphism, in particular, all minimal self-adjoint dilations

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are unitarily equivalent. The self-adjoint operator K[κa , κb , V ] is absolutely continuous and its spectrum coincides with the real axis, i.e. spec(K) = R. The multiplicity of its spectrum is two. The dilation space K and the dilation K[κa , κb , V ] can be explicitly given by K := L2 (R− , C2 ) ⊕ L2 (Ω) ⊕ L2 (R+ , C2 ). and (K[κa , κb , V ]f)(x) = −i

d d f− (x− ) ⊕ (l[V ]f )(x) ⊕ −i f+ (x+ ), dx− dx+

(3.2)

x := (x− , x, x+ ), for f := f− ⊕ f ⊕ f+ ∈ dom(K[κa , κb , V ]) where

b

b f− (x− ) f+ (x+ ) := f− := f + a a f− f+ (x− ) (x+ ) and

dom(K[κa , κb , V ]) :=

 f ∈ W 1,2 (R− , C2 ) ⊕ W 1,2 (Ω) ⊕ W 1,2 (R+ , C2 ) :      1 1,2  (Ω)   mf ∈ W    1 b  2m(b) f (b) − κb f (b) = αb f− (0)

            

           

           

1 a 2m(a) f (a) + κa f (a) = αa f− (0) 1 b 2m(b) f (b) − κb f (b) = αb f+ (0) 1 a 2m(b) f (a) + κa f (b) = αa f+ (0)

.

(3.3) For more details the reader is referred to [19]. Obviously, the closed symmetric operator L[V ], (L[V ]f)(x)

:=

−i dxd− f− (x− ) ⊕ (S[V ]f )(x) ⊕ −i dxd+ f+ (x+ )

f ∈ dom(L[V ]) :=

W01,2 (R− , C2 ) ⊕ dom(S[V ]) ⊕ W01,2 (R+ , C2 )

is a symmetric restriction of K[κa , κb , V ], where W01,2 (R± , C2 ) := {f± ∈ W 1,2 (R, C2 ) : f± (0) = 0}. The deﬁciency indices of L[V ] are (4, 4). The domain of the adjoint operator L[V ]∗ is given by dom(L[V ]∗ ) := W 1,2 (R− , C2 ) ⊕ dom(S[V ]∗ ) ⊕ W 1,2 (R+ , C2 ). Another self-adjoint extension of L[V ] is deﬁned by KD [V ], (KD [V ]f)(x)

:=

f ∈ dom(KD [V ]) :=

−i dxd− f− (x− ) ⊕ (HD [V ]f )(x) ⊕ −i dxd+ f+ (x+ ), {f ∈ dom(L[V ]∗ ) : f− (0) = f+ (0)} .

If we introduce the diﬀerentiation operator K0 d (K0 f0 )(x) := −i dx f0 (x), x ∈ R, f0 ∈ dom(K0 ) := W 1,2 (R, C2 )

(3.4)

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and using the decomposition K = L2 (Ω) ⊕ K0 ,

K0 := L2 (R, C2 ),

(3.5)

then the operator KD [V ] admits the representation KD [V ] = HD [V ] ⊕ K0 .

(3.6)

The wave operators W± [κa , κb , V ], W± [κa , κb , V ] := s − lim eitK[κa ,κb ,V ] e−itKD [V ] P ac (KD [V ]) t→±∞

can be identiﬁed with the Lax-Phillips wave operators, cf. [10, 22], because the absolutely continuous subspace Kac (KD [V ]) of KD [V ] coincides with K0 . We note ac [V ] of KD [V ] coincides with K0 . The wave that the absolutely continuous part KD operators exist by the Lax-Phillips scattering theory and are complete, cf. [22]. However, in our special situation there is an additional reason for the existence and completeness of the wave operators. Since K[κa , κb , V ] and KD [V ] are self-adjoint extensions of one and the same closed symmetric operator L[V ] with deﬁciency indices (4, 4) its turns out that the resolvent diﬀerence of K[κa , κb , V ] and KD [V ] is a four dimensional operator. Hence the wave operator exist and are complete by the trace class existence theorem, cf. [10, 20]. The Lax-Phillips scattering operator SLP [κa , κb , V ] is deﬁned by SLP [κa , κb , V ] := W+ [κa , κb , V ]∗ W− [κa , κb , V ]. It acts only on the subspace K0 and is unitary there. Further, the Lax-Phillips scattering operator commutes with KD [V ], in particular, with 0 ⊕ K0 . The Fourier transform F : L2 (R, C2 ) −→ L2 (R, C2 ), 1 (F f0 )(λ) := √ dxe−iλx f0 (x), f0 ∈ L2 (R, C2 ), 2π R deﬁnes a unitary operator such that F K0 F ∗ coincides with the multiplication operator M , (M f)(λ) := λf(λ),

λ ∈ R,

f ∈ dom(M ) := {f ∈ L2 (R, C2 ) : λf(λ) ∈ L2 (R, C2 ). Since Lax-Phillips scattering operator SLP [κa , κb , V ] commutes with K0 the transformed operator F SLP [κa , κb , V ]F ∗ commutes with M . Hence there is a measurable family {SLP [κa , κb , V ](λ)}λ∈R of unitary operators on C2 such that the F SLP [κa , κb , V ]F ∗ coincides with the multiplication operator induced by {SLP [κa , κb , V ](λ)}λ∈R . The family {SLP [κa , κb , V ](λ)}λ∈R is called the LaxPhillips scattering matrix. One of the main results of the Lax-Phillips scattering theory is that SLP [κa , κb , V ](λ) = Θ[κa , κb , V ](λ)∗ holds for a.e. λ ∈ R, see also [17].

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4. Phase shift The phase shift ω[κa , κb , V ](·) : R −→ R is deﬁned by e−2πiω[κa ,κb ,V ](λ) := det(SLP [κa , κb , V ](λ)),

λ ∈ R,

(4.1)

which is equivalent to e2πiω[κa ,κb ,V ](λ) = det(Θ[κa , κb , V ](λ)),

λ∈R

Notice that the phase shift is determined modulo Z. To eliminate this nonuniqueness of the deﬁnition we demand in the following that ω[κa , κb , V ](λ) is continuous in λ ∈ R and obeys lim ω[κa , κb , V ](λ) = 0

(4.2)

λ→−∞

which is in accordance with Lemma 2.2. Lemma 4.1. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then the phase shift ω[κa , κb , V ](·) is holomorphic in a neighborhood of R and satisfies ω [κa , κb , V ](λ) := =−

d ω[κa , κb , V ](λ) dλ

1 tr(T [κa , κb , V ](λ)T [κa , κb , V ](λ)∗ ) ≤ 0 2π

for λ ∈ R. Proof. For brevity we set H := H[κa , κb , V ], T (λ) := T [κa , κb , V ](λ), T∗ (λ) := (H[κa , κb , V ]∗ − λ)−1 and Θ(λ) := Θ[κa , κb , V ](λ) as well as T∗ [κa , κb , V ](λ) := α ω(λ) := ω[κa , κb , V ](λ). Since the characteristic function Θ(λ) is holomorphic in a neighborhood of R one gets that the phase shift ω(λ) is also holomorphic there. By T (λ)T (λ)∗ = α (H − λ)−1 − (H ∗ − λ)−1 T (λ)∗ + T∗ (λ)T (λ)∗ , λ ∈ R, and Lemma 3.1 of [19] we ﬁnd T (λ)T (λ)∗ = iαT∗ (λ)∗ T∗ (λ)T (λ)∗ + T∗ (λ)T (λ)∗ ,

λ ∈ R,

or T (λ)T (λ)∗ = {I + iαT∗ (λ)∗ }T∗ (λ)T (λ)∗ ,

λ ∈ R.

Using Formula (3.39) of [19] we obtain T (λ)T (λ)∗ = Θ(λ)∗ T∗ (λ)T (λ)∗ ,

λ ∈ R.

Using (2.8), a straightforward computation shows ∂ Θ(λ) = −iT∗ (λ)T (λ)∗ , ∂λ

λ ∈ R,

which gives T (λ)T (λ)∗ = iΘ(λ)∗

∂ Θ(λ), ∂λ

λ ∈ R.

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Taking into account formula (IV.1.14) of [15] we obtain d d ∂ ln (det(Θ(λ))) = −2π ω(λ) 0 ≤ tr(T (λ)T (λ)∗ ) = i tr(Θ(λ)∗ Θ(λ)) = i ∂λ dλ dλ for λ ∈ R. Lemma 4.1 shows that the phase shift is a non-increasing function. Since limλ→−∞ ω[κa , κb , V ](λ) = 0 the phase function is non-positive. In order to estimate the growth of −ω[κa , κb , V ](·) let us investigate the phase distribution function Φ[κa , κb , V ](λ) := card{s < λ : det(Θ[κa , κb , V ](s)) = 1},

λ ∈ R.

To estimate Φ[κa , κb , V ](λ) we consider the eigenvalue problem µ ∈ T,

Θ[κa , κb , V ](λ)x = µx,

x ∈ C2 ,

for each ﬁxed λ ∈ R. To treat this problem we introduce the family {Hθ [V ]}θ∈(0,2π) , Hθ [V ] := H[qa (θ), qb (θ), V ] and H0 [V ] := HD [V ] where the boundary coeﬃcients are given by α2b cot(θ/2) α2 cot(θ/2) and qa (θ) := qa − a . 2 2 The spectrum spec(Hθ [V ]) consists of simple eigenvalues spec(Hθ [V ]) {λk [V ](θ)}k∈N , −∞ < λ1 [V ](θ) < λ2 [V ](θ) < . . . . qb (θ) := qb −

=

Lemma 4.2. If V ∈ L∞ R (Ω), then Hθ [V ] ≥ Hθ [V ] for 0 ≤ θ ≤ θ < 2π.

Proof. The sesquilinear form tθ [V ] corresponding to Hθ [V ] is given by dom(tθ [V ]) = W 1,2 (Ω), tθ [V ](f, g)

= −qa (θ)f (a)g(a) − qb (θ)f (b)g(b) +

(4.3) b

dx a

1 f (x)g (x) + V (x)f (x)g(x), 2m(x)

f, g ∈ dom(tθ [V ]) = W 1,2 (Ω), θ ∈ (0, 2π). Since qa (θ ) ≤ qa (θ) and qb (θ ) ≤ qb (θ) for θ < θ we easily obtain tθ [V ] ≤ tθ [V ]. If θ = 0, then dom(t0 [V ]) = W01,2 (Ω) ⊆ W 1,2 (Ω) = dom(tθ [V ]) and tθ [V ](f, f ) ≤ t0 [V ](f, f ),

f ∈ dom(t0 [V ]),

θ ∈ (0, 2π)

which completes the proof. The min-max principle gives the following

Corollary 4.3. If V ∈ L∞ R (Ω), then the eigenvalue curves λn [V ](·) of Hθ [V ] satisfy λn [V ](θ ) ≤ λn [V ](θ),

0 ≤ θ ≤ θ < 2π,

n ∈ N.

Let us show that in fact the monotonicity of the eigenvalue curves is strict: Lemma 4.4. If V ∈ L∞ R (Ω), then λn [V ](θ ) < λn [V ](θ),

0 ≤ θ < θ < 2π,

n ∈ N.

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Proof. We note that {Hθ := Hθ [V ]}θ∈(0,2π) is not only a monotone family but also an analytic one of self-adjoint operators of type (B), cf. [20, Section VII.4.2]. This yields that the eigenvalues of λn (θ) := λn [V ](θ) depend analytically on θ ∈ (0, 2π). Assume now that there is a k ∈ N such that λk (θ ) = λk (θ ) for some 0 < θ < θ < 2π. In this case we get λk (θ ) = λk (θ) = λk (θ ) for θ ∈ [θ , θ ]. Since λk (θ) is analytic we ﬁnd λk (θ) = λk (0), θ ∈ (0, 2π), that is, λk (θ) is constant and equals the Dirichlet eigenvalue λk (0). Next we show that if for some k ∈ N we have λk (θ) = λk (0), θ ∈ (0, 2π), then for each j ∈ 1, 2, . . . , k one has λj (θ) = λj (0), θ ∈ (0, 2π). Indeed, let us assume that there is a θ ∈ (0, 2π) such that λk−1 (θ) < λk−1 (0). In this case there is a neighborhood U := (λk−1 (θ), λk (0)) of λk−1 (0) which contains no eigenvalue of Hθ for θ ∈ (θ, 2π). However, this is impossible by Lemma 2.1. In fact, if θ is suﬃciently close to 2π, then the neighborhood U has to contain an eigenvalue of Hθ . Hence the assumption λk−1 (θ) < λk−1 (0) was false which yields λk−1 (θ) = λk−1 (0) for θ ∈ (0, 2π). By induction we get that λj (θ) = λj (0), θ ∈ (0, 2π), holds for each j = 1, 2, . . . , k. In particular, this holds for the lowest eigenvalue λ1 (θ) = λ1 (0), θ ∈ (0, 2π), which is given by λ1 (θ) := inf{tθ [V ](f, f ) : f ∈ W 1,2 (Ω),

f L2 (Ω) = 1},

θ ∈ (0, 2π).

But (4.3) implies limθ↑2π λ1 (θ) = −∞ which contradicts the conclusion that λ1 (θ) remains unchanged for θ ∈ (0, 2π). Our next aim is to determine limθ↓0 λn [V ](θ) and limθ↑2π λk [V ](θ). Lemma 4.5. If V ∈ L∞ R (Ω), then the eigenvalue curves satisfy lim λn [V ](θ) = λn [V ](0), θ↓0

n ∈ N,

(4.4)

and lim λn [V ](θ) = λn−2 [V ](0),

θ↑2π

n ∈ N,

(4.5)

where λ−1 [V ](0) := λ0 [V ](0) := −∞. Proof. The family {Hθ [V ]}θ∈(0,π) is operator norm continuous in the resolvent sense. In particular, this yields that the eigenvalues λk [V ](θ), k ∈ N, are continuous in θ ∈ (0, 2π). Moreover, since limθ↓0 qa (θ) = limθ↓0 qb (θ) = ∞ and limθ↑2π qa (θ) = limθ↑2π qb (θ) = ∞ we get by Lemma 2.1 lim (Hθ [V ] − i)−1 − (HD [V ] − i)−1 B(H) θ↓0

= lim (Hθ [V ] − i)−1 − (HD [V ] − i)−1 B(H) = 0. θ↑2π

An application of Lemma 4.2 implies (4.4). It remains to show (4.5). First, by monotonicity the limits limθ↑2π λk [V ](θ), k ∈ N, exist, too. We introduce the intervals ∆1 := (−∞, λ1 [V ](0)) and ∆n := (λn−1 [V ](0), λn [V ](0)),

n = 2, 3, . . . ,

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that is, the sequence of spectral gaps of the Dirichlet operator HD [V ]. Further, we ] deﬁned by consider the symmetric operator S[V ]g S[V

:=

]) := dom(S[V

]), l[V ]g, g ∈ dom(S[V 1 g ∈ W 1,2 (Ω), g(a) = 0, 1,2 m g ∈ W (Ω) : . 1 2m(b) g (b) = g(b) = 0

] has the deﬁciency indices (1, 1). Obviously we The closed symmetric operator S[V θ [V ], θ ∈ (0, 2π), have S[V ] ≤ S[V ] ≤ HD [V ] where S[V ] is deﬁned by (2.1). By H we denote the self-adjoint operator θ [V ]g H θ [V ]) dom(H

:= l[V ]g, :=

θ [V ]), g ∈ dom(H

g ∈ W 1,2 (Ω) :

1 1,2 , g(a) m(x) g (x) ∈ W 1 2m(b) g (b) = qb (θ)g(b)

= 0,

,

0 [V ] := HD [V ]. Moreover, similar to Lemma 4.2 the family and we set H θ [V ]}θ∈(0,2π) is non-increasing, i.e. {H θ [V ] ≤ H θ [V ], H

0 ≤ θ ≤ θ < 2π,

k [V ](θ)}k∈N and analytic in sense of type B, cf. [20, Sect. VII.4.2]. Denoting by {λ the eigenvalues of Hθ [V ] we get similarly to Lemma 4.4 that k [V ](θ), k [V ](θ ) < λ λ

k ∈ N,

0 ≤ θ < θ < 2π.

(4.6)

] the open intervals ∆k are gaps for Since HD [V ] is a self-adjoint extension of S[V θ [V ] of S[V ]. Since S[V ] has deﬁciency indices (1, 1) the self-adjoint extension H ] has at most one eigenvalue in each gap ∆k . Taking into account (4.6) we ﬁnd S[V k [V ](θ) ∈ ∆k , λ

k ∈ N,

θ ∈ (0, 2π).

We set 1 [V ](θ)), 1 (θ) := (−∞, λ ∆

k−1 [V ](θ), λ k [V ](θ)), k (θ) := (λ ∆

k = 2, 3, . . . ,

θ ∈ (0, 2π). Obviously we have k (θ) ⊆ ∆k−1 ∪ {λk−1 [V ](0)} ∪ ∆k ∆

θ ∈ (0, 2π),

k ∈ N.

(4.7)

] deﬁned by Further, let us introduce the symmetric operator S[V   1 1,2 (Ω),   mg ∈ W 1 ]g := l[V ]g, dom(S[V ]) := g ∈ W 1,2 (Ω) : 2m(a) g (a) = g(a) = 0, S[V ,   1 g (b) = q (θ)g(b) b 2m(b) θ [V ], θ ∈ which has the deﬁciency indices (1, 1), too. Obviously, the operator H k (θ) [0, 2π), is a self-adjoint extension of S[V ]. Therefore, the open intervals ∆

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]. Moreover, the operator are spectral gaps of the closed symmetric operator S[V Hθ [V ], θ ∈ [0, 2π), is a self-adjoint extension of S[V ], too. As above we get k (θ), λk [V ](θ) ∈ ∆

k ∈ N,

θ ∈ (0, 2π).

Taking into account (4.7) we obtain λk [V ](θ) ∈ ∆k−1 ∪ {λk−1 [V ](0)} ∪ ∆k . Hence we have either lim λk [V ](θ) = λk−1 [V ](0) or

θ↑2π

lim λk [V ](θ) = λk−2 [V ](0)

θ↑2π

for k = 2, 3, . . . . Let us assume that for some j ≥ 2 we have lim λj [V ](θ) = λj−1 [V ](0).

θ↑2π

In this case, we ﬁnd that limθ↑2π λj−1 [V ](θ) = λj−3 [V ](0) is impossible. Indeed, if θ is suﬃciently close to 2π, then there is neighborhood of λj−2 [V ](0) which does not contain an eigenvalue of Hθ [V ]. However, this contradicts Lemma 2.1. Therefore, we obtain that limθ↑2π λk [V ](θ) = λk−1 [V ](0), k = 2, 3, . . . , j − 1. Furthermore, one gets that limθ↑2π λj+1 [V ](θ) = λj−1 [V ](0) is also impossible. In fact, for each suﬃciently small neighborhood of λj−1 [V ](0) there is a suﬃciently large θ ∈ (0, 2π) such that this neighborhood contains two eigenvalues of Hθ [V ] which contradicts again Lemma 2.1. Hence limθ↑2π λk [V ](θ) = λk−1 [V ](0), k = j + 1, j + 2, . . . . Therefore, we ﬁnd limθ↑2π λk [V ](θ) = λk−1 [V ](0) for k ∈ N. In particular, we have that the interval ∆1 contains only one eigenvalue of Hθ [V ] for each θ ∈ (0, 2π). However, this is impossible, too. To show this we introduce the self-adjoint operator hθ , θ ∈ (0, 2π), (hθ g)(x)

:=

dom(hθ ) :=

2

d −τ dx g ∈ dom(hθ ), 2 g(x) + V L∞ g(x), τ f (a) = −qa (θ)f (a) 2,2 f ∈ W (Ω) : τ f (b) = qb (θ)f (b)

and τ := 1/2mL∞ . Obviously, we have Hθ [V ] ≤ hθ , θ ∈ (0, 2π), which yields λk [V ](θ) ≤ µk (θ), k ∈ N, for θ ∈ (0, 2π), where {µk (θ)}k∈N are the eigenvalues of hθ . An involved but straightforward computation shows that the ﬁrst two eigenvalues µ1 (θ) and µ2 (θ) of hθ tend to −∞ as θ ↑ 2π. Hence the ﬁrst two eigenvalues λ1 [V ](θ) and λ2 [V ](θ) tend also to −∞ as θ ↑ 2π which shows that for suﬃciently large θ ∈ (0, 2π) one has λ1 [V ](θ) ∈ ∆1 and λ2 [V ](θ) ∈ ∆1 . Next we show that the eigenvalues of the characteristic function Θ[κa , κb , V ](λ) are intrinsically connected with the eigenvalues of the family {Hθ [V ]}θ∈[0,2π). Lemma 4.6. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then µ = eiθ ∈ spec(Θ[κa , κb , V ](λ)) ⇐⇒ λ ∈ spec(Hθ [V ]),

θ ∈ [0, 2π),

λ ∈ R.

Proof. Multiplying the relation (2.8) on the left by T [κa , κb , V ](λ)∗ we ﬁnd T [κa , κb , V ](λ)∗ ξ − iT [κa , κb , V ](λ)∗ αT [κa , κb , V ](λ)∗ ξ = µT [κa , κb , V ](λ)∗ ξ.

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Setting g := T [κa , κb , V ](λ)∗ ξ ∈ W 1,2 (Ω) we obtain g − iT [κa , κb , V ](λ)∗ αg = µg

or T [κa , κb , V ](λ)∗ αg = i(µ − 1)g.

Let h ∈ L2 (Ω). Then αg, T [κa , κb , V ](λ)h = i(µ − 1)(g, h) where we recall that ·, · denotes the scalar product of C2 . Setting f := (H[κa , κb , V ] − λ)−1 h ∈ dom(H[κa , κb , V ]) we get αg, αf = i(µ − 1)(g, (H[κa , κb , V ] − λ)f ). One has

(g, (H[κa , κb , V ] − λ)f ) =

b

(4.8)

dx g(x)((l[V ]f )(x) − λf (x)).

a

Since (l[V ] − λ)g = 0 we ﬁnd (g, (H[κa , κb , V ] − λ)f ) 1 1 1 1 g (b)f (b) − g (a)f (a). f (b) + g(a) f (a) + = −g(b) 2m(b) 2m(a) 2m(b) 2m(a) Since f ∈ dom(H[κa , κb , V ])) we get that (g, (H[κa , κb , V ] − λ)f ) = −g(b)κb f (b) − g(a)κa f (a) +

1 1 g (b)f (b) − g (a)f (a) 2m(b) 2m(a)

which yields (g, (H[κa , κb , V ] − λ)f ) 1 1 = g (b) − κb g(b) f (b) + − g (a) − κa g(a) f (a). 2m(b) 2m(a) Taking into account (4.8) one gets that the element g has to satisfy the boundary conditions 1 α2b g(b) = i(µ − 1) 2m(b) g (b) − κb g(b) , 1 α2a g(a) = i(µ − 1) − 2m(a) g (a) − κa g(a) which implies 1 g (b) = qb (θ)g(b), 2m(b)

and

1 g (a) = −qa (θ)g(a), 2m(a)

θ ∈ (0, 2π),

for µ = 1. If µ = 1, then g(a) = g(b) = 0. Hence, g ∈ dom(HD [V ]) and λ ∈ spec(HD [V ]) = spec(H0 [V ]), i.e θ = 0. Conversely, if λ ∈ spec(Hθ [V ]), θ ∈ [0, 2π), then the eigenfunction g, Hθ [V ]g = λg, satisﬁes the equation T [V ]∗ (λ)αg = i(µ − 1)g or

(I − iT [V ]∗ (λ)α)g = µg.

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Multiplying on the left by α we obtain (I − iαT [V ]∗ (λ))αg = µαg.

Setting ξ := αg and using (2.8) we complete the proof.

Lemma 4.7. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then we have spec(Hθ [V ]) ∩ spec(H2π−θ [V ]). (4.9) {λ ∈ R : det(Θ[κa , κb , V ](λ)) = 1} = θ∈(0,π)

Proof. At ﬁrst we note that det(Θ[κa , κb , V ](λ)) = 1 if and only if µ = eiθ ∈ spec(Θ[κa , κb , V ](λ)) and µ = ei(2π−θ) ∈ spec(Θ[κa , κb , V ](λ)), θ ∈ [0, 2π). It remains to show that the cases θ = 0 and θ = π are impossible: indeed, if θ = 0, then µ = 1. In this case the eigenvalue µ = 1 of Θ[κa , κb , V ](λ) has the multiplicity two. Hence, there are two mutually orthogonal eigenvectors ξ1 , ξ2 ∈ C2 such that that Θ[κa , κb , V ](λ)ξi = ξi , i = 1, 2. We set gi := T [κa , κb , V ](λ)∗ ξi ∈ W 1,2 (Ω),

i = 1, 2.

Both functions gi are eigenfunctions of HD [V ] with the eigenvalue λ. Since the spectrum of HD [V ] is simple there are constants Ci ∈ C such that C1 g1 +C2 g2 = 0. Hence T [κa , κb , V ](λ)∗ {C1 ξ1 + C2 ξ2 } = 0. For each h ∈ L2 (Ω) we have (C1 ξ1 + C2 ξ2 , T [κa , κb , V ](λ)h) = 0. Since ran(T [κa , κb , V ](λ)) = C2 we ﬁnd C1 ξ1 + C2 ξ2 = 0 which is impossible. The same holds for θ = π which yields µ = −1. By Lemma 4.6 we have µ = eiθ ∈ spec(Θ[κa , κb , V ](λ)) if and only if λ ∈ spec(Hθ [V ]) and µ = ei(2π−θ) ∈ spec(Θ[κa , κb , V ](λ)) if and only if λ ∈ spec(H2π−θ [V ]). Hence µ = eiθ , µ = ei(2π−θ) ∈ spec(Θ[κa , κb , V ](λ)) ⇐⇒ λ ∈ spec(Hθ [V ]) ∩ spec(H2π−θ [V ])

which proves (4.9). Let us introduce the spectral distribution function ND [V ](λ) := card{s < λ : s ∈ spec(HD [V ])},

λ ∈ R.

Theorem 4.8. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then phase and spectral distribution functions are related by ND [V ](λ) ≤ Φ[κa , κb , V ](λ) ≤ ND [V ](λ) + 1, Proof. Let us consider the sets spec(Hθ [V ]) ∩ spec(H2π−θ [V ]), Λn := ∆n ∩ θ∈(0,π)

λ ∈ R.

n ∈ N.

(4.10)

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By Lemma 4.7 one has {λ ∈ R : det(Θ[κa , κb , V ](λ)) = 1} =

Λn .

n∈N

By Proposition 4.5 only the eigenvalues λn [V ](θ), λn+1 [V ](θ), θ ∈ (0, 2π), belong to the interval ∆n , other eigenvalues cannot. Further, by Proposition 4.5 we have lim λn [V ](θ) = λn [V ](0) and θ↓0

lim λn+1 [V ](2π − θ) = λn−1 [V ](0), θ↓0

n ∈ N.

Since λn [V ](θ) is decreasing and λn+1 [V ](2π − θ) is increasing in θ ∈ (0, 2π), there is at most one θ ∈ (0, π) such that λn+1 [V ](2π − θ) = λn [V ](θ) which yields card{Λn } ≤ 1. Moreover, we have λn−1 [V ](0) < λn+1 [V ](θ) < λn+1 [V ](π),

θ ∈ (π, 2π),

and λn [V ](π) < λn [V ](θ) < λn [V ](0),

θ ∈ (0, π),

as well as λn [V ](π) < λn+1 [V ](π). Hence there is at least one θ ∈ (0, π) such that λn+1 [V ](2π − θ) = λn [V ](θ) which shows card{Λn } ≥ 1. Therefore card{Λn } = 1 which implies immediately (4.10). Corollary 4.9. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then 0 ≤ −ω[κa , κb , V ](λ) ≤ 2 + where (λ + V− L∞ )+ :=

1 2

1 2mL∞ |Ω| (λ + V− L∞ )+ , π

λ ∈ R,

(4.11)

(λ + V− L∞ + |λ + V− L∞ |) ≥ 0.

Proof. Obviously, we have −ω[κa , κb , V ](λ) ≤ 1 + Φ[κa , κb , V ](λ),

λ ∈ R.

Using Theorem 4.8 we ﬁnd −ω[κa , κb , V ](λ) ≤ 2 + ND [V ](λ),

λ ∈ R.

Further, we note that hD ≤ HD [V ], (hD g)(x) g

1 := − 2m L∞

∈

d2 dx2 g(x)

− V− L∞ g(x),

dom(hD ) := {f ∈ W 2,2 (Ω) : f (a) = f (b) = 0}.

The spectral distribution function nD (·) of hD can be estimated by nD (λ) ≤

1 2mL∞ |Ω| (λ + V− L∞ )+ , π

Since ND [V ](λ) ≤ nD (λ), λ ∈ R, one gets (4.11).

λ ∈ R.

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5. Spectral shift and trace formula Since H[κa , κb , V ] and HD [V ] are extensions of one and the same closed symmetric operator S[V ] with deﬁciency indices (2, 2) the resolvent diﬀerence obeys (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 ∈ L1 (H),

z ∈ C+ .

In fact, the diﬀerence is a two dimensional operator. Theorem 5.1. If V ∈ L∞ (Ω) and κa , κb ∈ C+ , then there is a real function ξ[κa , κb , V ](·) ∈ L1 (R, (1 + λ2 )−1 dλ) such that the trace formula tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 (5.1) = − (λ − z)−2 ξ[κa , κb , V ](λ)dλ R

holds for z ∈ C+ . Proof. Using formulas (3.13) of [19] one veriﬁes −iT [κa, κb , V ](i)∗ T [κa , κb , V ](i) =

(H[κa , κb , V ]∗ + i)−1 − (H[κa , κb , V ] − i)−1 +2i(H[κa, κb , V ]∗ + i)−1 (H[κa , κb , V ] − i)−1

which shows that Condition (4.2) of Theorem 4.1 of [1] is satisﬁed. Since HD [V ] is self-adjoint Condition (4.3) of [1] also holds. Applying Theorem 4.1 of [1] we complete the proof. A real function ξ[κa , κb , V ](λ) ∈ L1 (R, (1 + λ2 )dλ) is called a spectral shift of the pair {H[κa , κb , V ], HD [V ]} if the trace formula (5.1) is satisﬁed. Because K[κa , κb , V ] and KD [V ] are self-adjoint extensions of the same closed symmetric operator L[V ] with deﬁciency indices (4, 4) one has (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 ∈ L1 (H) for z ∈ C \ R. Using again Theorem 4.1 of [1] we ﬁnd that the pair {K[κa , κb , V ], KD [V ]} admits a spectral shift η[κa , κb , V ](·) ∈ L1 (R, (1+λ2 )−1 dλ), too. The trace formula then takes the form tr (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 = − (λ − z)−2 η[κa , κb , V ](λ)dλ, z ∈ C \ R. R

Let us clarify the relation between ξ[κa , κb , V ](·) and η[κa , κb , V ](·). Lemma 5.2. Assume V ∈ L∞ R (Ω) and κa , κb ∈ C+ . Then tr (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 = tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1

(5.2)

for z ∈ C+ . Consequently, any spectral shift ξ[κa , κb , V ](·) ∈ L1 (R, (1+λ2 )−1 dλ) of the pair {H[κa , κb , V ], HD [V ]} is a spectral shift of the pair {K[κa , κb , V ], KD [V ]} and vice versa.

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Proof. Using the terminology of Ch. 3 and taking into account (3.5) and (3.6) we ﬁnd that ((KD [V ] − z)−1 f)(x) x− = i dy ei(x− −y)z f− (y) ⊕ (HD [V ] − z)−1 f (x) −∞

⊕ i

0

x+

dy ei(x+ −y)z f+ (y) + i

0

−∞

(5.3)

dy ei(x+ −y)z f− (y),

f = f− ⊕ f ⊕ f+ and z ∈ C+ . From Theorem 4.2 of [19] one gets the representation ((K[κa , κb , V ] − z)−1 f)(x) x− dy ei(x− −y)z f− (y) = i

(5.4)

−∞

0 dy e−iyz f− (y) ⊕ (H[κa , κb , V ] − z)−1 f (x) + iT∗ [κa , κb , V ](z)∗ −∞ x+ i(x+ −y)z izx+ dy e T [κa , κb , V ](z)f ⊕ i f+ (y) + ie 0

+ iΘ[κa , κb , V ](z)∗

0

−∞

dy ei(x+ −y)z f− (y),

f = f− ⊕ f ⊕ f+ and z ∈ C+ . Denoting by P± the orthogonal projections form K onto the subspaces L2 (R± , C2 ) one easily obtains from (5.3) and (5.4) that (5.5) P± (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 P± = 0 for z ∈ C+ . Using the representation tr (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 = tr P− (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 P− + tr PHK (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 PHK + tr P+ (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 P+ and taking into account (5.5) we get tr (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 = tr PHK (K[κa , κb , V ] − z)−1 − (KD [V ] − z)−1 PHK for z ∈ C+ . Using that K[κa , κb , V ] is a self-adjoint dilation of the maximal dissipative operator H[κa , κb , V ] we have thus proved (5.2). The second assertion follows directly from the ﬁrst. Lemma 5.3. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then 1 d (EK[κa ,κb ,V ] (λ)PHK f, PHKg )K = T [κa , κb , V ](λ)f, T [κa , κb , V ](λ)g C2 (5.6) dλ 2π

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for a.e. λ ∈ R and f, g ∈ K where EK[κa ,κb ,V ] (·) denotes the spectral measure of the self-adjoint dilation K[κa , κb , V ]. Proof. We note that d (λ)PHK f, PHKg )K (E dλ K[κa ,κb ,V ] 1 lim ((K[κa , κb , V ] − λ − i)−1 )PHK f, PHKg)K = 2πi ↓0 − ((K[κa , κb , V ] − λ + i)−1 )P K f, P Kg)K H

H

for a.e. λ ∈ R. Since K[κa , κb , V ] is a dilation of H[κa , κb , V ] we ﬁnd d (EK[κa ,κb ,V ] (λ)PHK f, PHKg )K dλ 1 lim ((H[κa , κb , V ] − λ − i)−1 )f, g)H − ((H[κa , κb , V ]∗ − λ + i)−1 )f, g)H = 2πi ↓0 which yields d (EK[κa ,κb ,V ] (λ)PHK f, PHKg )K (5.7) dλ 1 ((H[κa , κb , V ] − λ)−1 )f, g)H − ((H[κa , κb , V ]∗ − λ)−1 )f, g)H = 2πi where we have used that the spectrum of H[κa , κb , V ] is non-real. Finally, Lemma 3.1 of [19] states the coincidence of the right hand sides of (5.7) and (5.6), what completes the proof. Theorem 5.4. If V ∈ L∞ R (Ω) and κa , κb ∈ C+ , then ξ0 [κa , κb , V ](λ) := ω[κa , κb , V ](λ) + ND [V ](λ),

λ ∈ R,

(5.8)

defines a spectral shift of the pair {H[κa , κb , V ], HD [V ]} and, hence, of the pair {K[κa , κb , V ], KD [V ]}. Proof. Using that K[κa , κb , V ] is a dilation of H[κa , κb , V ] we get −1 ((H[κa , κb , V ] − z) f, f ) = (λ − z)−1 d(EK[κa ,κb ,V ] (λ)f, f ), R

f ∈ H, for z ∈ C+ . Since K[κa , κb , V ] is absolutely continuous we obtain d ((H[κa , κb , V ] − z)−1 f, f ) = (λ − z)−1 (EK[κa ,κb ,V ] (λ)f, f ) dλ, dλ R f ∈ H, for z ∈ C+ . Using Lemma 5.3 we ﬁnd 1 ((H[κa , κb , V ] − z)−1 f, f ) = (λ − z)−1 (T [κa , κb , V ](λ)f, T [κa , κb , V ](λ)f ) dλ, 2π R (5.9) f ∈ H, for z ∈ C+ . Further, we have −1 (5.10) ((HD [V ] − z) f, f ) = (λ − z)−1 d(EHD [V ] (λ)f, f ), R

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f ∈ H, for z ∈ C+ . We note that tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 fn , fn =

427

(5.11)

n∈N

where{fn }n∈N is an orthonormal basis of H. Inserting (5.9) and (5.10) into (5.11) we get tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 1 = (λ − z)−1 (T [κa , κb , V ](λ)fn , T [κa , κb , V ](λ)fn ) dλ 2π R n∈N (λ − z)−1 d(EHD [V ] (λ)fn , fn ) − R

which leads to the relation tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 1 = (λ − z)−1 tr(T [κa , κb , V ](λ)∗ T [κa , κb , V ](λ)) dλ 2π R − (λ − z)−1 d tr(EHD [V ] (λ)). R

Since ND [V ](λ) = tr(EHD [V ] (λ)), one has

R

(λ − z)−1 d tr(EHD [V ] (λ)) =

R

λ ∈ R,

(λ − z)−1 d ND [V ](λ).

Integrating by parts and using that ND (λ) behaves like the square root of λ at +∞ we get (λ − z)−1 d tr(EHD [V ] (λ)) = (λ − z)−2 ND [V ](λ) dλ. R

R

Similarly, by Lemma 4.1 we get 1 (λ − z)−1 tr(T [κa , κb , V ](λ)∗ T [κa , κb , V ](λ)) dλ 2π R 1 = (λ − z)−1 tr(T [κa , κb , V ](λ)T [κa , κb , V ](λ)∗ ) dλ 2π R = − (λ − z)−1 ω [κa , κb , V ](λ) dλ. R

Again, integrating by parts and taking into account Theorem 4.8 we obtain 1 (λ − z)−1 tr(T [κa , κb , V ](λ)∗ T [κa , κb , V ](λ)) dλ = 2π R = − (λ − z)−2 ω[κa , κb , V ](λ) dλ. R

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Summing up we ﬁnd tr (H[κa , κb , V ] − z)−1 − (HD [V ] − z)−1 = − (λ − z)−2 {ω[κa , κb , V ](λ) + ND [V ](λ)} dλ R

for z ∈ C+ which proves (5.8).

Corollary 5.5. If V ∈ L∞ R (Ω) and κa , κb ∈ C, then the spectral shift ξ0 [κa , κb , V ](λ) of the pair {H[κa , κb , V ], HD [V ]} obeys lim ξ0 [κa , κb , V ](λ) = 0

(5.12)

λ→−∞

and −2 ≤ ξ0 [κa , κb , V ](λ) ≤ 0,

λ ∈ R.

(5.13)

Proof. The relation (5.12) follows from (4.2). To verify (5.13) we note that by deﬁnition one has Φ[κa , κb , V ](λ) ≤ −ω[κa , κb , V ](λ) ≤ Φ[κa , κb , V ](λ) + 1,

λ ∈ R.

Taking into account Theorem 5.4 we ﬁnd Φ[κa , κb , V ](λ) − ND [V ](λ) ≤ −ξ0 [κa , κb , V ](λ) ≤ Φ[κa , κb , V ](λ) + 1 − ND [V ](λ), λ ∈ R. Finally, using Theorem 4.8 we have 0 ≤ −ξ0 [κa , κb , V ](λ) ≤ 2,

λ ∈ R,

which yields (5.13).

Remark 5.6. We note that a weaker version of Corollary 5.5 can be obtained using abstract results on the spectral shift. Indeed, let us introduce the Cayley transforms U := (i − K[κa , κb , V ])(i + K[κa , κb , V ])−1 and UD := (i − KD [κa , κb , V ])(i + KD [κa , κb , V ])−1 where K[κa , κb , V ] and KD [κa , κb , V ] are given by (3.2)-(3.3) and (3.4). We note that U − UD is a four dimensional operator. This follows from the fact K[κa , κb , V ] and KD [V ] are self-adjoint extension of the symmetric operator L[V ] which has deﬁciency indices (4, 4). Since ξ0 [κa , κb , V ](λ) obeys the trace formula (5.1) one gets by a straightforward computation that η0 (t) := ξ0 [κa , κb , V ](tan(t/2)), obeys the trace formula tr((U − ζ)−1 − (UD − ζ)−1 ) = −i

π

−π

t = (−π, π),

η0 (t) eit dt, (eit − ζ)2

|ζ| = 1,

for the pair {U, UD }. The function η0 (·) is called a spectral shift of the pair {U, UD }. Any function η(t) := η0 (t) + c, t ∈ (−π, π], c ∈ R, is, of course, a spectral shift of the pair {U, UD }, too. Conversely, any spectral shift of the pair {U, UD } diﬀers

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from η0 (·) by a real constant. Among all spectral shifts there is a special normalized one ηn (·) obeying π

i −π

−1 ηn (t)dt = tr(ln0 (UD U ))

where ln0 (·) is a suitably chosen branch of ln(·), see [21, 29]. Please, notice that there is a real constant cn such that ηn (t) = η0 (t) + cn ,

t ∈ (−π, π].

Since U − UD is a four-dimensional operator one gets from [21] that |ηn (t)| ≤ 4, t ∈ (−π, π]. By limt→−π η0 (t) = 0 we obtain that |cn | ≤ 4. Hence, we ﬁnd |η0 (t)| ≤ 8, t ∈ (−π, π], which yields |ξ0 [κa , κb , V ](λ)| ≤ 8,

λ ∈ R.

(5.14)

We note that (5.14) is weaker than (5.13), however, the proof relies only on abstract results on the spectral shift. Remark 5.7. The result (5.14) immediately implies that ND [V ](λ) − 8 ≤ ω[κa , κb , V ](λ) ≤ ND [V ](λ) + 8,

λ ∈ R.

Remark 5.8. From (4.1) and (5.8) we get det(SLP [κa , κb , V ](λ)) = e−2πiξ0 [κa ,κb ,V ](λ)

(5.15)

for a.e. λ ∈ R. However, formula (5.15) is the well-known Birman-Krein formula for the pair {K[κa , κb , V ], KD [V ]} which relates the spectral shift to the scattering matrix, cf. [11, 29].

References [1] V. M. Adamjan and H. Neidhardt, On the summability of the spectral shift function for pair of contractions and dissipative operators, J. Oper. Theory 24 (1990), no. 1, 187–205. [2] B. P. Allakhverdiev, On dissipative extensions of the symmetric Schr¨ odinger operator in Weyl’s limit-circle case, Dokl. Akad. Nauk SSSR 293 (1987), 777-781. [3] B. P. Allakhverdiev, Schr¨ odinger type dissipative operator with a matrix potential, in ´ Spectral theory of operators and its applications No.9, 11-41, “Elm”, Baku, 1989. [4] B. P. Allakhverdiev, On dilation theory and spectral analysis of dissipative Schr¨ odinger operators in Weyl’s limit-circle case, Izv. Akad. Nauk SSSR, Ser. Mat. 54 (1990), No.2, 242-257. [5] B. P. Allakhverdiev, On the theory of non-selfadjoint operators of Schr¨ odinger type with a matrix potential, Izv. Ross. Akad. Nauk, Ser. Mat. 56 (1993), No. 2, 193-205. [6] B. P. Allakhverdiev, A. Canoglu, Spectral analysis of dissipative Schr¨ odinger operators, Proc. R. Soc. Edinb., Sect. A, Math 127 (1997), no. 6, 1113-1121. [7] B. P. Allakhverdiev, F. G. Maksudov, On the theory of the characteristic function and spectral analysis of a dissipative Schr¨ odinger operator, Dokl. Akad. Nauk SSSR 303 (1988), no.6, 1307-1309.

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[8] B. P. Allakhverdiev, S. Saltan, Spectral Analysis of non-self-adjoint Schr¨ odinger operators with a matrix potential, J. Math. Anal. Appl. 303 (2005), 208-219. [9] M. Baro, H.-Ch. Kaiser, H. Neidhardt, J. Rehberg, Dissipative Schr¨ odinger-Poisson systems J. Math. Phys. 45 (2004), no. 1, 21-43. [10] H. Baumg¨ artel, M. Wollenberg, Mathematical scattering theory, Akademie-Verlag, Berlin 1983. ˇ Birman, M. G. Kre˘ın, On the theory of wave operators and scattering operators, [11] M. S. Dokl. Akad. Nauk SSSR 144 (1962), 475-478. [12] 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. [13] V. A. Derkach, M. M. Malamud, Characteristic functions of almost solvable extensions of Hermitian operators, Ukr. Math. J.44 (1992) , No.4, 379-401. [14] C. Foias, B. Sz.-Nagy, B, Harmonic analysis of operators on Hilbert spaces, NorthHolland Publishing Company, Amsterdam-London 1970. [15] I. C. Gohberg, M. G. Kre˘ın, Introduction to the theory of linear non-selfadjoint operators, Translations of Mathematical Monographs, Vol. 18 , American Mathematical Society, Providence, R.I. 1969. [16] V. I. Gorbachuk, M. L. Gorbachuk, Boundary value problems for operator diﬀerential equations, Kluwer Academic Publishers Group, Dordrecht, 1991. [17] H.-Ch. Kaiser, H. Neidhardt, J. Rehberg, Density and current of a dissipative Schr¨ odinger operator, J. Math. Phys. 43 (2002), no.11, 5325-5350. [18] H.-Ch. Kaiser, H. Neidhardt, J. Rehberg, Macroscopic current induced boundary conditions for Schr¨ odinger-type operators, Integr. Equ. Oper. Theory 45 (2003), 3963. [19] H.-Ch. Kaiser, H. Neidhardt, J. Rehberg, On 1-dimensional dissipative Schr¨ odingertype operators, their dilations and eigenfunction expansions, Math. Nachr. 252 (2003), 51-69. [20] T. Kato, Perturbation theory for linear operators, Springer-Verlag, BerlinHeidelberg-New York 1966. [21] M. .G. Kre˘ın, On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268–271. [22] P. Lax, R. S. Phillips, Scattering theory, Academic Press, New York-London 1967. [23] H. Neidhardt, J. Rehberg, Uniqueness for dissipative Schr¨ odinger-Poisson systems, J. Math. Phys. 46 (2005), no. 11, 113513. [24] B. S. Pavlov, Self-adjoint dilation of the dissipative Schr¨ odinger operator and its resolution in terms of eigenfunctions, Math. USSR Sb. 102(144) (1977), 511-536. [25] B. S. Pavlov, Dilation theory and spectral analysis of non-selfadjoint diﬀerential operators, Transl., II. Ser., Am. Math. Soc. 115 (1981), 103-142; translation from Proc. 7th. Winter School, Drogobych 1974, 3-69 (1976). [26] B. S. Pavlov, Spectral theory of non-selfadjoint operators, In Proc. Int. Congr. Math., Warszawa 1983, Vol.2, 1011-1025 (1984). [27] B. S. Pavlov, Spectral analysis of a dissipative singular Schr¨ odinger operator in terms of a functional model, Partial Diﬀerential Equations VIII (M. A. Shubin, ed), Encyclopaedia of Mathematical Science, vol. 65, Springer, Berlin, 1966, 87-153.

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[28] B. S. Pavlov, Irreversibility, Lax-Phillips approach to resonance scattering and spectral analysis of non-selfadjoint operators in Hilbert space, Int. J. Theor. Phys. 38 (1999), no.1, 21-45. [29] D. R. Yafaev, Mathematical scattering theory. General theory, American Mathematical Society, Providence, RI, 1992. Hagen Neidhardt and Joachim Rehberg Weierstrass-Institute for Applied Analysis and Stochastics Mohrenstr. 39 D-10117 Berlin Germany e-mail: [email protected] [email protected] Submitted: March 17, 2006

Integr. equ. oper. theory 58 (2007), 433–446 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030433-14, published online April 14, 2007 DOI 10.1007/s00020-007-1496-y

Integral Equations and Operator Theory

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum Onur Yavuz Abstract. We consider a multiply connected domain Ω = D \ n j=1 B(λj , rj ) where D denotes the unit disk and B(λj , rj ) ⊂ D denotes the closed disk centered at λj ∈ D with radius rj for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and the operators T and rj (T − λj I)−1 are polynomially bounded, then there exists a nontrivial common invariant subspace for T ∗ and (T − λj I)∗−1 . Mathematics Subject Classification (2000). Primary 47A15; Secondary 47A60. Keywords. Invariant subspaces, polynomially bounded operators, multiply connected regions, functional calculus.

1. Introduction Using a technique introduced by Scott Brown in [4], Brown, Chevreau, and Pearcy proved that every Hilbert space contraction whose spectrum contains the unit circle has an invariant subspace [5]. Ambrozie and M¨ uller generalized this result to Banach space operators by proving that the adjoint of a polynomially bounded operator whose spectrum contains the unit circle has an invariant subspace [1]. Recall that a bounded linear operator T deﬁned on a complex Banach space X, is said to be polynomially bounded if there exists a constant K > 0 such that p(T ) ≤ K sup{|p(λ)| : |λ| ≤ 1} for all polynomials p, and the constant K is said to be the polynomial bound of T . In [12] we have proved an analogous result for operators with an annular spectrum. Indeed, we have considered an annulus deﬁned by A = {z ∈ C : r0 < |z| < 1} for some 0 < r0 < 1, and proved the following theorem.

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Theorem 1.1. Let T be an invertible bounded linear operator on a complex Banach space X whose spectrum contains ∂A, and such that T and r0 T −1 are polynomially bounded. Then there exists a nontrivial common invariant subspace for T ∗ and T ∗ −1 . In this paper, we extend this result to operators whose spectrum contains the boundary of a multiply connected region. We consider a multiply connected domain deﬁned as follows. Let λ1 , λ2 , . . . , λn be points in D and r1 , r2 , . . . , rn be positive numbers such that the cloed disks centered at λj with radius rj , which we will denote by B(λj , rj ) are contained in D and are pairwise disjoint. Let Cj be the boundary of B(λj , rj ) and Uj be the unbounded component of C \ Cj for j = 1, 2, . . . , n. We will write C0 for T, U0 for D, and λ0 for 0 when convenient. The multiply connected domain bounded by the circles (Cj )nj=0 will be called Ω. Theorem A. Let T be a bounded linear operator whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and such that the operators T and rj (T − λj I)−1 for j = 1, 2, . . . , n are polynomially bounded. Then there exists a −1 nontrivial common invariant subspace for the operators T ∗ and (T − λj I)∗ . The main tools Ambrozie and M¨ uller used in their paper are Apostol’s theorem, an improved version of Zenger’s theorem, and Carleson’s interpolation theorem. In order to adapt Ambrozie and M¨ uller’s result to the case of an annulus, besides the main tools used in [1], we basically needed some estimates concerning Poisson kernels in an annulus, which act as representing measures, and an inner function whose derivative does not vanish on the boundary. We can construct such a function using Blaschke products on the annulus as deﬁned in [11] (See [12]). An alternative, Ahlfors’s function, which is deﬁned on more general domains, and comparison results regarding harmonic measure allow us to extend our result to operators with a multiply connected spectrum. In the ﬁrst section we will state Apostol’s theorem and will make some reductions. The second section will be devoted to some estimates concerning Poisson kernels associated with the region Ω and the third section to some interpolation results. We prove our main result in the last section.

2. Preliminaries We modify the deﬁnition of Apostol set [1] to ﬁt our setting. Definition. A subset Λ of Ω is called an Ω-Apostol set if all points on ∂Ω but countably many are radial limits of Λ. Let us denote by L(X) the Banach algebra of bounded linear operators on X. The following result follows from the original result (Lemma 2.1 [2]) by Apostol applied to operators T and rj (T − λj I)−1 for j = 1, . . . , n.

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Theorem 2.1. Let T ∈ L(X) be an operator whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and such that the operators T and rj (T − λj I)−1 for j = 1, 2, . . . , n are polynomially bounded. Suppose that for some ε > 0 and k ≥ 1, the set Λε,k := {λ ∈ Ω : ∀˜ε > ε ∃ u ∈ X with u = 1 and T u − λu < ε˜(dist(λ, ∂Ω))k } is not an Ω-Apostol set. Then T has nontrivial hyperinvariant subspaces. Thus, we may assume that the set Λε,k := {λ ∈ Ω : ∀˜ε > ε ∃ u ∈ X with u = 1 and T u − λu < ε˜(dist(λ, ∂Ω))k } is an Ω-Apostol set for every ε > 0 and k ≥ 1. Let us denote by A(Ω) the Banach algebra of continuous functions in Ω which are analytic on Ω with sup norm and by H ∞ (Ω) the Banach algebra of bounded analytic functions on Ω. As in [12], by the following proposition (see [10] for a proof) and the fact that every function in A(Ω) can be approximated uniformly by rational functions whose poles are outside Ω (See [7, p.86]), we can extend the functional calculus f → f (T ) to the entire algebra A(Ω), and the resulting map will satisfy the inequality f (T ) ≤ Kf A(Ω) for f ∈ A(Ω). Proposition 2.2. Every h ∈ H ∞ (Ω) can be expressed uniquely as h = h0 + h1 + h2 + · · · + hn where hj ∈ H ∞ (Uj ) and hj (∞) = 0 for every j = 0, 1, 2, . . . , n. If h is rational, then hj is rational for j = 0, 1, 2, . . . , n. By Proposition 3.2 [3] we may assume that at least one of the sequences ∗n ∞ (T n )∞ )n=1 and one of the sequences ((rj (T − λj I)−1 )n )∞ n=1 or (T n=1 or −1 ∗n ∞ ((rj (T − λj I) ) )n=1 for each j = 1, 2, . . . , n converge to 0 strongly. Thus, by corresponding results in [3] we may assume that each of the operators rj (T −λj I)−1 has a functional calculus deﬁned on H ∞ (D). These assumptions, together with Proposition 2.2, will allow us to deﬁne an H ∞ (Ω) functional calculus for T . Indeed, if h ∈ H ∞ (Ω) and z ∈ Ω, we can write h(z) = h0 (z) + h1 (r1 (z − λ1 )−1 ) + h2 (r2 (z − λ2 )−1 ) + · · · + hn (rn (z − λn )−1 ) where hj ∈ H ∞ (D) and hj (0) = 0 for j = 1, 2, . . . , n. Let us deﬁne h(T ) by h(T ) = h0 (T ) + h1 (r1 (T − λ1 I)−1 ) + h2 (r2 (T − λ2 I)−1 ) + · · · + hn (rn (T − λn I)−1 ). We have h0 (T ) ≤ KT h∞ and hj ((rj (T − λj I)−1 )) ≤ Krj (T −λj I)−1 hj ∞ for every j = 1, 2, . . . , n. By a similar argument as in Lemma 2.2 [12], we deduce that h(T ) ≤ Kh∞ for some constant K > 0 and every h ∈ H ∞ (Ω) . Moreover, we can actually assume that the extended functional calculus is an isometry as is veriﬁed by the next two lemmas. We skip the proofs as they are identical to those of the corresponding lemmas in [12]. Lemma 2.3. If Λε,k is an Ω-Apostol set for every ε > 0 and k ≥ 1, then h∞ ≤ h(T ) for h ∈ H ∞ (Ω).

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Lemma 2.4. If T ∈ L(X) is such that Ω is a K-spectral set for T , then T is similar to an operator T on another Banach space X with the property that f (T ) ≤ f A(Ω) for all functions f in A(Ω). In other words, Ω is a spectral set for T . In the presence of a weakly-continuous H ∞ (Ω) functional calculus we also have h(T ) ≤ h∞ for every h ∈ H ∞ (Ω) when T is deﬁned as in the above proof. We will use the notation AΩ (X) to denote the set of bounded linear operators T on X which have an isometric functional calculus from H ∞ (Ω) to L(X) that maps 1(z) ≡ z to T . By the above discussions, from now on we may assume that T is in AΩ (X). The functionals x⊗T x∗ : H ∞ (Ω) → C deﬁned by (x⊗T x∗ )(h) = h(T )x, x∗ for x ∈ X, x∗ ∈ X ∗ , h ∈ H ∞ (Ω) will be of particular interest in constructing an invariant subspace for the operators T and (T − λj I)−1 . We have x⊗T x∗ ≤ xx∗ ; thus the functionals x⊗T x∗ are bounded. Note that the algebra H ∞ (Ω) carries a weak* topology viewed as the dual space of L1 (Ω)/⊥ H ∞ (Ω). For technical reasons we need the functionals x ⊗T x∗ to be weak∗ -continuous. Using Proposition 2.2 and the same arguments in [12] we may assume that these functionals are weak∗ -continuous for all x ∈ X and x∗ ∈ X ∗ . So it will be suﬃcient to prove the following theorem in order to prove our main result. Theorem B. Let T ∈ AΩ (X) be such that Λk,ε is an Ω-Apostol set for every ε > 0 and k ≥ 1. Assume the functional x ⊗T x∗ : H ∞ (Ω) → C is weak∗ -continuous for all x ∈ X and x∗ ∈ X ∗ . Then there exists a nontrivial common invariant subspace for the operators T ∗ and (T − λj I)∗ −1 .

3. Poisson kernels The fact that the Drichlet problem can be solved on the domain Ω implies the existence of harmonic measures. For a domain U for which the Drichlet problem can be solved, we denote by w(λ, I, U ) the harmonic measure of a Borel subset I of ∂U associated with λ ∈ U . If f is a continuous function on ∂U , then the integral f (ζ)dw(λ, ζ, U ) u(λ) = ∂U

is the solution to the Dirichlet problem on U for the boundary values f (ζ). One can consult [8] for a proof, and also [7] and [9] for a more detailed discussion of harmonic measure in general domains. Note that harmonic measure is a probability measure. One can prove easily the following comparison results. Theorem 3.1 (Comparison Theorem). If U1 ⊂ U2 w(λ, I, U1 ) ≤ w(λ, I, U2 ) for λ ∈ U1 .

and

I ⊂ ∂U1 ∩ ∂U2 , then

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Lemma 3.2. If ϕ is a bijective analytic function from a domain U1 to a domain U2 which extends continuously to the boundary, then for I ⊂ ∂U2 , we have w(λ, ϕ−1 (I), U1 ) = w(ϕ(λ), I, U2 ). When U is a multiply connected region bounded by ﬁnitely many circles, we denote arc length measure by ; when convenient we will write | · | for . We will denote angular measure on ∂U by m. More precisely, the angular measure of an interval on ∂U is equal to the measure of the angle with vertex at the center of the corresponding circle. Since for every λ ∈ U , the measure w(λ, ·, U ) is absolutely continuous with respect to the measure , there exist density functions which we will denote by Kλ,U so that u(ζ)Kλ,U (ζ)|dζ| u(λ) = ∂U

for every continuous function on U which is harmonic in U . There are also density λ,U , with functions with respect to angular measure, which we will denote by K ˜ λ,U (ζ)dm(ζ). u(λ) = u(ζ)K ∂U

λ,Ω (ζ) for every ζ ∈ ∂Ω where f (ζ) = 1/rj on Cj and Note that Kλ,Ω (ζ) = f (ζ)K f (ζ) = 1 on C0 . Lemma 3.3. If ϕ is a bijective analytic function from a domain U1 to a domain U2 with piecewise smooth boundaries which extends analytically to the boundary, we have the following change of variables formula: Kλ,U1 (ζ) = Kϕ(λ),U2 (ϕ(ζ)) · |ϕ (ζ)| for λ ∈ U1 and ζ ∈ ∂U1 . Proof. If u : U 2 → C is harmonic, then u ◦ ϕ is harmonic in U 1 , and we have: u(ϕ(ζ))Kλ,U1 (ζ)|dζ|. u(ϕ(λ)) = ∂U1

On the other hand we have: u(ϕ(λ)) = ∂U2 = ∂U1

u(ξ)Kϕ(λ),U2 (ξ)|dξ| u(ϕ(ζ))Kϕ(λ),U2 (ϕ(ζ))|ϕ (ζ)||dζ|.

We will deﬁne intervals Iλ,Ω analogous to ones deﬁned in [1]. Let us ﬁx a positive number η such that the circles centered at λj with radius rj + η and the circle centered at 0 with radius 1 − η are pairwise disjoint. Note that for λ ∈ Ω r with | λ − λj | < rj + η, we have | λ −jλj | > 1 − η for j = 1, 2, . . . , n. Let us also ﬁx a positive number a > 1.

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For λ = reiθ ∈ Ω we will write λ = rj eiθj + λj when |λ − λj | < rj + η. We now deﬁne Iλ,Ω as follows:  |λ−λ |  {rj eitj + λj : |tj − θj | < a( rj j − 1)} if | λ − λj | < η + rj for j ≥ 1 Iλ,Ω :=  {eit : |t − θ| < a(1 − r)} otherwise. We deﬁne a family of annuli in Ω as follows: For j = 1, 2, . . . , n, let Aj := {λ ∈ Ω : | λ − λj | < rj + η} and A0 := {λ ∈ Ω : | λ | > 1 − η}. Note that Aj is conformally equivalent to annuli rj A˜j = {λ ∈ D : rj +η < λ < 1}. Let us denote by ϕj the conformal mapping between j . A comparison of Ω to these annuli in terms of harmonic measure will Aj and A imply an analogue of Lemma 4.1 of [12]. For each of the annuli A˜j we deﬁne intervals Iµ,A˜j as in [12](See also ( ) which is to appear a few lines after Corollary 3.6). We reﬁne the constant a so that a > 2(n + 1) and the set inclusion Iλ,Ω ⊇ ϕ−1 ˜j ) holds for every j and j (Iϕ(λ),A λ ∈ Aj , and choose a constant ρ < η such that ϕj maps the set j for which the statement of Lemma {λ ∈ Ω : | λ − λj | < rj + ρ} into a subset of A 4.1 of [12] corresponding to Aj holds. Then by Lemma 3.2 and the Comparison theorem 3.1, we have the following: Lemma 3.4. Let λ ∈ Ω satisfy either | λ − λj | < rj + ρ for some j = 1, 2, . . . , n or | λ | > 1 − ρ. Then w(λ, Iλ , Ω) ≥ 0.77. We now deﬁne functions Qλ : ∂Ω → [0, ∞) as follows:   Kλ,Ω (ζ) if ζ ∈ Iλ,Ω Qλ (ζ) :=  0 otherwise Corollary 3.5. For λ ∈ Ω with | λ − λj | < rj + ρ or | λ | > 1 − ρ, 23 (Kλ,Ω (ζ) − Qλ (ζ)) |dζ| ≤ Qλ (ζ)|dζ|. 77 ∂Ω ∂Ω Proof.

∂Ω

(Kλ,Ω (ζ) − Qλ (ζ)) |dζ| Kλ,Ω (ζ)|dζ| 23 . = ∂Ω −1≤ 77 Q (ζ)|dζ| Q (ζ)|dζ| λ λ ∂Ω ∂Ω

As another consequence of Lemma 3.4, we have the following: Corollary 3.6. Let λ ∈ Ω satisfy either | λ − λj | < rj + ρ for some j = 1, 2, . . . , n or | λ | > 1 − ρ. Then max K(λ,Ω) (ζ) ≥

ζ∈∂Ω

0.77 . (Iλ )

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In what follows, we will show that (Iλ )·maxζ∈∂Ω K(λ,Ω) (ζ) also has an upper bound, and to this purpose we will compare Ω to the doubly connected regions deﬁned below: For every 1 ≤ j ≤ n, deﬁne Bj := {λ ∈ D : | λ − λj | > rj } and set B0 = B1 . Clearly, Bj ⊃ Ω for every j = 0, 1, 2, . . . , n. For every j = 0, 1, 2, . . . , n, there exists a conformal mapping ψj between the j bounded by the unit circle and doubly connected region Bj and some annuli B a circle centered at 0 with radius Rj for some Rj < 1. In fact these conformal λ −t λ mappings are in the form ψ(λ) = 1−t jλ jλ for some 0 < tj < 1 . Note that for j

j

| λ −µ|(1−|t λ |2 )

j j every λ, µ ∈ Bj , we have |ψj (λ) − ψj (µ)| ≤ := Nj . (1−|tj λj |)2 We choose positive constants αj for every j = 0, 1, 2, . . . , n so that for the intervals Iψj (λ),B˜j deﬁned below where ψj (λ) = reiθ

  Iψj (λ),B j :=



{eit : |t − θ| < αj (1 − r)} {Rj eit : |t − θ| < αj (1 −

Rj r )}

if r ≥

Rj

if r <

Rj

( )

we have m(Iλ,Bj ) ≤ m(Iψj (λ),B j ). By the Comparison theorem 3.1 and Lemma 3.3, we have the following for every j = 0, 1, . . . , n: max Kλ,Ω (ζ) ≤ max Kλ,Bj (ζ) ≤ Nj max Kψj (λ),B j (µ)

ζ∈Cj

ζ∈Cj

µ∈ψ(Cj )

=

Nj Rj

≤

Nj M j = Rj m(Iλ,Bj )

max K j (µ) ≤ ψj (λ),B

µ∈ψ(Cj )

Nj M j Rj (Iλ,Bj ) rj

Nj M j Rj m(Iψj (λ),B j )

=

rj Nj Mj Rj (Iλ,Bj )

where Mj is the constant provided by Lemma 4.3 of [12] for each j = 0, 1, . . . , n. The following lemma summarizes above observations. We will use the notation γλ = (maxζ∈∂Ω Kλ,Ω (ζ))−1 . Lemma 3.7. There exists a constant S > 1 such that if λ ∈ Ω satisﬁes either | λ − λj | < rj + ρ for some j = 1, 2, . . . , n, or | λ | > 1 − ρ, then (Iλ,Ω ) ≤ γλ ≤ S(Iλ,Ω ). S From now on since there will be no confusion of domain, we will write Iλ for Iλ,Ω . In order to prove our main result we need an inner function deﬁned on Ω whose derivative does not vanish on the boundary. The Ahlfors function which has been obtained as a solution to an extremal problem will serve our purpose. We refer the reader to [7] for a proof and a more detailed discussion of properties of the Ahlfors function.

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The Ahlfors function associated with Ω has nice properties which will be stated in the following theorem. Again we refer the reader to [7] for a proof. Theorem 3.8. Let u be the Ahlfors function associated with Ω and p ∈ Ω. Then 1. u maps Ω onto D exactly n + 1 times, 2. u extends analytically across each Cj , and maps each Cj homeomorphically onto T, 3. u is not zero on any Cj for j = 0, 1, 2, . . . , n. The Ahlfors function u plays the role of the function u(λ) = λ in the arguments of [1]. The following two results are proved in the same way as the corresponding ones of [12]. Let us ﬁx a constant N > 0 such that 1/N ≤ |u (z)| ≤ N for all z ∈ ∂Ω. Lemma 3.9. Let Λ ⊂ Ω be an Ω-Apostol set and I ⊂ ∂Ω be an open interval such that u is one-to-one on I. Then for suﬃciently large m ∈ N, there exists a separated subset F of Λ with the following properties: 1. Iλ ⊂ I for all λ ∈ F , m 2. |u(λ) − 1| < 1/9,1 for all λ ∈ F, 3. m λ∈F Iλ ≥ 40πN 2 (I), 4. λ∈F γλ ≤ S(I), 5. ∂Ω λ∈F γλ u(λ)m K(λ,Ω) (ζ) − χI (ζ) |dζ| ≤ c1 (I) 409 where c1 = 1 − 36,000SN 2π . Fix c2 ∈ (c1 , 1) . Theorem 3.10. Let f : ∂Ω → C be a nonnegative integrable function and Λ ⊂ Ω be an Apostol set. Then for all m suﬃciently large, there exist a separated subset F of Λ and positive numbers αλ (λ ∈ F ) such that: m 1. |u(λ) − 1| < 1/9 for λ ∈ F , 2. λ∈F α λ ≤ 2Sf 1, and 3. ∂Ω λ∈F αλ u(λ)m K(λ,Ω) (ζ) − f (ζ) |dζ| ≤ c2 f 1 .

4. Interpolation Results We will state below an improved version of Zenger’s theorem [1]. We ﬁrst recall the following deﬁnition from [1]: nonzero vectors in a Banach Definition. Let L > 0. A collection {u 1n, u2 , . . . , un } of n space X is said to be L-circled if j=1 βj uj ≤ L j=1 γj uj whenever |βj | ≤ |γj | for j = 1, 2, . . . , n. Theorem 4.1. Consider positive numbers n an L-circled set {w1 , w2 , . . . , wn } ⊂ X, ∗ α1 , α2 , . . . , αn with α = 1, and a functional ϕ ∈ X . Then there exist j=1 j √ n ∗ scalars s1 , s2 , . . . , sn and ψ ∈ X such that ϕ − ψ ≤ 1, j=1 sj wj ≤ L 2, and ψ(sj wj ) = αj for j = 1, 2, . . . , n.

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We say a subset F of Ω is separated if the intervals {Iλ : λ ∈ F } are pairwise disjoint. The following two lemmas are analogues of Lemma 4.3 of [1]. Lemma 4.2. Let c > 0 be given. There is a constant δ > 0 with the following property: If F is a separated ﬁnite subset of D with |λ| > c for every λ ∈ F , then

λ0 − λ 1 − λλ ≥ δ λ∈F \{λ0 }

0

for each λ0 ∈ F . r

j for j = 1, 2, . . . , n and f0 (z) = z. Note that Let us write fj (z) = z−λ j |fj (z)| > 1 − η for |z − λj | < rj + η and j > 0.

Lemma 4.3. There is a constant δ > 0 with the following property: If F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn is a separated ﬁnite subset of Ω with F0 ⊂ {λ ∈ Ω : | λ | > 1 − η} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj + η} for j = 1, 2, . . . , n, then

fj (µ) − fj (λ) ≥δ 1 − f j (λ)fj (µ) λ∈Fj

for each µ ∈ F \ Fj and j = 0, 1, 2, . . . , n. |µ−λ | Proof. We observe that for j = 0, the inequality |1−λµ| > η/2 holds for every λ ∈ F0 and µ ∈ F \ F0 . Also for j > 0, we have the following: f (µ) − f (λ) rj − rj r rj λ − λj µ−λj j j j /2 − = rj ≥ 1 − f j (λ)fj (µ) 1 − rj µ−λ µ − λj λ − λj j λ−λj

=

rj | λ − λj −µ + λj | ≥ rj dist(Ck , Cj )/8 2|µ − λj || λ − λj |

for every λ ∈ Fj and µ ∈ Fk where j = k. Then the proof for Lemma 4.3 [1] works in this case as well. The following result follows from Carleson’s interpolation theorem. See [1] and [6]. Proposition 4.4. Let c > 0 be given. There exists a constant b with the following property: If F is a separated ﬁnite subset of D, such that | λ | > c for every λ ∈ F , then given scalars {cλ : λ ∈ F }, there exists f ∈ H ∞ (D) such that f ∞ ≤ b supλ∈F |cλ | and f (λ) = cλ for λ ∈ F . We also need a special version of the above proposition which is an analogue of Lemma 3.5 of [12]. Lemma 4.5. There is a constant σ > 0 with the following property: If F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn is a separated ﬁnite subset of Ω with F0 ⊂ {λ ∈ Ω : | λ | > 1−η} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj +η} for j = 1, 2, . . . , n, then there exist functions gj ∈ H ∞ (Ω), with gj ∞ ≤ σ, such that gj ≡ 1 on Fj and gj ≡ 0 on Fk for 0 ≤ j, k ≤ n and k = j.

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Proof. Deﬁne Bj (z) =

IEOT

fk (z) − fk (λ) . 1 − f k (λ)fk (z) 0≤k≤n λ∈F k=j n

k

By the previous lemma Bj (µ) ≥ δ for all µ ∈ Fj . Since the intervals {Ifj (µ) : µ ∈ Fj } are pairwise disjoint and |fj (µ)| > 1−η for every µ ∈ Fj , it follows from Proposition 4.4 that there exist a constant b > 0 and a function hj ∈ H ∞ (Ω) with hj (fj (µ)) = 1/Bj (µ) and hj ∞ ≤ b/δ n . Let gj (z) := hj (fj (z))Bj (z) and σ = b/δ n . Then gj ≡ 1 on Fj and gj ≡ 0 on F \ Fj , and gj ∞ ≤ σ. Remark 1. In particular for λ0 ∈ F we can ﬁnd g ∈ H ∞ (Ω) with g(λ0 ) = 1 and g(λ) = 0 for λ ∈ F \ {λ0 }, and g∞ ≤ σ. The following lemma provides an upper bound for “approximate eigenvalues” of T . The letter σ denotes the universal constant obtained in the previous lemma. Lemma 4.6. There exists κ > 0 with the following property: If F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn is a separated ﬁnite subset of Ω such that F0 ⊂ {λ ∈ Ω : | λ | > 1−ρ} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj +ρ} for j = 1, 2, . . . , n, and {uλ : λ ∈ F } ⊂ X and {µλ : λ ∈ F } ⊂ C satisfy uλ = 1, 1 2 (dist(λ, ∂Ω)) , and (T − λI)uλ < µλ uλ ≤ 1, 2σπ λ∈F

then necessarily |µλ | ≤ κ. Proof. Let λ0 ∈ F satisfy |µλ0 | = maxλ∈F |µλ |. Then by Remark 1, there exists such that f (λ0 ) = 1, f (λ) = 0 for λ ∈ F \ {λ0 }, and f ∞ ≤ σ. f ∈ H ∞ (Ω) Setting u = λ∈F µλ uλ , we have f (T )u ≤ σu ≤ σ. Now, for λ ∈ F there exists a function gλ which is analytic on Ω with f (z) − f (λ) = gλ (z)(z − λ) and gλ ∞ ≤ 2f ∞ (dist(λ, ∂Ω))−1 ≤ 2σ(dist(λ, ∂Ω))−1 . We have σ ≥ f (T )u ≥ f (λ)µλ uλ − µλ (f (λ) − f (T ))uλ λ∈F λ∈F ≥ |µλ0 uλ0 | − |µλ | · gλ (T )(T − λI)uλ λ∈F

1 dist(λ, ∂Ω) π λ∈F 1 (Iλ ) = |µλ0 | 1 − π 2a λ∈F 2π(n + 1) n+1 ≥ |µλ0 | 1 − = |µλ0 | 1 − . 2aπ a ≥ |µλ0 | − |µλ0 |

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(Note that since the intervals {Iλ : λ ∈ F } are pairwise disjoint, we have λ∈F Iλ < 2(n + 1)π.) Taking into account that a > n + 1, we conclude that σ the condition of the lemma is true with κ = 1−(n+1)/a . We will show in the next three lemmas that any family of vectors {uλ : λ ∈ F } which satisfy the hypothesis of the previous lemma are 2b(n + 1)L-circled where L = 2(n + 1) + σ and b is as in Proposition 4.4. We ﬁrst state the corresponding result for every Fj separately which is a slight modiﬁcation of Proposition 6.2 in [1]. Proposition 4.7. If F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn is a separated ﬁnite subset of Ω such that F0 ⊂ {λ ∈ Ω : | λ | > 1 − ρ} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj + ρ} for j = 1, 2, . . . , n, and {uλ : λ ∈ F } ⊂ X satisfy uλ = 1, (T − λI)uλ <

1 (dist(λ, ∂Ω))2 , 2σπ

then each {uλ : λ ∈ Fj } for j = 0, 1, . . . , n is 2b-circled. Lemma 4.8. Let F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn be a separated ﬁnite subset of Ω such that F0 ⊂ {λ ∈ Ω : | λ | > 1 − ρ} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj + ρ} for j = 1, 2, . . . , n, and {uλ : λ ∈ F } ⊂ Xsatisfy uλ = 1, 1 (dist(λ, ∂Ω))2 . 2σπ Then given βλ ∈ D with λ∈F βλ uλ = 1 we have xj ≤ L where xj = λ∈Fj βλ uλ , for j = 0, 1, 2, . . . , n. (T − λI)uλ <

Proof. By Lemma 4.3, for j = 0, 1, . . . , n, there exists g ∈ H ∞ (Ω) with g(λ) = 1 for λ ∈ Fj , g(λ) = 0 for λ ∈ F \ Fj , and g∞ ≤ σ. Let x = x0 + x1 + · · · + xn . We have xj ≤ g(T )x − xj + g(T )x and β g(T )u − β u g(T )x − xj = λ λ λ λ λ∈F λ∈Fj = β g(λ)u + β (g(T ) − g(λ))u − β u λ λ λ λ λ λ λ∈F λ∈F λ∈Fj = βλ (g(T ) − g(λ))uλ . λ∈F

There exists qλ ∈ H ∞ (Ω) with qλ ∞ ≤ 2(dist(λ, ∂Ω))−1 g∞ and g(z) − g(λ) = qλ (z)(z − λ). Thus,

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1 (dist(λ, ∂Ω))2 2(dist(λ, ∂Ω))−1 g∞ + σx 2σπ λ∈F 1 = dist(λ, ∂Ω) + σ π λ∈F 1 m(Iλ ) + σ < π

xj ≤

λ∈F

1 2(n + 1)π + σ = L. π

Lemma 4.9. If F = F0 ∪ F1 ∪ F2 ∪ · · · ∪ Fn is a separated ﬁnite subset of Ω such that F0 ⊂ {λ ∈ Ω : | λ | > 1 − ρ} and Fj ⊂ {λ ∈ Ω : | λ − λj | < rj + ρ}, and {uλ : λ ∈ F } ⊂ X satisfy uλ = 1, 1 (dist(λ, ∂Ω))2 , 2σπ then the family {uλ : λ ∈ F } is 2(n + 1)bL-circled. (T − λI)uλ <

Proof. By Proposition 4.7 we know that {uλ : λ ∈ Fj } is 2b-circled for every j = 0, 1, 2, . .. , n. Assume without loss of generality that |βλ | ≤ |γλ | ≤ 1 and λ∈F uλ γλ = 1. Then n ≤ u β u β λ λ λ λ j=0 λ∈Fj λ∈F n ≤ 2b u γ λ λ j=0 λ∈Fj ≤ 2b(n + 1)L.

5. Main result We may assume that u(T )n → 0 for every x ∈ X. Indeed, for the hyperinvariant spaces M = {x : u(T )n x → 0} for T and M∗ = {x∗ : u(T )∗n x → 0} for T ∗ , if neither M = X nor M∗ = X ∗ holds, it follows from Theorem 3.2 [3] that u(T )∗ , and so T ∗ has hyperinvariant subspaces. Therefore, it will be enough to prove the following theorem to obtain our main result. Theorem C. Let T ∈ AΩ (X) be such that the set Λk,ε is an Apostol set for every ε > 0 and k ≥ 1. Assume that the functional x ⊗T x∗ : H ∞ (Ω) → C is weak∗ continuous for every x ∈ X and x∗ ∈ X ∗ and u(T )n x → 0 for all x ∈ X. Then there exists a nontrivial common invariant subspace for the operators T and (T − λj I)−1 for j = 1, 2, . . . , n.

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For f ∈ L1 (∂Ω) we will denote by Mf the functional deﬁned by f (ζ)h(ζ)|dζ| for h ∈ A(Ω). Mf (h) = ∂Ω

In particular, we will denote by Eλ the functionals corresponding to K(λ,Ω) deﬁned in Section 3 . We have Mf ≤ f 1 for all f ∈ L1 (∂Ω) and Eλ (h) = h(ζ)K(λ,Ω) (ζ)|dζ| = h(λ) , h ∈ A(Ω) . ∂Ω

The hypothesis of Theorem C implies that for given x ∈ X and x∗ ∈ X ∗ , there exists f ∈ L1 (∂Ω) such that h(T )x, x∗ = h(ζ)f (ζ)|dζ| , h ∈ A(Ω). ∂Ω

In the remainder of the section we show that for every g ∈ L1 (∂Ω), there exist x ∈ X and x∗ ∈ X ∗ such that Mg (h) = (x ⊗T x∗ )(h) for every h ∈ A(Ω). This will prove Theorem C. Indeed, it would imply that for λ0 ∈ ∂Ω there exist x and x∗ such that x ⊗T x∗ = Eλ0 . Thus f (T )x, x∗ = f (λ0 ) for all f ∈ A(Ω). We may assume (T − λ0 I)x = 0, since otherwise Ker(T − λ0 I) would be a hyperinvariant subspace for T . Thus, the closure of {f (T )x : f ∈ A(Ω), f (λ0 ) = 0} would be the desired nontrivial common invariant subspace for the operators T and (T − λj I)−1 for j = 1, 2, . . . , n. Since all of the results of this section are slight modiﬁcations of the corresponding ones in [12], we omit the proofs. The following proposition forms the most crucial step of factorization we desire. The vectors x and x∗ needed will come from Theorem 2.1 and the improved version, Theorem 4.1 of Zenger’s theorem. Let us ﬁx a constant c3 ∈ (c2 , 1). Proposition 5.1. Assume the hypothesis of Theorem C is satisﬁed. Fix a nonnegative function f ∈ L1 (∂Ω) with f 1 = 1 and y ∗ ∈√X ∗ . Then for m suﬃciently large, there exist x ∈ X, x∗ ∈ X ∗ such that x ≤ 4 2(n + 1)SbL, m x∗ ≤ 1, and x ⊗T (u(T )∗ x∗ + y ∗ ) − Mf < c3 . The assumptions that u(T )n x → 0 for every x ∈ X and that the functionals x ⊗T x∗ are weak∗ -continuous, are essential √ in the proofs of the lemmas to follow. From now on τ will denote the constant 4 2Sb(n + 1)L. Lemma 5.2. Assume the hypothesis of Theorem C is satisﬁed. Then for given y ∈ X, y ∗ ∈ X ∗ , ε > 0, and a nonnegative function f ∈ L1 (∂Ω) , there exist w ∈ X and w∗ ∈ X ∗ such that 1. w ⊗T (w∗ + y ∗ ) − Mf ≤ c3 f 1 , 2. y ⊗T w∗ < ε, 3. w ≤ τ f 1 , w∗ ≤ f 1 . Fix an integer N such that c3 +πN −1 < 1, and a positive constant c satisfying 1 − N −1 (1 − c3 − πN −1 ) < c < 1.

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Lemma 5.3. Assume the hypothesis of Theorem C is satisﬁed. Then for given y ∈ X, y ∗ ∈ X ∗ , and h ∈ L1 (∂Ω), there exist x ∈ X and x∗ ∈ X ∗ such that 1/2

1. y − x ≤ τ h 1 , ∗ ∗ 2. y − x ≤ h1 , 3. x ⊗T x∗ − y ⊗T y ∗ − Mh ≤ ch1 . Lemma 5.4. Assume the hypothesis of Theorem C is satisﬁed. Then for all g ∈ L1 (∂Ω), there exist x ∈ X and x∗ ∈ X ∗ such that Mg = x ⊗T x∗ . Acknowledgment. This paper is based on my dissertation which was submitted to Indiana University. I would like to thank my thesis advisor Hari Berocovici for his guidance and support.

References [1] C. Ambrozie and V. M¨ uller, Invariant subspaces for polynomially bounded operators, J. Functional Analysis, 213 (2004), 321–345. [2] C. Apostol, Utraweakly closed operator algebras, J. Operator Theory, 2 (1979), 49–61. [3] C. Apostol, Functional calculus and invariant subspaces, J. Operator Theory, 4 (1980), 159–190. [4] S. Brown, Some invariant subspaces for subnormal operators, Integral Equations Operator Theory, 1 (1978), 310–333. [5] S. Brown, B. Chevreau and C. Pearcy, On the structure of contraction operators. II, J. Functional Analysis, 76 (1988), 30–55. [6] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math, 80 (1958), 921-930. [7] S.D. Fischer, Function theory on planar domains, John Wiley & Sons, New York, 1983. [8] J.B. Garnett, Applications of Harmonic Measure, John Wiley & Sons, New York, 1986. [9] J.B. Garnett and D.M. Marshall, Harmonic Measure, Cambridge University Press, New York, 2005. [10] V.I. Paulsen, Completely bounded maps and dilations, Longman Scientiﬁc & Technical, Harlow, 1986. [11] D. Sarason, The H p spaces of an annulus, Mem. Amer. Math. Soc., 56 (1965). [12] O. Yavuz, Invariant subspaces for Banach space operators with an annular spectral set, to appear in Trans. Amer. Math. Soc. Onur Yavuz Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey e-mail: [email protected] Submitted: November 6, 2006 Revised: December 20, 2006

Integr. equ. oper. theory 58 (2007), 447 0378-620X/030447-1, DOI 10.1007/s00020-007-1499-8 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

Erratum to: “On the Range of the Aluthge Transform” [IEOT 57 (2) (2007), 209–215, DOI 10.1007/s00020-006-1452-2]

Guoxing Ji, Yongfeng Pang and Ze Li Abstract. In Theorem 3 of the article “On the Range of the Aluthge Transform” [IEOT 57 (2) (2007), 209–215, DOI 10.1007/s00020-006-1452-2] p > 2 has to be assumed. Mathematics Subject Classification (2000). Primary 47A15; Secondary 47B20. Keywords. Erratum, Aluthge transform, polar decomposition, range.

Lemma 1 in the article “On the Range of the Aluthge Transform” [IEOT 57 (2) (2007), 209–215, DOI 10.1007/s00020-006-1452-2] is incorrect. Thus we have to assume that p > 2 in Theorem 3. That is, Theorem 3. Let H = Cp for p > 2. Then R(∆) is neither closed nor dense in B(H). Guoxing Ji, Yongfeng Pang and Ze Li College of Mathematics and Information Science Shaanxi Normal University Xian, 710062 People’s Republic of China e-mail: [email protected]

Integr. equ. oper. theory 58 (2007), 449–475 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040449-27, published online June 27, 2007 DOI 10.1007/s00020-007-1517-x

Integral Equations and Operator Theory

The Algebra of Diﬀerential Operators Associated to a Weight Matrix F. Alberto Gr¨ unbaum and Juan Tirao Abstract. Given a weight matrix W (x) of size N on the real line one constructs a sequence of matrix valued orthogonal polynomials, {Pn }n≥0 . We study the algebra D(W ) of diﬀerential operators D with matrix coeﬃcients such that Pn D = Λn Pn , with Λn in the algebra A of N ×N complex matrices. We study certain representations of this algebra, prove that it is a *-algebra and give a precise description of its isomorphic image inside the algebra AN0 . Mathematics Subject Classiﬁcation (2000). Primary 33C45, 47L80; Secondary 47E05. Keywords. Matrix orthogonal polynomials, bispectral problem, algebra of differential operators, ad-conditions, adjoint operation.

1. Introduction This paper considers a non-commutative version of a problem studied in the scalar case in [6] and [13]. We give a self contained account starting in the next section, and for the beneﬁt of the reader we describe here the situation that arises in the much simpler scalar case. The main ingredients both in the scalar case, as well as in the present case are very similar. In [6] one starts with a given L which is taken to be a second order diﬀerential operator written in Scroedinger’s form Lf = −D2 f + V (x)f One considers a family of its eigenfunctions f (x, k), satisfying Lf (x, k) = kf (x, k) and studies the algebra (which in this paper will be called D(W )) of diﬀerential operators B(k, ∂k ) in the spectral variable k such that Bf (x, k) = Θ(x)f (x, k)

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for some function Θ(x). This is, in this case, a commutative algebra sharing a large collection of common eigenfunctions, parametrized by x. Given such a commutative algebra, there is a powerful theory, started by Burchnall and T. Chaundy around 1920, see the reference and a simple account in [20], and further developed by I. Krichever, see [23] in the 70’s in connection with soliton type equations. A very nice account is given in [27]. This theory associates to the commutative algebra D(W ) an algebraic curve and a bundle on it. In the case at hand we can be very concrete. In [6] one proves that the set of operators L for which the algebra is nontrivial consists of the Bessel and Airy operators, with V (x) = c/x2 and V (x) = cx, respectively, and then two large disjoint sets of L s: those connected with the Korteweg-deVries hierarchy and those refered to in [6] as the ”even family”. These last set was later shown in [25] to be related to the master symmetries of KdV. The description of the operators L going with the KdV hierarchy and having a nontrivial algebra, as well as the description of the algebra itself is summarized now: For the V (x) in the KdV family we have νp (νp + 1) V (x) = (x − p)2 p∈P

with P a ﬁnite subset of C, and νp ∈ Z>0 . The set P has to be chosen such that νq (νq + 1) = 0 for 1 ≤ j ≤ νp and each p ∈ P. (q − p)2j+1 q∈P q=p

One can also write

V (x) = −2

where θ(x) is given by θ(x) =

θ (x) θ(x)

,

1 (x − p) 2 νp (νp +1) . p

It is a fact, proved in [2], that these V (x) are the only rational functions, decaying at inﬁnity, which remain rational under the KdV ﬂows. It turns out that in the KdV case there are two families of eigenfunctions of L=-D2 +V that lead to nontrivial algebras. These families are characterized by their behaviour at inﬁnity. For these potentials we have the following characterization of the algebra of diﬀerential operators in the spectral variable: The eigenfunction φ± ∞ (x, k) satisﬁes an equation of the form ± B ± (k, ∂k )φ± ∞ (x, k) = Θ(x)φ∞ (x, k)

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if and only if the polynomial Θ has the property that Θ(2j−1) (p) = 0 for all 1 ≤ j ≤ νp , for each pole p ∈ C of V. Notice that the fact that we have only odd order derivatives makes this set of polynomials Θ into an algebra. Once Θ(x) is determined one can produce a diﬀerential operator in our algebra by the method described in section 3 of the present paper. The description of the operators L in the even family with a nontrivial algebra as well as the description of the algebra itself is similar to the one in the KdV case and is given in [6]. In summary, in all the cases where L is a Schroedinger type diﬀerential operator the curve is singular and it is described quite explicitly. The singularities of the curve corresponds to the points p ∈ P. For further developments related to the so called bispectral problem considered in [6], see [3], [17], [18], [28], [33] and [34]. Moving one step closer to the situation in this paper, in [13], one considers a scalar weight function W (x) and the corresponding family of orthogonal polynomials, {Pn }n≥0 . These polynomials satisfy a second order diﬀerence equation of the form LPn = xPn One then asks for those weights W (x) such that there is (at least one nontrivial) diﬀerential operator D such that DPn = Λn Pn for some numerical sequence Λn . If the diﬀerential operator is to have order two one has an old result proved by Routh around 1880, see [31], and then again by S. Bochner in a much better known paper from 1929, [4]. For higher order operators there is still no complete solution to the bispectral problem even in the scalar case, but there are some reasonable conjectures. The bispectral problem alluded to above comes in several ﬂavours. There is a continuous-continuous version, see [6], where the physical variable x as well as the spectral variable k run in part of the real line. In another important version of the problem, [13], physical space is replaced by the non-negative integers and one typically uses n for the old x and sticks to k for the spectral variable. In this paper, which can be read quite independently of these two references, [6, 13], and with apologies to the reader, we use n for the physical space variable and x for the spectral one. This should not create any confusion. The case to be studied in this paper starts with a matrix valued weight W (x) and then as in [21] and [22] one builds a sequence of matrix valued orthogonal polynomials. Just as in the scalar case one obtains a (block) tridiagonal semiinﬁnite matrix L such that LPn = xPn

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It is now natural to consider diﬀerential operators D with matrix valued coeﬃcients acting on Pn on the right-hand side. For details, see section 2. The relevant equation that supplements the previous one to get a bispectral problem is Pn D = Λn Pn

(1.1)

where each Λn is a matrix. We interject here two comments: a) we have two operators, namely L and D acting on our family of polynomials Pn . These operators act on diﬀerent variables, namely n and x, but since they have matrix coeﬃcients the only form to make them commute with each other is to have them acting on diﬀerent sides of the argument Pn . b) there is a long history behind the consideration of this bispectral problem, consisting of a pair of equations satisﬁed by Pn . This arises naturally in an eﬀort to understand some very important work by D. Slepian, H. Landau and H. Pollak, at Bell Labs, back around 1960. For a brief account of this motivation, the reader may want to consult, for instance [11, 12]. The set of those D satisfying (1.1) will be denoted by D(W ). Starting with [15], [16] and [10, 8] one has a growing collection of weight matrices W (x) for which the algebra D(W ) is not trivial, i.e. does not consist only of scalar multiples of the identity operator. The study of this question starts with [7]. A ﬁrst attempt to go beyond the issue of the existence of one nontrivial element in D(W ) and to study (experimentally, with assistance from symbolic computation) the full algebra is undertaken in [5]. An analytical proof of some of the results conjectured in [5] will appear in [32]. In the representation theory of a Lie group G the algebra D(G) of all right invariant diﬀerential operators on G, plays an important role. If π is a ﬁnite dimensional matrix representation of G then [πD](g) = [πD](e)π(g) for all g ∈ G. Thus π is a matrix valued eigenfunction of each D ∈ D(G). More generally, if K is a compact subgroup of G the algebra D(G)K of all diﬀerential operators in D(G) which are also left invariant under K, plays an important role in the study of the spherical functions on G of any K-type. If Φ is a matrix spherical function and D ∈ D(G)K , then [ΦD](g) = [ΦD](e)Φ(g) for all g ∈ G. Thus Φ is a matrix valued eigenfunction of each D ∈ D(G)K . Moreover the eigenvalue map ΛΦ : D → [ΦD](e) is a ﬁnite dimensional representation of the algebra D(G)K , and the family of representations ΛΦ separates the elements of D(G)K . Also D(G)K has a canonical *-operation. As we will see, there is a parallelism between these facts and the contents of this paper. This parallelism should be stronger when G/K is a compact rank one symmetric space as can be seen when G/K is the complex projective plane SU (3)/U (2). In this case the spherical functions can be expressed by means of a sequence of orthogonal polynomials with respect to a weight matrix derived from the Haar measure of G (see [15] and [30]).

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In the preliminary section the basic deﬁnitions and results about weight matrices and sequences of orthogonal polynomials are given. Also the algebra D(W ) is deﬁned and the interplay of its elements with the elements of a sequence of orthogonal polynomials is established. In particular the eigenvalue representations Λn of D(W ) into the algebra A of all N × N complex matrices is introduced. In Section 3 the ad-conditions coming from the bispectral pairs (L, D), where L is the diﬀerence operator associated to the three term recursion relation satisﬁed by the sequence of monic orthogonal polynomials and D ∈ D(W ), are used to described the image of D(W ) into AN0 , the direct product of N0 copies of the matrix algebra A, by the eigenvalue isomorphism Λ = ΠΛn . This gives a completely diﬀerent presentation of D(W ). In Section 4 we are concerned with the notion of the adjoint of a diﬀerential operator. In particular we prove that the algebra D(W ) has a canonical *-operation such that D ∈ D(W ) is symmetric, with respect to the inner product among matrix polynomials deﬁned by the weight matrix W , if and only if D = D∗ . In particular this shows that the problem of ﬁnding one or all symmetric diﬀerential operators (see Deﬁnition 4.4) is the same as to ﬁnd, respectively, one or all elements in D(W ). This paper concludes with a challenge that might be of interest to people working in noncommutative geometry.

2. Preliminaries Let W = W (x) be a weight matrix of size N on the real line. By this we mean a complex N × N -matrix valued integrable function on the interval (a, b) such that W (x) is positive deﬁnitive almost everywhere and with ﬁnite moments of all orders. From now on we shall denote by A the C ∗ -algebra of all N × N matrices over C with the operator norm, and A[x] will denote the algebra over C of all polynomials in the undetermined x with coeﬃcients in A. With the symbol I we will denote the identity of A. We introduce as in [21] and [22] the following Hermitian sesquilinear form in the linear space A[x]: b (P, Q) = P (x)W (x)Q(x)∗ dx. a

We observe that the map P, Q → (P, Q) from A[x] × A[x] into A has the following properties: for all P, Q, R ∈ A[x], a, b ∈ C and T ∈ A we have (aP + bQ, R) = a(P, R) + b(Q, R), (T P, Q) = T (P, Q), (P, Q)∗ = (Q, P ), (P, P ) ≥ 0;

if (P, P ) = 0 then P = 0.

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All these properties follow directly from the deﬁnition except the last one which will be established in the following lemma. In other words we have that A[x] is a left inner product A-module. More generally we could assume that W is an N × N matrix of complex Borel measures on the real line, such that: the numerical matrix W (X) is positive semideﬁnite for any Borel set X, W has ﬁnite moments of any order, and W is nondegenerate, that is for P ∈ A[x] P (x)dW (x)P (x)∗ = 0 (P, P ) = R

only when P = 0. We point out the following general fact: if E is a (left) inner product C ∗ module over a C ∗ algebra A then E has a scalar valued norm ||x|| = ||(x, x)||1/2 and an A-valued norm |x| = (x, x)1/2 . An inner product A-module which is complete with respect to its scalar valued norm is called a Hilbert A-module. Moreover, given an (incomplete) inner product A-module E0 , like our left A-module A[x], one can form its completion E just as in the case of an ordinary inner product space, and thus obtain a Hilbert A-module. See [24]. Although we will not use in this paper the Hilbert A-module completion of our A[x], we feel that it is convenient to be aware of its existence. For another nice and readable introduction to this subject, see [29]. For our next result we need to quote from [1] the following principle of measurable choice: Let X and Y be complete separable metric spaces and E, a closed σ-compact subset of X × Y . Then π1 (E) is a Borel set in X and there exists a Borel function φ : π1 (E) → Y whose graph is contained in E. Let H(N ), U (N ) and ∆(N ) denote respectively, the real vector space of all Hermitian N × N matrices, the unitary group of all N × N matrices and the real vector space of all diagonal N × N matrices. Proposition 2.1. There is a Borel function ψ : H(N ) → U (N ) associating with each Hermitian matrix H, a unitary matrix ψ(H) such that ψ(H)∗ Hψ(H) is real diagonal. Proof. Let E = {(H, U, ∆) ∈ H(N ) × U (N ) × ∆(N ) : U ∗ HU = ∆}. Clearly E is closed in H(N ) × U (N ) × ∆(N ) and, since E = ∪k≥1 Ek where Ek = {(H, U, ∆) ∈ E : ||H|| = ||∆|| ≤ k}, it is σ-compact. Since any Hermitian matrix is unitarily equivalent to a real diagonal matrix, π1 (E) = H(N ). Applying the principle of measurable choice with X = H(N ) and Y = U (N ) × ∆(N ) we obtain a Borel function φ : H(N ) → U (N ) × ∆(N ) whose graph is contained in E. The proof is completed by setting ψ = π1 ◦ φ.

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j Proposition 2.2. Let P = 0≤j≤n x Pj be an A-polynomial of degree n. Then ker (P, P ) = 0≤j≤n ker(Pj∗ ). In particular (P, P ) is nonsingular if Pj is nonsingular for some 0 ≤ j ≤ n. Moreover (P, P ) = 0 implies P = 0. N Proof. Let {ei (x)}N such that W (x)ei (x) = i=0 be an orthonormal basis of C αi (x)ei (x) for 0 ≤ i ≤ N . Using Proposition 2.1, we may assume that the functions ei (x) and αi (x) are measurable functions. If e ∈ CN let ai (x)ei (x). P ∗ (x)e = 0≤i≤N

Then

b b ((P, P )e, e) = (P W P ∗ e, e) dx = (W P ∗ e, P ∗ e) dx. a a Now W P ∗ e = i ai (x)αi (x)ei (x) and (W P ∗ e, P ∗ e) = i ai (x)αi (x)¯ ai (x). Hence b ((P, P )e, e) = ai (x)αi (x)¯ ai (x) dx. a

i

ai (x) = 0 a.e. and Since αi (x) > 0 a.e. (P, P )e = 0 implies that i ai (x)αi (x)¯ therefore ai (x)¯ ai (x) = 0 a.e. for all 0 ≤ i ≤ N . Thus P ∗ (x)e = 0 for all x which is equivalent to Pj∗ e = 0 for all 0 ≤ j ≤ n. This proves the proposition. Corollary 2.3. The even moments of W b M2n = x2n W (x) dx a

are nonsingular matrices for all n ≥ 0. Proof. Let P (x) = xn I. Then

(P, P ) =

b

x2n W (x) dx = M2n .

a

Hence the corollary follows from Lemma 2.2.

Proposition 2.4. Let Vn = {F ∈ A[x] : deg F ≤ n} for all n ≥ 0, V−1 = 0 and ⊥ ⊥ Vn−1 = {H ∈ Vn : (H, F ) = 0 for all F ∈ Vn−1 }. Then Vn−1 is a left A-module and ⊥ (i) Vn = Vn−1 ⊕ Vn−1 for all n ≥ 0. ⊥ 2 (ii) dim Vn−1 = N for all n ≥ 0. ⊥ (iii) There is a unique monic polynomial Pn in Vn−1 and it is of degree n for all n ≥ 0. Proof. We proceed by induction on n ≥ 0. For n = 0 the statements are all true from the deﬁnitions. For n = 1 we have V1 = xA ⊕ A and dim V1 = 2N 2 . We look for P1 = xI + A0 ∈ xA ⊕ A orthogonal to P0 = I: (P1 , P0 ) = (xI, P0 ) + A0 (P0 , P0 ).

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Thus we have to choose A0 = −(xI, P0 )(P0 , P0 )−1 . Then we have that P1 ∈ V0⊥ , since (P1 , T ) = (P1 , P0 )T ∗ = 0 for all T ∈ V0 . If P ∈ V1 then P = xB1 + B0 and P − B1 P1 ∈ V0 . Thus V1 = V0 + V0⊥ . On the other hand if P ∈ V0 ∩ V0⊥ then (P, P ) = 0 which implies that P = 0 by Lemma 2.2. Since dim V1 = 2N 2 and dim V0 = N 2 it follows that dim V0⊥ = N 2 . This completes the proof of the proposition for n = 1. Now let us assume that n > 1 and that the proposition is true for all m ≤ ⊥ n − 1. We look for Pn = xn I + An−1 Pn−1 + · · · + A0 P0 ∈ Vn−1 : (Pn , Pm ) = (xn I, Pm ) + Am (Pm , Pm ). Thus we have to choose Am = −(xn I, Pm )(Pm , Pm )−1 , 0 ≤ m ≤ n − 1. Then it is ⊥ easy to verify that Pn ∈ Vn−1 . If P ∈ Vn then P = xn Bn + · · · + B0 and P − Bn Pn ∈ Vn−1 . Thus Vn = ⊥ ⊥ . On the other hand if P ∈ Vn−1 ∩ Vn−1 then (P, P ) = 0 which implies Vn−1 + Vn−1 that P = 0 by Lemma 2.2. Since dim Vn = (n + 1)N 2 and dim Vn−1 = nN 2 it ⊥ = N 2 . This completes the proof of the proposition. follows that dim Vn−1 Corollary 2.5. Let W = W (x) be a weight matrix of size N . Then {Pn }n≥0 is the unique sequence of monic orthogonal polynomials in A[x]. Moreover any sequence {Qn }n≥0 of orthogonal polynomials in A[x] is of the form Qn = An Pn where An ∈ GLN (C) is arbitrary for each n ≥ 0. The following standard argument, given for instance in [K1,K2] shows that the sequence of monic orthogonal polynomials {Pn }n≥0 satisﬁes a three term recursion relation xPn (x) = An Pn−1 (x) + Bn Pn (x) + Pn+1 (x),

n≥0

(2.1)

where we put P−1 (x) = 0. In fact according to Proposition 2.4 we can write xPn = V0 P0 + · · · + Vn−2 Pn−2 + An Pn−1 + Bn Pn + Pn+1 . For all 0 ≤ j ≤ n − 2 we have 0 = (Pn , xPj ) = (xPn , Pj ) = Vj (Pj , Pj ), hence Vj = 0 for all 0 ≤ j ≤ n − 2 as we wanted to prove. We come now to the notion of a diﬀerential operator with matrix coeﬃcients acting on matrix valued polynomials, i.e. elements of A[x]. These operators could be made to act on our functions either on the left or on the right. One ﬁnds a discussion of these two actions in [7]. The conclusion there is that if one wants to have matrix weights W that are not direct sums of scalar one and that have matrix polynomials as their eigenfunctions, one should settle for right-hand side diﬀerential operators. We agree now to say that D given by D=

s i=0

∂ i Fi (x),

∂=

d , dx

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acts on P (x) by means of PD =

s

∂ i (P )(x)Fi (x)

i=0

Before going on we make two observations. The three term recursion mentioned above gives rise to a diﬀerence operator L, to be formally introduced in Section 3. This operator, which acts on the variable n of our family of polynomials Pn (x) is acting on the left. As in any bispectral situation there is another operator, acting in the variable x and its action should commute with that of L. In the matrix case the only way to get L and a diﬀerential operator D to commute for sure when acting on Pn (x) is to make them act on diﬀerent sides. One could make D act on P on the right as deﬁned above, and still write down the symbol DP for the result. The advantage of using the notation P D is that it respects associativity: if D1 and D2 are two diﬀerential operators we have P (D1 D2 ) = (P D1 )D2 . We have a right module. We are ready to continue. Proposition 2.6. Let W = W (x) be a weight matrix of size N and let {Pn }n≥0 be the sequence of monic orthogonal polynomials in A[x]. If s d D= , ∂ i Fi (x), ∂= dx i=0 is a linear right-hand side ordinary diﬀerential operator of order s such that Pn D = Λn Pn

for all

n≥0

(2.2)

with Λn ∈ A, then Fi = Fi (x) ∈ A[x] and deg Fi ≤ i. Moreover D is determined by the sequence {Λn }n≥0 . We could have written the eigenvalue matrix Λn to the right of the matrix valued polynomials Pn above. However, as shown in [7] this only leads to uninteresting cases where the weight matrix is diagonal. We are dealing with a bimodule and it is important to keep the scalars (matrices) in the appropriate place. Proof. If we put n = 0 in (2.2) we get F0 = Λ0 . If we put n = j ≥ 1 in (2.2) we get j j−1 i ∂ (Pj )Fi = j!Fj + ∂ i (Pj )Fi , Λj Pj = Pj D = i=0

therefore j!Fj = Λj Pj −

j−1

i=0

∂ i (Pj )Fi

for all n ≥ 0.

(2.3)

i=0

Now by induction on j it follows that Fj is a polynomial of degree less or equal to j for all j ≥ 0. Also from (2.3) it is clear that the sequence {Λn }n≥0 determines the diﬀerential operator D.

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To ease the notation if ν ∈ C let [ν]i = ν(ν − 1) · · · (ν − i + 1), [ν]0 = 1. s i Proposition 2.7. Let D = i=0 ∂ Fi (x) satisfy (2.2), with Fi (x) =

i

xj Fji (D).

(2.4)

j=0

Then Λn =

s

[n]i Fii (D)

for all

n ≥ 0.

(2.5)

i=0

Hence n → Λn is a matrix valued polynomial function of degree less or equal to ord(D). Proof. From (2.2) one gets s

∂ i (Pn )(x)Fi (x) = Λn Pn (x).

i=0

Comparing monomials of degree n we obtain s [n]i Fii (D) = Λn .

i=0

We are ready to introduce the main character of our tale. Given a sequence of orthogonal polynomials {Qn }n≥0 we shall be interested in the algebra D(W ) of all right-hand side diﬀerential operators with matrix valued coeﬃcients that have the polynomials Qn as their eigenfunctions. Notice that if Qn D = Γn Qn for some eigenvalue matrix Γn ∈ A, then Γn is uniquely determined by D. In such a case we write Γn (D) = Γn . Thus D(W ) = {D : Qn D = Γn (D)Qn , Γn (D) ∈ A for all n ≥ 0}.

(2.6)

First of all we observe that the deﬁnition of D(W ) depends only on the weight matrix W = W (x) and not on the sequence {Qn }n≥0 . This follows at once from Corollary 2.5. Proposition 2.8. Given a sequence {Qn }n≥0 of orthogonal polynomials let us con. sider the algebra D(W ) deﬁned in (2.6). Also let Γ(D, n) = Γn (D). Then D → Γ(D, n) is a representation of D(W ) into A, for each n ≥ 0. Moreover the sequence of representations {Γn }n≥0 separates the elements of D(W ). Proof. That D → Γ(D, n) is a linear map from D(W ) into A is obvious, and that it is not identically zero for each n ≥ 0 follows from the fact that Γ(I, n) = I, where I denotes the identity diﬀerential operator and also the identity matrix. If D1 and D2 are in D(W ) then from the deﬁnition (2.6) it follows that Pn (D1 D2 ) = (Γ(D1 , n)Pn )D2 = Γ(D1 , n)(Pn D2 ) = Γ(D1 , n)Γ(D2 , n)Pn .

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Hence Γ(D1 D2 , n) = Γ(D1 , n)Γ(D2 , n). Finally by writing Qn = An Pn we get Γn (D) = An Λn (D)A−1 n . Now the last assertion follows from Proposition 2.6. It is worth observing that each algebra D(W ) is a subalgebra of the Weyl algebra D over A of all linear right-hand side ordinary diﬀerential operators with coeﬃcients in A[x]: D= D= ∂ i Fi : Fi ∈ A[x] . i

It is also interesting to introduce the subalgebra D of the Weyl algebra D deﬁned by ∂ i Fi ∈ D : deg Fi ≤ i . D= D= i

Then from Proposition 2.6 it follows that D(W ) ⊂ D for any weight matrix W . Also this algebra D comes with a family {Λν }ν∈C of N -dimensional representations deﬁned by Λν (D) = [ν]i Fii , i

as we establish in the following proposition. Proposition 2.9. If D = si=0 ∂ i Fi ∈ D with Fi = ij=0 xj Fji , then Λν (D) =

s

[ν]i Fii

i=0

deﬁnes a representation Λν of D into A, for any ν ∈ C. Proof. It is clear that Λν is a linear map from D into A. Thus, to prove that it is a representation it is suﬃcient to establish that

Λν (∂ s Fs )(∂ r Gr ) = Λν (∂ s Fs )Λν (∂ r Gr ), for all s, r ≥ 0. The Leibnitz rule gives r r s+i (r−i) (∂ s Fs )(∂ r Gr ) = ∂ Fs Gr . i i=0 Therefore

r

r Λν (∂ s Fs )(∂ r Gr ) = [ν]s+i Fs(r−i) Gr s+i i i=0 r r = [ν]s+i [s]r−i Fss Grr . i i=0

Then, to ﬁnish the proof, it is enough to see that r r [ν]s+i [s]r−i = [ν]s [ν]r . i i=0

(2.7)

(2.8)

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This can be established by diﬀerentiating r- times the following identity (xν )(s) xs = [ν]s xν , which holds for all x ≥ 0. From the left-hand side we obtain r

ν (s) s (r) r (x ) x = (xν )(s+i) (xs )(r−i) i i=0 r r = [ν]s+i xν−s−i [s]r−i xs−r+i i i=0 r r = [ν]s+i [s]r−i xν−r . i i=0 From the right-hand side we get

(r) [ν]s xν = [ν]s [ν]r xν−r . This completes the proof of (2.8) and therefore the proposition is also proved. Proposition 2.10. If D ∈ D satisﬁes the symmetry condition (P D, Q) = (P, QD) for all P, Q ∈ A[x], then D ∈ D(W ). Proof. Since D ∈ D, D(Vn ) ⊂ Vn (see Proposition 2.4 for the deﬁnition of Vn ⊥ ⊥ ⊥ and Vn−1 ). Then, by the symmetry hypothesis, D(Vn−1 ) ⊂ Vn−1 . Therefore, by Proposition 2.4, Pn D = Λn Pn for some Λn ∈ A. This completes the proof of the proposition. It is worth observing that the diﬀerential operator of order zero D = xI satisﬁes the symmetry condition, but no polynomial can be an eigenvector of it.

3. The ad-conditions Proposition 2.8 gives for each diﬀerential operator D in D(W ) a sequence of representations {Λn }n≥0 of the algebra D(W ) into the algebra A of N × N matrices. In other words we have a homomorphism Λ : D(W ) → AN0 of D(W ) into the direct product of N0 copies of A. Moreover Λ is injective. In Theorem 3.1 we give a precise description of the range of this homomorphism. We start with a couple of basic remarks that do not involve the algebra D(W ). Recall that our starting point is a weight matrix W (x) on the real line and its unique sequence of monic orthogonal polynomials {Pn }n≥0 , together with the three-term recursion relation see (2.1) xPn (x) = An Pn−1 (x) + Bn Pn (x) + Pn+1 (x), where we put P−1 (x) = 0.

n ≥ 0,

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It is convenient to introduce the block tridiagonal matrix L   B0 I   L = A1 B1 I  .. .. .. . . . where all the matrices Ai , Bi are in A and I denotes the N × N identity matrix. The recursion relation now takes the form LP = xP where P stands for the vector

(3.1)

  P0 (x) P1 (x)   P (x) = P2 (x) .   .. .

The ﬁrst consequence of (3.1) is obtained by applying the diﬀerential operator ∂ i to both sides to yield, since L is independent of x, L∂ i (P ) = i∂ i−1 (P ) + x∂ i (P ), i.e., (L − xI)∂ i (P ) = i∂ i−1 (P ). We now consider the diﬀerence equation

(3.2)

LQ = xQ for a vector



 Q0 (x)   Q(x) = Q1 (x) .. .

(Q−1 (x) = 0)

where Q0 (x) is given and arbitrary. It is clear that Qn (x) is completely determined once L and Q0 (x) are given, i.e., we have Qn (x) = Mn (x)Q0 (x) where the N × N matrix Mn (x) is a polynomial in x of degree N that depends in a complicated way on the matrices Ai , Bi that make up L. In particular we have Pn (x) = Mn (x)P0 (x) = Mn (x) and thus Qn (x) = Mn (x)Q0 (x) = Pn (x)Q0 (x). This gives us the second important consequence of (3.1), namely: any solution Q of the equation LQ = xQ is given by Qn (x) = Pn (x)Q0 (x) (3.3)

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where Pn (x) are the monic orthogonal polynomials which give rise to L. After these two observations we are ready to go back to D(W ). Assume that D ∈ D(W ), i.e., Pn D = Λn Pn n ≥ 0. If Λ denotes the block diagonal matrix  Λ0  Λ=

  

Λ1 ..

.

we observe that from (3.1) we get, for any integer m ≥ 0, . (ad L)m (Λ)P = [L, . . . , [L, Λ] . . . ]P m times m m = (−1)m−j Lj ΛLm−j P j j=0 m m = (−1)m−j Lj Λxm−j P j j=0 m m−j m (−1) = Lj xm−j ΛP j

(3.4)

j=0

= (L − xI)m ΛP. We are now in a position to state the main result of this section. Theorem 3.1. Let W (x) be a weight matrix on the real line, {Pn }n≥0 the corresponding sequence of monic orthogonal polynomials and L the block tridiagonal matrix that gives LP = xP . If D ∈ D(W ) and Λ is the block diagonal matrix as above with Λn = Λn (D) we have (ad L)m+1 (Λ) = 0

(3.5)

for some m. Conversely, if Λ is a block diagonal matrix satisfying this condition for some m ≥ 0, then there is a unique diﬀerential operator D in D(W ) such that Λn = Λn (D) for all n ≥ 0. Moreover the order of D is equal to the minimum m satisfying (3.5). Remark 3.2. Our proof produces the diﬀerential operator D in D(W ) explicitly. Proof. We start by showing that condition (3.5) is suﬃcient. This is the hardest part of the proof. The argument here is based on the one in [DG] where necessity is proved easily but suﬃciency takes a few pages. In the matrix case necessity is shown in [GI] and suﬃciency is tackled in [M]. The ad-conditions that appear above

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were ﬁrst introduced in [DG] and used later in [13], [14], [27] and [33]. Assume now that, for some m ≥ 0, we have (ad L)m+1 (Λ) = 0. This gives, from (3.4), (ad L)m+1 (Λ)P = (L − xI)m+1 ΛP = 0. This means that the vector . Q(m) = (L − xI)m ΛP satisﬁes LQ(m) = xQ(m) and thus, by (3.3), we have Q(m) n (x) = Pn (x)S0 (x)

n≥0

for some S0 = S0 (x). The value of S0 is given by the ﬁrst component of Q(m) , S0 = ((L − xI)m ΛP )0 . Introduce now a new vector Q(m−1) by means of . Q(m−1) = (L − xI)m−1 ΛP − ∂(P )S0 . We claim that, just as in the previous step, (L − xI)Q(m−1) = 0. Indeed, we have (L − xI)Q(m−1) = (L − xI)m ΛP − (L − xI)∂(P )S0 . The ﬁrst term on the right is Q(m) and using (3.2) for i = 1, the last term gives −P S0 . The previous step shows that these two terms are the same and we can then conclude, by (3.3), that = Pn S1 Q(m−1) n for some S1 = S1 (x). Its explicit expression is given by looking at the ﬁrst component of the vector Q(m−1) , and we get S1

= ((L − xI)m−1 ΛP )0 − ∂(P0 )S0 = ((L − xI)m−1 ΛP )0 .

For the beneﬁt of the reader we show explicitly one more step and then give the general inductive argument. Introduce the vector Q(m−2) by means of S0 . − ∂(P )S1 . Q(m−2) = (L − xI)m−2 ΛP − ∂ 2 (P ) 2 Now S0 (L − xI)Q(m−2) = (L − xI)m−1 ΛP − (L − xI)∂ 2 (P ) − (L − xI)∂(P )S1 . 2

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The ﬁrst term on the right-hand side gives, from the deﬁnition of Q(m−1) , Q(m−1) + ∂(P )S0 . The second and third terms can be replaced (using (3.2) for i = 1 and 2) by S0 and − P S1 , −2∂(P ) 2 respectively. By using the conclusion of the previous step this gives a vanishing right-hand side and we can now conclude, see (3.3), that Qm−2 = Pn S2 n where the expression for S2 = S2 (x) is S2 = ((L − xI)m−2 ΛP )0 . Notice that here, and before, we have used that P0 (x) is independent of x. Now assume by induction that we have deﬁned, for 0 ≤ i < k, vectors Q

(m−i)

Si−r . = (L − xI)m−i ΛP − ∂ r (P ) r! r=1 i

and matrices S0 = S0 (x), . . . , Sk−1 = Sk−1 (x) such that (x) = Pn (x)Si (x), Q(m−i) n with Si = ((L − xI)m−i ΛP )0 . Introduce Q(m−k) by means of Sk−1 S0 S1 . − ∂ k−1 (P ) − · · · − ∂(P ) . Qm−k = (L − xI)m−k ΛP − ∂ k (P ) k! (k − 1)! 1 An application of (L − xI) to both sides gives (L − xI)Qm−k

=

(L − xI)m−(k−1) ΛP − (L − xI)∂ k (P )

−

(L − xI)∂ k−1 (P )

S0 k!

Sk−1 S1 − · · · − (L − xI)∂(P ) . (k − 1)! 1

The ﬁrst term yields, by induction, Q

m−(k−1)

+

k−1 r=1

∂ r (P )

Sk−1−r . r!

The remaining terms can be replaced by S0 S1 −∂ k−1 (P ) − ∂ k−2 (P ) − · · · − ∂(P )Sk−2 − P Sk−1 . (k − 1)! (k − 2)! All these terms except for the last one above cancel the terms in the previous summation and we are left with (L − xI)Q(m−k) = Qm−(k−1) − P Sk−1 . By the inductive hypothesis this vanishes and, using (3.3), we conclude Q(m−k) = P Sk

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for a matrix Sk = Sk (x) given by Sk = ((L − xI)m−k ΛP )0 . This process can be continued as long as k ≤ m. For k = m we get m

Q(0) = ΛP −

∂ r (P )

r=1

Sm−r . r!

Once again we get (L − xI)Q(0) = Q(1) − P Sm−1 which, by induction has been shown to vanish. A last appeal to (3.3) gives Q(0) = P Sm for a matrix Sm = Sm (x) given by Sm = (ΛP )0 = Λ0 . An alternative way of writing these identities gives Q(0) = P Sm = P Λ0 . Using the expression that deﬁnes Q(0) we have ΛP −

m

∂ r (P )

r=1

or

m

∂ r (P )

r=1

Sm−r = P Λ0 r!

Sm−r + P Λ0 = ΛP r!

which can be rewritten as m

∂ r (P )

r=0

Sm−r = ΛP. r!

This gives for the operator D advertised earlier the expression D=

m

∂ r (P )

r=0

Sm−r r!

and we have shown, as promised, that Pn D = Λn Pn

n ≥ 0.

Recall that we have explicit expressions for the coeﬃcient matrices Si = Si (x) in terms of L and Λ. Also observe that the order of D is less or equal to the minimum m satisfying (3.5). Now we shall establish that the condition (3.5)is necessary. We start from P D = ΛP

and LP = xP.

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Compute the action of . . ad(x)(D) = [x, D] = xD − Dx on the vector P P [x, D]

= P xD − P Dx = LP D − ΛP x = LΛP − ΛLP = (LΛ − ΛL)P = [L, Λ]P = ad L(Λ)P.

The point of this computation is that the (right) action of the diﬀerential operator A = [x, D] on P has been converted into the (left) action of the diﬀerence operator . B = [L, Λ] on the same vector P . So we have P A = BP . If we start from here and compute the action of [x, A] on P we get P [x, A] = P xA − P Ax = LP A − BP x = LBP − BLP = [L, B]P, i.e., P (ad x)2 (D) = (ad L)2 (Λ)P. If this is iterated we get P (ad x)j (D) = (ad L)j (Λ)P for any nonnegative integer j. If D has order m we observe that the diﬀerential operator (ad x)m+1 (D) vanishes identically and we conclude that (ad L)m+1 (Λ)P = 0. ﬁnite-band matrix (ad L)m+1 (Λ), then for If Aij denotes the ij-block of the each i Aij is zero for almost all j, and j Aij Pj = 0. Hence 0= Aij Pj , Pk = Aij (Pj , Pk ) = Aik (Pk , Pk ), j

j

because {Pn }n≥0 is a sequence of orthogonal polynomials, thus Aik = 0 for all i, k and we conclude that the matrix (ad L)m+1 (Λ) = 0. Observe that we have also established that the order of D is greater or equal to the minimum m satisfying (3.5). Since the correspondence D → Λ(D) is a bijection we now have ord(D) ≤ min{m : (3.5) holds} ≤ ord(D). This completes the proof of the theorem.

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4. The Adjoint of D in D(W ) It is well known that deﬁning the adjoint of a diﬀerential operator (even in the scalar case) is a delicate matter, which we will recall below. The reader should be reassured that we are only trying to deﬁne the adjoint for diﬀerential operators in D(W ). This will allow us to circumvent all the delicate issues dealing with domains of deﬁnitions, diﬀerent selfadjoint extensions of a symmetric operator, etc. We start with some comments about the adjoint of a diﬀerential operator on the closed interval [0, 1]. Let D=

n

fi (x)∂ i

i=0

be a linear diﬀerential operator with C ∞ coeﬃcients on the interval [0, 1]. Then a diﬀerential operator D∗ is call a formal adjoint of D if 1 1 Df (x)¯ g (x)dx = f (x)D∗ g(x)dx 0

0

∞

for all f, g ∈ C ([0, 1]) vanishing at the end points of the interval. Then the existence (integration by parts) and the uniqueness of the formal adjoint are easy to establish. This is a special case of a similar result for diﬀerential operators between Hermitian vector bundles over a diﬀerentiable manifold M, possibly with boundary, with a strictly positive smooth measure. A completely diﬀerent situation arise if we look for a diﬀerential operator D on [0, 1] such that 1 1 Df (x)¯ g (x)dx = f (x)Dg(x)dx 0

0

exits it is a formal adjoint for all f, g ∈ C ∞ ([0, 1]). Then it is clear that if such D and therefore unique. In such a case we refer to it as the adjoint of D. As an example let us see that the diﬀerential operator ∂ = d/dx on [0, 1] has no adjoint. The following argument is taken from [19]. In fact we will prove that there is no ∂ such that 1 1 ∂f (x)¯ g (x)dx = f (x)∂g(x)dx 0

0

for all f, g ∈ C[x]. First of all let us see that the linear functional on C[x] deﬁned by L(f ) = c1 f (z1 ) + · · · + cn f (zn ) for some c1 , . . . , cn ∈ C× and some z1 , . . . , zn ∈ C, can not be represented by any g ∈ C ∞ ([0, 1]). By the contrary suppose that there exists g ∈ C ∞ ([0, 1]) such that 1 f (x)¯ g (x) dx c1 f (z1 ) + · · · + cn f (zn ) = 0

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for all f ∈ C[x]. Let h(x) = (x − z1 ) · · · (x − zn ). Then 1 h(x)f (x)¯ g (x) dx = c1 (hf )(z1 ) + · · · + cn (hf )(zn ) = 0 0

¯ we get for all f ∈ C[x]. Then, taking f = hg 1 |h(x)|2 |g(x)|2 dx = 0, 0

which implies that g = 0 which is a contradiction. Then Now let us assume that ∂ on the interval [0, 1] had an adjoint ∂. 1 1 f (x)∂g(x) ∂f (x)¯ g (x)dx dx = 0

0

= f (1)¯ g(1) − f (0)¯ g(0) −

1

f (x)∂g(x)dx 0

for all f, g ∈ C[x]. Hence the linear functional L(f ) = f (1)¯ g(1) − f (0)¯ g(0) (g ﬁx) + ∂g which contradicts what we proved before. would be represented by ∂g Now something interesting happens with the diﬀerential operators of the algebra D(W ) associated to a weight matrix W = W (x) on the real line. In fact in ∈ D(W ) this section we shall establish that for any D ∈ D(W ) there is a unique D such that (P D, Q) = (P, QD) for all P, Q ∈ A[x]. For the purpose at hand this choice of a domain, namely A[x], for the operators D in D(W ) suﬃces. Even in the scalar case one can consider issues which are of no use to us in this paper: for example the familiar second order diﬀerential operator of Legendre allows for many diﬀerent selfadjoint extensions, since one is in the limit circle case-at each end point-in Weyl’s classiﬁcation. orthogonal polynomials Proposition 4.1. Let {Pn }n≥0 be the sequence of monic s associated to the weight matrix W = W (x). Given D = i=0 ∂ i Fi ∈ D(W ) and s i = D i=0 ∂ Gi ∈ D, then for all n, m ≥ 0 the following conditions are equivalent: for all P, Q ∈ A[x] with deg P ≤ n, deg Q ≤ m, (i) (P D, Q) = (P, QD) i i ∗ (ii) 0≤j≤i≤s [u]i Fj Mu+v+j−i = 0≤j≤i≤s [v]i Mu+v+j−i (Gj ) for all 0 ≤ u ≤ n, 0 ≤ v ≤ m, for all 0 ≤ u ≤ n, 0 ≤ v ≤ m. (iii) (Pu D, Pv ) = (Pu , Pv D) for Proof. If (i) holds, in particular we have that ((xu I)D, xv I) = (xu I, (xv I)D) all 0 ≤ u ≤ n, 0 ≤ v ≤ m. Thus b b s s ∗ v u−i x [u]i x Fi W dx = xu W [v]i xv−i Gi dx, a

i=0

a

i=0

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which is equivalent to b [u]i xu+v+j−i Fji W dx = 0≤j≤i≤s

a

[v]i a

0≤j≤i≤s

b

469

xu+v+j−i W (Gij )∗ dx.

This proves (ii). To see that (ii) implies (iii) we simply write q v (Pu D, Pv ) = xp Ppu D, x Pq =

p

q

Ppu ((xp I)D, xq I)(Pqv )∗

p,q

=

p

x

Ppu ,

p

=

v ∗ Ppu (xp I, (xq I)D)(P q)

p,q

q

x

Pqv

= (Pu , Pv D). D

q

Finally to prove that (iii) implies (i) we write P = 0≤v≤m Bv Pv with Au , Bv ∈ A. Then (P D, Q) = Au Pu D, Bv Pv =

u

v

Au (Pu D, Pv )Bv∗ =

u,v

=

u

Au Pu ,

0≤u≤n

Au Pu and Q =

∗ Au (Pu , Pv D)B v

u,v

Bv Pqv

= (Pu , Pv D). D

v

The proof of the proposition is complete. Lemma 4.2. For any b, c, d ∈ N we have b+d−j b−c j c+d (−1) = . j b d

(4.1)

0≤j≤d

Proof. A proof of this lemma is based on the following identity m (1 + w)m 1 dw, = 2πi wn+1 n

(4.2)

which holds for any m, n ∈ No , and can be carried out using the techniques described in [9]. polynomials asTheorem 4.3. Let {Pn }n≥0 be the sequence of monic orthogonal s sociated to the weight matrix W = W (x). Given D = ∂ i Fi ∈ D(W ) let i=0 = s ∂ i Gi ∈ D, where the Gi are deﬁned inductively by D i=0

(i) G0 = (P0 , P0 )Λ0 (D)∗ (P0 , P0 )−1 , and j−1 (ii) j!Gj = (Pj , Pj )Λj (D)∗ (Pj , Pj )−1 Pj − i=0 ∂ i (Pj )Gi for 1 ≤ i ≤ s. for all P, Q ∈ A[x]. Then (P D, Q) = (P, QD)

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Proof. The ﬁrst thing to be observed is that by deﬁnition if 0 ≤ m ≤ s then = Pm D

s

i

∂ (Pm )Gi = m!Gm +

m−1

i=0

∂ i (Pm )Gi

i=0

= (Pm , Pm )Λ(D)∗ (Pm , Pm )−1 Pm . Let 0 ≤ m ≤ s and n ≥ 0. Then (Pn D, Pm ) = Λn (D)(Pn , Pm ) = δnm Λm (D)(Pm , Pm ) and

= (Pn , Pm ) (Pm , Pm )Λm (D)∗ (Pm , Pm )−1 ∗ (Pn , Pm D) = δnm Λm (D)(Pm , Pm ). Therefore for all 0 ≤ m ≤ s and n ≥ 0 we have (Pn D, Pm ) = (Pn , Pm D).

Thus from Proposition 4.1 we know that for each 0 ≤ m ≤ s and n ≥ 0 the following equation En,m holds: [n]i Fji Mn+m+j−i = [m]i Mn+m+j−i (Gij )∗ . 0≤j≤i≤s

0≤j≤i≤s

Now we go on to prove that for each n ≥ 0 the equations En,m hold for all m > s by proving that each of these equations are linear combinations of the En,r with 0 ≤ r ≤ s. We start by looking for a solution a0 , a1 , . . . , as of the following system of linear equations a0 [0]i + a1 [1]i + · · · + as [s]i = [m]i This system is equivalent to 0 1 s m a0 + a1 + · · · + as = i i i i

0 ≤ i ≤ s.

0 ≤ i ≤ s,

whose coeﬃcient matrix is the Pascal matrix (pi,j ) where pi,j = ji , 0 ≤ i, j ≤ s.

The inverse (qi,j ) of the Pascal matrix is given by qi,j = (−1)i+j ji . Therefore the unique solution of our system is: s j m (−1)i+j 0 ≤ i ≤ s. ai = i j j=i The coeﬃcients a0 , a1 , . . . , as have been chosen in such a way that the righthand side of the linear combination a0 En+m,0 + a1 En+m−1,1 + · · · + as En+m−s,s

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is equal to the right-hand side of the equation En,m . In fact the right-hand side of the linear combination is ar [r]i Mn+m+j−i (Gij )∗ = ar [r]i Mn+m+j−i (Gij )∗ 0≤r≤s

0≤j≤i≤s

0≤j≤i≤s

=

0≤r≤s

[m]i Mn+m+j−i (Gij )∗ .

0≤j≤i≤s

Now we have to check that the same happens with the left-hand sides: ar [n + m − r]i Fji Mn+m+j−i = ar [n + m − r]i 0≤r≤s

0≤j≤i≤s

0≤j≤i≤s

×Fji Mn+m+j−i

=

0≤r≤s

[n]i Fji Mn+m+j−i .

0≤j≤i≤s

This will be established by proving the following identity: ar [n + m − r]i = [n]i . 0≤r≤s

If we divide both side by i! this is equivalent to n+m−r n ar = . i i

(4.3)

0≤r≤s

We just compute s s n+m−r m n+m−r r+j j ar (−1) = i r j i r=0 j=r 0≤r≤s

=

j s j=0 r=0

r+j

(−1)

j m n+m−r . r j i

(4.4)

If we let b = i, c = i − n − m + j, d = n + m − i and use Lemma 4.2 we get j n+m−r n+m−j r j (−1) = . r i i−j r=0 Thus if we go back to (4.4) we obtain s n+m−r m n+m−j ar (−1)j = . i j i−j j=0 0≤r≤s

If we let b = n + m − i, c = m − i, d = i and use again Lemma 4.2 we get s m n+m−j n (−1)j = , j i − j i j=0 which establishes (4.3) completing the proof of the theorem.

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Deﬁnition 4.4. A diﬀerential operator D ∈ D that satisﬁes (P D, Q) = (P, QD) for all P, Q ∈ A[x] will be called symmetric. Then, by Proposition 2.10, the set S(W ) of all symmetric diﬀerential operators is a real subspace of D(W ). Corollary 4.5. For any D ∈ D(W ) there exists a unique diﬀerential operator D∗ ∈ D(W ) such that (P D, Q) = (P, QD∗ ) for all P, Q ∈ A[x]. We shall refer to D∗ as the adjoint of D. The map D → D∗ is a *-operation in the algebra D(W ), and the orders of D and D∗ coincide. Moreover S(W ) is a real form of the space D(W ), i.e. D(W ) = S(W ) ⊕ iS(W ) as real vector spaces. If {Qn }n≥0 is a sequence of orthogonal polynomials and {Γn }n≥0 is the corresponding sequence of representations of D(W ) (see Proposition 2.8), then Γn (D∗ ) = (Qn , Qn )Γn (D)∗ (Qn , Qn )−1 for all D ∈ D(W ). In particular if {Qn }n≥0 is a sequence of orthonormal polynomials then D is symmetric if and only Γn (D) is Hermitian for all n ≥ 0. Proof. The existence of D∗ was established in the previous theorem. The uniqueness of D∗ and the fact that the map D → D∗ is a *-operation in the algebra D(W ) follow at once from Proposition 2.2. From Theorem 4.3 it follows that ord(D∗ ) ≤ ord(D) and since (D∗ )∗ = D we get ord(D) ≤ ord(D∗ ) ≤ ord(D), which proves that ord(D∗ ) = ord(D). That S(W ) is a real form of the space D(W ) follows from the fact that D → D∗ is an involutive linear map over R such that (iD)∗ = −iD∗ . The fourth assertion is a consequence of Γn (D)(Qn , Qn ) = (Qn D, Qn ) = (Qn , Qn D∗ ) = (Qn , Qn )Γn (D∗ )∗ , hence Γn (D∗ ) = (Qn , Qn )Γn (D)∗ (Qn , Qn )−1 . Finally if {Qn }n≥0 is a sequence of orthonormal polynomials then Γn (D∗ ) = Γn (D)∗ . Then D = D∗ implies Γn (D) = Γn (D∗ ) = Γn (D)∗ . Conversely if Γn (D) = Γn (D)∗ , then Γn (D) = Γn (D)∗ = Γn (D∗ ) and D = D∗ by Proposition 2.8. Corollary 4.6. If D ∈ S(W ) then there exists a sequence of orthonormal polynomials {Qn } in A[x] such that Γn (D) is diagonal for all n ≥ 0. Proof. If {Qn } is orthonormal, then Γn (D) is Hermitian and therefore there exists a sequence of unitary matrices Un such that Un Γn (D)Un−1 = ∆n (D) is a diagonal matrix for all n ≥ 0. Then {Un Qn } is a new orthonormal sequence such that (Un Qn )D = Un Γn (D)Qn = Un Γn (D)Un−1 (Un Qn ) = ∆n (D)(Un Qn ). This completes the proof of the corollary.

In conclusion, we observe that if one is given a weight matrix W (x), then the algebra D(W ) considered here is most likely going to be trivial, i.e. CI. In [8]and [16] one ﬁnds necessary and suﬃcient conditions on W such that some second order

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diﬀerential operator, symmetric in the sense deﬁned above, should exist. A similar result can, of course, be given for a symmetric diﬀerential operator of any order. Therefore one has, modulo the diﬃcult task of explicitly solving the equations in [8] and [16], a way of getting S(W ). An important consequence of the results above is that having determined S(W ) we have determined all the algebra D(W ). As far as getting explict results there is another way to proceed: solve the ad-conditions given in section 3. So far this has not been possible. We end with the remark that in the scalar case considered in [6], there is a description of the solutions of these equations: the corresponding operators L are obtained by succesive applications of the (rational) Darboux process starting form very special Bessel or Airy cases. To obtain any explicit results of this kind in this noncommutative situation remains an interesting challenge. An even larger, and rather blurry, challenge is that of ﬁnding the appropriate algebro-geometric objects that reduce in the abelian case to a curve and a bundle on it. Acknowledgment This paper is partially supported by NSF Grant DMS 0603901, CONICET Grant PIP 6304 and by FONCYT Grant PICT 314554.

References [1] E.A. Azoﬀ, Borel measurability in linear algebra, Proc. Math. Soc. 42 nr. 3 (1974), 346-350. [2] H. Airault, H. McKean and J. Moser, Rational and elliptic solutions of the Kortewegde Vries equation and a related many-body problem, Comm. Pure and Applied Math. 30 (1977), 95–148. [3] Y. Berest and G. Wilson, Classiﬁcation of rings of diﬀerential operators on aﬃne curves, Internat. Math. Res. Notices 2 (1999), 105–109. ¨ [4] S. Bochner, Uber Sturm–Liouvillesche polynomsysteme, Math Z. 29 (1929), 730–736. [5] M. Castro and F.A. Gr¨ unbaum, The algebra of matrix valued diﬀerential operators associated to a given family of matrix valued orthogonal polynomials: ﬁve instructive examples IMRN, 2006. [6] J.J. Duistermaat and F.A. Gr¨ unbaum, Diﬀerential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240. [7] A.J. Duran, Matrix inner product having a matrix symmetric second order diﬀerential operators, Rocky Mountain Journal of Mathematics 27, nr. 2 (1997), 585–600. [8] A.J. Duran and F.A. Gr¨ unbaum, Orthogonal matrix polynomials satisfying second order diﬀerential equations, International Math. Research Notices, 2004 : 10 (2004), 461–484. [9] G.P. Egorychev, Integral representation and the computation of combinatorial sums, Translations of Mathematical Monographs, AMS. 59 (1984), 1-286.

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[10] F.A. Gr¨ unbaum, Matrix valued Jacobi polynomials, Bull. Sciences Math 127 nr. 3 (2003), 207–214. [11] F.A. Gr¨ unbaum, The bispectral problem: an overview, In: Special Functions 2000: Current Perspective and Future Directions, Eds. J. Bustoz et al. (2001), 129–140. [12] F.A. Gr¨ unbaum, Some bispectral musings, In: The bispectral problem (Montreal, PQ, 1997), 11–30, CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, RI, 1998. [13] F.A. Gr¨ unbaum and L. Haine, A theorem of Bochner revisited, A.S. Fokas and I.M. Gelfand (eds.), Algebraic Aspects of Integrable Systems, 143–172 , Progr. Nonlinear Diﬀerential Equations 26, Birkh¨ auser, Boston, 1997. [14] F.A. Gr¨ unbaum and P. Iliev, A noncommutative version of the bispectral problem, J. of Computational and Appl. Math. 161 (2003), 99–118. [15] F.A. Gr¨ unbaum, I. Pacharoni and J.A. Tirao, Matrix valued spherical functions associated to the complex projective plane, J. Functional Analysis 188 (2002), 350–441. [16] F.A. Gr¨ unbaum, I. Pacharoni and J.A. Tirao, Matrix valued orthogonal polynomials of the Jacobi type, Indag. Mathem. 14 nrs. 3,4 (2003), 353–366. [17] L. Haine and P. Iliev, Commutative rings of diﬀerence operators and an adelic ﬂag manifold, Internat. Math. Res. Notices 6 (2000), 281–323. [18] J. Harnad and A. Kasman, editors, The bispectral problem Amer. Math. Soc., Providence, CRM proceedings and lectures notes 14 (1998). [19] K. Hoﬀman and R. Kunze, Linear Algebra, Prentice-Hall, 1971, New Jersey. [20] E.L. Ince, Ordinary Diﬀerential Equations, Dover 1928. [21] M.G. Krein, Fundamental aspects of the representation theory of hermitian operators with deﬁciency index (m, m), AMS Translations, Series 2, 97, Providence, Rhode Island (1971), 75–143. [22] M.G. Krein, Inﬁnite J-matrices and a matrix moment problem, Dokl. Akad. Nauk SSSR 69 nr. 2 (1949), 125–128. [23] I.M. Krichever, Algebraic curves and non-linear diﬀerence equations, (Russian) Uspekhi Mat. Nauk 33 (1978), 215–216, translation in Russ. Math. Surveys 33 (1978), 255-256. [24] E.C. Lance, Hilbert C*-modules Lectures Notes, London Math. Soc. 210, Cambridge University Press 1995. [25] F. Magri and J. Zubelli, Diﬀerential equations in the spectral parameter, Darboux transformations and a hierarchy of master equations for KdV, Comm. Math. Physics 141 (1991), 329–351. [26] L. Miranian, Matrix valued orthogonal polynomials, Thesis, UC Berkeley, 2005. [27] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related nonlinear equations, in: M. Nagata (ed.), Proceedings of International Symposium on Algebraic Geometry (Kyoto 1977), Kinokuniya Book Store, Tokyo, 1978, 115–153. [28] F. Nijhoﬀ and O. Chalykh, Bispectral rings of diﬀerence operators, Russian Math. Surveys, 54 (1999), 644–645. [29] M. Rieﬀel, Induced representations of C ∗ -Algebras, Advances in Mathematics 13, No. 2 (1974), 176–257.

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[30] P. Rom´ an and J. Tirao, Spherical functions, the complex hyperbolic plane and the hypergeometric operator, International J. of Math. 17, No. 10 (2006), 1151–1173. [31] E. Routh, On some properties of certain solutions of a diﬀerential equation of the second order, Proc. London Math. Soc., 16 (1884), 245–261. [32] J. Tirao, The algebra of diﬀerential operators associated to a weight matrix: a ﬁrst example, To appear. [33] G. Wilson, Bispectral commutative diﬀerential operators, J. Reine Angew. Math. 442 (1993), 177–204. [34] G. Wilson, Collisions of Calogero-Moser Particles and an Adelic Grassmanian, with an appendix by I.G. Macdonald), Invent. Math. 133 (1998), 1–41. F. Alberto Gr¨ unbaum Department of Mathematics University of California Berkeley, CA 94720 USA e-mail: [email protected] Juan Tirao CIEM-FaMAF Universidad Macional de C´ ordoba Argentina e-mail: [email protected] Submitted: March 5, 2007 Revised: April 19, 2007

Integr. equ. oper. theory 58 (2007), 477–486 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040477-10, published online June 27, 2007 DOI 10.1007/s00020-007-1500-6

Integral Equations and Operator Theory

Compressions of Stable Contractions L´aszl´o K´erchy and Vladim´ır M¨ uller Abstract. The stability of compressions of stable contractions is studied and a suﬃcient orbit condition is given. On the other hand, it is shown that there are non-stable compressions of the 1-dimensional backward shift and a complete characterization of weighted unilateral shifts with this property is provided. Dilations of bilateral weighted shifts to backward shifts are also considered. Mathematics Subject Classification (2000). Primary 47A20; Secondary 47A45. Keywords. Dilation, compression, stable contraction, weighted shift.

1. Introduction Let H be a complex Hilbert space, and let L(H) denote the C ∗ -algebra of all bounded linear operators acting on H. An operator T ∈ L(H) is called stable, if its positive powers converge to zero in the strong operator topology, that is when limn→∞ T nx = 0 for every x ∈ H. The Banach–Steinhaus Theorem shows that each stable operator T is power bounded, which means the boundedness of the norm-sequence {T n}n∈N , indexed by the set N of positive integers. Let P(H) stand for the set of all orthogonal projections in L(H). We are interested in the question whether the stability of T ∈ L(H) implies the stability of the operator TP := P T P ∈ L(H), for a projection P ∈ P(H). Let R(P ) denote the range of P . The operator P T |R(P ) ∈ L(R(P )) is called the compression of T to the subspace R(P ). The equations TPn = P (T P )n , (P T )n = TPn−1 T and (T P )n = T (P T )n−1P (n ∈ N) show that the operators TP , P T , T P and the compression of T to the subspace R(P ) are stable at the same time. If the Hilbert space H is non-separable, then it can be decomposed into an orthogonal sum of separable subspaces, which are reducing for both T and P . Hence, we can and shall assume that H is separable. The first author was partially supported by Hungarian NSRF (OTKA) grant no. T 035123, ˇ and by InstiT 049846. The second author was supported by grant no. 201/06/0128 of GA CR tutional research plan AV0Z10190503.

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Let P(T ) be the set of all projections in P(H), whose range is an invariant subspace of T . For P ∈ P(T ), the operator T |R(P ) ∈ L(R(P )) is called the restriction of T to its invariant subspace R(P ). The elements of the set Ps (T ) := {P1 − P2 : P1 , P2 ∈ P(T ), P1 ≥ P2 } are the projections whose ranges are semiinvariant subspaces of T . It can be easily seen that, for any P ∈ Ps (T ), the equality TPn = P T n P is true for every n ∈ N. Thus, the stability of T is inherited by TP in that case. Changing the viewpoint, passing from the compression to the operator on the larger space, another terminology is also in use. Let F and G be Hilbert spaces. We say that an operator A ∈ L(F ) can be dilated to an operator B ∈ L(G), in d

notation: A ≺ B, if there exists an isometry Z ∈ L(F , G) such that A = Z ∗ BZ. This happens precisely when A is unitarily equivalent to a compression of B to a pd

subspace of G. The operator A can be power dilated to B, in notation: A ≺ B, if there exists an isometry Z ∈ L(F , G) such that An = Z ∗ B n Z for every n ∈ N. It pd

is known that A ≺ B if and only if A is unitarily equivalent to the compression of B to a subspace semiinvariant with respect to B (see [S, Lemma 0]). If B is stable and A can be power dilated to B, then A is clearly stable. The question is whether the stability of B implies the stability of A, if A can be only dilated to B. We give a simple example which shows that the answer is negative in such a generality. An operator T is called uniformly stable, if limn→∞ T n = 0. This happens if and only if its spectral radius r(T ) is less than 1. In general, even the uniform stability of T does not imply the stability of TP . Indeed, let (e1 , e2 ) be an orthonormal basis in the Hilbert space H, and let T ∈ L(H) be deﬁned by T e1 := 2e2 , T e2 := 0. Then T 2 = 0, and so T is uniformly stable. On the other hand, if P ∈ P(H) is the projection onto the 1-dimensional subspace spanned by the vector e1 + e2 , then TP (e1 + e2 ) = P T (e1 + e2 ) = P (2e2 ) = e1 + e2 , and so TP is not stable. Therefore, the stability of TP could be expected only under additional conditions. It is natural to make the assumption that T ∈ L(H) is a contraction: T ≤ 1. Actually, the question whether compressions of stable contractions are also stable was posed to the ﬁrst named author by Rongwei Yang. If T is a strict contraction then the answer is obviously positive, since TP ≤ T < 1 implies the uniform stability of TP . It is also easy to verify that the stability of the contraction T is inherited by TP if the projection P has ﬁnite rank. Indeed, it is enough to show that r(TP ) < 1. Assuming r(TP ) = 1, there exist λ ∈ C, |λ| = 1 and 0 = x ∈ H such that TP x = λx. Since x = TP x ≤ T P x ≤ P x ≤ x, we infer that T x = λx, which contradicts to the stability of T . By a well-known theorem of C. Foias, restrictions of the inﬁnite-dimensional backward shift provide all stable contractions. To be more precise, for any 1 ≤ n ≤ ∞ (:= ℵ0 ) ﬁx an n-dimensional Hilbert space En , and let us consider the corresponding Hardy space H 2 (En ). The operator Sn ∈ L(H 2 (En )) of multiplication by

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the identical function χ(z) = z is the n-dimensional unilateral shift, and its adjoint Bn := Sn∗ ∈ L(H 2 (En )) is the n-dimensional backward shift. We recall that the 1 defect operators of a contraction T ∈ L(H) are deﬁned by DT := (I − T ∗ T ) 2 and 1 DT ∗ := (I − T T ∗) 2 . The defect spaces of T are the closures of the ranges of the defect operators: DT := (DT H)− , DT ∗ := (DT ∗ H)− , and dT := dim DT , dT ∗ := dim DT ∗ are the defect numbers of T . (For more information on the role of these objects in the study of Hilbert space contractions, we refer to the monograph [NF].) Let ST denote the operator of multiplication by χ on H 2 (DT ). The adjoint BT = ST∗ is unitarily equivalent to Bn , where n = dT . Ifthe contraction T is n n stable, then the transformation ZT : H → H 2 (DT ), h → ∞ n=0 χ DT T h is an ∗ isometry, whose range is invariant for BT . Since T = ZT BT ZT , we can see that T is unitarily equivalent to a restriction of B∞ . Taking into account that compressions of restrictions of B∞ are compressions of B∞ , we obtain that Yang’s question is equivalent to the problem whether all compressions of the inﬁnite-dimensional backward shift B∞ are stable. (We mention also that by a recent result of J.-C. Bourin in [B], for any sequence {An }n∈N of strict contractions with supn An < 1, ∞ there exists a decomposition H 2 (E∞ ) = n=1 ⊕ Mn such that An is unitarily equivalent to the compression of B∞ to Mn for all n.) In [TW] K. Takahashi and P.Y. Wu studied the question which contractions can be dilated to a unilateral shift. They proved that if at least one of the defect indices of the contraction T is ﬁnite, and if T can be dilated to B∞ , then T is stable. Another result due to C. Benhida and D. Timotin states that if T ∈ L(H) is a stable contraction with dT ∗ < ∞, and if for P ∈ P(H) the projection I −P has ﬁnite rank, then the operator TP is also stable (see [BT, Lemma 3.3]). In Section 2 we give an orbit condition yielding the stability for compressions of B∞ . In view of a general theorem on contractions, it can be easily justiﬁed that if a non-stable contraction T can be dilated to B∞ , then contractions similar to the unilateral shift S1 can also be dilated to B∞ . Indeed, T is necessarily completely non-unitary, and its residual set ρ(T ) is of positive Lebesgue measure. (We recall that the Borel subset ρ(T ) of the unit circle T is the support of the spectral measure of the canonical unitary operator associated with T ; for its detailed study we refer to [K2].) Choosing an appropriate sequence {αn }n∈N ∞on T, it can be attained that ρ(T1 ) = T holds for the orthogonal sum T1 = n=1 ⊕ αn T . Then, by [K1, Theorem 3] there exists a subspace M1 , invariant for T1 , such that the restriction T2 := T1 |M1 is similar to S1 . Taking into account that α1 B∞ ⊕ α2 B∞ ⊕ · · · is unitarily equivalent to B∞ , we infer that T2 can be dilated to B∞ . It is proved in Section 3 that there are indeed non-stable unilateral weighted shifts, similar to S1 , which can be dilated even to B1 , and so the answer to Yang’s question is negative. Actually, a complete characterization of such unilateral weighted shifts is given. Finally, in Section 4, dilations of bilateral weighted shifts into backward shifts are studied.

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2. Orbit condition We are going to show that a contraction, which is close to an isometry regarding the behaviour of the orbit of a vector, cannot be dilated to the inﬁnite-dimensional backward shift B∞ ∈ L(H 2 (E∞ )). The proof relies on some elementary inequalities. Let us ﬁx a projection P ∈ P(H 2 (E∞ )), and let us consider the operator BP := (B∞ )P = P B∞ P ∈ L(H 2 (E∞ )). We shall examine the orbit {BPn u}n∈N of an arbitrarily chosen vector u ∈ H 2 (E∞ ) under the action of BP . Let Z+ denote the set of non-negative integers. For any k ∈ Z+ , let Ek ∈ k P(H 2 (E∞ )) be the projection onto the subspace S∞ E∞ . (Here E∞ is identiﬁed with 2 the set of constant functions in H (E ).) The projections {Ek }∞ ∞ k=0 are pairwise ∞ orthogonal, and the series k=0 Ek converges to the identity operator I in the strong operator topology. Lemma 2.1. The constant components of the orbit vectors converge to zero: lim E0 BPn u = 0.

n→∞

Proof. The equation B∞ E0 = 0 implies that BPn+1 u = P B∞ (I − E0 )BPn u holds, for every n ∈ N. Thus BPn+1 u2 ≤ (I − E0 )BPn u2 = BPn u2 − E0 BPn u2 , whence E0 BPn u2 ≤ BPn u2 − BPn+1 u2 follows. Since BP is a contraction, the sequence {BPn u}n∈N converges decreasingly to a non-negative number. Hence the dominating sequence {BPn u2 − BPn+1 u2 }n∈N tends to zero, which yields the statement. By the next lemma each Fourier coeﬃcient of the vectors in the orbit converges to zero. Lemma 2.2. (a) For every n ∈ N, (I − P )B∞ BPn u2 ≤ BPn u2 − BPn+1 u2 . (b) For every n ∈ N, k ∈ Z+ , we have 1 Ek+1 BPn u ≤ Ek BPn+1 u + BPn u2 − BPn+1 u2 2 . (c) For every k ∈ Z+ , limn→∞ Ek BPn u = 0 is true. Proof. (a) It is immediate that (I − P )B∞ BPn u2 = B∞ BPn u2 − BPn+1 u2 ≤ BPn u2 − BPn+1 u2 .

(b) Taking into account that Ek B∞ = B∞ Ek+1 , we can write B∞ Ek+1 BPn = Ek B∞ BPn = Ek BPn+1 + Ek (I − P )B∞ BPn . Hence Ek+1 BPn u = B∞ Ek+1 BPn u ≤ Ek BPn+1 u + (I − P )B∞ BPn u is true, and an application of (a) yields the requested inequality. (c) This statement follows by induction on k, relying on (b) and Lemma 2.1. ∞ For any k ∈ Z+ , let us consider the projection Qk := j=k Ej ∈ P(H 2 (E∞ )).

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Lemma 2.3. For every n, l ∈ N and k ∈ Z+ , we have Qk BPn+l u ≤ Qk+l BPn u +

n+l−1 j=n

BPj u2 − BPj+1 u2

12

.

Proof. Since Qk B∞ = B∞ Qk+1 , we infer that Qk BPn+1 = Qk B∞ BPn − Qk (I − P )B∞ BPn = B∞ Qk+1 BPn − Qk (I − P )B∞ BPn . Hence Qk BPn+1 u ≤ Qk+1 BPn u + (I − P )B∞ BPn u, and so Lemma 2.2 (a) yields the required inequality for l = 1. Then the statement can be veriﬁed by induction on l. Now, we are ready to prove our theorem. n 2 1 n+1 u2 2 < ∞, then limn→∞ BPn u = 0. Theorem 2.4. If ∞ n=1 BP u − BP Proof. In view of Lemma 2.3, the inequality BPn+l u ≤

k−1 j=0

Ej BPn+l u + Qk+l BPn u +

∞ 12 BPj u2 − BPj+1 u2

j=n

holds, for any k, l, n ∈ N. Given a positive ε, let us choose n0 ∈ N so that ∞ 12 BPj u2 − BPj+1 u2 < ε/3. j=n0

Since limk→∞ Qk BPn0 u = 0, we can ﬁnd k0 ∈ N such that Qk0 +l BPn0 u < ε/3 is true for every l ∈ N. Finally, by Lemma 2.2 (c) there exists l0 ∈ N such that k0 −1 n0 +l u < ε/3 is valid, for every l ≥ l0 . Then BPn0 +l u < ε is fulﬁlled j=0 Ej BP for l ≥ l0 , which proves the statement. By the aforementioned theorem of Foias we obtain the following immediate consequence of Theorem 2.4. Corollary 2.5. Let T ∈ L(H) be a stable contraction, and let P ∈ P(H), x ∈ H be 1 ∞ n 2 n+1 given. If x2 2 < ∞, then limn→∞ TPn x = 0. n=1 TP x − TP

3. Dilation of unilateral weighted shifts The simplest examples for contractions similar to S1 can be found in the class of unilateral weighted shifts. Let {vk }k∈N be an orthonormal basis in the Hilbert space K. Given any bounded sequence {wk }k∈N of complex numbers, let us consider the operator W ∈ L(K), deﬁned by W vk := wk vk+1 (k ∈ N). The unilateral weighted shift W is a contraction precisely when |wk | ≤ 1 for every k ∈ N. All such contractions are obtained up to unitary equivalence assuming that wk belongs to the closed interval [0, 1], for every k ∈ N. Therefore we can assume that wk ∈ [0, 1] for all k ∈ N.

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It is easy to verify that W is non-stable if and only if ∞ k=k0 wk > 0 for ∞ some k0 ∈ N, which happens exactly when k=1 (1 − wk ) < ∞. Furthermore, this condition is equivalent to the decomposability of W in the form W = W0 ⊕ W1 , where W0 is a nilpotent operator on a ﬁnite-dimensional space, and W1 is a unilateral weighted shift similar to S1 . Let us assume that W is a non-stable contraction (that is {wk }k∈N ⊂ [0, 1] ∞ and k=1 (1 − wk ) < ∞), and that W can be dilated to the inﬁnite-dimensional backward shift B∞ . There exists k0 ∈ N such that ∞ k=k0 wk > 0. Since k0 +n−1 w > 0, we infer by Theorem 2.4 that limn→∞ W n vk0 = limn→∞ k=k k 0 ∞ 1 W n vk0 2 − W n+1 vk0 2 2 = ∞. n=1

Taking into account that, for every n ∈ N, k +n−1

0 1 1 1 n 2 n+1 2 2 W vk0 − W vk0 = wk (1 − wk20 +n ) 2 ≤ 2(1 − wk0 +n ) 2 , ∞

k=k0 1 2

we conclude that k=1 (1 − wk ) = ∞. We shall show that under these conditions W can be really dilated to B∞ , even more, it can be dilated to the 1-dimensional backward shift B1 . Namely, we are going to prove the following theorem. Theorem 3.1. Let W ∈ L(K) be the unilateral weighted shift corresponding to the ∞ weight sequence {wk }k∈N ⊂ [0, 1], satisfying the condition k=1 (1 − wk ) < ∞. The non-stable contraction W can be dilated to B if and only if it can be dilated ∞ ∞ 1 to B1 , which happens exactly when k=1 (1 − wk ) 2 = ∞. It is easy to ﬁnd sequences satisfying the previous conditions. For example, these are fulﬁlled if wk = 1 − εk −p (k ∈ N) with 1 < p ≤ 2 and 0 < ε < 1. Therefore, the answer for Yang’s question is negative: there are stable contractions having non-stable compressions. We note also that if ε is small, then the similarity constant s(W, S1 ) := inf{Q · Q−1 : QW = S1 Q} can be arbitrarily close to 1. Proof. Let {wk }k∈N ⊂ [0, 1] be a weight sequence satisfying ∞ k=1 (1 − wk ) < ∞ ∞ 1 and k=1 (1 − wk ) 2 = ∞. We have to show that the corresponding unilateral weighted shift W ∈ L(K), W vk := wk vk+1 (k ∈ N), can be dilated to B1 . π ∞ For any k ∈ N, let α(k) ∈ [0, 2 ] be deﬁned by cos α(k) = wk . The assumption k=1 (1 − wk ) < ∞ yields that limk→∞ α(k) = 0. Taking into account that (1 − 1 1 1 wk ) 2 ≤ (1 − wk2 ) 2 = sin α(k) ≤ 2(1 − wk ) 2 and π2 α(k) ≤ sin α(k) ≤ α(k) (k ∈ N), ∞ ∞ 1 the assumption k=1 (1−wk ) 2 = ∞ can be equivalently expressed as k=1 α(k) = j ∞. For any i, j ∈ N, i ≤ j, let us use the notation α(i, j) := k=i α(k). With {α(k)}k∈N we associate three sequences: {kj }∞ α(j)}∞ j=0 ⊂ Z+ , { j=0 ⊂ π ∞ (0) := 0, let us [0, 2 ] and {rj }j=0 ⊂ N in the following way. Setting k0 := 0 and α assume that {ki }ji=0 and { α(i)}ji=0 have already been deﬁned, for j ∈ Z+ . Then

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kj+1 is deﬁned as the minimum of the integers k satisfying the conditions k > kj ∞ and α (j)+α(kj +1, k) > 5π . (The assumption i=1 α(i) = ∞ ensures the existence 2 5π of such a k. Clearly, kj+1 > kj + 4.) Since 0 ≤ 2 − ( α(j) + α(kj + 1, kj+1 − 1)) < α(kj+1 ) ≤ π2 , we infer that

5π − ( α(j) + α(kj + 1, kj+1 − 1)) sin ( α(j) + α(kj + 1, kj+1 − 1)) = cos 2 > cos α(kj+1 ) ≥ 0, and so there exists a unique α (j + 1) ∈ [0, π2 ] such that −1

cos α (j + 1) = cos α(kj+1 ) (sin( α(j) + α(kj + 1, kj+1 − 1)))

.

{rj }∞ j=0

The sequence is deﬁned by rj := kj+1 − kj (j ∈ Z+ ). Note that rj > 4. Let us choose a sequence {nj }∞ j=0 of positive integers satisfying the conditions n0 > r0 , n1 > n0 + r1 , and nj > nj−1 + rj + rj−2 for every j ≥ 2. Fixing a unit vector e0 ∈ E1 , let us consider the orthonormal basis {e(n) := 2 S1n e0 }∞ n=0 in the Hardy space H (E1 ). For any j ∈ Z+ , let u(kj ) := (cos α (j))e(nj ) + (sin α (j))e(nj+1 + rj ), and, for any 1 ≤ i < rj , let u(kj + i) :=

(cos( α(j) + α(kj + 1, kj + i))) e(nj − i) + (sin( α(j) + α(kj + 1, kj + i))) e(nj+1 + rj − i).

The assumptions made at the choice of {nj }∞ j=0 ensure that the resulting sequence is orthonormal. {u(k)}∞ k=0 Exploiting the fact that

(cos ϕ)f + (sin ϕ)g, (cos ψ)f + (sin ψ)g = cos(ψ − ϕ) is valid whenever (f, g) forms an orthonormal system, we infer that

B1 u(kj + i − 1), u(kj + i) = cos α(kj + i) holds, for every j ∈ Z+ and 1 ≤ i < rj . Furthermore, it is easy to see that α(j)+α(kj +1, kj+1 −1)) cos α (j +1) = cos α(kj+1 )

B1 u(kj+1 − 1), u(kj+1 ) = sin( is true, for every j ∈ Z+ . Thus, we have obtained that the equation

B1 u(k − 1), u(k) = cos α(k) = wk is fulﬁlled, for every k ∈ N. Taking into account that the vector B1 u(k −1) is orthogonal to u(l) whenever l = k (k ∈ N), we conclude that the compression of B1 to the subspace M, spanned by the vectors {u(k)}∞ k=0 , is unitarily equivalent to the unilateral weighted shift W.

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4. Dilation of bilateral weighted shifts Let us consider the Hilbert space L2 (En ) of vector-valued functions, deﬁned with respect to the Lebesgue measure µ on T, where En is an n-dimensional Hilbert space. The operator Sˇn ∈ L(L2 (En )) of multiplication by the identical function χ is the n-dimensional bilateral shift. ˇ indexed by the Let {vk }k∈Z be an orthonormal basis in the Hilbert space K, set Z of all integers. Given a bounded sequence {wk }k∈Z of complex numbers, let ˇ ∈ L(K) ˇ be deﬁned by W ˇ vk := wk vk+1 (k ∈ Z). The bilateral weighted shift W ˇ W is a contraction precisely when |wk | ≤ 1 holds, for every k ∈ Z. We may assume without loss of generality that wk ∈ [0, 1] (k ∈ Z). ˇ is similar to the unitary operator Sˇ1 if and We note that the contraction W ∞ ∞ only if wk > 0 is true for every k ∈ Z, and k=1 (1−wk ) < ∞, k=1 (1−w−k ) < ∞ are valid. The following theorem shows that there are operators in the similarity class of unitaries, which can be dilated to B1 . ˇ ∈ L(K) ˇ be the bilateral weighted shift corresponding to the Theorem 4.1. Let W ∞ 1 1 2 2 weight sequence {wk }k∈Z ⊂ [0, 1]. If ∞ k=1 (1 − wk ) = k=1 (1 − w−k ) = ∞, ˇ can be dilated to the 1-dimensional backward shift B1 . then W Proof. For any k ∈ Z, let α(k) ∈ [0, π2 ] be deﬁned by cos α(k) = wk . Let us ∞ ∞ consider the sequences {kj }∞ α(j)}∞ j=0 , { j=0 and {rj }j=0 , associated with {α(k)}k=1 according to the proof of Theorem 3.1, with initial data k0 = 0 and α (0) = 0. ∞ ∞ Furthermore, let {k−j }∞ , { α (−j)} and {r } be the sequences associated −j j=0 j=0 j=0 with {α(−k)}∞ (−0) = α(0). (Here we make k=1 , with initial data k−0 = 0 and α diﬀerence between the indices 0 and −0.) The positive integers {n±j }∞ j=0 are chosen in the following way. We set n0 > r0 and n−0 := n0 + 1. Assuming that {n±i }ji=0 have already been deﬁned, for j ∈ Z+ , let nj+1 > n−j + r−j + rj+1 + 2 and n−(j+1) := nj+1 + rj + r−j + 2. The vectors {u(k)}∞ k=0 are deﬁned as in the proof of Theorem 3.1. On the other hand, for any j ∈ Z+ , let (−j))e(n−j ) + (sin α (−j))e(n−(j+1) − r−j ), u(−k−j ) := (cos α and, for any 1 ≤ i < r−j , let u(−k−j − i) := (cos( α(−j) + α(−k−j − 1, −k−j − i)) e(n−j + i) + (sin( α(−j) + α(−k−j − 1, −k−j − i)) e(n−(j+1) − r−j + i). l (For k, l ∈ Z+ , k ≤ l, α(−k, −l) := s=k α(−s).) The resulting set {u(±k)}∞ k=0 is an orthonormal system in H 2 (E1 ). Let us consider the subspace M := M− ⊕ M+ , ∞ where M+ := ∨{u(k)}∞ k=0 and M− := ∨{u(−k)}k=0 . It is easy to verify that PM B1 u(k − 1) = PM+ B1 u(k − 1) = wk u(k) (k ∈ N) is true. We obtain by symmetry that PM S1 u(−(k − 1)) = PM− S1 u(−(k − 1)) = w−k u(−k) (k ∈ N). Since, for any l ∈ Z, we have

PM B1 u(−k), u(l) = u(−k), PM S1 u(l) = δ(l, −(k − 1))w−k ,

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where δ(i, j) := 1 if i = j and δ(i, j) := 0 otherwise, it follows that PM B1 u(−k) = w−k u(−(k − 1)) (k ∈ N). Furthermore, the relation B1 e(−0) = e(0) implies that PM B1 u(−0) = =

PM B1 ((cos α(0))e(n−0 ) + (sin α(0))e(n−1 − r−0 )) (cos α(0))e(n0 ) = w0 u(0).

Therefore, the compression of B1 to M is unitarily equivalent to the bilateral ˇ. weighted shift W Keeping the previous notation, for any j ∈ Z+ , let aj := nj − rj . Let us also introduce the notation s0 := r0 + r−0 + 1, and sj := rj + r−j + rj−1 + r−(j−1) + 2 for j ∈ N. We can see that M is included in the subspace MH := ∨{e(n)}n∈H , where H = N ∩ ∪∞ j=0 [aj , aj + sj ] . In view of this observation, we can strengthen the statement of the previous theorem. ˇ be the bilateral weighted shift correˇ i ∈ L(K) Corollary 4.2. For any i ∈ N, let W 1 sponding to the weight sequence {wi,k }k∈Z ⊂ [0, 1]. If ∞ i,k ) 2 = ∞ and k=1 (1 − w ∞ 1 ∞ ˇ 2 k=1 (1 − wi,−k ) = ∞ for every i ∈ N, then the orthogonal sum i=1 ⊕ Wi can be dilated to B1 . ∞ Proof. For every i ∈ N, let {ri,±j }∞ j=0 and {si,j }j=0 be the sequences corresponding to the weight sequence {wi,k }k∈Z . Let τ : N → N × Z+ be a bijection. We set aτ (1) ∈ N arbitrarily. Assuming that {aτ (l) }m l=1 have already been deﬁned, for m ∈ N, let us choose aτ (m+1) ∈ N so that aτ (m+1) > aτ (m) + sτ (m) + 2 hold. Having introduced the positive integers {ai,j : i ∈ N, j ∈ Z+ }, we can deﬁne the numbers {ni,±j : i ∈ N, j ∈ Z+ } as follows: ni,j := ai,j + ri,j (j ∈ Z+ ), ni,−0 := ni,0 + 1 and ni,−j := ni,j + ri,j−1 + ri,−(j−1) + 2 (j ∈ N). For every i ∈ N, let Mi be the subspace of H 2 (E1 ) constructed with these data in the way described in the proof of Theorem 4.1. If i1 = i2 , then the subspaces M i1 and B1 Mi1 are orthogonal ∞ to Mi2 . Thus, taking the orthogonal sum M := i=1 ⊕ Mi , we conclude that ∞ ˇ i=1 ⊕ Wi is unitarily equivalent to the compression of B1 to M.

Since the unilateral weighted shifts are restrictions of bilateral weighted shifts, an analogous extension of Theorem 3.1 is also valid. 1 2 The assumption ∞ k=1 (1−w−k ) = ∞ in Theorem 4.1 was made for technical reasons. It can be dropped if we increase the dimension of the backward shift. ˇ ∈ L(K) ˇ be the bilateral weighted shift corresponding to the Theorem 4.3. Let W ∞ 1 ˇ can be dilated weight sequence {wk }k∈Z ⊂ [0, 1]. If k=1 (1 − wk ) 2 = ∞, then W to the 3-dimensional backward shift B3 . Proof. Fixing an orthonormal basis (e1 , e2 , e3 ) in E3 , the system {ei (n) := S3n ei : 1 ≤ i ≤ 3, n ∈ Z+ } will be an orthonormal basis in H 2 (E3 ). For any k ∈ Z, let α(k) ∈ [0, π2 ] be deﬁned by cos α(k) = wk . Let us consider the sequences ∞ ∞ {kj }∞ α(j)}∞ j=0 , { j=0 , {rj }j=0 and {nj }j=0 associated with {α(k)}k∈N in the proof (0) = 0. of Theorem 6, with k0 = 0 and α

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The orthonormal sequence {u(k)}k∈Z is deﬁned as follows. Let u(0) := e1 (n0 ), and for any 1 ≤ i < r0 , let u(i) := (cos α(1, i))e1 (n0 − i) + (sin α(1, i))e3 (n1 + r0 − i). For any j ∈ N, let u(kj ) := (cos α (j))e3 (nj ) + (sin α (j))e3 (nj+1 + rj ), and, for any 1 ≤ i < rj , let u(kj + i) :=

(cos( α(j) + α(kj + 1, kj + i))) e3 (nj − i) + (sin( α(j) + α(kj + 1, kj + i))) e3 (nj+1 + rj − i).

Finally, for any k ∈ N, let u(−k) := (cos α(0, −(k − 1))) · e1 (n0 + k) + (sin α(0, −(k − 1))) · e2 (n0 + k). It is easy to verify that the compression of B3 to the subspace M := ∨{u(k)}k∈Z ˇ. is unitarily equivalent to the bilateral weighted shift W In the light of the previous theorems a transparent characterization of all contractions, which can be dilated to B∞ , seems to be out of reach.

References J.-C. Bourin, Compressions and pinchings J. Operator Theory, 50 (2003), 211–220. C. Benhida and D. Timotin, Finite rank perturbations of contractions Integral Equations Operator Theory 36 (2000), 253–268. [K1] L. K´erchy Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes Acta Sci. Math. (Szeged), 61 (1995), 443–476. [K2] L. K´erchy On the hyperinvariant subspace problem for asymptotically nonvanishing contractions Operator Theory: Advances and Applications, 127 (2001), 399–422. [NF] B. Sz.-Nagy and C. Foias Harmonic analysis of operators on Hilbert space North Holland – Akad´emiai Kiad´ o, Amsterdam – Budapest, 1970. [S] D. Sarason On spectral sets having connected complement Acta Sci. Math. (Szeged), 26 (1966), 289–299. [TW] K. Takahashi and P.Y. Wu Dilation to the unilateral shifts Integral Equations Operator Theory 32 (1998), 101–113. [B] [BT]

L´ aszl´ o K´erchy Bolyai Institute, University of Szeged, Aradi v´ertan´ uk tere 1, H-6720 Szeged, Hungary e-mail: [email protected] Vladim´ır M¨ uller Institute of Mathematics, Academy of Sciences of Czech Republic, ˇ a 25, 115 67 Praha 1, Czech Republic Zitn´ e-mail: [email protected] Submitted: November 10, 2004 Revised: May 16, 2007

Integr. equ. oper. theory 58 (2007), 487–502 c 2007 Birkhäuser Verlag Basel/Switzerland 0378-620X/040487-16, published online June 27, 2007 DOI 10.1007/s00020-007-1503-3

Integral Equations and Operator Theory

Composition Operators and Vector-valued BMOA Jussi Laitila Abstract. Analytic composition operators Cϕ : f → f ◦ ϕ are studied on Xvalued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reﬂexive and Cϕ is compact on BMOA, then Cϕ is weakly compact on the X-valued space BMOAC (X) deﬁned in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA. Mathematics Subject Classification (2000). Primary 47B33; Secondary 30D50, 46E40. Keywords. Composition operator, bounded mean oscillation.

1. Introduction Let ϕ be an analytic self-map of the unit disk D = {z ∈ C : |z| < 1}. Compactness properties of the composition operators Cϕ : f → f ◦ ϕ have been extensively studied on Banach spaces of analytic functions on D. We refer to [19] and [10] for the basic results related, for example, to the classical Hardy spaces. Recently the question of which composition operators are weakly compact has been studied on various spaces of X-valued analytic functions, where X is some complex inﬁnite-dimensional Banach space; see [16], [7], [14], [15], for example. Since the operator Cϕ ﬁxes the constant functions, such a space usually supports no compact composition operators. In the present paper we study the weak compactness of Cϕ on certain vector-valued BMOA spaces, which are Xvalued generalizations of the classical space BMOA of analytic functions on D that have bounded mean oscillation on the unit circle T = ∂D. The author was supported in part by the Finnish Academy of Science and Letters (Vilho, Yrjö and Kalle Väisälä Foundation) and the Academy of Finland, projects 53893 and 210970.

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Compactness and weak compactness of Cϕ on the scalar-valued BMOA space have been studied in several recent papers, such as [22], [8], [20], [17], [9] and [23]. In [14] some of these results were extended to the vector-valued space BMOA(X), which is deﬁned as a Möbius invariant version of the Hardy space H 1 (X). There are also other interesting possibilities of approaching vector-valued BMOA; see [3], [4] and [6], for example. One alternative arises by considering the weak vectorvalued space wBMOA(X), which consists of the analytic functions f : D → X such that x∗ ◦ f ∈ BMOA for all x∗ ∈ X ∗ . Weak compactness of Cϕ on a large class of such weak spaces, including wBMOA(X), has been studied by Bonet, Domański and Lindström [7]; see also [14]. In this paper we consider composition operators on BMOAC (X), a vectorvalued version of BMOA deﬁned in terms of Carleson measures, which was introduced by Blasco [4] in connection with vector-valued multipliers. It is known that the three spaces BMOA(X), wBMOA(X) and BMOAC (X) have quite diﬀerent properties if X is inﬁnite dimensional. In fact, it was shown in [4] that BMOA(X) and BMOAC (X) coincide (and the respective norms are equivalent) only if X is isomorphic to a Hilbert space. We will observe below that the spaces BMOAC (X) and wBMOA(X) never coincide if X is inﬁnite dimensional. Our main result states that if ϕ induces a compact composition operator on BMOA and X is reﬂexive, then Cϕ is weakly compact on BMOAC (X). This result complements the earlier ones from [14] and [7]. The proof will be based on a new function-theoretic condition which characterizes the compact composition operators on the scalar-valued BMOA space. The necessity part of this characterization will be established in Section 2. In Section 3 we provide some preliminary results about the space BMOAC (X) and composition operators. Our main result will be proved in Section 4. As a consequence, a characterization is obtained of the weakly compact composition operators on BMOAC (X) under some restrictions on ϕ, where X is a reﬂexive Banach space.

2. Compactness of composition operators on BMOA The space BMOA consists of the analytic functions f : D → C, which are Poisson integrals of functions that have bounded mean oscillation on T. We recall the following equivalent reformulation of BMOA as a Möbius invariant version of the Hardy space H 2 ; see [2]. An analytic function f : D → C belongs to BMOA if and only if f ∗ = sup f ◦ σa − f (a)H 2 < ∞, a∈D

where σa (z) = (a − z)/(1 − az) for a, z ∈ D and · H p denotes the usual norm 2π dθ . on the Hardy space H p (1 ≤ p < ∞) given by f pH p = sup0
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on BMOA for any 1 ≤ p < ∞. We refer to [12] and [13] for further properties of BMOA. It is known that for every analytic map ϕ : D → D, the composition operator Cϕ : f → f ◦ ϕ is bounded on BMOA; see [21, Thm. 3], [1, Thm. 12], for example. There are several equivalent characterizations of the compact composition operators on BMOA; see [8], [20], [23]. The following result is due to Smith [20, Thm. 1.1]: the operator Cϕ is compact on BMOA if and only if lim

sup

sup |w|2 N (σϕ(a) ◦ ϕ ◦ σa , w) = 0

(2.1)

m({ζ ∈ T : |(ϕ ◦ σa )(ζ)| > t}) = 0,

(2.2)

r→1 {a∈D : |ϕ(a)|>r} 0<|w|<1

and lim

sup

t→1 {a∈D : |ϕ(a)|≤R}

for every R ∈ (0, 1). Here m is the Lebesgue measure on T and N (ϕ, ·) is the Nevanlinna counting function given by N (ϕ, z) = w∈ϕ−1 (z) log(1/|w|) for z ∈ D \ {ϕ(0)}, where each point in the preimage ϕ−1 (z) is counted according to its multiplicity. We next state a new necessary condition for the compactness of Cϕ on BMOA. This result will be later used for studying the weak compactness of Cϕ on vectorvalued BMOA spaces. Theorem 2.1. Let ϕ : D → D be analytic. If Cϕ is compact on BMOA, then N (ϕ ◦ σa , w) = 0, |w|→1 {a∈D : |ϕ(a)|≤R} log(1/|w|) lim

sup

(2.3)

for every R ∈ (0, 1), where σa (z) = (a − z)/(1 − az) for a, z ∈ D. It will follow from our results in Section 4, that (2.3) and (2.1) together are also suﬃcient for the compactness of Cϕ on BMOA. In other words, (2.2) can be replaced by (2.3) in the above characterization of the compact composition operators on BMOA (see Corollary 4.5). The main idea for the proof of Theorem 2.1 comes from the work of Bourdon, Cima and Matheson [8, Thm. 4.1], where it is shown that the compactness of Cϕ on BMOA implies its compactness on H 2 . The proof in [8] is based on an integral criterion ([8, Thm. 3.1]), which we replace for convenience by a related equivalent criterion due to Wirths and Xiao [23]. The counting function will be controlled by applying certain methods from the proof due to Shapiro [18, Thm. 2.3] of the fact that Cϕ is compact on the Hardy space H 2 if and only if N (ϕ, w) = 0. |w|→1 log(1/|w|) lim

(2.4)

Note that, by [8, Thm. 4.1] and the result of Shapiro, the compactness of Cϕ on BMOA implies (2.4). Theorem 2.1 can be viewed as a slightly stronger version of this result.

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We need some auxiliary results before the proof of Theorem 2.1. We will frequently use the following known identities concerning the automorphisms σa : z → (a − z)/(1 − az): for all a, z ∈ D, we have (σa ◦ σa )(z) = z and 1 − |σa (z)|2 = (1 − |z|2 )|σa (z)|; see [12, §I.1]. The relevance of the Nevanlinna counting function for composition operators comes from the change of variables formula 1 dA(z) = (λ ◦ ϕ)(z)|ϕ (z)|2 log λ(z)N (ϕ, z) dA(z), (2.5) |z| D D for positive measurable functions λ : D → R, where dA is the Lebesgue measure on D; see [18, §4.3]. Combined with the Littlewood–Paley identity (see [12, Lemma VI.3.1] or [10, Thm. 2.30]) 1 dA(z) 2 , (2.6) f − f (0)H 2 = 2 |f (z)|2 log |z| π D formula (2.5) yields the useful identity f ◦ ϕ − f (ϕ(0))2H 2 = 2

D

|f (z)|2 N (ϕ, z)

dA(z) , π

for analytic functions f : D → C and ϕ : D → D. We will also need the following estimate for the integral in (2.6): there is a constant c > 0 such that 1 2 dA(z) ≤ c |f (z)|2 (1 − |z|2 ) dA(z), |f (z)| log (2.7) |z| D D for all analytic functions f : D → C; see [12, Lemma VI.3.2]. On the other hand, since 1−|z|2 ≤ 2 log(1/|z|) for z ∈ D, also a reverse estimate holds (with a diﬀerent constant). Finally, we need the “only if” part of the following result from [23]. Theorem 2.2 ([23, Thm. 5.1]). Let ϕ : D → D be analytic. The composition operator Cϕ is compact on BMOA if and only if lim sup sup |(f ◦ ϕ) (z)|2 (1 − |σa (z)|2 ) dA(z) = 0. r→1 f

BMOA ≤1

a∈D

{z∈D : |ϕ(z)|>r}

We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. Assume that Cϕ is compact on BMOA. Let R ∈ (0, 1) and ε > 0. It is known that supw∈D fw BMOA < ∞, where the functions fw ∈ BMOA are given by fw (z) = log(1 − wz) for w ∈ D; see [13, Thm. 11.4], for example. By Theorem 2.2, there is a number t0 ∈ (0, 1) such that |(fw ◦ ϕ) (u)|2 (1 − |(σa ◦ σb )(u)|2 ) dA(u) < ε, sup a,b,w∈D

{z∈D : |ϕ(z)|>t0 }

since |(σa ◦ σb )(u)| = |σc (u)| on D for some c ∈ D. We next abbreviate Ω(b) = {z ∈ D : |(ϕ ◦ σb )(z)| > t0 } for b ∈ D. By applying the change of variable u = σb (z) and

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the identities (σb ◦ σb )(z) = z and 1 − |σa (z)|2 = (1 − |z|2 )|σa (z)|, we get ε > sup |(fw ◦ ϕ) (σb (z))|2 (1 − |σa (z)|2 )|σb (z)|2 dA(z) a,b,w∈D

= sup b,w∈D

Ω(b)

sup a∈D

Ω(b)

|(fw ◦ ϕ ◦ σb ) (z)|2 (1 − |z|2 )|σa (z)| dA(z) .

Hence the measures µb,w , given by |w|2 |(ϕ ◦ σb ) (z)|2 (1 − |z|2 ) dA(z), |1 − w(ϕ ◦ σb )(z)|2 for b, w ∈ D, are Carleson measures, so there is a constant C > 0 such that sup |g|2 dµb,w ≤ Cεg2H 2 , (2.8) dµb,w (z) = 1Ω(b)

b,w∈D

D

2

for all g ∈ H ; see [12, Lemma VI.3.3] or [10, Thm. 2.33] Consider next b ∈ D such that |ϕ(b)| ≤ R. For w ∈ D, let kw denote the analytic function kw (z) = 1 − |w|2 /(1 − wz), so that kw H 2 = 1. Recall that for any analytic map ψ : D → D, the composition operator Cψ is bounded on H 2 with Cψ : H 2 → H 2 2 ≤ 2/(1 − |ψ(0)|2 ); see [19, p. 16] or [10, Cor. 3.7]. Hence kw ◦ ϕ ◦ σb 2H 2 ≤ 2/(1 − R2 ) for all w, b ∈ D such that |ϕ(b)| ≤ R. By choosing g = kw ◦ ϕ ◦ σb in (2.8) and abbreviating dν(z) = (1 − |z|2 ) dA(z), we get |w|2 (1 − |w|2 )|(ϕ ◦ σb ) (z)|2 |(kw ◦ ϕ ◦ σb ) (z)|2 dν(z) = dν(z) |1 − w(ϕ ◦ σb )(z)|4 Ω(b) Ω(b) |(kw ◦ ϕ ◦ σb )(z)|2 dµb,w (z) (2.9) = D

≤ Cεkw ◦ ϕ ◦ σb 2H 2 ≤

2Cε , 1 − R2

for b, w ∈ D such that |ϕ(b)| ≤ R. Choose next a number r0 ∈ (0, 1) so that |w|2 (1 − |w|2 )/(1 − |w|t0 )4 < ε for all w ∈ D with |w| > r0 . Then |(kw ◦ ϕ ◦ σb ) (z)|2 ≤ ε|(ϕ ◦ σb ) (z)|2 for such w and z ∈ D \ Ω(b) = {z ∈ D : |(ϕ ◦ σb )(z)| ≤ t0 }. Since ϕ ◦ σb − ϕ(b)2H 2 ≤ 1, we get from (2.6) that 1 dA(z) ≤ πε, |(kw ◦ ϕ ◦ σb ) (z)|2 dν(z) ≤ 2ε |(ϕ ◦ σb ) (z)|2 log |z| D\Ω(b) D for all w ∈ D such that |w| > r0 . On the other hand, by applying (2.5) to the function λ(z) = |kw (z)|2 , and using (2.7), (2.9) and the above estimate, we get 2C |kw (z)|2 N (ϕ ◦ σb , z) dA(z) ≤ c |(kw ◦ ϕ ◦ σb ) (z)|2 dν(z) ≤ c( + π)ε, 1 − R2 D D for all b, w ∈ D such that |ϕ(b)| ≤ R and |w| > r0 . We conclude that sup |kw (z)|2 N (ϕ ◦ σb , z) dA(z) = 0. (2.10) lim |w|→1 {b : |ϕ(b)|≤R}

D

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We ﬁnally recall how condition (2.3) can be obtained from (2.10) by applying some methods from [18, 5.4]; see also [10, p. 138]. Put s = (R + 1)/2 and h = −1 (1 − R)/4, so that 0 < s, h < 1. Since σw = σw , we get w − ϕ(b) |w| − |ϕ(b)| s−R −1 ≥ > = h, (2.11) ((ϕ ◦ σb )(0))| = |σw 1 − wϕ(b) 2 2 for all w, b ∈ D such that |w| > s and |ϕ(b)| ≤ R. Fix next w ∈ D such that |w| > s. By using the identity (1 − |w|2 )|kw (z)|2 = |w|2 |σw (z)|2 and the change of variable u = σw (z), we get |w|2 dA(z) dA(z) 2 = |kw (z)| N (ϕ ◦ σb , z) N (ϕ ◦ σb , z)|σw (z)|2 2 π 1 − |w| π D D 2 dA(u) |w| . N (ϕ ◦ σb , σw (u)) = 2 1 − |w| D π Moreover, (2.11) and the sub-mean value property of N (ϕ, ·) (see [18, 4.6] or [10, p. 137]) yield dA(u) ≥ h2 N (ϕ ◦ σb , w). N (ϕ ◦ σb , σw (u)) π hD Thus |w|2 h2 N (ϕ ◦ σb , w) h2 N (ϕ ◦ σb , w) dA(z) ≥ ≥ , |kw (z)|2 N (ϕ ◦ σb , z) π (1 − |w|2 ) 8 log(1/|w|) D for all w ∈ D such that |w| > s and |ϕ(b)| ≤ R. Condition (2.3) follows now from (2.10).

3. Vector-valued BMOA and composition operators In the sequel X = (X, · X ) will always be a complex Banach space. We will consider the following versions of X-valued BMOA; see [3], [4], [14], for instance. Definition 3.1. (1) The space BMOA(X) consists of the analytic functions f : D → X such that f ∗,X = supa∈D f ◦ σa − f (a)H 1 (X) < ∞, where · H 1 (X) denotes the norm on the X-valued Hardy space H 1 (X) given by f H 1 (X) = 2π dθ sup0
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(3) The space BMOAC (X) consists of the analytic functions f : D → X such that f 2C,X

= sup a∈D

D

f (z)2X (1 − |σa (z)|2 )

dA(z) < ∞. π

We equip BMOAC (X) with the complete norm f BMOAC (X) = f (0)+f C,X . The space BMOAC (X) can also be characterized in terms of Carleson measures. In fact, by using the identity 1 − |σa (z)|2 = (1 − |z|2 )|σa (z)| and a theorem of Carleson, we get that f ∈ BMOAC (X) if and only if the measure dµf (z) = f (z)2X (1 − |z|2 ) dA(z) is a Carleson measure; see [12, Lemma VI.3.3] or [10, Thm. 2.33]. In the special case X = C the seminorms · ∗,C and · C,C are comparable (one checks this fact from (2.6) and (2.7) using a change of variables). Hence BMOA = BMOA(C) = wBMOA(C) = BMOAC (C) with equivalent norms. If X is inﬁnite dimensional, then the X-valued BMOA spaces diﬀer from each other considerably. For example, it is known that BMOA(X) and BMOAC (X) coincide, and the respective norms are equivalent, if and only if X is isomorphic to a Hilbert space [4, Cor. 1.1]. (In fact, if BMOAC (X) ⊂ BMOA(X), then X has type 2, and if BMOA(X) ⊂ BMOAC (X), then X has cotype 2; see [4, Thm. 1.2].) It not diﬃcult to verify that BMOA(X) ⊂ wBMOA(X) for all Banach spaces X. However, the spaces BMOA(X) and wBMOA(X) coincide only if X is ﬁnite dimensional [14, p. 743]. The following observation complements these facts. Proposition 3.2. The spaces BMOAC (X) and wBMOA(X) coincide, and the respective norms are equivalent, if and only if X is finite dimensional. Proof. Let X be any complex Banach space. We get from (2.6), (2.7) and the change of variables w = σa (z) that x∗ ◦ f ◦ σa − x∗ (f (a))2H 2 ≤ 2c |(x∗ ◦ f ◦ σa ) (z)|2 (1 − |z|2 ) dA(z) D = 2c |(x∗ ◦ f ) (w)|2 (1 − |σa (w)|2 ) dA(w) ≤ 2cx∗ 2X ∗ f 2C,X , D

for f ∈ BMOAC (X) and x∗ √ ∈ X ∗ , where we also used the identity (σa ◦ σa )(w) = w. Thus f wBMOA(X) ≤ 2cf BMOAC (X) for f ∈ BMOAC (X). Moreover, if dim(X) = n < ∞, then it is not diﬃcult to ﬁnd a constant C(n) such that f BMOAC (X) ≤ C(n)f wBMOA(X) for all f ∈ wBMOA(X). Assume next that X is inﬁnite dimensional. Let n ∈ N. By Dvoretzky’s theorem, there exist an n-dimensional subspace En ⊂ X and a linear isomorphism Tn : n2 → En so that Tn ≤ 2 and Tn−1 = 1; see [11, Thm. 19.1]. We deﬁne the analytic function fn : D → X by fn (z) =

n (Tn ek ) k √ z , k k=1

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for z ∈ D, where (e1 , . . . , en ) is an orthonormal basis of n2 . Then an argument in [14, p. 744] shows that supn∈N fn wBMOA(X) < ∞. On the other hand, since fn (z)2X =

n √ n √ n k(Tn ek )z k−1 2X ≥ kek z k−1 2n2 = k|z|2(k−1) , k=1

k=1

k=1

we get fn 2C,X ≥ 2

n k k=1

1 0

r2(k−1) (1 − r2 )rdr =

n k=1

log n 1 ≥ . k+1 2

Thus fn BMOAC (X) → ∞, as n → ∞. This shows that the two norms are not equivalent. Moreover, by the open mapping theorem, we get BMOAC (X) wBMOA(X). We next consider the composition operators Cϕ : f → f ◦ ϕ on the space BMOAC (X). It is known that for every analytic map ϕ : D → D, the operator Cϕ is bounded on BMOA(X) and wBMOA(X); see [14, Prop. 3] and [15, Thm. 5.2], for example. We sketch here for completeness a proof that Cϕ is bounded on BMOAC (X) for any complex Banach space X. We ﬁrst need the following vector-valued version of (2.7): it holds that 1 2 dA(z) ≤ c f (z)2X (1 − |z|2 ) dA(z), f (z)X log (3.1) |z| D D for any complex Banach space X and analytic function f : D → X. In fact, the proof of (3.1) in [12, Lemma VI.3.2] remains valid also in the vector-valued setting, since the map z → f (z)2X is subharmonic. Moreover, by the change of variable w = σa (z) and the identity (σa ◦ σa )(z) = z, we have f (w)2X (1 − |σa (w)|2 ) dA(w) = (f ◦ σa ) (z)2X (1 − |z|2 ) dA(z), (3.2) D

D

for all analytic functions f : D → X. Using the estimate 1 − |z|2 ≤ 2 log(1/|z|) we get from (3.1) and (3.2) that 1 dA(z) f 2C,X ≤ 2 sup (f ◦ σa ) (z)2X log ≤ 2cf 2C,X . (3.3) |z| π a∈D D Recall next that the Nevanlinna counting function N (ψ, z) satisﬁes the Littlewood inequality N (ψ, z) ≤ N (σψ(0) , z) = log(1/|σψ(0) (z)|),

(3.4)

for all analytic maps ψ : D → D and z ∈ D \ {ψ(0)}; see [18, p. 380] or [10, p. 33]. The fact that Cϕ is bounded on BMOAC (X) can be seen from (3.3), (3.4) and the

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formula (2.5) applied to the function λ(z) = f (z)2X . Indeed, 1 dA(z) f ◦ ϕ2C,X ≤ sup (f ◦ ϕ ◦ σa ) (z)2X log |z| π a∈D D dA(z) = 2 sup f (z)2X N (ϕ ◦ σa , z) π a∈D D dA(z) ≤ 2 sup f (z)2X N (σϕ(a) , z) π a∈D D 1 dA(z) ≤ 2cf 2C,X , = 2 sup (f ◦ σϕ(a) ) (z)2X log |z| π a∈D D for all f ∈ BMOAC (X). The upper bound

1 + |ϕ(0)| Cϕ : BMOAC (X) → BMOAC (X) ≤ C 1 + log , 1 − |ϕ(0)|

(3.5)

where C > 0 is a constant, can now be deduced from the following lemma, which will also be useful in the sequel. Lemma 3.3. For all f ∈ BMOAC (X) and z ∈ D, we have √ 2f C,X f (z)X ≤ 1 − |z|2

(3.6)

and 1 1 + |z| f (z)X ≤ f (0)X + √ f C,X log . 1 − |z| 2

(3.7)

Proof. By the mean value property, 2π dθ (f ◦ σz ) (0)2X 1 ≤ (f ◦ σz ) (reiθ )2X , 2 2 2 2 (1 − |z| ) (1 − |z| ) 0 2π 1 for every r ∈ (0, 1). Since 4 0 (1 − r2 )r dr = 1, we get f (z)2X =

f

(z)2X

4 ≤ (1 − |z|2 )2

0

1

0

2π

(f ◦ σz ) (reiθ )2X

dθ 2f C,X (1 − r2 )r dr ≤ , 2π (1 − |z|2 )2

by (3.2). This proves (3.6). Since f (z) − f (0) = eiθ |z|eiθ ∈ D, we get f (z) − f (0)X which proves (3.7).

√ ≤ 2f C,X

0

|z|

|z| 0

f (teiθ ) dt for every z =

1 1 + |z| 1 , dt = √ f C,X log 1 − t2 1 − |z| 2

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4. Weakly compact composition operators on BMOAC (X) A bounded linear map T on a Banach space E is weakly compact if T BE is a weakly compact set, where BE is the closed unit ball of E. Recall that the weakly compact operators on a Banach space form a closed operator ideal. We refer to [24, §II] for the notion of weak compactness and the properties of weakly compact operators. We next record some easy observations about (weakly) compact composition operators on BMOAC (X). First, if the composition operator Cϕ : f → f ◦ ϕ is weakly compact on BMOAC (X), then X is reﬂexive and Cϕ is weakly compact also on BMOA. Indeed, since Cϕ (fx ) = fx for the constant functions fx ≡ x (where x ∈ X) and x → fx is an isometric embedding of X into BMOAC (X), the weak compactness of Cϕ on BMOAC (X) yields that BX is a weakly compact set. This means that X is reﬂexive. Moreover, given some non-zero x0 ∈ X, we get that Cϕ is weakly compact on the closed subspace x0 BMOAC (C) = {x0 f : f ∈ BMOAC (C)} of BMOAC (X). Since BMOA is obviously isomorphic to x0 BMOAC (C), we deduce that Cϕ is weakly compact on BMOA. Secondly, the above argument also shows that in the interesting case where X is inﬁnite dimensional (that is, BX is not compact), Cϕ cannot be compact on BMOAC (X). Our main result provides a suﬃcient condition for the weak compactness of composition operators on BMOAC (X). Theorem 4.1. Let X be a reflexive Banach space and suppose that ϕ : D → D is an analytic map such that Cϕ is compact on BMOA. Then Cϕ is weakly compact on BMOAC (X). Theorem 4.1 complements the corresponding results for composition operators on BMOA(X) and wBMOA(X). For wBMOA(X) the weak compactness of Cϕ follows from a general result for a large class of vector-valued function spaces of weak type [7, Prop. 11]; see also [14, §5]. In the case of BMOA(X) the proof is essentially a vector-valued modiﬁcation of a scalar-valued argument due to Smith; see [14, Thm. 7]. The starting point for our proof of Theorem 4.1 is the necessary condition for the compactness of composition operators on BMOA, which we obtain by combining (2.1) and (2.3) from Section 2. For convenience, we restate this result here: if Cϕ is compact on BMOA, then lim

sup

sup |w|2 N (σϕ(a) ◦ ϕ ◦ σa , w) = 0

r→1 {a∈D : |ϕ(a)|>r} 0<|w|<1

and

N (ϕ ◦ σa , w) = 0, |w|→1 {a∈D : |ϕ(a)|≤R} log(1/|w|) lim

sup

(4.1)

(4.2)

for every R ∈ (0, 1). We will show that if X is reﬂexive, then (4.1) and (4.2) imply the weak compactness of Cϕ on BMOAC (X). The argument is essentially contained in the following two lemmas which will be proved below. Here Cr denotes

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the linear operator given by (Cr f )(z) = f (rz) for analytic functions f : D → X and r ∈ (0, 1). Lemma 4.2. The operators Cr : BMOAC (X) → BMOAC (X) satisfy the following properties for r ∈ (0, 1). 1. sup0
sup max{(f − Cr f ) (z)X , (f − Cr f )(z)X } → 0,

f BMOAC (X) ≤1 |z|≤R

as r → 1. 3. If X is reflexive, then Cr is weakly compact on BMOAC (X). Lemma 4.3. Let ϕ : D → D be an analytic map such that conditions (4.1) and (4.2) hold. Then Cϕ − Cϕ Cr : BMOAC (X) → BMOAC (X) → 0, as r → 1. Note that the proof of Theorem 4.1 is easy to complete by using Lemmas 4.2 and 4.3. Indeed, assume that X is reﬂexive and Cϕ is compact on BMOA, so (4.1) and (4.2) hold. Let rn = n/(n + 1) and consider the linear operators Tn = Cϕ Crn for n ∈ N. By parts (1) and (3) of Lemma 4.2, every Tn is bounded and weakly compact on BMOAC (X). Since Cϕ − Tn → 0 as n → ∞, by Lemma 4.3, the operator Cϕ is weakly compact on BMOAC (X). This proves Theorem 4.1. We next prove Lemmas 4.2 and 4.3. Proof of Lemma 4.2. The assertion (1) follows from (3.5) and the fact that Cr is the composition operator induced by the mapping z → rz. We next prove (2). Let 0 < r, R < 1. Consider an analytic function f : D → X and a point z ∈ D. Put ρ = (|z| + 1)/2 so that |rz| < |z| < ρ < 1. From the Cauchy integral formula we get 2π iθ

ρf (ρe ) ρrf (ρeiθ ) dθ − f (z) − rf (rz)X = ρ − ze−iθ ρ − rze−iθ 2π X 0 ≤

(1 − r)f (ρeiθ )X . −iθ ||ρ − rze−iθ | θ∈[0,2π) |ρ − ze sup

Using Lemma 3.3 and the identity 2(ρ − |z|) = 1 − |z| = 2(1 − ρ) we ﬁnd a constant c > 0 such that cf C,X (f − Cr f ) (z)X ≤ (1 − r) . (4.3) (1 − |z|)3 |z| Moreover, since (f − Cr f )(z) = eiθ 0 (f − Cr f ) (teiθ ) dt, where z = |z|eiθ , we get |z| dt cf C,X (f − Cr f )(z)X ≤ c(1 − r)f C,X ≤ (1 − r) . (4.4) 3 (1 − t) 2(1 − |z|)2 0

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We obtain (2) by taking the supremum over all z ∈ D and f satisfying |z| ≤ R and f BMOAC (X) ≤ 1 in (4.3) and (4.4), and letting r → 1. We ﬁnally prove (3).We will approximate C r using the truncation operators n ∞ k k Pn , where (Pn f )(z) = x z for f (z) = k=0 k k=0 xk z in BMOAC (X) and n ≥ 0. We ﬁrst note that the operators Pn are bounded ∞ on BMOAC (X). Indeed, for any analytic function f : D → X with f (z) = k=0 xk z k , we have x0 X = f (0)X ≤ f BMOAC (X) . Moreover, there is a constant c > 0 such that xk X ≤ c · supz∈D (1 − |z|2 )f (z)X for all f ∈ BMOAC (X) and k ≥ 1 (here one may apply the familiar√ scalar-valued argument; see [5, p. 101], for example). Hence supk≥1 xk X ≤ 2cf BMOAC (X) , by Lemma 3.3. Since z n BMOAC (C) ≤ 1 for √ n ≥ 1, we obtain Pn ≤ 1 + 2cn. k Let next ε > 0 and ﬁx n0 so that ∞ k=n0 +1 kr < ε. Then, for any z ∈ D and ∞ k f ∈ BMOAC (X) with f (z) = k=0 xk z , we get ∞

((Cr − Pn0 Cr )f ) (z)X ≤

xk X rk k|z|k−1 ≤

√ 2cεf BMOAC (X) .

k=n0 +1

Since (Cr − Pn0 Cr )f BMOAC (X) ≤ supz∈D ((Cr − Pn0 Cr )f ) (z)X , by the deﬁnition of the BMOAC (X) norm, we get Cr − Pn Cr → 0, as n → ∞. The proof of (3) is completed by noting that, for every n ∈ N, the operator Pn is weakly (X); compact on BMOAC (X) since it factors through the reﬂexive direct sum n+1 2 see the proof of [16, Prop. 2]. For the proof of Lemma 4.3 we need a reﬁnement of (3.4) due to Smith [20, Lemma 2.1]. For convenience, we use the following technical modiﬁcation of this result from [14]. Lemma 4.4 ([14, Lemma 10]). Let ψ : D → D be an analytic function with ψ(0) = 0. Suppose that there is ε ∈ (0, 1e ) such that sup |w|2 N (ψ, w) ≤ ε2 .

0<|w|<1

Then N (ψ, z) ≤ 2ε log(1/|z|) for all z ∈ D with

√ ε ≤ |z| < 1.

We are now ready to prove Lemma 4.3. Proof of Lemma 4.3. For r ∈ (0, 1), let Sr denote the linear operator f → f −Cr f , so that K := sup0
r→1 f

sup

BMOAC (X) ≤1

(f − Cr f )(ϕ(0))X = 0,

by Lemma 4.2(2), it suﬃces to show that lim

r→1 f

sup

BMOAC (X) ≤1

where we denote

Ma (g) =

D

sup Ma (Cϕ Sr f ) = 0, a∈D

g (z)2X (1 − |σa (z)|2 )

dA(z) , π

(4.5)

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for g ∈ BMOAC (X) and a ∈ D. Let ε ∈ (0, 1e ) and f ∈ BMOAC (X) be arbitrary. We will abbreviate ϕa = σϕ(a) ◦ ϕ ◦ σa and gr,a = (Sr f ) ◦ σϕ(a) for all a ∈ D and r ∈ (0, 1). By (4.1), there is R ∈ (0, 1) such that sup0<|w|<1 |w|2 N (ϕa , w) < ε2 for all a ∈ D with |ϕ(a)| > R. Hence Lemma 4.4 implies that N (ϕa , z) ≤ 2ε log(1/|z|), (4.6) √ for all a, z ∈ D such that |ϕ(a)| > R and ε ≤ |z| < 1. Using (3.2) and the identity (Cϕ Sr f ) ◦ σa = gr,a ◦ ϕa we get dA(z) Ma (Cϕ Sr f ) = . (gr,a ◦ ϕa ) (z)2X (1 − |z|2 ) π D Thus the estimate (1 − |z|2 ) ≤ 2 log(1/|z|) and the change of variables formula (z)2X yield (2.5) applied to the function λ(z) = gr,a dA(z) , (4.7) Ma (Cϕ Sr f ) ≤ 2 gr,a (z)2X N (ϕa , z) π D for all r ∈ (0, 1). By (4.6) and (3.1), we have dA(z) 1 dA(z) 2 ≤ 2ε g (z) N (ϕ , z) gr,a (z)2X log a r,a X √ π |z| π ε≤|z|<1 D dA(z) ≤ 2cε ((Sr f ) ◦ σϕ(a) ) (z)2X (1 − |z|2 ) ≤ 2cεSr f 2C,X , π D for a ∈ D such that |ϕ(a)| > R. Moreover, by applying the identities gr,a (z)X = 2 2 (Sr f ) (σϕ(a) (z))X · |σϕ(a) (z)| and 1 − |σϕ(a) (z)| = (1 − |z| )|σϕ(a) (z)|, the Littlewood inequality (3.4) and Lemma 3.3, we get dA(z) 1 dA(z) 2 ≤ g (z) N (ϕ , z) gr,a (z)2X log a r,a X √ √ π |z| π |z|< ε |z|< ε 1 1 dA(z) ≤ 2Sr f 2C,X log √ (1 − |z|2 )2 |z| π |z|< ε √ 4 ε ≤ Sr f 2C,X . (1 − ε)2

By combining these estimates with (4.7), we get sup {a∈D : |ϕ(a)|>R}

Ma (Cϕ Sr f ) ≤ Cεf 2BMOAC (X) .

(4.8)

for all r ∈ (0, 1), where C is a constant. We next consider a ∈ D such that |ϕ(a)| ≤ R. By (4.2), there is t0 ∈ (0, 1) such that N (ϕ ◦ σa , z) ≤ ε log(1/|z|),

(4.9)

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for every a, z ∈ D satisfying |ϕ(a)| ≤ R and |z| > t0 . Using Lemma 4.2(2) we choose r0 ∈ (0, 1) so that sup (Sr f ) (z)2X ≤ εf 2BMOAC (X) ,

(4.10)

|z|≤t0

for all r ≥ r0 . By (3.2), the estimate 1 − |z|2 ≤ 2 log(1/|z|), and the formula (2.5) applied to the function λ(z) = (Sr f ) (z)2X , we get dA(z) Ma (Cϕ Sr f ) = ((Sr f ) ◦ ϕ ◦ σa ) (z)2X (1 − |z|2 ) π D (4.11) dA(z) ≤ 2 (Sr f ) (z)2X N (ϕ ◦ σa , z) . π D Hence dA(z) 1 dA(z) 2 ≤ ε (Sr f ) (z)2X log (Sr f ) (z)X N (ϕ ◦ σa , z) π |z| π t0 <|z|<1 D ≤ K 2 cεf 2BMOAC (X) ,

by (4.9) and (3.1). Moreover, since 2 D N (ϕ ◦ σa , z) dA(z) = ϕ ◦ σa − ϕ(a)2H 2 ≤ 1, π we get from (4.10) that dA(z) 2 (Sr f ) (z)2X N (ϕ ◦ σa , z) ≤ εf 2BMOAC (X) , π |z|≤t0 for r ≥ r0 . By combining the preceding estimates with (4.11), we get sup {a∈D : |ϕ(a)|≤R}

Ma (Cϕ Sr f ) ≤ (2K 2 c + 1)εf 2BMOAC (X) ,

for all r ≥ r0 . Finally, by taking (4.8) together with the above estimate, we obtain (4.5). This proves the lemma and ﬁnishes the proof of Theorem 4.1. In the special case X = C, the operators Cr are compact on BMOA for all r ∈ (0, 1). Thus we have the following characterization. Corollary 4.5. The composition operator Cϕ is compact on BMOA if and only if (4.1) and (4.2) hold. A complete characterization of the weakly compact composition operators on BMOAC (X) depends on the question whether all weakly compact composition operators on BMOA are compact or not. Unfortunately this question still seems to be open for arbitrary composition operators on BMOA. However, there are some partial positive results in the literature, which in combination with Theorem 4.1 lead to characterizations of weakly compact composition operators on BMOAC (X) in some cases. By applying [20, Thm. 4.1], [9, Thm. 1] and [17, Cor. 5.4], we obtain the following partial characterization. Assume that ϕ : D → D is analytic and satisﬁes one of the following conditions: 1. ϕ is univalent, or 2. ϕ ∈ VMOA and ϕ(D) lies inside a polygon inscribed in the unit circle.

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Then Cϕ is weakly compact on BMOAC (X) if and only if X is reﬂexive and Cϕ is compact on BMOA. See [14, p. 741] for the details. Acknowledgement I thank my supervisor Hans-Olav Tylli for his many valuable suggestions and comments.

References [1] J. Arazy, S.D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145. [2] A. Baernstein II, Analytic functions of bounded mean oscillation, in: Aspects of contemporary complex analysis, Proc. NATO Adv. Study Inst., Durham, 1979, Academic Press, London-New York, 1980, pp. 3–36 [3] O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces, Illinois J. Math. 41 (1997), 532–558. [4] O. Blasco, Remarks on vector-valued BMOA and vector-valued multipliers, Positivity 4 (2000), 339–356. [5] O. Blasco, On coeﬃcients of vector-valued Bloch functions, Studia Math. 165 (2004), 101–110. [6] O. Blasco and M. Pavlović, Complex convexity and vector-valued Littlewood-Paley inequalities, Bull. London Math. Soc. 35 (2003), 749–758. [7] J. Bonet, P. Domański and M. Lindström, Weakly compact composition operators on analytic vector-valued function spaces, Ann. Acad. Sci. Fenn. Math. 26 (2001), 233–248. [8] P.S. Bourdon, J.A. Cima and A.L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183–2196. [9] J.A. Cima and A.L. Matheson, Weakly compact composition operators on VMO, Rocky Mountain J. Math. 32 (2002), 937–951. [10] C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [11] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. [12] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York-London, 1981. [13] D. Girela, Analytic functions of bounded mean oscillation, in: Complex function spaces, Mekrijärvi, 1999, Univ. Joensuu Dept. Math. Rep. Ser. 4, Univ. Joensuu, Joensuu, 2001, pp. 61–170. [14] J. Laitila, Weakly compact composition operators on vector-valued BMOA, J. Math. Anal. Appl. 308 (2005), 730–745. [15] J. Laitila and H.-O. Tylli, Composition operators on vector-valued harmonic functions and Cauchy transforms, Indiana Univ. Math. J. 55 (2006), 719–746. [16] P. Liu, E. Saksman and H.-O. Tylli, Small composition operators on analytic vectorvalued function spaces, Paciﬁc J. Math. 184 (1998), 295–309.

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[17] S. Makhmutov and M. Tjani, Composition operators on some Möbius invariant Banach spaces, Bull. Austral. Math. Soc. 62 (2000), 1–19. [18] J.H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404. [19] J.H. Shapiro, Composition Operators and Classical Function Theory, SpringerVerlag, New York, 1993. [20] W. Smith, Compactness of composition operators on BMOA, Proc. Amer. Math. Soc. 127 (1999), 2715–2725. [21] K. Stephenson, Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), 565–577. [22] M. Tjani, Composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University, 1996. [23] K.J. Wirths and J. Xiao, Global integral criteria for composition operators, J. Math. Anal. Appl. 269 (2002), 702–715. [24] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, Cambridge, 1991. Jussi Laitila Department of Mathematics and Statistics University of Helsinki P.O. Box 68 FIN-00014 University of Helsinki Finland e-mail: [email protected] Submitted: December 14, 2005 Revised: May 19, 2007

Integr. equ. oper. theory 58 (2007), 503–549 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040503-47, published online June 27, 2007 DOI 10.1007/s00020-007-1502-4

Integral Equations and Operator Theory

An Operator-valued Berezin Transform and the Class of n-Hypercontractions Anders Olofsson Abstract. We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction T ∈ L(H) we associate a positive L(H)-valued operator measure dωn,T supported on the ¯ in a way that generalizes the above notion of operatorclosed unit disc D valued Berezin transform. This construction of positive operator measures dωn,T gives a natural functional calculus for the class of n-hypercontractions. We revisit also the operator model theory for the class of n-hypercontractions. The new results here concern certain canonical features of the theory. The operator model theory for the class of n-hypercontractions gives information about the structure of the positive operator measures dωn,T . Mathematics Subject Classiﬁcation (2000). Primary 47A20, 47A25; Secondary 47A45, 47B20. Keywords. Berezin transform, n-hypercontraction, functional calculus, standard weighted Bergman space, operator model theory, moment problem.

Contents 0. 1. 2. 3. 4. 5. 6.

Introduction Invariance properties of n-hypercontractions The Berezin transform for a general radial measure Construction of the operator measure dωn,T Relations with the operator measure dωT The space An (E) and its shift operator Sn Operator model theory. General considerations

504 510 516 520 525 527 530

Research supported by the M.E.N.R.T. (France) and the G. S. Magnuson’s Fund of the Royal Swedish Academy of Sciences.

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7. Operator model theory. n-hypercontractions 8. Structure properties of the operator measure dωn,T 9. Subnormal contractions and the Hausdorﬀ moment problem References

533 538 541 548

0. Introduction Let H be a general not necessarily separable complex Hilbert space and denote by L(H) the space of all bounded linear operators on H. Let n ≥ 1 be an integer. An operator T ∈ L(H) is called an n-hypercontraction if the operator inequality m k m (−1) T ∗k T k ≥ 0 in L(H) k k=0

holds true for every 1 ≤ m ≤ n. In this terminology a 1-hypercontraction is a contraction, but for n ≥ 2 the class of n-hypercontractions is a more restricted class of operators. The class of n-hypercontractions was ﬁrst introduced by Agler [1, 2] whereas the study of contractions on Hilbert space is a classical topic of which the book [27] by Sz.-Nagy and Foias is a standard reference. Let n ≥ 1 be an integer. We shall need the Hilbert space An (D) of analytic functions in the unit disc D with reproducing kernel 1 Kn (z, ζ) = ¯ n , (z, ζ) ∈ D × D. (1 − ζz) The space A1 (D) is just the standard Hardy space H 2 (D), and for n ≥ 2 the space An (D) is the standard weighted Bergman space of square integrable analytic functions in D corresponding to the weighted area measure dµn (z) = (n − 1)(1 − |z|2 )n−2 dA(z),

z ∈ D;

here dA(z) = dxdy/π, z = x + iy, is the usual planar Lebesgue area measure normalized so that the unit disc D is of unit area. For notational reasons we let also dµ1 denote the normalized Lebesgue arc length measure on the unit circle T = ∂D. For n ≥ 1, an analytic function f in D belongs to the space An (D) if and only if the norm f 2An = lim

r→1

¯ D

|f (rz)|2 dµn (z)

is ﬁnite. Notice that this norm can also be written |ak |2 µn;k , f 2An = k≥0

where ak is the k-th Taylor coeﬃcient f ∈ An (D) (see (0.5) below) and {µn;k }k≥0 is the sequence of moments of the measure dµn deﬁned by k+n−1 2k µn;k = |z| dµn (z) = 1/ , k ≥ 0. (0.1) k ¯ D

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A standard reference for Bergman spaces on the unit disc is the recent book [19] by Hedenmalm, Korenblum and Zhu. The function Bn deﬁned by Bn (z, ζ) =

(1 − |z|2 )n |Kn (z, ζ)|2 = ¯ 2n , Kn (z, z) |1 − ζz|

¯ (z, ζ) ∈ D × D,

is called the Berezin kernel associated to the kernel function Kn . The corresponding transform deﬁned by Bn [f ](z) = Bn (z, ζ)f (ζ)dµn (ζ), z ∈ D, ¯ D

for, say, f ∈ L1 (µn ) is called the Berezin transform. Notice that for n = 1 the so-called Poisson transform is obtained. It is well-known that the Berezin transform reproduces harmonic functions. In the literature the Berezin transform has attracted some attention because of its use in the study of Toeplitz operators; see for instance [3, 7]. In this paper we shall consider certain related operator-valued Berezin transforms that we now proceed to deﬁne. Let T ∈ L(H) be an operator with spectral radius r(T ) = maxz∈σ(T ) |z| strictly less than 1. The operator-valued Berezin kernel Bn (T, ·) is the function deﬁned by n n ∗k k ¯ )−n , ζ ∈ D. ¯ (−1)k (0.2) Bn (T, ζ) = (I − ζT ∗ )−n T T (I − ζT k k=0

Notice that Bn (T, ζ) ≥ 0 in L(H) if T is an n-hypercontraction. We have an associated operator-valued Berezin transform deﬁned by Bn [f ](T ) = Bn (T, ζ)f (ζ)dµn (ζ) ¯ D

¯ Throughout the paper we denote by C(D) ¯ the space of continfor, say, f ∈ C(D). ¯ uous functions on the closed unit disc D. We shall associate to an n-hypercontraction T ∈ L(H) a positive L(H)¯ The Berezin valued operator measure dωn,T supported on the closed unit disc D. transform of a monomial ζ¯j ζ k , j, k ≥ 0, has the power series expansion Wn;m;j,k |z|2m z k−min(j,k) , z ∈ D. Bn [ζ¯j ζ k ](z) = z¯j−min(j,k) m≥0

This equality clearly determines the numbers Wn;m;j,k uniquely. The operator measure dωn,T is deﬁned by its action on monomials by the requirement that Wn;m;j,k T ∗m T m T k−min(j,k) (0.3) ζ¯j ζ k dωn,T (ζ) = T ∗(j−min(j,k)) ¯ D

m≥0

for j, k ≥ 0 (see Theorem 3.1). We remark that there is a decay estimate Wn;m;j,k = O(m−(n+1) ) as m → ∞

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so that formula (0.3) makes sense (see Lemma 3.1). Notice also that since the ¯ (Stone-Weierstrass) the space C[z, z¯] of polynomials in z and z¯ is dense in C(D) operator measure dωn,T is uniquely determined by its action on monomials. The operator measure dωn,T extends the above notion of operator-valued Berezin transform in the sense that the equality ¯ f (ζ)dωn,T (ζ) = Bn [f ](T ), f ∈ C(D), ¯ D

¯ denotes the space of holds when r(T ) < 1 (see Corollary 3.1). Recall that C(D) ¯ continuous functions on the closed unit disc D. We remark that for n = 1 the operator measure dω1,T obtained in this way is supported by the unit circle T and coincides with a certain operator measure on T denoted by dωT (see Proposition 4.1). We mention that the operator measure dωT is closely related to the unitary dilation of the contraction T and was called the harmonic spectral measure by Foias [15, 16] in the 1950’s (see Section 4). The operator measures dωn,T have the continuity property that the map ¯ C(D) × Cn (f, T ) → f (ζ)dωn,T (ζ) ∈ L(H) ¯ D

is continuous; here Cn denotes the set of all n-hypercontractions in L(H) and L(H) is equipped the uniform operator topology (see Theorem 3.2). The operator measures dωn,T are also shown to have a property of conformal invariance with respect to conformal automorphisms of the unit disc (see Corollary 3.2). This property of conformal invariance is inherited from corresponding conformal invariance properties of the class of n-hypercontractions (see Section 1). The above considerations yield also a natural functional calculus for the class of n-hypercontractions. Let the function u in D be the Berezin transform of f ∈ ¯ u = Bn [f ]. The function u is real-analytic in D and has a power series C(D): expansion u(z) = cjk z¯j z k , z ∈ D. j,k≥0

The function u now operates on the class of n-hypercontractions T ∈ L(H) in the sense that j+k ∗j k u(T ) := lim r cjk T T = f (ζ)dωn,T (ζ) in L(H) (0.4) r→1

j,k≥0

¯ D

(see Theorem 3.3). We emphasize that the limit in (0.4) is computed in the uniform operator topology, that is, in operator norm. A basic property coming from the ¯ = I is the norm inequality positivity of the operator measure dωn,T and ωn,T (D) ¯ f (ζ)dωn,T (ζ) ≤ f ∞ , f ∈ C(D), ¯ D

which by (0.4) extends the classical von Neumann inequality [25].

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For reasons of modeling a general n-hypercontraction we shall need to consider also Hilbert space valued versions of the spaces An (D). Let E be a Hilbert space and denote by An (E) = An (D, E) the space of all E-valued analytic functions f (z) = ak z k , z ∈ D; (0.5) k≥0

here ak ∈ E for k ≥ 0, with ﬁnite norm ak 2 µn;k , f 2An = k≥0

where {µn;k }k≥0 is the sequence of moments of dµn given by (0.1). Notice that this is consistent with the previous description of the space An (D). On the space An (E) we have a natural shift operator S = Sn deﬁned by ak−1 z k , z ∈ D, (Sn f )(z) = zf (z) = k≥1

for f ∈ An (E) given by (0.5). It turns out that the shift operator Sn acts boundedly on the space An (E) in such a way that the adjoint operator Sn∗ is an nhypercontraction with the property that limk→∞ Sn∗k = 0 in the strong operator topology (see Proposition 5.1). In Sections 6 and 7 we shall revisit some operator model theory relating to the class of n-hypercontractions. Recall that an operator A ∈ L(H) is part of an operator B ∈ L(K) if H is a B-invariant subspace of K and A = B|H ; the operator B is then called an extension of A. As pointed out in the previous paragraph the adjoint shift operator Sn∗ is an n-hypercontraction with the property that limk→∞ Sn∗k = 0 in the strong operator topology. It is also clear that every isometry is an n-hypercontraction. The principal modeling result of n-hypercontractions due to Agler [1, 2] asserts that an operator T ∈ L(H) is an n-hypercontraction if and only if it is part of an operator of the form Sn∗ ⊕ U , where U is an isometry. As a byproduct of this result one has that an operator T ∈ L(H) is part of an adjoint shift operator Sn∗ if and only T is an n-hypercontraction such that limk→∞ T k = 0 in the strong operator topology. This result by Agler was ﬁrst proved using C ∗ algebra methods. The purpose of our presentation in Sections 6 and 7 is to show that there is a certain uniqueness property and associated canonical construction behind this modeling result of n-hypercontractions. To describe our results we need some more notation. For an operator T ∈ L(H) such that the limit limk→∞ T k x2 exists for every x ∈ H we consider the operator 1/2 in L(H), Q = lim T ∗k T k k→∞

where the positive square root is used and the limit is computed in the weak operator topology. We denote by Q the closure in H of the range of Q, that is, Q = Q(H). On the space Q we have a natural isometry U deﬁned by U : Qx → QT x for x ∈ H and continuity.

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In Section 6 we consider the more general problem of modeling an operator T ∈ L(H) as part of an operator of the form T1∗ ⊕ T2 , where Tj ∈ L(Hj ) (j = 1, 2) are operators such that limk→∞ T1∗k = 0 in the strong operator topology and T2 is an isometry. This more general modeling problem amounts to that of ﬁnding an isometry V = (V1 , V2 ) : H → H1 ⊕ H2 (0.6) of H into H1 ⊕ H2 satisfying the intertwining relation V T = (T1∗ ⊕ T2 )V.

(0.7)

It turns out that there is a canonical choice of V2 and (T2 , H2 ) given by V2 = Q, T2 = U and H2 = Q, and that the general modeling problem (0.6) and (0.7) reduces to that of ﬁnding a bounded linear operator V1 : H → H1 satisfying the norm equality x2 − T x2 = V1 x2 − T1∗ V1 x2 ,

x ∈ H,

T1∗ V1

(see Theorems 6.1 and 6.2). and the intertwining relation V1 T = For an n-hypercontraction T ∈ L(H) we consider the defect operators m m ∗k k 1/2 (−1)k in L(H) Dm,T = T T k k=0

for 1 ≤ m ≤ n, where the positive square root is used. We have an associated defect space Dn,T deﬁned as the closure in H of the range of Dn,T , that is, Dn,T = Dn,T (H). In Section 7 we specialize the modeling problem (0.6) and (0.7) further to the case when H1 = An (E) and T1 = Sn is the shift operator acting on this space. It turns out that there is a canonical choice of coeﬃcient space E and operator V1 : H → An (E) given by E = Dn,T and V1 x = V1,n x for x ∈ H, where for x ∈ H the Dn,T -valued analytic function V1,n x is deﬁned by the formula k + n − 1 (Dn,T T k x)z k , z ∈ D (0.8) (V1,n x)(z) = Dn,T (I − zT )−nx = k k≥0

(see Theorem 7.1). To some extent formula (0.8) is also motivated by the explicit form of the operator-valued Berezin kernel (0.2). In the case of an n-hypercontraction T ∈ L(H) we show that the map V1 = V1,n : x → V1,n x given by (0.8) is admissible for the above modeling problem (0.6) and (0.7) in the sense that the map V = (V1,n , Q) : H → An (Dn,T ) ⊕ Q deﬁned by V x = (V1,n x, Qx) for x ∈ H is an isometry of H into An (Dn,T ) ⊕ Q satisfying the interwining relation V T = (Sn∗ ⊕ U )V (see Theorem 7.2).

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In Section 8 we use the operator model theory for the class of n-hypercontractions to give some more detailed results describing the structure of the operator measures dωn,T . Let us denote by S the σ-algebra of planar Borel sets. The operator measure dωn,T naturally decomposes as ∗ ωn,Sn∗ (S)V1,n + QωU (S)Q, ωn,T (S) = V1,n

and for n ≥ 2 we further have that ∗ V1,n ωn,Sn∗ (S)V1,n =

S ∈ S,

D

S

Bn (T, ζ)dµn (ζ),

S∈S

(see Theorems 8.1 and 8.2). Notice that this gives that ωn,T (S) = 0 in L(H) if S is a Borel subset of D of planar Lebesgue area measure zero. In particular, we have that ¯ dωn,T (ζ) = Bn (T, ζ)dµn (ζ), ζ ∈ D, if n ≥ 2 and T ∈ L(H) is an n-hypercontraction such that limk→∞ T k = 0 in the strong operator topology (see Corollary 8.1). Invoking a classical theorem of Sz.Nagy and Foias we deduce that the operator measure dωU is absolutely continuous with respect to Lebesgue arc length measure on T if T ∈ L(H) is a completely non-unitary contraction. The method of construction of operator models used here goes back at least to work of de Branges and Rovnyak [10, Theorem 1] in the 1960’s; see also [27, Section I.10.1]. In this context we also want to mention more recent related work by M¨ uller [22], Vasilescu [28, 29], M¨ uller and Vasilescu [23], and Curto and Vasilescu [13, 14] concerned with modeling of operators in terms of weighted shifts, and also the papers Ambrozie, Engliˇs and M¨ uller [4] and Arazy and Engliˇs [5]. Also, operator models of this type form an integral part in recent work on constrained von Neumann inequalities by Badea and Cassier [8]. It was shown by Agler [2, Theorem 3.1] that an operator T ∈ L(H) is a subnormal contraction if and only if it is an n-hypercontraction for every n ≥ 1. In Section 9 we derive this characterization of subnormal contractions as a limit case of our study of operator-valued Berezin transforms (see Theorem 9.1). As an application of this result by Agler we consider two operator-valued moment problems of Hausdorﬀ type (see Theorem 9.2 and Proposition 9.3). At several places in this paper we encounter operators T such that limk→∞ T k = 0 in the strong operator topology. We mention that an operator T ∈ L(H) is said to belong to the class C0· if limk→∞ T k = 0 in the strong operator topology (see [27, Section II.4]). In a recent paper [26] we have studied a related positive operator measure dωT on the unit n-torus Tn associated to an n-tuple T = (T1 , . . . Tn ) of commuting contractions in L(H) having a so-called regular unitary dilation. The more involved construction of the operator measure dωn,T in this paper using the numbers Wn;m;j,k and the decay of these numbers as m → ∞ (see Lemma 3.1) is due to a more complicated regularity behavior of Berezin transforms compared to the case of the Poisson transform. Eventhough f ∈ C[z, z¯] is a polynomial, the Berezin

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transform B2 [f ] is in general not C 2 -smooth up to the boundary T = ∂D (see Remark 3.2 for an example). A similar regularity behavior is known to present itself in the study of the Dirichlet problem for the so-called invariant Laplacian (the Laplace-Beltrami operator) for the unit ball in Cn (see [21, Chapter 6]). Preliminaries. Let us recall the notions of weak, strong and uniform operator topology. The uniform operator topology on L(H) is the usual topology on L(H) deﬁned by the operator norm. The strong operator topology (SOT) on L(H) is the topology on L(H) deﬁned by the semi-norms L(H) T → T x ∈ [0, ∞),

x ∈ H.

Notice that Tk → T (SOT) means that Tk x → T x in H for every x ∈ H. The weak operator topology (WOT) on L(H) is the topology on L(H) deﬁned by the semi-norms T → | T x, y| for x, y ∈ H. In the paper we shall need some facts from the theory of integration in Hilbert space. Let S be the σ-algebra of planar Borel sets. A ﬁnitely additive set function µ : S → L(H) is called a positive operator measure if µ(S) ≥ 0 in L(H) for every S ∈ S and the set functions µx,y , x, y ∈ H, deﬁned by µx,y (S) = µ(S)x, y for S ∈ S, are all complex regular Borel measures. A positive operator measure µ is of ﬁnite semi-variation |µ|(S) :=

sup x,y≤1

|µx,y |(S) = µ(S),

S ∈ S,

where |µx,y | is the total variation of the complex measure µx,y . The integral S f dµ is deﬁned as an operator in L(H) by the duality requirement that S f dµx, y = S f dµx,y for all x, y ∈ H. An important property of the integral is the norm inequality f (s)dµ(s) ≤ |µ|(S)f ∞ , S

where · ∞ denotes the norm of essential supremum on S. We refer to [26] for some more details. We shall use the following operator version of the F. Riesz representation ¯ into L(H) which is positive in the sense theorem: If Λ is a linear map from C(D) ¯ then there exists a positive L(H)-valued that Λ(f ) ≥ 0 in L(H) if f ≥ 0 in D, ¯ which represents Λ in the sense that operator measure dλ on D ¯ f (z)dλ(z), f ∈ C(D) Λ(f ) = ¯ D

(see [26]).

1. Invariance properties of n-hypercontractions The purpose of this section is to discuss some invariance properties of the class of n-hypercontractions and the related operator-valued Berezin kernel. Let n ≥ 1 be

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an integer, and recall that an operator T ∈ L(H) is called an n-hypercontraction if m k m (−1) (1.1) T ∗k T k ≥ 0 in L(H) k k=0

for 1 ≤ m ≤ n. Notice that the deﬁning property (1.1) of an n-hypercontraction is equivalently formulated that m k m (−1) T k x2 ≥ 0, x ∈ H, k k=0

for 1 ≤ m ≤ n. Let us consider the backward shift operator λ acting on sequences a = {ak }∞ k=0 by (λa)k = ak+1 for k ≥ 0. We notice that an operator T ∈ L(H) is an n-hypercontraction if and only if (I − λ)m a ≥ 0 for 1 ≤ m ≤ n and every k 2 sequence a = {ak }∞ k=0 of the form ak = T x for k ≥ 0, where x ∈ H. It is known that if T ∈ L(H) is an n-hypercontraction, then so is rT for every 0 ≤ r < 1 (see [2, Lemma 1.9]). For the sake of completeness we include a proof of this fact. Proposition 1.1. If T ∈ L(H) is an n-hypercontraction, then so is rT for every 0 ≤ r < 1. Proof. We consider the backward shift operator λ acting on sequences a = {ak }∞ k=0 by (λa)k = ak+1 for k ≥ 0. By the binomial theorem we have that m m (I − r2 λ)m = (1 − r2 )m−k r2k (I − λ)k . (1.2) k k=0

k 2 Consider now a sequence a = {ak }∞ k=0 of the form ak = T x for k ≥ 0, where x ∈ H. Since the operator T is an n-hypercontraction we have that (I − λ)m a ≥ 0 for 1 ≤ m ≤ n. By the binomial identity (1.2) we conclude that (I − r2 λ)m a ≥ 0 for 1 ≤ m ≤ n. This yields the conclusion that the operator rT is an n-hypercontraction.

The following lemma gives a kind of stability property of n-hypercontractions. Lemma 1.1. Let n ≥ 2. Let T ∈ L(H) be an operator such that n k n T ∗k T k ≥ 0 in L(H), (−1) k k=0

and T k x2 = o(k) as k → ∞ for every x ∈ H. Then the operator T is an n-hypercontraction, that is, inequality (1.1) holds for 1 ≤ m ≤ n. Proof. Let us ﬁrst recall a simple fact about convex sequences: If a = {ak }∞ k=0 is a is decreasing. In terms convex sequence and lim supk→∞ ak /k ≤ 0, then {ak }∞ k=0 of the backward shift λ this fact gives that (I − λ)2 a ≥ 0 and ak = o(k) implies (I − λ)a ≥ 0.

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k 2 We now consider a sequence a = {ak }∞ k=0 of the form ak = T x for k ≥ 0, n where x ∈ H. By assumption we know that (I − λ) a ≥ 0 and ak = o(k). By repeated applications of the observation in the previous paragraph, we conclude that (I − λ)m (a) ≥ 0 for 1 ≤ m ≤ n. This yields the conclusion of the lemma.

We mention in passing that Lemma 1.1 relaxes the growth assumption of T power bounded used in [22, Corollary 3.6]. Lemma 1.1 gives the following converse to Proposition 1.1. Corollary 1.1. Let T ∈ L(H) be an operator such that n j n (−1) r2j T ∗j T j ≥ 0 in L(H) j j=0 for r = rk → 1, 0 ≤ rk < 1, and lim supk→∞ T k x2/k ≤ 1 for every x ∈ H. Then the operator T is an n-hypercontraction. Proof. By an application of Lemma 1.1 we conclude that the operator rk T is an n-hypercontraction. Now letting rk → 1 the conclusion of the corollary follows. We shall now consider some properties of invariance with respect to conformal automorphisms of the unit disc. First we need a lemma. Lemma 1.2. Let n ≥ 1 be an integer, and let T ∈ L(H) an operator such that r(T ) ≤ 1. Then the equality n k n (−1) ϕα (T )∗k ϕα (T )k k k=0 n n ∗k k ¯ T )−n = (1 − |α|2 )n (I − αT ∗ )−n (−1)k T T (I − α k k=0

holds for every conformal automorphism ϕα of the unit disc of the form ϕα (z) = (z − α)/(1 − αz) ¯ for z ∈ D, where α ∈ D. Proof. To simplify some formulas in the proof we shall write n k n Sn (T ) = (−1) T ∗k T k . k k=0

With this notation the assertion of the lemma reads as ¯ T )−n . Sn (ϕα (T )) = (1 − |α|2 )n (I − αT ∗ )−n Sn (T )(I − α

(1.3)

We shall prove formula (1.3) by induction on n ≥ 0. Notice that by the standard n−1 formula nk = n−1 + k k−1 for binomial coeﬃcients we have Sn (T ) = Sn−1 (T ) − T ∗ Sn−1 (T )T.

(1.4)

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Let us now turn to the proof of (1.3). For n = 0 there is nothing to prove. Assume n ≥ 1 and that formula (1.3) holds true with n replaced by n − 1. Using (1.4) and the induction hypothesis we compute that Sn (ϕα (T )) = Sn−1 (ϕα (T )) − ϕα (T )∗ Sn−1 (ϕα (T ))ϕα (T ) ¯ T )−n+1 = (1 − |α|2 )n−1 (I − αT ∗ )−n+1 Sn−1 (T )(I − α − (1 − |α|2 )n−1 ϕα (T )∗ (I − αT ∗ )−n+1 Sn−1 (T )(I − α ¯ T )−n+1 ϕα (T ) ¯T ) = (1 − |α|2 )n−1 (I − αT ∗ )−n (I − αT ∗ )Sn−1 (T )(I − α

− (T ∗ − α ¯ I)Sn−1 (T )(T − αI) (I − α ¯ T )−n = (1 − |α|2 )n (I − αT ∗ )−n Sn−1 (T ) − T ∗ Sn−1 (T )T (I − α ¯ T )−n = (1 − |α|2 )n (I − αT ∗ )−n Sn (T )(I − α ¯ T )−n . By the principle of induction this completes the proof of formula (1.3).

Let us denote by Aut(D) the set of all conformal automorphisms ϕ of the unit disc D. It is well-known that every ϕ ∈ Aut(D) can be written in the form ϕ(z) = eiθ ϕα (z),

z ∈ D,

iθ

¯ z). where e ∈ T, α ∈ D and ϕα (z) = (z − α)/(1 − α We can now conclude that the conformal automorphisms operate on the class of n-hypercontractions. Corollary 1.2. If T ∈ L(H) is an n-hypercontraction, then so is ϕ(T ) for every ϕ ∈ Aut(D).

Proof. This is clear by Lemma 1.2.

We mention here that Corollary 1.2 is contained in the statement of [12, Theorem 2.1]. Recall that the operator-valued Berezin kernel is the function deﬁned by the formula n n ∗k k ¯ )−n , ζ ∈ D; ¯ (−1)k T T (I − ζT Bn (T, ζ) = (I − ζT ∗ )−n k k=0

here T ∈ L(H) is an operator such that r(T ) < 1. We shall now prove a property of invariance of this operator-valued Berezin kernel. Proposition 1.2. Let n ≥ 1, and let T ∈ L(H) be an operator such that r(T ) < 1. Then the operator-valued Berezin kernel has the invariance property that Bn (ϕ(T ), ϕ(ζ))(1 − |ϕ(ζ)|2 )n = Bn (T, ζ)(1 − |ζ|2 )n , for every ϕ ∈ Aut(D).

¯ ζ ∈ D,

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Proof. It is easy to see that it suﬃces to consider ϕ of the form ϕ = ϕα . We ﬁrst compute that 1 − αζ¯ n ¯ )−n . (I − ϕα (ζ)ϕα (T ))−n = (I − α ¯ T )n (I − ζT 1 − |α|2 Using Lemma 1.2 we now compute that 1−α ¯ ζ n (I − αT ∗ )n (I − ζT ∗ )−n Bn (ϕα (T ), ϕα (ζ)) = 1 − |α|2 n n ∗k k ¯ T )−n × (1 − |α|2 )n (I − αT ∗ )−n (−1)k T T (I − α k k=0 1 − αζ¯ n ¯ )−n × (I − α ¯ T )n (I − ζT 1 − |α|2 n |1 − α ¯ ζ|2n ∗ −n k n ∗k k ¯ )−n = (I − ζT ) (−1) T (I − ζT T k (1 − |α|2 )n k=0

|1 − α ¯ ζ|2n = Bn (T, ζ). (1 − |α|2 )n By the well-known formula 1 − |ϕα (ζ)|2 =

(1 − |α|2 )(1 − |ζ|2 ) |1 − α ¯ ζ|2

the conclusion of the proposition now follows.

We remark that in Proposition 1.2 we have r(ϕ(T )) < 1 by the spectral mapping theorem. Associated to the Berezin kernel Bn (T, ·) we have the operator-valued Berezin transform deﬁned by ¯ Bn (T, ζ)f (ζ)dµn (ζ), f ∈ C(D). (1.5) Bn [f ](T ) = ¯ D

We shall now prove that this operator-valued Berezin transform (1.5) commutes with the action of conformal automorphisms. Theorem 1.1. Let T ∈ L(H) be an operator such that r(T ) < 1. Then the operatorvalued Berezin transform has the invariance property that Bn [f ◦ ϕ](T ) = Bn [f ](ϕ(T )),

¯ f ∈ C(D),

for every ϕ ∈ Aut(D). Proof. We assume that n ≥ 2. The case n = 1 is handled similarly. By a change of variables we see that Bn [f ◦ ϕ](T ) = (n − 1) Bn (T, ϕ−1 (ζ))f (ζ)|(ϕ−1 ) (ζ)|2 (1 − |ϕ−1 (ζ)|2 )n−2 dA(ζ). D

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Notice that T = ϕ−1 (ϕ(T )). By an application of Proposition 1.2 we now conclude that |(ϕ−1 ) (ζ)|2 (1 − |ζ|2 )n dA(ζ). Bn [f ◦ ϕ](T ) = (n − 1) Bn (ϕ(T ), ζ)f (ζ) −1 (ζ)|2 )2 (1 − |ϕ D By an invariance property of the Bergman kernel function we know that 1 |(ϕ−1 ) (ζ)|2 = , (1 − |ϕ−1 (ζ)|2 )2 (1 − |ζ|2 )2

ζ ∈ D.

This completes the proof of the theorem.

We remark that in Theorem 1.1 we have r(ϕ(T )) < 1 by the spectral mapping theorem. We shall consider also the variant of the operator-valued Berezin kernel deﬁned by n ∗ −n k n ¯ )−n , ζ ∈ D, T ∗k T k (I − ζT Bn (T, ζ) = (I − ζT ) (−1) (1.6) k k=0

where T ∈ L(H) is an operator such that r(T ) ≤ 1. Notice that this modiﬁed operator-valued Berezin kernel given by (1.6) has the corresponding invariance property that Bn (ϕ(T ), ϕ(ζ))(1 − |ϕ(ζ)|2 )n = Bn (T, ζ)(1 − |ζ|2 )n ,

ζ ∈ D,

for ϕ ∈ Aut(D) (see Proposition 1.2). Notice also that the Berezin kernel Bn (T, ζ) given by (1.6) is positive in L(H) if T is an n-hypercontraction. We shall need the following lemma. Lemma 1.3. Let n ≥ 2, and let T ∈ L(H) be an n-hypercontraction. Then the function Bn (T, ·) deﬁned by (1.6) is integrable with respect to the measure dµn . Furthermore, we have that | Bn (T, ζ)x, y|dµn (ζ) ≤ xy, x, y ∈ H. D

Proof. Let 0 ≤ r < 1. By Proposition 1.1 the Berezin kernel Bn (rT, ζ) is positive in L(H). By Corollary 2.1 in the next section with f = 1 we have that

Bn (rT, ζ)x, xdµn (ζ) = x2 , x ∈ H. D

Letting r → 1 an application of Fatou’s lemma gives that

Bn (T, ζ)x, xdµn (ζ) ≤ x2 . D

By the Cauchy-Schwarz inequality we now have that | Bn (T, ζ)x, y|dµn (ζ) ≤ Bn (T, ζ)x, x1/2 Bn (T, ζ)y, y1/2 dµn (ζ) D

D

≤ xy,

x, y ∈ H.

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This completes the proof of the lemma.

For T ∈ L(H) an n-hypercontraction and n ≥ 2, the Berezin transform Bn [f ](T ) deﬁned by (1.5) and (1.6) is well-deﬁned by Lemma 1.3. The invariance property of Theorem 1.1 remains true in this context. Proposition 1.3. Let n ≥ 2, and let T ∈ L(H) be an n-hypercontraction. Then the operator-valued Berezin transform deﬁned by (1.5) and (1.6) has the invariance property that ¯ Bn [f ◦ ϕ](T ) = Bn [f ](ϕ(T )), f ∈ C(D), for every ϕ ∈ Aut(D). Proof. See the proof of Theorem 1.1. We omit the details.

We remark that in Proposition 1.3 the operator ϕ(T ) is an n-hypercontraction by Corollary 1.2. We wish to point out that similar conformal invariance properties of operators have been studied in the context of the unit ball in Cn by Curto and Vasilescu [12]. The principal object of study in [12] is the operator-valued M-harmonic Poisson kernel introduced in [28].

2. The Berezin transform for a general radial measure In this section we shall derive some formulas for the Berezin transform in the con¯ We assume throughout text of a general radial measure µ on the closed unit disc D. the section that the associated kernel function Kµ is non-vanishing in D × D. ¯ such that Let µ be a ﬁnite positive radial measure on the closed unit disc D ¯ µ(D \ rD) > 0 for every 0 ≤ r < 1. The Bergman space Aµ (D) is the space of all analytic functions f (z) = ak z k , z ∈ D, k≥0

with ﬁnite norm f 2Aµ = lim

r→1

¯ D

|f (rz)|2 dµ(z) =

|ak |2 µk ,

k≥0

where {µk }k≥0 is the sequence of moments of µ deﬁned by µk = |z|2k dµ(z), k ≥ 0. ¯ D

1/k limk→∞ µk

= 1. Notice that The function Kµ deﬁned by 1 ¯ k, Kµ (z, ζ) = (ζz) µk k≥0

(z, ζ) ∈ D × D,

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is known as the kernel function for the Bergman space Aµ (D). The corresponding function Bµ deﬁned by Bµ (z, ζ) =

|Kµ (z, ζ)|2 , Kµ (z, z)

¯ (z, ζ) ∈ D × D,

is called the Berezin kernel. In what follows we assume that the kernel function Kµ is non-vanishing in D × D, that is, Kµ (z, ζ) = 0 for (z, ζ) ∈ D2 . We write 1 ¯ k , (z, ζ) ∈ D × D. = ck (ζz) Kµ (z, ζ) k≥0

The assumption of non-vanishing of the kernel function Kµ is a non-trivial assumption which means that lim supk→∞ |ck |1/k ≤ 1, so that the above series is convergent. Notice that n cn−k /µk = 0 (2.1) c0 /µ0 = 1 and k=0

for n ≥ 1. We now proceed to deﬁne the Berezin transform for operator-valued arguments. Let T ∈ L(H) be an operator such that r(T ) < 1. We set 1 1 ¯ Bµ (T, ζ) = ζ k T ∗k ck T ∗k T k ζ¯k T k , ζ ∈ D. (2.2) µk µk k≥0

k≥0

k≥0

Notice that by the spectral radius formula the sums in (2.2) are absolutely convergent in L(H). We shall be interested in operator-valued Berezin transforms of the type Bµ [f ](T ) =

¯ D

Bµ (T, ζ)f (ζ)dµ(ζ),

¯ where, say, the function f is in C(D).

Lemma 2.1. Let T ∈ L(H) be an operator such that r(T ) < 1. Then ¯ Bµ (T, ζ) = prs (ζ) T ∗r T s , ζ ∈ D,

(2.3)

r,s≥0

where the polynomials min(r,s)

prs (ζ) =

l=0

1 cl ζ r−l ζ¯s−l µr−l µs−l 1/(r+s)

satisfy the growth bound lim supr,s→∞ prs C(D) ≤ 1. In particular, we have that ¯ Bµ (T, ζ)f (ζ)dµ(ζ) = prs (ζ)f (ζ)dµ(ζ) T ∗r T s (2.4) ¯ D

¯ for, say, f ∈ C(D).

r,s≥0

¯ D

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Proof. Expanding formula (2.2) for Bµ (T, ζ) we have that

Bµ (T, ζ) =

j,k,n≥0

1 cn ζ j ζ¯k T ∗(j+n) T k+n . µj µk

Notice that by the spectral radius formula this sum is absolutely convergent uni¯ By a change of order of summation we obtain the series expanformly in ζ ∈ D. sion (2.3). Indeed, if we set r = j + n, s = k + n and l = n, then r, s ≥ 0 and 0 ≤ l ≤ min(r, s), which gives (2.3). The growth bound for the polynomials prs 1/k follows by limk→∞ µk = 1 and lim supk→∞ |ck |1/k ≤ 1. The last formula (2.4) follows by termwise integration of (2.3). We shall now compute the Berezin transform of a monomial. Proposition 2.1. Let T ∈ L(H) be an operator such that r(T ) < 1 and ﬁx j, k ≥ 0. Then Bµ (T, ζ)ζ¯j ζ k dµ(ζ) = T ∗(j−min(j,k)) Wm;j,k T ∗m T m T k−min(j,k) , ¯ D

m≥0

where Wm;j,k =

m l=0

µj+k−min(j,k)+l cm−l . µj−min(j,k)+l µk−min(j,k)+l

Proof. Recall formula (2.4) in Lemma 2.1. Notice that the polynomial prs has the homogeneity property that prs (eiθ ζ) = ei(r−s)θ prs (ζ) for eiθ ∈ T. Since the measure µ is radial we have that prs (ζ)ζ¯j ζ k dµ(ζ) = 0 ¯ D

whenever r + k = s + j. Assume now that r = j − min(j, k) + m and s = k − min(j, k) + m, where m ≥ 0. For such r, s we have that prs (ζ)ζ¯j ζ k dµ(ζ) ¯ D

=

m l=0

cl µj−min(j,k)+m−l µk−min(j,k)+m−l

¯ D

|ζ|2(j+k−min(j,k)+m−l) dµ(ζ) = Wm;j,k ,

where the last equality follows by a change of order of summation. Going back to formula (2.4) we have that Bµ (T, ζ)ζ¯j ζ k dµ(ζ) = T ∗(j−min(j,k)) Wm;j,k T ∗m T m T k−min(j,k) . ¯ D

This completes the proof of the lemma.

m≥0

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We remark that Proposition 2.1 gives the power series expansion for the Berezin transform of a monomial Bµ (z, ζ)ζ¯j ζ k dµ(ζ) = z¯j−min(j,k) Wm;j,k |z|2m z k−min(j,k) , z ∈ D, ¯ D

m≥0

where j, k ≥ 0. Lemma 2.2. Assume that j = 0 or k = 0. Then Wm;j,k =

m l=0

µj+k−min(j,k)+l cm−l = δm,0 , µj−min(j,k)+l µk−min(j,k)+l

where δ0,0 = 1 and δm,0 = 0 for m ≥ 1 is the Kronecker’s delta. Proof. We have that Wm;j,k =

m l=0

m

µj+k−min(j,k)+l cm−l = cm−l /µl , µj−min(j,k)+l µk−min(j,k)+l l=0

and the conclusion follows by (2.1).

We now conclude that the Berezin transform reproduces harmonic polynomials. Corollary 2.1. Let T ∈ L(H) be an operator such that r(T ) < 1. Then for every ¯ we have that harmonic function f = ck r|k| eikθ (z = reiθ ) in C(D) ∞ Bµ (T, ζ)f (ζ)dµ(ζ) = ck T (k), ¯ D

k=−∞

k

where T (k) = T for k ≥ 0 and T (k) = T

∗|k|

for k < 0.

Proof. By Proposition 2.1 and Lemma 2.2 we have that Bµ (T, ζ)f (ζ)dµ(ζ) = lim Bµ (T, ζ)f (rζ)dµ(ζ) ¯ D

r→1

= lim

r→1

¯ D

∞

ck r|k| T (k) =

k=−∞

∞

ck T (k) in L(H),

k=−∞

where the limits are computed in the uniform operator topology.

We wish to point out that the assumption of non-vanishing of the kernel function Kµ in D × D is of a non-trivial nature even for simple measures µ. Let dµ = c dδ0 + dµn , where c ≥ 0 is a positive parameter and dδ0 is the unit Dirac mass at 0. A straightforward computation shows that 1 1 Kµ (z, ζ) = −1 + ¯ n , (z, ζ) ∈ D × D. c+1 (1 − ζz)

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It is a straightforward matter to verify that for c large the function Kµ has zeroes in D × D. This example has been communicated to the author by Carl Sundberg (private discussion). On the other hand, Hedenmalm and Perdomo [20] have shown that if the weight function w : D → (0, ∞) is such that the function

D z → log w(z)/(1 − |z|2 ) is subharmonic, then the corresponding kernel function Kw is non-vanishing in D × D.

3. Construction of the operator measure dωn,T The purpose of this section is to construct the operator measure dωn,T and discuss some of its properties. Let n ≥ 1 and let T ∈ L(H) be an operator such that r(T ) < 1. The operator-valued Berezin kernel is deﬁned by the formula n n ∗k k ¯ )−n , ζ ∈ D. ¯ Bn (T, ζ) = (I − ζT ∗ )−n (−1)k T T (I − ζT k k=0

Recall from Section 2 that Bn (T, ζ)ζ¯j ζ k dµn (ζ) = T ∗(j−min(j,k)) Wn;m;j,k T ∗m T m T k−min(j,k) ¯ D

m≥0

for j, k ≥ 0, where µn;j+k−min(j,k)+m−l n (−1)l l µn;j−min(j,k)+m−l µn;k−min(j,k)+m−l l=0

k+n−1 (see Proposition 2.1); here µn;k = 1/ for k ≥ 0 are the moments of dµn . k We shall need the following lemma. min(m,n)

Wn;m;j,k =

Lemma 3.1. Let n ≥ 1 and j, k ≥ 0 be integers. Then

Wn;m;j,k = O m−(n+1) as m → ∞. Proof. We set al = Since 1 µn;k we have that al =

1 (n − 1)!

=

n−1

s=1 (j

µn;j+k−min(j,k)+l . µn;j−min(j,k)+l µn;k−min(j,k)+l

n−1 k+n−1 1 = (k + s), k (n − 1)! s=1 − min(j, k) + l + s) n−1 s=1 (k − min(j, k) + l + s) . n−1 s=1 (j + k − min(j, k) + l + s)

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In particular, the coeﬃcient al is a rational function in l. By the division algorithm, we see that al has the form q(l) , (3.1) al = p(l) + r(l) where p, q, r are polynomials with deg(p) ≤ n − 1 and deg(q) < deg(r) ≤ n − 1. We now consider the shift operator σ acting on sequences a = {ak }∞ k=0 by (σa)k = ak−1 for k ≥ 1 and (σa)0 = 0. In terms of this shift operator we have that min(m,n)

l n Wn;m;j,k = (−1) am−l = (I − σ)n a m , m ≥ 0. l l=0

Now using the above algebraic form (3.1) of the coeﬃcients al we see that

(I − σ)n a m = O m−(n+1) as m → ∞.

The conclusion of the lemma follows.

Remark 3.1. For n = 1 the numbers Wn;m;j,k are easily computable. We have that W1;0;j,k = 1 and W1;m;j,k = 0 for m ≥ 1. Remark 3.2. Let us also consider the case when n = 2 and j = k = 1. A computation gives that W2;0;1,1 = 1/2, W2;1;1,1 = 1/3 and W2;m;1,1 =

2 1 2 1 − + = m+2 m+1 m (m + 2)(m + 1)m

for m ≥ 2. In particular, we see that decay estimate in Lemma 3.1 gives the right order of magnitude in this case. A further computation using Proposition 2.1 gives that 1 1 1 2 , z ∈ D, W2;m;1,1 |z|2m = 2 − 2 + 1 − 2 log B2 [|ζ|2 ](z) = |z| |z| 1 − |z|2 m≥0

which is not C 2 -smooth up to the boundary T = ∂D. We are now ready to construct the operator measure dωn,T . Theorem 3.1. Let n ≥ 1 be an integer, and let T ∈ L(H) be an n-hypercontraction. Then there exists a positive L(H)-valued operator measure dωn,T on the closed unit ¯ such that disc D Wn;m;j,k T ∗m T m T k−min(j,k) , ζ¯j ζ k dωn,T (ζ) = T ∗(j−min(j,k)) ¯ D

m≥0

where

µn;j+k−min(j,k)+m−l n (−1)l l µn;j−min(j,k)+m−l µn;k−min(j,k)+m−l l=0

k+n−1 for j, k ≥ 0; here µn;k = 1/ . Furthermore, the operator measure dωn,T is k uniquely determined by this action on monomials. min(m,n)

Wn;m;j,k =

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IEOT

Wn;m;j,k T ∗m T m T k−min(j,k)

m≥0 j k

for monomials z¯ z , j, k ≥ 0, and extend this map Λ linearly to a linear map Λ from the space C[z, z¯] of polynomials in z and z¯ into L(H). We shall show below ¯ into L(H) that this map Λ extends uniquely to a bounded linear map from C(D) of norm less than or equal to 1 with the property that Λ(f ) ≥ 0 in L(H) for ¯ 0 ≤ f ∈ C(D). By an operator version of the F. Riesz representation theorem (see the preliminaries in the introduction) it then follows that there exists a positive L(H)-valued ¯ such that operator measure dωn,T on D ¯ f (z)dµn,T (z), f ∈ C(D). Λ(f ) = ¯ D

Clearly, this operator measure dωn,T has the action on monomials described in the ¯ (Stone-Weierstrass) it theorem. Since the polynomials in C[z, z¯] is dense in C(D) is clear that the operator measure dωn,T is uniquely determined by its action on monomials. We now proceed to prove the estimates needed. Let f (z) = j,k≥0 cjk z¯j z k be a polynomial in C[z, z¯] and 0 ≤ r < 1. By Proposition 2.1 we have that Bn (rT, ζ)f (ζ)dµn (ζ) (3.2) ¯ D cjk (rT )∗(j−min(j,k)) Wn;m;j,k (rT )∗m (rT )m (rT )k−min(j,k) . = m≥0

j,k≥0

By Proposition 1.1 the Berezin kernel Bn (rT, ζ) is positive in L(H). We have now that the left-hand side in (3.2) is of norm less than or equal to f C(D) ¯ (see the preliminaries in the introduction) and is positive in L(H) if f ≥ 0 in D. Also, the right-hand side in (3.2) tends to Λ(f ) in L(H) as r → 1. Passing to the limit as ¯] and that Λ(f ) ≥ 0 in r → 1 we conclude that Λ(f ) ≤ f C(D) ¯ for f ∈ C[z, z L(H) if f ∈ C[z, z¯] is such that f ≥ 0 in D. Since the polynomials in C[z, z¯] is ¯ (Stone-Weierstrass), the map Λ extends uniquely to a continuous dense in C(D) ¯ → L(H) of norm less than or equal to 1. Let us verify the linear map Λ : C(D) ¯ positivity property that Λ(f ) ≥ 0 in L(H) if 0 √ ≤ f ∈ C(D). ¯ also the function f is in C(D) ¯ Since 0 ≤ f ∈ C(D), can ﬁnd a √ and we ¯ Now the sequence {pj } of polynomials in C[z, z¯] such that pj → f in C(D). ¯ Now Λ(f ) = polynomial fj = |pj |2 is positive and we have that fj → f in C(D). limj→∞ Λ(fj ) ≥ 0 in L(H). This completes the proof of the theorem. ¯ = dωn,T = I. By these The operator measure dωn,T is positive and ωn,T (D) properties we have the inequality f (ζ)dωn,T (ζ) ≤ f ∞ , f ∈ C(D) ¯ (3.3) ¯ D

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(see the preliminaries in the introduction). We also have that f dωn,T ≥ 0 in ¯ L(H) if f ≥ 0 in D. We next observe that the operator measure dωn,T generalizes the notion of operator-valued Berezin transform. Corollary 3.1. Let T ∈ L(H) be an n-hypercontraction such that r(T ) < 1. Then dωn,T (ζ) = Bn (T, ζ)dµn (ζ),

¯ ζ ∈ D.

Proof. By the formulas stated in the ﬁrst paragraph in this section the operator measures Bn (T, ·)dµn and dωn,T have the same action on monomials. The corollary follows by the uniqueness assertion of Theorem 3.1. We remark that in terms of action on test functions the assertion of Corollary 3.1 means that ¯ f (ζ)dωn,T (ζ) = Bn (T, ζ)f (ζ)dµn (ζ), f ∈ C(D). ¯ D

¯ D

The operator measures dωn,T enjoy the following continuity property. Theorem 3.2. Denote by Cn the set all n-hypercontractions in L(H), and let Cn and L(H) be equipped with the uniform operator topology. Then the map ¯ × Cn (f, T ) → C(D) f (ζ)dωn,T (ζ) ∈ L(H) ¯ D

is continuous. Proof. Let {Tm } be a sequence of n-hypercontractions such that Tm → T0 in L(H). Using Lemma 3.1 it is straightforward to check that j k ¯ ζ ζ dωn,Tm (ζ) → ζ¯j ζ k dωn,T0 (ζ) in L(H) ¯ D

¯ D

as m → ∞. By linearization we see that P dωn,Tm → P dωn,T0 in L(H) for every polynomial P in C[z, z¯]. The proof is now completed by a standard approximation argument. Let also ¯ and ﬁx ε > 0. By approximation (Stone-Weierstrass) we can fm → f0 in C(D), ﬁnd a polynomial P in C[z, z¯] such that f0 − P C(D) ¯ < ε/4. Recall the inequality (3.3). We now have that

fm (ζ)dωn,Tm (ζ) − f0 (ζ)dωn,T0 (ζ) ≤ fm (ζ) − f0 (ζ) dωn,Tm (ζ) ¯ ¯ ¯ D D D

f0 (ζ) − P (ζ) dωn,Tm (ζ) + P (ζ)dωn,Tm (ζ) − P (ζ)dωn,T0 (ζ) + +

¯

D ¯ D

for m large.

P (ζ) − f0 (ζ) dωn,T0 (ζ) < ε

¯ D

¯ D

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The previous results yield in particular a uniform functional calculus for the ¯ class of n-hypercontractions. Let u = Bn [f ] be the Berezin transform of f ∈ C(D). The function u is real-analytic in D and has a power series expansion cjk z¯j z k , z ∈ D. u(z) = j,k≥0

For an operator T ∈ L(H) such that r(T ) < 1 we set u(T ) := cjk T ∗j T k in L(H).

(3.4)

j,k≥0

Notice that by the spectral radius formula the series in (3.4) is absolutely convergent in L(H). This functional calculus extends naturally to the class of nhypercontractions. Theorem 3.3. Let T ∈ L(H) be an n-hypercontraction and let u = Bn [f ] be the ¯ Then Berezin transform of f ∈ C(D). lim u(rT ) = f (ζ)dωn,T (ζ) in L(H). r→1

¯ D

Proof. We consider ﬁrst the case when T ∈ L(H) is an operator such that r(T ) < 1. By Lemma 2.1 we have that Bn (T, ζ)f (ζ)dµn (ζ) = prs (ζ)f (ζ)dµn (ζ) T ∗r T s , (3.5) ¯ D

r,s≥0

where

min(n,r,s)

prs (ζ) =

l=0

¯ D

n r−l ¯s−l 1 (−1)l ζ ζ . l µn;r−l µn;s−l

If we substitute zI, z ∈ D, for T in (3.5) we obtain the power series expansion of u, that is, u(z) = crs z¯r z s , z ∈ D, where crs =

r,s≥0

prs f dµn . We now conclude that u(T ) = Bn (T, ζ)f (ζ)dµn (ζ), ¯ D

where u(T ) is deﬁned by (3.4). Let us now consider the case when T ∈ L(H) is an n-hypercontraction. By the result of the previous paragraph we have that Bn (rT, ζ)f (ζ)dµn (ζ) = f (ζ)dωn,rT (ζ), u(rT ) = ¯ D

¯ D

where the last equality follows by Corollary 3.1. Notice that the operator rT is an n-hypercontraction by Proposition 1.1, so that dωn,rT exists by Theorem 3.1. The conclusion of the theorem now follows by Theorem 3.2.

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Remark 3.3. We remark that in the ﬁrst paragraph in the proof of Theorem 3.3 we showed that u(T ) = Bn (T, ζ)f (ζ)dµn (ζ) ¯ D

when u(T ) is deﬁned by (3.4) and T ∈ L(H) is an arbitrary operator such that r(T ) < 1. The operator measures dωn,T have a property of invariance with respect to conformal automorphisms of the unit disc. Corollary 3.2. Let T ∈ L(H) be an n-hypercontraction. Then we have the invariance property that ¯ (f ◦ ϕ)(ζ)dωn,T (ζ) = f (ζ)dωn,ϕ(T ) (ζ), f ∈ C(D), ¯ D

¯ D

for every ϕ ∈ Aut(D). Proof. Let 0 ≤ r < 1. By Theorem 1.1 we have that Bn (rT, ζ)(f ◦ ϕ)(ζ)dµn (ζ) = Bn (ϕ(rT ), ζ)f (ζ)dµn (ζ), ¯ D

¯ D

which we can restate as (f ◦ ϕ)(ζ)dωn,rT (ζ) = f (ζ)dωn,ϕ(rT ) (ζ). ¯ D

¯ D

Notice that the operators rT and ϕ(rT ) are n-hypercontractions by Proposition 1.1 and Corollary 1.2. Letting r → 1 the conclusion of the corollary now follows by Theorem 3.2. Notice that ϕ(T ) is an n-hypercontraction by Corollary 1.2 so that the operator measure dωn,ϕ(T ) exists by Theorem 3.1.

4. Relations with the operator measure dωT It is well-known that every contraction T ∈ L(H) has a unitary dilation, that is, there exists a unitary operator U ∈ L(K) on some larger Hilbert space K containing H as a closed subspace such that T k = P U k |H ,

k ≥ 0,

where P is the orthogonal projection of K onto H. This unitary dilation U ∈ ∞ L(K) can be chosen minimal in the sense that K = k=−∞ U k (H) and is then uniquely determined up to isomorphism. We refer to [27, Chapter I] for details of the construction. Let now U ∈ L(K) be a unitary dilation of a contraction T ∈ L(H). By the spectral theorem the unitary operator U ∈ L(K) has an L(K)-valued spectral measure dE supported by the unit circle T. Compressing the spectral measure dE

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down to H we obtain a positive L(H)-valued operator measure dωT on T by the requirement that ωT (S) = P E(S)|H , S ∈ S; here as above P is the orthogonal projection of K onto H and S is the σ-algebra of planar Borel sets. This operator measure dωT does not depend on the particular choice of unitary dilation U ∈ L(K) of T ∈ L(H) (see the formula for the Fourier coeﬃcients ω ˆ T (k) below). In terms of Fourier coeﬃcients the operator measure dωT is characterized by the requirement that ∗k T for k ≥ 0, ω ˆ T (k) = e−ikθ dωT (eiθ ) = T |k| for k < 0. T In the case of a contraction T ∈ L(H) such that r(T ) < 1 the operator measure dωT has the explicit form dωT (eiθ ) = P (T, eiθ ) dθ/2π,

eiθ ∈ T,

where P (T, ·) is the operator-valued Poisson kernel given by the formula P (T, eiθ ) = (I − eiθ T ∗ )−1 (I − T ∗ T )(I − e−iθ T )−1 ,

eiθ ∈ T.

We refer to the paper [26] for the similar construction in the context of the unit polydisc Dn in Cn . In the terminology of Foias [15, 16] the operator measure dωT is called the harmonic spectral measure for the contraction T with respect to the ¯ spectral set D. We next observe that dω1,T = dωT . Proposition 4.1. Let T ∈ L(H) be a contraction. Then ¯ dω1,T (ζ) = dωT (ζ), ζ ∈ D. In particular, the operator measure dω1,T is supported by the unit circle T. Proof. By Remark 3.1 we know that W1;0;j,k = 1 and W1;m;j,k = 0 for m ≥ 1. By Theorem 3.1 we now have that ˆ T (j − k) = ζ¯j ζ k dω1,T (ζ) = T ∗(j−min(j,k)) T k−min(j,k) = ω ζ¯j ζ k dωT (ζ) ¯ D

T

for all j, k ≥ 0. By linearization we see that P dω1,T = P dωT for every polyno mial P ∈ C[z, z¯] and an approximation argument (Stone-Weierstrass) gives that ¯ This completes the proof of the proposif dω1,T = f dωT for every f ∈ C(D). tion. We remark that in terms of action on test functions the assertion of Proposition 4.1 means that ¯ f (ζ)dω1,T (ζ) = f (eiθ )dωT (eiθ ), f ∈ C(D) ¯ D

(compare Proposition 4.2 below).

T

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We shall need the following lemma. Lemma 4.1. Let the numbers Wn;m;j,k be as in Section 3. Then Wn;m;j,k = 1. m≥0

Proof. We shall use a property of the Berezin transform of a continuous function. ¯ then Bn [f ] ∈ C(D) ¯ and Bn [f ] = f on T. For a proof of this Namely, if f ∈ C(D), fact we refer to [19, Proposition 2.3]. Let us now turn to the proof of the lemma. Notice ﬁrst that the sum in the lemma is absolutely convergent by Lemma 3.1. By Proposition 2.1 we have the power series expansion Bn (z, ζ)ζ¯j ζ k dµn (ζ) = z¯j−min(j,k) Wn;m;j,k |z|2m z k−min(j,k) , z ∈ D. ¯ D

m≥0

Now letting z → 1 using the property of the Berezin transform quoted in the previous paragraph the conclusion of the lemma follows. We shall now consider the case of an isometry. Proposition 4.2. Let T ∈ L(H) be an isometry. Then ¯ dωn,T (ζ) = dωT (ζ), ζ ∈ D, for n ≥ 1. In particular, the operator measure dωn,T does not depend on n ≥ 1 and is supported by the unit circle T. Proof. By the construction of the operator measure dωn,T in Theorem 3.1 we know that Wn;m;j,k T ∗m T m T k−min(j,k) . (4.1) ζ¯j ζ k dωn,T (ζ) = T ∗(j−min(j,k)) ¯ D

m≥0

Since T is an isometry, meaning that T ∗ T = I, we have by (4.1) and Lemma 4.1 that ζ¯j ζ k dωn,T (ζ) = T ∗(j−min(j,k)) T k−min(j,k) ¯ D

for all j, k ≥ 0. The conclusion of the proposition now follows by a linearization and approximation argument (see the proof of Proposition 4.1).

5. The space An (E) and its shift operator Sn In this section we shall discuss some properties of the shift operator Sn and its adjoint Sn∗ acting on the space An (E). Let E be a Hilbert space. We denote by An (E) the Hilbert space of all E-valued analytic functions f (z) = ak z k , z ∈ D; (5.1) k≥0

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here ak ∈ E for k ≥ 0, with ﬁnite norm ak 2 µn;k , f 2An = where µn;k = 1/

k+n−1 k

k≥0

for k ≥ 0. The norm of An (E) can also be written 2 f An = lim f (rz)2 dµn (z). r→1

¯ D

The measure dµ1 is the normalized (mass 1) Lebesgue arc length measure on the unit circle T, and for n ≥ 2 the measure dµn is the weighted area measure given by dµn (z) = (n − 1)(1 − |z|2 )n−2 dA(z), z ∈ D, where dA(z) = dxdy/π, z = x + iy, is the normalized Lebesgue area measure. On the space An (E) we have a natural shift operator S = Sn deﬁned by ak−1 z k , z ∈ D, (Sn f )(z) = zf (z) = k≥1

for f ∈ An (E) given by (5.1). In fact, by the formula n−1 k+n−1 1 1 = = (k + j) k µn;k (n − 1)! j=1 we see that the weight sequence {µn;k }k≥0 is decreasing in k ≥ 0 and that the ratio µn;k+1 /µn;k tends to 1 as k → ∞. Therefore the operator Sn is bounded on An (E) of norm equal to 1. The adjoint operator Sn∗ of Sn has the form µn;k+1 (Sn∗ f )(z) = ak+1 z k , z ∈ D, (5.2) µn;k k≥0

where f ∈ An (E) is given by (5.1) above. For later use it will be convenient to have available the following lemma. Lemma 5.1. Let Sn be as above and let f ∈ An (E) be given by (5.1). Then m k + n − m − 1 m (−1)j Sn∗j f 2An = µ2n;k ak 2 j k j=0 k≥0

for 1 ≤ m < n, and

n j=0

(−1)j

n Sn∗j f 2An = a0 2 . j

Proof. By formula (5.2) we have that Sn∗j f 2An =

µ2n;k+j k≥0

µn;k

ak+j 2 .

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Let 1 ≤ m ≤ n. A computation shows that 2 m m µn;k+j j m ∗j 2 j m (−1) (−1) ak+j 2 Sn f An = j j µ n;k j=0 j=0 k≥0

=

min(m,k) k≥0

(−1)j

j=0

529

(5.3)

m 1 2 µ ak 2 , j µn;k−j n;k

where the last equality follows by a change of order of summation. We now notice that the sum min(m,k) m 1 (−1)j j µ n;k−j j=0 equals the k-th coeﬃcient in the power series expansion of the function (1 − z)m

1 1 = . (1 − z)n (1 − z)n−m

We conclude that min(m,k)

j=0

m 1 k+n−m−1 (−1) = j µn;k−j k j

for 1 ≤ m < n, and that min(n,k)

j=0

(−1)j

n 1 = δk,0 , j µn;k−j

where δ0,0 = 1 and δk,0 = 0 for k ≥ 1 is the Kronecker’s delta. Substituting the values of these last two sums into (5.3) we arrive at the formulas in the lemma. The following proposition establishes two basic properties of the adjoint shift operator Sn∗ . Proposition 5.1. The operator Sn∗ : An (E) → An (E) is an n-hypercontraction such that limk→∞ Sn∗k = 0 in the strong operator topology. Proof. Let us ﬁrst verify that Sn∗k → 0 (SOT). If f ∈ An (E) is a polynomial, then clearly Sn∗k f = 0 for k large. Since Sn∗k ≤ 1 we conclude by an approximation argument that Sn∗k f → 0 in An (E) for every f ∈ An (E). The assertion that the operator Sn∗ is an n-hypercontraction is evident by Lemma 5.1. We shall now compute the operator measure dωn,Sn∗ . Proposition 5.2. For n ≥ 2, we have that dωn,Sn∗ (ζ) = Bn (Sn∗ , ζ)dµn (ζ),

¯ ζ ∈ D.

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Proof. By Lemma 1.3 we know that the function Bn (Sn∗ , ·) is integrable with respect to dµn . To prove the proposition it suﬃces to show that j k ¯ Bn (Sn∗ , ζ)ζ¯j ζ k dµn (ζ) (5.4) ζ ζ dωn,Sn∗ (ζ) = ¯ D

D

for j, k ≥ 0. The conclusion of the proposition then follows by a linearization and approximation argument (see the proof of Proposition 4.1). Let now f, g ∈ An (E) be polynomials and 0 ≤ r < 1. Since Sn∗k f = 0 for k large for such an f , the resolvent sum 1 ¯ ∗ )−n f = (I − rζS rk ζ¯k Sn∗k f n µn;k k≥0

is ﬁnite. Therefore, the function n n k ∗k ¯ ∗ )−n f, (I − rζS ¯ ∗ )−n g (I − rζS

Bn (rSn∗ , ζ)f, g =

(−1)k r2k S S n n k n n k=0

¯ By Corollary 3.1 we now have that is a polynomial in r, ζ and ζ. j k ¯

ζ ζ dωn,rSn∗ (ζ)f, g = Bn (rSn∗ , ζ)ζ¯j ζ k dµn (ζ)f, g ¯ D D ∗ = Bn (rS , ζ)f, gζ¯j ζ k dµn (ζ) → Bn (S ∗ , ζ)f, gζ¯j ζ k dµn (ζ) D

n

D

as r → 1. Since also ζ¯j ζ k dωn,rSn∗ (ζ) → ζ¯j ζ k dωn,Sn∗ (ζ) ¯ D

¯ D

n

in L(H)

as r → 1 by Theorem 3.2, we conclude that

Bn (Sn∗ , ζ)ζ¯j ζ k dµn (ζ)f, g = ζ¯j ζ k dωn,Sn∗ (ζ)f, g ¯ D

D

for f, g polynomials in An (E). By approximation (5.4) follows.

We remark that in terms of action on test functions the assertion of Proposition 5.2 means that ¯ f (ζ)dωn,Sn∗ (ζ) = Bn (Sn∗ , ζ)f (ζ)dµn (ζ), f ∈ C(D). ¯ D

D

6. Operator model theory. General considerations In this section we consider the problem of modeling an operator T ∈ L(H) as part of an operator of the form T1∗ ⊕ T2 , where Tj ∈ L(Hj ) (j = 1, 2) are operators such that T1∗k → 0 in the strong operator topology and T2 is an isometry. The principal results in this section are Theorems 6.1 and 6.2 below. Let us begin with some general considerations.

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Let T ∈ L(H) be an operator such that the limit lim T k x2

k→∞

exists for every x ∈ H. Since T k x2 = T ∗k T k x, x for x ∈ H, we have by polarization that the operator limit limk→∞ T ∗k T k exists in the weak operator topology. We can now introduce the operator 1/2 Q = lim T ∗k T k in L(H), k→∞

where the positive square root is used. Notice that Qx2 = lim T k x2 , k→∞

x ∈ H.

(6.1)

Associated to the operator Q we have the range space Q deﬁned as the closure in H of the range of Q, that is, Q = Q(H). By (6.1) we see that the formula U : Qx → QT x gives a well-deﬁned map which by continuity extends uniquely to an isometry U on Q satisfying the intertwining relation QT = U Q. We have the following theorem. Theorem 6.1. Let T ∈ L(H) and let Tj ∈ L(Hj ) (j = 1, 2) be operators such that T1∗k → 0 in the strong operator topology and T2 is an isometry on H2 . Assume that we have an isometry V = (V1 , V2 ) : H → H1 ⊕ H2 of H into H1 ⊕ H2 satisfying the intertwining relation V T = (T1∗ ⊕ T2 )V.

(6.2)

Then • the limit limk→∞ T k x2 exists for every x ∈ H, • the map V2 : H → H2 factorizes as V2 = Vˆ2 Q, where Vˆ2 : Q → H2 is an isometry, in such a way that the intertwining relation Vˆ2 U = T2 Vˆ2 holds, • the operator V1 : H → H1 satisﬁes the norm equality x2 − T x2 = V1 x2 − T1∗ V1 x2 ,

x ∈ H.

Furthermore, the operator limit Q2 = V2∗ V2 = lim T ∗k T k k→∞

exists in the strong operator topology. Proof. Since the map V is an isometry, we have by the intertwining relation (6.2) and the isometry property of T2 that T k x2 = T1∗k V1 x2 + V2 x2 ,

x ∈ H,

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for k ≥ 0. By this equality and the assumption that T1∗k → 0 (SOT), we have that the limit limk→∞ T k x2 exists and equals V2 x2 for every x ∈ H. We thus have that Qx2 = lim T k x2 = V2 x2 , x ∈ H. k→∞

By this last equality we see that the formula Vˆ2 : Qx → V2 x gives a well-deﬁned map which by continuity extends uniquely to an isometry Vˆ2 : Q → H2 such that Vˆ2 Q = V2 . Also Vˆ2 U Qx = Vˆ2 QT x = V2 T x = T2 V2 x = T2 Vˆ2 Qx, x ∈ H, which gives that Vˆ2 U = T2 Vˆ2 . Let us now verify the norm equality for the operator V1 . Since the operators V = (V1 , V2 ) and T2 ∈ L(H2 ) are isometries we have using the intertwining relations T2 V2 = V2 T and V1 T = T1∗ V1 that x2 − V1 x2 = V2 x2 = T2 V2 x2 = V2 T x2 = T x2 − V1 T x2 = T x2 − T1∗ V1 x2 ,

x ∈ H.

This gives the norm equality for V1 . Let us now turn to the last limit assertion of the theorem. By the intertwining relation (6.2) we have V T k = (T1∗k ⊕ T2k )V, and passing to the adjoint operator we see that T ∗k V ∗ = V ∗ (T1k ⊕ T2∗k ). We now have that T ∗k T k = T ∗k V ∗ V T k = V ∗ (T1k T1∗k ⊕ T2∗k T2k )V = V ∗ (T1k T1∗k ⊕ IH2 )V, where in the last step we used that T2∗ T2 = I. Since T1∗k → 0 (SOT) we have that also T1k T1∗k → 0 (SOT). Passing to the limit we now conclude that lim T ∗k T k = V ∗ (0 ⊕ IH2 )V = V2∗ V2

k→∞

in the strong operator topology.

We remark that in the statement of Theorem 6.1 and also in Theorem 6.2 below the existence of the limit limk→∞ T k x2 for every x ∈ H is included merely to ensure the existence of the operator Q. As pointed out in the paragraph preceding Theorem 6.1 the limit assertion (6.1) can be rephrased saying that Q2 = lim T ∗k T k k→∞

(6.3)

in the weak operator topology. The last conclusion of Theorem 6.1 says that the limit (6.3) holds also in the stronger sense of convergence in the strong operator topology. By Theorem 6.1 we see that there is a natural choice of space H2 and operator T2 given by H2 = Q and T2 = U . We shall now show that this choice (T2 , H2 ) = (U, Q) does the job.

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Theorem 6.2. Let T ∈ L(H) and let T1 ∈ L(H1 ) be an operator such that T1∗k → 0 in the strong operator topology. Assume that there exists a bounded linear operator V1 : H → H1 satisfying the norm equality x2 − T x2 = V1 x2 − T1∗ V1 x2 ,

x ∈ H,

(6.4)

as well as the intertwining relation V1 T = T1∗ V1 . Then the limit limk→∞ T k x2 exists for every x ∈ H and the map V = (V1 , Q) : H → H1 ⊕ Q deﬁned by V : x → (V1 x, Qx) for x ∈ H is an isometry of H into H1 ⊕ Q satisfying the intertwining relation V T = (T1∗ ⊕ U )V ; here Q, Q and U are as in the discussion preceding Theorem 6.1. Proof. Substituting T j x for x in (6.4) we obtain using the intertwining relation V1 T = T1∗ V1 that ∗(j+1)

T j x2 − T j+1 x2 = T1∗j V1 x2 − T1

V1 x2

for j ≥ 0. Summing these equalities for j = 0, . . . , k − 1 we see that x2 − T k x2 = V1 x2 − T1∗k V1 x2 ,

x ∈ H.

Now since T1∗k → 0 (SOT), we see that the limit limk→∞ T k x2 exists for every x ∈ H. Furthermore, by a passage to the limit we conclude that x2 = V1 x2 + lim T k x2 = V1 x2 + Qx2 , k→∞

x ∈ H.

This last equality shows that the map V = (V1 , Q) is an isometry of H into H1 ⊕Q. The intertwining relation V T = (T1∗ ⊕ U )V follows by V1 T = T1∗ V1 and QT = U Q. This completes the proof of the theorem.

7. Operator model theory. n-hypercontractions In this section we continue the study of operator model theory from Section 6. Of particular concern here is the modeling of a general n-hypercontraction T ∈ L(H) as part of an operator of the form Sn∗ ⊕ U ; here U ∈ L(Q) is the canonical isometry associated to T described in Section 6 and Sn is the shift operator on a space An (E). In the notation of Section 6 we have that H1 = An (E) and T1 = Sn . Recall that the adjoint shift operator Sn∗ on An (E) is an n-hypercontraction such that limk→∞ Sn∗k = 0 in the strong operator topology (see Proposition 5.1). Let T ∈ L(H) be an n-hypercontraction, and consider the defect operators m 1/2 k m Dm,T = (−1) in L(H) T ∗k T k k k=0

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for 1 ≤ m ≤ n, where the positive square root is used. We write Dn,T for the defect space deﬁned as the closure in H of the range of Dn,T , that is, Dn,T = Dn,T (H). Theorem 7.1. Let T ∈ L(H), and let V1 : H → An (E) be a bounded linear operator satisfying the norm equality x2 − T x2 = V1 x2An − Sn∗ V1 x2An ,

x ∈ H,

(7.1)

as well as the intertwining relation V1 T = Sn∗ V1 . Then the operator T is an nhypercontraction and there exists an isometry Vˆ1 : Dn,T → E such that the operator V1 admits the representation (V1 x)(z) = Vˆ1 Dn,T (I − zT )−n x, z ∈ D, for x ∈ H. Proof. By Theorem 6.2 with T1 = Sn and H1 = An (E) the operator T is part of the operator Sn∗ ⊕ U , where U is an isometry. Therefore the operator T is an n-hypercontraction. The operator Vˆ1 in the theorem is deﬁned by the formula Vˆ1 : Dn,T x → (V1 x)(0) for x ∈ H. We shall show that this formula gives a well-deﬁned map which by continuity extends uniquely to an isometry Vˆ1 : Dn,T → E. To accomplish this it 2 suﬃces to prove the norm equality Dn,T x2 = x ∈ H.

n(V 1 x)(0)

n−1 for n−1 Notice ﬁrst that the standard identity k = + k−1 for binomial k coeﬃcients gives us the formula n n−1 n n−1 k 2 (−1)k (−1)k T k x2 = T x − T k+1 x2 , x ∈ H, (7.2) k k k=0

k=0

which is valid for an arbitrary operator T ∈ L(H). Let us now turn to the proof of the norm equality Dn,T x2 = (V1 x)(0)2 for x ∈ H. Let x ∈ H and write f = V1 x, where f ∈ An (E) is given by (5.1). Substituting T k x for x in the norm equality (7.1) we obtain using the intertwining relation V1 T = Sn∗ V1 that T k x2 − T k+1 x2 = Sn∗k f 2An − Sn∗(k+1) f 2An . By formula (7.2) we have that n n−1

k 2 2 k n k 2 k n−1 T x − T k+1 x2 Dn,T x = (−1) (−1) T x = k k k=0 k=0 n−1

n−1 Sn∗k f 2An − Sn∗(k+1) f 2An = (−1)k k k=0 n n = (−1)k Sn∗k f 2An . k k=0

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By Lemma 5.1 we have that n

n (−1) Sn∗k f 2An = a0 2 . k k

k=0

We now conclude that Dn,T x2 = a0 2 = (V1 x)(0)2 . This gives the asserted norm equality. We now turn our attention to the representation formula for the operator V1 . By the intertwining relation V1 T = Sn∗ V1 we have that Vˆ1 Dn,T T k x = (V1 T k x)(0) = (Sn∗k V1 x)(0) = µn;k ak , where in the last step we have used (5.2); here V1 x = f ∈ An (E) is given by (5.1). A computation now gives that 1 f (z) = ak z k = (Vˆ1 Dn,T T k x)z k = Vˆ1 Dn,T (I − zT )−n , z ∈ D. µn;k k≥0

k≥0

This completes the proof of the theorem.

Let T ∈ L(H) be an n-hypercontraction. For an element x ∈ H we consider the Dn,T -valued analytic function V1,n x deﬁned by the formula k + n − 1 (Dn,T T k x)z k , z ∈ D. (7.3) (V1,n x)(z) = Dn,T (I − zT )−n x = k k≥0

The explicit form of this function V1,n x given by (7.3) is of course strongly suggested by Theorem 7.1. The formula (7.3) is also to some extent motivated by the explicit form of the operator-valued Berezin kernel (0.2) studied earlier. Our next task is to model a general n-hypercontraction using the map V1,n : x → V1,n x. The following proposition gives a norm bound for this operator V1,n . Proposition 7.1. Let T ∈ L(H) be an n-hypercontraction. Then the above map V1,n : x → V1,n x deﬁned by (7.3) maps H into An (Dn,T ) in such a way that V1,n x2An ≤ x2 ,

x ∈ H,

and the intertwining relation V1,n T = Sn∗ V1,n holds. Proof. Let us ﬁrst verify the intertwining relation V1,n T = Sn∗ V1,n . By (5.2) and (7.3) we have that µn;k+1 1 (Sn∗ V1,n x)(z) = (Dn,T T k+1 x)z k µn;k µn;k+1 k≥0

=

1 (Dn,T T k+1 x)z k = (V1,n T x)(z), µn;k

k≥0

This gives the conclusion that Sn∗ V1,n = V1,n T .

z ∈ D.

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Let us now turn our attention to the norm bound of the operator V1,n . Let 0 ≤ r < 1 and ﬁx x ∈ H. By Corollary 2.1 with f = 1 we have that

Bn (rT, ζ)x, xdµn (ζ) = x2 . ¯ D

Recall that also the operator rT is an n-hypercontraction (see Proposition 1.1). In particular, the defect operator Dn,rT is deﬁned, and by the deﬁning formula for the operator-valued Berezin kernel (0.2) we have that ¯ )−n x2 .

Bn (rT, ζ)x, x = Dn,rT (I − ζrT A change of variables and the Parseval formula now shows that 1 Dn,rT (rT )k x2 = Dn,rT (I − ζrT )−n x2 dµn (ζ) = x2 . µn;k ¯ D k≥0

Now letting r → 1 an application of Fatou’s lemma gives that 1 Dn,T T k x2 ≤ x2 , x ∈ H. µn;k k≥0

This completes the proof of the proposition. We shall need the following lemma.

Lemma 7.1. Let T ∈ L(H) be an n-hypercontraction. Then we have the norm equality 1 x2 − T x2 = Dn,T T k x2 , x ∈ H. µn−1;k k≥0

Proof. Let us ﬁrst make a few preparatory remarks. Recall that m 2 j m Dm,T x = (−1) T j x2 , x ∈ H, j j=0 for 1 ≤ m ≤ n. Using a standard formula for binomial coeﬃcients it is a straightforward matter to verify that Dm+1,T x2 = Dm,T x2 − Dm,T T x2 ,

x ∈ H, k

(7.4) 2

for 1 ≤ m < n. We also notice that since the limit limk→∞ T x exists (the operator T is a contraction) we have that limk→∞ Dm,T T k x2 = 0 for 1 ≤ m ≤ n. Let us now turn to the proof of the lemma. Substituting T j x for x in formula (7.4) we see that Dm+1,T T j x2 = Dm,T T j x2 − Dm,T T j+1 x2 ,

x ∈ H,

for j ≥ 0. Summing these equalities for j = 0, . . . , k − 1 we obtain that k−1 j=0

Dm+1,T T j x2 =

k−1

Dm,T T j x2 − Dm,T T j+1 x2

j=0

= Dm,T x2 − Dm,T T k x2 .

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Now letting k → ∞, using that Dm T k x2 → 0 (see the previous paragraph), we conclude that Dm,T x2 = Dm+1,T T k x2 , x ∈ H, k≥0

for 1 ≤ m < n. Iterating this last equality, we arrive at D1,T x2 = Dn,T T k1 +···+kn−1 x2 = k1 ,...,kn−1 ≥0

k≥0

1 Dn,T T k x2 , µn−1;k

where in the last step we have used a standard property of binomial coeﬃcients. This completes the proof of the lemma. We can now model a general n-hypercontraction. Theorem 7.2. Let T ∈ L(H) be an n-hypercontraction, and consider the map V1,n : x → V1,n x given by formula (7.3). Then the map V = (V1,n , Q) : H → An (Dn,T ) ⊕ Q deﬁned by V x = (V1,n x, Qx) for x ∈ H is an isometry of H into An (Dn,T ) ⊕ Q satisfying the intertwining relation V T = (Sn∗ ⊕ U )V ; here Q, Q and U are as in the discussion preceding Theorem 6.1. Proof. We shall apply Theorem 6.2 with V1 = V1,n and T1 = Sn acting on the space H1 = An (Dn,T ). Recall that Sn∗k → 0 (SOT) (see Proposition 5.1). By Proposition 7.1 the map V1,n : H → An (Dn,T ) is bounded of norm less than or equal to 1 and the intertwining relation V1,n T = Sn∗ V1,n holds. It remains to verify the norm equality (6.4) in our case. Let x ∈ H. We have that 1 1 V1,n x2An − Sn∗ V1,n x2An = Dn,T T k x2 − Dn,T T k+1 x2 µn;k µn;k k≥0 k≥0 1 1 Dn,T T k x2 . = Dn,T x2 + − µn;k µn;k−1 k≥1

n n−1 n−1 By the standard identity k = k + k−1 for binomial coeﬃcients we further conclude that 1 V1,n x2An − Sn∗ V1,n x2An = Dn,T T k x2 . µn−1;k k≥0

By Lemma 7.1 this last sum on the right hand-side equals x2 − T x2. This completes the proof of (6.4) in our case. In the proof of Theorem 7.2 above we needed the boundedness of V1,n as an operator from H into An (Dn,T ). The proof of this boundedness property we gave in Proposition 7.1 used properties of the operator-valued Berezin kernel studied earlier. Adapting the argument from the proof of Theorem 6.2 instead, we can prove

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this boundedness of V1,n directly without reference to operator-valued Berezin kernels. In fact, we have the following proposition. Proposition 7.2. Let T ∈ L(H) be an n-hypercontraction. Then we have the norm equality 1 x2 = Dn,T T k x2 + lim T k x2 , x ∈ H. k→∞ µn;k k≥0

Proof. Notice ﬁrst that since T is a contraction the limit limk→∞ T k x2 exists. Substituting T j x for x in Lemma 7.1 we obtain that 1 T j x2 − T j+1 x2 = Dn,T T k+j x2 , x ∈ H, µn−1;k k≥0

for j ≥ 0. Summing these equalities for j = 0, . . . , l − 1 we conclude that x2 − T l x2 =

l−1 ∞ j=0 k=0

1 µn−1;k

Dn,T T k+j x2 ,

x ∈ H.

Now letting l → ∞, noticing that 1 1 Dn,T T k+j x2 = Dn,T T k x2 , µn−1;k µn;k j,k≥0

k≥0

the conclusion of the proposition follows.

Notice that the conclusion of Proposition 7.2 can be rephrased saying that the map V = (V1,n , Q) in Theorem 7.2 is an isometry of H into An (Dn,T ) ⊕ Q. Theorem 7.2 has the following corollary when the operator T is also in the class C0· . Corollary 7.1. Let T ∈ L(H) be an n-hypercontraction such that limk→∞ T k = 0 in the strong operator topology. Then the map V1,n : x → V1,n x deﬁned by (7.3) is an isometry of H into An (Dn,T ) satisfying the intertwining relation V1,n T = Sn∗ V1,n .

Proof. The operator Q in Theorem 7.2 vanish.

8. Structure properties of the operator measure dωn,T The purpose of this section is to discuss some results describing the structure of the operator measure dωn,T in some more detail. We denote by S the σ-algebra of planar Borel sets. Theorem 8.1. Let T ∈ L(H) be an n-hypercontraction, and let V = (V1,n , Q) : H → An (Dn ) ⊕ Q be the isometry in Theorem 7.2. Then ∗ ωn,T (S) = V1,n ωn,Sn∗ (S)V1,n + QωU (S)Q,

S ∈ S;

here the operator U is as in the discussion preceding Theorem 6.1.

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Proof. A computation using the intertwining relation V T = (Sn∗ ⊕ U )V shows that ∗ Snr Sn∗s V1,n + QU ∗r U s Q T ∗r T s = V1,n

(8.1)

for r, s ≥ 0. By Theorem 3.1 we have that ζ¯j ζ k dωn,T (ζ) = T ∗(j−min(j,k)) Wn;m;j,k T ∗m T m T k−min(j,k) ¯ D

∗ = V1,n

m≥0

¯ D

ζ¯j ζ k dωn,Sn∗ (ζ) V1,n + Q ζ¯j ζ k dωn,U (ζ) Q ¯ D

for j, k ≥ 0. An approximation argument gives that ∗ ∗ f (ζ)dωn,T (ζ) = V1,n f (ζ)dωn,Sn (ζ) V1,n + Q f (ζ)dωn,U (ζ) Q ¯ D

¯ D

¯ D

¯ (see the proof of Proposition 4.1). By Proposition 4.2 we know that for f ∈ C(D) dωn,U = dωU . This completes the proof of the theorem. We remark that in terms of action on test functions the assertion of Theorem 8.1 means that ∗ f (ζ)dωn,T (ζ) = V1,n f (ζ)dωn,Sn∗ (ζ) V1,n + Q f (eiθ )dωU (eiθ ) Q ¯ D

¯ D

T

¯ for f ∈ C(D). Theorem 8.2. Let T ∈ L(H) be an n-hypercontraction for some n ≥ 2, and let V = (V1,n , Q) be as in Theorem 8.1. Then ωn,T (S) = Bn (T, ζ)dµn (ζ) + QωU (S)Q, S ∈ S; D

S

here the operator U is as in the discussion preceding Theorem 6.1. Proof. By Proposition 5.2 we have that dωn,Sn∗ (ζ) = Bn (Sn∗ , ζ)dµn (ζ),

¯ ζ ∈ D,

which by Theorem 8.1 allows us to conclude that ∗ ∗ f (ζ)dωn,T (ζ) = V1,n Bn (Sn , ζ)f (ζ)dµn (ζ) V1,n + Q f (eiθ )dωU (eiθ ) Q ¯ D

D

T

¯ for f ∈ C(D). We shall now consider the Berezin kernel n ∗ −n k n ¯ )−n , Bn (T, ζ) = (I − ζT ) (−1) T ∗k T k (I − ζT k k=0

(8.2)

ζ ∈ D,

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in some more detail. By formula (8.1) and Lemma 2.1 we have that Bn (T, ζ) = prs (ζ)T ∗r T s r,s≥0 ∗ = V1,n

=

prs (ζ)Snr Sn∗s V1,n + Q prs (ζ)U ∗r U s Q

r,s≥0 ∗ V1,n Bn (Sn∗ , ζ)V1,n

r,s≥0

+ QBn (U, ζ)Q,

ζ ∈ D.

∗

Now since U is an isometry, that is, U U = I, we have that Bn (U, ζ) = 0 for all ζ ∈ D. We conclude that ∗ Bn (Sn∗ , ζ)V1,n , Bn (T, ζ) = V1,n

ζ ∈ D.

By formulas (8.2) and (8.3) the conclusion of the theorem follows.

(8.3)

Notice that by Theorem 8.2 we have that ωn,T (S) = 0 for every Borel subset S of D of planar Lebesgue measure zero. We remark that the operator measure Bn (T, ζ)dµn (ζ) appearing in Theorem 8.2 has an invariance property with respect to conformal automorphisms of the unit disc (see Proposition 1.3). We have the following corollary when the operator T is in the class C0· . Corollary 8.1. Let n ≥ 2, and let T ∈ L(H) be an n-hypercontraction such that limk→∞ T k = 0 in the strong operator topology. Then ¯ dωn,T (ζ) = Bn (T, ζ)dµn (ζ), ζ ∈ D. Proof. In this case the operator U is not present (see Corollary 7.1). The corollary follows by Theorem 8.2. We remark that in terms of action on test functions the assertion of Corollary 8.1 means that ¯ f (ζ)dωn,T (ζ) = Bn (T, ζ)f (ζ)dµn (ζ), f ∈ C(D). ¯ D

D

We recall that a contraction T ∈ L(H) is said to be completely non-unitary (c.n.u.) if for every element x ∈ H the equalities T ∗k x2 = x2 = T k x2 ,

k ≥ 0,

imply that x = 0 (see [27, Section I.3]). A classical result of Sz.-Nagy and Foias asserts that the spectral measure for the minimal unitary dilation of a c.n.u. contraction is absolutely continuous with respect to Lebesgue arc length measure on the unit circle (see [27, Theorem II.6.4]). Let T ∈ L(H) be a completely non-unitary contraction, and let U be as in the discussion preceding Theorem 6.1. Using the result of Sz.-Nagy and Foias qouted in the previous paragraph it is straightforward to see that the operator measure dωU is absolutely continuous with respect to Lebesgue arc length measure on T, that is, ωU (S) = 0 in L(H) for every planar Borel set S such that µ1 (S) = 0. We omit the details.

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9. Subnormal contractions and the Hausdorﬀ moment problem A theorem of Agler [2, Theorem 3.1] asserts that an operator T ∈ L(H) is a subnormal contraction if and only if it is an n-hypercontraction for every n ≥ 1. In this section we shall reconsider this characterization of subnormal contractions and derive it as a limit case of our study of operator-valued Berezin transforms (see Theorem 9.1 below). As an application of this result we shall also consider two operator-valued moment problems of Hausdorﬀ type (see Theorem 9.2 and Proposition 9.3). First we need a lemma. Lemma 9.1. Let n ≥ 2, and let T ∈ L(H) be an operator such that r(T ) < 1. Then j k ¯ ¯ j (I − ζT ∗ )−k (T − ζI)k (I − ζT ¯ )−j dµn (ζ) Bn (T, ζ)ζ ζ dµn (ζ) = (T ∗ − ζI) D

D

for all integers j, k ≥ 0. Proof. We have a series expansion ¯ )−j = (T − ζI)k (I − ζT

pj,k;l (ζ) T l ,

l≥0

¯ A more detailed analysis shows where the pj,k;l (ζ)’s are polynomials in C[ζ, ζ]. that the maximum max|ζ|≤1 |pj,k;l (ζ)| grows at most like a polynomial in l. We now have the series expansion ¯ j (I − ζT ∗ )−k (T − ζI)k (I − ζT ¯ )−j = ¯ j,k;l (ζ) T ∗l1 T l2 , (T ∗ − ζI) pk,j;l1 (ζ)p 2 l1 ,l2 ≥0

which we integrate term by term to obtain that ¯ j (I − ζT ∗ )−k (T − ζI)k (I − ζT ¯ )−j dµn (ζ) (T ∗ − ζI) (9.1) D ¯ j,k;l (ζ)dµn (ζ) T ∗l1 T l2 . = pk,j;l1 (ζ)p 2 l1 ,l2 ≥0

D

We shall now consider in some more detail the integrals appearing on the right-hand side in (9.1). Substituting zI, z ∈ D, for T in (9.1) we obtain that j k ¯ j,k;l (ζ)dµn (ζ) z¯l1 z l2 , z ∈ D, ϕz (ζ) ϕz (ζ) dµn (ζ) = pk,j;l1 (ζ)p 2 D

l1 ,l2 ≥0

D

where ϕz (ζ) = (z − ζ)/(1 − z¯ζ) is a conformal automorphism of the unit disc. The change of variables w = ϕz (ζ) gives that j k ϕz (ζ) ϕz (ζ) dµn (ζ) = Bn (z, w)w¯j wk dµn (w) D

D

(see [19, Section 2.1]). By Proposition 2.1 we have the power series expansion Bn (z, ζ)ζ¯j ζ k dµn (ζ) = z¯j−min(j,k) Wn;m;j,k |z|2m z k−min(j,k) , z ∈ D. D

m≥0

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Comparing coeﬃcients we conclude that ¯ j,k;l (ζ)dµn (ζ) = Wn;m;j,k pk,j;l1 (ζ)p 2 D

for l1 = j − min(j, k) + m, l2 = k − min(j, k) + m, m ≥ 0, and that ¯ j,k;l (ζ)dµn (ζ) = 0 pk,j;l1 (ζ)p 2 D

for all other values of l1 , l2 ≥ 0. Going back to (9.1) we conclude that ¯ j (I − ζT ∗ )−k (T − ζI)k (I − ζT ¯ )−j dµn (ζ) (T ∗ − ζI) D ∗(j−min(j,k)) ∗m m k−min(j,k) =T Wn;m;j,k T T T = Bn (T, ζ)ζ¯j ζ k dµn (ζ), D

m≥0

where the last equality holds by Proposition 2.1. This completes the proof of the lemma. Let us recall the notion of a subnormal operator. An operator T ∈ L(H) is called subnormal if there exists a normal operator N ∈ L(K) on some larger Hilbert space K containing H as a closed subspace such that T = N |H ; the operator N is then called a normal extension of T . A normal extension N of T can be chosen minimal in the sense that K = j,k≥0 N ∗j N k (H) and is then uniquely determined up to unitary equivalence. By the spectral theorem the normal operator N has a spectral measure dE supported on the spectrum σ(N ) of N . If we set µ(S) = P E(S)|H ,

S ∈ S;

(9.2)

here S is the σ-algebra of planar Borel sets and P is the orthogonal projection of K onto H, then dµ is a positive L(H)-valued operator measure on K = σ(N ) which represents the operator T in the sense that T ∗j T k = z¯j z k dµ(z), j, k ≥ 0. (9.3) K

Notice that (9.3) determines the operator measure dµ uniquely (Stone-Weierstrass). Conversely, assume that (9.3) holds for some (compactly supported) positive operator measure dµ. By a theorem of Naimark [24] there exists an L(K)valued spectral measure dE also supported by K such that (9.2) holds. This spectral measure dE is then the spectral measure for the normal extension N = zdE(z) of T with σ(N ) ⊂ K. A standard reference for subnormal operators is K the book Conway [11]; see also Bram [9]. Theorem 9.1. Let T ∈ L(H) be an operator such that the inequality n k n (−1) T ∗k T k ≥ 0 in L(H) k k=0

(9.4)

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holds for every n ≥ 1. Then there exists a positive L(H)-valued operator measure ¯ such that dµ on D ∗j k T T = z¯j z k dµ(z), j, k ≥ 0. ¯ D

In particular, the operator T is a subnormal contraction. Proof. We consider the map Λ with values in L(H) deﬁned for polynomials f (z) = ¯j z k in C[z, z¯] by j,k≥0 cjk z Λ(f ) = cjk T ∗j T k . j,k≥0

We shall show below that Λ(f ) ≥ 0 in L(H) if f (z) ≥ 0 for all z ∈ D. By approximation the map Λ then extends uniquely to a continuous linear map Λ ¯ into L(H) such that Λ(f ) ≥ 0 in L(H) if 0 ≤ f ∈ C(D) ¯ (see the proof from C(D) of Theorem 3.1). By an operator version of the F. Riesz representation theorem it then follows that that there exists a positive L(H)-valued operator measure dµ on ¯ such that D ¯ f (z)dµ(z), f ∈ C(D) Λ(f ) = ¯ D

(see the preliminaries in the introduction). This gives then the conclusion of the theorem. Let us prove the estimate needed. We consider ﬁrst the case of an operator T ∈ L(H) such that r(T ) < 1 satisfying (9.4) for n ≥ 1. Notice that the sequence ¯ = {µn } of probability measures converges weak∗ to the Dirac measure δ0 in M (D) ∗ ¯ C(D) , that is, ¯ f (z)dµn (z) = f (0), f ∈ C(D). lim n→∞

D

By Lemma 9.1 we have that ¯ j (I − ζT ∗ )−k (T − ζI)k (I − ζT ¯ )−j dµn (ζ) lim ζ¯j ζ k dωn,T (ζ) = lim (T ∗ − ζI)

n→∞

¯ D

n→∞ ∗j

=T T

D k

in L(H)

for j, k ≥ 0. Taking linear combinations we see that lim f (ζ)dωn,T (ζ) = Λ(f ) in L(H) n→∞

¯ D

¯ Since the dωn,T ’s are positive operator measures, we conclude for every f ∈ C[ζ, ζ]. that Λ(f ) ≥ 0 in L(H) if f ∈ C[z, z¯] is positive in D. Let us now consider the case of an arbitrary operator T ∈ L(H) satisfying (9.4) for n ≥ 1. Let 0 ≤ r < 1. By Proposition 1.1 the operator rT satisﬁes (9.4) for n ≥ 1, and by the result of the previous paragraph we have that cjk rj+k T ∗j T k ≥ 0 in L(H) j,k≥0

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for every f (z) = j,k≥0 cjk z¯j z k in C[z, z¯] such that f ≥ 0 on D. Letting r → 1 we conclude that Λ(f ) ≥ 0 in L(H) for every polynomial f in C[z, z¯] which is positive in D. This completes the proof of the theorem. It is known that a subnormal contraction is an n-hypercontraction for every n ≥ 1. For the sake of completeness we include some details of proof. Proposition 9.1. If T ∈ L(H) is a subnormal operator such that T ≤ 1, then T is an n-hypercontraction for every n ≥ 1. Proof. Let dµ be as in (9.3). For c > 1 we have that ∗k k |z|2k dµ(z) ≥ c2k µ({z ∈ K : |z| > c}) in L(H) I ≥T T = K

for k ≥ 1. Letting k → ∞ we see that µ({z ∈ K : |z| > c}) = 0. This shows that ¯ We now have that the operator measure dµ is supported by D. n n ∗k k (−1)k T T = (1 − |z|2 )k dµ(z) ≥ 0 in L(H), k ¯ D k=0

which shows that the operator T is an n-hypercontraction for every n ≥ 1.

We want to mention here that the relationship between the spectrum of a subnormal operator and the spectrum of its minimal normal extension has been studied by Halmos [17] and Bram [9] in the 1950’s; see also [11, Theorem II.2.11]. Notice that the spectrum of the minimal normal extension N of a subnormal operator T ∈ L(H) equals the support of the operator measure dµ given by (9.3) (see the discussion preceding Theorem 9.1). We shall now turn to a discussion of some related moment problems. We say that an inﬁnite matrix {Ljk }j,k≥0 with entries Ljk in L(H) is positive deﬁnite if N

Ljk xj , xk ≥ 0

j,k=0

for every choice of x0 , . . . , xN ∈ H. As an application of Theorem 9.1 we have the following variation of a moment problem considered by Atzmon [6]; see also [30, Theorem 3.7]. Theorem 9.2. Let {Ljk }j,k≥0 be an inﬁnite matrix with entries Ljk in L(H) such that the matrices n

n (−1)m , n ≥ 0, (9.5) Lj+m,k+m m j,k≥0 m=0 are all positive deﬁnite. Then there exists a positive L(H)-valued operator measure ¯ such that dλ on D Ljk = z j z¯k dλ(z), j, k ≥ 0. (9.6) ¯ D

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Proof. By (9.5) with n = 0 we have that the matrix {Ljk }j,k≥0 is positive deﬁnite. On the space of H-valued analytic polynomials ak z k ; (9.7) f (z) = k≥0

here ak ∈ H for k ≥ 0, we consider the semi-norm deﬁned by

Ljk aj , ak . f 2 = j,k≥0

This semi-norm induces in a natural way a Hilbert space AL in which the equivalence classes of H-valued polynomials form a dense subset. We now consider the shift operator S deﬁned by ak−1 z k (9.8) (Sf )(z) = zf (z) = k≥1

for f a polynomial given by (9.7). A computation shows that S m f 2 =

Ljk aj−m , ak−m =

Lj+m,k+m aj , ak , j,k≥m

and that n

(−1)m

m=0

j,k≥0

n n n

(−1)m S m f 2 = Lj+m,k+m aj , ak ; m m m=0 j,k≥0

here f is given by (9.7) and n ≥ 1. By (9.5) with n = 1 we have that the shift operator S induces a well-deﬁned contraction on the space AL which we also denote by S. Invoking the full strength of (9.5) for n ≥ 1 we have that the induced operator S on AL is an n-hypercontraction for every n ≥ 1. By Theorem 9.1 we conclude ¯ such that that there exists a positive L(AL )-valued operator measure dµ on D S ∗j S k = z¯j z k dµ(z), j, k ≥ 0. ¯ D

We have a natural map A0 mapping the element x ∈ H to the corresponding constant element x in AL , that is, A0 x = f , where f is given by (9.7) with a0 = x and ak = 0 for k ≥ 1. We now set λ(F ) = A∗0 µ(F )A0 for F ∈ S (Borel sets). This ¯ We proceed to show gives us a positive L(H)-valued operator measure dλ on D. that (9.6) holds with this choice of dλ. We have that z j z¯k dλ(z) = A∗0 z j z¯k dµ(z) A0 = A∗0 S ∗k S j A0 , ¯ D

¯ D

and A∗0 S ∗k S j A0 x, y = S j A0 x, S k A0 y = Ljk x, y for x, y ∈ H, which gives that A∗0 S ∗k S j A0 = Ljk . This completes the proof of the theorem.

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We remark that the method of proof of Theorem 9.2 is adapted from Atzmon [6]. We also remark that an operator measure dλ is uniquely determined by (9.6) (Stone-Weierstrass); the same uniqueness remark applies in the context of Proposition 9.3 below. Theorem 9.2 has the following converse. Proposition 9.2. Let the inﬁnite matrix {Ljk }j,k≥0 with entries Ljk in L(H) be a Hausdorﬀ moment sequence in the sense that (9.6) holds for some positive L(H)¯ Then the inﬁnite matrices (9.5) are all positive valued operator measure dλ on D. deﬁnite. Proof. By a theorem of Naimark [24] there exists an L(K)-valued spectral measure ¯ and a bounded linear operator A : H → K such that dE on D λ(S) = A∗ E(S)A,

S ∈ S;

here S is the σ-algebra of planar Borel sets. A computation shows that n n n n j+m k+m dλ(z) (−1)m (−1)m z¯ Lj+m,k+m = z Lj,k;n = m m ¯ D m=0 m=0 ∗ =A z j z¯k (1 − |z|2 )m dE(z) A. ¯ D

Let x0 , . . . , xN ∈ H. We now have that

Lj,k;n xj , xk =

A∗ z j z¯k (1 − |z|2 )m dE(z) Axj , xk j,k≥0

¯ D

j,k≥0

j 2 zdE(z) (1 − |z|2 )m/2 dE(z) Axj ≥ 0. = j≥0

¯ D

¯ D

This completes the proof of the proposition.

We can adapt the proof of Theorem 9.2 to yield also the following version of the operator-valued Hausdorﬀ moment problem. Proposition 9.3. Let {Lk }k≥0 be a sequence of operators in L(H) such that n n Lj+k ≥ 0 in L(H) (−1)k (9.9) k k=0

for all integers n, j ≥ 0. Then there exists a radial L(H)-valued positive operator ¯ such that measure dλ on D Lk = |z|2k dλ(z), k ≥ 0. (9.10) ¯ D

Furthermore, by a change of variables, we have a positive L(H)-valued operator measure dν on the closed unit interval [0, 1] such that Lk = xk dν(x), k ≥ 0. (9.11) [0,1]

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Proof. By (9.9) with n = 0, the operators Lk are all positive. On the space of Hvalued analytic polynomials f of the form (9.7) we consider the semi-norm deﬁned by f 2 =

Lk ak , ak . k≥0

This semi-norm induces in a natural way a Hilbert space AL in which the equivalence classes of H-valued polynomials form a dense subset. We consider the shift operator S deﬁned by (9.8). By (9.9) with n = 1, the shift operator S induces a contraction on AL which we also denote by S. A computation shows that

Lk+j aj , aj S k f 2 = j≥0

and therefore we have that n n n n (−1)k

(−1)k S k f 2 = Lk+j aj , aj ≥ 0 k k k=0

j≥0

k=0

for f given by (9.7) and n ≥ 1. We thus have that the induced operator S on AL is an n-hypercontraction for every n ≥ 1, and by Theorem 9.1 we conclude that ¯ such that there exists a positive L(AL )-valued operator measure dµ on D z¯j z k dµ(z), j, k ≥ 0. S ∗j S k = ¯ D

Consider the natural map A0 mapping the element x ∈ H to the corresponding constant element x in AL , that is, A0 x = f , where f is given by (9.7) with a0 = x and ak = 0 for k ≥ 1. We now set λ(F ) = A∗0 µ(F )A0 for F ∈ S (Borel sets). A computation shows that 0 for j = k, j k ∗ j k ∗ ∗k j z z¯ dλ(z) = A0 z z¯ dµ(z) A0 = A0 S S A0 = L for j=k ¯ ¯ k D D (see the proof of Theorem 9.2). We conclude that dλ is a radial positive operator measure satisfying (9.10). The last conclusion (9.11) of the proposition is evident by a change of variables. We mention that a sequence {Lk }k≥0 satisfying (9.9) for n, j ≥ 0 is sometimes called totally monotone (see [18, Section 11.6]). Notice that if {Lk }k≥0 is a Hausdorﬀ moment sequence in the sense of (9.10) or (9.11), then the sequence {Lk }k≥0 is totally monotone. Indeed, if (9.10) holds we have that n n k n k n (−1) (−1) Lj+k = |z|2(j+k) dλ(z) k k ¯ D k=0 k=0 = (1 − |z|2 )n |z|2j dλ(z) ≥ 0 in L(H) ¯ D

for n, j ≥ 0, and similarly in the case of (9.11).

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References [1] J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982) 608–631. [2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985) 203– 217. [3] P. Ahern, On the range of the Berezin transform, J. Funct. Anal. 215 (2004) 206–216. [4] C.-G Ambrozie, M. Engliˇs and V. M¨ uller, Operator tuples and analytic models over general domains in Cn , J. Operator Theory 47 (2002) 287–302. [5] J. Arazy and M. Engliˇs, Analytic models for commuting operator tuples on bounded symmetric domains, Trans. Amer. Math. Soc. 355 (2003) 837–864. [6] A. Atzmon, A moment problem for positive measures on the unit disc, Paciﬁc J. Math. 59 (1975) 317–325. [7] S. Axler and D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998) 387–400. [8] C. Badea and G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2002) 260–297. [9] J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94. [10] L. de Branges and J. Rovnyak, Appendix on square summable power series, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics, Wiley, 1966, 347–392. [11] J. B. Conway, The theory of subnormal operators, American Mathematical Society, 1991. [12] R. E. Curto and F.-H. Vasilescu, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993) 65–78. [13] R. E. Curto and F.-H. Vasilescu, Standard operator models in the polydisc, Indiana Univ. Math. J. 42 (1993) 791–810. [14] R. E. Curto and F.-H. Vasilescu, Standard operator models in the polydisc II, Indiana Univ. Math. J. 44 (1995) 727–746. [15] C. Foias, La mesure harmonique-spectrale et la th´eorie spectrale des op´erateurs g´en´eraux d’un espace de Hilbert, Bull. Soc. Math. France 85 (1957) 263–282. [16] C. Foias, Some applications of spectral sets. I harmonic spectral measure, Acad. R. P. Romˆıne. Stud. Cerc. Mat. 10 (1959) 365–401; Amer. Math. Soc. Translations Series 2 61 (1967) 25–62. [17] P. R. Halmos, Spectra and spectral manifolds, Ann. Soc. Polon. Math. 25 (1952) 43–49. [18] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. [19] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Springer, 2000. [20] H. Hedenmalm and Y. Perdomo, Mean value surfaces with prescribed curvature form, J. Math. Pures Appl. 83 (2004) 1075–1107. [21] S. G. Krantz, Partial Diﬀerential Equations and Complex Analysis, CRC Press, 1992. [22] V. M¨ uller, Models for operators using weighted shifts, J. Operator Theory 20 (1988) 3–20.

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[23] V. M¨ uller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993) 979–989. [24] M. A. Naimark, On a representation of additive operator set functions, C. R. (Doklady) Acad. Sci. URSS (N.S.) 41 (1943) 359–361. [25] J. von Neumann, Eine Spektraltheorie f¨ ur allgemeine Operatoren eines unit¨ aren Raumes, Math. Nachr. 4 (1951) 258–281. [26] A. Olofsson, Operator-valued n-harmonic measure in the polydisc, Studia Math. 163 (2004) 203–216. [27] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, NorthHolland, 1970. [28] F.-H. Vasilescu, An operator-valued Poisson kernel, J. Funct. Anal. 110 (1992) 47–72. [29] F.-H. Vasilescu, Positivity conditions and standard models for commuting multioperators, Multivariable operator theory (Seattle, WA, 1993), Contemp. Math. 185, Amer. Math. Soc. 1995, 347–365. [30] F.-H. Vasilescu, Moment problems for multi-sequences of operators, J. Math. Anal. Appl. 219 (1998) 246–259. Anders Olofsson Falugatan 22 1tr SE-113 32 Stockholm Sweden e-mail: [email protected] Submitted: November 2, 2005 Revised: May 25, 2007

Integr. equ. oper. theory 58 (2007), 551–562 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040551-12, published online June 27, 2007 DOI 10.1007/s00020-007-1505-1

Integral Equations and Operator Theory

Quadratically Hyponormal Recursively Generated Weighted Shifts Need Not Be Positively Quadratically Hyponormal Yiu T. Poon and Jasang Yoon Abstract. We study a class of weighted shifts Wα deﬁned by a recursively generated sequence α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ and characterize the diﬀerence between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisﬁes a ﬁnite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, Wα is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift Wα (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift Wα need not be positively quadratically hyponormal. Mathematics Subject Classification (2000). Primary 47B20, 47B37; Secondary 47-04. Keywords. Recursively generated weighted shifts, positively quadratically hyponormal, quadratically hyponormal.

1. Introduction Let H and K be separable, inﬁnite dimensional, complex Hilbert spaces, let L(H, K) be the set of bounded linear operators from H to K and write L(H) := L(H, H). Recall that a bounded linear operator T ∈ L(H) is normal if T ∗ T = T T ∗, and subnormal if T = N |H , where N is normal and N (H) ⊆ H. An operator T such that T ∗ T ≥ T T ∗ is said to be hyponormal. For k ≥ 1, T is k-hyponormal if

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(I, T, · · · , T k ) is (jointly) hyponormal (cf. [3]) and T is weakly k-hyponormal if   k   αj T j : α ≡ (α1 , · · · , αk ) ∈ Ck   j=1

consists entirely of hyponormal operators, that is, p(T ) is hyponormal for every polynomial p of degree at most k (cf. [3] and [5]). If k = 2 then T is said to be quadratically hyponormal and if k = 3 then T is said to be cubically hyponormal. It is well known that k-hyponormal ⇒ weakly k-hyponormal. On the other hand, results in ([10], [3] and [14]) show that weakly 2-hyponormal operators are not necessarily 2-hyponormal. For α ≡ {αk }∞ k=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 ek := αk ek+1 (all k ≥ 0), where {ek }∞ n=0 is the canonical orthonormal basis in 2 (Z+ ). The moments of α are given as 1, if n = 0 γn ≡ γn (α) := . (1.1) α20 · ... · α2n−1 , if n > 0 It is easy to see that Wα is never normal, and that it is hyponormal if and only if α0 ≤ α1 ≤ · · · . We now recall recursively generated weighted shifts. Given 0 < α0 ≤ α1 < · · · < αm < αm+1 , we are going to deﬁne αn for n > m + 1 recursively and denote the corresponding sequence by α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ (m ≥ 2). Given α ≡ (α0 , α1 , α2 )∧ with 0 ≤ α0 ≤ α1 ≤ α2 , Wα is trivially subnormal (see [3]). For α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ with 0 < αi = αi+1 (1 ≤ i ≤ m and m ≥ 2), Wα is quadratically hyponormal if and only if Wα is subnormal (see [9]). For these reasons, we do not consider recursively generated weighted shifts Wα of these types. Deﬁne γn+1 := ϕ0 γn−1 + ϕ1 γn , (n > m + 1), (1.2) where − α2m ) α2 α2 (α2 α2m (α2m+1 − α2m−1 ) ϕ0 := − m−1 2m m+1 , ϕ := . (1.3) 1 αm − α2m−1 α2m − α2m−1 Then we have αn = γn+1 γn for n > m + 1. We note that the choice of ϕ0 and ϕ1 implies that (1.2) also holds for n = m + 1. For n, t ≥ 0, let  un := α2n − α2n−1      vn := α2n α2n+1 − α2n−1 α2n−2 wn := α2n (α2n+1 − α2n−1 )2   qn := u   √n + vn t  rn := wn t.

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Here, for notational convenience, we set α−2 = α−1 := 0. Let dn be deﬁned by the following 2-step recursive formula: d0 = q0 ,

d1 = q0 q1 − r02 ,

2 dn+2 = qn+2 dn+1 − rn+1 dn .

(1.4)

Then Wα is quadratically hyponormal if and only if dn (t) ≥ 0 for all t > 0 and n ≥ 0 (see [3]). We observe that dn is a polynomial in t of degree n + 1, that is, n+1 c(n, i)ti , dn (t) ≡ i=0

where the coeﬃcients c(n, i) satisfy the following recursive relation for n ≥ 0, i ≥ 1. c(n + 2, i) = un+2 · c(n + 1, i) + vn+2 · c(n + 1, i − 1) − wn+1 · c(n, i − 1), c(n, 0) = u0 · · · un ,

c(n, n + 1) = v0 · · · vn ,

c(1, 1) = u1 v0 + v1 u0 − w0

Recall thatfor recursively generated α, Wα is quadratically hyponormal if and n+1 only if dn (t) = i=0 c(n, i)ti > 0, for all n ≥ 0 and t > 0 ([5, Lemma 4.1]), and is positively quadratically hyponormal if c(n, i) ≥ 0 for all n ≥ 0 and 0 ≤ i ≤ n+1 (cf. [4]). It follows from deﬁnitions and c(n, n + 1) = v0 · · · vn that positive quadratic hyponormality implies quadratic hyponormality, but the converse is not true in general (cf. [1]). Recently, I. B. Jung and S. S. Park [12, Theorem 4.6] showed that a recursively generated weighted shift Wα , where α ≡ α0 , (α1 , α2 , α3 )∧ with 0 < α0 ≤ α1 < α2 < α3 is quadratically hyponormal if and only if it is positively quadratically hyponormal. This leads to the following open problem: Problem 1.1. ([5, Problem 4.7], [11, Problem 2.4] and [13, Problem 4.35]) For m ≥ 3, let α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ with 0 < α0 ≤ α1 < · · · < αm < αm+1 . Is it true that the weighted shift Wα is quadratically hyponormal if and only if it is positively quadratically hyponormal? In Theorem 2.5, we give a new suﬃcient condition of quadratic hyponormality of recursively generated weighted shift Wα (m ≥ 2). Also, we characterize the diﬀerence between quadratic hyponormality and positive quadratic hyponormality for a class of recursively generated weighted shifts Wα (m ≥ 2). A shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisﬁes a ﬁnite number (dependent on m) of conditions. For m = 2, these conditions are automatically satisﬁed. This gives a simpler proof (Corollary 2.8) of ([12, Theorem 4.6]). Finally, by combining the tools in Theorems 2.5 and 2.6, and Corollary 2.8, we give a negative answer (Example 2.9) for Problem 1.1.

2. Main Results We begin by listing several lemmas which are needed in the proofs of Theorems 2.5 and 2.6, and Corollary 2.8, and Example 2.9.

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Lemma 2.1. Let α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ (m ≥ 2). Then un vn+1 = wn for all n ≥ m.

Proof. See the proof of [5, Theorem 4.3].

Lemma 2.2. Let α ≡ α0 , · · · , αm−2 , (αm−1 , αm , αm+1 )∧ with 0 < α0 ≤ α1 < · · · < m+j vi αm < αm+1 (m ≥ 2). Let p0 := 0 and for j ≥ 1 pj := ui . For ≥ 0, let P (t) :=

i=m+1

pj tj . Then for n ≥ m, we have

j=0

dn (t) =

n

us

· [dm (t) + Pn−m (t) · a(t)] ,

(2.1)

s=m+1

where a(t) := dm (t) − um dm−1 (t) (m ≥ 1). Here, we set

n

us

= 1 when

s=m+1

n = m.

Proof. Since α0 ≤ α1 < α2 < · · · , we have ui > 0 for i ≥ 2. We prove it by induction on n. We observe that (2.1) holds for n = m. For n = m + 1, we have dm+1 (t)

2 = qm+1 dm (t) − rm dm−1 (t) (by (1.4))

= (um+1 ) dm (t) + tvm+1 dm (t) − tvm+1 um dm−1 (t) (by Lemma 2.1) vm+1 vm+1 = (um+1 ) dm (t) + t dm (t) − t um dm−1 (t) um+1 um+1 vm+1 = (um+1 ) dm (t) + t · {dm (t) − um dm−1 (t)} um+1 = (um+1 ) [dm (t) + P1 (t) · a(t)] . Therefore, (2.1) also holds for n = m + 1. Suppose (2.1) holds for n = m + and n = m + − 1 with ≥ 1. Then we have m+ dm+ (t) = us · [dm (t) + P (t) · a(t)] and s=m+1

dm+−1 (t) =

m+−1 s=m+1

us

· [dm (t) + P−1 (t) · a(t)] .

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Hence, we have dm++1 (t) = =

2 qm++1 dm+ (t) − rm+ dm+−1 (t)

(um++1 + tvm++1 ) dm+ (t) − tvm++1 um+ dm+−1 (t)

=

((um++1 ) + tvm++1 ) −tvm++1 um+

=

m++1

m+−1

· dm (t) +

s=m+1

us

· [dm (t) + P (t) · a(t)]

s=m+1

us

· [dm (t) + P−1 (t) · a(t)]

s=m+1

us

m+

m++1

us

· P (t) · a(t)

s=m+1

m++1

vm++1 + us · t · P (t) · a(t) um++1 s=m+1 m++1 vm++1 − us · t · P−1 (t) · a(t) um++1 s=m+1 =

m++1

us

s=m+1

=

m++1

· dm (t) + P (t) + p+1 t+1 · a(t)

us

· [dm (t) + P+1 (t) · a(t)] .

s=m+1

Therefore, (2.1) also holds for n = m + + 1. Hence, (2.1) holds for all n ≥ m. √ ϕ1 + ϕ21 +4ϕ0 ϕ2 and K := − ϕ10 L. Lemma 2.3. Let α be as in Lemma 2.2, and let L := 2 Then we have (i) For n ≥ m ≥ 2, ϕ0 = −

(ii)

vm+1 um+1

<

α2n−1 α2n (α2n+1 − α2n ) α2n (α2n+1 − α2n−1 ) and ϕ = 1 2 α2n − αn−1 α2n − α2n−1

vm+2 um+2

ϕ2

= − ϕ01 α2n−2 . Therefore, the sequence

increasing and limn→∞ uvnn = K.

(iii) For n ≥ m + 2,

vn un

vn un

∞

is

n=m+1

Proof. (i) is a straightforward calculation from (1.3). For (ii) and (iii), see the proof of [5, Theorem 4.3].

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Lemma 2.4. Suppose b(t) := n=1 b t is a polynomial with degree n and there exist 1 ≤ r < n and t0 > 0 such that (i) b ≤ 0 for all 1 ≤ ≤ r, b ≥ 0 for all r + 1 ≤ ≤ n − 1 and bn > 0, and (ii) b(t0 ) ≥ 0. tn . Then Let t > 0 ( = 1, · · · , n) such that t0 < tt21 ≤ tt32 ≤ · · · ≤ tn−1 n =s b t > 0 for all 1 ≤ s ≤ n. Proof. If s > r, then we have b ≥ 0 for all s ≤ ≤ n − 1 and bn > 0, therefore n =r+1 b t > 0. n n If s ≤ r, then we have =s b t ≥ =1 b t because b ≤ 0 for all 1 ≤ ≤ s. Thus, it suﬃces to prove the case when s = 1.

n r−1 n tr r t r t + b t = b t b t r 0 tr =1 =r t0 =1 0 tr

r n ti r−1 ti−1 tr r r r = + br t0 + =r+1 b t0 =1 b t0 tr0 ti ti−1 i=+1

i=r+1 r−1 tr > b tr t−r + br tr0 + n=r+1 b tr0 t−r r 0

t0 =1 0 0 n tr b t = =1 0 tr0 ≥ 0.

Therefore, we have the desired result.

The following result provides a new conditions for quadratically hyponormal. √ ϕ2

For the reader’s convenience, we recall that K = − ϕ10 L, where L =

ϕ1 +

ϕ21 +4ϕ0 . 2

Theorem 2.5. Let α be as in Lemma 2.2 and a(t) ≡ dm (t) − um dm−1 (t). Then we have (i) If Wα is quadratically hyponormal then (a) dk (t) > 0 for all 0 ≤ k ≤ m and t > 0. 1 . (b) a(t) ≥ 0 for all t ≥ K ∞ 1 k (c) dm (t) + pk t · a(t) ≥ 0 for all 0 < t < K . k=1

Kt · a(t) ≥ 0 (ii) Suppose conditions (a) and (b) in (i) are satisﬁed. If dm (t) + 1−Kt 1 for all 0 < t < K , then Wα is quadratically hyponormal.

Proof. To prove (i), suppose Wα is quadratically hyponormal. (a) follows from deﬁnition. pk 1 1 k+1 . We note that pk+1 = uvk+1 → K as k → ∞. Therefore, For (b), suppose t > K ∞ the series pk tk diverges to ∞. Thus, it follows from (2.1) that a(t) ≥ 0. k=1

1 , it follows from Lemma 2.2 that For every n > m and 0 < t < K n−m dn (t) k dm (t) + pk t ≥ 0. · a(t) = n k=1 uk k=m+1

(2.2)

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Hence, (c) holds. Kt · To prove (ii), suppose conditions (a) and (b) in (i) are satisﬁed and dm (t) + 1−Kt 1 a(t) ≥ 0 for all 0 < t < K . Let n > m and t > 0. If a(t) ≥ 0, then by (2.1), 1 dn (t) > 0. Suppose a(t) < 0. Then 0 < t < K . By Lemma 2.3, pk < K k . Therefore, we have dn (t) = dm (t) + n uk

n−m

pk t

k

· a(t) ≥ dm (t) +

k=1

Kt · a(t) ≥ 0 . 1 − Kt

k=m+1

Hence, Wα is quadratically hyponormal. We now present our main Theorem. Theorem 2.6. Let α be as in Lemma 2.2 and dm (t)−um dm−1 (t) ≡ a(t) := Let dn (t) :=

n+1

m+1

a i ti .

i=1

c(n, k)tk (for the convenience of notation, we let c(n, k) = 0 for

k=0

k < 0 or k > n + 1.). Suppose Wα is quadratically hyponormal and there exists 1 ≤ r < m+1 such that ai ≤ 0 for all 1 ≤ i ≤ r and ai > 0 for all r+1 ≤ i ≤ m+1. Then we have (i) c(n, k) > 0 for all n ≥ m + 1 and m + 1 ≤ k ≤ n + 1. (ii) Wα is positively quadratically hyponormal if and only if dn (t) have nonnegative coeﬃcients for 0 ≤ n ≤ m and c(2m − 1, k) ≥ 0 for all 2 ≤ k ≤ m. Proof. To prove (i), observe that ai = c(m, i) − um c(m − 1, i) for all 1 ≤ i ≤ m and . am+1 = c(m, m + 1) = v0 v1 · · · vm > 0. For n ≥ m + 1, let e(n, k) := c(n,k) n

Then, it follows from Lemma 2.2 that for n ≥ m + 1 we have

dn (t) n us

= dm (t) +

dn (t) n us

= dm (t) +

ai Pn−m (t)ti

m+1 n−m

ai · p · t+i

(2.3)

i=1 =1

s=m+1

⇒

us

i=1

s=m+1

⇒

m+1

s=m+1

k−1

e(n, k) = c(m, k) +

ai pk−i ,

for 2 ≤ k ≤ n + 1 ,

i=k−n+m

where ai = 0 for i ≤ 0 or i > m + 1. Since am+1 = c(m, m + 1), we have e(n, m + 1) = c(m, m + 1) +

m i=2m−n+1

ai pm+1−i

= am+1 + am p1 + · · · + as pm+1−s ,

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where s = max(1, 2m + 1 − n). For k > m + 1, c(m, k) = 0. Thus, we have e(n, k) = =

ak−1 p1 + ak−2 p2 + · · · + ak−n+m pn−m am+1 pk−m−1 + am pk−m + · · · + as pk−s ,

where s = max(1, k − n + m). By Lemma 2.2 and Lemma 2.3, we have um+i+1 pi pi−1 p2 1 um+1 1 < = < < ··· < < = . K vm+i+1 pi+1 pi p1 p1 vm+1 Since Wα is quadratically hyponormal and there exists r such that ai ≤ 0 for all 1 ≤ i ≤ r and ai > 0 for all r + 1 ≤ i ≤ n, we can apply Theorem 2.5 (i) (b) and Lemma 2.4 to get the results in (i). To prove (ii), suppose dn (t) have nonnegative coeﬃcients for 0 ≤ n ≤ m and c(2m − 1, k) ≥ 0 for all 2 ≤ k ≤ m. Let n ≥ m + 1. For k = 0, 1, we have e(n, k) = c(m, k) ≥ 0 (by (2.3) and a−1 = a0 = 0). For 2 ≤ k ≤ m, we have e(n, k) = =

c(m, k) + ak−1 p1 + ak−2 p2 + · · · + ak−n+m pn−m c(m, k) + ak−1 p1 + ak−2 p2 + · · · + as pk−s ,

where s = max{1, k − (n − m)}. In particular, e(2m − 1, k) = c(m, k) + ak−1 p1 + ak−2 p2 + · · · + a1 pk−1 ≥ 0. Thus, for 2 ≤ k ≤ m, we have   ≥ e(2m − 1, k) ≥ 0, e(n, k) =  ≥ e(m, k) ≥ 0,

if s ≤ r if s > r.

Hence, Wα is positively quadratically hyponormal. The other direction of (ii) follows from deﬁnition of positive quadratic hyponormality. As an application of Theorem 2.6, we can obtain a new proof (Corollary 2.8) of [12, Theorem 4.6]. We begin with: Lemma 2.7. Let α : α0 , (α1 , α2 , α3 )∧ , where 0 < α0 ≤ α1 < α2 < α3 . If Wα is quadratically hyponormal, then (i) c(4, 3) > 0. (ii) c(3, 2) > 0. Proof. Without loss of generality, we can assume that α0 = 1. 3 To prove (i), let d2 (t) − u2 d1 (t) =: i=1 ai ti . Then a direct calculation shows that c(4, 3) = a1 (v3 v4 ) + a2 (v3 u4 ) + a3 (u3 u4 ) α21 α22 (α23 α21 )(α22

− 1), a3 = α41 α22 (α22 α13 − α21 ) > 0. Also, a2 = If a1 ≥ 0, then c(4, 3) > 0.

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Suppose a1 < 0. By Theorem 2.6 (i), the quadratic hyponormality of Wα implies that c(4, 3) > 0. To prove (ii), let a = α21 , h = u2 = α22 − α21 and k = u3 = α23 − α22 . By direct (a + h)(F3 − G3 ) and computation, using Mathematica [15], we have c(3, 2) = h (a + h)k(F4 − G4 ) with c(4, 3) = h3 (a + h + k) = f31 k + f32 k 2 + f33 k 3 = g30 + g31 k + g32 k 2 + g33 k 3 = f41 k + f42 k 2 + f43 k 3 + f44 k 4 + f45 k 5 = g40 + g41 k + g42 k 2 + g43 k 3 + g44 k 4 + g45 k 5

F3 G3 F4 G4 where

f31 f32 f33 and

2 2

= 3a h + 2ah = 3a2 h + 3ah2 = a2 + ah

3

g30 g31 g32 g33

= h4 = 2ah2 + 3h3 = 3ah + 3h2 =a+h

f41 f42 f43 f44 f45

= 2a4 h4 + 5a3 h5 + 4a2 h6 + ah7 = a5 h2 + 9a4 h3 + 19a3 h4 + 15a2 h5 + 4ah6 = 10a4 h2 + 25a3 h3 + 21a2 h4 + 6ah5 = 5a4 h + 14a3 h2 + 13a2 h3 + 4ah4 = a4 + 3a3 h + 3a2 h2 + ah3

g40 g41 g42 g43 g44 g45

= a2 h6 + 2ah7 + h8 = 2a3 h4 + 8a2 h5 + 11ah6 + 5h7 = a4 h2 + 7a3 h3 + 21a2 h4 + 24ah5 + 10h6 = 9a3 h2 + 25a2 h3 + 26ah4 + 10h5 = 5a3 h + 14a2 h2 + 14ah3 + 5h4 = a3 + 3a2 h + 3ah2 + h3 .

This gives F3 G4 − F4 G3

⇒ G4 (F3 − G3 )

= ah3 k(a5 hk 2 + a4 hk(h + k)2 + h2 (h + k)6 +a2 (h + k)5 (3h + k) + ah(h + k)5 (3h + 2k) +a3 (h + k)2 (h3 + 4h2 k + 3hk 2 + k 3 )) > 0 = F3 G4 − F4 G3 + G3 (F4 − G4 ) > 0.

Hence, we have c(3, 2) > 0.

By combining Theorem 2.6 and Lemma 2.7 we have the following result: Corollary 2.8. (cf. [12, Theorem 4.6]) Let Wα be as in Lemma 2.7. Then Wα is quadratically hyponormal if and only if Wα is positively quadratically hyponormal.

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Proof. Without loss of generality, we can assume α0 = 1. Suppose Wα is quadratically hyponormal. Then a straightforward calculation shows that d0 (t) = d1 (t) = d2 (t) =

1 + α21 t (α21 − 1) + (α22 − 1)α21 t + α22 α41 t2 (α22 − α21 )(α21 − 1) + α22 (α23 − α21 )(α21 − 1)t +α21 α22 ((α23 − α21 )(α21 − 1) + α23 (α22 − α21 ))t2 + α41 α22 (α22 α23 − α21 )t3

Therefore, dn (t) has non-negative coeﬃcients for 0 ≤ n ≤ 2. From the proof of Lemma 2.7, we have d2 (t) − u2 d1 (t) = a1 t + a2 t2 + a3 t3 where a2 , a3 > 0. If a1 ≥ 0, then all the coeﬃcients of dm (t), Pn−m (t) and a(t) are nonnegative. n+1 Therefore, by (2.1), all coeﬃcients of dn (t) = c(n, k)tk are non-negative and k=0

Wα is positively quadratically hyponormal. Suppose a1 < 0. From Lemma 2.7, we have c(3, 2) > 0. Therefore, the result follows from (ii) in Theorem 2.6. We are now ready to give an example which gives a negative answer to Problem 1.1. 2009 809 Example 2.9. Let α ≡ 1, 1, (α2 , α3 , α4 )∧ where α2 = 11 10 , α3 = 1800 and α4 = 720 . Then Wα is quadratically hyponormal but not positively quadratically hyponormal.

Proof. A direct computation shows that d0 (t) d1 (t) d3 (t) a(t)

= 1+t t(1 + 11t) 11t2 (2009 + 4099t) , d2 (t) = , = 10 180000 2 11t (10497600 + 140792729t + 818492419t2) and = 233280000000 11t2 (−31450320 + 55205609t + 818492419t2) ≡ d3 (t) − u3 d2 (t) = . 233280000000

Thus, dk (t) > 0 for all 0 ≤ k ≤ 3 and t > 0, and a(t) ≥ 0 for all t ≥ Moreover, we have d3 (t) + =

Kt 1−Kt

· a(t)

t2 (c−bt+at2 ) 26302786560000000

≈ 0.000495 − 0.00382273t + 0.0172499t2 > 0

1 K

0.189072.

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√ for all t > 0, where a := 4099(148512423467 − 4064207 86604505), √ b := 2009(12226827733 + 4064207 86604505) and c := 13019879347200. Therefore, by Theorem 2.5 (ii), Wα is quadratically hyponormal. Since d4 (t) =

11t2 (1183625395200 − 2385425462112t + 124339089018097t2) , 3507038208000000000

Wα is not positively quadratically hyponormal. Thus the proof is complete.

References [1] Y.B. Choi, J.K. Han and W.Y. Lee, One-step extension of the Bergman shift, Proc. Amer. Math. Soc. 128(12) (2000), 3639–3646. [2] J. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, 1991. [3] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13(1990), 49–66. [4] R. Curto, An operator-theoretic approach to truncated moment problems, in Linear Operators, Banach Center Publ., 38(1997), 75–104. [5] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations Operator Theory, 18(1994), 369–426. [6] R. Curto and I. B. Jung, Quadratically hyponormal weighted shifts with two equal weights, Integral Equations Operator Theory, 37(2000), 208–231. [7] R. Curto and S.H. Lee, Quartically hyponormal weighted shifts need not be 3hyponormal, J. Math. Anal. Appl., 314(2006), 455–463. [8] R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. 115(1993), 480–497. [9] R. Curto and J. Yoon, Propagation phenomena for hyponormal 2-variable weighted shifts, J. Operator Theory, to appear. [10] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Operator Theory: Adv. Appl. 35(1988), 1–22. [11] I.B. Jung, Bridges between hyponormality and subnormality operators, Trends in Math. 4(2001), 119–126. [12] I.B. Jung and S.S. Park, Quadratically hyponormal weighted shifts and their examples, Integral Equations Operator Theory, 36(2000), 2343–2351. [13] I.B. Jung and W.Y. Lee, Gap theory of operators (in Korean), Comm. Korean Math. Soc. 16(2001), 25–66. [14] S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107(1989), 187–195. [15] Wolfram Research, Inc. Mathematica, Version 4.2, Wolfram Research Inc., Champaign, IL, 2002.

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Yiu T. Poon Department of Mathematics Iowa State University Ames, Iowa 50011 USA e-mail: [email protected] Jasang Yoon Department of Mathematics The University of Texas-Pan American Edinburg, Texas 78539 USA e-mail: [email protected] Submitted: April 2, 2006 Revised: May 21, 2007

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Integr. equ. oper. theory 58 (2007), 563–571 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040563-9, published online June 27, 2007 DOI 10.1007/s00020-007-1501-5

Integral Equations and Operator Theory

Schr¨ odinger Operators with Rapidly Oscillating Potentials Itaru Sasaki ˆ = −∆ + V with rapidly oscillating poAbstract. Schr¨ odinger operators H tentials V such as cos |x|2 are considered. Such potentials are not relatively compact with respect to the free Schr¨ odinger operator −∆. We show that the oscillating potential V do not change the essential spectrum of −∆. Moreover ˆ we derive upper bounds for negative eigenvalue sums of H. Mathematics Subject Classification (2000). Primary 35J10; Secondary 35P15. Keywords. Oscillating potential, Lieb-Thirring inequality.

1. Introduction ˆ = −∆+V with rapidly oscillatIn this paper, we consider Schr¨ odinger operators H d ing potentials V in d-dimensional space R . Typical examples of rapidly oscillating potentials are V = cos(|x|2 ), and V = (1 + |x|2 )−1 e|x| sin(e|x| ). Note that the second example is singular at inﬁnity. In this paper, we use the Friedrichs extension ˆ for the self-adjointness of H. ˆ is bounded from below, We show that, for a class of oscillating potentials V , H and V does not change the essential spectrum of −∆, i.e., σess (−∆ + V ) = [0, ∞). This means that the negative part spectrum of −∆ + V is discrete. It is well known that the moment of the eigenvalues of the Schr¨ odinger operator −∆ + V has the following estimate: ∞ γ |en | ≤ Lγ,d V− (x)γ+d/2 dx, (d = 1, 2, 3, . . .), n=1

Rd

where V− (x) := − min{0, V (x)}, e1 ≤ e2 ≤ e3 ≤ · · · are negative eigenvalues of −∆+V and Lγ,d is a universal constant [1, 2, 4]. In general, for a rapidly oscillating This work was supported by Japan Society for the Promotion of Science(JSPS).

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potential V (x), the negative part V− (x)γ+d/2 is not integrable: V− (x)γ+d/2 dx = ∞,

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(1.1)

Rd

∞ In this paper, we show that n=1 |en |γ is finite for rapidly ∞ oscillating potentials while (1.1) holds. Moreover, we give new criteria for n=1 |en |γ < ∞ and derive ˆ upper bounds for negative eigenvalue sums of H. In analysis of the Schr¨ odinger operator with an oscillating potential, the positive part of the potential is important. Because, for a low energy state u, the expectation value |u, V u| becomes small by the oscillation of the potential. But |u, V u| is not small when the positive part of V is cut oﬀ. In [5], V. B. Matveev and M. M. Skriganov discussed the Schr¨odinger operators with rapidly oscillating potentials, and proved the existence of the wave operators. In [10], M. M. Skriganov showed that, for a class of (unbounded) rapidly oscillating potential, the Schr¨ odinger operators are essentially self-adjoint. He also ˆ = [0, ∞), the number of the negative spectrum of H ˆ is ﬁnite, proved that σess (H) ˆ has no positive eigenvalue under some conditions. Another discussion for and H rapidly oscillating potential is written in the book [8, Appendix 2 to XI.8]. In the methods of [10], the continuity of potential and the diﬀerentiability of the angular part of the potential is needed. In our analysis, we don’t use the continuity and the diﬀerentiability of the oscillating potentials.

2. Main Results We consider the Schr¨ odinger operator acting on L2 (Rd ): H := (−∆ + V )C02 (Rd ), where ∆ is the d-dimensional Laplacian and V ∈ L2loc (Rd ) is a real-valued function. Let Sd−1 be the d-dimensional unit sphere, and let Θ be the standard measure on Sd−1 . We write x ∈ Rd as x = rθ, r = |x|, θ ∈ Sd−1 . For a constant R > 0, we denote by χR the characteristic function of the ball {x ∈ Rd ||x| < R}. We deﬁne V := {g ∈ L2loc (Rd )|for all R > 0, χR g is − ∆-compact}. Now we present our Hypothesis for the potential V : [V.1] The potential V can be written as V = Vosc + Vc , where the real-valued function Vc ∈ L2loc (Rd ) is −∆-compact, and Vosc ∈ V satisﬁes the following condition: The limit R lim Vosc (rθ)dr, R→+∞

0

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exists uniformly in θ ∈ Sd−1 , i.e., for any > 0 there exists a constant M > 0 such that R R Vosc (rθ)dr − Vosc (rθ)dr < , 0 0 for all R, R ∈ [M, ∞), and almost every θ ∈ Sd−1 . Example. We set V1 (r) := a sin(br ), V2 (r) := a cos(br ), 2 −1 r

V3 (r) := (1 + r )

a, b ∈ R \ {0}, > 1

r

e sin(e ).

Let Y ∈ L∞ (Sd−1 ) be a bounded function. Then the functions V (x) = Vi (r)Y (θ), (i = 1, 2, 3) satisfy the condition [V.1]. Proposition 2.1. Assume the condition [V.1]. Then the symmetric operator H is bounded from below. We give the proof of Proposition 2.1 in the last section. By the Proposition ˆ the Friedrichs 2.1, we can consider the Friedrichs extension of H. We denote by H ˆ extension. H is self-adjoint and bounded from below. Theorem 2.2. Assume the condition [V.1]. Then the potential V does not change the essential spectrum of −∆ : ˆ = [0, ∞). σess (H) Remark 2.3. The above potentials Vi Y , i = 1, 2, 3, are not relatively compact with respect to −∆. Therefore Theorem 2.2 is non-trivial. We do not assume that the continuity and twice integrability of V . Compare the condition [V.1] with the Skriganov’s condition [10]. Now we set the second condition for the potential V : [V.2] For almost every θ ∈ Sd−1 , there exists a constant R(θ) ≥ 0 such that r Vosc (sθ)ds ∈ [0, ∞), lim r→∞

R(θ)

and Ω := {rθ ∈ R |0 ≤ r ≤ R(θ), θ ∈ Sd−1 } is Borel measurable and bounded. d

Example. For any θ ∈ Sd−1 , we choose a positive-valued non-decreasing sequence 2 d {an (θ)}∞ n=0 such that an (θ) → ∞ (n → ∞). Let V4 ∈ Lloc (R ) be a function such that ≥ 0, for a2n (θ) < r < a2n+1 (θ), n ∈ N ∪ {0}, V4 (rθ) = ≤ 0, for a2n+1 (θ) < r < a2n+2 (θ), n ∈ N, and |Sn (θ)| ≥ |Sn+1 (θ)|,

for almost every θ ∈ Sd−1 ,

n ∈ N ∪ {0},

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a (θ) where Sn (θ) := ann+1 V4 (rθ)dr. By using a fact on the alternative series, one (θ) can show that the integral R lim V4 (rθ)dr R→∞

converges for a.e. θ ∈ S , and R V4 (rθ)dr ≥ 0, lim

0

d−1

R→∞

a2n (θ)

R

lim

R→∞

a2n+1 (θ)

V4 (rθ)dr ≤ 0,

n ∈ N ∪ {0}.

Therefore the potential V4 satisﬁes the condition [V.2], if the set {a0 (θ)θ ∈ Rd |θ ∈ Sd−1 } is Borel measurable and bounded. Hence, it is easy to see that, the potentials Vi Y, (i = 1, 2, 3) in Example 2 satisfy the condition [V.2] as Vosc = Vi Y . For the potential Vosc which satisﬁes [V.2], we set M V (r, θ) := lim Vosc (sθ)ds (1 − χΩ ), M→∞ r where χΩ is the characteristic function of Ω. For a real function f , we write the negative part as f− := − min{0, f }. In ˆ we need additional condiorder to show the Lieb-Thirring type inequality of H, tions: [V.3] For a constant γ ≥ 0, the estimates (Vc )− ∈ Lγ+d/2 (Rd ), (d − 1)r−1 V ∈ Lγ+d/2 (Rd ),

(Vosc )− χΩ ∈ Lγ+d/2 (Rd ), V ∈ L2γ+d(Rd )

hold. ˆ is discrete. Under the condition [V.1], the negative part of the spectrum of H ˆ Let en be the n-th eigenvalue of H counting multiplicity. Theorem 2.4. Assume [V.1]–[V.3]. Then the Lieb-Thirring type estimate ∞ |en |γ ≤Lγ,d inf (1 − )−d/2 0<<1

n=1

×

Rd

d − 1 V 2 V + (Vc )− + (Vosc )− χΩ + r

<∞, holds for γ and d with 1 , 2 γ > 0,

γ≥

γ ≥ 0,

for d = 1, for d = 2, for d ≥ 3,

where Lγ,d is a universal constant (see [1, 2, 3, 4]).

γ+d/2 dx

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Example. For the potentials Vi (i = 1, 2, 3) in Example 2, we check the condition [V.3] as Vosc = Vi (i = 1, 2, 3). One can show that there exist strictly positive constants Ri (i = 1, 2, 3) such that r Vi (s)ds ≥ 0, i = 1, 2, 3. (2.1) lim r→∞

Deﬁne

Ri

M Vi (r) := lim Vi (s)ds (1 − χRi ), M→∞ r

i = 1, 2, 3.

One can show that const. Vi ≤ (1 − χRi ) −1 , (i = 1, 2), r

const. V3 ≤ . 1 + r2

Hence, the following estimates hold: V1 , V2 ∈ L2γ+d (Rd ) if ( − 1)(2γ + d) > d, r−1 V1 , r−1 V2 ∈ Lγ+d/2 (Rd ) if (2γ + d) > 2d, V3 ∈ L2γ+d (Rd ),

r−1 V3 ∈ Lγ+d/2 (Rd ) for all γ ≥ 0.

3. Proof of Proposition 2.1, Theorems 2.2 and 2.4 In this section, we assume the condition [V.1]. Let Λd be the Laplace-Beltrami operator on Sd−1 . Lemma 3.1. For all R > 0, there exist constants a(R), b(R) such that u, Vosc u ≤ u, Vosc χR u + a(R) u 2 + b(R)u, −∆u, u, Vosc χR u − a(R) u 2 − b(R)u, −∆u ≤ u, Vosc u,

u ∈ C02 (Rd ),

lim a(R) = lim b(R) = 0.

R→∞

R→∞

Proof. For all u ∈ C02 (Rd ), we have u, Vosc u = u, χR Vosc u + u, (1 − χR )Vosc u = u, χR Vosc u + dΘ(θ) rd−1 drVosc (rθ)|u(rθ)|2 . Sd−1

We deﬁne

[R,∞)

W (R, r; θ) := [R,r]

Vosc (sθ)ds.

By integration by parts, for almost every θ ∈ Sd−1 , we have ∂

rd−1 drVosc (rθ)|u(rθ)|2 = − W (R, r; θ) |u(rθ)|2 rd−1 dr. (3.1) ∂r [R,∞) [R,∞)

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Hence we obtain

d−1 |u(rθ)|2 rd−1 dr |(3.1)| ≤ 1 + sup |W (R, r; θ)| R r≥R [0,∞) ∂u(rθ) 2 d−1 + sup |W (R, r; θ)| dr, ∂r r r≥R [0,∞)

By the deﬁnition of Λd we have u, −∆u = dΘ(θ) and − Therefore, for all u

∞

Sd−1

0

∞

dΘ(θ) Sd−1 ∈ C02 (Rd ),

du(rθ) 2 u(rθ) − (Λd u)(rθ) rd−1 dr, dr r2

0

u(rθ)(Λd u)(rθ)rd−1 dr ≥ 0.

R > 0, we have

|u, (1 − χR )Vosc u| ≤ a(R) u 2 + b(R)u, −∆u, where

d−1 a(R) := 1 + sup |W (R, r; θ)|, R r≥R θ∈Sd−1

b(R) := sup |W (R, r; θ)|. r≥R θ∈Sd−1

By the condition [V.1], one can easily show that lim a(R) = lim b(R) = 0.

R→∞

R→∞

Proof of Proposition 2.1. Using Lemma 3.1, we have H = −∆ + V ≥ −(1 − b(R))∆ + Vc + χR Vosc − a(R), in the sense of the quadratic forms on C02 (Rd ) = D(H). Since χR Vosc is −∆compact, Vc −|χR Vosc | is inﬁnitesimally small with respect to −∆. For a suﬃciently large R > 0, the coeﬃcient 1−b(R) is positive. Hence H is bounded from below. For a self-adjoint operator A bounded from below, acting on a inﬁnite dimensional Hilbert space, we denote Σ(A) the bottom of the essential spectrum inf σess (A). If σess (A) = ∅, we deﬁne Σ(A) = +∞. Lemma 3.2. Let A, B be self-adjoint operators bounded from below, acting on a infinite dimensional Hilbert space, such that Q(A) ⊆ Q(B) and B ≤ A,

on Q(A),

where “Q(·)” denotes the form domain. Then Σ(B) ≤ Σ(A).

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Proof. By the min-max principle, we have µn (B) ≤ µn (A),

n = 1, 2, 3, . . . ,

where µn (A) is the n-th eigenvalue of A, or the bottom of the essential spectrum of A. It is easy to see that limn→∞ µn (A) = Σ(A), and limn→∞ µn (B) = Σ(B). Therefore Σ(B) ≤ Σ(A). ˆ ⊆ [0, ∞) and 0 ∈ σess (H). ˆ Lemma 3.3. σess (H) Proof. Using the estimate of Lemma 3.1, for all large R > 0, we have −(1 − b(R))∆ + Vc + χR Vosc − a(R) ≤ H,

(3.2)

H ≤ −(1 + b(R))∆ + Vc + χR Vosc + a(R)

(3.3)

C02 (Rd )

in the sense of quadratic forms on = D(H). We take the Friedrichs ˆ = Q(−∆) = extension of the quadratic forms (3.2),(3.3). Hence we get Q(H) 1/2 D((−∆) ), and the inequalities ˆ −(1 − b(R))∆ + Vc + χR Vosc − a(R) ≤ H, (3.4) ˆ ≤ −(1 + b(R))∆ + Vc + χR Vosc + a(R), H

(3.5)

ˆ Since Vc +χR Vosc is −∆-compact, hold in the sense of the quadratic forms on Q(H). Vc + χR Vosc does not change the essential spectrum of −∆. By using Lemma 3.2 with (3.4),(3.5), we have ˆ ≤ a(R). −a(R) ≤ Σ(H) ˆ = 0. Therefore σess (H) ˆ ⊆ [0, ∞) and 0 ∈ σess (H). ˆ Taking R → ∞, we obtain Σ(H) ˆ Proof of Theorem 2.2. By Lemma 3.3, it is suﬃcient to show that [0,∞) ⊆ σess (H). ∞ ˆ ˆ Since 0 ∈ σess (H), there exists a sequence {vn }n=1 ⊂ D(H) such that w ˆ n → 0 (n → ∞).

vn = 1, vn → 0 (n → ∞), Hv Using the inequality (3.4), there exist constants C1 > 0, C2 ∈ R such that ˆ Hence, the sequence (−∆)1/2 vn is uniformly bounded. By C1 (−∆) + C2 ≤ H. this fact, a suitable subsequence {(−∆)1/2 vnj }∞ j=1 has a weak limit. The weak limit w w 1/2 vn → 0 lead to that (−∆) vnj → 0(j → ∞). Thus, by using the compactness in the deﬁnition [V.1], we have lim vnj , Vc vnj = lim vnj , χR Vosc vnj = 0.

j→∞

j→∞

By (3.4), for all large R > 0 we obtain (1 − b(R)) lim sup (−∆)1/2 vnj 2 ≤ a(R). j→∞

s Thus we get the strong limit (−∆)1/2 vnj → 0 (j → ∞). For each k ∈ Rd , we set wj (x) := eik·x vnj (x),

j = 0, 1, 2, . . . .

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ˆ Since H ˆ is the Friedrichs extension of H, we note that We show that wj ∈ D(H). ∗ ˆ ˆ ˆ ⊆ Q(−∆) implies D(H) = Q(H) ∩ D(H ) = Q(−∆) ∩ D(H ∗ ), and vnj ∈ D(H) 2 d wj ∈ Q(−∆). For all u ∈ C0 (R ), we have ˆ nj + u, k 2 eik·x vnj + u, eik·x 2k · (−i∇)vnj Hu, wj = u, eik·x Hv ˆ Also Hence, we obtain wj ∈ D(H ∗ ), and so wj ∈ D(H). ˆ j = eik·x Hv ˆ nj + k 2 eik·x vnj + eik·x 2k · (−i∇)vnj . Hw By the above arguments, the last term eik·x 2k·(−i∇)vnj converges to zero strongly ˆ nj = 0. Therefore, we obtain the as j → ∞. By the deﬁnition of vnj , s-limj→∞ Hv limit ˆ − k 2 )wj = 0, lim (H j→∞

2

ˆ Since k ∈ Rd is arbitrary, [0, ∞) ⊆ σess (H). ˆ which implies that k ∈ σess (H).

Proof of Theorem 2.4. Assume that the conditions [V.1], [V.2], and [V.3]. Remember the deﬁnition of W (R, r; θ). Deﬁne ¯ (θ) := lim W (R(θ), r; θ). W r→∞

¯ (θ) ≥ 0. For almost every θ ∈ Sd−1 , and for all u ∈ By the condition [V.2], W 2 d C0 (R ), we have ∞ ∂ W (R(θ), r; θ) (rd−1 |u(rθ)|2 )dr − ∂r R(θ) ∞ ¯ (θ) − W (R(θ), r; θ)) ∂ (rd−1 |u(rθ)|2 )dr + W ¯ (θ)R(θ)d−1 |u(R(θ)θ)|2 = (W ∂r R(θ) ∞ ¯ (θ) − W (R, r; θ)) ∂ (rd−1 |u(rθ)|2 )dr ≥ (W ∂r R(θ) ∞ ≥− V (rθ) (d − 1)rd−2 |u(rθ)|2 + 2rd−1 |∂u(rθ)/∂r||u(rθ)| dr R(θ)

≥ −

∞

0

∂u(rθ) 2 d−1 r dr − ∂r

0

∞

d−1 −1 2 V (rθ) + V (rθ) |u(rθ)|2 rd−1 dr, r

ˆ where > 0 is arbitrary. Hence we obtain the lower bound of H: ˆ ≥ −(1 − )∆ − (Vc )− − χΩ (Vosc )− − (d − 1)r−1 V − −1 V 2 , H in the sense of the quadratic forms on C02 (Rd ). Finally, by using the Lieb-Thirring inequality ([3, Theorem 12.4]), we obtain the conclusion of Theorem 2.4.

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Acknowledgment The author is grateful to Professor A. Arai of Hokkaido University for discussions and helpful comments.

References [1] D. Hundertmark, A. Laptev and T. Weidl, New bounds on the Lieb-Thirring constants, Invent. math. 140 (2000), 693–704. [2] A. Laptev and T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions, Acta Math. 184 (2000), 87–111. [3] E. H. Lieb and M. Loss, Analysis, Graduate studies in mathematics, American Mathematical Society, 2nd edition, 2001. [4] E. H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schr¨ odinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Princeton University Press, (1976), 269–303. [5] V. B. Matveev and M. M. Skriganov, Wave operators for the Schr¨ odinger equation with rapidly oscillating potential, Soviet Math. Dokl Vol. 13 (1972), 185–188. [6] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. I, Academic Press, New York, 1972. [7] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. II, Academic Press, New York, 1975. [8] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. III, Academic Press, New York, 1979. [9] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV, Academic Press, New York, 1978. [10] M. M. Skriganov, On the spectrum of the Schr¨ odinger operator with a rapidly oscillating potential, Proc. Steklov Inst. Math. 125 (1973), 177–185. Itaru Sasaki Department of Mathematics Hokkaido University Sapporo 060-0810 Japan e-mail: [email protected] Submitted: January 20, 2005 Revised: March 19, 2007

Integr. equ. oper. theory 58 (2007), 573–589 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040573-17, published online June 27, 2007 DOI 10.1007/s00020-007-1515-z

Integral Equations and Operator Theory

Spectrum of the One-dimensional Schr¨ odinger Operator With a Periodic Potential Subjected to a Local Dilative Perturbation Leonid Zelenko Abstract. We study the spectrum of the one-dimensional Schr¨ odinger operator with a potential, whose periodicity is violated via a local dilation. We obtain conditions under which this violation preserves the essential spectrum of the Schr¨ odinger operator and an inﬁnite number of isolated eigenvalues appear in a gap of the essential spectrum. We show that the considered perturbation of the periodic potential is not relative compact in general. Mathematics Subject Classification (2000). Primary 47F05, Secondary 47E05, 35Pxx. Keywords. Schr¨ odinger operator, perturbation of a periodic potential, essential spectrum, discrete part of the spectrum.

1. Introduction In the papers [Z1] and [Z2] we described the essential spectrum σe (H) of a multidimensional Schr¨ odinger operator H = −∆ + V (x)· acting in the space L2 (Rd ), in terms of a family of Schr¨odinger operators {H y }y∈Rd with periodic potentials Vy (x) which approximate the potential V (x) at inﬁnity in some sense (see also [Z3]-[Zel-Rof]). In this case we call {Vy (x)}y∈Rd the asymptotic family for the potential V (x) (see Deﬁnition 2.1 of this paper for the case d = 1). Under some natural conditions it was proved that the essential spectrum σe (H) coincides with the set Γ{Vy } of the points λ ∈ C, for which the family of norms {Rλ (H y )}y∈Rd is unbounded at inﬁnity. Under some conditions the set Γ{Vy } coincides with the set Σ{Vy } of limit points of the spectra σ(H y ) of the operators H y for |y| → ∞. For instance, the latter fact holds if all the potentials Vy (x) are real-valued. Supported by KAMEA Project for scientific Absorption in Israel and partially by the GermanIsraeli Foundation (GIF), grant number I-619-17.6/2001.

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In the present paper we consider the one-dimensional case (d = 1) and apply the above approach to the study of the spectrum of Schr¨ odinger operator H with the potential V (x) = α2 (x)V˜ (α(x)x), (1.1) ˜ where V (x) is a continuous real-valued periodic function, α(x) ∈ (0, 1) for any x ∈ R and lim|x|→∞ α(x) = 1. Physically V˜ (x) is the electric potential of a periodic atomic lattice and V (x) is the electric potential of this lattice subjected to a local dilation (for instance, via some local heating) with the dilation coeﬃcient (α(x))−1 at each point x ∈ R. Consider the family of periodic potentials Vy (x) = α2 (y)V˜ (α(y)x) (y ∈ R), (1.2) obtained from the potential V (x) by “freezing” the dilation coeﬃcient at a point ˜ be the Schr¨odinger operators with potentials Vy (x) and y ∈ R. Let H y and H V˜ (x) respectively. Since the operator x 1 (Wy u)(x) = u α(y) α(y) ˜ we have: establishes a unitary equivalence between the operators H y and α2 (y)· H, ˜ σ(H y ) = α2 (y) · σ(H).

(1.3)

In other words, the spectrum σ(H y ) of the operator H y is a contraction of the ˜ of the operator H ˜ with the coeﬃcient α2 (y). Recall that the specspectrum σ(H) trum of the Schr¨ odinger operator with a continuous real-valued potential is the union of a ﬁnite or countable number of isolated closed intervals having non-zero length (see [Kuch], [Wil]). We show that, under additional conditions for α(x), the family of potentials {Vy (x)}x∈R is asymptotic for the potential V (x), hence, by [Z1] and (1.3), the essential spectrum σe (H) of the operator H coincides with the ˜ of the operator H ˜ (Theorem 3.3), which, in turn, coincides with spectrum σ(H) ˜ its essential spectrum σe (H) (because the potential V˜ (x) is periodic). In other words, we have proved that some wide class of local dilative perturbations of the ˜ Notice that potential V˜ (x) preserves the essential spectrum of the operator H. these perturbations are not relatively compact in general (see Example 2 in Section 3). Furthermore, we have found conditions under which an inﬁnite number ˜ of the of eigenvalues of the operator H appear in a gap of the spectrum σ(H) ˜ (Theorem 4.3). unperturbed operator H The present paper is divided into four sections. After this Section 1 (Introduction), in Section 2 (Preliminaries) we recall some concepts and results from [Z1], used in the present paper. In Section 3 we investigate the essential spectrum of the operator H. In Section 4 we study the discrete spectrum of the operator H in a gap of its essential spectrum. We shall use the following notation:

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R is the ﬁeld of real numbers; C is the ﬁeld of complex numbers; Cl(A) is the closure of a set in a topological space; C l (I, C) is the set of continuous complex-valued functions deﬁned on an interval I ⊆ R and having there continuous derivatives of the order ≤ l. If l = 0, we write C(I, C) instead of C 0 (I, C); C l (I) is the part of C l (I, C), consisting of all real-valued functions. If l = 0, we write C(I) instead of C 0 (I); L2 (I) is the set of complex-valued functions deﬁned on an interval I ⊆ R such that they are square integrable on I. f I is the L2 -norm of a function f ∈ L2 (I). If I = R, we write f instead of f R ; DH is the domain of a linear operator H acting in a Hilbert space H; σ(H), σe (H) and R(H) are the spectrum, the essential spectrum and the resolvent set of a linear operator H acting in a Hilbert space H.

2. Preliminaries First of all, let us recall some concepts and results from [Z1]. A central concept there was the concept of asymptotic family of potentials for a potential V (x). In this paper we need only the one-dimensional version (d = 1) of this concept. Definition 2.1. Consider a family of complex-valued measurable locally bounded on R potentials {Vy (x)}y∈R , parameterized by the points y of the real line R, and the corresponding family of Schr¨ odinger operators d2 + Vy (x) · . dx2 This family of potentials is called an asymptotic family for a measurable locally bounded complex-valued potential V (x), if there exists M > 0 and a nondecreasing function h : [M, ∞) → R such that limt→∞ h(t) = ∞ and the following relation holds: lim sup |V (x) − Vy (x)| = 0, Hy = −

|y|→∞ x∈Qh y

where

Qhy =

[y, y + h(y)], if y ≥ M, [y − h(|y|), y], if y ≤ −M.

(2.1)

In this case we call the family {H y }y∈R an asymptotic family of Schr¨ odinger opd2 + V (x)· and we call the function erators for the Schr¨ odinger operator H = − dx 2 h(t) a supporting function for the asymptotic family of potentials {Vy (x)}y∈R . As was shown in [Z1], it is possible to construct the essential spectrum of a Schr¨ odinger operator with a potential V (x) via the spectra of Schr¨ odinger operators with periodic potentials Vy (x) which are asymptotic for the potential V (x).

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In this paper we need only the following simple version of this description for the one-dimensional case (see [Z1], Theorems 4.6, 5.1 and Proposition 5.7): Theorem 2.2. Assume that for a measurable locally bounded potential V (x) there exists an asymptotic family of real-valued periodic potentials Vy (x) (y ∈ R) with periods T (y) > 0 and a supporting function h(t), such that the condition is satisﬁed: lim

|y|→∞

T (y) = 0. h(|y|)

Furthermore, assume that the family of potentials {Vy (x)}y∈R is uniformly bounded from below, that is there exists γ > 0 such that ∀ x, y ∈ R : Vy (x) > −γ. Then the essential spectrum σe (H) of the operator d2 + V (x)· dx2 is constructed via the spectra σ(H y ) of the operators H=−

Hy = −

d2 + Vy (x)· dx2

in the following manner: σe (H) =

∞

(2.2)

 Cl 

N =1

(2.3) 

σ(H y ) .

(2.4)

y:|y|≥N

3. Essential spectrum of the operator H We now return to the case of a local dilative perturbation of a periodic potential described in Introduction. We consider the one-dimensional Schr¨ odinger operator H, deﬁned by (2.2), with the potential of the form (1.1), where V˜ (x) is a real-valued periodic function, α ∈ C(R), ∀ x ∈ R : 0 < α(x) < 1 (3.1) and lim α(x) = 1.

|x|→∞

(3.2)

Along with the potential V (x) we consider the family of periodic potentials Vy (x), deﬁned by (1.2) and the corresponding Schr¨ odinger operators H y deﬁned by (2.3). In the sequel we shall need the following Lemma 3.1. Let α(x) be a real-valued continuously diﬀerentiable function deﬁned on [M, ∞) (M > 0) such that the function xα (x) is positive and non-increasing on [M, ∞) and there exists a ﬁnite limit limx→∞ α(x). Then lim xα (x) = 0.

x→∞

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Proof. Assume, on the contrary, that limx→∞ xα (x) = A = 0. Since xα (x) is positive and non-increasing, this assumption implies that ∀ x ≥ M : xα (x) ≥ A > 0. Then, if x ≥ M , we have the inequality

x , M which contradicts the existence of a ﬁnite limit limx→∞ α(x). The lemma is proven. α(x) ≥ α(x0 ) + A ln

The following lemma yields conditions, under which the family of potentials Vy (x), deﬁned by (1.2), is asymptotic for the potential V (x). Lemma 3.2. Assume that, along with the conditions (3.1) and (3.2), the function α(x) satisﬁes the following conditions: (a) α(x) is continuously diﬀerentiable on R \ (−M, M ) for some M > 0 and the function xα (x) is positive and non-increasing on [M, ∞) and it is positive and non-decreasing on (−∞, M ]. Furthermore, assume that the function V˜ (x) satisﬁes the condition: (b) V˜ (x) is periodic and it satisﬁes the Lipschitz condition, that is ∃ L > 0 ∀ x1 , x2 ∈ R : |V˜ (x1 ) − V˜ (x2 )| ≤ L|x1 − x2 |. Then the family of potentials {Vy (x)}y∈R , deﬁned by (1.2), is asymptotic for the potential V (x), deﬁned by (1.1), with the supporting function

1 1 , (t ≥ M ). (3.3) h(t) = min tα (t) t|α (−t)| Proof. Observe that, in view of condition (a) and Lemma 3.1, the function h(t), deﬁned by (3.3), is non-decreasing and limt→∞ h(t) = ∞. Let Qhy be the interval deﬁned by (2.1). Assume that y ≥ M and estimate for x ∈ Qhy = [y, y + h(y)], making use of conditions (3.1), (3.2) and (a), (b): |V (x) − Vy (x)| = |α2 (x)V (α(x)x) − α2 (y)V (α(y)x)| ≤ |α2 (x) − α2 (y)||V (α(x)x)|

+|α (y)||V (α(x)x) − V (α(y)x)| ≤ 2V¯ + Lx |α(x) − α(y)| ≤ 2V¯ + L(y + h(y)) α (y)h(y),

(3.4)

2

where V¯ = supx∈R |V˜ (x)|. Thus, taking into account deﬁnition (3.3) of the function h(t), we have aV¯ α (y) L + L yα (y) + . sup |V (x) − Vy (x)| ≤ √ y y h x∈Qy

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In view of condition (a) and Lemma 3.1, we obtain from this estimate: lim sup |V (x) − Vy (x)| = 0.

y→∞ x∈Qh y

The case y ≤ −M is treated analogously. So, by Deﬁnition 2.1, the family {Vy (x)} is asymptotic for the potential V (x). The lemma is proven. We now turn to the main theorem of this section. Theorem 3.3. If all the conditions of Lemma 3.2 are satisﬁed, then the essential odinger operator H deﬁned by (2.2), with the potenspectrum σe (H) of the Schr¨ ˜ of the Schr¨ tial V (x) deﬁned by (1.1), coincides with the spectrum σ(H) odinger operator 2 ˜ = − d + V˜ (x) · . H (3.5) dx2 Proof. By Lemma 3.2, the family {Vy (x)}y∈R of periodic potentials, deﬁned by (1.2), is asymptotic for the potential V (x). Furthermore, in view of conditions (3.1), (3.2) and condition (b) of Lemma 3.2, this family is uniformly bounded on R and each potential Vy (x) has a period T (y) > 0 such that the function T (y) is bounded on R. Thus, all the conditions of Theorem 2.2 are satisﬁed for this family, hence formula (2.4) is valid for σe (H). Taking into account formula (1.3) ˜ The theorem is proven. and condition (3.2), we obtain that σe (H) = σ(H). Example 1. As is not diﬃcult to see, a function α(x), satisfying conditions (3.1), (3.2) and condition (a) of Lemma 3.2, can have an arbitrary fast rate of convergence to 1 as |x| → ∞. For instance, the functions α(x) = 1 − α(x) = 1 −

1 2(1 + |x|)γ

(γ > 0),

1 exp{−|x|γ } (γ > 0), 2

1 exp {− exp{|x|γ }} (γ > 0), 2 and so on, satisfy all the conditions mentioned above. On the other hand, the functions (ln 2)γ α(x) = 1 − γ (γ > 0), 2 (ln(2 + |x|)) α(x) = 1 −

(ln(ln 3))γ γ (γ > 0), 2 (ln(ln(3 + |x|))) and so on, also satisfy all the conditions mentioned above. In other words, a function α(x), satisfying these conditions, can have an arbitrary slow rate of convergence to 1 as |x| → ∞. These examples show that condition (a) of Lemma 3.2 is a restriction on the regularity of the growth of the function α(x) as |x| → ∞, but not on the rate of convergence of it to 1. α(x) = 1 −

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Remark 3.4. In the literature ones use the following generalization of H. Weyl Theorem on preservation of essential spectrum of an operator H under a compact perturbation (see, for instance, [Gl], Ch.I, §1, no 4, no 5 and [H-S], Ch. 14): Proposition 3.5. Let H and K be self-adjoint operators acting in a Hilbert space H such that DH ⊆ DK . Assume that K is compact with respect to H in the sense that K (H − λ0 I)−1 is a compact operator in H for some λ0 ∈ R(H). Then σe (H + K) = σe (H). The following example shows that Theorem 3.3 cannot be proved with the help of Proposition 3.5, because in general, under conditions of this theorem, the ˜ is not compact with respect to H. ˜ operator K = H − H Example 2. Assume that the potential V (x) has the form (1.1) with α(x) = 1 − δ(x), V˜ (x) = cos(πx), where

1 δ(x) = for |x| ≥ 1, 2 |x| δ(x) is continuous on R and 0 < δ(x) < 1 for any x ∈ [−1, 1], Thus, the functions α(x) and V˜ (x) satisfy all the conditions of Lemma 3.2, that is they satisfy all the ˜ is a multiplication operator conditions of Theorem 3.3. Observe that K = H − H by the function K(x) = V (x) − V˜ (x). Since V (x) and V˜ (x) are bounded on R, the operator K is bounded in L2 (R). As is easy to see, in order to prove that the ˜ it is enough to construct such a operator K is not compact with respect to H, sequence {fn }n≥N ⊂ DH˜ (N > 0) that ˜ n − λ0 fn ≤ 1 ∀ n ≥ N : Hf (3.6) ˜ but the sequence {Kfn }n≥N is not compact in L2 (R). Since for some λ0 ∈ R(H), ˜ is bounded from below, we can choose λ0 to be a real number, the operator H ˜ Denote φ(x) = xα(x) − x = −δ(x)x, hence such that λ0 < min σ(H). x φ(x) = − for |x| ≥ 1. 2 |x| Consider the sequence of points xn = 4(2n + 1)2 (n = 1, 2, . . . ), hence φ(xn ) = −2n − 1. Then we have: K(xn ) = (α2 (xn ) − 1)V˜ (α(xn )xn ) + V˜ (xn + φ(xn )) − V˜ (xn ) = (α2 (xn ) − 1) cos(α(xn )xn ) + cos 4(2n + 1)2 π − (2n + 1)π − cos 4(2n + 1)2 π = −2 + (α2 (xn ) − 1) cos(α(xn )xn ). Then ∃ N > 0, ∀ n ≥ N : |K(xn )| > 1. (3.7) ˜ Observe that, by the deﬁnition of the potentials V (x) and V (x), the function K(x) = V (x) − V˜ (x) satisﬁes the Lipschitz condition on [1, ∞), that is ∃ l > 0, ∀ x, x ∈ [1, ∞) : |K(x) − K(x )| ≤ l|x − x |.

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This fact and (3.7) imply that there exists > 0 such that 1 . (3.8) 2 Let us construct the sequence fn ∈ DH˜ in the following manner: fn (x) = f (x−xn ), where the function f (x) is not identically zero, it belongs to the class C 2 (R) and supp(f ) ⊆ [−, ]. As is easy to see, f can be chosen such that the sequence fn satisﬁes the condition (3.6). On the other hand, in view of (3.8), 1 1 |K(x)|2 |fn (x)|2 dx > fn 2 = f 2 . ∀ n ≥ N : Kfn2 = 2 2 R ∀ n ≥ N, ∀ x ∈ [xn − , xn + ] : |K(x)| >

Since supp(Kfn ) ⊆ [xn − , xn + ], then the latter estimate implies that the sequence {Kfn }n≥N is not compact in L2 (R). Thus, the operator K is not compact ˜ with respect to H.

4. Discrete part of the spectrum of the operator H In this section we study the discrete part of the spectrum of the operator H, deﬁned by (2.2) and (1.1). For estimation of the number of points of spectrum of an operator H, lying in an interval, we shall be based on the following well known Glazman Principle ([Gl], Ch. I, §1, Theorems 9bis and 13bis ): Proposition 4.1. Let H be a linear self-adjoint operator acting in a Hilbert space H. Then: (i) The number of points of the spectrum of H lying in a semi-axis (−∞, µ0 ) coincides with the maximal dimension of linear subspaces 1 G ⊆ DH , for which the property is valid: ∀ f ∈ G : (Hf − µ0 f, f ) < 0; (ii) The number of points of the spectrum of H lying in an interval of the form (λ0 − δ, λ0 + δ) (δ > 0) coincides with the maximal dimension of linear subspaces G ⊆ DH , for which the property is valid: ∀ f ∈ G : Hf − λ0 f < δf . In the sequel we shall use a result from [Z1] (Proposition A1.1) on an estimate odinger operator. for the L2 -norm of the gradient of a function by means of a Schr¨ We need only the following one-dimensional version of this result: Lemma 4.2. Let I and I˜ be intervals of the form: I = [a, b], 1A

I˜ = [a − 1, b + 1]

“linear subspace” G of a Hilbert space H is a subspace of H, if we mean the latter only as a linear space, that is in general G is not closed in the topology of H.

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˜ C). If a measurable complex-valued and u(x) be a function from the class C 2 (I, ˜ potential V (x), deﬁned on I, is bounded there and the operator H is deﬁned by (2.2), then the following estimate is valid: u 2I ≤ K Hu2I˜ + γu2I˜ ,

where

γ = max 1, sup (− (V (x))) x∈I˜

and a positive constant K does not depend on u, a, b and the potential V (x). We now turn to the main result of this section. Theorem 4.3. Let H be the Schr¨ odinger operator deﬁned by (2.2), with the potential ˜ be the Schr¨ V (x) deﬁned by (1.1), and H odinger operator deﬁned by (3.5). Assume that for the functions α(x) and V˜ (x) conditions (3.1), (3.2) and conditions (a), (b) of Lemma 3.2 are satisﬁed. ˜ > 0 and the condition (i) If µ0 = min σ(H) 1 lim x=0 (4.1) |x|→∞ β(x) is satisﬁed, where

β(x) = 1 − α2 (x), (4.2) then the set σ(H) ∩ (−∞, µ0 ) consists of an inﬁnite number of isolated eigenvalues of the operator H, it is bounded from below and µ0 is its unique accumulation point. ˜ of the operator H ˜ such that λ > (ii) If (λ , µ ) is a gap of the spectrum σ(H) −∞, and the conditions 1 lim x = 0, (4.3) |x|→∞ β 2 (x) 1

4

lim β(x)|β (x)| 3 |x| 3 = ∞

|x|→∞

(4.4)

are satisﬁed, then the set σ(H) ∩ (λ , µ ) consists of an inﬁnite number of isolated eigenvalues of the operator H, which can accumulate only to the endpoints of the gap (λ , µ ). Moreover, if µ > 0, then µ is an accumulation point of these eigenvalues, and if λ < 0, then λ is an accumulation point of these eigenvalues. Proof. Recall that, since all the conditions of Theorem 3.3 are satisﬁed, then ˜ σe (H) = σ(H). Let us prove assertion (i). If we shall construct a sequence of functions un ∈ DH , such that supp(un ) ∩ supp(um ) = ∅, if n = m (4.5) and ∃ N > 0 ∀ n ≥ N : (Hun − µ0 un , un ) < 0, (4.6)

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then, if L = span {un }n≥N , the following property will be valid: ∀ u ∈ L : (Hu − µ0 u, u) < 0. Since the set L is an inﬁnite-dimensional linear subspace of DH , then by assertion (i) of Proposition 4.1 the semi-axis (−∞, µ0 ) will contain an inﬁnite number of points of spectrum of the operator H. Since (−∞, µ0 ) ∩ σe (H) = ∅, then all these points are isolated eigenvalues of the operator H and they cannot accumulate to the points of this semi-axis. Observe that, in view of (1.1), the potential V (x) is bounded on R, hence the operator H is bounded from below. This means that the set of eigenvalues of the operator H lying in (−∞, µ0 ) will be bounded from below and has only µ0 as an accumulation point. So, in this way we shall prove assertion (i). In order to construct such a sequence un , consider the family of periodic odinger potentials {Vy (x)}y∈R , deﬁned by (1.2), and corresponding family of Schr¨ operators H y , deﬁned by (2.3). It is clear that T (y) =

T0 α(y)

(4.7)

is a period of Vy (x), if T0 > 0 is a period of V˜ (y). Denote µ0 (y) = min σ(H y ). Then, in view of (1.3) and (4.2), µ0 (y) = α2 (y)µ0 = (1 − β(y))µ0 .

(4.8)

Let ψy (x) be a Bloch function of the Hamiltonian H y , corresponding to the minimal energy level µ0 (y) (a “ground state” of H y ). which is a non-trivial T (y)periodic solution of the equation H y ψ − µ0 (y)ψ = 0

(4.9)

(see [Kuch], [Wil]). We can choose it such that the normalization condition T (y) |ψy (x)|2 dx = 1 (4.10) 0

is satisﬁed. Consider a sequence of points yn ∈ R and intervals In = [yn , yn + ∆n ] such that yn > 0,

lim yn = ∞, ∆n > 2,

n→∞

lim ∆n = ∞.

n→∞

(4.11)

A dependence of ∆n on yn will be speciﬁed in the sequel. Furthermore, the intervals In will be constructed such that In ∩ Im = ∅, if n = m.

(4.12)

Let us construct the desired sequence of functions un ∈ DH in the following manner: un (x) = ψn (x)θn (x),

(4.13)

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where ψn (x) = ψyn (x) and each of θn (x) deﬁned in the following manner:  1, if    κ(x − yn ), if θn (x) = χ(x − y − ∆ + 1), if  n n   0, if

583

is a compactly supported C 2 -function, x ∈ [yn + 1, yn + ∆n − 1], x ∈ [yn , yn + 1], x ∈ [yn + ∆n − 1, yn + ∆n ], x∈ / In ,

(4.14)

where the functions κ(x) and χ(x) are deﬁned on the interval [0, 1] such that κ, χ ∈ C 2 [0, 1], κ(0) = κ (0) = κ (0) = 0, κ(1) = 1, κ (1) = κ (1) = 0, χ(0) = 1, χ (0) = χ (0) = 0, χ(1) = χ (1) = χ (1) = 0.

(4.15)

As is clear, supp(un ) ⊆ In .

(4.16)

H yn un = θn H yn ψn − 2ψn θn − ψn θn .

(4.17)

We have from (2.3) and (4.13):

Then, taking into account selfadjointness of the operators H y , the fact that ψn is a solution of equation (4.9) with y = yn and using formula (4.8), we get: (H yn un − µ0 un , un ) = (µ0 (yn ) − µ0 )un 2 − 2 (ψn ψ¯n )θn θn dx R 2 − |ψn |2 θn θn dx = −µ0 β(yn )un 2 + |ψn |2 (θn ) dx. (4.18) R

R

Thus, we have the equality: 2

(Hun − µ0 un , un ) = −µ0 β(yn )un + 2 + (V (x) − Vyn (x)) |un (x)| dx,

R

2

|ψn |2 (θn ) dx

R

which, in view of (4.16) implies the estimate: (Hun − µ0 un , un ) ≤ −µ0 β(yn )un 2 + = −µ0 β(yn )(1 − Rn )un 2 ,

R

2

|ψn |2 (θn ) dx + ωn un 2 (4.19)

where ωn = and

Rn =

R

sup x∈[yn ,yn +∆n ]

|V (x) − Vyn (x)|

(4.20)

2

|ψn (x)|2 (θn (x)) dx ωn + µ0 β(yn )un 2 µ0 β(yn )

(4.21)

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(recall that, by the assumption, µ0 > 0 and, by (4.2), (3.1), β(yn ) > 0). Observe that, in view of (4.7), continuity of the function α(x) and conditions (3.1), (3.2) the family of periods {T (y)}y∈R of potentials Vy (x) has the following properties: sup T (y) < ∞

(4.22)

y∈R

and inf T (y) > 0.

(4.23)

y∈R

Denote Tn = T (yn ). Then, taking into account Tn -periodicity of ψn (x), the normalization condition (4.10), deﬁnition (4.14), (4.15) and conditions (4.11), we obtain the following estimates with some constants c1 > 0, c2 > 0: yn +1 2 2 2 |ψn (x)| (θn (x)) dx ≤ max |κ (x)| |ψn (x)|2 dx R

2

x∈[0,1]

yn +∆n

yn

2

|ψn (x)| dx + max |χ (x)| x∈[0,1] yn +∆n −1 1 2 2 ≤ max |κ (x)| + max |χ (x)| + 1 ≤ c1 Tn x∈[0,1] x∈[0,1]

(4.24)

and ∃ N > 0, ∀ n ≥ N : yn +∆n −1 un 2 = |ψn (x)|2 θn2 (x) dx ≥ |ψn (x)|2 dx yn +1 R ∆n − 2 ≥ ≥ c2 ∆n . Tn

(4.25)

In the same manner, as the estimate (3.4), we obtain from (4.20), (1.1), (1.2) and (4.2) the following estimate with some constant c3 > 0: ωn ≤ c3 (1 + yn + ∆n ) |β (yn )|∆n .

(4.26)

Thus, we get from (4.21) and (4.24)-(4.26) that ∃ c4 > 0, ∃ N > 0, ∀ n ≥ N : 1 (1 + yn + ∆n ) |β (yn )|∆n |Rn | ≤ c4 + . β(yn )∆n β(yn )

(4.27)

Now we choose the sequence ∆n in the following manner: 1 ∆n = 2 + . |β (yn )|yn Observe that, in view of (4.2) and Lemma 3.1, limn→∞ ∆n = ∞. Furthermore, selecting, if it is necessary, a subsequence from the sequence yn , we can satisfy also

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the condition (4.12). After this choice of ∆n the estimate (4.27) takes the form: ∃ c5 > 0, ∃ N > 0, ∀ n ≥ N : |β (yn )|yn |β (yn )| 1 + . |Rn | ≤ c5 2 √ + β(yn ) β(yn ) yn β(yn )yn

(4.28)

Condition (4.1) implies that

|β (y)|y 1 lim = lim y=0 y→∞ y→∞ β(y) β(y)

Furthermore, by d’Hospital rule, we obtain from (4.1) that 1 1 = lim = 0. lim y→∞ β(y)y y→∞ β(y)

(4.29)

(4.30)

Thus, we have from (4.28)-(4.30) that limn→∞ Rn = 0. Therefore, in view of estimate (4.19), condition (4.6) is satisﬁed for the sequence {un }n≥N , if N > 0 is large enough. Furthermore, in view of (4.16) and (4.12), the condition (4.5) is satisﬁed for the functions un . So, we have proved assertion (i). ˜ We now turn to the proof of assertion (ii). Let (λ , µ ) be a gap of σ(H), such that λ > −∞. We shall consider only the right endpoint µ of the gap in the case µ > 0, because the left endpoint λ in the case λ < 0 is treated analogously. Consider a point ˜ ∈ λ + µ , µ . λ 2 Then ! ˜ − (µ − λ), ˜ µ ⊂ (λ , µ ). λ If we shall construct a sequence of functions un ∈ DH , such that (4.5) is valid and ˜ n < (µ − λ)u ˜ ∃ N > 0, ∀ n ≥ N : Hun − λu n ,

(4.31)

then using the same arguments as above, we shall obtain from assertion (ii) of ˜ ˜ Proposition 4.1 that the set σ(H) ∩ λ − (µ − λ), µ consists of an inﬁnite number of points. Since (λ , µ ) is a gap of σe (H), the set σ(H) ∩ (λ , µ ) consists of isolated eigenvalues of H, which cannot accumulate to the points of (λ , µ ). ˜ − (µ − λ) ˜ ∈ (λ , µ ) imply that µ is an These circumstances and the fact that λ accumulation point of these eigenvalues. So, in this way we shall prove assertion (ii). In view of (1.3) and (4.2), the interval (λ (y), µ (y)), where λ (y) = (1 − β(y))λ and (4.32) µ (y) = (1 − β(y))µ , is a gap of the spectrum of the operator H y deﬁned by (2.3). Like above, consider a Bloch function ψy (x) of the Hamiltonian H y corresponding to the energy level

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µ (y), which is a non-trivial solution of the equation Hψ − µ (y)ψ = 0,

(4.33)

ψ(x + T (y)) = exp(iτ ) · ψ(x),

(4.34)

satisfying the condition y

where, since µ (y) is an endpoint of a gap of σ(H ), τ ∈ {0, π}.

(4.35)

Recall that T (y) is a period of the potential Vy (x), expressed by the formula (4.7), in which T0 is a period of the potential V˜ (x). Assume that the Bloch function ψy (x) satisﬁes also the normalization condition (4.10). Like above, consider a sequence of intervals In = [yn , yn +∆n ] satisfying the condition (4.11), denote ψn (x) = ψyn (x), and deﬁne the sequence of functions un ∈ DH by the formula (4.13), where the functions θn (x) are deﬁned by (4.14) and (4.15). A dependence of ∆n on yn will be speciﬁed in the sequel. Taking into account (4.17), (4.32), (4.2), (3.1), (3.2) and the fact that ψn (x) satisﬁes the equation (4.33) with y = yn , we obtain: ∃ N > 0, ∀ n ≥ N : ! " ˜ n ≤ (µ − λ) ˜ − µ β(yn )(1 − R ˜ n ) un , Hun − λu

(4.36)

where

ωn ˜ n = 2ψn θn + ψn θn + R (4.37) µ β(yn )un µ β(yn ) and ωn is deﬁned by (4.20). Observe that, in view of (4.34) and (4.35), each function |ψn (x)| is Tn -periodic (Tn = T (yn )). Then, making use of (4.22), (4.14) and the normalization condition (4.10), we obtain the estimate (4.25) with some constant c2 > 0. In the similar manner, as the estimate (4.24), we obtain the estimate

ψn θn ≤ c6

(4.38)

with some constant c6 > 0, making use of (4.23). Furthermore, in view of (4.14), we have: # yn +1 ψn θn 2 ≤ max max |κ (x)|2 , max |χ (x)|2 |ψn |2 dx x∈[0,1] x∈[0,1] yn yn +∆n

+ yn +∆n −1

|ψn |2 dx .

(4.39)

Taking into account the equality H yn − µ(yn )ψn = 0 and the uniform boundedness on R of the sequence of functions {Vyn (x) − µ(yn )}∞ n=1 , we obtain by Lemma 4.2 (with u = ψn and H = H yn − µ(yn )I) that the estimate is valid yn +∆n yn +∆n +1 yn +1 yn +2 |ψn |2 dx+ |ψn |2 dx ≤ c7 |ψn |2 dx + |ψn |2 dx yn

yn +∆n −1

yn −1

yn +∆n −2

with some constant c7 > 0. From this estimate, estimate (4.39), the normalization condition (4.10) and property (4.23), we obtain the estimate ψn θn ≤ c8 with

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some constant c8 > 0. Thus, from the latter estimate, estimates (4.38), (4.25) and deﬁnition (4.37), we obtain the estimate: ∃ c9 > 0, ∃ N > 0, ∀ n ≥ N : 1 (1 + yn + ∆n )|β (yn )|∆n ˜ √ |Rn | ≤ c9 + . β(yn ) β(yn ) ∆n

(4.40)

Let us choose the sequence ∆n in the following manner: 1 ∆n = 2 + 2 . (|β (yn )|yn ) 3 Like above, we obtain that limn→∞ ∆n = ∞. Without loss of generality we can also assume that the condition (4.12) is satisﬁed. After this choice of ∆n the estimate (4.40) takes the form: ∃ c10 > 0, ∃ N > 0, ∀ n ≥ N : 1 1 (|β (yn )) 3 1 (|β (yn )|yn ) 3 ˜ + |Rn | ≤ c10 2 + . 4 4 1 β(yn ) β(yn )yn3 β(yn ) (|β (yn )) 3 yn3 ˜ n = 0. Then, by In view of (4.3) and (4.4), this estimate implies that limn→∞ R (4.36), the condition (4.31) is satisﬁed for the sequence of functions {un }n≥N , if N > 0 is large enough. Furthermore, the condition (4.5) is satisﬁed for these functions. So, we have proved assertion (ii). The theorem is proven. Example 3. Let us deﬁne the function β(x), taking part in Theorem 4.3, in the following manner: (ln 2)ν β(x) = ν, 2 (ln(2 + |x|)) where ν > 0. As is easy to check, the function α(x) = 1 − β(x) satisﬁes the conditions (3.1), (3.2) and condition (a) of Lemma 3.2, that is it satisﬁes all the conditions of Theorem 3.3. Let us clear up the following question: for which values of ν the function β(x) satisﬁes the conditions of Theorem 4.3? We have: 2lν 1 lν−1 sign(x) (l = 1, 2) = (ln(2 + |x|)) (β(x))l (ln 2)ν 2 + |x| We see from this equality that condition (4.1) is satisﬁed for ν ∈ (0, 1) and it is not satisﬁed for ν ≥ 1, and condition (4.3) is satisﬁed for ν ∈ 0, 12 and it is not satisﬁed for ν ≥ 12 . This means that these conditions impose restrictions on the rate of convergence of β(x) to zero as |x| → ∞. Let us check the condition (4.4). We have: 1 4 1 4 (ln 2)ν ν 3 |x| 3 . β(x) |β (x)| 3 |x| 3 = 4 1 ν+ 1 2 (ln(2 + |x|)) 3 3 (2 + |x|) 3 Hence condition (4.4) is satisﬁed for any ν > 0. So, condition (4.1) of assertion (i) of Theorem 4.3 is satisﬁed if and only if ν ∈ (0, 1), and both of conditions (4.3) and (4.4) of assertion (ii) of this theorem are satisﬁed if and only if ν ∈ 0, 12 .

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1 Observe that the power function β(x) = 2(1+|x|) σ (σ > 0) does not satisfy conditions (4.1) and (4.3). The above examples show that these conditions impose on the function β(x) a very slow (less than logarithmic) rate of convergence to zero as |x| → ∞.

The following conjecture appears in connection with Theorem 4.3: ˜ of the operator H. ˜ Conjecture. Assume that (λ , µ ) is a gap of the spectrum σ(H) We conjecture that under some conditions (stronger than conditions of Theorem 4.3) the eigenvalues of the operator H, lying in (λ , µ ), do not accumulate to µ , if µ < 0, and they do not accumulate to λ , if λ > 0. We think that the cases µ = 0 or λ = 0 are indeﬁnite: the eigenvalues can accumulate or not accumulate to such endpoints.

References [Gl]

I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Diﬀerential Operators. I.P.S.T., Jerusalem, 1965.

[H-S]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory with Applications to Schr¨ odinger Operator. Applied math. sciences, Springer-Verlag New York Inc., v. 113, 1995

[Kuch]

P. Kuchment, Floquet Theory for Partial Diﬀerential Equations. Birkh¨ auser, Basel, 1993.

[Wil]

C. H. Wilcox, Theory of Bloch waves. J. Anal. Math. 33 (1978), 146–167.

[Z1]

L. Zelenko, Construction of the essential spectrum for a multidimensional non-self-adjoint Schr¨ odinger operator via the spectra of operators with periodic potentials, Part I, Integral Equations and Operator Theory, 46 (2003), 11–68.

[Z2]

L. Zelenko, Construction of the essential spectrum for a multidimensional non-self-adjoint Schr¨ odinger operator via the spectra of operators with periodic potentials, Part II, Integral Equations and Operator Theory, 46 (2003), 69– 124.

[Z3]

L. Zelenko, On a generic topological structure of the spectrum to onedimensional Schr¨ odinger operators with complex limit-periodic potentials, Integral Equations and Operator Theory, 50 (2004), 393–430.

[Zel-Kuch] L. B. Zelenko and P. A. Kuchment, On the Floquet representation of exponentially increasing solutions of elliptic equations with periodic coeﬃcients. Soviet Math. Dokl., 19 (1978), No 2, 506–507. [Z4]

L. B. Zelenko, Spectrum of Schr¨ odinger’s equation with a complex pseudoperiodic potential, I. Diﬀerential Equations, 12 (1976), 563–569.

[Z5]

L. B. Zelenko, Spectrum of Schr¨ odinger’s equation with a complex pseudoperiodic potential, II. Diﬀerential Equations, 12 (1976), 999–1006.

[Z6]

L. B. Zelenko, Asymptotic distribution of eigenvalues in a lacuna of continuous spectrum of the perturbed Hill operator. Mathematical Notes, 20 (1976), 750– 755.

Vol. 58 (2007) [Z7]

[Zel-Rof]

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L. B. Zelenko, The limit spectrum of a non-self-adjoint second order diﬀerential operator with slowly varying coeﬃcients. Mathematical Notes, 13 (1973), 80–86. L. B. Zelenko and F. S. Rofe-Beketov, The limit spectrum of systems of ﬁrst order diﬀerential equations with slowly varying coeﬃcients. Diﬀerential Equations, 7 (1971), No 11, 1498–1505.

Leonid Zelenko Department of Mathematics University of Haifa Haifa, 31905 Israel e-mail: [email protected] Submitted: December 1, 2006 Revised: May 25, 2007

Integr. equ. oper. theory 58 (2007), 591–596 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040591-6, published online April 14, 2007 DOI 10.1007/s00020-007-1490-4

Integral Equations and Operator Theory

Powers of Hypercyclic Functions for Some Classical Hypercyclic Operators R. M. Aron, J. A. Conejero, A. Peris and J. B. Seoane–Sep´ ulveda Abstract. We show that no power of any entire function is hypercyclic for Birkhoﬀ’s translation operator on H(C). On the other hand, we see that the set of functions whose powers are all hypercyclic for MacLane’s diﬀerentiation operator is a Gδ -dense subset of H(C). Mathematics Subject Classiﬁcation (2000). Primary 47A16; Secondary 30D15. Keywords. Hypercyclic vectors, universal functions.

1. Introduction and preliminaries Let X be a separable, inﬁnite-dimensional F -space. A linear and continuous operator T deﬁned on X is said to be hypercyclic if there exists x ∈ X such that its orbit under T , {T n x : n ∈ N}, is dense in X. The ﬁrst two examples of hypercyclic operators were given in the space H(C) of entire functions endowed with the compact-open topology. In 1929, Birkhoﬀ saw that the translation operator is hypercyclic on H(C) [3]. Later, MacLane proved that the derivative operator is also hypercyclic [16]. In both cases, the authors provide the construction of a hypercyclic function. A revised proof of both results can be found in [1] (see also [10].) For further information about the construction and properties of these hypercyclic entire functions see [17, 4, 5, 9, 12, 7, 15, 2, 6] The Baire Category Theorem provides a Gδ -dense set of hypercyclic vectors for both of them, see [13, 14] for a exhaustive survey of results concerning hypercyclicity of operators. Besides, they also share a common dense manifold of hypercyclic vectors (see the proof of Theorem 5.1 in [11]). However, nothing more is known concerning the structure of the set of hypercyclic vectors for these operators. The second author was supported in part by GVA, grant CTESPP/2005. The second and third authors were supported in part by MEC and FEDER, Project MTM200402262 and Research Net MTM2006-26627-E.

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Our purpose in this note is to study the behaviour of the powers of the hypercyclic vectors of both the Birkhoﬀ translation and the MacLane diﬀerentiation operators. Despite the fact that these operators share many properties, we will see that the results obtained for powers are completely diﬀerent.

2. Birkhoﬀ ’s operator In [3], Birkhoﬀ constructed a universal entire function, f , for the operator τ1

: H(C) f (z)

−→ →

H(C) f (z + 1)

One could wonder what kind of structure the set HC(τ1 ) = {f ∈ H(C) : f is hypercyclic for τ1 } has. As we have mentioned this is a Gδ -dense set. Here we see that, if we denote Bk := {f ∈ H(C) : f k ∈ HC(τ1 )}, k ∈ N, then Bk = ∅ for k > 1, in particular if f ∈ HC(τ1 ), then no power of f can be hypercyclic for τ1 . In order to do this, we will use a well known result by Hurwitz related to zeros of limits of entire functions (see, e.g. [8, p. 152]). Our main theorem in this section characterizes the closure of orbits of the powers of the hypercyclic functions for Birkhoﬀ’s translation operator: Theorem 2.1. Let 1 < p ∈ N, f ∈ HC(τ1 ), and g ∈ H(C). If the order of each zero of g is a multiple of p, then g ∈ Orb(τ1 , f p ). Proof. Suppose that the order of each zero of g is a multiple of p. Let us call (an )n the sequence of non-zero zeros of g. Each an has multiplicity pmn with mn ∈ N, n ∈ N, and 0 has multiplicity pm, for some m ∈ N ∪ {0}. By Weierstrass’s theorem [8, Ch. VII, Th. 5.13], there is a sequence (pn )n of integers, and an entire function ϕ, such that g(z) = z pm eϕ(z)

∞ i=1

i Eppm (z/ai ), i

where E0 (z) := 1 − z, and Eq (z) := (1 − z) exp z + z 2 /2 + . . . + z q /q , for q ≥ 1. Also, by Weierstrass’s theorem, the sequence (pn )n can be chosen in order to have ∞ that i=1 Epmi i (z/ai ) is also an entire function. Let us deﬁne m ϕ(z)/p

g˜(z) = z e

∞ i=1

Epmi i

z ai

.

Next, since f ∈ HC(τ1 ), for any compact set K ⊂ C, there is a sequence (nj )j ∈ N with f (z + nj ) − g˜(z)K → 0,

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as j → ∞. It follows that f p (z + nj ) − g(z)K ≤ R(K, p) · f (z + nj ) − g˜(z)K → 0, as j → ∞, where R(K, p) > 0 is a constant that only depends on K and p, and g ∈ Orb(τ1 , f p ). Theorem 2.2. Let 1 < p ∈ N, f, g ∈ H(C). If g ∈ Orb(τ1 , f p ), then the order of each zero of g is a multiple of p. Proof. Suppose that we consider a zero z0 of g. Take a closed disk D centered at z0 with no other zeros of g. By hypothesis there is a sequence (nj )j ⊂ N verifying f p (z + nj ) → g(z) as j → ∞, uniformly on D. By Hurwitz’s theorem, there is some n ∈ N such that the total number of zeros (counting multiplicity) of f p (z + n) and g(z) in D coincide. Therefore the order of z0 is a multiple of p. From the previous theorems we have the following corollaries: Corollary 2.3. Let p, q ∈ N, p > 1, and f ∈ HC(τ1 ). Then z q ∈ Orb(τ1 , f p ) ⇐⇒

q ∈ N. p

Corollary 2.4. The set Bk := {f ∈ H(C) : f k ∈ HC(τ1 )} = ∅ for every k > 1. Clearly, all the previous results also hold for any general Birkhoﬀ operator, namely τt (f )(z) = f (z + t), with t ∈ C \ {0}.

3. MacLane’s operator In [16], MacLane constructed a universal entire function for the diﬀerentiation operator D : H(C) −→ H(C) f (z) → f (z) on H(C). Now we consider the following set, for k ∈ N, Mk := {f ∈ H(C) : f k ∈ HC(D)}. As we did in the previous section, one could ask what kind of structure the sets Mk have. In contrast to the results for τt , we have the following: Theorem 3.1. For every k ∈ N, Mk is a Gδ -dense set.

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Proof. Fix k ∈ N and let (Un )n be a countable basis of open sets in H(C). For every n ∈ N, we deﬁne the set Gn,k := {f ∈ H(C) : there exists j ∈ N such that Dj f k ∈ Un }. Fix n ∈ N. Clearly, Gn,k is open and non-void. We will show that it is also m i dense. Consider ε > 0, an arbitrary polynomial p(z) = i=0 ai z and q(z) = m i i=0 bi z ∈ Un . Without loss of generality, bi = 0 for 0 ≤ i ≤ m. For any ε > 0 and any compact set K ⊂ C we will prove that there exist f (z) ∈ H(C) and j ∈ N f k (z) = q(z), and we will be done. such that for ||f (z) − p(z)||K < ε and Dj m Let f (z) := p(z) + r(z) with r(z) := i=0 ci z i+n , for some n > m and the ci will be determined in order to obtain Dj f k (z) = q(z). Let j := (k − 1)m + kn. The derivative Dj f k (z) = Dj rk (z) is a polynomial of degree m, where the coeﬃcient of z m−l , 0 ≤ l ≤ m, is  (km + kn − l)!  (m − l)!

(sm ,...,sm−l )∈Al

k sm sm−1 . . . sm−l

 sm−l  sm−1 , (3.1) . . . cm−l csmm cm−1

with

Al :=

(sm , . . . , sm−l ) ∈

Nl+1 0

m

:

i=m−l

si = k,

m

isi = km − l ,

(3.2)

i=m−l

since the powers of z accompanying the ci have to add up to km − l, and k sm sm−1 . . . sm−l is the corresponding multinomial coeﬃcient. If we identify the coeﬃcients of q(z) with the coeﬃcients of Dj f k (z), we have a non-linear triangular system that can be easily solved if we begin with the coeﬃcients of higher degree and go down. Besides, we have to show that if n is big enough, then the coeﬃcients ci are small enough in order to have that f (z) is as close to p(z) as we want. Comparing the m-th coeﬃcients gives that 1/k m!bm cm = . (3.3) (km + kn)! After identifying the (m − 1)-st coeﬃcients of q(z) and Dj f k (z),we have n bm−1 1+ . (3.4) bm m Each of the coeﬃcients, ci , depends on n, i.e. each ci can be seen as a sequence k = o(1/ (km + kn)!) and in n. Thus, from (3.3) and (3.4), we can say that c m cm−1 = o(n/ k (km + kn)!). To conclude analogous statements for the rest of the cm−1 = cm

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coeﬃcients we proceed as follows. Take 2 ≤ l ≤ m, and suppose that cm−i = i k o(n / (km + kn)!) for 0 ≤ i ≤ l − 1, so that (km + kn − l)! sm sm−1 sm−l+1 cm cm−1 . . . cm−l+1 = o(1), (m − l)! for any choice (sm , sm−1 , . . . , sm−l+1 , 0) ∈ Al (see (3.2)). Thus, (km + kn − l)! k−1 cm cm−l = o(1), (3.5) (m − l)! and then we obtain that cm−l = o(nl / k (km + kn)!). We should observe that, to solve the system, we need that all the ci in the previous process have to be non-zero. If it was not the case, a suitable modiﬁcation of q(z) solves the problem. Finally, the set ∩∞ n=1 Gn,k is a second category set in H(C), which coincides with Mk , and this concludes the proof of the theorem. As we have previously seen, the sets Mk (k ≥ 1) are Gδ -dense sets. From this fact it follows that ∩∞ k=1 Mk is a Gδ -dense set as well. To summarize, we can give the following result: Theorem 3.2. There exists f ∈ H(C) such that f k ∈ HC(D) for every k ∈ N. Moreover, this behaviour is generic, i.e. the following set is residual {f ∈ H(C) : f k ∈ HC(D) for every k ∈ N}. It is also interesting to notice that B1 ∩ ∩∞ j=1 Mk is a Gδ -dense set as well. Acknowledgment We would like to thank Luis Bernal, Antonio Bonilla, and K. Grosse-Erdmann for pointing out to us a gap in the proof of Theorem 2.1 and for several discussions and helpful comments. We also want to thank the referee for helpful comments and remarks. The second author acknowledges the hospitality he received from the Department of Mathematical Sciences at Kent State University during NovemberDecember, 2005, while this paper was being written.

References [1] R. Aron and D. Markose, On universal functions. J. Korean Math. Soc. 41 (2004), 65–76. [2] L. Bernal-Gonz´ alez and A. Bonilla, Exponential type of hypercyclic entire functions. Archiv Math. 78 (2002), 283–290. [3] G.D. Birkhoﬀ, D´emonstration d’un th´eor`eme ´el´ementaire sur les fonctions enti` eres. C. R. Acad. Sci. Paris 189 (1929), 473–475. [4] C. Blair and L.A. Rubel, A universal entire function. Amer. Math. Monthly 90 (1983), 331–332.

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[5] C. Blair and L.A. Rubel, A triply universal entire function. Enseign. Math. 30 (1984), 269–274. [6] A. Bonilla and K.G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro. Int. Equat. Oper. Theory 56 (2006), 151-162. [7] K.C. Chan and J.H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40 (1991), 1421–1449. [8] J.B. Conway, Functions of One Complex Variable. Springer-Verlag, Berlin/New York, 1978. [9] S.M. Duyos Ruiz, Universal functions and the structure of the space of entire functions. Dokl. Akad. Nauk. SSSR 279 (1984), 792–795. [10] G. Fern´ andez and A.A. Hallack, Remarks on a result about hypercyclic nonconvolution operators. J. Math. Anal. Appl. 309 (2005), 52–55. [11] G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98 (1991), 229–269. [12] K.G. Grosse-Erdmann, On the universal functions of G.R. MacLane. Complex Variables Theory Appl. 15 (1990), 193–196. [13] K.G. Grosse-Erdmann, Universal families and hypercyclic operators. Bull. Amer. Math. Soc. 36 (1999), 345–381. [14] K.G. Grosse-Erdmann, Recent developments in hypercyclicity. Rev. R. Acad. Cien. Serie A Mat. 97 (2003), 273–286. [15] W. Luh, V.A. Martirosian, and J M¨ uller, Universal entire functions with gap power series. Indag. Math. (N.S.) 9 (1998), 529–536. [16] G.R. MacLane, Sequences of derivatives and normal families. J. Analyse Math. (1952), 72–87. [17] W. Seidel and J.L. Walsh, On approximation by euclidean and non-euclidean translations of an analytic function. Bull. Amer. Math. Soc. 47 (1941), 916–920. R. M. Aron Department of Mathematical Sciences, Kent State University, Kent, OH44242, USA e-mail: [email protected] J. A. Conejero Departament de Matem` atica Aplicada and IMPA-UPV, F. Inform` atica Universitat Polit`ecnica de Val`encia, E-46022 Val`encia, Spain e-mail: [email protected] A. Peris Departament de Matem` atica Aplicada and IMPA-UPV, E.T.S. Arquitectura Universitat Polit`ecnica de Val`encia, E-46022 Val`encia, Spain e-mail: [email protected] J. B. Seoane–Sep´ ulveda Facultad de Ciencias Matem´ aticas, Departamento de An´ alisis Matem´ atico Universidad Complutense de Madrid, Plaza de las Ciencias 3, E-28040 Madrid, Spain e-mail: [email protected] Submitted: April 19, 2006 Revised: December 7, 2006

Integr. equ. oper. theory 58 (2007), 597–601 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040597-5, published online April 14, 2007 DOI 10.1007/s00020-007-1495-z

Integral Equations and Operator Theory

Stability Theorems for Linear Combinations of Idempotents J. J. Koliha and V. Rakoˇcevi´c Abstract. We prove a stability theorem for the nullity of a linear combination c1 P1 + c2 P2 of two idempotent operators P1 , P2 on a Banach space provided c1 , c2 and c1 +c2 are nonzero. We then show that for c1 P1 +c2 P2 the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of c1 , c2 , and that the nullity, defect and index of c1 P1 + c2 P2 are stable. Mathematics Subject Classification (2000). 47A53, 47B99, 46H99, 15A99. Keywords. Linear combinations of idempotents, nullity, stability theorems.

1. Introduction and preliminaries In [5] we studied the nonsingularity of the diﬀerence and sum of two idempotent matrices. Baksalary and Baksalary [1] then proved that, for idempotent matrices P1 , P2 , the nonsingularity of P1 + P2 is equivalent to the nonsingularity of any linear combination c1 P1 + c2 P2 , where c1 , c2 = 0 and c1 + c2 = 0. Recently, Du et al. [3] gave a rather complicated proof of this result for two idempotent operators on a Hilbert space. In [7] we extended the Baksalary and Baksalary result [1] by proving the stability of the nullity and rank of c1 P1 + c2 P2 under the choice of c1 and c2 , and posed the following question motivated by results of [1] and [6]: If P1 , P2 are idempotent operators in a Hilbert space, is it true that P1 + P2 is Fredholm if and only if any linear combinations c1 P1 + c2 P2 is Fredholm, where c1 , c2 ∈ C \ {0} and c1 + c2 = 0? In this note we give an aﬃrmative answer to this problem extended to Banach space operators using simple arguments based on the stability of the nullity of linear combinations of two idempotent operators. The main result of [3] then follows as a special case. Let X be an inﬁnite-dimensional complex Banach space and let B(X) be the set of all bounded linear operators on X. An operator P ∈ B(X) is idempotent

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if P 2 = P . Throughout this paper N (T ) and R(T ) will denote the nullspace and the range of T ∈ B(X), respectively. Set α(T ) = dim N (T ), the nullity of T , and β(T ) = dim X/R(T ), the defect of T . An operator T ∈ B(X) is semi-Fredholm if R(T ) is closed and at least one of α(T ) and β(T ) is ﬁnite. For such an operator we deﬁne the index of T by i(T ) = α(T ) − β(T ). Let Φ+ (X) (Φ− (X)) denote the set of upper (lower) semi-Fredholm operators, that is, the set of all semi-Fredholm operators with α(T ) < ∞ (β(T ) < ∞). An operator T ∈ B(X) is Fredholm if T ∈ Φ(X) := Φ+ (X) ∩ Φ− (X). If T ∈ B(X), we write T ∈ B(X ) for the adjoint of T . Recall that R(T ) is closed if and only if R(T ) is closed, and that in this case α(T ) = β(T ) and β(T ) = α(T ). Furthermore, T ∈ Φ+ (X) if and only if T ∈ Φ− (X ), and T ∈ Φ− (X) if and only if T ∈ Φ+ (X ) (see [4]). Recall that Sadovskii [8] and later (independently) Buoni, Harte and Wickstead [2] introduced the following useful functorial construction known as the essential enlargement of a Banach space. For any Banach space X we set ∞ (X) = {x = (xn ) : xn ∈ X,

sup xn < ∞}. n

∞

Clearly, (X) is a Banach space equipped with the supremum norm, and m(X) = {x = (xn ) ∈ ∞ (X) : {xn : n ∈ N} is totally bounded in X} is a closed subspace of ∞ (X). Hence the quotient space = ∞ (X)/m(X) X deﬁned by is a Banach space. Any T ∈ B(X) determines an operator T ∈ B(X) T((xn ) + m(X)) = (T xn ) + m(X),

(xn ) ∈ ∞ (X).

(1.1)

The mapping T → T is a continuous algebra homomorphism of B(X) to B(X). The following result whose proof can be found in [2], [4] or [8], will play a crucial role in the proof of our main Theorem 3.1. Theorem 1.1. If T ∈ B(X), then T is upper semi-Fredholm if and only if T is injective. Let A be a Banach algebra. For any a ∈ A we deﬁne the left regular representation of a by for all x ∈ A. La (x) = ax, Then La ∈ B(A), the mapping a → La is an algebra monomorphism of A to B(A) with La = a. We will need the following known fact whose proof we include for completeness. Lemma 1.2. Let a ∈ A. Then a is invertible in A if and only if La is invertible in B(A).

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Linear Combinations of Idempotents

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Proof. Let a be invertible with the inverse b ∈ A. Then Lb is the inverse of La . −1 Conversely, if La is invertible and b = L−1 a (1), then ab = La La (1) = 1, and −1 La Lb = Lab = L1 = I, that is, Lb = La . Thus a is invertible in A with the inverse b.

2. The nullity of c1 P1 + c2 P2 We start our observations with the following result which for matrices was proved in [5, Theorem 2.2]. For convenience, we deﬁne a subset Γ of C2 by Γ = {(c1 , c2 ) ∈ C : c1 = 0, c2 = 0, c1 + c2 = 0}. Theorem 2.1. Let P1 , P2 be two idempotents in B(X) and let (c1 , c2 ) ∈ Γ. Then dim [N (c1 P1 + c2 P2 )] = dim [N ((I − P1 )P2 ) ∩ N (P1 )].

(2.1)

Proof. First we prove that (2.1) holds with ≤ in place of equality. For this suppose that x ∈ N (c1 P1 + c2 P2 ). Then [(I − P1 )P2 ](I − P1 )x = c−1 1 [(I − P1 )P2 ](c1 I − c1 P1 − c2 P2 + c2 P2 )x = c−1 1 [(I − P1 )P2 ](c1 I + c2 P2 )x = (c1 + c2 )(c1 c2 )−1 (I − P1 )(c1 P1 + c2 P2 )x = 0. Thus, since (I − P1 )x ∈ N (P1 ), we conclude that (I − P1 )N (c1 P1 + c2 P2 ) ⊂ N ((I − P1 )P2 ) ∩ N (P1 ).

(2.2)

Suppose that that x ∈ N (c1 P1 + c2 P2 ) and (I − P1 )x = 0. Then x = P1 x and (c1 +c2 )P2 x = P2 (c1 P1 x+c2 P2 x) = 0. So P2 x = 0, and x = P1 x = −c−1 1 c2 P2 x = 0. Thus I − P1 embeds N (c1 P1 + c2 P2 ) injectively into N ((I − P1 )P2 ) ∩ N (P1 ), and the inequality ≤ in (2.1) is proved. To complete the proof of the theorem we prove the reverse inequality in (2.1). Towards this end we set c = c1 c−1 2 and prove that ((1 + c)I − P2 )[N ((I − P1 )P2 ) ∩ N (P1 )] ⊂ N (c1 P1 + c2 P2 ).

(2.3)

Suppose that x ∈ N ((I − P1 )P2 ) ∩ N (P1 ). Then P1 x = 0 and P2 x = P1 P2 x. Thus (c1 P1 + c2 P2 ) (1 + c)I − P2 x = (c1 (1 + c)P1 − c1 P1 P2 + (c1 + c2 )P2 − c2 P2 )x = −c1 P1 P2 x + (c1 + c2 )P2 x − c2 P2 x = 0, and we obtain (2.3). Let x ∈ N ((I − P1 )P2 ) ∩ N (P1 ) and ((1 + c)I − P2 )x = 0. Then P1 x = 0, P2 x = P1 P2 x = (1 + c)x. Thus (1 + c)x = P2 x = P1 (1 + c)x = 0, that is, x = 0. Hence (1+c)I −P2 embeds N ((I −P1 )P2 )∩N (P1 ) injectively into N (c1 P1 +c2 P2 ), and (2.1) holds with ≥ in place of equality. This completes the proof.

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Corollary 2.2. For any two idempotents P1 , P2 ∈ B(X) the nullity of c1 P1 + c2 P2 is constant on Γ, that is, α(c1 P1 + c2 P2 ) = α(P1 + P2 )

for all (c1 , c2 ) ∈ Γ.

3. Fredholm properties of c1 P1 + c2 P2 The main result of this note is the following stability theorem. Theorem 3.1. Let P1 , P2 ∈ B(X) be idempotents. Then: (i) If c1 P1 +c2 P2 is upper semi-Fredholm for some (c1 , c2 ) ∈ Γ, then it is upper semi-Fredholm for all (c1 , c2 ) ∈ Γ, and α(c1 P1 + c2 P2 ) is constant on Γ. (ii) If c1 P1 + c2 P2 is lower semi-Fredholm for some (c1 , c2 ) ∈ Γ, then it is lower semi-Fredholm for all (c1 , c2 ) ∈ Γ, and β(c1 P1 + c2 P2 ) is constant on Γ. (iii) If c1 P1 + c2 P2 is Fredholm for some (c1 , c2 ) ∈ Γ, then it is Fredholm for all (c1 , c2 ) ∈ Γ, and α(c1 P1 + c2 P2 ), β(c1 P1 + c2 P2 ) and i(c1 P1 + c2 P2 ) are constant on Γ. Proof. (i) Let c1 P1 + c2 P2 ∈ Φ+ (X) for some (c1 , c2 ) ∈ Γ, and let (λ1 , λ2 ) ∈ Γ. Under the algebra homomorphism T → T deﬁned by (1.1), (c1 P1 +c2 P2 ) = c1 P1 + By Theorem 1.1, c2 P2 , and the operators P1 and P2 are idempotents in B(X). N (λ1 P1 + N (c1 P1 + c2 P2 ) = {0}, and then by Corollary 2.2 (in the space X), λ2 P2 ) = {0}. Thus λ1 P1 + λ2 P2 is upper semi-Fredholm by Theorem 1.1. Finally, by Theorem 2.1, we have α(c1 P1 + c2 P2 ) = α(λ1 P1 + λ2 P2 ), and (i) is proved. (ii) Let c1 P1 + c2 P2 ∈ Φ− (X) for some (c1 , c2 ) ∈ Γ. This implies that c1 P1 + c2 P2 ∈ Φ+ (X) and β(c1 P1 + c2 P2 ) = α(c1 P1 + c2 P2 ). Further, P1 and P2 are idempotents in B(X ). Thus (ii) follows from (i). (iii) This follows from (i) and (ii). As a corollary to Theorem 3.1 we obtain the following result. Corollary 3.2. Let P1 and P2 be two idempotents in B(X). Then the invertibility of c1 P1 + c2 P2 is independent of the choice of (c1 , c2 ) ∈ Γ. Proof. Let c1 P1 + c2 P2 be invertible for some choice of (c1 , c2 ) ∈ Γ. Then c1 P1 + c2 P2 is Fredholm with the nullity and defect equal to zero. By Theorem 3.1 (iii), λ1 P1 + λ2 P2 is invertible for any choice of (λ1 , λ2 ) ∈ Γ. Remark 3.3. Corollary 3.2 was recently proved for Hilbert space operators as the main result in [3, Theorem 1] by Du et al. In constrast with our arguments, their proof is applicable only in a Hilbert space, and is rather long and complicated. Our ﬁnal application is to idempotent elements in a Banach algebra. Corollary 3.4. Let p1 , p2 be two idempotents in a Banach algebra A. Then the invertibility of c1 p1 + c2 p2 is independent of the choice of (c1 , c2 ) ∈ Γ.

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Proof. Suppose that the element c1 p1 +c2 p2 is invertible for some pair (c1 , c2 ) ∈ Γ. According to Lemma 1.2, the operator c1 Lp1 + c2 Lp2 is invertible in B(A) with Lp1 and Lp2 idempotent. By Corollary 3.2, λ1 Lp1 + λ2 Lp2 is invertible for any choice of (λ1 , λ2 ) ∈ Γ. Then by Lemma 1.2, λ1 p1 + λ2 p2 is invertible in A.

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. J. Buoni, R. Harte and T. Wickstead, Upper and lower Fredholm spectra, Proc. Amer. Math. Soc. 66 (1977), 309–314. [3] H. Du, X. Yao and C. Deng, Invertibility of linear combinations of two idempotents, Proc. Amer. Math. Soc. 134 (2006), 1451–1457. [4] R. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York and Basel, 1988. [5] J. J. Koliha, V. Rakoˇcevi´c and I. Straˇskraba, The diﬀerence and sum of projectors, Linear Algebra Appl. 388 (2004), 279–288. [6] J. J. Koliha and V. Rakoˇcevi´c, Fredholm properties of the diﬀerence of orthogonal projections in a Hilbert space, Integral Equations Operator Theory 52 (2005), 125– 134. [7] J. J. Koliha and V. Rakoˇcevi´c, The nullity and rank of linear combinations of idempotent matrices, Linear Algebra Appl., in press. [8] B. N. Sadovskii, Limit-compact and condensing operators, Uspekhi Mat. Nauk. 27 (1972), 81–146 (in Russian). J. J. Koliha Department of Mathematics University of Melbourne Melbourne VIC 3010 Australia e-mail: [email protected] V. Rakoˇcevi´c Faculty of Science and Mathematics University of Niˇs 18000 Niˇs Serbia and Montenegro e-mail: [email protected] Submitted: April 30, 2006 Revised: May 21, 2006