Schatten-class Perturbations of Toeplitz Operators

Transcription

1 Schatten-class Perturbations of Toeplitz Operators Dominik Schillo Saarbrücken, Schatten-class Perturbations of Toeplitz Operators 1

2 Definition Let T C be the unit circle with the canonical probability measure m. The Hardy space with respect to m will be denoted by { } H 2 (m) = f L 2 (m) ; ˆf (n) = 0 for all n < 0 L 2 (m). Let f L (m). We call T f : H 2 (m) H 2 (m), g P H 2 (m)(fg), the Toeplitz operator with symbol f. Theorem (Brown-Halmos, 1964) For X B(H 2 (m)), the following are equivalent: 1 there exists f L (m) such that X = T f, 2 T z XT z X = 0, where z L (m) is the identity map. Schatten-class Perturbations of Toeplitz Operators 2

3 Definition We define H (m) = L (m) H 2 (m) L (m) and we call I m = {f H (m) ; f = 1 m-a.e.} the set of inner functions with respect to m. Schatten-class Perturbations of Toeplitz Operators 3

4 Theorem (Brown-Halmos, 1964) For X B(H 2 (m)), the following are equivalent: 1 there exists f L (m) such that X = T f, 2 Tu XT u X {0} for all u I m. Schatten-class Perturbations of Toeplitz Operators 4

5 Theorem (Gu, 2004) For X B(H 2 (m)), the following are equivalent: 1 there exist f L (m) and F F(H 2 (m)) such that X = T f + F, 2 T u XT u X F(H 2 (m)) for all u I m. Schatten-class Perturbations of Toeplitz Operators 5

6 Theorem (Xia, 2009) For X B(H 2 (m)), the following are equivalent: 1 there exist f L (m) and K K(H 2 (m)) such that X = T f + K, 2 T u XT u X K(H 2 (m)) for all u I m. Schatten-class Perturbations of Toeplitz Operators 6

7 Definition Let p [1, ), and let H be a Hilbert space. An operator S B(H) is a Schatten-p-class operator if Denote by S p (H) = S p p = tr( S p ) <. { } S B(H) ; S p < the set of all Schatten-p-class operators. Remark We have for all 1 p q <. F(H) S p (H) S q (H) K(H) B(H) Schatten-class Perturbations of Toeplitz Operators 7

8 Let T C be the unit circle with the canonical probability measure m M + 1 (T). Theorem (Didas-Eschmeier-S., 2017) Let p [1, ). For X B(H 2 (m)), the following are equivalent: 1 there exist f L (m) and S S p (H 2 (m)) such that X = T f + S, 2 T u XT u X S p (H 2 (m)) for all u I m. Schatten-class Perturbations of Toeplitz Operators 8

9 Definition Let D C be the unit disc, and let O(D) be the set of all scalar-valued analytic functions on D. We denote by 1 A(D) = { f C(D) ; f D O(D) } the disc algebra, 2 A(D) the Shilov boundary of A(D) (i.e., the smallest closed subset of D such that for all f A(D)). sup f (z) = sup f (z) z D z A(D) Proposition We have A(D) = D = T. Schatten-class Perturbations of Toeplitz Operators 9

10 Definition Let D C d be a bounded domain, and let O(D) be the set of all scalar-valued analytic functions on D. We denote by 1 A(D) = { f C(D) ; f D O(D) } C(D) the domain algebra of D, 2 A(D) the Shilov boundary of A(D) (i.e., the smallest closed subset of D such that for all f A(D)). sup f (z) = sup f (z) z D z A(D) Schatten-class Perturbations of Toeplitz Operators 10

11 Let D C d be a bounded strictly pseudoconvex or a bounded symmetric and circled domain. We denote by µ M + 1 ( A(D)) the canonical probability measure on A(D). Example 1 D = B d : A(Bd ) = B d = S d and µ = σ. 2 D = D d : A(D d ) = T d and µ = d m. Schatten-class Perturbations of Toeplitz Operators 11

12 We have H 2 τ L 2 (m) = A(D) (m) T and H τ w (m) = A(D) T. Definition We define H 2 τ L 2 (µ) = A(D) (µ) A(D) L 2 (µ) and H τ w (µ) = A(D) A(D) L (µ). Furthermore, we denote by I µ = {f H (µ) ; f = 1 µ-a.e.} the set of inner functions with respect to µ. Schatten-class Perturbations of Toeplitz Operators 12

13 Definition Let f L (µ). We call T f : H 2 (µ) H 2 (µ), g P H 2 (µ)(fg), the Toeplitz operator with symbol f. Theorem (Didas-Eschmeier-Everard, 2011) For X B(H 2 (µ)), the following are equivalent: 1 there exists f L (µ) such that X = T f, 2 Tu XT u X = 0 for all u I µ. Schatten-class Perturbations of Toeplitz Operators 13

14 Definition We denote by H (D) O(D) the set of all bounded analytic functions on D. Theorem The map r m : H (D) H (m), θ τ w - lim r 1 [θ(r )] =: θ is an isometric algebra isomorphism and a weak* homeomorphism with r m (f D ) = [f T ] for all f A(D). Schatten-class Perturbations of Toeplitz Operators 14

15 Definition We denote by H (D) O(D) the set of all bounded analytic functions on D. Theorem There exists a map r µ : H (D) H (µ) which is an isometric algebra isomorphism and a weak* homeomorphism with r µ (f D ) = [f A(D) ] for all f A(D). We write θ = r µ (θ) for θ H (D). Schatten-class Perturbations of Toeplitz Operators 15

16 Let D C d be a bounded strictly pseudoconvex or bounded symmetric and circled domain and µ M + 1 ( A(D)) be the probability measure on the Shilov boundary A(D). Theorem (Didas-Eschmeier-S., 2017) Let p [1, ). For X B(H 2 (µ)), the following are equivalent: 1 there exist f L (µ) and S S p (H 2 (µ)) such that X = T f + S, 2 T u XT u X S p (H 2 (µ)) for all u I µ. Schatten-class Perturbations of Toeplitz Operators 16

17 Proposition Let (α k ) k N be a sequence in H (µ) with τ w - lim k α k = α [0, 1), and let X B(H 2 (µ)) be an operator such that Y = τ WOT - lim k T α k XT αk B(H 2 (µ)) exists. If T u XT u X K(H 2 (µ)) for all u I µ, then there exists a function f L (µ) such that X = T f + 1 (X Y ). 1 α2 Schatten-class Perturbations of Toeplitz Operators 17

18 Proposition (Hiai, 1997) Let p [1, ). The map p : (B(H 2 (µ)), τ WOT ) [0, ], S S p is lower semi-continuous. In the setting of the proposition on the last slide, we obtain τ WOT- lim X T α k k XT αk lim inf X T αk XT p αk. p k Schatten-class Perturbations of Toeplitz Operators 18

19 We denote by I D = r 1 µ (I µ ) = {θ H (D) ; θ I µ } the set of inner functions with respect to D and µ. Proposition (Aleksandrov, 1984) Let α [0, 1). Then there exists a sequence (α k ) k N in I D such that τ w - lim k α k = α. Schatten-class Perturbations of Toeplitz Operators 19

20 Proposition (Xia, 2009) Let p [1, ). Suppose that X B(H 2 (µ)) is an operator such that T u XT u X S p (H 2 (µ)) for all u I µ. Then, for all ε > 0, there exists δ = δ(ε) > 0 such that Tθ XT θ X p ε for all θ I D with 1 θ dµ A(D) δ. Schatten-class Perturbations of Toeplitz Operators 20

21 Proof of the main theorem. There exists 0 < δ < 1 such that Tθ XT θ X p 1 for all θ I D with 1 θ dµ A(D) δ. Set α = 1 δ/2. = There exists (α k ) k N in I D with τ w - lim k αk = α. By passing to a subsequence, we can achieve that 1 α A(D) k dµ δ for all k N and that at the same time the limit Y = τ WOT - lim T α k XT α B(H2 (µ)) k k exists. Hence, τ WOT- lim X T α k XT α k k lim inf X Tα XT α 1. k k p p = X Y S p (H 2 (µ)). k Schatten-class Perturbations of Toeplitz Operators 21

22 Remark The following ingredients are essential for the proof: 1 the triple (A(D) A(D), A(D), µ) is regular (in the sense of Aleksandrov), 2 the measure µ is a faithful Henkin measure. Schatten-class Perturbations of Toeplitz Operators 22

23 Theorem (Brown-Halmos, 1964) For X B(H 2 (m)), the following are equivalent: 1 there exists f H (m) such that X = T f, 2 XT g T g X {0} for all g H (m). Theorem (Davidson, 1977) For X B(H 2 (m)), the following are equivalent: 1 there exist f H (m) + C(T) and K K(H 2 (m)) such that X = T f + K, 2 XT g T g X K(H 2 (m)) for all g H (m). Schatten-class Perturbations of Toeplitz Operators 23

24 Theorem (Hartman, 1958) We have H (m) + C(T) = {f L (m) ; H f is compact} with H f = ( id L 2 (m) P H (m)) 2 Mf H 2 (m) for f L (m). Theorem (Davidson, Hartman) For X B(H 2 (m)), the following are equivalent: 1 there exist f L (m) such that H f is compact and K K(H 2 (m)) such that X = T f + K, 2 XT g T g X K(H 2 (m)) for all g H (m). Schatten-class Perturbations of Toeplitz Operators 24

25 Let d 1 and D = B d. Theorem (Ding-Sun, 1997) For X B(H 2 (µ)), the following are equivalent: 1 there exist f L (µ) such that H f is compact and K K(H 2 (µ)) such that X = T f + K, 2 XT g T g X K(H 2 (µ)) for all g H (µ). Schatten-class Perturbations of Toeplitz Operators 25

26 Let d 1 and D = B d or D = D d. Theorem (Guo-Wang, 2006) For X B(H 2 (µ)), the following are equivalent: 1 there exist f L (µ) such that H f is of finite rank and F F(H 2 (µ)) such that X = T f + F, 2 XT g T g X F(H 2 (µ)) for all g H (µ). Schatten-class Perturbations of Toeplitz Operators 26

27 Let d 1, and let D C d be a bounded strictly pseudoconvex or bounded symmetric and circled domain and µ M + 1 ( A(D)) be the probability measure on the Shilov boundary A(D). Theorem (Didas-Eschmeier-S., 2017) Let p [1, ). For X B(H 2 (µ)), the following are equivalent: 1 there exist f L (µ) such that H f is in the Schatten-p-class and S S p (H 2 (µ)) such that X = T f + S, 2 XT g T g X S p (H 2 (µ)) for all g H (µ). Schatten-class Perturbations of Toeplitz Operators 27