The Essential Norm of Operators on the Bergman Space

Transcription

1 The Essential Norm of Oerators on the Bergman Sace Brett D. Wick Georgia Institute of Technology School of Mathematics ANR FRAB Meeting 2012 Université Paul Sabatier Toulouse May 26, 2012 B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

2 This talk is based on joint work with: Daniel Suárez University of Buenos Aires Mishko Mitkovski Georgia Institute of Technology B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

3 Motivations for the Problem Weighted Bergman Saces on B n Let B n := {z C n : z < 1}. For α > 1, we let dv α (z) := c α (1 z 2 ) α dv(z), with c α := Γ(n + α + 1) n! Γ(α + 1). The choice of c α gives that v α (B n ) = 1. For 1 < < the sace A α is the collection of holomorhic functions on B n such that f A α := B n f (z) dv α (z) <. For λ B n let k (,α) n+1+α λ (z) = (1 λ 2 ) q A comutation shows: A 1. α k (,α) λ (1 λz) n+1+α. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

4 Motivations for the Problem Toelitz Oerators and the Toelitz Algebra The rojection of Lα 2 onto A 2 α is given by the integral oerator f (w) P α (f )(z) := (1 zw) n+1+α dv α(w). B n This oerator is bounded from L α to A α when 1 < < and 1 < α. Let M a denote the oerator of multilication by the function a, M a (f ) := af. The Toelitz oerator with symbol a L is the oerator given by T a := P α M a. It is immediate to see that T a L(A α ) a L. More generally, for a measure µ we will define the oerator f (w) T µ f (z) := dµ(w), (1 wz) n+1+α which will define an analytic function for all f H. B n B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

5 Motivations for the Problem Toelitz Oerators and the Toelitz Algebra For symbols in L we let T,α be the C subalgebra of L(A α) generated by T a. An imortant class of oerators in T,α are those that are finite sums of finite roducts of Toelitz oerators. Namely, for symbols a jk L with 1 j J and 1 k K we will need to study the oerators: J K Additionally, T ajk j=1 k=1 L(A α) J K T,α = T ajk : a jk L 1 j J 1 k K j=1 k=1 B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

6 Motivations for the Problem Geometry of the Ball For z B n, ϕ z will denote the automorishm of B n such that ϕ z (0) = z. The seudohyerbolic and hyerbolic metrics are defined by ρ(z, w) := ϕ z (w) and β(z, w) := 1 2 log 1 + ρ(z, w) 1 ρ(z, w). The hyerbolic disc centered at z of radius r is denoted by D(z, r) := {w B n : β(z, w) r} = {w B n : ρ(z, w) tanh r}. Lemma (Lattices on B n ) Given r > 0, there is a family of Borel sets D m B n and oints {w m } m=1 such that (i) D ( w m, r 4) Dm D (w m, r) for all m; (ii) D k D l = if k l; (iii) m D m = B n. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

7 Motivations for the Problem Geometry of the Ball Note that for these sets: If w D m then (1 w 2 ) (1 w m 2 ) and 1 zw 1 zw m uniformly in z B n. Lemma (Whitney Decomositions) There is a ositive integer N = N (n) such that for any σ > 0 there is a covering of B n by Borel sets {B j } that satisfy: (i) B j B k = if j k; (ii) Every oint of B n is contained in at most N sets Ω σ (B j ) = {z : β(z, B j ) σ}; (iii) There is a constant C(σ) > 0 such that diam β B j C(σ) for all j. Idea of Proof: Via the Whitney Decomosition of the unit ball B n, artition into dyadic cubes. This then gives (i) immediately. The remaining oints are then well known geometric facts. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

8 Motivations for the Problem Geometry of the Ball Whitney Decomosition of D (Taken from Classical and Modern Fourier Analysis by Grafakos) B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

9 Motivations for the Problem Geometry of the Ball Let σ > 0 and k a non-negative integer. Let {B j } be the covering of the ball from the revious Lemma with (k + 1)σ instead of σ. For 0 i k and j 1 write F 0,j = B j and F i+1,j = {z : β(z, F i,j ) σ}. Corollary Let σ > 0 and k be a non-negative integer. For each 0 i k the family of sets F i = {F i,j : j 1} forms a covering of B n such that (i) F 0,j1 F 0,j2 = if j 1 j 2 ; (ii) F 0,j F 1,j F k+1,j for all j; (iii) β(f i,j, Fi+1,j c ) σ for all 0 i k and j 1; (iv) Every oint of B n belongs to no more than N elements of F i ; (v) diam β F i,j C(k, σ) for all i, j. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

10 Motivations for the Problem Carleson Measures for A α A measure µ on B n is a Carleson measure for A α if f (z) dµ(z) f (z) dv α (z) f A α. B n B n Lemma (Characterizations of A α Carleson Measures) Suose that 1 < < and α > 1. Let µ be a measure on B n and r > 0. The following quantities are equivalent, with constants that deend on n, α and r: (1 z (1) µ CM := su 2 ) n+1+α z Bn (2) ı := inf { C : B n dµ(w); 1 zw 2(n+1+α) ( B n f (z) dµ(z)) 1 (3) µ Geo := su z Bn µ(d(z,r)) (1 z 2 ) n+1+α ; (4) T µ L(A α ). C ( B n f (z) dv α (z)) 1 B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29 } ;

11 Motivations for the Problem The Berezin Transform For S L(A α), we define the Berezin transform by B : L(A α) L (B n ): B(S)(z) := B(S)(z) S L(A α ) Sk (,α) z, k (q,α) z A 2 α k (,α) k (q,α) λ A λ α. A q α S L(A α ). If S is comact, then B(S)(z) 0 as z 1: B(S)(z) Sk (,α) k (q,α) λ A λ α A q Sk (,α) λ α A. α However, k (,α) λ 0 as z 1 and so Sk (,α) λ A 0. α B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

12 Motivations for the Problem The Berezin Transform The Berezin transform is one-to-one: Enough to show that B(S)(z) = 0 S = 0. Set F(z, w) =, k (q,α). Sk (,α) z w Then F(z, z) = 0 and F is analytic in the first variable and anti-analytic in the second variable. This imlies that F is identically zero. So we have that Sk (,α) z = 0 for all z B n, or S = 0. Range of B is not closed: B 1 : B(L(A α)) L(A α) is not bounded. A 2 α B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

13 Motivations for the Problem Related Results Theorem (Axler and Zheng, Indiana Univ. Math. J. 47 (1998)) Suose that a jk L (D) with 1 j J and 1 k K. Let S = J Kk=1 j=1 T ajk The following are equivalent: (a) The oerator S is comact on A 2 (D); (b) B(S)(z) 0 as z 1; (c) Sk z A 2 α 0 as z 1. The interesting imlication is (b) (a); The same roof works in the case of the unit ball, but was done by Raimondo. Theorem (Engliš, Ark. Mat. 30 (1992)) Let 1 < < and α > 1. If S is a comact oerator on A α, then S T,α. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

14 Motivations for the Problem Main Question of Interest From the revious Theorem and simle functional analysis we have that if S is comact on A α then S T,α B(S)(z) 0 as z 1. Question (Characterizing the Comacts) If S T,α and B(S)(z) 0 as z 1, then is S is comact? Yes! Shown to be true by Suárez for A when 1 < < and α = 0. Extended to α > 1 by Suárez, Mitkovski and BDW using some maximal ideal theory and aroriate modifications of the original roof. Alternate roof obtained by Mitkovski and BDW, but removing the maximal ideal theory. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

15 Main Results Characterizations of Comactness and Essential Norm Theorem (D. Suárez, M. Mitkovski and BDW) Let 1 < < and α > 1 and S L(A α). Then S is comact if and only if S T,α and lim z 1 B(S)(z) = 0. We can actually obtain much more recise information about the essential norm of an oerator. For S L(A α) recall that } S e = inf { S Q L(A α ) : Q is comact. We need to define other measures of the size of an oerator S L(A α): b S := su lim su M 1D(z,r) S r>0 z 1 L(A α,lα) M1rB c S := lim c S. r 1 n L(A α,lα) In the last definition, we have that rb c n := B n \ rb n. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

16 Main Results Characterizations of Comactness and Essential Norm Let r > 0 and let {w m } and D m be the sets that form the lattice in B n. Define the measure µ r = m v α (D m )δ wm (1 w m 2 ) α+n+1 δ wm. m It is well know that µ r is a A α Carleson measure, so T µr bounded. Lemma T µr Id on L(A α) when r 0. : A α A α is Let r > 0 be chosen so that T µr Id L(A α ) < 1 4, and µ := µ r. Then set { } a S (ρ) := lim su su Sf A α : f T µ1d(z,ρ) (A α), f A α 1 z 1 and define a S := lim ρ 1 a S (ρ). B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

17 Main Results Characterizations of Comactness and Essential Norm Theorem (D. Suárez, M. Mitkovski and BDW) Let 1 < < and α > 1 and let S T,α. Then there exists constants deending only on n,, and α such that: a S b S c S S e. For the automorhism ϕ z such that ϕ z (0) = z define the ma U (,α) z f (w) := f (ϕ z (w)) (1 z 2 ) n+1+α (1 wz) 2(n+1+α) A standard change of variable argument and comutation gives that f A = f A α f A α. α U z (,α). B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

18 Main Results Characterizations of Comactness and Essential Norm For z B n and S L(A α) we then define the ma S z := U z (,α) S(U z (q,α) ). One should think of the ma S z in the following way. This is an oerator on A α and so it first acts as translation in B n, then the action of S, then translation back. Theorem (D. Suárez, M. Mitkovski and BDW) Let α > 1 and 1 < < and S T,α. Then S e su lim su S z f A α. f A =1 z 1 α B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

19 Sketch of Proofs Main Ingredients Connecting the Geometry and Oerator Theory Lemma (D. Suárez, M. Mitkovski and BDW) Let S T,α, µ a Carleson measure and ɛ > 0. Then there are Borel sets F j G j B n such that (i) B n = F j ; (ii) F j F k = if j k; (iii) each oint of B n lies in no more than N (n) of the sets G j ; (iv) diam β G j d(, S, ɛ) and ST µ M 1Fj ST 1Gj µ < ɛ. j=1 L(A α,lα) B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

20 Sketch of Proofs Main Ingredients A Uniform Algebra and its Maximal Ideal Sace Let A denote the bounded functions that are uniformly continuous from the metric sace (B n, ρ) into the metric sace (C, ). Associate to A its maximal ideal sace M A which is the set of all non-zero multilicative linear functionals from A to C. Since A is a C algebra we have that B n is dense in M A. The Toelitz oerators associated to symbols in A are useful to study the Toelitz algebra T,α. Theorem (D. Suárez, M. Mitkovski and BDW) The Toelitz algebra T,α is equal to the closed algebra generated by {T a : a A}. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

21 Sketch of Proofs Main Ingredients A Uniform Algebra and its Maximal Ideal Sace For an element x M A \ B n choose a net z ω x. Form S zω and look at the limit oerator obtained when z ω x, denote it by S x. Lemma (D. Suárez, M. Mitkovski and BDW) Let S L(A α). Then B(S)(z) 0 as z 1 if and only if S x = 0 for all x M A \ B n. We can extend this to comute the essential norm of an oerator S in terms of S x where x M A \ B n. Theorem (D. Suárez, M. Mitkovski and BDW) Let S T,α. Then there exists a constant C(, α, n) such that su x M A \B n S x L(A α ) S e. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

22 Sketch of Proofs Main Ingredients The Hilbert Sace Case Theorem (D. Suárez, M. Mitkovski and BDW) For S T 2,α we have and S e = su r(s x ) lim x M A \B n k su x M A \B n S x L(A 2 α ) ( su x M A \B n with equality when S is essentially normal. ) Sx k 1 k L(A 2 α) Theorem (D. Suárez, M. Mitkovski and BDW) Let α > 1 and S T 2,α. Then S e = su lim su S z f A 2 α. f A 2 =1 z 1 α = r e (S) B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

23 Sketch of Proofs Main Ingredients The Hilbert Sace Case Theorem (D. Suárez, M. Mitkovski and BDW) Let S T 2,α. The following are equivalent: (1) λ / σ e (S); (2) λ / σ(s x ) and su (S x λi ) 1 L(A < ; 2 x M A \B x M n A \B n α ) (3) There is a number t > 0 deending only on λ such that (S x λi )f A 2 α t f A 2 α for all f A 2 α and x M A \ B n. and (Sx λi )f t f A 2 A 2 α α B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

24 Sketch of Proofs Main Ingredients Removing the Maximal Ideal Sace Lemma (M. Mitkovski and BDW) Let T be a finite sum of finite roducts of Toelitz oerators on A 2 α. For every ɛ > 0 there exists r > 0 such that for the covering F r = {F j } associated to r M 1Fj TP α M 1G c j < ɛ. j L(A 2 α ) Proosition (M. Mitkovski and BDW) Let T be a finite sum of finite roducts of Toelitz oerators on A 2 α. Then T e su lim su T z f A 2 α. f A 2 =1 z 1 α B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

25 Other Directions and Oen Questions Other Directions The Fock sace F is the collection of holomorhic functions f on C n such that f 2 F := f (z) 2 e π z 2 dv(z) <. C n This is a reroducing kernel Hilbert sace with k λ (z) = e πzλ as kernel. Similar results are true for the Fock Sace. Theorem (W. Bauer and J. Isralowitz) Let S L(F). Then S is comact if and only if S T 2 and lim z B(S)(z) = 0. Corollary (W. Bauer and J. Isralowitz) Let S T 2, then S e su lim su S z f F. f F =1 z B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

26 Other Directions and Oen Questions Oen Questions If S L(A 2 α) then B(S) L. Similarly, if S K(A 2 α) then B(S) 0 as z 1. And, even better, S K(A 2 α) if and only if S T 2,α and B(S) 0 as z 1. Question Can we characterize the Schatten class oerators on A 2 α as those that belong to the Toelitz algebra T 2,α and an integrability condition on the Berezin transform B(S)(z)? One can show that if S S then B(S) L (B n;λ n) := ( B n B(S)(z) dλ n (z) ) 1 S S. Proosition (M. Mitkovski and BDW) Let 1 < < and α > 1. If S S, then B(S) L (B n ; λ n ) and S T 2,α. B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

27 Other Directions and Oen Questions Oen Questions Let Ω be a bounded symmetric domain in C n. These are Hermitian symmetric saces with a comlete Riemannian metric given by the Bergman metric. For each a Ω there is a biholomorhic automorhism ϕ a that interchanges 0 and a. Let A 2 (Ω) denote the Bergman sace of analytic functions on Ω that are square integrable with resect to volume measure. This sace has a reroducing kernel K a, and relates to the automorhisms by K w (z) = K ϕ(w) (ϕ(z))j c ϕ(w)j c ϕ(w) Conjecture (M. Mitkovski and BDW) Let S T 2, then S e su lim su S z f A 2 (Ω). f A 2 (Ω) =1 z Ω B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

28 Conclusion (Modified from the Original Dr. Fun Comic) Thanks to Pascal and Stefanie for Organizing the Worksho! B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29

29 Conclusion Thank You! B. D. Wick (Georgia Tech) Essential Norm on Bergman Saces May 26, / 29