Closed range composition operators on BMOA

Transcription

1 Closed range composition operators on BMOA University of Arkansas Joint work with Kevser Erdem University of Arkansas CAFT 2018 University of Crete July 2-6, 2018

2 Notation D = {z C : z < 1} T = {z C : z = 1} ϕ : D D analytic self map of D α q (z) = q z 1 qz, q D Mobius transformation D(q, r) = {z D : α q (z) < r}, r (0, 1) H(D) is the set of analytic functions on D

3 Composition operators C ϕ f = f ϕ C ϕ is a linear operator Let ϕ be a non constant self map of D By the Open Mapping Theorem for analytic functions, ϕ(d) is an open subset of D C ϕ : H(D) H(D) is one to one

4 The Hardy space H 2 H 2 is the Hilbert space of analytic functions f on D f (z) = a n z n, f 2 H = 2 n=0 a n 2 < n=0

5 The Hardy space H 2 H 2 is the Hilbert space of analytic functions f on D f (z) = a n z n, f 2 H = 2 n=0 a n 2 < n=0 an equivalent norm on H 2 is given by f 2 H f (0) 2 + (1 z 2 ) f (z) 2 da(z) 2 D

6 BMOA BMOA is the Banach space of analytic functions on D f G = sup f α q f (q) H 2 < q D

7 BMOA BMOA is the Banach space of analytic functions on D f G = sup f α q f (q) H 2 < q D f 2 = sup q D D f (z) 2 (1 α q (z) 2 ) da(z)

8 BMOA BMOA is the Banach space of analytic functions on D f G = sup f α q f (q) H 2 < q D f 2 = sup q D D The norm we will use in BMOA is: f (z) 2 (1 α q (z) 2 ) da(z) f BMOA = f (0) + f

9 Composition operators on BMOA C ϕ f = f ϕ C ϕ is always a bounded operator on BMOA

10 Composition operators on BMOA C ϕ f = f ϕ C ϕ is always a bounded operator on BMOA f ϕ 2 = sup q D = sup q D D D (f ϕ) (z) 2 (1 α q (z) 2 )da(z) f (ϕ(z)) 2 ϕ (z) 2 (1 α q (z) 2 )da(z)

11 Composition operators on BMOA C ϕ f = f ϕ C ϕ is always a bounded operator on BMOA f ϕ 2 = sup q D = sup q D = sup q D D D D (f ϕ) (z) 2 (1 α q (z) 2 )da(z) f (ϕ(z)) 2 ϕ (z) 2 (1 α q (z) 2 )da(z) f (ζ) 2 (1 α q (z) 2 ) da(ζ) ϕ(z)=ζ

12 Counting functions on BMOA For each q D we define the BMOA counting function by N q,ϕ (ζ) = (1 α q (z) 2 ) if ζ ϕ(d), N q,ϕ (ζ) = 0 ϕ(z)=ζ

13 Counting functions on BMOA For each q D we define the BMOA counting function by N q,ϕ (ζ) = (1 α q (z) 2 ) if ζ ϕ(d), N q,ϕ (ζ) = 0 ϕ(z)=ζ C ϕ f 2 = sup q D D const. f 2 f (ζ) 2 N q,ϕ (ζ) da(ζ)

14 C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA C > 0 such that f BMOA f BMOA C f ϕ BMOA

15 C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA C > 0 such that f BMOA f BMOA f ϕ BMOA

16 C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA C > 0 such that f BMOA f BMOA f ϕ BMOA C ϕ is bounded below on BMOA f BMOA f ϕ f

17 C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA C > 0 such that f BMOA f BMOA f ϕ BMOA C ϕ is bounded below on BMOA f BMOA f ϕ f We say that C ϕ is closed range in BMOA if C ϕ (BMOA) is closed in BMOA C ϕ is closed range C ϕ is bounded below

18 C ϕ closed range C ϕ : H 2 H 2 J. Cima, J. Thomson, W. Wogen (1973) ν(e) := m(ϕ 1 (E)), for E any Borel subset of T C ϕ closed range on H 2 dν dm essentially bounded away from 0

19 C ϕ closed range C ϕ : H 2 H 2 J. Cima, J. Thomson, W. Wogen (1973) ν(e) := m(ϕ 1 (E)), for E any Borel subset of T C ϕ closed range on H 2 dν dm essentially bounded away from 0 N. Zorboska (1994), K. Luery (2013) and P. Ghatage, MT (2014)

20 Sampling sets in BMOA We define H D to be a sampling set for BMOA if for all f BMOA f (z) 2 (1 α q (z) 2 ) da(z) f 2. sup q D H

21 Sampling sets in BMOA We define H D to be a sampling set for BMOA if for all f BMOA f (z) 2 (1 α q (z) 2 ) da(z) f 2. sup q D H For each ε > 0 and q D G ε,q = {ζ : N q,ϕ (ζ) > ε (1 α q (ζ) 2 )} C ϕ is closed range on BMOA ε > 0 such that q D G ε,q is a sampling set for BMOA

22 Sampling sets in BMOA We define H D to be a sampling set for BMOA if for all f BMOA f (z) 2 (1 α q (z) 2 ) da(z) f 2. sup q D H For each ε > 0 and q D G ε,q = {ζ : N q,ϕ (ζ) > ε (1 α q (ζ) 2 )} C ϕ is closed range on BMOA ε > 0 such that q D G ε,q is a sampling set for BMOA q D G ε,q is a sampling set for BMOA C ϕ is closed range on BMOA

23 Carleson measures Let µ be a finite positive Borel measure on D. We say that µ is a (Bergman space) Carleson measure on D if there exists c > 0 such that for all f A 2 f (z) 2 dµ(z) c f (z) 2 da(z) D Let 0 < r < 1. Then µ is a Carleson measure if and only if there exists c r > 0 such that for all q D, µ(d(q, r)) c r A(D(q, r)) D

24 Carleson measures The Berezin symbol of µ is µ(q) = α q(z) 2 dµ(z), q D. D µ is Carleson measure if and only if µ is a bounded function on D

25 Carleson measures The Berezin symbol of µ is µ(q) = α q(z) 2 dµ(z), q D. D µ is Carleson measure if and only if µ is a bounded function on D µ q, q D, is a collection of uniformly Carleson measures if and only if the Berezin symbols of µ q, for all q D, are uniformly bounded in D. Recall C ϕ f 2 = sup q D D f (ζ) 2 N q,ϕ (ζ) da(ζ) const. f 2 The measures N q,ϕ (ζ) da(ζ) are uniformly Carleson measures

26 Dan Luecking and the RCC Let µ be a finite positive Carleson measure on D. We say that µ satisfies the reverse Carleson condition if r (0, 1) such that µ(d(q, r)) A(D(q, r)), q D G D satisfies the reverse Carleson condition if the Carleson measure χ G (z) da(z) satisfies the reverse Carleson condition. Luecking f (z) 2 da(z) C f (z) 2 da(z), f A 2 D A(G D(q, r)) A(D(q, r)), q D G

27 G ε,q,q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA.

28 G ε,q,q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA. For each ε > 0 and q, q D G ε,q,q = {ζ : N q,ϕ(ζ) > ε (1 α q (ζ) 2 )}

29 G ε,q,q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA. For each ε > 0 and q, q D G ε,q,q = {ζ : N q,ϕ(ζ) > ε (1 α q (ζ) 2 )} k > 0 q D α q ϕ k ε > 0, r (0, 1) such that q D, q D such that G ε,q,q D(q, r) D(q, r) 1. (1)

30 RCC - Geometry of disks - Luecking 1981 Given a measurable set F, the following are equivalent: δ > 0, r (0, 1) such that disks D with centers on T, F D > δ D D. δ 0 > 0, η (0, 1) such that q D F D (q, η(1 q )) > δ 0 D (q, η(1 q )). δ 1 > 0, r (0, 1) such that q D F D(q, r) > δ 1 D(q, r).

31 RCC like sets - Geometry of disks Given a collection of measurable sets F q, q D, the following are equivalent: δ > 0, r (0, 1) such that q D and disks D with centers on T, q D such that F q D > δ D D. δ 0 > 0, η (0, 1) such that q D q D such that F q D (q, η(1 q )) > δ 0 D (q, η(1 q )). δ 1 > 0, r (0, 1) such that q D q D such that F q D(q, r) > δ 1 D(q, r).

32 A reminder G D satisfies the reverse Carleson condition if the Carleson measure χ G (z) da(z) satisfies the reverse Carleson condition. Luecking f (z) 2 da(z) C f (z) 2 da(z), f A 2 D A(G D(q, r)) A(D(q, r)), q D G

33 A reminder G D satisfies the reverse Carleson condition if the Carleson measure χ G (z) da(z) satisfies the reverse Carleson condition. Luecking f (z) 2 da(z) C f (z) 2 da(z), f A 2 D A(G D(q, r)) A(D(q, r)), q D G k > 0 q D α q ϕ k ε > 0, r (0, 1) such that q D, q D such that G ε,q,q D(q, r) D(q, r) 1. is closed range?

34 B The Bloch space B is the set of functions f analytic on D such that f B := f (0) + f B = f (0) + sup f (z) (1 z 2 ) < z D

35 B The Bloch space B is the set of functions f analytic on D such that f B := f (0) + f B = f (0) + sup f (z) (1 z 2 ) < z D f B sup f α q f (q) A 2 q D BMOA B f sup f α q f (q) H 2 q D

36 C ϕ : B BMOA C ϕ : B BMOA is closed range ε > 0, r (0, 1) such that q D, q D such that G ε,q,q D(q, r) D(q, r) 1.

37 C ϕ : B BMOA C ϕ : B BMOA is closed range ε > 0, r (0, 1) such that q D, q D such that G ε,q,q D(q, r) D(q, r) 1. C ϕ : B BMOA is closed range k > 0 such that q D α q ϕ k

38 , etc. is closed range k > 0 q D α q ϕ k

39 , etc. is closed range k > 0 q D α q ϕ k

40 , etc. is closed range k > 0 q D α q ϕ k Assuming that C ϕ : X X is a bounded operator, C ϕ : X X is closed range k > 0 q D α q ϕ X k, where X = B, Besov type spaces, Q p

41 C ϕ : B B versus J. Akeroyd, P. Ghatage, M.T: C ϕ is closed range on B α q ϕ B 1, q D

42 C ϕ : B B versus J. Akeroyd, P. Ghatage, M.T: C ϕ is closed range on B α q ϕ B 1, q D C ϕ is closed range on B C ϕ is closed range on BMOA.

43 C ϕ : B B versus J. Akeroyd, P. Ghatage, M.T: C ϕ is closed range on B α q ϕ B 1, q D C ϕ is closed range on B C ϕ is closed range on BMOA. P. Ghatage, D. Zheng, N. Zorboska: for ϕ univalent C ϕ closed range on BMOA C ϕ closed range on B We conclude: C ϕ is closed range on B C ϕ is closed range on BMOA

44 C ϕ : H 2 H 2 versus N. Zorboska (1994): C ϕ is closed range on H 2 ε > 0 such that the set G ε,0,0 = {ζ D : (1 z 2 ) > ε(1 ζ 2 )} satisfies the RCC ϕ(z)=ζ

45 C ϕ : H 2 H 2 versus N. Zorboska (1994): C ϕ is closed range on H 2 ε > 0 such that the set G ε,0,0 = {ζ D : (1 z 2 ) > ε(1 ζ 2 )} ϕ(z)=ζ satisfies the RCC K. Luery (2013), P. Ghatage and MT (2014) C ϕ is closed range on H 2 q D q α q ϕ H 2 1

46 C ϕ : H 2 H 2 versus N. Zorboska (1994): C ϕ is closed range on H 2 ε > 0 such that the set G ε,0,0 = {ζ D : (1 z 2 ) > ε(1 ζ 2 )} ϕ(z)=ζ satisfies the RCC K. Luery (2013), P. Ghatage and MT (2014) C ϕ is closed range on H 2 q D q α q ϕ H 2 1 C ϕ is closed range on H 2 C ϕ is closed range on BMOA

47 α q ϕ 1, q D J. Laitila (2010) isometries among composition operators on BMOA using the seminorm f G, f G = sup f α q f (q) H 2 < q D

48 α q ϕ 1, q D J. Laitila (2010) isometries among composition operators on BMOA using the seminorm f G, f G = sup f α q f (q) H 2 < q D Below we give another characterization of closed range composition operators on BMOA: k (0, 1] such that q D, α q ϕ k k (0, 1] such that q D q D with α q (q ) 2 1 k 2, a sequence (q n ) in D such that ϕ(q n ) q and lim n ϕ q n H 2 k, where n, ϕ qn = α ϕ(qn ) ϕ α qn

49 Recall ε > 0 and q, q D G ε,q,q = {ζ : N q,ϕ(ζ) > ε (1 α q (ζ) 2 )}

50 Main theorem: The following statements are equivalent: (a) k > 0 q D α q ϕ k (b) ε > 0, r (0, 1) such that q D, q D such that G ε,q,q D(q, r) D(q, r) 1. (c) C ϕ : B BMOA is closed range (d) is closed range (e) k (0, 1] such that q D q D with α q (q ) 2 1 k 2, a sequence (q n ) in D such that ϕ(q n ) q and lim n ϕ q n 2 k where n, ϕ qn = α ϕ(qn ) ϕ α qn

51 Thank you