Weighted composition operators between Hilbert spaces of analytic functions in the operator norm and Hilbert–Schmidt norm topologies

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.1 (1-13)

J. Math. Anal. Appl. ••• (••••) •••–•••

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Weighted composition operators between Hilbert spaces of analytic functions in the operator norm and Hilbert–Schmidt norm topologies ✩ Takuya Hosokawa a , Kei Ji Izuchi b , Shûichi Ohno c a b c

College of Engineering, Ibaraki University, Hitachi, Ibaraki 316-8511, Japan Department of Mathematics, Niigata University, Niigata 950-2181, Japan Nippon Institute of Technology, Miyashiro, Minami-Saitama 345-8501, Japan

a r t i c l e

i n f o

Article history: Received 9 May 2014 Available online xxxx Submitted by E. Saksman Keywords: Weighted composition operator Path connected component Hilbert–Schmidt operator Hardy space Weighted Bergman space Dirichlet space

a b s t r a c t We here investigate the topological structure of the space of weighted composition operators boundedly acting between two Hilbert spaces of analytic functions on the open unit disk, satisfying some natural hypotheses. We will consider the operator norm topology and the Hilbert–Schmidt norm topology respectively. These results will be involved in the investigation for the explicit cases of the classical Hardy– Hilbert space, the weighted Bergman spaces and the Dirichlet space. Furthermore we will estimate the Hilbert–Schmidt norms of difference of two composition operators acting from the Dirichlet space to the Hardy and the weighted Bergman spaces. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let H(D) be the space of analytic functions on the open unit disk D := {|z| < 1} and H ∞ the space of bounded analytic functions on D with the supremum norm ·∞ . Let S(D) be the set of analytic self-maps of D. Denote by H a Hilbert space of analytic functions on D with the norm ·H satisfying the following conditions: (#1) For any α ∈ D, the point evaluation τα : H  f → f (α) is a bounded linear functional on H, and sup|α|≤r τα H < ∞ for every 0 < r < 1. ✩ The first author is partially supported by Grant-in-Aid for Young Scientists (B), Japan Society for the Promotion of Science (No. 25800055). The second author is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 24540164). The third author is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 24540190). E-mail addresses: [email protected] (T. Hosokawa), [email protected] (K.J. Izuchi), [email protected] (S. Ohno).

http://dx.doi.org/10.1016/j.jmaa.2014.07.043 0022-247X/© 2014 Elsevier Inc. All rights reserved.

JID:YJMAA 2

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.2 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

(#2) H ∞ · H ⊂ H and f gH ≤ f ∞ gH for every f ∈ H ∞ and g ∈ H. (#3) 1H = 1 and {z n /z n H : n ≥ 0} is an orthonormal basis in H. (#4) For every f ∈ H and 0 ≤ r ≤ 1, we have fr (z) := f (rz) ∈ H. By (#3), H contains all analytic polynomials. By (#2), for f ∈ H ∞ we have f H = f ·1H ≤ f ∞ 1H = f ∞ . Many classical Hilbert spaces of analytic functions on D satisfy conditions (#1)–(#4). For ϕ ∈ S(D), we define the composition operator Cϕ : H → H(D) by Cϕ f = f ◦ ϕ for f ∈ H. If Cϕ f ∈ H for every f ∈ H, then Cϕ : H → H is a bounded linear operator. We denote by C(H) the space of bounded composition operators Cϕ : H → H with the operator norm topology. We refer to [5,14] for an overview of composition operators. Let ϕ ∈ S(D) and u ∈ H. We may define the weighted composition operator Mu Cϕ : H → H(D) by Mu Cϕ f = u · (f ◦ ϕ) for every f ∈ H. If u · (f ◦ ϕ) ∈ H for every f ∈ H, then Mu Cϕ : H → H is bounded and we denote by Mu Cϕ H its operator norm. We note that Mu Cϕ = 0 if and only if u = 0. Let Cw (H) be the space of nonzero bounded weighted composition operators on H with the operator norm topology, that is, Cw (H) = {Mu Cϕ : Mu Cϕ : H → H is bounded, u = 0}. In the study of (weighted) composition operators, one of the main subjects is determining the set {u ∈ H : Mu Cϕ ∈ Cw (H)} for a given ϕ ∈ S(D) and the other is determining the topological structure in Cw (H). By (#2), if Cϕ ∈ Cw (H) then Mu Cϕ ∈ Cw (H) for every u ∈ H ∞ with u = 0. The boundedness of Mu on range of a composition operator was investigated in [1,10]. The boundedness and compactness of weighted composition operators on the Hardy and Bergman spaces have been characterized. For example, see [4,6,7]. About the topological structure, Berkson [2] first studied the component structure of the set of all composition operators on the Hardy–Hilbert space H 2 in the topology induced by the operator norm. Further MacCluer [11] and Shapiro and Sundberg [15] investigated and the latter authors raised the problems on the component structure in the topologies induced by the operator norm and the essential operator norm and explicitly gave the conjecture that two composition operators would lie in the same component if and only if they have compact difference, that is, the difference of the two composition operators is compact. This conjecture was answered in the negative by [13] and Bourdon [3] provided an example of two composition operators induced by linear fractional self-maps of D which are in the same component but do not have compact difference. Also see [8]. In general, it seems fairly difficult to describe all path connected components in Cw (H). We would like to mention that Cw (H) ∪ {0} is a path connected set. The reason is that for Mu Cϕ ∈ Cw (H), we have Mtu Cϕ ∈ Cw (H) ∪ {0} and Mt0 u Cϕ − Mtu Cϕ H = |t0 − t|Mu Cϕ H for every 0 ≤ t0 , t ≤ 1, so Mu Cϕ and 0 are in the path connected set in Cw (H) ∪ {0}. By this fact, the condition 0 ∈ / Cw (H) is a key to study path connected components in Cw (H). By condition (#3), Mu Cϕ ∈ Cw (H) is Hilbert–Schmidt if and only if Mu Cϕ 2H,HS :=

∞  uϕn 2H < ∞. z n 2H n=0

We denote by Cw,HS (H) the space of Hilbert–Schmidt operators Mu Cϕ in Cw (H) with the Hilbert–Schmidt norm topology. The topology on Cw,HS (H) is stronger than the operator norm one. So a path connected set in Cw,HS (H) is so in Cw (H).

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.3 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

3

Let H1 be another Hilbert space of analytic functions on D with H1 ⊂ H satisfying conditions (#1), (#3) and (#4). We note that H1 needs not satisfy (#2). Furthermore we assume that (#5) f H ≤ f H1 for every f ∈ H1 . For a bounded linear operator T : H1 → H, we write T H1 ,H its operator norm. For Mu Cϕ ∈ Cw (H), we have Mu Cϕ f H ≤ Mu Cϕ H f H ≤ Mu Cϕ H f H1 for every f ∈ H1 . Hence Mu Cϕ : H1 → H is bounded and Mu Cϕ H1 ,H ≤ Mu Cϕ H

for every Mu Cϕ ∈ Cw (H).

(1.1)

Restricting Mu Cϕ ∈ Cw (H) on H1 , we may consider that Mu Cϕ is also a bounded linear operator from H1 to H. We denote by Cw (H1 , H) the space of Mu Cϕ : H1 → H, Mu Cϕ ∈ Cw (H), with the operator norm topology. For the non-weighted case, we write C(H1 , H). We note that Cw (H1 , H) = Cw (H) as sets, so if Mu Cϕ ∈ Cw (H1 , H), then u ∈ H. By (1.1), the topology of Cw (H) is stronger than the one of Cw (H1 , H). Hence a path connected set in Cw (H) is so in Cw (H1 , H). We have that Mu Cϕ ∈ Cw (H1 , H) is Hilbert–Schmidt if and only if Mu Cϕ 2H1 ,H,HS

∞  uϕn 2H := < ∞. z n 2H1 n=0

We denote by Cw,HS (H1 , H) the space of Mu Cϕ ∈ Cw (H1 , H) which are Hilbert–Schmidt. We consider the Hilbert–Schmidt norm topology on Cw,HS (H1 , H). The topology on Cw,HS (H1 , H) is stronger than the operator norm one. So a path connected set in Cw,HS (H1 , H) is so in Cw (H1 , H). Since Mu Cϕ 2H1 ,H,HS ≤

∞  uϕn 2H = Mu Cϕ 2H,HS n 2 z H n=0

by (#5),

we have Cw,HS (H) ⊂ Cw,HS (H1 , H). This paper is organized as follows. In Section 2, we shall prove that if z n H /z n H1 → 0 as n → ∞, then Cw (H1 , H) is a path connected space. In Section 3, we shall prove that Cw,HS (H1 , H) is a path connected space. As applications of these results, in Section 4 we study the cases that H is either the classical Hardy– Hilbert space H 2 or the weighted Bergman spaces L2α , −1 < α < ∞, on D, and H1 is either H 2 or L2α or the Dirichlet space D on D. In Section 5, we study the Hilbert–Schmidt norms of differences of composition operators in C(D, H 2 ) and C(D, L2α ) for −1 < α < ∞. We shall show that CHS (D, L2α ) = {Cϕ : ϕ ∈ S(D)} as sets. For ϕ, ψ ∈ S(D), let ϕt = tϕ + (1 − t)ψ for 0 ≤ t ≤ 1. We also prove that {Cϕt : 0 ≤ t ≤ 1} is a continuous path in CHS (D, L2α ). 2. Path connectedness of Cw (H1 , H) Let H and H1 be the spaces satisfying conditions given in the introduction.

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.4 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

4

Lemma 2.1. If ϕ ∈ S(D) and ϕ∞ < 1, then Cϕ f ∈ H ∞ for every f ∈ H and Cϕ f ∞ ≤ f H

sup |α|≤ϕ∞

τα H .

Proof. For f ∈ H and z ∈ D, by (#1) we have      (Cϕ f )(z) = f ϕ(z)  ≤ f H τϕ(z) H ≤ f H

sup |α|≤ϕ∞

τα H ,

so we get the assertion. 2 Theorem 2.2. If z n H /z n H1 → 0 as n → ∞, then Cw (H1 , H) is a path connected space. Proof. Let Mu Cϕ ∈ Cw (H1 , H). Since Cw (H1 , H) = Cw (H) as sets, we have u ∈ H and Mu Cϕ H < ∞. Let 0 ≤ r < 1. For f ∈ H, by Lemma 2.1 we have f ◦ rϕ ∈ H ∞ and   Mu Crϕ f H = u(f ◦ rϕ)H ≤ f ◦ rϕ∞ uH

by (#2)

≤ f H uH sup τα H . |α|≤r

By (#1), Mu Crϕ ∈ Cw (H), so Mu Crϕ ∈ Cw (H1 , H). We shall show that {Mu Crϕ : 0 ≤ r ≤ 1} is a path connected set in Cw (H1 , H). Fix 0 ≤ r0 ≤ 1. It ∞ is sufficient to show that Mu Cr0 ϕ − Mu Crϕ H1 ,H → 0 as r → r0 . Let g = n=0 an z n ∈ H1 . For each 0 ≤ r ≤ 1, let g[r] (z) =

∞ 

  an r0n − rn z n .

n=1

Since H1 ⊂ H and H satisfies (#4), we have g[r] ∈ H. Hence  ∞ 2    2  n   n n  − Mu Crϕ )g H = u an r 0 − r ϕ  0ϕ  

 (Mu Cr

H

n=1

=

Mu Cϕ g[r] 2H

= Mu cϕ 2H

∞ 

≤ Mu Cϕ 2H g[r] 2H  2   2 |an |2 r0n − rn  z n H

by (#3)

n=1

 ∞ 2 z k 2H   2  |an |2 z n H ≤ Mu Cϕ 2H sup r0k − rk  2 k 1 z H1 n=1 k≥1  2  k  k 2 k  z H  ≤ Mu Cϕ H sup r0 − r g2H1 . z k H1 k≥1 Then 

Mu Cr0 ϕ − Mu Crϕ H1 ,H For any positive integer n, we have

 k  k k  z H  . ≤ Mu Cϕ H sup r0 − r z k H1 k≥1

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.5 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

5

 n−1    k  k  k z k H k  z H k k  z H   ≤ r + sup sup r0 − r − r . 0 k z k H1 z k H1 k≥1 k≥n z H1 k=1

Hence z k H . k k≥n z H1

lim sup Mu Cr0 ϕ − Mu Crϕ H1 ,H ≤ Mu Cϕ H sup r→r0

Therefore by the assumption, we get Mu Crϕ → Mu Cr0 ϕ as r → r0 in Cw (H1 , H). This shows that {Mu Crϕ : 0 ≤ r ≤ 1} is a path connected set in Cw (H1 , H). Thus Mu Cϕ and Mu C0 are in the same path connected set in Cw (H1 , H). Let Mu Cϕ , Mv Cψ ∈ Cw (H1 , H). We have   (Mu C0 − Mv C0 )f 

H

  ≤ u − vH f (0) ≤ u − vH τ0 H1 f H1

for every f ∈ H1 . Hence Mu C0 − Mv C0 H1 ,H ≤ u − vH τ0 H1 . It is not difficult to show that there is a continuous path {ut : 0 ≤ t ≤ 1} in H such u0 = u, u1 = v and ut = 0 for every 0 ≤ t ≤ 1. For 0 ≤ t0 ≤ 1, we have Mut0 C0 − Mut C0 H1 ,H ≤ ut0 − ut H τ0 H1 . Letting t → t0 , we have Mut C0 → Mut0 C0 in Cw (H1 , H). Hence Mu C0 and Mv C0 are in the same path connected component in Cw (H1 , H). Thus by the last paragraph, Cw (H1 , H) is a path connected space. 2 Lemma 2.3. If ϕ∞ < 1 and u ∈ H, then Mu Cϕ ∈ Cw (H) and is compact. Proof. By the first paragraph of the proof of Theorem 2.2, we have Mu Cϕ ∈ Cw (H). To show that Mu Cϕ is compact, let {fn }n be a sequence in H such that there is a positive constant K satisfying fn H < K for every n. By (#1), we may assume that fn converges to some f ∈ H(D) uniformly on any compact subset of D. By the assumption, fn ◦ϕ → f ◦ϕ in H ∞ . Hence by (#2), u(fn ◦ϕ), u(f ◦ϕ) ∈ H and   Mu Cϕ fn − u(f ◦ ϕ) ≤ uH fn ◦ ϕ − f ◦ ϕ∞ → 0, H Thus Mu Cϕ ∈ Cw (H) is compact.

n → ∞.

2

Corollary 2.4. If z n H /z n H1 → 0 as n → ∞, then any Mu Cϕ ∈ Cw (H1 , H) is compact. Proof. For 0 < r < 1, by Lemma 2.3 Mu Crϕ ∈ Cw (H) is compact. By (#5), id : H1 → H is bounded. Hence Mu Crϕ : H1 → H is compact. By the proof of Theorem 2.2, we get the assertion. 2 3. Path connectedness of Cw,HS (H1 , H) Let H and H1 be the spaces satisfying conditions given in the introduction. We note that Cw,HS (H1 , H) ⊂ Cw (H1 , H) = Cw (H) as sets and the topology on Cw,HS (H1 , H) is induced by the Hilbert–Schmidt norm.

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.6 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

6

Lemma 3.1. (i) {Mu Cϕ : ϕ ∈ H ∞ , ϕ∞ < 1, u ∈ H, u = 0} ⊂ Cw,HS (H). (ii) Cw,HS (H) ⊂ Cw,HS (H1 , H). Proof. (i) Let ϕ ∈ H ∞ with ϕ∞ < 1 and u ∈ H with u = 0. We have Mu Cϕ 2H,HS =

∞ ∞   uϕn 2H ϕ2n ∞ 2 ≤ u H n 2 n 2 z z H H n=0 n=0

by (#2)

= u2H Cϕ∞ 2H,HS . Since Cϕ∞ is rank one, so is Hilbert–Schmidt. Hence Mu Cϕ ∈ Cw,HS (H). (ii) Let Mu Cϕ ∈ Cw,HS (H). Since id : H1 → H is bounded and Mu Cϕ |H1 = Mu Cϕ · id, we get (ii).

2

Theorem 3.2. Cw,HS (H1 , H) is a path connected space. Proof. Let Mu Cϕ ∈ Cw,HS (H1 , H). By (#3), ∞  uϕn 2H = Mu Cϕ 2H1 ,H,HS < ∞. n 2 z H1 n=0

(3.1)

We shall show that {Mu Crϕ : 0 ≤ r ≤ 1} is a path connected set in Cw,HS (H1 , H). By Lemma 3.1, Mu Crϕ ∈ Cw,HS (H1 , H) for every 0 ≤ r ≤ 1. Let us fix 0 ≤ r0 ≤ 1. We shall show that Mu Cr0 ϕ −Mu Crϕ H1 ,H,HS → 0 as r → r0 . For any positive integer N , we have Mu Cr0 ϕ − Mu Crϕ 2H1 ,H,HS =

∞ N −1 ∞ n 2     n  u(r0n − rn )ϕn 2H uϕn 2H r0 − rn 2 uϕ H + ≤ . 2 2 n n z H1 z H1 z n 2H1 n=0 n=0 n=N

Take ε > 0 arbitrarily. Then by (3.1), we may take N large enough so that ∞  uϕn 2H < ε. z n 2H1

n=N

Hence Mu Cr0 ϕ − Mu Crϕ 2H1 ,H,HS < ε +

N −1 

n 2  n  r0 − rn 2 uϕ H . n z 2H1 n=0

Letting r → r0 , we have lim sup Mu Cr0 ϕ − Mu Crϕ 2H1 ,H,HS ≤ ε. r→r0

Thus we get Mu Cr0 ϕ − Mu Crϕ 2H1 ,H,HS → 0 as r → r0 . Hence Mu Cϕ and Mu C0 are in the same path connected component in Cw,HS (H1 , H). Let Mv Cψ ∈ Cw,HS (H1 , H) be another operator. By the last paragraph, Mv Cψ and Mv C0 are in the same path connected component in Cw,HS (H1 , H). Let {ut : 0 ≤ t ≤ 1} be a continuous path in H such that u0 = u, u1 = v and ut = 0 for every 0 ≤ t ≤ 1. We have

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.7 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

Mut0 C0 − Mut C0 2H1 ,H,HS = ut0 − ut 2H → 0

7

as t → t0 .

Hence Mu C0 and Mv C0 are in the same path connected component in Cw,HS (H1 , H). Therefore Mu Cϕ and Mv Cψ are in the same path connected component in Cw,HS (H1 , H). This completes the proof. 2 4. Applications Let H 2 be the classical Hardy–Hilbert space on D. It is well known that H 2 satisfies conditions (#1)–(#4). For each f ∈ H 2 , it is known that there is the radial limit f ∗ almost everywhere on ∂D with respect to the normalized Lebesgue measure m. By Littlewood’s subordination theorem, Cϕ : H 2 → H 2 is bounded for every ϕ ∈ S(D) (see [14, p. 13]). For a given ϕ ∈ S(D), it is not known the characterization of the set of u ∈ H 2 such that Mu Cϕ : H 2 → H 2 is bounded [1,10]. But it is known that Cw (H 2 ) has many path connected components [3,8]. For −1 < α < ∞, the weighted Bergman space L2α on D is the space of f ∈ H(D) satisfying

f 2L2α :=

  f (z)2 dAα (z) < ∞,

D

where dAα = (α + 1)(1 − |z|2 )α dA and A stands for the normalized Lebesgue measure on D. When α = 0, L20 is the classical Bergman space. It is known that L2α satisfies conditions (#1)–(#4) and Cϕ : L2α → L2α is bounded for every ϕ ∈ S(D) (see [14, p. 17]). We have  n 2 z  2 = n!Γ(2 + α) , Lα Γ(n + 2 + α) where Γ(s) stands for the usual Gamma function (for example, see [17]). Then H 2  L2α and f L2α ≤ f H 2 for f ∈ H 2 . Also we have that for −1 < α1 < α2 < ∞, L2α1  L2α2 and f L2α ≤ f L2α for f ∈ L2α1 . For 2 1 a given ϕ ∈ S(D), it is not known the characterization of the set of u ∈ L2α such that Mu Cϕ : L2α → L2α is bounded. Refer to [6,7] for the boundedness and compactness of weighted composition operators on the weighted Bergman spaces. Moorhouse [12] partially answered the question of when two composition operators lie in the same component of Cw (L2α ). Let D be the Dirichlet space on D. For f ∈ H(D), we have that f ∈ D if and only if f 2D

2  := f (0) +



  2 f (z) dA(z) < ∞.

D

It is known that D satisfies conditions (#1), (#3) and (#4). But D does not satisfy condition (#2). For ∞ ∞ f = n=0 an z n ∈ D, we have f 2D = |a0 |2 + n=1 n|an |2 . Then D  H 2 and f H 2 ≤ f D for f ∈ D. It is also known that there is an analytic self-map ϕ ∈ S(D) such that Cϕ : D → D is not bounded (see [14, p. 18]). We note that 12D = 1 and z n 2D = n for every n ≥ 1. We have 1 z n H 2 = √ → 0 as n → ∞. z n D n By Stirling’s formula, we have Γ(n + λ) ∼ (n + 1)λ−1 , n!Γ(λ)

λ > 0.

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.8 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

8

Hence for −1 < α < ∞, we have z n 2L2

α

=

z n 2D

1 n!Γ(2 + α) ∼ → 0 as n → ∞ nΓ(n + 2 + α) n(n + 1)1+α

and z n 2L2

α

z n 2H 2

=

1 n!Γ(2 + α) ∼ →0 Γ(n + 2 + α) (n + 1)1+α

as n → ∞.

For −1 < α1 < α2 < ∞, we also have z n 2L2

α2

z n 2L2

α1

=

n!Γ(2 + α2 ) Γ(n + 2 + α1 ) Γ(n + 2 + α2 ) n!Γ(2 + α1 )

∼ (n + 1)−(2+α2 −1) (n + 1)2+α1 −1 = (n + 1)α1 −α2 → 0 as n → ∞. Hence by Theorem 2.2, we have the following. Corollary 4.1. Cw (D, H 2 ), Cw (D, L2α ), Cw (H 2 , L2α ) for −1 < α < ∞, and Cw (L2α1 , L2α2 ) for −1 < α1 < α2 < ∞ are path connected spaces. By Theorem 3.2, we have the following. Corollary 4.2. Cw,HS (D, H 2 ), Cw,HS (D, L2α ), Cw,HS (H 2 , L2α ) for −1 < α < ∞, and Cw,HS (L2α1 , L2α2 ) for −1 < α1 < α2 < ∞ are path connected spaces. √ Since {1, z n / n : n ≥ 1} is an orthonormal basis of D, in the same way as in Shapiro and Taylor [16] we have Mu Cϕ 2D,H 2 ,HS

=

u2H 2

∞   2 uϕn 2H 2 1 2 = uH 2 + u∗  log + dm. n 1 − |ϕ∗ |2 n=1 ∂D

Hence Mu Cϕ ∈ Cw (D, H 2 ) is Hilbert–Schmidt if and only if

 ∗ 2 u  log

∂D

1 dm < ∞. 1 − |ϕ∗ |2

Similarly, Mu Cϕ 2D,L2α ,HS = u2L2α +

∞ uϕn 2  L2

α

n=1

n

= u2L2α + D

  u(z)2 log

1 dAα (z) 1 − |ϕ(z)|2

and Mu Cϕ 2H 2 ,L2α ,HS

∞   n 2   uϕ L2 = = n=0

α

Hence Mu Cϕ ∈ Cw (D, L2α ) is Hilbert–Schmidt if and only if

D

|u(z)|2 dAα (z). 1 − |ϕ(z)|2

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.9 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••



  u(z)2 log

D

9

1 dAα (z) < ∞, 1 − |ϕ(z)|2

and Mu Cϕ ∈ Cw (H 2 , L2α ) is Hilbert–Schmidt if and only if

D

|u(z)|2 dAα (z) < ∞. 1 − |ϕ(z)|2

When H = H1 in Theorem 3.2, we have the following. Corollary 4.3. If H satisfies all (#1)–(#4), then Cw,HS (H) is a path connected space. 5. Differences of composition operators In this section, we study the Hilbert–Schmidt norms of differences of composition operators in C(D, H 2 ) and C(D, L2α ) for −1 < α < ∞. We note that      C D, H 2 = C D, L2α = Cϕ : ϕ ∈ S(D) as sets. By the same way as in the proof of Theorem 2.2, C(D, H 2 ) and C(D, L2α ) are path connected spaces with respect to the operator norm topologies. As in Section 4, we have the following. Lemma 5.1. (i) Cϕ ∈ CHS (D, H 2 ) if and only if

log ∂D

1 dm < ∞. 1 − |ϕ∗ |2

(ii) Cϕ ∈ CHS (D, L2α ) if and only if

log D

1 dAα (z) < ∞. 1 − |ϕ(z)|2

For ϕ ∈ S(D), we write E(ϕ) = {eiθ ∈ ∂D : |ϕ∗ (eiθ )| = 1}. In case (i) of Lemma 5.1, we have m(E(ϕ)) = 0 and by [9, p. 138] we may say that Cϕ ∈ CHS (D, H 2 ) if and only if ϕ is a non-extreme point of the closed unit ball of H ∞ . Applying the same way as in the proof of Theorem 3.2, CHS (D, H 2 ) and CHS (D, L2α ) are path connected spaces with respect to the Hilbert–Schmidt norm topologies. Theorem 5.2. CHS (D, L2α ) = {Cϕ : ϕ ∈ S(D)} as sets for every −1 < α < ∞. Proof. We have

log D

1 1 dAα (z) = < ∞. 1 − |z|2 (α + 1)2

Let ϕ ∈ S(D) with ϕ(0) = 0. Then |ϕ(z)| ≤ |z| on D and

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.10 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

10

D

1 log dAα (z) ≤ 1 − |ϕ(z)|2

log D

1 dAα (z) < ∞. 1 − |z|2

By Lemma 5.1(i), we have Cϕ ∈ CHS (D, L2α ). Let ϕ ∈ S(D) with ϕ(0) = 0. Put a = ϕ(0) and ψ(z) =

ϕ(z) − a , 1 − aϕ(z)

z ∈ D.

Then ψ ∈ S(D) with ψ(0) = 0. By the last paragraph,

log

1 dAα (z) < ∞, 1 − |ψ(z)|2

log

1 dAα (z) < ∞. 1 − |ψ(z)|

D

so

D

We have ϕ(z) =

ψ(z) + a , 1 + aψ(z)

z ∈ D.

Since   |ψ(z)| + |a| (1 − |a|)(1 − |ψ(z)|) = , 1 − ϕ(z) ≥ 1 − 1 + |a||ψ(z)| 1 + |a||ψ(z)| we have 2 1 ≤ . 1 − |ϕ(z)| (1 − |a|)(1 − |ψ(z)|) Hence

D

1 dAα (z) ≤ log 1 − |ϕ(z)|

D

2 dAα (z) + log 1 − |a|

so, by Lemma 5.1(ii) we have ϕ(z) ∈ CHS (D, L2α ).

log D

1 dAα (z) < ∞, 1 − |ψ(z)|

2

By Lemma 5.1 and Theorem 5.2, we have the following. Corollary 5.3. CHS (D, H 2 )  CHS (D, L2α ) = {Cϕ : ϕ ∈ S(D)} for every −1 < α < ∞. Corollary 5.4. For any ϕ, ψ ∈ S(D), Cϕ − Cψ : D → L2α is Hilbert–Schmidt for every −1 < α < ∞. For ϕ, ψ ∈ S(D), we have Cϕ − Cψ 2D,H 2 ,HS =

∞ ∞     1 1 ϕn − ψ n 2 2 ≥ ϕn − ψ n 2 2 = Cϕ − Cψ 2 2 D,Lα ,HS . H Lα n n n=1 n=1

For z, w ∈ D with z = w, let ρ(z, w) = |z − w|/|1 − wz|.

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.11 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

11

Theorem 5.5. For ϕ, ψ ∈ S(D), we have

Cϕ − Cψ 2D,L2α ,HS =

log D

1 dAα (z). 1 − ρ(ϕ(z), ψ(z))2

Proof. We have that ∞   1 ϕn − ψ n 2 2 Lα n n=1

 ∞ n |ϕ(z)|2n + |ψ(z)|2n − ϕn (z)ψ(z)n − ϕ(z)n ψ(z) dAα (z) = n n=1

Cϕ − Cψ 2D,L2α ,HS =

D



= D

1 1 1 1 − log log + log − log 1 − |ϕ(z)|2 1 − |ψ(z)|2 1 − ϕψ(z) 1 − ϕ(z)ψ(z)

=

log

|1 − ϕ(z)ψ(z)|2 dAα (z) (1 − |ϕ(z)|2 )(1 − |ψ(z)|2 )

log

1 dAα (z). 1 − ρ(ϕ(z), ψ(z))2

D

= D

dAα (z)

2

Corollary 5.6. For ϕ, ψ ∈ S(D), we have

log D

1 dAα (z) < ∞. 1 − ρ(ϕ(z), ψ(z))2

Proof. By Theorem 5.2, Cϕ D,L2α ,HS + Cψ D,L2α ,HS < ∞. By Theorem 5.5,

log D

1 dAα (z) = Cϕ − Cψ 2D,L2α ,HS 1 − ρ(ϕ(z), ψ(z))2  2 ≤ Cϕ D,L2α ,HS + Cψ D,L2α ,HS < ∞.

2

Corollary 5.7. For ϕ, ψ ∈ S(D) and 0 ≤ t ≤ 1, let ϕt = tϕ + (1 − t)ψ ∈ S(D). Then {Cϕt : 0 ≤ t ≤ 1} is a continuous path in CHS (D, L2a ) connecting Cϕ with Cψ . Proof. By Theorem 5.2, Cϕt ∈ CHS (D, L2a ) for every 0 ≤ t ≤ 1. It is sufficient to show that lim Cϕt − Cϕ 2D,L2α ,HS = 0.

t→1

Since ρ(ϕ(z), ϕt (z)) ≤ ρ(ϕ(z), ψ(z)) for z ∈ D, log By Corollary 5.6,

1 1 ≤ log 1 − ρ(ϕ, ϕt )2 1 − ρ(ϕ, ψ)2

on D.

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.12 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

12

log D

1 dAα (z) < ∞. 1 − ρ(ϕ(z), ψ(z))2

Since ρ(ϕ(z), ϕt (z)) → 0 as t → 1, by the Lebesgue dominated convergence theorem,

log D

1 dAα (z) → 0 1 − ρ(ϕ(z), ϕt (z))2

as t → 1. By Theorem 5.5, we get the assertion. 2 Next, we shall study the structure of CHS (D, H 2 ). For ϕ, ψ ∈ S(D) with ϕ = ψ, we have that ϕ∗ (eiθ ) = ψ (eiθ ) for almost every eiθ ∈ ∂D. Hence we may define ρ(ϕ∗ (eiθ ), ψ ∗ (eiθ )) for almost every eiθ ∈ ∂D. In the same way as in the proof of Theorem 5.5, we have the following. ∗

Theorem 5.8. For ϕ, ψ ∈ S(D) with ϕ = ψ, we have

Cϕ − Cψ 2D,H 2 ,HS =

log ∂D

1 dm. 1 − ρ(ϕ∗ , ψ ∗ )2

Proof. We have that ∞   1 ϕn − ψ n 2 2 H n n=1

 ∞ |ϕ∗ |2n + |ψ ∗ |2n − ϕn ψ ∗ n − ϕ∗ n ψ ∗ n dm = n n=1

Cϕ − Cψ 2D,H 2 ,HS =

∂D



= ∂D

1 1 1 1 log + log − log − log ∗ ψ∗ ∗ ∗ 1 − |ϕ∗ |2 1 − |ψ ∗ |2 1 − ϕ 1−ϕ ψ



log

= ∂D

|1 − ϕ∗ ψ ∗ |2 dm = (1 − |ϕ∗ |2 )(1 − |ψ ∗ |2 )

log ∂D

1 dm. 1 − ρ(ϕ∗ , ψ ∗ )2

dm

2

Corollary 5.9. For ϕ, ψ ∈ S(D) with Cϕ , Cψ ∈ CHS (D, H 2 ) and 0 ≤ t ≤ 1, let ϕt = tϕ + (1 − t)ψ ∈ S(D). Then {Cϕt : 0 ≤ t ≤ 1} is a continuous path in CHS (D, H 2 ) connecting Cϕ with Cψ . Proof. By the fact mentioned in the below of Lemma 5.1, ϕ and ψ are non-extreme points of the closed unit ball of H ∞ . Hence Cϕt ∈ CHS (D, H 2 ). Using Theorem 5.8, in the same way as in the proof of Corollary 5.7 we may prove the assertion. 2 / CHS (D, H 2 ). Then Cϕ − Cψ D,H 2 ,HS = ∞ for every ψ ∈ S(D) Theorem 5.10. Let ϕ ∈ S(D) satisfy Cϕ ∈ with ψ = ϕ. Proof. Suppose that Cϕ − Cψ D,H 2 ,HS < ∞ for some ψ ∈ S(D) with ψ = ϕ. By Theorem 5.8, we have m(E(ϕ)) = m(E(ψ)) = 0 and

log ∂D

1 dm < ∞. 1 − ρ(ϕ∗ , ψ ∗ )2

Let η = (ϕ + ψ)/2. We have that ρ(ϕ∗ , η ∗ ) ≤ ρ(ϕ∗ , ψ ∗ ) almost everywhere on ∂D. Hence

JID:YJMAA

AID:18716 /FLA

Doctopic: Complex Analysis

[m3L; v 1.134; Prn:18/08/2014; 16:10] P.13 (1-13)

T. Hosokawa et al. / J. Math. Anal. Appl. ••• (••••) •••–•••

∂D

1 log dm ≤ 1 − ρ(ϕ∗ , η ∗ )2

log ∂D

13

1 dm < ∞. 1 − ρ(ϕ∗ , ψ ∗ )2

By Theorem 5.8 again, Cϕ − Cη D,H 2 ,HS < ∞. Since η is a non-extreme point of the closed unit ball of H ∞ , by Lemma 5.1 we have Cη ∈ CHS (D, H 2 ). Then Cϕ ∈ CHS (D, H 2 ). This is a contradiction. Thus we get the assertion. 2 Corollary 5.11. Let ϕ, ψ ∈ S(D). If Cϕ − Cψ D,H 2 ,HS < ∞, then both Cϕ and Cψ are in CHS (D, H 2 ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

K.R.M. Attele, Multipliers of the range of composition operators, Tokyo J. Math. 15 (1992) 185–198. E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981) 230–232. P.S. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003) 228–245. M.D. Contreras, A.G. Hernández-Díaz, Weighted composition operators between different Hardy spaces, Integral Equations Operator Theory 46 (2003) 165–188. C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. Ž. Čučković, R. Zhao, Weighted composition operators on the Bergman space, J. Lond. Math. Soc. (2) 70 (2004) 459–511. Ž. Čučković, R. Zhao, Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math. 51 (2007) 479–498. E.A. Gallardo-Gutiérrez, M.J. González, P.J. Nieminen, E. Saksman, On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008) 986–1001. K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, New Jersey, 1962. K.J. Izuchi, Q.D. Nguyen, S. Ohno, Composition operators induced by analytic maps to the polydisk, Canad. J. Math. 64 (2012) 1329–1340. B.D. MacCluer, Components in the space of composition operators, Integral Equations Operator Theory 12 (1989) 725–738. J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005) 70–92. J. Moorhouse, C. Toews, Differences of composition operators, Contemp. Math. 321 (2003) 207–213. J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. J.H. Shapiro, C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990) 117–152. J.H. Shapiro, P.D. Taylor, Compact, nuclear, and Hilbert–Schmidt composition operators on H 2 , Indiana Univ. Math. J. 23 (1973) 471–496. K. Zhu, Operator Theory on Function Spaces, Marcel Dekker, New York, 1990, second edition, Amer. Math. Soc., Providence, 2007.