Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

Introduction Let D be the unit disk {z ∈ D : |z| < 1} in the complex plane. Let H(D) be the space of all analytic functions on D. Given a positive integrable function ω ∈ C[0, 1), we extend ω on D by setting ω(z) = w(|z|) for each z ∈ D. Let ω be the weight function such that ω(z)dm(z) defines a finite measure on D; that is, ω ∈ L(D, dm). For such a weight ω, the weighted Hilbert space Hω consists of all analytic functions f on D such that


Introduction
Let D be the unit disk {z ∈ D : |z| < 1} in the complex plane. Let H(D) be the space of all analytic functions on D. Given a positive integrable function ω ∈ C 2 [0, 1), we extend ω on D by setting ω(z) = w(|z|) for each z ∈ D. Let ω be the weight function such that ω(z)dm(z) defines a finite measure on D; that is, ω ∈ L 1 (D, dm). For such a weight ω, the weighted Hilbert space H ω consists of all analytic functions f on D such that For example, consider the weighted Hilbert space H α associated with the weight ω α (r) = (1 − r 2 ) α where α > −1. The weighted Hilbert space H 1 is the Hardy space H 2 . The Dirichlet space D α is H α for 0 ≤ α < 1. The weight Bergman space A 2 α is the weighted Hilbert space H α+2 .
Let φ be an analytic function maps D into itself, the composition operator induced by φ is defined on the space H(D) of all analytic functions on D by operator-theoretic properties of C φ in terms of the function-theoretic properties of the induced map φ. We refer the reader to the monographs ( [6], [8], [9], [23], [29], [30]), the papers ( [3]- [5]) and the references therein for the overview of the field as of the early 1990s. Composition operators on the weighted Hilbert space H ω have been studied by many authors, see for example [11], [22] and the related references therein.
In this paper, we consider the admissible weight ω. We say ω is admissible weight if it is nonincreasing and ω(r)(1 − r) −(1+δ) is non-decreasing for some δ > 0. We characterize the boundedness and compactness of the generalized composition operators on the space H ω using the ω-Carleson measures. Moreover, we give a lower bound for the essential norm of the generalized composition operators.

Preliminaries
In this section we provide some useful definitions and auxiliary results that are crucial for the paper's main results. For a fixed a ∈ D, the Möbius transformation is defined as ϕ a (z) = a − z 1 − az , for all z ∈ D. Furthermore, it is well known that for all a, z ∈ D, we have For r ∈ (0, 1) and a ∈ D, the pseudohyperbolic metric ρ on D is defined as ρ(z, a) = |ϕ a (z)|. Moreover, the pseudohyperbolic disk is defined as E(a, r) = {z ∈ D : ρ(z, a) < r}. It is well known that E(a, r) = ϕ a (rD), and for every z ∈ E(a, r) Carleson measures were first introduced by Carleson [2], who studied positive Borel measures µ on the unit disk that satisfy for any function f in the Hardy space H p (D) the condition as a tool to study interpolating sequences and the corona problem. These measures have been extended and found many applications in the study of composition operators in various spaces of functions, for example see [6], [8], [9], [29] and [30] for the study of Carleson measures in spaces of analytic functions. Following similar notation, we define (vanishing) ω-Carleson measures on the weighted Hilbert spaces.

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Definition 1. Let µ be a positive Borel measure. We say µ is a ω-Carleson measure if there exists a constant C > 0 such that for all f ∈ H ω , ∫ Moreover, we say µ is a vanishing ω-Carleson measure if for any bounded sequence {f k } in H ω that converges to zero uniformly on compact subsets of D as k → ∞.
The essential norm of a bounded operator T , denoted by ∥T ∥ e , is its distance in the operator norm from the space of compact operators. Thus, for any g ∈ H(D) and a self-analytic function φ of D we define where K is the space of compact operators on the space H ω . It is well known that a bounded operator T is compact if and only if ∥T ∥ e = 0, so that estimates of essential norm lead for I g,φ to be compact. The essential norm has been studied by many authors in spaces of analytic functions, see for example [1], [4], [17], [24] and the related references therein.
Let µ be a positive Borel measure on D, the averaging function of µ is µ(E(z, r)) m(E(a, r)) .
Since m(E(z, r)) ≈(1 − |z| 2 ) 2 , we setμ Let ω be the admissible weight. For convenience, we will use the notatioñ Definition 2. Let r ∈ (0, 1), and let {a n } be a sequence in D. We say the sequence {a n } is r-lattice, if there exits a positive integer m such that 1. the disk D is covered by the sequence {E(a n , r)}; 2. every point in D belongs to at most m sets in {E(a n , 2r)}.
For more information about r-lattice sequences we refer the reader to the monograph [29]. The following lemma is a combination of Lemmas 2.4 and 2.5 in Kellay's and Lefèvre's paper [11].

Lemma 3. Let ω be the admissible weight function. Then for any
Then ∥f a ∥ Hω ≈ 1.

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Let f be an analytic function on D. Then using the Mean Value Property for the analytic function f ′ , we get where D(z, r) is the Euclidean disk centered at a with radius r. Apply this to f ′ (ϕ a ) then use change of variables, we get for some constant C > 0 Using Equation (1) and Estimate (2), we get Now replacing |f ′ | by |f ′ |ω 1/2 ω −1/2 in the last inequality, then using Hölder's inequality we get Squaring the last inequality, we get Note that for 0 < r 1 < r 2 < 1, we have ω(r 2 ) ≤ ω(r 1 ) because ω is non-increasing. Also since Hence, using Equation (4) then Estimate (2), Inequality (3) becomes The above argument gives the following point-evaluation in D for any function f ∈ H ω .

Lemma 4.
Let ω be the admissible weight function, and r ∈ (0, 1). Then there exists a positive constant C (depending only on r) such that for any function f ∈ H ω , we have for any a ∈ D

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Suppose that g is an analytic function on D such that g ∈ L 2 (D, ωdm), and φ is an analytic self-map of D. We define the positive Borel measure on D by where E is a Borel subset of D. Hence, by (Theorem III.10.4, [7]), we get the following change of variable formula where f is an arbitrary measurable positive function on D.

Main Results
In this section, we characterize the boundedness and compactness of the generalized composition operators on the space H ω using the ω-Carleson measures. We also give a lower bound for the essential norm of the generalized composition operators. The following theorem gives a characterization of the ω-Carleson measure on the weighted Hilbert spaces H ω .

Theorem 5.
Let ω be the admissible weight function, r ∈ (0, 1) and µ be a positive Borel measure on D. Then the following are equivalent.

There exists a constant C > 0 such that for any
Proof. First, we show that (1) implies (2). Suppose that µ is an ω-Carleson measure. For any a ∈ D, consider the function f a that is defined in Lemma 3. Then we get f a ∈ H ω and ∥f a ∥ Hω ≈ 1. Since µ is an ω-Carleson measure, using Definition 1, there exists a constant C > 0 such that Since ∥f a ∥ Hω ≈ 1, we get condition (2).

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where the first inequality comes from the fact that for all z ∈ E(a, r), 1 − |a| 2 ≈ |1 − az|. Now, using our hypothesis condition (2), we have ∫ On one hand, for any a ∈ D with |a| ≤ 1/2 we have ∫ On the other hand, for any a ∈ D with |a| > 1/2 we have ∫ Using the Inequalities (5), (6), and (7) there exists a constant C 1 > 0 such that which gives condition (3), as desired.
By Lemma 4, there exists a constant C > 0 such that

Now, using our hypothesis condition (3), there exists a positive constant C
where m is the integer in Definition 2. Therefore, µ is an ω-Carleson measure. This completes the proof.

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The following theorem characterizes the vanishing ω-Carleson measure on the weighted Hilbert spaces H ω .

Theorem 6.
Let ω be the admissible weight function, r ∈ (0, 1) and µ be a positive Borel measure on D. Then the following are equivalent.
Proof. First, we show that (1) implies (2). Let {a k } be a sequence in D such that |a k | → 1 as k → ∞. For this sequence {a k }, we consider the function By using Lemma 3, {f k } is a bounded sequence in H ω and ∥f k ∥ Hω ≈ 1.
Since ω is a non-increasing function and a k ∈ D, we get This gives, for any z ∈ D Hence {f k } is a bounded sequence in H ω that converges to zero uniformly on compact subsets of D as k → ∞. Since µ is a vanishing ω-Carleson measure, using Definition 1, we have this gives condition (2), as desired. Second, we show that (2) implies (3). For any a ∈ D, using Estimate (2), we have Now, using our hypothesis condition (2), we get lim |a|→1 µ (E(a, r)) ω(a)(1 − |a| 2 ) 2 = 0, which gives condition (3), as desired.
Finally, we show that (3) implies (1). Let {f n } be a bounded sequence in H ω that converges to zero uniformly on compact subsets of D. Then there exists a constant M > 0 such that ∥f n ∥ Hω ≤ M . Now, for a fixed r > 0, we consider the r-lattice {a k } in D that is defined in Definition 2. Since a k → ∂D as k → 1, using our hypothesis condition (3), we get lim k→∞ µ (E(a k , r)) Then for a given ϵ > 0, there is a positive integer N 0 such that for all k ≥ N 0 Using similar argument to that in the proof of Theorem 5, there exists a constant C > 0 such that where m is the positive integer in Definition 2.
On the other hand, Let D 0 denote the union of the closure of E(a k , r) for k = 1, 2, 3, ..., N 0 − 1. Then D 0 is a compact subset of D. Since {f n } converges to zero uniformly on compact subsets of D, there exists a positive integer N 1 > N 0 such that for all n ≥ N 1 By Cauchy estimate, we know that {f ′ n } converges to zero uniformly on compact subsets of D as n → ∞. Therefore, last inequality gives

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Using Inequality (8) and Equation (9), we get Because ϵ > 0 is arbitrary, we have which gives that µ is a vanishing ω-Carleson measure, as desired. The proof is finished.
The following theorem characterizes the boundedness of the generalized composition operators I (g,φ) on the weighted Hilbert spaces H ω . Proof. Suppose that I (g,ω) is bounded on H ω . This is equivalent to for any f ∈ H ω , there exists a constant C > 0 such that ∥I (g,ω) f ∥ Hω ≤ C∥f ∥ Hω . Since I (g,ω) f (0) = 0, using change of variable formula, we get By Definition 1, we get the measure µ ω,g,φ is an ω-Carleson measure. Using Theorem 5, this is equivalent to sup Using the change of variable formula one more time, we get which gives the desired result.
The following corollary is an immediate consequence of Theorem 5 and Theorem 7. 2. The measure µ ω,g,φ is an ω-Carleson measure.
The next theorem gives a lower bound of the essential norm of the generalized composition operators I (g,φ) on the weighted Hilbert spaces H ω .
Theorem 9. Let g ∈ H(D), φ be an analytic self-map of D, and ω be the admissible weight function. If the operator I (g,φ) is bounded on H ω , then there exists a constant C > 0 such that Proof. Suppose that I (g,φ) is bounded on H ω . Let f a be the function defined in Lemma 3, then f a ∈ H ω and ∥f a ∥ Hω ≈ 1. Moreover, using similar argument to that in the proof of Theorem 6, we get f a converges to zero uniformly on compact subsets of D as |a| → 1. Now, for a fixed compact operator K on H ω , we have ∥Kf a ∥ Hω → 0 as |a| → 1. Hence, there exists a constant C > 0 such that Taking infimum over all compact operators K, we get the desired result.
Second, we show (2) is equivalent to (3). For any a ∈ D, we have By Theorem 6, we get that (2) is equivalent to (3).
Let {f k } be a bounded sequence in H ω that converges to zero uniformly on compact subsets of D.