Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

Waleed Al-Rawashdeh

doi:10.18052/www.scipress.com/ijarm.10.1

Abstract

Letφbe an analytic self-map of the open unit disk D andgbe an analytic function on D. The generalized composition operator induced by the mapsgandφis defined by the integral operatorI(g,φ)f(z) =∫0zf′(φ(ς))g(ς)dς. Given an admissible weightω, the weighted Hilbert spaceHωconsists of all analytic functionsfsuch that ∥f∥2Hω= |f(0)|2+∫D|f′(z)|2ω(z)dA(z) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the spaceHωusing theω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

Highlights

Let D be the unit disk {z ∈ D : |z| < 1} in the complex plane
Let ω be the weight function such that ω(z)dm(z) defines a finite measure on D; that is, ω ∈ L1(D, dm). For such a weight ω, the weighted Hilbert space Hω consists of all analytic functions f on D such that
In this paper we use similar techniques to study this integral operator on the weighted Hilbert spaces of analytic functions

Summary

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. 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. It is well known that the composition operator Cφf = f ◦ φ defines a linear operator Cφ which acts boundedly on various spaces of analytic or harmonic functions on D. These operators have been studied on many spaces of analytic functions. Composition operators on the weighted Hilbert space Hω have been studied by many authors, see for example [11], [22] and the related references therein. In this paper we use similar techniques to study this integral operator on the weighted Hilbert spaces of analytic functions. We provide some useful definitions and auxiliary results that are crucial for the a−z paper’s main results

For a fixed a

Main Results

It follows that