On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

Karim Hedayatian,Lotfollah Karimi

doi:10.1155/2009/931020

Abstract

A bounded linear operatorTon a Hilbert spaceℋ, satisfying‖T2h‖2+‖h‖2≥2‖Th‖2for everyh∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

Highlights

Introduction and PreliminariesWe denote by B H the space of all bounded linear operators on a Hilbert space H
It is easy to see that for every convex operator T, the sequence T ∗n ΔT T n n forms an increasing sequence
Proposition 1.3 implies that Cφ∗ ΔφCφ ≥ Δφ ≥ 0 if and only if Cφ is an isometry

Summary

Introduction and Preliminaries

We denote by B H the space of all bounded linear operators on a Hilbert space H. Isometries of the Hardy space H2 among composition operators are characterized in 17, page 444 , 18 and 12, page 66. Bayart 5 generalized this result and showed that every composition operator on H2 which is similar to an isometry is induced by an inner function with a fixed point in the unit disc. We are interested in studying the convexity of composition and multiplication operators acting on a weighted Hardy space H2 β. We will offer necessary and sufficient conditions under which a convex composition operator may be isometry on a large class of weighted Hardy spaces containing Hardy, Bergman, and Dirichlet spaces. It is easy to see that for every convex operator T , the sequence T ∗n ΔT T n n forms an increasing sequence We use this fact to prove the following theorem.

Composition Operators

Multiplication Operators