Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

Valentin Matache

doi:10.1515/conop-2016-0009

Abstract

AbstractOperators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Highlights

Given a linear space L consisting of complex valued functions with common domain of definition D and a selfmap of D, we call the composition operator with symbol and denote C the linear mapC f Df ı f 2 L: If is an analytic map on D, the linear functionT ; fD fı f 2L is called the weighted composition operator with symbols and '
; The unitary invariant properties of the composition operators on H 2....C/ can be reduced to the study of the corresponding properties for weighted composition operators on H 2.U/, as follows
One of the tools in the study of composition operators on H 2....C/ is the fact that C is unitarily equivalent to T ;', .z/ D .1 '.z//=.1 z/, z 2 U, a weighted composition operator acting on H 2.U/ [11]

Summary

Introduction

Given a linear space L consisting of complex valued functions with common domain of definition D and a selfmap of D, we call the composition operator with symbol and denote C the linear map. We are able to characterize the Hermitian composition operators on H 2....C/ and find their spectra, essential spectra, and numerical ranges (Theorem 3.1). We determine which Möbius, respectively which inner maps induce normal composition operators (Theorem 3.4 and Proposition 3.5). In the sequel we use the notation .C /, p.C /, e.C /, and W .C / for: the spectrum, point spectrum, essential spectrum, and numerical range of composition operators C , respectively

Invertible composition operators

Normal composition operators