The Spectrum of Operators on C(K) with the Grothendieck Property and Characterization of J-Class Operators which are adjoints

TL;DR

This paper explores the spectral properties of operators on C(K)-spaces with the Grothendieck property, characterizes J-class operators that are adjoints, and provides criteria for their existence and structure.

Contribution

It establishes a characterization of the spectrum boundary for operators on C(K) with the Grothendieck property and classifies J-class operators that are adjoints, including non-existence results.

Findings

C(K) has the Grothendieck property iff the spectrum boundary of every operator consists of eigenvalues of its adjoint.

No invertible J-class operators exist on C(K) with the Grothendieck property.

Complete characterization of J-class operators on l^{} that are adjoints from l^1 operators.

Abstract

This article deals with properties of spectra of operators on C(K)-spaces with the Grothendieck property (e.g. l^{\infty}) and application to so called J-class operators introduced by A. Manoussos and G. Costakis. We will show that C(K) has the Grothendieck property if and only if the boundary of the spectrum of every operator on C(K) consists entirely of eigenvalues of its adjoint. As a consequence we will see that there does not exist invertible J-class operators on C(K) with the Grothendieck property. In the third section we will give a quantitative and qualitative characterization of all J-class operators on l^{\infty} which are adjoints from operators on l^1.