Compact operators without extended eigenvalues

TL;DR

This paper constructs specific compact quasinilpotent operators on a separable Hilbert space that have only the extended eigenvalue 1, challenging previous assumptions about their spectral properties.

Contribution

It demonstrates the existence of compact quasinilpotent operators with a singleton set of extended eigenvalues, specifically {1}, which is a novel finding.

Findings

Existence of such operators with only extended eigenvalue 1

Extended eigenvalues can be very limited for certain compact operators

Provides new insights into the spectral theory of compact operators

Abstract

A complex number $\lambda$ is called an extended eigenvalue of a bounded linear operator $T$ on a Banach space $\B$ if there exists a non-zero bounded linear operator $X$ acting on $\B$ such that $XT=\lambda TX$. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set ${1}$.