Extended spectrum and extended eigenspaces of quasi-normal operators

TL;DR

This paper characterizes the sets of extended eigenvalues and eigenvectors for products of positive and self-adjoint operators, as well as for normal operators, expanding understanding of their spectral properties.

Contribution

It provides a detailed description of extended spectra and eigenspaces for specific classes of operators, including products of positive, self-adjoint, and normal operators.

Findings

Characterization of extended eigenvalues for operator products

Description of extended eigenvectors for these operators

Extension of spectral theory to broader classes of operators

Abstract

We say that a complex number $\lambda$ is an extended eigenvalueof a bounded linear operator T on a Hilbert space H if there exists anonzero bounded linear operator X acting on H, called extended eigen-vector associated to $\lambda$, and satisfying the equation T X = $\lambda$XT . In thispaper we describe the sets of extended eigenvalues and extended eigen-vectors for the product of a positive and a self-adjoint operator whichare both injective. We also treat the case of normal operators.