On Quasinormality of compact perturbations of the isometries

TL;DR

This paper characterizes when compact perturbations of isometries are quasinormal, providing classifications for rank-one perturbations of unilateral shifts and Hardy shifts in Hilbert spaces.

Contribution

It offers a complete characterization and classification of quasinormality for compact perturbations of isometries, especially rank-one perturbations of shifts.

Findings

Complete characterization of quasinormality for compact perturbations of isometries.

Classification of rank-one perturbations of unilateral shifts as quasinormal.

Extension of results to Hardy shifts and general separable Hilbert spaces.

Abstract

We study the compact perturbations of an isometry on a separable Hilbert space and provide a complete characterization of when they are quasinormal. Based on that, we present a complete classification for a rank-one perturbation of a unilateral shift of finite multiplicity to be quasinormal in the setting of the Hardy space. The result can also be generalized for a separable Hilbert space. As an application, we provide a complete characterization for quasinormality of a rank-one perturbation of the Hardy shift.