On Quasinormality of compact perturbations of the isometries

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.