Invariant subspaces of analytic perturbations

TL;DR

This paper classifies invariant subspaces of finite rank analytic perturbations of the unilateral shift on Hardy spaces, revealing distinctive properties and providing illustrative examples.

Contribution

It introduces a natural class of finite rank operators for analytic perturbations and offers a complete classification of their invariant subspaces.

Findings

Complete classification of invariant subspaces for certain finite rank perturbations.

Identification of properties like cyclicity, essential normality, and hyponormality in these perturbations.

Examples illustrating the theoretical results and properties.

Abstract

By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, we first isolate a natural class of finite rank operators for which the corresponding perturbations are analytic, and then we present a complete classification of invariant subspaces of those analytic perturbations. We also exhibit some instructive examples and point out several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations.