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Perturbations of Toeplitz operators on vector-valued Hardy spaces

Arshad Khan, Sneh Lata, Dinesh Singh

Abstract

In this article, we completely classify invariant subspaces of finite-rank perturbations of a class of Toeplitz operators on vector-valued Hardy spaces. As a consequence, in the vector-valued setting, we characterize invariant and almost invariant subspaces of a class of Toeplitz operators, as well as nearly invariant subspaces associated with certain Blaschke-based operators. We further treat the finite defect case for these nearly invariant subspaces.

Perturbations of Toeplitz operators on vector-valued Hardy spaces

Abstract

In this article, we completely classify invariant subspaces of finite-rank perturbations of a class of Toeplitz operators on vector-valued Hardy spaces. As a consequence, in the vector-valued setting, we characterize invariant and almost invariant subspaces of a class of Toeplitz operators, as well as nearly invariant subspaces associated with certain Blaschke-based operators. We further treat the finite defect case for these nearly invariant subspaces.

Paper Structure

This paper contains 5 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statements of the main results
  4. Proofs of Theorems \ref{['Th3.2']} & \ref{['Th3.6']}, and their Applications I, II & III
  5. Proofs of Theorems \ref{['Th4.2']} & \ref{['Th4.4']}, and their general cases

Key Result

Lemma 2.2

BT Suppose $T$ is a bounded operator on a Hilbert space $\mathcal{H}$. Let $T$ be a $C._{0}$ contraction such that the rank of $I-T^*T$ is finite, and let $\mathcal{M}\subset \mathcal{H}$ be a subspace of finite codimension. Then $TP_\mathcal{M}$ is a $C._{0}$ contraction, where $P_\mathcal{M}$ is t

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 3.6
  • Theorem 3.7
  • Definition 3.8
  • ...and 16 more