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Hyperinvariant subspaces for normaloid essential isometric operators

Neeru Bala, Ramesh Golla

Abstract

In this article, we prove the existence of a non-trivial hyperinvariant subspace for a subclass of compact perturbations of scalar multiple of a partial isometry. Later, we illustrate that this class contains several important classes of operators. As a consequence, we prove that a Schatten class perturbation of a partial isometry with finite-dimensional null space has a non-trivial hyperinvariant subspace.

Hyperinvariant subspaces for normaloid essential isometric operators

Abstract

In this article, we prove the existence of a non-trivial hyperinvariant subspace for a subclass of compact perturbations of scalar multiple of a partial isometry. Later, we illustrate that this class contains several important classes of operators. As a consequence, we prove that a Schatten class perturbation of a partial isometry with finite-dimensional null space has a non-trivial hyperinvariant subspace.

Paper Structure

This paper contains 4 sections, 18 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Normaloid essentially isometric opertors
  4. Absolutely norm attaining operators

Key Result

Lemma 3.1

Let $T\in\mathcal{B}(H)$ and $N(T)=N(T^*)$. Then $\sigma_{\text{ess}}(T^*T)=\sigma_{\text{ess}}(TT^*)$.

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 27 more