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Cauchy dual and Wold-type decomposition for bi-regular covariant representations

Dimple Saini

Abstract

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for {regular} completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation $(σ, V)$ modulo $N(\wV)$ is hyponormal modulo $N(\wV)$.

Cauchy dual and Wold-type decomposition for bi-regular covariant representations

Abstract

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for {regular} completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation modulo is hyponormal modulo .
Paper Structure (7 sections, 31 theorems, 74 equations)

This paper contains 7 sections, 31 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Properties of generalized inverse of regular covariant representations
  4. Wold-type decomposition for bi-regular covariant representations
  5. Cauchy dual of bi-regular covariant representations
  6. Powers of Moore-Penrose inverse
  7. Cauchy dual of concave covariant representations

Key Result

Theorem 1.2

$($Olofsson$)$ Let $V$ be a bounded linear operator on a Hilbert space $\mathcal{H}$ such that Then $V$ has the wandering subspace property.

Theorems & Definitions (70)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Lemma 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 60 more