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A note on the $A$-numerical range of semi-Hilbertian operators

Anirban Sen, Riddhick Birbonshi, Kallol Paul

Abstract

In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed.

A note on the $A$-numerical range of semi-Hilbertian operators

Abstract

In this paper we explore the relation between the -numerical range and the -spectrum of -bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of -normal operator and prove that closure of the -numerical range of an -normal operator is the convex hull of the -spectrum. We further prove Anderson's theorem for the sum of -normal and -compact operators which improves and generalizes the existing result on Anderson's theorem for -compact operators. Finally we introduce strongly -numerically closed class of operators and along with other results prove that the class of -normal operators is strongly -numerically closed.
Paper Structure (4 sections, 33 theorems, 62 equations)

This paper contains 4 sections, 33 theorems, 62 equations.

Table of Contents

  1. Introduction and preliminaries
  2. $A$-spectrum and $A$-numerical range
  3. Anderson's theorem on semi-Hilbertian operators
  4. strongly $A$-numerically closed classes of $\mathcal{B}_{{A}^{1/2}}(\mathcal{H})$

Key Result

Lemma 2.1

arias 1feki_LAA_2020 (i) Let $T \in \mathcal{B}(\mathcal{H}).$ Then $T \in \mathcal{B}_{A^{1/2}}(\mathcal{H})$ if and only if there exists a unique $\widetilde{T} \in \mathcal{B}(\textbf{R}(A^{1/2}))$ such that $Z_AT=\widetilde{T}Z_A.$ Moreover, if $T,S \in \mathcal{B}_{A^{1/2}}(\mathcal{H})$ and $\

Theorems & Definitions (65)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Proposition 2.7
  • proof
  • ...and 55 more