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A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces

Arup Majumdar, P. Sam Johnson

Abstract

In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |λ|: λ\in σ_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |λ|: λ\in σ_{A}(T)\} = r_{A}(T), \text{ when } T \in B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte spectrum $σ_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}= (T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that $r_{A_{h}}(\mathbf{T}) \leq \sup \{\|λ\|_{2}: λ\in σ_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte spectral radius of $\mathbf{T}$.

A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces

Abstract

In this paper, we prove the relation , where is a positive semidefinite operator (not necessarily to have a closed range) and is the -spectral radius of in . Also we prove that . By introducing -Harte spectrum of a -tuple operator , we prove that , where is the -Harte spectral radius of .
Paper Structure (4 sections, 33 theorems, 14 equations)

This paper contains 4 sections, 33 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some characterizations of the $A$-spectral radius of $A^{\frac{1}{2}}$-adjoint operators
  4. Characterization of $A$-Harte spectral radius for commuting $A^{\frac{1}{2}}$-adjoint operators

Key Result

Theorem 2.2

MR0203464 Let $E, F \in B(H)$. The following conditions are equivalent: If one of these conditions holds, then there exists a unique operator $D \in B(H)$ such that $ED = F$ with $R(D) \subset \overline {R(E^{*})}$. Furthermore, $N(D) = N(F)$. The unique operator, $D$, is called the reduced solution of $EX = F$.

Theorems & Definitions (52)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 42 more