Spectral radius of semi-Hilbertian space operators and its applications

TL;DR

This paper introduces the spectral radius concept for operators on semi-Hilbertian spaces, establishing inequalities and characterizations that extend classical spectral theory to this broader setting.

Contribution

It defines the $A$-spectral radius, compares it with the $A$-numerical radius, and characterizes $A$-normaloid and $A$-spectraloid operators, extending spectral analysis in semi-Hilbertian spaces.

Findings

Proves $r_A(T) \,\leq\, \omega_A(T)$ for $A$-bounded operators.

Shows $r_A(T) = \omega_A(T) = \|T\|_A$ for $A$-normaloid operators.

Provides characterizations of $A$-normaloid and $A$-spectraloid operators.

Abstract

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq \omega_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $\omega_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=\omega_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover, some characterizations of $A$-normaloid and $A$-spectraloid operators are given.