$G_1$ class elements in a Banach algebra

TL;DR

This paper studies elements called $G_1$-class in Banach algebras, characterizing their spectral properties and eigenstructure, and providing examples and properties of such elements and operators.

Contribution

It introduces and analyzes the class of $G_1$-class elements, establishing their spectral and eigenvalue properties, and characterizes when they are scalar multiples of the identity.

Findings

$G_1$-class elements have spectra with specific properties.

Isolated spectral points of $G_1$-class operators are eigenvalues.

Finite spectrum $G_1$-class operators decompose into eigenspaces.

Abstract

Let $A$ be a complex unital Banach algebra with unit $1$. An element $a\in A$ is said to be of \textit{$G_{1}$-class} if $$\|(z-a)^{-1}\|=\frac{1}{\text{d}(z,\sigma(a))} \quad \forall z\in \mathbb{C}\setminus \sigma(a).$$ Here $d(z, \sigma(a))$ denotes the distance between $z$ and the spectrum $\sigma(a)$ of $a$. Some examples of such elements are given and also some properties are proved. It is shown that a $G_1$-class element is a scalar multiple of the unit $1$ if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if $T$ is a $G_1$ class operator on a Banach space $X$, then every isolated point of $\sigma(T)$ is an eigenvalue of $T$. If, in addition, $\sigma(T)$ is finite, then $X$ is a direct sum of eigenspaces of $T$.