$G_1$ class elements in a Banach algebra

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 $$\Vert (z-a{)}^{-1}\Vert =\frac{1}{\text{d}(z,\sigma (a))}\forall z\in \mathbb{C}\setminus \sigma (a).$$ 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 $T$.

13 pages