$$G_{1}$$-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 $$G_{1}$$ -class if $$\begin{aligned} \Vert (z-a)^{-1}\Vert =\frac{1}{\text {d}(z,\sigma (a))} \quad \forall z\in {\mathbb {C}}\setminus \sigma (a). \end{aligned}$$ 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.