DOUBLY ABEL BOUNDED OPERATORS WITH SINGLE SPECTRUM

Abstract

In a unitary Banach algebra conditions are sought for an element, the spectrum of which is the singleton {1}, to be the identity of the algebra. Our main result is that this is true if the element is Abel bounded and doubly (N)-uniformly Abel bounded. This condition is fulfilled if the element is doubly power bounded in which case the theorem was proved by I. Gelfand [4] (1941). If the element is doubly Cesàro bounded, the result is due to M. Mbekhta and J. Zemánek [7, Theorem 2]. We present an example of a matrix which is Abel bounded and doubly (2)-Abel bounded, but not Cesàro bounded.