SPECTRAL CHARACTERIZATIONS OF ELEMENTS IN SEMISIMPLE BANACH ALGEBRAS

We recall some definitions and properties of spectrum and spectral radius in Banach algebras and introduce some spectral characterizations of elements in semisimple Banach algebras. We investigate under what conditions, in a semisimple Banach algebra �� the number of elements in the spectrum of ���� is less than or equal to the number of elements in the spectrum of �� for all �� in an arbitrary neighbourhood of the identity. We also investigate the conditions that an element �� is of rank one and idempotent.