Convergence of certain subsequences of the power sequence in a Banach algebra

Abstract

We consider the existence of limits‎

‎$\limn a^{qn}$ and‎

‎$\limn a^{q^n}$‎,

‎where $a$ is an arbitrary element of a complex Banach algebra with the‎

‎unit‎, ‎and‎

‎$q$ is an integer‎, ‎$q\geq 2$‎.