Power-regularity of weighted shift operators

Abstract

A linear bounded operator T on a complex Banach space X is said to be power-regular if the sequence \(\{\Vert T^n x\Vert ^{\frac{1}{n}}\}_{n=1}^{\infty }\) is convergent for every \(x\in X\). For unilateral weighted shift S, we give a sufficient condition that S is power-regular. As an application, we construct a class of power-regular operators. Moreover, we show that there exist invertible power-regular bilateral weighted shifts, whose inverses are not power-regular.