Essential norm of Cesaro operators on Lp and Cesaro spaces

In this paper, we consider the Cesaro-mean operator Gamma acting on some Banach spaces of measurable functions on (0, 1), as well as its discrete version on some sequences spaces. We compute the essential norm of this operator on L-P([0, 1]), for p is an element of(1, +infinity] and show that its value is the same as its norm: p/(p-1). The result also holds in the discrete case. On Cesaro spaces the essential norm of Gamma turns out to be 1. At last, we introduce the Muntz-Cesaro spaces and study some of their geometrical properties. In this framework, we also compute the value of the essential norm of the Cesaro operator and the multiplication operator restricted to those Muntz-Cesaro spaces. (C) 2018 Elsevier Inc. All rights reserved.