Essential norm of Cesà ro operators on Lp and Cesà ro spaces

In this paper, we consider the Cesàro-mean operator Γ 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 Lp([0,1]), for p∈(1,+∞] and show that its value is the same as its norm: p/(p−1). The result also holds in the discrete case. On Cesàro spaces the essential norm of Γ turns out to be 1. At last, we introduce the Müntz-Cesàro spaces and study some of their geometrical properties. In this framework, we also compute the value of the essential norm of the Cesàro operator and the multiplication operator restricted to those Müntz-Cesàro spaces.