Essential norm of Cesàro operators on L p and Cesàro spaces

In this paper, we consider the Cesaro-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 L p ([0, 1]), for p ∈ (1, +∞] and show that its value is the same as its norm, namely p/(p − 1). The result also holds in the discrete case. On Cesaro spaces, the essential norm of Γ turns out to be equal to 1. Lastly, we introduce the Muntz-Cesaro spaces and study some of their geometrical properties. In this framework, we also compute the essential norm of the Cesaro and multiplication operators restricted to those Muntz-Cesaro spaces.