Essential norm of Cesàro operators on L p and Cesàro spaces

Ihab Al Alam (1) , Loïc Gaillard , Georges Habib (2) , Pascal Lefèvre (3) , Fares Maalouf (4)

1 UNIVERSITE LIBANAISE Faculté de Sciences II
2 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]
3 LML - Laboratoire de Mathématiques de Lens
4 USJ - Université Saint-Joseph de Beyrouth

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 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 Cesàro spaces, the essential norm of Γ turns out to be equal to 1. Lastly, we introduce the Müntz-Cesàro spaces and study some of their geometrical properties. In this framework, we also compute the essential norm of the Cesàro and multiplication operators restricted to those Müntz-Cesàro spaces.

Mots clés: Cesàro spaces; Cesàro operator; Müntz spaces; compact operator; essential norm; Multiplication operator; 47B07; 47B38

Domaines: Analyse fonctionnelle [math.FA]

Ihab Al Alam, Loïc Gaillard, Georges Habib, Pascal Lefèvre, Fares Maalouf. Essential norm of Cesàro operators on L p and Cesàro spaces. 2017. ⟨hal-01648668⟩