Espaces de Bergman-Orlicz, espaces de fonctions holomorphes à croissance lente, et leurs opérateurs de composition

Stéphane Charpentier
I2M, Aix-Marseille Université
/user/stephane.charpentier/

Date(s) : 13/06/2016 iCal
10h00 - 11h00

Bergman-Orlicz spaces, spaces of slowly growing holomorphic functions, and their composition operators

We show that the weighted Bergman-Orlicz space $A_{α}^{ψ}$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $ψ$ satisfies the so-called $Δ^{2}$ –condition. In addition we prove that this condition characterizes those $A_{α}^{ψ}$ on which every composition operator is bounded or order bounded into the Orlicz space $L_{α}^{ψ}$ . This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $ψ$ satisfies the $Δ^{2}$ –condition, a composition operator is compact on $A_{α}^{ψ}$ if and only if it is order bounded into the so-called Morse-Transue space $M_{α}^{ψ}$ . Our results stand in the unit ball of $C^{N}$ .