Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions

Abstract

For a fixed nonnegative integer m, an analytic map ϕ and an analytic function ψ, the generalized integration operator I (m) ϕ,ψ is defined byfor X-valued analytic function f , where X is a Banach space. Some estimates for the norm of the operator I (m) ϕ,ψ : wA p α (X) → A p α (X) are obtained. In particular, it is shown that the Volterra operator J b : wA p α (X) → A p α (X) is bounded if and only ifα is in the Schatten class S p (A 2 α ) for 2 ≤ p < ∞ and α > −1. Some corresponding results are established for X-valued Hardy spaces and X-valued Fock spaces.