In this paper, we completely characterize the compactness of the Volterra type integration operators J b acting from weighted Bergman spaces A p α to Hardy spaces H q for all 0 < p, q < ∞. Furthermore, we give some estimates for the essential norms of J b : A p α → H q in the case 0 < p ≤ q < ∞. We finally describe the membership in the Schatten(-Herz) class of the Volterra type integration operators.

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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.

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