Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology

Abstract

Let X and Y be Banach spaces and (Ω, Σ, μ) a finite measure space. In this note we introduce the space L ^p/μ; ℒ(X, Y)] consisting of all (equivalence classes of) functions Φ:Ω↦ℒ(X, Y) such that ω↦Φ(ω)x is strongly μ-measurable for all x∈X and ω↦Φ(ω)f(ω) belongs to L ¹(μ; Y) for all f∈L ^p′ (μ; X), 1/p+1/p′=1. We show that functions in L ^p/μ; ℒ(X, Y)] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all ℒ(X, Y)-valued multipliers acting boundedly from L ^p(μ; X) into L ^q(μ; Y), 1≤q<p<∞.