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

Oscar Blasco, Jan van Neerven
2009 Vector Measures, Integration and Related Topics  
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
more » ... c 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 < ∞. (2000) . 28B05, 46G10. Mathematics Subject Classification
doi:10.1007/978-3-0346-0211-2_6 fatcat:2emzjejpuve5fhwcnofpnhlmai