Vol. 92, No. 2, 1981

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Representation of compact and weakly compact operators on the space of Bochner integrable functions

Kevin T. Andrews

Vol. 92 (1981), No. 2, 257–267
Abstract

If X has the Radon-Nikodym property, then for every compact operator T : L1(μ,X) Y there is a bounded function g : Ω L(X,Y ) that is measurable for the uniform operator topology on L(X,Y ) such that

      ∫
T(f) =   fgdμ
Ω

for all f in L1(μ,X). The same result holds for weakly compact operators if X is separable Schur space. These representations yield Radon-Nikodym theorems for operator valued measures and a generalization of a theorem of D. R. Lewis.

Mathematical Subject Classification 2000
Primary: 47B38
Secondary: 46E40, 47B05
Milestones
Received: 24 September 1979
Revised: 15 February 1980
Published: 1 February 1981
Authors
Kevin T. Andrews