A full-invariant theorem and some applications |
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Authors: | Wu Junde Lu Shijie |
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Institution: | Department of Mathematics, Zhejiang University, Yuquan Campus, Hang Zhou, 310027, People's Republic of China |
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Abstract: | Let (X,τ1) and (Y,τ2) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑iTi in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ1)→(λβ,σ(λβ,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true. |
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Keywords: | Locally convex space Sequence space Compact operator Full-invariant |
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