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Tytuł artykułu

On Some Classes of Operators on C(K,X)

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Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X) → Y is a strongly bounded operator with representing measure m:Σ → L(X,Y).
We show that if T is a strongly bounded operator and T̂:B(K,X) → Y is its extension, then T is limited if and only if its extension T̂ is limited, and that T* is completely continuous (resp. unconditionally converging) if and only if T̂* is completely continuous (resp. unconditionally converging).
We prove that if K is a dispersed compact Hausdorff space and T is a strongly bounded operator, then T is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) whenever m(A):X → Y is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) for each A ∈ Σ.
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  • Department of Mathematics, University of Wisconsin--River Falls, River Falls, WI 54022-5001, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-ba7997-1-2016
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