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

100%

Ghenciu I.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

tom Vol. 63, no. 3

261--274

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 T : B(K,X) → Y is its extension, then T is limited if and only if its extension T is limited, and that T∗ is completely continuous (resp. unconditionally converging) if and only if T∗ 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.

Weakly precompact operators on $C_{b}(X,E)$ with the strict topology

88%

Stochmal J.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2016

tom 36

nr 1

65-77

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b}(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T:C_{b}(X,E) → F$. In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T:C_{b}(X,E) → F$ is weakly precompact.