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

Ioana Ghenciu. "On Some Classes of Operators on C(K,X)." Bulletin of the Polish Academy of Sciences. Mathematics 63.3 (2015): 261-274. <http://eudml.org/doc/281265>.

@article{IoanaGhenciu2015,
abstract = { 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. },
author = {Ioana Ghenciu},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Limited operators; weakly precompact operators; spaces of continuous functions},
language = {eng},
number = {3},
pages = {261-274},
title = {On Some Classes of Operators on C(K,X)},
url = {http://eudml.org/doc/281265},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Ioana Ghenciu
TI - On Some Classes of Operators on C(K,X)
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 3
SP - 261
EP - 274
AB - 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.
LA - eng
KW - Limited operators; weakly precompact operators; spaces of continuous functions
UR - http://eudml.org/doc/281265
ER -