Weak precompactness, strong boundedness, and weak complete continuity

Abstract

Weak precompactness in spaces of vector measures and in the space of Bochner integrable functions is studied. Uniform countable additivity and uniform integrability are characterized in terms of weak precompactness. Through this, a connection between strongly bounded operators and operators having weakly precompact adjoints on abstract continuous function spaces is established. These operators are compared with weakly completely continuous operators.

Ideal Exhaustiveness, Weak Convergence and Weak Compactness in Banach Spaces

“…Remark 4.18. Observe that the thesis of Proposition 4.17 can be interpreted as a weak I-Cauchy-type condition in the space L ∞ (λ): indeed we know from the classical literature that, by virtue of the Riesz representation theorem, the dual of L ∞ (λ) is isomorphic to the space of all countable additive λ-continuous measures, and the integral is a functional which realizes such an isomorphism (see [1,24,25,23,35,36]). The topics and the tools about Proposition 4.17 are related also with some properties of precompactness of sets of measures and uniform integrability (see also [1,24,25,23,35,36]).…”

Strongly Bounded Representing Measures and Convergence Theorems

“…(ii) Every unconditionally converging operator on C(K, X) is strongly bounded [3,19,28], and thus weakly compact.…”

“…Using the existence of a control measure for m and Lusin's theorem, we can find a compact subset K 0 of K such thatm(K \ K 0 ) < /2 and φ n = h n | K 0 is continuous for each n ∈ ‫.ގ‬ Let H = [φ n ] be the closed linear span of (φ n ) in C(K 0 , X) and S : H → C(K, X) be the isometric extension operator given by Theorem 1 of [13]. Let ψ n = S(φ n ), n ∈ ‫.ގ‬ Since (φ n (t)) is weakly Cauchy for each t ∈ K 0 , the sequence (φ n ) is weakly Cauchy in C(K 0 , X) (Theorem 9 of [19], Lemma 3.2 of [3]). Then (ψ n ) is weakly Cauchy in C(K, X) and (T(ψ n )) is weakly Cauchy.…”

“…T is unconditionally converging and m is strongly bounded [3,19,28]. Let (f n ) be a sequence in the unit ball of C(K, X).…”

A Riesz representation theory for completely regular Hausdorff spaces and its applications

“…L.E; F 00 / has been studied by Foias and Singer [18], Dinculeanu [1,2], Goodrich [19], Dobrakov [20] and Shuchat [3]. Different classes of operators on C.X; E/ have been studied intensively; see [1,2], [20][21][22][23][24][25][26][27][28][29][30][31][32]. The study of the relationship between operators and their representing operator measures is a central problem in the theory.…”