Fernando Mayoral

Compact sets of compact operators in absence of

We characterize the compactness of a subset of compact operators between Banach spaces when the domain space does not have a copy of 11. In a recent paper, F. Galaz-Fontes [6] characterizes the (relative) compactness of a set of compact operators from a reflexive and separable Banach space X into another Banach space Y. Here we prove that such characterization is also valid when X does not contain a copy of 11. In order to clarify our terminology we say that a set of bounded linear operators H C L(X, Y) is sequentially weaknorm equicontinuous (called uniformly w-continuous in [6]) if for each weakly-null sequence (xn) in X the sequences (h(Xn)) converge in norm to 0 uniformly in h c H, i.e. sup {fIh(xn)I: h e H} converges to 0. Theorem 1. Let X be a Banach space without a copy of 11 and let H be a subset of bounded operators from X to a Banach space Y. Then H is a relatively compact subset in the space of compact operators K(X, Y) in the uniform topology of operators if and only if it verifies the following two conditions: (1) H is pointwisely relatively compact, i.e. for each x c X the set H(x) = {h(x): h c H} is relatively compact in Y. (2) H is sequentially weak-norm equicontinuous. Note that if X does not contain a copy of 11, then by applying the RosenthalDor Theorem (see [11] and [4] or [3, Ch. IX]) every bounded sequence in X has a weakly-Cauchy subsequence and, therefore, a bounded linear operator h: X Y is compact if and only if it is completely continuous, that is, if and only if it takes weakly convergent sequences in X to convergent ones in Y. This tells us that condition (2) applied to a singleton characterizes compact operators when the space X does not have a copy of 11 [7, 17.7]. Our notation is standard: L(X, Y) is the Banach space of bounded linear operators from the Banach space X to the Banach space Y, endowed with the topology of the uniform convergence on the unit ball Bx of X, K(X, Y) is its closed subspace Received by the editors April 20, 1998. 2000 Mathematics Subject Classification. Primary 47B07, 46B25.

On Compactness in Spaces of Bochner Integrable Functions

We show that a bounded subset K of Lp(μ,X) is relatively norm compact if and only if K is p-uniformly integrable, scalarly relatively compact, and either tight or flatly concentrated. The scalar relative compactness can be also replaced by several oscillation criteria.

Complex interpolation of spaces of integrable functions with respect to a vector measure

Let (Ω, Σ) be a measurable space andm: Σ →X be a vector measure with values in the complex Banach space X: We apply the Calderón interpolation methods to the family of spaces of scalarp-integrable functions with respect tom with 1≤p≤∞. Moreover we obtain a result about the relation between the complex interpolation spaces [X0,X1][θ] and [X0,X1][θ] for a Banach couple of interpolation (X0,X1) such thatX1 ⊂X0 with continuous inclusion.

Inductive limits of spaces of vector- valued integrable functions

For an order-continuous Banach function space Λ and a separated inductive limit E:= indnEn, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then Lp {E} is barrelled for 1 ≤ p ≤ ∞.

Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure

We study properties of compactness of multiplication operators between spaces of p-power integrable scalar functions with respect to a vector measure m.

Norming sets and integration with respect to vector measures

Abstract Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ℝ be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X*. In this paper we show that this is the case when A is a James boundary for B X * (respectively, Λ is weak*-thick). Some examples and applications are given as well.

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract We study Dieudonne-Kothe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.