Santiago Díaz

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.

An Abstract Banach-Steinhaus Theorem and Applications to Function Spaces

We give an abstract Banach-Steinhaus theorem for locally convex spaces having suitable algebras of linear projections modelled on a σ-finite measure space. This theorem is applied to deduce barrelledness results for the space L∞ (μ, E) of essentially bounded and μ-measurable functions from a Radon measure space (Ω, σ, μ) into a locally convex space E and also for B (μ, E), the closure of the space of simple functions. Sample: if μ is atomless, then B (μ, E) is barrelled if and only if E is quasi-barrelled and E′(β (E′, E)) has the property (B) of Pietsch.

COMPLEMENTED COPIES OF c 0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.

QUOTIENTS OF VECTOR-VALUED FUNCTION SPACES

We study when c 0 is isomorphic to a quotient of L p (μ, X ) (1 p <∞). The results are extended to order continuous Kothe function spaces.

Grothendieck's property in L p (μ, X )

We prove that, for non purely atomic measures, L p (μ, X ) is a Grothendieck space if and only if X is reflexive.

Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions

Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.

Some strong forms of the local duality of ultraproducts

We obtain two refinements of the so called local duality of ultrapowers, that is, the ultrapower version of the well-known principle of local reflexivity. 1991 Mathematics Subject Classification . 46B07, 46B08.

ON THE DUNFORD-PETTIS PROPERTY IN SPACES OF VECTOR-VALUED BOUNDED FUNCTIONS

X) denotes the Banach space of (equivalence classes of) essentially bounded,measurable and X-valued functions denned over a finite measure spac Ee (fl, fi)., Whendoes L°°(ii,X) have the Dunford-Pettis property? In general, this property does notlift from X to L°°(fi,X) p.56] O [6. n th,e other hand, the only non-trivial positiveresult, as far as we know, i

Weakly compact operators onH

We prove that a weakly compact operator fromH∞ or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH∞ or any of its even duals either fixes a copy ofl∞ or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH∞ or any of its even duals either contains a copy ofl∞ or is super-reflexive; (3) every subspace ofL1/H01 or any of its even duals either contains a complemented copy ofl1 or is super-reflexive.