On Grothendieck Sets

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is μ ∈ b a A such that μ n A → μ A for every A ∈ M then μ n → μ weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikodým set for b a ( A ) if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n ∈ N of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n ∈ N of subsets, there is p ∈ N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space ℓ ∞ Σ is a Grothendieck space. Some applications to classic Banach space theory are given.

Complemented Subspaces of Linear Bounded Operators

We study the complementation of the space W(X, Y) of weakly compact operators, the space K(X, Y) of compact operators, the space U(X, Y) of unconditionally converging operators, and the space CC(X, Y) of completely continuous operators in the space L(X, Y) of bounded linear operators from X to Y. Feder proved that if X is infinite-dimensional and c0 ↪ Y, then K(X, Y) is uncomplemented in L(X, Y). Emmanuele and John showed that if c0 ↪ K(X, Y), then K(X, Y) is uncomplemented in L(X, Y). Bator and Lewis showed that if X is not a Grothendieck space and c0 ↪ Y, then W(X, Y) is uncomplemented in L(X, Y). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.

Grothendieck's property in Lp(μ, X)

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

Sur les cones (faiblement complets) contenus dans le dual d'un espace de Banach non-reflexif

Let B be a Banach space. consider the convex proper weakly complete cones X contained in B′ with σ(B′, B) such that X ∩ B′, is conic in the sense of Asimow: that is, there exists α ≥ 0 and f ∈ B″ such that ‖ ‖B ≤ f ≤ α·‖ ‖B on X. This class arises in the theory of integral representations.If B is reflexive, such a cone has a weakly-compact basis. This paper considers the converse problem:- if one requires that X ∩ B′1 be σ(B′, B) metrisable, the existence of X (without a compact σ(B′, B) basis) is equivalent to the statement that B is not a Grothendieck space.However, in every space C(K) with infinitely compact K, one can find such a cone X. If two such cones in B′ are not too far apart, their sum belongs to this class.

Well-bounded operators of type (B) in a class of Banach spaces

It is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of C0-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature.