Raffaella Cilia

Polynomial characterization of L∞-spaces

It is shown that, given an index m, a Banach space E is an L∞-space if and only if every 1-dominated m-homogeneous polynomial on E is integral. This extends a result for linear operators due to Stegall and Retherford.

NUCLEAR AND INTEGRAL POLYNOMIALS

Let E be a Banach space whose dual E has the approximation property, and let m be an index. We show that E has the Radon-Nikod´ ym property if and only if every m-homogeneous integral polynomial from E into any Banach space is nuclear. We also obtain factorization and composition results for nuclear polynomials.

Polynomials on banach spaces whose duals are isomorphic to ℓ 1 (Γ)

We prove that the dual of a Banach space E is isomorphic to an l 1 (Γ) space if and only if, for a fixed integer m , every m -homogeneous 1-dominated polynomial on E is nuclear. This extends a result for linear operators due to Lewis and Stegall. The same techniques used for this result allow us to prove that, if every m -homogeneous integral polynomial between two Banach spaces is nuclear, then every integral (linear) operator between the same spaces is nuclear.

Pelczynski's property (V) and weak* basic sequences

Abstract In this note we study the property (V) of Pelczynski, in a Banach space X, in relation with the presence, in the dual Banach space X*, of suitable weak* basic sequences. We answer negatively to a question posed by John and we prove that, if X is a Banach space with the Property (V) of Pelczynski and the Gelfand Phillips property, then X is reflexive if and only if every quotient with a basis is reflexive. Moreover, we prove that, if X is a Banach space with the property (V) of Pelczynski, then either X is a Grothendieck space or W (X, Y) is uncomplemented in L(X, Y) provided that Y is a Banach space such that W (X, Y) ≠ L(X, Y).

POLYNOMIAL FACTORIZATION THROUGH L γ (μ) SPACES

We give conditions so that a polynomial be factorable throu- gh an Lr(µ) space. Among them, we prove that, given a Banach space X and an index m, every absolutely summing operator on X is 1-factorable if and only if every 1-dominated m-homogeneous polynomial on X is right 1-factorable, if and only if every 1-dominated m-homogeneous polynomial on X is left 1-factorable. As a consequence, if X has local unconditional structure, then every 1-dominated homogeneous polynomial on X is right and left 1-factorable.

Weak compactness and compactness of dominated operators onC(Ω,X)

In questo lavoro presentiamo alcuni risultati relativi alla debole compattezza e alla compattezza degli operatori dominati sugli spaziC(Ω,X).