T. Figiel

The approximation property does not imply the bounded approximation property

There is a Banach space which has the approxi- mation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck (2).

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p moment of a sum of independent symmetric random variables to that of the p and 2 moments of the individual variables, are computed in the range 2 < p ≤ 4. This complements the work of Utev who has done the same for p > 4. The qualitative nature of the extreme cases turns out to be different for p < 4 than for p > 4. The method developed yields results in some more general and other related moment inequalities.

Large subspaces ofl ∞ n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant

Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results are obtained for subspaces ofl∞n, while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition, a new isometric characterization ofl∞n is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern1/2.

On hilbertian subsets of finite metric spaces

The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, dY) embeds (1 +ε)-isomorphically into the Hilbert spacel2.

On the structure of non-weakly compact operators on Banach lattices

In this paper we study extensions of theorems of Hagler and Stegall on LP~ spaces [11] and of Rosenthal [24, 25], and Pelczynski [21] on C(K) spaces to more general Banach spaces with some lattice structure which do not contain complemented copies of I r We show that i fX is a separable Banach space such that Co does not embed into X* and T is a bounded linear operator from X into some Banach space Y such that T* Y* is not separable then (i) if X has local unconditional structure h la Gordon and Lewis [7].

On hereditarily indecomposable Banach spaces

Abstract A set-theoretical proof of Gowers’ Dichotomy Theorem is presented together with its application to another dichotomy concerning asymptotic l 2 basic sequences.