Stanisław Prus

Measure of weak noncompactness and real interpolation of operators

A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T : A i → B i , i = 0, 1 is weakly compact, then so is T : A θ, p → B θ, p for all 0 P

Asymptotical smoothness and its applications

In this paper we introduce the notion of asymptotical smoothness of a Banach space and show that it is strongly related to the Kadec-Klee property. This notion is then applied to obtain new theorems about weak convergence of almost orbits of three various types of semigroups of mappings.

On Milman’s moduli for Banach spaces

We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.

Fixed point property for general topologies in some Banach spaces

We study the fixed point property with respect to general vector topologies in L-embedded Banach spaces. Considering a class of topologies in l1 such that the standard basis is convergent, we characterize those of them for which the fixed point property holds. We show that in c0-sums of some Banach spaces the weak topology is in a sense the coarsest topology for which the fixed point property holds.

Uniform nonsquareness of direct sums of Banach spaces

An inequality between James constants of Banach spaces

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

X_s

Weak compactness and fixed point property for affine mappings

and the James constant of their direct sum is obtained. This gives a characterization of uniform nonsquareness of sums of Banach spaces.

Measure of weak noncompactness under complex interpolation

It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.