Jiří Spurný

Quantitative Dunford–Pettis property

Abstract We investigate possible quantifications of the Dunford–Pettis property. We show, in particular, that the Dunford–Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford–Pettis property. We prove that L 1 spaces and C ( K ) spaces possess both of them. We also show that several natural measures of weak non-compactness are equal in L 1 spaces.

ON QUANTIFICATION OF WEAK SEQUENTIAL COMPLETENESS

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.

Integral representation theory : applications to convexity, Banach spaces and potential theory

This ambitious and substantial monograph, written by prominent experts in the field, presents the state of the art of convexity, with an emphasis on the interplay between convex analysis and potential theory; more particularly, between Choquet theory and the Dirichlet problem. The book is unique and self-contained, and it covers a wide range of applications which will appeal to many readers.

On approximation of affine Baire-one functions

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces.In particular, we investigate the function spaceH(U), consisting of the functions continuous on the closure of a bounded open setU⊂ℝm and harmonic onU, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the spaceB1b(H(U)) of bounded functions which are pointwise limits of functions from the spaceH(U) and thatB1b(H(U)) is not uniformly closed. On the other hand, every Baire-oneH(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging toH(U).It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space.

On a difference between quantitative weak sequential completeness and the quantitative Schur property

We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of

Baire classes of Banach spaces and strongly affine functions

\ell_1

Quantification of the reciprocal Dunford-Pettis property

holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property but fails the quantitative form of the Schur property.

Choquet like sets in function spaces

We construct a metrizable simplex X and a Baire-two function f on X satisfying the barycentric formula such that f is not of affine class two; i.e., there is no bounded sequence of affine Baire—one functions on X converging to f. This provides an example of a Banach L ∞ -space E such that E ** 2 ≠ E ** B2 .

QUANTIFICATION OF THE BANACH-SAKS PROPERTY

We prove in particular that Banach spaces of the form

THE DIRICHLET PROBLEM FOR BAIRE-TWO FUNCTIONS ON SIMPLICES

C_0(\Omega)