REMARKS ON COMPLETELY CONTINUOUS POLYNOMIALS

Abstract

We present some results pointing out pairs E, F of Band spaces for which any polynomial P: E → F is completely continuous. Hence we study local complete continuity of holomorphic functions.     

ON A THEOREM OF A.A. LYAPUNOV

Abstract

In this note we obtain some extensions and an approximation of the Lyapunov convexity theorem by means of the bilinear integration of a set-valued function. The integration is performed successively with respect to a non-atomic, a direct sum and a Darboux vector measure. The necessary counterexamples are provided.     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

HOLOMORPHIC MAPS ON DOMAINS WITH EMPTY INTERIOR

Abstract

We bring to light further infinite dimensional domains with empty interior on which holomorphic maps can be constructed.     

THE WEAK RADON-NIKODYM PROPERTY IN SPACES OF NUCLEAR OPERATORS

Abstract

We prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the Radon-Nikodym property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has the WRNP. If moreover X* has the RNP, then N(X,Y) has the RNP.     

Two new properties of ideals of polynomials and applications

In this paper we introduce two properties for ideals of polynomials between Banach spaces and showhow useful they are to deal with several a priori different problems. By investigating these properties we obtain, among other results, new polynomial characterizations of L_∞ spaces and characterizations of Banach spaces whose duals are isomorphic to f 1 (Λ).     

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c₀ if and only if X contains a copy of c₀. In this case, if the probability μ has infinite range, a copy of c₀ consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.     

THE LIMIT SET OF RIEMANN SUMS OF A VECTOR STIELTJES INTEGRAL

Abstract

Conditions for convexity of the limit set of Riemann sums of a vector integral are investigated.     

Characterization of reflexivity by equivalent renorming

A new rotundity property of Day's norm on c₀(Γ) is introduced. This property provides in particular a renorming characterization of the class of all reflexive Banach spaces.     

Fixed point theorems for holomorphic mappings and operator theory in indefinite metric spaces

In this paper we establish several new results on the existence and uniqueness of a fixed point for holomorphic mappings and one-parameter semigroups in Banach spaces. We also present an application to operator theory on spaces with an indefinite metric.     

Almost summing mappings

We introduce a general definition of almost p-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost p-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. Received: 17 June 2002     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Radon–Nikodým indexes and measures of weak noncompactness

The aim of this paper is to present a quantitative version of the Radon–Nikodým property and some other results related to it. This approach gives an extra insight to the classical results. We introduce two indexes: an index of representability of measures and an index of dentability. We review classic results in order to obtain relationships between these and other indexes.     

On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

On the Riesz representation theorem and integral operators

We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.