A characterisation of minimal subdifferential mappings of locally Lipschitz functions

In this paper we characterise, in terms of the upper Dini derivative, the Clarke subdifferential mapping being a minimal weak* cusco, and we show that on any Banach space the Clarke subdifferential mapping of a pseudo-regular or semi-smooth locally Lipschitz function is always a minimal weak* cusco.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Search for properties of the missing Moore graph

In the degree-diameter problem, the only extremal graph the existence of which is still in doubt is the Moore graph of order 3250, degree 57 and diameter 2. It has been known that such a graph cannot be vertex-transitive. Also, certain restrictions on the structure of the automorphism group of such a graph have been known in the case when the order of the group is even. In this paper we further investigate symmetries and structural properties of the missing Moore (57, 2)-graph(s) with the help of a combination of spectral, group-theoretic, combinatorial, and computational methods. One of the consequences is that the order of the automorphism group of such a graph is at most 375.     

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 geometry of the set of square roots of elements inC *-algebras

This research was partially supported by the CONICET and the University of Buenos Aires.     

Unconditional families in Banach spaces

It is shown that for every separable Banach space X with non-separable dual, the space contains an unconditional family of size . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.     

The Radon‐Nikodym property,generalized bounded variation and the average range

A generalized bounded variation characterization of Banach spaces possessing the Radon‐Nikodym property is given in terms of the average range. We prove that a Banach space X has the Radon‐Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .     

Schauder type theorems for differentiable and holomorphic mappings

Denoting byC _wu ^p (E) the algebra of allC ^p-real-valued functions on the real Banach spaceE such that the functions and the derivatives are weakly uniformly continuous on bounded subsets, it is known that the algebra homomorphismsA:C _wu ^q (F)C _wu ^p (E) are induced by differentiable mappingsg:EF ^**. We prove that, for 1p+1q, the following are equivalent: (a)A is compact; (b)g and its derivatives are compact; (c)gC _wu ^p (E,F ^**) (the authors had proved that, forp=q<,A is [weakly] compact if and only ifg is a constant mapping, and it is known that ifq<p, thenA is always induced by a constant mapping and is therefore compact). Also, for an entire function of bounded typegH _b (U,F), where is a balanced open subset, andE,F are complex Banach spaces, lettingA:H _b (F)H _b (U) be the homomorphism given byA(f)=fg for allfH _b (F), we prove thatA is compact if and only ifg is compact.Supported in part by DGICYT Grant PB 94-1052 (Spain).Supported in part by DGICYT Grant PB 93-0452 (Spain).     

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).     

Differential calculus founded on an isomorphism

This paper founds calculus on a natural isometric linear isomorphism. Once this Foundational Isomorphism is proved (with elementary Banach space methods) several familiar calculus properties of continuous curves drop out as quick corollaries. Then the calculus can be further developed in very general setting via categorical methods. VSERC aided     

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.     

Global inversion theorems via coercive functionals on metric spaces

Let X$X$ and Y$Y$ be metric spaces. We give sufficient metric conditions for a local homeomorphism f:X→Y$f:X\to Y$ to be a global one. We achieve this by means of auxiliary coercive functionals; several expected global inversion theorems are obtained by choosing different functionals.     

Cone monotone mappings: Continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces, and we define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K$K$-monotone dominated and cone-to-cone monotone mappings. K$K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K$K$-monotone functions defined by Borwein, Burke and Lewis to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zají?ek, we show some relationships between these classes. Then, we show that every K$K$-monotone function f:X→R$f:X\to \mathbb{R}$, where X$X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K$K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w^∗$w^{\ast }$-senses) of these mappings.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

Projective spaces of aC *-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C^*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection =2p–1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.Partially supported by UBACYT TW49 and TX92, PIP 4463 (CONICET) and ANPCYT PICT 97-2259 (Argentina)     

Complex calculus via foundation theorems

It was shown recently that continuous differentiation theory can be founded on a natural isometric linear isomorphism. We develop in this paper an analogous theory for complex differentiation; however, this requires different foundational isomorphism.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.