-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Classes of Hardy spaces associated with operators, duality theorem and applications

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space and a space associated with the operator were introduced and studied. In this paper we define a class of spaces associated with the operator for a range of acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical spaces. We then establish a duality theorem between the spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on and give the inclusion between the classical spaces and the spaces associated with operators.

     

On decompositions in homotopy theory

We first describe Krull-Schmidt theorems decomposing spaces and simply-connected co- spaces into atomic factors in the category of pointed nilpotent -complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic spaces and homotopy types of atomic co- spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed.

     

Bases for vector spaces over the two-element field and the axiom of choice

It is shown that the axiom of choice follows in a weaker form than the Zermelo - Fraenkel set theory from the assertion: every set of generators G of a vector space V over the two element field includes a basis L for V. It is also shown that: for every family of non empty sets there exists a family of odd sized sets such that, for every , iff in every vector space over the two-element field every subspace has a complementarysubspace iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.

     

Domaine numérique du produit et de la bimultiplication

In this paper, we present an extension of Bouldin's result (1970) concerning the numerical range of the product of two operators and that are commuting and for which one of the set or consists of positive numbers. We also prove that if or is a subnormal operator on a separable Hilbert space, then

where is the operator bimultiplication and is the convex hull.

RÉSUMÉ. Dans ce travail, nous améliorons un résultat de Bouldin (1970) concernant la localisation de le domaine numérique du produit de deux opérateurs et sur un espace de Hilbert lorsque et commutent et est constitué de réels strictement positifs. Dans le cas où ou est un opérateur sous normal sur un espace de Hilbert séparable, nous montrons que

où est l'opérateur produit ou bimultiplication et est l'enveloppe convexe.

     

Isometries for the Carathéodory metric

Given two open unit balls and in complex Banach spaces, we consider a holomorphic mapping such that and is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that is a complex closed analytic submanifold of .

     

On certain co-H spaces related to Moore spaces

We show that certain co- spaces, constructed by Anick and Gray, carry a homotopy co-associative and co-commutative co- structure.

     

The Jung Theorem in metric spaces of curvature bounded above

The classical Jung Theorem states in essence that the diameter of a compact set in satisfies where is the circumradius of . The theorem was extended recently to the hyperbolic and the spherical -spaces. Here, the estimate above is extended to a class of metric spaces of curvature introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form where is a positive integer suitably defined for the set and its circumcenter. It can be that is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian -dimensional space, an integer always exists and satisfies . Then . In case of , one has .

     

Relative boundedness and relative compactness for linear operators in Banach spaces

If and are linear operators acting between Banach spaces, we show that compactness of relative to does not in general imply that has -bound zero. We do, however, give conditions under which the above implication is valid.

     

Universal spaces for almost -dimensionality

We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.

     

The ``fundamental theorem' for the algebraic -theory of spaces. III. The nil-term

In this paper we identify the ``nil-terms' for Waldhausen's algebraic -theory of spaces functor as the reduced -theory of a category of equivariant spaces equipped with a homotopically nilpotent endomorphism.

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

Factorization of holomorphic mappings on -spaces

We prove a universal mapping theorem for a large class of holomorphic mappings on a -space, stating that can be locally written in the form where and are bounded linear operators on certain Banach spaces consisting of functions on , and the division is taken pointwise.

     

Cesàro function spaces fail the fixed point property

The Cesàro sequence spaces , are reflexive but they have the fixed point property. In this paper we prove that in contrast to these sequence spaces the corresponding Cesàro function spaces on both and for are not reflexive and they fail to have the fixed point property.

     

-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field are studied. It is proven that the behaviour of these maps when is not locally compact is very different from the case of real- or complex-valued functions: in general, for -compact spaces and , the existence of a (weakly) separating additive map implies that and are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of and . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

     

Compact composition operators on the Smirnov class

We show that a composition operator on the Smirnov class is compact if and only if it is compact on some (equivalently: every) Hardy space for . Along the way we show that for composition operators on both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

     

Codimension 1 linear isometries on function algebras

Let be a function algebra on a locally compact Hausdorff space. A linear isometry is said to be of codimension if the range of has codimension in . In this paper, we provide and study a classification of codimension 1 linear isometries on function algebras in general and on Douglas algebras in particular.

     

An index theory for -actions

This paper concerns an index theory for -actions induced by a homeomorphism of a compact space. We give a definition of a genus for uniform spaces and prove that the genus for compact spaces is an index. To this end we show a -version of the Borsuk-Ulam theorem and the existence of a continuous equivariant extension for these -actions.

     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.