Some approximation properties of Banach spaces and Banach lattices

The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ _∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ ₁-space and Z is a ℒ _p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ ⁿ .     

Chaotic polynomials on Banach spaces

We show the existence of chaotic (in the sense of Devaney) polynomials on Banach spaces of q-summable sequences. Such polynomials P consist of composition of the backward shift with a certain fixed polynomial p of one complex variable on each coordinate. In general we also prove that P is chaotic in the sense of Auslander and Yorke if and only if 0 belongs to the Julia set of p.     

Scalar-valued dominated polynomials on Banach spaces

It is well known that 2-homogeneous polynomials on -spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space such that every scalar-valued homogeneous polynomial on is -dominated for every .

     

Bases in spaces of homogeneous polynomials on Banach spaces

In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces.     

Ideals of polynomials between Banach spaces revisited

Ideals of polynomials and multilinear operators between Banach spaces have been exhaustively investigated in the last decades. In this paper, we introduce a unified (and more general) approach and propose some lines of investigation in this new framework. Among other results, we prove a Bohnenblust–Hille inequality in this more general setting.     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Minimal ideals of n-homogeneous polynomials on Banach spaces

The minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.     

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.     

A note on geometrical properties of Banach spaces using ψ-direct sums

Let X be a Banach space and ψ a continuous convex function on [0,1] satisfying certain conditions. Let Xψ_⊕X be the ψ-direct sum of X. In this note, we characterize the strict convexity, uniform convexity and uniformly non-squareness of Banach spaces using ψ-direct sums, which extends the well-known characterization of these spaces.     

Some geometric and topological properties of Banach spaces via ball coverings

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that Gδ property of points in a Banach space X endowed with the ball topology is equivalent to the space X admitting the ball-covering property. Moreover, smoothness, uniform smoothness of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.     

Some descriptive set-theoretic properties of the isomorphism relation between Banach spaces

Consider the space of closed linear subspaces of a separable Banach space equipped with the standard Effros Borel structure. The isomorphism relation between Banach spaces being elements of  determines a partition of  . In this note we prove a result describing the complexity of analytic subsets of  intersecting a large enough number of the above-mentioned parts of  .

     

Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.