Banach spaces not having copies of 1 ∞ and Z1.
Banach spaces of bounded Szlenk index
For a countable ordinal α we denote by the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each admits a separable, reflexive universal space. We also show that spaces in the class embed into spaces of the same class with a basis. As a consequence we deduce that each is analytic in the Effros-Borel structure of subspaces of C[0,1].
Banach spaces of bounded Szlenk index II
For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is and which is universal for all separable Banach spaces whose Szlenk index does not exceed . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.
Banach Spaces of Compact Multipliers and Their Dual Spaces.
Banach spaces of homogeneous polynomials without the approximation property
We present simple proofs that spaces of homogeneous polynomials on and provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).
Banach Spaces of Type p have Arbitrarily Distortable Subspaces.
Banach spaces on which every unconditionally converging operator is completely continuous
Banach spaces quasi-reflexive of order one
Banach spaces which admit a norm with the uniform Kadec-Klee property
Several results are established about Banach spaces Ӿ which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from Ӿ to the Lebesgue-Bochner space if and only if Ӿ is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.
Banach spaces which are -ideals in their bidual have property
We show that every Banach space which is an -ideal in its bidual has the property of Pelczynski. Several consequences are mentioned.
Banach spaces which are proper M-ideals
Banach spaces which are semi-L-summands in their biduals.
Banach spaces which embed into their dual
We use Birkhoff-James' orthogonality in Banach spaces to provide new conditions for the converse of the classical Riesz representation theorem.
Banach spaces whose bounded sets are bounding in the bidual.
Banach spaces widely complemented in each other
Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that can be decomposed into a direct sum of and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then,...
Banach spaces with a supershrinking basis
We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the -theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in .
Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces
Banach spaces with small Calkin algebras
Let X be a Banach space. Let 𝓐(X) be a closed ideal in the algebra ℒ(X) of the operators acting on X. We say that ℒ(X)/𝓐(X) is a Calkin algebra whenever the Fredholm operators on X coincide with the operators whose class in ℒ(X)/𝓐(X) is invertible. Among other examples, we have the cases in which 𝓐(X) is the ideal of compact, strictly singular, strictly cosingular and inessential operators, and some other ideals introduced as perturbation classes in Fredholm theory. Our aim is to present some...
Banach Spaces with the Condition of Mazur.
Banach spaces with the differentiable norms