A twisted sum Z of two Banach spaces X and Y is defined (see [11]) through a short exact sequence: 0 −→ X −→ Z −→ Y −→ 0. These short exact sequences in the category of (quasi-)Banach spaces are considered naturally in the investigation of three space properties (a property P in the category of quasi-Banach spaces is called a three space property if for every short exact sequence as above, Z ha...

We study the complexity of Banach space valued integration in the randomized setting. We are concerned with r-times continuously differentiable functions on the d-dimensional unit cube Q, with values in a Banach space X, and investigate the relation of the optimal convergence rate to the geometry of X. It turns out that the n-th minimal errors are bounded by cn−r/d−1+1/p if and only if X is of ...

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following: let X be a Banach space with an unconditional basis {ei}i∈N; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A,B ∈ P, [ei]i∈A ≇ [ei]i∈B , or for a residual set of infinite subsets A of N, [ei]i∈A is isomorphic to X, and in that case, X is is...

Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I+f(A)B is invertible for every analytic function f on sigma(A). Also we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space ell^2 for solving some PDEs which is extended to general operator space under some ...

An atomic decomposition is considered in Banach space.  A method for constructing an atomic decomposition of Banach  space, starting with atomic decomposition of  subspaces  is presented. Some relations between them are established. The proposed method is used in the  study  of the  frame  properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

A type of Strassen's Theorem for measures taking values in the positive cone of a Banach lattice is proved. An application is given to metrics for convergence of vector measures. 1. Preliminaries: vector measures Let y be a field of subsets of a set X, and let (B, || • ||) be a Banach space. (All vector spaces we consider are assumed to have real scalars.) Then a(Sr, B) is the set of all additi...

Abstract The author discusses necessary and sufficient conditions of the complete convergence for sums of B-valued independent but not necessarily identically distributed r.v.′s in Banach space of type p, and obtain characterization of Banach space of type p in terms of the complete convergence. A series of classical results on iid real valued r.v.′s are extended. As application author gives th...