Let A + B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h : X → A + B splits into a sum h = f + g of continuous mappings f : X → A and g : X → B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear ...

Abstract: This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator T on a Hilbert space with an orthogonal basis, we define the isomorphic structure Σ(T ) as the family of all subsets of the basis so that T restricted to their span is a nice isomorphism. Our main result is a dimension-free optimal estimate of the size of Σ(T ). It improve...

If a and b are elements of an algebra, then we show that ab is Drazin invertible if and only if ba is Drazin invertible. With this result we investigate products of bounded linear operators on Banach spaces. 1. Drazin inverses Throughout this section A is a real or complex algebra with identity e 6= 0. We denote the group of invertible elements of A by A. We call an element a ∈ A relatively reg...