Operators in tight by support Banach spaces

TL;DR

This paper constructs a specific bounded operator on a subspace of Gowers unconditional space that challenges previous assumptions, and explores properties of operators in tight by support Banach spaces.

Contribution

It provides a counterexample to a question by Gowers and analyzes the structure of operators in tight by support Banach spaces.

Findings

Existence of a bounded operator not a strictly singular perturbation of a diagonal restriction.

In tight by support Banach spaces, no two isomorphic infinite-dimensional subspaces form a direct sum.

Abstract

We answer the question of W.T. Gowers, giving an example of a bounded operator on a subspace of Gowers unconditional space which is not a strictly singular perturbation of a restriction of a diagonal operator. We make some observations on operators in arbitrary tight by support Banach space, showing in particular that in such space no two isomorphic infinitely dimensional subspaces form a direct sum.