A weak Hilbert space with few symmetries

TL;DR

This paper constructs a special weak Hilbert space where all operators are essentially diagonal plus strictly singular, leading to subspaces that are not isomorphic to their proper subspaces, revealing unique symmetry properties.

Contribution

It introduces a weak Hilbert space with a restricted operator structure, demonstrating novel asymmetry in its subspace isomorphisms.

Findings

All bounded operators are of the form D+S with D diagonal and S strictly singular

Block subspaces are not isomorphic to any of their proper subspaces

The space exhibits minimal symmetry among weak Hilbert spaces

Abstract

We construct a weak Hilbert Banach space such that for every block subspace $Y$ every bounded linear operator on Y is of the form D+S where S is a strictly singular operator and D is a diagonal operator. We show that this yields a weak Hilbert space whose block subspaces are not isomorphic to any of their proper subspaces.