Diagonals of operators and Blaschke's enigma

TL;DR

This paper develops new techniques for constructing diagonals of bounded operators on Hilbert spaces under Blaschke-type conditions, broadening the understanding of possible diagonals and generalizing existing results in operator theory.

Contribution

It introduces a novel framework for constructing operator diagonals under Blaschke assumptions, extending prior work and identifying large subsets of diagonals for bounded operators.

Findings

New techniques for diagonal construction under Blaschke conditions

Identification of large subsets of possible diagonals

Generalizations of Bourin, Herrero, and Stout's results

Abstract

We introduce new techniques allowing one to construct diagonals of bounded Hilbert space operators and operator tuples under "Blaschke-type" assumptions. This provides a new framework for a number of results in the literature and identifies, often large, subsets in the set of diagonals of arbitrary bounded operators (and their tuples). Moreover, our approach leads to substantial generalizations of the results due to Bourin, Herrero and Stout having assumptions of a similar nature.