Diagonals of self-adjoint operators II: Non-compact operators

Abstract

Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\mathcal{D}(T)$ of all possible diagonals of $T$. For operators $T$ with at least two points in their essential spectrum ${\sigma }_{ess}(T)$, we give a complete characterization of $\mathcal{D}(T)$ for the class of self-adjoint operators sharing the same spectral measure as $T$ with a possible exception of multiplicities of eigenvalues at the extreme points of ${\sigma }_{ess}(T)$. We also give a more precise description of $\mathcal{D}(T)$ for a fixed self-adjoint operator $T$, albeit modulo the kernel problem for special classes of operators. These classes consist of operators $T$ for which an extreme point of the essential spectrum ${\sigma }_{ess}(T)$ is also an extreme point of the spectrum $\sigma (T)$. Our results generalize a characterization of diagonals of orthogonal projections by Kadison, Blaschke-type results of Müller and Tomilov, and Loreaux and Weiss, and a characterization of diagonals of operators with finite spectrum by the authors.

This is the second part of the paper which was originally submitted as arXiv:2212.08182v1