Diagonals of self-adjoint operators I: compact operators

TL;DR

This paper characterizes the possible diagonals of self-adjoint compact operators on infinite-dimensional Hilbert spaces, extending previous results and addressing the kernel problem for positive compact operators.

Contribution

It provides a complete characterization of diagonals for compact self-adjoint operators modulo the kernel, generalizing earlier work on trace class and positive operators.

Findings

Characterization of diagonals for compact operators sharing eigenvalues

Determination of diagonals modulo the kernel for fixed compact operators

Extension of previous diagonal characterization results

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 compact operators $T$, we give a complete characterization of diagonals modulo the kernel of $T$. That is, we characterize $\mathcal D(T)$ for the class of operators sharing the same nonzero eigenvalues (with multiplicities) as $T$. Moreover, we determine $\mathcal D(T)$ for a fixed compact operator $T$, modulo the kernel problem for positive compact operators with finite-dimensional kernel. Our results generalize a characterization of diagonals of trace class positive operators by Arveson and Kadison and diagonals of compact positive operators by Kaftal, Loreaux, and Weiss. The proof uses the technique of diagonal-to-diagonal results, which was pioneered in the earlier joint work of the authors with Siudeja.