Diagonals of self-adjoint operators

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. 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 $\sigma_{ess}(T)$. As in the case of compact operators, 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 compact positive operators by Kaftal, Loreaux, and Weiss, Blaschke-type results of M\"uller and Tomilov, and a characterization of diagonals of operators with finite spectrum by the authors.