  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 M\"uller and
Tomilov, and Loreaux and Weiss, and a characterization of diagonals of
operators with finite spectrum by the authors.
