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 [38, 39], Blaschke-type results of Müller and Tomilov [51] and Loreaux and Weiss [48], and a characterization of diagonals of operators with finite spectrum by the authors [15].
