Diagonals of self-adjoint operators II: non-compact operators: Diagonals of self-adjoint operators...: M. Bownik, J. Jasper.

Bownik, Marcin; Jasper, John

Given a self-adjoint operator T on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set D (T) of all possible diagonals of T. For operators T with at least two points in their essential spectrum σ ess (T) , we give a complete characterization of 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 σ ess (T) . We also give a more precise description of 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 σ ess (T) is also an extreme point of the spectrum σ (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].

ORTHOGRAPHIC projection; HILBERT space; EIGENVALUES; SELFADJOINT operators; MULTIPLICITY (Mathematics)

Mathematische Annalen, 2025, Vol 391, Issue 1, p431

0025-5831

Academic Journal

10.1007/s00208-024-02910-z