Diagonals of self-adjoint operators I: Compact operators

Abstract

Given a self-adjoint operator T on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $D\left(T\right)$ 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 $D\left(T\right)$ for the class of operators sharing the same nonzero eigenvalues (with multiplicities) as T. Moreover, we determine $D\left(T\right)$ 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 [5] and diagonals of compact positive operators by Kaftal and Weiss [24] and Loreaux and Weiss [28]. The proof uses the technique of diagonal-to-diagonal results, which was pioneered in the earlier joint work of the authors with Siudeja [12].