Diagonals of self-adjoint operators I: compact 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. Our results generalize a characterization of diagonals of trace class positive operators by Arveson and Kadison and diagonals of compact positive operators by Kaftal, Loreaux, and Weiss. The proof uses the technique of diagonal-to-diagonal results, which was pioneered in the earlier joint work of the authors with Siudeja.