Diagonals of self-adjoint operators I: compact operators

Marcin Bownik; John Jasper

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arXiv · 2212.08182

Diagonals of self-adjoint operators I: compact operators

Marcin Bownik·John Jasper

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Abstract

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.

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BibTeX RIS

Marcin Bownik, John Jasper. 2022-12-15. Diagonals of self-adjoint operators I: compact operators. https://arxiv.org/abs/2212.08182

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