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arXiv · 1410.7441

Diagonality and idempotents with applications to problems in operator theory and frame theory

Jireh Loreaux·Gary Weiss
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Abstract

We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.

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Jireh Loreaux, Gary Weiss. 2014-10-27. Diagonality and idempotents with applications to problems in operator theory and frame theory. https://doi.org/10.7900/jot.2014nov05.2054

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