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

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.Comment: To appear in the Journal of Operator Theor

Diagonals of operators with specified singular values

Thompson\u27s theorem provides a characterization of the diagonals of finite matrices with specified singular values. This theorem is part of the broader study of diagonals of operators including the Schur-Horn theorem. Here we present an extension of Thompson\u27s theorem to compact operators and show how the techniques can also be used to characterize diagonals of unitary operators. This is joint work with John Jasper and Gary Weiss

Traces on ideals and the commutator property

We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the spectrum and algebraic multiplicities, possess this property and we suspect others do, but we know of no other traces that do. This paper is intended to be part survey. We provide here a brief overview of some facts concerning traces on ideals, especially involving Lidskii formulas and spectral traces. We pose the central question: whenever the relevant products, $AB$, $BA$ lie in an ideal, do bounded operators $A$, $B$ always commute under any trace on that ideal, i.e.,$\tau(AB) = \tau(BA)$? And if not, characterize which traces/ideals do possess this property.Comment: 6 page