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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, , lie
in an ideal, do bounded operators , always commute under any trace on
that ideal, i.e.,? And if not, characterize which
traces/ideals do possess this property.Comment: 6 page
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