Some trace class commutators of trace zero

Kittaneh, Fuad

doi:10.1090/S0002-9939-1991-1086332-X

Some trace class commutators of trace zero
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Proc. Amer. Math. Soc. 113 (1991), 655-661 Request permission

Abstract:

It is shown that if $T$ is an operator on a separable complex Hilbert space and $X$ is a Hilbert-Schmidt operator such that $TX - XT$ is a trace class operator, then the trace of $TX - XT$ is zero provided one of the two conditions holds: (a) ${T^2}$ is normal; (b) ${T^n}$ is normal for some integer $n > 2$ and ${T^*}T - T{T^*}$ is a trace class operator. Related results involving essentially unitary operators and Cesàro operators are also given.

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