Some Trace Class Commutators of Trace Zero

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.