Derivations and the trace-class operators

Weber, R. E.

doi:10.1090/S0002-9939-1979-0512062-0

Derivations and the trace-class operators
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Proc. Amer. Math. Soc. 73 (1979), 79-82 Request permission

Abstract:

Let $\mathcal {R}({\Delta _A})$ represent the range of the derivation generated by $A \in \mathcal {B}(\mathcal {H})$. It is shown that for each $n \geqslant 2,{\text {tr}}({T^n}) = 0$ for any trace-class operator $T \in \{ A\} ’$ which is either (a) the weak limit of a sequence in $\mathcal {R}({\Delta _A})$ or (b) a finite rank operator in the weak closure of $\mathcal {R}({\Delta _A})$. From this it follows that if $K \in \{ A\} ’$ is a compact operator in the weak closure of $\mathcal {R}({\Delta _A})$, then K is quasinilpotent.

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