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Derivations and the trace-class operators

Author: R. E. Weber
Journal: Proc. Amer. Math. Soc. 73 (1979), 79-82
MSC: Primary 47B47
MathSciNet review: 512062
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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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Keywords: Derivation ranges, trace-class, compact operator, quasinilpotent operator
Article copyright: © Copyright 1979 American Mathematical Society

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