Abstract
In this paper we obtain some results concerning the set \(\mathcal{M} = \cup \left\{ {\overline {R(\delta _A )} \cap \{ A\} ':A \in \mathcal{L}(\mathcal{H})} \right\}\), where \(\overline {R(\delta _A )} \) is the closure in the norm topology of the range of the inner derivation δ A defined by δ A (X) = AX − XA. Here \(\mathcal{H}\) stands for a Hilbert space and we prove that every compact operator in \(\overline {R(\delta _A )} ^{\omega } \cap \{ A^* \} '\) is quasinilpotent if A is dominant, where \(\overline {R(\delta _A )} ^{\omega } \) is the closure of the range of δ A in the weak topology.
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References
J. H. Anderson: On normal derivations. Proc. Amer. Math. Soc. 38 (1973), 135–140..
J. H. Anderson: Derivation ranges and the identity. Bull. Amer. Math. Soc. 79 (1973), 705–708..
D. C. Kleïnecke: On operator commutators. Proc. Amer. Math. Soc. 8 (1957), 535–536..
J. G. Stampfli, B. L. Wadhwa: On dominant operators. Monatsh. Math. 84 (1977), 143–153.
R. E. Weber: Derivations and the trace class operators. Proc. Amer. Math. Soc. 73 (1979), 79–82.
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Mecheri, S. Commutants and derivation ranges. Czechoslovak Mathematical Journal 49, 843–847 (1999). https://doi.org/10.1023/A:1022465420571
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DOI: https://doi.org/10.1023/A:1022465420571