# Article

 Title: Commutants and derivation ranges (English) Author: Mecheri, Salah Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 49 Issue: 4 Year: 1999 Pages: 843-847 Summary lang: English . Category: math . Summary: In this paper we obtain some results concerning the set ${\mathcal M} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal L(H)}\bigr \rbrace$, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal H$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta _A)}^w\cap \lbrace A^*\rbrace ^{\prime }$ is quasinilpotent if $A$ is dominant, where $\overline{R(\delta _A)}^w$ is the closure of the range of $\delta _A$ in the weak topology. (English) MSC: 47A10 MSC: 47A65 MSC: 47B47 idZBL: Zbl 1008.47038 idMR: MR1746710 . Date available: 2009-09-24T10:28:23Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127534 . Reference: [1] J.H.Anderson: On normal derivations.Proc. Amer. Math. Soc. 38 (1973), 135–140.. Zbl 0255.47036, MR 0312313, 10.1090/S0002-9939-1973-0312313-6 Reference: [2] J.H.Anderson: Derivation ranges and the identity.Bull. Amer. Math. Soc. 79 (1973), 705–708.. Zbl 0269.47021, MR 0322518, 10.1090/S0002-9904-1973-13271-9 Reference: [3] D.C.Kleïnecke: On operator commutators.Proc. Amer. Math. Soc. 8 (1957), 535–536.. Zbl 0079.12904, MR 0087914, 10.1090/S0002-9939-1957-0087914-4 Reference: [4] J.G.Stampfli, B.L.Wadhwa: On dominant operators.Monatsh. Math. 84 (1977), 143–153. Zbl 0374.47010, MR 0458225, 10.1007/BF01579599 Reference: [5] R.E.Weber: Derivations and the trace class operators.Proc. Amer. Math. Soc. 73 (1979), 79–82. Zbl 0372.47019, MR 0512062, 10.1090/S0002-9939-1979-0512062-0 .

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