Geodesic
Commutants and derivation ranges
Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 843-847

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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.
Classification : 47A10, 47A65, 47B47 
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     author = {Mecheri, Salah},
     title = {Commutants and derivation ranges},
     journal = {Czechoslovak Mathematical Journal},
     pages = {843--847},
     publisher = {mathdoc},
     volume = {49},
     number = {4},
     year = {1999},
     mrnumber = {1746710},
     zbl = {1008.47038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_1999__49_4_a15/}
}
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Mecheri, Salah. Commutants and derivation ranges. Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 843-847. http://geodesic.mathdoc.fr/item/CMJ_1999__49_4_a15/