Czechoslovak Mathematical Journal, Vol. 49, No. 4, pp. 843-847, 1999

Commutants and derivation ranges

Salah Mecheri

Universite de Tebessa, Departement des sciences exactes, 12000 Tebessa, Algerie

Abstract: In this paper we obtain some results concerning the set ${\cal M} = \cup\bigl\lbrace\overline{R(\delta_A)}\cap\lbrace A\rbrace' A\in{\cal L(H)}\bigr\rbrace$, 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 $\cal H$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta_A)}^w\cap\lbrace A^*\rbrace'$ 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.

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