Geodesic
A remark on the range of elementary operators
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 4, pp. 1065-1074 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
Classification : 47A30, 47B10, 47B20, 47B47
Keywords: elementary operators; ultraweak closure; weak closure; quasi-adjoint operator 
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Bouali, Said; Bouhafsi, Youssef. A remark on the range of elementary operators. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 4, pp. 1065-1074. http://geodesic.mathdoc.fr/item/CMJ_2010_60_4_a14/

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