Geodesic
On the range of some elementary operators
Commentationes Mathematicae Universitatis Carolinae, Tome 65 (2024) no. 1, pp. 53-62
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Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L(H)$, the generalized derivation $\delta_{A,B}$ and the multiplication operator $M_{A,B}$ are defined on $L(H)$ by $\delta_{A,B}(X)=AX-XB$ and $M_{A,B}(X)=AXB$. In this paper, we give a characterization of bounded operators $A$ and $B$ such that the range of $M_{A,B}$ is closed. We present some sufficient conditions for $\delta_{A,B}$ to have closed range. Some related results are also given.
Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L(H)$, the generalized derivation $\delta_{A,B}$ and the multiplication operator $M_{A,B}$ are defined on $L(H)$ by $\delta_{A,B}(X)=AX-XB$ and $M_{A,B}(X)=AXB$. In this paper, we give a characterization of bounded operators $A$ and $B$ such that the range of $M_{A,B}$ is closed. We present some sufficient conditions for $\delta_{A,B}$ to have closed range. Some related results are also given.
DOI : 10.14712/1213-7243.2025.004
Classification : 47A16, 47A30, 47B07, 47B20, 47B47
Keywords: generalized derivation; elementary operator; generalized inverse; Kato spectrum 
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El Mouadine, Hamza; Faouzi, Abdelkhalek; Bouhafsi, Youssef. On the range of some elementary operators. Commentationes Mathematicae Universitatis Carolinae, Tome 65 (2024) no. 1, pp. 53-62. doi: 10.14712/1213-7243.2025.004

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