Said Bouali, Department of Mathematics, Faculty of Science, Ibn Tofail University, B.P. 133, 24000 Kénitra, Morocco, e-mail: said.bouali@yahoo.fr; Youssef Bouhafsi, Department of Mathematics, Faculty of Science, Chouaib Doukkali University, Iben Maachou Street, P.O.Box 20, 24000 El Jadida, Morocco, e-mail: ybouhafsi@yahoo.fr
Abstract: Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A, B\in L(H)$, the generalized derivation $\delta_{A,B}$ and the elementary operator $\Delta_{A,B}$ are defined by $\delta_{A,B}(X)=AX-XB$ and $\Delta_{A,B}(X)=AXB-X$ for all $X\in L(H)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of $\delta_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $\Delta_{A,B}$ with respect to the wider class of unitarily invariant norms on $L(H)$.
Keywords: derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property
Classification (MSC 2010): 47A30, 47A63, 47B15, 47B20, 47B47, 47B10
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