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 $L(H)$.
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