Hamza El Mouadine

Let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ into itself. Given $A,B\in \mathcal{L}(H)$, the generalized derivation $\delta_{A,B}\in \mathcal{L}(\mathcal{L}(H))$ is defined by $\delta_{A,B}(X)=AX-XB$. For hyponormal operators $A$ and $B$, L.A. Fialkow gave necessary conditions for which the range of $\delta_{A,B}$ is norm closed in $\mathcal{L}(H)$ [13]. In this paper, we present necessary or sufficient conditions under which the generalized derivation $\delta_{A,B}$ has closed range, where $A$ and $B$ are $p$-hyponormal operators. We give a characterization for the range of $\delta_{A,B}$ to have closed range in the case when $A$ and $B$ are cyclic subnormal operators.