A Remark on the Range Closures of an Elementary Operator
A Remark on the Range Closures of an Elementary Operator
Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ into itself. For $A,B\in L(H)$, the elementary operator $\tau_{A,B}\in L(L(H))$ is defined by $\tau_{A,B}(X)=AXB-X$. An operator $A\in L(H)$ is said to be generalized quasi-adjoint if $ATA=T$ implies $A^{\ast}TA^{\ast}=T$ for every $T\in C_{1}(H)$ (trace class operators). …