Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δ A,B and the elementary operator δ A,B are defined by δ A,B(X)=AX-XB and δ 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 δ 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 Δ A,B with respect to the wider class of unitarily invariant norms on L(H).