Finite operators and orthogonality (Q950648)

For \(A,B\) in the algebra \(B(H)\) of all bounded linear operators on a complex Hilbert space \(H\), the generalized derivation \(\delta_{A,B} : B(H) \to B(H)\) is defined to be \(\delta_{A,B}(X) = AX - XB\). An operator \(A \in B(H)\) is called finite if \(\| I-\delta_{A,A}(X)\| \geq 1\) for all \(X \in B(H)\) [cf.\ \textit{J.\,P.\thinspace Williams}, Proc.\ Am.\ Math.\ Soc.\ 26, 129--136 (1970; Zbl 0199.19302)]. An operator \(A\) is said to be paranormal if \(\| Ax\|^2 \leq \| A^2x\|\| x\|\) for all \(x\in H\). In the paper under review, the author proves that a paranormal operator is finite and investigates the inequalities of type \(\| T-\delta_{A,B}(X)\|\geq\| T\|\).