Some inequalities in \(B(H)\) (Q5932796)

Let \(H\) denote a separable Hilbert space and let \(B(H)\) be the space of linear bounded operators from \(H\) to \(H\). For \(A, B\in B(H)\), satisfying \(\|A\|\leq 1\) and \(\|B\|\leq 1\), the set \(\Delta(A,B)\) is defined by NEWLINE\[NEWLINE \Delta(A,B)=\{T\in B(H): ATB=T\}. NEWLINE\]NEWLINE The main results of the paper are the following inequalities NEWLINE\[NEWLINE d(T,\Delta(A,B))\leq\sup_{n=1,2,\dots} \|A^nTB^n-T\|\leq 2 d(T,\Delta(A,B)) NEWLINE\]NEWLINE for every operator \(T\in B(H)\) where \(d(T,\Delta(A,B))\) is the distance of \(T\) to \(\Delta(A,B)\).