Another generalization of Anderson’s theorem

Abstract:

In this paper, we prove that if A and B are normal operators on a Hilbert space H, then, for every operator S satisfying $ASB = S, \left \| {AXB - X + S} \right \| \geq {\left \| A \right \|^{ - 1}}{\left \| B \right \|^{ - 1}}\left \| S \right \|$ for all operators $X \in B(H)$, and that if A and B are contractions, then, for every operator S satisfying $ASB = S$ and ${A^ \ast }S{B^ \ast } = S,\left \| {AXB - X + S} \right \| \geq \left \| S \right \|$ for all operators $X \in B(H)$, where $B(H)$ denotes the set of all bounded linear operators on H.