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 A S B = S , ‖ A X B − X + S ‖ ≥ ‖ A ‖ − 1 ‖ B ‖ − 1 ‖ S ‖ ASB = S, \left \| {AXB - X + S} \right \| \geq {\left \| A \right \|^{ - 1}}{\left \| B \right \|^{ - 1}}\left \| S \right \| for all operators X ∈ B ( H ) X \in B(H) , and that if A and B are contractions, then, for every operator S satisfying A S B = S ASB = S and A ∗ S B ∗ = S , ‖ A X B − X + S ‖ ≥ ‖ S ‖ {A^ \ast }S{B^ \ast } = S,\left \| {AXB - X + S} \right \| \geq \left \| S \right \| for all operators X ∈ B ( H ) X \in B(H) , where B ( H ) B(H) denotes the set of all bounded linear operators on H.