A sharp estimate in an operator inequality

Abstract:

Let $\mathcal {H}$ and $\mathcal {K}$ be Hilbert spaces, and suppose $A \in \mathcal {B}(\mathcal {H})$ and $B \in \mathcal {B}(\mathcal {K})$ are selfadjoint operators with $\operatorname {dist} (\sigma (A),\sigma (B)) \geq \delta > 0$. It is known that for any $Q \in \mathcal {B}(\mathcal {K},\mathcal {H})$ we must have $\tfrac {\pi }{2}||AQ - QB|| \geq \delta ||Q||$. In this paper we give examples proving that $\pi /2$ is sharp in this inequality.