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A sharp estimate in an operator inequality

Author: R. McEachin
Journal: Proc. Amer. Math. Soc. 115 (1992), 161-165
MSC: Primary 47A30; Secondary 15A45, 47A55, 47B15
MathSciNet review: 1081093
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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}\vert\vert AQ - QB\vert\vert \geq \delta \vert\vert Q\vert\vert$. In this paper we give examples proving that $ \pi /2$ is sharp in this inequality.

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Article copyright: © Copyright 1992 American Mathematical Society

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