Closing the gap in a subspace perturbation bound
Let \(X\) and \(Y\) be bounded self-adjoint operators on a Hilbert space \(\mathcal H\) and \(K_ X\subseteq \sigma(X)\), \(K_ Y\subseteq \sigma(Y)\) closed sets bounded apart by \(\delta>0\) with corresponding spectral projectors \(E\) and \(F\). From the work [\textit{R. Bhatia}, \textit{C. Davis} and \textit{P. Koosis}, J. Funct. Anal. 82, 138-150 (1989; Zbl 0674.42002)] it is known that \({\pi\over 2}||| X-Y|||\geq \delta||| EF|||\) for any unitary invariant norm. In the article under review it is shown that \({\pi\over 2}\) is best possible in this equality. The analogous question for normal operators is discussed.
scientific article; zbMATH DE number 179131