A characterization of the class of partial isometries
Let \(S\) be a self-adjoint invertible operator acting on a Hilbert space. The Corach-Porta-Recht inequality states that \(\|SXS^{-1} + S^{-1}XS\| \geq 2\|X\|\) holds true for every bounded linear operator~\(X\). Several authors have considered the case of equality and, more generally, characterized subclasses of normal operators by inequalities or equalities in the spirit of the Corach-Porta-Recht inequality. The following is part of the main result of this paper. Suppose that \(S\) is a closed range operator such that the range of the adjoint \(S^{\ast}\) is the range of \(S\). Denote by \(P\) the orthogonal projection onto the range of \(S\). Then \(S\) is a nonzero real multiple of a normal partial isometry if and only if \(\|S^{\ast}XS^{+} + S^{+}XS^{\ast}\| = 2\|PXP\|\) for all \(X\), if and only if \(\|S^{\ast}\oplus S^{+} + S^{+}\oplus S^{\ast}\|_{\lambda} = 2\). Here, \(S^{+}\) is the Moore-Penrose inverse of \(S\) and \(\|\cdot\|_{\lambda}\) denotes the injective norm on the tensor product.
scientific article; zbMATH DE number 6063153
scientific article