Abstract
Let \(\mathcal{H}\) be a Hilbert space over the field \(\mathbb{C}\), and let \(\mathcal{B}(\mathcal{H})\) be the \(\ast\)-algebra of all linear bounded operators on \(\mathcal{H}\). Sufficient conditions for the positivity and invertibility of operators from \(\mathcal{B}(\mathcal{H})\) are found. An arbitrary symmetry from a von Neumann algebra \(\mathcal{A}\) is written as the product \(A^{-1}UA\) with a positive invertible \(A\) and a self-adjoint unitary \(U\) from \(\mathcal{A}\). Let \(\varphi\) be the weight on a von Neumann algebra \(\mathcal{A}\), let \(A\in \mathcal{A}\), and let \(\|A\|\le 1\). If \(A^*A-I\in \mathfrak{N}_{\varphi}\), then \(|A|-I\in \mathfrak{N}_{\varphi}\) and, for any isometry \(U\in \mathcal{A}\), the inequality \(\|A-U\|_{\varphi,2}\ge \||A|-I\|_{\varphi,2}\) holds. If \(U\) is a unitary operator from the polar decomposition of the invertible operator \(A\), then this inequality becomes an equality.
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Funding
This work was carried out within the framework of the Development Program of the Scientific and Educational Mathematical Center of the Volga Federal Region (agreement no. 075-02-2022-882).
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 350–359 https://doi.org/10.4213/mzm13548.
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Bikchentaev, A.M. Invertibility of the Operators on Hilbert Spaces and Ideals in \(C^*\)-Algebras. Math Notes 112, 360–368 (2022). https://doi.org/10.1134/S0001434622090036
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DOI: https://doi.org/10.1134/S0001434622090036