On invertibility of some operator sums
type: Item
instance of: scholarly article
title: On invertibility of some operator sums
zbMATH Open document ID: 1279.47011
DOI: 10.1134/S1995080212030055
published in: Lobachevskii Journal of Mathematics
publication date: 2013-01-23T00:00:00Z
review text: Let \(\mathcal H\) be a complex Hilbert space and \(\mathcal B\)\((\mathcal H)\) denote the algebra of bounded linear operators on \(\mathcal H\). Among other results, the author proves the following: Let \(\{A_1, A_2, \dots, A_n\} \subseteq \mathcal B\)\((\mathcal H)\) and \(p_1, p_2, \dots, p_n\) be positive real numbers. If \(\sum_{k=1}^n A_k\) is invertible, then so is \(\sum_{k=1}^n {| A_k |}^{p_k}\), where for \(X \in \mathcal B\)\((\mathcal H)\), \(| X | = \sqrt{X^*X}\). As consequences, the following are shown:{\parindent=6mm \begin{itemize}\item[(1)] If \(A,B \in \mathcal B\)\((\mathcal H)\) are nonnegative and \(A-B\) is invertible, then \(A+B\) is invertible. \item[(2)] Let \(A,B \in \mathcal B\)\((\mathcal H)\) be self-adjoint operators such that \(A \geq 0\) and \(-A \leq B \leq A\). If \(B\) is invertible, then \(A\) is invertible. \end{itemize}} It is also shown that, if \(U,V\) are unitary operators, then \(U+V\) is invertible if and only if \(\parallel U-V \parallel <2\).
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6128340
zbMATH Keywords: Hilbert space
zbMATH Keywords: linear bounded operator
zbMATH Keywords: isometry
zbMATH Keywords: unitary operator
zbMATH Keywords: projection
zbMATH Keywords: invertibility
zbMATH Keywords: unilateral shift
author: Airat Bikchentaev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6128340