On invertibility of some operator sums
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 \(\parallel U-V \parallel <2\).
scientific article; zbMATH DE number 6128340