Spectral properties for some extensions of isometric operators
The author obtains some properties of \(2\)-isometric and quasi-\(2\)-isometric operators, shows that the spectrum is continuous on the set of all \(2\)-isometric operators and deduces that the Weyl and Browder spectrum have the same properties. He also proves that, if \(T , S^{*}\) are \(2\)-isometric and invertible isometric operators, respectively, and \(X\) is a Hilbert-Schmidt operator such that \(T X = X S\), then \(T^{*}X = X S^{*}\). Moreover, the author shows that every Riesz projection \(E\) of a \(2\)-isometric operator \(T \) is self-adjoint and satisfies \( \mathcal{R}(E) = \mathcal{N}(T -\lambda) = \mathcal{N} (T -\lambda)^{*}\), where \(\mathcal{N}(T), \mathcal{R}(T)\) denote the kernel and the range of \(T\in B(\mathcal{H})\), respectively. It is shown that a quasi-\(2\)-isometric operator satisfies Bishop's property (\(\beta\)). Finally, Weyl-type theorems are proved for \(f(d_{T,S})\), with quasi-\(2\)-isometric operator entries \(T\) and \(S^{*} \).
scientific article; zbMATH DE number 7222415