Spectrum of quasi-class \((A,k)\) operators (Q642826)

Summary: An operator \(T \in B (\mathcal H)\) is called quasi-class \((A, k)\) if \(T^{{\ast}k}(|T^2| - |T|^2)T^k \geq 0\) for a positive integer \(k\), which is a common generalization of class \(A\). In this paper, firstly we consider some spectral properties of quasi-class \((A, k)\) operators; it is shown that, if \(T\) is a quasi-class \((A, k)\) operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of \(T\) are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class \(A\) operators. Particularly, we show that, if \(T\) is a class \(A\) operator and either \(\sigma(|T|)\) or \(\sigma(|T^{\ast}|)\) is not connected, then \(T\) has a nontrivial invariant subspace.