Weyl's theorem for \(p\)-hyponormal or \(M\)-hyponormal operators (Q2777593)

m7bounded linear operator \(T\) on a complex Hilbert space is said to satisfy Weyl's theorem if the set of points \(z\) in its spectrum for which \(T-zI\) is a Fredholm operator of index 0 equals the set of isolated eigenvalues of finite multiplicity. It is shown that Weyl's theorem holds if \(T\) is \(p\)-hyponormal (i.e., \((T^*T)^p \geq(TT^*)^p\) for some \(p>0)\) [see \textit{M. Cho}, \textit{M. Itoh} and \textit{S. Oshiro}, Glasg. Math. J. 43, 375-381 (2001; Zbl 0902.47021)] or \(M\)-hyponormal (i.e., \((T-zI) (T-zI)^*\leq M^2(T-z I)^* (T-zI)\) for some \(M>0\) and each \(z\in\mathbb{C})\), and if additionally \(T-xI\) is a Fredholm operator of index 0 for each \(z\neq 0\), then \(T\) is compact and normal.