Weyl's theorems in the class of algebraically \(p\)-hyponormal operators (Q2752299)

A bounded linear operator \(A\) on a complex separable Hilbert space is called \(p\)-hyponormal for \(0< p\leq 1\) if \((A^* A)^p- (AA^*)^p\geq 0\), and \(A\) is called algebraically \(p\)-hyponormal if \(q(A)\) is \(p\) hyponormal for some polynomial \(q\). The Weyl spectrum \(\sigma_w(A)\) is the set of those points \(\lambda\) of \(\sigma(A)\) for which \(A-\lambda\) is not a Fredhom operator of index \(0\), and the set \(\sigma_{eac}(A)\) is defined to consist of all complex numbers \(\lambda\) such that \(A-\lambda\) is not a semi-Fredholm operator with non-positive index. It is shown that algebraically \(p\)-hyponormal operators satisfy Weyl's theorem, i.e. their Weyl spectrum equals the spectrum except for those isolated points which are eigenvalues of finite multiplicity. Spectral mapping theorems with respect to polynomials are proven for \(\sigma_w(A)\) and \(\sigma_{ea}(A)\).