Weyl type theorem and spectrum for \((p,k)\)-quasiposinormal operators (Q1943742)

Let \({\mathcal H}\) be an infinite-dimensional separable Hilbert space, and let \(B({\mathcal H})\) be the algebra of all bounded linear operators on \({\mathcal H}\). For given real numbers \(0< p\leq 1\) and \(c>0\) and a positive integer \(k\), we say that \(T\in B({\mathcal H})\) is a \((p,k)\)-quasiposinormal operator if \(T^{*k}[cA^2(T^*T)^p- (TT^*)^p]T^k\geq 0\). In this paper, the authors consider Weyl type theorems about \((p,k)\)-quasiposinormal operators. They show that, if \(T\) is a \((p,k)\)-quasiposinormal operator, then it satisfies Weyl's theorem and the spectral mapping theorem for the Weyl spectrum. Moreover, they verify that, if \(T^*\) is \((p,k)\)-quasiposinormal, then generalized \(a\)-Weyl's theorem holds for \(T\). Finally, it is proved that the spectral relation \(\sigma_{jap}(T)\setminus\{0\}=\sigma_{ap}(T)\setminus\{0\}\) holds for any \((p,k)\)-quasiposinormal operator \(T\), where \(\sigma_{ap}(T)\) and \(\sigma_{jap}(T)\) denote the approximate point spectrum and the joint approximate spectrum of \(T\), respectively.