On \(p\)-quasihyponormal operators (Q2496416)

A bounded linear operator \(T\) on a complex Hilbert space is said to be a \(p\)-quasihyponormal if \(T^*(| T| ^{2p}-| T^*| ^{2p})T\geq0\). If \(p=1\), the operator \(T\) is simply called quasihyponormal. For a \(p\)-quasihyponormal operator \(T\) with the polar decomposition \(T=:U| T|\), the author shows that the operator \(T_p:=U| T| ^p\) is quasihyponormal and its spectrum is given by \(\sigma(T_p)=\{r^pe^{i\theta}:r^pe^{i\theta}\in\sigma(T)\}\). He also proves that \[ \| | T| ^{2p}-| T^*| ^{2p}\| \leq2\| T\| ^p\left(\frac{p}{\pi}\int\int r^{2p-1}drd\theta\right)^{\frac{1}{2}}, \] that every Riesz idempotent \(E\) with respect to an isolated point \(\lambda\) of the spectrum of \(T\) is self-adjoint and satisfies \(\text{ran}(E)=\ker (T-\lambda)=\ker(T-\lambda)^{*}\), and that Weyl's theorem holds for \(T\).