Properties of \(p\)-\(\omega\)-hyponormal operators (Q2369374)

Let \(T\) be a bounded linear operator on a complex Hilbert space and let \(T^{\ast}\) denote the adjoint operator of \(T\), \(T=U|T|\) be the polar decomposition of \(T\), where \(|T| =(T^{\ast}T)^{1/2}\). Set \(\tilde{T} = |T|^{1/2}U|T|^{1/2}\). For \(p > 0\), \(T\) is said to be \(p\)-\(\omega\)-hyponormal if \(|\tilde{T}|^p\geq |T|^p\geq |(\tilde{T})^\ast|^p\). In particular, if \(p=1\), \(T\) is said to be \(\omega\)-hyponormal. In this paper, the authors discuss the approximate spectrum and the numerical range of \(p\)-\(\omega\)-hyponormal operators. It is also shown that for a \(p\)-\(\omega\)-hyponormal operator \(T\), if \(\tilde{T}\) is normal, then \(T= \tilde{T}\).