Normality of products of \(p\)-hyponormal operators and their Aluthge transforms (Q1885313)

For a compact operator \(T\), let \(\{\lambda_{j}(T)\}\) denote the eigenvalues of \(T\) enumerated (taking into account multiplicity) according to their decreasing modulus, and let \(\{s_{j}(T)\}\) denote the singular values of \(T\) enumerated (taking into account multiplicity) according to their decreasing order. Let \(A\) and \(B^{*}\) be \(p\)-hyponormal operators for \(\frac{1}{2}\leq p\leq 1\) such that \(AB\) is compact. The author discusses the relations among \(| \lambda_{j}(\check{B}\tilde{A})| \), \(| \lambda_{j}(\tilde{A}\check{B})| \), \(s_{j}(\check{B}\tilde{A})\), \(s_{j}(AB)\) and \(s_{j}(\tilde{A}\check{B})\), where \(\tilde{A}\) means the Aluthge transform of \(A\) and \(\check{B}\) means \(\widetilde{(B^{*})}^{*}\).