Normality of Products of p-Hyponormal Operators and Their Aluthge Transforms

Abstract.

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform $\widetilde{A}\,\,\textrm{of} \,A$ is defined to be the operator $\widetilde{A}=|A|^{\frac{1}{2}}U|A|^{\frac{1}{2}}$. We say that A $\in$ B(H) is p-hyponormal, $0 < p, \,\textrm{if}\,|A^{\ast}|^{2p}\leq |A|^{2p}$. Let $\check{B} = \tilde{B}\ast^{\ast}$. Given p-hyponormal $A, B^{\ast}, \frac{1}{2}\leq p < 1$ , such that AB is compact, this note considers the relationship between $|\lambda_{j}(AB)|, s_{j}(AB), |\lambda_{j}(\tilde{A}\check{B})|,s_{j}(\tilde{A}\check{B})\,\textrm{and}\,s_{j}(\check{B}\tilde{A}),\, \textrm{where}\,|\lambda_{j}(T)|\,(\textrm{respectively}, s_{j}(T))$ denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that $s_{j}(AB)$ is bounded above by $s_{j}(\tilde{A}\check{B})$ and below by $s_{j}(\check{B}\tilde{A})$ for all j = 1, 2, . . . and that if also $\tilde{A}\check{B}$ is normal, then there exists a unitary U ¹ such that $|\lambda_{j}(U_{1}AB)| = |\lambda_{j}(\tilde{A}\breve{B})| = |\lambda_{j}(\breve{B}\tilde{A})| = s_{j}(\tilde{A}\check{B}) = s_{j}(\check{B}\tilde{A}) = s_{j}(AB) = s_{j}(U_{1}AB)$ for all j = 1, 2, . . ..