Binormal operator and *-Aluthge transformation

Abstract

Let T = U|T| be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation $\tilde T^{( * )} = |T^ * |^{\tfrac{1} {2}} U|T^ * |^{\tfrac{1} {2}}$ is called the Aluthge transformation and $\tilde T_n$ means the n-th Aluthge transformation. Similarly, the transformation $\tilde T^{( * )} = |T^ * |^{\tfrac{1} {2}} U|T^ * |^{\tfrac{1} {2}}$ is called the *-Aluthge transformation and $\tilde T_n^{( * )}$ means the n-th *-Aluthge transformation. In this paper, firstly, we show that $\tilde T^{( * )} = UV|\tilde T^{( * )} |$ is the polar decomposition of $\tilde T^{( * )}$, where $|T|^{\tfrac{1} {2}} |T^ * |^{\tfrac{1} {2}} = V||T|^{\tfrac{1} {2}} |T^ * |^{\tfrac{1} {2}} |$ is the polar decomposition. Secondly, we show that $\tilde T^{( * )} = U|\tilde T^{( * )} |$ if and only if T is binormal, i.e., [|T|, |T*|]=0, where [A,B] = AB ‒ BA for any operator A and B. Lastly, we show that $\tilde T_n^{( * )}$ is binormal for all non-negative integer n if and only if T is centered, and so on.