New characterizations of SD operator and its generalized Alutghe transform

Abstract

We investigate the class of star-dagger operators for which $A^*$ and $A^\dag$ commute.‎‎

‎Let $A=U|A|$ be the polar decomposition, $\widetilde{A}(s,t) =|A|^{s} U|A|^{t}$ be the generalized Aluthge transformation, and $\widetilde{A} ^{(*)}(s,t) =|A^*|^{s} U|A^*|^{t}$ be the generalized $*$-Aluthge transformation of $A$, respectively.‎

We have discovered new characterizations for star-dagger operators, specifically that $A$ is a star-dagger operator if and only if $U$ and $A$ commute. In this particular case, we have proven that $\widetilde{A}(s,t) = P_{\mathscr{R}(A^*)} A$ and $\widetilde{A}^{(*)}(s,t) = A P_{\mathscr{R}(A)}$ when $s, t > 0$ and $s+t=1$.