On commutator of generalized Aluthge transformations and Fuglede–Putnam theorem

Abstract

Let $A=U|A|$ be the polar decomposition of A on a complex Hilbert space ${\mathscr {H}}$ and $0<s,t$. Then ${\widetilde{A}}_{s, t}=|A|^sU|A|^t$ and ${\widetilde{A}}_{s, t}^{(*)}=|A^*|^sU|A^*|^t$ are called the generalized Aluthge transformation and generalized $*$-Aluthge transformation of A, respectively. A pair (A, B) of operators is said to have the Fuglede–Putnam property (breifly, the FP-property) if $AX=XB$ implies $A^*X=XB^*$ for every operator X. We prove that if (A, B) has the FP-property, then $({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})$ and $(({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})$ has the FP-property for every $s,t>0$ with $s+t=1$. Also, we prove that $({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})$ has the FP-property if and only if $(({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})$ has the FP-property, where A, B are invertible and $0 < s, t$ with $s + t =1$. Moreover, we prove that if $0 < s, t$ and ${\widetilde{A}}_{s, t}$ is positive and invertible, then $\left\| {\widetilde{A}}_{s, t}X-X{\widetilde{A}}_{s, t}\right\| \le \left\| A\right\| ^{2t}\left\| ({\widetilde{A}}_{s, t})^{-1}\right\| \left\| X\right\|$ for every operator X. Also, if $0 <s, t$ and X is positive, then $\left\| |{\widetilde{A}}_{s, t}|^{2r} X-X|{\widetilde{A}}_{s, t}|^{2r}\right\| \le \frac{1}{2}\left\| |A|\right\| ^{2r}\left\| X\right\|$ for every $r>0$.