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 $$r>0$$ .