Fuglede–Putnam type theorems via the generalized Aluthge transform

Abstract

Let $T=U|T|$ and $S=V|S|$ be the polar decompositions of $T\in {\fancyscript{L}}({\fancyscript{H}})$ and $S\in {\fancyscript{L}}({\fancyscript{K}})$ and let $Com (T,S)$ stand for the set of operators $X\in \fancyscript{L}(\fancyscript{K},\fancyscript{H})$ such that $TX=XS.$ A pair $(T,S)$ is said to have the Fuglede–Putnam property if $Com(T,S)\subseteq Com(T^*,S^*).$ Let $\widetilde{Z}(s,t)$ denote the generalized Aluthge transform of a bounded operator $Z$. We show that (i) if $T$ is invertible class $A(s,t)$ operator with $s+t=1$ and $S^*$ is a class ${\mathcal {Y}}$, then $Com(T,S)\subseteq Com(T^*,S^*);$ (ii) if the pair $(T,S)$ have the Fuglede–Putnam property, then the range of $\delta _{T,S}$ is orthogonal to the kernel of $\delta _{T,S}$; (iii) if the pair $(T,S)$ have the Fuglede–Putnam property, then $Com(T,S)\subseteq Com(\widetilde{T}(s,t),\widetilde{S}(s,t))$, furthermore, if $T$ is invertible, then $Com(T,S)= Com(\widetilde{T}(s,t),\widetilde{S}(s,t))$. Finally, if $Re(U|T|^s)\ge a>0$ and $Re(V|S|^s)\ge a>0$ and $X$ is an operator such that $U^*X=XV,$ then we prove that $\left\| (\widetilde{T}(s,s))^*X-X\widetilde{S}(s,s)\right\| _p\ge 2a\left\| |T|^sX-X|S|^s\right\| _p$ for any $1\le p\le \infty$ such that $0<s<1.$