Fuglede-Putnam type theorems via the Aluthge transform

Let $A=U\mathrm{\mid }A\mathrm{\mid }$ and $B=V\mathrm{\mid }B\mathrm{\mid }$ be the polar decompositions of $A\in $\mathbb{B}$(\mathscr{H}_1)$and$B\in $\mathbb{B}$(\mathscr{H}_2)$andlet$Com(A,B)$standforthesetofoperators$X\in$\mathbb{B}$(\mathscr{H}_2,\mathscr{H}_1)$suchthat$AX=XB$\mathrm{.}Apair$(A,B)$issaidtohavetheFP-propertyif$Com(A,B)\subseteqCom(A^\ast,B^\ast)$\mathrm{.}Let$\tilde{C}$denotetheAluthgetransformofaboundedlinearoperator$C$. We show that (i) if $A$ and $B$ are invertible and $(A,B)$ has the FP-property, then so is $(\tilde{A},\tilde{B})$; (ii) if $A$ and $B$ are invertible, the spectrums of both $U$ and $V$ are contained in some open semicircle and $(\tilde{A},\tilde{B})$ has the FP-property, then so is $(A,B)$; (iii) if $(A,B)$ has the FP-property, then $Com(A,B)\text{subseteqCom}(\tilde{A},\tilde{B})$, moreover, if $A$ is invertible, then $Com(A,B)=Com(\tilde{A},\tilde{B})$. Finally, if $Re(U\mathrm{\mid }A{\mathrm{\mid }}^{\frac{1}{2}})\ge a>0$ and $Re(V\mathrm{\mid }B{\mathrm{\mid }}^{\frac{1}{2}})\ge a>0$ and $X$ is an operator such that $U^{\ast }X=XV$, then we prove that $|\tilde{A}^* X-X\tilde{B}|_p\geq 2a|,|B|^{1\over2}X-X|B|^{1\over2}|_p$forany$1 \leq p \leq \infty$.