Let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A\\in \\mathbb{B}(\\mathscr{H}_1)$ and $B\\in \\mathbb{B}(\\mathscr{H}_2)$ and let $\\mbox{Com}(A,B)$ stand for the set of operators $X\\in\\mathbb{B}(\\mathscr{H}_2,\\mathscr{H}_1)$ such that $AX=XB$. A pair $(A,B)$ is said to have the FP-property if $\\mbox{Com}(A,B)\\subseteq\\mbox{Com}(A^\\ast,B^\\ast)$. Let $\\tilde{C}$ denote the Aluthge transform of a bounded linear operator $C$. 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 $\\mbox{Com}(A,B)\\subseteq\\mbox{Com}(\\tilde{A},\\tilde{B})$, moreover, if $A$ is invertible, then $\\mbox{Com}(A,B)=\\mbox{Com}(\\tilde{A},\\tilde{B})$. Finally, if $\\mbox{Re}(U|A|^{1\\over2})\\geq a>0$ and $\\mbox{Re}(V|B|^{1\\over2})\\geq a>0$ and $X$ is an operator such that $U^* X=XV$, then we prove that $\\|\\tilde{A}^* X-X\\tilde{B}\\|_p\\geq 2a\\|\\,|B|^{1\\over2}X-X|B|^{1\\over2}\\|_p$ for any $1 \\leq p \\leq \\infty$.

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