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Given self-adjoint operators $A, B\in\mathbb{B}(\mathscr{H})$ it is said $A\leq_uB$ whenever $A\leq U^*BU$ for some unitary operator $U$. We show that $A\leq_u B$ if and only if $f(g(A)^r)\leq_uf(g(B)^r)$ for any increasing operator convex function $f$, any operator monotone function $g$ and any positive number $r$. We present some sufficient conditions under which if $B\leq A\leq U^*BU$, then $B=A=U^*BU$. Finally we prove that if $A^n\leq U^\ast A^nU$ for all $n\in\mathbb{N}$, then $A=U^\ast AU$., Comment: To appear in C. R. Math. Acad. Sci. Paris

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 $Com(A,B)$ 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 $Com(A,B)\subseteqCom(A^\ast,B^\ast)$. 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 $Com(A,B)\subseteqCom(\tilde{A},\tilde{B})$, moreover, if $A$ is invertible, then $Com(A,B)=Com(\tilde{A},\tilde{B})$. Finally, if $Re(U|A|^{1\over2})\geq a>0$ and $Re(V|B|^{1\over2})\geq a>0$ 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 $1 \leq p \leq \infty$., Comment: 13 pages; to appear in Positivity

Let $T\in\mathbb{B}(\mathscr{H})$ and $T=U|T|$ be its polar decomposition. We proved that (i) if $T$ is log-hyponormal or $p$-hyponormal and $U^n=U^\ast$ for some $n$, then $T$ is normal; (ii) if the spectrum of $U$ is contained in some open semicircle, then $T$ is normal if and only if so is its Aluthge transform $\widetilde{T}=|T|^{1\over2}U|T|^{1\over2}$.