Fuglede-Putnam type theorems via the Aluthge transform

Abstract

Let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A\in \mathbb{B}({\mathcal{H}}_1)$ and $B\in \mathbb{B}({\mathcal{H}}_2)$ and let $Com(A,B)$ stand for the set of operators $X\in \mathbb{B}({\mathcal{H}}_2,{\mathcal{H}}_1)$ 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 $Re(U|A{|}^{\frac{1}{2}})\ge a>0$ and $Re(V|B{|}^{\frac{1}{2}})\ge a>0$ and $X$ is an operator such that $U^{\ast }X=XV$, then we prove that $\Vert {\tilde{A}}^{\ast }X-X\tilde{B}{\Vert }_p\ge 2a\Vert |B{|}^{\frac{1}{2}}X-X|B{|}^{\frac{1}{2}}{\Vert }_p$ for any $1\le p\le \infty$.

arXiv comment13 pages; to appear in Positivity