Positivity, ( ISI ), Volume (17), No (1), Year (2013-1) , Pages (151-162)

Title : ( Fuglede–Putnam type theorems via the Aluthge transform )

Authors: Mohammad Sal Moslehian , sadegh nabavi ,

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Abstract

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 $1 \\leq p \\leq \\infty$.

Keywords

, Aluthge transformation, polar decomposition, norm inequality, Fuglede-Putnal theorem
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@article{paperid:1025105,
author = {Sal Moslehian, Mohammad and Nabavi, Sadegh},
title = {Fuglede–Putnam type theorems via the Aluthge transform},
journal = {Positivity},
year = {2013},
volume = {17},
number = {1},
month = {January},
issn = {1385-1292},
pages = {151--162},
numpages = {11},
keywords = {Aluthge transformation; polar decomposition; norm inequality; Fuglede-Putnal theorem},
}

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%0 Journal Article
%T Fuglede–Putnam type theorems via the Aluthge transform
%A Sal Moslehian, Mohammad
%A Nabavi, Sadegh
%J Positivity
%@ 1385-1292
%D 2013

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