Fuglede–Putnam type theorems via the generalized Aluthge transform
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- 作者：M. H. M. Rashid (1)
- 关键词：Fuglede–Putnam theorem ; Generalized Aluthge transform ; Class $$A(s ; t)$$ A ( s ; t ) operators ; Generalized derivation ; 47B20 ; 47A63
- 刊名：Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
- 出版年：2014
- 出版时间：September 2014
- 年：2014
- 卷：108
- 期：2
- 页码：1021-1034
- 全文大小：222 KB
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1. Department of Mathematics and Statistics, Faculty of Science, Mu’tah University, P.O.Box (7), Alkarak, Jordan - ISSN：1579-1505
文摘
Let \(T=U|T|\) and \(S=V|S|\) be the polar decompositions of \(T\in {\fancyscript{L}}({\fancyscript{H}})\) and \(S\in {\fancyscript{L}}({\fancyscript{K}})\) and let \(Com (T,S)\) stand for the set of operators \(X\in \fancyscript{L}(\fancyscript{K},\fancyscript{H})\) such that \(TX=XS.\) A pair \((T,S)\) is said to have the Fuglede–Putnam property if \(Com(T,S)\subseteq Com(T^*,S^*).\) Let \(\widetilde{Z}(s,t)\) denote the generalized Aluthge transform of a bounded operator \(Z\) . We show that (i) if \(T\) is invertible class \(A(s,t)\) operator with \(s+t=1\) and \(S^*\) is a class \({\mathcal {Y}}\) , then \(Com(T,S)\subseteq Com(T^*,S^*);\) (ii) if the pair \((T,S)\) have the Fuglede–Putnam property, then the range of \(\delta _{T,S}\) is orthogonal to the kernel of \(\delta _{T,S}\) ; (iii) if the pair \((T,S)\) have the Fuglede–Putnam property, then \(Com(T,S)\subseteq Com(\widetilde{T}(s,t),\widetilde{S}(s,t))\) , furthermore, if \(T\) is invertible, then \(Com(T,S)= Com(\widetilde{T}(s,t),\widetilde{S}(s,t))\) . Finally, if \(Re(U|T|^s)\ge a>0\) and \(Re(V|S|^s)\ge a>0\) and \(X\) is an operator such that \(U^*X=XV,\) then we prove that \(\left\| (\widetilde{T}(s,s))^*X-X\widetilde{S}(s,s)\right\| _p\ge 2a\left\| |T|^sX-X|S|^s\right\| _p\) for any \(1\le p\le \infty \) such that \(0