Fuglede-Putnam type theorems via the generalized Aluthge transform
type: Item
instance of: scholarly article
title: Fuglede-Putnam type theorems via the generalized Aluthge transform
zbMATH Open document ID: 1314.47033
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2014-09-09T00:00:00Z
review text: Let \(H\) be a separable complex Hilbert space and \(L(H)\) be the algebra of all bounded linear operators on \(H\). For \(S, T\in L(H)\), let \(\mathrm{Com}(T,S)=\{ X\in L(H):\; TX=XS\}\). A pair \((T,S)\) is said to have the Fuglede-Putnam property if \(\mathrm{Com}(T,S) \subseteq \mathrm{Com}(T^*,S^*)\). Several results related to pairs with the Fuglede-Putnam property are presented. We mention a result which is close to the classical Fuglede-Putnam theorem. Theorem. If \(T\) is an invertible operator in the class \(A(s,t)\), where \(s+t=1\), and \(S^*\) is in the class \({\mathcal Y}\), then \(\mathrm{Com}(T,S) \subseteq \mathrm{Com}(T^*,S^*)\). Recall that \(T\in L(H)\) belongs to the class \(A(s,t)\) if \((|T^*|^t|T|^{2s}|T^*|^t)^{t/(t+s)} \geq |T^*|^{2t}\). If, for some \(\alpha\geq 1\), there exists a positive number \(k_\alpha\) such that \(|TT^*-T^*T|^\alpha\leq k_{\alpha}^{2}(T-\lambda)^*(T-\lambda)\) for all complex numbers \(\lambda\), then \(T\) is in the class \({\mathcal Y}\).
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 6341689
zbMATH Keywords: Fuglede-Putnam theorem
zbMATH Keywords: generalized Aluthge transform
zbMATH Keywords: class \(A(s,t)\) operators
zbMATH Keywords: generalized derivation
author: Mohammad H. M. Rashid
MaRDI profile type: Publication
full work available at URL: s13398-013-0158-y
OpenAlex ID: W2064877390
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DOI: 10.1007/S13398-013-0158-Y
description: scientific article; zbMATH DE number 6341689