Some properties of the generalized Aluthge transform
type: Item
instance of: scholarly article
title: Some properties of the generalized Aluthge transform
zbMATH Open document ID: 1092.47003
author: Xiumei Liu
author: Guoxing Ji
published in: Nihonkai Mathematical Journal
publication date: 2005-04-18T00:00:00Z
review text: If \(T\) is a bounded operator on a complex separable Hilbert space \(\mathcal H\), the Aluthge transform is defined by \(\widetilde {T} = | T| ^{\frac{1}{2}} U | T| ^{\frac{1}{2}}\) where \(T = UT\) is the polar decomposition of \(T\). The generalized Aluthge transform is defined by \(\widetilde {T}(t) = | T| ^{t} U | T| ^{1-t}\) for \(0< t<1\), \(T(0) = U^*UU| T| \), and \(T(1) = | T| U\). The authors' main result is that if \(\rho(A) = \{\lambda \in \mathbb C : R(A-\lambda I)\) is closed\(\} \setminus \{0 \} \), then \(\rho(T) = \rho(\widetilde {T}(t))\) for \(0\leq t \leq 1\). Some additional results on the numerical range of the generalized Aluthge transform are also obtained.
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 2157946
zbMATH Keywords: generalized Aluthge transform
zbMATH Keywords: polar decomposition
zbMATH Keywords: closed range
zbMATH Keywords: numerical range
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2157946