On weakly unitarily invariant norm and the \(\lambda\)-Aluthge transformation for invertible operator
type: Item
instance of: scholarly article
title: On weakly unitarily invariant norm and the \(\lambda\)-Aluthge transformation for invertible operator
zbMATH Open document ID: 1108.15028
author: Kazuyoshi Okubo
published in: Linear Algebra and its Applications
publication date: 2006-12-07T00:00:00Z
review text: For any bounded linear operator \(T\) on a Hilbert space and \(0\leq\lambda\leq 1\), the \(\lambda\)-Aluthge transform \(T_\lambda\) of \(T\) is by definition the operator \(P^\lambda UP^{1-\lambda}\), where \(U\) and \(P\) are the operators in the polar decomposition \(T= UP\) of \(T\), with \(P= (T*T)^{1/2}\) and \(U\) a partial isometry satisfying \(U*UP= P\). In this paper, the author first proves that if \(T\) acts on a space \(H\) of dimension 2, then \(\| T_\lambda\|=\| T_{1- \lambda}\|\) for any \(0\leq \lambda\leq 1\), where \(\|\cdot\|\) denotes the spectral norm. By way of an example, he also shows that this is no longer the case if \(\dim H\geq 3\). Next, he proves that if \(T\) is invertible and \(B\) commutes with \(T\), then \(|||P^\lambda BUP^{1-\lambda}|||\leq |||BT|||\) for any weakly unitarily invariant norm \(|||\cdot|||(|||W*XW|||= |||X|||\) for any operators \(X\) and \(U\) with \(U\) unitary). This generalizes a previous result by the same author that \(|||p(T_\lambda)|||\leq |||p(T)|||\) for any polynomial \(p\).
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 5077726
zbMATH Keywords: Aluthge transform
zbMATH Keywords: \(\lambda\)-Aluthge transform
zbMATH Keywords: weakly unitarily invariant norm
zbMATH Keywords: \(\rho\)-radius
zbMATH Keywords: Hilbert space
zbMATH Keywords: spectral norm
MaRDI profile type: Publication
full work available at URL: j.laa.2006.04.003
OpenAlex ID: W2038186889
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cites work: On weakly unitarily invariant norm and the Aluthge transformation
DOI: 10.1016/J.LAA.2006.04.003
description: scientific article; zbMATH DE number 5077726