\(\lambda\)-Aluthge transforms and Schatten ideals
type: Item
instance of: scholarly article
title: \(\lambda\)-Aluthge transforms and Schatten ideals
zbMATH Open document ID: 1074.47006
author: Jorge Antezana
author: P. G. Massey
author: Demetrio Stojanoff
published in: Linear Algebra and its Applications
publication date: 2005-08-01T00:00:00Z
full work available at URL: http://sedici.unlp.edu.ar/handle/10915/83154
review text: Let \(T\) be a bounded linear operator on a Hilbert space and let \(T=U| T| \) be the polar decomposition of \(T\). Then for every \(\lambda \in [0,1]\) the \(\lambda\)-Aluthge transform is defined by \(\Delta_\lambda(T)=| T| ^\lambda U | T| ^{1-\lambda}\). For \(\lambda=1/2\), this gives the usual Aluthge transform introduced in [\textit{A. Aluthge}, Integral Equations Oper. Theory 13, No. 3, 307--315 (1990; Zbl 0718.47015)] to study \(p\)-hyponormal operators. The authors prove generalizations of well-known properties of the Aluthge transform to \(\lambda\)-Aluthge transforms, as well as several results which are new even for the usual Aluthge transform. In particular, they show that for \(\lambda \in (0,1)\), the map \(T \mapsto \Delta_\lambda(T)\) is continuous at every closed range operator \(T\). Moreover, for \(\lambda \in (0,1)\), they prove that the Schatten \(p\)-norms of the \(\lambda\)-Aluthge transforms decrease with respect to the Schatten \(p\)-norms of the original operator, and that equality holds if and only if \(T\) is normal. The relationship to Riesz's functional calculus is also considered as well as the Jordan structure of the iterated \(\lambda\)-Aluthge transforms in the finite-dimensional case.
reviewed by: Carsten Michels
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 2189256
zbMATH Keywords: Aluthge transform
zbMATH Keywords: Schatten norms
zbMATH Keywords: Riesz calculus
zbMATH Keywords: polar decomposition
Wikidata QID: Q88554048
MaRDI profile type: Publication
OpenAlex ID: W2078577708
cites work: On p-hyponormal operators for \(0<p<1\)
cites work: Aluthge Transforms and the Convex Hull of the Eigenvalues of a Matrix
cites work: The iterated Aluthge transforms of a 2-by-2 matrix converge.
cites work: Young's inequality in trace-class operators
cites work: Generalized inverses. Theory and applications.
cites work: Some Inequalities for Norms of Commutators
cites work: Q4147133
cites work: Q3999065
cites work: Q3349311
cites work: Q4878601
cites work: Complete contractivity of maps associated with the Aluthge and Duggal transforms.
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: Matrix Young inequalities for the Hilbert-Schmidt norm
cites work: Aluthge transforms of operators
cites work: The iterated Aluthge transform of an operator
cites work: On weakly unitarily invariant norm and the Aluthge transformation
cites work: Q4066072
cites work: Q3857325
cites work: Heinz and McIntosh inequalities, Aluthge transformation and the spectral radius
cites work: Numerical range of Aluthge transform of operator
cites work: An expression of spectral radius via Aluthge transformation
cites work: Characterizations of \(\log A\geq\log B\) and normaloid operators via Heinz inequality
DOI: 10.1016/J.LAA.2005.03.016
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description: scientific article