The iterated Aluthge transforms of a matrix converge

Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

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15 Citations (Scopus)

Abstract

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by. Δ(T)=|T|1/2U|T|1/2. Let Δn(T) denote the n-times iterated Aluthge transform of T, i.e., Δ0(T)=T and Δn(T)=Δ(Δn-1(T)), nεN. We prove that the sequence {Δn(T)}nεN converges for every r×r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function. © 2010 Elsevier Inc.
Original languageEnglish
Pages (from-to)1591-1620
JournalAdvances in Mathematics
Volume226
Issue number2
DOIs
Publication statusPublished - 30 Jan 2011

Keywords

  • Aluthge transform
  • Polar decomposition
  • Primary
  • Secondary
  • Similarity orbit
  • Stable manifold theorem

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