Iterated Aluthge transforms: a brief survey

JORGE ANTEZANA; ENRIQUE PUJALS; DEMETRIO STOJANOFF

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Año: 2008 vol. 49 p. 29 - 41

0041-6932

Given an rxr complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by Delta(T)= |T|^{1/2} U |T |^{1/2}. Let Delta^{n}(T) denote the n-times iterated Aluthge transform of T, i.e. Delta^{0}(T)=T and $Delta^{n}(T)=Delta(Delta^{n-1}(T)), where n is a positive integer. We prove that the sequence {Delta^{n}(T)} converges for every rxr  matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003.