Iterated Aluthge transforms: a brief survey

J. ANTEZANA, E. PUJALS Y D. STOJANOFF

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UMA

Lugar: Buenos Aires; Año: 2008 vol. 49 p. 29 - 41

0041-6932

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then, the Aluthge transform is defined by
D(T) = |T|1/2U|T|1/2. Let D
n(T) denote the n-times iterated Aluthge transform of T, i.e.
D0(T) = T and
Dn(T) =
D(D
n−1(T)), n 2 N. In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the fact that the sequence {
n(T)}n2N converges for every r×r matrix T. This result was conjecturated by Jung, Ko and Pearcy in 2003.