Iterated Aluthge transforms: a brief survey

given an r × r complex matrix t, if is the polar decomposition of t, then the aluthge transform is defined by let denote the n-times iterated aluthge transform of t, i.e. and , . 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 converges for every r × r matrix t. this result was conjectured by jung, ko and pearcy in 2003.