Convergence of iterated Aluthge transform sequence for diagonalizable matrices

JORGE ANTEZANA; ENRIQUE PUJALS; DEMETRIO STOJANOFF

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2007 p. 255 - 278

0001-8708

Given an $r imes r$ complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by Deltaleft(T ight)= |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 number. We prove that the sequence {Delta^{n}(T)} converges for every rxr diagonalizable matrix T. We show that the limit Delta^{infty}( cdot) is a map of class $C^infty$ on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of rxr matrices with $r$ different eigenvalues.