Convergence of iterated Aluthge transform sequence for diagonalizable matrices

J. ANTEZANA, E. PUJALS Y D. STOJANOFF

ADVANCES IN MATHEMATICS

Elsevier

Año: 2007 vol. 216 p. 255 - 278

0001-8708

Given an r x r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by  A(T) = |T|^{1/2} U |T |^{1/2}. Let A^{n}(T) denote the n-times iterated Aluthge transform of T, i.e. A^{0}(T)=T and A^{n}(T)=A(A^{n-1}(T)), n inN. We prove that the sequence {A^{n}(T)} converges for every r x r diagonalizable  matrix T. We show that the limit A^{infty}(.) is a map of class C^infty on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r x r matrices  with r different eigenvalues.