Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: λ-Aluthge transform

JORGE ANTEZANA; ENRIQUE PUJALS; DEMETRIO STOJANOFF

INTEGRAL EQUATIONS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Lugar: BASEL; Año: 2008 vol. 62 p. 465 - 488

0378-620X

Let $lambda in (0,1)$ and let $T$ be a $r imes r$ complex matrix with polar decomposition $T=U|T|$. Then, the $la$- Aluthge transform is defined by $$Delta_lambda left(T ight)= |T|^{lambda} U |T |^{1-lambda}.$$ Let $Delta_lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$,  $ninmathbb{N}$. We prove that the sequence ${Delta_lambda^{n}(T)}_{ninmathbb{N}}$ converges for every $r imes r$ {f diagonalizable}  matrix $T$. We show regularity results for the two parameter map $(la , T) mapsto alulit{infty}{T}$,  and we study for which matrices the map $(0,1) i lambda  mapsto Delta_lambda^{infty}( T)$ is constant.