Convergence of iterated Aluthge transform sequence for diagonalizable matrices

Demetrio Stojanoff | Antezana, Jorge | Pujals, Enrique R.

Keywords: Aluthge transform | stable manifold theorem | similarity orbit | polar decomposition.

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by (null) Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf 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 $r\times r$ matrices with $r$ different eigenvalues.

Anexos:

• Aluthge.pdf