Convergence of iterated Aluthge transform sequence for diagonalizable matrices

Details Convergence of iterated Aluthge transform sequence for diagonalizable matrices Convergence of iterated Aluthge transform sequence for diagonalizable matrices

2006-04-12

arxiv.org/abs/math/0604283v1

Mathematics

Functional Analysis

25 pages

Scientific paper

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 $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. 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 %of class $C^\infty$ on the (open and dense) set of $r\times r$ matrices with $r$ different eigenvalues.

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