Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ-Aluthge Transform.

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U| T|. Then the λ-Aluthge transform is defined by . Let $$\Delta^{n}_{\lambda} (T)$$ denote the n-times iterated Aluthge transform of T, $$n\, \in \, {\mathbb{N}}$$ . We prove that the sequence $${\{\Delta^{n}_{\lambda} (T)\}}_{n \in {\mathbb{N}}}$$ converges for every r × r diagonalizable matrix T. We show regularity results for the two parameter map $$(\lambda, T) \longmapsto \Delta^{\infty}_ {\lambda} (T)$$ , and we study for which matrices the map $$(0, 1) \ni \lambda \longmapsto \Delta^{\infty}_ {\lambda} (T)$$ is constant. [ABSTRACT FROM AUTHOR]