Iterated Aluthge transforms of composition operators on H²

In this paper, we study various properties of the iterated Aluthge transforms of the composition operators Cψ and Cσ where ψ(z) = az + (1-a) and σ(z) + az (1-a)z+1 for 0 < a < 1. We express the iterated Aluthge transforms e C(n) . and C^∼ψ(n) as weighted composition operators with linear fractional symbols. As a corollary, we prove that C^∼ψ(n) . and C^∼ψ(n) are not quasinormal but binormal. In addition, we show that C^∼ψ(n) . and e C(m) . are quasisimilar for all non-negative integers n and m. Finally, we show that C^∼ψ(n) and C^∼ψ(n) converge to normal operators in the strong operator topology.