Iterated Aluthge transforms of composition operators on H-2

Abstract

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