For φ(z)=az+(1-a) with 0<a<1, the n-fold Aluthge iterates of C_φ on A_α²(D) have closed forms whose norms and numerical radii are computed explicitly, and the sequence converges strongly; the same is done for the adjoint and linked to a weighted Hardy space operator.
Complex Analysis and Operator Theory 7(1), 69-99 (2013)
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Iterated Aluthge transforms of some composition operators on weighted Bergman spaces
For φ(z)=az+(1-a) with 0<a<1, the n-fold Aluthge iterates of C_φ on A_α²(D) have closed forms whose norms and numerical radii are computed explicitly, and the sequence converges strongly; the same is done for the adjoint and linked to a weighted Hardy space operator.