Singular values and Schmidt pairs of composition operators on the Hardy space

We find the singular values and corresponding Schmidt pairs of a compact composition operator Cϕ induced by ϕ(z) = az+b, where |a|+|b| < 1, on the classical Hardy space.We do so by solving a functional equation that is a generalization of Schröder’s equation: find a function f , holomorphic on the open unit disc, and a complex number λ such that G(z)f (ψ(z)) = λf (ψ(z)), where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator MGCψ.  2004 Published by Elsevier Inc.