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

Abstract

We find the singular values and corresponding Schmidt pairs of a compact composition operator C φ induced by φ ( z ) = a z + 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 ) ) = λ f ( 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 M G C ψ .