Analytic Toeplitz and Composition Operators

This paper is a continuation of 1 where we began the study of intertwining analytic Toeplitz operators. Recall that X intertwines two operators A and B if XA = BX. Let H 2 be the Hilbert space of analytic functions in the open unit disk D for which the functions f r (θ) = f (re iθ) are bounded in the L 2 norm, and H ∞ be the set of bounded functions in H 2. For φ ∊ H φ, T φ (or T φ (z) ) is the analytic Toeplitz operator defined on H 2 by the relation (T φ f) (z) = φ (z) f (z). For φ ∊ H ∞, we shall denote φ (z): |z| < 1 by Range (φ) or φ (D). Then where and σ (T φ) = Closure (φ (D) ) 1. If φ ∊ H ∞ maps D into D, then we define the composition operator C φ on H 2 by the relation (C φ f) (z) = f (φ (z) ). J. Ryff has shown 11, Theorem 1 that C φ, is a bounded linear operator on H 2.