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 .