BOUNDED COMPOSITION OPERATORS FROM THE BERGMAN SPACE TO THE HARDY SPACE

Abstract

Abstract. Let φ be an analytic self map of the open unit disc D. In thispaper, we study the composition operator C φ from the Bergman spaceon D to the Hardy space on D. 1. IntroductionLet Dbe the open unit disc in the complex plane. L 2a and H 2 denote theBergman space and the Hardy space on D, respectively. Then H 2 is containedin L 2a . If H ∞ is a set of all bounded analytic functions, then H ∞ is containedin H 2 . For an analytic self map φof D, the composition operator C φ is definedby (C φ f)(z) = f(φ(z)) (z∈ D) for fin H, the set of all analytic functions onD. The Nevanlinna counting function of φ, is defined on D\{φ(0)} byN φ (w) =P φ(z)=w log1|z.T. Nakazi [4, Theorem 4] gives a necessary and sufficient condition for anisometric operator C φ from L 2a to H 2 . That is, C φ is isometric from L 2 to H 2 if and only if N φ (w) = 2R 1|w| log r|w| rdrfor nearly all w∈ D\{0}.W. Smith [6, Theorem 1.1] gives a necessary and sufficient condition for abounded composition operator C φ from L 2a