Differentiation and composition on the Hardy and Bergman spaces

Banach spaces of analytic functions are defined by norming a collection of these functions defined on a set X. Among the most studied are the Hardy and Bergman spaces of analytic functions on the unit disc in the complex plane. This is likely due to the richness of these spaces. An analytic self-map of the unit disc induces a composition operator on these spaces in the natural way. Beginning with independent papers by E. Nordgren and J. V. Ryff in the 1960\u27s, much work has been done to relate the properties of the composition operator to the characteristics of the inducing map. Every composition operator induced by an analytic self-map of the unit disc is bounded on the Hardy and Bergman spaces. Differentiation is another linear operation which is natural on spaces of analytic functions. Unlike the composition operator, the differentiation operator is poorly behaved on the Hardy and Bergman spaces; that is, it is not a bounded operator. We define a linear operator, possibly unbounded, by applying composition followed by differentiation; that is, for f in a Hardy or Bergman space and an analytic self-map of the disk, $\phi$,$$DC\sb\phi(f)=(f\circ\phi)\prime.$$We have found a characterization for the boundedness of this operator on the Hardy space in terms of the inducing map. The operator is bounded exactly when the image of the self-map of the disc is contained in a compact subset of the disc. In contrast, we have found a self-map of the disc with supremum norm equal to one that induces a bounded operator on the Bergman spaces. In this setting we have found conditions necessary for boundedness, and conditions sufficient to imply boundedness. These conditions are closely related. The techniques used involve Carleson-type measures on the unit disc. A very general question arising out of this work involves relating boundedness of the differentiation operator to characteristics of these measures