On function-theoretic conditions characterizing compact composition operators on $H^2$

Abstract

For a holomorphic self-map $\varphi$ of the unit disk of the complex plane, the compactness of the composition operator $C_{\varphi}(f) = f\circ \varphi$ on the Hardy spaces is known to be equivalent to the various function theoretic conditions on $\varphi$, such as Shapiro's Nevanlinna counting function condition, MacCluer's Carleson measure condition, Sarason condition and Yanagihara-Nakamura condition, etc. A direct function-theoretic proof of Shapiro's condition and Sarason's condition was recently given by Cima and Matheson. We give another direct function-theoretic proof of the equivalence of these conditions by use of Stanton's integral formula.