Let denote the unit disk of the complex plane. If is an analytic function on and , then is defined by
and the space is defined by . If is analytic, then the composition operator is defined by .
The main statements of this paper indicate that
(i) is bounded if and only if
(ii) if , then is -compact.
Notes indicated as added in the proof of the paper state that similar conclusions have been derived by different procedures in papers of R. C. Roan [Rocky Mt. J. Math., 10, 371-379 (1980; Zbl 0433.46023)] and J. H. Shapiro [Proc. Am. Math. Soc. 100, 49-57 (1987; Zbl 0622.47028)].