Angular derivatives and compact composition operators on the Hardy and Bergman spaces. (English) Zbl 0608.30050

MacCluer, Barbara D.; Shapiro, Joel H.

Can. J. Math. 38, 878-906 (1986).

With \(\alpha >-1\) and \(d\lambda\) Lebesgue measure in the unit disc U, define the Bergman \(A^ p_{\alpha}\) space by \[ A^ p_{\alpha}=\{f\quad holomorphic\quad U| \int_{U}| f|^ p(1-| z|)^{\alpha}d\lambda <\infty \}. \] If \(\phi\) is a holomorphic map of U into U the composition operator \(C\phi f=f\circ \phi\) is a bounded operator on \(A^ p_{\alpha}\), and on the Hardy spaces \(H^ p\). The principal result of the paper is to characterize the compact composition operators on \(A^ p_{\alpha}\) with \(\alpha >-1\), \(0<p<\infty\). The key to their characterization is the existence (or nonexistence) of the angular derivative \[ \lim_{z\to \zeta}\frac{(\phi (z)-\omega)}{(z-\zeta)} \] where \(| \zeta | =1=| \omega |\) and the limit is taken as z in U tends non-tangentially to \(\zeta\).
Suppose \(0<p<\infty\) and \(\alpha >-1\) are given and \(\phi\) : \(U\to U\). Then \(C_{\phi}\) is a compact operator on \(A^ p_{\alpha}\) if and only if \(\phi\) has no angular derivative at any point of \(\partial U\). The paper contains a unifying approach to other results in this area. Also the last section studies the problem for spaces of functions holomorphic on the ball in \({\mathbb{C}}^ M\). The latter situation is more complicated and the results less sharp.

Reviewer: J.A.Cima

MSC:

30H05

Spaces of bounded analytic functions of one complex variable

Keywords:

Bergman space; composition operators; compact operator; angular derivative