Fixed points and invariant domains of holomorphic mappings of the Hilbert ball

Abstract

THIS paper is an extension of the joint work of the present author, T. Sekowski and A. Stachura [4]. There, some methods of the theory of nonexpansive mappings were used to prove fixed point theorems for holomorphic self-mappings of the unit ball in complex Hilbert space. Earlier results on the existence of fixed points of holomorphic self-mappings of domains in Banach spaces may be found in works of Earle and Hamilton [2], Hayden and Suffridge [6, 7, 111, and Rudin [9]. All the results depend strongly on the consequences of the Schwartz-Pick lemma and properties of so-called invariant hyperbolic metrics. Detailed discussion of invariant metrics is contained in the long article by Harris [5] and in the recent book by Franzoni and Vesentini [3]. Hyperbolic geometry of the unit ball in C” is clearly described in a very recent book by Rudin [lo]. In spite of the fact that we are dealing with the infinite dimensional case we very often follow the notation, terminology and some methods from this book. A very interesting discussion of the one-dimensional case has been given recently by Burckel [l]. Invariant “ellipsoids” which we discuss here have their origin in the old papers by Julia [S] and Wolff [2].