Biholomorphic maps in Hilbert space have a fixed point

Abstract

The results in this paper reveal a dichotomy in regard to the existence of fixed points for smooth real maps and biholomorphic maps in Hubert space. Kakutani has shown that there exists a homeomorphism of the closed unit sphere of Hubert space onto itself which has no fixed point. A slight modification of his example shows that there is a diffeomorphism having the same property. Our results show that in the complex case every biholomorphic map of the unit ball onto itself in Hubert space has a fixed point. A function h defined on an open subset D of a Banach space into a Banach space is called holomorphic in D if h has a Frechet derivative at each point of D. Standard results about holomorphic maps may be found in [4]. By biholomorphic we mean a holomorphic map with a holomorphic inverse. It is a known result that in Cn an injective holomorphic map is biholomorphic. The corresponding result does not seem to be known in infinite dimensions even assuming the range is an open set. Our proofs make use of some results obtained recently for holomorphic mappings in Banach spaces. One knows that in the plane every bijective holomorphic map of the unit disk which takes zero to zero is given by a rotation. In Hubert Space the analogous result is that every biholomorphic map of the unit ball onto the unit ball which leaves zero fixed is given by a unitary operator. This result follows easily from the work of R. S. Phillips [6]. Also L. Harris [3] has obtained more general results in this direction. Our result is the following theorem for a complex Hubert space H. Theorem: Suppose B = {ze H: \\z < 1} and h is a biholomorphic map of B onto B. Then h is biholomorphic in a larger region, maps B onto B, and has a fixed point in B. In § 2 we give a proof of the above result and in § 3 we show that the fixed points are either isolated points or the fixed point sets are affine subspaces.