Biholomorphic maps in Hilbert space have a fixed point

Thomas Lee Hayden and Ted Joe Suffridge

Vol. 38 (1971), No. 2, 419–422

DOI: 10.2140/pjm.1971.38.419

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 Hilbert space. Kakutani has shown that there exists a homeomorphism of the closed unit sphere of Hilbert 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 Hilbert space has a fixed point.

Mathematical Subject Classification 2000

Primary: 32A99

Secondary: 46G20

Milestones

Received: 28 May 1970

Revised: 22 October 1970

Published: 1 August 1971