Earle–Hamilton fixed-point theorem: Difference between revisions

Revision as of 14:57, 19 June 2012

Latest revision as of 07:05, 31 December 2024

(31 intermediate revisions by 9 users not shown)

Line 1:

In [[mathematics]], the '''Earle–Hamilton fixed point theorem''' is a result in [[geometric function theory]] giving sufficient conditions for a [[holomorphic mapping]] of an open ___domain in a complex [[Banach space]] into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and [[Richard S. Hamilton]] by showing that, with respect to the [[Carathéodory metric]] on ''D''the ___domain, ''f''the holomorphic mapping becomes a [[contraction mapping]] to which the contraction[[Banach mappingfixed-point theorem]] can be applied.

==Statement==

Let ''D'' be a connected open subset of a complex [[Banach space]] ''X'' and let ''f'' be a homolomorphicholomorphic mapping of ''D'' into itself such that:

*the image ''f''(''D'') is bounded in norm;

Line 9 ⟶ 10:

==Proof==

Replacing ''D'' by an ε-neighbourhood of ''f''(''D''), it can be assumed that ''D'' is itself bounded in norm.

For ''z'' in ''D'' and ''v'' in ''X'', set

Line 23 ⟶ 26:

:<math>\displaystyle{d(x,y)=\inf_{\gamma(0)=x,\gamma(1)=y} \ell(\gamma)}</math>

for ''x'' and ''y'' in ''D''. It is a continuous function on ''D'' x ''D'' for the norm topology.

If the diameter of ''D'' is less than ''R'' then, by taking suitable holomorphic functions ''g'' of the form

:<math>\displaystyle{g(z)=a(z) + b}</math>

with ''a'' in ''X''* and ''b'' in '''C''', it follows that

:<math>\displaystyle{\alpha(z,v)\ge \|v\|/R,}</math>

and hence that

:<math>\displaystyle{d(x,y)\ge \|x-y\|/R.}</math>

In particular ''d'' defines a metric on ''D''.

The chain rule

:<math>\displaystyle{(g\circ f)^\prime(z)=g^\prime(f(z))f^\prime(z)}</math>

implies that

:<math>\displaystyle{\alpha(f(z),f^\prime(z)v)\le \alpha(z,v)}</math>

and hence ''f'' satisfies the following generalization of the [[Schwarz lemma#Schwarz–Pick theorem|Schwarz-Pick inequality]]:

:<math>\displaystyle{d(f(x),f(y))\le d(x,y).}</math>

For δ sufficiently small and ''y'' fixed in ''D'', the same inequality can be applied to the holomorphic mapping

:<math>\displaystyle{f_{\delta,y}(z)=f(z) +\delta (f(z)-f(y))}</math>

and yields the improved estimate:

:<math>\displaystyle{d(f(x),f(y))\le (1+\delta)^{-1} d(x,y).}</math>

The Banach fixed-point theorem can be applied to the restriction of ''f'' to the closure of ''f''(''D'') on which ''d'' defines a complete metric, defining the same

topology as the norm.

==Other holomorphic fixed point theorems==

In finite dimensions the existence of a fixed point can often be deduced from the [[Brouwer fixed point theorem]] without any appeal to holomorphicity of the mapping. In the case of [[bounded symmetric ___domain]]s with the [[Bergman metric]], {{harvtxt|Neretin|1996}} and {{harvtxt|Clerc|1998}} showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies. The bounded symmetric ___domain ''D'' = ''G'' / ''K'' is a complete metric space for the Bergman metric. The open semigroup of the complexification ''G''_''c'' taking the closure of ''D'' into ''D'' acts by [[contraction mapping]]s, so again the Banach fixed-point theorem can be applied. Neretin extended this argument by continuity to some infinite-dimensional bounded symmetric domains, in particular the Siegel generalized disk of symmetric Hilbert-Schmidt operators with operator norm less than 1. The Earle-Hamilton theorem applies equally well in this case.

==References==

*{{citation|lastlast1=Earle|firstfirst1= Clifford J.|last2=Hamilton|first2= Richard S.|

title=A fixed point theorem for holomorphic mappings|year= 1970|series=Proc. Sympos. Pure Math.|volume= XVI|pages= 61–65|publisher =American MathemeticalMathematical Society}}

*{{citation|last=Neretin|first= Y. A.|title=Categories of symmetries and infinite-dimensional groups|series= London Mathematical Society Monographs|volume=16|publisher= Oxford University Press|year= 1996|isbn=0-19-851186-8}}

*{{citation|last=Harris|first= Lawrence A.|title=Fixed points of holomorphic mappings for domains in Banach spaces|journal=Abstr. Appl. Anal.|year= 2003|volume= 5| pages=261–274|url=http://www.hindawi.com/journals/aaa/2003/121329/abs/}}

*{{citation|last=Clerc|first=Jean-Louis|title=Compressions and contractions of Hermitian symmetric spaces|journal=Math. Z.|volume= 229|year=1998|pages=1–8|doi=10.1007/pl00004648|s2cid=122333415}}

{{DEFAULTSORT:Earle-Hamilton fixed-point theorem}}

{{mathematics-stub}}

[[Category:FunctionalTheorems in complex analysis]]

[[Category:ComplexFixed-point analysistheorems]]

[[Category:Mathematical theorems]]