Fixed point theorem by Ryll-Nardzewski

The fixed point theorem by Ryll-Nardzewski , named after Czesław Ryll-Nardzewski , is a theorem from the mathematical branch of functional analysis . The proposition ensures the existence of a common fixed point of a family of certain images of a compact , convex set within itself.

Formulation of the sentence

Let be a locally convex space , for example a normalized space , and be a non-empty weakly - compact convex set. Furthermore, let us be a non-empty family of maps with the following properties: ${\displaystyle E}$ ${\displaystyle C\subset E}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle T:C\rightarrow C}$

1. ${\displaystyle {\mathcal {S}}}$is a semi-group , which means: applies to everyone . ${\displaystyle T_{1},T_{2}\in {\mathcal {S}}}$ ${\displaystyle T_{1}\circ T_{2}\in {\mathcal {S}}}$
2. Each is weakly continuous and affine , that is, for and holds . ${\displaystyle T\in {\mathcal {S}}}$ ${\displaystyle x,y\in C}$ ${\displaystyle \alpha \in [0,1]}$ ${\displaystyle T(\alpha x+(1-\alpha )y)=\alpha T(x)+(1-\alpha )T(y)}$
3. ${\displaystyle {\mathcal {S}}}$is non-contracting , that is, for two different points , 0 is not at the end of . ${\displaystyle x,y\in C}$ ${\displaystyle \{T(x)-T(y);T\in {\mathcal {S}}\}}$

Then there is at least one common fixed point of , that is: there is one , so that for all . ${\displaystyle {\mathcal {S}}}$ ${\displaystyle x_{0}\in C}$ ${\displaystyle T(x_{0})=x_{0}}$ ${\displaystyle T\in {\mathcal {S}}}$

Remarks

• To prove it, we first show that every finite subset of has a fixed point, and then we conclude the claim with a compactness argument. ${\displaystyle {\mathcal {S}}}$
• The requirement that it should be non-contracting is automatically fulfilled if all elements are from isometrics of a standardized space. This special case is also called the fixed point theorem of Ryll-Nardzewski : Every semigroup of weakly continuous affine isometries of a weakly compact convex set has a fixed point in itself. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$

application

The best known application is the derivation of the existence of the hair measure on a compact group . The space of the finite Borel measures on is the dual space of the space of the continuous functions on , and therefore carries the weak - * - topology , which makes a locally convex space, the weak topology of which is precisely this weak - * - topology. The convex set is taken . For and are explained by the formulas . Define further through ${\displaystyle G}$ ${\displaystyle E=M(G)}$ ${\displaystyle G}$ ${\displaystyle C(G)}$ ${\displaystyle G}$ ${\displaystyle M(G)}$ ${\displaystyle C:=\{\mu \in M(G);\mu \geq 0,\mu (1)=1\}}$ ${\displaystyle f\in C(G)}$ ${\displaystyle x\in G}$ ${\displaystyle {\tilde {f}},{}_{x}f,f_{x}\in C(G)}$ ${\displaystyle {\tilde {f}}(y):=f(y^{-1}),{}_{x}f(y):=f(xy),f_{x}(y):=f(yx)}$ ${\displaystyle S_{0},S_{1},L_{x},R_{x}:M(G)\rightarrow M(G)}$

• ${\displaystyle S_{0}\,:=\,id_{M(G)}}$
• ${\displaystyle S_{1}(\mu )(f)\,:=\,\mu ({\tilde {f}})}$
• ${\displaystyle L_{x}(\mu )(f)\,:=\,\mu ({}_{x}f)}$
• ${\displaystyle R_{x}(\mu )(f)\,:=\,\mu (f_{x})}$

Then there is a semi-group of isometrics that depicts in itself. If one applies the fixed point theorem of Ryll-Nardzewski to this situation, one obtains a measure that can easily be proven as a hair measure . ${\displaystyle {\mathcal {S}}:=\{S_{i}L_{x}R_{y};i\in \{0,1\},x,y\in G\}}$ ${\displaystyle C}$

swell

• John B. Conway: A Course in Functional Analysis , Springer-Verlag (1994), ISBN 0387972455
• C. Ryll-Nardzewski: On fixed points of semigroups of endomorphisms of linear spaces , Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, Univ. California Press, Berkeley (1967), pp. 55-61