Birkhoff-Kellogg's theorem

The set of Birkhoff Kellogg ( English Birkhoff Kellogg theorem ) is a theorem of the mathematical branch of nonlinear functional analysis , the a in 1922 by the two mathematicians George David Birkhoff and Oliver Dimon Kellogg presented scientific work back. It deals with the question of under what conditions for certain operators on infinite - dimensional Banach spaces , the eigenvalue problem is solved. The theorem turns out to be an analogue of the classical Poincaré-Brouwer theorem in topology .

Formulation of the sentence

The sentence can be summarized as follows:

Given a infinite - dimensional Banach space and in a limited open subset which the zero contained . ${\displaystyle X}$ ${\displaystyle G\subset X}$ ${\displaystyle 0\in X}$

On the closed envelope of a compact ( linear or nonlinear) operator is given that satisfies the condition ${\displaystyle {\overline {G}}}$ ${\displaystyle G}$ ${\displaystyle \mathrm {K} \colon {\overline {G}}\to X}$

${\displaystyle \inf _{x\in \partial {G}}{\|\mathrm {K} (x)\|}>0}$.

meet.

Then:

The eigenvalue problem is solvable for. There is an edge point and a real number that satisfy the equation . ${\displaystyle \mathrm {K} }$ ${\displaystyle x_{0}\in \partial {G}}$ ${\displaystyle \lambda \in \mathbb {R} \setminus \{0\}}$ ${\displaystyle \mathrm {K} (x_{0})=\lambda x_{0}}$

background

The proof of the Birkhoff-Kellogg theorem is essentially based on a general eigenvalue principle , for the derivation of which the Leray-Schauder degree of mapping is used, as well as the following approximation theorem for compact operators ( English Approximation Theorem for Compact Operators ):

Given are two Banach spaces (over with ) as well as a bounded, non-empty subset and then an arbitrary operator . ${\displaystyle X,Y}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle M\subset X}$ ${\displaystyle \Psi \colon M\to Y}$

Then:

${\displaystyle \Psi }$is a compact operator if and only if there is a sequence of operators such that the following three conditions are always fulfilled: ${\displaystyle {\Psi }_{n}\colon M\to Y\,(n=1,2,3,\ldots )}$ ${\displaystyle n=1,2,3,\ldots }$

(i) is compact. ${\displaystyle {\Psi }_{n}}$

(ii) . ${\displaystyle \sup _{x\in M}{\|\Psi (x)-{\Psi }_{n}(x)\|_{Y}}\leq {\frac {1}{n}}}$

(iii) The of the image set (above ) spanned linear subspace has finite dimension . ${\displaystyle {\Psi }_{n}(M)}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \operatorname {span} \left({\Psi }_{n}(M)\right)}$

literature

• GD Birkhoff, OD Kellogg: Invariant points in function space. In: Transactions of the American Mathematical Society . tape 23 , 1922, pp. 96-115 ( MR1501192 ).
• Eberhard Zeidler : Lectures on nonlinear functional analysis I: Fixed point theorems . BG Teubner Verlagsgesellschaft , Leipzig 1976 ( MR0473927 ).
• Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems . Translated by Peter R. Wadsack. Springer Verlag , New York, Berlin, Heidelberg, Tokyo 1986, ISBN 0-387-90914-1 ( MR0816732 ).

Individual evidence

1. ↑ Eberhard Zeidler: Lectures on nonlinear functional analysis I 1976, p. 12, pp. 152–153
2. ^ Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I 1986, p. 557 ff
3. ^ Zeidler (1976), p. 153
4. ^ Zeidler (1986), p. 559
5. ↑ is the set of boundary points of . ${\displaystyle \partial {G}}$ ${\displaystyle G}$
6. ↑ Zeidler (1976), p. 25, pp. 152-153
7. ^ Zeidler (1986), p. 55, pp. 558-559