Krasnoselskii genus

In nonlinear functional analysis, the Krasnoselskii genus generalizes the notion of dimension for vector spaces. The Krasnoselskii genus of a linear space $A$ is the smallest natural number $n$ for which there exists a continuous odd function of the form $f:A\to {\mathbb{ℝ}}^n\setminus 0$. The genus was introduced by Mark Aleksandrovich Krasnoselskii in 1964, and an equivalent definition was provided by Charles Coffman in 1969.

Krasnoselskii Genus

We follow the definition given by Coffman.

Let

• $E$ be a Banach space,
• ${𝒜}=\{A\subset E:A\text{ closed},;A=-A\}$ be the collection of symmetric closed subsets of $E$,
• $C(A,{\mathbb{ℝ}}^n)$ the space of continuous functions $A\to {\mathbb{ℝ}}^n$.

For $A\in {𝒜}$ define the set

$K_A=\{n\in \mathbb{\mathbb{N}}:\mathrm{\exists }f\in C(A,{\mathbb{ℝ}}^n\setminus 0),;f(-x)=-f(x)\}$

Then the Krasnoselskii genus of $A$ is defined as

$\gamma (A)=\{\begin{array}{ll}\inf K_A& \text{if }{K}_A\ne \mathrm{\varnothing },\\ \mathrm{\infty }& \text{if }{K}_A=\mathrm{\varnothing },\\ 0& \text{if }A=\mathrm{\varnothing }.\end{array}$

In other words, if $\gamma (A)=n$ then there exists a continuous odd function $\phi :A\to {\mathbb{ℝ}}^n$ such that $0\notin \phi (A)$. Moreover $n$ is the minimal possible dimension, i.e. there exists no such function $\theta :A\to {\mathbb{ℝ}}^d$ with $d<n$.

Properties

• Let $\mathrm{\Omega }\subset {\mathbb{ℝ}}^n$ be a bounded symmetric neighborhood of $0$ in ${\mathbb{ℝ}}^n$. Then the genus of its boundary is $\gamma (\partial \mathrm{\Omega })=n$.
• For $A,B\in {𝒜}$, the following holds:

1. If there exists an odd function $f\in C(A,B)$, then $\gamma (A)\le \gamma (B)$.
2. If $A\subset B$, then $\gamma (A)\le \gamma (B)$.
3. If there exists an odd homeomorphism between $A$ and $B$, then $\gamma (A)=\gamma (B)$.

Combining these statements, it follows immediately that if there exists an odd homeomorphism between $A$ and $\partial \mathrm{\Omega }$ then $\gamma (A)=n$.