### An invariant subspace problem for multilinear operators on finite dimensional spaces

#### Abstract

We introduce the notion of invariant subspaces

for multilinear operators from which the invariant

subspace problems for multilinear and polynomial

operators arise. We prove that polynomial

operators acting in a finite dimensional complex

space and even polynomial operators acting in a

finite dimensional real space have eigenvalues.

These results enable us to prove that polynomial

and multilinear operators acting in a finite

dimensional complex space, even polynomial and

even multilinear operators acting in a finite

dimensional real space have nontrivial invariant

subspaces. Furthermore, we prove that odd polynomial

operators acting in a finite dimensional real space

either have eigenvalues or are homotopic to scalar

operators; we then use this result to prove that odd

polynomial and odd multilinear operators acting in a

finite dimensional real space may or may not have

invariant subspaces.

for multilinear operators from which the invariant

subspace problems for multilinear and polynomial

operators arise. We prove that polynomial

operators acting in a finite dimensional complex

space and even polynomial operators acting in a

finite dimensional real space have eigenvalues.

These results enable us to prove that polynomial

and multilinear operators acting in a finite

dimensional complex space, even polynomial and

even multilinear operators acting in a finite

dimensional real space have nontrivial invariant

subspaces. Furthermore, we prove that odd polynomial

operators acting in a finite dimensional real space

either have eigenvalues or are homotopic to scalar

operators; we then use this result to prove that odd

polynomial and odd multilinear operators acting in a

finite dimensional real space may or may not have

invariant subspaces.

#### Keywords

Invariant subspaces; multilinear operators; polynomial operators; topological degree; admissible operators

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.