An invariant subspace problem for multilinear operators on finite dimensional spaces

John Emenyu

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.

Keywords


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

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