Topological Methods in Nonlinear Analysis

An invariant subspace problem for multilinear operators on finite dimensional spaces

John Emenyu

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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.

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Topol. Methods Nonlinear Anal., Volume 44, Number 1 (2014), 1-10.

First available in Project Euclid: 11 April 2016

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Emenyu, John. An invariant subspace problem for multilinear operators on finite dimensional spaces. Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 1--10.

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