Topological Methods in Nonlinear Analysis

An invariant subspace problem for multilinear operators on finite dimensional spaces

John Emenyu

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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.

Article information

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

Dates
First available in Project Euclid: 11 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1460381466

Mathematical Reviews number (MathSciNet)
MR3289004

Zentralblatt MATH identifier
1352.47034

Citation

Emenyu, John. An invariant subspace problem for multilinear operators on finite dimensional spaces. Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 1--10. https://projecteuclid.org/euclid.tmna/1460381466


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