An invariant subspace problem for multilinear operators on Banach spaces and algebras

John Emenyu

doi:10.1186/s13660-016-1120-2

Abstract

This paper is concerned with the study of invariant subspace problems for nonlinear operators on Banach spaces/algebras. Our study reveals that one faces unprecedented challenges such as lack of vector space structure and unbounded spectral sets when tackling invariant subspace problems for nonlinear operators via spectral information. To bypass some of these challenges, we modified an eigenvalue problem for nonlinear operators to cater for the structural properties of nonlinear operators and then established that nonlinear operators of finite type on a complex Banach algebra have nontrivial invariant subspaces.

Highlights

1 Introduction The aim of this paper is to study invariant subspace problems for polynomial and multilinear operators on infinite dimensional Banach spaces
A closed linear subspace M of E is invariant for p ∈ P(mE; E) if p(M) ⊆ M; it is invariant for T ∈ L(mE; E) if p(M) ⊆ M; it is strongly invariant for T ∈ L(mE; E) if T(M, . . . , M) ⊆ M
5 Results and discussion we limit the study of Problem . via Problem . to the ideals of finite type generated by {T T . . . Tm : Tj ∈ L(A)} where m ∈ N is fixed and A is a Banach algebra. This ideal class is larger than Pf and the ideals coincide only if A has an rn-property, see [ ]; if Tj ∈ L(A) are linear endomorphisms the nonlinear operators they generate are known to preserve the structures of Banach algebras, a property most suitable for our research framework

Summary

Introduction

The aim of this paper is to study invariant subspace problems for polynomial and multilinear operators on infinite dimensional Banach spaces.Throughout this paper, we denote Banach spaces by E and F, and the dual space of E by E. ) and mimicking the proof of Theorem in [ ] clearly p ∈ P(mE; E) has nontrivial invariant subspaces of the form Ma = {f ∈ C[ , ] : f (s) = , s ∈ [ , a] ⊂ [ , ]} so that by equation This ideal class is larger than Pf (mA; A) and the ideals coincide only if A has an rn-property, see [ ]; if Tj ∈ L(A) are linear endomorphisms the nonlinear operators they generate are known to preserve the structures of Banach algebras, a property most suitable for our research framework.

Objectives

Results

Conclusion