1 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
\(E'\). A map
\(T:E\times\cdots\times E\to F\) is
m-linear if it is linear in each of the
m-variables and for this map, a map
\(p:E\to F: p(x)=T(x,\ldots,x)\) for all
\(x\in E\) is called an
m-homogeneous polynomial map. We denote the space of continuous multilinear and polynomial maps by
\(\mathcal{L}({}^{m}E; F)\) and
\(\mathcal{P}({}^{m}E; F)\), respectively, see [
1,
2]. The term
operator will be restricted to the elements of
\(\mathcal{L}({}^{m}E; E)\); notice 1-linear operators are just elements of
\(\mathcal{L}(E)\).
Problems
1.3 and
1.4 are referred to, in the sequel, as
invariant subspace problem (ISP) and
eigenvalue problem for multilinear operators, respectively.
The notion of invariant subspaces introduced in [
3] provided a basis for extending the underlying notion to nonlinear operators in a natural way; however, a natural entry point to delve into the study of such notion in infinite dimensions is first to deal with ideals of compact nonlinear maps generated by
\(\{T_{1}T_{2} \ldots T_{m}:T_{j}\in\mathcal{L}(E)\}\) where
\(m\in \mathbb{N}\) is fixed. These maps are known as maps of finite type and they inherit certain intrinsic properties of linear maps such as approximability by finite dimensional maps and bounded point spectra that are useful in the study of their associated ISPs. Notice
\(T\in\mathcal{L}({}^{m}E; F)\) is finite dimensional if its multirange
\(T(E, \ldots, E)\) is contained in a finite dimensional subspace of
F; see Section 3 in [
4]. In general, structural properties of
\(T_{j}\in\mathcal{L}(E)\) such as compactness were key in the study of their ISP; see [
5,
6]. On the one hand, the work of Bényi and Torres [
7] embodies different kinds of notions of compactness of a class of bilinear operators of finite type. Indeed, separate compactness of each section map of a bilinear map of finite type does not guarantee its compactness thus lending a study of Problem
1.3 a more general and broader consideration rather than restricting it to only compact nonlinear operators.
In Section
2, we will give a review of nonlinear spectral theory and Section
3 contains a review of techniques for tackling Problem
1.3, the modified form of Problem
1.4 and some tools such as Lemma
3.4 for establishing our results in Section
5. Section
4 stipulates problems associated with tacking Problem
1.3 whereas Section
5 comprises our major results.
2 An overview of nonlinear spectral theory
The study of Problem
1.3 via Problem
1.4 requires an in-depth knowledge of solvability properties of the equation
\(\lambda z-p(z)=0\), hence, the structural properties of
\(p\in\mathcal{P}({}^{m}E; E)\); namely, injectivity of
\(\lambda I-p\) and whether it maps some bounded neighborhood of 0 onto a neighborhood of 0, continuity of
\((\lambda I-p)^{-1}\), boundedness and nontriviality of the nullset
\(\mathcal{N} (\lambda I-p)\). These underlying structural properties yielded several variants of spectra of
\(p\in\mathcal{P}({}^{m}E; E)\), each having different notions of eigenvalues; see Chapters 6-9 in [
8]. Among several others, the Furi-Martelli-Vignoli spectrum
\(\sigma_{\mathrm{FMV}}(p, I)\), see [
9], 1978, and its modified form called Appell-Giorgieri-Väth spectrum
\(\sigma_{\mathrm{AGV}}(p,I)\), see [
8], p.150, account for asymptotic properties of stably nonlinear solvable continuous operators; the Feng spectrum
\(\sigma_{F}(p, I)\) accounts for global properties of epi and
k-epi operators, see [
10], 1997; the small Väth spectrum
\(\phi(p, I)\) and large Väth spectrum
\(\Phi(p, I)\) account for local properties of strictly and properly epi operators, respectively, see [
11], 2001. However, none of the known nonlinear spectra adheres to the minimal requirements; specifically most of their associated eigenvalue notions are incompatible with the general notion of the classical eigenvalues [
12], Definitions 1 and 2. Further, these spectra may be disjoint from point spectra, see Example 6.6 in [
8], [
11], are not discrete and unbounded even if their underlying operators are bounded and compact, see Example 3.5 in [
12], Theorem 3.5 in [
13]; [
8] gives extensive and exhaustive literature on this topic.
3 An overview of invariant subspace techniques for nonlinear maps
An attempt of a
de facto investigation of Problem
1.3 is by Donoghue, see Example 4 in [
14] and [
3], which comprise a more systematic study via Problem
1.4 or more generally via elementary invariant subspaces. The challenge to the underlying technique in [
3] is the limitation in knowledge of topological degrees in infinite dimensions. Moreover, the nondiscreteness and disjointness of nonlinear spectra from point spectra hinder the adaptation of linear approaches to Problem
1.3. For instance, extending perturbation methods [
15‐
17] modeled on the assumption
\(\sigma(T+K)\subset \sigma(T)\) is tantamount to the consideration of solvability properties of the eigenvalue equation
\(\lambda z-p(z)=0\) for some
\(\lambda \in\Bbbk\) or alternatively structural properties of
\(p\in\mathcal{P}({}^{m}E; E)\). This in turn is tantamount to the application of
\(\sigma_{\mathrm{FMV}} (p, I)\),
\(\sigma_{\mathrm{AGV}}(p, I)\),
\(\sigma_{F}(p, I)\) but due to the underlying problems they usually lead to either unsuccessful or partial solutions to the Problem
1.3; they are marginally successful in very limited circumstances such as when
\(p\in\mathcal{P}({}^{m}E; E)\) is epi, see Chapter 7 in [
8].
In general, any robust approach to the Problem
1.3 should reflect structural properties of
\(p\in\mathcal{P}({}^{m}E; E)\) such as unboundedness and lack of a vector space structure of
\(\mathcal{N} (\lambda I-p)\), noncommutativity of
\(p\in\mathcal{P} ({}^{m}E; E)\) with
\(\lambda I-p\) or generally
\(\operatorname{lin}\{p^{n}: n\in\mathbb{N}\}\), and the homogeneity degree of
\(p\in\mathcal{P}({}^{m}E; E)\), and one must cater for the classical notion of eigenvalues in the sense of linear operators. The unboundedness of eigenvalues of the pair
\((p, I)\) arises from their structural differences, particularly homogeneity degrees, and the nondiscreteness of their spectrum is due to the lack of a Fredholm alternative or broadly an application of Borsuk’s theorem for odd linear and nonlinear maps; see [
9]. These pitfalls, except for the unboundedness of
\(\mathcal{N} (\lambda J_{m}-p)\), in certain cases can be bypassed while preserving the structure of
\(p\in\mathcal{P} ({}^{m}E; E)\) by replacing the pair
\((p, I)\) with
\((p, J_{m})\) where
\(J_{m}\) is some well behaved function in the sense that it allows the properties of the pair
\((p, J_{m})\) to be made compatible;
\(J_{m}\) may be chosen to cater for the disparity in homogeneity degrees and discreteness of the spectrum of the pair
\((p, J_{m})\). Specifically, we will choose the function
\(J_{m}\) to be the homeomorphism defined by
\(J_{m}(z)=\|z \|^{m-1}z\) where
m is the homogeneity degree of
\(p\in\mathcal{P}({}^{m}E; E)\), and we study Problem
1.3 via the modified eigenvalue Problem
3.1.
Lemma
3.4 below is the modified form of Lemma 2.6 in [
3].
The modification of the pair
\((p, I)\) with the pair
\((p, J_{m})\) for
\(m\in\{2n-1: n\in\mathbb{N}\}\) yields the following particular form of Theorem 4 in [
18] (also see Theorem 9.12 in [
8]).
In the situation of Theorem
3.5, the spectrum and resolvent set of the pair
\((p,J_{m})\) are denoted by
\(\sigma(p, J_{m})\) and
\(\rho(p, J_{m}) =\mathbb{C}\setminus\sigma(p, J_{m})\) respectively. In general, the spectral radius,
\(r(p, J_{m}):=\sup\{|\lambda|: \mathcal{N}(\lambda J_{m}-p)\neq\{0\}\}\) and in our case,
\(r(p,J_{m}):=\sup\{|\lambda|:\lambda\in\sigma (p, J_{m})\}\). We will say
\(\sigma_{\mathrm{ev}}(p,J_{m})\), the set of eigenvalues of the pair
\((p, J_{m})\), is bounded if
\(r(p, J_{m})<\infty\) and
\(T\in \mathcal{L}({}^{m}E; E)\) is
quasinilpotent if
\(r(p, J_{m})=0\).
Theorem
3.6 is a consequence of Theorem
3.5
Theorem
3.6 demonstrates that certain choices of the pair
\((p, J_{m})\) yield spectra with desirable properties analogous to the classical spectra of linear maps. The reason for this nice behavior is that the local, asymptotic, and global properties of the pair
\((p, J_{m})\) with equal homogeneity degrees are the same.
5 Results and discussion
In this section, we limit the study of Problem
1.3 via Problem
3.1 to the ideals of finite type generated by
\(\{T_{1} T_{2}\ldots T_{m}:T_{j}\in\mathcal{L}(\mathcal{A})\}\) where
\(m\in\mathbb{N}\) is fixed and
\(\mathcal{A}\) is a Banach algebra. This ideal class is larger than
\(\mathcal{P}_{f}({}^{m}\mathcal{A}; \mathcal{A})\) and the ideals coincide only if
\(\mathcal{A}\) has an
\(r_{n}\)-property, see [
20]; moreover, if
\(T_{j}\in\mathcal{L}(\mathcal{A})\) are linear endomorphisms then the nonlinear operators they generate are known to preserve the structures of Banach algebras, a property most suitable for our research framework.
The literature in [
21], Chapter VI and [
22], Chapters 10, 11, on Banach algebras is sufficient for our purpose. A normed linear complex space
\((\mathcal{A},\|\cdot\|)\) equipped with a multiplication
\((a, b)\to ab\) from
\(\mathcal{A}\times\mathcal{A}\) into
\(\mathcal{A}\) is a normed algebra if it is an algebra and
\(\|a b\|\leqslant\|a\| \|b\|\) for all
\(a,b\in\mathcal{A}\). A normed algebra
\(\mathcal{A}\) is a Banach algebra if the normed space
\((\mathcal{A}, \|\cdot\|)\) is a Banach space. A Banach algebra
\(\mathcal{A}\) is commutative if
\(ab=ba\) for all
\(a, b\in\mathcal{A}\); it is unital if there is
\(1\in\mathcal{A}\) such that
\(1a=a=a1\) for all
\(a\in\mathcal{A}\) and such an element is called a unit. Here, we take the unit
\(1\in\mathcal{A}\) to have norm one. An element
\(a\in\mathcal{A}\) is called an
idempotent if
\(a^{2}=a\).
A vector subspace \(\mathcal{B}\) of \(\mathcal{A}\) is a subalgebra if itself is an algebra with respect to the operations of \(\mathcal{A}\). A subalgebra \(\mathcal{U}\) of an algebra \(\mathcal{A}\) is a right ideal if \(\mathcal{U}\mathcal{A}=\{ua: u\in\mathcal{U}, a\in\mathcal{A}\}\subset\mathcal{A}\) and it is a left ideal if \(\mathcal{A} \mathcal{U}=\{au:u\in\mathcal{U}, a\in \mathcal{A}\}\subset\mathcal{A}\); it is a two-sided ideal if it is both left and right ideal. A proper ideal \(\mathcal{M} \subset\mathcal{A}\) is called a maximal ideal if it is not properly contained in any other proper ideal. In a commutative unital algebra every proper ideal is contained in some maximal ideal and the same holds for noncommutative algebras with the observation that left and right ideals should be treated separately (the corresponding maximal ideals are also left or right); the proofs of these facts are premised on Zorn’s lemma.
The central point of our results, as envisaged in Theorems
5.4,
5.7 and
5.6 is that structural properties of operators and spaces are key in the development of techniques for tackling nonlinear ISP. Indeed, an attempt to generalize the techniques [
5] to nonlinear invariant subspace Problem
1.3 fails due to the lack of approximation step [
5], Statement II. Besides, the application of the methods [
6,
25] to nonlinear invariant subspace Problem
1.3 is hampered by the challenges outlined in Remark
4.6. More precisely, the unboundedness and non-vector space structure of the set
\(\mathcal{N}(\lambda J_{m}-p)\) devoid of the analogs [
26], Lemma 10.2, Theorem 10.18, that are key to the techniques in [
6]. Similarly, the application of the method [
25] to nonlinear invariant subspace Problem
1.3 shows that quasinilpotent compact multilinear operators have nontrivial invariant subset. Further, to determine invariant subspaces for
\(T\in\mathcal{L} ({}^{m}E; E)\) via the modified nonlinear eigenvalue Problem
3.1 generally requires determining common coincidence points of a family of commuting maps
\(\{q_{i} \in\mathcal{P}({}^{m}E; E): q_{i}q_{j}=q_{j}q_{i} \text{ for all } i, j\in\mathbb{N}\}\) with a homeomorphism
\(J_{m}\); this results into problems associated with
\(\sigma_{\mathrm{FMV}} (p, J_{m})\) spectrum.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The work as a whole is a contribution of the author.