NON-SVEP, Right-Inversion Point Spectrum and Chaos

We discuss some relations between the local existence of analytic selections of eigenvectors (LSP =\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} “NON-SVEP”) for an operator in Banach space and some chaoticity properties of linear dynamical system (with discrete or continuous time) generated by this operator. Our main goal is to prove the existence of a strong connection of some results known for many years in the local spectral theory to some important problems (which seem to be not solved so far) in the linear chaos theory. We also find a simple particular solution of the problem formulated in “Eigenvectors Selection Conjecture” (Banasiak and Moszyński in Discrete Contin Dyn Syst A 20(3):577–587, 2008, Conjecture 4.3, p. 585) and we formulate a new convenient spectral criterion for linear chaos. To make the assumptions more clear we introduce some special parts of the point spectrum of a closed operator, including the right-inversion point spectrum. Using this new criterion we prove chaoticity of a large class of super-upper-triangular operators in lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p$$\end{document} and c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document} spaces and also of some strongly continuous semi-groups generated by such operators.

B := U for discrete time case iR for continuous time case, (0. 1) and to recall this criterion, let us remind first the notion of selection of eigenvectors ("a potential single element" of spanning eigenvector field, see e.g., [17]). Let A be a linear operator in X, and let f : Ω −→ X, where ∅ = Ω ⊂ C.
Definition 0.1. f is a selection of eigenvectors for A (on Ω) iff f (λ) ∈ Ker (A − λI) for any λ ∈ Ω. We usually abbreviate this name to: e.v. selection (for A, on Ω). An e.v. selection f is: • non-trivial iff f = 0 0 (where 0 0 denotes the constant-zero function on Ω); • analytic iff Ω is an open set and f is an X-vector-valued analytic function.
Using this terminology we can shortly formulate the spectral chaoticity criterion mentioned previously as follows.

Criterion 0.2. Assume that X is a Banach space, A is the generator of a linear dynamical system T in X and Ω ⊂ C is open and connected. If there exists a non-trivial analytic selection f of eigenvectors of
A on Ω such that Ω ∩ B = ∅, then T is sub-chaotic. Moreover, for f as above linf (Ω) is a space of chaoticity for T .
This formulation is a joint reformulation of several particular results. In the continuous time case, see [4,Criterion 3.3] (being a "sub-chaotic extension" of a "full-chaoticity case" from [10]). The discrete time case was not explicitly formulated in similar form, but one can easily obtain it, e.g., using the Godefroy and Shapiro result [17,Theorem 3.1] and repeating almost the same arguments as in the continuous case.
In fact, with such assumptions also stronger assertions can be easily obtained. For continuous time case one can prove the mixing property of the system restricted to this subspace, using [17,Theorem 7.32]. For discrete time case one can prove the mixing and the frequent hypercyclicity property of the restricted system using [17,Theorem 9.22], see the proof of Remark 4.4. Note that there exist also some similar criteria in which some weaker than analyticity regularity conditions on the selection are assumed (see [5,8,12]). A question related to such kinds of criteria was posed in-let's call it-"Eigenvectors Selection Conjecture" [5,Conjecture 4.3,p. 585]. It concerned some sufficient "general" conditions for the closed operator A, which could guarantee that the "richness" of a set for which we know that it is contained in the point spectrum, gives automatically the existence of a non-trivial selection of eigenvectors being regular in a sense (eg. measurable, continuous etc.). In the present paper we partially solve this problem in Theorem 1.6 ("Eigenvectors Selection Theorem I"). Our solution sounds surprisingly simple: it suffices to know only that one point λ 0 is in the point spectrum of the 4 M. Moszyński IEOT operator A and that A − λ 0 I is surjective, and we obtain the existance of a non-trivial selection of eigenvectors with very strong regularity-being analytic! More precisely, we explain that in some sense the problem was solved many years before it was actually posed. . . -However, this solution was hidden in a different part of operator theory-the local spectral theory-usually not being associated with linear chaos theory. The paper consists of two parts. Our main goal in the first part (Sects. 1 and 2) is to show an essential intersection of the two seemingly distant areas of operator theory. Starting from a classical Finch result of local spectral theory and making several simple steps-remarks, we get interesting new results for linear chaos theory! These simple remarks, however, show some of the concepts known for a long time in quite a new light. In Sect. 1 we recall some necessary notions and results of local spectral theory, which are related to our selection existence problems, including a reformulation of the Finch result of 1975 mentioned above [15]. In Sect. 2 we simply formulate the subchaos results directly following from Sect. 1. To make our considerations less abstract we illustrate them several times by the simplest classical example of linear chaotic operator-Rolewicz operator ( [20] and, e.g., [17]).
In the second part of the paper (Sects. 3 and 4) we formulate a weaker version of the solution of the "Eigenvectors Selection Conjecture" mentioned above-Corollary 3.3 ("Eigenvectors Selection Theorem II"). Yet, this time it is obtained in a different manner-without the use of local spectral theory. We study here some special parts of the point spectra of closed operators which are related to the problems investigated above. The most important here is the part of σ p (A) which we call the right-inversion point spectrum (RIPS), denoted here by σ p* (A) [see (3.3)]. The single point of the RIPS of the generator A of a linear dynamical system T can guarantee sub-chaos or even chaos for T . The appropriate result-"RIPS Chaoticity Criterion" (Theorem 3.4) seems to be a new and convenient tool in linear chaos theory. Its assumptions and the assertion turn out to be a good compromise for some interesting applications (compare, e.g., with Theorem 2.2 and see the comments in Example 2.4). Our main application of this tool is presented in Sect. 4, where we prove the chaoticity of a large class of super-upper-triangular operators in l p and c 0 spaces (Theorem 4.1). Note that those operators can be treated as generalizations (and also as perturbations) of "classical" weighted left-shifts which in turn generalize the mentioned Rolewicz type operators. We also study here discrete and continuous dynamical systems generated by some related operators and we make remarks on the more detailed "chaotic properties"-mixing and frequent hypercyclicity.

Notation
The symbols L(X), C(X), B(X) denote the sets of linear, closed, and bounded operators on the normed space X, respectively (here the domain D(A) of A need not be dense for A ∈ L(X) or C(X), but D(A) = X for A ∈ B(X)). If A ∈ L(X) and X = {0}, then σ(A) is its spectrum and σ p (A) is the set of its all eigenvalues (the point spectrum).
Vol. 88 (2017) NON-SVEP, Right-Inversion Point Spectrum 5 If X is a Banach space and Ω is an open subset of C, then A(Ω, X) := {f : f : Ω −→ X and is an analytic X-vector-valued function}; for f ∈ A(Ω, X), λ 0 ∈ Ω, n ∈ N by f λ0,n we denote the n-th coefficient in X in the power series expansion of f with the center at λ 0 , i.e.
for λ in a neighborhood of λ 0 .

Local Selection Property (LSP), SVEP and NON-SVEP
We are interested here in such linear operators acting in a Banach space, that possess a non-trivial analytic e.v. selection on a neighborhood of a given For r > 0 let D(λ 0 , r) be the open disc in C of radius r centered at λ 0 .

Definition 1.1. A has the local selection property at
. The above will be abbreviated to: A has the LSP at λ 0 . Proof. Using the definition of the LSP choose > 0 and g ∈ S(D(λ 0 , ), A), g = 0 0. So, we have , and obviously the expansion is also convergent. Now let us define f : the closedness of A and by the continuity of f at λ 0 . Hence f ∈ S(D(λ 0 , ), A) and f (λ 0 ) = 0. So, by the continuity of f we get λ 0 ∈ Int σ p (A). IEOT As one can easily see, the LSP is closely related to the well-known SVEP (single-valued extension property). The SVEP is mostly defined only for bounded operators (see e.g., [1,18]), and we need it for more general case (in fact-for some closed operators mainly). But the classical definition itself can be extended to any A ∈ L(X), with no difficulty (for closed operator case see e.g., [14,15] in general Banach spaces, and [2] in Hilbert space).
Calling the NON-SVEP the opposite to the SVEP, more precisely, defining: A has the NON-SVEP at λ 0 iff A does not have the SVEP at λ 0 , and using the fact that two analytic functions on a open connected set Ω coincide, if they take the same values on a subset possessing accumulation point in Ω, we obviously get: The result below is in fact a reformulation of a theorem by J.K. Finch [15,Th. 2,p. 61]. It can be also treated as one of possible and natural solutions of Eigenvectors Selection Conjecture of [5].
Proof. It suffices to use the above remark, and the above mentioned original Finch result to the operator A − λ 0 I. Example 1.7. (The Rolewicz type operators) Let X = l p = l p (N)-the standard power p-summable sequences space with 1 ≤ p < +∞ and consider A = μT −1 , where μ ∈ C \ {0} and T −1 denotes the backward shift operator, given by (T −1 f ) n = f n+1 for f ∈ l p and n ∈ N. Then A satisfies the assumptions of Theorem 1.6 with λ 0 = 0 and so A has the LSP at 0. Surely, in this simple particular case one can easily obtain the LSP result without the above theorem, just by constructing the appropriate selection of eigenvalues. Note that in the case p = 2 and |μ| > 1 this operator is known in the literature as Rolewicz operator (see [20] and, e.g., [17]).

LSP and Sub-chaos
Assume that X is a Banach space and A is the generator of a linear dynamical system T in X (discrete or continuous system, with the sense of the generator described in Sect. 0 and with B defined by (0.1)).
Using the terminology of Sect. 1 we can again shortly formulate the first part of Criterion 0.2 as follows. As an immediate corollary following this criterion we get a convenient result allowing to prove sub-chaoticity with a relatively simple way for many examples of dynamical systems.
Proof. Observe first that in both cases of the time set the operator A is closed, because it is a generator of T (see e.g., [19] for the cont. time case, being the C 0 semi-group case). Now we get the assertion by Criterion 2.1 and Theorem 1.6.
We illustrate the above theorem by two examples-of the discrete and of the continuous dynamical system generated by the same operator. In both cases sub-chaoticity, and in fact even chaoticity, is a well-known result (see e.g., [17]), but our goal here is only to illustrate a new method. Note also an important technical difference related to the use of the above theorem as a practical tool in these two cases. Example 2.4. (Rolewicz type discrete system) Consider now the analogous discrete dynamical systems T generated by the Rolewicz type operators as above. For this system we must assume that |μ| > 1. Then 1 μ ∈ D(0, 1) = σ p (T −1 ), and we can try to use Theorem 2.2, taking e.g., λ 0 = 1 ∈ U ∩ σ p (A) in this case. So, we would obtain sub-chaos of T , if we only proved that μT −1 − I is surjective. Although this is true for any p ∈ [1; +∞), checking it is not just trivial and it needs some "technical work". However, as we shall show in the next section, this work can be replaced by the fast use of a certain convenient "tool", which is formulated in Theorem 3.4.

Right-Inversion Point Spectrum and the RIPS Chaoticity Criterion
We introduce here three special parts of the point spectrum of a closed operator in Banach space. The first two are related directly to problems considered in the previous sections, and the third corresponds to a new idea, playing a crucial role here. The results of this section, excluding Proposition 3.1 (i), are obtained independently of the local spectral theory results of Sect. 1, i.e. the appropriate analytic selections of eigenvectors are constructed here in some simple, elementary ways, in particular, without using Eigenvectors Selection Theorem 1.6. For a Banach space X and A ∈ C(X) we define: which gives Af (λ) = λf (λ). Thus we proved that f ∈ S(U, A). Observe that (3.4) means exactly that for any λ ∈ U the second part of (iii) is proved and in particular U ⊂ σ p (A). The proof of the first part is completed by using the following simple lemma.
Obviously, the proposition above allows to formulate a more particular solution of Eigenvectors Selection Conjecture of [5].
Note that (3.4) is the explicit power expansion for the selection f and the coefficients of this expansion are expressed in terms of B 0 and f 0 . We can use this fact to get one more result of linear chaos theory. Recall that a vector x ∈ X is cyclic for C ∈ B(X) iff lin{C n x : n ∈ N} = X.
Assume, as in the previous section, that A is the generator of a linear dynamical system T in X, and B is given by (0.1). We can now formulate a convenient tool-a new formulation of an abstract chaoticity criterion for T . Observe that the choice of the above operator B 0 -the right inverse to (A − λ 0 I)-is not unique, and thus also the value B 0 can be not uniquely determined by A ∈ C(X) and λ ∈ C. Therefore, it is convenient to define: Using the above notation we can slightly reformulate the above criterion for chaos. IEOT

Chaotic Super-Upper-Triangular Operators
In this section X = l p = l p (N) with 1 ≤ p < +∞ or X = c 0 -the standard space of complex sequences on N converging to 0. We study here a large class of super-upper-triangular operators in X and we prove a generalization of the chaoticity result for Rolewicz type operators from Example 3.6.
By T −1 , T +1 we denote the backward and the forward shift operator in X, respectively. For A ∈ B(X) we consider its matrix terms A(k, l) ∈ C given by A(k, l) := (Ae l ) k for k, l ∈ N, where e l ∈ X is the l-th standard "base" vector, i.e. (e l ) n = 0 for n = l, (e l ) l = 1. Recall that A ∈ B(X) is upper-triangular iff A(k, l) = 0 for any k > l, k, l ∈ N and A ∈ B(X) is super-upper-triangular iff A(k, l) = 0 for any k ≥ l, k, l ∈ N (i.e. A posseses only zero matrix terms below the first super diagonal). By A sup-dia we denote the superdiagonal part of A (i.e. "the first super diagonal" of A) and by A off -its "off-superdiagonal part": Proof. Denote μ j := A(j, j + 1) for j ∈ N, Λ := A sup-dia T +1 and R := A off T +1 . By (4.1) Λ is the diagonal operator with the sequence {μ n } n≥0 on the main diagonal, i.e.
Note that the proof of Theorem 4.1 can be slightly generalized, by admitting adding of some multiples of the identity to the generator, and by considering also the continuous system case. One just needs to consider A + μI instead of A and λ 0 := μ instead of 0. We have then (A + μI) − λ 0 I = A, and the above proof works with the same operator B 0 , providing dist (B, μ) ≤ 1. Hence we get the following result. and μ ∈ C. The linear dynamical system T in X (discrete or continuous one) generated by A + μI is chaotic, providing |Re μ| ≤ 1 at the continuous case, |μ| ≤ 2 at the discrete case.