NONWANDERING OPERATORS IN BANACH SPACE

We introduce nonwandering operators in infinite-dimensional separable Banach space. They are new linear chaotic operators and are relative to hypercylic operators, but different from them. Firstly, we show some examples for nonwandering operators in some typical infinite-dimensional Banach spaces, including Banach sequence space and physical background space. Then we present some properties of nonwandering operators and the spectra decomposition of invertible nonwandering operators. Finally, we obtain that invertible nonwandering operators are locally structurally stable.


Introduction
In the research field of operator, hypercyclic operators and linear chaotic operators have been intensively studied recently.The first observation of hypercyclic operators was by Birkhoff (see [7]).Since then, many researchers discussed this kind of operators (see [1,2,3,4,5,6,8,9,10,11,12,16,17,18,19,20,22,23,24,35]).In 1991, for the first time, Godefroy and Shapiro [16] connected the research of hypercyclic operators and linear chaotic operators and pointed out that some hypercyclic operators are chaotic under the definition of Devaney (see [13,28]).From then on, most hypercyclic operators in the literature have been proved to be chaotic.This implies that almost all hypercyclic operators are linear chaotic.It is well known that linear operators in finite-dimensional linear spaces can not be chaotic but the nonlinear operators may be.Only in infinitedimensional linear spaces can linear operators have chaotic properties.This has attracted wide attention (see [12,13,15,20,26,27,31]).
While in the research field of differential dynamical system, Axiom A system is an important subject.It requires that the nonwandering set Ω( f ) possesses hyperbolic structure and density of periodic points, where hyperbolic structure is based on Whitney's decomposition and the hyperbolic property of the tangent cluster at each point.However, Axiom A system is restricted in finite-dimensional compact Riemann space.Due to the linear property of operators, its tangent bundle at each point is linear operator itself.
On the basis of the above work, we introduce nonwandering operators in infinitedimensional Banach space, which are the generalization of Axiom A dynamic system but different from it.They are new linear chaotic operators and are relative to hypercylic operators, but different from them.
The paper is organized as follows.In Section 2, we list the basic notations and definitions.Then in Section 3, the existence of nonwandering operators on Banach sequence space and physical space is shown.In Section 4, we obtain some basic properties of nonwandering operators.In Section 5, the spectra decomposition of invertible nonwandering operators is completed.Finally, we discuss the local structural stability of invertible nonwandering operators in Section 6.

Basic notations and definitions
Let (X, • ) be an infinite-dimensional separable Banach space on real number field or complex number field K. Let L(X) be the set of all bounded linear operators over X.We will refer to N, Z, Q, R, and C as the sets of positive integers, integers, rational numbers, and the real and complex scalar fields, respectively.Definition 2.1 (see [6,18]).Suppose T ∈ L(X).If there is a vector x ∈ X such that Orb(T,x) = {x, Tx,T 2 x,...} is dense in X, then call x a hypercyclic vector and T a hypercyclic operator.Definition 2.2 (see [3,13]).Suppose T ∈ L(X), call T a linear chaotic operator or a linear chaotic map if it satisfies the following two conditions: (1) T is topologically transitive, that is, T has a dense orbit in X; (2) the set of periodic points Per(T) for T is dense in X.
Remark 2.3.The definition of chaotic map in the sense of Devaney needs another condition: (3) T has a sensitive dependence on initial conditions (see [3]).In fact, conditions (1) and (2) imply condition (3) (see [13]).Thus condition (3) can be omitted from Definition 2.2.Because of the complexity of infinite-dimensional dynamic systems, research of their chaotic properties is usually changed to the study of attractors and inertial manifolds (see [21]).Attractors and inertial manifolds in infinite-dimensional linear space are restricted to be closed invariant linear subspaces (see [21]).So we give the definition of nonwandering operators on closed invariant linear subspace.Definition 2.4.Suppose E ⊂ X is a closed linear subspace of X, and E 1 ⊂ E, E 2 ⊂ E are also closed linear subspaces in X.For arbitrary x ∈ E, if there is a unique decomposition such that , then E is called the direct sum of E 1 and E 2 , and written as E = E 1 ⊕ E 2 , where ⊕ represents direct sum.Definition 2.5.Suppose T ∈ L(X).(1) Assume that there exists a closed subspace E ⊂ X, which has hyperbolic structure: , where E u , E s are closed subspaces.In addition, there exist constants τ(0 < τ < 1) and c > 0, such that for any 2) Assume also that Per(T) is dense in E. Then T is said to be a nonwandering operator relative to E. Remark 2.6.(1) T may be invertible or not.When T is invertible, the spectral property of nonwandering operators is different from that of hypercyclic operators (see Theorem 4.2), but when T is not invertible, the case is much complicated.We give an example for such case.(See Remark 3.5.) (2) If T is a nonwandering operator, then Per(T) E = Φ.In fact, we can easily get it from the hyperbolic structure of E.
(3) Because T is a linear operator, the tangent bundle at each point in E is T itself.Therefore, the definition of nonwandering operators is the natural generalization of Axiom A dynamic system in finite-dimensional differentiable dynamical systems to infinitedimensional space.And these operators are meaningful.Definition 2.7.Suppose T ∈ L(X) and {e i } ∞  1 is a basis in X, then T is called a unilateral backward shift operator relative to Definition 2.9.Suppose that (X, • 1 ), (Y , • 2 ) are two Banach spaces.f : X → Y is called a Lipschitz mapping if there exists α > 0, such that for any x, y ∈ X, f (x) − f (y) 2 ≤ α x − y 1 , where the smallest α is written as Lip( f ).Definition 2.10.Let T i : X i → X i (i = 1,2) be two operators in Banach space X i .If there exists a homeomorphism ϕ : Definition 2.11.Let T ∈ L(X) be a nonwandering operator relative to E. T is called locally structural stable in L(X) if there is a neighborhood U of T and a nonempty open subset V ⊂ E, such that for each linear operator S ∈ U, S is topologically conjugate to T on V .Proposition 3.1 (see [19]).Let X be a Banach sequence space on countable infinite index set, consider the following assertions:

Existence of nonwandering operators in
(1) (e i ) i∈I is an unconditional basis; (2) (e i ) i∈I is a basis in some ordering, and if (x i ) ∈ X, then also is a bounded order of scalars.Then the following implications hold: (1)⇔( 2
Proof.Let X be an infinite-dimensional separable Banach sequence space, in which {e i } ∞ 1 is an unconditional basis, then for the unilateral backward shift operator T on X, λT is a nonwandering operator on X whenever In fact, we can construct a closed invariant subspace E ⊂ X such that E has hyperbolic structure.Let l = λ/2, then we have 0 < |l| < 1. Suppose y 0 = ∞ i=1 b i e i such that λT y = ly, then we get a vector y 0 = {b 1 ,(1/2)b 1 ,(1/2) 2 b 1 ,...}.Let E s = span{y 0 }, then E s is a closed invariant subspace of eigenvectors corresponding to the eigenvalue l = λ/2 for λT.Thus, for each x ∈ E s , there is x = my 0 , (λB)x = λmB y 0 = |l| my 0 = |l| x , where 0 i=1 c i e i , which satisfies λT y = ky, then we get a vector Next, we prove Per(λT) is dense in X.For each n ∈ N, λT(|λ| > 1) has n-period points, such as x ={x 1 ,x 2 ,...,x n ,x 1 /λ n ,x 2 /λ n ,...,x n /λ n ,x 1 /λ 2n ,x 2 /λ 2n ,...,x n /λ 2n ,...}, where {x i } n i=1 ⊂ R. Let y = (y n ) ∈ X be an n 0 -period point for λT, then there exist n 0 ∈ N and i ∈ N, such that (λT) n0 y = y and y i = 0.By Proposition 3.1, for each i ∈ N and n 0 ∈ N the series Therefore, each y (i,n0) (i ≤ n 0 ) is a periodic point for λT.Now we will prove that λT has dense set of periodic points.For each z ∈ span{e n : n ∈ N}, we suppose z = m i=1 z i e i and where C is a constant.Since (e n ) is an unconditional basis and the series ∞ n=1 λ −n e n converge in X, there exist where (ε n ) takes 0 or 1.By (3.2), y = m i=1 z i y (i,n0) is a periodic point for T in X, and (3.5) Then by (3.4), we have y − z ≤ ε, namely, there exists a periodic point y arbitrarily close to z.We obtain that Per(λT) is dense in X, and so is in E. Thus λT is a nonwandering operator relative to E.
Remark 3.3.In this theorem, closed invariant subspaces E s , E u are finite dimensional.In the next, we present an example in physics, in which E s , E u are infinite dimensional.

Nonwandering operator in physical background system.
There are examples for hypercyclic and linear chaotic operators in physics (see [14,20,26,31]).Similarly, nonwandering operators can occur in systems of concrete physical background.Consider a very small frictionless mass-spring system whose evolution is determined by Schrodinger equation: with displacement x, mass m, stiffness k, natural frequency ω = √ k/m, and wave function ψ to be determined in the complex separable Hilbert space X = L 2 (−∞,∞).It is easy to see that (3.6) can be rewritten as The stationary states ψ satisfy and so do the polynomial where H n (x) = (−1) n e x 2 (d/dx n )e x 2 is the n-Hermite polynomial.
Noting that H n (x) = 2nH n−1 (x), we have the following iteration: The unobservable differential (annihilation, lowering) operator B of (3.10) is an unbounded densely defined and weighted backward shift operator in X = L 2 (−∞,+∞).The natural space for the quantum harmonic oscillator is the Banach space Fof all rapidly decreasing functions, that is, The norm • of F is defined as (r ≥ 0) (see [26]). (3.12) Under the norm, B is continuous on space F (see [26]).So B is bounded operator on space F.
Theorem 3.4.The annihilation operator B on Banach space F is a nonwandering operator.
Proof.For each λ ∈ R, it is easy to obtain that (3.13) In the following we will prove that E u and E s are invariant under the operator B.
On one hand, for each In conclusion, we get BE u = E u .Similarly, BE s = E s holds.
Finally we prove that Per(B) is dense in F. Let φ λ be an eigenvector corresponding to λ, where λ are roots of unity.Then φ λ initiate periodic orbits of B. Thus φ λ are dense in F. If not, then there is some function α = ∞ n=0 a n ψ n in F which is orthogonal to each such φ λ , that is, But the zeros of analytic functions are isolated, so Per(B) is dense in F, thus B is a nonwandering operator.(2) Although nonwandering operators are relative to hypercyclic operators, some hypercyclic operators are not nonwandering operators.For example, the "Bergman" backward shift operator B (see [11,Section3.8]),corresponding to weight sequence β(n) = 1/n + 1, is hypercyclic (see [34]), but is not a nonwandering operator because it does not possess dense set of periodic points (see [34]).
(3) There exists a nonwandering operator, but it is not hypercyclic.For example, let (X, • ) be a Banach space, and let B be a nonwandering operator relative to E = E s ⊕ E u given in Theorem 3.4.But B is not a hypercyclic operator on space E. Otherwise, there exists a vector x ∈ E such that {B n x : n = 0,1,2,...} is dense in E (see Definition 2.1).Suppose x = x 1 + x 2 , x 1 ∈ E u , x 2 ∈ E s .For each y ∈ E, there exist n i such that Thus it is contrary to the density of the orbit under B, and then B is not hypercyclic on E.

Properties of nonwandering operators
Proposition 4.1.Suppose T ∈ L(X, • ) and E ⊂ X is a closed subspace, then T is a nonwandering operator relative to E if and only if the following conditions hold: (1)

and there exists some norm
By [37], the following spectral properties of nonwandering operators are obtained.(2) For hypercyclic operator T ∈ L(X), we have σ(T) ∂D = Φ (see [24]).However, the above Theorem 4.2 shows that nonwandering operators differ from hypercyclic operator when it is an invertible operator.Hence they have completely different properties, although they are actually both connected to linear chaotic operators (see Remark 3.5).
Ansari [1] obtained the following result: if T is a hypercydic operator on complex separable Banach space, then so is T m ; moreover, T and T m have the same hypercyclic vectors.Similarly we obtain the following results for nonwandering operators.Theorem 4.4.Suppose T ∈ L(X) and T is an invertible nonwandering operator relative to closed subspace E, then so are T m and T −m for each m ∈ N.
Proof.Obviously, T m and T −m satisfy condition (1) in Definition 2.5.We have that periodic points of T are also the ones of T m and T −m .Because Per(T) is dense in E, then Per(T m ) and Per(T −m ) are also dense in E. Therefore T m and T −m are also nonwandering operators relative to E. Theorem 4.5.Let (X, • ) be an infinite-dimensional separable Banach space, and let E 1 , E 2 be closed subspaces in X and E 1 E 2 = {0}.If the restrictions T| E1 ,T| E2 ∈ L(X) are invertible nonwandering operators relative to E 1 , E 2 , respectively, then T| E is a nonwandering operator relative to Proof.Since T| Ei (i = 1,2) is a nonwandering operator relative to E i , then E i has hyperbolic structure: , where E s i , E u i are also closed subspaces.Furthermore, there exist 0 < τ i < 1 and constant c i > 0, such that, for each we define the following norm on E u : for all Lemma 5.3).Namely, for all x ∈ E u , there exist constants c i > 0 (i = 3,4), such that c 4 x ≤ x 0 ≤ c 3 x .
For each Thus T| E is a nonwandering operator relative to E.

Spectra decomposition of nonwandering operators
In this section, we give the spectra decomposition of invertible nonwandering operators T relative to infinite-dimensional closed subspace.
Theorem 5.1.Suppose T ∈ L(X) is an invertible nonwandering operator relative to infinitedimensional closed subspace E, then there exist closed disjoint nonempty subsets

and for arbitrary nonempty open sets
In order to prove the theorem, we firstly introduce the following notations.For y ∈ X, let where s ∈ E, p ∈ Per(T).Thus we have X p ⊂ B η (W p ) ⊂ B η (X p ).Now we need the following lemmas.
Lemma 5.2.Let X be a Banach space, • and let • 0 be two different norms on it.If there exist some constant a > 0, such that x ≤ a x 0 , then Proof.By equivalent norm theorem and Banach inverse operator theorem, we can easily obtain this result.
Lemma 5.3.Let E = E u ⊕ E s be closed subspace with the norm • , we define a new norm Proof.we can easily prove that (X, • 0 ) is a Banach space.Furthermore, for all x ∈ X, we have x ≤ 2 x 0 .Then by Lemma 5.2, Lemma 5.3 holds.
Proof.(1) Obviously, X p ⊂ B η (X p ).In the following we will prove that B η (X p ) ⊂ X p .Firstly, for ε > 0 small enough and 0 < η < ε, let x ∈ B η (X p ) Per(T).Since X p ⊂ B η (W p ) ⊂ B η (X p ), there exists ω ∈ W p , such that x − ω < η < ε.By Lemma 5.4, there exists a unique point y such that (2) Since X p is the invariant set of T, then for each y ∈ X q , there exists z ∈ X q such that for each l ∈ N, y = T lm z, where m is the period of periodic point q.Furthurmore, according to the fact that W u (q) = X q , there exist z i ∈ W u (q) such that lim i→+∞ z i = z, and y = lim i→+∞ T lm z i .Since z i ∈ W u (q), there exist n 0 ∈ N and some constant η > 0, such that when n > n 0 , T n (z i − q) > η holds.If l is large enough, such that lm > n 0 , then T lm (z i − q) > η.Then for any n ∈ N, T n (T lm (z i − q)) > η holds, that is, T n+lm z i − T n q > η.Then T n (y − q) = T n y − T n q = T n y − T n+lm z i + T n+lm z i − T n q ≥ T n+lm z i − T n q − T n y − T n+lm z i ≥ η − T n y − T lm z i > η, and we have y ∈ W u η (q).Therefore X q ⊂ W u η (q).For q ∈ X p , then W u η (q) ⊂ B η (X p ) = X p , thus X q ⊂ X p .For 0 < η < δ and y ∈ X p X q = B η (W p ) X q , then there exists x ∈ W p such that x − y < η, so x ∈ B η (X q ) = X q .Suppose the periods of p, q are l and m, respectively.Note that x ∈ X q and X q is a closed invariant set of T m , so lim k→+∞ T −klm (q − x) = 0 holds.In addition, x ∈ W p = W u (p) E, so lim k→∞ p − T −klm x = 0 stands.Similarly, for arbitrary z ∈ W p , lim k→∞ q − T −klm z = 0. Hence z ∈ T klm X q = X q .Then W p ⊂ X q and X p = W p ⊂ X q .Similarly, we can get X q ⊂ X p .In conclusion, we get X p = X q .
Proof of Theorem 5.1.Because X is separable, there exists a countable series {x i } i∈N , which is dense in X.For each x i , we construct an open ball B(x i ), which centers at x i with radius ε/3 (ε = δ/2, and δ is discussed in Lemma 5.5( 1)), such that E ⊂ i∈N B(x i ).Now that Per(T) = E, there exist countable periodic points p Obviously, we can suppose that any two of X pi (i = 1,2,...) are disjoint (if not, let X pi be the subtraction of the combination of the preceding sets from X pi ).Note that TW pi = W T pi , TX pi = X T pi , so we can let E i be the combination of all X pi wherein p i have the same period.Therefore we separate E into the combination of closed sets E i (i = 1,2,...), and any two of For any two different sets X pi ,X pj ⊂ E i , there exists l ∈ N, such that T l X pi = X pj .Now, in order to prove that for arbitrary nonempty open sets U,V ⊂ E i , there exists n ∈ N, such that (T n U) V = Φ, we only have to prove that for arbitrary nonempty open sets U,V ⊂ X pi ⊂ E i , there exists n ∈ N, such that (T n U) V = Φ.In fact, since Per(T) = E, there exists a periodic point q in U X pi , then by Lemma 5.5(2), both X q = X pi and V X q = V X pi = V = Φ hold.Therefore, there exists x ∈ V X q .Suppose that the period of q is m, then lim k→+∞ T −km x − q = lim k→+∞ T −km x − T −km q = 0, lim k→+∞ T −km x = q ∈ U. Since q is an inner point in open set U, then for constant k large enough, T −km x ∈ U stands.Then x ∈ T km U V .Let n = km, thus (T n U) V =Φ.The proof is finished.
Remark 5.6.(1) E i in this theorem cannot be the second countable Baire set.Otherwise, T is topologically transactive in E i , that is, there exists a dense orbit of T in E i .Hence from [10], there exists a dense orbit of T in E, that is, T is a hypercyclic operator in E. This is contrary to Remark 4.3 (2).
(2) Since E i in this theorem is not second countable Baire set, T is impossibly hypercyclic in E i .

Local structural stability of nonwandering operators
Structural stability is the key subject in the differentiable dynamical systems (see [25,29,30,32,33,36]). It is well known that hyperbolic linear shift operators can keep their hyperbolic invariant properties under small perturbation, which inspires us to make an attempt to study the local structural stability of the nonwandering operators.

Remark 3 . 5 .
(1) The annihilation operator B is not an invertible nonwandering operator.Here σ(B) ∂D = Φ, where σ(B) is the spectrum of B, and ∂D is unit circle.In fact, we have σ(B) = C.