Syndetic Sensitivity and Mean Sensitivity for Linear Operators

: We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system ( X × Y , T × S ) such that ( X × Y , T × S ) is coﬁnitely sensitive but ( X , T ) and ( Y , S ) are not syndetically sensitive. For the mean sensitivity aspect, we show that if ( Y , S ) is sensitive and not mean sensitive, where Y is a complex Banach space, the spectrum of T meets the unit circle. We also obtain some results regarding mean sensitive perturbations.


Introduction
By a linear dynamical system (l.d.s.) (Y, S), we mean a Banach space Y and a bounded linear operator S : Y → Y. Throughout the manuscript, we let 0 Y be the zero element of the Banach space Y.We denote by I the identity operator.The collection of all positive integers (nonnegative integers, real numbers, respectively) is denoted by N (Z + , R, respectively).
Ruelle [1] first used the notion of sensitivity.According to the works by Guckenheimer [2], Auslander and Yorke [3] introduced the notion of sensitivity on the compact metric space.The sensitivity of linear dynamics has been discussed in [4][5][6][7].Recall that an l.d.s.(Y, S) is sensitive if there is δ > 0 such that, for any y ∈ Y and any neighborhood W of y, there are x ∈ W and v ∈ N with ||S v x − S v y|| > δ.Let δ > 0. For any nonempty open subset W ⊂ Y, define N S (W, δ) = {v ∈ Z + : there exist y 1 , y 2 ∈ W with ||S v y 1 − S v y 2 || > δ}.
Then, it can be verified that an l.d.s.(Y, S) is sensitive if and only if there is δ > 0 such that N S (W, δ) = ∅ for any nonempty open subsets W ⊂ X.
Before proceeding, let us recall some notions related to the families.Denote by thick if, for each u ∈ N, there exists a u ∈ Z + such that {a u , a u + 1, • • • , a u + u} ⊂ S; 2.
syndetic if there exists M ∈ Z + such that a u+1 − a u ≤ M for any u ∈ N.
Moothathu [8] introduced three stronger forms of sensitivity: cofinite sensitivity, multisensitivity and syndetic sensitivity.Subsequently, F -sensitivity for some families was studied in [9][10][11][12][13].We adapt the notions of syndetic sensitivity and cofinite sensitivity for the linear systems.An l.d.s.(Y, S) is called syndetically sensitive (cofinitely sensitive, respectively) if there exists δ > 0 such that N S (W, δ) is syndetic (cofinite, respectively) for any nonempty open subset W ⊂ Y. Inspired by [8], we have the following query: Is there a sensitive l.d.s.(Y, S) that is not syndetically sensitive?The answer is yes (see Example 1).
Let (Y, S) be an l.d.s.The collection of all continuous linear functionals on Y is denoted by B(Y, F).Notice that B(Y, F) is the dual space of Y and is denoted by Y * .If y * ∈ Y * , then we write y * (y) = y, y * , y ∈ Y. Let S * : Y * → Y * be defined by S * y * = y * • S for any y * ∈ Y * .Then, S * is called the adjoint of S and (Y * , S * ) is an l.d.s.(see, for instance, ( [5] Appendix A)).
Inspired by the approach in ( [5], Chapter 10), and [14,15], the operator S induces a bounded operator L S : K → K defined by There are many different notions of sensitivity on the compact space, such as mean sensitivity [16], diam-mean sensitivity [17] and mean n-sensitivity [18].Let (Y, S) be an l.d.s.For every y ∈ Y and any ε > 0, define If there is δ > 0 such that, for any y ∈ Y and any ε > 0, there is x ∈ B(y, ε) satisfying lim sup Naturally, we have the following query: Is there a mean sensitive l.d.s. that is not mean b-sensitive?The answer is no (see Theorem 6).
Matache [19] showed that if (Y, S) is hypercyclic, then σ(S) has a nonvoid intersection with the unit circle.We have established that σ(S) intersects the unit circle for the sensitive l.d.s (Y, S), where (Y, S) is not mean sensitive and Y is a complex Banach space (Theorem 7), and that there is a sensitive l.d.s.(Y, S) that is neither hypercyclic nor mean sensitive (Example 2).
The paper is organized as follows.In Section 2, we recall some results in linear dynamics, which will be used later.In Section 3, we study the adjoint operator.In Section 4, we study the left multiplication operators.In Section 5, we show that there exists an l.d.s.such that 1.
In Section 6, we study the mean sensitive system.Let Y = l p , 1 ≤ p < ∞, or c 0 .We prove that (Y, I + B ω ) and (Y, I + F ω ) are mean sensitive (see Propositions 1, 2 and 4).In Section 7, we study the spectrum property of linear dynamical systems.

Preliminaries
In this section, we recall some results in linear dynamics, used in the later discussion.Let Z 1 and Z 2 be Banach spaces over F.  [20]).For y ∈ Y, we call orb(y, S) = {y, Sy, S 2 y, • • • } the orbit of y under S.An l.d.s (Y, S) is hypercyclic if there is some y ∈ Y such that orb(y, S) = Y.
Let W 1 W 2 be the subspaces of Then, we say that Y is the direct sum of the subspaces W 1 and W 2 , and write Y = W 1 ⊕ W 2 (see, for instance, [21], p. 68).
We state some theorems that will be used in the following.
Similar to Theorem 4, we have the following.Proof.Necessity.Let ε > 0 and v ∈ N.Then, diam(S v B(0 Y , ε)) > 0 by the diammean sensitivity of (Y, S), and so there exists Similarly to the proof of the necessity of Theorem 4, there is . By linearity, one has (Y * , S * ) is diam-mean sensitive.Sufficiency.Let ε > 0 and v ∈ N.Then, diam((S * ) v B(0 Y * , ε)) > 0 by the diammean sensitivity of (Y * , S * ), and so there is . Similarly to the proof of the sufficiency of Theorem 4, there exists Since y ∈ Y and ε > 0 are arbitrary, we find that (Y, S) is diam-mean sensitive.

Left Multiplication Operators
In this section, we study the left multiplication operators.
Similar to Theorem 5, we have the following.
Remark 2. Let (Y, S) be an l.d.s.Then, the cofinite sensitivities of (Y, S) and (K, L S ) are equivalent properties.
Corollary 2. Let (Y, S) be an l.d.s.Then, (Y, S) is diam-mean sensitive if and only if (K, L S ) is diam-mean sensitive.
Proof.Necessity.Let ε > 0 and v ∈ N.Then, diam(S v B(0 Y , ε)) > 0 by the diam-mean sensitivity of (Y, S), and so there exists Similarly to the proof of the necessity of Theorem 5, there exists . By linearity, we find that (K, L S ) is diam-mean sensitive.Sufficiency.Let ε > 0 and v ∈ N.Then, diam((L S ) v B(0 K , ε)) > 0 by the diam-mean sensitivity of (K, L S ), and so there is Similarly to the proof of the sufficiency of Theorem 5, there exists Thus, (Y, S) is diam-mean sensitive.

Sensitivity but Not Cofinite Sensitivity
In this section, we show that there exists an l.d.s.(X × Y, T × S) such that (X, T) is not syndetically sensitive; 3.
defined for t ≥ 1, then we set and and The sequences {a u } u∈N , {c u } u∈N satisfy the following conditions: 1. 3. 4.
Let T : X → X be defined by for any (x 1 , Let S : Y → Y be defined by for any (y 1 , y 2 , y 3 , • • • ) ∈ Y. Now, let us check that the l.d.s.(X, T) has the properties via Claims 4 and 5.We need firstly the following three claims.Claim 1.For any v ≥ 3, we have a v+1 > 2 v+1 a 1 .
Proof of Claim 1.Let v ≥ 3.By the construction of {a u } u∈N ,{c u } u∈N , one has . and so one has a 1 ( by the construction of {a u } u∈N , {c u } u∈N ).
Case 3: If Proof of Claim 3. Let s > 1 and take a s Consider the following three cases.
Case 1: If (a u + u), then l ≤ s, and so and by the construction of (ω u ) u∈N ).
Case 3: by the construction of (ω u ) u∈N .

Claim 4. (X, T) is sensitive.
Proof of Claim 4. Let s > 4 and take By Lemma 1, (X, T) is sensitive.
and by the construction of (µ v ) v∈N ).
Case 3: by the construction of (µ v ) v∈N .
In summary, where s > 1.
and so Thus, (Y, S) is sensitive by Lemma 1.

Mean Sensitivity
In this section, we study mean sensitive systems.We obtain some results regarding mean sensitive perturbations.
Recall that (Y, S) is absolutely Cesàro bounded if there exists a constant C > 0 such that sup for every u ≥ 0, one has for every v ∈ N, which implies that (Y, S) is mean b-sensitive.Sufficiency.The proof is trivial.

Spectrum Property
In this section, we study the spectrum property for sensitive operators.Y = M 1 ⊕ M 2 ; 3.
σ(S| M 2 ) = σ 2 .Note that σ(S| M 2 ) = σ 2 .Similarly, we have that (M 2 , S| M 2 ) is mean sensitive by Lemma 2. Thus, there is δ > 0 such that, for any ε > 0, there exists y ∈ B(0 Y , ε) such that lim sup Recall that a linear dynamical system (Y, S) is Li-Yorke sensitive if there is δ > 0 such that, for any y ∈ Y and any ε > 0, there exists x ∈ B(y, ε) with lim sup Example 2. There is a non hypercyclic, sensitive l.d.s.(Y, S) that is not mean sensitive.
If L 1 , L 2 , • • • , L 2t−1 , L 2t are are well defined for t ≥ 1, then we set In order to obtain the desired properties, we further require {l t } t∈N ⊂ N, ε 0 to satisfy the following conditions: for all t ≥ 1.

Theorem 4 .
Let (Y, S) be an l.d.s.Then, (Y, S) is syndetically sensitive if and only if (Y * , S * ) is syndetically sensitive.

Remark 1 .
Let (Y, S) be an l.d.s.Then, the cofinite sensitivities of (Y, S) and (Y * , S * ) are equivalent properties.Corollary 1.Let (Y, S) be an l.d.s.Then, (Y, S) is diam-mean sensitive if and only if (Y * , S * ) is diam-mean sensitive.

Theorem 6 .||S u y 0
≤ C y for all y ∈ Y.An l.d.s.(Y, S) is mean sensitive if and only if (Y, S) is mean b-sensitive.Proof.Necessity.Since (Y, S) is mean sensitive, one knows that (Y, S) is not absolutely Cesáro bounded, and so there exists y 0 ∈ Y such that sup || = ∞.Let b ≥ 2 and take ε > 0.Then, there exists a strictly increasing sequence {m v } v∈N with1