Éphane St

Karl Charpentier

Quentin Grosse-Erdmann

Menet

Karl Grosse-Erdmann
email: kg.grosse-erdmann@umons.ac.be

Quentin Menet
email: quentin.menet@umons.ac.be

CHAOS AND FREQUENT HYPERCYCLICITY FOR WEIGHTED SHIFTS

Keywords: 2010 Mathematics Subject Classification. 47A16, 47B37, 46A45 Frequently hypercyclic operator, chaotic operator, weighted shift operator, Köthe sequence space, power series space

] have recently shown that every frequently hypercyclic weighted shift on p is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Trans. Amer. Math. Soc. 358 (2006)] who constructed a non-chaotic frequently hypercyclic weighted shift on c 0 . We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on H(D) is chaotic, while H(C) admits a non-chaotic frequently hypercyclic weighted shift.

For these operators, under suitable assumptions, chaos does imply frequent hypercyclicity.

Introduction

Chaos and frequent hypercyclicity are two of the most important notions in linear dynamics. An operator T on a separable Fréchet space X is called chaotic if it admits a dense orbit (i.e., it is hypercyclic) and if it has a dense set of periodic points. If one demands from an orbit that it combines both aspects of chaos, i.e., that it is dense and that it returns often, one arrives at the notion of frequent hypercyclicity: A vector x ∈ X is called frequently hypercyclic for T if, for any non-empty open set U in X, dens{n ≥ 0 : T n x ∈ U } > 0; recall that, for a subset A ⊂ N 0 , dens(A) = lim inf N →∞ 1 N +1 card{n ≤ N : n ∈ A} denotes its lower (asymptotic) density. The operator T is called frequently hypercyclic if it admits a frequently hypercyclic vector. This notion was introduced by Bayart and Grivaux [START_REF] Bayart | Hypercyclicité: le rôle du spectre ponctuel unimodulaire[END_REF], [START_REF] Bayart | Frequently hypercyclic operators[END_REF]. For more information on linear dynamics we refer to [START_REF] Bayart | Dynamics of linear operators[END_REF] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF].

The precise relationship between chaos and frequent hypercyclicity has been intriguing researchers for the last decade. Despite appearances, these notions have turned out to be independent. First, Bayart and Grivaux [START_REF] Bayart | Invariant Gaussian measures for operators on Banach spaces and linear dynamics[END_REF] showed that a frequently hypercyclic operator need not be chaotic. Indeed, they constructed a frequently hypercyclic weighted backward shift on c 0 that does not even have a single non-trivial periodic point; see also [START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF]. Recently, the third author [START_REF] Menet | Linear chaos and frequent hypercyclicity[END_REF] constructed operators on any of the spaces p , 1 ≤ p < ∞, and c 0 that are chaotic but not frequently hypercyclic.

One of the best understood classes of operators in linear dynamics is that of weighted (backward) shifts on sequence spaces. Given a sequence w = (w n ) n of non-zero scalars, the corresponding weighted shift operator is formally given by Indeed, let X be a Fréchet sequence space, that is, a Fréchet space of (real or complex) sequences x = (x n ) n≥0 such that each coordinate functional x → x n , n ≥ 0, is continuous. By the closed graph theorem, a weighted shift B w defines a (continuous) operator on X as soon as it maps the space X into itself. Let e n , n ≥ 0, denote the unit sequences. Under the assumption that (e n ) n is an unconditional basis in the Fréchet sequence space X then a weighted shift B w on X is known to be chaotic if and only if the series (1.1)

n≥0 1 w 1 • • • w n e n
converges in X; see [START_REF] Grosse-Erdmann | Hypercyclic and chaotic weighted shifts[END_REF]. It then follows from the Frequent Hypercyclicity Criterion that, in this case, B w is also frequently hypercyclic; see [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Corollary 9.14].

We know from the result of Bayart and Grivaux cited above that not every frequently hypercyclic weighted shift on c 0 is chaotic. In light of this, the following recent result of Bayart and Ruzsa [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF] came as a surprise.

Theorem 1.1 (Bayart, Ruzsa). Let 1 ≤ p < ∞ and let w = (w n ) n be a bounded sequence of non-zero scalars. Then the weighted shift B w is frequently hypercyclic on p if and only if it is chaotic.

So what do the spaces p possess that c 0 does not? This question was the starting point of our present investigation. In Section 2 we will show that the result of Bayart and Ruzsa extends to all Banach sequence spaces for which (e n ) n is a boundedly complete unconditional basis. Going beyond the class of Banach spaces we then turned our attention to Köthe sequence spaces, which are important generalizations of the spaces p and c 0 . The Köthe sequence space of order p is defined as λ p (A) = x = (x n ) n≥0 : for all m ≥ 1, n≥0 |x n | p a m,n < ∞ , while the Köthe sequence space of order 0 is given by c 0 (A) = x = (x n ) n≥0 : for all m ≥ 1, lim n→∞ |x n |a m,n = 0 , where A = (a m,n ) m≥1,n≥0 is a matrix of strictly positive numbers such that, for all m ≥ 1, n ≥ 0, a m,n ≤ a m+1,n . In particular, if a m,n = 1 for every m, n then λ p (A) = p and c 0 (A) = c 0 . Thus one could naively imagine that on Köthe sequence spaces of order p each frequently hypercyclic weighted shift is chaotic, while on Köthe sequence spaces of order 0 there exists a frequently hypercyclic weighted shift that is not chaotic. However, it is not seldom that λ p (A) = c 0 (A) for any 1 ≤ p < ∞. This is, for instance, the case for the space H(C) of entire functions and for the space H(D) of holomorphic functions on the unit disk. The link between chaotic and frequently hypercyclic weighted shifts on these spaces is therefore particularly intriguing.

In order to be able to deal with these spaces, the main part of the paper is devoted to Fréchet sequence spaces, in varying degrees of generality. Two interesting phenomena arise. First, as we will see in Section 3, there are simple Fréchet sequence spaces that do not support any weighted shift, and there are natural Köthe sequence spaces that do not support any hypercyclic weighted shift, so that our study may become vacuous. Secondly, bounded completeness is no longer sufficient for ensuring that frequent hypercyclicity implies chaos. Still, in Section 2 we obtain a version of the Bayart-Ruzsa theorem for a class of Fréchet sequence spaces. In Section 3 we spell out this result in the case of Köthe sequence spaces, and we state sufficient conditions for the existence of a frequently hypercyclic weighted shift that is not chaotic. This will allow us to obtain the puzzling result that every frequently hypercyclic weighted shift on H(D) is chaotic, while there exists a frequently hypercyclic weighted shift on H(C) that is not chaotic.

Finally, in Section 4, we generalize these two examples by studying the particularly important class of power series spaces (of finite or infinite type). We will obtain a full characterization of when every frequently hypercyclic weighted shift is chaotic on a class of power series spaces that satisfy some regularity assumption. In particular, it will appear that, on the one hand, there are Köthe sequence spaces c 0 (A), different from any λ p (A), 1 ≤ p < ∞, on which any frequently hypercyclic weighted shift is chaotic; on the other hand, there are Köthe sequence spaces λ p (A), different from c 0 (A), on which there exists a non-chaotic frequently hypercyclic weighted shift.

2.

When frequent hypercyclicity implies chaos for weighted shifts 2.1. The Bayart-Ruzsa theorem on Banach sequence spaces. The purpose of this subsection is to extend Theorem 1.1 to a large class of Banach sequence spaces. Before stating this result we remark that Bayart and Ruzsa obtained a stronger version of their result: even U-frequent hypercyclicity implies chaos (see [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF]Theorem 4]). An operator T on a Fréchet space X is called U-frequently hypercyclic if there is vector x ∈ X, also called U-frequently hypercyclic, such that, for any non-empty open set U in X,

dens{n ≥ 0 : T n x ∈ U } > 0,
where, for a subset A ⊂ N 0 , dens(A) = lim sup N →∞ 1 N +1 card{n ≤ N : n ∈ A} denotes its upper (asymptotic) density. This notion is obviously weaker than frequent hypercyclicity.

In our result, as is usual in the context of weighted shifts, we will demand that (e n ) n is an unconditional basis, which is the case both for c 0 and the spaces p . What distinguishes these spaces, however, is that (e n ) n is boundedly complete in p but not in c 0 . Recall that a basis (f n ) n in a Fréchet space X is called boundedly complete if, for any sequence of scalars x = (x n ) n , whenever the sequence

N n=0 x n f n N ≥0
is bounded in X then it converges in X; see [START_REF] Albiac | Topics in Banach space theory[END_REF].

Theorem 2.1. Let X be a Banach sequence space for which (e n ) n is a boundedly complete unconditional basis. Let B w be a weighted shift on X. Then the following assertions are equivalent:

(i) B w is U-frequently hypercyclic on X; (ii) B w is frequently hypercyclic on X; (iii) B w is chaotic on X; (iv) the series n≥0 1 w 1 •••wn e n is convergent in X.
As we have mentioned in the introduction, the implications (iv)=⇒(iii)=⇒(ii) hold in all Banach sequence spaces in which (e n ) n is an unconditional basis, while the implication (ii)=⇒(i) is trivial. It remains therefore to show that (i) implies (iv) under bounded completeness.

The following lemma will be crucial in the proof. A subset A ⊂ N 0 is called syndetic if it is infinite and of bounded gaps, that is, sup n∈A inf m∈A,m>n (m -n) < ∞. Lemma 2.2. Let X be a Banach space with an unconditional basis (f n ) n≥0 . Let (α n ) n≥0 be a sequence of scalars and A ⊂ N 0 a subset of positive upper density such that, for some R > 0, the family

1 N + 1 n∈A n≤N m∈A n≤m≤n+M α m-n f m-n M,N ≥R is bounded in X.
Then there is a syndetic set F ⊂ N 0 such that the sequence n∈F, n≤N

α n f n N ≥0 is bounded in X.
The proof is based on Theorem 2.3 below, which is an improvement of a result due to Erdős and Sárközy that was implicitly obtained by Bayart and Ruzsa in order to prove Theorem 1.1. Note that these authors only state a weaker version [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF]Theorem 8].

Theorem 2.3 (Bayart, Ruzsa). Let A ⊂ N 0 be a set of upper density δ := dens(A) > 0, and let 0 < ε < δ 2 . Then there exists a strictly increasing sequence (N j ) j of positive integers such that the set

k ∈ N 0 : lim j→∞ card{n ≤ N j : n ∈ A ∩ (A -k)} N j + 1 > δ 2 -ε is syndetic.
Proof of Lemma 2.2. Let (α n ) n and A ⊂ N 0 be as in the statement of the lemma, with δ := dens(A) > 0, and fix 0 < ε < δ 2 . By Theorem 2.3 there exists a strictly increasing sequence (N j ) j of positive integers such that the set

F := k ∈ N 0 : lim j→∞ card{n ≤ N j : n ∈ A ∩ (A -k)} N j + 1 > δ 2 -ε is syndetic. For N, M ≥ 0 we set y N,M = 1 N + 1 n∈A n≤N m∈A n≤m≤n+M α m-n f m-n .
Reordering the (finite) sum, we get (2.1)

y N,M = 1 N + 1 M k=0 n∈A, n∈A-k n≤N α k f k = M k=0 card{n ≤ N : n ∈ A ∩ (A -k)} N + 1 α k f k .
Since (f n ) n is an unconditional basis there exists a constant C > 0 such that, whenever

x = n≥0 x n f n ∈ X and b = (b n ) n is a bounded sequence of scalars, then n≥0 b n x n f n ∈ X and (2.2) n≥0 b n x n f n ≤ C b ∞ n≥0 x n f n , see [2, Proposition 3.1.3].
Taking for b a suitable 0-1-sequence we obtain from (2.1) and (2.2) that

(2.3) y N,M ≥ 1 C k∈F, k≤M card{n ≤ N : n ∈ A ∩ (A -k)} N + 1 α k f k .
By definition of F and another application of unconditionality we have that, for any M ≥ 0 and for all sufficiently large j,

k∈F, k≤M card{n ≤ N j : n ∈ A ∩ (A -k)} N j + 1 α k f k ≥ δ 2 -ε C k∈F, k≤M α k f k . (2.4)
It then follows from (2.3) and (2.4) that for any M ≥ 0 and all sufficiently large j, k∈F, k≤M

α k f k ≤ C 2 δ 2 -ε y N j ,M .
The hypothesis now implies that k∈F, k≤M

α k f k M ≥R is bounded in X.
Remark 2.4. (a) Obviously, the hypothesis of the lemma holds in particular if the family

m∈A n≤m≤n+M α m-n f m-n n∈A,M ≥0
is bounded in X. This is the condition that will be satisfied in our application of the lemma.

(b) Using essentially the same proof one can show that the lemma also holds for Fréchet spaces. However, in that setting the hypothesis is too strong for our intended application, see the proof of Theorem 2.9.

We will apply Lemma 2.2 to the unit sequence (e n ) n in a Banach sequence space X and to a sequence (α n ) n that arises in a natural way from the weights of a weighted shift B w . In fact, as is usual in this context, we will define a sequence

v = (v n ) n≥0 by (2.5) v n = 1 w 1 • • • w n , n ≥ 0;
note that v 0 = 1. Conversely, given a sequence v = (v n ) n≥0 of non-zero scalars with v 0 = 1, we recover the sequence w by setting (2.6)

w n = v n-1 v n , n ≥ 1.
Lemma 2.5. Let X be a Banach sequence space for which (e n ) n is an unconditional basis. Let B w be a weighted shift on X, and let v be the sequence associated to w by (2.5). If A ⊂ N 0 is a subset of positive upper density such that, for some R > 0, the family

1 N + 1 n∈A n≤N m∈A n≤m≤n+M v m-n e m-n M,N ≥R is bounded in X, then the sequence n≤N v n e n N ≥0
is bounded in X.

Proof. We know from Lemma 2.2 that n∈F, n≤N

v n e n N ≥0
is bounded in X for some syndetic set F ⊂ N 0 . By the definition of B w we have that

B w n∈F, 0≤n≤N v n e n = n∈F, 0≤n≤N v n B w e n = n∈F, 1≤n≤N v n w n e n-1 = n∈F, 1≤n≤N v n-1 e n-1 .
Now, since F is syndetic there is some K ≥ 0 such that

N 0 = k≤K (F -k), which implies that K k=0 B k w n∈F, 0≤n≤N v n e n = 0≤n≤F N m n,N v n e n
with certain integers m n,N ≥ 1, where

F N = max{n ∈ F : n ≤ N }, with max ∅ = -1.
The result now follows by the continuity of B w , the unconditionality of the basis and the fact that F N → ∞ as N → ∞.

Remark 2.6. Instead of filling the gaps of F via a backward shift, as done in the proof of the lemma, one may also fill them by a forward shift. Let w = (w n ) n≥0 be a sequence of non-zero scalars. The corresponding forward shift F w is defined by F w x = ∞ n=1 w n-1 x n-1 e n . Then the preceding lemma also holds if F w is an operator on X and v satisfies w n = v n+1 vn , n ≥ 0.

We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. As mentioned before, it suffices to show that (i) implies (iv).

Let x = (x n ) n ∈ X be a U-frequently hypercyclic vector for B w . Let C be a constant due to unconditionality appearing in (2.2). Then the set

A := n ∈ N 0 : B n w x -e 0 ≤
e 0 2C has positive upper density. Since (2.7)

B n w x = k≥0 w k+1 • • • w k+n x k+n e k = k≥0 v k v k+n x k+n e k ,
where v is the sequence associated to w by (2.5), we have that for every n ∈ A

1 v n x n e 0 -e 0 ≤ C B n w x -e 0 ≤ e 0 2 .

Hence

(2.8)

x n v n ≥ 1/2
for every n ∈ A. By (2.7) and (2.8), we note that for every n ∈ A and every integer M ≥ 0,

m∈A n≤m≤n+M v m-n e m-n ≤ 2C m∈A n≤m≤n+M v m-n x m v m e m-n = 2C 0≤k≤M k+n∈A v k x k+n v k+n e k ≤ 2C 2 B n w x ≤ 2C 2 ( B n w x -e 0 + e 0 ) ≤ C(1 + 2C) e 0 .
Applying now Lemma 2.5, taking account of Remark 2.4(a), we obtain that

n≤N 1 w 1 • • • w n e n N ≥0 = n≤N v n e n N ≥0
is bounded in X. Since the basis (e n ) n is boundedly complete we can finally deduce (iv).

Remark 2.7. Since the proof uses bounded completeness of the basis only in the last step, we have also proved the following. Let X be a Banach sequence space for which (e n ) n is an unconditional basis. If a weighted shift B w on X is U-frequently hypercyclic then ( n≤N

1 w 1 •••wn e n ) N ≥0 is bounded in X.

Can one go beyond bounded completeness?

There is an indication that this might be difficult: it is known that if (e n ) n is an unconditional basis in a Banach sequence space X, then it fails to be boundedly complete if and only if X contains a copy of c 0 (see [START_REF] Albiac | Topics in Banach space theory[END_REF]Theorem 3.3.2] for the precise statement), and we know from Bayart and Grivaux [START_REF] Bayart | Invariant Gaussian measures for operators on Banach spaces and linear dynamics[END_REF] that Theorem 2.1 fails on c 0 . But it is not clear how to exploit these facts.

Question 2.8. Is there a Banach sequence space X for which (e n ) n is a non-boundedly complete unconditional basis so that every frequently hypercyclic weighted shift on X is chaotic? 2.2. The Bayart-Ruzsa theorem on Fréchet sequence spaces. Let us now turn to Fréchet sequence spaces in which (e n ) n is an unconditional basis. We will see that in this setting bounded completeness of the basis no longer suffices to have all frequently hypercyclic weighted shifts chaotic; see the summary at the end of the paper. The reason is that with any application of the continuity of the weighted shift we potentially lose quality of the seminorm. To be more precise, let us fix an increasing sequence ( • m ) m≥1 of seminorms defining the topology of X. Then continuity of an iterate B n w , n ≥ 1, of B w means that, for any m ≥ 1, B n w x m is majorized by a multiple of x q but where q may depend on n and m; this makes our Banach space proof break down. It turns out that it suffices to impose the property of topologizability on the weighted shifts, see [START_REF] Bonet | Topologizable operators on locally convex spaces[END_REF]: (T) Any weighted shift B w on X is topologizable, that is, for any m ≥ 1 there is some q(m) ≥ 1 such that for any n ≥ 0 there is some constant C m,n > 0 such that, for any x ∈ X,

B n w x m ≤ C m,n x q(m) .
The property (T) is a condition on the space X, and it is independent of the particular sequence ( • m ) m≥1 chosen. We then have the following generalization of Theorem 2.1. Theorem 2.9. Let X be a Fréchet sequence space that satisfies (T) and for which (e n ) n is a boundedly complete unconditional basis. Let B w be a weighted shift on X. Then the following assertions are equivalent:

(i) B w is U-frequently hypercyclic on X;

(ii) B w is frequently hypercyclic on X;

(iii) B w is chaotic on X;

(iv) the series n≥0

1 w 1 •••wn e n is convergent in X. Proof.
First, the unconditionality of the basis implies that, whenever x = n≥0 x n e n ∈ X and b = (b n ) n is a bounded sequence of scalars, then x = n≥0 b n x n e n ∈ X; in addition, for any m ≥ 1, there exist a constant K m > 0 and some p(m) ≥ 1 such that, for all such x and b, (2.9)

n≥0 b n x n e n m ≤ K m b ∞ n≥0
x n e n p(m) ; see [START_REF] Albiac | Topics in Banach space theory[END_REF]Proposition 3.1.3] in the case of a Banach space, but the proof works as well for Fréchet spaces, using [START_REF] Kamthan | Sequence spaces and series[END_REF]Theorem 3.3.9].

It suffices again to prove that (i) implies (iv). Thus, let x = (x n ) n ∈ X be a U-frequently hypercyclic vector for B w and m ≥ 1. We fix s ≥ 1 such that e 0 s > 0 and we let u = q(p(m)), r = p(p(u)) and t = max(p(s), p(p(r)))

where q(•) comes from property (T). Since x is U-frequently hypercyclic and e 0 s > 0, the set

A := n ∈ N 0 : B n w x -e 0 t ≤
e 0 s 2K s has positive upper density. By (2.9) we obtain for any n ∈ A

1 v n x n e 0 -e 0 s = w 1 • • • w n x n e 0 -e 0 s ≤ K s B n w x -e 0 p(s) ≤ K s B n w x -e 0 t ≤ e 0 s 2 ,
where v is the sequence associated to w by (2.5). We continue as in the proof of Theorem 2.1, obtaining first that | xn vn | ≥ 1/2 for any n ∈ A, and then for any integer M ≥ 0,

m∈A n≤m≤n+M v m-n e m-n r ≤ 2K r K p(r) B n w x p(p(r)) ≤ 2K r K p(r) ( B n w x -e 0 t + e 0 t ) ≤ 2K r K p(r) ( e 0 s
2Ks + e 0 t ). This shows that the family

m∈A n≤m≤n+M v m-n e m-n n∈A,M ≥0
is bounded with respect to the seminorm • r ; note however that A depends on r. Writing for N, M ≥ 0

y N,M = 1 N + 1 n∈A n≤N m∈A n≤m≤n+M v m-n e m-n ,
we obtain that also the family (y N,M ) N,M ≥0 is bounded with respect to the seminorm • r .

Let δ := dens(A) and 0 < ε < δ 2 . Then we obtain exactly as in the proof of Lemma 2.2 by a double application of unconditionality that there is a syndetic set F ⊂ N 0 so that

C := sup M ≥0 k∈F, k≤M v k e k u ≤ sup j≥1,M ≥0 K u K p(u) δ 2 -ε y N j ,M p(p(u)) = sup j≥1,M ≥0 K u K p(u) δ 2 -ε y N j ,M r < ∞.
In particular, there is some K ≥ 0 such that

N 0 = k≤K (F -k)
and by property (T) there is a constant C > 0 such that, for any x ∈ X and 0

≤ n ≤ K, B n w x p(m) ≤ C x q(p(m)) = C x u .
Following now exactly the proof of Lemma 2.5 we obtain, together with an application of unconditionality, that for any N ≥ 0

0≤n≤F N v n e n m ≤ K m (K + 1) CC,
where

F N = max{n ∈ F : n ≤ N }, with max ∅ = -1.
Note that m ≥ 1 is arbitrary here. Since F N → ∞ as N → ∞, another application of unconditionality shows that the sequence

0≤n≤N 1 w 1 • • • w n e n N ≥0 = 0≤n≤N v n e n N ≥0
is bounded in X, so that the bounded completeness of the basis implies (iv).

Remark 2.10. (a) Of course, one could just demand that condition (T) holds for a given weighted shift B w on X, and then we could conclude that, for this shift, frequent hypercyclicity implies chaos. But given the simplicity of the characterization of chaos, it would be much easier to show directly that B w is chaotic. (b) Remark 2.7 also applies to Fréchet sequence spaces.

Chaotic and frequently hypercyclic weighted shifts on Köthe sequence spaces

In this section, we aim to apply Theorem 2.9 to a particularly interesting class of Fréchet sequence spaces, the Köthe sequence spaces, also called Köthe echelon spaces, see [START_REF] Meise | Introduction to functional analysis[END_REF]Chapter 27] or [START_REF] Bierstedt | Some aspects of the modern theory of Fréchet spaces[END_REF].

Let A = (a m,n ) m≥1,n≥0 be a matrix of strictly positive numbers such that, for all m ≥ 1, n ≥ 0, a m,n ≤ a m+1,n . Such a matrix is called a Köthe matrix.

Let 1 ≤ p < ∞. We recall that the Köthe sequence space of order p is defined as

λ p (A) = x = (x n ) n≥0 : for all m ≥ 1, n≥0 |x n | p a m,n < ∞ ,
while the Köthe sequence spaces of order 0 and ∞ are given by

c 0 (A) = x = (x n ) n≥0 : for all m ≥ 1, lim n→∞ |x n |a m,n = 0 , λ ∞ (A) = x = (x n ) n≥0 : for all m ≥ 1, sup n≥0 |x n |a m,n < ∞ .
The topologies are respectively induced by the (semi-)norms

x m = n≥0 |x n | p a m,n 1/p , m ≥ 1, for 1 ≤ p < ∞, x m = sup n≥0 |x n |a m,n , m ≥ 1,
for order 0 and ∞.

We note that, for each of the spaces λ p (A), 1 ≤ p < ∞, and c 0 (A), the sequence (e n ) n is an unconditional basis. On the other hand, the space λ ∞ (A) only has (e n ) n as a basis if it coincides with c 0 (A) (which can happen, see Proposition 3.2); hence we will only study weighted shifts on Köthe sequence spaces of finite order. Remark 3.1. It is easy to see that a Köthe sequence space X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) is normable and hence a Banach space if and only if

∃µ ≥ 1, ∀m ≥ 1 : sup n≥0 a m,n a µ,n < ∞.
In the first case, X = {(x n ) n : n≥0 a µ,n |x n | p < ∞} is a weighted p -space, which implies that any frequently hypercyclic weighted shift on X is chaotic. In the second case, X is a weighted c 0 -space, so that there is a frequently hypercyclic weighted shift on X that is not chaotic. Now, in order to apply Theorem 2.9 we need to ensure that the sequence (e n ) n is boundedly complete. For the Köthe sequence spaces of order p ∈ [1, ∞), this is clearly always the case. For the simplest Köthe sequence space of order 0, c 0 , or more generally, when c 0 (A) is a Banach space, then the sequence (e n ) n is not boundedly complete. But this is not so for all Köthe sequence space of order 0. Let us assume that (e n ) n is boundedly complete in c 0 (A), and let x = (x n ) n be an arbitrary sequence in λ ∞ (A). Then the sequence ( N n=0

x n e n ) N ≥0 is bounded in c 0 (A) and thus x belongs to c 0 (A). Therefore c 0

(A) = λ ∞ (A). It is just as easy to see that if c 0 (A) = λ ∞ (A) then (e n ) n is boundedly complete in c 0 (A).
Now, the identity c 0 (A) = λ ∞ (A) can be characterized in terms of the entries of A, see [START_REF] Meise | Introduction to functional analysis[END_REF]Theorem 27.9]. Thus we have the following. Proposition 3.2. Let A be a Köthe matrix. Then the following assertions are equivalent:

(i) (e n ) n is a boundedly complete basis for c 0 (A); (ii) c 0 (A) = λ ∞ (A); (iii) A satisfies the condition (BC) ∀I ⊂ N infinite, ∀m ≥ 1, ∃µ ≥ 1 : inf n∈I a m,n a µ,n = 0.
We mention in passing that Chapter 27 of Meise and Vogt [START_REF] Meise | Introduction to functional analysis[END_REF] contains characterizations of various other interesting properties of Köthe sequence spaces in terms of the Köthe matrix A. We single out the following result that will be used later; see [START_REF] Meise | Introduction to functional analysis[END_REF]Proposition 27.16] and use that λ

p (A) ⊂ λ q (A) ⊂ c 0 (A) ⊂ λ ∞ (A) if 1 ≤ p ≤ q < ∞.
Proposition 3.3. Let A be a Köthe matrix. Then the following assertions are equivalent:

(i) for some p ∈ [1, ∞), c 0 (A) = λ p (A); (ii) for some p ∈ [1, ∞), λ ∞ (A) = λ p (A); (iii) for some p = q in [1, ∞), λ p (A) = λ q (A); (iv) for all p ∈ [1, ∞), λ p (A) = c 0 (A) = λ ∞ (A); (v) A satisfies the condition (N) ∀m ≥ 1, ∃µ ≥ 1 : n≥0 a m,n a µ,n < ∞.
Of course, (N) implies (BC). Note that (N) also characterizes when c 0 (A) (or any λ p (A), p ∈ [1, ∞]) is nuclear, see [START_REF] Meise | Introduction to functional analysis[END_REF]Proposition 28.16].

It is now an easy matter to apply Theorem 2.9 to (certain) Köthe sequence spaces and thus obtain conditions under which any frequently hypercyclic weighted shift on such a space is chaotic. However, before stating this result in Subsection 3.2, we will observe a rather unexpected phenomenon: there are some Köthe sequence spaces that do not support any hypercyclic weighted shift.

3.1. Existence of (chaotic or frequently) hypercyclic weighted shifts on Köthe sequence spaces. Let us pause briefly in order to reflect on the question we are trying to discuss. On the positive side we want to show that, on a given Fréchet sequence space, every frequently hypercyclic weighted shift is chaotic. Now it might be that the problem is meaningless because there might not be any frequently hypercyclic weighted shifts on the space. Indeed, for (frequent) hypercyclicity it helps if the weights are big, while the continuity of the weighted shift limits their growth. There is then a trade-off.

Even more, it is known that there are separable infinite-dimensional Banach spaces that do not support any chaotic or any frequently hypercyclic operator, see [START_REF] Bonet | A Banach space which admits no chaotic operator[END_REF], [START_REF] Shkarin | On the spectrum of frequently hypercyclic operators[END_REF]. For shifts, we do have a characterization of frequently hypercyclic weighted shifts on an arbitrary Fréchet sequence space in which (e n ) n is an unconditional basis; see [START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF]Theorem 6.2]. However, the conditions are rather involved. So let us approach the problem from above (are there any weighted shifts? are there hypercyclic weighted shifts?) and from below (are there chaotic weighted shifts?).

3.1.1. Existence of weighted shifts. The existence of weighted shifts on (most) Banach or Fréchet sequence spaces is easily characterized. Proposition 3.4. (a) Any Banach sequence space that contains all finite sequences admits a weighted shift.

(b) Let X be a Fréchet sequence space in which (e n ) n≥0 is a basis. Let ( • m ) m be an increasing sequence of seminorms that generates the topology of X. Then X admits a weighted shift if and only if

(3.1) ∀m ≥ 1, ∃µ ≥ 1, ∀n ≥ 0 : e n m = 0 =⇒ e n+1 µ = 0.
Proof. We first show sufficiency of (3.1) in (b). Let, for any m ≥ 1, µ m =: µ ≥ 1 be given by this condition. Since (e n ) n is a basis, the Banach-Steinhaus theorem implies that there are ν m ≥ 1 and C m > 0 such that, for all x ∈ X, n ≥ 0 and m ≥ 1,

(3.2)
x n e n µm ≤ C m x νm .

Let

a n = max m≤n C m e n m e n+1 µm , n ≥ 0;
note that this is well-defined by (3.1) if we let 0 0 = 0. Now let w = (w n ) n≥1 be a sequence of non-zero scalars such that

n≥0 |w n+1 |a n < ∞.
Then we have that, for any m ≥ 1 and x ∈ X, The necessity of (3.1) follows directly from the continuity of any given weighted shift on X.

For (a) note that, in the Banach space case, (3.1) is automatic. Moreover, inequality (3.2) holds even without the basis assumption when one allows the constant C m to depend on n since the projections x → x n e n , n ≥ 1, are continuous; the proof can then be continued as before.

In particular, any Köthe sequence space of finite order admits a weighted shift.

Example 3.5. We give here an example of a Fréchet sequence space in which (e n ) n is a basis but that admits no weighted shift. Indeed, let

X = (x n ) n : n≥0 |x 2n | < ∞ .
This turns into a Fréchet sequence space under the seminorms

x m = max 0≤n≤m |x 2n+1 | + n≥0 |x 2n |, m ≥ 1.
Then X does not admit any weighted shift because no weight can turn all sequences into 1 -sequences.

3.1.2. Existence of hypercyclic weighted shifts. For a general Banach or Fréchet sequence space, the existence of hypercyclic weighted shifts seems to be a complicated matter. Thus we restrict our attention to Köthe sequence spaces. Note that a weighted shift B w defines an operator on a Köthe sequence space λ p (A) or c 0 (A) if and only if

∀m ≥ 1, ∃µ ≥ 1, C > 0, ∀n ≥ 1 : |w n | p a m,n-1 ≤ Ca µ,n , on λ p (A), |w n |a m,n-1 ≤ Ca µ,n , on c 0 (A).
For later use we note here that, in terms of the associated sequence v (see (2.5) and (2.6)), the conditions can be written as

(3.3) ∀m ≥ 1, ∃µ ≥ 1, C > 0, ∀n ≥ 1 : |v n-1 | p a m,n-1 ≤ C|v n | p a µ,n , on λ p (A), |v n-1 |a m,n-1 ≤ C|v n |a µ,n , on c 0 (A). Proposition 3.6. Let X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) be a Köthe sequence space.
Then there exists a hypercyclic weighted shift on X if and only if there exist a strictly increasing sequence (m k ) k of positive integers and a positive sequence

(C k ) k such that ∀n ≥ 1, ν n := inf k≥1 C k a m k+1 ,n a m k ,n-1 > 0 and ∀m ≥ 1, lim inf N →∞ 1 N n=1 ν n a m,N = 0.
Proof. Let B w be a hypercyclic weighted shift on X. In this proof, let p = 1 for the space c 0 (A). By continuity, there exist a strictly increasing sequence (m k ) k of positive integers and a positive sequence (C k ) k such that, for every n ≥ 1 and k ≥ 1, we have

|w n | p a m k ,n-1 ≤ C k a m k+1 ,n
hence, for any n ≥ 1,

ν n := inf k≥1 C k a m k+1 ,n a m k ,n-1 ≥ |w n | p > 0.
Moreover, since B w is hypercyclic, we have that, for every m ≥ 1, lim inf

N →∞ 1 N n=1 ν n a m,N ≤ lim inf N →∞ 1 N n=1 |w n | p a m,N = 0,
see [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 4.8]. This shows necessity.

The sufficiency is easily seen by considering the weighted shift B w with w n = ν 1/p n , n ≥ 1, and using the fact that a m,n ≤ a m+1,n for all m ≥ 1, n ≥ 0.

Under an additional assumption on the Köthe matrix A, the characterising condition can be simplified.

Corollary 3.7. Let X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) be a Köthe sequence space. Suppose that

(3.4) ∀m, j ≥ 1, ∃J ≥ 1, C > 0, ∀n ≥ 0 : a m,n+1 a 1,n ≤ C a J,n+1 a j,n .
Then there exists a hypercyclic weighted shift on X if and only if

(3.5) ∃µ ≥ 1, C > 0, ∀m ≥ 1 : lim inf N →∞ N n=1 a 1,n-1 C N N n=1 a µ,n a m,N = 0.
Proof. The condition is even necessary without assumption (3.4) on A. Indeed, if X supports a hypercyclic weighted shift then by the previous proposition there is a strictly positive sequence (ν n ) n as well as m 1 , µ := m 2 ≥ 1 and C > 0 such that

ν n ≤ C a µ,n a m 1 ,n-1 ≤ C a µ,n a 1,n-1
, n ≥ 1, and lim inf

N →∞ 1 N n=1 ν n a m,N = 0, m ≥ 1.
This implies (3.5).

On the other hand, suppose that (3.5) holds with some µ ≥ 1 and C > 0. Then one obtains by (3.4) a strictly increasing sequence (m k ) k of positive integers and a positive sequence (C k ) k such that, for all k ≥ 1 and n ≥ 0,

a µ,n+1 a 1,n ≤ C k a m k+1 ,n+1 a m k ,n .
In particular, setting C k = CC k , we deduce that for any n ≥ 1,

ν n := inf k≥1 C k a m k+1 ,n a m k ,n-1 ≥ C a µ,n a 1,n-1 > 0
and that for every m ≥ 1, lim inf

N →∞ 1 N n=1 ν n a m,N ≤ lim inf N →∞ 1 C N N n=1 aµ,n a 1,n-1 a m,N = 0.
A Köthe sequence space without hypercyclic weighted shifts can arise quite naturally.

Example 3.8. Let X be the sequence space given by

X = x = (x n ) n≥0 : n≥0 |x n |ρ 2 n < ∞ for all ρ > 0 .
This can be considered as the space of entire functions with certain lacunary power series. Then X is the Köthe sequence space λ 1 (A) with a m,n = m 2 n , m ≥ 1, n ≥ 0. It is easily verified that (3.4) holds but (3.5) does not. Thus, X supports no hypercyclic weighted shift. This example will be generalized in Example 4.3(a).

3.1.3. Existence of chaotic weighted shifts. This case can be treated exactly as the existence of hypercyclic shifts. Note, however, that the characterizing condition of chaos,

1 N n=1 wn N ≥0
∈ X (see condition (1.1)) depends on the order of the Köthe sequence space.

Proposition 3.9. Let X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) be a Köthe sequence space.

Then there exists a chaotic weighted shift on X if and only if there exist a strictly increasing sequence (m k ) k of positive integers and a positive sequence

(C k ) k such that ∀n ≥ 1, ν n := inf k≥1 C k a m k+1 ,n a m k ,n-1 > 0 and 1 N n=1 ν 1/p n N ≥0 ∈ X,
where p = 1 if X = c 0 (A).

Corollary 3.10. Let X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) be a Köthe sequence space such that (3.4) holds. Then there exists a chaotic weighted shift on X if and only if

∃µ ≥ 1, C > 0 : N n=1 a 1/p 1,n-1 C N N n=1 a 1/p µ,n N ≥0 ∈ X,
where p = 1 if X = c 0 (A).

3.2.

When frequent hypercyclicity implies chaos on Köthe sequence spaces. Let us spell out Theorem 2.9 in the case of Köthe sequence spaces.

Theorem 3.11. Let X be a Köthe sequence space λ p (A), or a Köthe sequence space c 0 (A) that satisfies (BC). Suppose that the following condition holds:

(TK) for any strictly positive sequence

v = (v n ) n , if ∀m ≥ 1, ∃µ ≥ 1, C > 0, ∀n ≥ 1 : v n-1 a m,n-1 ≤ Cv n a µ,n , then (3.6) ∀m ≥ 1, ∃µ ≥ 1, ∀j ≥ 1, ∃ C j > 0, ∀n ≥ j : v n-j a m,n-j ≤ C j v n a µ,n .
Let B w be a weighted shift on X. Then the following assertions are equivalent: (i) B w is U-frequently hypercyclic on X;

(ii) B w is frequently hypercyclic on X;

(iii) B w is chaotic on X;

(iv) the series n≥0

1 w 1 ••
•wn e n is convergent in X. Indeed, since B w being an operator on X can be expressed by condition (3.3), with a similar condition for B j w , it is straightforward to see that condition (T) for the space X is equivalent to condition (TK) for the Köthe matrix A.

In the case of Köthe sequence spaces λ p (A) we have already examples to which the theorem is applicable: these are given by the classical p -spaces since they are Banach spaces and thus satisfy (T) trivially. But there also are non-Banach examples and even examples of order 0. Proposition 3.12. Every frequently hypercyclic weighted shift on H(D) is chaotic.

Proof. The space H(D) of holomorphic functions on the unit disk D can be identified with the Köthe sequence spaces c 0 (A) or λ p (A) for any 1 ≤ p < ∞ by letting a m,n = 1 (1+ 1 m )

n . These spaces coincide since A satisfies (N), and it suffices to show that A satisfies (TK). Let (v n ) n be a strictly positive sequence such that

∀m ≥ 1, ∃µ ≥ 1, C > 0, ∀n ≥ 1 : v n-1 a m,n-1 ≤ Cv n a µ,n .
We can first show that lim sup n v n-1 vn

1 n ≤ 1. Indeed, given m ≥ 1, µ ≥ 1, C > 0 such that, for every n ≥ 1, v n-1 a m,n-1 ≤ Cv n a µ,n , we get v n-1 v n ≤ C a µ,n a m,n-1 = C 1 + 1 m 1 + 1 m 1 + 1 µ n and thus lim sup n v n-1 vn 1 n ≤ 1+ 1 m 1+ 1 µ ≤ 1 + 1 m .
Since this inequality holds for every m ≥ 1, we deduce that lim sup n v n-1 vn

1 n ≤ 1.
We now show that if lim sup n v n-1 vn

1 n ≤ 1 then (3.6) holds. Let m ≥ 1, µ = m + 1, j ≥ 1.
We need to find a constant C j > 0 such that, for every n ≥ j,

v n-j v n ≤ C j (1 + 1 m ) j 1 + 1 m 1 + 1 m+1 n .
We remark that such a constant will exist if lim sup n v n-j vn 1 n ≤ 1, and since lim sup

n v n-j v n 1 n ≤ j-1 l=0 lim sup n v n-l-1 v n-l 1 n ≤ j-1 l=0 lim sup n max 1, v n-l-1 v n-l 1 n-l ≤ 1,
we get the desired result.

Note that the above result is not vacuous since H(D) supports chaotic weighted shifts such as the differentiation operator [START_REF] Shapiro | Simple connectivity and linear chaos[END_REF]. Examples of Köthe matrices A to which Theorem 3.11 applies and where all the spaces c 0 (A) and λ p (A), 1 ≤ p < ∞, are different can also be found (see Example 4.15(a)).

3.3.

When frequent hypercyclicity does not imply chaos on Köthe sequence spaces. We now show that some Köthe sequence spaces, even of order p ∈ [1, ∞), support frequently hypercyclic, non-chaotic weighted shifts.

For this we recall the following result; see [START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF]Theorem 6.2]. If X is a Fréchet sequence space in which (e n ) n is an unconditional basis, then a weighted shift B w on X is frequently hypercyclic if and only if there exist a sequence (ε r ) r≥1 of positive numbers with ε r → 0 as r → ∞ and a sequence (A r ) r≥1 of pairwise disjoint sets of positive lower density such that (i) for any r ≥ 1, n∈Ar v n+r e n+r converges in X;

(ii) for any r, s ≥ 1, any m ∈ A s , and any j = 0, . . . , r,

n∈Ar n>m v n-m+j e n-m+j s ≤ min(ε r , ε s ),
where v is the sequence associated to w via (2.5).

Recall also that, by (1.1), B w is chaotic if and only if

(3.7) n≥0
v n e n converges in X.

For the existence of a non-chaotic frequently hypercyclic weighted shift we will modify the construction of such a shift in c 0 as given in [START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF]Theorem 7.3]. The following was shown in the proof there.

Lemma 3.13 ([11]

). There exists a strictly increasing sequence (N k ) k≥0 of non-negative integers with N 0 = 0 and pairwise disjoint sets A r , r ≥ 1, of positive lower density with the following properties:

(a) For r ≥ 1, if n ∈ A r , then for any k ≥ 0,

N k ≤ n < N k+1 =⇒ N k + k ≤ n < N k+1 -r. (b) For r, s ≥ 1, if n ∈ A r , m ∈ A s , n > m, then for any k ≥ 0, N k ≤ n -m < N k+1 =⇒ N k + max(r, s) ≤ n -m < N k+1 -max(r, s).
A frequently hypercyclic, non-chaotic weighted shift on c 0 was then given by the weight w that is associated, via (2.6), to the following sequence v = (v n ) n :

v n = 1 2 n-N k for N k ≤ n < N k+1 , k ≥ 0.
For general spaces it seems natural to consider

v n = 1 b n-N k ,n for N k ≤ n < N k+1 , k ≥ 0,
for suitable numbers b m,n . This leads us to the following condition for Köthe sequence spaces X; note that part (γ) depends on which space λ p (A) or c 0 (A) one considers.

(B) There is a strictly positive upper triangular matrix

B = (b m,n ) m≥0,n≥m
with increasing columns such that the following properties hold:

(α) ∀m ≥ 1, ∃µ ≥ 1, C > 0, ∀j ≥ 0, n ≥ j : a m,n a µ,n+1 ≤ C b j,n b j+1,n+1 ; (β) ∃m ≥ 1 : inf n≥0 a m,n b 0,n > 0;
and either (γ), where

(γ) ∃(j m ) m≥1 :      ∀m ≥ 1 : n≥jm a m,n b jm,n < ∞, if X = λ p (A), lim m→∞ lim sup n→∞ a m,n b jm,n = 0, if X = c 0 (A); or ( γ), where ( γ)
               ∀m ≥ 1, ∃N ≥ 0, ρ < 1, ∀j ≥ 0, n ≥ max(N, j) : a m,n+1 a m,n ≤ ρ b j+1,n+1 b j,n ; ∃(n k ) k≥1 , (d k ) k≥1 , ∀m ≥ 1 : lim k→∞ sup n≥max(d k ,n k ) a m,n b d k ,n = 0; ∃(j m ) m≥1 , ∀m ≥ 1 : lim n→∞ a m,n b jm,n = 0.
Here, (j m ) m , (n k ) k and (d k ) k are strictly increasing sequences of positive integers. Also note that the n-th column of the matrix B has (only) n + 1 entries, and these are supposed to be increasing.

We have introduced the alternative and rather technical condition ( γ) in order to deal with some interesting spaces that do not satisfy (γ). One example is given by the power series space of order p ≥ 1 and infinite type with α = (log(log(n + 3)) n , see Theorem 4.6. Note, however, that the third condition in ( γ) implies (γ) when X = c 0 (A) so that, for these spaces, condition ( γ) is of no interest. Theorem 3.14. Let X = λ p (A), 1 ≤ p < ∞, or X = c 0 (A) be a Köthe sequence space. If condition (B) holds then X supports a chaotic weighted shift, and there exists a frequently hypercyclic weighted shift that is not chaotic.

Proof. We start by showing that under condition (B) the hypotheses of Proposition 3.9 are satisfied. By (α), there exist a strictly increasing sequence (m k ) k≥1 of positive integers and a positive sequence (C k ) k≥1 such that, for every k ≥ 1 and n ≥ j ≥ 0,

a m k ,n a m k+1 ,n+1 ≤ C k b j,n b j+1,n+1
.

Therefore, we have for every n ≥ 1,

ν n := inf k≥1 C k a m k+1 ,n a m k ,n-1 ≥ max j≤n-1 b j+1,n b j,n-1 > 0.
It remains to show that if (γ) or ( γ) holds then

1 N n=1 ν 1/p n
N ≥0 ∈ X, where p = 1 for the space X = c 0 (A). First, let (j m ) m be a strictly increasing sequence of positive integers. We then deduce that, for every m ≥ 1 and every N ≥ j m , we have

1 N n=1 ν n a m,N ≤ 1 jm-1 n=1 bn,n b n-1,n-1 1 N n=jm b jm,n b jm-1,n-1 a m,N = 1 b jm-1,jm-1 b 0,0 b jm-1,jm-1 N -1 n=jm b jm-1,n N -1 n=jm b jm,n a m,N b jm,N ≤ b 0,0 a m,N b jm,N (3.8)
since B has increasing columns. If (γ) holds, then (3.8) implies that

1 N n=1 ν 1/p n N ≥0
∈ X; note that a sequence x = (x n ) n belongs to c 0 (A) if and only if there is an increasing sequence (k m ) m of positive integers such that sup k≥km |x k |a m,k → 0 as m → ∞.

On the other hand, if ( γ) holds then for every m ≥ 1 there exist J m ≥ 0 and ρ m < 1 such that, for every n ≥ J m , a m,n+1 am,n ≤ ρ m b 1,n+1 b 0,n . We then deduce that, for every N > J m ,

1 N n=1 ν n a m,N ≤ N n=1 b 0,n-1 b 1,n a m,N ≤ Jm n=1 b 0,n-1 b 1,n N n=Jm+1 ρ m a m,n-1 a m,n a m,N = Jm n=1 b 0,n-1 b 1,n a m,Jm ρ N -Jm m .
Since ρ m < 1, it follows again that

1 N n=1 ν 1/p n N ≥0 ∈ X.
We can thus deduce in each case from Proposition 3.9 that there exists a chaotic weighted shift on X.

We now show that under condition (B), the space X also supports a weighted shift which is frequently hypercyclic but not chaotic. We first consider a Köthe sequence space λ p (A).

Let B = (b m,n ) m,n be a strictly positive upper triangular matrix that satisfies condition (B), where we begin by the variant (α), (β) and (γ). Let (N k ) k≥0 be a strictly increasing sequence of non-negative integers with N 0 = 0 and A r , r ≥ 1, pairwise disjoint sets of positive lower density such that (a) and (b) of Lemma 3.13 hold.

We then define a weighted shift B w with weight w associated, via (2.6), to the sequence v given by (3.9)

v n = 1 b 1/p n-N k ,n , N k ≤ n < N k+1 , k ≥ 0;
note the power 1/p that is required by the form of the seminorms in λ p (A). Then we have for

N k ≤ n < N k+1 -1, k ≥ 0, v n+1 v n p = b n-N k ,n b n-N k +1,n+1
,

while for n = N k+1 -1, k ≥ 0, we have that v n+1 v n p = b n-N k ,n b 0,n+1 ≥ b n-N k ,n b n-N k +1,n+1
,

where we have used the fact that the columns of the matrix B are increasing. Thus, in view of (3.3), condition (α) implies that B w is an operator on λ p (A).

Next, let m ≥ 1 be such that (β) holds. Then

inf k≥0 v p N k a m,N k = inf k≥0 a m,N k b 0,N k > 0,
which, by (3.7), implies that B w is not chaotic on λ p (A).

It remains to show that B w is frequently hypercyclic on λ p (A). It follows from (γ) that for any m ≥ 1 there are j m ≥ 1 and n m ≥ j m such that

(γ ) n≥nm a m,n b jm,n < 1 2 m ,
where we may assume that (j m ) m and (n m ) m are strictly increasing. We will then consider the sets

A r = A nr , r ≥ 1.
It suffices to show that conditions (i) and (ii) in the characterization of frequent hypercyclicity stated above hold for the sequence (A r ) r≥1 of pairwise disjoint sets of positive lower density. Thus, let r ≥ 1 and m ≥ 1. Using condition (a) of Lemma 3.13 we see that

n∈A r v p n+r a m,n+r = k≥0 n∈A r N k ≤n<N k+1 a m,n+r b n+r-N k ,n+r .
Also by (a) we have that if n ∈ A r , N k ≤ n < N k+1 then n ≥ N k + k, so that for all but finitely many k we have that n + r -N k ≥ j m , hence b n+r-N k ,n+r ≥ b jm,n+r , which together with (γ ) implies (i). Now, let r, s ≥ 1, m ∈ A s , and j = 0, . . . , r. Using condition (b) of Lemma 3.13 we see that

n∈A r n>m v p n-m+j a s,n-m+j = k≥0 n∈A r ,n>m N k ≤n-m<N k+1 a s,n-m+j b n-m+j-N k ,n-m+j ≤ k≥0 n∈A r ,n>m N k ≤n-m<N k+1 a max(r,s),n-m+j b j max(r,s) ,n-m+j ,
where we have used the fact that if

n ∈ A r = A nr , m ∈ A s = A ns , n > m, N k ≤ n-m < N k+1
for k ≥ 0, and j = 0, . . . , r then, by (b), n-m+j-N k ≥ max(n r , n s ) ≥ max(j r , j s ) = j max(r,s) . Now, for these n, m, j we have that n-m+j ≥ max(n r , n s ) = n max(r,s) . Hence, with condition (γ ), we have that

n∈A r n>m v p n-m+j a s,n-m+j ≤ ν≥n max(r,s) a max(r,s),ν b j max(r,s) ,ν < 1 2 max(r,s) = min( 1 2 r , 1 2 s ).
This shows (ii), which completes the proof for λ p (A) under condition (B) with (α), (β) and (γ).

The proof for c 0 (A) is very similar if one takes p = 1 in (3.9) and notes again that x = (x n ) n ∈ c 0 (A) if and only if there is an increasing sequence (k m ) m of positive integers such that sup k≥km |x k |a m,k → 0 as m → ∞.

Next suppose that (B) holds with (α), (β) and ( γ), where it suffices to consider the case when X = λ p (A); see the remark before the statement of the theorem. Choose again (N k ) k≥0 and (A r ) r≥1 according to Lemma 3.13. Throughout the remainder of the proof, for m ≥ 1, let ν m = N and ρ m = ρ be given by the first condition in ( γ). We may assume that (ν m ) m is strictly increasing. Also, let (d k ) k and (j m ) m be the sequences of integers given by the second and third conditions in ( γ). Then we can find inductively strictly increasing sequences (l k ) k≥1 , (l k ) k≥1 of strictly positive integers such that, if k ≥ 1 and 1 ≤ m ≤ k,

(3.10) a m,N l k +d l k b d l k ,N l k +d l k ≤ 1 2 k and (3.11) a m,N l k +jm b jm,N l k +jm ≤ (1 -ρ m ) 1 2 k ,
and we set l 0 = l 0 = d 0 = 0. We may also assume that

l k ≥ d l k , k ≥ 1.
We then set

N k = N l k , d k = d l k , k ≥ 0.
It follows from (a) and (b) of Lemma 3.13 that we have (a') For r ≥ 1, if n ∈ A r , then for any k ≥ 0,

N k ≤ n < N k+1 =⇒ N k + d k ≤ n < N k+1 -r. (b') For r, s ≥ 1, if n ∈ A r , m ∈ A s , n > m, then for any k ≥ 0, N k ≤ n -m < N k+1 =⇒ N k + max(r, s) ≤ n -m < N k+1 -max(r, s).
We now define a weighted shift B w via a sequence v exactly as in (3.9), however with (N k ) k replaced by (N k ) k . Conditions (α) and (β) imply again that B w is an operator on X that is not chaotic. It remains to show that conditions (i) and (ii) in the characterization of frequent hypercyclicity hold.

Let r ≥ 1, m ≥ 1. It follows with (a') that, for k ≥ 0, (3.12)

n∈Ar N k ≤n<N k+1 v p n+r a m,n+r ≤ n∈Ar n≥N k +d k a m,n+r b n+r-N k ,n+r ≤ n≥N k +d k a m,n b n-N k ,n
.

By the first condition in ( γ) we have for all

n ≥ N k a m,n+1 b n+1-N k ,n+1 ≤ ρ m a m,n b n-N k ,n provided that k is so big that N k ≥ ν m . If, in addition, k ≥ m,
n≥N k +d k a m,n b n-N k ,n ≤ 1 1 -ρ m a m,N k +d k b d k ,N k +d k ≤ 1 1 -ρ m 1 2 k .
Altogether it follows from (3.12) and (3.13) that, for any r ≥ 1 and m ≥ 1,

n∈Ar v p n+r a m,n+r = k≥0 n∈Ar N k ≤n<N k+1 v p n+r a m,n+r < ∞.

This shows condition (i).

Finally, for any m ≥ 1, let δ m be an integer such that

(3.14) δ m > N k+1 -N k , 0 ≤ k < m.
We may assume that (δ m ) m is strictly increasing. We will then consider the sets A r = A max(νr,jr,δr) , r ≥ 1.

It suffices now to show that condition (ii) above holds for the sequence (A r ) r≥1 .

To this end, let r, s ≥ 1. By (b') and the definition of the A r we have that if

n ∈ A r , m ∈ A s and N k ≤ n -m < N k+1 (k ≥ 0) then (3.15) n -m ≥ max(ν r , ν s ) = ν max(r,s) and (3.16) n -m ≥ N k + max(j r , j s ) = N k + j max(r,s) ; moreover, we can remark that k ≥ max(r, s) since n -m ≥ N k + max(δ r , δ s ) = N k + δ max(r,s)
and since it follows from (3.14) that N k + δ max(r,s) > N k+1 for every k < max(r, s). Now let m ∈ A s and j = 0, . . . , r. Then we have for k ≥ 0, using (b') once more and also (3.16), that

n∈A r N k ≤n-m<N k+1 v p n-m+j a s,n-m+j ≤ n∈A r N k ≤n-m<N k+1 a s,n-m+j b n-m+j-N k ,n-m+j ≤ n-m≥N k +j max(r,s) a max(r,s),n-m+j b n-m+j-N k ,n-m+j , ≤ n-m≥N k +j max(r,s) a max(r,s),n-m b n-m-N k ,n-m . (3.17)
Moreover, by (3.15) and the first condition in ( γ), we have

a max(r,s),n+1-m b n+1-m-N k ,n+1-m ≤ ρ max(r,s) a max(r,s),n-m b n-m-N k ,n-m
whenever n -m ≥ N k . Thus, with (3.17) and (3.11), we obtain after summing a geometric series that

n∈A r N k ≤n-m<N k+1 v p n-m+j a s,n-m+j ≤ 1 1 -ρ max(r,s) a max(r,s),N k +j max(r,s) b j max(r,s) ,N k +j max(r,s) ≤ 1 2 k whenever k ≥ max(r, s).
Altogether we have that

n∈A r n>m v p n-m+j a s,n-m+j = k≥max(r,s) n∈A r N k ≤n-m<N k+1 v p n-m+j a s,n-m+j ≤ 2 2 max(r,s) = min 2 2 r , 2 2 s .
This shows condition (ii), which completes the proof for λ p (A) also under condition (B) with (α), (β) and ( γ).

Let us mention that Theorem 3.14 can be applied to the space c 0 (take A = (1) m≥1,n≥0 and B = (m + 1) m≥0,n≥m ) to recover the result of Bayart and Grivaux [START_REF] Bayart | Invariant Gaussian measures for operators on Banach spaces and linear dynamics[END_REF]. An example of a non-Banach space to which it is applicable is the space H(C) of entire functions, which can be identified with the Köthe sequence spaces c 0 (A) or λ p (A) for any 1 ≤ p < ∞ by letting a m,n = m n . Proposition 3.15. There exists a frequently hypercyclic weighted shift on H(C) that is not chaotic.

Proof. Let A be the Köthe matrix given by a m,n = m n . We consider the upper triangular matrix B = (b m,n ) m≥0,n≥m given by b m,n = 2 mn and we show that condition (B) holds. We first remark that B has increasing columns. Condition (α) is satisfied by considering µ = 4m and C = 1 since for every n ≥ j ≥ 0,

a m,n a µ,n+1 = 1 4 n+1 m ≤ 1 2 n+j+1 = b j,n b j+1,n+1
.

Condition (β) is trivially satisfied for any m since b 0,n = 1 for every n ≥ 0; finally, condition (γ) is satisfied for j = m since, for every n ≥ j, am,n b j,n = m 2 m n . It follows from Theorem 3.14 that there exists a frequently hypercyclic weighted shift on H(C) that is not chaotic.

In view of Propositions 3.12 and 3.15, we see that among Köthe sequence spaces satisfying (N), there are examples for which frequent hypercyclicity implies chaos and others for which this is not the case. Actually, this is also the case among Köthe sequence spaces not satisfying (N) (see Example 4.15).

Thus we have now a sufficient and a necessary condition for frequent hypercyclicity to imply chaos concerning weighted shifts on Köthe sequence spaces. We are still quite far from a characterization. For instance, one may ask the following. Question 3.16. (a) Is there a Köthe matrix A so that, on c 0 (A), every frequently hypercyclic weighted shift is chaotic (and that there are such shifts) while λ p (A) admits a frequently hypercyclic, non-chaotic weighted shift? This would inverse the known behaviour for c 0 and p . (b) Is there a Köthe matrix A so that, for some p ≥ 1, every frequently hypercyclic weighted shift is chaotic on λ p (A) (and that there are such shifts) while, for some q = p, λ q (A) admits a frequently hypercyclic, non-chaotic weighted shift? Note that, so far, all our conditions are blind to the order p once it is not zero.

In summary, there exist a Köthe sequence space that does not support any chaotic or frequently hypercyclic weighted shift (Example 3.8), one where every frequently hypercyclic weighted shift is chaotic and where there are such shifts ( p or H(D) (see [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF] and Proposition 3.12)), and one that supports a chaotic weighted shift and a frequently hypercyclic, non-chaotic shift (c 0 or H(C) (see [START_REF] Bayart | Invariant Gaussian measures for operators on Banach spaces and linear dynamics[END_REF] and Proposition 3.15)). The only remaining case is the following. Question 3.17. Is there a Köthe sequence space supporting frequently hypercyclic weighted shifts but no chaotic weighted shift?

Chaotic and frequently hypercyclic weighted shifts on power series spaces

In this section, we will apply the results of the previous section to so-called power series spaces which generalize the spaces H(D) and H(C), see [START_REF] Meise | Introduction to functional analysis[END_REF]Chapter 29]. Let α = (α n ) n≥0 be an increasing sequence of strictly positive numbers that tends to infinity, and r ∈ R ∪ {∞}.

Let 1 ≤ p < ∞. Then the power series space of order p and type r is defined as

Λ p,r (α) = x = (x n ) n≥0 : for all t < r, n≥0 |x n | p e tαn < ∞ ,
while the power series space of order 0 and type r is given by

C 0,r (α) = x = (x n ) n≥0 : for all t < r, lim n→∞ |x n |e tαn = 0 ,
which are topologized by the obvious (semi-)norms. If r = ∞ then the space is said to be of infinite type, otherwise of finite type. Power series spaces are particular Köthe sequence spaces.

Many interesting spaces are or can be considered as power series spaces, for example the space s of rapidly decreasing sequences (any p, r = ∞, α = (log(n + 2)) n ), the space H(C) of entire functions (any p, r = ∞, α = (n + 1) n ) and the space H(D R ) of holomorphic functions on the disk of radius R > 0 (any p, r = log R, α = (n + 1) n ). A recent addition is the space ces(p+), 1 ≤ p < ∞, which was studied in [START_REF] Albanese | The Fréchet spaces ces(p+), 1 < p < ∞[END_REF] and shown to coincide with Λ 1, 1 p -1 (α) for α = (log(n + 2)) n .

In the literature, authors are often content with considering one particular order, like p = 2 (see [16, Chapter 29]) or p = 1; but see, for example, [START_REF] Dubinsky | Inclusion theorems for absolutely λ-summing maps[END_REF] for the full family.

a m,n = m αn , m ≥ 1, n ≥ 0.
The following is easily verified, see Remark 3.1, Propositions 3.2 and 3.3, and the proof of [START_REF] Meise | Introduction to functional analysis[END_REF]Proposition 29.6].

Proposition 4.1. Let X = Λ p,∞ (α), 1 ≤ p < ∞, or X = C 0,∞ (
α) be a power series space of infinite type. Then it is a non-Banach Fréchet space for which the basis (e n ) n is boundedly complete. In addition, its Köthe matrix satisfies condition (N) if and only if

sup n≥1 log(n) α n < ∞.
Moreover we know from Subsection 3.1 that power series spaces always support weighted shifts. As for the existence of (frequently) hypercyclic or chaotic weighted shifts we have the following.

Proposition 4.2. Let X = Λ p,∞ (α), 1 ≤ p < ∞, or X = C 0,∞ (α) be a power series space of infinite type.

(a) Then X supports a hypercyclic weighted shift if and only if lim sup

N →∞ N -1 n=0 α n α N = ∞.
(b) Moreover, X supports a chaotic weighted shift if and only if

lim N →∞ N -1 n=0 α n α N = ∞.
Proof. It is readily verified that the Köthe matrix (m αn ) m,n satisfies condition (3.4).

(a) Suppose that lim sup N →∞

N -1 n=0 αn α N = ∞. Choose µ ≥ 2 and C ≥ 1.
Writing any given m ≥ 1 as m = µ ρ with some ρ ≥ 0 we see that lim inf

N →∞ m α N C N µ N n=1 αn = lim inf N →∞ 1 C N µ α N 1 α N N -1 n=1 αn+1-ρ = 0.
Hence, by Corollary 3.7, there exists a hypercyclic weighted shift on X.

On the other hand, if lim sup N →∞ Then we have that, for all N ≥ 0,

m α N C N µ N n=1 αn = m C N α N µ 1 α N N n=1 αn α N ≥ 1.
Hence, by Corollary 3.7, X admits no hypercyclic weighted shift. (b) The proof, based this time on Corollary 3.10, is very similar and therefore omitted.

In the sequel we will be led to study properties of power series spaces based on the behaviour of the quotients α n+1 α n as n → ∞; note that these quotients are all at least 1. Thus the following is of interest. Then the corresponding power series spaces of infinite type do not support any hypercyclic weighted shifts. Indeed, there is then some n 0 ≥ 0 and ρ > 1 such that α n+k αn ≥ ρ k for all n ≥ n 0 and k ≥ 0. Hence, for any N ≥ n 0 , N -1

n=n 0 α n α N ≤ N -n 0 k=1 1 ρ k ≤ 1 ρ -1 ,
so that the assertion follows from the previous proposition.

(b) On the other hand, if

lim n→∞ α n+1 α n = 1
then any corresponding power series spaces of infinite type supports a chaotic weighted shift.

In fact, in that case, for every ε > 0, there exists some n 0 ≥ 0 such that αn α n+k ≥ 1 (1+ε) k for every n ≥ n 0 and k ≥ 0, and thus we have that lim inf

N →∞ N -1 n=n 0 α n α N ≥ lim inf N →∞ N -n 0 k=1 1 (1 + ε) k = 1 ε .
Since this inequality holds for every ε > 0, the assertion follows again from the previous proposition.

We return to the question whether every frequently hypercyclic weighted shift is chaotic. In the context of power series spaces of infinite type, condition (TK) turns out to be too strong and does not allow the existence of hypercyclic weighted shift, as shown by the following result.

Theorem 4.4. Let X = Λ p,∞ (α), 1 ≤ p < ∞, or X = C 0,∞ (
α) be a power series space of infinite type. If condition (TK) holds then X does not support hypercyclic weighted shifts.

Proof. Let v n = 1 2 α 1 +...+αn for n ≥ 0. Then, for any m ≥ 1, when choosing µ = 2m, we have for any n ≥ 1 that

v n-1 a m,n-1 = 1 2 α 1 +...+α n-1 m α n-1 = v n 2 αn m α n-1 ≤ v n (2m) αn = v n a µ,n .
Thus, by (TK), taking m = 1, there is some µ ≥ 1 such that for any j ≥ 1 there is some constant C j > 0 such that, for any n ≥ j, 1 2 α 1 +...+α n-j ≤ C j 1 2 α 1 +...+αn µ αn , hence 2 α n-j+1 +...+αn ≤ C j µ αn , and thus jα n-j ≤ α n-j + . . . + α n-1 ≤ α n-j+1 + . . . + α n ≤ log 2 C j + α n log 2 µ.

We then have that, for any j ≥ 1, lim sup

n→∞ α n-j α n ≤ lim sup n→∞ log 2 C j jα n + log 2 µ j = log 2 µ j .
We choose j ≥ 1 such that log 2 µ j < 1 2 . Then there is some n 0 ≥ j such that, for every n ≥ n 0 ,

α n-j α n ≤ 1 2 .
It follows that lim sup

N →∞ N -1 n=0 α n α N ≤ lim sup N →∞ 1 α N N -n 0 j ν=0 N -νj n=N -(ν+1)j+1 α n ≤ lim sup N →∞ N -n 0 j ν=0 j α N -νj α N ≤ lim sup N →∞ j N -n 0 j ν=0 1 2 ν < ∞.
We then deduce from Proposition 4.2 that X does not support hypercyclic weighted shifts.

This result is surely disappointing. It shows that our sufficient condition for frequent hypercyclicity to imply chaos obtained in Subsection 2.2 is not good enough in the present context; recall that, for Köthe sequence spaces, (TK) is equivalent to (T). Question 4.5. Does there exist a power series space of infinite type that supports chaotic weighted shifts and for which every frequently hypercyclic weighted shift is chaotic?

In the opposite direction we have the following result.

Theorem 4.6. Let X = Λ p,∞ (α), 1 ≤ p < ∞, or X = C 0,∞ (α) be a power series space of infinite type. Suppose that either      ∃ε ∈ (0, 1), ∀C > 0, ∃j ≥ 1 :

n≥j ε α n-j +...+α n-1 -Cαn < ∞, if X = Λ p,∞ (α), ∀C > 0, ∃j ≥ 1 : lim inf n→∞ (α n-j + . . . + α n-1 -Cα n ) > -∞, if X = C 0,∞ (α); or lim n→∞ α n+1 α n = 1.
Then X supports a chaotic weighted shift, and there exists a frequently hypercyclic weighted shift on X that is not chaotic.

Proof. We will apply Theorem 3. b j,n = (m/ε) αn µ α n+1 ≤ 1, so that part (α) of condition (B) holds. Part (β) is trivially satisfied. Now suppose that the first hypothesis holds, and let X = Λ p,∞ (α). Then choose as ε a corresponding value. Let m ≥ 1, and let C > 0 be such that m = ε -C . Then a m,n b j,n = ε α n-j +...+α n-1 -Cαn , n ≥ j ≥ 0.

Hence the hypothesis implies that also part (γ) of condition (B) holds in this case. For X = C 0,∞ (α), one first needs to note that the hypothesis also holds for 2C, so that we have, for the j ≥ 1 given by the hypothesis,

α n-j + . . . + α n-1 -Cα n = (α n-j + . . . + α n-1 -2Cα n ) + Cα n → ∞,
which implies part (γ) also in this case by the same argument as above.

Finally suppose that the second hypothesis is satisfied, where now the space X is arbitrary. For simplicity let us take ε = 1 2 . Let m ≥ 1, and choose η > 0 such that m 1+η < 2m. Then there is some N ≥ 0 such that α n+1 ≤ (1 + η)α n for all n ≥ N . Hence we have for all j ≥ 0 and n ≥ max(N, j) that

a m,n+1 a m,n b j,n b j+1,n+1 = m α n+1 (2m) αn ≤ m 1+η 2m αn ≤ m 1+η 2m α 0 < 1,
which shows the first condition in ( γ).

Next we fix a strictly increasing sequence (d k ) k≥1 of positive integers such that, for all

k ≥ 1, 2 d k ≥ k 2 (1 + k). We then choose η k > 0 such that (1 + η k ) d k ≤ 2 for k ≥ 1.
Finally, there are ν k ≥ 0 such that α n+1 ≤ (1 + η k )α n for all k ≥ 1 and n ≥ ν k , and we can assume that (ν k ) k is strictly increasing. Thus we have for k

≥ 1, n ≥ ν k and 1 ≤ m ≤ k that a m,n+d k b d k ,n+d k = m α n+d k 2 αn+...+α n+d k -1 ≤ m (1+η k ) d k αn 2 d k αn ≤ m 2 2 d k αn ≤ 1 k + 1 αn ≤ 1 k + 1
4.7. Let X = Λ p,∞ (α), 1 ≤ p < ∞, or X = C 0,∞ (
α) be a power series space of infinite type.

(a) If

lim inf n→∞ α n+1 α n > 1 then X supports no hypercyclic weighted shift. (b) If lim n→∞ α n+1 α n = 1
then X supports a chaotic weighted shift, and there exists a frequently hypercyclic weighted shift on X that is not chaotic.

Corollary 4.7(b) applies in particular to the space H(C) of entire functions, see also Proposition 3.15, and the space s.

Proposition 4.8. The space s of rapidly decreasing sequences admits a frequently hypercyclic weighted shift that is not chaotic.

We end our study of power series spaces of infinite type by giving an example of such a space supporting chaotic weighted shift but to which Theorem 4.6 is not applicable. Example 4.9. Consider the sequence space

X = (x n ) n : ∀m ≥ 1, sup 2 k-1 <n≤2 k |x n | m k! → 0 as k → ∞ .
This is a power series space C 0,∞ (α) with α given by

α n = k! if 2 k-1 < n ≤ 2 k , k ≥ 1,
and α 0 = 1. Then we have for 2

k-1 < N ≤ 2 k , k ≥ 2, that N -1 n=0 α n α N ≥ 2 k-1 n=2 k-2 +1 α n k! = 2 k-2 (k -1)! k! → ∞
as N → ∞, so that by Proposition 4.2 the space C 0,∞ (α) admits chaotic and hence frequently hypercyclic weighted shifts.

On the other hand, we have for any j ≥ 1 and any k sufficiently large that

α 2 k +1-j + . . . + α 2 k -α 2 k +1 = jk! -(k + 1)!, so that lim inf n→∞ (α n-j + . . . + α n-1 -α n ) = -∞.
And, of course, lim sup n→∞ α n+1 αn > 1. Theorem 4.6 is thus not applicable, and we do not know if every frequently hypercyclic weighted shift on C 0,∞ (α) is chaotic or not.

4.2.

Power series spaces of finite type. We turn to power series spaces of finite type. We first note that, since e tαn = e rαn e (t-r)αn , one can change the finite type r by a diagonal transform into any other finite type; hence any weighted shift on a power series spaces of finite type is conjugate to a weighted shift on a power series spaces of type 0. In the sequel it will therefore suffice to do the proofs in the case of r = 0. Now, power series spaces of type 0 are special Köthe sequence spaces with Köthe matrix A = (m, n) m,n given by

a m,n = 1 1 + 1 m αn , m ≥ 1, n ≥ 0.
We again have the following, see Remark 3.1, Propositions 3.2 and 3.3, and the proof of [START_REF] Meise | Introduction to functional analysis[END_REF]Proposition 29.6]. Proposition 4.10. Let X = Λ p,r (α), 1 ≤ p < ∞, or X = C 0,r (α) be a power series space of finite type. Then it is a non-Banach Fréchet space for which the basis (e n ) n is boundedly complete. In addition, its Köthe matrix satisfies condition (N) if and only if

lim n→∞ log(n) α n = 0.
Moreover, by Subsection 3.1, these spaces always support weighted shifts. The characterizing conditions of Propositions 3.6 and 3.9 for the existence of hypercyclic or chaotic weighted shifts do not seem to simplify easily. Thus we will be content with the following. Then the corresponding power series spaces of finite type do not support any hypercyclic weighted shifts. We will show this for Λ p,r (α), the argument for C 0,r (α) being similar.

Proof. Since α is increasing, the first limsup in the hypothesis is a supremum. Thus, the first assertion follows from Example 4.11(b). In addition, there is some δ > 0 such that, for all j ≥ 1, we have that (4.1) δα n ≤ α n-j holds for all sufficiently large n.

To complete the proof it suffices, by Theorem 3.11, to show that condition (TK) is satisfied. To this end, let (v n ) n≥0 be a strictly positive sequence so that, for any m ≥ 1, there are µ m ≥ 1 and C m > 0 such that, for all n ≥ 1,

v n-1 a m,n-1 ≤ C m v n a µm,n , hence v n-1 ≤ C m v n (1 + 1 m ) α n-1 (1 + 1 µm ) αn ≤ C m v n (1 + 1 m ) αn (1 + 1 µm ) αn . Let (n m
) m be a strictly increasing sequence of positive integers such that, for all n ≥ n m , C m (1 + 1 µm ) αn ≤ 1. Thus, if we define M n = log(1 + 1 m ) for n m ≤ n < n m+1 , m ≥ 1, with M n = log 2 for 0 ≤ n < n 1 , then (M n ) n is a decreasing positive sequence that converges to 0 and such that, for all n ≥ n 1 , v n-1 ≤ v n e Mnαn . Now let m ≥ 1. Then there is some µ ≥ 1 such that This shows that (TK) holds.

As for the existence of non-chaotic frequently hypercyclic weighted shifts we have the following.

Theorem 4.13. Let X = Λ p,r (α), 1 ≤ p < ∞, or X = C 0,r (α) be a power series space of finite type. Suppose that lim sup Then X supports a chaotic weighted shift, and there exists a frequently hypercyclic weighted shift on X that is not chaotic.

Proof. We will show that the hypothesis implies condition (B) under the alternative (γ) in Theorem 3.14. We first consider the case when X = Λ p,r (α). Let ε ∈ (0, 1) be given by the hypothesis. We Next, choose m so that ε(1 + 1 m ) ≤ 1. Then a m,n b 0,n = 1 ε αn (1 + 1 m ) αn ≥ 1 for all n ≥ 0, so that also part (β) holds.

Finally, let m ≥ 1, and define δ > 0 by ε δ = 1 1+ 1 m . Then we have for n ≥ j ≥ 1, a m,n b j,n = ε -α n-j +δαn , so that by considering the integer j given in the hypothesis for ε and δ, we can deduce that also (γ) holds. This completes the proof for X = Λ p,r (α). If X = C 0,r (α), it is easy to see that the hypothesis implies that ∀δ > 0, ∃j ≥ 1 : lim n→∞ ε -α n-j +δαn = 0, where ε ∈ (0, 1) can be chosen arbitrarily; see also the proof of the following corollary. Then we proceed as in the case of X = Λ p,r (α).

The following is immediate from the previous results.

Corollary 4.14. Let X = Λ p,r (α), 1 ≤ p < ∞, or X = C 0,r (α) be a power series space of finite type. then X supports a chaotic weighted shift, and there exists a frequently hypercyclic weighted shift on X that is not chaotic.

Proof. In view of Example 4.11 and the Theorems 4.12 and 4.13, we need only show that, for X = Λ p,r (α), the hypothesis in (c) implies the one in Theorem 4.13. Thus, let ε ∈ (0, 1) be arbitrary, and let δ > 0, where we may assume that δ ≤ 1 2 . Then the hypothesis in (c) implies that there are some j ≥ 1 and some N j ≥ j such that, for all n ≥ N j , α n-j ≤ δα n . This implies that n≥N j ε -α n-j +2δαn ≤ n≥N j ε δαn .

But since we also have that α n+j αn ≥ 2 for all large n, the latter series converges, which had to be shown. Corollary 4.14(b) applies in particular to the space H(D) of holomorphic functions on the unit disk D, see also Proposition 3.12.

The following table summarizes our findings for power series spaces Λ p,r (α), 1 ≤ p < ∞, and C 0,r (α) of finite or infinite type when α satisfies

ρ := lim n→∞ α n+1 α n ∈ [1, +∞],
see Corollaries 4.7 and 4.14. In this table, '=' stands for the fact that every frequently hypercyclic weighted shift is chaotic (and that there are chaotic and hence frequently hypercyclic weighted shifts), ' =' for the fact that some frequently hypercyclic weighted shifts are not chaotic, and '×' indicates that there are no hypercyclic (and hence no frequently hypercyclic) weighted shifts.

r < ∞ example r = ∞ example ρ = 1 = H(D) = H(C), s 1 < ρ < ∞ = × ρ = ∞ × ×
The specific examples in the table are all nuclear spaces. We end by adding some nonnuclear examples. This will support the claim made at the end of the Introduction. Then the corresponding Köthe sequence spaces are power series spaces of type 0 with α n = log(n + 1), n ≥ 1. Thus, on any such space, there are frequently hypercyclic weighted shifts, and they are all chaotic since lim n α n+1 αn = 1. Since condition (N) is not satisfied, see Proposition 4.10, all the spaces c 0 (A) and λ p (A), 1 ≤ p < ∞, are different by Proposition 3.3. Note that λ 1 (A) is the space ces(1+) studied in [START_REF] Albanese | The Fréchet spaces ces(p+), 1 < p < ∞[END_REF], see our discussion at the beginning of this section.

(b) Let the Köthe matrix A be given by a m,n = (log(n + 1)) m , m ≥ 1, n ≥ 0.

Then the corresponding Köthe sequence spaces are power series spaces of infinite type with α n = log(log(n + 1)), n ≥ 2. On any such space there are frequently hypercyclic weighted shifts that are not chaotic since lim n α n+1 αn = 1. Since condition (N) is not satisfied, see Proposition 4.1, all the spaces c 0 (A) and λ p (A), 1 ≤ p < ∞, are different by Proposition 3.3.

n≥0w

  n+1 x n+1 e n m ≤ n≥0 |w n+1 | e n m e n+1 µm x n+1 e n+1 µm ≤ n≥0 |w n+1 |C m e n m e n+1 µm x νm ≤ m-1 n=0 |w n+1 |C m e n m e n+1 µm + n≥m |w n+1 |a n x νm < ∞, so that B w defines a weighted shift on X.

  then summing a geometric series gives us with (3.10) that(3.13)

4. 1 .

 1 Power series spaces of infinite type. Power series spaces of infinite type are special Köthe sequence spaces with Köthe matrix A = (a m,n ) m,n given by

< 0 N - 1

 01 ∞ then, since α is an increasing sequence, we also have lim sup N →∞ N α N ≤ lim sup N →∞ 1 α n=0 αn α N < ∞. Let µ ≥ 1 and C > 0, and choose m ≥ sup

14 .

 14 First of all, for an arbitrary ε ∈ (0, 1), define the upper triangular matrix B = (b m,n ) by b m,n = 1 ε α n-m +...+α n-1 n ≥ m ≥ 0, where we interpret b 0,n = 1, n ≥ 0. Then B has increasing columns. Let m ≥ 1, and choose µ ≥ m/ε. Then we have for n ≥ j ≥ 0, a m,n a µ,n+1 b j+1,n+1

α 0 .m + 1 αn.

 01 This shows the second condition in ( γ) withn k = ν k + d k , k ≥ 1.Taking j m = d m for m ≥ 1, and choosing m = k above we have that, for all m ≥ 1 andn ≥ n m , a m,n+jm b jm,n+jm ≤1Thus also the third condition in ( γ) holds.Thus we have by Example 4.3 and Theorem 4.6 the following.

Corollary

Example 4 .

 4 11. (a) Suppose that lim n→∞ α n+1 α n = ∞.

1 µ

 1 ) < δ log(1 + 1 m ). Let j ≥ 1. Then we have for all n ≥ n 1 + j thatv n-j a m,n-j a µ,n ≤ v n 1 (1 + 1 m ) α n-j exp(M n-j+1 α n-j+1 + . . . + M n α n )(1 + 1 µ ) αn ≤ v n exp(-log(1 + 1 m )α n-j + jM n-j α n + log(1 + 1 µ )α n ).Since (M n ) n tends to 0, we have by (4.2) that for all n sufficiently largev n-j a m,n-j a µ,n ≤ v n exp(-log(1 + 1 m )α n-j + δ log(1 + 1 m )α n );hence, by (4.1) we have for all large n that v n-j a m,n-j a µ,n ≤ v n .

  ∈ (0, 1), ∀δ > 0, ∃j ≥ 1 :n≥j ε -α n-j +δαn < ∞, if X = Λ p,r (α), lim sup j→∞ lim inf n→∞ α n+j α n = ∞, if X = C 0,r (α).

  then define the upper triangular matrix B = (b m,n ) by bm,n = ε α n-m , n ≥ m ≥ 0.Note that B has increasing columns. Let m ≥ 1, and choose µ ≥ 1 such that(1 + 1 µ ) M ≤ 1 + 1 m , where M = sup n≥0 α n+1αn . Then we have for n ≥ j ≥ 0, a m,n a µthat part (α) of condition (B) holds.

  X supports a chaotic weighted shift, and every frequently hypercyclic weighted shift on X is chaotic. (c) If lim sup n→∞ α n+1 α n < ∞ and lim sup j→∞ lim inf n→∞ α n+j α n = ∞

Example 4 .

 4 15. (a) Let the Köthe matrix A be given bya m,n = 1 (n + 1) 1/m , m ≥ 1, n ≥ 0.

Stéphane Charpentier and Quentin Menet were supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). Karl Grosse-Erdmann was supported by FNRS grant PDR T.0164.16. Quentin Menet is a Research Associate of the Fonds de la Recherche Scientifique -FNRS.

Thus, let B w be a weighted shift on Λ p,r (α). By continuity there then exist µ ≥ 1 and C > 1 such that, for every n ≥ 1, |w n | p a 1,n-1 ≤ Ca µ,n and thus |w n | p ≤ C2 α n-1 (1+ 1 µ ) αn . Therefore we get for n ≥ 1 that

αn → ∞ and therefore n αn → 0 as n → ∞ there is some N ≥ 0 so that, for all

Thus we have for all n ≥ N ,

which contradicts [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Theorem 4.8]. We conclude that B w is not hypercyclic. (b) On the other hand, if

then any corresponding power series space of finite type supports a chaotic weighted shift. Specifically, we will show that the weighted shift B w with w n = 2 for all n ≥ 1 is chaotic on any such space, which we will prove again only for Λ p,r (α). To this end, let m ≥ 1. It then follows from the hypothesis that there are n m ≥ 0 and µ ≥ 1 such that, for every n ≥ n m ,

Thus there is a constant C > 0 such that, for all n ≥ 1, w p n a m,n-1 ≤ Ca µ,n , so that B w is an operator on Λ p,r (α). Moreover, for every m ≥ 1, we have that

Concerning the condition (TK), power series spaces of finite type behave quite differently from their counterparts of infinite type, see Theorem 4.4. Theorem 4.12. Let X = Λ p,r (α), 1 ≤ p < ∞, or X = C 0,r (α) be a power series space of finite type. Suppose that

Then X supports a chaotic weighted shift. Moreover, for any weighted shift B w on X, the following assertions are equivalent: (i) B w is U-frequently hypercyclic on X;

(ii) B w is frequently hypercyclic on X;

(iii) B w is chaotic on X;

(iv) the series n≥0 1 w 1 •••wn e n is convergent in X.


LAMA Evgeny Abakumov
email: evgueni.abakoumov@univ-eiffel.fr

Anton Baranov
email: anton.d.baranov@gmail.com

Éphane St

Andrei Charpentier

Lishanskii

Stéphane Charpentier
email: stephane.charpentier.1@univ-amu.fr

Andrei Lishanskii
email: lishanskiyaa@gmail.com

NEW CLASSES OF HYPERCYCLIC TOEPLITZ OPERATORS

Keywords: Subject Classification. 47A16, 47B35, 30H10 hypercyclic operator, Toeplitz operator, univalent function

We study hypercyclicity of Toeplitz operators in the Hardy space H 2 (D) with symbols of the form R(z) + ϕ(z), where R is a rational function and ϕ ∈ H ∞ (D). We relate this problem to cyclicity of certain families of functions for analytic Toeplitz operators and give new sufficient conditions for hypercyclicity based on deep results of B. Solomyak.

Introduction and main results

Toeplitz operators are among the most important objects in operator theory. They have numerous applications in complex analysis, theory of orthogonal polynomials, mathematical physics, etc. Recall that for a function ψ ∈ L ∞ (T) the Toeplitz operator T ψ : H 2 → H 2 with the symbol ψ is defined as T ψ f = P + (ψf ), where P + stands for the orthogonal projection from L 2 (T) onto H 2 . As usual, D and T denote the unit disc and the unit circle, respectively. We recall that the Hardy space H 2 of the unit disc D consists of functions analytic in D whose Taylor coefficients are square summable, and H ∞ is the space of all bounded analytic functions in D.

In the present paper we study the hypercyclicity property for a class of Toeplitz operators on H 2 . A continuous linear operator T on a separable Banach space X is said to be hypercyclic if there exists x ∈ X such that the set {T n x : n ∈ N 0 } is dense in X (here N 0 = {0, 1, 2, . . . }). Toeplitz operators with antianalytic symbols were among the basic examples of hypercyclic operators. In 1969, S. Rolewicz [START_REF] Rolewicz | On orbits of elements[END_REF] showed that the operator T αz (a multiple of the backward shift) is hypercyclic on H 2 whenever |α| > 1. Later, G. Godefroy and J. Shapiro [START_REF] Godefroy | Operators with dense, invariant cyclic vector manifolds[END_REF] showed that for a function ϕ ∈ H ∞ the antianalytic Toeplitz operator T ϕ is hypercyclic if and only if ϕ(D) ∩ T = ∅. On the other hand, it is obvious that there are no hypercyclic Toeplitz operators with analytic symbols (i.e., among multiplication operators).

However, the problem of describing hypercyclic Toeplitz operators in the general case seems to be largely open. This problem was explicitly stated by S. Shkarin [START_REF] Shkarin | Orbits of coanalytic Toeplitz operators and weak hypercyclicity[END_REF] who described hypercyclic Toeplitz operators with symbols of the form Φ(z) = az+b+ cz (i.e., with tridiagonal matrix). A. Baranov and A. Lishanskii [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF] gave necessary and (separately) sufficient conditions for hypercyclicity of Toeplitz operators with polynomial antianalytic part of the symbol, that is, for symbols Φ of the form Φ(z) = P 1 z + ϕ(z), where P is a polynomial and ϕ ∈ H ∞ . It turned out that the valence of the symbol played the central role in this study.

In the present paper we will consider more general symbols. Let R be a rational function without poles in D, i.e., ( 1)

R(z) = P (z) + r l=1 k l j=1 α l,j (z -η l ) j , where P (z) = N 1 k=0 c k z k is a polynomial of degree N 1 , α l,j ∈ C, and η l ∈ D := C \ D, 1 ≤ l ≤ r, are the (distinct) poles of R of multiplicities k l . We put N 2 = r
l=1 k l and denote by N = N 1 + N 2 the degree of the rational function R. We do not exclude the cases when N 1 = 0 (i.e., P = const) or N 2 = 0 (R = P ). Finally, let ϕ ∈ H ∞ , and put

(2) Φ(z) = R 1 z + ϕ(z).
Thus, Φ is analytic in D except the poles 1/η 1 , . . . , 1/η r . The aim of this note is to describe new classes of hypercyclic Toeplitz operators T Φ with the symbols of the form (2).

As in [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF], our conditions will be formulated in terms of the valence of Φ. Recall that an analytic function h in a domain D is said to be n-valent in D if the equation h(z) = w has at most n solutions in D counting multiplicities for any w ∈ C. It is said to be exactly n-valent in D if this equation has exactly n solutions (counting multiplicities) for any w ∈ h(D). Note that Φ (given by ( 2)) has a pole of order N 1 at zero and poles of multiplicities k l at 1/η l , whence the equation Φ(z) = w has exactly N solutions when |w| is sufficiently large.

Let us first illustrate the connection between the valence of Φ and the hypercyclicity of T Φ through a necessary condition. For the case of symbols with polynomial antianalytic part the statement was proved in [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF].

Proposition 1.1. Let Φ be given by (2). If T Φ is hypercyclic, then the function Φ is N -valent in D.
Let us also note that if T Φ is hypercyclic, then we have

σ(T Φ ) ∩ T = ∅, σ(T Φ ) ∩ D = ∅.
The first condition is a basic property that holds for any hypercyclic operator [START_REF] Grosse-Erdmann | Linear Chaos[END_REF]Theorem 5.6], while the second one follows from the fact that, for any Toeplitz operator, its spectral radius coincides with its norm [START_REF] Nikolski | Operators, Functions, and Systems: an Easy Reading[END_REF]Part B,Ch. 4].

It can be proven that if Φ is N -valent, then σ(T Φ ) = C \ Φ(D, N ), where Φ(D, N ) consists of those w ∈ C for which Φ(z) = w has exactly N solutions in D. We refer to [1, Proposition 2.2] for the details. We do not know whether the hypercyclicity of

T Φ implies that σ(T Φ ) ∩ D = ∅.
The key step of the proof of our sufficient conditions for hypercyclicity is an application of the following principle, already used in [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF]: Hypercyclicity of T Φ can be reduced to standard cyclicity of some analytic Toeplitz (i.e., multiplication) operator. We recall that a finite (or countable) family U of functions in H 2 is said to be cyclic

for T h , h ∈ H ∞ (D), if the family {h k u : u ∈ U, k ≥ 0} is complete in H 2 . More specifically, for λ ∈ C \ Φ(D), let us set (3) h λ (z) = 1 Φ(z) -λ .
Clearly, h λ is analytic and bounded in D.

Proposition 1.2. Let Φ be given by (2). We assume that

(4) D ∩ (C \ Φ(D)) = ∅ and D ∩ (C \ Φ(D)) = ∅,
and that, for any λ ∈ C \ Φ(D), the family {1, z, . . . , z N -1 } is cyclic for T h λ with h λ given by (3). Then T Φ is hypercyclic.

Based on this important observation, we will then formulate several sufficient conditions for the hypercyclicity of T Φ . From now on, we assume that Φ is given by (2) and satisfies [START_REF] Grosse-Erdmann | Linear Chaos[END_REF]. We denote by A(D) the disc algebra, i.e., the space of functions continuous in D and analytic in D.

The following result concerning the case of constant and maximal valence of the symbol is a direct extension of [1, Theorem 1.2].

Theorem 1.3. Assume that ϕ ∈ A(D) and Φ satisfies the following Maximal Valence Condition:

(MVC) the function Φ is exactly N -valent on D \ {0, 1/η 1 , . . . , 1/η r }, i.e., for any w ∈ Φ(D) the equation Φ(z) = w has exactly N solutions in D counting multiplicities.
Then T Φ is hypercyclic.

Theorem 1.3, however, does not apply if Φ has varying valence in D.

A novel feature of the present note is that we make use of deep results of B. Solomyak about cyclic families for analytic Toeplitz operators. Together with Proposition 1.2, this allows us to significantly enlarge the class of symbols inducing hypercyclic Toeplitz operators and, in particular, to include symbols with varying valence. These results are new even for symbols with polynomial antianalytic part studied in [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF].

Cyclicity of families of functions for analytic Toeplitz operator is very far from being understood; deep results in this direction were obtained by B. Solomyak [START_REF] Solomyak | Cyclic families of functions for analytic Toeplitz operators[END_REF] and B. Solomyak and A. Volberg [START_REF] Solomyak | Multiplicity of analytic Toeplitz operators[END_REF] in 1980-s. We use the geometric conditions on the valence of the function Φ given in [START_REF] Solomyak | Cyclic families of functions for analytic Toeplitz operators[END_REF] to distinguish two more classes of hypercyclic Toeplitz operators.

Theorem 1.4. Assume that ϕ ∈ A(D) and, for some λ ∈ C \ Φ(D), the function h λ given by (3) satisfies the Increasing Argument Condition:

(IAC) The set h λ (T) is a finite union of C 2 -smooth Jordan arcs and some continuous branch of the function t → arg h λ (e it ) is (strictly) increasing on [0, 2π].

Then T Φ is hypercyclic.

To state the last (and, in a sense, more general) result, we need to introduce some notations (see [START_REF] Solomyak | Cyclic families of functions for analytic Toeplitz operators[END_REF]Paragraph 1.3]). Let h be a meromorphic function in D (i.e., meromorphic in {|z| < ρ} for some ρ > 1) with poles η 1 , . . . , η s in D. Then the set h(T) splits the plane into finitely many connected components. Let Ω (k) h be the union of the connected components of C \ h(T) in which the number of pre-images in D \ {η 1 , . . . , η s } of h is equal to k (counting multiplicity). The bounded connected components of the set C \ h(D) =: Ω (0) h will be called holes. We say that h is a function of general position if it is analytic in a neighborhood of T, the curve t → h(e it ) has only finitely many points of self-intersection which are simple and transversal, and h | T = 0.

k 1 = k 2 -1 or k 1 = k 2 + 1.
Moreover, if for some λ ∈ Ω 0 h , h λ is given by (3), then h λ is of general position if and only if so Φ is. Theorem 1.5. Assume that Φ is of general position and satisfies the following Decreasing Valence Condition:

(DVC) For some λ 0 ∈ D ∩ Ω (0) Φ and λ 1 ∈ D ∩ Ω (0)
Φ , for any j ∈ {0, 1}, and for any component andG 1 is adjacent to the hole which contains λ j .

G ⊂ Ω (k) Φ , k ≥ 1, there exist connected components G i of Ω (i) Φ , 1 ≤ i ≤ k, such that G = G k , the components G i and G i-1 are adjacent for 1 < i ≤ k,
Then T Φ is hypercyclic.

We shall observe that under the assumption of the previous theorem, condition (DVC) also reads as follows: For some λ 0 ∈ D ∩ Ω (0) Φ and some λ 1 ∈ D ∩ Ω (0) Φ , and for any w ∈ C \ Φ(D), there exist continuous paths γ 0 , γ

1 : [0, 1] → C with γ i (0) = w and γ i (1) = λ i such that γ i ∩ Ω (k)
Φ is connected for any k ≥ 0 and i = 0, 1.

Remark 1.6. Let λ 0 ∈ D ∩ Ω (0) Φ and λ 1 ∈ D ∩ Ω (0)
Φ . Then it is easily seen that Φ satisfies (DVC) for λ 0 and λ 1 if and only if both functions h = 1 Φ-λ 0 and h = 1 Φ-λ 1 are analytic in D, are of general position, and satisfy the following condition:

(DVC') For any component G ⊂ Ω (k) h , k ≥ 1, there exist connected components G i of Ω (i) h , 1 ≤ i ≤ k, such that G = G k , the components G i and G i-1 are adjacent for 1 < i ≤ k, and G 1 is adjacent to the unbounded connected component of Ω (0) h .
Pictures of domains having this property or not can be found in [8, Fig. 1, Page 812] and in Figures 1 and2 below.

In the next section we provide examples of symbols Φ of the form (2) satisfying (MVC), (IAC) or (DVC). The proofs of Propositions 1.1-1.2 and Theorems 1.3-1.5 are postponed to Sections 3 and 4.

Examples

Example 2.1. Let Ψ be a univalent function in A(D) which maps D onto a Jordan domain Ω with 0 ∈ Ω and let B be a finite Blaschke product of degree N . Let γ ∈ D be such that Ψ(γ) = 0. Then

Φ(z) = 1 Ψ • B(z) = r l=1 k l j=1 a l,j (z -λ l ) j + ϕ(z),
where λ 1 , . . . λ l are such that B(λ l ) = γ with multiplicity k l and ϕ ∈ A(D). Thus, Φ is a symbol of the form (2) which satisfies (MVC).

Example 2.2. Another example of a symbol satisfying (MVC) can be obtained as follows. Let Ω be a Jordan domain invariant under rotation by 2π/N , for some N ∈ N. and let Ψ : D → Ω be the conformal mapping with Ψ(0) = 0. Then it is clear that Φ = 1/Ψ N will be exactly N -valent in D.

Example 2.3. It is easy to provide with examples of Φ satisfying (IAC) as small perturbation of z n . Indeed, let ε > 0, λ ∈ C and Ψ ∈ A(D) ∩ C 1 (D). Then one defines

Φ(z) = λ + [z n (1 + εΨ(z))] -1 .
Clearly, Φ is of the form P (1/z) + ϕ(z), and a direct computation gives that the derivative of t → arg h(e it ) is equal to n + O(ε), where h = (Φ -λ) -1 . So Φ satisfies (IAC) for any ε small enough.

Example 2.4. Let us first build a 2-valent function h ∈ H(D), of general position, which satisfies (DVC') and has a zero of order 2 at 0. To do so, let us first (the steps are illustrated in Figure 1) consider the conformal map Ψ which sends D onto the sector of annulus Γ given by

Γ = z ∈ C : r < |z| < R, - π 4 + α < arg z < π -α , with 0 < r < R < (2 √ 2 -1)r, 0 < α < π 8 .
Then step 2 consists in composing Ψ with the map (1 + εz)z 2 -β(1 + i), where ε > 0 is very small and β is such that 0 < c < c < ã < a < b < b < d < d (see Figure 1 for the definitions of a, ã, b, b, c, c, d, d). Note that such β exists because of the choice of r and R. Thus, the resulting map is g(z) = (1 + εΨ(z))Ψ(z) 2 -β(1 + i). There is, however, a small problem that g is not analytic in D. To overcome it consider g(ρz) for some ρ ∈ (0, 1) close enough to 1. Finally, put

h(z) = g 2 (ρz) = (1 + εΨ(ρz))Ψ(ρz) 2 -β(1 + i) 2 .
The resulting domain h(D) is shown in the last part of Figure 1.

It is easily seen that h is 2-valent, of general position, has zero of order 2 at 0 and satisfies (DVC'). Note also that (h -µ) -1 also satisfies (DVC') for any µ / ∈ h(D). If we put Φ = 1/h, then it is clear that Φ(z) = a 1 z -2 + a 2 z -1 + ϕ(z), where ϕ is analytic in D. Thus, we can take λ 0 ∈ D so that h λ 0 has (DVC') (for e.g., λ 0 = 0). It remains to find λ 1 . Note, that

h λ 1 = 1 Φ -λ 1 = - 1 λ 1 - 1 λ 2 1 (h -λ -1 1 )
.

Thus, if we can choose λ 1 ∈ D so that 1/λ 1 / ∈ h(D) (which is possible for most of the choices of r and R) we conclude that Φ is of general position and has (DVC). By construction, Φ satisfies neither (MVC) nor (IAC).

Example 2.5. Choosing the parameter β in a good way we can modify the construction in Example 2.4 so that 0 < ã < a < c < c < b < b < d < d. Then, repeating the construction, we build Φ of general position, which does not satisfy the condition (DVC') (see Figure 2).

Necessary conditions

In this section we prove Proposition 1.1. The proof goes along the same lines as the proofs in [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF] and we include it for the sake of completeness only. Assume that Φ is not N -valent and so, for some µ, the equation Φ(z) = µ has N +1 solutions z 1 , z 2 , . . . z N +1 in D counting multiplicities. We assume that these solutions are distinct. If they are not distinct the argument requires an obvious modification (see [START_REF] Baranov | Hypercyclic Toeplitz operators[END_REF] for details). We will show that the adjoint operator TΦ has an eigenvector and so T Φ is not hypercyclic, a contradiction. We construct this eigenvector of TΦ as a linear combination of Cauchy kernels k zm , where k λ (z) = 1 1-λz . Recall that for any antianalytic Toeplitz operator we have

0 D Γ r R 0 α α Ψ z 2 (1 + z) -β(1 + i) 0 d b a -c ic -iã -i b -i d h(z) := Ψ 2 (z)(1 + Ψ(z)) -β(1 + i) 2 z 2 h(D) 0 -d2 c 2 -c 2 d 2 b 2 a 2 -ã 2 -b2 : Ω (2) h : Ω (1) h : Ω (0) h Figure 1. Example of h satisfying (DVC') h(D) 0 -d2 a 2 -ã 2 d 2 b 2 c 2 -c 2 -b 2 : Ω (2) h : Ω (1) h : Ω (0) h
T ϕ k λ = ϕ(λ)k λ . Put f = N +1 m=1 β m k zm
, where β m are some complex coefficients. Note that T Φ = T R * (z)+ϕ(z) , where

R * (z) = N 1 k=0 ck z k + r l=1 k l j=1 ᾱl,j (z -ηl ) j
is analytic and bounded in D. Hence, using the fact that R(1/z m ) + ϕ(z m ) = µ, we get ( 5)

T Φ f (z) = N +1 m=1 β m R * (z) 1 -z m z + ϕ(z m ) 1 -z m z = μf (z) + N +1 m=1 β m R * (z) -R(1/z m ) 1 -z m z .
Clearly,

R * (z) -R(1/z m ) 1 -z m z ∈ span z k , 1 (z -ηl ) j : 0 ≤ k ≤ N 1 -1, 1 ≤ l ≤ r, 1 ≤ j ≤ k l .
Since the dimension of this span is N , we can find nontrivial β m such that the last sum in ( 5) is identically zero, and so T Φ f = μf .

Sufficient conditions

The proofs of Theorems 1.3-1.5 are each based on Proposition 1.2, which in turns is based on an application of the Godefroy-Shapiro Criterion (see [START_REF] Godefroy | Operators with dense, invariant cyclic vector manifolds[END_REF], [2, Corollary 1.10] or [4, Theorem 3.1]).

Theorem (Godefroy-Shapiro Criterion). If, for a bounded linear operator T on a separable Banach space X, both ∪ |λ|<1 Ker (T -λI) and ∪ |λ|>1 Ker (T -λI) span a dense subspace in X, then T is hypercyclic.

We will then first search for the eigenvectors of T Φ . 4.1. Computation of eigenfunctions. Let us denote by S n (g) the nth partial sum of the Taylor series of g ∈ H 2 at 0. Put Q(z) = r l=1 (1 -η l z) k l . We recall that T z is the backward shift S * on H 2 , i.e.,

T z f = S * f = f (z) -f (0) z and T zk f = (S * ) k f = f (z) -S k-1 (f ) z k . Then, for η ∈ D, k ≥ 1 and g ∈ H 2 , (S * -ηI) k g = k j=0 k j (-η) k-j (S * ) j g = g k j=0 k j (-η) k-j z j - k j=1 k j (-η) k-j S j-1 (g) z j = (1 -ηz) k z k g - 1 z k k j=1 k j (-ηz) k-j S j-1 (g).
Thus, for f ∈ H 2 , applying the previous to g = [(S * -ηI) k ] -1 f , we obtain

[(S * -ηI) k ] -1 f = z k f + p k (1 -ηz) k ,
where p k is some polynomial of degree at most k -1. It follows that

T Φ f = N 1 k=0 c k (S * ) k f + r l=1 k l j=1 α l,j [(S * -η l ) j ] -1 f + ϕf = N 1 k=0 c k f + q k z k + r l=1 k l j=1 α l,j z j f + p l,j (1 -η l z) j + ϕf = Φf + N 1 k=1 q k z k + r l=1 k l j=1 α l,j p l,j (1 -η l z) k l ,
where p l,j is a polynomial of degree at most k l -1 and q k is a polynomial of degree at most k -1 (while q 0 = 0). Therefore, the equation T Φ f λ = λf λ is equivalent, for some polynomials p and q with degree at most N 2 -1 and N 1 -1 respectively, to

f λ = z N 1 p + Qq z N 1 Q(Φ -λ) .
Clearly, f λ ∈ H 2 for any λ ∈ C \ Φ(D), and so f λ is an eigenvector of T Φ . Note that this holds true for any choice of the polynomials p and q with degree at most N 2 -1 and N 1 -1 respectively. Expanding f λ around λ 0 , for λ close to λ 0 , we get

f λ = z N 1 p + Qq Qz N 1 (Φ -λ 0 ) k≥0 h k (λ -λ 0 ) k , with h = 1 Φ-λ 0 . Thus, if f ∈ H 2 is orthogonal to all such f λ , then f ⊥ z N 1 p + Qq Qz N 1 (Φ -λ 0 ) h k , k ≥ 0
for any polynomials p and q with degree at most N 2 -1 and N 1 -1 respectively. Note that any polynomial of degree at most N -1 can be represented as z N 1 p + Qq for some p and q, whence

f ⊥ (z N 1 Q(Φ -λ 0 )) -1 z j h k , 0 ≤ j ≤ N -1, k ≥ 0.
Now, by assumption, the family {1, z, . . . , z N -1 } is a cyclic set for the operator T h , i.e., the family {z j h k : 0

≤ j ≤ N -1, k ≥ 0} is complete in H 2 . Since z N 1 Q(Φ -λ 0 ) is a nonvanishing function in A(D), it is invertible in A(D)
, and so the family

{(z N 1 Q(Φ -λ 0 )) -1 z j h k : 0 ≤ j ≤ N -1, k ≥ 0}
is also complete. Thus, f = 0. Since the same argument works for U 1 , we see that the families {f λ } λ∈U 0 and {f λ } λ∈U 1 are complete in H 2 . Hence, by the Godefroy-Shapiro Criterion, T Φ is hypercyclic.

Remark 4.1. Note that for the conclusion of Proposition 1.2 to hold it is sufficient to assume that there exist λ 0 ∈ D ∩ (C \ Φ(D)) and λ 1 ∈ D ∩ (C \ Φ(D)) such that for h 0 = 1 Φ-λ 0 and h 1 = 1 Φ-λ 1 the family {1, z, . . . , z N -1 } is cyclic for T h 0 and for T h 1 .

Before giving the proofs of Theorems 1.3-1.5, let us recall Solomyak's results that we shall apply. 4.3. Solomyak's theorems. Even if we use only a very special case of the results from [START_REF] Solomyak | Cyclic families of functions for analytic Toeplitz operators[END_REF], we find it appropriate to give a short survey of them. Recall that a matrixvalued function of size n × p whose entries are in H 2 is said to be outer if p ≥ n and the greatest common inner divisor of its minors of order n is 1. In this definition it is possible that p = ∞.

[u i (ζ j ), u i (ζ j ), . . . u (K j -1) i (ζ j )] 1≤i≤m 1≤j≤l = l j=1 K j ;
(P2) For any z ∈ T and for sufficiently small neighbourhood V z such that in V z ∩ D there exist k one-to-one branches ψ 1 = id, ψ 2 , . . . , ψ k of the function h

-1 • h, the matrix-function [(u i • ψ j )(ζ)] 1≤i≤m, 1≤j≤k is outer in V z ∩ D.
Note that this theorem applies also to the case when {u i } i is an infinite sequence, i.e., m = ∞. Proof of Theorem 1.4. Since Φ satisfies (IAC), there is λ ∈ C \ Φ(D) such that Theorem 4.2 applies to the function h = 1 Φ-λ . Thus, the family {z j h k : 0 ≤ j ≤ N -1, k ≥ 0} is complete in H 2 . Let us show that for any other µ ∈ C \ Φ(D) and h µ = 1 Φ-µ the family {z j h k µ : 0 ≤ j ≤ N -1, k ≥ 0} is also complete in H 2 . Note that h 1 = c 1 + c 2 h-c 3 , where c 1 , c 2 , c 3 are some constants (which can be written explicitly in terms of λ and µ, see (3)) and c 3 / ∈ h(D). By condition (IAC), the closed domain Ω = h(D) has no holes, and so the function z can be approximated by functions in span{(z -c 3 ) -k : k ≥ 0} uniformly in Ω. Thus, h ∈ span{(h -c 3 ) -k : k ≥ 0} in H 2 , whence

span{z j h k 1 : 0 ≤ j ≤N -1, k ≥ 0} = span{z j h k : 0 ≤ j ≤ N -1, k ≥ 0} = H 2 .
Now, T Φ is hypercyclic by Proposition 1.2.

Proof of Theorem 1.5. Let λ 0 ∈ D ∩ (C \ Φ(D)) and λ 1 ∈ D ∩ (C \ Φ(D)). By Remark 1.6, both functons h 0 = 1 Φ-λ 0 and h 1 = 1 Φ-λ 1 belong to A(D), are of general position and satisfy (DVC'). Then, by Theorem 4.3, the families {z j h k 0 : 0 ≤ j ≤ N -1, k ≥ 0} and {z j h k 1 : 0 ≤ j ≤ N -1, k ≥ 0} are complete in H 2 . Therefore, by Proposition 1.2 and Remark 4.1, T Φ is hypercyclic.

  Finally, let us say that two disjoint open sets U and V in C \ h(T) are adjacent if U ∩ V contains h(I) for some open arc I ⊂ T. Then we remark that if h is meromorphic in D and of general position, then a connected component of Ω (k 1 ) h and a connected component of Ω (k 2 ) h are adjacent if and only if

Figure 2 .

 2 Figure 2. Example of h which does not satisfy (DVC')

4. 2 .

 2 Proof of Proposition 1.2. By (4), we can find two open sets U 0 ⊂ D ∩ (C \ Φ(D)) and U 1 ⊂ D ∩ (C \ Φ(D)) consisting of eigenvalues of T Φ . Let λ 0 ∈ U 0 .

Theorem 4 . 2

 42 (B. Solomyak, 1987). Let h ∈ A(D) and let the set h(T) be a finite union of C 2 -smooth Jordan arcs. If h satisfies the property (IAC), then the set {u 1 , . . . , u m } ⊂ H 2 is cyclic for the operator T h if and only if the following two conditions hold:(P1) For any ζ 1 , ζ 2 , . . . , ζ l ∈ D such that h(ζ 1 ) = . . . = h(ζ l ) = a,and the value a is taken at the point ζ j with multiplicity K j , we have rank

Theorem 4 . 3 4 . 4 .

 4344 (B. Solomyak, 1987). Let h be analytic in D. If h is of general position and satisfies the property (DVC'), then the set {u 1 , . . . , u m }, m ∈ N, is cyclic for the operator T h if and only if the conditions (P1), (P2) and (P3) hold, where(P3) For any hole G ⊂ Ω (0) h there exist i, j ≤ m such that (u i • ν -1 )/(u j • ν -1 )| ∂G / ∈ N (G), where ν = h| T and N (G) is the Nevanlinna class in G.We shall apply Solomyak's theorems only to the set {1, z, . . . , z N -1 } which, by[8, Sect. 1.3, Rem. 3], satisfies the conditions (P1), (P2) and (P3) for any N -valent h satisfying the conditions of Solomyak's theorems 4.2 and 4.3. Proof of Theorems 1.3-1.5. Proof of Theorem 1.3. If Φ satisfies (MVC), then also for any λ ∈ C \ Φ(D) the function h = 1 Φ-λ has the property that for any w ∈ h(D) the equation h(z) = w has exactly N solutions in D. By [1, Proposition 3.1] the family {z j h k : 0 ≤ j ≤ N -1, k ≥ 0} is complete in H 2 . Thus, by Proposition 1.2, T Φ is hypercyclic.

The results of Sections 2 and 3 were obtained with the support of Ministry of Science and Higher Education of the Russian Federation, agreement No 075-15-2019-1619. The results of Section 4 were obtained with the support of Russian Science Foundation grant 19-11-00058.