William Alexandre

Clifford Gilmore

Sophie Grivaux

Typicality

Frédéric Bayart

Typicality of operators on Fréchet algebras admitting a hypercyclic algebra

Keywords: 2010 Mathematics Subject Classification. 47A16, 46B87, 46E05, 54E52 Hypercyclic operators, hypercyclic algebras, Fréchet spaces of holomorphic functions, differentiation operator, eigenvector fields

come

Introduction and main results

Our main aim in this work is to show that within certain natural classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence, a typical operator possesses a hypercyclic algebra. Its set of hypercyclic vectors thus has one of the richest structures for which one can hope.

Let us recall some pertinent definitions: given a continuous operator T on a topological vector space X, a vector x is said to be a hypercyclic vector for T if its orbit tT n x ; n ě 0u under the action of T is dense in X. The set of hypercyclic vectors for T is denoted by HCpT q, and it is dense in X as soon as it is nonempty. Whenever X is a second-countable Baire space, HCpT q is a G δ set (i.e. a countable intersection of open sets), so it is residual in X as soon as it is nonempty. The study of the linear structure of this set HCpT q has been the object of many interesting and deep studies: it is known that whenever HCpT q is nonempty, the set HCpT q Y t0u contains a dense linear manifold [START_REF] Bourdon | Invariant manifolds of hypercyclic vectors[END_REF]. For certain classes of hypercyclic operators T , HCpT q Y t0u contains a closed infinite-dimensional subspace. See for instance [START_REF] González | Semi-Fredholm theory: hypercyclic and supercyclic subspaces[END_REF] for a characterization in spectral terms of operators acting on complex separable Banach spaces with this property. We mention also the work [START_REF] Menet | Hypercyclic subspaces and weighted shifts[END_REF].

When X is a topological algebra, it makes sense to ask whether HCpT q Y t0u contains a non-trivial subalgebra of X. Such an algebra will be called a hypercyclic algebra, and whenever it exists, we will say that T admits a hypercyclic algebra. This question of the existence of a hypercyclic algebra was first considered by Bayart and Matheron in [4, Chapter 8], and independently by Shkarin in [START_REF] Shkarin | On the set of hypercyclic vectors for the differentiation operator[END_REF]: they showed that the differentiation operator D : f Þ Ñ f 1 on the algebra HpCq of entire functions on the complex plane admits a hypercyclic algebra. On the other hand, the translation operators T a : f Þ Ñ f p ¨`aq, a ‰ 0, acting on HpCq do not admit hypercyclic algebras [START_REF] Aron | Powers of hypercyclic functions for some classical hypercyclic operators[END_REF]. The study of the existence of hypercyclic algebras has by now developed into a flourishing branch of linear dynamics. See for instance the works [START_REF] Bès | Hypercyclic algebras for convolution and composition operators[END_REF], [START_REF] Bayart | Hypercyclic algebras[END_REF], [START_REF] Bernal-González | Hypercyclic algebras for D-multiples of convolution operators[END_REF], [START_REF] Bès | Algebrable sets of hypercyclic vectors for convolution operators[END_REF], [START_REF] Bès | Hypercyclic algebras for convolution operators of unimodular constant term[END_REF], [START_REF] Falcó | Algebrability of the set of hypercyclic vectors for backward shift operators[END_REF] and [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF] among many other relevant references.

Several of these papers deal with the important question of characterizing the entire functions φ of exponential type such that φpDq admits a hypercyclic algebra. The paper [START_REF] Bayart | Hypercyclic algebras[END_REF] by Bayart introduces an approach to this problem based on the study of the eigenvectors of the operator φpDq, which will be of particular importance in this paper.

Our aim is to study the question of the existence of hypercyclic algebras from the Baire Category point of view. Recall that if pE, τ q is a Polish space (i.e. separable and completely metrizable), and (P) is a certain property of elements of E, we say that (P) is typical (or equivalently, that a typical x P E has property (P)) if the set tx P E ; x has (P)u is comeager in E. A comeager set in E is a set which contains a dense G δ set, i.e. which is large in E in the sense of Baire Category. Thus given a particular Polish space pE, τ q, it may be of interest to determine whether some natural properties of elements of E are typical or not.

Given a separable Fréchet algebra X (i.e. a separable completely metrizable topological algebra), our general goal is to define some natural spaces L of continuous linear operators on X, which are Polish spaces when endowed with the topology of pointwise convergence on X, and to determine whether a typical operator T in such a space L possesses a hypercyclic algebra. When pX, } ¨}q is a separable Banach algebra, natural spaces L to consider are the closed balls B M pXq consisting of bounded operators T on X with }T } ď M , with M ą 1 (so as to have a chance that a typical T P B M pXq for the topology of pointwise convergence on X is hypercyclic). In this Banach space setting, we denote by BpXq the algebra of bounded operators on X. The topology on BpXq defined as the topology of pointwise convergence on X is usually called the Strong Operator Topology (SOT): if T α is a net of elements of BpXq, and if T P BpXq, T α Ý Ñ T for the SOT if and only if T α x } ¨} Ý Ý Ñ T x for every x P X. When X is a separable Banach space, pB M pXq, SOTq is a Polish space for every M ą 0. The study of typical properties of operators T P pB M pXq, SOTq was initiated in the Hilbertian setting by Eisner and Mátrai [START_REF] Eisner | On typical properties of Hilbert space operators[END_REF] and continued in the works [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF] and [START_REF] Grivaux | Generic properties of p-contractions and similar operator topologies[END_REF] in the case where X " p pNq, 1 ď p ă `8, or X " c 0 pNq. Typical properties of operators for other Polish topologies on closed balls B M pXq were also studied in [START_REF] Eisner | A "typical" contraction is unitary[END_REF] and [START_REF] Grivaux | Local spectral properties of typical contractions on p-spaces[END_REF], as well as in the monograph [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF].

When X is a separable Fréchet space, we denote by LpXq the algebra of continuous linear operators on X, and by SOT the topology on LpXq of pointwise convergence on X. Let pN j q jě1 be a sequence of semi-norms on X defining its topology. Given a sequence pM j q jě1 of positive real numbers and a sequence pk j q jě1 of positive integers, we define L pM j q,pk j q pXq :" T P LpXq ; @j ě 1, @x P X, N j pT xq ď M j N k j pxq ( ¨

We will show in Proposition 2.1 below that pL pM j q,pk j q pXq, SOTq is a Polish space. Of course, this space may be very small for certain choices of the sequences pM j q jě1 and pk j q jě1 , but in general, it makes sense to study SOT-typical properties of elements of L pM j q,pk j q pXq. Observe that the space L pM j q,pk j q pXq depends on the choice of the sequence of semi-norms pN j q jě1 . The main result of this paper deals with the case where X " HpCq, the space of entire functions on C endowed with the topology of uniform convergence on compact sets. This topology on HpCq can be defined via the following sequence pN j q jě1 of semi-norms: for every f P HpCq, f pzq " ÿ mě0 a m z m , z P C, let N j pf q :"

ÿ mě0 |a m | j m .
There are of course plenty of different choices of sequences of semi-norms defining the topology of HpCq, but for our purposes this choice is the most convenient. To simplify the notation, we will write L pM j q,pk j q :" L pM j q,pk j q pHpCqq, subordinated to this choice of the sequence pN j q jě1 . We are now ready to state our main result.

Theorem 1.1. Let the sequences pM j q jě1 and pk j q jě1 be such that (i) for all α ě 1, j α " opM j q as j tends to infinity; (ii) for all j ě 1, M j ě j `1; (iii) for all j ě 1, k j ě j `2;

(iv) M 1 ă k 1 and k j ě k 1 for all j ě 1.

Then a typical operator T P pL pM j q,pk j q , SOTq admits a hypercyclic algebra.

Choosing the sequences pM j q jě1 and pk j q jě1 in such a way that for all j ě 1, M j ě j `1 and k j ě j `2 ensures that the differentiation operator D -the crucial example of an operator on HpCq which admits a hypercyclic algebra -belongs to L pM j q,pk j q . The operators D and 1 n! D n will be of special importance in the proof of Theorem 1.1; they will play a role analogous to the left shift operator and its iterates in the Banach space case.

Before going further, we notice that the differentiation operator is a so-called tame operator, and that the spaces L pM j q,pk j q in Theorem 1.1 include spaces of tame operators. They are thus natural spaces to be considered in the context of Fréchet spaces. We recall briefly the definition of a tame operator, and we refer to the survey [START_REF] Hamilton | The inverse function theorem of Nash and Moser[END_REF] for more on tame Fréchet spaces. Let X be a Fréchet space whose topology is given by a certain increasing family of semi-norms p Ñj q jě1 , and let T : X Ñ X be a linear map. The map T is called tame if there exist two integers b, r ě 1 such that for every j ě b, the following is true: there exists a constant C j ą 0 such that Ñj p T xq ď C j Ñj`r pxq for every x P X. A tame linear map is automatically continuous. It can easily be observed that the differentiation operator D is tame, and the operators belonging to the space L pM j q,pk j q considered in Theorem 1.1 are tame provided that there exists an integer r ě 2 such that k j " j `r for all j sufficiently large.

In order to prove Theorem 1.1, we must exhibit a dense set of operators T belonging to L pM j q,pk j q admitting a hypercyclic algebra. These operators will be of the form

T " A `1 pn `1q! D n`1
for some integer n ě 0, where A acts on C n rzs (the vector space of complex polynomials of degree at most n) as a nilpotent endomorphism, and Apz j q " 0 for every j ą n. We note that these operators are in general not of the form φpDq, where φ is an entire function of exponential type (compare to [START_REF] Bayart | Hypercyclic algebras[END_REF], [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF], [START_REF] Bès | Hypercyclic algebras for convolution and composition operators[END_REF]); thus they provide a new family of operators acting on HpCq that admit hypercyclic algebras.

We also investigate the case of the complex Fréchet algebras X " p pNq, 1 ď p ă `8 or X " c 0 pNq endowed with the coordinatewise product: if x " px n q ně1 and y " py n q ně1 are two sequences of complex numbers belonging to X, we define x ¨y " px n y n q ně1 . Then }x ¨y} ď }x} }y}, where } ¨} denotes the classical p -or c 0 -norm on X. As X is a Banach space, we place ourselves in the closed balls B M pXq of BpXq, M ą 1, and we prove the following theorem.

Theorem 1.2. Let X " p pNq, 1 ď p ă `8, or X " c 0 pNq, endowed with the coordinatewise product. Let M ą 1. A typical operator T P pB M pXq, SOTq admits a hypercyclic algebra.

The proofs of Theorems 1.1 and 1.2 build on the approach via eigenvectors introduced by Bayart in [START_REF] Bayart | Hypercyclic algebras[END_REF], and further developed by Bayart et al. in [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF]. These results show, in two different classical contexts, that having a hypercyclic algebra is a quite common phenomenon, at least from the Baire Category point of view.

However, at this point a word of caution is in order: as explained in [18, Proposition 3.2], hitherto the properties (P) of operators (on X " p pNq or c 0 pNq) which have been studied from the typicality perspective are either typical or "atypical" in pB M pXq, SOTq; in other words, for such a property (P), either a typical T P pB M pXq, SOTq has (P), or a typical T P pB M pXq, SOTq does not possess (P). This relies on the topological 0 ´1 law (cf. [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 8.46], and [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF]Proposition 3.2] for its application in our context) and on the fact that for the properties under consideration, T has (P) if and only if JT J ´1 has (P) for every surjective isometry J of X. In our setting, this argument fails: having a hypercyclic algebra is not a property which is a priori stable by conjugation by invertible isometries (cf. [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF]Remark 4.7], where it is observed that admitting a hypercyclic algebra is not a property which is preserved by similarity).

The paper is organized as follows: Sections 2, 3, 4 and 5 are devoted to the proof of our main result, Theorem 1.1. We introduce in Section 2 our Polish spaces of operators on the algebra of entire functions; then a technical result regarding the density of certain classes of operators in pL pM j q,pk j q , SOTq is proved in Section 3; in Section 4 we show, using the Godefroy-Shapiro Criterion, that hypercyclic operators form a dense G δ subset of pL pM j q,pk j q , SOTq; finally in Section 5, after recalling the Baire Category Criterion from [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF]Corollary 2.4] which we use to prove the existence of a hypercyclic algebra for the operators under consideration, we prove Theorem 1.1. Theorem 1.2 is proved in Section 6. Section 7 contains the additional result that when X " p pNq, 1 ď p ă `8, or X " c 0 pNq, and M ą 1, a typical T P pB M pXq, SOTq admits a closed infinite-dimensional hypercyclic subspace (Theorem 7.2).

2. Polish Space of operators on the algebra of entire functions.

We present in this section some general facts which will be of use in the sequel. We begin by proving the following proposition for a general separable Fréchet space X. Proposition 2.1. Let X be a separable Fréchet space, and let pN j q jě1 be a sequence of semi-norms defining its topology. Let pM j q jě1 be a sequence of positive real numbers and let pk j q jě1 be a sequence of positive integers. Then L pM j q,pk j q pXq :" T P LpXq ; @j ě 1, @x P X, N j pT xq ď M j N k j pxq ( is a Polish space when endowed with the SOT.

Proof. The proof is similar to that of [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF]Lemma 3.1]. We recall it here briefly for the sake of completeness. We denote by Z a countable dense subset of X. Then let LpM j q,pk j q pZq :" tT : Z Ñ X; T is linear and @j ě 1, @x P Z, N j pT xq ď M j N k j pxqu.

Then LpM j q,pk j q pZq is closed in X Z , where X Z is endowed with the product topology. Indeed, since X Z is metrizable, let pT k q k be a sequence of elements of LpM j q,pk j q pZq which converges to T . Since the product topology in X Z is the pointwise topology, T is necessarily linear; since for all j ě 1, and all x P Z, the sequence pN j pT k xqq k converges to N j pT xq, we also have that N j pT xq ď M j N k j pxq, and so T belongs to LpM j q,pk j q pZq. Since X Z is a Polish space, LpM j q,pk j q pZq is a Polish space as a closed subset of a Polish space. Now, we observe that the map Φ : pL pM j q,pk j q pXq, SOTq Ñ LpM j q,pk j q pZq defined by ΦpT q " T | Z is a homeomorphism, and we conclude that pL pM j q,pk j q pXq, SOTq is also a Polish space.

From now on, we will consider the case where X " HpCq, the space of entire functions on the complex plane endowed with the topology of uniform convergence on compact sets. We denote by pz k q k the monomial basis of HpCq and for z P C and r ą 0, we let Dpz, rq be the open disk of center z and radius r. We record here the following property. Property 2.2. For every f P HpCq and every j ě 1, we have

sup Dp0,jq |f | ď N j pf q ď pj `1q sup Dp0,j`1q |f |.
Proof. The lefthand-side inequality is trivial, and the righthand-side inequality follows from Cauchy's inequalities: for all k P N and all r ą 0, we have

1 k! ˇˇˇB k f Bz k p0q ˇˇˇ" 1 2π ˇˇˇˇż |ζ|"r f pζq ζ k`1 dζ ˇˇˇˇď 1 r k sup Dp0,rq |f |.
This implies that

N j pf q ď `8 ÿ k"0 ˆj j `1 ˙k sup Dp0,j`1q |f | " pj `1q sup Dp0,j`1q |f |.
Therefore a sequence pf n q n of functions in HpCq converges to f in HpCq if and only if the sequence pN j pf n ´f qq n converges to 0 for every j ě 1; thus the family pN j q j indeed induces the topology of HpCq.

The next property provides a useful neighborhood basis of an element of L pM j q,pk j q for the SOT.

Property 2.3. Let T 0 belong to L pM j q,pk j q . A neighborhood basis of T 0 for the SOT is given by the family of sets V T 0 ε,r,K " tT P L pM j q,pk j q ; @k " 0, . . . , K, N r ppT ´T0 qz k q ă εu, where ε ą 0, and K, r are integers with K ě 0 and r ě 1.

Proof. It suffices to show that if pT k q k is a sequence of elements of L pM j q,pk j q such that, for all ε ą 0, all K ě 0 and all r ě 1, there exists κ 0 P N such that for all k ě κ 0 , T k belongs to V 0 ε,r,K , then pT k q k converges SOT to 0. Let f be an entire function and let j ě 1 be a positive integer. There exists an analytic nonzero polynomial P such that N k j pf ´P q ă ε 2M j . Denote by d the degree of P , write P as P pzq " ř d l"0 p l z l , and set ε 1 "

ε 2 ř d l"0 |p l |
. By assumption, there exists κ 0 P N such that for all k ě κ 0 , T k belongs to V 0 ε 1 ,d,j . Therefore we have

N j pT k f q ď N j pT k pf ´P qq `Nj pT k P q ď M j N k j pf ´P q `d ÿ l"0 |p l |N j pT k z l q ď M j ε 2M j `ε1 d ÿ l"0 |p l | " ε.
This implies that the sequence pT k f q k converges to 0 uniformly on any compact subset of C, and thus pT k q k converges SOT to 0.

In the forthcoming proof of Theorem 1.1, the operators S n :" 1 n! D n , n ě 1, will play a crucial role. It is thus important to be able to demonstrate that they belong to the space L pM j q,pk j q . In the next proposition, we provide conditions on the sequences pM j q jě1 and pk j q jě1 which ensure that this is indeed the case. Proposition 2.4. Let pM j q jě1 be a sequence of positive real numbers and let pk j q jě1 be a sequence of positive integers such that for every j ě 1, M j ě j `1 and k j ě j `2. Then S n belongs to L pM j q,pk j q for every n ě 1.

Proof. Let n ě 1, j ě 1, and let z P Dp0, j `1q. By Cauchy's formula, we have:

1 n! B n f Bz n pzq " 1 2iπ ż |ζ´z|"1
f pζq pζ ´zq n`1 dζ so that |S n pf qpzq| ď sup Dp0,j`2q |f |. We then deduce from Property 2.2 that N j pS n f q ď pj `1q sup Dp0,j`1q

|S n pf q| ď pj `1q sup Dp0,j`2q |f | ď pj `1qN j`2 pf q, and this proves that S n belongs to L pM j q,pk j q when pM j q jě1 and pk j q jě1 satisfy the assumptions of this proposition.

3. Dense families of operators in L pM j q,pk j q For every n ě 0, we define the operator T n : HpCq Ñ HpCq by setting

T n f pzq " n ÿ j"0 1 j! B j f
Bz j p0qz j , where f P HpCq and we note that for all j ě 1, N j pT n f q ď N j pf q.

Given A P L pM j q,pk j q , our aim is to approximate A with an operator supporting a hypercyclic algebra. We first approximate A by an operator of the form B n " T n AT n . When n goes to infinity, pB n q n converges SOT to A, and for all n ě 1, B n belongs to L pM j q,pk j q . Indeed, we have for every f P HpCq and every n, j ě 1 that

N j pB n f q ď N j pAT n f q ď M j N k j pT n f q ď M j N k j pf q.
When M j ě j `1 and k j ě j `2 for all j ě 1, it follows from Proposition 2.4 that S n`1 belongs to L pM j q,pk j q for every n ě 0. Hence, for every δ P r0, 1s, the operator p1 ´δqB n `δS n`1 also belongs to L pM j q,pk j q . Moreover, pp1 ´δqB n `δS n`1 q δPp0,1q SOT converges to B n as δ goes to 0. Thus we have proved the following. Proposition 3.1. Let pM j q jě1 be a sequence of positive real numbers, and let pk j q jě1 be a sequence of positive integers such that M j ě j `1 and k j ě j `2 for every j ě 1. Then the following holds: for every A P L pM j q,pk j q and every SOT-neighborhood V of A, there exist n ě 0, an operator B P L pM j q,pk j q satisfying B " T n BT n , and δ P p0, 1q such that B `δS n`1 belongs to V.

We will prove in Section 4 below that operators of the form B `δS n`1 are hypercyclic, where δ ą 0, n ě 0 and B is such that B " T n BT n . So the reader mainly interested in the density of hypercyclic operators in L pM j q,pk j q can proceed directly to Section 4.

On the other hand, it does not seem to be trivial that an operator of the form B `δS n`1 with δ P p0, 1q, n ě 0 and B such that B " T n BT n supports a hypercyclic algebra. What we will be able to prove in Section 5 is that if for every polynomial P , the sequence pB j P q j converges to 0 in HpCq, then B `δS n`1 indeed supports a hypercyclic algebra (cf. Theorem 5.1). With this result in view, we now show that the family of nilpotent operators B satisfying B " T n BT n for some n ě 0 is SOT dense in L pM j q,pk j q provided that the sequences pM j q j and pk j q j satisfy some suitable assumptions, which in fact ensure that L pM j q,pk j q contains sufficiently many operators. Proposition 3.2. Let pM j q jě1 be a sequence of positive real numbers, and let pk j q jě1 be a sequence of positive integers such that (i) for all α ě 1, j α " opM j q as j tends to infinity; (ii) k j ą j for every j ě 1; (iii) M 1 ă k 1 and k j ě k 1 for every j ě 1.

Let A be an operator belonging to L pM j q,pk j q and let V be an SOT-neighborhood of A in L pM j q,pk j q . Then there exist B P V and n ě 1 such that B " T n BT n (in particular, for all f P HpCq, Bf is a polynomial of degree at most n) and B n`1 " 0. Remark 3.3. Conditions (i) and (ii) of Proposition 3.2 ensure that L pM j q,pk j q contains sufficiently many operators. For example, for every n ě 0 and every a P C close enough to 0 (how close depends on n), the operator B n defined by B n f " az n f , f P HpCq, belongs to L pM j q,pk j q . Indeed, for every j ě 1 and every i ě 1 we have

N j pB n z i q " N j paz i`n q " |a|j i`n .
Thus B n belongs to L pM j q,pk j q if and only if |a|j i`n ď M j k i j for every j ě 1. Condition (ii) ensures in particular that j ď k j , so that N j pB n z i q ď |a|j n k i j for every i, j ě 1. Condition (i) ensures the existence of j 0 ě 1 such that j n ď M j for every j ě j 0 ; if |a| ď 1, N j pB n z i q ď M j k i j for every i ě 1 and every j ě j 0 . Now, if |a| is small enough, depending on n, we can ensure that |a|j n ď M j for every 1 ď j ď j 0 and thus N j pB n z i q ď M j k i j for every i ě 1 and every j ď j 0 . Hence B n belongs to L pM j q,pk j q .

Condition (iii) is a technical condition which will be needed in the proof of Proposition 3.2. It does not appear to be overly restrictive.

Remark 3.4. In the Hilbertian setting, it is known that nilpotent contractions on a complex separable Hilbert space H are SOT ˚-dense in B 1 pHq, so in particular SOT-dense ([16, Proposition 4.6]). It seems to be unknown whether this result can be extended to

p -spaces, 1 ă p ă `8, p ‰ 2.
The rest of this section is devoted to the proof of Proposition 3.2.

Proof of Proposition 3.2. Without loss of generality (cf. Property 2.3), we assume that V has the form V " V A ε 0 ,r,K " tB P L pM j q,pk j q ; N r pBz j ´Az j q ă ε 0 , j " 0, . . . , Ku, where ε 0 ą 0, and r, K ě 1 are fixed.

Let n 0 ě K be a sufficiently large integer and δ P p0, 1q a sufficiently small positive number. We set A 0 " p1 ´δqT n 0 AT n 0 .

If n 0 is sufficiently large and δ is sufficiently small, then A 0 belongs to V. We now fix the parameters n 0 ě K and δ P p0, 1q. We then consider an operator B defined as follows:

Bz j " $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % A 0 z j `εz n 0 `1`j , 0 ď j ď n 0 εz n 0 `1`j , n 0 `1 ď j ď pn 0 `1qpn 1 `1q ´1 z n 0 `1`j , pn 0 `1qpn 1 `1q ď j ď pn 0 `1qn 2 ´1 D n 0 ,n 1 ,n 2 pz j q, pn 0 `1qn 2 ď j ď pn 0 `1qpn 2 `1q ´1 0, j ě pn 0 `1qpn 2 `1q
where

D n 0 ,n 1 ,n 2 pz j q :" ´n1 ÿ m"0 1 ε n 1 ´m`1 z pn 0 `1qm A n 2 `1´m 0 z j´n 2 pn 0 `1q ´n2 ÿ m"n 1 `1 z pn 0 `1qm A n 2 `1´m 0 z j´n 2 pn 0 `1q
and where n 1 , n 2 ě 1, ε ą 0 have to be determined in order to ensure that B belongs to L pM j q,pk j q . Our first goal is to show that with this definition, B is nilpotent -more precisely that B n 2 `1 " 0. Fact 3.5. For every P P C n 0 rzs and every 1 ď n ď n 2 , we have

B n P " minpn,n 1 q ÿ k"0 ε k z kpn 0 `1q A n´k 0 P `εn 1 `1 n ÿ k"n 1 `1 z kpn 0 `1q A n´k 0 P. (3.1)
Proof. Equality (3.1) is true for n " 1, since by definition BP " A 0 P `εz n 0 `1P . If we assume that (3.1) is true for some n ě 1, then, for every 1 ď k ď n 1 , z kpn 0 `1q A n´k 0 P is a sum of monomials whose degrees lie between kpn 0 `1q and pk `1qpn 0 `1q ´1; since kpn 0 `1q ě n 0 `1 and pk `1qpn 0 `1q ´1 ď pn 1 `1qpn 0 `1q ´1, it follows that

Bpz kpn 0 `1q A n´k 0 P q " εz pk`1qpn 0 `1q A n´k 0 P for 1 ď k ď n 1 .
Analogously, for n 1 `1 ď k ď n 2 ´1, z kpn 0 `1q A n´k 0 P is a sum of monomials whose degrees lie between kpn 0 `1q and pk `1qpn 0 `1q ´1, with kpn 0 `1q ě pn 1 `1qpn 0 `1q and pk `1qpn 0 `1q ď n 2 pn 0 `1q ´1. Thus

Bpz kpn 0 `1q A n´k 0 P q " z pk`1qpn 0 `1q A n´k 0 P for every n 1 ď k ď n 2 ´1.
We now consider separately two cases.

Case 1: If n ă n 1 , we have B n`1 P " BA n 0 P `n ÿ k"1 ε k B ´zkpn 0 `1q A n´k 0 P " A n`1 0 P `εz n 0 `1A n 0 P `n ÿ k"1 ε k`1 z pk`1qpn 0 `1q A n´k 0 P " A n`1 0 P `εz n 0 `1A n 0 P `n`1 ÿ k"2 ε k z kpn 0 `1q A n`1´k 0 P " minpn`1,n 1 q ÿ k"0 ε k z kpn 0 `1q A n`1´k 0 P.
Hence (3.1) still holds true at rank n `1 when n ă n 1 .

Case 2: If n 1 ď n ă n 2 , we have

B n`1 P " BA n 0 P `n1 ÿ k"1 ε k B ´zkpn 0 `1q A n´k 0 P ¯`ε n 1 `1 n ÿ k"n 1 `1 B ´zkpn 0 `1q A n´k 0 P " A n`1 0 P `εz n 0 `1A n 0 P `n1 ÿ k"1 ε k`1 z pk`1qpn 0 `1q A n´k 0 P `εn 1 `1 n ÿ k"n 1 `1 z pk`1qpn 0 `1q A n´k 0 P.
This gives that

B n`1 P " A n`1 0 P `εz n 0 `1A n 0 P `n1 `1 ÿ k"2 ε k z kpn 0 `1q A n`1´k 0 P `εn 1 `1 n`1 ÿ k"n 1 `2 z kpn 0 `1q A n`1´k 0 P " A n`1 0 P `εz n 0 `1A n 0 P `n1 ÿ k"2 ε k z kpn 0 `1q A n`1´k 0 P `εn 1 `1 n`1 ÿ k"n 1 `1 z kpn 0 `1q A n`1´k 0 P.
Hence (3.1) is still true at rank n `1 in this second case where n 2 ą n ě n 1 , which ends the proof of Fact 3.5.

Fact 3.6. For every P P C n 0 rzs, we have B n 2 `1P " 0.

Proof. Fact 3.5 applied to n " n 2 yields that

B n 2 `1P " B ˜n1 ÿ k"0 ε k z kpn 0 `1q A n 2 ´k 0 P `εn 1 `1 n 2 ÿ k"n 1 `1 z kpn 0 `1q A n 2 ´k 0 P " A n 2 `1 0 P `εz n 0 `1A n 2 0 P `n1 ÿ k"1 ε k`1 z pk`1qpn 0 `1q A n 2 ´k 0 P `εn 1 `1 n 2 ´1 ÿ k"n 1 `1 z pk`1qpn 0 `1q A n 2 ´k 0 P ´εn 1 `1 n 1 ÿ m"0 1 ε n 1 ´m`1 z mpn 0 `1q A n 2 `1´m 0 P ´εn 1 `1 n 2 ÿ m"n 1 `1 z mpn 0 `1q A n 2 `1´m 0 P.
Writing the sum

ε n 1 `1 n 1 ÿ m"0 1 ε n 1 ´m`1 z mpn 0 `1q A n 2 `1´m 0 P as A n 2 `1 0 P `n1 ´1 ÿ m"0 ε m`1 z pm`1qpn 0 `1q A n 2 ´m 0 P,
and the sum

ε n 1 `1 n 2 ÿ m"n 1 `1 z mpn 0 `1q A n 2 `1´m 0 P as ε n 1 `1z pn 1 `1qpn 0 `1q A n 2 ´n1 0 P `εn 1 `1 n 2 ´1 ÿ m"n 1 `1 z pm`1qpn 0 `1q A n 2 ´m 0 P,
we obtain that B n 2 `1P " 0, which proves Fact 3.6.

We are now ready to prove the next fact. Fact 3.7. We have B n 2 `1 " 0.

Proof. Let us first check that the family p1, . . . , z n 0 , B1, . . . , Bz n 0 , . . . , B n 2 1, . . . , B n 2 z n 0 q is a basis of C pn 0 `1qpn 2 `1q´1 rzs. Suppose that ř n 2 k"0 ř n 0 j"0 λ k,j B k z j " 0 for some complex numbers λ k,j . Applying B n 2 to this equality, we obtain by Fact 3.6 that ř n 0 j"0 λ 0,j B n 2 z j " 0. Fact 3.5 then implies that

n 0 ÿ j"0 λ 0,j ˜n1 ÿ k"0 ε k z kpn 0 `1q A n 2 ´k 0 z j `εn 1 `1 n 2 ÿ k"n 1 `1 z kpn 0 `1q A n 2 ´k 0 z j ¸" 0 (3.2)
so that the two polynomials

n 0 ÿ j"0 λ 0,j ˜n1 ÿ k"0 ε k z kpn 0 `1q A n 2 ´k 0 z j `εn 1 `1 n 2 ´1 ÿ k"n 1 `1 z kpn 0 `1q A n 2 ´k 0 z j and ´εn 1 `1 n 0 ÿ j"0
λ 0,j z n 2 pn 0 `1q z j are equal. For 0 ď k ď n 2 ´1 and 0 ď j ď n 0 , the degree of z kpn 0 `1q A n 2 ´k 0 z j is less than n 2 pn 0 `1q ´1 while the degree of z n 2 pn 0 `1q`j is strictly greater than n 2 pn 0 `1q ´1. It follows that ř n 0 j"0 λ 0,j z n 2 pn 0 `1q`j " 0, and so that λ 0,0 " ¨¨¨" λ 0,n 0 . Suppose now that λ 0,0 " ¨¨¨" λ 0,n 0 " ¨¨¨" λ l,0 " ¨¨¨" λ l,n 0 for some 0 ď l ă n 2 . We know that

ř n 2 k"l`1
ř n 0 j"0 λ k,j B k z j " 0, and we apply B n 2 ´pl`1q to this equality. We then obtain as previously that λ l`1,0 " ¨¨¨" λ l`1,n 0 . By induction, λ j,k " 0 for all j and k, and p1, . . . , z n 0 , B1, . . . , Bz n 0 , . . . , B n 2 1, . . . , B n 2 z n 0 q is indeed a basis of C pn 0 `1qpn 2 `1q´1 rzs. Now, writing any polynomial P P C pn 0 `1qpn 2 `1q´1 rzs as P " ř n 2 k"0 ř n 0 j"0 µ k,j B k z j , we obtain B n 2 `1P " 0. This proves Fact 3.7.

The last step in our proof of Proposition 3.2 is the following Fact 3.8, which shows that the operator B belongs to V for suitably chosen n 1 , n 2 and ε ą 0. Fact 3.8. There exist integers n 1 and n 2 large enough and ε ą 0 small enough such that the operator B belongs to V.

Proof. Since n 0 ě K, we have Bz j " A 0 z j `εz j for every j " 0, . . . , K, and thus N r pBz j ´Az j q ď N r pBz j ´A0 z j q `Nr pA 0 z j ´Az j q ď εr j `Nr pA 0 z j ´Az j q.

Since A 0 belongs to V, we have N r pA 0 z j ´Az j q ă ε 0 for every j " 0, . . . , K. One can choose ε ą 0 small enough, so as to ensure that N r pBz j ´Az j q ă ε 0 also holds for every j " 0, . . . , K. Therefore, to demonstrate that B belongs to V, the main problem is to show that B belongs to L pM j q,pk j q for suitable choices of the integers n 1 and n 2 (which will have to be large) and of the real number ε ą 0 (which must be sufficiently small). Since M j j n 0 `1 Ñ `8 as j Ñ `8, there exists an integer j 0 P N such that j n 0 `1 ď M j for all j ě j 0 , and provided that j 0 is large enough, we also have

M 1 k n 0 1 1 j n 0 `1 0 ă 1.
Once j 0 is fixed, we choose n 1 , n 2 and ε as follows:

(i) Since k j ą j for all j, we can first choose an integer n 1 sufficiently large, so that j n 0 `1 0 ˆj k j ˙pn 1 `1qpn 0 `1q ď M j for every 1 ď j ď j 0 .

(ii) We next choose ε ą 0 satisfying the following two properties:

0 ă ε ď δ and εj 2n 0 `1 0 ď δ min 1ďjăj 0 M j . (iii) Let c 0 :" ř `8 m"0 ˆM1 k n 0 1 1 j n 0 `1 0 ˙m ă `8.
We choose an integer j 1 ą j 0 such that

c 0 1 ε n 1 `1 M 1 k n 0 1 1 j 1 ă 1.
(iv) Lastly, using again the assumption that k j ą j for all j ě 1 and that M 1 ă k 1 , we choose an integer n 2 sufficiently large, such that (a) c 0 1

ε n 1 `1 M 1 k n 0 1 ´j k j ¯n2 pn 0 `1q 1 j ă 1 for every j 0 ď j ă j 1 ; (b) n 2 `1 ε n 1 `1 ´j k j ¯n2 pn 0 `1q pmax 1ďiďj 0 k i q n 0 ă 1 for every 1 ď j ď j 0 ; (c) n 2 `1 ε n 1 `1 k n 0 1 ´M1 k 1 ¯n2 ă 1.
For these choices of the parameters n 1 , n 2 and ε ą 0, we will prove that N j pBz m q ď M j k m j for every integer m with 0 ď m ď pn 2 `1qpn 0 `1q ´1 and every j ě 1. Since Bz m " 0 for all m ě pn 2 `1qpn 0 `1q, this will imply that B belongs to L pM j q,pk j q .

We fix j ě 1, and separate the proof into four cases.

Case 1 : 1 ď m ď n 0 .

We have Bz m " A 0 z m `εz n 0 `1`m so N j pBz m q ď N j pA 0 z m q `εN j pz n 0 `1`m q. Notice that N j pA 0 z m q " p1 ´δqN j pT n 0 AT n 0 z m q ď p1 ´δqN j pAz m q ď p1 ´δqM j k m j from which we deduce that N j pBz m q ď p1 ´δqM j k m j `εj n 0 `1`m . If j ě j 0 , then j n 0 `1 ď M j so N j pBz m q ď p1 ´δ `εqM j k m j and since δ ě ε, we get N j pBz m q ď M j k m j . If 1 ď j ă j 0 , N j pBz m q ď p1 ´δqM j k m j `εj 2n 0 `1 0 and by (ii) we also have that

N j pBz m q ď M j k m j . Case 2: pn 1 `1qpn 0 `1q ď m ď n 2 pn 0 `1q ´1.
In this case we have Bz m " z m`n 0 `1 so N j pBz m q " j m`n 0 `1. If j ě j 0 , we have j n 0 `1 ď M j and k j ą j, so N j pBz m q ď M j k m j . If 1 ď j ă j 0 ,

N j pBz m q " j n 0 `1 ˆj k j ˙m k m j ď j n 0 `1 0 ˆj k j ˙pn 1 `1qpn 0 `1q k m j ď M j k m j
where the last inequality follows from (i).

Case 3: pn 0 `1q ď m ď pn 1 `1qpn 0 `1q ´1.

For these values of m we have Bz m " εz m`n 0 `1 so N j pBz m q " εj m`n 0 `1.

If j ě j 0 , we have j n 0 `1 ď M j , k j ą j and ε ă 1, so N j pBz m q ď M j k m j . If 1 ď j ă j 0 ,

N j pBz m q ď εj n 0 `1 0 j m ď M j k m j
where the last inequality follows from (ii) and from the fact that j ă k j .

Case 4: n 2 pn 0 `1q ď m ď pn 2 `1qpn 0 `1q ´1.

We write m as m " n 2 pn 0 `1q `l with 0 ď l ď n 0 . In this case we have that

Bz m " ´n1 ÿ i"0 1 ε n 1 ´i`1 z pn 0 `1qi A n 2 ´i`1 0 z l ´n2 ÿ i"n 1 `1 z pn 0 `1qi A n 2 ´i`1 0 z l
and since 0 ă ε ă 1, it follows that

N j pBz m q ď 1 ε n 1 `1 n 2 ÿ i"0 N j ´zpn 0 `1qi A n 2 ´i`1 0 z l ¯" 1 ε n 1 `1 n 2 ÿ i"0 j pn 0 `1qi N j ´An 2 ´i`1 0 z l ¯.
Next we will estimate the norms of the iterates of A 0 . The main difficulty with this estimate comes from the fact that given a polynomial P , the quantity N j pA 0 P q can be controlled by N k j pP q only, and not by N j pP q.

We remark that if P is a polynomial of degree at most d, we have N j pP q ď j d N 1 pP q. Thus for any polynomial P of degree at most n 0 , we have N 1 pA 0 P q ď M 1 k n 0 1 N 1 pP q. It follows that N 1 pA n 0 P q ď pM 1 k n 0 1 q n N 1 pP q for every n ě 0. (3.3) -We consider first the case where j ě j 0 . For every 0 ď i ď n 2 and every 0 ď l 1 ď n 0 , (3.3) gives

N j ´An 2 ´i`1 0 z l 1 ¯ď j n 0 N 1 ´An 2 ´i`1 0 z l 1 ¯ď j n 0 pM 1 k n 0 1 q n 2 ´i`1 . It then follows that N j pBz m q ď 1 ε n 1 `1 n 2 ÿ i"0 j pn 0 `1qi j n 0 pM 1 k n 0 1 q n 2 ´i`1 " j pn 0 `1qpn 2 `1q`n 0 ε n 1 `1 n 2 ÿ i"0 ˆM1 k n 0 1 1 j n 0 `1 ˙n2 ´i`1 .
By the definition of c 0 given in (iii), we get

N j pBz m q ď j pn 0 `1qpn 2 `1q`n 0 ε n 1 `1 c 0 M 1 k n 0 1 1 j n 0 `1 Since k j ě 1 and m ě n 2 pn 0 `1q it follows that N j pBz m q ď M j k m j c 0 1 ε n 1 `1 M 1 k n 0 1 ˆj k j ˙n2 pn 0 `1q j n 0 M j ,
and using the fact that M j ě j n 0 `1 (because j ě j 0 ), it follows that

N j pBz m q ď M j k m j c 0 1 ε n 1 `1 M 1 k n 0
1 ˆj k j ˙n2 pn 0 `1q 1 j Ïf j ě j 1 , using that j k j ď 1 as well as property (iii), we get that N j pBz m q ď M j k m j . If j 0 ď j ă j 1 , (iv)-(a) gives the inequality N j pBz m q ď M j k m j .

-It now remains to consider the integers j such that 1 ď j ă j 0 . For all polynomials P of degree at most n 0 we have N j pA 0 P q ď p1 ´δqM j N k j pP q ď p1 ´δqM j k j n 0 N 1 pP q, from which it follows that

N j pBz m q ď 1 ε n 1 `1 n 2 ÿ i"0 j pn 0 `1qi N j ´An 2 ´i`1 0 z l ď 1 ε n 1 `1 n 2 ÿ i"0 j pn 0 `1qi p1 ´δqM j k j n 0 N 1 ´An 2 ´i 0 z l ¯.
We use p3.3q in order to get N 1 ´An 2 ´i 0 z l ¯ď pM 1 k n 0 1 q n 2 ´i and

N j pBz m q ď 1 ε n 1 `1 M j k j n 0 n 2 ÿ i"0 j pn 0 `1qi pM 1 k n 0 1 q n 2 ´i " 1 ε n 1 `1 M j k j n 0 j n 2 pn 0 `1q n 2 ÿ i"0 ˆM1 k n 0 1 1 j n 0 `1 ˙n2 ´i . If M 1 k n 0 1 1 j n 0 `1 ď 1, we obtain N j pBz m q ď 1 ε n 1 `1 M j k j n 0 j n 2 pn 0 `1q pn 2 `1q " M j k j m n 2 `1 ε n 1 `1 j n 2 pn 0 `1q k n 2 pn 0 `1q`l´n 0 j ď M j k j m n 2 `1 ε n 1 `1 ˆj k j ˙n2 pn 0 `1q ˆmax 1ďiďj 0 k i ˙n0 .
Then condition (iv)-(b) implies that N j pBz m q ď M j k m j . In the case where M 1 k n 0 1 1 j n 0 `1 ě 1, we have

N j pBz m q ď 1 ε n 1 `1 M j k j n 0 j n 2 pn 0 `1q pn 2 `1q ˆM1 k n 0 1 1 j n 0 `1 ˙n2 " M j k j m n 2 `1 ε n 1 `1 k n 0 ´l 1 ˆM1 k 1 ˙n2 k n 2 pn 0 `1q`l´n 0 1 k n 2 pn 0 `1q`l´n 0 j ď M j k j m n 2 `1 ε n 1 `1 ˆM1 k 1 ˙n2 k n 0 ´l 1
where the last inequality comes from the fact that k j ě k 1 . Now (iv)-(c) gives that N j pBz m q ď M j k m j . This shows for every j ě 1 and every m ě 0 that N j pBz m q ď M j k m j , and Fact 3.8 is thus proved.

Combining Facts 3.7 and 3.8 concludes the proof of Proposition 3.2.

What we actually need in the proof of Theorem 1.1 is that operators of the form B `δS n`1 are dense in L pM j q,pk j q , for δ P p0, 1q and B P L pM j q,pk j q nilpotent such that B " T n BT n . The following corollary shows that this is an easy consequence of Proposition 3.2. Corollary 3.9. Let pM j q jě1 be a sequence of positive real numbers, and let pk j q jě1 be a sequence of positive integers such that (i) for all α ě 1, j α " opM j q as j tends to infinity; (ii) k j ą j for every j ě 1;

(iii) M 1 ă k 1 and k j ě k 1 for every j ě 1.

Let A be an operator belonging to L pM j q,pk j q and let V be an SOT-neighborhood of A in L pM j q,pk j q . Then there exist B P L pM j q,pk j q , δ P p0, 1q and n ě 1 such that B " T n BT n , B n`1 " 0 and B `δS n`1 belongs to V.

Proof. We denote by B 1 P V a nilpotent operator with B 1 " T n B 1 T n for some integer n ě 1, given by Proposition 3.2. If δ P p0, 1q is sufficiently small, p1 ´δqB 1 `δS n`1 belongs to V, B " p1 ´δqB 1 belongs to L pM j q,pk j q and it satisfies B " T n BT n and B n`1 " 0.

Density of hypercyclic operators

Our aim in this section is to prove the following theorem. Theorem 4.1. Let pM j q jě1 be a sequence of positive real numbers, and let pk j q jě1 be a sequence of positive integers such that M j ě j `1 and k j ě j `2 for every j ě 1. Then the set of hypercyclic operators is a dense G δ subset of pL pM j q,pk j q , SOTq.

The statement of Theorem 4.1 is not strictly necessary for the proof of Theorem 1.1. However, we will subsequently make use of many of the ingredients of its proof, in particular of the expression and the properties of the eigenvectors of operators of the form B `δS n`1 , where B " T n BT n (cf. Proposition 3.1).

Proof. Let us begin by proving that the set of hypercyclic operators is G δ in L pM j q,pk j q . The argument for this is classical (cf. [17, Corollary 2.2]). Let pU p q pě1 be a countable basis of open sets of HpCq. The set of hypercyclic operators in L pM j q,pk j q can be expressed as G "

!

T P L pM j q,pk j q ; @p, q ě 1 Du P U p DN ě 1 such that

T N u P U q ) " č p,qě1 ď N ě1
! T P L pM j q,pk j q ; Du P U p such that T N u P U q ) .

To establish that G is a G δ set, it suffices to prove that, for all p, q and N , the set G p,q,N :" ! T P L pM j q,pk j q ; Du P U p such that T N u P U q

)

is SOT-open. This is a consequence of the following claim:

Claim 4.2. For every integer N ě 1 and every function u P HpCq, the map φ N,u : T Þ Ñ T N u from L pM j q,pk j q into HpCq is SOT-continuous.

We take the claim for granted for the moment. Let T 0 belong to G p,q,N . Choose u P U p such that T N 0 u belongs to U q . Then φ ´1 N,u pU q q is an SOT-open set contained in G p,q,N containing T 0 . Therefore G p,q,N is indeed SOT-open, and G is an SOT-G δ set, provided the claim holds.

Proof of Claim 4.2. Let pT n q be a sequence of elements of L pM j q,pk j q that SOT-converges to T . For every j ě 0, define κpjq :" k j . Note that for all S P L pM j q,pk j q , all l ě 1, all f P HpCq and all j ě 1, we have

N j pS l f q ď ˜l ź m"1 M κ m´1 pjq ¸Nκ l pjq pf q, (4.1)
where κ m pjq " κ ˝¨¨¨˝κ pjq, j ě 0, is the composition of the function κ with itself m times. Now we express φ N,u pT n q ´φN,u pT q as follows:

φ N,u pT n q ´φN,u pT q " N ÿ l"1 ´T l n T N ´lu ´T l´1 n T N ´l`1 u ¯" N ÿ l"1 T l´1 n pT n ´T qT N ´lu.
Thus, by (4.1), we have for all j ě 1 that

N j pφ N,u pT n q ´φN,u pT qq ď N ÿ l"1 N j ´T l´1 n pT n ´T qT N ´lu ď N ÿ l"1 ˜l´1 ź m"1 M κ m´1 pjq ¸Nκ l´1 pjq ´pT n ´T qT N ´lu ¯.
Since N ě 1 and u P HpCq are fixed, and since the sequence pT n q n SOT-converges to T , we have N κ l´1 pjq `pT n ´T qT N ´lu ˘ÝÑ 0 as n Ñ `8 for every l " 1, . . . , N . Thus N j pφ N,u pT n q ´φN,u pT qq ÝÑ 0, proving the claim.

Next we prove that hypercyclic operators are SOT-dense in L pM j q,pk j q . To this end, by Proposition 3.1 it suffices to prove that the operators B `δS n`1 , with n ě 0 and B " T n BT n , are hypercyclic.

The general strategy of the proof is the following: we will show (cf. Lemma 4.5 below) that if a subset A of C has an accumulation point in C, then the linear span of entire functions f which satisfy pB `δS n`1 qf " λf for some λ P A is dense in HpCq. The Godefroy-Shapiro Criterion (cf. for instance [20, Theorem 3.1]) will then imply that B δS n`1 is hypercyclic, and thus Theorem 4.1 will follow. Our first task is to describe the eigenvectors of B `δS n`1 . For n ě 0, we denote by Π n`1 : C n`1 Ñ C the canonical projection defined by Π n`1 pz 0 , . . . , z n q " z n for every pz 0 , . . . , z n q P C n`1 . For λ P C, let M λ be the pn `1q-square matrix defined as

M λ " ¨0 ¨¨¨¨¨¨0 pn`1q! δ λ 1 . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 ¨¨¨0 1 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' .
We begin by proving the following lemma.

Lemma 4.3. Let B be a continuous linear operator on HpCq and let δ ą 0. Suppose that there exists n ě 0 such that B " T n BT n . Then for every polynomial P P C n rzs and every λ P C, the entire function f λ,P defined by f λ,P pzq "P pzq `Πn`1 ˜żr0,zs pn `1q! δ expppz ´tqM λ q `p´B `λIqP ptq, 0, . . . , 0 ˘dt is the only eigenvector of B `δS n`1 associated to the eigenvalue λ such that T n f λ,P " P . In particular, dim kerpB `δS n`1 ´λIq " n `1.

Proof of Lemma 4.3. Let f belong to HpCq. We set P " T n f and g " f ´P . Since B " T n BT n and T n g " 0, we have that Bg " 0. Therefore pB `δS n`1 qf " λf if and only if BP `δS n`1 g " λg `λP .

Thus g is a solution of the ordinary differential equation pEq :

# y n`1 ´λ pn`1q! δ y " pn`1q! δ pλI ´BqP y pnq p0q " ¨¨¨" yp0q " 0.

Hence there exists a unique eigenvector f λ,P of B `δS n`1 associated to the eigenvalue λ such that T n f λ,P " P . It has the form f λ,P " P `g, where g is the unique solution of pEq.

As usual, in order to solve the differential equation pEq, one sets Y " py pnq , . . . , yq. Then y is a solution of pEq if and only if Y is a solution of the system

pE 1 q : $ & % Y 1 " M λ Y `pn`1q! δ ´p´B `λIqP ptq, 0, . . . , 0 Ȳ p0q " 0.
The unique solution of pE 1 q is given by Y pzq " ż r0,zs pn `1q! δ expppz ´tqM λ q `p´B `λIqP ptq, 0, . . . , 0 ˘dt which yields the expression of f λ,P as given in the statement of Lemma 4.3.

We will need a more explicit expression of f λ,P when P describes the canonical basis P 0 , . . . , P n of C n rzs, with P l pzq " z l , l " 0, . . . , n. For i, j " 0, . . . , n, let a ij P C be such that

Bz j " n ÿ i"0 a ij z i for every 0 ď j ď n.
We also set, for j " 0, . . . , n and l ě 1,

α pjq l " j ÿ m"0 ˆj m ˙p´1q m pn `1ql
`m and α pjq 0 " 1.

Lemma 4.4. Let B be a continuous linear operator on HpCq and let δ ą 0. Suppose that there exists n ě 0 such that B " T n BT n . Then we have for every i " 0, . . . , n f λ,P i pzq " z i ``8 ÿ

l"0 ˜ˆpn `1q! δ ˙l`1 1 ppn `1ql `nq! ˜λl`1 α piq l`1 z i`pn`1qpl`1q
´λl n ÿ j"0 a ji z pn`1qpl`1q`j α pjq l`1 ¸¸.

Proof of Lemma 4.4. We define the matrices J pjq " ´Jpjq m,l ¯1ďm,lďn`1 , for j " 0, . . . , n, by setting J pjq m,l " # 1, for m " j `1, . . . , n `1 and l " m ´j 0, otherwise and N pjq " ´N pjq m,l ¯1ďm,lďn`1 , for j " 1, . . . , n, by setting N pjq m,l " # 1, for m " 1, . . . , n `1 ´j and l " j `m 0, otherwise with N pn`1q " 0.

We then have, for every j " 0, . . . , n and every l ě 0 that

M pn`1ql`j λ " ˆλ pn `1q! δ ˙l ˆJpjq `λ pn `1q! δ N pn`1´jq
˙, which implies that exp ppz ´tqM λ q " `8 ÿ l"0 ˜n ÿ j"0 pz ´tq pn`1ql`j ppn `1ql `jq! ˆλ pn `1q! δ ˙l ˆJpjq `λ pn `1q! δ N pn`1´jq ˙¸.

It then follows from Lemma 4.3 that f λ,P i pzq " z i `żr0,zs `8 ÿ l"0 pz ´tq pn`1ql`n ppn `1ql `nq! ˆpn `1q! δ ˙l`1 λ l ˜λt i ´n ÿ j"0 a ji t j ¸dt.

In order to compute the above integral, we decompose t j as t j " ř j m"0 `j m ˘zj´m pt ´zq m and thus obtain that the integral in the expression above is equal to This then yields that

ż r0,zs `8 ÿ l"0 ˜p´1q pn`1ql`n ppn `1ql `nq! ˆpn `1q! δ ˙l`1 λ l ˜λ i ÿ m"0 ˆi m ˙zi´m pt ´zq pn`1ql`n`m
f λ,P i pzq " z i ``8 ÿ l"0 ˜1 ppn `1ql `nq! ˆpn `1q! δ ˙l`1 ˜αpiq l`1 λ l`1 z i`pn`1qpl`1q ´λl n ÿ j"0 α pjq l`1 a ji z pn`1qpl`1q`j ¸¸,
which is exactly the expression given in Lemma 4.4.

Our last task is to prove the density of the vector space spanned by the functions f λ,P , where P ranges over P P C n rzs and λ ranges over a certain subset A of C having an accumulation point in C. Proof of Lemma 4.5. We proceed as in the proof of [START_REF] Godefroy | Operators with dense, invariant, cyclic vector manifolds[END_REF]Theorem 5.1]. Let Λ be a continuous linear functional on HpCq that vanishes on each function f λ,P , λ P A, P P C n rzs. The lemma will be proved as soon as we show that Λ " 0. As in [START_REF] Godefroy | Operators with dense, invariant, cyclic vector manifolds[END_REF], we use the fact that there exists a complex Borel measure µ on C, supported on a disk of radius r ą 0 and centered at the origin, such that for every f P HpCq Λpf q " ż C f dµ.

For 0 ď j ď n and l ě 0, we set

u pjq l " ż C z pn`1ql`j dµpzq.
We aim to prove for all j and l, that u pjq l " 0. The continuity of Λ will then imply that Λ " 0.

For every polynomial P P C n rzs, we define F P : C Ñ C by setting

F P pλq " ż C f λ,P dµ, λ P C.
On the one hand, since Λpf λ,P q " 0 " ş C f λ,P dµ for every λ P A and every P P C n rzs, we have F P pλq " 0 for every λ P A.

On the other hand, since f λ,P depends holomorphically on λ by Lemma 4.3, and since µ is compactly supported, differentiation under the integral sign shows that F P is an entire function. Since A has an accumulation point, we deduce that F P " 0. In particular, for every P P C n rzs and every l ě 0,

F plq P p0q " ż C B l f λ,P Bλ l ˇˇˇλ "0
dµ " 0.

We will now show that the values of the u pjq l 's are linked to those of the derivatives F plq z j p0q. In order to do this, we need to compute the successive derivatives of f λ,z j with respect to λ at the point λ " 0.

We continue to denote by B the pn`1q-square matrix B " pa ij q 1ďi,jďn`1 , i.e. the matrix of the restriction of the operator B to C n rzs with respect to the canonical basis of C n rzs. We also denote by Ũl the vector C n`1 defined by Ũl " pα From Lemma 4.4 we deduce that

F P i p0q " u piq 0 ´pn `1q! δ 1 n! n ÿ j"0 a ji α pjq 1 u pjq 1
for every i " 0, . . . n, and since F P i p0q " 0 for i " 0, . . . , n we get that

Ũ0 " pn `1q! δn! B Ũ1 ,
where the notation B denotes the transpose of the matrix B. For l ě 1, we differentiate with respect to λ the expression given in Lemma 4.4 and obtain that for every i " 0, . . . , n,

B l F P i Bλ l p0q " ˆpn `1q! δ ˙l ˆl! ppn `1qpl ´1q `nq! α piq l u piq l ´pn `1q! δ l! ppn `1ql `nq! n ÿ j"0 a ji α pjq l`1 u pjq l`1 ¸" 0 from which we deduce that Ũl " pn `1q! δ ppn `1qpl ´1q `nq! ppn `1ql `nq! B Ũl`1 .
It follows that for every m ě 0 and every l ě 0, Ũl " ˆpn `1q! δ ˙m ppn `1qpl ´1q `nq! ppn `1qpl `m ´1q `nq! B m Ũl`m , with the convention that p´1q! " 1.

If we denote by } ¨} the sup norm on C n`1 and by |||¨||| the norm on pn `1q square matrices subordinated to this norm, we get that

› › › Ũl › › › ď ˆpn `1q! δ ˙m ppn `1qpl ´1q `nq! ppn `1qpl `m ´1q `nq! ||| B||| m › › › Ũl`m › › › for every l, m ě 0.
We now observe that for every j " 0, . . . , n and every l ě 1, we have

α pjq l " j ÿ m"0 ˆj m ˙p´1q m pn `1ql `m " ż 1 0 j ÿ m"0 ˆj m ˙p´1q m t pn`1ql`m´1 dt " ż 1 0
t lpn`1q´1 pt ´1q j dt, so that 0 ď α pjq l ď 1 for every l ě 1. Recall also that α pjq 0 " 1.

On the other hand, remembering that µ is supported on the disc centered at 0 and of radius r (which may be assumed to be bigger than 1), we have ˇˇu pjq l ˇˇ" ˇˇˇż C z pn`1ql`j dµpzq ˇˇˇď r pn`1ql`j |µ|pCq for every j " 0, . . . , n and l ě 0.

It thus follows that

› › › Ũl › › › ď ˆpn `1q! δ ˙m ppn `1qpl ´1q `nq! ppn `1qpl `m ´1q `nq! ||| B||| m r pn`1qpl`mq`n |µ|pCq.
Letting m go to infinity, we conclude that Ũl " 0. Thus u pjq l

" 0 for all j, l ě 0, from which it follows that Λ " 0. Lemma 4.5 is thus proved.

As explained at the beginning of the proof of Theorem 4.1, Lemma 4.5 combined with the Godefroy-Shapiro Criterion allows us to conclude that operators in L pM j q,pk j q of the form B`δS n`1 , with δ ą 0, n ě 0 and B " T n BT n , are hypercyclic. Hypercyclic operators are thus dense in pL pM j q,pk j q , SOTq. We have already observed that the set of hypercyclic operators is G δ in pL pM j q,pk j q , SOTq, so this concludes the proof of Theorem 4.1.

Operators supporting a hypercyclic algebra: proof of Theorem 1.1

A last crucial step in the proof of Theorem 1.1 is the following result.

Theorem 5.1. Let B be a continuous linear operator on HpCq such that there exists n ě 0 which satisfies B " T n BT n . Suppose also for every j ě 0 that the sequence pB m z j q m converges to 0 in HpCq as m goes to infinity. Then for every δ P p0, 1q, the operator A " B `δS n`1 supports a hypercyclic algebra.

To prove Theorem 5.1, we will apply [2, Lemma 1.6] (cf. also [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF]Corollary 2.4]), which provides a useful "Birkhoff-type Criterion" for proving that an operator admits a hypercyclic algebra.

Lemma 5.2 ([2]

). Let A be a continuous operator on a separable, metrizable and complete topological algebra X. Assume that for any pair pU, Vq of nonempty open sets in X, any neighborhood W of 0 in X, and for any integer m ě 1, one can find a vector u P U and an integer N ě 0 such that A N pu j q belongs to W for every 1 ď j ă m and A N pu m q belongs to V. Then A supports a hypercyclic algebra.

In [START_REF] Bayart | Hypercyclic algebras[END_REF], this criterion is applied in the following way: the vector u is found in a dense linear subspace of the space X spanned by holomorphic vector fields pE λ q λPC of eigenvectors of A. To compute A N pu j q, we require that these eigenvector fields be multiplicative, i.e. that E λ`µ " E λ ¨Eµ for every λ and µ in C. In our case, the product of two eigenvectors is not necessarily an eigenvector. To overcome this obstacle, we will introduce multiplicative vector fields whose powers are close to powers of eigenvectors fields. It is here that the assumption that the sequences pB m z j q m , j ě 0, converge to 0 in HpCq plays a crucial role.

We begin by giving a simpler expression for some of the eigenvectors exhibited in Lemma 4.3. Let B be an operator satisfying the hypothesis of Theorem 5.1 and let A be the operator A " B `δS n`1 , with δ P p0, 1q. For every α P C, we denote by e α the function defined by e α pzq " exppαzq, z P C. We also set

x 0 " ˆpn `1q! δ ˙1 n`1 ,
and we denote by φ the function defined by φpzq " ˆz x 0 ˙n`1 " δ pn `1q! z n`1 for every z P C.

For every β P Czφ ´1 pσ pBqq, where σ pBq denotes the spectrum of the operator B acting on HpCq, we put ε β " e β `pφpβqI ´Bq ´1 Be β .

We note that σ pBq is finite. Keeping in mind that Bε β is a polynomial of degree at most n, we have

Aε β " δ pn `1q! β n`1 e β `Be β `B pφpβqI ´Bq ´1 Be β " φpβqe β `φpβq pφpβqI ´Bq ´1 Be β `Be β ´pφpβqI ´Bq pφpβqI ´Bq ´1 Be β " φpβqε β ,
i.e. ε β is an eigenvector of A associated to the eigenvalue φpβq.

We now prove the following density lemma.

Lemma 5.3. Let B be a subset of C which has an accumulation point and is such that βe 2iπj n`1 belongs to B for every 0 ď j ď n and every β P B. Then the linear vector space span " ε β ; β P Bzφ ´1 pσ pBqq ‰ is dense in HpCq.

Proof. We will show that for every β P Bzφ ´1 pσ pBqq, we have span

" ε β 1 ; β 1 " βe 2iπj n`1 , j " 0, . . . , n ı " kerpA ´φpβqIq.
Then, since for every β P Bzφ ´1 pσ pBqq and every 0 ď j ď n the complex numbers βe Since φpBqzσ pBq has an accumulation point in C, Lemma 4.5 will imply that the linear vector space spanned by the family ε β ; β P Bzφ ´1 pσ pBqq ( is dense in HpCq.

So, let β belong to Bzφ ´1 pσ pBqq, and set β j " βe 2iπj n`1 , j " 0, . . . , n. We will check that the functions ε β 0 , . . . , ε βn are linearly independent in HpCq. Since for every j " 0, . . . , n the function ε β j is an eigenvector of A associated to the eigenvalue φpβq, and since, by Lemma 4.3, dim kerpA ´φpβqIq " n `1, Lemma 5.3 will thus be proven.

Let γ 0 , . . . , γ n be n`1 complex numbers such that ř n j"0 γ j ε β j " 0. Since for every j the function ε β j is the sum of e β j and a polynomial of degree at most n, differentiating n `1 times the latter sum we get that ř n j"0 β n`1 γ j e β j " 0. Since φp0q " 0 is an eigenvalue of B, β n`1 ‰ 0, and we thus have ř n j"0 γ j e β j " 0. Now since e β 0 , . . . , e βn are linearly independent functions in HpCq, γ 0 " ¨¨¨" γ j " 0, and ε β 0 , . . . , ε βn are linearly independent. Lemma 5.3 is thus proved.

The vectors ε β , β P B, thus span a dense linear subspace of HpCq provided that B has an accumulation point and is invariant by the rotation of angle 2π n`1 . However, in general it does not hold that ε α ¨εβ " ε α`β . In order to overcome this difficulty, we will take advantage of the fact that, of course, e α ¨eβ " e α`β , and of the link between A m e β and A m ε β given by the following Fact 5.4.

Fact 5.4. For every β P Czφ ´1 pσ pBqq and every m ě 0, we have

A m e β " φpβq m ε β ´pφpβqI ´Bq ´1 B m`1 e β . (5.1)
Proof. By definition of ε β , we have the equality e β " ε β ´pφpβqI ´Bq ´1 Be β . If we assume for some integer m ě 0 that (5.1) is true, then, since pφpβqI ´Bq ´1 B m`1 e β is a polynomial of degree at most n, we have

A m`1 e β " φpβq m Aε β ´B pφpβqI ´Bq ´1 B m`1 e β " φpβq m`1 ε β ´pφpβqI ´Bq ´1 B m`2 e β .
Therefore, by induction, (5.1) is valid for every m ě 0.

When pB m P q m converges to 0 for every polynomial P of degree at most n, Fact 5.4 implies that A m e β is close to A m ε β " φpβq m ε β . This will be a key element in showing that A satisfies the assumptions of Lemma 5.2, cf. Lemma 5.6 below. In the proof of this lemma, we will need Lemma 5.5, which can be compared to the beginning of the proof of Theorem 2.1 in [START_REF] Bayart | Hypercyclic algebras[END_REF]. Lemma 5.5. Let m be a positive integer, and let η and ε be positive real numbers. To the parameters η and ε we associate two subsets A and B of C defined as follows: A " Dp0, ηq is the open disk of center 0 and of radius η in C, and B is the set B " n ď j"1 tζ j p1 `itq ; t P r´η, ηsu where ζ j " x 0 p1 `εqe 2iπj n`1 for every 0 ď j ď n. Then there exist η ą 0 and ε ą 0 sufficiently small such that for every j " 1, . . . , m, every d " 0, . . . , j with d ‰ m, every ξ1 , . . . , ξj´d P A and every ξ 1 , . . . , ξ d P B, we have

ˇˇˇφ ˆξ 1 `¨¨¨`ξ j´d `ξ1 `¨¨¨`ξ d m ˙ˇˇˇă 1, (5.2)
and for every l " 1, . . . , m and every ξ 1 , . . . , ξ l P B which are not all equal, we have ˇˇˇφ ˆξ1 `¨¨¨`ξ l m

˙ˇˇˇă |φpξ 1 q| 1 m ¨¨¨|φpξ l q| 1 m . (5.3)
Proof. We first prove (5.3). We consider first the case where l ă m: in this case we have ˇˇˇξ 1 `¨¨¨`ξ l m ˇˇˇď m ´1 m x 0 p1 `εqp1 `ηq ă x 0 as soon as ε and η are sufficiently small. Thus ˇˇˇφ ˆξ1 `¨¨¨`ξ l m ˙ˇˇˇă φpx 0 q " 1.

On the other hand, for every i " 1, . . . , l, |φpξ i q| " p1 `εq n`1 ą 1 for every ε ą 0. Thus ˇˇˇφ ˆξ1 `¨¨¨`ξ l m

˙ˇˇˇă |φpξ 1 q| 1 m ¨¨¨|φpξ l q| 1 m .
In the case where l " m, we denote by i j the unique integer with 0 ď i j ď n such that ξ j " ζ i j p1 `it j q " x 0 p1 `εqe 2iπ n`1 i j p1 `it j q for some t j P r´η, ηs. We first assume that the sequence pi j q 1ďjďm is not constant. Since the unit disc is strictly convex, we have

α :" sup ˇˇˇˇ1 m m ÿ p"1 e 2iπ n`1 jp ˇˇˇˇă 1
where the upper bound is taken over all the m-tuples of integers pj 1 , . . . , j m q lying between 0 and n which are not all equal. Hence if ε is sufficiently small

ˇˇˇζ i 1 `¨¨¨`ζ im m ˇˇˇď αx 0 p1 `εq ă x 0 .
By continuity, if η is sufficiently small (depending on ε but not on the ξ j 's), we have ˇˇˇξ 1 `¨¨¨`ξ m m ˇˇˇă x 0 , and as before this yields that ˇˇˇφ ˆξ1 `¨¨¨`ξ m m

˙ˇˇˇă 1 ď |φpξ 1 q| 1 m ¨¨¨|φpξ m q| 1 m .
Next, we assume that the sequence pi j q 1ďjďm is constant, i.e. i j " i 1 for every 1 ď j ď m, and we consider the function ψ defined on R by ψptq " ln |φ pζ i 1 p1 `itqq| " n`1 2 lnp1 `t2 q `pn `1q lnp1 `εq. We have ψ 2 ptq " pn `1q 1´t 2 p1`t 2 q 2 and if 0 ă η ă 1, the function ψ is strictly convex on r´η, ηs. It follows that ˇˇˇφ ˆξ1 `¨¨¨`ξ m m

˙ˇˇˇă |φpξ 1 q| 1 m ¨¨¨|φpξ m q| 1 m ,
and so (5.3) is shown in this case as well.

We now establish (5.2). Provided ε and η are sufficiently small, we have ˇˇˇξ 1 `¨¨¨`ξ j´d `ξ1 `¨¨¨`ξ d m ˇˇˇď pj ´dqη `dx 0 p1 `εq m ď mη `m ´1 m x 0 p1 `εq ă x 0 and (5.2) readily follows from these inequalities.

Next, we finally prove that the operator A satisfies the hypothesis of Lemma 5.2.

Lemma 5.6. Let B be a continuous linear operator on HpCq such that there exists n ě 0 which satisfies B " T n BT n . We suppose that the sequence pB m z j q m converges to 0 in HpCq as m tends to infinity for every j ě 0. Let δ P p0, 1q and we set A " B `δS n`1 .

Then A fulfills the assumptions of Lemma 5.2.

Proof. Let U and V be two open sets in HpCq, let W be a neighborhood of 0 in HpCq and let m be a positive integer. Let A and B be the subsets of C given by Lemma 5.5. Since A has accumulation points in C, the linear vector space span re α ; α P As is dense in HpCq.

Thus there exist a 1 , . . . , a p P C and α 1 , . . . , α p P A such that the function ř p j"1 a j e α j belongs to U.

Since B is invariant by the rotation of angle 2π n`1 , and since B has accumulation points, Lemma 5.3 asserts that span rε β ; β P Bs is dense in HpCq. Thus there exist b 1 , . . . , b q P C and β 1 , . . . , β q P B such that the function ř q j"1 b j ε β j belongs to V. During the forthcoming computations, quantities of the form φ ˜p ÿ j"0 r j α j `1 m q ÿ j"0 s j β j will appear, where the r j 's and s j 's are positive integers such that ř p j"1 r j `řq j"1 s j ď m. We will need to be able to ensure that the sums ř p j"0 r j α j `1 m ř q j"0 s j β j do not belong to the spectrum of B. In order to ensure this condition, we notice that for all tuples r " pr 1 , . . . , r p q P N p and s " ps 1 , . . . , s q q P N q such that ř p j"1 r j `řq j"1 s j ď m, the set E r,s " # ppα 1 , . . . , α p q, pβ 1 , . . . , β q qq P A p ˆBq ; φ ˜p ÿ j"0 r j α j `1 m q ÿ j"0 s j β j ¸P σ pBq + is a closed subset of A p ˆBq with empty interior. Since ř p j"1 r j `řq j"1 s j ď m, there are only finitely many such sets. Therefore, the set (5.4)

A p ˆBq z ¨ď rPN p ,sPN q ř p j"1 r j `řq j"1 s j ďm E r,s ‹ ‹ ' is dense in A p ˆBq .

We now observe that e α 1 tends to e α in HpCq as α 1 tends to α, and that ε β 1 tends to ε β in HpCq as β 1 tends to β, with β, β 1 P Czφ ´1pσ pBqq. The first assertion is obvious. In order to prove the second assertion, let r be a positive integer. We equip C n rzs with the norm N r and denote by N r the norm on the space BpC n rzsq of linear continuous operators on C n rzs induced by N r . We notice that for every w P Czσ pBq, for every polynomial P , pwI ´Bq ´1P is still a polynomial, and that the map Θ : w Þ Ñ pwI ´Bq ´1B from Czσ pBq into BpC n rzsq is well defined and continuous. Now for all β, β 1 P Czφ ´1pσ pBqq, we have Therefore, the continuity of Θ implies that N r `εβ 1 ´εβ ˘tends to zero as β 1 P Czφ ´1pσ pBqq tends to β P Czφ ´1pσ pBqq. Hence, ε β 1 tends to ε β in HpCq when β 1 P Czφ ´1pσ pBqq tends to β P Czφ ´1pσ pBqq.

Therefore, by choosing `pα 1 1 , . . . , α 1 p q, pβ 1 1 , . . . , β 1 q q ˘in the set defined in (5.4) close enough to ppα 1 , . . . , α p q, pβ 1 , . . . , β q qq, we can assume that for every r P N p and every s P N q such that ř p j"1 r j `řq j"1 s j ď m, the complex number φ ´řp j"0 r j α j `1 m ř q j"0 s j β j ¯does not belong to σ pBq.

For N ě 0 and j " 1, . . . , q, let c j pN q be a complex number such that

c j pN q m " b j φpβ j q N Ẅe set u N " p ÿ l"1 a l e α l `q ÿ j"1 c j pN qe β j m .
For every β P B, we have |β| ą x 0 , thus |φpβq| ą 1 and the sequence ˜q ÿ j"1 c j pN qe β j m ¸N converges to 0 in HpCq as N goes to infinity. Consequently, u N belongs to U if N is large enough.

We now compute the quantities u k N and A N u k N for every k P t1, . . . , mu. For every d-tuple L " pl 1 , . . . , l d q P t1, . . . , pu d , we put a L " a l 1 ¨¨¨a l d when d ě 1 and a H " 1. For every J " pj 1 , . . . , j k´d q P t1, . . . , qu k´d , we put c J pN q " c j 1 pN q ¨¨¨c j k´q pN q when k ´d ě 1, and c H pN q " 1. We have

u k N " k ÿ d"0 ÿ LPt1,...,pu k´d JPt1,...,qu d hpL, Jqa L c J pN qe α l 1 `¨¨¨`α l k´d `1 m pβ j 1 `¨¨¨`β j d q
where the hpL, Jq's are constants depending only on L and J but not on N (cf. the proof of [2, Theorem 2.1]).

Setting λ " α l 1 `¨¨¨`α l k´d `1 m pβ j 1 `¨¨¨`β j d q, we get from Fact 5.4 that c J pN qA N e λ " c J pN q ´φpλq N ε λ ´pφpλqI ´Bq ´1 B N `1e λ ¯.

(a) If k ă m, for every d P t0, . . . , ku it follows from (5.2) that |φpλq| ă 1. Since the sequence pc J pN qq N is bounded, it follows that `cJ pN qφpλq N ε λ ˘N converges to 0 in HpCq. Since `BN`1 e λ ˘N also converges to 0 in HpCq, `cJ pN qA N e λ ˘N converges to 0, and so there exists N sufficiently large such that for every k P t1, . . . , m ´1u, the function A N u k N belongs to W. (b) If k " m and d ă m, again from (5.2) in Lemma 5.5 we have |φpλq| ă 1, and since pc J pN qq N is a bounded sequence, `cJ pN qA N e λ ˘N converges to 0 in HpCq.

(c) If k " d " m and if the β j 's are not all equal, then

ˇˇc J pN qφpλq N ˇˇ" |b j 1 ¨¨¨b jm | 1 m ˇˇˇˇˇφ ´βj 1 `¨¨¨`β jm m φ pβ j 1 q 1 m ¨¨¨φ pβ jm q 1 m ˇˇˇˇˇN .
Assertion (5.3) in Lemma 5.5 yields that

ˇˇˇˇˇφ ´βj 1 `¨¨¨`β jm m φ pβ j 1 q 1 m ¨¨¨φ pβ jm q 1 m ˇˇˇˇˇă 1,
so that pc J pN qφpλq N ε λ q N converges to 0 in HpCq. Since `Bn`1 e λ ˘N converges to 0 in HpCq, it follows again that `cJ pN qA N e λ ˘N converges to 0 in HpCq.

(d) Finally, if k " m " d and if all the β j 's are equal, then λ " β j 1 , J " pj 1 , . . . , j 1 q and

c J pN qA N e λ " c j 1 pN q m φ pβ j 1 q N ε β j 1 ´pφpβ j 1 qI ´Bq ´1 B N `1e β j 1 " b j 1 ε j 1 ´pφ pβ j 1 q I ´Bq ´1 B N `1e β j 1 .
Since pB N `1e β j 1 q N converges to 0 in HpCq, it follows that `cJ pN qA N e λ ˘N converges to b j 1 ε j 1 .

From assertions (b)-(d) above, we finally deduce that the sequence `AN u m N q ˘N converges to ř q j"1 b j ε j , and so there exists N large enough such that A N u m N belongs to V. This ends the proof of Lemma 5.6.

Proof of Theorem 1.1: Let pU p q pě1 be a basis of open sets of HpCq, and let pW r q rě1 be a basis of neighborhoods of 0 in HpCq. Now let G :" ! A P L pM j q,pk j q ; @p, q, r ě 1, @m ě 1, Du P U p , DN ě 1, such that @0 ď n ă m, T N pu n q P W r and T N pu m q P U q ( . Therefore, in order to prove Theorem 1.1, it suffices to prove that G is a dense G δ set.

We can rewrite G as G " č p,q,r,mě1

ď uPUp N ě1 G p,q,r,m,u,N ,
where G p,q,r,m,u,N :" tT P L pM j q,pk j q ; @0 ď n ă m, T N u m P W r and T N u m P U q u.

It suffices to show that G p,q,r,m,u,N is an open set in L pM j q,pk j q . Let T 0 belong to G p,q,r,m,u,N .

If T is sufficiently SOT-close to T 0 , then Claim 4.2 implies that for every 0 ď n ă m, T N u n

Since M 1 ą 1, we have › › ›v pkq › › › ÝÑ 0 as k ÝÑ `8. For n " m 0 , m 0 `1, . . . , m 1 we have

" v pkq ‰ n " N ÿ i"0 ´αi M 1k´1 δ ¯n{m 0 e kpN `1q`i `d ÿ i"N `1´α i M 1k
¯n{m 0 e kpN `1q`i (recall that the product considered on X is the coordinatewise product with respect to the canonical basis). We thus have T k " v pkq ‰ m 0 " v by (6.1). Since M 1 ą 1, we also have

› › › " v pkq ‰ m 0 › › › ÝÑ 0 as k ÝÑ `8. On the other hand, if m 0 ă n ď m 1 , we have T k " v pkq ‰ n " N ÿ i"0 ´αi M 1k´1 δ ¯n{m 0 M 1k´1 δ e i `d ÿ i"k`1 ´αi M 1k ¯n{m 0 M 1k e i by (6.1). So T k " v pkq ‰ n " N ÿ i"0 α n{m 0 i ¨1 `M 1k´1 δ ˘n{m 0 ´1 ¨ei `d ÿ i"N `1 α n{m 0 i ¨1 `M 1k ˘n{m 0 ´1 ¨ei and since n m 0 ´1 ą 0, we obtain that › › ›T k " v pkq ‰ n › › › ÝÑ 0 as k ÝÑ `8
for every m 0 ă n ď m 1 . Summarising, we have shown that u `vpkq ÝÑ u as k ÝÑ `8 (6.4) T k `r u ˚m0 `"v pkq ‰ m 0 ˘ÝÑ v as k ÝÑ `8 (6.5)

T k `r u ˚n `"v pkq ‰ n ˘ÝÑ 0 as k ÝÑ `8 for every m 0 ă n ď m 1 . (6.6)

We now define a bounded operator L on X by setting

Le j " # f j , if 0 ď j ď J e j , if j ą J.
Then we have

}Lx ´x} " › › › › › J ÿ j"0 x j pf j ´ej q › › › › › ď max 0ďjďJ |x j | J ÿ j"0 }f j ´ej } ď δ 1 ¨}x} for every x P X, so that }L ´I} ď δ 1 ă ε. If 0 ă δ 1 ă 1, L is invertible, and › › L ´1 ´I› › ď δ 1 1 ´δ1 ď 2δ 1
as soon as 0 ă δ 1 ă 1{2. Let now x k " u `vpkq and y k " L ´1pu `vpkq q. If k is sufficiently large, the support of v pkq is disjoint from the interval r0, Js. We have

y k " J ÿ j"0 γ j e j `N ÿ i"0 ´αi M 1k´1 δ ¯1{m 0 e kpN `1q`i `d ÿ i"N `1´α i M 1k
¯1{m 0 e kpN `1q`i so that for every n ě 1,

y n k " J ÿ j"0 γ n j e j `N ÿ i"0 ´αi M 1k´1 δ ¯n{m 0 e kpN `1q`i `d ÿ i"N `1´α i M 1k
¯n{m 0 e kpN `1q`i .

It follows that (6.7)

Lpy n k q " r u ˚n `"v pkq ‰ n for all k sufficiently large.

By (6.4), (6.5), and (6.6), combined with (6.7), we have, as k goes to infinity,

x k ÝÑ u and y k ÝÑ L ´1u (6.8)

T k Lpy m 0 k q ÝÑ v T k Lpy n k q ÝÑ 0 for every m 0 ă n ď m 1 . Set now T 1 :" L ´1T L. Then }T 1 ´T } " › › L ´1T L ´T › › ď › › L ´1T pL ´Iq › › `› › pL ´1 ´IqT › › ď }T } δ 1 p1 `2 δ 1 q `}T } . 2 δ 1 ă 4}T } δ 1 , so that }T 1 } ă M and dpT 1 , T q ă ε 2
, provided that δ 1 is sufficiently small. We also deduce from the properties above that, as k goes to infinity,

T k 1 py m 0,1 k
q ÝÑ L ´1v (6.9)

T k 1 py n k q ÝÑ 0 for every m 0 ă n ď m 1 . (6.10) Now, L ´1u " z belongs to U and if δ 1 is sufficiently small, L ´1v belongs to V. If we choose k sufficiently large, and then set b " y k , we eventually obtain that b P U (by (6.8)), T k 1 `bm 0 ˘P V (by (6.9)), and T k 1 `bn ˘P W for every m 0 ă n ď m 1 (by (6.10)) which is exactly what is required by Lemma 6.1.

Returning to the proof of Theorem 1.2, we now consider an operator S P B M pXq with }S} ă M , and ε ą 0. Our aim is to construct an operator T P G M pXq with dpT, Sq ă ε. Let pε l q lě1 be a sequence of positive numbers such that ř lě1 ε l ă ε. Using Lemma 6.1, we can construct by induction on l ě 1 ' a sequence pT l q lě0 of operators on X, with T 0 " S, ' a sequence pb l q lě1 of vectors of X, ' a sequence pk l q lě1 of positive integers such that for every l ě 1, (a) dpT l , T l´1 q ă ε l ;

(b) b l P U q l Ď U q l ; (c) @ 1 ď l 1 ď l, T k l 1 l `bm 0,l 1 l 1 ˘P U r l 1 and @ m 0,l 1 ă n ď m 1,l 1 , T k l 1 l `bn l 1 ˘P W s l 1 .
The fact that we can ensure, at each step l ě 1, that condition (c) holds for every 1 ď l 1 ď l, and not only for l, relies on the observation that the maps T Þ ÝÑ T k , k ě 1, are SOT-continuous from B M pXq into BpXq. Thus if T l is constructed sufficiently close to T l´1 , i.e. if dpT l , T l´1 q is sufficiently small, our induction assumption implies that (c) is true for every 1 ď l 1 ă l.

Since pB M pXq, dq is a complete metric space, the sequence of operators pT l q lě1 converges in pB M pXq, dq to an operator T P B M pXq such that dpT, Sq ă ε. Moreover, letting l tend to infinity in (c) above, and using again the SOT-continuity of the maps T Þ ÝÑ T k , we obtain that for all l ě 1, T k l `bm 0,l l ˘P U r l Ď U r l , and T k l `bn l ˘P W s l 1 Ď W s l for all m 0,l ă n ď m 1,l . Hence T belongs to G M pXq.

We have thus proved that G M pXq is dense in pB M pXq, SOTq. This finishes the proof of Theorem 1.2.

There are other interesting topologies which can turn the operator balls B M pXq into Polish spaces. One of the most relevant is the so-called Strong ˚Operator Topology (SOT ˚): if pT α q is a net of operators in BpXq, T P BpXq and T ˚P BpX ˚q is the adjoint of T , then we say that T α SOT ˚/ / T if and only if T α SOT / / T in X and T α SOT / / T ˚in X ˚.

When X is a Banach space with separable dual, the balls B M pXq, M ą 0, are Polish spaces when endowed with the SOT ˚topology. See the works [START_REF] Eisner | On typical properties of Hilbert space operators[END_REF], [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF], [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF], [START_REF] Grivaux | Generic properties of p-contractions and similar operator topologies[END_REF], and [START_REF] Grivaux | Local spectral properties of typical contractions on p-spaces[END_REF] for a study of typical properties of operators for the SOT ˚, as well as explanations on the relevance of this topology. Theorem 1.2 admits the following analogue for the SOT ˚topology. Theorem 6.2. Let X " p pNq, 1 ă p ă `8, or X " c 0 pNq, endowed with the coordinatewise product. Let M ą 1. A typical operator T P pB M pXq, SOT ˚q admits a hypercyclic algebra.

Proof. The proof is similar to that of Theorem 1.2. Since the set G M pXq is SOT-G δ , it is also SOT ˚-G δ . And a look at the proof of Theorem 1.2 shows that G M pXq is in fact SOT ˚-dense in B M pXq.

We say that T P BpXq is dual hypercyclic if both T and its adjoint T ˚are hypercyclic.

The first examples of such operators were obtained by Salas [START_REF] Salas | A hypercyclic operator whose adjoint is also hypercyclic[END_REF][START_REF] Salas | Dual disjoint hypercyclic operators[END_REF] and Petersson [START_REF] Petersson | Spaces that admit hypercyclic operators with hypercyclic adjoints[END_REF], and it is an immediate consequence of the remark after [17, Proposition 2.3] (see also [START_REF] Grivaux | Local spectral properties of typical contractions on p-spaces[END_REF]Fact 2.1]) that when X " p pNq, 1 ă p ă `8, or X " c 0 pNq, an SOT ˚-typical T P B M pXq is dual hypercyclic. Using the same argument, we can deduce from Theorem 6.2 the following result. Proposition 6.3. If X " p pNq, 1 ă p ă `8, or X " c 0 pNq, and M ą 1, a typical operator T P pB M pXq, SOT ˚q is such that both T and T ˚admit a hypercyclic algebra (with respect to the coordinatewise product).

The existence of T P BpXq such that both T and T ˚admit a hypercyclic algebra can also be deduced directly from [START_REF] Bayart | Baire theorem and hypercyclic algebras[END_REF]Corollary 4.11] and [START_REF] Salas | A hypercyclic operator whose adjoint is also hypercyclic[END_REF].

Typicality of Admitting a Hypercyclic Subspace

Let X be a complex separable infinite-dimensional Banach space. Our aim in this final section is to study whether, given M ą 1, a typical operator T P pB M pXq, SOTq admits a closed infinite-dimensional hypercyclic subspace. When it makes sense, we also consider this question for T P pB M pXq, SOT ˚q.

For a hypercyclic operator T P BpXq, a hypercyclic subspace is defined as a closed infinite-dimensional subspace Z Ď X such that every nonzero vector in Z is hypercyclic for T . Since the early work of Bernal González and Montes-Rodríguez [START_REF] González | Universal functions for composition operators[END_REF], the investigation of hypercyclic subspaces has amassed a vast literature, as documented in [START_REF] Bayart | Dynamics of linear operators[END_REF]Chapter 8] and [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Chapter 10]. In contrast to the generic property that the set HC pT q Y t0u always contains a dense linear manifold of hypercyclic vectors, it turns out that there exist hypercyclic operators that do not admit a hypercyclic subspace. Examples of operators that support hypercyclic subspaces in the Fréchet space setting include the differentiation and translation operators acting on the space HpCq of entire functions (cf. [START_REF] Grosse-Erdmann | Linear chaos[END_REF]Examples 10.12 and 10.13]). For the Banach spaces X " p pNq, 1 ď p ă 8, or X " c 0 pNq, a weighted backward shift B w P BpXq admits a hypercyclic subspace if The following characterization of operators that satisfy the Hypercyclicity Criterion and admit a hypercyclic subspace was identified by León-Saavedra and Montes-Rodríguez [START_REF] León-Saavedra | Spectral theory and hypercyclic subspaces[END_REF] in the Hilbert space setting, and by González et al. in [START_REF] González | Semi-Fredholm theory: hypercyclic and supercyclic subspaces[END_REF] for separable complex Banach spaces.

Theorem 7.1 ([15], [START_REF] León-Saavedra | Spectral theory and hypercyclic subspaces[END_REF]). Let X be a separable complex Banach space. Suppose that T P BpXq satisfies the Hypercyclicity Criterion. The following assertions are equivalent.

(i) T possesses a hypercyclic subspace.

(ii) There exists some closed infinite-dimensional subspace Z 0 Ď X and an increasing sequence of integers pn k q such that T n k x Ñ 0 for all x P Z 0 .

(iii) There exists some closed infinite-dimensional subspace Z 0 Ď X and an increasing sequence of integers pn k q such that sup k T n k |Z 0 ă 8. (iv) The essential spectrum σ e pT q of T intersects the closed unit disk.

We recall that an operator T P BpXq is said to be Fredholm if both the dimension of its kernel dim kerpT q and the codimension of its range codim ranpT q are finite. Equivalently, T is Fredholm if and only if it has closed range and dim kerpT q ă 8 and dim kerpT ˚q ă 8. The essential spectrum σ e pT q of the operator T P BpXq is defined to be σ e pT q :" tλ P C : T ´λ is not Fredholmu.

The following theorem is a straightforward consequence of results from [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF] and [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF]. Theorem 7.2. Let X " p pNq, 1 ď p ă `8, or X " c 0 pNq. For every M ą 1, a typical T P pB M pXq, SOTq admits a hypercyclic subspace.

Proof. An elementary adaptation of the proofs of [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF]Proposition 2.3] and [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF]Proposition 2.16] shows that a typical operator T P pB M pXq, SOTq satisfies the Hypercyclicity Criterion.

  m ˙zj´m pt ´zq pn`1ql`n`m ¸¸dt " ˆj m ˙zj´m p´zq pn`1qpl`1q`m pn `1qpl `1q `m ¸¸.

Lemma 4 . 5 .

 45 Let B be a continuous linear operator on HpCq such that B " T n BT n for some n ě 0 and let δ ą 0. For every subset A of C which has an accumulation point in C, the vector space span « ď λPA kerpB `δS n`1 ´λIq ff is dense in HpCq.

2iπj n` 1 still

 1 belong to Bzφ ´1 pσ pBqq, it will follow that span β P Bzφ ´1 pσ pBqq ‰ .

N

  r `εβ 1 ´εβ ˘"N r `eβ 1 ´eβ `Θpφpβ 1 qqT n e β 1 ´ΘpφpβqqT n e β ďN r `eβ 1 ´eβ ˘`N r `Θpφpβ 1 qq ´Θpφpβqq ˘Nr `eβ 1 Nr pΘpφpβqq N r `eβ 1 ´eβ ˘.

  |w j`k | ă `8,cf.[START_REF] Grosse-Erdmann | Linear chaos[END_REF] Example 10.10]. However, it is well known that scalar multiples cB of the backward shift for |c| ą 1 do not possess hypercyclic subspaces (cf.[START_REF] Grosse-Erdmann | Linear chaos[END_REF] Example 10.26]).

All three authors were supported by the Labex CEMPI (ANR-11-LABX-0007-01). This work began when the second author was a CEMPI Postdoctoral Fellow at the University of Lille. We are grateful to

belongs to W r and T N u m belongs to U q . Thus G p,q,r,m,u,N is indeed SOT-open and G is a G δ set.

We now check that G is dense in L pM j q,pk j q . Let A belong to L pM j q,pk j q and let V be a SOT-neighborhood of A in L pM j q,pk j q . It follows from Corollary 3.9 that there exist B P L pM j q,pk j q , δ P p0, 1q and n ě 1 such that B `δS n`1 belongs to V, B " T n BT n and B n`1 " 0. By Lemma 5.6, the operator B `δS n`1 satisfies the hypothesis of Lemma 5.2 and thus belongs to G. Hence G is dense in L pM j q,pk j q . This concludes the proof of Theorem 1.1.

Proof of Theorem 1.2 and related results

In this section, the setting is the following: X is one of the (complex) Banach spaces p pNq, 1 ď p ă `8, or c 0 pNq. We endow X with the coordinatewise product, so that it becomes a Banach algebra. For every M ą 1, a typical operator T P pB M pXq, SOTq is hypercyclic, see the proof of [START_REF] Grivaux | Linear dynamical systems on Hilbert spaces: typical properties and explicit examples[END_REF]Proposition 2.3] which can be adapted in a straightforward way from the Hilbertian setting to the case where X " p pNq, 1 ď p ă `8, or X " c 0 pNq. Theorem 1.2 augments this statement by showing that, in fact, a typical T admits a hypercyclic algebra.

Proof of Theorem 1.2. Let pU q q qě1 be a basis of the topology of X consisting of open balls. We write U q " Bpx q , ρ q q for each q ě 1, where x q is the center of the ball and ρ q its radius. For each s ě 1, we set W s " Bp0, 2 ´sq, so that pW s q sě1 is a basis of neighborhoods of 0 in X. Set also U q " Bpx q , 1 2 ρ q q and W s " Bp0, 2 ´ps`1q q, so that U q Ď U q and W s Ă W s . Let `Uq l , U r l , W s l , m 0,l , m 1,l ˘lě1 be an enumeration of all the tuples pU, V, W, m 0 , m 1 q, where U and V belong to the set tU q ; q ě 1u, W belongs to the set tW s ; s ě 1u, and pm 0 , m 1 q is a pair of positive integers with m 0 ă m 1 .

Proceeding as in the proof of Theorem 1.1, we consider the set

By [3, Corollary 2.4] (or by Lemma 5.2 in Section 5 above), every operator belonging to G M pXq admits a hypercyclic algebra. In order to show that the property of having a hypercyclic algebra is typical in pB M pXq, SOTq, it thus suffices to show that G M pXq is SOT-dense in B M pXq.

Let d be a distance on B M pXq which induces the SOT and turns B M pXq into a complete separable metric space. The main step in order to show the density of G M pXq in pB M pXq, SOTq is the following lemma. Lemma 6.1. Let S P B M pXq with }S} ă M , and let ε ą 0. Let also U and V be two nonempty open subsets of X, let W be an open neighborhood of 0, and let m 0 ă m 1 be two positive integers. Then there exists an operator T P B M pXq with }T } ă M , an integer k ě 1 and a vector b P X such that:

Proof. Denote by pe n q ně0 the canonical basis of X, and let E N :" re 0 , e 1 , . . . , e N s be the linear span of the first N `1 basis vectors in X, N ě 0. Let also P N denote the canonical projection of X onto E N , and set S N " P N SP N . We have }S N } ď }S} ă M . Let δ ą 0 and γ ą 0 be such that maxp1, }S} `δq ă M ´γ ă M .

Setting M 1 :" M ´γ, we consider the operator T acting on X defined by

Then T is a bounded operator on X with }T } " M 1 ă M . Moreover, T can be made as close to S as we wish with respect to the SOT, provided that N is chosen sufficiently large. We fix N so large that dpT, Sq ă ε 2 Öperators of this form were introduced in [17, Proposition 2.10], where it is shown that for every complex number λ with |λ| ă M 1 , the eigenspace kerpT ´λIq is pN `1qdimensional. Whenever Λ is a subset of the open disk Dp0, M 1 q Ă C, which has an accumulation point in Dp0, M 1 q, the eigenspaces kerpT ´λIq, λ P Λ, span a dense linear subspace of X. In particular, the vector space H ´pT q :" span r kerpT ´λIq ; |λ| ă 1s is dense in X. For every k ě 1 and every n ě 0, we have:

T k e kpN `1q`n "

We choose a vector z P U, with finite support, which we write as z " ř J j"0 γ j e j . Since H ´pT q is dense in X, there exists for each 0 ď j ď J a vector f j P H ´pT q with }f j ´ej } ă δ 1 J`1 , where 0 ă δ 1 ă ε is sufficiently small (how small will be specified in the sequel). Letting u " ř J j"0 γ j f j , for δ 1 sufficiently small, u belongs to U. We now set for every n ě 1 (6.2) r u ˚n :"

Observe that r u ˚n belongs to H ´pT q for every n ě 1, so that

Let now v be a vector belonging to V, of the form v " ř d i"0 α i e i , with d ą N . For every k ě 1, we set (6.3) v pkq :"

¯1{m 0 e kpN `1q`i .

It thus suffices to show that the essential spectrum of a typical T P pB M pXq, SOTq is equal to the closed disk Dp0, M q. As observed in [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF]Remark 4.5], it follows from [12, Lemma 5.13] that a typical T P pB M pXq, SOTq is not Fredholm. Consider now the set A :" tT P B M pXq ; @λ P Dp0, M q, T ´λ is surjectiveu.

By the topological 0 ´1 law [18, Proposition 3.2], A is either meager or comeager in pB M pXq, SOTq.

-Suppose first that A is SOT-comeager in pB M pXq, SOTq (which is known to happen when X " 1 pNq or when X " 2 pNq). Since a typical T P pB M pXq, SOTq is not Fredholm, the continuity of the Fredholm index on Dp0, M q implies that for all λ P Dp0, M q, T ´λ is not Fredholm (see the argument just before Remark 4.5 in [START_REF] Grivaux | Does a typical p-space contraction have a non-trivial invariant subspace?[END_REF]). Hence σ e pT q " Dp0, M q.

-Suppose now that A is SOT-meager in pB M pXq, SOTq (which is known to happen when X " p pNq for p ą 2). As the set of operators T P pB M pXq, SOTq such that T ´λ has dense range for every λ P Dp0, M q is comeager in pB M pXq, SOTq (cf. [18, Proposition 3.9]), a typical T P pB M pXq, SOTq is such that for every λ P Dp0, M q, T ´λ does not have closed range. Hence T ´λ is not Fredholm, and σ e pT q " Dp0, M q.

The following shows that we have a similar statement for SOT ˚-typical operators.

Theorem 7.3. Let X " p pNq, 1 ă p ă `8. For every M ą 1, a typical T P pB M pXq, SOT ˚q admits a hypercyclic subspace.

Proof. This is a direct consequence of the fact that a typical operator T P pB M pXq, SOT ˚q satisfies the Hypercyclicity Criterion, and of [START_REF] Grivaux | Local spectral properties of typical contractions on p-spaces[END_REF]Theorem 3.1], which states that a typical T P pB M pXq, SOT ˚q is such that σ e pT q " Dp0, M q.

We can also deduce from Theorem 7.3 the following result. Proposition 7.4. Let X " p pNq, 1 ă p ă `8. For every M ą 1, a typical operator T P pB M pXq, SOT ˚q is such that T and its adjoint T ˚both admit hypercyclic subspaces.

(S. Grivaux) Univ. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille,

France

Email address: william.alexandre@univ-lille.fr Email address: clifford.gilmore@uca.fr Email address: sophie.grivaux@univ-lille.fr