Invariant subspaces for Fréchet spaces without continuous norm
aa r X i v : . [ m a t h . F A ] J u l INVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUTCONTINUOUS NORM
QUENTIN MENET
Abstract.
Let p X, p p j qq be a Fréchet space with a Schauder basis and with-out continuous norm, where p p j q is an increasing sequence of seminorms in-ducing the topology of X . We show that X satisfies the Invariant SubspaceProperty if and only if there exists j ě such that ker p j ` is of finite codi-mension in ker p j for every j ě j . Introduction
A Fréchet space X satisfies the Invariant Subspace Property if every operator T P L p X q possesses a non-trivial invariant subspace, i.e. a closed subspace M different from t u and X such that T M Ă M . It is an important question infunctional analysis to determine if a space X satisfies this property. Thanks toEnflo and Read, we know that there exist separable infinite-dimensional Banachspaces that do not satisfy the Invariant Subspace Property [4, 5, 14]. For instance,the spaces ℓ p N q and c p N q do not satisfy this property [15, 16]. On the other hand,there exist separable infinite-dimensional Banach spaces that satisfy the InvariantSubspace Property [1]. However, a characterization of Banach spaces satisfyingthe Invariant Subspace Property is still far from being found. In particular, itis not known if separable infinite-dimensional Hilbert spaces satisfy the InvariantSubspace Property and this problem is called the Invariant Subspace Problem.Fréchet spaces are a natural generalization of Banach spaces where the topologyis not necessarily given by one norm but can be given by an increasing sequence ofnorms or, more generally, by an increasing sequence of seminorms. These spacescan be divided into three families: ‚ Banach spaces; ‚ Non-normable Fréchet spaces with continuous norm; ‚ Fréchet spaces without continuous norm.Moreover, Fréchet spaces without continuous norm p X, p p j qq can still be dividedinto two subfamilies regarding the codimension of ker p j ` in ker p j .We can therefore wonder which Fréchet spaces satisfy the Invariant SubspaceProperty. Both behaviors can also be found among (non-normable) Fréchet spaces.There exist separable infinite-dimensional non-normable Fréchet spaces that satisfythe Invariant Subspace Property [11, 13] and a classical example is given by ω ,which is the space of all complex sequences endowed with the seminorms p j p x q “ Mathematics Subject Classification.
Key words and phrases.
Invariant subspaces, Fréchet spaces.Quentin Menet is a Research Associate of the Fonds de la Recherche Scientifique - FNRS andwas supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR(project Front). max n ď j | x n | , and, on the other hand, there exist separable infinite-dimensional non-normable Fréchet spaces that do not satisfy the Invariant Subspace Property [2, 3].Classical examples of such spaces are the space of entire functions H p C q and theSchwartz space of rapidly decreasing functions s [6, 7, 8].A first study trying to determine which non-normable Fréchet spaces satisfy theInvariant Subspace Property was recently carried out [12]. For separable infinite-dimensional non-normable Fréchet spaces with continuous norm, it was shown thata lot of them do not satisfy the Invariant Subspace Property and for Fréchet spaceswithout continuous norm p X, p p j qq , it was proved that if for every j ě j , ker p j ` is a subspace of finite codimension in ker p j then X satisfies the Invariant SubspaceProperty. This last result means that if Y is a Fréchet space with a continuousnorm then ω ‘ Y satisfies the Invariant Subspace Property. In particular, thespaces ω ‘ ℓ p N q and ω ‘ ℓ p N q satisfy the Invariant Subspace Property. However,we could not determine if a Fréchet space without continuous norm that do notmeet the above condition satisfies the Invariant Subspace Property. The goal ofthis paper consists in investigating these spaces in order to characterize Fréchetspaces without continuous norm that satisfy the Invariant Subspace Property.A first result concerning this subfamily of Fréchet spaces without continuousnorm was obtained by Goliński and Przestacki [9]. Indeed, they proved that thespace of smooth functions on the real line does not satisfy the Invariant SubspaceProperty. We will use several key ideas of this paper to show the following result. Theorem 1.1.
Let p X, p p j qq be a Fréchet space with a Schauder basis and withoutcontinuous norm, where p p j q is an increasing sequence of seminorms inducing thetopology of X . The space X satisfies the Invariant Subspace Property if and onlyif there exists j ě such that ker p j ` is of finite codimension in ker p j for every j ě j . In view of Theorem 2.1 in [12], one of the two implications is already known, andwe only need to prove that if for every j ě , there exists j ě j such that ker p j ` is of infinite codimension in ker p j then X does not satisfy the Invariant SubspaceProperty. In other words, if for every j ě , there exists j ě j such that ker p j ` is of infinite codimension in ker p j , we have to be able to construct an operator on X without non-trivial invariant subspaces. This will be done in Section 3.An interesting application of Theorem 1.1 concerns Fréchet spaces X N where if p X, p q j qq is a Fréchet space, we endow the space X N with the seminorms p j pp x n q n q “ max t q j p x n q : n ď j u . Corollary 1.2.
Let X be a Fréchet space with a Schauder basis. The space X N satisfies the Invariant Subspace Property if and only if X is finite-dimensional orisomorphic to ω . In particular, we deduce that the space p ℓ p N qq N does not satisfy the InvariantSubspace Property. 2. Properties of Schauder basis
The goal of this paper consists in showing that if p X, p p j qq is a Fréchet spacewith a Schauder basis and without continuous norm such that ker p j ` is of infinitecodimension in ker p j for infinitely many j then there exists an operator T on X such that each non-zero vector x in X is cyclic for T , i.e. span t T n x : n ě u is densein X . We will always assume in this paper that the sequence p p j q is increasing. NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 3
The construction of such an operator will rely on the ideas of Read and requiresthe existence of a Schauder basis. Therefore we first recall the following inequalitiessatisfied by Schauder basis.
Theorem 2.1 ([10, Theorem 6 (p. 298)]) . Let p X, p p j q j q be a Fréchet space with aSchauder basis p e n q n ě . Then for every j ě , there exist C j ą and J ě suchthat for every M ď N , for every x , . . . , x N P K , p j ´ M ÿ n “ x n e n ¯ ď C j p J ´ N ÿ n “ x n e n ¯ . These inequalities will be mainly used to get that if x “ ř n “ x n e n then p j p x n e n q “ p j p n ÿ m “ x m e m ´ n ´ ÿ m “ x m e m q ď p j p n ÿ m “ x m e m q` p j p n ´ ÿ m “ x m e m q ď C j p J p x q . The values p j p e n q are thus directly affected by the properties of kernels of ourseminorms. Theorem 2.2.
Let p X, p p j q j ě q be a Fréchet space with a Schauder basis p e n q n ě .If for every j , there exists j ě j such that ker p j ` is a subspace of infinitecodimension in ker p j then for every j , there exists j ě j such that |t n ě p j p e n q “ and p j p e n q ą u| “ 8 . Proof.
Let j ě . It follows from Theorem 2.1 that there exists J ą j and C j such that for every n ě , every x P X with x “ ř k “ x k e k , we have p j p x n e n q ď C j p J p x q . By assumption, there also exists j ě J such that ker p j ` is of infinite codimensionin ker p j . Therefore, for every N ě , there exists x p N q “ ř n “ N x p N q n e n P X suchthat p j p x p N q q “ and p j ` p x p N q q ą . Finally, since p j ` p x p N q q ą , there exists n ě N such that x p N q n ‰ and p j ` p e n q ą , and we get p j p e n q “ p j p x p N q n e n q| x p N q n | ď C j p J p x p N q q| x p N q n | ď C j p j p x p N q q| x p N q n | “ . In other words, the set t n ě p j p e n q “ and p j ` p e n q ą u is infinite. (cid:3) Construction of an operator without invariant subspaces
Let p X, p p j qq be a Fréchet space with a Schauder basis p e n q n ě and withoutcontinuous norm such that ker p j ` is of infinite codimension in ker p j for infinitelymany j . In view of Theorems 2.1 and 2.2, we can assume without loss of generalitythat p p j q is an increasing sequence of seminorms such that for every j ě , thereexists C j ě such that for every M ď N , every x , . . . , x N P K ,(3.1) p j ´ M ÿ n “ x n e n ¯ ď C j p j ` ´ N ÿ n “ x n e n ¯ , and such that for every j ě ,(3.2) |t n ě p j p e n q “ and p j ` p e n q ą u| “ 8 . Q. MENET
Moreover, we will assume that p C j q is an increasing sequence and for every j ě ,we will denote by E j the set t n ě p j p e n q “ and p j ` p e n q ą u and by E “ t n ě p p e n q ą u .Our construction of an operator without non-trivial invariant subspaces will bebased on the construction of Read applied to multiples of the basis p e n q n ě whichwe will reorder in a suitable way. We will denote by p u n q n ě the sequence obtainedby this reordering and we will then consider the map T : span t u j : j ě u Ñ span t u j : j ě u defined by T j u “ " u j ` T j ´ a n u if j P r a n , a n ` ∆ n q ; u j otherwise.The sequences p ∆ n q n ě and p a n q n ě will be determined by induction such that ∆ “ , ∆ “ and ∆ n ` “ a n ` ∆ n for every n ě and we will assume that thesequence p a n q grows sufficiently rapidly to get(3.3) n ` ă a n ` . For technical reasons, we will also need to consider a sequence p N n q n ě of positiveintegers such that for every n ě , N n ď n and such that for every i P N , thereexists an infinity of integers n such that N n “ i .The goal of this section consists in showing that for a convenient reordering p u n q of multiples of the basis p e n q and for a good choice of the sequence p a n q , the operator T can be extended on X and does not possess non-trivial invariant subspaces. Wewill thus assume that e n “ λ σ p n q u σ p n q where σ is a bijection on Z ` and λ m ‰ forevery m . Moreover, we will select the elements p u j q so that for every n ě , thereexists s n ě such that(3.4) t λ u , . . . , λ ∆ n ´ u ∆ n ´ u Ą t e , . . . , e s n u and(3.5) t λ u , . . . , λ ∆ n ´ u ∆ n ´ uzt e , . . . , e s n u Ă ker p p n ` q . Note that (3.4) is equivalent to require that r , s n s Ă σ ´ pr , ∆ n qq and that (3.5)means that σ ´ pr , ∆ n qqzr , s n s Ă Ť m ě n ` E m . Moreover, since σ is a bijection,we have that p s n q tends to . The construction of the reordering p u j q j ě will bedivided along the intervals r ∆ n , ∆ n ` q .As usual, we note that u is a cyclic vector for T and that we can compute thevectors T u j for every j ě from the iterates of u under the action of T as follows: T u j “ $&% u a n ` u if j “ a n ´ ; u a n ` ∆ n ´ u ∆ n if j “ a n ` ∆ n ´ ; u j ` otherwise.The map T can be extended continuously on X under the following conditions. Lemma 3.1 (Continuity) . Assume that(1) for every ď l ď n , every j P r ∆ n , a n q , we have (3.6) p l ` p u j q ě j ` p l p u j ` q , and for every ď l ď n , every j P r a n , ∆ n ` ´ q , we have (3.7) p l ` p u j q ě ∆ n ∆ n p l p u j ` q , NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 5 (2) for every n ě , we have (3.8) p p u a n ´ q ą a n p n p u q (3) for every n ě , we have (3.9) p n p u ∆ n q “ , p n ` p u ∆ n q ą and p n ` p u a n ` ∆ n ´ q ą . (4) for every j ě , every l ě , (3.10) p l ` p u j q “ ñ p l p u j ` q “ . Then the map T can be uniquely extended to a continuous operator on X satisfyingfor every N ě , every x P X , p N p T x q ď C N ` L N p N ` p x q , where C N ` is given by (3.1) and L N only depends on t u , . . . , u ∆ N ` ´ u .Proof. Let N ě and L N “ max t j p N p T u j q{ p N ` p u j q : p N ` p u j q ‰ , j ă ∆ N ` ´ u ` . It follows that for every j ă ∆ N ` ´ such that p N ` p u j q ‰ , we get p N p T u j q ď ´ j L N p N ` p u j q . In particular, this is the case for j “ a n ´ with n ď N in view of (3.8) and for j “ a n ` ∆ n ´ with n ă N in view of (3.9). Moreover, it follows from (3.10) thatfor every j ă ∆ N ` ´ such that T u j “ u j ` , we have p N p T u j q ď ´ j L N p N ` p u j q . We can thus conclude that for every j ă ∆ N ` ´ , p N p T u j q ď ´ j L N p N ` p u j q . On the other hand, if j ě ∆ N ` ´ and T u j “ u j ` , it follows from (3.6) thatif j P r ∆ n , a n ´ q for some n ě N ` then p N p T u j q ď ´ j p N ` p u j q ď ´ j L N p N ` p u j q and it follows from (3.7) that if j P r a n , ∆ n ` ´ q for some n ě N ` then p N p T u j q ď ∆ ´ n ´ ∆ n p N ` p u j q ď ∆ ´ n ´ ∆ n L N p N ` p u j q . Finally, if j “ a n ´ with n ě N ` , it follows from (3.6) and (3.8) that p N p T u a n ´ q “ p N p u a n ` u q ď ´ a n p N ` p u a n ´ q ` ´ a n p p u a n ´ qď ´ a n ` L N p N ` p u a n ´ q and if j “ a n ` ∆ n ´ with n ě N , it follows from (3.9) that p N p T u a n ` ∆ n ´ q “ p N p u ∆ n ` ´ u ∆ n q “ . Q. MENET
Therefore, if we consider x “ ř m “ x m e m , we can then let T x “ ř m “ x m T e m and we have p N p T x qď ÿ m “ p N p T p x m e m qqď L N ¨˝ ÿ m “ σ p m q p N ` p x m e m q ` ÿ n “ N ` ÿ m : σ p m qPr a n , ∆ n ` ´ q n ∆ n p N ` p x m e m q ˛‚ ď L N C N ` ¨˝ ÿ m “ σ p m q p N ` p x q ` ÿ n “ N ` ÿ m : σ p m qPr a n , ∆ n ` ´ q n ∆ n p N ` p x q ˛‚ ď L N C N ` p N ` p x q . (cid:3) Before continuing the construction of our operator T , it is necessary to do someremarks concerning the above theorem. We first notice that for every n ě , every j P r ∆ n , a n q , if σ ´ p j ` q P E n then (3.6) is satisfied. Such a choice will be donefor every j P r ∆ n , n q while we will choose σ ´ p a n ´ q P E so that (3.8) canbe satisfied. This will not contradict (3.10) since this last condition means that forevery j , if σ ´ p j q P E m j and σ ´ p j ` q P E m j ` then m j ` ě m j ´ . The indexes m j can thus decrease from n to on each block r ∆ n , a n q if a n is sufficiently big.Moreover, (3.10) for j “ a n ` ∆ n ´ follows from (3.9) since p n ` p u a n ` ∆ n ´ q ą and p n ` p u a n ` ∆ n q “ . This fact will be important because we will fix the elements u j block by block along the intervals r ∆ n , ∆ n ` q and we want that the conditionsto be satisfied by the elements in r ∆ n , ∆ n ` q do not depend from the elements u j for j ě ∆ n ` . It is also interesting to precise that we will fix the elements u j for j P r a n , ∆ n ` q before knowing the exact value of a n . This explains why thecondition (3.7) is different to the condition (3.6). Finally, in comparison to the caseof Fréchet spaces with continuous norm, we needed to add (3.10) since we do notdeal with norms.Given P p T q “ ř dn “ ρ n T n , we will denote by | P | the sum ř dn “ | ρ n | . Thefollowing key lemma (see for instance [8, Lemma 3]) gives the existence of a finitefamily of polynomials p P l q l ď L allowing to approach u by P l p T qp y q for every y in asuitable compact set. In the case of Fréchet spaces with continuous norm [12], theconsidered compact sets were given by K n “ ! y P span p u , . . . , u t n ´ q : p p y q ď { and p p τ n y q ě { ) for some map τ and some increasing sequence p t n q . Unfortunately, if p is nota norm on span p u , . . . , u t n ´ q , these sets K n are not necessarily compact. Thisexplains why in the paper [12], we have restricted our study to Fréchet spaces withcontinuous norm. Lemma 3.2.
Let ε ą , let a and t be positive integers with t ą a and p γ , . . . , γ t ´ q be a perturbed canonical basis of span p u , . . . , u t ´ q satisfying γ “ u and γ a “ εu a ` u . NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 7
Let } ¨ } be a norm on span t u j : j ě u and K Ă span p u , . . . , u t ´ q be a compactset in the induced topology such that ν : “ a ´ val γ p K q ě .Then there is a number D ě satisfying t ´ ÿ j “ | λ j | ď D for every y “ t ´ ÿ j “ λ j γ j P K and a finite family of polynomials p P l q Ll “ satisfying for every ď l ď L , val P l ě ν, deg P l ă t and | P l | ď D such that for any y P K there is ď l ď L such that for each perturbed forwardshift T : span t u j : j ě u Ñ span t u j : j ě u satisfying T j u “ γ j for every ď j ď t ´ , we have } P l p T q y ´ u } ď ε } u a } ` D max t ď j ă t } T j u } . Remark . We recall that by a perturbed canonical basis of span p u , . . . , u t ´ q ,we mean a family p γ , . . . , γ t ´ q satisfying for every ď j ď t ´ , γ j “ ř jl “ µ j,l u l with µ j,j ‰ and by a forward shift T : span t u j : j ě u Ñ span t u j : j ě u , wemean an operator T satisfying for every j ě , T u j “ ř j ` l “ λ j,l u l with λ j,j ` ‰ .Note that the norm } ¨ } in the above lemma has only to be a norm on span t u j : j ě u and not on X . This fact was successfully used by Golinski and Przestackiin their study of the space of smooth functions on the real line [9]. We adapt thisidea and consider the sequence of compact sets p K n q n ě given by K n “ ! y P span p u , . . . , u ∆ n ` ´ q : } y } ď { and } τ n y } ě { ) where, for every x P span t u j : j ě u “ span t e n : n ě u , if x “ ř n “ x n e n , } x } “ p p x q ` ÿ j ě C j ` ÿ n P E j n p j ` p x n e n q and where for every n ě , τ n ´ ∆ n ` ´ ÿ j “ y j T j u ¯ “ a n ´ ÿ j “ y j T j u . We remark that } ¨ } is a well-defined norm on span t u j : j ě u “ span t e n : n ě u since Ť j ě E j “ Z ` and each set K n is thus compact.Lemma 3.2 will be applied to sets K n and to norms }¨} N n where for every N ě ,the norm } ¨ } N is defined on span t e n : n ě u by(3.11) ››››› ÿ n “ x n e n ››››› N “ p N p x q ` ÿ j ě C j ` ÿ n P E j n p j ` p x n e n q . This choice of norms allows us to get for every x P span t u j : j ě u “ span t e n : n ě u , p N p x q ď } x } N but also to control } e n } N by considering n P E N with n sufficiently big.Since a n ´ val p K n q ě in the basis t u , . . . , T ∆ n ` ´ u u in view of the definitionof τ n , we can apply Lemma 3.2 to K n in order to get the following corollary. Q. MENET
Corollary 3.4.
For every n ě , there exist D n ě satisfying ∆ n ` ´ ÿ j “ | y j | ď D n for every y “ ∆ n ` ´ ÿ j “ y j T j u P K n and a family of polynomials P n “ p P n,k q k n k “ satisfying val p P n,k q ě , deg p P n,k q ă ∆ n ` and | P n,k | ď D n such that for each y P K n , there exists ď k ď k n such that } P n,k p T q y ´ u } N n ď } u a n } N n ` D n max ∆ n ` ď j ă n ` } u j } N n . Moreover, D n and P n only depend on t u , . . . , u ∆ n ` ´ u . We recall that our goal consists in showing that under some conditions on thesequences p a n q and p u n q , the operator T has no non-trivial invariant subspace or,in other words, that every non-zero vector x is cyclic under the action of T . Since,by definition of T , we already know that u is cyclic, it will suffice to prove that forevery N ě , every ε ą , there exists a polynomial Q such that p N p Q p T q x ´ u q ă ε . This polynomial Q will be obtained thanks to Corollary 3.4. In fact, we will showthat every non-zero vector x can be divided into two parts so that some multipleof the first part belongs to K n for some n (Lemma 3.5) and the second part doesnot perturb the desired estimation (Lemma 3.6). Lemma 3.5 (Sets K n ) . If for every n ě , every j P r a n , a n ` ∆ n q , (3.12) σ ´ p j q P E N n and p N n ` p u j q ě ∆ n } T j ´ a n u } , then for every x P X zt u , there exists M ą such that for every integer N ě ,every increasing sequence p n k q such that N n k “ N , we have that for all but finitelymany k , there exists M k ě M such that π p u qr , ∆ nk ` q xM k P K n k where π p u qr , ∆ nk ` q x “ ř l P σ ´ pr , ∆ nk ` qq x l e l if x “ ř l “ x l e l .Proof. We recall that K n “ ! y P span p u , . . . , u ∆ n ` ´ q : } y } ď { and } τ n y } ě { ) and that τ n ´ ∆ n ` ´ ÿ j “ y j T j u ¯ “ a n ´ ÿ j “ y j T j u . Let n ě . We start by computing } τ n u j } for every j P r , ∆ n ` q . ‚ If j ă a n then τ n u j “ u j . ‚ If j P r a n , a n ` ∆ n q then τ n u j “ ´ T j ´ a n u and thus by (3.12) } τ n u j } “ } T j ´ a n u } ď ∆ n p N n ` p u j q . NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 9
Let y “ ř ∆ n ` ´ j “ y j u j . We deduce that ››››› τ n ˜ ∆ n ` ´ ÿ j “ a n y j u j ¸››››› ď a n ` ∆ n ´ ÿ j “ a n } y j τ n u j } ď a n ` ∆ n ´ ÿ j “ a n ∆ n p N n ` p y j u j qď a n ` ∆ n ´ ÿ j “ a n C N n ` ∆ n p N n ` p y q ď n C N n ` ∆ n p N n ` p y q where C N n ` is given by (3.1). On the other hand, since N n ě and σ ´ p j q P E N n for every j P r a n , a n ` ∆ n q , it follows from (3.4) that ››››› ∆ n ` ´ ÿ j “ a n y j u j ››››› “ C N n ` a n ` ∆ n ´ ÿ j “ a n σ ´ p j q p N n ` p y j u j qď C N n ` a n ` ∆ n ´ ÿ j “ a n C N n ` σ ´ p j q p N n ` p y qď ÿ k ě min p σ ´ r a n ,a n ` ∆ n qq k ´ p N n ` p y qď s n ´ p N n ` p y q . Let x P X zt u with x “ ř l “ x l e l . We let π p e qr ,s s p x q “ ř sl “ x l e l for every s ě . Let M “ } π p u qr , ∆ m ` q x } where the index m is chosen so that M ą . Let N ě andlet p n k q be an increasing sequence such that N n k “ N . We let M k “ } π p u qr , ∆ nk ` q x } and we remark that M k ě M when n k ě m . If π p u qr , ∆ nk ` q x “ ř ∆ nk ` ´ i “ y j u j , itthen follows from (3.4) and (3.5) that } τ n k π p u qr , ∆ nk ` q x } “ ›››››› τ n k ¨˝ a nk ´ ÿ i “ y j u j ˛‚ ` τ n k ¨˝ ∆ nk ` ´ ÿ i “ a nk y j u j ˛‚›››››› ě ›››››› a nk ´ ÿ i “ y j u j ›››››› ´ ›››››› τ n k ¨˝ ∆ nk ` ´ ÿ i “ a nk y j u j ˛‚›››››› ě } π p u qr , ∆ nk ` q x } ´ ›››››› ∆ nk ` ´ ÿ i “ a nk y j u j ›››››› ´ n k C N ` ∆ nk p N ` p π p u qr , ∆ nk ` q x qě M k ´ s nk ´ p N ` p π p u qr , ∆ nk ` q x q ´ n k C N ` ∆ nk p N ` p π p u qr , ∆ nk ` q x q“ M k ´ s nk ´ p N ` p π p e qr ,s nk ` s x q ´ n k C N ` ∆ nk p N ` p π p e qr ,s nk ` s x q since N “ N n k ď n k by definition of p N n q . Finally, since p p m p π p e qr ,s nk ` s x qq k isbounded for every m , we conclude that for all but finitely many k , M k ě M and } τ n k π p u qr , ∆ nk ` q x } ě M k , and therefore that π p u qr , ∆ nk ` q xM k P K n k . (cid:3) We recall that at the beginning of this section, we have assumed that the sequence p C j q given by (3.1) is increasing. This assumption was done in order to simplifythe statement of the following lemma. For the same reason, we will also assumethat the sequence p D n q n ě given by Corollary 3.4 is increasing. Lemma 3.6 (Tails) . Assume that for every n ě , every j P r ∆ n , ∆ n ` q ,(1) for every l ď n , every ď r ă ∆ n , if j P r ∆ n , a n q then (3.13) p l ` p u j q ě j ` C n ` D n ´ p l p u j ` r q ; and if j P r a n , ∆ n ` q and j ` r ă ∆ n ` then (3.14) p l ` p u j q ě ∆ n ∆ n ` C n ` D n ´ p l p u j ` r q ; (2) if j P r ∆ n , n q then (3.15) p n p u j q “ (3) if j P r a n ´ ∆ n , a n q then (3.16) p p u j q ě j ` C n ` D n ´ sup m ă ∆ n p n p T m u q . Then for every n ě , every polynomial P with val p P q ě , deg p P q ă ∆ n ` and | P | ď D n , every x P span t e j : σ p j q ě ∆ n ` u , we have p N n p P p T q x q ď p N n ` p x q . Proof.
Let n ě and j P r ∆ n , ∆ n ` q . Let ď m ď n and ď r ă ∆ m . If j P r ∆ n , a n ´ r q then it follows from (3.13) that p N m ´ p T r u j q “ p N m ´ p u j ` r q ď j ` C N m ´ ` D m ´ p N m ´ ` p u j q . If j P r a n ´ r, a n q then it follows from (3.13) and (3.16) that p N m ´ p T r u j q “ p N m ´ p u j ` r ` T j ` r ´ a n u qď j ` C N m ´ ` D m ´ p N m ´ ` p u j q ` j ` C N m ´ ` D m ´ p p u j qď j C N m ´ ` D m ´ p N m ´ ` p u j q . On the other hand, if j P r a n , a n ` ∆ n ´ r q then it follows from (3.14) that p N m ´ p T r u j q “ p N m ´ p u j ` r q ď n ∆ n ` C N m ´ ` D m ´ p N m ´ ` p u j q . Finally, if j P r a n ` ∆ n ´ r, a n ` ∆ n q then j ` r P r a n ` ∆ n , a n ` q since a n ` n ă n ` ă a n ` by (3.3) and it follows from (3.15) that p N m ´ p T r u j q “ p N m ´ p u j ` r ´ u j ` r ´ a n q “ since j ` r P r ∆ n ` , n ` q and j ` r ´ a n P r ∆ n , n q .Let n ě and let P be a polynomial with val p P q ě , deg p P q ă ∆ n ` and | P | ď D n . It follows from the previous inequalities that if x “ ř j : σ p j qě ∆ n ` x j e j NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 11 then p N n p P p T q x q ď | P | max ď r ă ∆ n ` p N n p T r x qď D n ÿ j : σ p j qě ∆ n ` | x j | max ď r ă ∆ n ` p N n p T r e j qď D n ¨˝ ÿ j : σ p j qě ∆ n ` σ p j q C N n ` D n | x j | p N n ` p e j q` ÿ m ě n ` ÿ j : σ p j qPr a m ,a m ` ∆ m q m ∆ m ` C N n ` D n | x j | p N n ` p e j q ˛‚ ď ¨˝ ÿ j : σ p j qě ∆ n ` σ p j q´ ` ÿ m ě n ` ∆ m ˛‚ p N n ` p x q ď p N n ` p x q . (cid:3) Remark . Since for condition (3.7) in Lemma 3.1, the reason to consider condi-tion (3.14) relies on the fact that the value of a n will not be known when we willfix the elements u j for j P r a n , ∆ n ` q .Thanks to all these lemmas, we are now able to write a list of conditions suchthat if the sequence p u n q n ě satisfies each of these conditions then T admits nonon-trivial invariant subspace. Lemma 3.8 (Final result) . Under the assumptions (3.4) - (3.5) , (3.6) - (3.10) , (3.12) and (3.13) - (3.16) , if for every n ě , we have (3.17) } u a n } N n ď n and for every ∆ n ` ď j ă n ` , we have (3.18) } u j } N n ď n D n then every non-zero vector in X is cyclic for T and thus T has no non-trivialinvariant subspace.Proof. Let x P X zt u . Let N ě . Since u is a cyclic vector for T , it suffices to showthat there exists a sequence of polynomial p Q k q k such that p N p Q k p T q x ´ u q Ñ in order to deduce that x is cyclic for T .Let p n k q be an increasing sequence such that N n k “ N . By Lemma 3.5, thereexists M ą such that for all but finitely many k , there exists M k ě M such that y : “ π p u qr , ∆ nk ` q xM k P K n k . By Corollary 3.4, there then exists a polynomial P n k ,l with val p P n k ,l q ě , deg p P n k ,l q ă ∆ n k ` and | P n k ,l | ď D n k such that p N p P n k ,l p T q y ´ u q ď } P n k ,l p T q y ´ u } N ď } u a nk } N ` D n k max ∆ nk ` ď j ă nk ` } u j } N ď n k by (3.17) and (3.18) . On the other hand, it follows from Lemma 3.6 that p N p P n k ,l p T q π p u qr ∆ nk ` , `8q x q ď p N ` p π p u qr ∆ nk ` , `8q x q where π p u qr ∆ nk ` , `8q x is given by x ´ π p u qr , ∆ nk ` q x . Since N n ď n for every n ě , weconclude thanks to (3.5) that p N p M k P n k ,l p T q x ´ u q ď p N p P n k ,l p T q y ´ u q ` p N p M k P n k ,l p T qp π p u qr ∆ nk ` , `8q x qqď n k ` M p N ` p π p u qr ∆ nk ` , `8q x q“ n k ` M p N ` p π p e qp s nk ` , `8q x q Ñ , where π p e qp s nk ` , `8q x “ ř `8 l “ s nk ` ` x l e l if x “ ř l “ x l e l . Each non-zero vector in X is thus cyclic for T and therefore T does not possess any non-trivial invariantsubspace. (cid:3) We are now able to conclude the proof of Theorem 1.1.
Proof of Theorem 1.1.
We first recall that if there exists j ě such that ker p j ` is a subspace of finite codimension in ker p j for every j ě j then X satisfies theInvariant Subspace Property ([12, Theorem 2.1]).On the other hand, if we assume that p X, p p j qq is a Fréchet space with a Schauderbasis p e n q n ě and without continuous norm such that ker p j ` is of infinite codi-mension in ker p j for infinitely many j , then we can assume without loss of general-ity that (3.1) and (3.2) are satisfied and in view of Lemma 3.8, it remains to provethat it is possible to construct a sequence p u j q j ě satisfying (3.4)-(3.5), (3.6)-(3.10),(3.12), (3.13)-(3.16) and (3.17)-(3.18).To this end, we first let u “ e i and u “ e i with i ‰ i and i , i P E so that(3.9) is satisfied for n “ and (3.10) is satisfied for j “ . We recall that ∆ “ and that ∆ “ . We then select an index i a R t i , i u with i a P E N “ E sufficiently big so that by letting u a “ ∆1 } u } p N ` p e ia q e i a , we get (3.9), (3.12) and(3.17), i.e. p p u a q ą , p N ` p u a q ě ∆ } u } and } u a } N “ } u a } “ ∆ } u } i a C ď . Note that the choice of the index i a is independent of the value of a . We nowselect distinct indices K “ t k , k , k , k u different to i , i and i a with k l P E l .Let s “ max t i , i , i a , k , k , k , k u . We can complete the family K so that ‚ K X t i , i , i a u “ H‚ K Y t i , i , i a u Ą r , s s , ‚ if j P K and j ą s then j P E l for some l ě , NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 13 ‚ for every l ě , if K X E l ‰ H then K X E m ‰ H for every m ă l .We then let a “ | K | ` and for each n P p , a q , we take for u n a non-zero multipleof e s with s P K so that (3.6), (3.8) and (3.10) are satisfied. More precisely, weconsider an enumeration p i j q ă j ă a of K such that if i j P E s then i j ` P E s Y E s ´ and we let u j “ α j e i j . We can then deduce that (3.10) is satisfied for every j ă a and we can choose α j sufficiently big to get (3.6) for every ∆ ď j ď a ´ and toget (3.8) for n “ since i a ´ P E . We conclude that the family t u , . . . , u ∆ ´ u satisfies (3.4)-(3.5), (3.6)-(3.10), (3.12) and (3.17). Note that conditions (3.13)-(3.16) and (3.18) only concern the indices bigger than ∆ .We continue the construction of p u k q k ě by induction along the intervals r ∆ n , ∆ n ` q .We thus assume that u , ...u ∆ n ´ have been chosen for some n ě so that (3.4)-(3.5), (3.6)-(3.10), (3.12), (3.13)-(3.16) and (3.17)-(3.18) are satisfied. We will di-vide the selection of elements u j for j P r ∆ n , ∆ n ` q into three steps: r ∆ n , n q , r a n , a n ` ∆ n q and r n , a n q .Let D n ´ be the real number given by Corollary 3.4. We first select ∆ n distinctindices p i j q j Pr ∆ n , n q different from the indices already selected such that i j P E n for every j P r ∆ n , n q and we let u j “ α j e i j with α j ‰ . It follows that(3.6) and (3.10) are satisfied for every j P r ∆ n , n ´ q , that (3.15) and the firsttwo conditions of (3.9) are satisfied and that (3.13) is satisfied for j ` r ă n .Moroever, we can choose α j sufficiently small so that (3.18) is satisfied.We complete by choosing the elements u j for j P r a n , a n ` ∆ n q . We first select ∆ n ´ distinct indices p i j q j Pp a n ,a n ` ∆ n q different from the indices already selectedsuch that i j P E N n and we let u j “ α j e i j for every j P p a n , a n ` ∆ n q . It followsthat the last condition of (3.9) is satisfied and that (3.10) is satisfied for every j P p a n , a n ` ∆ n ´ q . Moreover, we can choose α j sufficiently big so that (3.12) issatisfied, that (3.7) is satisfied for j P p a n , ∆ n ` ´ q and that (3.14) is satisfied for j P p a n , ∆ n ` q (even if the value of a n has not still be determined). We then selecta sufficiently big index i a n P E N n such that by letting u a n “ ` ∆ n ∆ n ` C n ` D n ´ max t p n p u j q : j P p a n , ∆ n ` qu` ∆ n } u } ˘ e i an p N n ` p e i an q , we get (3.7), (3.12), (3.14) and (3.17), i.e. for every ď l ď n , every j P p a n , ∆ n ` q , p l ` p u a n q ě ∆ n ∆ n p l p u a n ` q , p N n ` p u a n q ě ∆ n } u } ,p l ` p u a n q ě ∆ n ∆ n ` C n ` D n ´ p l p u j q and } u a n } N n “ ` ∆ n ∆ n ` C n ` D n ´ max t p n p u j q : j P p a n , ∆ n ` qu ` ∆ n } u } ˘ i an C N n ` ď n . Moreover, (3.10) is satisfied for j “ a n since i a n and i a n ` belong to E N n .We are now looking for the elements u j for j P r n , a n q . Let I n be the set of al-ready selected indices. We select p n ` q ∆ n distinct indices K “ t k , ..., k p n ` q ∆ n ´ u such that K X I n “ H and such that k l ∆ n ` r P E l for every ď l ď n ` and ď r ă ∆ n . Let s n ` be the maximum of K Y I n . We can now complete the family K so that ‚ K X I n “ H , ‚ K Y I n Ą r , s n ` s , ‚ if j P K and j ą s n ` then j P E l for some l ě n ` , ‚ for every l ě , if K X E l ‰ H then K X E m ‰ H for every m ă l . We then let a n “ | K | ` n and for each j P r n , a n q , we take for u j a multipleof e s where s P K so that (3.6), (3.8), (3.10), (3.13) and (3.16) are satisfied. Moreprecisely, we consider an enumeration p i j q n ď j ă a n of K such that if i j P E s then i j ` P E s Y E s ´ and we let u j “ α j e i j . Since for every s ď n ` , the set K containsat least ∆ n indices in E s , we can deduce that i j P E for every j P r a n ´ ∆ n , a n q and that i j P Ť s ě n ` E s for every j P r n , n q . It follows that for any choice of p α i q i Pr n ,a n q , (3.10) is satisfied for every j ă ∆ n ` , (3.6) is satisfied for j “ n ´ and (3.13) is satisfied for every j P r ∆ n , n q . We can then choose α i sufficientlybig for every i P r n , a n q so that we get (3.8) and (3.16) but also (3.6) and (3.13)for every j P r ∆ n , a n q . We conclude that the family t u , . . . , u ∆ n ` ´ u satisfies(3.4)-(3.5), (3.6)-(3.10), (3.12), (3.13)-(3.16) and (3.17)-(3.18).It follows from Lemma 3.8 that X does not satisfy the Invariant Subspace Prop-erty. (cid:3) We end this paper by showing how Theorem 1.1 leads to the identification ofspaces X N satisfying the Invariant Subspace Property. Proof of Corollary 1.2.
Let p X, p q j q j q be a Fréchet space with a Schauder basiswhere we assume that p q j q is an increasing sequence. It follows that X N endowedwith the seminorms p j pp x n q n q “ max t q j p x n q : n ď j u is a Fréchet space with aSchauder basis and without continuous norm since none of the seminorms p j is anorm. Moreover, the sequence p p j q is increasing and we can thus apply Theorem 1.1to the space X N .If X is finite-dimensional or if X is isomorphic to ω then for every j ě , ker p j ` is of finite codimension in X and thus in particular in ker p j and it follows fromTheorem 1.1 that X N satisfies the Invariant Subspace Property. On the other hand,if X is infinite dimensional but is not isomorphic to ω then there exists j ě suchthat ker q j is of infinite codimension in X and it follows that for every j ě j , ker p j ` is of infinite codimension in ker p j . We can then conclude by applyingTheorem 1.1. (cid:3) Remark . Corollary 1.2 means that if X is a Fréchet space with a Schauder basisthen X N satisfies the Invariant Subspace Property if and only if X N is isomorphicto ω . References [1] Argyros, S. A.; Haydon, R. G. A hereditarily indecomposable L -space thatsolves the scalar-plus-compact problem. Acta Math. 206 (2011), no. 1, 1–54.[2] Atzmon, A. An operator without invariant subspaces on a nuclear Fréchetspace. Ann. of Math. (2) 117 (1983), no. 3, 669–694.[3] Atzmon, A. Nuclear Fréchet spaces of entire functions with transitive differ-entiation. J. Anal. Math. 60 (1993), 1–19.[4] Enflo, P. On the invariant subspace problem in Banach spaces. Sémi-naire Maurey-Schwartz (1975-1976) Espaces L p , applications radonifiantes et NVARIANT SUBSPACES FOR FRÉCHET SPACES WITHOUT CONTINUOUS NORM 15 géométrie des espaces de Banach, Exp. Nos. 14-15, 7 pp. Centre Math., ÉcolePolytech., Palaiseau, 1976.[5] Enflo, P. On the invariant subspace problem for Banach spaces. Acta Math.158 (1987), no. 3-4, 213–313.[6] Goliński, M. Invariant subspace problem for classical spaces of functions. J.Funct. Anal. 262 (2012), no. 3, 1251–1273.[7] Goliński, M. Operators on Fréchet spaces without nontrivial invariant sub-spaces, Thesis, Adam Mickiewicz University in Poznań, Poland, 2013.[8] Goliński, M. Operator on the space of rapidly decreasing functions with allnon-zero vectors hypercyclic. Adv. Math. 244 (2013), 663–677.[9] Goliński, M.; Przestacki, A. The invariant subspace problem for the space ofsmooth functions on the real line, J. Math. Anal. Appl. 482 (2020), no. 2.[10] Jarchow, H. Locally convex spaces. Mathematische Leitfäden, Stuttgart, 1981.[11] Körber, Karl-Heinz. Die invarianten Teil räume der stetigen Endomorphismenvon ω . (German) Math. Ann. 182 (1969) 95–103.[12] Menet, Q. Invariant subspaces for non-normable Fréchet spaces, Adv. Math.339 (2018), 495–539.[13] Shields, Allen L. A note on invariant subspaces. Michigan Math. J. 17 (1970),231–233.[14] Read, C. J. A solution to the invariant subspace problem. Bull. London Math.Soc. 16 (1984), no. 4, 337–401.[15] Read, C. J. A solution to the invariant subspace problem on the space l . Bull.London Math. Soc. 17 (1985), no. 4, 305–317.[16] Read, C. J. The invariant subspace problem for a class of Banach spaces. II.Hypercyclic operators. Israel J. Math. 63 (1988), 1–40. Quentin Menet, Département de Mathématique, Université de Mons, 20 Place duParc, 7000 Mons, Belgique
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