Cosine Families in GDP Quojection-Fréchet Spaces

x ∈ X . This notion was introduced by M.Sova in 1966 [12], which associates to each C0-cosine family an operator called the infinitesimal generator. It is well-known, in the classical case (where X is a Banach space), that a C0-cosine family {C(t)}t≥0 is exponentially equicontinuous in X [4,8,10]. Therefore the family {C(t)}t≥0 is strongly continuous in X [12], i.e. the map C(.)x is continuous in R, for all x ∈ X . This means that the notions ”strongly continuous cosine family” and ”C0-cosine family” coincide. The infinitesimal generator A of a C0-cosine family is a closed operator and is densely defined. The link between the family {C(t)}t≥0 and its generator is given by the relation: λR(λ, A) = ∫ ∞ 0 eC(t)dt, for some λ ∈ C, with R(λ, A) = (λI − A) [4,8]. If in addition the family {C(t)}t≥0 is uniformly continuous (i.e. ‖ C(t) − I ‖−→ 0, as t −→ 0 ), then the infinitesimal generator is a bounded operator in X [9,13]. If X is a Hausdorff locally convex space, we say that {C(t)}t≥0 is a strongly continuous cosine family if it satisfies the D’Alembert functional equation, C(0) = I and C(t) −→ C(t0) in Ls(X) as t −→ t0, for all t0 ≥ 0 [3]. If instead of the last condition, the family {C(t)}t≥0 verifies C(t) −→ I in Ls(X) as t −→ 0, we say that {C(t)}t≥0 is a C0-cosine family. In the second section, we are interested studying in general the C0-cosine families in Hausdorff locally convex spaces and we rely on the work of M.Sova, on the Banach spaces [12], to show that any locally equicontinuous C0-cosine family in X (h.l.c.s.) is strongly continuous. By a well-known analogy of K.Yosida [13] for semi-groups, we show in the third section the existence of the resolvent of the infinitesimal generator of an exponentially equicontinuous C0-cosine family on X . In the case of Fréchet spaces, we have given in [3] an example of a uniformly continuous cosine family whose infinitesimal generator is not defined everywhere in the space, and we showed that if the space is a Quojection, the infinitesimal generator of all uniformly continuous cosine family is a continuous operator. In the fourth section, we begin with a proposition that gives an answer of Conejero’s question [6] in the case of cosine families in the space ω = C. Finally, we show that if X is a GDP Quojection-Fréchet space, then all C0-cosine family in X is uniformly continuous, therefore its infinitesimal generator is a continuous linear operator on X .


Introduction
A C 0 -cosine family {C(t)} t≥0 in a Banach space X is a family of bounded linear operators in X satisfying the D'Alembert functional equation (see Definition 1.1), C(0) = I and lim t−→0 + C(t)x = x, for all x ∈ X. This notion was introduced by M.Sova in 1966 [12], which associates to each C 0 -cosine family an operator called the infinitesimal generator.
It is well-known, in the classical case (where X is a Banach space), that a C 0 -cosine family {C(t)} t≥0 is exponentially equicontinuous in X [4,8,10]. Therefore the family {C(t)} t≥0 is strongly continuous in X [12], i.e. the map C(.)x is continuous in R + , for all x ∈ X. This means that the notions "strongly continuous cosine family" and "C 0 -cosine family" coincide. The infinitesimal generator A of a C 0 -cosine family is a closed operator and is densely defined. The link between the family {C(t)} t≥0 and its generator is given by the relation: λR(λ 2 , A) = ∞ 0 e −λt C(t)dt, for some λ ∈ C, with R(λ 2 , A) = (λ 2 I − A) −1 [4,8]. If in addition the family {C(t)} t≥0 is uniformly continuous (i.e. C(t) − I −→ 0, as t −→ 0 + ), then the infinitesimal generator is a bounded operator in X [9,13].
If X is a Hausdorff locally convex space, we say that {C(t)} t≥0 is a strongly continuous cosine family if it satisfies the D'Alembert functional equation, C(0) = I and C(t) −→ C(t 0 ) in L s (X) as t −→ t 0 , for all t 0 ≥ 0 [3]. If instead of the last condition, the family {C(t)} t≥0 verifies C(t) −→ I in L s (X) as t −→ 0 + , we say that {C(t)} t≥0 is a C 0 -cosine family.
In the second section, we are interested studying in general the C 0 -cosine families in Hausdorff locally convex spaces and we rely on the work of M.Sova, on the Banach spaces [12], to show that any locally equicontinuous C 0 -cosine family in X (h.l.c.s.) is strongly continuous.
By a well-known analogy of K.Yosida [13] for semi-groups, we show in the third section the existence of the resolvent of the infinitesimal generator of an exponentially equicontinuous C 0 -cosine family on X.
In the case of Fréchet spaces, we have given in [3] an example of a uniformly continuous cosine family whose infinitesimal generator is not defined everywhere in the space, and we showed that if the space is a Quojection, the infinitesimal generator of all uniformly continuous cosine family is a continuous operator. In the fourth section, we begin with a proposition that gives an answer of Conejero's question [6] in the case of cosine families in the space ω = C N . Finally, we show that if X is a GDP Quojection-Fréchet space, then all C 0 -cosine family in X is uniformly continuous, therefore its infinitesimal generator is a continuous linear operator on X.

C 0 -Cosine family of operators in locally convex spaces:
Let X be a sequentially complete locally convex Hausdorff space and Γ X a system of continuous seminorms determining the topology of X. The strong topology τ s in the space L(X), of all continuous linear operators from X into itself, is determined by the family of seminorms: for each x ∈ X and q ∈ Γ X , in this case we write L s (X).
Denote by B(X) the collection of all bounded subsets of X. The topology τ b of uniform convergence on the elements of B(X) is defined by the family of seminorms: for each B ∈ B(X) and q ∈ Γ X , in this case we write L b (X). For a Banach space X, τ b is the operator norm topology in L(X). If Γ X is countable and X is complete, then X is called a Fréchet space.

It's clear that every exponentially equicontinuous family is necessarily locally equicontinuous.
It is known that every C 0 -cosine family in a Banach space is necessary exponentially equicontinuous [8,12]. For Fréchet spaces this need not be the case. For example, in the sequence space ω = C N , the family: ∈ ω, and t ≥ 0, defines a C 0 -cosine family in ω which is not exponentially equicontinuous.
Based on M. Sova's theorem ( [12], Theorem 2.7), which shows that for every C 0 -cosine family, in a Banach space, the map C(.)x is continuous in R + for every x ∈ X. The following proposition generalizes this result in locally convex spaces.

Proposition 2.3. Let {C(t)} t≥0 be a locally equicontinuous C 0 -cosine family in locally convex space X.
Then, for all x ∈ X, and t 0 ≥ 0: lim For each n ∈ N * , put: is a decreasing sequence of positive numbers, therefore there exists K ∈ R + such that K n −→ K. According to the hypothesis, we obtain that K > 0, hence the result. 2 o / There exist (τ n ) n ⊂ R + and (σ n ) n ⊂ R + , with τ n < σ n , for all n ∈ N * , such that: Let s, t ∈ R + with s ≤ t, we have: On the other hand, for all p ∈ Γ X , and x ∈ X we have: Now, for each n ∈ N * we take: t = τ 4n , s = σ 4n − τ 4n , x = x 0 , and p = q, we obtain: According to 2 o / and 4 o / we have: Since {C(t)} t≥0 is locally equicontinuous in X, there exist q * ∈ Γ X , and M ≥ 0 such that: As lim Which is contradict 1 o /, hence the result.
The previous proposition show that if the C 0 -cosine family {C(t)} t≥0 is locally equicontinuous, then the following conditions are equivalent: In this case, the notions "C 0 -cosine family" and "strongly continuous cosine family" coincide. By the same procedure, and under the condition of locally equicontinuity, we have the equivalence between ii) of Definition 1. and the condition: It follows that Proposition 2.6. Let {C(t)} t≥0 be a locally equicontinuous C 0 -cosine family in X and A its infinitesimal generator. And let x, y ∈ X, then:

The Resolvent of infinitesimal generator:
In this section we study some spectral properties of the infinitesimal generator, which are important for our main result.
We begin with the following lemma for ease of reading.

2
, ∀λ ∈ C. Then lim h−→0 + B h = −λ. On the other hand, let p ∈ Γ X and x ∈ X, we have: For semigroups, the two next Theorems is due to K.Yosida [13].

Theorem 3.2. Let {C(t)} t≥0 be an exponentially equicontinuous C 0 -cosine family of order ω in X.
For all λ > ω, consider the linear operator C λ : Then, the following properties hold: 1. C λ is a continuous linear operator in X.

Im(C λ ) ⊆ D(A)
, and for all x ∈ X we have: 3. for all x ∈ X: lim Proof.

C 0 -Cosine family in a GDP Quojection-Fréchet space:
It is known that a Fréchet space X is a projective limit of a sequence of Banach spaces (X k ) k with respect to the projective operators P k : X k+1 −→ X k . A Fréchet space is a quojection if it is isomorphic to a projective limit of Banach spaces with surjective projective operators [5].
Recall that a locally convex Hausdorff space is a Grothendieck space, if every sequence in X ′ , which convergent for σ(X ′ , X), is also convergent for σ(X ′ , X ′′ ) [1].
A locally convex Hausdorff space X is said to have the Dunford-Pettis property if for all T ∈ L(X, Y ), for Y any quasicomplete locally convex Hausdorff space, which transforms elements of B(X) into relatively σ(Y, Y ′ )-compact subsets of Y , also transforms σ(X, X ′ )-compact subsets of X into relatively compact subsets of Y [1,9].
A Grothendieck locally convex Hausdorff space X with the Dunford-Pettis property is called, briefly, a GDP space.
We begin this section with the following lemma [Lemma 2.4 [2]] which plays an important role in the main results:  If, in addition, X is a Quojection Fréchet space and there exists a fundamental sequence {q j } j≥1 , of seminorms generating the locally convex topology of X which satisfy: For each j ∈ N there exists c j > 0 such that: for all x ∈ X and n ∈ N * . Then: We recall our Theorem [3] about the cosine family in Quojection space.

Theorem 4.2. Let X be a quojection. The infinitesimal generator of every uniformly continuous cosine family is continuous, (i.e. A ∈ L(X)).
Moreover, for all x ∈ X and t ≥ 0 we have: On the other hand, we consider the space ω = C N of all sequence equipped with its topology of coordinates convergence (i.e. p k (x) = max 0≤j≤k |x j |, (x j ) j∈N ∈ ω, for each k ∈ N). It's well known that, the space ω is a Quojection, because it is a product of countable copies of the Banach space C, Moreover ω is a Montel space, therefore the strong convergence and the uniform convergence coincide on bounded sets, then every C 0 -cosine family is actually uniformly continuous.
The following proposition gives a version of the previous result in the case of cosine families.

Proposition 4.3.
Every locally equicontinous C 0 -cosine family on the space ω = C N has a continuous infinitesimal generator A and is of the form: , for all t ≥ 0.
Let {C(t)} t≥0 be an exponentially equicontinous C 0 -cosine family of order ω in locally convex Hausdorff space X, i.e. for all p ∈ Γ X , there exist q ∈ Γ X and M ≥ 0 such that: , for all t ≥ 0, and x ∈ X.
For each p ∈ Γ X , we definep in X by: Eachp is a continuous seminorm in X. Moreover, since {C(t)} t≥0 is exponentially equicontinuous of order ω in X,p satisfies: Then, the family of seminormsΓ X = {p, p ∈ Γ } is also a system of continuous seminorms generating the locally convex topology of X. 1. For all t ≥ 0 and x ∈ X:p (C(t)x) ≤ 2e ωtp (x).
1. Let t ≥ 0, x ∈ X, then for eachp ∈Γ X we have: 2p(C(t)x) = 2 sup Then, for all seminormp ∈Γ X we have: In particular if λ > ω + 1, then for eachp ∈Γ X we obtain: For semigroups, the following Theorem is due to A.A. Albanese, J. Bonet, W.J. Ricker [2]. Proof. According to the discussion before Proposition 4.2 there is a fundamental increasing sequence {q j } j∈N * of continuous seminorms on X such that for all j ∈ N * : q j (C(t)x) ≤ 2e ωt q j (x), for all x ∈ X, and t ≥ 0.