Integrated Semigroups on Fréchet Spaces I : Bochner Integral , Laplace Integral and Real Representation Theorems

Abstract In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

In this articles we will study the integration of the vectorial functions in Fréchet spaces.Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.
(1.2) Definition 2. A complex linear topological spaces S is called a locally convex, linear topological space, or, in short, a locally convex space, if any if its open sets contains a convex, balanced and absorbing open set.
Definition 3. A complex linear space F is called a Quasi-normed linear space if, for every x ∈ F, there is associated a real number |x| F , the quasi-norm of the vector x, which satisfies |x| F ≥ 0 and |x| F = 0 ⇔ x = 0; (1.3) The topology of a quasi-normed linear space F is thus defined by the distance d (x, y) = |x − y| F . (1.8) We say that the sequence {x n } n ⊂ F converges strongly to x ∈ F, x n → x for n → +∞ in F, or (1.10)

Fréchet Space
Definition 4. A quasi-normed linear space F is called a Fréchet space if it is complete, i.e., if every Cauchy sequence of F converges strongly to a point of F.
Remark 1.Let the topology of a locally convex space F be defined by a countable number of semi-norms p n (x), with n = 1, 2, .... Then F is a quasi-normed linear space by the quasi-norm . (1.11) For, the convergence lim h→∞ p n (x h ) = 0, with n = 1, 2, ...is equivalent to F − lim h→∞ x h = 0 with respect to the quasi-norm |x| F above.
Definition 5. (Fréchet in Bourbaki sence) A locally convex space F is called a Fréchet space if it is quasi-normed and complete.
Let X, Y be locally convex spaces on the same scalar field; we denote by L (X, Y) the totality of continuous linear operators on X into Y; L (X, Y) is a linear space.
Definition 6. (Simple Convergence Topology).This is the topology of the convergence at each point of X and thus it is defined by the family of semi-norms of the form p (T ) = sup j=1,...,r { q ( T x j )} (1.12) where are an arbitrary finite system of point of X and q an arbitrary continuous semi-norm on Y. L (X, Y) endowed with this topology will be denoted by L s (X, Y) and it is a locally convex linear topological space.
Definition 7. (Bounded Convergence Topology).This is the topology of uniform convergence on bounded sets of X and thus it is defined by the family of semi-norms of the form p (T ) = sup x∈B {q (T x)} (1.13) where B is an arbitrary bounded set of X and q an arbitrary continuous semi-norm on Y. L (X, Y) endowed with this topology will be denoted by L b (X, Y) and it is a locally convex linear topological space.
Remark 2. Since any finite set of X is bounded, the simple convergence topology is weaker than the bounded convergence topology.

Bochner Integral
In this section we will introduce the concept of integration of Bochner for vectorial functions in spaces of Fréchet.The properties here exposed are limited instead to the quasi-norm that induces the metric structure of the space of Fréchet.
Definition 8. Let F be a complex Fréchet space and let I be an interval in R; a function f : for some n ≥ 1, x j ∈ F and Lebesgue measurable sets Ω j ⊂ I with finite Lebesgue measure.In the representation of a simple function, the sets Ω j may always be arranged to be disjoint, and then Definition 9. Let F be a complex Fréchet space and let I be an interval in R; a function f : I → F is measurable if there is a sequence of simple functions g h such that for almost all t ∈ I.
Theorem 1.Let F be a complex Fréchet space and let I be an interval in R; if f : Definition 10.Let F be a complex Fréchet space and let I be an interval in R; we say that f : Definition 11.Let F be a complex Fréchet space and let I be an interval in R; we say that f : Definition 12. Let F be a complex Fréchet space and let I be an interval in R; we say that f : Definition 13.For a simple function g : , we define the Bochner's integral where L 1 (Ω i ) denote the Lebesgue measure of Ω i .
Definition 14.Let F be a complex Fréchet space, I be an interval in R and f : I → F is weakly measurable; we say that (2.9) Theorem 2. Let F be a complex Fréchet space and let I be an interval in R; f : (2.10) Proof.If f is Bochner integrable, then there exists an approximating sequence of simple functions g n .Thus f and (2.12) To prove the converse statement, let {h n } n∈N be a sequence of simple functions approximating f pointwise on I − Ω 0 , where Define simple functions by (2.16) we can apply the scalar dominated convergenge theorem and obtain that lim (2.17) where x ∈ F. We see that f is Bochner integrable but where x ∈ F; then f is Bochner integrable but not weakly integrable.
Example 3. Let be x : R → R the function defined by x (s) = 1 for all s ∈ R, then x ∈ L 2 loc (R).We define the semi-norms on L 2 loc (R) by, We have the same results if we choose B k such that lim inf Theorem To study the various relationships among the weak integral, definition ( 14) and the integral in Bochner sense, definition (15), we introduce a particular class of linear and continuous functionals.Then we will give a new definition of "weak" integral related to this class of linear and continuous functionals.
Definition 16.Let F be a complex Fréchet space, then if l is linear and there exists a constant C l such that Definition 18.Let F a complex Fréchet space, I an interval in R and L B (F, R) {0}.Let f : I → F weakly measurable.
We say that f is As a further comment we observe that it is possible to introduce other functional classes that have some connection with the integration.
Definition 19.Let {∥•∥ k } k≥1 be the the family of semi-norms that induces the topology on F and f : I → F a Bochner measurable function; for 1 ≤ p < ∞ and for all k ≥ 1, we define ) . (3.9) ) is a metric space.
As a final observation we can say that here the considered spaces of Fréchet show a wide range of possible definitions of integral, each of which has interesting peculiarity.Substantially we will use the definition ( 14) but as shown in immediately the following proposition sometimes we will use the possible relationschip with ( 14), ( 15) and ( 18).
Definition 22.Let F and G be a complex Fréchet space, then if l is linear and there exists a constant C l such that Proposition 2. Let T ∈ L B (F, G) be a linear continuous operator between the Fréchet spaces F and G. Let f : is Bochner integrable and Proof.Let's take F × F with the quasi-norm By theorem (2) g is Bochner integrable, moreover and applying proposition (2) to the two projection maps of F × F onto F shows that this give the result.
is the abscissa of convergence of f .
and we must proof that From (4.10) we have if Re {λ} > Re {λ 0 }; then the (4.11) gives (4.12) Definition 26.For f : [0, +∞) → F the exponential growth bound is given by We have Let abs ( f ) > 0 then F ∞ = 0 and, for λ 0 > abs ( f ), integration by parts gives The set for all s ≥ 0; thus the set and Remark is holomorphic for Re {λ} > abs ( f ) and, for all n ∈ N and Re {λ} > abs ( f ), as an improper Bochner integral.
Proof.Let's define q h : C → F for every h ∈ N by and q h, j : C → F for every h, j ∈ N by We see that for every k ∈ N and j > i Fix ε > 0 then exists j k,ε ∈ N such that q h, j (λ) − q h,i (λ) k < ε for all i, j > j k,ε then we have The limits exist uniformly for λ in a bounded subset of C. By the Weiesrtass convergence theorem, the functions q h are entire and q if Re {λ} > Re {λ 0 } and we have e −λ 0 s f (s) ds; integration by parts gives It follows that q h converges to f uniformly on compact subset of {λ : Re {λ} > abs ( f )}.By the Weiestrass convergence theorem, f is holomorphic and Proof.Results follow from the formulae: The second statement follows from (4.19) letting τ → +∞.
Theorem 7. Let f ∈ L 1 loc ([0, +∞) , F), suppose that abs ( f ) < +∞ and that t > 0 is a Lebesgue point of f ; then ) and max {abs ( f ) , 0} < w < +∞.By remark ( 14) w > ϖ (F), where Since w > ϖ (F), then the set {e −ws G (s) : s ≥ 0} is bounded in F and there exists a sequence of positive real numbers, if we put t = λs, where λ > 0, we have for all λ > 0. By theorem (5) we have Let thus by (4.22) we get Let's take moreover, since n > wt, we have If s t = u; then Let's define b n , where n 0 > 1 δ , converges; therefore b n → 0 for n → +∞ and we get Let's define Let's define M k e wtu du, we get d n converges and we obtain d n → 0 for n → +∞ and 5. The FsLip 0 ([0, +∞) , F) Space and the L B (X, F) Space Let F a Fréchet space and f : [0, +∞) → F, we define Proposition 9.
) is a quasi-normed metric space.
) is a quasi-normed metric space.
Let F a Fréchet space, X a Banach space and f : X → F a linear continuous operator; then for all k ∈ N there exists (5.6) Definition 28.Let F a Fréchet space, X a Banach space and f ∈ L(X, F) we define ) . (5.8) and for ε ↓ 0 we have |T x| F ≤ C ∥x∥ X .By (5.22) we obtain ) .