����������� ������ 49 (1997), 153–162

UDK 517.982.4

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α-TIMES INTEGRATED SEMIGROUPS (α ∈ R^{−})

Milorad Mijatović and Stevan Pilipović

Abstract. The α-times integrated semigroups, α ∈ R^{−} = (−∞, 0], are introduced and analyzed as extensions of 0-integrated semigroups.

0. Introduction

We introduce and analyze α-times integrated semigroups, α ∈ R^{−}. With α ∈ N this type of semigroups is extensively investigated in many papers, see for example [1], [2], [4], [5], [6], [11], [17]; for α > 0 we refer to [6], [10].

In this paper we apply results concerning 0-integrated semigroups [9] and analyze families of operators on the test space K_{1} with values in L(E, E) which are n-th distributional derivatives of α-times integrated semigroup for α > 0 sufficiently large and n > α.

As an application, we consider the Cauchy problem u′ = Au + T, T ∈ K′_{1} in the setting of α-times integrated semigroups α < 0.

1. Preliminaries

By L(E) = L(E, E) is denoted the space of bounded linear operators from a Banach space (E, ∥ · ∥) into itself and C(R, L(E)) is the space of continuous mappings from R into L(E). We refer to [15] and [18] for the definitions of spaces D(R), E(R), S(R), their strong duals S′(E) = L(S(R), E) and to [20] for the space S_{+} = {φ; |t^{k}φ^{(ν)}(t)| < C_{k,ν}, t ∈ [0, ∞), k, ν ∈ N_{0}} (N_{0} = N ∪ {0}) and its dual S′_{+} which consists of tempered distributions supported by [0, ∞).

The space of exponentially decreasing test functions on the real line R is defined by K_{1}(R) = {φ; |e^{k|t|}φ^{(ν)}(t)| < C_{k,ν}, t ∈ R, k, ν ∈ N_{0}} ([3]). This space has the same topological properties as S(R). The space K_{1}(R^{2}) is defined in an appropriate way. The strong dual of K_{1}(R), K′_{1}(R) is the sapce of exponential distributions. The space K′_{1+} ⊂ K′_{1}(R) consists of distributions which are supported by [0, ∞).

AMS Subject Classification: 46 F 99

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997.

It is the dual space to K_{1+} = {φ; |e^{k|t|}φ^{(ν)}(t)| < C_{k,ν}, t ∈ [0, ∞), k, ν ∈ N_{0}} which has the same topological properties as S_{+}. Note,

f ∈ K′_{1}(R) if and only if e^{−r|x|}f ∈ S′(R) for some r ∈ R.  (1)

The space K′_{1}(E) consists of continuous linear mappings S : K_{1} → E with the strong topology. Similarly K′_{1+}(E) is defined; we have K′_{1+}(E) ⊂ K′_{1}(E).

The convolution of f ∈ K′_{1+}(E) and g ∈ K′_{1+} is defined by ⟨f ∗ g, φ⟩ = ⟨f, ǧ ∗ φ⟩, φ ∈ K_{1}(R) (ǧ(t) = g(−t)). One can prove easily that f ∗ g = g ∗ f ∈ K′_{1+}(E).

Let T: [0, ∞) → L(E) be strongly continuous. Then it is exponentially bounded at infinity if there exist M ⩾ 0 and ω ⩾ 0 such that

∥T (t)∥ ⩽ Me^{ωt},  t ⩾ 0.  (2)

In this case φ ↦ ∫_{0}^{∞} T (t)φ(t) dt, φ ∈ K_{1}(R), defines an element of K′_{1+}(L(E)). The structure of K′_{1+}(L(E)) is given in the following theorem.

Theorem 1. [9] Let S ∈ K′_{1+}(L(E)).

a) There exists n_{0} ∈ N such that for every n ⩾ n_{0} there exist a strongly continuous function F_{n}: R → L(E), supp F_{n} ⊂ [0, ∞) and positive constants m_{n} and C_{n}, such that

∥F_{n}(t)∥ ⩽ C_{n}e^{m_{n}t}, t ⩾ 0, S = F_{n}^{(n)} (^{(n)} is the distributional n-th derivative).

b) Let S ∈ K′_{1+}(L(E)) and ψ, φ ∈ K_{1}(R). Then

⟨S(t, ⟨S(s, x), ψ(s)⟩), φ(t)⟩ = ∫ F_{n_{0}}(t, F_{n_{0}}(s, x))ψ^{(n_{0})}(s)φ^{(n_{0})}(t) ds dt. (3)

c) Let φ(t, s) ∈ K_{1}(R^{2}) and φ_{ν}(t), ψ_{ν}(s) be sequences in D(R) such that the product sequence φ_{ν}(t) · ψ_{(ν)}(s) converges to φ(t, s) in K_{1}(R^{2}) as ν → ∞. Then the limit

lim_{ν→∞} ⟨S(t, ⟨S(s, x), ψ_{ν}(s)⟩), φ_{ν}(t)⟩

exists and defines an element of K′_{1}(R^{2}) which we denote by S(t, S(s, x)), i.e.

⟨S(t, S(s, x)), φ(t, s)⟩ = lim_{ν→∞} ⟨S(t, ⟨S(s, x), ψ_{ν}(s)⟩), φ_{ν}(t)⟩ , φ ∈ K_{1}(R^{2}). (4)
d) Also, for φ ∈ K_{1}(R^{2}) and r, p ∈ N, we have
(i) ⟨\frac{∂^{r}}{∂t^{r}} S(t, S(s, x)), φ(t, s)⟩ = (−1)^{r} ⟨S(t, S(s, x)), \frac{∂^{r}}{∂t^{r}} φ(t, s)⟩;
(ii) ⟨\frac{∂^{p}}{∂s^{p}} S(t, S(s, x)), φ(t, s)⟩ = ⟨S (t, \frac{∂^{p}}{∂s^{p}} S(s, x)) , φ(t, s)⟩

= (−1)^{p} ⟨S(t, S(s, x)), \frac{∂^{p}}{∂s^{p}} φ(t, s)⟩ .

α-times integrated semigroups (α ∈ R^{−}) 155

As in the case of ordinary distributions (1) we have

f ∈ K′_{1+}(L(E)) if and only if e^{−r|x|}f ∈ S′_{+}(L(E)) for some r ⩾ 0.  (5)

The Laplace transformation of an f satisfying (5) is defined by

L(f)(λ) = f̂(λ) = ⟨f(t), e^{−λt}η(t)⟩ , Re λ > r,

where η ∈ C^{∞}(R), supp η = [−ε, ∞), ε > 0 and η ≡ 1 on [0, ∞). This definition does not depend on η (cf. [20]). If f ∈ L^{1}([0, ∞), E) (which means ∥ ∫_{0}^{∞} f(t) dt∥_{E} < ∞), then

f̂(λ) = ∫_{0}^{∞} e^{−λt}f(t) dt = ⟨f(t), e^{−λt}⟩ , Re λ > 0,

where the integral is taken in Bochner’s sense.

2. α-times integrated semigroup

Let T: (0, ∞) → L(E) be strongly continuous, integrable in a neighborhood of 0 (i.e. integrable on (0, ε) for some ε > 0) and exponentially bounded at infinity, which means that (2) holds on (ε, ∞) for some ε > 0. The operator R: {λ ∈ C : Re λ > ω} → L(E) defined by

R(λ) = ∫_{0}^{∞} e^{−λt}T (t) dt, Re λ > ω,

where the integral is understood in Bochner’s sense, is the Laplace transformation of T.

The family R(λ) = ∫_{0}^{∞} e^{−λt}T (t) dt, Re λ > ω, where T: [0, ∞) → L(E) is a strongly continuous and exponentially bounded function, is a pseudoresolvent iff T (t)T (s) = T (t + s), t, s ⩾ 0. Let α > 0 and S: (0, ∞) → L(E) be strongly continuous, integrable in a neighborhood of 0, exponentially bounded at infinity and

R(λ) = λ^{α} ∫_{0}^{∞} e^{−λt}S(t) dt, ℜλ > ω.

Then (R(λ))_{Re λ>ω} is a pseudoresolvent iff

S(t)S(s) = \frac{1}{Γ(α)}[∫_{t}^{t+s}(t + s − r)^{α−1}S(r) dr − ∫_{0}^{s}(t + s − r)^{α−1}S(r) dr], t, s ⩾ 0

(6)

(cf. [2], [10]).

Recall,

f_{α}(t) = \{ \begin{matrix}\frac{H(t)t^{α−1}}{Γ(α)} , & t ∈ R, α > 0, \\ f_{α+n}^{(n)}, & t ∈ R, α ⩽ 0, α + n > 0, n ∈ N, \end{matrix}  (7)

(H is Heaviside’s function).

Theorem 2. Let α ∈ R^{−}, S_{α} ∈ K′_{1+}(L(E)) and R(λ) = λ^{α}L(S_{α})(λ). Then (R(λ))_{Re λ>ω} is a pseudoresolvent iff there exists n_{0} ∈ N such that n_{0} + α > 0 and

S_{n_{0}+α}(t, ·) = (S_{α} ∗ f_{n_{0}})(t, ·), t ⩾ 0,

is continuous, supp S_{n_{0}+α} ⊂ [0, ∞) and satisfies

⟨S_{α}(t, S_{α}(s, x)), φ(t)ψ(s)⟩ = ⟨(S_{n_{0}+α}(t, S_{n_{0}+α}(s, x)))^{(n_{0},n_{0})}, φ(t)ψ(s)⟩

= ⟨\frac{1}{Γ(n_{0} + α)}(∫_{t}^{t+s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr−

− ∫_{0}^{s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr)^{(n_{0},n_{0})}, φ(t)ψ(s)⟩ (8)

for every φ, ψ ∈ K_{1}(R).

Moreover, (8) holds with S_{n+α} = S_{α} ∗ f_{n}, for every n ⩾ n_{0}.

Remark. [9] If α = 0, then (8) is equivalent to

⟨S_{0}(t, S_{0}(s, x)), φ(t, s)⟩ = ⟨S_{0}(t + s, x), φ(t, s)⟩ , φ ∈ K_{1}(R^{2}).

Proof. We have S_{α} = S_{n_{0}+α}^{(n_{0})}. Let x ∈ E. Relation (8) implies

(S_{α}(t, S_{α}(s, x))) = (S_{n_{0}+α}(t, S_{n_{0}+α}(s, x)))^{(n_{0},n_{0})} =

(\frac{H(t)H(s)}{Γ(n_{0} + α)}[∫_{t}^{t+s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr−

∫_{0}^{s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr])^{(n_{0},n_{)})}, (9)

t, s > 0, in the distributional sense. Since both sides are supported by [0, ∞) × [0, ∞), it follows that

(S_{α}(t, S_{α}(s, x))) = S_{n_{0}+α}(t, S_{n_{0}+α}(s, x)) =

\frac{1}{Γ(n_{0} + α)}[∫_{t}^{t+s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr−

∫_{0}^{s}(t + s − r)^{n_{0}+α−1}S_{n_{0}+α}(r, x) dr]

holds true for every t, s ⩾ 0. Thus, R(λ, ·) = λ^{n_{0}+α}L(S_{n_{0}+α})(λ, ·) is a pseudoresolvent. Let n ⩾ n_{0}. Since S_{n+α} = S_{n_{0}+α+(n−n_{0})}^{n−n_{0}}, it follows that (8) holds for every n ⩾ n_{0}.

Definition 1. Let (S(t))_{t⩾0} be a strongly continuous exponentially bounded family in L(E) and α > 0. Then it is called an α-times integrated semigroup if (6) is satisfied and S(0) = 0 ([10]).

Let S_{α} ∈ K′_{1+}(L(E)) and α ∈ R^{−}. Then, S_{α} is called an α-times integrated semigroup if there exists n_{0} ∈ N, such that n_{0} + α > 0, S_{n_{0}+α} = S_{α} ∗ f_{n_{0}} is continuous on R, supported by [0, ∞), exponentially bounded and satisfies (8). This is equivalent to say that, for some n_{0} and every n ⩾ n_{0}, it is an n-th distributional derivative of an n + α-times integrated semigroup.

We will use the symbol (S(t))_{t⩾0} or (S_{α}(t))_{t⩾0} for an α-times integrated semigroup if it is not specified whether α > 0 or α ⩽ 0, although for α ⩽ 0 it is an element of K′_{1+}(L(E)) and the above expression is formal.

Definition 2. Let α > 0. Then, (S(t))_{t⩾0} with the above properties is called non-degenerate if S(t)x = 0 for all t ⩾ 0, implies x = 0 ([10]). Let α ⩽ 0. Then S ∈ K′_{1+}(L(E)) is called non-degenerate if ⟨S(t, x), φ(t)⟩ = 0 for all φ ∈ K_{1} implies x = 0.

Note, C_{0}-semigroup is a 0-integrated semigroup ([9]). Also, if (S(t))_{t⩾0} is an n-times integrated semigroup, then n-th distributional derivative S^{(n)} is a 0integrated semigroup.

Definition 3. Let α ∈ R. An operator A is the generator of an α-times integrated semigroup (S(t))_{t⩾0} iff (a, ∞) ⊂ ρ(A) for some a ∈ R and the function

λ ↦ \frac{(λI − A)^{−1}}{λ^{α}} = L(S_{α})(λ), Re λ > a, is injective, where the Laplace transformation is understood in ordinary sense for α > 0 and in distributional sense for α ⩽ 0.

Part b) of Theorem 1 and the above definition directly imply the next Proposition.

Proposition 1. a) Let S_{α}, α ∈ R be an α-times integrated semigroup. Then S_{α} ∗ f_{−α} is a 0-integrated semigroup.

b) Let α < 0. Then A is the generator of an α-times integrated semigroup S_{α} iff A is the generator of a 0-integrated semigroup S_{α} ∗ f_{−α}.

3. The properties of A

Let A be the generator of an α-times integrated semigroup (S(t))_{t⩾0}, α > 0. Recall ([2], [10]), for all x ∈ D(A) and t ⩾ 0, S(t)x ∈ D(A), AS(t)x = S(t)Ax,

S(t)x = \frac{t^{α}}{Γ(α + 1)}x + ∫_{0}^{t} S(s)Ax ds. Moreover, ∫_{0}^{t} S(s)x dx ∈ D(A) for all x ∈ E,

t ⩾ 0 and

A ∫_{0}^{t} S(s)x ds = S(t)x − \frac{t^{α}}{Γ(α + 1)}x.

Theorem 3. Let α ∈ R^{−} and A be a generator of an α-times integrated semigroup (S(t))_{t⩾0}, S ∈ K′_{1+}(L(E)). Then, for all φ ∈ K_{1}, we have

a) A ⟨S(t, x), φ(t)⟩ = ⟨S(t, Ax), φ(t)⟩ for every x ∈ D(A).
b) ⟨S(t, x), φ(t)⟩ ∈ D(A) for every x ∈ E.

c) ⟨S(t, x), φ(t)⟩ = ⟨f_{α+1}(t, x), φ(t)⟩ + ⟨(f_{1} ∗ S)(t, Ax), φ(t)⟩, x ∈ D(A) and

A ⟨(f_{1} ∗ S)(t, x), φ(t)⟩ = ⟨S(t, x), φ(t)⟩ − ⟨f_{α+1}(t, x), φ(t)⟩ , x ∈ E.  (10)

Remark. if α = −1, then (10) with S = S_{−1}, implies

A ⟨(f_{1} ∗ S_{−1})(t, x), φ(t)⟩ = ⟨S_{−1}(t, x), φ(t)⟩ − ⟨δ(t, x), φ(t)⟩ ,

i.e.

A ⟨S_{0}(t, x), φ(t)⟩ = ⟨S_{0}(t, x), φ′(t)⟩ − φ(0)x, x ∈ E, φ ∈ K_{1}.

We will use also the notation A ⟨S(t, x), φ(t)⟩ = ⟨AS(t, x), φ(t)⟩.

Proof. We will also use notation S_{α} for S. Let φ ∈ D(R) and x ∈ D(A). Then ⟨S_{α}(t, x), φ(t)⟩ = (−1)^{n_{0}} ⟨S_{n_{0}+α}(t, x), φ^{(n_{0})}(t)⟩ , n_{0} + α > 0, n_{0} ∈ N

and Proposition 3.3 in [2] implies S_{n_{0}+α}(t, x) ∈ D(A) and AS_{n_{0}+α}(t, x) = S_{n_{0}+α}(t, Ax). This and the continuity of A imply

A ⟨S_{α}(t, x), φ(t)⟩ = (−1)^{n_{0}}A ⟨S_{n_{0}+α}(t, x), φ^{(n_{0})}(t)⟩

= (−1)^{(n_{0})}A ∫ S_{n_{0}+α}(t, x)φ^{(n_{0})}(t) dt = (−1)^{n_{0}}A lim_{ν→∞} ∑_{j=1}^{ν} S_{n_{0}+α}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}

= (−1)^{n_{0}} lim_{ν→∞} ∑_{j=1}^{ν} AS_{n_{0}+α}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}

= (−1)^{n_{0}} ⟨AS_{n_{0}+α}(t, x), φ^{(n_{0})}(t)⟩ = ⟨S_{α}(t, Ax), φ(t)⟩ , x ∈ E, φ ∈ D,

where (∑_{j=1}^{ν} S_{n_{0}+α}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}) is a sequence of integral sums for ∫ S_{n_{0}+α}(t, x)φ^{(n_{0})}(t) dt.

Let φ ∈ K_{1} and φ_{ν} be a sequence in D which converges to φ in K_{1}. Then

A ⟨S(t, x), φ(t)⟩ = lim_{ν→∞} ⟨S(t, Ax), φ_{ν}(t)⟩ = ⟨S(t, Ax), φ(t)⟩ .

This implies the assertion.

b) Proposition 3.3 in [2] implies ∫_{0}^{t} S_{n_{0}}(s, x) ds ∈ D(A) for every x ∈ E. Thus, ⟨∫_{0}^{t} S_{n_{0}}(s, x) ds, φ(t)⟩ ∈ D(A) for every φ ∈ K_{1} and x ∈ E. We know that ⟨S_{n_{0}+α}, φ(t)⟩ ∈ D(A) for every φ ∈ K_{1}. By putting φ^{(n_{0})} instead of φ, we obtain ⟨S(·, x), φ⟩ ∈ D(A) for every φ ∈ K_{1}.

c) Similarly, using Proposition 3.3 in [2], we obtain ⟨S_{α}(t, x), φ(t)⟩ = (−1)^{n_{0}} ⟨S_{n_{0}+α}(t, x), φ^{(n_{0})}(t)⟩

= (−1)^{(n_{0})} ⟨f_{n_{0}+α+1}(t, x), φ^{(n_{0})}(t)⟩ + (−1)^{n_{0}} ⟨(f_{1} ∗ S_{n_{0}+α})(t, Ax), φ^{(n_{0})}(t)⟩

= ⟨f_{α+1}(t, x), φ(t)⟩ + ⟨(f_{1} ∗ S_{n_{0}+α}^{(n_{0})})(t, Ax), φ(t)⟩

= ⟨f_{α+1}(t, x), φ(t)⟩ + ⟨(f_{1} ∗ S_{α})(t, Ax), φ(t)⟩ , x ∈ D(A), φ ∈ K_{1},

which gives the first assertion.

Again by using the quoted Proposition 3.3 in [2], it follows

A ⟨(f_{1} ∗ S_{α})(t, x), φ(t)⟩ = (−1)^{(n_{0})} ⟨A(f_{1} ∗ S_{n_{0}+α})(t, x), φ^{(n_{0})}(t)⟩

= (−1)^{(n_{0})} ⟨S_{n_{0}+α}(t, x), φ^{(n_{0})}(t)⟩ − (−1)^{n_{0}} ⟨f_{n_{0}+α+1}(t, x), φ^{(n_{0})}(t)⟩

= ⟨S_{n_{0}+α}^{(n_{0})}(t, x), φ(t)⟩ − ⟨f_{α+1}(t, x), φ(t)⟩ = ⟨S_{α}(t, x), φ(t)⟩ − ⟨f_{α+1}(t, x), φ(t)⟩

which gives (10).

Arendt ([2]) has obtained the characterization of a generator A of an (n + 1)- times integrated semigroup (S(t))_{t⩾0}, n ∈ N if A is a non-densely defined linear operator.

Theorem 4. Let α ∈ R, ω ∈ R, M ⩾ 0 and n ∈ N such that α + n > 0 if α ∈ (−∞, 0]. If α > 0 we take n = 0.

a) Let A be a (non-densely defined) linear operator on a Banach space E such that (a, ∞) ⊂ ρ(A) for some a ⩾ 0 and ω ∈ (−∞, a]. The following statements are equivalent:

(i) A generates an α + n + 1-times integrated semigroup (S(t))_{t⩾0} satisfying

lim_{h ↓ 0} sup \frac{1}{h}∥S(t + h) − S(t)∥ ⩽ Me^{ωt}, t ⩾ 0.

(ii) ∥\frac{1}{k!} (\frac{R(λ, A)}{λ^{α+n}})^{(k)}∥ ⩽ M (\frac{1}{λ − ω} )^{k+1}, for all Re λ > a, k ∈ N_{0}.

b) If A satisfies the equivalent conditions of (a), then the part of A on \overline{D(A)} is the generator of an (α + n)-times integrated semigroup.

c) Let A in (a) be a densely defined linear operator. Then (ii) in (a) is equivalent with the following condition:

A generates an (α+n)-times integrated semigroup (S(t))_{t⩾0} satisfying ∥S(t)∥ ⩽ Me^{ωt}, t ⩾ 0.

Remark. The case α = 0 in Theorem 2c) is the Hille-Yosida theorem.

Corollary 1. Let α ⩽ 0 and α + n > 0. If a densely defined linear operator A generates an (α + n)-times integrated semigroup, then its adjoint A^{∗} generates an (α + n + 1)-times integrated semigroup.

This directly follows from Theorem 4 since R(λ, A)^{∗} = R(λ, A^{∗}) for λ real.

4. Relations with distributional semigroup

We follow the definition of an exponentially bounded distributional semigroup, SGDE, given in [7], Definition 6.1. Note, instead of S(R), we use the space K_{1}(R) (cf. [9]). As in [7], we put D_{0} = {φ ∈ C_{0}^{∞}; supp φ ∈ [0, ∞)}.

If (T (t))_{t⩾0} is a C_{0}-semigroup and S_{α} = T ∗ f_{α}, α ∈ R then we define
S_{α}(φ, x) = (S_{α}(·, x) ∗ φ̌)(0) = ((T ∗ f_{α}(·, x)) ∗ φ̌)(0), x ∈ E, φ ∈ K_{1}. (11)

One can show that S_{α} is an α-times integrated semigroup.

Theorem 5. Let (S_{α}(t))_{t⩾0}, α ∈ R, be an α-times integrated semigroup. Assume that its infinitesimal generator A is densely defined. Then,

S_{α}(φ, x) = (S_{α} ∗ φ̌)(0)(x),  φ ∈ K_{1},  (12)

defines an element of K′_{1+}(L(E)) which is an SGDE iff α = 0.

Proof. Let (S(t))_{t⩾0} be an SGDE. As it was remarked by Arendt, Theorem 4.3 in [2] and Theorem 3.2 in [13] imply that there exists an n-times integrated semigroup (S_{n}(t))_{t⩾0}, n ∈ R such that

S(φ, x) = ⟨S_{n}^{(n)}(t, x), φ(t)⟩ = (S_{n}^{(n)}(·, x) ∗ φ̌)(0), φ ∈ D, x ∈ E.

This implies S_{n}^{(n)} = S_{n} ∗ f_{−n} = S_{0}, where S_{0} is a 0-integrated semigroup equal to S.

Now we will prove that for α ∈ R \ {0}, (12) does not define an SGDE. If it happened for some α ∈ R \ {0}, then (S_{α}(t))_{t⩾0} and ((S_{α} ∗ f_{−α})(t))_{t⩾0} would determine different SGDE’s which is impossible by the uniqueness of an SGDE with the given infinitesimal generator A.

Let A be an operator on E and T ∈ K′_{1+}(E). Then u ∈ K′_{1+}(E) is a solution to

u′ = Au + T in K′_{1}(E)  (13)

if ⟨u(t), φ(t)⟩ ∈ D(A) for every φ ∈ K_{1}(R) and (13) holds.

Let (S_{0}(t))_{t⩾0} be a 0-integrated semigroup with an infinitesimal generator which is not necessarily densely defined. We recall: if for some x ∈ E

S_{0}(φ, x) = ∫ S_{0}(t, x)φ(t) dt = 0 for every φ ∈ D_{0},  (14)

then x = 0.

As in [7], we extend (S_{0}(t))_{t⩾0} on T ∈ E′(R), supp T ⊂ [0, ∞) by using δsequences {ρ_{ν}} in D_{0}, (ρ_{ν} → δ): S_{0}(T, x) = lim_{ν→∞} S_{0}(T^{∗}ρ_{ν}, x) for those x ∈ E for which this limit exists. Because of (14), we can define the closure of S_{0}(T, ·) which will be denoted by \overline{S_{0}(T, ·)}. Theorem 4b) implies that a 0-integrated semigroup has the same properties as an SGDE except the set {S_{0}(φ, x); φ ∈ D_{0}, x ∈ E} is dense in E (cf. [7]).

Let U ∈ K′_{1+}(L(E, D(A))), V ∈ K′_{1+}(L(D(A), E)) and supp U ⊂ [a, ∞), supp V ⊂ [b, ∞), a, b ∈ R. Then U^{∗}V and V^{∗}U are defined as in [15]. Moreover, they are elements of K′_{1+}(L(D(A))) and K′_{1+}(L(E)), respectively, and their supports are bounded from the left by a + b.

Theorem 6. Let α ∈ R^{−} and S_{α} ∈ K′_{1+} be an α-times integrated semigroup with the infinitesimal generator A, such that S_{α} ∗ f_{−α} be a 0-integrated semigroup. Then

a) (−A + \frac{∂}{∂t}) ∗ S_{α} = f_{α} ⊗ I\overline{D(A)}, S_{α} ∗ (−A + \frac{∂}{∂t}) = f_{α} ⊗ I_{D(A)}, where

−A + \frac{∂}{∂t} = −δ ⊗ A + δ′ ⊗ I.

b) Let T ∈ K′_{1}(L(\overline{D(A)})). Then u = S_{α} ∗ f_{−α} ∗ T is the unique solution of (13).

Proof. a) Put S = S_{α} ∗ f_{−α}. Then, as in [7] Theorem 4.1, one can prove

(−A + \frac{∂}{∂t}) ∗ S_{0} = δ ⊗ I\overline{D(A)}.  (15)

Since D(A) is not dense in E, in general, we apply both sides of (15) on x ∈ D(A). Then, by making convolution with f_{α} we obtain the first assertion of a). In a similar way we prove the second one.

b) This simply follows from a).

Theorem 7. Let A be an infinitesimal generator of an α-times integrated semigroup (S_{α}(t))_{t⩾0}, α ∈ R^{−}. Then S_{α} ∗ f_{−α} determines an SGDE with the infinitesimal generator A on E_{0} × K_{1}, where E_{0} = \overline{\{S_{0}(φ, x); φ ∈ D_{0}, x ∈ E\}} and

(−A + \frac{∂}{∂t}) ∗ S_{α} = f_{α} ⊗ I_{E_{0}}, S_{α} ∗ (−A + \frac{∂}{∂t}) = f_{α} ⊗ I_{D(A)∩E_{0}}.

Let T ∈ K′_{1+}(E_{0}). Then u = S_{α} ∗ f_{−α} ∗ T is the unique solution of (13).

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(received 02.09.1997.)

Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia