����������� ������ 54 (2002), 117–124

UDK 517.982.4

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INTEGRATED SEMIGROUPS OF UNBOUNDED LINEAR OPERATORS AND C_{0}-SEMIGROUPS ON SUBSPACES

Ratko Kravarušić, Milorad Mijatović and Stevan Pilipović

Abstract. We give conditions under which an infinitesimal generator of an integrated semigroups of unbounded linear operator becomes an infinitesimal generator of a perturbated semigroup of bounded linear operators. Also, we analyze when a linear operator in a Banach space is an infinitesimal generator of an integrated semigroups of unbounded linear operators.

0. Introduction

Integrated semigroups of unbounded linear operators in Banach spaces have been studied in [8], [9], [10]. This paper is a continuation of such investigations. Here we use also some results of [17] for exponentially bounded integrated semigroups.

We proved in [9] that any semigroup of unbounded linear operators under additional conditions is an exponentially bounded semigroup on a subspace with a possibly stronger norm. On the other hand, in [17] is showed that every exponentially bounded integrated semigroup is an integrated C_{0}-semigroup on a subspace (with a possibly stronger norm). We obtain this result for semigroups of unbounded operators. Some assertions from the perturbation theory for bounded operators are also used.

1. Preliminaries

Let E be a Banach space with the norm ∥ · ∥ and let (S(t))_{t≥0} be a family of unbounded linear operators in E. We denote by D(S(t)) the domain od S(t) and set

D = { x ∈ D(S(s)S(t)) : S(0)x = 0, S(t)x is strongly continuous for t ≥ 0,

S(s)S(t)x = ∫_{0}^{s}(S(r + t) − S(r))x dr = S(t)S(s)x, for s, t ≥ 0. }

AMS Subject Classification: 47 D 06, 47 D 40

Keywords and phrases: C_{0}-semigroups, integrated semigroups, infinitesimal generator, perturbed operator.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

118 R. Kravarušić, M. Mijatović, S. Pilipović

If D ≠ {0}, then (S(t))_{t≥0} is said to be 1-integrated semigroup of unbounded linear operators in E or an integrated semigroup of unbounded linear operators.

Differentiation spaces C^{n}, n ∈ ℕ_{0} = ℕ ∪ {0} are defined as follows:

(i) C^{0} = D, (ii) C^{n} = { x ∈ D : S(t)x is n-times continuously differentiable function of t ≥ 0 }.

If x ∈ D, S(t)x ∈ C^{1}, then S′(s)S(t)x = S(s + t)x − S(s)x, s, t ≥ 0. This implies

S′(s)S′(t)x = S′(s + t)x, x ∈ C^{1}, s, t ≥ 0.

Let N = { x ∈ D : S(t)x = 0, t ≥ 0}. A semigroup (S(t))_{t≥0} is called nondegenerate if N = {0}. We shall observe only non-degenerate semigroups.

Let ω ∈ ℝ^{+} = (0, ∞). Define

∥x∥_{ω} := sup_{t≥0} e^{−ωt}∥S(t)x∥, x ∈ ⋂_{t≥0} D(S(t)),

E_{ω} := {x ∈ D : ∥x∥_{ω} < ∞}

We assume in this paper an additional condition:

For every ω > 0 exists C_{ω} > 0, such that ∥ · ∥_{ω} ≥ C_{ω}∥ · ∥.

Proposition 1. ([8]) a) If ω_{1} ≤ ω_{2} and x ∈ D, then ∥x∥_{ω_{2}} ≤ ∥x∥_{ω_{1}}. Hence, if ω_{1} ≤ ω_{2}, then E_{ω_{1}} ⊆ E_{ω_{2}}.

b) If x ∈ E_{ω}, then S(t)x ∈ E_{ω} and ∥S(t)x∥_{ω} ≤ \frac{2e^{ωt}}{ω} ∥x∥_{ω}.

Let \overline{E}_{ω} denote the closure of the set E_{ω} under norm ∥ · ∥ and S(t)|\overline{E}_{ω} be the part of S(t), i.e.

D(S(t)|\overline{E}_{ω}) = \{ x ∈ \overline{E}_{ω} : x ∈ D(S(t)) and S(t)x ∈ \overline{E}_{ω} \}.

Theorem 1. ([8]) Let (S(t))_{t≥0} be an integrated semigroup of unbounded linear operators in E.

a) Let ω > 0 be fixed. Suppose that for every t ≥ 0, S(t)|\overline{E}_{ω} is a closed operator in \overline{E}_{ω}. Then (E_{ω}, ∥ · ∥_{ω}) is a Banach space.

b) If S(t) is a closed operator in E, then S(t)|\overline{E}_{ω} is closed in \overline{E}_{ω} for t ≥ 0 and ω > 0.

For fixed ω > 0 and λ ∈ ℂ, Re λ > ω, define

R^{ω}(λ)x := λ ∫_{0}^{∞} e^{−λt}S(t)x dt, x ∈ E_{ω}.

Observe that as an operator in (E_{ω}, ∥ · ∥_{ω}), R^{ω}(λ) is bounded but in general unbounded in (E, ∥ · ∥).

Theorem 2. ([8)] The family (R^{ω}(λ))_{Re λ>ω} is the resolvent of a closed operator A^{ω} in E_{ω} (closed in the ∥ · ∥_{ω} norm topology), ω > 0.

Let D(A) = ⋃_{ω>0} D(A^{ω}) and x ∈ D(A). Then there exists ω ∈ ℝ^{+} such that x ∈ D(A^{ω}) and there exists y ∈ E_{ω} such that x = R^{ω}(λ)y for Re λ > ω. Put Ax = λ−y. Then, we call A the infinitesimal generator of the integrated semigroup (S(t))_{t≥0}.

Theorem 3. ([10]) a) For x ∈ D(A), S(t)x is a differentiable function of t for t ≥ 0 and S′(t)x − x = S(t)Ax, or, equivalently,

S(t)x − tx = ∫_{0}^{t} S(s)Ax ds.

b) If x ∈ D(A), then there exists ω′ ∈ ℝ^{+} such that R^{ω}(λ)Ax = AR^{ω}(λ)x, Re λ > ω, ω ≥ ω′.

Let ω ∈ ℝ^{+} and set

D(A_{1}^{ω}) = {x ∈ E_{ω} : (i) S(t)x is differentiable for t ≥ 0 with respect to ∥ · ∥;

(ii) ∃y ∈ E_{ω} such that S′(t)x − x = S(t)y }

For x ∈ D(A_{1}^{ω}) define y := A_{1}^{ω}x. Let D(A_{1}) = ⋃_{ω>0} D(A_{1}^{ω}) and for x ∈ D(A_{1}^{ω}) we define A_{1}x := A_{1}^{ω}x.

Theorem 4. ([10]) Let A be the infinitesimal generator of the integrated semigroup of unbounded linear operators (S(t))_{t≥0} . Then A = A_{1}.

For all ω ∈ ℝ^{+}, let D_{ω} := \overline{D(A^{ω})}^{∥·∥_{ω}}.

Theorem 5. ([9]) Let ω > 0 be fixed. Then, for t ≥ 0, S(t)D_{ω} ⊂ D_{ω} and (S^{ω}(t))_{t≥0} = (S(t)|D_{ω})_{t≥0} is an exponentially bounded integrated semigroup on (D_{ω}, ∥ · ∥_{ω}) with the infinitesimal generator equal to A^{ω}|D_{ω}.

2. Integrated semigroups and C_{0}-semigroups

Let ω > 0, and

D_{ω}^{1} = { x ∈ D_{ω} : S(t)x is strongly continuously differentiable function of t ≥ 0 with respect to the norm ∥ · ∥_{ω}. }

Then (S^{′ω}(t))_{t≥0} = (S′(t)|D_{ω}^{1})_{t≥0} is strongly continuous but not exponentially bounded. For x ∈ D_{ω}^{1} and ν ∈ ℝ^{+}, let

∥x∥_{ω,ν} = sup_{t≥0} e^{−νt}∥S^{′ω}(t)x∥, D̃_{ω,ν} = {x ∈ D_{ω}^{1} : ∥x∥_{ω,ν} < ∞}.

It is easy to prove that D̃_{ω,ν} is a Banach space (with respect to the norm ∥ · ∥_{ω,ν}) and

∥S′(t)x∥_{ω,ν} ≤ e^{νt}∥x∥_{ω,ν}, t ≥ 0, x ∈ D̃_{ω,ν}.

Moreover, if ν_{1} ≤ ν_{2}, then ∥ · ∥_{ω,ν_{2}} ≤ ∥ · ∥_{ω,ν_{1}} and D̃_{ω,ν_{1}} ⊂ D̃_{ω,ν_{2}}. But, on D̃_{ω,ν} we have lost the strong continuity of (S^{′ω}(t))_{t≥0}. In order to enforce the strong continuity, we consider the following subspace

D_{ω,ν} = { x ∈ D̃_{ω,ν} : ∥S^{′ω}(t)x − x∥_{ω,ν} → 0, for t ↓ 0 }.

It is clear that D_{ω,ν} is a closed subspace of D̃_{ω,ν} with the norm ∥·∥_{ω,ν}. Further, S^{′ω}(t)D_{ω,ν} ⊂ D_{ω,ν} and S^{′ω}(t) is strongly continuous on D_{ω,ν} (under the norm ∥ · ∥_{ω,ν}). The question is, whether after these restrictions, the space D_{ω,ν} contains D(Ã^{ω}) = D(A^{ω}|D_{ω})? Since ∥S^{ω}(t)∥_{ω} ≤ \frac{2}{ω} e^{ωt}, t ≥ 0, we have the next theorem which is an appropriate modification of Theorem 4.1 ([17]).

Theorem 6. Let ν > ω. Then:

a) (D_{ω,ν}, ∥ · ∥_{ω,ν}) is a Banach space. Moreover, D(Ã^{ω}) ⊂ D_{ω,ν} ⊂ D_{ω}^{1}, and

∥x∥_{ω} ≤ ∥x∥_{ω,ν}, x ∈ D_{ω,ν}, ∥x∥_{ω,ν} ≤ M̃_{ω}∥x∥_{ω,Ã^{ω}}, x ∈ D(Ã^{ω}).

Here ∥x∥_{ω,Ã^{ω}} = ∥x∥_{ω}+∥Ã^{ω}x∥_{ω} denotes the graf ω-norm of Ã^{ω} and M̃^{ω} is a positive constant.

b) D_{ω,ν} is invariant under S^{′ω}(t) and the restriction (T^{ω}(t))_{t≥0} = (S^{′ω}(t)|D_{ω,ν})_{t≥0} is a strongly continuous semigroup on (D_{ω,ν}, ∥ · ∥_{ω,ν}) with the infinitesimal generator Ã^{ω,ν} which is the part of Ã^{ω} in D_{ω,ν}.

Thus, for an integrated semigroup of unbounded linear operators (S(t))_{t≥0} which satisfies Theorem 1.a), there exists a family of subspaces with the stronger norms and the restriction (S(t))_{t≥0} on these subspaces are integrated C_{0}-semigroup (Theorem 5. and Theorem 6.).

3. Perturbations. Characterization of generators of integrated semigroups of unbounded linear operators

We will use some results of perturbation theory [1], [7], [14], [17].

Let (S(t))_{t≥0} be an integrated semigroup of unbounded linear operators in a Banach space E, with an infinitesimal generator A. The first question is, what are the conditions which we have to impose on an integrated semigroup of bounded linear operators such that its infinitesimal generator is “nearly” close to A?

The second question is, when a linear operator in E is an infinitesimal generator of an integrated semigroup of bounded linear operators?

In Theorem 8 ([10]), we have given an answer to these questions. But, now, we use Theorem 4.5 from [17] and give conditions under which a linear operator is the infinitesimal generator of an exponentially bounded integrated semigroup on the spaces (D_{ω}, ∥ · ∥_{ω}), ω > 0.

Theorem 7. (see Theorem 4.5 in [17]) Let Ã^{ω} generate a non-degenerate exponentially bounded integrated semigroup on (D_{ω}, ∥·∥_{ω}) and let B^{ω,ν} be a bounded linear operator on (D_{ω,ν}, ∥ · ∥_{ω,ν}), ν > ω. Then:

a) A_{0}^{ω} = Ã^{ω} + B^{ω,ν} with D(A_{0}^{ω}) = D(Ã^{ω}) generates a non-degenerate exponentially bounded integrated semigroup on (D_{ω}, ∥ · ∥_{ω}).

b) The part A_{0}^{ω,ν} of A_{0}^{ω} in D_{ω,ν} generates a strongly continuous semigroup on (D_{ω,ν}, ∥ · ∥_{ω,ν}).

Theorem 8. Let (S(t)_{t≥0} be an integrated semigroup of unbounded linear operators in a Banach space E with an infinitesimal generator A and (S^{ω}(t))_{t≥0} be the restriction of (S(t))_{t≥0} on the subspace (D_{ω}, ∥·∥_{ω}) with the infinitesimal generators Ã^{ω}, ω > 0. If there exists a family of bounded linear operators (B^{ω,ν})_{ν>ω>0} defined on (D_{ω,ν}, ∥ · ∥_{ω,ν}) such that ω_{1} ≤ ω_{2} implies B^{ω_{1},ν_{1}} ⊂ B^{ω_{2}ν_{2}} (for some v_{1} > ω_{1} and for some v_{2} > ω_{2}), then there exists an integrated semigorup of unbounded linear operators defined on a space “no less than” D = ⋃_{ω>0} D_{ω}.

Proof. By Theorem 7, for every ω > 0 the operator A_{0}^{ω} = Ã^{ω} + B^{ω,ν} with D(A_{0}^{ω}) = D(Ã^{ω}), generates an exponentially bounded integrated semigroup (T^{ω}(t))_{t≥0} on (D_{ω}, ∥ · ∥_{ω}). Then ∥T^{ω}(t)∥_{ω} ≤ M′e^{ω′t}, t ≥ 0.

We will prove that ω_{1} ≤ ω_{2} implies T^{ω_{1}}(t) ⊂ T^{ω_{2}}(t). Indeed, by Proposition 3.1 ([17])

R(λ, Ã^{ω_{1}}+B^{ω_{1}}) = λ ∫_{0}^{∞} e^{−λt}T^{ω_{1}}(t) dt, R(λ, Ã^{ω_{2}}+B^{ω_{2},ν_{2}}) = λ ∫_{0}^{∞} e^{−λt}T^{ω_{2}}(t) dt.

Here ν_{1} > ω_{1}, ν_{2} > ω_{2},

∥T^{ω_{1}}(t)∥_{ω_{1}} ≤ M′_{1}e^{ω′_{1}t} and ∥T^{ω_{2}}(t)∥_{ω_{2}} ≤ M′_{2}e^{ω′_{2}t}, t ≥ 0.

For λ > max{ω′_{1}, ω′_{2}} the integrals exist in the norms ∥ · ∥_{ω_{1}} and ∥ · ∥_{ω_{2}} and therefore in the norm ∥ · ∥. We have Ã^{ω_{1}} + B^{ω_{1},ν_{1}} ⊂ Ã^{ω_{2}} + B^{ω_{2},ν_{2}} and for x ∈ D_{ω_{1}}

R(λ, Ã^{ω_{1}} + B^{ω_{1},ν_{1}})x = R(λ, Ã^{ω_{2}} + B^{ω_{2},ν_{2}})x.

This implies T^{ω_{1}}(t)x = T^{ω_{2}}(t)x, t ≥ 0. For x ∈ D = ⋃_{ω>0} D_{ω}, set

T (t)x = T^{ω}(t)x , t ≥ 0, if x ∈ D_{ω}.

It easy to prove that family of operators (T (t))_{t≥0} on D satisfies all conditions for the integrated semigroup.

Example 1. Let (S(t)_{t≥0} be the given integrated semigroup of unbounded linear operators in E. Fix t_{0} > 0. Then the family (S′^{ω,ν}(t_{0}))_{ν>ω>0} satisfies the assumptions of Theorem 8. Namely, for ω_{1} ≤ ω_{2} we choose ν_{1} > ω_{1} and ν_{2} ≥ ν_{1}. Clearly, S′^{ω_{1},ν_{1}}(t_{0}) ⊂ S′^{ω_{2},ν_{2}}(t_{0}). Moreover, the operators (S′^{ω_{1},ν_{1}}(t_{0}))_{ν>ω>0} are bounded on (D_{ω}, ∥ · ∥_{ω}). Then, we can take B^{ω,ν} = Ã^{ω} + S′^{ω_{1},ν_{1}}(t_{0}).

Let D′_{ω} = \overline{D(Ã_{0}^{ω})} = \overline{D(Ã^{ω})}. Then D_{ω} ⊂ D′_{ω} ⊂ \overline{E}_{ω} because D_{ω} ⊂ E_{ω}.

Theorem 9. Let (T (t))_{t≥0} be an integrated semigroup on D = ⋃_{ω>0} D_{ω} by Theorem 8. Then there exists an extension (T̃(t))_{t≥0} of (T (t))_{t≥0} defined on D′_{ω} = ⋃_{ω>0} D′_{ω} if one of two equivalent conditions holds:

a) The operator J^{ω}(t) = ∫_{0}^{t} T (s)A_{0}^{ω} ds, t ≥ 0, defined on D(A_{0}^{ω}) = D(Ã), is closable (in the norm ∥ · ∥).

b) The operator I^{ω}(t) = \frac{1}{2πi} ∫_{0}^{t} ∫_{γ−i∞}^{γ+i∞} e^{λt}R^{ω}(λ)A_{0}^{ω} \frac{dλ}{λ} ds, t ≥ 0, defined on D((A_{0}^{ω})^{3}), is closable (in the norm ∥ · ∥).

In the case that one of the equivalent conditions holds, we have that T̃^{ω}(t) = T̃(t)|D′_{ω}, t ≥ 0 are closed operators on D′_{ω} for ω > 0.

Proof. a)⇒ b) Let J^{ω}(t) be closable on D(A_{0}^{ω}). Let (x_{n})_{n} be a sequence on D(A_{0}^{ω}) such that x_{n} → 0 and J^{ω}(t)x_{n} → y as n → ∞. Then y = 0. Moreover, we have

T^{ω}(t)x_{n} − tx_{n} = ∫_{0}^{t} T (t)A_{0}^{ω}x_{n} ds.

If x_{n} → 0 and T^{ω}(t)x_{n} → z as n → ∞, then T^{ω}(t)x_{n} − tx_{n} → z and this implies z = 0. Hence T^{ω}(t) is closable.

Let (x_{n})_{n} be a sequence on D((A_{0}^{ω})^{3}) such that x_{n} → 0 and I^{ω}(t)x_{n} → u as n → ∞. Then I^{ω}(t)x_{n} = T^{ω}(t)x_{n} − tx_{n}. We have then that T^{ω}(t) is closable and it follows u = 0 and I^{ω}(t) is closable. The converse is also true (we use the fact that \overline{D(A_{0}^{ω})} = \overline{D((A_{0}^{ω})^{3})}.

If conditions in a) or b) hold, then the operator T^{ω}(t) is closable. Denote the smallest closure by \overline{T}^{ω}(t) and define

T̃(t)x = \overline{T}^{ω}(t)x, if x ∈ D(\overline{T}^{ω}(t)) = D′_{ω}.

Then T̃(t) is well defined and (T̃(t))_{t≥0} is an extension of (T (t))_{t≥0}. For x ∈ D(A_{0}) = ⋃_{ω>0} D(A_{0}^{ω}), define

A_{0}x = A_{0}^{ω}x, if x ∈ D(A_{0}^{ω}).

We will prove that the infinitesimal generator A_{1} is an extension of the infinitesimal generator A_{0}. It is sufficient to prove A_{0}^{ω} ⊂ A_{1}^{ω}.

Let x ∈ D(A_{0}^{ω}). Then

T^{′ω}(t)x − x = T^{ω}(t)A_{0}^{ω}x.

Thus T^{ω}(t)x is differentiable and there exists y = A_{0}^{ω}x ∈ E_{ω} such that T^{′ω}(t)x−x = T^{ω}(t)y. This implies x ∈ D(A_{1}^{ω}) and A_{1}^{ω}x = A_{0}^{ω}x. Hence, A_{0}^{ω} ⊂ A_{1}^{ω}.

Let A be a closed linear operator in E. Then, by Theorem 4.2 [17], A generates a non-degenerate exponentially bonunded integrated semigroup if and only if the following conditions hold:

a) There exists some λ ∈ ℝ such that R(λ, A) exists as an everywhere defined bounded linear operator on E.

b) There exists a norm ∥ · ∥_{ω} on D(A) such that

(i) ∥x∥ ≤ c_{1}∥x∥_{ω} ≤ c_{2}∥x∥_{A} for x ∈ D(A),

(ii) the part A^{ω} of A in E_{ω} = \overline{D(A)}^{∥·∥_{ω}} generates a strongly continuous semigrup (T^{ω}(t))_{t≥0} on E_{ω}. This leads to the next theorem.

Theorem 10. Let E be a Banach space and let (E_{ω}, ∥ · ∥^{ω})_{ω>0} be a nested family of nontrivial subspaces of E such that ω_{1} ≤ ω_{2} implies E_{ω_{1}} ⊂ E_{ω_{2}}. Let A be a linear operator with the domain and range in E′ = ⋃_{ω>0} E_{ω}. Let A^{ω} denote the part of A in E_{ω}. In addition, suppose:

(i) For every ω > 0 there exists K_{ω} > 0 such that ∥x∥^{ω} ≥ K_{ω}∥x∥, x ∈ E_{ω}. (ii) If ω_{1} ≤ ω_{2} and x ∈ E_{ω_{1}}, then ∥x∥^{ω_{2}} ≤ ∥x∥^{ω_{1}}. (iii) For every ω > 0 there exists λ_{ω} ∈ ℝ such that R(λ_{ω}, A^{ω}) exists as an

everywhere defined bounded linear operator on E_{ω}.

(iv) There exists a norm ∥ · ∥^{ω,ν} on D(A^{ω}) such that

∥x∥^{ω} ≤ c_{1}(ω)∥x∥^{ω,ν} ≤ c_{2}(ω)∥x∥_{A^{ω}}^{ω},

and the part of A^{ω} in E_{ω,ν} = \overline{D(A^{ω})}^{∥·∥_{ω,ν}} generates a strongly continuous semigroup (T^{ω,ν}(t))_{t≥0} on E_{ω,ν}.

Then there exists an integrated semigroup (S(t))_{t≥0} of (in general unbounded) linear operators defined “no less than” on E′ = ⋃_{ω>0} E_{ω}.

Proof. By Theorem 4.2 [17], for every ω > 0, A^{ω} generates an exponentially bounded integrated semigroup (S^{ω}(t))_{t≥0}. Then A^{ω_{1}} ⊂ A^{ω_{2}} implies S^{ω_{1}}(t) ⊂ S^{ω_{2}}(t) (see the proof of Theorem 8) and we can define an integrated semigorup (S(t))_{t≥0} on E′ = ⋃_{ω>0} E_{ω}.

Remarks. The norms ∥ · ∥^{ω} and ∥ · ∥^{ω,ν} in Theorem 10 are different from the norms ∥ · ∥_{ω} and ∥ · ∥_{ω,ν}. Namely, the norms ∥ · ∥_{ω} and ∥ · ∥_{ω,ν} are defined in the case where we have an integrated semigroups but in Theorem 10 we have not got such an assumption.

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(received 31.12.2002)

R. Kravarušić, Faculty of Economics, Banja Luka, Republic Serbian M. Mijatović and S. Pilipović, Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia