1. Introduction
If
A is the infinitesimal generator of a linear, strongly continuous semigroup
in a Banach space
X, then for all
, there exists a unique, strongly continuous solution to the integral equation
Of course, this equation can (at least formally) be considered as the integrated version of the differential equation
There are cases when (
1) admits a solution only if
f is sufficiently regular. One may require regularity in space, for instance:
where
is a bounded linear operator. In the context of partial differential equations, one may think of an operator
C whose range consists of functions that are sufficiently regular in space. On the other hand, one may require time regularity, such as:
which means that
f is the fractional integral of order
of an
-function
g.
In the case of spatial regularity given by Equation (
3), one arrives at the concept of a
C-regularized semigroup (see, e.g., [
1]). In the case of time regularity described by Equation (
4), we obtain an
-times-integrated semigroup (see, e.g., [
2,
3] for integer
and [
4,
5,
6,
7] for fractional
). If both types of regularization are to be combined, we finally obtain an
-times-integrated,
C-regularized semigroup, see [
8,
9,
10,
11,
12]. For deeper insights into the properties of the resolvent families of the semigroup, we recommend exploring the works on resolvent families and abstract Volterra equations in locally convex spaces [
13,
14]. These studies offer particularly relevant and insightful perspectives on the corresponding resolvent families. Now, let
denote the dual space of
X, and let
be the adjoint operator of
A. The dual operators
of a strongly continuous linear semigroup generated by
A satisfy the semigroup property again, but
depends on
t continuously only with respect to the weak* topology on
. The properties of such dual semigroups are well established [
15,
16,
17,
18,
19]. In particular, there is a weakly* dense, closed subspace
such that the restriction of
to
is strongly continuous in
t. The generator of this semigroup is simply the part of
with values in
. Moreover,
is the closure in the norm of
of the domain
. If
X is reflexive, then
and
coincide, and
is a strongly continuous semigroup on
, generated by
. Dual semigroups play a crucial role when numerics and control problems involving semigroups are considered.
In this paper, we generalize this concept to -times-integrated C-regularized semigroups . It is not surprising that this is possible. The interesting part is which additional assumptions are needed to make the machinery work. In order to define a single-valued generator of the -times-integrated C-regularized semigroups, we require that be nondegenerate (i.e., only if ). The adjoint family is nondegenerate if and only if both and are dense subspaces of X. We can define the subspace of strong continuity . Again, contains the closure of , and also we have . If is the part of in , and , then is a subset of the generator of . To prove equality, we require the additional assumption that be dense in with respect to the graph norm of A. This condition, of course, holds always for strongly continuous semigroups. We do not know whether this condition is necessary for equality.
The following sections of this paper provide a comprehensive exploration of these topics.
Section 2 introduces the definition and basic properties of the adjoint family
, as well as the properties of
-times-integrated
C-regularized semigroups in terms of the weak* topology.
Section 3 explores whether the adjoint family can become nondegenerate. In
Section 4, we discuss the relations between the generator of
and the adjoint
of
A. Finally, the theory of the subspace of strong continuity
and its implications for reflexive spaces are given in
Section 5.
2. Strongly Continuous -Times-Integrated C-Regularized Semigroups
We begin by introducing the definition and properties of
-times-integrated
C-regularized semigroups. In this paper,
X will be a Banach space, and the space
will denote the space of bounded linear operators on
X. This definition has been introduced by several investigators; for further details, see [
8,
9,
20].
Definition 1 ([
8,
9,
20])
. Let and . A linear family of operators is called an α-times-integrated C-regularized semigroup on X if it satisfies:- (1)
For all ,
- (2)
for .
- (3)
is continuous for each .
- (4)
for all and .
Moreover, is said to be nondegenerate if for all implies .
The lemma referenced in Theorem 5 [
8], Proposition 2.2 [
21], and in the work by [
10] can be found below.
Lemma 1 ([
8,
10,
21])
. Suppose is a nondegenerate α-times-integrated C-regularized semigroup. Then, C is injective. Furthermore, for to be nondegenerate, it is necessary (and sufficient in the case of ) for C to be injective. The next definition outlines the characterization of the generator of the nondegenerate
-times-integrated C-regularized semigroup as presented in Definition 6 [
8].
Definition 2 ([
8])
. Let , and be a nondegenerate α-times-integrated C-regularized semigroup. The generator A of is defined by the following property: and if and only ifholds for all . The assumption that
is nondegenerate ensures that the operator
A is well defined. The well-known properties of the generator of a nondegenerate
-times-integrated
C-regularized semigroup
can be found in Theorems 7, 8 [
8].
Lemma 2 ([
8])
. Let A be the generator of a nondegenerate α-times-integrated C-regularized semigroup . Then,- (a)
A is a closed linear operator.
- (b)
For any and , and .
- (c)
.
3. Nondegeneracy of the Adjoint Family
Now, we turn to the adjoint family. In the subsequent analysis,
will denote the dual space of
X. We will utilize the concept of the weak*-integral: if
is a function such that
is integrable for all
, then the weak*-integral of
is defined by the property
If
is a closed, densely defined operator on
X, then
will denote the adjoint operator. The following properties of the adjoint operator are well known, see, for example, [
19,
22].
Lemma 3 ([
19,
22])
. Let be a closed, densely defined operator, and let be its adjoint. Then,- (a)
is weakly*-closed.
- (b)
is closed with respect to the norm topology in .
- (c)
is dense with respect to the weak*-topology in .
- (d)
If X is reflexive, then is dense with respect to the norm topology in .
In the forthcoming discussion, we will explain the details of finding the adjoint family for the semigroup . We will carefully look at its properties and explain why they are important for our mathematical analysis.
Definition 3. Let be an α-times-integrated C-regularized semigroup on a Banach space X. The family is called the adjoint family of .
The following lemma can be easily proven through straightforward calculation.
Lemma 4. Let , be an α-times-integrated C-regularized semigroup on a Banach space X and let be the adjoint family. Then,
- (a)
For all ,
- (b)
for all .
- (c)
For each , the map is continuous with respect to the weak*-topology in .
- (d)
For and ,
Let us define the nondegenerate adjoint of an -times-integrated C-regularized semigroup.
Definition 4. Consider an α-times-integrated C-regularized semigroup on a Banach space X, and let be its adjoint family. We say that is nondegenerate if for all implies that .
However, it is worth noting that the adjoint of a nondegenerate -times-integrated C-regularized semigroup may not always be nondegenerate, as illustrated in the following example:
Example 1. Let . For , we defineand Then, forms a nondegenerate one-time-integrated C-regularized semigroup. Moreover, and for ,and In this case, is a 1-times-integrated -regularized semigroup on the Banach space . However, it is degenerate because there exists in such that for all .
Remark 1. This example can be extended to the case where as follows: , where the operator C is defined by (6). To characterize integrated regularized semigroups with nondegenerate adjoints, we need to introduce the following lemma.
Lemma 5. Suppose , and is a nondegenerate α-times-integrated C-regularized semigroup on a Banach space X with generator A. Let and define Then, is a dense subspace of X if and only if both the domain of A and the range of C are dense in X.
Proof. Suppose
is dense for any fixed
. Let
and
be arbitrary. Then, there exist
,
,
such that
Now, for each
, let
. Then, there exist
such that
Thus, we conclude that x can be approximated by a sequence in .
To prove that
is dense, it is sufficient to show that for
and
, the vector
can be approximated by elements in
. This implies that every vector in the dense subspace
can be approximated by elements in
. We choose a sequence of functions
with supports contained in
such that
. By the strong continuity of
with respect to
t, we obtain
All we have to show is that
. If we define
we notice that
By utilizing the properties of convolution and Laplace transforms, as demonstrated in Theorem 10 [
8], we conclude that
solves the equation
In particular, .
Conversely, assuming that
and
are dense, let
and
. Let
. Pick
, and
such that
In the upcoming theorem, we provide a necessary and sufficient condition for the adjoints of -times-integrated C-regularized semigroups on a Banach space X to be nondegenerate.
Theorem 1. Let and be a nondegenerate C-regularized α-times-integrated semigroup on a Banach space X with generator A. Let be its adjoint. Then, is nondegenerate if and only if both the domain of A and the range of C are dense.
Proof. Let . Then, for all if and only if for all , and , which is equivalent to for all , where is taken from Lemma 5. Therefore, is nondegenerate if and only if is the only functional annihilating all of . This is equivalent to the assertion that is dense, and by using Lemma 5, the result follows. □
4. The Adjoint of the Generator
In this section, we will examine the relationship between the adjoint of the generator of an -times-integrated C-regularized semigroup and the weak* generator of the adjoint family. It is important to note that the adjoint operator of the generator operator A of is well defined because the domain of A is densely defined, given our assumption that the adjoint semigroup is nondegenerate.
Theorem 2. Let and be a nondegenerate α-times-integrated C-regularized semigroup, such that the adjoint is also nondegenerate. Let A be the generator of and be its adjoint. Then,
- (a)
If and , then and .
- (b)
If , then and .
Moreover, if is dense in with respect to the graph norm of A, then
- (c)
If and , then and .
Proof. - (a)
Let
and
be arbitrary. Then, for any fixed
, we have
This implies
and
.
- (b)
Similarly as (a).
- (c)
Let
. Choose a sequence
such that
and
. Note that
and
, as shown in [
8]. We have
In the limit,
, implying
.
□
Theorem 3. Let and be a nondegenerate α-times-integrated C-regularized semigroup, with its adjoint also being nondegenerate. Let A denote the generator of and its adjoint.
- (a)
If and , then for all , - (b)
Suppose is dense in with respect to the graph norm of A. If such that (7) holds for all , then with .
Proof. First, let
. Take any
. Then,
Since this holds for all
x in the dense subspace
, Equation (
7) follows.
To prove (b), assume that
is dense in
with respect to the graph norm of
A. Let
and
satisfy (
7). If
, we have
Consequently, for all
and
, we have
. Now, let
be arbitrary. Take a sequence
such that
and
. Fix some
. Then,
Taking the limit for , we obtain . Therefore, . □
5. The Subspace of Strong Continuity
The adjoint of a semigroup, which combines two mathematical operations, is typically continuous over time only in relation to a specific type of topology. We introduce the concept of a special subspace, known as the “sun space”, to address this in the context of semigroup adjoints. The adjoint of an -times-integrated C-regularized semigroup typically exhibits continuity over time solely concerning the weak* topology in . To address this, we incorporate the concept of the subspace of strong continuity, denoted as or sometimes referred to as the “sun space”, from the theory of adjoint of strongly continuous semigroups.
Definition 5. Let be a nondegenerate, C-regularized, α-times-integrated semigroup with generator A. Assume that and are dense in X. Let be the adjoint family. We defineMoreover, denotes the restriction , and denotes the part of in , where , i.e., if and . The following theorem explains important properties of nondegenerate semigroups that are -times-integrated C-regularized, along with their adjoints. It introduces a special space called , which shows how the adjoint family remains continuous over time. This theorem also shows that stays the same under specific operations and describes how the generator of the adjoint semigroup, denoted as B, equals the adjoint of the generator A, denoted as . Furthermore, it clarifies the conditions when matches the weak*-closure of the domain of . Overall, this theorem provides a thorough understanding of how adjoint semigroups behave and their structure concerning -times-integrated C-regularized semigroups in Banach spaces, where domains and ranges are dense.
Theorem 4. Let and be a nondegenerate, α-times-integrated C-regularized semigroup with generator A. Assume that and are dense in X, where . Let be the adjoint family, and let and be the adjoints of A and C, respectively. Then,
- (a)
is (norm-)closed and is a weakly*-dense, linear subspace of .
- (b)
is invariant under and .
- (c)
The restriction is a strongly continuous, α-times-integrated, -regularized semigroup. If B is the generator of , then in the sense that for all we have and .
- (d)
If is dense in with respect to the graph norm of A, then is the generator of , and we have . Moreover,
Proof. - (a)
It is clear that is a linear subspace of . The closedness of follows easily from the uniform boundedness of the operators for t in compact intervals. The weak* density will follow from (to be proven in (d)) and the weak* density of by using Lemma 3(c).
- (b)
To prove invariance under , note that , which is continuous in t if is continuous. Invariance under follows similarly by using Lemma 4(d).
- (c)
Since
is continuous in
t for
, the weak* integrals in Lemma 4 and in (
7) are in fact Bochner integrals. Lemma 4 implies then that
is an
-times-integrated,
C-regularized semigroup. If
, then by Theorem 3, the pair
satisfies (
7) with a Bochner integral. This is the defining equation for the generator of
, so that
.
- (d)
Now, let
be dense in
with respect to the graph norm of
A. Then, by using Theorem 3(b), we will have that, if
,
(i.e.,
), if and only if (
7) holds. The latter is equivalent to
, and we have
.
To prove that
. Let us consider
and
. Then, there exists constants
and
such that, for any
, such that
Then, for
, we have
As
, the estimate above goes to 0. Thus,
and hence
. Since
is closed in
, then
. By using the first part of (a), the fact that
is dense in
, and
, then
.
□
Corollary 1. Let and be an α-times-integrated, C-regularized semigroup on a reflexive Banach space X with a densely defined generator A and with dense range . Then, ; in particular, the adjoint family is a strongly continuous, α-times-integrated, -regularized semigroup on . Moreover, let be the adjoint operator of A and let B denote the generator of . Then, for all , we have with . If is dense in with respect to the graph norm of A, then .
Proof. For reflexive spaces, the weak and weak* topologies are the same. Hence, is a weakly dense subspace. However, for convex sets, the weak and the norm closures are the same, and is closed in the norm topology. Thus, . The remaining part of the corollary is a direct application of Theorem 4. □
Remark 2. We have the following remark:
If A has a nonempty resolvent, the hypothesis that is dense can be replaced by the weakest hypothesis that is dense. In fact, let us say ; it would follow that for , take converging to and then:and also we will haveand we can note that The condition is dense can be replaced by the condition that the range of C is dense. This can be immediate by the fact that C is bounded and is dense. In fact, for any , one has and . By using the strong continuity, we have , and then the result follows.
Let and be a nondegenerate α-times-integrated C-regularized semigroup such that the adjoint is also nondegenerate. Let A be the generator of and be its adjoint. If and , thenIn fact, if we pick an arbitrary , then Let and be a β-times integrated, C-regularized semigroup on a reflexive Banach space X with a densely defined generator A and with dense range . For any , we definewhere is the fractional integral of of order α (see, for instance, [8,23]). Then, we have that is an -times-integrated C-regularized semigroup on Banach space with generator . In fact, from Theorem 15 [8], we haveand also:Then, by using the fractional integral definition, the result follows.