Relative Demicompactness Properties for Exponentially Bounded C 0-Semigroups

Abstract

Let C be an invertible bounded linear operator in Banach space X. In this paper, we use the concept of relative demicompactness in order to study some properties of an exponentially bounded C-semigroup (T(t))t ≥ 0. More precisely, we prove that the relative demicompactness of T(t) at some positive values of t is equivalent to relative demicompactness of C – A where A is the infinitesimal generator of (T(t))t ≥ 0. Besides, we study the relative demicompactness of the resolvent. Finally, we present some conditions on exponentially bounded C-semigroups in Hilbert space guaranteeing the relative demicompactness of AC.

Full text

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ISSN 1066-369X, Russian Mathematics, 2022, Vol. 66, No. 6, pp. 1–7. © Allerton Press, Inc., 2022.
Russian Text © The Author(s), 2022, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2022, No. 6, pp. 3–12.
Relative Demicompactness Properties
for Exponentially Bounded C0-Semigroups
Hedi Benkhaleda,*, Asrar Elleucha,**, and Aref Jeribia,***
a Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, 3000 Tunisia
*e-mail: hedi.benkhaled13@gmail.com
**e-mail: asrar_elleuch@yahoo.fr
***e-mail: Aref.Jeribi@fss.rnu.tn
Received August 3, 2021; revised August 3, 2021; accepted September 29, 2021
Abstract—Let C be an invertible bounded linear operator in Banach space X. In this paper, we use the
concept of relative demicompactness in order to study some properties of an exponentially bounded
C-semigroup (T(t))t≥0. More precisely, we prove that the relative demicompactness of T(t) at some
positive values of t is equivalent to relative demicompactness of C – A where A is the infinitesimal gen-
erator of (T(t))t≥0. Besides, we study the relative demicompactness of the resolvent. Finally, we pres-
ent some conditions on exponentially bounded C-semigroups in Hilbert space guaranteeing the rela-
tive demicompactness of AC.
Keywords: C-semigroup, relative demicompact linear operator, Hilbert space
DOI: 10.3103/S1066369X22060019
INTRODUCTION
Here and below, X will denote a Banach space. The set of all closed densely defined (respectively,
bounded) linear operators on X will be denoted by (X) (respectively, (X)). For operator A, we denote
by (A) its definition domain and by R(A) its domain of values. If λ belongs to the resolvent set ρ(A) of
operator A, then by R(λ, A) we denote its resolvent (λI – A)–1.
The notion of an exponentially bounded C-semigroup was proposed by Davies and Pang [1] and then
studied by many authors (see [2–4]). This theory makes it possible to study abstract Cauchy problems.
Let C be an injective operator from (X). A family (T(t))t≥0 ⊆ (X) is called an exponentially
bounded C-semigroup on X if
(i) T(t + s)C = T(t)T(s) for t, s ≥ 0 and T(0) = C.
(ii) T(.)x : [0,+ ∞) → X is a continuous function for any x ∈ X.
(iii) There are M ≥ 0 and ω ∈ such that ||T(t)|| ≤ Meωt at t ≥ 0.
An exponentially bounded C-semigroup (T(t))t≥0 is thought to be uniformly continuous if function
T(.) : → (X) is continuous in the operator norm.
Infinitesimal generator A for (T(t))t≥0 is defined by formula:
and in this case,
Remark 1. Classical C0-semigroups are C-semigroups, where C is the identity operator I.
We recall [5] that operator A : (A) ⊆ X → X is called demicompact if for any bounded sequence {xn}
from domain (A) such that xn – Axn converges to x ∈ X, there exists a convergent subsequence in {xn}.
The class of all demicompact operators acting in a Banach space contains the class of all compact opera-
tors. The concept of demicompactness develops since 1966 when it was proposed by Petryshyn [5] in con-
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2
RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
HEDI BENKHALED et al.
nection with the discussion of fixed points. Moreover, Chaker et al. [6], as well as Jeribi [7], utilized dem-
icompact operators to study the essential spectra of closed linear operators. In a recent paper, Benkhaled
et al. [8], used demicompact operators to obtain some results from the theory of C0-semigroups. More
precisely, they obtained that for a C0-semigroup (T(t))t≥0 in Banach space X with infinitesimal
generator A, under certain restrictions, the following statements are equivalent:
(a) ∆ = ]0,+ ∞[.
(b) I – A is demicompact.
(c) λR(λ, A) is demicompact at some (and then at any) λ > ω; here, ∆ = {t > 0 such that T(t) is demi-
compact}.
In 2014, Krichen [9] proposed a generalization of the concept of demicompactness by introducing a
class of relatively demicompact linear operators with respect to a given linear operator as follows.
Definition 1 [9, Definition 2.1]. Let X be a Banach space and A : (A) ⊆ X → X and C : (C) ⊆ X → X
be densely defined closed linear operators such that (A) ⊆ (C). Operator A is called C-demicompact
(or relatively demicompact with respect to C) if for any bounded sequence {xn} from (A) such that Cxn – Axn
converges to x ∈ X, there exists a convergent subsequence in {xn}.
The class of all relatively demicompact operators is not as small as it might seem, it contains many other
operators. As examples of C-demicompact operators, we note those operators A, for which (C – A)–1 exists
and is continuous on its domain R(C – A). Note also that if C is invertible and C–1A is compact, then A is
a C-demicompact operator. In [7, 9], in the theory of Fredholm and perturbations, the authors obtained
some results concerning the class of relatively demicompact linear operators, and the results were applied
to the study of the relation between some results on perturbation and the behavior of the relative essential
spectra of unbounded linear operators acting in Banach spaces. Note that the sum and product with
respect to relatively demicompact operators, as well as the product of a complex number by a relatively
demicompact operator, is no longer necessarily relatively demicompact. The next result states that a com-
pact perturbation with respect to a relatively demicompact operator is relatively demicompact.
Lemma 1. Let X be a Banach space and A, C ∈ (X), where (A) ⊆ (C). If A is C-demicompact and B
is compact, then A + B is C-demicompact.
Proof. Let {xn} be a bounded sequence from (A) such that Cxn – (A + B)xn converges to y. Since B is
compact, there is a subsequence { } of {xn} such that converges to z. Since representation:
is valid, – converges to y + z. Using the C-demicompactness of A, we obtain the fact that { }
has a convergent subsequence, hence A + B is C-demicompact.
The purpose of this paper is to continue the analysis started in [8] and extend it to more general classes
by developing some aspects of the theory of C-semigroups with the participation of the concept of relative
demicompactness. In Section 1, some definitions and results needed in what follows are provided. In Sec-
tion 2, the relation between the relative demicompactness of T(t), C – A and λR(λ, A)C is discussed. In
Section 3, we present some results from the perturbation theory for the relative demicompactness of the
resolvent of the generator of an exponentially bounded C-semigroup. In Section 4, it is shown that if A
generates an exponentially bounded contracting C-semigroup in a Hilbert space, then AC is relatively
demicompact; this result is illustrated by an example.
1. PRELIMINARY INFORMATION
We present the most important basic results used later here.
Lemma 2 ([2]). Let (T(t))t≥0 be an exponentially bounded C-semigroup in X with infinitesimal generator A.
The following statements are true:
(a) A is a closed linear operator, and ⊇ (C).
(b) For any x ∈ (A), t ≥ 0, we have T(t)Ax = AT(t)x and
(c) For any x ∈ X, t ≥ 0 we have ∈ (A) and
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Txd
$

RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
RELATIVE DEMICOMPACTNESS PROPERTIES 3
(d) x ∈ (A), T(t)x ∈ (A), t ≥ 0, we have T(t)Ax = AT(t)x and CT(t)x is a differentiable function of vari-
able t, and
(e) Interval (ω, ∞) is contained in ρ(A), and for any λ > ω, x ∈ X, we have
Theorem 1 [4, Theorem 2.1.47]. Let B ∈ L(X) and A be the generator of an exponentially bounded C-semi-
group (T(t))t≥0 while BC = CB and BA ⊆ AB. We suppose that there exists injective operator ∈ (X) such that
Then, (A + B) generates an exponentially bounded -semigroup.
Definition 2 ([10]). Let A, C ∈ (X) while CA ⊆ AC. Linear operator A is called C-dissipative if
Definition 3 ([10]). We call a C-semigroup contractive if ||T(t)x|| ≤ ||Cx||, x ∈ X, and t ≥ 0.
Everywhere below, C is an invertible operator in (X) and (T(t))t≥0 is an exponentially bounded
C-semigroup in Banach space X satisfying the condition ||T(t)|| ≤ Mewt for some constants M ≥ 0, w ∈ .
The infinitesimal generator for (T(t))t≥0 is denoted by A.
2. EXPONENTIALLY BOUNDED C-SEMIGROUP IN WHICH T(t)
IS C-DEMICOMPACT AT SOME (RESPECTIVELY, ANY) t > 0
This section is devoted to C-semigroups, in which T(t) is C-demicompact at some positive values of
t> 0. In this connection, we denote by Θ the following set:
Theorem 2. We assume that (T(t))t≥0 is uniformly continuous. The following statements are equivalent:
(1) C – A is C-demicompact.
(2) There exists ξ > 0 such that ]0, ξ[ ⊂ Θ.
(3) λR(λ, A)C is C-demicompact at some (and hence for any) λ > ω.
Proof. (1) ⇒ (2). Let {xn} be a bounded sequence in X such that Cxn – T(t)xn converges to y at some
t> 0. By statement (c) of Lemma 2, we have
Assuming yn = , we obtain the fact that Ayn converges to y. It follows from the C-demicompact
property of operator C – A that {yn} has a subsequence { } that converges strongly to z. On the other
hand, since T(t) is uniformly continuous at t ≥ 0, for any ε > 0, there exists ξ > 0 such that for all t < ξ, we
have
Therefore, at any t < ξ, the inequality holds:
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tt t

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RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
HEDI BENKHALED et al.
Hence, for sufficiently small ε, we obtain the fact that is invertible at any t < ξ. Then,
It means that { } has a convergent subsequence. Therefore, T(t) is C-demicompact for any 0 < t < ξ and
some ξ > 0.
(2) ⇒ (3). We suppose that there exists ξ > 0 such that ]0, ξ[ ⊂ Θ. We take t0 ∈ ]0, ξ[ such that
is invertible, and let {xn} be a bounded sequence in X such that Cxn – λR(λ, A)Cxn converges to y for some
λ > ω. Due to the uniform continuity of T(t) and statement (b) of Lemma 2, we can write
Hence, converges to C–1(A – λ)y while (T(t0) – C)xn converges to
– λ)y. Since t0 ∈ ]0, ξ[ ⊂ Θ, then T(t0) is C-demicompact. Thus, {xn} contains a strongly
convergent subsequence.
(3) ⇒ (1). Let λ > ω and {xn} be a bounded sequence in X such that Axn converges to y. From the notation:
it follows that Cxn – λR(λ, A)Cxn converges to –R(λ, A)Cy. Thus, the C-demicompactness of λR(λ, A)C
guarantees that {xn} has a strongly convergent subsequence.
We now present a sufficient condition, under which embedding ]0, ξ[ ⊂ Θ at some ξ > 0 implies Θ =]0, ∞[.
Proposition 1. Let ξ > 0. We assume that for any t ≥ ξ, there exists t0 ∈ ]0, ξ[ such that C2 – T(t0)T(t) is
compact. If ]0, ξ[ ⊂ Θ, then Θ =]0, ∞[.
Proof. Let t ≥ ξ and {xn} be a bounded sequence in X such that Cxn – T(t)xn converges to y. It is needed
to show that {xn} contains a convergent subsequence. Since t ≥ ξ, by our assumption there exists t0 ∈ ]0, ξ[
such that C2 – T(t0)T(t) is compact. Therefore, {xn} contains a subsequence { } such that {(C2 –
T(t0)T(t)) } converges to z. We set = . From the relations:
it follows that (C – T(t0)) converges to z – T(t0)y. Since T(t0) is C-demicompact, { } contains a strongly con-
vergent subsequence, which is also { = }. Whence T(t) is C-demicompact at any t ≥ ξ.
Remark 2. Note that under the condition of Proposition 1, quantities t and t0 are separated. However,
if t tends to t0, then C2 – T2(t0) is compact. Identity
implies that C(C – T(2t0)) is compact, as is C – T(2t0). From
taking Lemma 1 into account, we c onclu de t hat C is C-demicompact, which is only possible if X is of f init e
dimension.
Theorem 2 and Proposition 1 directly imply
Corollary 1. Let (T(t))t≥0 be a uniformly continuous exponentially bounded C-semigroup and ξ > 0.
We suppose that for any t ≥ ξ, there exists t0 ∈ ]0, ξ[ such that C2 – T(t0)T(t) is compact. The following
statements are equivalent:
(1) C – A is C-demicompact.
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RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
RELATIVE DEMICOMPACTNESS PROPERTIES 5
(2) Θ =]0, ∞[.
(3) λR(λ, A)C is C-demicompact at some (and hence for any) λ > ω.
3. RELATIVE DEMICOMPACTNESS OF THE RESOLVENT
In this section, we study the properties of relative demicompactness for the resolvent of a closed
densely defined linear operator under the action of a compact perturbation.
Proposition 2. Let A ∈ (X), B be a compact operator, and λ ∈ ρ(C–1AC) ∩ ρ(C–1(A + B)C). Then,
R(λ,C–1AC) is C-demicompact if and only if R(λ, C–1(A + B)C is C-demicompact.
Proof. Let λ ∈ ρ(C–1AC) ∩ ρ(C–1(A + B)C) and {xn} be a bounded sequence in X such that Cxn –
R(λ,C–1(A + B)C)xn converges to y. Since B is compact, {R(λ, C–1(A + B)C)C–1BCR(λ, C–1AC)xn} has
subsequence {R(λ, C–1(A + B)C)C–1BCR(λ, C–1AC) } that converges to z. Then, from relation:
it follows that – R(λ, C–1AC) converges to y + z. Since operator R(λ, C–1AC) is C-demicompact,
sequence { } contains a strongly convergent subsequence. Hence, R(λ, C–1(A + B)C) is C-demicompact.
In the opposite direction, the reasoning is similar.
Next, we reformulate the previous proposition to study the relative demicompactness properties of the resol-
vent of the generator of an exponentially bounded C-semigroup under the action of a compact perturbation.
Corollary 2. Let B be a compact operator and A be a generator of an exponentially bounded C-semigroup
(T(t))t≥0 while BC = CB and BA ⊆ AB. We assume that there exists an invertible operator ∈ (X) such that
Let λ ∈ ρ() ∩ ρ(). Then, R(λ, ) is -demicompact if and only if
R(λ,) is -demicompact.
Proof follows from Theorem 1 and Proposition 2.
4. C-DISSIPATIVE OPERATORS AND CONTRACTING C-SEMIGROUPS
The following proposition establishes the relative demicompactness of operator AC under the condi-
tion that operator A is C-dissipative.
Proposition 3. Let A ∈ (X). We assume that CA ⊆ AC. If A is C-dissipative, then AC is C-demicompact.
Proof. Let {xn} be a bounded sequence in (A) such that {Cxn – ACxn} is a strongly convergent sequence
and {Cxn – ACxn} is a fundamental sequence. Let yn = Cxn. Since A is a C-dissipative operator, then for any
λ > 0, we have
Now, for λ = 1, we obtain
Since {Cxn – ACxn} is a fundamental sequence, {Cxn} is also fundamental, and since X is a Banach space,
{Cxn} converges, hence {xn = C–1yn} also converges.
Example 1. Let X = ([0, ∞)) be the space of all bounded continuous functions on [0, ∞) with
supremum norm. We define operator (A, (A)) as follows:
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Af s sf s f A

6
RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
HEDI BENKHALED et al.
Let (Cf )(s) = , where s > 0. At λ > 0, we have
for any s > 0 and f ∈ (A). Hence, we obtain
for any f ∈ (A) and λ > 0. Thus, A is a C-dissipative operator. Hence, by Proposition 3, AC is C-demi-
compact.
Next, we present conditions imposed on exponentially bounded C-semigroups in a Hilbert space and
guaranteeing the relative demicompactness of operator AC, where A is its infinitesimal generator. Let H be
a Hilbert space over = or . We denote by ⟨., .⟩H the inner product in H, and by ||.||H the correspond-
ing norm.
Proposition 4. Let (A, (A)) be a generator of exponentially bounded C-semigroup (T(t))t≥0 in H.
If (T(t))t≥0 is contractive, then AC is C-demicompact.
Proof. We divide the proof into two steps.
Step 1. We show that 2Re⟨AC2x, C2x⟩H ≤ 0 for all x ∈ (A).
Let x ∈ (A). Then, (T(t))t≥0 is contractive, and we have
Taking into account that T(s)x ∈ (A) by Lemma 2 statement (d), we obtain
Hence, decreases and ≤ 0.
On the other hand, by statement (d) of Lemma 2, we have
Thus, we obtain = ≤ 0 for all x ∈ (A).
Step 2. We show that AC is C-demicompact.
Let {xn} be a bounded sequence in (A) such that {Cxn – ACxn} is a strongly convergent sequence. Let
yn = C(Cxn – ACxn). Since {Cxn – ACxn} is a strongly convergent sequence and C ∈ (X), we obtain the
fact that {yn} is also a strongly convergent sequence. Therefore, this sequence is fundamental. From
it follows that {zn = C2xn} is also a fundamental sequence and, therefore, converges. Hence, sequence {xn=
C–2zn} converges.
Example 2. Let Ω be a domain in and H = L2(Ω). Let also Mq be the multiplication operator defined
in L2(Ω) by the formulas:
for measurable function q : Ω → . Then, operators
+
()
1
s
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s
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λ− = ≥ =λ
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11
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$
+= ≤ ≥
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H
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2
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2Re ( ) , ( ) 2Re ( ) , ( ) 0.
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HH
dd
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=
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22
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22
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≥−+ −
22
22
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R
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$$
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=∈∈R
() , and ,
tq
Tt f e f t f H

RUSSIAN MATHEMATICS Vol. 66 No. 6 2022
RELATIVE DEMICOMPACTNESS PROPERTIES 7
define a contracting strongly continuous semigroup on space H (see [11] for details). The generator
(A,(A)) for (T(t))t≥0 in space H is given by formulas:
On the other hand, we consider operator Cf = . Setting S(t) = T(t)C, it is easy to verify that (S(t))t≥0
is a contracting C-semigroup generated by A (see [3, Example 1]). Applying Proposition 4, we obtain the
fact that AC is C-demicompact.
ACKNOWLEDGMENTS
The authors thank Professor S. Piskarev for valuable comments and advice on this paper.
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
REFERENCES
1. E. B. Davies and M. M. H. Pang, “The Cauchy problem and a generalization of the Hille–Yosida theorem,”
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