A NEW CHARACTERIZATION OF B-BOUNDED SEMIGROUPS WITH APPLICATION TO IMPLICIT EVOLUTION EQUATIONS

Definition 1.1. Let X and Z be Banach spaces and suppose that A and B are two linear operators such that (i) A :D(A)→X, B :D(B)→ Z, with D(A), D(B)⊂X, (ii) D(A) ⊂ D(B) and for some ω ∈ R the resolvent set of A satisfies ρ(A) ⊃ ]ω,∞[. A one-parameter family of linear operators (Y (t))t≥0, which satisfies (1) Y (t) : → Z, with X ⊃ ⊇D(B), and for any t ≥ 0 and f ∈D(B) ∥∥Y (t)f ∥∥ ≤M exp(ωt)‖Bf ‖, (1.1)


Introduction
The study of the transport equation with multiplying boundary conditions recently prompted A. Belleni Morante to introduce a new class of evolution operators that he called B-bounded semigroups. The original definition of this class was given in [7] and generalized first by Belleni himself in [6], and then by Banasiak in [3]. Banasiak's definition reads as follows.
is called a B-quasi bounded semigroup generated by A and B.
Here and in what follows we denote by · the norm in the Banach space Z. In the case B is bounded, D(B) = X = Z, M = 1, ω = 0, one obtains the original definition, while the case X = Z gives Belleni's generalization.
It is immediate to see that, if B = I , B = 0 then the family (Y (t)) t≥0 is neither a C 0 -semigroup nor an integrated semigroup [1,9,10]. In general, the family (Y (t)) t≥0 is not even a C-existence family [8]. A deep analysis of this topic, that is, on the link of B-bounded semigroups with C-existence families and C-regularized semigroups can be found in [5].
However, one can see that the notion of B-bounded semigroup is strictly related to the notion of empathy [4]. Furthermore, both the empathy theory and the B-bounded semigroup theory can be used for solving implicit evolution equations. In such a context the B-bounded semigroup method seems to generalize the method used by Showalter [11].
The above considerations show how important is to obtain a full characterization of generators of B-bounded semigroups. A first characterization, under the assumption that operators A and B satisfy all the conditions of Definition 1.1, has been obtained in [3]. For his study Banasiak introduced the extrapolation space X B , which represents the completion of X with respect to the (semi)norm · B = B · . Later, without using the Banach space X B , it has been proved in [2] that the Banasiak characterization theorem holds even if the assumption (ii) of Definition 1.1 is replaced by the weaker assumption: (iii) there exists ω ∈ R such that for each λ > ω, the map Recently, it has been shown that in many cases even this assumption is too restrictive and can be relaxed even further. The relaxation suggested in [5] is the following: . We assume that the operator A B is closable in X B , that is, we assume that if the sequence (f n ) n∈N of elements of D B (A) is such that Bf n → 0 and BAf n → h in Z as n → ∞, then h = 0. Denoting -=Ā X B B , we assume further that there exist subspaces: ᐄ and D satisfying In all the mentioned cases it has been proved that, if A ∈ B-G(M, ω, X, Z), then there exists a C 0 -semigroup (T (t)) t≥0 in . Such a semigroup is uniquely determined by the given pair (A, B).
In my opinion the achieved results are not completely satisfying yet. Indeed each paper contains assumptions on the pair (A, B) which are weaker than those previously considered, but are sufficient to guarantee the existence in Z B of the above mentioned C 0 -semigroup (T (t)) t≥0 .
According to me the approach has to be completely reversed. In other words, first one has to determine all the properties possessed by a couple (A, B) of operators if they generate a B-quasi bounded semigroup (Y (t)) t≥0 . Then one has to determine which further assumption can assure that It is the purpose of this paper to answer both questions.
In Section 2, we determine all the properties that two linear operators A and B possess if the following holds: (a) assumption (i) of Definition 1.1 is satisfied; , that is, there exists a family (Y (t)) t≥0 such that conditions (1), (2), and (3) of Definition 1.1 hold; (c) family (Y (t)) t≥0 satisfies the further condition It is easy to see that assumption (c) is independent from assumptions (1), (2), and (3) of Definition 1.1, which only imply Y (0)f = Bf for all f ∈ D B (A). However, if we want to obtain the above-mentioned representation for the family (Y (t)) t≥0 , then we have to suppose condition (c) be satisfied too.
In Section 3, we show that, if (a), (b), and (c) hold, then the simplest further condition that can assure the required representation of (Y (t)) t≥0 is the following: (v) for all f ∈ D(B), λ > ω there exists a sequence (f n ) n∈N of elements of D B (A) such that Bf n → g and BAf n → h in Z as n → ∞ with λg − h = Bf .
In Section 4 we compare our result with the previous ones and show that condition (v) is weaker than (iv).
Finally, in Section 5 we show how our results allow to improve the theory developed in [4] for solving implicit evolution equations.

Properties of a pair (A, B) generating a B-bounded semigroup
In this section, we want to characterize all the properties that two linear operators A and B possess if the above conditions (a), (b), and (c) are satisfied.
In order to achieve our aim, we consider the mapping Ꮽ defined by putting, for all g ∈ B(D B (A)), (2.1) Clearly the properties of mapping Ꮽ and those of the pair (A, B) are strictly linked with each other; indeed mapping Ꮽ only depends on the given operators A and B. Concerning this we can prove the following theorem.
Such a property has been proved in [1]. For the sake of completeness we report here the considerations contained in [1], but some of them in a slightly different form.
Consider f ∈ D B (A) such that Bf = 0. From properties of Definition 1.1 we obtain, for all t ≥ 0, But then Y (t)Af = 0 for any t ≥ 0. In particular, BAf = Y (0)Af = 0. This shows that Ꮽ is a single valued mapping from B(D B (A)) to Z B . The linearity is easily verified. In order to show that Ꮽ is closable in Z B observe that we can define a one-parameter family of linear operators (T (t)) t≥0 from R(B) to Z by putting, for f ∈ D(B), t ≥ 0, Thanks to property (1) of Definition 1.1 we obtain for any g = Bf ∈ R(B), t ≥ 0 Therefore, for each t ≥ 0 operator T (t) is bounded. By the extension principle, it can be uniquely extended to a bounded linear operator with domain Z B . Denote by T (t) the extension too. Then inequality (2.5) holds for all t ≥ 0, g ∈ Z B . Moreover, definition (2.4) and condition Y (0) = B yield T (0) = I . From property (2) of Definition 1.1 we deduce that, fixed g ∈ R(B), t → T (t)g is a continuous function of t for 0 ≤ t < ∞. Inequality (2.5) allows to extend the property for any g ∈ Z B . Thus for g ∈ Z B the function t → T (t)g ∈ C([0, ∞), Z).
Finally, as a consequence of property (3) of Definition 1.1 we have for all g = Bf ∈ B(D B (A)) Now we are able to show that operator Ꮽ is closable in Z B . Indeed, consider a sequence Now writing (2.7) for each g n and taking the limit as n → ∞ we obtain for all t ≥ 0, Consider the closureᏭ of Ꮽ. We remember that From this and formula (2.7) we immediately obtain for all g ∈ D(Ꮽ ), t ≥ 0, Formula (2.11) allows to prove the stated properties.

Remark 2.3.
It is interesting to consider the case X = Z and B = I .
Thanks to Theorem 2.1 in such a case we can state the following. Suppose that A : D(A) → X is a linear operator in X. If there exists a one-parameter family of linear operators (T (t)) t≥0 , which satisfies the following: (1 ) T (t) : X → X, and for any t ≥ 0 and f ∈ X, (4 ) T (0) = I , then operator A is closable. Its closureĀ has the following properties: (α ) for any λ > ω, one can define a linear bounded operator R(λ) : X → X such that

New characterization and generation theorems
In Section 2, we have established which properties a pair (A, B) of operators must possess if they generate a B-bounded semigroup, that is, if there exists a family (Y (t)) t≥0 which satisfies conditions (1), (2), and (3) of Definition 1.1. We have also shown that to each family (Y (t)) t≥0 satisfying conditions (1), (2), and (3) of Definition 1.1 we can associate a strongly continuous family of linear operators (T (t)) t≥0 from Z B to Z so that the formula In this section, we first look for the simplest further property of the pair (A, B) able to guarantee that the family (T (t)) t≥0 is a strongly continuous semigroup in Z B . Considering (2.2) and (2.12), we immediately see that such a property can be obtained by requiringᏭ to be the generator of a strongly continuous semigroup in Z B with the Hille constants M and ω, that is,Ꮽ ∈ G(M, ω, Z B ). Indeed, in such a case we have T (t) = exp(tᏭ ). The study of the above-mentioned topic is carried out in Theorem 3.1.
Then we deduce a new generation theorem (Theorem 3.5) and a new characterization theorem (Theorem 3.6). Now in order to prove Theorem 3.1 a preliminary consideration is necessary. Thanks to (2.9) and (2.10), condition (v) can be rewritten in the equivalent form: for all f ∈ D(B), λ > ω there exists g ∈ D(Ꮽ ) such that Considering this we can prove the following theorem.
. Now, we prove that all these conditions are satisfied whenever (v) holds.
(j) We know that assumption (v) is equivalent to condition (3.2). Since R(λI −Ꮽ ) is a closed subset of Z B , we have Now inequality (2.13) gives Therefore, ρ(Ꮽ ) ⊇]ω, +∞[. (jj) For any λ > ω, n ∈ N and g ∈ Z B put (3.8) The properties of the family (T (t)) t≥0 assure that for each λ > ω and n ∈ N, formula (3.8) defines a bounded linear operator from Z B to Z with (3.9) 234 Characterization of B-bounded semigroups Therefore, we only have to prove that such an operator is given by This can be shown by induction. Formula (3.8) holds for n = 1 because (2.12) and (3.8) give Suppose that formula (3.10) holds for n = m. Then we obtain from (2.11), for λ > ω, g ∈ D(Ꮽ ), that We easily deduce that Note that Because the function t → T (t)h is continuous at t = 0, for any > 0 there exists δ > 0 such that for any 0 ≤ t < δ, Therefore, for λ > ω 1 = max{0, ω}, for λ sufficiently large.

Luisa Arlotti 235
Thus we have shown thatᏭ ∈ G(M, ω, Z B ). Now, it is immediate to recognize that for any t ≥ 0 operators exp(tᏭ ) and T (t) coincide. Indeed, we know that for given s ≥ 0 and g ∈ D(Ꮽ ), we have exp(sᏭ )g ∈ D(Ꮽ ) ⊆ Z B . Therefore, for each t > 0 we can consider the function s → h(s), 0 ≤ s ≤ t defined by By virtue of formula (2.11) we have This shows that h(s) = h(0) for 0 ≤ s ≤ t. In particular, we have h(t) = h(0), that is, Because t is an arbitrary positive number and g is an arbitrary element of D(Ꮽ ), which is dense in Z B , formula (3.3) is proved for any t ≥ 0.   Proof. Suppose that the assumptions of Theorem 2.1 and condition (v) hold. Then also (3.4) holds and R(λI −Ꮽ ) = R(µI −Ꮽ ) for any λ, µ > ω. Moreover, thanks to point (jjj) of Theorem 3.1, we can state that B(D B (A)), which is dense in D(Ꮽ ), is dense in Z B too. Thus we can conclude that condition (v ) is satisfied.
Conversely, suppose that the assumptions of Theorem 2.1 and condition (v ) hold. Then for given λ,µ > ω and f ∈ D(Ꮽ ) there exists g ∈ D(Ꮽ ) such that and therefore, This shows that

The pair (A, B) generates a B-quasi bounded semigroup (Y (t)) t≥0 with the property Y (0)f = Bf for all f ∈ D(B), and satisfies either condition (v) or condition (v ) if and only if (2.1) defines a single-valued mapping Ꮽ : B(D B (A)) → Z B which is closable in Z B and whose closureᏭ generates a C 0 -semigroup in Z B . The B-quasi bounded semigroup (Y (t)) t≥0 is such that Y (t)f = exp(tᏭ )Bf for any t ≥ 0 and f ∈ D(B).
Similarly we can state this new characterization theorem.

Comparison with the previous results
This section is devoted to compare our new characterization and generation theorems (Theorems 3.5 and 3.6) with Theorem 3.1 and Proposition 3.2 in [5].
To this aim we have to introduce the Banach space X B and to provide some results proved in [3].
Definition 4.1. We consider the set X of sequences (f n ) n∈N such that f n ∈ D(B) for n ∈ N and (Bf n ) n∈N is a Cauchy sequence. We define X B to be the space of all classes of equivalence of sequences (f n ) n∈N ∈ X with respect to the following relation: We also recall the Banasiak characterization theorem proved in [5]. It states the following: suppose that the operators A and B satisfy assumption (i) of Definition 1.1 and assumption (iv). Then the pair (A, B) generates a B-quasi bounded semigroup (Y (t)) t≥0 , with Y (0) = B, if and only if the following conditions hold: there exist M > 0 and ω ∈ R such that for any η ∈ ᐄ, λ > ω, and n ∈ N, (4.5) Finally, we recall that, according to Proposition 3.2 of [5], the family (Y (t)) t≥0 can be represented in the form if and only if the further following assumption is satisfied: (iv ) there exists some λ > ω such that the operator λI −is injective.
Now we can compare our new characterization theorem with the previous results. Such a comparison is carried out in the following propositions.

Proposition 4.4. Condition (iv) implies condition (v).
Proof. Suppose that condition (iv) is satisfied. Consider f ∈ D B (A) such that Bf = 0 and take the sequence (f n ) n∈N of elements of D B (A) such that f n = f for any n. Then Bf n = Bf = 0 and BAf n = BAf for any n. Since operator A B is closable in X B , we have BAf = 0. This shows that (2.1) defines a single-valued mapping from B(D B (A)) to Z B . Moreover, considering Definition 4.3, we obtain (4.7) The properties of the linear operator Ꮾ assure that, if where, according to (iv), -=Ā X B B . We remember that we can state that there exist subspaces ᐅ and E satisfying R(B) ⊆ ᐅ ⊆ Z B , and that is, condition (v). This proves the statement.
Proof. Using the operator Ꮽ, it is immediate to recognize that condition (2 ) can be rewritten in the following equivalent form: (2 ) there exist M > 0 and ω ∈ R such that for any g ∈ ᐅ, λ > ω, and n ∈ N, (4.15) But we know that ᐅ ⊇ R(B) and that M cannot be less than 1. Therefore, we deduce that if (2 ) holds then condition (2) of Theorem 3.6 holds too.  [5] imply (iv ) and therefore they are sufficient to guarantee the representation (4.6). Such a property holds however we choose the set D. Finally, using (4.9) and Definition 4.2, we can easily see that the two representations (3.22) and (4.6) coincide. On the other hand, if the assumptions of our characterization theorem allow to write the family (Y (t)) t≥0 in the form (3.22), the stronger assumptions of the Banasiak characterization theorem must give the same representation, that is, (4.6), whatever the set D is.

Application to implicit evolution equations
Recently Banasiak, [4], proved that the B-bounded semigroups theory can be applied for solving implicit evolution equations. Here we want to show how our technique can simplify such an application.
Indeed consider the Cauchy problem for the implicit evolution equation Here X is a linear space and Z is a Banach space. Assume that the operator K is one-to-one. Define B = K −1 and to such an operator associate the Banach space Z B = R(B) = D(K). To B associate also the Banach space X B and the operators P and Ꮾ given by Definitions 4.1, 4.2, and 4.3, respectively. If operators K and L are closable in X B , then, according to [4], we introduce the following definition.

Definition 5.1. A Z-valued function t → u(t) is called an X B -solution to problem (5.1) if it is a classical solution of the problem
that is, t →K X B u(t) is continuously differentiable in X B , the differential equation holds for all t > 0 in X B , and the initial condition holds as a limit in the topology of X B .
Sufficient conditions for the existence of an X B -solution to problem (5.1) are obtained in the following theorem. Assume that the operator K is one-to-one. Define the operator In order to prove such a theorem some preliminary considerations are necessary. First of all we can remark the following. Proposition 5.4. If the operator K is one-to-one, then it is closable in X B , with Proof. Take g ∈ Z B . Because D(K) = Z B there exists a sequence (g n ) n∈N of elements of D(K) such that g n → g in Z B as n → ∞. Put f n = Kg n and = Ꮾ −1 g. Then K −1 f n → Ꮾ in Z B as n → ∞, and therefore, according to Remark 5.3, PKg n = Pf n → = Ꮾ −1 g in X B as n → ∞.
This proves the statement, since g ∈ D(K X B ) andK X B g = Ꮾ −1 g.
Proposition 5.5. Suppose that the operator K is one-to-one and that the operator Ꮽ is closable in Z B . Then L| D(Ꮽ) is closable in X B , with as n → ∞. Because Ꮽ = K −1 L is supposed to be closable in Z B , we have Ꮾ = 0 and therefore = 0. This shows that L| D(Ꮽ) is closable in X B . Now D L X B = g ∈ Z B : ∃ g n n∈N in D(Ꮽ) and ∈ X B : g n → g, PLg n → = g ∈ Z B : ∃ g n n∈N in D(Ꮽ) and ∈ X B : g n → g, K −1 Lg n → Ꮾ = D Ꮽ , By virtue of Proposition 5.5, we obtain for all t > 0, Moreover, taking into account definitions (4.7) and (4.8), we have Thus the assertion is completely proved.
Remark 5.6. Suppose that the assumptions of Theorem 5.2 are satisfied and consider the family (Y (t)) t≥0 defined by putting for f ∈ R(K), t ≥ 0 Such a family has the properties (1) and (2) of Definition 1.1. Moreover, for all t > 0 and f ∈ K(D(Ꮽ)), we have we see that also property (3) of Definition 1.1 is satisfied with A = LK −1 . Therefore, the family (Y (t)) t≥0 is a B-quasi bounded semigroup generated by the pair (A, B). Moreover, the relation between such a pair and the operator Ꮽ is the same as in Section 4.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009