aa r X i v : . [ m a t h . A P ] F e b SEMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS
CHRISTIAN BUDDE
Abstract.
We use a version of the Trotter–Kato approximation theorem for strongly continu-ous semigroups in order to study flows on growing networks. For that reason we use the abstractnotion of direct limits in the sense of category theory.
Introduction
Transport of goods is nowadays of extreme importance and indispensable considering what mankindneeds for daily life. Now imagine a start-up company shipping special goods all over the world. Ofcourse, the company starts with a small network of customers. However, assuming the companygrows and retains the already existing routes and customers, the ship network grows and grows.It might come to the point in the development of the company, that one actually lost the view onall specific routes but only knows how the network works since it becomes too big. However, onestill wants to know how the transport is going on the whole network.Mathematically speaking, the routes and customers can be described through edges and verticesof a graph, respectively. By giving the graph a metric structure one obtains so-called quantumgraphs or networks. Transport can be modeled simply by the linearized flow equation ∂∂t w ( t, x ) = c ∂∂x w ( t, x ) , where c > C -semigroups. The techniques are also used frequently by other authors [7, 8, 11]even for the situation one asks for weaker solutions [15] by means of bi-continuous semigroups.Notice that the motivating example given above is not the only one, actually, one can imagine alot more scenarios, e.g., social networks [5, 3] or brain connections [27], just to name a few.That a network is growing, through adding vertices and edges, means that one has a sequence ofat first finite graphs, i.e., one knows the exact structure as described in the situation above. Inthis case, each finite graph of the sequence yields a phase space L ([0 , , C m ), where m ∈ N isthe number of edges of the graph. It is important to notice, that hence the phase space changesdepending on number of edges m . One assumes that each graph is a subgraph of the subsequentgraph in the sequence, describing the above mentioned situation of growing networks. The abovementioned situation, where the network becomes to big in order to know all the network routeswill be modelled by an infinite graph. Then we work on the Banach space L (cid:0) [0 , , ℓ (cid:1) . Thetransition from finite to infinite graphs will be modeled by direct limits in a certain category.The approximation of the transport process on the direct limit graph, is done by a version ofthe Trotter–Kato approximation theorem, which is originally due to T. Kato [20, Chapter IX,Thm. 3.6] and H.F. Trotter [28, Thm. 5.2 & 5.3] and modified by a version by K. Ito and F. Kappel[19, Thm. 2.1] which we are going to use. Actually this is related to the first Trotter–Kato theorem,cf. [18, Chapter III, Thm. 4.8]. In the present paper we extend the work of Ito and Kappel by Mathematics Subject Classification.
Key words and phrases. strongly continuous semigroups, Trotter–Kato theorems, transport problems, networks,category theory. another approximation theorem, which is an extension of the second Trotter–Kato theorem, cf.[18, Chapter III, Thm. 4.9]. We notice, that this paper deals with categorical limits of graphs.However, there is another notion of graph limits due to L. Lovász [23] by means of graphons orgraphings. This notion of limits totally differs from what we consider within this paper but isworth to mention since this is a interesting topic which is supposed to yield future research onnetworks and dynamical systems. In fact, graph limits in the sense of L. Lovász are in the focusof forthcoming papers.The structure of the paper is as follow: in the first section we recall all fundamentals on networks,flows on it and category theory. In Section 2 we apply the first Trotter–Kato theorem for our modelof growing networks. The following section consists of the second Trotter–Kato approximationtheorem in the style of Ito and Kappel.1.
Preliminaries
Graphs and networks.
In order to talk about finite and infinite networks we make use ofthe notation used in [21], [16] or [17]. A network is modeled by a finite or infinite directed graphs G = (V( G ) , E( G )), where V( G ) = { v i : i ∈ I } is the set of vertices and E( G ) = { e j : j ∈ J } ⊆ V × V is the set of directed edges for some at most countable sets
I, J ⊆ N . For a directed edgee = (v i , v k ), i , k ∈ I , we call v i the tail and v k the head of e. Further, the edge e is an outgoingedge of the vertex v i and an incoming edge for the vertex v k . Recall that a graph G is called simple if there are neither loops nor multiple edges in G . This means in particular, that there are noedges of the form e = (v i , v i ), i ∈ I (i.e., the tail and the head of the edge coincide and so an edgeconnects a vertex with itself) and no several edges connecting two vertices in the same direction.We also assume that the graph G is uniformly locally finite meaning that each vertex has onlyfinitely many outgoing edges and that the number of outgoing edges is uniformly bounded fromabove.The structure of a graph can be described by its incidence or its adjacency matrix. The outgoingincidence matrix Φ − = (Φ − ij ) is defined by(1.1) Φ − ij := ( i e j −→ . , By v i e j −→ we mean that the vertex v i is the tail of the edge e j . The incoming incidence matrix Φ + = (Φ + ij ) is defined by(1.2) Φ + ij := ( e j −→ v i , . Here e j −→ v i means that the vertex v i is the head of the edge e j . The incidence matrix Φ ofthe directed graph G , describing the structure of the network completely, is then defined byΦ := Φ + − Φ − . There are two other important matrices associated to a general graph and whichare needed in what follows. The transposed adjacency matrix of the graph G is defined by A := Φ + (cid:0) Φ − (cid:1) ⊤ . The nonzero entries of A correspond exactly to the edges of the graph, cf. [9, p. 280]. In fact, A can be described explicitly as(1.3) A ij := ( j e k −→ v i , . Last but not least we use the so-called (transposed) adjacency matrix of the line graph B = ( B ij )defined by B := (Φ − ) ⊤ Φ + . One can also give an explicit entrywise description as(1.4) B ij := ( e j −→ v k e i −→ , . EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 3
Notice, that by the assumption of the uniform locally finiteness of the graph the matrix B is abounded operator on ℓ := ℓ ( J ).In what follows, we stick to the following mathematical setting. We identify every edge of ourgraph with the unit interval, e j ≡ [0 ,
1] for each j ∈ J , and parametrize it contrary to the directionof the flow, if c j > j ∈ J , so that it is assumed to have its tail at the endpoint 1 and its headat the endpoint 0, i.e., the material flows from 1 to 0. With this assumption, we stay within theframework introduced by B. Dorn, M. Kramar Fijavž and E. Sikolya, see for example [21, 16]. Forsimplicity we use the notation e j (1) and e j (0) for the tail and the head, respectively. In this waywe obtain a metric graph .1.2. Category theory.
By taking all simple locally finite directed graphs together, one obtainsa rich mathematical structure by means of a category. We recall the most important definitionshere as they can be found for example in the monographs by S. Mac Lane [24] or S. Awodey [6].We first recap the basic definition of a category.
Definition 1.1.
A category C consists of objects A, B, C, . . . and arrows f, g, h, . . . (also called morphisms ). For each arrow f there are given objects dom( f ) and cod( f ) called the domain and codomain of f . We write f : A → B to indicate that A = dom( f ) and B = cod( f ). Given arrows f : A → B and g : B → C , that is, with cod( f ) = dom( g ) there is given an arrow g ◦ f : A → C called the composite of f and g . Furthermore, for each object A there is given an arrow 1 A : A → A called the identity arrow of A . These arrows are required to satisfy the following axiomas:(a) Associativity , i.e., for f : A → B , g : B → C and h : C → D one has h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f, (b) Unit law , i.e., for each f : A → B one has f ◦ A = f = 1 B ◦ f. As said above, the objects we are interested in are simple and locally finite graphs. In order toform a category, we need to specify what the arrows in the category are. For that reason we recallthe definition of the so-called graph homomorphisms.
Definition 1.2. A (graph)-homomorphism between two graphs G = (V( G ) , E( G )) and H =(V( H ) , E( H )) is a map ϕ : V( G ) → V( H ) such that (v , v ) ∈ E( G ) implies that ( ϕ (v ) , ϕ (v )) ∈ E( H ). If such an homomorphism is injective, then G is a subgraph of H .By taking together graphs and their homomorphisms we obtain a category. Definition 1.3.
The category C := SimpLocFinG consists of simple and locally finite graphsas objects and graph homomorphisms as arrows.In category theory constructions on categories, e.g., products of categories or free categories, aswell as universals and limits play a central role. For the purpose of this paper we recall thefollowing definition of a direct limit in a category. Notice that we simplified the original definitionto sequences of objects instead of directed systems of objects, cf. [24, Chapter V, Sect. 1] or [6,Def. 5.17 & 5.18], in order to fit in our framework.
Definition 1.4.
Let C be a category and ( A n ) n ∈ N a sequence of objects in C such that there existmaps ϕ n : A n → A n +1 for each n ∈ N , i.e., we have the following diagram A ϕ −→ A ϕ −→ A ϕ −→ A −→ · · · We say that an objects A in C is the direct limit of the sequence ( A n ) n ∈ N if for any n ∈ N thereexists an arrow ψ n : A n → A such that ψ n +1 ◦ ϕ n = ψ n for each n ∈ N , i.e., the following diagramscommute for each n ∈ N : A n ψ n " " ❋❋❋❋❋❋❋❋❋ ϕ n / / A n +1 ψ n +1 (cid:15) (cid:15) A EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 4
Moreover, A is universal in the sense that if another object B such that there exist arrows ϑ n : A n → B such that ϑ n +1 ◦ ϕ n = ϑ n for all n ∈ N , then there exists a unique arrow α : A → B suchthat α ◦ ψ n = ϑ n for each n ∈ N .We will use the concept of direct limits for our special category C = SimpLocFinG for the specialcase that the arrows ( ϕ n ) n ∈ N are all injective, i.e., that we consider a sequence ( G n ) n ∈ N of simpleand locally finite graphs which is growing in the sense that G n is a subgraph of G n +1 for each n ∈ N . As a matter of fact, each element in the sequence ( G n ) n ∈ N is supposed to be a finite simplegraph and hence locally finite. The direct limit itself does not have to be finite anymore.1.3. Flows in networks.
So far we considered the algebraic components of this paper. We willnow turn to the analytical structure, i.e., we discuss transport processes on networks. Notice thatwe only treat the infinite graph case here, since the finite one is included and only needs somesmall modifications. Let G = (V( G ) , E( G )) be a simple and locally finite graph, then we studythe following partial differential equation on the graph G for i ∈ I , j ∈ J : ∂∂t w j ( t, x ) = c j ∂∂x w j ( t, x ) , x ∈ (0 , , t ≥ ,w j ( x,
0) = f j ( x ) , x ∈ (0 , , Φ − ij w j (1 , t ) = X k ∈ N Φ + ik w k (0 , t ) , t ≥ . (PDE)How this equation relates to the classical linear Boltzmann equation is described in [17, Sect. 1].In what follows, we assume that all velocities c j , j ∈ J on the edges stay away from zero and arebounded from above, i.e., there exist m, M > m ≤ c j ≤ M, j ∈ J. (1.5)The boundary conditions of the equation depend on the structure of the network which is intro-duced by the incidence matrices. Now consider the Banach space X := L (cid:0) [0 , , ℓ (cid:1) equippedwith the norm given by k f k := Z k f ( s ) k ℓ d s and introduce the (unbounded) operator ( A, D( A )) on X defined by A := diag (cid:18) dd x (cid:19) , D( A ) := (cid:8) f ∈ W , (cid:0) [0 , , ℓ (cid:1) : f (1) = B C f (0) (cid:9) , (1.6)where B C := C − B C and C := diag ( c j ) . (1.7)Notice, that (1.5) assures, that the operator B C is bounded. It is well-known that the correspondingabstract Cauchy problem given by ( ˙ u ( t ) = Au ( t ) , t ≥ u (0) = f, (ACP)on the Banach space X = L (cid:0) [0 , , ℓ (cid:1) is equivalent to the partial differential equation (PDE),i.e., a solution of (ACP) gives rise to a solution of (PDE) and vice versa, cf. [16, Prop. 3.1] and [21]for the finite graph case. We now need the notions of well-posedness of abstract Cauchy problemsand C -semigroups, cf. [18, Chapter II, Thm. 6.7]. Definition 1.5.
A function u : R + → X is called a (classical) solution of (ACP) if u is continu-ously differentiable with respect to X , u ( t ) ∈ D( A ) for all t ≥ Definition 1.6.
The abstract Cauchy problem (ACP) is called well-posed if for every f ∈ D( A ),there exists a unique solution u ( · , f ) of (ACP), D( A ) is dense in X and if for every sequence ( f n ) n ∈ N in D( A ) with lim n →∞ f n = 0, one has lim n →∞ u ( t, f n ) = 0 uniformly in compact intervals. Definition 1.7.
A family of bounded linear operators ( T ( t )) t ≥ is called strongly continuous one-parameter semigroup of linear operators , or C -semigroup , if the following properties are satisfied: EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 5 (i) T ( t + s ) = T ( t ) T ( s ) and T (0) = I for all t, s ≥ t ց k T ( t ) f − f k = 0 for each f ∈ X .Each C -semigroup ( T ( t )) t ≥ gives rise to an operator ( A, D( A )) called the generator. This oper-ator is defined as follows. Ax := lim t ց T ( t ) x − xt , D( A ) := (cid:26) x ∈ X : lim t ց T ( t ) x − xt exists (cid:27) . The converse question, which operator ( A, D( A )) is the generator of a C -semigroups is moreinvolving. As a matter of fact, this question is answered by the so-called Hille–Yosida theorem, cf.[18, Chapter II, Thm. 3.8], [29]. If a given operator ( A, D( A )) generates a C -semigroup ( T ( t )) t ≥ on a Banach space X satisfying k T ( t ) k ≤ M e ωt for some M ≥ ω ∈ R and for all t ≥
0, then wewill denote this by A ∈ G ( M, ω, X ). The most important fact is, that by [18, Chapter II, Cor. 6.9]the abstract Cauchy problem (ACP) is well-posed in the sense of Definition 1.6 if and only ifthe operator ( A, D( A )) is the generator of a C -semigroup. The following result shows, that ourexplicit abstract Cauchy problem (ACP) associated to (PDE) is well-posed, cf. [16, Thm. 3.4].For the finite network case, we refer to [21, Prop. 2.5]. Theorem 1.8.
The operator ( A, D( A )) on X = L (cid:0) [0 , , ℓ (cid:1) defined by (1.6) , generates a C -semigroup. Therefore, (ACP) is well-posed. Approximation of flows on direct limit graphs
We now consider the situation as described earlier. Let ( G n = (V( G n ) , E( G n ))) n ∈ N be a growingsequence of finite simple graphs, i.e., there exist injective graph homomorphisms ϕ n : G n → G n +1 for each n ∈ N . Notice that by [26, Def. 8.1] the limit of such a sequence ( G n ) n ∈ N exists. Let usdenote this limit by G = (V( G ) , E( G )). In particular, one has G = S n ∈ N G n . For each n ∈ N wehave a strongly continuous semigroup ( T n ( t )) t ≥ solving (ACP) on the space L ([0 , , C ) | E( G n ) | .Moreover, we have a C -semigroup ( T ( t )) t ≥ on L (cid:0) [0 , , ℓ (cid:1) . The clue is, that the semigroups( T n ( t )) t ≥ approximate the semigroup ( T ( t )) t ≥ is a certain sense. To make this more precise, werefer to the work of K. Ito and F. Kappel [19]. In fact, we will use the following theorem. Theorem 2.1. [19, Thm. 2.1]
Let X and X n , n ∈ N , be Banach spaces and such that for each n ∈ N there exist bounded linear operators P n : X → X n and E n : X n → X such that sup n ∈ N k P n k < ∞ , sup n ∈ N k E n k < ∞ and P n E n = I n , where I n denotes the identity operator on X n , n ∈ N . Let A ∈ G ( M, ω, X ) and A n ∈ G ( M, ω, X n ) for each n ∈ N and let ( T ( t )) t ≥ and ( T n ( t )) t ≥ be thesemigroups generated by A and A n on X and X n , respectively. Then the following statements areequivalent. (a) There exists λ ∈ ρ ( A ) ∩ T n ∈ N ρ ( A n ) , such that for all x ∈ X , lim n →∞ k E n R ( λ , A n ) P n x − R ( λ , A ) x k = 0 . (b) For every x ∈ X and t ≥ n →∞ k E n T n ( t ) P n x − T ( t ) x k = 0 uniformly on bounded t -intervals. In order to apply this theorem we have to specify all required data for our situation. The choicesfor the Banach spaces are clear, i.e., one chooses X n := L ([0 , , C ) | E( G n ) | = L (cid:16) [0 , , C | E( G n ) | (cid:17) , n ∈ N , and X := L (cid:0) [0 , , ℓ (cid:1) . Now we have to specify what the operators ( P n ) n ∈ N and ( E n ) n ∈ N have to be in our case. Since G = (V( G ) , E( G )) is the direct limit of the sequence ( G n ) n ∈ N there exist graph homomorphisms ψ n : G n → G , n ∈ N , which by [26, Prop. 8.3] are injective, too. These maps yield maps E n : X n → X for n ∈ N . Actually, the operator E n intuitively extend the functions on G by EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 6 infinitely many zeros. Without loss of generality, we may assume that the labeling of the edgesof G n and G coincide on E( G ) \ E( G n ). To be more detailed, the operator E n has the followingaction E n ( f , f , . . . , f | E ( G n ) | ) = ( f , f , . . . , f | E( G n ) | , , , . . . ) . The operators P n : X → X n , n ∈ N , are just the restrictions to the smaller subspace, i.e., itis a cut-off operator. Again, by assuming that labeling of the edges of G n and G coincide onE( G ) \ E( G n ) we can describe P n as follows P n ( f , f , f , . . . ) = ( f , f , . . . , f | E( G n ) | )By construction it is clear that k P n k ≤ k E n k ≤ n ∈ N . Remark 2.2.
The sequence of (injective) graph homomorphisms ( ϕ n ) n ∈ N corresponding to thesequence ( G n ) n ∈ N of graphs extends, similar to the maps ( ψ n ) n ∈ N , the a sequence of (Φ n ) n ∈ N oflinear maps Φ n : X n → X n +1 . It is easy to verify that L (cid:0) [0 , , ℓ (cid:1) is the direct limit of theBanach spaces ( X n ) n ∈ N in the category of Banach spaces with contractions as morphisms. Moredetails regarding functors and categories of Banach spaces can for example been found in themonograph by P.W. Michor [25].Let us set the observation of Remark 2.2 in a bigger picture by means of category theory. For thatreason, we recall the following definition, cf. [24, Chapter I, Sect. 3] or [6, Def. 1.2]. Definition 2.3.
Let C and C be two categories. A functor F : C → C between the categories C and C is a mapping of objects to objects and arrows to arrows, such that(i) F ( f : A → B ) = F ( f ) : F ( A ) → F ( B ),(ii) F (1 A ) = 1 F ( A ) ,(iii) F ( f ◦ g ) = F ( f ) ◦ F ( g ).In other words, a functor F preserves domains and codomains, identity arrows, and composition.Now let us apply the concept of functors for our situation. In particular, let us denote thecategory of Banach spaces together with linear bounded operator between them as arrows by B .Then there exists a functor F : C → B by F ( G = (V( G ) , E( G ))) = L ([0 , , C ) | E( G ) | if G is finiteand F ( G = (V( G ) , E( G ))) = L (cid:0) [0 , , ℓ (cid:1) if G is infinite. Moreover, for the graph homomorphism ϕ n : G n → G n +1 one defines F ( ϕ n ) = Φ n .Let us now come back to our transport problem. By the previous section, we know that theoperator A and A n are in fact generators of C -semigroups ( T ( t )) t ≥ and ( T n ( t )) t ≥ , respectively.The following result shows, that the semigroups ( T n ( t )) t ≥ „ converge “ to ( T ( t )) t ≥ in the sense ofTheorem 2.1(b). Proposition 2.4.
Let ( G n ) n ∈ N be a increasing sequence of graphs with limit G . By (( A n D( A n )) n ∈ N and ( A, D( A )) we denote the operators defined by (1.6) associated to the transport problems on X n and X , respectively. Then, for every x ∈ X and t ≥ one has that k E n T n ( t ) P n x − T ( t ) x k → for n → ∞ uniformly on bounded t -intervals. Intuitively spoken, the semigroups ( T n ( t )) t ≥ , n ∈ N ,approximate ( T ( t )) t ≥ along the growing sequence of graphs.Proof. In order to prove the result, we make use of Theorem 2.1. By [9, Prop. 18.12] one has that λ ∈ ρ ( A ) if Re( λ ) > λ ∈ ρ ( A ) one has R ( λ, A ) = (cid:0) I + E λ ( · )(1 − B C,λ ) − B C,λ ⊗ δ (cid:1) R λ , where δ denotes the point evaluation at 0, E λ ( s ) := diag (cid:0) e ( λ/c j ) s (cid:1) , B C,λ := E λ ( − B C , see also(1.7), and ( R λ f )( s ) := Z s E λ ( s − t ) C − f ( t ) d t. Hence, it is clear that ρ ( A ) ∩ S n ∈ N ρ ( A n ) = ∅ . By the explicit description of the operator( E n ) n ∈ N , ( P n ) n ∈ N and the resolvents it is clear that k E n R ( λ , A n ) P n x − R ( λ , A ) x k → n → ∞ . Therefore, by Theorem 2.1 we conclude that for every x ∈ X and t ≥ k E n T n ( t ) P n x − T ( t ) x k → n → ∞ uniformly on bounded t -intervals. (cid:3) EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 7
Remark 2.5.
We mentioned in the introduction, that the theory for flows in networks has beengeneralized by M. Kramar Fijavž and the author in [15] to a bigger class of operator semigroups onthe phase space L ∞ (cid:0) [0 , , ℓ (cid:1) , the so-called bi-continuous semigroups. These objects have a richstructure and have been introduced by F. Kühnemund [22] and further developed by B. Farkasand the author [13, 14, 12]. We will not go into the details of this theory since this is not thetopic of this paper. Nevertheless, it is worth to mention that even in the case of bi-continuoussemigroups, there are Trotter–Kato approximation theorems [1, 2] in the spirit of [18, ChapterIII, Thm. 4.8 & 4.9]. Moreover, there is a recent paper on the first Trotter–Kato theorem whichis closely related to the work of Ito and Kappel, cf. [4]. We would like to notice, that even ifthere are these approximation theorems for bi-continuous semigroups, the procedure describedabove is not imitable. This is due to the fact, that one has to assume that the network is finite ifone allows velocities on the edges of the network which are not rational (and linear dependent).Unfortunately, this is due to the absence of a Lumer–Phillips type generation theorem for bi-continuous semigroups. However, if one assumes that even the direct limit is finite, then theprocedure from above just works out.3. A second Trotter–Kato type theorem
The assumption in Theorem 2.1 is that we know that there exists a C -semigroup on the spaces X and in some sense we know how to approximate them by means of resolvents. However, anotherimportant question is, if there exists a C -semigroup as a limit on X if I only know that thereexists a sequence of semigroups on the spaces X n . The following theorem is related to the secondTrotter–Kato theorem on a single Banach space [18, Chapter III, Thm. 4.9]. We now formulatethis theorem such that it fits into the framework of K. Ito and F. Kappel. Theorem 3.1.
Let X and X n , n ∈ N , be Banach spaces and such that for each n ∈ N thereexist bounded linear operators P n : X → X n and E n : X n → X such that sup n ∈ N k P n k < ∞ , sup n ∈ N k E n k < ∞ and P n E n = I n , where I n the the identity operator on X n , n ∈ N . Let A n ∈ G ( M, ω, X n ) for each n ∈ N and let ( T n ( t )) t ≥ , n ∈ N , be the semigroups generated by A n on X n . Then the following statements are equivalent. (a) There exists λ ∈ ρ ( A ) ∩ T n ∈ N ρ ( A n ) and a bounded operator R ∈ L ( X ) with dense range,such that for all x ∈ X , lim n →∞ k E n R ( λ , A n ) P n x − Rx k = 0 . (b) There exists a strongly continuous semigroup ( T ( t )) t ≥ on X such that for every x ∈ X and t ≥ n →∞ k E n T n ( t ) P n x − T ( t ) x k = 0 uniformly on bounded t -intervals.Proof. The implication (b) ⇒ (a) is just an application of Theorem 2.1. For the converse, assumethat the assertion (a) holds. First of all, we notice that { R ( λ ) : Re( λ ) > } with R ( λ ) x := lim n →∞ E n R ( λ, A n ) P n x, x ∈ X, is a pseudoresolvent on X such that (cid:13)(cid:13) λ k R ( λ ) k (cid:13)(cid:13) ≤ M, k ∈ N . (3.1)Recall from [18, Chapter III, Def. 4.3] that { R ( λ ) : Re( λ ) > } is a pseudoresolvent if R ( λ ) is abounded linear operator for all λ > λ ) > R ( λ ) − R ( µ ) = ( λ − µ ) R ( λ ) R ( µ ) , holds for all λ, µ ∈ C with Re( λ ) > µ ) > { R ( λ ) : Re( λ ) > } is indeed a pseudoresolvent, we slightly modify [18, Chapter III,Prop. 4.4]. In fact, consider the setΓ := n λ ∈ C : Re( λ ) > , lim n →∞ E n R ( λ, A n ) P n x exists for all x ∈ X o . EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 8
By the assumptions of assertion (a) we have that Γ = ∅ . By [18, Chapter IV, Prop. 1.3], one hasthat for a given µ ∈ Γ E n R ( λ, A n ) P n = X k ∈ N ( µ − λ ) k E n R ( µ, A n ) k +1 P n , whenever | µ − λ | < Re( µ ), where the convergence is with respect to the operator norm anduniform in { λ ∈ C : | µ − λ | < α Re( µ ) } for each α ∈ (0 , n ∈ N k E n k < ∞ and sup n ∈ N k P n k < ∞ we see that E n R ( λ, A n ) P n x converges for all λ satisfying | µ − λ | < α Re( µ )whenever n → ∞ . We conclude, that Γ is an open set in C + := { λ ∈ C : Re( λ ) > } . On theother hand side, let λ with Re( λ ) > α ∈ (0 ,
1) one can find µ ∈ Γ such that | µ − λ | < α Re( µ ). By what we have seen before, λ ∈ Γ showing that Γ is alsoclosed in C + . Since C + is a connected space, we conclude that the only subsets which are bothopen and closed are ∅ and C + . Since we observed that Γ = ∅ , we have Γ = C + .Finally, we have to show that (3.1) holds. To do so, we observe that we have the following estimatefor λ > A n ∈ G ( M, ω, X n ) for each n ∈ N k λR ( λ ) k ≤ lim n →∞ k E n k · k λR ( λ, A n ) k · k P n k ≤ M k R ( λ, A n ) k ≤ M M , where M := sup n ∈ N k E n k · sup n ∈ N k P n k ≥ M ≥ k λR ( λ, A n ) k ≤ M which exists since A n ∈ G ( M, ω, X n ) for each n ∈ N as an application of the Hille–Yosidageneration theorem for strongly continuous semigroups, cf. [18, Chapter II, Thm. 3.8]. Thisfinally leads to the fact that (3.1) is satisfied.Since by construction Ran( R ( λ )) = Ran( R ), which is dense by assumption, we conclude that thereexists a densely defined operator ( B, D( B )) on X such that R ( λ ) = R ( λ, B ) for λ >
0, cf. [18,Chapter III, Cor. 4.7]. Hence, the operator ( B, D( B )) satisfies the following estimate (cid:13)(cid:13) λ k R ( λ, B ) k (cid:13)(cid:13) ≤ M, k ∈ N , yielding a bounded strongly continuous semigroup ( T ( t )) t ≥ on X . By a second application ofTheorem 2.1 we conclude that assertion (b) has to be true. (cid:3) Remark 3.2.
Notice that Theorem 2.1 allows to approximate a given semigroup by other semi-groups. However, the assertion of Theorem 3.1 is stronger in the sense, that one has not to knowthat there exists a semigroup on the space X but that one has approximants which behaves wellin the sense that they eventually converge to one semigroup.4. Example
In this final section we consider an example of a growing sequence ( G n ) n ∈ N of networks. We onlyshow the first two elements G and G of the sequence since it has an obvious pattern. v v v v e e e e e Figure 1.
Graph of G v v v v v v e e e e e e e e e Figure 2.
Graph of G We assume that the velocities are all equal to 1. The corresponding weighted (transposed) adja-cency matrix of the line graph B and B of the graphs G and G are given by B = EMIGROUPS FOR FLOWS ON LIMITS OF GRAPHS 9 and B = = B Since we assumed that the velocities are all equal to one, we can make use of the explicit expressionof the transport semigroups ( T ( t )) t ≥ and ( T ( t )) t ≥ on the networks G and G , respectively,cf. [16, Sect. 3] or [10, Sect. 18.2]. In particular, one has T i ( t ) f ( s ) = B ki f ( x + t − k ) , for k ∈ N such that k ≤ t + x < k + 1 and f ∈ L (cid:0) [0 , , C | E( G i ) | (cid:1) where i = 1 ,
2. By this, oneobserves that ( T ( t )) t ≥ is indeed a restriction of ( T ( t )) t ≥ . Acknowledgement
This article is based upon work from COST Action CA18232 MAT-DYN-NET, supported byCOST (European Cooperation in Science and Technology). The author is very grateful to theanonymous referee suggestions. They helped to improve the article. Especially, the style of thearticle overall has been improved.
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