LUMPABILITY OF LINEAR EVOLUTION EQUATIONS IN BANACH SPACES

. We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction opera- tor onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics deﬁned by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factorization. We indicate several applications to particular systems, including delay diﬀerential equations.


1.
Introduction. Consider a linear dynamical system defined on a Banach space X: ẋ(t) = Ax(t), with A : D(A) ⊆ X → X. We assume that the dynamics (1) is well defined, in the sense that for every x 0 ∈ D(A) there exists a unique classical solution x ∈ C 1 ([0, +∞), D(A)) that depends continuously on the initial condition x 0 . In addition, consider a linear bounded map M : X → Y where Y is another Banach space. We view the operator M as representing a reduction of the state space: it is surjective but not an isomorphism. The question we are interested in is whether the variable y = M x also satisfies a well-posed and self-contained linear dynamics on Y , sayẏ (t) = Ay(t), y = M x.
If this is the case, then we refer to M as a reduction or lumping operator. Diagrammatically, the system (1) is said to be lumpable by the operator M if there exists a linear operator A : Y → Y such that the following diagram commutes that is, Typically, the operator A here is required to have analogous properties as A; for example, A should be bounded if A is bounded, or they should both be generators of strongly continuous semigroups on X and Y , respectively, in case they are unbounded (and defined on proper subsets of their respective spaces; see Diagram 8).
The term lumping originates from chemical reaction systems, where the aim is to aggregate the species involved in the reaction into a few groups, called lumps of chemical reagents, and describe the reaction with a reduced number of equations. [29]. A similar concept of aggregation of states has been used in the theory of Markov chains, where the question is whether the newly-formed aggregates also admit a Markovian description for the state transitions [10,19,21], as well as in population dynamics [3]. Diagram 2, however, is more general, as the operator M can also represent other types of reduction, for example projections or averages. It can also be interpreted in the context of multi-level systems, where X and Y are sometimes referred to as micro (lower) and macro (upper) levels, respectively. Here the question is, given some dynamics A on the micro states X, finding the conditions on M such that Y represents a new level with its own autonomous dynamics. We also mention the connection to the notion of semi-conjugacy in nonlinear dynamical systems, where two flows φ t and ψ t , generated by the nonlinear operators A and A defined on topological spaces X and Y , respectively, are called semi-conjugate if there exists a surjection M : X → Y such that M (φ t ) = ψ t (M ) [8,9]. However, the interpretation of Diagram 2 is then rather different, because in semi-conjugacy the flow ψ t is already given and the question is the existence of a surjection M , while the lumping problem starts from a given surjection M and asks whether there exists a reduced flow on the space Y . In fact, a more commonly used property in dynamical systems is conjugacy, which is obtained when M is invertible, in which case there is no reduction at all.
An interesting connection exists between lumpability and factorization of operators: given two linear operators E and D on a Banach space, D is said to be a left multiple of E if there exists another linear operator C such that With D = M A, E = M , and C = A, (4) corresponds to the lumping relation (3). Factorization has been studied for bounded operators on Hilbert [13] and Banach spaces [5,14]. Some generalizations to unbounded operators can be found in [17], under the assumption of a pseudoinverse operator for E. The operator C in (4) exists if and only if E majorizes D [5,14], i.e. there exists some k > 0 such that: In this context the operator E need not to be surjective, and C is then defined on the range of E. However, in the lumping analysis one considers mostly surjective lumping operators M , because all the reduction operators used in the lumping literature, like averages or projections, are indeed surjective. It is worth noting that, if we relax our assumption on the range of M , we do not fall in the setting considered in [5,14] and the analysis does not generalize in a straightforward way.
On the other hand, we do consider unbounded operators A, without assuming the existence of a pseudoinverse for the lumping operator, but focusing on generators of strongly continuous semigroups, which may be unbounded but have some interesting spectral properties. Indeed, our aim is to study lumpability from the dynamical systems point of view, in order to obtain well-posed reduced dynamics. In passing, we mention the related notion of B-bounded semigroups [6,4,2] and the successive reflection method [7] for proving the existence of a semigroup. Before proceeding to operators on generic Banach spaces, it is instructive to look at the situation in finite-dimensional Euclidean spaces. In the notation of diagram (2), let X = R n and Y = R k , let M be a matrix with full row rank and A be a (k × k) matrix. If k < n, M represents a reduction of the state space dimension. Lumpability of finite-dimensional systems has been studied by, e.g., Li and Rabitz in application to chemical kinetics [22,25] and by Gurvits and Ledoux in the setting Markov chains [19]. In this finite dimensional context the following result is known (e. g., [22]). We also note the relation of lumpability to the notion of observability in control theory. Indeed, the action of the lumping operator M can be viewed as yielding a system observable y = M x, or the output of a linear time-invariant control system where typically k < n. Recall that the system is called observable if every initial state x 0 ∈ R n can be uniquely reconstructed from the system output y. This happens if and only if the observability matrix has full rank n. It is easy to see that if the system is lumpable by M , then Thus, in this case lumpability implies that the control system (6) is not observable [11]. Our aim in this paper is to extend these results to infinite-dimensional systems involving both bounded and unbounded operators. Previous work in this area was carried out for bounded operators by Coxson [11], and by Zoltan and Toth in the context of Hilbert spaces [24], both requiring the existence of a continuous pseudoinverse of the lumping operator. We shall obtain more general conditions for lumpability in abstract Banach spaces that apply to dynamics generated by unbounded operators, such as partial and delay differential equations. In particular, the pseudoinverse of the lumping operator is not involved in our method, so we don't need additional hypotheses to guarantee its existence. Our approach is based on the theory of strongly continuous semigroups in Banach spaces and holds under quite general conditions, requiring only that the dynamics be well posed, in the sense of the Hille and Yosida theorem.
We prove that a necessary and sufficient condition for lumpability is the invariance of the kernel of the lumping operator under the whole semigroup of the solution operators. In particular, if this kernel is invariant under the semigroup, then one can construct a new strongly continuous semigroup on the reduced state space whose generator makes Diagram 2 commute. In case the semigroup of solutions is not known a priori, we give necessary and sufficient conditions for lumpability directly on the infinitesimal generator: here a condition is needed on the resolvent set of the generator to guarantee the invariance of the kernel of the lumping operator under the generated semigroup.
Furthermore, we complement the analysis by describing lumpability with respect to the dual space, dealing with the adjoints of the evolution operators. This represents an alternative view of the problem, allowing a different interpretation of lumpability and exploiting some interesting properties of adjoint operators. To obtain lumpability, the range of the lumping operator adjoint must be invariant under the adjoint semigroup. The adjoint of a strongly continuous semigroup need not be strongly continuous on the whole dual Banach space, but it is continuous with respect to the weak star topology. For this reason we use the notion of weak star generator and we analyze how the lumping operator adjoint acts on the sun dual space of the reduced state space, i.e. the closed subspace on which the reduced semigroup preserves strong continuity.
The paper is organized as follows. In Section 2 we discuss some known results about lumpability for bounded operators and indicate a relation to operator factorization. In Section 3 we extend the lumpability analysis to unbounded operators using semigroup theory. We also discuss the case of non-surjective lumping operators. Section 4 presents dual conditions for lumpability using sun dual spaces. We supplement the theory with several applications, including delay differential equations.
2. Lumpability for bounded operators. Let X be a Banach space, and let B(X) denote the Banach algebra of linear bounded operators from X to itself with norm A = sup x ≤1 Ax . We first consider system (1) when A ∈ B(X). Since A is bounded, (1) is well defined and the solutions are given by x(t) = e At x(0), where e At = ∞ k=0 A k t k /k! , and the series is convergent in the topology of B(X). We consider the diagram (2) where M : X → Y is a linear, bounded and surjective operator between Banach spaces X, Y . The main lumpability result in this setting is the following. This result was proved by Barnes in [5] in the context of factorization of operators. A proof in the context of lumping can be found in [11], which uses the pseudoinverse of a bounded operator under the additional assumption that the kernel of M is topologically complemented in X.

Remark 1.
A basic kind of lumping is obtained by the familiar quotient projection operation. Consider a closed subset C ⊂ X such that AC ⊆ C , and take Y = X C . By the invariance of C , we can define the bounded linear operator A[x] := [Ax]. Then, for x ∈ X, πAx = [Ax] = A[x] = Aπx, so that the following diagram commutes: Remark 2. As in the finite-dimensional case, we can view the system as a control system with output y = M x. In this context, (7) is said to be observable if [26] +∞ k=0 ker(M A k ) = {0}.
The following result related to factorization of operators can be seen as a connection to lumpability of bounded operators.
Theorem 2.2 ( [14], Thm. 1). Let D and E be bounded linear operators from a Banach space X to itself. Then the following conditions are equivalent: With D = M A and E = M , (i) corresponds to the lumping relation (3), with C = A. Unlike the case of lumping, in the context of factorization the operator E need not be surjective, and the operator C is then defined on the range of E. Here in most cases we assume the surjectivity of M , but at the same time we relax (ii) and we only ask an invariance condition for the kernel of M . A condition for lumpability in the case of a non-surjective lumping operator M is discussed in Section 3.
3. Lumpability for unbounded operators. We now turn to the case when the operator generating the dynamics is unbounded; thus, we consider the abstract Cauchy problem (1) where A : D(A) ⊂ X → X is a linear unbounded operator. It is a classical result in semigroup theory [20,15,23] that the dynamics (1) is well posed if and only if A is the generator of a strongly continuous semigroup {T (t)} t≥0 on X, and in that case, for every x 0 ∈ D(A), the unique classical solution of (1) is The lumpability problem in the unbounded case can be expressed as the commutativity of the diagram We assume that the linear operator M : X → Y is bounded and surjective, while A and A are defined on a proper subset of X and Y , respectively. Suppose that A generates a strongly continuous semigroup on X, which we denote by {T (t)} t≥0 . We want the operator A to be again the generator of a strongly continuous semigroup in order to obtain a well-defined dynamics on the upper level. Thus, we need the lumping relation M A = AM to hold on D(A).
Theorem 3.1. The following statements are equivalent.
There exists a linear operator A on M (D(A)) such that A generates a strongly continuous semigroup on Y , and AM = M A (i.e., system (1) is lumpable by the operator M ).
For each t ≥ 0, T (t) is well defined due to the invariance of the kernel, and, applying theorem 2.1, one can see that it is bounded. Moreover, the family (9) is a strongly continuous semigroup on Y because: In particular, letting ω denote the growth bound of T (t), we will show that ω is less or equal than the growth bound ω of T (t). To this end, we consider the quotient Banach space X/ ker(M ) with the quotient norm Define the following operators from X/ ker(M ) to Y : By the Banach-Schauder theorem, M is a homeomorphism. By the boundedness of T (t), it follows that M T (t) is bounded: It follows that showing that ω ≤ ω.
Let A be the generator of the new semigroup T (t). Consider an element y = M x in M (D(A)). By the definition of a generator and the continuity of M on X, Hence, A is defined on M (D(A)), which is a dense subset of Y because A is densely defined and M is bounded and surjective. On this subset the lumping relation also holds between the two generators: AM x = M Ax. We have thus obtained the inclusion M (D(A)) ⊂ D( A). We next show that the domain of A is exactly M (D(A)). For this purpose, we take λ ∈ C that belongs both to the resolvent set of A and of A, and use the integral representation of the resolvent operator. Given an arbitrary element y for which A is defined, there exists s = M x ∈ Y such that y = (λI − A) −1 s. Hence one can write where z belongs to D(A). Therefore, D(A) = M (D(A)). 2 ⇒ 1. We will show that the invariance of ker(M ) under the semigroup is a necessary condition to have a well-defined dynamics on Y . Suppose that the operator Ay := M Ax defined on M (D (A)) generates a strongly continuous semigroup on Y . Consider the following maps from R + to Y : . These two maps are both solutions of the abstract Cauchy problem ẏ(t) = Ay(t), In fact, the first map is a solution by definition, while for the second map we have We note that if a closed subspace is invariant under T (t) for all t ≥ 0, then by definition it is invariant under the infinitesimal generator A; however, the converse is not true. As a simple counterexample, let X be the Banach space C 0 (R) of all continuous functions on R that tend to zero at infinity, endowed with the supremum norm. The differentiation operator generates the strongly continuous semigroup of left translations Clearly, the closed subspace C = {f ∈ X : f (s) = 0, ∀s ≤ 0} is invariant under A but not invariant under translations. We mention the following characterization of closed invariant subspaces (see, e.g., [30]), which will be used in subsequent proofs.
Then the following are equivalent: It is typically the case in applications that one knows the generator A but not the associated semigroup. Therefore, it is necessary to find conditions on M that give the invariance of its kernel under the semigroup without knowing the semigroup itself. The next result gives conditions on the operator A for lumpability.
If (13)  Proof. Suppose that 1 and 2 hold. By 1, the operator A is well-defined. By 2, We know that R(λ) is bounded. Moreover, R(λ) is the inverse operator of ( A−λI); indeed, for y = M x, (1)   which is an holomorphic function from ρ(A) to C. It is known that for |λ − λ 0 | sufficiently small (to be precise,  . This is the case for, e.g., infinitesimal generators with discrete spectrum having a connected resolvent set.

Remark 5 (Observability with unbounded operators). Let
A be the unbounded generator of a strongly continuous semigroup with growth bound ω. It can be shown that the system ẋ(t) = Ax(t), y(t) = M x(t) is observable if and only if, for any µ ∈ ρ(A) satisfying Re (µ) > ω, the following system is observable: where the resolvent operator R(µ, A) = (µI − A) −1 is indeed bounded [16,27]. Hence the condition for observability is reduced to If the system is lumpable by M then ker(M ) is invariant under the semigroup, and hence also invariant under the resolvent operators for Re (µ) > ω [30]. Since ker(M ) = 0, this implies that (15) is not satisfied and the system is non-observable. Hence, the observation stated in [11] for bounded operators holds also in the unbounded case.
Example 1 (Quotient semigroup). Let C be a closed subspace that is invariant under a semigroup {T (t)} t≥0 (or, equivalently, satisfying statement 2 of Proposition 3.2). As in the bounded case, the quotient projection yields a lumping on the system associated with the generator A. The semigroup induced on the quotient space is  . . , f (x k )) , which simply evaluates a given function at the k points. We can write where "diag" denotes a diagonal matrix. Thus M yields a lumping on the system associated with A. Note that the kernel of M is invariant under A, but not fully contained in D(A); hence (13) is not satisfied. Since the new operator A is a diagonal matrix, we pass from an infinite dimensional dynamical system to a system defined on a k-dimensional space. On the other hand, the resolvent condition given in statement 2 of Proposition 3.2 is satisfied. This can be easily seen considering that the resolvent set of A is the complementary set of 1. since λ ∈ ρ(A), h(x) λ−h(x) is bounded, so that h(x)f (x) tends to zero at infinity; 2. since g vanishes at the points x i and the previous property holds, f also vanishes on this set of points. Hence, we can take every element in ρ(A) that is greater than ω as λ of statement 2 of Proposition 3.2.
where x t ∈ X is the function given by
Keeping the notation as above, we have the following result.
Evaluating at x = 0 yields Since λ ∈ ρ( L), (19) holds iff M c 0 = 0, i.e, c 0 ∈ ker(M ). We have proved that system (16) is lumpable by M . For every h = M f , f ∈ D(A), the generator of the semigroup on the upper level is which is again the differentiation operator, but defined on the set This operator is exactly the generator of the semigroup associated with the delayed system (17).
To give an example of functionals L on X which satisfy the hypothesis of the previous proposition, take where q i ∈ R and α i ∈ (0, r). It is easy to verify that L acts the same way as L but on a space of scalar-valued functions, Example 4. The following example illustrates that σ( A) is generally not contained in σ(A). Consider again the Banach space C 0 (R) and the semigroup of left translations (12) generated by the derivative operator Af = f , as given in (11) . The spectrum of A is the imaginary axis; σ(A) = iR, i being the imaginary unit [1, A-III,2.4]. Indeed, for every λ = iα, α ∈ R, there exists a sequence f n (x) := e −|x|/n e iαx such that f n = 1 and lim n→+∞ Af n − λf n = 0. A sequence of this kind is called an approximated eigenvector and its existence implies that (A − λI) is not bounded below, i.e. not invertible. It follows that ρ(A) is a disconnected subset of the complex plane. Consider now the lumping operator which acts as the restriction to R + . The operator M linear, bounded, and surjective by the Tietze extension theorem. Furthermore, ker(M ) is the ideal of functions vanishing on R + and it is invariant under A. If f ∈ D(A), it is clear that M Af = f | R + = (f | R + ) . It follows that the reduced operator A is again a derivative generating the semigroup of left translations on C 0 (R + ): T (t)g(s) = g(s + t), s ∈ R + , t ≥ 0, g ∈ C 0 (R + ).
It is known that the spectrum of A is Indeed, the functions e λx are eigenfunctions for Re(λ) < 0, and f n (x) := e −x/n e iαx is an approximated eigenfunction for Re(λ) = 0 [1]. In this case σ( A) is larger than the spectrum of the original operator A. Note that the growth bound of the semigroup T (t) is ω = 0 (indeed, T (t) is a contraction semigroup). In this case sup λ∈σ(A) {Re(λ)} = ω(T ) = 0. The largest connected component of ρ(A) containing an interval [r, +∞) is Hence, ( A − λI) is invertible for all λ ∈ ρ ∞ (A) by Proposition 3.4.
Although the lumping operators in the literature are surjective, it is interesting to discuss lumpability in the case when ran(M ) = Y . A condition for the existence of a reduced operator in the bounded case is given by Theorem 2.2 in the context of operator factorization. Here we prove the following result.
Proposition 3.6. Let T (t) be a strongly continuous semigroup on X generated by A. Let M be linear and continuous from X to Y such that the following condition holds: (j) For {x n } ⊂ X, M x n → 0 implies M T (t)x n → 0 ∀t ≥ 0. Remark 6. Note that condition (j) is stronger than assuming that If, in addition to (20), M has closed range, then condition (j) follows (see [14]). However, (j) does not follow from (20) if the range of M is not closed. Hence, Proposition 3.6 does not generalize Theorem 3.1, but rather gives another version of lumpability with a different assumption.
Proof of Proposition 3.6. It is not hard to see that (j) is equivalent to the following statement: Indeed, (21) clearly implies (j), and the converse follows since M · is a seminorm. Now, by (21) and Theorem 2.2, for every t ≥ 0 one can construct a family of linear and bounded operators on ran(M ): By the boundedness of T (y), these operators can be extended to ran(M ) in the following way: T (t)y := lim n→∞ M T (t)x n for M x n → y.
It can be verified that T (t) is a strongly continuous semigroup of operators on ran(M ). Note also that the value of k t in (21) Then by (23), It follows that lim Condition (jj) implies that a reduced operator A can be constructed on the dense subspace M D(A) in such a way that AM = M A. But from (jj) it follows that there exists k > 0 such that M Ax ≤ k M x , ∀x ∈ D(A) (this fact can be proved in the same way as for bounded operators; see e.g. [5]). Therefore, the reduced operator A can be extended to a bounded operator on ran(M ). Condition (jj) is stronger than the hypotheses of Theorem 3.3. Indeed, not every lumping leads to a bounded reduced operator. Note also that, T (t) being the semigroup generated by A, condition (jj) cannot be obtained from the analogous condition (j), unless one assumes stronger hypotheses such as the boundedness of A.

4.
Dual conditions for lumpability. We now consider the lumpability problem from a dual perspective. As a motivation, first consider the problem in finite dimensions. Let X = R n and Y = R k with k < n. Since the matrix A exists if and only if ker(M ) is A-invariant [11], an equivalent condition for lumpability is the invariance of ran(M ) under A . This dual characterization has been utilized for studying lumpability in finite-dimensional systems and Markov chains, e. g., in [19] and [22]. Our aim is to generalize these results to infinite-dimensional systems, for both bounded and unbounded operators.
4.1. Background in adjoint operators and semigroups. Before going into details of lumping analysis, we briefly describe the setting and introduce the notation; for further details we refer to [28] and [18]. Let X * denote the dual space of a Banach space X, namely the set of linear and bounded functionals from X to C. Let j denote the canonical inclusion in the double dual X * * defined by j : X → X * * , j(x)(x * ) := x * (x), ∀x * ∈ X * , x ∈ X.

FATIHCAN M. ATAY AND LAVINIA RONCORONI
For two subspaces C and S of X and X * , respectively, we denote the annihilators If C is closed then C = C ⊥⊥ , while S ⊥⊥ coincides with the weak* closure of S . For a linear operator A between two Banach spaces X and Y whose domain D(A) is dense in X, we also consider the adjoint operator A * : Y * → X * defined by A * (y * )(x) = y * (Ax) on the domain Let {T (t)} t≥0 be a strongly continuous semigroup on X generated by A. The family of the adjoint operators T * (t) : X * → X * is again a semigroup of bounded operators on X * and is a continuous semigroup with respect to the weak star topology. In fact, it is the semigroup generated by the operator A * , which is closed and densely defined with respect to the weak* topology, and is given by Although the semigroup {T * } t≥0 may fail to be strongly continuous, one can find a closed subspace of X * in which strong continuity holds. Thus, the sun dual of X is the closed subspace X ⊂ X * defined by The sun dual semigroup of {T (t)} t≥0 is the strongly continuous semigroup obtained by restricting the adjoint semigroup to the sun dual space, We denote the generator of the sun dual semigroup by A . It is the restriction of the adjoint operator A * to the domain It is known that A * is the weak* closure of A and D(A * ) = X [20]. As an example of a sun dual space we mention that, for the semigroup of left translations (12) on X = L 1 (R), X is the space C ub (R) of uniformly continuous and bounded functions on the real line [28]. One can iterate the construction of the sun dual space and define the double sun dual X as the closed subspace of X * on which the adjoint semigroup T * (t) is strongly continuous. We call X sun-reflexive if X is isomorphic to X .
Finally, we recall that there are some cases in which the passage to the adjoint semigroup preserves strong continuity. This always happens when X is a reflexive space: since in this case the weak and the weak* topologies on the dual space coincide, the adjoint semigroup is weakly continuous and thus strongly continuous [20]. Similarly, if the semigroup is uniformly continuous, then its adjoint will also be uniformly continuous, because

4.2.
Dual lumpability for bounded operators. Consider system (1) generated by a bounded operator A ∈ B(X). We have seen in Theorem 2.1 that a lumping of this system through a bounded and surjective map M : X → Y can be obtained if and only if ker(M ) is invariant under A. Similarly to the finite-dimensional case, we give an equivalent condition for lumpability in terms of adjoint operators. . Given x * = M * y * , we have A * x * = A * M * y * = M * A * y * ∈ ran(M * ); i.e., statement 2 holds.
2 ⇒ 1. Note that statement 2 is equivalent to ran(A * M * ) ⊆ ran(M * ). Thus, which is the condition for lumpability.
Example 5. Consider the lumping operation corresponding to the quotient projection of Remark 1. By definition, the adjoint of the quotient projection is It is known that the range of π * can be identified with the annihilator C ⊥ , which is invariant under A * . (This can be seen by taking φ ∈ C ⊥ , applying A * , and using the invariance of C to obtain A * φ(x) = φ(Ax) = 0 ∀x ∈ C .) The reduction of A to A through π can indeed be identified with the restriction of A * to the closed subspace C ⊥ .

4.3.
Dual lumpability for unbounded operators. We will obtain the dual conditions for lumpability in the general case of dynamics generated by an unbounded operator A. Since the family {T (t)} t≥0 is made up of bounded operators, we have the following result.
Proposition 4.2. The following statements are equivalent: 1. There exists an operator A defined on M (D(A)) such that A generates a strongly continuous semigroup on Y and AM = M A (i.e. the system is lumpable by the operator M ); 2. ran(M * ) is invariant under T * (t) for every t ≥ 0.
Proof. 1 ⇒ 2. Let T (t) be the strongly continuous semigroup generated by A. We have shown (see proof of Theorem 3.1) that T satisfies the lumping relation T (t)M x = M T (t)x, x ∈ X. This implies that the kernel of M is T (t)-invariant. Statement 2 then follows through the following implications (considering that the surjectivity of M implies that the range of its adjoint is star-weakly closed):  which is the necessary and sufficient condition for lumpability. Example 6. We give a dual interpretation of the lumping through the evaluation operator described in Example 2. Let h : R → C be a continuous function such that sup x∈R Re(h(x)) < ∞. Then the family of bounded operators T (t) given by T (t)f (x) = e th(x) f (x) is a strongly continuous semigroup on the Banach space X = C 0 (R), with generator Af (x) = h(x)f (x). We consider the lumping operator M : C 0 (R) → C k defined by M f = (f (x 1 ), . . . , f (x k )) , which evaluates a given function at the k points x 1 , . . . , x k ∈ R. By the Riesz-Markov theorem, C 0 (R) * can be identified with the Banach space M (R) of all complex, regular, Borel measures on the real line. If φ ∈ C 0 (R) * and µ φ is the measure associated with φ, ∀f ∈ C 0 (R), then φ(f ) = f (x) dµ φ (x). Consider now the adjoint of the lumping operator M , M * : (C k ) * → M (R), M * (α 1 , . . . , α k ) = α 1 δ(x 1 ) + · · · + α k δ(x k ). This is an injective operator whose range is the closed subspace of all linear combinations of δ(x 1 ), . . . , δ(x k ) with complex coefficients (which is clearly isomorphic to C k ). It is easy to obtain T * (t)M * (α 1 , . . . , α k ) = α 1 e th(x1) δ(x 1 ) + · · · + α k e th(x k ) δ(x k ), which implies that the range of M * is invariant under T * (t). In particular: where T (t) is the reduced semigroup on C k given by This construction shows the advantages of the dual approach, because M * is indeed an invertible operator on a finite dimensional space.
We now establish a dual condition for lumpability in terms of the adjoint of the generator. To this end, recall that the adjoint operator A * is a generator only in the sense of the weak* topology. Fortunately, many properties of strongly continuous semigroups hold also for the adjoint semigroup where the same limits are considered in the weak* topology; in fact, it is easy to verify (see [28] for more details) that 1) for every x * ∈ X * , t > 0, T * (t)x * := weak*-lim where R (λ; A * ), λ ∈ ρ(A * ), is the resolvent operator of A * , and 2) R (λ; A * ) = weak*-∞ 0 e −λs T * (s)x * ds, where the right side is the weak* integral, defined as the unique element such that for every x ∈ X ∞ 0 e −λs T * (s)x * ds (x) = lim k→∞ k 0 e −λs T * (s)x * (x)ds.
Since the range of M * is weak* closed, by the above results it is easy to verify that ran(M * ) is invariant under the adjoint semigroup T * (t) if and only if it is