Positive Perturbations of Dual and Integrated Semigroups

Transcription

1 Positive Perturbations of Dual and Integrated Semigroups Horst R. Thieme Department of Mathematics Arizona State University Tempe, AZ , USA Abstract Positive perturbations of generators of locally Lipschitz continuous increasing integrated semigroups on an abstract L space are again generators of locally Lipschitz continuous increasing integrated semigroups. Positive perturbations of generators of positive dual semigroups on a dual abstract L space are generators of semigroups that are weakly right continuous. These results are reformulated in terms of cumulative outputs and then applied to age-structured models for population dynamics. Research partially supported by The Bank of Sweden Tercentenary Foundation, The Swedish Council for Forestry and Agricultural Research, and the Swedish Cancer Foundation Research partially supported by NSF grant DMS

2 1. Introduction Let X be an abstract L space. Under which conditions, if at all, are certain classes of semigroups or integrated semigroups preserved under positive perturbation of their generators? This question has been addressed for the class of increasing integrated semigroups by Arendt (1987) and for the class of positive C -semigroups by Desch (unpublished) and Voigt (1989). We address it for the class of locally Lipschitz continuous increasing integrated semigroups and the class of dual semigroups Scenario We recall that an abstract L-space X is a Banach lattice whose norm satisfies (1.1) x + y = x + y, x, y X +, where X + denotes the cone of non-negative elements. Let D be a linear subspace of X and A, B : D X be linear operators. We assume that B is a positive operator, i.e., (1.2) x D X + = Bx X +. We recall that any positive everywhere defined linear operator on X is automatically bounded (Schaefer, 1974; Chapter II, Theorem 5.3). We further assume that A is resolvent positive, i.e., that some interval (ω, ) is contained in the resolvent set of A and (λ A) 1 is a positive operator for λ > ω. It follows from the resolvent identity that, for x X +, (λ A) 1 x is non-increasing in λ and so is B(λ A) 1 x because B is positive. B(λ A) 1 is an everywhere defined positive operator and thus bounded. As X + is generating (i.e., X = X + X + ), B(λ A) 1 is non-increasing in λ and so is the spectral radius of B(λ A) 1, spr B(λ A) 1. We assume that, for some λ > ω, (1.3) spr B(λ A) 1 < 1. By our previous remarks (1.3) holds for all sufficiently large λ >. Apparently B(λ A) 1 maps B(D) and B(D) into itself. So (1.3) is in particular satisfied if the operator norm of the restriction of B(λ A) 1 to B(D) is smaller than 1, (1.4) B(λ A) 1 B(D) < 1.

3 1.2. Review This note supplements perturbation results by Arendt (1987a) and Desch (unpublished), Voigt (1988). (For unknown terminology see Sections 2 and 3.) Theorem 1.1 [Arendt, 1987a]. If A generates an increasing (once) integrated semigroup, so does A + B. Theorem 1.2 [Voigt, 1988]. If A generates a positive C -semigroup, so does A + B. Arendt (1987a, Theorem 3.1, Theorem 5.7, Proposition 6.5) formulates and proves Theorem 1.1 more generally for ordered Banach spaces X where X + is generating and normal and X is an order ideal in its bidual X. Actually Theorem 1.1 holds in an ordered Banach space with a generating and regular cone. Recall that the cone X + is called regular if any decreasing sequence in X + has a (strong) limit. The proof is the same as in Arendt (1987) with the modification that Arendt s Theorem 5.6 is replaced by our Theorem 2.5 which is essentially the version proved by Bochner (1942). Arendt s Example 3.3 shows that Theorem 1.2 may fail in X = L p (Ω) for 1 < p <. Voigt (1988) observes that condition (1.3) is necessary for both Theorem 1.1 and Preview of main results A particularly interesting class of (once) integrated semigroups are those that are locally Lipschitz continuous (in the operator norm). Their generators are characterized by the fact that all sufficiently large λ > are contained in the resolvent set of A and by the resolvent estimates (1.5) (λ A) n M(λ ω) n, λ > ω, n N, with some M 1, ω R (Arendt, 1987b, Theorem 4.1; Kellermann, Hieber, 1989, Theorem 2.4). It is well known that any operator A satisfying these properties is characterized by the fact that there exists an equivalent norm which makes A ωi an m-dissipative operator. Hence we call such an operator A para-m-dissipative. This is preferable over the historically incorrect name Hille-Yosida operator that has recently been used in the literature. If a para-m-dissipative operator is densely defined, then it actually generates a strongly continuous semigroup. Anyway, the part of A in X = D generates a strongly continuous semigroup on X (with the same norm). It is a natural question (which we answer positively in this paper) whether a perturbation result like Theorem 1.1 or 1.2 also holds for generators of locally Lipschitz continuous integrated semigroups.

4 Theorem 1.3. If A generates a locally Lipschitz continuous increasing (once) integrated semigroup, so does A+B. Moreover the part of A+B in D generates a strongly continuous semigroup on D. Notice that we recover the C -semigroup result in Theorem 1.2 as a special case because then D = X. So Arendt s Example 3.3 shows that Theorem 1.3. may fail if X is an order ideal in X, but not an abstract L space. If X = Z is the dual space of a Banach space Z and A is the dual of the generator of a strongly continuous semigroup Ṡ on Z, then A is the generator of the dual semigroup Ṡ in an appropriate sense. Even if B is a bounded linear operator from D (with the norm of X) to X, A + B will not be a generator of a dual semigroup in general, but generate (in an appropriate sense) a semigroup T that is only weakly continuous (Clément, Diekmann, Gyllenberg, Heijmans, Thieme, 1989), i.e. z, T (t)x is a continuous function of t for any z Z, x X. Actually T is continuous in a somewhat stronger sense: T (t)z is a (strongly) continuous function of t (with values in X = Z ) for any z Z. The second notion of continuity is better behaved than the first: If T (t)z is (strongly) continuous at t =, then it is automatically continuous at any t. If z, T (t)x is continuous at t =, then we can only conclude that it is right continuous at an arbitrary t. This motivates us to make the following Definition 1.4. Let X = Z be the dual of a Banach space Z and T a semigroup of bounded linear operators T (t), t, on X. T is called a weakly measurable semigroup if z, T (t)x is a Borel measurable function of t for any z Z, x X. T is called a W+-semigroup if z, T (t)x is a right continuous function of t for any z Z, x X. Equivalently, we sometimes call these semigroups weakly right continuous. T is called a W -semigroup if z, T (t)x is a continuous function of t for any z Z, x X. Equivalently, we sometimes call these semigroups weakly continuous. T is called a C -semigroup if T (t)z is a strongly continuous function of t with values in X = Z for any z Z. If B is unbounded, but positive (as we assume), we get the following results. Theorem 1.5. Let X = Z be the dual of an abstract M space Z, X + = Z+, and A the dual of the generator of a positive C -semigroup on Z. a) Then the operator A + B with domain D is the generator of a weakly measurable exponentially bounded positive semigroup T on X in the following sense:

5 (i) For any pair x, y X we have that x D and y = (A + B)x if and only if T (t)x = x + T (s)yds t. (ii) For any x X, t, we have that T (s)xds D and (A + B) T (s)xds = T (t)x x. b) For any x X, there are only countably many times t at which T (t)x is not weakly continuous. Moreover, for z, T (t)x, z Z, x X, the limits from the left and from the right always exist in t > and t respectively. c) The weak integrals T (t)x = T (s)xds provide the integrated semigroup T on X which is generated by A according to Theorem 1.3. The part of A + B in X = D(A) generates a strongly continuous non-negative semigroup on X which coincides with the restriction of T to X. As for the continuity properties of T we have the following characterizations: Theorem 1.6. Under the assumptions of Theorem 1.5, the following holds for the semigroup T generated by A + B: a) T is a W+-semigroup if and only if (1.7) z, B(λ A) 1 x, λ, z Z, x X, b) T is a C -semigroup if and only if (B(λ A) 1 ) z for λ, z Z. c) T is a dual semigroup provided that (B(λ A) 1 ) maps Z into Z and (B(λ A) 1 ) z, λ, z Z. If Z is an order ideal in Z, these conditions are also necessary. The integrals in Theorem 1.5 have to be interpreted in the weak sense, i.e., T (s)yds is the continuous functional on Z defined by z, T (s)yds = z, T (s)y ds, z Z. The second integral makes sense because the semigroup T is weakly right continuous, i.e., z, T (t)y is a right continuous function of t for any z Z, y X = Z. Theorem 1.5 a (ii) states that, for any y X, the differential equation du dt = (A + B)u, t >, u() = y

6 can be solved in an integral sense. Theorem 1.5 a) implies that for sufficiently large λ. (λ A B) 1 x = e λt T (t)xdt, x X, The proofs of Theorem 1.5 and 1.6 rely on Theorem 1.3 in so far as it establishes estimate (1.5) for the resolvent of A + B. The problem lies in the fact that, for general unbounded B satisfying (1.3), one can establish (1.5) with A + B replacing A only for n = 1, but not for arbitrary n. Even when we assume that X is an abstract L space and that all positivity assumptions are satisfied we have not managed to show (1.5) directly. If X is an abstract L space one can use a perturbation procedure involving Stieltjes integral equations which we developed for perturbing semigroups and evolutionary systems by stepresponses and cumulative outputs (Diekmann, Gyllenberg, Thieme, 1993, preprint). This approach is motivated by the fact that, for concrete problems, it is often difficult to figure out the domain of A and the exact action of B. Therefore we take 1.4. An alternative point view: cumulative outputs and resolvent outputs If A, B : D X with A being the generator of a locally Lipschitz integrated semigroup, then the operator family (1.8) F (λ) = B(λ A) 1, λ > ω, satisfies the relation (1.9) F (λ)(µ A) 1 = 1 ( ) F (λ) F (µ), λ, µ > ω. µ λ This is a consequence of the resolvent identity. We now take the converse point of view. Any family F (λ) satisfying (1.9) is of the form (1.8) because one can recover B by setting (1.1) Bx = F (λ)(λ A)x, x D. (1.9) implies that definition (1.1) does not depend on the choice of λ. In many applications the operator B in (1.8) describes an output operation of the system under consideration. The motivates the following name: A family of bounded operators satisfying (1.9) is called a resolvent output family for A.

7 Strangely enough, though any positive operator B induces a positive resolvent output family for A the converse is not necessarily true unless A is densely defined. Example 8.7 presents a positive resolvent output family where the associated operator B is not positive. This is why we make the following statements. Theorem 1.7. Let F (λ), λ > ω, be a positive resolvent output family for the generator A of an increasing integrated semigroup on an ordered Banach space X with regular and generating cone X +, spr F (λ) < 1 for some λ > ω. Let B be the operator induced by F via (1.1). Then A+B is the generator of an increasing integrated semigroup on X. If X is an abstract L space and A generates a locally Lipschitz continuous increasing integrated semigroup, so does A + B. Moreover the part of A + B in D generates a strongly continuous semigroup on D. Theorem 1.8. Let the abstract L space X = Z be the dual of an abstract M space Z, X + = Z +, and A the generator of a positive dual semigroup. Let F (λ), λ > ω, be a resolvent output family for the generator A of an increasing integrated semigroup on an ordered Banach space X with regular and generating cone X +, spr F (λ) < 1 for some λ > ω. Let B be the operator induced by F via (1.1). a) Then A + B generates a weakly right continuous positive semigroup T on X in the sense of Theorem 1.6 a) (i), (ii), if and only if F (λ)x, λ, x X. b) If one (and then both) of the two equivalent conditions in a) hold, then B is positive and T has the properties in Theorem 1.6 b). c) If z Z, then T (t)z is continuous in t if and only if F (λ)z, λ. Typically F (λ) is the Laplace Stieltjes transform of cumulative outputs for a semigroup or an integrated semigroup: Let T be a semigroup. Then a family of bounded operators V (t), t, with V () =, is called a cumulative output for T if (1.11) V (t) T (r) = V (t + r) V (r), t, r. Relation (1.11) explains the term cumulative output. If V (t) describes the cumulative output of a system the dynamics of which are described by the semigroup T, then the cumulative output between t and t + r, given by the difference on the right hand side of (1.11), is obtained by letting the state of the system evolve for r time units and then collecting the cumulative output over a time interval of length t.

8 Cumulative outputs for an integrated semigroup T are obtained by formally integrating relation (1.11) over r: (1.12) V (t)t (r) = +r V (s)ds V (s)ds r V (s)ds. Taking Laplace transforms in t and r we find that the Laplace Stieltjes transforms F (λ) = ˇV (λ) of a cumulative output family V satisfy (1.9), where A is the generator of the strongly continuous integrated semigroup T. If the family V (t) is strongly continuous in t, then, in return, (1.9) implies (1.12) by the uniqueness property of the Laplace transform. If V (t) and T (t) are strongly continuous, then (1.9) implies (1.11). If V is a cumulative output of the integrated semigroup S, then the criterion in Theorem 1.8 a) is equivalently rewritten as V (t)x, t, x X. Under an additional technical condition one finds that T (t) is weakly continuous in t if and only if V is strongly continuous in t. See Theorem 7.1. In applications one typically uses cumulative outputs rather than resolvent outputs (or even perturbations of the generator), because one has the natural explanation mentioned above, and (1.11) is easier to check than (1.12) or (1.9) and typically implies (1.9). Further a necessary and sufficient condition for obtaining a weakly continuous semigroup can apparently be found in terms of the cumulative output, but not of the corresponding resolvent output. We can use resolvent outputs to characterize exponentially bounded solutions of the Cauchy problem (1.13) du dt in terms of their Laplace transform = (A + B)u, u() = x, û(λ) = Taking the Laplace transform of (1.13) we obtain e λt u(t)dt. λû(λ) x = (A + B)û(λ). This is equivalent to û(λ) = (λ A) 1 (Bû(λ) + x). Setting f(λ) = Bû(λ) we obtain (1.14) û(λ) = (λ A) 1 (f(λ) + x) f(λ) = F (λ)(f(λ) + x).

9 Hence we can define a function u : [, ) X to be the Laplace solution solution to (1.13) if and only if its Laplace transform exists and satisfies (1.14) for sufficiency large λ >. We will illustrate the use of cumulative outputs and of step responses (which are the dual concept) in models for age-structured populations dynamics (Section 8). These models are formulated in terms of survival probabilities and cumulative per capita births. The solutions of these models lead to strongly continuous semigroups on the space of integrable functions. On the space of measures, however, the type of semigroup that is induced depends on the properties of the functions describing the survival probabilities and cumulative births. We give a complete characterization. These models demonstrate that, in a natural way, positive perturbations of dual semigroups on an abstract L space lead to all kind of operator semigroups on spaces of Borel measures: semigroups that are weakly measurable, but not weakly right continuous; semigroups that are weakly right continuous, but not weakly continuous; semigroups that are weakly continuous, but no C -semigroups, C -semigroups that are no dual semigroups. This paper is organized as follows. In Section 2 we discuss some aspects of integration in ordered Banach spaces that will be used later. In Section 3 we summarize some wellknown and some less known facts concerning semigroups and integrated semigroups with an emphasis on locally Lipschitz continuous integrated semigroups and dual semigroups. Properties of output families are studied in Section 4. In Section 5 we prove Theorem 1.3 and Theorem 1.7, whereas Theorems 1.4 to 1.6 and Theorem 1.8 are proved in Section 6. In Section 7 we prove the criterion for finding a weakly continuous perturbed semigroup in terms of cumulative outputs. Section 8 contains applications to age-structured population models including cell populations. 2. Integration and Laplace transforms of monotone functions More generally than in the Introduction we assume that X is an ordered Banach space with a generating and normal cone X + (i.e., X = X + X +, X = X+ X+). Let u : [a, b] X be a monotone non-decreasing function, i.e., u(t) u(s) if a s t b. As in Arendt (1987a, Section 4) it is not difficult to see that v is of weakly bounded variation in the sense of Hille, Phillips (1957, Definition 3.2.4). Theorem in Hille, Phillips (1957) implies that the Riemann Stieltjes integral b tu(dt) exists. By the integration of a parts formula for Riemann Stieltjes integrals we have the following: Lemma 2.1. Let u : [a, b] X be a monotone non-decreasing function. Then u is Riemann integrable and the Riemann Stieltjes integral b tu(dt) exists and a b a u(t)dt = bu(b) au(a) b a tu(dt).

10 Lemma 2.1 implies that the Riemann integral of a monotone non-decreasing function commutes with a positive linear operator even if this one is not bounded or closed. Proposition 2.2. Let B : D(B) X be a positive linear operator from a subspace D(B) of X into X. Let [a, b] be an interval in R and u : [a, b] D(B) a monotone increasing function such that b u(t)dt D(B). Then u(t) and Bu(t) are Riemann integrable and a B b a u(t)dt = b a Bu(t)dt. Proof: As B is positive and linear, Bu(t) is monotone non-decreasing and thus, by our previous remarks, Riemann integrable. By Lemma 2.1 it is sufficient to show that B b a tu(dt) = b a t(bu(dt)). Let a = t < t n+1 = b be an arbitrary partition of [a, b]. Then, since B is linear and positive, b n n B tu(dt) B t j+1 (u(t j+1 ) u(t j )) = t j+1 (Bu(t j+1 ) Bu(t j )). a j= j= As b t(bu(dt)) exists as a Riemann-Stieltjes integral, the last sums converge towards b a a t(bu(dt)) if a suitable sequence of partitions is chosen. Hence B b a tu(dt) b a t(bu(dt)). The other inequality is shown similarly. This implies the assertion. Riemann integrals are not so nice as Lebesgue integrals. We recall that the cone X + is called regular if any monotone non-increasing sequence in X + has a (strong) limit. A regular cone is automatically normal (Krasnosel skii, 1964, Theorem 1.6). If the cone X + is regular, Riemann integration of non-decreasing functions can be replaced by Lebesgue integration: Lemma 2.3. Let u : [a, b] X be a monotone non-decreasing function and X + be generating and regular. Then there exists a monotone non-decreasing function v : [a, b] X that is Borel measurable and continuous from the right such that r u(s)ds = r v(s)ds whenever a r t b.

11 The integral on the left hand side is a Riemann integral whereas the integral on the right hand side can be interpreted both as Riemann and Lebesgue integral. Proof: Extend u by u(t) = u(b) for t and define w(h, s) = 1 h s+h s u(σ)dσ = 1 u(s + hτ)dτ, h >, s a. As u is bounded, w is continuous and monotone non-decreasing in both s a, h >. In particular the limit w(s) = lim h w(h, s) exists as X + is regular, and defines a monotone non-decreasing and Borel measurable function of s a. We claim that r u(s)ds = r w(s)ds. To see this we apply some functional x X + to both integrals. Then u(s)ds, x = r r u(s), x ds, where both integrals are taken in the sense of Riemann. As x X +, the function u(s), x is monotone non-decreasing and thus, since scalar, Borel measurable with the Riemann integral being identical to the Lebesgue integral. It is well-known that our claim holds if u were scalar. Hence we have u(s)ds, x = w(s)ds, x. r As X + is generating, the claim follows. In order to achieve continuity from the right we set v(s) = lim h w(s + h), s a. Again the limit exists because w is non-decreasing and X + is regular. v is Borel measurable and, more strongly, continuous from the right. It follows from Lebesgue dominated convergence theorem that r r +h v(s)ds = lim w(s + h)ds = lim w(s)ds = w(s)ds. h r h r+h r

12 This completes the proof. If u : [, ) X is monotone non-decreasing, then the Riemann integrals b e λt u(t) dt and the Riemann Stieltjes integrals b e λt u(dt) exist for all λ C and b > by Theorem in Hille, Phillips (1957) and integration by parts. The Laplace transform and the Laplace Stieltjes transform of u are defined by b (2.1) û(λ) = lim e λt u(t)dt, b and b (2.2) ǔ(λ) = lim e λt u(dt) b in case that the respective limits exist. Arendt (1987a, Proposition 5.5) proves the following result. Proposition 2.4. Assume that ω R and that the limit in (2.1) exists for λ = ω in the weak topology. Then the limit in (2.1) exists in norm, if Rλ > ω. Moreover u is exponentially bounded, i.e., there exist M >, ω ω such that u(t) Me ωt, t, and ǔ(λ) = λû(λ), Rλ > ω. Finally ǔ(λ) is a completely monotonic function of λ > ω, i.e., it is infinitely often differentiable and (2.3) ( 1) n ǔ (n) (λ) X +, n N. Complete monotonicity can also be defined for functions f : (a, ) X that are not a priori differentiable, namely, a monotone non-decreasing function f : (a, ) X is called completely monotonic if (2.4) ( 1) n n hf(t), n N, t > a, h > where 1 hf(t) = f(t + h) f(t), n+1 h = 1 h( n hf).

13 (2.3) and (2.4) are equivalent provided that f is infinitely often differentiable. Definition (2.4) has the advantage that one immediately realizes that complete monotonicity is preserved under pointwise convergence. The following converse of Proposition 2.4, a vector-valued version of the Hausdorff- Bernstein-Widder theorem, is essentially shown by Bochner (1942). Proposition 2.5. Assume that X + is generating and regular. Let f : (a, ) X be a completely monotonic function in the sense of (2.4). Then there exists a uniquely determined non-decreasing function u : [, ) X that is right continuous at any t >, u() =, and (2.5) f(λ) = ǔ(λ). Moreover u is Borel measurable and exponentially bounded and (2.6) f(λ) = λû(λ). If f(λ) for λ, then u can be chosen to be right continuous at any t. Proof: Existence of a function ũ, ũ() =, that is non-decreasing and satisfies (2.5) follows from Bochner (1942). Proposition 5.5 in Arendt (1987) provides exponential boundedness. Integration by parts yields (2.6). Proposition 2.3 now implies that u can be chosen to be Borel measurable and continuous from the right at t >. In general we cannot get right continuity at t = without violating the equivalence of (2.5) and (2.6). As f(+) = lim λ f(λ), f is right continuous at if the latter limit is. A function u with all these properties is uniquely determined as can be easily seen by testing with an element x X + and using that X + is generating. Corollary 2.6. Let X be an abstract L space. Then the function u in Proposition 2.5 is of locally bounded variation and has at most countably many discontinuities. Proof: First of all, any abstract L space has its cone regular. See Schaefer (1974, Theorem II.5.1, Proposition II.8.3). So Proposition 2.5 applies. The additional properties of u are immediate consequences of the fact that u is non-decreasing and (1.1). In fact, for any finite sequence, t t n+1, (2.7) n u(t j+1 ) u(t j ) = u(t n+1 ) u(t ). j= It is useful to introduce an order between bounded linear operators, let us say B 1, B 2 : (2.8) B 1 B 2 B 2 x B 1 x X + x X +.

14 A family of bounded linear operators V (t) on X is called increasing if (2.9) V (s) V (t), t s. Proposition 2.7. Let X be a Banach lattice and Y an abstract L space and V (t), t, be a increasing family of positive linear operators from X to Y. Then V (t)x is of locally bounded variation for any x X. In particular, for any finite sequence t t 1 t n+1 we have (2.1) n V (t j+1 )x V (t j )x (V (t n+1 V (t )) x j= V (t n+1 ) V (t ) x, x X. Proof: As V (t j+1 ) V (t j ) are positive operators and X, Y are Banach lattices we have V (t j+1 )x V (t j )x V (t j+1 )x V (t j )x (V (t j+1 ) V (t j )) x = V (t j+1 ) x V (t j ) x By (2.7), with u(t) = V (t) x taking values in the abstract L space Y, n V (t j+1 )x V (t j )x (V (t n+1 V (t )) x V (t n+1 ) V (t ) x. j= As x = x in a Banach lattice, the assertion follows. Finally we obtain the following results for operator families. Proposition 2.8. Assume that X and Y are ordered Banach spaces and that the positive cone X + of X is generating and the positive cone Y + of Y is generating and regular. Let F (λ), λ > a, be a completely monotonic family of bounded linear operators from X to Y in the sense of (2.4). Then there exists a uniquely determined non-decreasing family V (t), t, of positive linear operators from X to Y that is strongly right continuous at t >, V () =, and F (λ) = ˇV (λ), λ > a. Moreover, for any x X, V (t)x is Borel measurable and exponentially bounded in t and F (λ) = λ ˆV (λ), λ > a.

15 If F (λ)x for λ, x X, then V (t) can be chosen to be strongly right continuous at t. If Y is an abstract L space, V (t) has the properties in Proposition 2.7. Proof: Let x X +, f(t) = F (t)x. Then f is completely monotonic and takes values in the generating and regular cone Y +. By Proposition 2.5, there exists a unique non-decreasing function t v(t, x) that is continuous from the right, v(, x) =, The uniqueness implies that F (λ)x = ˇv(, x)(λ). v(t, x + y) = v(t, x) + v(t, y), v(t, αx) = αv(t, x), x, y X +, α. Since X = X + X +, there exists a unique linear operator V (t) on X such that V (t)x = v(t, x) for x X +. Since V (t) is a positive operator from an ordered Banach space with generating cone into an ordered Banach space with normal cone, V (t) is bounded (Schaefer, 1971; Chapter V: 5.6, 5.5, 3.3 Corollary 3; Chapter IV: 3.4). 3. Some remarks on semigroups and integrated semigroups In the following we call a family S(t), t >, of bounded linear operators non-degenerate if S(t)x = occurs for all t > only if x = Semigroups Most references on operator semigroups define a (one-parameter) semigroup to be a family of linear bounded operator Ṡ(t), t, on a Banach space X such that (3.1) Ṡ(t)Ṡ(r) = Ṡ(t + s), t, s ; Ṡ() = I; with I being the identity operator on X. Interestingly enough, Hille, Phillips (1957), in their classic, develop the theory for Ṡ(t), t >, and consider Ṡ() = I as a special case. Actually any one-parameter semigroup which is only defined for t > can be extended to t this way, but applications where I should be replaced by another projection P with P Ṡ(t) = Ṡ(t) = Ṡ(t)P, t >, are well conceivable. The infinitesimal generator of a semigroup is defined by Ax = lim(ṡ(t)x x) t

16 with the domain of A, D(A), being the linear subspace of those elements x for which this limit exists. The most frequently studied one-parameter semigroups are C -semigroups where Ṡ(t)x x, t, x X. which indeed makes the extension Ṡ() = I the natural one. For C -semigroups we have the property that, for any pair x, y X, (3.2) x D(A), Ax = y Ṡ(t)x x = Ṡ(r)ydr Integrated semigroups (Once) integrated semigroups are motivated by formally defining S(t) = and discovering that Ṡ(s)ds (3.3) S(t)S(r) = S() =. +r S(s)ds S(s)ds r S(s)ds, t, r, As topological property of S one generally chooses that S(t) is strongly continuous in t, i.e., S(t)x is a continuous function of t for any x X. We mention that n times integrated semigroups have been considered (see Arendt (1987b), Neubrander (1988) for two pioneering papers, Ahmed (1991) for a short account available in a text-book and Hieber (thesis) for the case that n is a positive, but not necessarily natural number). One is mainly interested in non-degenerate integrated semigroups, i.e., S(t)x = for all t > occurs only for x =. The generator A of a non-degenerate integrated semigroups is given by requiring that, for x, y X, (3.4) x D(A), y = Ax S(t)x tx = S(s)yds, t. Formally this definition can be obtained by integrating (3.2). Notice that this definition makes sense and defines a closed operator A, even if S is not an integrated semigroup. Actually one has the following result:

17 Theorem 3.1. Let S(t), t, be a non-degenerate strongly continuous family of bounded linear operators on X and let the closed linear operator A be defined by (3.4). Then S is an integrated semigroup if and only if S(s)ds D(A) for all t and A S(s)ds = S(t)x tx t. Proof: The only if part follows from Thieme (199a, Lemma 3.5). The if part part follows from the proof of Thieme (199a, Theorem 6.2). We will also need the following relation (Thieme, 199a, Lemma 3.4): Proposition 3.2. Let S(t), t, be a non-degenerate integrated semigroup and A its generator. Let x D(A). Then S(t)x D(A), S(t)x is differentiable and d S(t)x = x + AS(t)x = x + S(t)Ax. dt If S(t) is exponentially bounded, i.e., there exist M, ω > such that S(t) Me ωt t, one has the following useful characterization which is a combination of Theorem 3.1 in Arendt (1987b) and Proposition 3.1 in Thieme (199a). Theorem 3.3. Let S(t), t, be a strongly continuous exponentially bounded family of bounded linear operators on X and A : D(A) X be a linear operator in X. Then S is a non-degenerate integrated semigroup and A its generator if and only if there exists some ω > such that any λ > ω is contained in the resolvent set of A and (3.5) (λ A) 1 = λŝ(λ). Here the Laplace transform Ŝ(λ) is defined pointwise: Ŝ(λ)x = e λt S(t)xdt. Actually formula (3.5) can be used to define the generator A in the case of exponentially bounded integrated semigroups (Arendt, 1987b, p. 338; Neubrander, 1988, Definition 4.1). A particularly interesting family of (once) integrated semigroups are those that are locally Lipschitz continuous.

18 Theorem 3.4. The following statements are equivalent for a linear closed operator A in a Banach X: (i) A is the generator of an integrated semigroup S that is locally Lipschitz continuous in the sense that, for any b >, there exists a constant Λ > such that S(t) S(r) Λ t r ; r, t b. (ii) A is the generator of an integrated semigroup S and there exist constants M 1, ω R such that S(t) S(r) M r e ωs ds, r t <. (iii) There exist constants M 1, ω R such that (ω, ) is contained in the resolvent set of A and (λ A) n M(λ ω) n, n = 1, 2,... The constants M, ω in (ii), (iii) can be chosen to be identical. Moreover, if one (and then all) of (i), (ii), (iii) holds, D(A) coincides with those x X for which S(t)x is continuously differentiable. The derivatives S (t)x, t, x D(A), provide bounded linear operators S (t) from X = D(A) into itself forming a C -semigroup on X which is generated by the part of A in X, A. Proof: The statements follow from combining the results by Arendt (1987 b, Theorem 4.1) with those by Kellermann, Hieber (1989, Proposition 2.2, Theorem 2.4, and their proofs). It follows from Pazy (1983), Lemma 5.1 in Section 1.5, that an closed operator satisfying Theorem 3.4 (iii) is characterized by the fact that A ωi is m-dissipative after equivalent renormalization of the space X. Therefore we suggest to call such operators para-m-dissipative. The estimates in Theorem 3.4 (iii) (identical with (1.5)) have been called Hille-Yosida estimates in Clément, Diekmann, Gyllenberg, Heijmans, Thieme (1989b) where an extrapolation theory was developed (Section 5). Historically even more wrongly, they were called estimates of the Hille-Yosida theorem, by one of us (Thieme, 199b) when nonlinear perturbations of A were considered. It was pointed out to us by the late Stavros Busenberg that the historically correct name would have been Hille-Yosida-Feller-Miyadera-Phillips estimates (see Pazy 1983, p. 254, Section 1.5). Unfortunately the above name has been adopted and operators satisfying the estimates (1.5) are now sometimes studied under the name Hille-Yosida operators (van Neerven, 1992), Nagel, Sinestrari, preprint). Without this unfortunate name and without the connection to integrated semigroups, this class of operators was investigated earlier by Da Prato, Sinestrari (1987) and Bénilan, Crandall,

19 Pazy (1988) who studied the associated non-homogeneous Cauchy problem. Lumer (1991, Theorem 3.5) considers the situation where a dissipative closed operator does not generate a locally Lipschitz continuous integrated semigroup and gives an interesting characterization in terms of the non-existence of generalized solutions to the associated homogeneous Cauchy problem. It is a natural question under which conditions the semigroup S (t) can be extended to a semigroups on all of X. This is possible if S(t) maps X into D(A). A sufficient condition for the latter is given in Clément, Diekmann et al. (1989), Section 5, hypothesis (H2). Proposition 3.5. Let A be the generator of a locally Lipschitz continuous (once) integrated semigroup S such that S(t) maps X into D(A) for any t. Then (3.6) Ṡ(t)x := x + AS(t)x, x X, defines a non-degenerate semigroup with the following properties: a) For x D(A), S(t)x is continuously differentiable and Ṡ(t)x = S (t)x, t. b) Ṡ(t) maps D(A) into itself and Ṡ(t)Ax = AṠ(t). c) For any sufficiently large λ >, (λ A) 1 Ṡ(t) is locally Lipschitz continuous in t with respect to the operator norm. d) For all t, r, Ṡ(t)S(r) = S(t + r) S(t) = S(r)Ṡ(t). Proof: As S(t) is bounded and A is closed, AS(t) is a closed everywhere defined operator. Hence Ṡ(t) is a bounded operator on X by the closed graph theorem. Using the fact that S is an integrated semigroup, we show that Ṡ is a semigroup. Ṡ(t)Ṡ(r)x = x + AS(r)x + AS(t)x + A2 S(t)S(r)x (+r = x + AS(r)x + AS(t)x + A 2 S(s)xds S(s)xds r ) S(s)xds =x + AS(r)x + AS(t)x + A (S(t + r)x (t + r)x S(t)x + tx S(r)x + rx) = x + AS(t + r)x = Ṡ(r + t)x. Obviously Ṡ() = I. In order to show a) we first realize that Ṡ(t)x = x + S(t)Ax = d dt S(t)x, x D(A), by Proposition 3.2 such that Ṡ(t)x is continuous in t for x D(A). As S(t) is locally Lipschitz continuous (in the operator norm), we have that Ṡ(t)x M x, t r, x D(A),

20 with the constant M depending on r. The same estimate holds for x D(A). The continuity of Ṡ(t)x for x D(A) now follows by approximating x by elements in D(A). Part b) and c) follow from the following consideration: (3.7) (3.8) (λ A) 1 Ṡ(t) = (λ A) 1 (I + AS(t)) = (λ A) 1 S(t) + λ(λ A) 1 S(t) = (λ A) 1 S(t) + S(t)λ(λ A) 1 = Ṡ(t)(λ A) 1. Part b) follows from (3.8) and part c) from (3.7) and the fact that S(t) is locally Lipschitz continuous. Part d): By (3.3) and Theorem 3.1, Ṡ(t)S(r) = S(r) + AS(t)S(r) = S(r) + A (+r r ) S(s)ds = S(r) + S(t + r) (t + r)i S(t) + ti S(r) + ri = S(t + r) S(t). Further S(r)Ṡ(t) = S(r) + S(r)AS(t) = S(r) + AS(r)S(t) because S(r) commutes with A (Proposition 3.2). In order to see that Ṡ is non-degenerate, let x X such that Ṡ(t)x = for all t >. Then Ṡ(t)(λ A) 1 x = by (3.8). As Ṡ(t) is strongly continuous in t on D(A) we have that (λ A) 1 x = which implies x =. Unfortunately one does not seem to obtain any information about continuity properties of Ṡ(t) in any meaningful topology that is independent of A and even local boundedness generally remains in the dark. Lemma 3.6. Let the assumptions of Proposition 3.5 hold. Assume that there exists a constant K > such that x K lim sup λ (λ A) 1 x x X. λ Then the semigroups Ṡ is exponentially bounded. Proof: By (3.8) and Part a) of Proposition 3.5 we have that (λ A) 1 Ṡ(t) = S (t)(λ A) 1.

21 But S (t) is a strongly continuous semigroup on D(A) and thus exponentially bounded. The assertion now follows from the estimates (1.5) Duality Let A : D(A) X be a linear operator in X. Then the dual operator A : X 2 X is defined by (3.9) A (x ) = {y X ; Ax, x = x, y x D(A)} D(A ) = {x X, A (x ) }. Assume that A is closed and has a non-empty resolvent set. Then, for λ in the resolvent set of A, (λ A) 1 plays the role of the resolvent for A. In particular (λ A) 1 maps X into D(A ) and (3.1) x + λ(λ A) 1 x A (λ A) 1 x, x X, x + λ(λ A) 1 x = (λ A) 1 y x D(A ), y A x. We define (3.11) X = D(A ). Finally we assume that A satisfies one (and then all) of the equivalent conditions (i), (ii), (iii) of Theorem 3.4. Then (3.12) X = {x X ; λ(λ A) 1 x x, λ }. By definition, X is invariant under (λ A) 1. It is easy to see that, for x D(A ), A x X contains at most one element. So we can define the part of A in X by (3.13) x D(A ) x D(A ), A x X A x X = {A x }, x D(A ). One easily realizes that (λ A ) 1 = (λ A) 1 with the latter denoting the restriction of (λ A) 1 to X. In particular (1.5) holds with A replacing A. Moreover D(A ) is dense in X and so A generates a C -semigroup on X. Theorem 3.7. Let the closed linear operator A satisfy (1.5). Then the part of A in X = D(A ), A, is the infinitesimal generator of a C -semigroup Ṡ. Further the

22 following holds for the integrated semigroup S on X and the C -semigroup S on X that are associated with A: a) S (t)x = Ṡ (r)x dr x X, t. b) S (t)x, x = x, Ṡ (t)x, x X, x X. c) If S(t) maps X into D(A) for any t and Ṡ is the semigroup (3.6), then Ṡ (t)x = Ṡ (t)x, x X. S (t) and Ṡ (t) denote the dual operators associated with S(t) and Ṡ(t) respectively. Proof: Part a) can easily been seen by taking Laplace transforms. Part b) then follows by differentiation. For part c), let x D(A ).Then, by (3.6), Ṡ(t)x, x = x, x + S(t)x, A x = As D(A ) is dense in X, this holds for all x X. x, x + Ṡ (r)a x dr = x, Ṡ (t)x Semigroups on dual spaces Let X = Z be the dual of a Banach space Z and let Ṡ (t), t, be a C -semigroup on Z with generator A. Then we can consider the dual semigroup Ṡ(t) = Ṡ (t) on X which is formed by the dual operators associated with Ṡ (t). In the following a dual semigroup is meant to be the dual of a C -semigroup. Ṡ is a weakly continuous semigroup in the sense that z, Ṡ(t)x is a continuous function of t for any z Z, x X. The dual operator A = A of A has the following remarkable properties (Butzer, Behrens, 1967): Proposition 3.8. Let A = A. Then the following holds: a) For any x, y X we have that x D(A) and y = Ax if and only if Ṡ(t)x x = Ṡ(r)ydr t.

23 b) For any x X, t we have that Ṡ(r)xdr D(A) and ( ) A Ṡ(r)xds = Ṡ(t)x x. The integrals in Proposition 3.8 have to be interpreted in the weak sense. Part a) implies that A is the generator of the locally Lipschitz continuous integrated semigroup (3.14) S(t) = Ṡ(r)dr which consists of the dual operators associated with the integrated semigroup Part b) recovers (3.6). S (t) = Ṡ (r)dr. It is worth noting that the integration of general weakly continuous (or even locally bounded weakly measurable) semigroups like in (3.9) does not necessarily produce integrated semigroups. See Clément, Diekmann et al. (1989), Section 1, where we have introduced the term integral w semigroup for a weakly continuous or an exponentially bounded weakly measurable semigroup Ṡ with (3.14) providing an integrated semigroup. Being an integral w semigroup is actually equivalent to Part b) of Proposition 3.8, provided one has defined A using Part a) as a definition. This explains our formulations in Theorem 1.5. More on dual and integral w semigroups can also been found in Clément, Heijmans et al. (1987) and van Neerven (1992). We conclude this section by the following results elaborating Proposition 3.5 for dual Banach spaces. Proposition 3.9. Let X = Z be the dual of a Banach space Z and A be the generator of a locally Lipschitz continuous (once) integrated semigroup S on X such that S(t) maps X into D(A) for any t. Let Ṡ be the non-degenerate semigroup defined by (3.6). Then the following holds: a) The condition z, λ(λ A) 1 x z, x for λ, z Z, x X, is sufficient for Ṡ to be exponentially bounded and weakly measurable, and it is necessary for Ṡ(t) to be weakly continuous at t =. Further this condition implies that Ṡ(r)xds = S(t)x

24 and e λt Ṡ(t)xdt = (λ A) 1. b) For any z Z, λ(λ A) 1 z z for λ if and only if Ṡ (t)z is continuous in t. c) Ṡ is a dual semigroups if and only if λ(λ A) 1 z z for λ, z Z, and (λ A) 1 maps Z into Z. Proof: a) Let x X. Choose some z Z, z = 1 such that z, x (1/2) x. Then x 2 lim z, λ(λ λ A) 1 x 2 lim sup λ (λ A) 1 x. λ Hence Ṡ is exponentially bounded by Lemma 3.6. The weak measurability and the other statements follow from Proposition 3.5 and Theorem 3.3: Let z X, x X = Z. Then z, Ṡ(r)x dr = lim z, λ(λ λ A) 1 Ṡ(r)x dr = lim λ z, S (r)λ(λ A) 1 x dr = lim λ z, S(t)λ(λ A) 1 x = z, S(t)x. b) Let us first assume that T (t)z is continuous in t. By part a), λ(λ A) 1 z = λ e λt Ṡ (t)zdt z, λ. Let us now assume that λ(λ A) 1 z z for λ. This implies that z X and Ṡ (t)z = Ṡ (t)z is continuous in t by Theorem 3.7 c). c) The only if part is obvious. To see the if part, Let J λ be the restriction of (λ A) 1 to X. Then the J λ form a family of pseudoresolvents on X for sufficiently large λ >, i.e., the J λ satisfy the resolvent identity. Further λj λ z z for λ. By Corollary 9.5 in Pazy (1983), there exists a closed densely defined operator A such that J λ = (λ A ) 1. As A generates a locally Lipschitz continuous integrated semigroup, the estimates (1.5) are satisfied. Hence the estimates (1.5) are satisfied for A replacing A. Thus A is the infinitesimal generator of a C -semigroup on Z. Apparently (λ A ) 1 = (λ A ) 1 = (λ A) 1, hence A = A and Ṡ is a dual semigroup by the uniqueness properties of the Laplace transform. 4. Perturbations and outputs

25 Let A be a linear closed operator in the Banach space X such that (ω, ) is contained in the resolvent set of A for some ω R. If B : D(A) Y is a linear operator into a Banach space Y, then the family F (λ) = B(λ A) 1 satisfies (4.1) F (λ)(µ A) 1 = 1 (F (λ) F (µ)), λ, µ > ω, λ µ. µ λ 4.1. Resolvent outputs As the operator B does not show up in (4.1) explicitly, we can generalize the above situation by calling a family F (λ), λ > ω, of bounded linear operators from X into a Banach space Y a resolvent output family for A if and only if (4.1) is satisfied. In the following we assume that F (λ) is a resolvent output family of A. Lemma 4.1. norm) and F (λ) is infinitely often differentiable in λ (with respect to the operator F (n) (λ) = ( 1) n n!f (λ)(λ A) n. Proof: For n = 1, the statement follows from (4.1) by taking the limit µ λ. For n > 1 it follows by induction. Lemma 4.2. For x D(A), F (λ)(λ A)x does not depend on λ > ω. Proof: Differentiation with respect to λ yields by Lemma 4.1. Indeed, for x D(A),: d ( ) F (λ)(λ A)x = F (λ)(λ A)x + F (λ)x = F (λ)(λ A) 1 (λ A)x + F (λ)x =. dλ By Lemma 4.2 we can define the following operator B : D(A) X: (4.2) Bx = F (λ)(λ A)x, x D(A). Lemma 4.3. F (λ) = B(λ A) 1. Lemma 4.4. Let lim sup λ F (λ) <. Then the following holds: a) F (λ)x for λ, x D(A). b) Bx = lim λ λf (λ)x, x D(A ), where A is the part of A in D(A).

26 Proof: By Lemma 4.2, for x D(A), λf (λ)x is bounded. Hence F (λ)x, λ. This also holds for x in the closure of D(A). If x D(A ), then Ax D(A), hence F (λ)ax for λ. Lemma 4.5. Assume that F (λ)x, λ, x X. Then Bx = lim λf (λ)x λ x D(A). Proof: By (4.1) we have λf (λ)(µ A) 1 x F (µ), λ. Hence, for x D(A), λf (λ)x = λf (λ)(µ A) 1 (µ A)x F (µ)(µ A)x = Bx, λ. For the following result we recall that X = D(A ). See Section 3.3. Lemma 4.6. If x X and F (λ)x for λ, then F (λ)x X for λ > ω. Proof: By (4.1) we have that µ(µ A) 1 F (λ)x = µ µ λ (F (λ)x F (µ)x ) F (λ)x, µ. Hence F (λ)x X by (3.12) Positive resolvent outputs We now consider the resolvent output families in ordered Banach spaces with regular cones. Definition 4.7. A resolvent output family F (λ), λ > ω, of an operator A is called a positive resolvent output family if both (λ A) 1 and F (λ) are positive operators for λ > ω. Lemma 4.1 has the following immediate consequence: Proposition 4.8. A positive resolvent output family is completely monotonic.

27 In particular a positive output family F (λ) is decreasing in λ, and, if X + is regular, the operator F ( ) = lim λ F (λ) is a positive (hence bounded) linear operator on X. Proposition 4.9. Let the cone of X be regular. If F is a positive resolvent output family, then λf (λ)x converges for λ, x D(A), and, for the operator B defined in (4.2), Bx = lim λf (λ)x F ( )Ax. λ If F (λ)x, λ, x X, then B is a positive operator. Proof: After the remarks preceeding this proposition, this is immediate from (4.2). This proposition suggests that it is possible for F (λ) to be a positive resolvent output family without B being a positive operator if F ( ). See Example Resolvent outputs and cumulative outputs Let A satisfy (1.5), i.e., be the generator of a locally Lipschitz continuous integrated semigroup S. See Theorem 3.5. Definition 4.1. A family V (t), t, of bounded linear operators from X to some Banach space Y is called a cumulative output family for S if the following holds: V (t) is strongly right continuous at any t >, V () =, and the right limit V (+) exists strongly, (4.3) V (t)s(r) = +r V (s)ds V (s)ds r V (s)ds, t, r. Note that (4.3) implies V (t)s(r) = V (r)s(t). If x X = D(A), then S(t)x can be strongly differentiated in t and we obtain Lemma Let V be a cumulative output family for S. Then V (t)s (r)x = V (t + r)x V (r)x, x X, t, r. The formula in Lemma 4.11 explains the name cumulative output. Let S (r) describe the state of a system at time r. If we let the system develop till time r and then take the

28 cumulative output of that system over a time period of length t, then this amounts to the difference of the cumulative outputs up to times t + r and time r. Proposition Let V (t), t, be a family of bounded linear operators that is strongly right continuous at any t >, exponentially bounded, and satisfies V () =. Then V is a cumulative output family for S if and only if the family F (λ) = λ ˆV (λ) is a resolvent output family for the generator A of S. Proof: Assume that V is a cumulative output family for S. Take Laplace transforms in (4.3) such that the left hand side becomes λ e λt V (t)dt µ e µr S(r)dr. This yields (4.1). Now assume that F (λ) is a resolvent output for A. Reversing the previous step we see that the Laplace transforms over t, r are equal for both sides of (4.3). The uniqueness properties of the Laplace transform and the requirement that V (t) is strongly right continuous at t > yields (4.3) for r, t >. As V () =, (4.3) holds automatically for t =. Proposition Let V be a cumulative output family for S that is locally bounded. Then V (t)x + V (+)x = lim λ V (t)λ(λ A) 1 x, t >, x X.

29 Proof: Hence V (t)λ(λ A) 1 x V (t)x + V (+)x V (t)λ2 e λr S(r)xdr V (t)x + V (+)x r 1 V (t) λ 2 e λr Me ωs dsdr + λ2 e λr V (t)s(r)x V (t)x + V (+)x 1 V (t) λ 2 e λr M 1 ω eωr dr 1 ( r ) + λ2 e λr (V (t + s) V (s))xds dr V (t)x + V (+)x Mλ 2 V (t) (λ ω)ω e(ω λ) 1 ( 1 ) + λ λe λr dr (V (t + s) V (s))xds V (t)x + V (+)x s Mλ 2 V (t) (λ ω)ω e(ω λ) 1 + λ ( e λs e λ) (V (t + s) V (s))xds V (t)x + V (+)x Mλ 2 V (t) (λ ω)ω e(ω λ) 1 + λ e λs (V (t + s) V (t) V (s) + V (+))xds 1 + λe λ V (t + s) V (s) x + V (t)x V (+)x λe λs ds. 1 lim sup V (t)λ(λ A) 1 x V (t)x + V (+)x λ lim sup λ λ e s (V (t + s/λ) V (t) V (s/λ))x + V (+)x ds. Lebesgue s theorem of dominated convergence now implies the assertion because V (t)x is right continuous at every t >. Lemma If V (t) is a locally bounded cumulative output family for S, then it is exponentially bounded. Proof: By Lemma 4.11, V (n + 1) = V (1) + V (n)s (1) on X.

30 By induction V (n + 1) ( S (1) + 1) n V (1). Let t = n + s with s < 1, x X. Then V (t)x V (s)x + V (n) S (s)x. S is a strongly continuous semigroup on X, hence S (s)x c x, V (s) c, s 1, x X. Thus This implies V (t)x + V (+)x (c + ( Ṡ(1) + 1)(n 1) c) x. V (t)x Ne αt x, t, x X. By Proposition 4.13, for x X, V (t)x = lim V (t)λ(λ λ A) 1 x lim sup λ Ne αt Mλ λ ω x NMeαt. Proposition 4.12 and Lemma 4.14 combine to the following Theorem Let V (t), t, be a family of bounded linear operators that is locally bounded and strongly right continuous at any t >, V () =. Then V is a cumulative output family for S if and only if V is exponentially bounded and the family F (λ) = λ ˆV (λ) is a resolvent output family for A. Proposition Assume that S(t) maps X into D(A) and let Ṡ be the semigroup defined by Ṡ(t)x = x + AS(t)x. Let V be a locally bounded cumulative output family for S. Then (4.4) V (t)ṡ(r) = V (t + r) V (r). Proof: Formula (4.4) automatically holds whenever t = or r =. So we assume that t, r >. By Lemma 4.11 we have that V (t)s (r)(λ A) 1 = (V (t + r) V (r))(λ A) 1.

31 Hence, by Proposition 4.13 and (3.8), V (t)ṡ(r)x = lim λ V (t)λ(λ A) 1 Ṡ(r)x = lim λ V (t)λs (r)(λ A) 1 x = lim λ (V (t + r) V (r))λ(λ A) 1 x =(V (t + r) V (r))x Positive cumulative output families In this subsection we again assume that X is an ordered Banach space with regular and generating cone X +. Combining Proposition 4.8, Proposition 2.8, and Proposition 4.12, we obtain Proposition Let F (λ) be a positive resolvent output for A. Then there exists a uniquely determined non-decreasing family V (t), t, of positive linear operators that is a cumulative output family for S and F (λ) = ˇV (λ), λ > ω. Moreover, for any x X, V (t)x is Borel measurable and exponentially bounded in t and F (λ) = λ ˆV (λ), λ > ω. If X is an abstract L space, V (t) has the properties in Proposition 2.7. Finally V (t) is strongly continuous at t= if and only if F (λ)x for λ, x X. Proposition Let X be an ordered Banach space with generating and regular cone X +. Assume that A is resolvent positive and S(t) maps X into D(A). Let V be a positive cumulative output family for S that is strongly right continuous in t. Then V is exponentially bounded and λ ˆV (λ) is a resolvent output for A. If B is the linear operator associated with F (λ) = λ ˆV (λ) via B = F (λ)(λ A), then V (t) = BS(t). Proof: By Lemma 4.11, V (t) is increasing on X. As A is resolvent positive, V (t) is increasing on X by Proposition As X + is generating and normal, V is locally bounded and hence exponentially bounded by Lemma As V (t) is strongly right continuous and V () =, we have that F (λ)x = λ ˆV (λ)x, λ, x X.