Construction of dynamical semigroups by a functional regularisation `a la Kato 7 A.F.M. ter Elst and Valentin A. Zagrebnov 1 0 2 n In memory of Tosio Kato on the 100th anniversary of his birthday a J 2 1 Abstract. A functional version of the Kato one-parametric regularisa- ] A tion for the construction of a dynamical semigroup generator of a relative F bound one perturbation is introduced. It does not require that the minus . h generator of the unperturbedsemigroup is a positivity preserving operator. t a The regularisation is illustrated by an example of a boson-number cut-off m regularisation. [ 1 v 6 0 Contents 5 3 0 1 Introduction 1 . 1 0 2 The regularisation theorem 4 7 1 : 3 Example 14 v i 3.1 Open boson system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 X r 3.2 A particle-number cut-off regularisation . . . . . . . . . . . . . . . . . . . 17 a 3.3 Core property and trace-preserving . . . . . . . . . . . . . . . . . . . . . . 20 1 Introduction The majority of papers concerning the construction of dynamical semigroups use the Kato regularisation method [Kat54], which goes back to 1954. This method allows to treat the caseofpositive unbounded perturbationswithrelative boundoneandtoconstruct minimal Markov dynamical semigroups [Dav76], [Dav77]. Mathematics Subject Classification. 47D05, 47A55, 81Q15, 47B65. Keywords. Dynamical semigroup, perturbation, Kato regularisation,positivity preserving. Later a similar tool of the one-parametric regularisation allowed Chernoff [Che72] to prove the following abstract result in a Banach space X for a perturbation of a contraction C -semigroup with generator A and domain D(A). Let B be a dissipative operator with 0 domain D(B) D(A) and relative bound one, that is, there exists a a 0 such that ⊃ ≥ Bx X Ax X +a x X, k k ≤ k k k k for all x D(A). If the domain D(B∗) of the adjoint operator B∗ is dense in the dual space ∈ X′, then the closure of the operator A+B is the generator of a C -semigroup. Note that 0 the hypothesis on B∗ is superfluous if the Banach space X is reflexive. See also Okazawa [Oka71] Theorem 2 for the reflexive case. The aim of the present paper is to put this tool into an abstract setting that covers the Kato regularisation method as a particular case. Our main result is a functional version of the Kato regularisation for the construction of generators when perturbations are with relative bound equal to one. Toproduceanapplicationofthisresult, weconstruct thegeneratorofaMarkovdynam- ical semigroup for an open quantum system of bosons [TZ16a] [TZ16b]. For this system the abstract Kato regularisation corresponds to the particle-number cut-off in the Fock space. Let be a Hilbert space over C. Consider the Banach space of bounded operators H ( ) and the subspace C = C ( ) of all trace-class operators. Let u,v ( ). We say 1 1 L H H ∈ L H that an operator u is positive, in notation u 0, if (ux,x) 0 for all x . We write H ≥ ≥ ∈ H u v if v u 0. Let C+ = u C : u 0 . Then C+ is a closed cone with trace-norm ≤ − ≥ 1 { ∈ 1 ≥ } 1 kukC1 = Tru for all u ∈ C+1. Let Csa be the Banach space over R of all self-adjoint operators of C . An operator 1 1 A: D(A) Csa with domain D(A) Csa is called positivity preserving if Au 0 for all → 1 ⊂ 1 ≥ u D(A)+, where D(A)+ := D(A) C+. A semigroup (S ) on Csa is called positivity ∈ ∩ 1 t t>0 1 preserving if the map S is positivity preserving for all t > 0. t Let D Csa be a subspace and let A,B: D Csa be two maps. Then we write A 0 ⊂ 1 → 1 ≥ if A is positivity preserving and we write A B if B A 0. Obviously is a partial ≤ − ≥ ≤ ordering on (Csa). L 1 Let H be the generator of a positivity preserving contraction C -semigroup (e−tH) 0 t>0 − on Csa. Let K: D(H) Csa be a positivity preserving operator and suppose that 1 → 1 Tr(Ku) Tr(Hu) ≤ for all u D(H)+. We shall prove in Lemma 2.2 that this implies that the operator K is ∈ H-bounded, but with relative bound equal to one. Hence it is an open problem whether operator (H K) with D(H K) = D(H), or a closed extension of this operator, is − − − again the generator of a C -semigroup. 0 2 Kato [Kat54] solved this problem for Kolmogorov’s evolution equations when the op- erator H is a positivity preserving map. To this end he proposed a regularisation of the perturbation K by replacing it by the one-parametric family (rK) and by taking r∈[0,1) finally the limit r 1. ↑ The aim of the present paper is twofold. First, we wish to consider a more general (functional) regularisation `a la Kato. Secondly, we aim to remove the condition that the operator H is positivity preserving and merely assume the condition that H is the gen- − erator of a positivity preserving semigroup. It is the positivity preserving of the quantum dynamical semigroup which is indispensable in applications. We require that the pertur- bation K of H admits the following type of regularisation. Definition 1.1. Let (K ) be a net such that K : D(H) Csa for all α J. We call α α∈J α → 1 ∈ the family (K ) a functional regularisation of the operator K if the following four α α∈J conditions are valid. (I) K is positivity preserving for all α J. α ∈ (II) For all α J there exist a [0, ) and b [0,1) such that α α ∈ ∈ ∞ ∈ Tr(K u) a Tru+b Tr(Hu) α α α ≤ for all u D(H)+. ∈ (III) K K K for all α,β J with α β. α β ≤ ≤ ∈ ≤ (IV) Forallu D(H)+ thereexistsadensesubspaceV of suchthatlim ((K u)x,x) = α α H ∈ H ((Ku)x,x) for all x V. H ∈ As an example one can take J = [0,1) and K = rK for all r J, i.e. a = 0 and r α ∈ b = r. This was used in [Kat54] under the additional assumption that H is positivity α preserving. The main theorem of this paper is the following. Theorem 1.2. Let H be the generatorof a positivity preservingcontraction C -semigroup 0 − on Csa. Let K: D(H) Csa be a positivity preserving operator and suppose that 1 → 1 Tr(Ku) Tr(Hu) (1.1) ≤ for all u D(H)+. Let (K ) be a functional regularisation of K. Set L = H K for α α∈J α α ∈ − all α J. Then one has the following. ∈ (a) For all α J the operator L is the generator of a positivity preserving contraction α ∈ − C -semigroup (Tα) on Csa. 0 t t>0 1 (b) If t > 0, then lim Tαu exists in Csa for all u Csa. α t 1 ∈ 1 3 For all t > 0 define T : Csa Csa by T u = lim Tαu. t 1 → 1 t α t (c) The family (T ) is a positivity preserving contraction C -semigroup on Csa for t t>0 0 1 which the generator is an extension of the operator (H K) with domain D(H). − − As a corollary we obtain the regularisation theorem of Kato invented in [Kat54] and which was extended to dynamical semigroups with unbounded generators by Davies in [Dav77]. Forcompleteness we recall that the concept of thedynamical semigroups was motivated by mathematical studies of the states dynamics of quantum open systems, see [Dav76]. In a certain approximation it can be described on an abstract (Banach) space of states by a C -semigroup of positive preserving maps. These semigroups are often called quantum 0 semigroups if in addition the Kossakowski–Lindblad–Davies Ansatz (see [AJP06]) is satis- fied. In this paper a dynamical semigroup is defined to be a positivity preserving contrac- tion C -semigroup on the Banach space Csa. The abstract space-states which we consider 0 1 in this paper consist of self-adjoint trace-class operators over a complex Hilbert space . H In Section 3 this Hilbert space is the boson Fock space F. A semigroup (T ) on Csa is t t>0 1 called trace preserving if Tr(T u) = Tru for all u Csa and t > 0. Then a Markov t ∈ 1 dynamical semigroup is a dynamical semigroup which is trace preserving. We prove Theorem 1.2 in Section 2. It turns out that the semigroup (T ) constructed t t>0 in Theorem 1.2 is minimal in the sense of Kato [Kat54]. We conclude Section 2 with sufficient conditions for (T ) being a Markov dynamical semigroup. t t>0 In Section 3 we present an example where the functional regularisation of the operator K is a particle-number cut-off in the Fock space F. We show that the semigroup which is constructed by this regularisation method is a Markov dynamical semigroup and that it is minimal. Moreover, the operator H is not positivity preserving. 2 The regularisation theorem We start with a lemma concerning bounded positivity preserving operators on Csa. 1 Lemma 2.1. (a) Let u Csa. Then there are unique v,w C+ such that u = v w and u = v +w, ∈ 1 ∈ 1 − | | where u is the absolute value of u. | | (b) Let A ∈ L(Cs1a) be positivity preserving. Then kAukC1 ≤ kA|u|kC1 for all u ∈ Cs1a. (c) Let A,B (Csa) be positivity preserving. Moreover, suppose that TrAu TrBu ∈ L 1 ≤ for all u C+. Then A B . ∈ 1 k k ≤ k k (d) Let A (Csa) be positivity preserving and let M 0. Suppose that Tr(Au) ∈ L 1 ≥ ≤ M Tru for all u C+. Then A M. ∈ 1 k k ≤ 4 (e) Let A,B (Csa) be positivity preserving and suppose that A B. Then An Bn ∈ L 1 ≤ ≤ for all n N. ∈ (f) Let (u ) be a net in C+. Suppose that u u for all α,β J with α β. α α∈J 1 α ≤ β ∈ ≤ Moreover, suppose that sup Tru : α J < . Then the net (u ) is convergent α α α∈J { ∈ } ∞ in C . 1 Proof. Statement (a) follows from the spectral representation of the self-adjoint operator u Csa. ∈ 1 (b). Let u Csa. Let v,w C+ be as in Statement (a). Then Av,Aw C+. So ∈ 1 ∈ 1 ∈ 1 Au C = Av Aw C Av C + Aw C = TrAv +TrAw = TrA u = A u C . k k 1 k − k 1 ≤ k k 1 k k 1 | | k | |k 1 (c) Let u Csa. Note that u C+ and Tr(B A) u 0 by assumption. Therefore ∈ 1 | | ∈ 1 − | | ≥ (b) gives Au C TrA u +Tr(B A) u = TrB u = B u C B u C = B u C k k 1 ≤ | | − | | | | k | |k 1 ≤ k kk| |k 1 k kk k 1 and the statement follows. (d). Choose B = M I and use Statement (c). (e). The proof is by induction. Let n N and suppose that An Bn. Then ∈ ≤ Bn+1u AnBu = An(B A)u+An+1u An+1u ≥ − ≥ for all u C+, since An is positivity preserving and (B A)u 0. ∈ 1 − ≥ (f). Let M = sup Tru : α J < . Let x . Then α (u x,x) is increasing α α H { ∈ } ∞ ∈ H 7→ and bounded above by M x 2 . So lim (u x,x) exists. By the polarisation identity k kH α α H lim (u x,y) exists for all x,y . Define the operator u: such that α α H ∈ H H → H (ux,y) = lim(u x,y) H α H α for all x,y . It is easy to see that u is symmetric and is an element of ( ). Clearly ∈ H L H (ux,x) = lim (u x,x) 0 for all x . So u 0. H α α H ≥ ∈ H ≥ Obviously 0 Tru Tru M for all α,β J with α β. So lim Tru M. α β α α ≤ ≤ ≤ ∈ ≤ ≤ Let N N and let e : n 1,...,N be an orthonormal set in . Then n ∈ { ∈ { }} H N N N (ue ,e ) = lim(u e ,e ) = lim (u e ,e ) limTru M. n n H α n n H α n n H α X X α α X ≤ α ≤ n=1 n=1 n=1 So u C+ and Tru lim Tru . Clearly Tru Tru for all α J and hence Tru = ∈ 1 ≤ α α α ≤ ∈ limαTruα. Sinceu−uα ≥ 0forallα ∈ J,itfollowsthatlimαku−uαkC1 = limαTr(u−uα) = 0. Therefore lim u = u in C . α α 1 A trace inequality together with positivity preserving gives H-boundedness of a per- turbation. 5 Lemma 2.2. Let H be the generator of a positivity preserving contraction C -semigroup 0 − on Csa. Let K: D(H) Csa be a positivity preserving operator. Suppose that Tr(Ku) 1 → 1 ≤ Tr(Hu) for all u D(H)+. Then K(λI +H)−1 is bounded and K(λI +H)−1 1 for ∈ k k ≤ all λ > 0. Moreover, Ku C Hu C for all u D(H) and in particular the operator k k 1 ≤ k k 1 ∈ K is H-bounded with relative bound one. Proof. Let λ > 0. Then the resolvent ∞ (λI +H)−1 = e−λtS dt Z t 0 is a positivity preserving bounded operator on Csa. Therefore by composition the operator 1 K(λI +H)−1: Csa Csa is positivity preserving, hence bounded by [Dav76] Lemma 2.1. 1 → 1 Moreover, TrK(λI +H)−1u TrH(λI +H)−1u = Tru λTr(λI +H)−1u Tru ≤ − ≤ for all u ∈ C+1. So kK(λI + H)−1k ≤ 1 by Lemma 2.1(d). Therefore kKukC1 ≤ k(λI + H)u C λ u C + Hu C for all u D(H) and the lemma follows. k 1 ≤ k k 1 k k 1 ∈ Inequalities between positivity preserving contraction C -semigroups are equivalent to 0 inequalities between the resolvents. Lemma 2.3. Let (S ) and (T ) be two positivity preserving bounded C -semigroups t t>0 t t>0 0 with generators H and L respectively. Then the following are equivalent. − − (i) S T for all t > 0. t t ≤ (ii) (λI +H)−1 (λI +L)−1 for all λ > 0. ≤ If, in addition, D(H) D(L), then (i) is also equivalent to ⊂ (iii) The operator H L is positivity preserving. − Proof. ‘(i) (ii)’. This follows from a Laplace transform. ⇒ ‘(ii) (i)’. It follows from Lemma 2.1(e) that (λI +H)−n (λI +L)−n for all n N. ⇒ ≤ ∈ Let t > 0. Then the Euler formula yields S u = lim(I + t H)−nu lim(I + t L)−nu = T u t n→∞ n ≤ n→∞ n t for all u C+. So S T . ∈ 1 t ≤ t ‘(i) (iii)’. Write K = H L. Let u D(H)+ and x . Then ⇒ − ∈ ∈ H (((I S )u)x,x) (((I T )u)x,x) ((Ku)x,x) = lim − t H − t H H t↓0 t − t (((T S )u)x,x) = lim t − t H 0. t↓0 t ≥ 6 So Ku 0 and K is positivity preserving. ≥ ‘(iii) (ii)’. Let λ > 0. Since the product of positivity preserving maps is positivity ⇒ preserving, we obtain that (λI +L)−1 (λI +H)−1 = (λI +L)−1(H L)(λI +H)−1 0. − − ≥ So (λI +L)−1 (λI +H)−1. ≥ Our first result is a perturbation theorem where the relative bound is less than one. We emphasise that we do not assume that the operator H is positivity preserving. Proposition 2.4. Let H be the generator of a positivity preserving contraction C - 0 − semigroup on Csa. Let K: D(H) Csa be a positivity preserving operator. Suppose there 1 → 1 exist a [0, ) and b [0,1) such that Tr(Ku) aTru+bTr(Hu) for all u D(H)+. ∈ ∞ ∈ ≤ ∈ Define L = H K. Then one has the following. − (a) The operator L is quasi-m-accretive. Moreover, the semigroup generated by L is a − positivity preserving semigroup. (b) If in addition Tr(Ku) Tr(Hu) for all u D(H)+, then L is m-accretive. So L ≤ ∈ − is the generator of a contraction semigroup. (c) If again Tr(Ku) Tr(Hu) for all u D(H)+, then ≤ ∈ N n lim (λI +H)−1 K(λI +H)−1 u = (λI +L)−1u N→∞X (cid:16) (cid:17) n=0 for all u Csa and λ > 0. ∈ 1 Proof. First suppose in addition that Tr(Ku) Tr(Hu) (2.1) ≤ for all u D(H)+. ∈ Let (S ) be the semigroup generated by H. Let λ > 0. Then H(λI + H)−1 = t t>0 − I λ(λI +H)−1 I since (λI +H)−1 is positivity preserving. Hence − ≤ Tr(K(λI +H)−1u) aTr((λI +H)−1u)+bTr(H(λI +H)−1u) ≤ a a (λI +H)−1u C +bTr(H(λI +H)−1u) +b Tru ≤ k k 1 ≤ (cid:16)λ (cid:17) for all u C+, where we used that (λI + H)−1u D(H)+. Moreover, K(λI + H)−1 is ∈ 1 ∈ positivity preserving as a composition of two positivity preserving maps. Therefore a K(λI +H)−1 +b k k ≤ λ 7 by Lemma 2.1(d). Let λ R and suppose that λ > a . Then λI +L = (I K(λI +H)−1)(λI +H) ∈ 1 b − is invertible and − ∞ n (λI +L)−1 = (λI +H)−1 K(λI +H)−1 . (2.2) X (cid:16) (cid:17) n=0 n If n N , then (λI + H)−1 K(λI + H)−1 (Csa) is positivity preserving. Hence ∈ 0 (cid:16) (cid:17) ∈ L 1 (λI +L)−1 is positivity preserving. Moreover, if u C+ then (2.2) yields (λI +L)−1u ∈ 1 ∈ D(H)+. Now by the addition assumption (2.1) one obtains Tru = Tr(λI +L)(λI +L)−1u = λTr(λI +L)−1u+Tr(H K)(λI +L)−1u − λTr(λI +L)−1u. ≥ Therefore Tr((λI+L)−1u) λ−1Tru. Since (λI+L)−1 is positivity preserving, it follows ≤ from Lemma 2.1(d) that (λI + L)−1 λ−1 for all λ > a . Hence the operator L is k k ≤ 1−b m-accretive and L is the generator of a contraction C -semigroup. 0 − Let (T ) be the semigroup generated by L. If t > 0, then the operator (I + t L)−1 t t>0 − n is positivity preserving for all large n N. Hence by the Euler formula one obtains ∈ that T u = lim (I + t L)−nu C+ for all u C+. Therefore the semigroup (T ) t n→∞ n ∈ 1 ∈ 1 t t>0 is positivity preserving. This proves Statements (a) and (b) of the the proposition if in addition (2.1) is valid. Note that in particular we have proved Statement (b). We next prove Statement (a) without the additional assumption (2.1). We may assume that b > 0. Choose ω = a. Then b Tr(Ku) aTru+bTr(Hu) = bTr (ωI +H)u Tr (ωI +H)u ≤ (cid:16) (cid:17) ≤ (cid:16) (cid:17) for all u D(H)+. So by the above the operator (ωI + H) K is m-accretive and is ∈ − the minus generator of a positivity preserving semigroup. Therefore L is quasi-m-accretive and it is the minus generator of a positivity preserving semigroup. Finally we prove Statement (c). The proof is inspired by the proof of Lemma 7 in [Kat54]. Fix λ > 0. Let N N and r (0,1). Then rK(λI+H)−1 r by Lemma 2.2. ∈ ∈ k k ≤ So the Neumann series gives N ∞ n n (λI +H)−1 rK(λI +H)−1 (λI +H)−1 rK(λI +H)−1 X (cid:16) (cid:17) ≤ X (cid:16) (cid:17) n=0 n=0 = (λI +H rK)−1 (λI +H K)−1, − ≤ − where we use Lemma 2.3 in the last step. Let u C+. Taking the limit r 1 gives ∈ 1 ↑ N n (λI +H)−1 K(λI +H)−1 u (λI +H K)−1u = (λI +L)−1u. X (cid:16) (cid:17) ≤ − n=0 8 In particular, N n Tr (λI +H)−1 K(λI +H)−1 u Tr (λI +L)−1u . (cid:16)X (cid:16) (cid:17) (cid:17) ≤ (cid:16) (cid:17) n=0 n Then Lemma 2.1(f) gives that v = lim N (λI +H)−1 K(λI +H)−1 u exists in N→∞ n=0 (cid:16) (cid:17) C . Then v (λI +L)−1u. Conversely, if NP N and r (0,1), then 1 ≤ ∈ ∈ N N n n (λI +H)−1 rK(λI +H)−1 u (λI +H)−1 K(λI +H)−1 u v. X (cid:16) (cid:17) ≤ X (cid:16) (cid:17) ≤ n=0 n=0 So ∞ n (λI +H rK)−1u = (λI +H)−1 rK(λI +H)−1 u v (2.3) − X (cid:16) (cid:17) ≤ n=0 If µ > a , then it follows from (2.2) that lim (µI+H rK)−1 = (µI+H K)−1 in the 1−b r↑1 − − strong operator topology. Since (H rK) is the generator of a contraction semigroup − − for all r (0,1], it follows from [Dav80] Theorem 3.17 that lim (µI + H rK)−1 = r↑1 ∈ − (µI + H K)−1 in the strong operator topology for all µ > 0. Then taking the limit − r 1 in (2.3) gives (λI + H K)−1u v. So v = (λI + H K)−1u and the proof is ↑ − ≤ − complete. We are now able to prove Theorem 1.2 regarding the functional regularisation of the perturbation of H and we shall prove that the perturbed semigroup is a dynamical semi- group. Theorem 2.5. Let H be the generatorof a positivity preservingcontraction C -semigroup 0 − on Csa. Let K: D(H) Csa be a positivity preserving operator and suppose that 1 → 1 Tr(Ku) Tr(Hu) ≤ for all u D(H)+. Let (K ) be a functional regularisation of K. Set L = H K for α α∈J α α ∈ − all α J. Then one has the following. ∈ (a) If α J, then the operator L is m-accretive and the semigroup (Tα) generated ∈ α t t>0 by L is a positivity preserving contraction semigroup. α − (b) If α,β J and α β, then Tα Tβ for all t > 0. ∈ ≤ t ≤ t (c) If t > 0, then lim Tαu exists in Csa for all u Csa. α t 1 ∈ 1 For all t > 0 define T : Csa Csa by T u = lim Tαu. t 1 → 1 t α t (d) If t > 0, then the map T is positivity preserving. t (e) (T ) is a contraction C -semigroup on Csa. t t>0 0 1 9 Now let L be the generator of the C -semigroup (T ) . 0 t t>0 − (f) Let λ > 0. Then lim (λI +L )−1u = (λI +L)−1u in C for all u Csa. α α 1 ∈ 1 (g) The operator L is an extension of operator H K. − Proof. (a). Condition (1.1) and Definition 1.1(III) imply that Tr(K u) Tr(Hu) α ≤ for all u D(H)+. Using Definition 1.1(I) and (II), we may apply Proposition 2.4 to H ∈ and K in order to obtain the statement. α (b). Let α,β J with α β. Then L L = K K 0. Moreover, D(L ) = α β β α α ∈ ≤ − − ≥ D(L ). Now the statement follows from Lemma 2.3(iii) (i). β ⇒ (c). Fix t > 0. Let u C+. Then (b) yields 0 Tαu Tβu for all α,β J with ∈ 1 ≤ t ≤ t ∈ α ≤ β. Moreover, Tr(Ttαu) = kTtαukC1 ≤ kukC1 for all α ∈ J, since Ttα is a contraction by Statement (a). So lim Tαu exists in C+ by Lemma 2.1(f). Then the statement for all α t 1 u Csa follows from Lemma 2.1(a). ∈ 1 (d). Since the semigroup (Tα) is positivity preserving for all α J by Proposi- t t>0 ∈ tion 2.4, the assertion follows from (c) and from the limit T u = lim Tαu for all u C+. t α t ∈ 1 (e). Let t > 0. Then TrTtu = limαTr(Ttαu) = limαkTtαukC1 ≤ kukC1 = Tru for all u C+. Since T is positivity preserving by Statement (d), it follows from Lemma 2.1(d) ∈ 1 t that T is a contraction. Next, taking the limit (c) one verifies the semigroup property of t the family (T ) . t t>0 To check the strong continuity of the semigroup (T ) , let u C+, t > 0 and α J. t t>0 ∈ 1 ∈ Then S Tα by Lemma 2.3(iii) (i) and Definition 1.1(I). So Tα S 0. Since Tα is a t ≤ t ⇒ t − t ≥ t contraction, it follows that kTtαu−StukC1 = Tr((Ttα −St)u) ≤ Tru−TrStu = Tr((I −St)u). Taking the limit over α one gets kTtu−StukC1 ≤ Tr((I−St)u). Since (St)t>0 is a strongly continuous semigroup on Csa and Tr is continuous from Csa into R, one deduces that 1 1 limt↓0kTtu−StukC1 = 0. But limt↓0Stu = u in Cs1a. So limt↓0Ttu = u in Cs1a. The extension of the last limit to all u Csa follows from Lemma 2.1(a). ∈ 1 (f). Let u C+. Let α,β J with α β. Then (b) and the definition of T give ∈ 1 ∈ ≤ 0 Tαu Tβu T u for all t > 0. Hence ≤ t ≤ t ≤ t 0 (λI +L )−1u (λI +L )−1u (λI +L)−1u. (2.4) α β ≤ ≤ ≤ Therefore by Lemma 2.1(f) it follows that lim (λI +L )−1u exists in C . We next show α α 1 that the limit is equal to (λI +L)−1u. Let x and N (1, ). For all α J define f ,f: [0,N] [0, ) by α ∈ H ∈ ∞ ∈ → ∞ f (t) = e−λt((Tαu)x,x) and f(t) = e−λt((T u)x,x) . α t H t H 10