The adjoint of a positive semigroup

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1 1 The djoint of positive semigroup J.M.A.M. vn Neerven Centre for Mthemtics nd Computer Science P.O. Box 479, 19 AB Amsterdm, The Netherlnds B. de Pgter Delft Technicl University P.O. Box 351, 26 GA Delft, The Netherlnds We study the properties of the djoint of positive semigroup T (t of opertors on Bnch lttice E. The min results re: (i If x E, then lim sup T (tx x 2 x ; (ii If x E nd either E hs order continuous norm or E hs qusi-interior point, then T (tx x for lmost ll t; (iii If E hs order continuous norm, then E is projection bnd; (iv If T (t is lttice semigroup, then the disjoint complement of E is T (t-invrint. 198 Mthemtics Subject Clssifiction: 47D5, 46B55, 46B3 Keywords & phrses: Positive semigroup, djoint semigroup, quotient semigroup.. Introduction Let {T (t} t be C -semigroup of bounded liner opertors on (rel or complex Bnch spce X. By defining T (t := (T (t for ech t, one obtines semigroup {T (t} t on the dul spce X. Throughout this pper, we will denote the semigroups {T (t} t nd {T (t} t by T (t nd T (t, respectively, nd it will be cler from the context when we men the semigroup or the single opertor. The djoint semigroup T (t fils in generl to be strongly continuous gin. Therefore it mkes sense to define X = {x X : lim T (tx x = }. This is the mximl subspce of X on which T (t cts in strongly continuous wy. The spce X ws introduced by Phillips in 1955 [Ph]. Recently, this spce hs been studied extensively by vrious uthors (e.g., [Ne], [NP], [P], in prticulr in connection with pplictions to certin evolution equtions (e.g., [Cl]. The purpose of this pper is to study the properties of E in cse E is Bnch lttice nd T (t is positive C -semigroup. Virtully nothing is known bout the Bnch lttice properties of E nd one of the most obvious questions, viz. under wht conditions E is

2 2 sublttice of E, is wide open. If T (t is lttice semigroup, in prticulr if T (t extends to positive group, then E is sublttice [Cl, Prt IV]; this follows from T (t(x + (x + = (T (tx + (x + T (tx x nd the lttice property of the norm. Recently, Grbosch nd Ngel [GN] constructed positive C -semigroup on n AL-spce E for which E is not sublttice of E. In fct, in this exmple the spce E, with the order inherited from E, even fils to be Bnch lttice in its own right. In order to motivte our min results, we strt by considering in some detil the trnsltion group T (t on C (R, given by T (tf(s = f(t + s. This semigroup hs some fetures which turn out to be representtive for the bstrct sitution. Theorem.1. Let T (t be the trnsltion group on E = C (R. (i ([Pl] µ E if nd only µ is bsolutely continuous with respect to the Lebesgue mesure m. (ii ([MG],[WY] If µ E is singulr with respect to m, then T (tµ µ for lmost ll t R. In prticulr, for ny ν E we hve lim sup T (tν ν = 2 ν s, where ν s is the singulr prt of ν. (iii The spce of singulr mesures is T (t-invrint. Note tht T (tν is just the trnslte in the opposite direction of ν in the sense tht for mesurble set G we hve (T (tν(g = ν(g t. Also, by (i it is cler tht mesure µ is singulr if nd only if µ E in the Bnch lttice sense. Versions of Theorem.1 for commuttive loclly compct groups (insted of R cn be found in [GM, Chpter 8]. In [P2], the Wiener-Young theorem ((ii bove hs been nlysed in detil in the context of djoint semigroups. There, extensions hve been obtined for the djoints of positive semigroups essentilly on C(K-spces. In the present pper, most of the results in [P2] will be extended to positive semigroups on rbitrry Bnch lttices. For the convenience of the reder, we include full proofs. Although severl proofs re completely different, this cuses smll overlp with [P2]. We will prove the following Bnch lttice versions of (i-(iii. C -semigroup on Bnch lttice E. Then: (i E is projection bnd if E hs order continuous norm (Theorem 2.1. Let T (t be positive The most importnt clss of (non-reflexive Bnch lttices whose duls hve order continuous norm is the clss of AM-spces. This clss contins C (R. In contrst, note tht the dul of n AL-spce does not hve order continuous norm unless E is finite-dimensionl. (ii Suppose x E. Then we hve lim sup T (tx x 2 x (Theorem 4.4. If moreover E hs order continuous norm or E hs qusi-interior point, then T (tx x for lmost ll t (Corollry 3.4. (iii The disjoint complement of E is T (t-invrint if T (t is lttice semigroup (Corollry 4.8. We use (iii to show tht if T (t is lttice semigroup, then the quotient E /(E dd is either zero or else very lrge (Theorem 4.1. Here (E dd is the bnd generted by E. We ssume the reder to be fmilir with some stndrd theory of Bnch lttices. For more informtion s well s the terminology we refer to [M], [AB], [S], [Z]. Throughout this pper, ll Bnch spces nd lttices my be either rel or complex.

3 3 1. Some preliminry informtion In this section we recll some of the bsic fcts bout djoint semigroups which will be used in the sequel. Proof cn be found e.g. in [BB]. Let T (t be C -semigroup (i.e., strongly continuous semigroup on Bnch spce X. Its genertor will be denoted by A with domin D(A. Considering the djoint semigroup T (t on the dul spce X, we define X = {x X : lim T (tx x = }, the domin of strong continuity of T (t. Then X is T (t-invrint, norm closed, wek - dense subspce of X (hence X = X if X is reflexive. The spce X is precisely the norm closure of D(A, the domin of the djoint of A. In prticulr, for λ ϱ(a = ϱ(a we hve R(λ, A x X for ll x X, where R(λ, A = R(λ, A = (λ A 1 is the resolvent. For ll x X we hve lim λ λr(λ, A x = x, where the limit is in the wek -sense. An lterntive description of X is given by X = {x X : lim λ λr(λ, A x x = }. If T (t extends to C -group, then the spce X with respect to the semigroup {T (t} t is equl to the domin of strong continuity of the group {T (t} t R. Exmples of spces X for vrious semigroups cn be found in [BB], [Ne], [NP]. In prticulr we mention tht if T is the trnsltion group on C (R or L 1 (R, the spce X cn be identified cnoniclly with L 1 (R nd BUC(R (the spce of ll bounded, uniformly continuous functions on R, respectively. We will hve the occsion to use the so-clled wek -integrls (or Gelfnd integrls of X -vlued functions. Let [, b] R nd f : [, b] X wek -continuous function (or, more generlly, wek -mesurble function such tht t f(t, x L 1 [, b] for ll x X. The wek -integrl wek f(t dt X is then defined by the formul wek f(t dt, x = f(t, x dt, x X. In this sitution, the function t f(t is Borel function on [, b] nd we hve the estimte wek f(t dt f(t dt. If T (t is C -semigroup on X, then for ech x X the mp t T (tx is wek -continuous on [, nd for ll < b R we hve wek T (tx dt D(A X. Finlly we sy few words bout the Bnch lttice sitution. Let E be Bnch lttice nd T (t positive C -semigroup on E. Suppose tht M, ω re such tht T (t Me ωt for ll t. If λ R is such tht λ > ω, then λ ϱ(a nd R(λ, A (for the bsic theory

4 4 of positive semigroups we refer to [N]. As mentioned in the introduction, E need not be sublttice of E. As usul, we denote by (E d the disjoint complement of E in E, i.e., (E d = {x E : x y for ll y E }. Here x y mens tht x y =. Then (E dd, the disjoint complement of (E d, is equl to the bnd generted by E. Since E = D(A, it is cler tht (E dd = (D(A dd. In generl, (E d is not T (t-invrint (see Exmple 3.7. However, the subspce (E dd is lwys T (t-invrint. Indeed, if x E is such tht x R(λ, A y for some y E nd λ > ω, then T (tx R(λ, A T (t y. This shows tht the (order idel generted by R(λ, A (E = D(A is T (t-invrint. Since T (t, being the djoint of positive opertor, is order continuous, this implies tht the bnd (D(A dd = (E dd is T (t-invrint s well. 2. The structure of E In this section we will ssume tht T (t is positive C -semigroup on Bnch lttice E. Theorem 2.1. If E is contined in sublttice of E with order continuous norm, then E is n idel in E. In prticulr, if E hs order continuous norm, then E is projection bnd. Proof: Let F be sublttice of E with order continuous norm, contining E. Step 1. First let x y with y E. We will show tht x E. Choose λ > be such tht R(λ, A for λ λ. Put G := {λr(λ, A y : λ λ }. Since y E, this set is reltively compct subset of F, hence certinly reltively wekly compct in F. Let sol F G := {f F : g G with f g} be the solid hull of G in F. Since F hs order continuous norm, sol F G is reltively wekly compct in F [M, Prop (iv]. Since E F nd λr(λ, A x R(λ, A x λr(λ, A y for ll λ λ, it is cler tht H := {λr(λ, A x : λ λ } sol F G. In prticulr, H is reltively wekly compct in F. Let z be ny σ(f, F -ccumultion point of H s λ. Then z is lso wek- nd hence wek -ccumultion point of H. But on the other hnd, wek lim λ λr(λ, A x = x. Therefore necessrily z = x. Since λr(λ, A x E for ech λ λ, it follows tht x belongs to the wek closure of E. Hence x E. Step 2. Suppose x y with y E. We will show tht x E. By Step 1 it suffices to show tht x E. Therefore we my ssume tht x. For λ λ put Then, since x nd λr(λ, A, z λ := λr(λ, A y x. z λ λr(λ, A y λr(λ, A y,

5 nd since λr(λ, A y is positive element in E, it follows from Step 1 tht z λ E. But since y E we hve lim λ λr(λ, A y = y, nd therefore lim λ z λ = lim λr(λ, λ A y x = y x = x. Since E is closed it follows tht x E. This proves tht E is n idel. The second sttement is consequence of the fct tht every closed idel in Bnch lttice with order continuous norm is projection bnd. //// In [NP] we observed tht if E is σ-dedekind complete Bnch lttice, then the bnd generted by E is the whole E. In fct, this follows from wek lim λ λr(λ, A x = x nd the fct tht every bnd projection in the dul of σ-dedekind complete Bnch lttice is wek -sequentilly continuous [AB, Thm ] (consider the bnd projection onto the bnd generted by E. Corollry 2.2. If E is σ-dedekind complete Bnch lttice whose dul hs order continuous norm, then E = E. An exmple of such Bnch lttice is E = c. The following corollry is converse of Theorem 2.1 in cse R(λ, A is wekly compct for some λ ϱ(a (hence for ll λ ϱ(a. This is the cse if nd only if E is -reflexive with respect to T (t; see [P1]. Corollry 2.3. If R(λ, A is wekly compct, then the following ssertions re equivlent: (i E is n idel; (ii E is contined in sublttice with order continuous norm; (iii E is σ-dedekind complete sublttice. Proof: (iii (ii: If E is σ-dedekind complete then, by the wek compctness of R(λ, A, E ctully hs order continuous norm [NP]. (ii (i follows from Theorem 2.1 nd (i (iii follows from the fct tht the dul of Bnch lttice is lwys Dedekind complete. //// 5 3. Disjointness lmost everywhere Throughout this section, let T (t be positive C -semigroup on Bnch lttice E. Fix rel λ ϱ(a with λ > ω, with ω R such tht T (t Me ωt for suitble constnt M 1. We strt with the simple observtion tht x {R(λ, Ax} dd for ll x E. Indeed, suppose y E such tht y R(λ, Ax =. Since R(µ, Ax R(λ, Ax for ll µ λ, this implies tht y (µr(µ, Ax =. Now it follows from lim µ µr(µ, Ax = x tht y x =. This shows tht {R(λ, Ax} d {x} d, nd hence x {R(λ, Ax} dd. For the djoint semigroup the sitution is different. It cn hppen tht x R(λ, A x for some x X. For exmple, let T (t be trnsltion group on E = C (R nd let x

6 6 be mesure which is singulr with respect to the Lebesgue mesure. Then x L 1 (R, here identifying bsolutely continuous mesures with their L 1 -densities. But R(λ, A x E = L 1 (R, so indeed x R(λ, A x. As one of the results of this section we will chrcterize these functionls x s the elements of (E d. The following lemm is first step towrds this chrcteriztion. We will use repetedly the formul ( x y, x = inf{ x, u + y, v : u, v [, x], u + v = x}, vlid for rbitrry x, y E nd x E (see e.g. [Z], Theorem Lemm 3.1. Suppose x E, x E nd y E stisfy R(λ, A x y, x =. Then, for lmost ll t (with respect to the Lebesgue mesure we hve T (tx y, x =. Proof: The formul (* pplied to T (tx y shows tht for x the function f(t := T (tx y, x is mesurble, being the infimum of continuous functions. We must show tht f =.e. Fix ε >. By (*, pplied to R(λ, A x y, it is possible to choose u, v [, x] such tht u + v = x nd R(λ, A x, u < ε, y, v < ε. Then e λt T (tx y, x dt e λt T (tx, u dt + e λt y, v dt = R(λ, A x, u + λ 1 y, v (1 + λ 1 ε. Since ε > is rbitrry it follows tht e λt T (tx y, x dt =. The lemm now follows from the fct tht the integrnd is positive function. //// Thus, if R(λ, A x y =, then by the lemm for ll x we hve T (tx y, x =, except for t belonging to set of mesure zero. This exceptionl set, however, my vry with x nd therefore one cnnot conclude tht T (tx y = for lmost ll t. The following exmple shows tht indeed this need not be the cse. Exmple 3.2. Let T be the unit circle in the complex plne, which will be identified with the intervl [, 2π, nd let C(T denote the Bnch lttice of continuous functions on T. Let E = l 1 ([, 2π; C(T. With the pointwise order, E is Bnch lttice. Note tht E = l ([, 2π; M(T, where M(T = C(T is the spce of bounded Borel mesures on T. Define n element x E by x (α = δ + δ α, where δ α is the Dirc mesure concentrted t α. Let R(t be the rottion group on C(T nd define positive C -group T (t on E by (T (tx(α := R(t(x(α. Then, using the fct tht the lttice opertions on E re defined pointwise, for ny t [, π we hve T (tx x (T (tx x (t = R(t(x (t x (t = (δ t + δ 2t (δ + δ t = δ t = 1.

7 Theorem 3.3. Suppose tht E hs qusi-interior point, or tht E hs order continuous norm. Then R(λ, A x y = ( x, y E implies tht T (tx y = for lmost ll t. Proof: Suppose first tht u > is qusi-interior. We hve by Lemm 3.1 tht T (tx y, u =,. t. Since u is qusi-interior point, this implies tht T (tx y =,. t. If E hs order continuous norm, then for ll z E the closed unit bll B E is pproximtely z -order bounded [M, Prop ], i.e. for ll ε > nd z E there is n x such tht B E [ x, x] + εb z. Here B z is the closed unit bll of the seminorm p z defined by p z (x = z, x. Choose x n such tht B E [ x n, x n ] + n 1 B y. By Lemm 3.1, there is set F n R > of full mesure such tht for ll t F n we hve T (tx y, x n =. Fix ny t F n. Let y B E rbitrry. Write y = y 1 + y 2 with y 1 [ x n, x n ], y 2 n 1 B y. Then 7 T (tx y, y T (tx y, y 1 + T (tx y, y 2 + y, y 2 1 n. It follows tht T (tx y, y = for ll t F := n F n. Since y is rbitrry nd the F n do not depend on y, it follows tht T (tx y = for t F. //// Corollry 3.4. Suppose x E, y (E d nd either E hs order continuous norm or E hs qusi-interior point. Then T (tx y for lmost ll t. Proof: y E implies y E, so in prticulr R(λ, A x y =. Therefore T (t x y = for lmost ll t. But T (tx T (t x, hence for lmost ll t lso T (tx y =. //// The following theorem gives the chrcteriztion of functionls in (E d, mentioned t the beginning of this section. Theorem 3.5. For x E the following sttements re equivlent: (i x (E d ; (ii R(λ, A x x = ; (iii For ll x E we hve T (tx x, x = for lmost ll t ; (iv For ll x E we hve lim inf T (tx x, x =. Proof: The implictions (i (ii nd (iii (iv re trivil, nd (ii (iii follows from Lemm 3.1. So only (iv (i needs proof. Tke x E stisfying (iv. Since E = D(A = R(λ, A E, it is sufficient to prove tht x R(λ, A y for ll y E. Moreover, since R(λ, A y R(λ, A y E, ll we hve to show is tht x z = for ll z E. To this end, fix z E nd let x 1 E be ny vector such tht x 1 nx z for some number n N. It follows from x 1 nx tht T (tx 1 x 1 T (tnx nx = n(t (tx x,

8 8 so x 1 stisfies (iv s well. Fix ε > nd x E with x = 1. There exists δ > such tht T (tz z < ε for ll t < δ. Furthermore, since lim inf T (tx 1 x 1, x =, there exists < t < δ such tht T (t x 1 x 1, x < ε. By the formul (*, there exist u, v E such tht u + v = x nd Then nd This implies tht T (t x 1, u < ε, x 1, v < ε. x 1, u = x 1, x x 1, v > x 1, x ε T (t x 1, v = x 1, x + T (t x 1 x 1, x T (t x 1, u > x 1, x 2ε. z, v = T (t z, v T (t z z, v T (t x 1, v T (t z z v ( > x 1, x 2ε ε v x 1, x 3ε. Hence ( z, x = z, u + z, v > x 1, u + x 1, x 3ε > 2x 1, x 4ε. Since ε is rbitrry it follows tht z, x 2x 1, x for ll x, i.e. 2x 1 z. Hence, 2x 1 2nx z nd we cn repet the bove rgument. After doing so k times we find tht 2 k x 1 z. Hence this holds for ll k N, so x 1 =. In prticulr, letting x 1 = x z, it follows tht x z =. This completes the proof. //// Next we will study the behviour of T (t on the disjoint complement (E d. In generl, (E d need not be T (t-invrint. It my even hppen tht T (te E for ll t >, e.g. if T (t is nlytic semigroup. Using the bove theorem we obtin the following result. Corollry 3.6. If T (t is lttice semigroup, then (E d is T (t-invrint. Proof: If x (E d, then R(λ, A x x =. Hence lso R(λ, A T (tx T (tx = T (t(r(λ, A x x =, so T (tx (E d by Theorem 3.5. //// We note tht, in prticulr, if T (t extends to positive group, then T (t is lttice semigroup nd the bove corollry pplies. Furthermore we note tht, s observed before, if T (t is lttice semigroup, then E is sublttice of E. The following exmple shows tht Corollry 3.6 (nd some results to follow fil if T (t is not lttice semigroup. Exmple 3.7. Let T (t be the semigroup on E = C[, 1] defined by { f(t + s, t + s 1; T (tf(s = f(1, else. Then one esily verifies the following fcts: (i E = L 1 [, 1] R δ 1 ; (ii δ E nd T (tδ = δ 1 E for ll t 1.

9 In view of Corollry 3.6 we will restrict our ttention in the lst prt of this section minly to the sitution in which T (t is lttice semigroup. We will study the occurrence of mutully disjoint elements in the orbits {T (tx : t }, where x (E d. The first result in this direction is simple consequence of Theorem 3.3. Proposition 3.8. Assume tht E hs order continuous norm, or tht E hs qusi-interior point. Furthermore, ssume tht T (t is lttice semigroup. Then for x (E d we hve: (i If s is fixed, then T (tx T (sx for lmost ll t ; (ii T (tx T (sx for lmost ll pirs (t, s (with respect to the Lebesgue mesure on R + R +. Proof: (i Tke s. It follows from Corollry 3.6 tht T (sx (E d. Now the result follows from Theorem 3.3 (with y = T (sx. (ii This follows vi Fubini s theorem from (i. //// Suppose tht (E d {} nd let < x (E d be fixed. We define t := inf{t > : T (tx = }. If T (tx for ll t we put t =. If t <, it follows from the wek -continuity of t T (tx tht T (t x = ; in prticulr t >. Hence T (tx > for ll t < t nd T (tx = for ll t t. We will sy tht set H [, t supports disjoint system (for x if {T (tx : t H} is disjoint system in E, i.e. T (tx T (sx for ny two t s H. In view Proposition 3.8 one might sk whether there exist lrge sets supporting disjoint system. Observe lredy tht, by Zorn s lemm, ny set supporting disjoint system is contined in mximl one. Let m denote the outer Lebesgue mesure. Lemm 3.9. Suppose tht E hs order continuous norm, or E hs qusi-interior point. Suppose T (t is lttice semigroup nd let x (E d. (i If H [, t is countble set supporting disjoint system, nd if J [, t is n open intervl, then there exists s J\H such tht H {s} supports disjoint system. (ii If H [, t is mximl set supporting disjoint system, then H is uncountble. (iii Let H [, t support disjoint system. If T (tx T (sx > for some < s < t, then m ([, t]\h 1 2 s. Proof: (i For t H let F t = {h : T (hx T (tx = }. By Proposition 3.8(i we know tht m(r + \F t =. Since H is countble, the set F = {F t : t H} stisfies m(r + \F = s well, nd hence F J. Now tke ny s F J. (ii Follows immeditely from (i. (iii Since T (tx T (sx >, lso T (t s + hx T (hx > for ll h s. Hence, if h H [, s], then h + t s H, i.e. so m ([, s] H m ([, t]\h. Now so m ([, t]\h 1 2s. //// ([, s] H + t s [, t]\h, s m ([, s] H + m ([, s]\h 2m ([, t]\h, 9

10 1 We do not know whether mximl set supporting disjoint system must be mesurble. This is the reson for tking the outer Lebesgue mesure rther thn the Lebesgue mesure. Exmple 3.1. Let T (t be the trnsltion group on E = C(T. Let x = δ + δ 1 nd let n N. Then T (2nx T (2n + 1x. The set H = [, 1 [2, 3... [2n, 2n + 1 is mximl set supporting disjoint system for x. By letting n we see tht the constnt in Lemm 3.9(iii is optiml. 1 2 Theorem Suppose tht E hs order continuous norm, or E hs qusi-interior point. Suppose T (t is lttice semigroup nd let x (E d. (i There exists n uncountble dense set H [, t supporting disjoint system. (ii If T (t extends to positive group, then either the orbit {T (tx : t R } is disjoint system, or m (R \H = for ech set H R supporting disjoint system. Proof: (i Let (J n n=1 be n enumertion of the open intervls in [, t with rtionl endpoints. Using Lemm 3.9(i we inductively construct sequence (t n n=1 supporting disjoint system with t n J n for ll n. This sequence (t n is contined in some mximl H supporting disjoint system. Clerly H is dense in [, t, nd by Lemm 3.9(ii H is uncountble. (ii Now ssume in ddition tht T (t extends to positive group, nd tht H R supports disjoint system with m (R + \H = K <. Then lso H + := H R + supports disjoint system nd m (R + \H + K. It follows from Lemm 3.9(iii tht T (tx T (sx = for ll s t > 2K. Therefore, if s t in R, then for n so lge tht s + n > 2K, t + n > 2K we hve T (n(t (tx T (sx = T (t + nx T (s + nx =. Since T (n is injective, this implies tht T (tx T (sx =. //// In the sitution of Theorem 3.11, it is cler from (i tht (E d is not norm seprble. So in this sitution we hve either (E d = {} or (E d is non-seprble. In this direction we cn prove more, under much weker ssumptions, using different method of proof. This is wht we will do next. First we recll some fcts. Let E be Bnch lttice nd J E n idel. The nnihiltor J = {x E : x, x =, x J} is bnd in E, nd hence we hve the bnd decomposition E = J (J d. Let P J be the bnd projection in E onto (J d. Lemm Let J E be n idel nd T : E E be positive opertor such tht T (J J. Then P J T T P J. Proof: Since T (J J implies tht T (J J, it follows tht T (I P J = (I P J T (I P J, nd so P J T P J = P J T. Hence P J T = P J T P J T P J. //// In the following theorem, T (t is ny positive C -semigroup on E. We do not ssume tht T (t be lttice semigroup. Theorem If (E d contins wek order unit, then T (t(e (E dd for ll t >. Proof: Let w (E d be wek order unit. Fix x E nd x E. Let J be the closed idel in E generted by the orbit {T (tx : t }. Then J is T (t-invrint nd hs qusi-interior point u J. By Lemm 3.1, w (E d implies tht T (tx w, u = for lmost ll t. Since P J (T (tx w T (tx w,

11 it follows tht P J (T (tx w, u =.e., nd hence P J (T (tx w J.e. But lso P J (T (tx w (J d, so P J (T (tx w =.e., hence P J (T (tx (E dd.e. Now observe tht, if t is such tht P J (T (tx (E dd, then by Lemm 3.12, P J (T (t + sx = P J (T (st (tx T (sp J (T (tx. Also, s observed in Section 1, (E dd is T (t-invrint. Combining these fcts, we conclude tht P J (T (tx (E dd for ll t >. Therefore, P J (T (tx w =, i.e., T (tx w J for ll t >, which implies in prticulr tht T (tx w, x = for ll t >. Since x E ws rbitrry, it follows tht T (tx w = for ll t >, i.e., T (tx (E dd for ll t >. //// Together with Theorem 2.1 this implies: Corollry Suppose E hs order continuous norm. If (E d contins wek order unit, then T (t(e E for ll t >, i.e. T (t is strongly continuous for t >. Corollry Suppose E hs order continuous norm nd suppose T (t extends to (not necessrily positive group. Then either E = E or (E d does not contin wek order unit. Corollry Suppose T (t is lttice semigroup. Then either (E d = {} or (E d does not contin wek order unit. Proof: Suppose (E d contins wek order unit. By Theorem 3.13, T (t(e (E dd for ll t >. It follows from Corollry 3.6 tht (E d is T (t-invrint, nd hence T (t((e d = {} for ll t >. From the wek -continuity of t T (tx it now follows tht (E d = {}. //// The preceding results cn be regrded s lttice versions of the following result proved in [Ne]: If T (t is C -semigroup on Bnch spce X such tht X /X is seprble, then T (t(x X for ll t >, i.e. T (t is strongly continuous for t >. In prticulr, if T (t extends to group, then either X = X or X /X is non-seprble. In the setting of Corollry 3.15, one might wonder when exctly one hs E = E. In this direction, we cn prove: Proposition Let E = C (Ω with Ω loclly compct Husdorff, nd let T (t be positive C -group on E. If E = E then T (t is multipliction group. Proof: Since ech opertor T (t is lttice isomorphism, toms in M(Ω = (C (Ω re mpped to toms. Hence, for ech ω Ω we hve T (tδ ω = φ ω (tδ ω(t, sy. Here δ ω is the Dirc mesure t ω. By the strong continuity of t T (tδ ω, we must hve tht ω(t = ω, so T (tδ ω = φ ω (tδ ω. For f C (Ω one then hs //// (T (tf(ω = δ ω, T (tf = φ ω (t f, δ ω = φ ω (tf(ω. Every multipliction group on rel Bnch lttice E hs bounded genertor [N, Proposition. C-II.5.16]. If E is complex, then positive semigroup leves invrint the rel prt of E. Therefore, both in the rel nd complex cse, from the bove results we conclude: Corollry Let T (t be positive C -group with unbounded genertor on the Bnch lttice E = C (Ω. Then (E d does not contin wek order unit. 11

12 12 4. Limes superior estimtes We strt in this section with n rbitrry C -semigroup T (t on Bnch spce X. We choose M 1 nd ω R such tht T (t Me ωt. It is our objective to study the quntity T (tx x s t for x X. Our first results re generl limes superior estimtes, which we will improve lter in the context of positive semigroups. For x X define ρ(x := lim sup T (tx x. It is cler tht ρ defines seminorm on X. Note tht ρ(x + x = ρ(x for ll x X nd x X. In prticulr, ρ(x = if nd only if x X. Furthermore, ( ρ(x lim sup T (t + 1 x (M + 1 x for ll x X. Since X is closed subspce of X, the quotient spce X /X is Bnch spce. Let q : X X /X be the quotient mp. The following result shows tht the seminorm ρ is ctully equivlent to the quotient norm on X /X. Theorem 4.1. For ll x X we hve qx ρ(x (M + 1 qx. Proof: For rbitrry x X nd x X we hve ρ(x = ρ(x + x (M + 1 x + x. By tking the infimum over ll x X we obtin ρ(x (M + 1 qx. For the converse, we recll tht for ny τ > we hve wek τ T (tx dt X. Therefore, Hence, //// qx 1 τ τ wek 1 τ τ T (tx dt x = 1 τ T (tx x dt sup T (tx x. t τ ( qx inf τ> sup t τ τ wek T (tx x dt T (tx x = ρ(x. We mention n immedite consequence of the bove theorem. Corollry 4.2. Let X Y, with Y complemented subspce of X, sy X = Y Z. Then there is constnt C > such tht for ll x Z we hve lim sup T (tx x C x. Proof: Since X Y, the formul x := qx defines norm on Z which stisfies x = inf x X x x inf y Y x y. But X /Y Z nd consequently x C x. Now we cn pply Theorem 4.1. ////

13 On the quotient spce X /X we cn define quotient semigroup Tq (t vi the formul T q (tqx := q(t (tx. Using the equivlence in Theorem 4.1, we cn investigte some properties of this quotient semigroup vi the seminorm ρ. For this purpose, the following result turns out to be useful. Lemm 4.3. Let [, b] R be closed intervl nd f : [, b] X wek -continuous function. Then t ρ(f(t is bounded Borel function on [, b] nd Proof: For n N, n >, define ρ (wek ρ n (x := f(t dt ρ(f(t dt. sup T (tx x, x X. t 1 n Ech ρ n is seminorm on X nd ρ n (x ρ(x for ll x X. Note tht ρ n (x = sup t 1 n = sup t 1 n ( sup ( x 1 sup x 1 T (tx x, x x, (T (t Ix = sup{ x, y : y D n }, where D n = t 1 n (T (t IB X, B X being the closed unit bll of X. Hence, ρ n (f(t = sup x Dn f(t, x for ll t b. Being the pointwise supremum of continuous functions, ρ n (f( is lower semi-continuous. Since ρ n (f(t ρ(f(t for ll t b, it follows tht ρ(f( is Borel function. For x D n we hve nd so wek f(t dt, x = f(t, x dt ρ (wek f(t dt ρ n (wek = sup x D n wek f(t dt f(t, x dt f(t dt, x Finlly, it follows from the monotone convergence theorem tht This concludes the proof. //// ρ n (f(t dt ρ(f(t dt. ρ n (f(t dt, ρ n (f(t dt. The bove lemm cn be used to prove the following property of the seminorm ρ. 13

14 14 Proposition 4.4. For ll x X we hve ρ(x lim sup ρ(t (tx x. In prticulr, if x X is such tht lim t ρ(t (tx x =, then ρ(x =, i.e., x X. Proof: For ll x X nd τ > we hve, using Lemm 4.3, ( 1 τ ρ(x = ρ τ wek 1 τ τ T (tx x dt ρ(t (tx x dt lim sup ρ(t (tx x. A combintion of this result with Theorem 4.1 yields the following: Corollry 4.5. If lim t T q (tqx qx =, then qx =. Thus the only element in X /X whose T q (t-orbit is strongly continuous, is the zero element. This result ws first proved in [Ne]. The (more complicted proof given there shows tht in fct the following stronger result is true: if the T q (t-orbit of some qx is norm-seprble in X /X, then it is identiclly zero for t >. We now return to the cse of positive C -semigroup on Bnch lttice. In Theorem.1, E is complemented in E nd therefore we cn lredy conclude from Theorem 4.1 tht the limes superior estimte must hold with some constnt C. In generl E is not complemented, but we lwys hve direct sum decomposition of E into the bnd generted by E nd the disjoint complement of E (which of course my be {}. Therefore Corollry 4.2 cn be pplied nd we get constnt C > such tht for ll x E we hve lim sup T (tx x C x. The following theorem shows tht in fct we cn chieve C = 2. Theorem 4.6. Let T (t be positive C -semigroup on Bnch lttice E. If x (E d, then lim sup T (tx x 2 x. Proof: First we observe tht for x E nd x E, Indeed, if y x, then nd hence lim inf T (tx, x x, x. lim inf T (t x, x lim inf T (tx, y = lim T (tx, y = x, y, lim inf T (tx, x sup{ x, y : y x} = x, x. Now tke x (E d nd x E with x = 1. From Lemm 3.1 we know tht T (t x x, x = for lmost ll t, nd hence T (tx x, x =.e. Using the lttice identity [AB, Theorem 1.4(4] ( 2 T (tx x = T (tx + x T (tx x,

15 15 we see tht, for lmost every t, T (tx x T (tx x, x T (tx x, x = T (tx, x + x, x. This implies tht lim sup T (tx x lim inf T (tx, x + x, x 2 x, x. Since x E of norm one is rbitrry, the result follows. //// If E hs order continuous norm, then by Theorem 2.1 E is projection bnd. Let π be the bnd projection onto its disjoint complement. Corollry 4.7. If E hs order continuous norm, then 2 πx lim sup T (tx x (M + 1 πx. In prticulr, if M = 1, i.e., if lim T (t = 1, then lim sup T (tx x = 2 πx. If x contined in the bnd generted by E but not contined in E itself, then the limes superior cn be nything between nd 2, s is shown by the following exmple. Exmple 4.8. Let E = L 1 (R, T (t the trnsltion group on E. Let f C (R be of norm one such tht f = on [ 1, 1]. Let α 1 nd define g E = L (R by f(s, s > 1; g(s := α, s [, 1]; α, s [ 1, α. Then g = 1, g belongs to the bnd generted by E, nd lim sup T (tg g = 2α. 5. References [AB] C.D. Aliprntis, O. Burkinshw, Positive Opertors, Pure nd Applied Mth. 119, Acdemic Press, [BB] P.L. Butzer, H. Berens, Semigroups of opertors nd pproximtion, Springer-Verlg New York (1967. [Cl] Ph. Clément, O. Diekmnn, M. Gyllenberg, H.J.A.M. Heijmns, H.R. Thieme, Perturbtion theory for dul semigroups, Prt I: The sun-reflexive cse, Mth. Ann. 277, (1987; Prt II: Time-dependent perturbtions in the sun-reflexive cse, Proc. Roy. Soc. Edinb. 19A, (1988; Prt III: Nonliner Lipschitz perturbtions in the sunreflexive cse, In: G. D Prto, M. Innelli (eds, Volterr Integro Differentil Equtions in Bnch Spces nd Applictions, Longmn, (1989; Prt IV: The intertwining formul nd the cnonicl piring, in: Semigroup theory nd Applictions, Lecture Notes in Pure nd Applied Mthemtics, Vol. 116, Mrcel Dekker Inc., New York-Bsel (1989; Prt V: Vrition of constnts formuls, in: Semigroup theory nd Evolution Equtions,

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