STABILITY AD DICHOTOMY OF POSITIVE SEMIGROUPS O L Stehen Montgomery-Smith Abstract A new roof of a result of Lutz Weis is given, that states that the stability of a ositive strongly continuous semigrou e ta t on L may be determined by the quantity sa We also give an examle to show that the dichotomy of the semigrou may not always be determined by the sectrum σa Consider a strongly continuous semigrou e ta t acting on a Banach sace X with unbounded generator A It has long been known that the sectral maing theorem e tσa σe ta \ {} does not necessarily hold Here σa denotes the sectrum of an oerator A Indeed, let sa su ReσA, and let ωa su Relogσe A inf{λ : e ta M λ e λt } Then there are examles of semigrous for which sa ωa see [] The urose of this aer is to give one situation in which it is true that sa ωa This next result has already been roved by Lutz Weis [We] We will give a different, shorter roof We refer the reader to [We] for a history of the roblem Theorem 1 Let e ta be a strongly continuous ositive semigrou on L Ω, F, µ, where Ω, F, µ is a sigma-finite measure sace, and 1 < Then ωa sa In order to show this result, we will make use of the following lemmas The first result may be derived from [C], Theorem 74 the reader may like to know that a roof of the Pringsheim-Landau Theorem used in [C] may be found on age 59 of [Wi] Lemma 2 Let e ta be a strongly continuous ositive semigrou on a Banach lattice X, and let g X Then for any λ > sa we have that λ A 1 g e sa λ g ds Here the right hand side is taken in the sense of an imroer integral The next result may be found in [LM1] and [LM2] 1991 Mathematics Subject Classification Primary 47-2, 47D6, Secondary 35B4 Research suorted in art by SF Grant DMS 921357 1 Tyeset by AMS-TEX
2 STEPHE MOTGOMERY-SMITH Lemma 3 Let e ta be a strongly continuous semigrou on a Banach sace X, and let 1 < Then 1 / σe 2πA if and only if iz σa and there is a constant c > such that for any v n, v n+1,, v n X we have ik A 1 v k e ikt dt c v k e ikt For the next result, we secialize to a Banach lattice of functions on a sigmafinite measure sace In fact, this is really no loss of generality, and the interested reader should find no trouble making sense of this result for a general Banach lattice by alying the ideas in [LT] Chater 14 Lemma 4 Let P be a ositive oerator on X, a Banach lattice of functions on a sigma-finite measure sace, such that g f X imlies that g X Let 1 < If f : [, 2π] X is a measurable, simle function, then P ft dt 1/ P 1/ ft dt dt Proof Let us set f v kχ Ak, where v k X, and the sets A k [, 2π] are disjoint Then, letting f k v k A k 1/, the result reduces to showing that 1/ n 1/ P f k P f k However, we know that 1/ f k lub Rea k f k Here, lub denotes the least uer bound in the lattice ow, since P is ositive, we have that P lub Rea k f k is an uer bound for Rea kp f k whenever a k q 1 Hence 1/ P f k lub P lub Rea k P f k Rea k f k n 1/ P f k
STABILITY AD DICHOTOMY OF POSITIVE SEMIGROUPS O L 3 Proof of Theorem 1 It is well known that sa ωa see [] Thus by simle rescaling arguments, we see that it is sufficient to show that if sa <, then T σe 2πA We will show, under the assumtion that sa <, that if f : R L is a bounded, measurable function that is eriodic with eriod 2π, then for each > we have e sa ft s ds L 1/ A 1 ft L dt 1/ In order to show this, we may assume without loss of generality that f restricted to [, 2π] is a simle function Fix > By the ositivity of e sa, and Fubini s Theorem, we have that 1/ 1/ e sa ft s ds e sa ft s ds L L 2π 1/ e sa ft s ds dt L By the integral version of Minkowski s Theorem see [HLP], Section 23, it follows that for each ω Ω 2π 1/ e sa ft sω ds dt e sa ft sω 1/ dt ds e sa ftω dt 1/ ds Finally, from Lemma 4, we see that 1/ 1/ e sa ft dt e sa ft dt Putting all of these together, and alying Lemma 2, we obtain 1/ 1/ e sa ft s ds e sa ft dt ds L 1/ e sa ft dt ds 1/ ft dt A 1 A 1 L ft dt 1/ L 1/ A 1 ft L dt, L L
4 STEPHE MOTGOMERY-SMITH where the last equality uses Fubini s theorem ow, if ft e iβt v ke ikt for some β R, then by Lemma 2, we see that e sa ft s ds ik + iβ A 1 v k e ikt uniformly in t as Hence by Lemma 3 it follows that e iβ / σe 2πA One might conjecture that the sectrum of the generator of a ositive semigrou e td on an L sace might characterize the dichotomy of the semigrou, that is, if a is any real number, then a + ir σd if and only if e ta T σe td However, this is not the case, as the next result shows Theorem 5 There is a ositive semigrou e td acting on an L 2 sace such that 1 + ir σd, but e 2π σe 2πD Proof For each M, let C M be the contraction acting on l M 2 1 1 1 C M ote that if λ, then λ C M 1 M 1 j by the matrix λ 1 λ 2 λ 3 λ 4 λ M λ 1 λ 2 λ 3 λ M+1 λ 1 j C j M λ 1 λ 2 λ M+2 λ 1 λ M+3 λ 1 Thus, if λ 1, then λ C M 1 M Also, if λ > 1, then λ CM 1 M 1 j λ 1 j 1/ λ 1 In articular, if λ 2, then λ CM 1 1 Also note that 1 t t 2 /2 t 3 /6 t M 1 /M 1! 1 t t 2 /2 t M 2 /M 2! 1 t t M 3 /M 3! e tc M 1 t M 4 /M 4! 1 Thus we see that e tc M is a ositive oerator Clearly e tc M e t C M e t Consider the ositive semigrou acting on L 2 [, 2π] by e ta M fx e 4t 1 fx dx + fx + Mt, 2π
STABILITY AD DICHOTOMY OF POSITIVE SEMIGROUPS O L 5 so that its generator is the closure of A M fx 4 fx dx 2π + M d dx fx ote that e ta M e 4t ow consider the ositive semigrou e tb M e ta M e tc M acting on X M L 2 [, 2π] l M 2 L 2 [, 2π] {1, 2,, M}, We see that this semigrou is generated by B M A M I + I C M Also, e tb M e 5t Consider a tyical element of X M given by fx n v ne inx X M, where v n l M 2, and f 2 X M 2π n v n 2 2 If λ 4 and λ / MZ \ {}, then λ / σb M, and λ B M 1 fx λ 4 C M 1 v + λ inm C M 1 v n e inx Thus { } λ B M 1 max λ 4 C M 1, su λ inm C M 1 n In articular, if Reλ 1 and λ M 2, then λ B M 1 1, whereas if λ 1 + im, then λ B M 1 M ow consider the semigrou e td e tb M M1 acting on X M L 2 [, 2π] {1, 2,, M} M1 M1 ote that e td really is a strongly continuous semigrou, with e td e 5t The generator D is the closure of B M, and hence its resolvent set consists of those λ such that M1 λ D 1 su λ BM 1 <, M 1 that is, σd {z : z 4 1} iz \ {} In articular, if Reλ 1, then λ / σd However, su λ 1+iZ λ D 1, and hence, by Gerhard s Theorem see [], 95, e 2π σe 2πD References [C] Ph Clement, HJAM Heijmans, et al, One Parameter Semigrous, orth-holland, 1987 [HLP] GH Hardy, JE Littlewood and G Pólya, Inequalities, Cambridge University Press, 1952 [LT] J Lindenstrauss and L Tzafriri, Classical Banach Saces Volume II, Sringer-Verlag, 1979 [LM1] Y Latushkin and SJ Montgomery-Smith, Lyaunov theorems for Banach saces, Bull AMS 31 1994, 44 49 [LM2] Y Latushkin and SJ Montgomery-Smith, Evolutionary semigrous and Lyaunov theorems in Banach saces, J Func Anal 127 1995, 173 197 [] R agel ed, One Parameter Semigrous of Positive Oerators, Sringer-Verlag, 1984 [We] L Weis, The stability of ositive semigrous on L saces, Proc AMS to aear [Wi] D Widder, The Lalace Transform, vol 6, Princeton Math Series, 1946 Math Det, University of Missouri, Columbia MO 65211, USA