Relation between the Boundary Point Spectrum of a Generator and of Its Adjoint

F . ANDREU AND J .M . MAZóN In this paper we get some relations between the boundary point spectrum of the generator A of a Cp-semigroup and the generator A* of the dual semigroup . These relations combined with the results due to Lyubich-Phóng and Arendt-Batty, yield stability results on positive Cosemigroups . After the classical stability theorem due to Liapunov, a lot of contributions have been made in order to get generalizations to infinite dimension of this theorem. One of the more recent results in this context is the following stability theorem due to Arendt-Batty [1] and Lyubich-Phóng [7] . (I) " Leí (T(t))t>o be a bounded Co -semigroup on a Banach space E with generator A . Denote by u(A) the spectrum of A. If u(A)fliR is countable and no eigenvalues of A* lies on the imaginary axis, then (T(t))t>o is uniformly stable (i .e ., lim t-<>~ T(t)f = 0 for all f E E) . " If in (I) we assume that the Banach space E is reflexive, then we can reformulate it in a more suitable form, requiring the absence of imaginary eigenvalues of A instead of its adjoint A* . More precisely, we have : (II) " Le¡ (T(t))t>o be a bounded Co-semigroup on a reflexive Banach space E with generator A. If a(A) fl iR is countable and no eigenvalues of A lies on ¡he imaginary axil, then (T(t»t>o is uniformly stable . " The above result points out the importante of knowing the relationship be~ tween the boundary point spectrum of the generator A of a Co-semigroup and the boundary point spectrum of its adjoint A* . This note is devoted to study this problem. Let E be a Banach lattice with positive cone E+ . The principal ideal in E generated by some f E E+ is denoted by Ef . Recall that f E E+ is called a quasi-interior point of E+ if Ef is dense in E (see [11]) . For each f E E+ the 390 F. ANDREU, J.M . MAZÓN principal ideal Ef endowed with the gauge function pf of [-f, f] is an AMspace with unit f, and the canonical embedding Ef -> E is continuous (see, [11, 11 .7 .2]) . If A is the generator of a Co-semigroup (T(t))t>o, we denote the spectrum (point, residual spectrum) of A by u(A)(pu(A), ru(A)), the resolvent set by p(A) := C a(A) and the resolvent by R(A, A) := (A A)-1 for A E p(A) . The spectral bound of A is defined by s(A) := sup{ReA : A E o(A)}, and the boundary point spectrum of A by pab(A) = {A E pu(A) : ReA -s(A)} . If (T(t)) t>o is C o -semigroup on E with generator A, it is known that the adjoint operator (T(t)*)t>o need not be a Co-semigroup on E* . We define EO to be the subspace of E* on which (T(t)*) t >o is strongly continuous, T(t)0 := T(t)1_o and AO the generator of the Co-semigroup (T(t)O) t>o (see [3] and [8]) . By adapting the proof of Sata 3.2 in [12] to the semigroup context one can obtain the following result . We include the proof for sake of completeness . Proposition 1 . Let (T(t)) t>o be a positive Co -semigroup on ¡he Banach lattice E with generator A. If the resolvent of A grows slowly (i .e ., the set {(A A)R(A, A) : A > s(A)} is bounded in £(E», then pab(A) c pub(A* ) ., Proof.. In view of the rescaling procedure we assume that s(A) = 0. Let a E R such that Af = iaf with 0 qÉ f E E. Let {an } be a sequence of strictly positive real numbers converging to 0 . Since f qÉ 0, by the Hahn-Banach theorem we can find an element 0 E E*, 1011 = 1 and (0,f) = 1. Then, R an +¡a, A w e T t dt