PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 10, Pages 2871{2879 S 0002-9939(01)05614-3 Article electronically published on May 10, 2001 ON STABILITY OF C0-SEMIGROUPS VU QUOC PHONG (Communicated by David R. Larson) Abstract. We prove that if T (t)isaC0-semigroupR on a Hilbert space E, 2 fk t f g k ≥ then (a) 1 ρ(T (!)) if and only if sup 0 exp (2πik)=! T (s)xds : t 2 g 1 2 0;k Z R < ,forallx E,and(b)T (t) is exponentially stable if and only fk t f g k ≥ 2 g 1 2 if sup 0 exp iλt T (s)xds : t 0,λ R < , for all x E. Analogous, but weaker, statements also hold for semigroups on Banach spaces. 1. Introduction and main results Let T (t) be a strongly continuous one-parameter semigroup (C0-semigroup) of bounded linear operators on a Banach space E, with generator A. The spectrum, resolvent set, and domain of A will be denoted by σ(A), ρ(A)andD(A), respec- tively. The semigroup is called exponentially stable if there exist M ≥ 1and!>0 such that kT (t)k≤Me−!t; 8t ≥ 0; and it is called asymptotically stable if limt!1 kT (t)xk =0,8x 2 E. In this paper, we prove the following theorems. Theorem 1. Suppose E is a Hilbert space and T (t), t ≥ 0,isaC0-semigroup on E. If, for some !>0, Z t 2πik s (1) sup e ! T (s)xds < 1; 8x 2 E; t≥0;k2Z 0 then 1 2 ρ(T (!)). Conversely, if T (t) is uniformly bounded and 1 2 ρ(T (!)),then (1) holds. Theorem 2. Suppose E is a Hilbert space and T (t), t ≥ 0,isaC0-semigroup on E.ThenT (t) is exponentially stable if and only if Z t iλs (2) sup e T (s)xds < 1; 8x 2 E: λ2R;t≥0 0 Theorem 3. Suppose E is a Banach space and T (t), t ≥ 0,isaC0-semigroup on E, with generator A.If Z t iλs (3) sup e T (s)xds < 18x 2 E; λ 2 R; t≥0 0 then Received by the editors February 20, 1998 and, in revised form, May 26, 1999. 2000 Mathematics Subject Classification. Primary 47D06. c 2001 American Mathematical Society 2871 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2872 VU QUOC PHONG (i) kT (t)xk!0 as t !1for all x 2 D(A2). (ii) If, in addition to (3), T (t) is uniformly bounded, then T (t) is asymptotically stable. (iii) If, in addition to (3), T (t) is analytic, then T (t) is exponentially stable. The proof of Theorems 1{3 is based on Lemma 1, which states that if Z t T (s)xds ≤ Mkxk; 8x 2 E; 0 then 0 2 ρ(A), and moreover, kA−1k≤M. This fact, and the Principle of Uniform Boundedness, imply that if the condition (2) holds, then the resolvent R(λ)=(λ − A)−1 is uniformly bounded in Re λ>0. Theorem 2 will follow imme- diately from this and a well-known theorem that for semigroups on Hilbert space the uniformly boundedness of the resolvent in fλ 2 C:Reλ>0g implies the exponential stability.1 Theorems 1 and 3 also follow from Lemma 1 in a similar way. Despite that the results and their proofs as given in this paper are elementary and very natural, to our knowledge they are new and may have some interest in the asymptotic behavior for C0-semigroups. In fact, after the first version of this paper was complete and circulated, van Neerven has kindly pointed out to the author in a private communication that the fact that the condition (2) implies the uniform boundedness of the resolvent in Re λ>0 was obtained in [7, Corollary 5],2 earlier and independently. The method of [7] is completely different (and more complicated) and, in our opinion, does not allow to obtain Theorems 1 and 2. Discrete analogs of our results also hold and are presented in Section 3. 2. Proofs The following lemma plays the key role in our argument. Lemma 1. Assume that E is a Banach space and T (t), t ≥ 0,isaC0-semigroup on E, with generator A.If Z t (4) sup T (s)xds < 1; 8x 2 E; t≥0 0 then 0 2 ρ(A). Moreover, if Z t (5) T (s) ds ≤ M; 8t ≥ 0; 0 then kA−1k≤M. Proof. From (4) and the Principle of Uniform Boundedness it follows that there exists M>0 such that (5) holds. Let us define, for every λ>0, the following operator R : λ Z Z 1 τ R x ≡ e−λtT (t)xdt≡ lim e−λtT (t)xdt; 8x 2 E: λ !1 0 r 0 1This theorem is usually attributed to Gearhart who proved it for contraction semigroups [2]. For general C0-semigroups it was obtained independently by Herbst [3], Howland [4] and Pr¨uss [11]. See also [6]. 2and hence the exponential stability, though this latter fact was not stated explicitly in [7], nor in [8]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON STABILITY OF C0-SEMIGROUPS 2873 From (4) and Z Z Z 1 1 t e−λtT (t)xdt= e−λtd T (s)xds 0 0 Z 0 Z Z t 1 1 t −λt −λt = e T (s)xds + λ T (s)xds e dt; 0 0 0 0 it follows that Rλ is well defined (for λ>0) and is a bounded linear operator on E,and Z Z 1 t −λt (6) Rλx = λ T (s)xds e dt: 0 0 −1 We show that Rλ =(λ − A) . From the well-known formula Z t A T (s)xds= T (t)x − x; x 2 E; 0 it follows that Z Z 1 t −λt ARλx = λ e A T (s)xds dt Z0 0 (7) 1 = λ e−λt(T (t)x − x)dt 0 = λRλx − x; and hence (λ − A)Rλx = x.SinceT (t)Ax = AT (t)x for all x in D(A), we have Rλ(λ − A)x =(λ − A)Rλx = x: −1 This implies, together with (7), that Rλ =(λ − A) . Now from (6) we have Z 1 −1 −λt (8) k(A − λ) k = kRλk≤λM e dt = M; 8λ>0: 0 This implies that 0 2 ρ(A), Z 1 A−1x = lim e−λtT (t)xdt; ! + λ 0 0 and (9) kA−1k≤M: The following lemma is in a sense a converse to Lemma 1. It must be a well- known fact but we include a proof because we could not find a convenient reference. Lemma 2. If kT (t)k≤M, 8t ≥ 0,and0 2 ρ(A),then Z t −1 sup T (s)ds ≤ (M +1)kA k: t≥0 0 Proof. We have Z Z t t T (s)xds= T (s)AA−1xds 0 Z0 t = d(T (s)A−1x)=T (t)A−1x − A−1x: 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2874 VU QUOC PHONG Therefore, Z t −1 −1 T (s)xds ≤kT (t) − IkkA kkxk≤(M +1)kA kkxk: 0 Lemmas 1 and 2 imply the following statement. Lemma 3. Assume that E is a Banach space and T (t), t ≥ 0,isaC0-semigroup on E, with generator A. Consider the following statements: (i) For every x 2 E Z t iλs sup e T (s)xds < 1; λ2R;t≥0 0 (ii) iR ⊂ ρ(A) and −1 (10) M1 ≡ sup k(A − iλ) k < 1: λ2R Then (i) implies (ii). If, in addition, T (t) is uniformly bounded, then (ii) implies (i). Proof. Assume that (i) holds. By the Principle of Uniform Boundedness, there exists a constant M such that Z t iλs sup e T (s) ds ≤ M: λ2R;t≥0 0 Applying Lemma 1 (and estimate (11)) to the semigroup e−iλtT (t), we have iλ 2 ρ(A)andk(A − iλ)−1k≤M, 8λ 2 R. k k≡ 1 Conversely, suppose that (ii) holds and supt≥0 T (t) M< .Let −1 M1 ≡ sup k(A − iλ) k: λ2R Applying Lemma 2 to the semigroup eiλtT (t), we have Z t iλt −1 sup e T (s) ds ≤ (M +1)k(A + iλ) k≤M1(M +1): t≥0,λ2R 0 Remark. If (10) holds, then it follows from the well-known equality 1 (11) dist(iλ; σ(A)) ≥ k(A − iλ)−1k that the spectral bound s(A)ofA is negative, where s(A) ≡ supfRe λ: λ 2 σ(A)g: In fact, (10) and (11) imply 1 s(A) ≤− : M1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON STABILITY OF C0-SEMIGROUPS 2875 As mentioned in the Introduction, the implication (i) ) (ii) in Lemma 3 was obtained recently by van Neerven [7, Theorem 4] (see also [8, Proposition 4.5.3]), by an entirely different method (even though he does not formulate it in that way).3 The following lemma can be proved completely analogously to Lemma 3. Lemma 4. Assume that E is a Banach space and T (t), t ≥ 0,isaC0-semigroup on E, with generator A. Consider the following statements: (i) For every x 2 E Z t 2πik s sup e ! T (s)xds < 1; k2Z;t≥0 0 (ii) 2πik 2 ρ(A), 8k 2 Z and ! − 2πik 1 (12) M1 ≡ sup A − < 1: k2Z ! Then (i) implies (ii).