INFINITESIMAL GENERATORS OF INVERTIBLE EVOLUTION FAMILIES

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1 Method of Functional Analyi and Topology Vol. 23 (207), no., pp INFINITESIMAL GENERATORS OF INVERTIBLE EVOLUTION FAMILIES YORITAKA IWATA Abtract. A logarithm repreentation of operator i introduced a well a a concept of pre-infiniteimal generator. Generator of invertible evolution familie are repreented by the logarithm repreentation, and a et of operator repreented by the logarithm i hown to be aociated with analytic emigroup. Conequently generally-unbounded infiniteimal generator of invertible evolution familie are characterized by a convergent power erie repreentation.. Introduction The logarithm of an injective ectorial operator wa introduced by Nollau [2] in 969. After a long time, the logarithm of ectorial operator were tudied again from 990 [2, 3, 4], and it utility wa etablihed with repect to the definition of the logarithm of operator [4, ] (for a review of ectorial operator, ee Hae [5]). While the ectorial operator ha been a generic framework to define the logarithm of operator, the ectorial property i not generally atified by the evolution family of operator, where the evolution family correpond to the exponential of operator in abtract Banach pace framework (for the definition of evolution family in thi paper, ee Sec. 2.). In thi paper we characterize infiniteimal generator of invertible evolution familie that ha not been ettled o far. Firt of all, by introducing a kind of imilarity tranform, a logarithm repreentation i obtained for uch generator. The logarithm repreentation i utilized to how a convergent power erie repreentation of invertible evolution familie generated by certain unbounded operator, although the validity of uch a repreentation i not etablihed for any evolution familie generated by unbounded operator. In thi context, the concept of pre-infiniteimal generator i introduced. 2. Mathematical etting 2.. Evolution family on Banach pace. Let X and B(X) be a Banach pace with a norm and a pace of bounded linear operator on X. The ame notation i ued for the norm equipped with B(X), if there i no ambiguity. For a poitive and finite T, let element of evolution family {U(t,)} T t, T be mapping: (t,) U(t,) atifying the trong continuity for T t, T (for review or textbook, ee [, 6, 0, 5, 7]). The emigroup propertie () U(t,r) U(r,) U(t,) and (2) U(,) I 2000 Mathematic Subject Claification. 47D03, 35A20, 34K30. Key word and phrae. Invertible evolution family, operator theory, maximal regularity. 26

2 INFINITESIMAL GENERATORS are aumed to be atified, where I denote the identity operator of X. Both U(t,) and U(,t) are aumed to be well-defined to atify (3) U(,t) U(t,) U(,) I, where U(,t) correpond to the invere operator of U(t,). Since U(t,) U(,t) U(t,t) I i alo true, the commutation between U(t,) and U(,t) follow. Operator U(t, ) i a generalization of exponential function; indeed the propertie hown in Eq. () (3) are atified by taking U(t,) a e t. Evolution family i an abtract concept of exponential function valid for both finite and infinite dimenional Banach pace. Let Y be a dene Banach ubpace of X, and the topology of Y be tronger than that of X. The pace Y i aumed to be U(t,)-invariant. Following the definition of C 0 - (emi)group (cf. the aumption H 2 in Sec. 5.3 of Pazy [5] or correponding dicuion in Kato [7, 8]), U(t,) i aumed to atify the boundedne [6, 5]; there exit real number M and β uch that (4) U(t,) B(X) Me βt, U(t,) B(Y) Me βt. Inequalitie (4) are practically reduced to U(t,) B(X) Me βt, U(t,) B(Y) Me βt, when t-interval i retricted to be finite [ T,T] Pre-infiniteimal generator. The counterpart of the logarithm in the abtract framework i introduced. There are two concept aociated with the logarithm of operator; one i the pre-infiniteimal generator and the other i t-differential of U(t, ). For T t, T, the weak limit wlim h 0 h (U(t+h,) U(t,)) u wlim h 0 h (U(t+h,t) I) U(t,) u i aumed to exit for u, which i an element of a dene ubpace Y of X. A linear operator A(t) : Y X i defined by (5) A(t)u : wlim h 0 h (U(t+h,t) I)u for u Y and T t, T, and then let t-differential of U(t,) in a weak ene be (6) t U(t,) u A(t)U(t,) u. Equation (6) i regarded a a differential equation atified by U(t,)u that implie a relation between A(t) and the logarithm: A(t) t U(t,) U(,t). Let u call A(t) defined by Eq. (5) for a whole family {U(t,)} T t, T the pre-infiniteimal generator. Note that pre-infiniteimal generator are not necearily infiniteimal generator; e.g., in t-independent cae, A defined by Eq. (5) i not necearily a denelydefined and cloed linear operator, while A mut be a denely-defined and cloed linear operator with it reolvent et included in {λ C : Reλ > β}. On the other hand, infiniteimal generator are necearily pre-infiniteimal generator. In the following a et of pre-infiniteimal generator i denoted by G(X) A principal branch of logarithm. The logarithm i defined by the Dunford integral in thi paper. Two difficultie of dealing with logarithm are it ingularity at the origin and it multi-valued property. By introducing a contant κ C, the ingularity can be handled. Let arg be a function of complex number, which give the angle between the poitive real axi to the line including the point to the origin. For the multi-valued

3 28 YORITAKA IWATA property, a principle branch (denoted by Log ) of the logarithm (denoted by log ) i choen for any complex number z C, a branch of logarithm i defined by Logz log z +iargz, where Z i a complex number choen to atify Z z, π < argz π, and argz argz +2nπ for a certain integer n. Lemma 2.. Let t and atify 0 t, T. For a given U(t,) defined in Sec. 2, it logarithm i well defined; there exit a certain complex number κ atifying (7) Log(U(t,)+κI) Logλ (λ U(t,) κ) dλ, where an integral path, which exclude the origin, i a circle in the reolvent et of U(t,)+κI. Here i independent of t and. Log(U(t,)+κI) i bounded on X. Proof. The logarithm Log hold the ingularity at the origin, o that it i neceary to how a poibility of taking a imple cloed curve (integral path) excluding the origin in order to define the logarithm by mean of the Dunford-Riez integral. It i not generally poible to take uch a path in cae of κ 0. Firt, U(t,) i aumed to be bounded for 0 t, T (Eq. (4)), and the pectral et of U(t,) i a bounded et in C. Second, for κ atifying κ > Me βt the pectral et of U(t,)+κI i eparated with the origin. Conequently it i poible to take an integral path including the pectral et of U(t,)+κI and excluding the origin. Equation (7) follow from the Dunford-Riez integral [3]. Furthermore, by adjuting the amplitude of κ, an appropriate integral path alway exit independent of t and. Log(U(t,)+κI) i bounded on X, ince i included in the reolvent et of (U(t,)+ κi). According to thi lemma, by introducing nonzero κ, the logarithm of U(t, ) + κi i well-defined without auming the ectorial property to U(t, ). On the other hand Eq. (7) i valid with κ 0 only for limited cae. 3. Main reult 3.. Logarithm repreentation of pre-infiniteimal generator. Theorem 3.. Let t and atify T t, T, and Y be a dene ubpace of X. For U(t,) defined in Sec. 2, let A(t) G(X) and t U(t,) be determined by Eq. (5) and (6) repectively. If A(t) and U(t,) commute, an evolution family {A(t)} T t T i repreented by mean of the logarithm function; there exit a certain complex number κ 0 uch that (8) A(t) u (I +κu(,t)) t Log (U(t,)+κI) u, where u i an element in Y. Note that U(t,) defined in Sec. 2 i aumed to be invertible. Proof. For U(t,) defined in Sec. 2, operator Log (U(t,)+κI) and Log (U(t+h,)+κI) are well defined for a certain κ (Lemma 2.). The t-differential in a weak ene i formally written by

4 INFINITESIMAL GENERATORS (9) wlim h 0 h {Log (U(t+h,)+κI) Log (U(t,)+κI)} Logλ{(λ U(t+h,) κ) (λ U(t,) κ) }dλ h Logλ wlim h 0 wlim h 0 {(λ U(t+h,) κ) U(t+h,) U(t,) (λ U(t,) κ) }dλ, h where, which i poible to be taken independent of t, and h for a ufficiently large certain κ, denote a circle in the reolvent et of both U(t,)+κI and U(t+h,)+κI. A part of the integrand of Eq. (9) i etimated a {(λ U(t+h,) κ) U(t+h,) U(t,) (λ U(t,) κ) }v h X (0) {(λ U(t+h,) κ) B(X) U(t+h,) U(t,) (λ U(t,) κ) }v, h X for v X. There are two tep to prove the validity of Eq. (9). In the firt tep, the former part of the right hand ide of Eq. (0) atifie {(λ U(t+h,) κ) B(X) <, ince λ i taken from the reolvent et of U(t+h,) κi. In the ame way the operator (λ U(t,) κ) i bounded on X and Y. Then the continuity of the mapping t (λ U(t,) κ) a for the trong topology follow: (λ U(t+h,) κ) (λ U(t,) κi) B(X) (λ U(t+h,) κ) B(X) (U(t+h,) U(t,))(λ U(t,) κ) B(X). In the econd tep, the latter part of the right hand ide of Eq. (0) i etimated a U(t+h,) U(t,) (λ U(t,) κ) u h X t+h () A(τ)U(τ,)(λ U(t,) κ) u dτ h h t +h t X A(τ)U(τ,) B(Y,X) (λ U(t,) κ) B(Y) u Y dτ for u Y. Becaue A(τ)U(τ,) B(Y,X) < i true by aumption, the right hand ide of Eq. () i finite. Equation () how the uniform boundedne with repect to h, then the uniform convergence (h 0) of Eq. (9) follow. Conequently the weak limit proce h 0 for the integrand of Eq. (9) i jutified, a well a the commutation between the limit and the integral. According to Eq. (9), interchange of the limit with the integral lead to t Log(U(t,)+κI) u dλ [ ( ) ] U(t+h,) U(t,) (Logλ)(λ U(t,) κ) wlim (λ U(t,) κ) u h 0 h

5 30 YORITAKA IWATA for u Y. Becaue we are alo allowed to interchange A(t) with U(t,), t Log(U(t,)+κI) u (Logλ)(λ U(t,) κ) A(t) U(t,) (λ U(t,) κ) dλ u (Logλ) (λ U(t,) κ) 2 U(t,) dλ A(t) u for u Y. A part of the right hand ide i calculated a (Logλ) (λ U(t,) κ) 2 U(t,) dλ λ (λ U(t,) κ) U(t,) dλ λ (λ U(t,) κ) {λ κ (λ U(t,) κ)} dλ (λ U(t,) κ) dλ κ λ (λ U(t,) κ) dλ (λ U(t,) κ) dλ κ λ (λ U(t,) κ) dλ (λ U(t,) κ) dλ { } κ(u(t,)+κi) λ (U(t,)+κI)(λ U(t,) κ) dλ (λ U(t,) κ) dλ { κ(u(t,)+κi) (λ U(t,) κ) dλ } λ dλ (I κ(u(t,)+κi) ) (λ U(t,) κ) dλ (I κ(u(t,)+κi) U(t,) n ) ν n+ dν I κ(u(t,)+κi), ν r n λ dλ due to the integration by part, where λ κ ν r i a properly choen circle large enough to include. () λ dλ 0 i een by applying dlogλ/dλ /λ. () ν r n U(t,)n ν n dν I follow from the ingularity of ν n. Conequently we have for u Y. A(t) u {I κ(u(t,)+κi) } t Log (U(t,)+κI) u (U(t,)+κI)U(t,) t Log (U(t,)+κI) u (I +κu(,t)) t Log (U(t,)+κI) u What i introduced by Eq. (8) i a kind of reolvent approximation of A(t) t Log (U(t,)+κI) (I +κu(,t)) A(t), in which A(t) i approximated by the reolvent of U(,t). A een in the following it i notable that there i no need to take κ 0. Thi point i different from the uual

6 INFINITESIMAL GENERATORS... 3 treatment of reolvent approximation. On the other hand, it i alo een by Eq. (8) that t Log (U(t,)+κI) (U(t,)+κI) κ0 A(t)(U(t,)+κI) how a tructure of imilarity tranform, where (U(t,) + κ) κ0 mean U(t,) + κ atifying a condition κ 0. Under the validity of Theorem 3., for T t, T, let a(t,) be defined by a(t,) : Log(U(t,)+κI), then Eq. (8) i written a A(t) (I +κu(,t)) t a(t,). Since κ i choen to eparate the pectral et of U(t,)+κI from the origin, the invere operator of (I +κu(,t)) U(,t)(U(t,)+κI) i well defined a (I+κU(,t)) (U(t,)+κI) U(t,). It alo enure that t a(t,) i well defined. Corollary 3.2. Let t and atify 0 t, T. For U(t,) and A(t) atifying the aumption of Theorem 3., the exponential of a(t,) i repreented by a convergent power erie (2) e a(t,) a(t,) n with a relation e a(t,) exp(log(u(t,)+κi)) U(t,)+κI. If a(t,) with different t and are further aumed to commute, (3) t e a(t,) u t a(t,) e a(t,) u i atified for u Y, where t denote a t differential in a weak ene. Proof. Since a(t,) i a bounded operator on X (Lemma 2.), the exponential of a(t,) i repreented by a convergent power erie [9]: (4) e a(t,) a(t,) n, where e a(t,) exp(log(u(t,)+κi)) e Logλ (λ U(t,) κ) dλ U(t,)+κI i atified. Since a(t,) with different t and commute, lead to t {a(t,)} n n{a(t,)} n t a(t,) (5) t e a(t,) u e a(t,) ( t a(t,))u for u Y. Thi i a linear evolution equation atified by e a(t,) u. Further calculi on Eq. (5) lead to t e a(t,) u t (U(t,)+κI)u t U(t,)u, and e a(t,) ( t a(t,))u (U(t,)+κI) t (Log(U(t,)+κI))u (U(t,)+κI)(U(t,)+κI) U(t,)A(t)u U(t,)A(t)u A(t)U(t,)u,

7 32 YORITAKA IWATA where Theorem 3. i applied. A a reult t U(t,)u A(t)U(t,)u i obtained. Note that e a(t,) doe not atify the emigroup property, while U(t,) atifie it. 4. Abtract Cauchy problem 4.. Autonomou cae. Logarithmic repreentation i utilized to olve autonomou Cauchy problem { t u(t) A(t)u(t), (6) u() u in X, where A(t) G(X) : Y X i aumed to be an infiniteimal generator of U(t,), T t, T i atified, Y i a dene ubpace of X permitting the repreentation hown in Eq. (8), and u i an element of X. A een in Eq. (3), under the aumption of commutation, a related Cauchy problem i obtained a { t v(t,) ( t a(t,)) v(t,), (7) v(,) e a(,) u in X, where t a(t,) t Log(U(t,) + κi) i well-defined. It i poible to olve rewritten Cauchy problem, and the olution i repreented by v(t,) e a(t,) a(t,) n u u for u X (cf. Eq. (4)). Theorem 4.. Operator e a(t,) i holomorphic. Proof. According to the boundedne of a(t,) on X (Lemma 2.), t n e a(t,) [8] i poible to be repreented a (8) t n e a(t,) λ n e λ (λ a(t,)) dλ, for a certain κ, where λ n e λ doe not hold any ingularity for any finite λ. Following the tandard theory of evolution equation, n t e a(t,) C θ,n π(tinθ) n i true for a certain contant C θ,n (n 0,,2,...), where θ (0π/2) and argt < π/2 are atified (for the detail, e.g., ee [7]). It follow that C θ,n (9) lim t +0 uptn t n e a(t,) lim t +0 uptn π(tinθ) n <. Conequently, for z t < tinθ, the power erie expanion (z t) n t n e a(t,) i uniformly convergent in a wider ene. Therefore e a(t,) i holomorphic.

8 INFINITESIMAL GENERATORS Theorem 4.2. For u X there exit a unique olution u(t) C([ T,T];X) for (6) with a convergent power erie repreentation ( ) (20) u(t) U(t,)u (e a(t,) a(t,) n κi)u κi u, where κ i a certain complex number. Proof. The unique exitence follow from the aumption for A(t). e a(t,) i holomorphic function (Theorem 4.) with the convergent power erie repreentation (Eq. (4)). The olution of the original Cauchy problem i obtained a u(t) (I +κu(,t)) v(t,) (I +κu(,t)) a(t,) n for the initial value u X. Note that A(t) i not aumed to be a generator of analytic evolution family, but only a generator of invertible evolution family. For I λ denoting the reolvent operator of A(t), the evolution operator defined by the Hille-Yoida approximation i written by ( ) u(t) lim exp I λ A(τ) dτ u, λ 0 o that more informative repreentation i provided by Theorem 4.2 compared to the tandard theory baed on the Hille-Yoida theorem Non-autonomou cae. Serie repreentation in autonomou part lead to the enhancement of the olvability. Let Y be a dene ubpace of X permitting the repreentation hown in Eq. (8), and u i an element of X. Let u conider non-autonomou Cauchy problem { t u(t) A(t)u(t)+f(t), (2) u() u in X, where A(t) G(X) : Y X i aumed to be the infiniteimal generator U(t,), f L ( T,T;X) i locally Hölder continuou on [ T,T] f(t) f() C H t γ for a certain poitive contant C H, γ and T t, T. The olution of non-autonomou problem doe not necearily exit in uch a etting (in general, f C([ T,T];X) i neceary). Theorem 4.3. Let f L ( T,T;X) be locally Hölder continuou on [ T,T]. For u X there exit a unique olution u(t) C([ T,T];X) for (2) uch that [ a(t,) n ] [ a(t,τ) n ] u(t) κi u + κi f(τ)dτ uing a certain complex number κ. Proof. Let u tart with cae with f C([ T,T];X). The unique exitence follow from the tandard theory of evolution equation. The repreentation follow from that of U(t,) and the Duhamel principle (22) u(t,) U(t,)u + (e a(t,) κi)u + U(t,τ)f(τ)dτ [e a(t,τ) κi]f(τ)dτ, u

9 34 YORITAKA IWATA where the convergent power erie repreentation of e a(t,) i valid (cf. Eq. (4)). Next let u conider cae with the locally Hölder continuou f(t). According to the linearity of Eq. (2), it i ufficient to conider the inhomogeneou term. For ǫ atifying 0 < ǫ << T, [e a(t,τ) κi]f(τ)dτ i true by taking ǫ 0. On the other hand, (23) A(t) [e a(t,τ) κi]f(τ)dτ A(t)U(t,τ)(f(τ) f(t))dτ + A(t)U(t,τ)(f(τ) f(t))dτ [e a(t,τ) κi]f(τ)dτ A(t)U(t,τ)f(τ)dτ A(t)U(t,τ)f(t)dτ τ U(t,τ)f(t)dτ A(t)U(t,τ)(f(τ) f(t))dτ U(t,t+ǫ)f(t)+U(t,)f(t) (+κu(,t)) t a(t,)[e a(t,τ) κi](f(τ) f(t))dτ U(t,t+ǫ)f(t)+U(t,)f(t), where τ U(t,τ) A(τ)U(t,τ) i utilized. The Hölder continuity and Eq. (9) lead to the trong convergence of the right hand of Eq. (23) A(t) [e a(t,τ) κi]f(τ)dτ (+κu(,t))( t a(t,))[e a(t,τ) κi](f(τ) f(t))dτ +(U(t,) I)f(t) (due to ǫ 0) for f L (0,T;X). A(t) i aumed to be an infiniteimal generator, o that A(t) i a cloed operator from Y to X. It follow that and A(t) [e a(t,τ) κi]f(τ)dτ [e a(t,τ) κi]f(τ)dτ Y (+κu(,t))( t a(t,))[e a(t,τ) κi](f(τ) f(t))dτ +(U(t,) I)f(t) X. The right hand ide of thi equation i trongly continuou on [ T,T]. A a reult, t [e a(t,τ) κi]f(τ)dτ [e a(t,t+ǫ) κi]f(t+ǫ)+ f(t)+ f(t)+ f(t)+a(t) ( t a(t,τ))e a(t,τ) f(τ)dτ (+κu(τ,t)) A(t)(U(t,τ)+κI)f(τ)dτ A(t)U(t,τ)f(τ)dτ [e a(t,τ) κi]f(τ)dτ.

10 INFINITESIMAL GENERATORS We ee that [ea(t,τ) κi]f(τ)dτ atifie Eq. (2), and that it i ufficient to aume f L (0,T;X) a Hölder continuou. Thi reult hould be compared to the tandard theory of evolution equation in which the inhomogeneou term f i aumed to be continuou on [ T,T]. If the inhomogeneou term f L p ([0,T];X) i further aumed to be atified for < p <, and Y D(A(t)) D(A(0)) and A( ) C([0,T],L(Y,X)), Eq. (7) with uch an inhomogeneou term correpond to the equation exhibiting the maximal regularity of type L p [6]. 5. Concluding remark A for the applicability of the theory, the condition to obtain the logarithmic repreentation (condition hown in Sec. 2) are not o retrictive; indeed, they can be atified by C 0 -group generated by t-independent infiniteimal generator. The mot retrictive condition to obtain Eq. (8) i the commutation between A(t) and U(t,). Such a commutation i trivially atified by t-independent A(t) A, and alo atified when the variable t i eparable (i.e., for an integrable function g(t), A(t) g(t)a). In thi ene the operator pecified in Theorem 3. correpond to a moderate generalization of t-independent infiniteimal generator. Acknowledgment. The author i grateful to Prof. Emeritu Hiroki Tanabe for valuable comment. Reference. W. Arendt, Semigroup and evolution equation: functional calculu, regularity and kernel etimate, Evolutionary equation. Vol. I, Handb. Differ. Equ., North-Holland, Amterdam, 2004, pp K.N. Boyadzhiev, Logarithm and imaginary power of operator on Hilbert pace, Collect. Math. 45 (994), no. 3, N. Dunford, Spectral theory. I. Convergence to projection, Tran. Amer. Math. Soc. 54 (943), M. Haae, Spectral propertie of operator logarithm, Math. Z. 245 (2003), no. 4, M. Haae, The functional calculu for ectorial operator, Operator Theory: Advance and Application, vol. 69, Birkhäuer Verlag, Bael, T. Kato, Perturbation theory for linear operator, Die Grundlehren der mathematichen Wienchaften, Band 32, Springer-Verlag New York, Inc., New York, T. Kato, Linear evolution equation of hyperbolic type, J. Fac. Sci. Univ. Tokyo Sect. I 7 (970), T. Kato, Linear evolution equation of hyperbolic type. II, J. Math. Soc. Japan 25 (973), T. Kato, A hort introduction to perturbation theory for linear operator, Springer-Verlag, New York-Berlin, S.G. Kreĭn, Linear differential equation in Banach pace, American Mathematical Society, Providence, R.I., 97.. C. Martínez Carracedo and M. Sanz Alix, The theory of fractional power of operator, North- Holland Mathematic Studie, vol. 87, North-Holland Publihing Co., Amterdam, V. Nollau, Über den Logarithmu abgechloener Operatoren in Banachchen Räumen, Acta Sci. Math. (Szeged) 30 (969), N. Okazawa, Logarithmic characterization of bounded imaginary power, Semigroup of operator: theory and application (Newport Beach, CA, 998), Progr. Nonlinear Differential Equation Appl., vol. 42, Birkhäuer, Bael, 2000, pp N. Okazawa, Logarithm and imaginary power of cloed linear operator, Integral Equation Operator Theory 38 (2000), no. 4, A. Pazy, Semigroup of linear operator and application to partial differential equation, Applied Mathematical Science, vol. 44, Springer-Verlag, New York, J. Prü and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equation with coefficient continuou in time, J. Math. Anal. Appl. 256 (200), no. 2,

11 36 YORITAKA IWATA 7. H. Tanabe, Equation of evolution, Monograph and Studie in Mathematic, vol. 6, Pitman (Advanced Publihing Program), Boton, Ma.-London, 979, Tranlated from the Japanee by N. Mugibayahi and H. Haneda. 8. A.E. Taylor, Spectral theory of cloed ditributive operator, Acta Math. 84 (95), Intitute of Innovative Reearch, Tokyo Intitute of Technology; Department of Mathematic, Shibaura Intitute of Technology addre: iwata Received 3/0/206; Revied 7/2/206

FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS

FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS Nguyen Thanh Lan Department of Mathematic Wetern Kentucky Univerity Email: lan.nguyen@wku.edu ABSTRACT: We ue Fourier erie to find a neceary