Strongly continuous semigroups

Transcription

1 Capter 2 Strongly continuous semigroups Te main application of te teory developed in tis capter is related to PDE systems. Tese systems can provide te strong continuity properties only. 2.1 Closed operators In tis introductory section we recall some definitions from te teory of (unbounded) linear operators (see [39] or [31] and [14] for more details). Let X be a (complex or real) Banac space. By linear operator A we mean te pair (D,A) consisting of a lineal D in X and a linear mapping A : D X. Te lineal D is called te domain of A and is denoted by D(A). Let two linear operators A : D(A) X X and B : D(B) X X be given. Ten te operator A is an extension of B (B A) if D(B) D(A) and Ax = Bx for x D(B). Tese operators are equal if we ave in addition D(B) = D(A). If te operator A is injective (i.e., Ax = if and only if x = ), ten we can define te inverse operator A 1 wit te domain D(A 1 ) = ImageA AD(A) by te formula A 1 y=x if y=ax. A linear operator A wit domain D(A) on a Banac space X is closed if for any sequence {x n } D(A) suc tat x n x an Ax n y for some x,y X strongly as n we ave tat x D(A) and y=ax. Te smallest closed extension of A, if it exists, is called te closure of A and is denoted by Ā. Operators aving a closure are called closable Exercise. An operator A is closed if and only if te grap is closed in X X. Γ(A)={(x;Ax) : x D(A)} X X 9

2 1 2 Strongly continuous semigroups Exercise. An operator A is closed if and only if A λi is closed, were λ is a scalar Exercise. Let A be closed. Sow tat operator A 1 is closed if A 1 exists Exercise. An operator A is closed if and only if it is weakly closed. Te latter means tat { } { } {xn } D(A), x D(A), implies. x n x, Ax n y weakly as n y=ax Hint: any lineal L in a Banac space is closed if and only if it is weakly closed. Te following property of closed operators is important in our considerations (for more general version we refer to Proposition A Proposition. Let A be a closed operator on X and (t) C([a,b],X). Assume tat (t) D(A) for every t and A(t) C([a, b], X). Ten Proof. b a b b (t)dt D(A) and A (t)dt = A(t)dt. a a On te Riemann sums for te integrals above we ave te relation A n i=1 (t i )(t i t i 1 )= Te result follows since A is closed. n i=1 A(t i )(t i t i 1 ). We also recall (see [31, 39]) tat te resolvent set of a linear operator A is { } ρ(a)= λ C λ A : D(A) X is bijective, (λ A) 1 : X X is a bounded operator Here and below we write λ instead of λi, were I is te identity operator on X. If A is closed, ten ρ(a)={λ C λ A : D(A) X is bijective}. Te resolvent set in open in C. Te spectrum σ(a) of A is te set C\ρ(A). Te operator R(λ,A)=(λ A) 1 is called te resolvent (operator) of A. 2.2 Semigroup generators Definition. Let S = {S t } t R+ be a strongly continuous semigroup on a Banac space X. Te infinitesimal operator (or generator) of S is te linear operator A defined on te domain

3 2.2 Semigroup generators 11 { } S t x x D(A)= x X : lim exists in te strong topology of X X t + t wic maps D(A) into X according to te relation Ax= lim t + S t x x, x D(A). t It is obvious tat D(A) is lineal and A is a linear operator Exercise. Sow tat if A is te generator of a semigroup S t, ten (A) A+cI is te generator for te semigroup S t := e ct S t for every c R. (B) ca is te generator for te semigroup S t := S ct for every c>. Te following assertions are standard and can be found at any book from te list [16, 18, 29, 38] in tat or anoter form Proposition. Let A : D(A) X X be a generator of strongly continuous semigroup S={S t } t R+. Ten te following properties old. (1) For every t > and x X, one as V(t;x) S τ xdτ D(A) and S t x x=a ( ) S τ xdτ. (2) If x D(A), ten S t x D(A) for every t and te mapping t S t x is strongly continuously differentiable. In addition, d dt S tx=as t x=s t Ax t, x D(A). (3) For eac x D(A) and τ t one as S t x S τ x= AS ξ xdξ = S ξ Axdξ. τ τ Proof. One can see from Exercise tat S I V(t,x)= 1 { + t S ξ xdξ S ξ xdξ Tis proves te first statement. Te second statement follows from te obvious relation S I S I S t x=s t x. } S t x x,. Te tird one can be obtained by te integration of te relation in te second statement.

4 12 2 Strongly continuous semigroups Teorem. Te generator A of a strongly continuous semigroup S t is a closed and densely defined linear operator tat determines te semigroup uniquely. Te latter property means tat if A and B are generators of te C -semigroups S t and T t on te same Banac space X, ten te equality A=B olds if and only if S t = T t for all t R +. Proof. Te domain D(A) is dense in X because by Proposition 2.2.3(1) for every fixed x X we ave z 1 V(,x)= 1 S τ xdτ D(A) and by (1.1.6) z x as. To prove tat A is closed we consider a sequence {x n } D(A) suc tat x n x an Ax n y strongly for some x,y X as n. It follows from Proposition 2.2.3(3) tat S t x n x n = S ξ Ax n dξ, t >, n=1,2,... Terefore passing to te limit we obtain tat S t x x= S ξ ydξ, t >. Tus by (1.1.6) we conclude tat tat x D(A) and y=ax. To prove te uniqueness of te generator for every x D D(A) = D(B) we consider te te function F(τ)=S t τ T τ x, τ [,t]. Using Proposition 2.2.3(2) one can sow tat F(τ) is strongly differentiable and d dτ F(τ)=S t τ[a B]T τ x τ [,t]. Tus F()=F(t) wic yields tat S t = T t for every t R +. Te following assertion establises relation between te resolvent of of generator and te corresponding semigroup Teorem (Integral representation of te resolvent). Let A : D(A) X X be a generator of a strongly continuous semigroup S={S t } t R+ suc tat S t Me ωt for all t R +, (2.2.1) for some M 1 and ω R. Ten te semiaxis(ω,+ ) lies in te resolvent set ρ(a) and for every λ > ω and x X we ave te relation

5 2.2 Semigroup generators 13 R(λ,A)x (λ A) 1 x= e λt S t xdt, (2.2.2) were we understand te integral as an improper Riemann integral. Proof. We first note tat te integral in (2.2.2) exists for every λ > ω and x X. Indeed, te function f(t) = e λt S t x is strongly continuous in X and tus te integral over every interval [, n] exists. By (2.2.1) we ave te exponentially decaying bound f(t) M exp{ (λ ω)t} ence te corresponding limit of finite integrals is also exists. Moreover, te integral in (2.2.2) defines a linear bounded operator in X wic we denote by R(λ). Now we sow tat R(λ) maps X into D(A). Indeed S I R(λ)x= 1 e λt [S t+ x S t x]dt = eλ 1 e λt S t xdt eλ e λt S t xdt. Terefore using te same argument as in Exercise we can conclude tat S I R(λ)x λr(λ)x x as. Tis means tat R(λ)x D(A) and AR(λ)x = λr(λ)x x for every x X. We obviously ave tat S I R(λ)x=R(λ) S I x R(λ)Ax for every x D(A). Tis yields R(λ)Ax=λR(λ)x x for x D(A). Tus we ave te relations (λ A)R(λ)x=x x X and R(λ)(λ A)x=x x D(A). Since A is closed, tey imply te inclusion (ω,+ ) ρ(a) and also te representation in (2.2.2) for every λ > ω and x X Exercise. Let S t be a strongly continuous semigroup on a Banac space X and ω be its te growt bound (see (1.1.3)). Ten ρ(a) {Reλ > ω } and σ(a) {Reλ ω } Exercise. Let A be te generator of a strongly continuous semigroup S t on a Banac space X. Sow tat x D(A) satisfies equation Ax = λx for some λ C if and only if S t x = e λt x for all t >. Hint: if x is eigen-element for A, ten from Proposition 2.2.3(3) we ave tat S t x x=λ S τxdτ. Tis exercise sows tat te eigenvalues of te generator A produces te eigenvalues of te semigroup via exponential mapping. Tis is not true for te wole spectrum witout additional ypoteses concerning a semigroup. We can only prove te following assertion.

6 14 2 Strongly continuous semigroups Teorem (Spectral inclusion). For te generator A of a strongly continuous semigroup S t on a Banac space X, we ave te inclusion for te spectrums e tσ(a) σ(s t )\{} for all t. (2.2.3) Tere is numerous examples wic sows tat te equality in tis relation is not true in general (see, e.g., te discussion in [15]). Proof. It follows from Proposition applied to te semigroup S t = e λt S t we obtain tat ( ) S t x e λt x=(a λ) e λ(t τ) S τ xdτ = e λ(t τ) S τ (A λ)xdτ for every x D(A). Terefore S t e λt is not bijective, if(a λ) fails to be bijective.

7 Capter 3 Generation Teorems We discuss conditions wic guarantee tat te operator given generates a strongly continuous semigroup. 3.1 Hille-Yosida Teorem We start wit Hille-Yosida Teorem wic deals wit contraction semigroups. We recall tat a semigroup S ={S t } t R+ is said to be contraction semigroup if (1.1.2) olds wit M = 1 and ω =, i.e., we ave tat S t 1 for all t Teorem (Hille, Yosida, 1948). An operator A in a Banac space X is te generator of a C contraction semigroup if and only if A is a closed densely defined operator; for eac λ > we ave tat λ ρ(a) and (λ A) 1 1, λ >. (3.1.1) λ Proof. Below we follow te line of argument presented in [29]. NECESSITY: If A : D(A) X X is te generator of a C contraction semigroup, ten by Teorem A is closed and densely defined. To prove te second statement we use Teorem wic olds wit M= 1 and ω = and tus implies tat(,+ ) ρ(a) and also te relation (λ A) 1 x e λt S t x dt e λt dt = 1, λ >. λ SUFFICIENCY: As in [29] we use te Yosida approximation metod Lemma. Let A be a closed densely defined operator satisfying (3.1.1). Ten [ λ(λ A) 1 x ] = x, x X. (3.1.2) lim λ + 15

8 16 3 Generation Teorems Proof. Let x D(A). Ten λ(λ A) 1 x x = (λ A) 1 Ax 1 Ax as λ. λ By (3.1.1) we ave λ(λ A) 1 1. Tus we can obtain (3.1.2) by te density argument. In te next lemma we define te Yosida approximation of te operator A Lemma. Let te ypoteses of Lemma be in force. Ten A λ = λa(λ A) 1 = λ + λ 2 (λ A) 1, (3.1.3) is a bounded linear operator in X for every λ > suc tat lim λ + A λ = Ax, x D(A). (3.1.4) Moreover, te operators A λ commute for different λ >. Proof. We apply Lemma wit Ax instead of x. Lemma implies tat te operators A λ generates a family {e ta λ} λ> of uniformly continuous semigroups. Tis family is uniformly bounded. Indeed using (3.1.1) and also Teorem 1.2.6(1) we ave We also note tat e ta λ =e tλ e tλ 2 (λ A) 1 exp{ tλ +tλ 2 (λ A) 1 } 1. 1 e ta λ e ta µ = =t 1 d dξ [ ] e tξ A λ e t(1 ξ)a µ dξ e tξ A λ [ A λ A µ ] e t(1 ξ)a λ dξ, Since te operators e ta λ commute for te different λ >, te relation above yields e ta λ x e ta µ x t A λ x A µ x, x X Tis and also Lemma imply tat tere exists te strong limit S t x= lim λ + eta λ x x X. Tis limit defines a family{s t } of linear operators on X suc tat S t 1 and S t x e ta λ x t Ax A λ x, x D(A). Tis relation allows us to sow tat S t is C -semigroup of contractions. Now prove tat A is te generator for S t. For tis we first note tat

9 3.1 Hille-Yosida Teorem 17 [ S t x x= lim e ta λ x x ] [ ] = lim e τa λ A λ xdτ, x X. λ + λ + Tis implies S t x x= S τ Axdτ, x D(A). If S t as a generator B, ten te relation above implies tat A B, i.e., D(A) D(B) and Ax=Bx for x D(A). Tus (I B)D(A)=(I A)D(A)=X. Since due to te necessity part of te teorem te operator (I B) 1 exists, we obtain tat D(A)=(I B) 1 X. Tis implies tat D(B)=D(A) and ence B=A, i.e., we ave constructed a C -semigroup wit generator A. Tis completes te proof of te teorem. Te following assertion gives us a caracterization of generators of arbitrary strongly continuous semigroups Teorem (Feller, Miyadera, Pillips, 1952). An operator A in a Banac space X is te generator of a C -semigroup possessing te estimate if and only if S t Me ωt for all t R + and for some M 1 and ω R (3.1.5) A is a closed densely defined operator; for eac λ > ω we ave tat λ ρ(a) and (λ A) n M (λ ω) n, λ >, n=1,2,... (3.1.6) Idea of te proof. Te argument involves te following observations. First, by considering (if it is necessary) scaled semigroup S t = e ωt S t we can reduce te proof to te case of bounded semigroups (ω = ). Next we can define te (equivalent) norm in X: x = sup S t x. t It is easy to see tat S t is contraction wit respect to tis norm. Tis allows to derive te estimate (3.1.6) from te corresponding statement of Teorem In te inverse statement te following norm x = lim µ + { [ sup µ n (µ A) n x ] } n Z + is involved. We refer for details to any source from te list [16, 18, 29, 38].

10 18 3 Generation Teorems 3.2 Lumer-Pillips teorem In tis section we consider a convenient caracterization of generators of contraction semigroups wic does not require explicit knowledge of te resolvent. Tis caracterization became very useful in various PDE applications. We start wit te following definition Definition. A linear operator A on a Banac space X is called dissipative if (λ A)x λ x for every λ > and x D(A). (3.2.1) If A is dissipative, ten A is called accretive. Tis definition is motivated by te following assertion wic represents te caracteristic property of dissipative operators in Hilbert spaces Proposition. An operator A is dissipative in a Hilbert space X if and only if Re(Ax,x) 1 [(Ax,x)+(x,Ax)] for every x D(A). (3.2.2) 2 Proof. Let A be dissipative. Ten λ 2 x 2 (λ A)x 2 = λ 2 x 2 2λRe(Ax,x)+ Ax 2 for every λ > and x D(A). Tus In te limit λ + we obtain (3.2.2). If (3.2.2) olds, ten 2λRe(Ax,x)+λ 1 Ax 2. (λ A)x 2 = λ 2 x 2 2λRe(Ax,x)+ Ax 2 λ 2 x 2 + Ax 2. Tis yields dissipativity Teorem (Lumer, Pillips, 1961). Let A : D(A) X X be a densely defined operator on a Banac space X. Ten A generates a strongly continuous contraction semigroup if and only if (i) A is dissipative; (ii) tere exists λ > suc tat λ A is surjective. Moreover, if A is te generator, ten λ A is surjective for every λ >. Proof. We use te argument presented in [29]. 1. Let A be te generator. Ten by Hille-Yosida Teorem (,+ ) ρ(a) and tus λ A is surjective for every λ >. To prove te dissipativity relation in (3.2.1) we note tat (3.1.1) implies tat

11 3.2 Lumer-Pillips teorem 19 λ (λ A) 1 y y for every λ > and y X. If we take y=(λ A)x, ten we obtain (3.2.1). 2. Let te conditions in (i) and (ii) be in force. Prove tat A is closed. By dissipativity relation (3.2.1) te operator λ A is injective. Tus tere exists (λ A) 1 defined on X and suc tat (λ A) 1 x λ 1 x for every λ > and x X, i.e., (λ A) 1 is a bounded operator. Tis implies (see Exercise 2.1.3) tat λ A is closed wic is equivalent to te closeness of A. Tus it remains to establis tat(,+ ) ρ(a) and relation (3.1.1) old. It is sufficient to prove tat(λ A)D(A)=X for every λ > (if tis is true, ten (,+ ) ρ(a) and (3.1.1) follows from (3.2.1)). Let Λ ={λ > : (λ A)D(A)=X}. It is clear tat λ Λ if and only if λ ρ(a) R +. Since ρ(a) is open, we can conclude tat Λ is an open set in te semi-axisr + \{}. To sow tat Λ =R + \{} it is sufficient to sow tat Λ is closed inr + \{}. Let λ n Λ and λ n λ >. Since λ n Λ, we ave tat y X x n D(A) : λ n x n Ax n = y. It follows from (3.2.1) tat x n λn 1 y C. Te same relation also yields Tus we ave λ m x n x m λ m (x n x m ) A(x n x m ) = (λ m λ n )x n + λ n x n Ax }{{} n (λ m x m Ax m ) }{{} y y = λ m λ n x n λ m λ n λn 1 y. x n x m λ m λ n y. λ n λ m Tis implies tat {x n } is a Caucy sequence, i.e., tere exists te strong limit x = lim n x n X. We also ave Ax n = λ n x n y λx y as n. Since A is closed, tis yields tat x D(A) and(λ A)x=y. Tus(λ A)D(A)= X. Tis means tat te set Λ is closed in R + \{} and tus te proof of Teorem is complete Remark. In te case wen X is a Hilbert space 1 one can prove tat te conditions (i) and (ii) in Teorem imply tat A is a densely defined operator (see 1 Even reflexivity of X would be enoug.

12 2 3 Generation Teorems [29, Teorem 4.6., p.16]). Tis means tat in applications of te Lumer-Pillips teorem tere is no need to ceck te density condition in te explicit form for te generation in tis case. 3.3 Some generation results in Hilbert spaces Now we consider te case wen X is a Hilbert space wit te inner product(, ). We recall te following definition Definition (Adjoint operator). Let A : D(A) X X be a densely defined operator on X. Ten an operator A : D(A ) X X is said to be adjoint of A if te domain D(A ) consists of elements z X suc tat x (Ax,z) is a bounded linear functional on D(A) (wic can be extended by continuity to X); (Ax,z)=(x,A z) for all x D(A) and z D(A ). We collect some elementary properties of adjoint operators in te following exercise Exercise. Sow tat (A) If A is bounded, ten A is also bounded and A = A. (B) If A,B are bounded, ten (AB) = B A. (C) If A is closed and densely defined, so is A. Furtermore, A = A. (D) Using Exercise and te relation between te graps of te operators A and A sow tat te adjoint operator A is closed (even if A is not closed). Hint: see, e.g., [31] Teorem (Adjoint semigroup). Let A generate a C semigroup {S t } t R+ on a Hilbert space X. Ten {S t} t R+ is a C semigroup on X wose generator is A. Proof. Te argument is standard (see, e.g.,[18, p.3]. Since S t S τ = (S τ S t ), te family {S t} t R+ of adjoint operators forms a semigroup and S t = S t Me ωt, t >. (3.3.1) For x D(A) and z D(A ) we also ave tat Tus (x,st z z)=(s t x x,z)= (AS τ x,z)dτ = (S τ x,a z)dτ = (x,sτa z)dτ. (3.3.2)

13 3.3 Some generation results in Hilbert spaces 21 Tis implies tat (x,st z z) x Sτ A z dτ Mte ωt x A z. S t z z Mte ωt A z, z D(A ). Hence using (3.3.1) we conclude tat{s t} t R+ is a strongly continuous semigroup. Let B denote its generator. Ten for every x D(A) and y D(B) we ave (Ax,y)= lim 1 (S t x x,y)= lim 1 (x,s t y y)=(x,by) Tus B A. One can also see from (3.3.2) tat Tis implies tat A B. St z z= SτA zdτ. Te following assertion is a direct consequence of Lumer-Pillips Teorem Teorem. Let A : D(A) X X be a densely defined closed operator on a Hilbert space X. If bot A and its adjoint A are dissipative, ten A generates a contraction semigroup on X. Proof. By Lumer-Pillips Teorem it suffices to prove tat(i A)D(A) = X. Since A is dissipative and closed, one can see tat(i A)D(A) is a (closed) subspace of X. Tus if(i A)D(A) X, ten tere exists z X, z, suc tat(z,x Ax)= for any x D(A). Tis implies tat z D(A ) and z A z=. Dissipativity of A implies tat z = and tus (I A)D(A) = X. Te following result deals wit strongly continuous groups of linear operators (see Definition 1.1.1). By Exercise we can extend a semigroup {S t } t to a group provided Sτ 1 exists for some τ >. We also recall tat a group {S t } t R is called unitary if S t x = x for all t R and x X. One can see tat in tis case S t are unitary operators and St = S t. Te main result in te teory of unitary groups is te following teorem Teorem (Stone, 1932). Let A be a densely defined operator on a Hilbert space X. Ten A generates a unitary C -group {S t } t R on X if and only if A is skew-adjoint, i.e., A = A. Proof. If A is a generator unitary C -group S t, ten by Teorem A is te generator for St = St 1 = S t. Tus a simple calculation sow tat A = A. If A = A, ten A is closed (see Exercise 3.3.2(D)) and we ave tat Re(Ax,x)= 1 [(Ax,x)+(x,Ax)]= for x D(A). 2 It is clear tat Re(A x,x) =. Terefore by Teorem bot operators A and A generate contraction C -semigroups{s t + } t and{st =(S t + ) } t respectively. One can see tat

14 22 3 Generation Teorems d dt S+ t S t x= d dt S t S + t x= for all x D(A). Terefore S t + = (St ) 1 = (St) 1 for every t. Hence te operators are unitary and we can define unitary C -groups by te formula { S + S t = t, for t ; S t, for t <. It is clear tat A is te generator for S t.