Semigroups of Operators

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1 Lecture 11 Semigroups of Operators In tis Lecture we gater a few notions on one-parameter semigroups of linear operators, confining to te essential tools tat are needed in te sequel. As usual, X is a real or complex Banac space, wit norm. In tis lecture Gaussian measures play no role Strongly continuous semigroups Definition Let {T (t) : t } be a family of operators in L(X). We say tat it is a semigroup if T () = I, T (t + s) = T (t)t (s) t, s. A semigroup is called strongly continuous (or C -semigroup) if for every x X te function T ( )x : [, ) X is continuous. Let us present te most elementary properties of strongly continuous semigroups. Lemma Let {T (t) : t } L(X) be a semigroup. Te following properties old: (a) if tere exist δ >, M 1 suc tat ten, setting ω = (log M)/δ we ave T (t) M, t δ, T (t) Me ωt, t. (11.1.1) Moreover, for every x X te function t T (t)x is continuous in [, + ) iff it is continuous at. (b) If {T (t) : t } is strongly continuous, ten for any δ > tere is M δ > suc tat T (t) M δ, t [, δ]. 133

2 134 Lecture 11 Proof. (a) Using repeatedly te semigroup property in Definition we get T (t) = T (δ) n 1 T (t (n 1)δ) for (n 1)δ t nδ, wence T (t) M n Me ωt. Let x X be suc tat t T (t)x is continuous at, i.e., lim + T ()x = x. Using again te semigroup property in Definition it is easily seen tat for every t > te equality lim + T (t + )x = T (t)x olds. Moreover, T (t )x T (t)x = T (t )(x T ()x) Me ω(t ) (x T ()x), < < t, wence lim + T (t )x = T (t)x. It follows tat t T (t)x is continuous in [, + ). (b) Let x X. As T ( )x is continuous, for every δ > tere is M δ,x > suc tat T (t)x M δ,x, t [, δ]. Te statement follows from te Uniform Boundedness Principle, see e.g. [Br, Capter 2] or [DS1, II.1]. If (11.1.1) olds wit M = 1 and ω = ten te semigroups is said semigroup of contractions or contractive semigroup. From now on, {T (t) : t } is a fixed strongly continuous semigroup. Definition Te infinitesimal generator (or, sortly, te generator) of te semigroup {T (t) : t } is te operator defined by { T () I } T () I D(L) = x X : lim x, Lx = lim x. + + By definition, te vector Lx is te rigt derivative of te function t T (t)x at t = and D(L) is te subspace were suc derivative exists. In general, D(L) is not te wole X, but it is dense, as te next proposition sows. Proposition Te domain D(L) of te generator is dense in X. Proof. Set a+t M a,t x = 1 T (s)x ds, a, t >, x X t a (tis is a X-valued Bocner integral). As te function s T (s)x is continuous, we ave (see Exercise 11.1) lim M a,tx = T (a)x. t In particular, lim t + M,t x = x for every x X. Let us sow tat for every t >, M,t x D(L), wic implies tat te statement olds. We ave T () I M,t x = 1 ( ) T ( + s)x ds T (s)x ds t = 1 ( +t ) T (s)x ds T (s)x ds t = 1 ( +t ) T (s)x ds T (s)x ds t t = M t,x M, x. t

3 Semigroups of Operators 135 Terefore, for every x X we ave M,t x D(L) and LM,t x = T (t)x x. (11.1.2) t Proposition For every t >, T (t) maps D(L) into itself, and L and T (t) commute on D(L). If x D(L), ten te function T ( )x is differentiable at every t and d T (t)x = LT (t)x = T (t)lx, t. dt Proof. For every x X and for every > we ave T () I T (t)x = T (t) T () I x. If x D(L), letting we obtain T (t)x D(L) and LT (t)x = T (t)lx. Fix t and let >. We ave T (t + )x T (t )x = T (t ) T () I x T (t )Lx as +. Tis sows tat T ( )x is rigt differentiable at t. Let us sow tat it is left differentiable, assuming t >. If (, t ) we ave T (t )x T (t )x = T (t ) T () I x T (t )Lx as +, as T (t ) T () I ( T () I x T (t )Lx T (t ) x Lx) + (T (t ) T (t ))Lx and T (t ) sup t t T (t) < by Lemma It follows tat te function t T (t)x is differentiable at all t and its derivative is T (t)lx, wic is equal to LT (t)x by te first part of te proof. Using Proposition we prove tat te generator L is a closed operator. Terefore, D(L) is a Banac space wit te grap norm x D(L) = x + Lx. Proposition Te generator L of any strongly continuous semigroup is a closed operator. Proof. Let (x n ) be a sequence in D(L), and let x, y X be suc tat x n x, Lx n =: y n y. By Proposition te function t T (t)x n is continuously differentiable in [, ). Hence for < < 1 we ave (see Exercise 11.1) T () I x n = 1 LT (t)x n dt = 1 T (t)y n dt,

4 136 Lecture 11 and ten T () I x y T () I (x x n ) + 1 T (t)(y n y)dt + 1 C + 1 x x n + C y n y + 1 T (t)ydt y, T (t)ydt y were C = sup <t<1 T (t). Given ε >, tere is suc tat for < we ave T (t)ydt/ y ε/3. For (, ], take n suc tat x x n ε/3(c + 1) and y n y ε/3c: we get T () I x y ε and terefore x D(L) and y = Lx, i.e., te operator L is closed. Proposition implies tat for any x D(L) te function u(t) = T (t)x is differentiable for t and it solves te Caucy problem u (t) = Lu(t), t, (11.1.3) u() = x. Lemma For every x D(L), te function u(t) := T (t)x is te unique solution of (11.1.3) belonging to C([, + ); D(L)) C 1 ([, + ); X). Proof. From Proposition we know tat u (t) = T (t)lx for every t, and ten u C([, + ); X). Terefore, u C 1 ([, + ); X). Since D(L) is endowed wit te grap norm, a function u : [, + ) D(L) is continuous iff bot u and Lu are continuous. In our case, bot u and Lu = u belong to C([, + ); X), and ten u C([, + ); D(L)). Let us prove tat (11.1.3) as a unique solution in C([, + ); D(L)) C 1 ([, + ); X). If u C([, + ); D(L)) C 1 ([, + ); X) is any solution, we fix t > and define te function v(s) := T (t s)u(s), s t. Ten (Exercise 11.2) v is differentiable, and v (s) = T (t s)lu(s) + T (t s)u (s) = for s t, wence v(t) = v(), i.e., u(t) = T (t)x. Remark If {T (t) : t } is a C -semigroup wit generator L, ten for every λ C te family of operators S(t) = e λt T (t), t, is a C -semigroup as well, wit generator L + λi : D(L) X. Te semigroup property is obvious. Concerning te generator, for every x X we ave and ten iff x D(L). S()x x lim + S()x x λ T () x = e + eλ x x T () x = lim eλ + eλ x x = Lx + λx +

5 Semigroups of Operators 137 Let {T (t) : t } be a strongly continuous semigroup. Caracterising te domain of its generator L may be difficult. However, for many proofs it is enoug to know tat good elements x are dense in D(L). A subspace D D(L) is called a core of L if D is dense in D(L) wit respect to te grap norm. Te following proposition gives an easily ceckable sufficient condition in order tat D is a core. Lemma If D D(L) is a dense subspace of X and T (t)(d) D for every t, ten D is a core. Proof. Let M, ω be suc tat T (t) Me ωt for every t >. For x D(L) we ave 1 Lx = lim t t T (s)lx ds. Let (x n ) D be a sequence suc tat lim n x n = x. Set y n,t = 1 t T (s)x n ds = 1 t T (s)(x n x) ds + 1 t T (s)x ds. As te D(L)-valued function s T (s)x n is continuous in [, + ), te vector T (s)x nds belongs to D(L). Moreover, it is te limit of te Riemann sums of elements of D (see Exercise 11.1), ence it belongs to te closure of D in D(L). Terefore, y n,t belongs to te closure of D in D(L) for every n and t. Furtermore, y n,t x 1 t T (s)(x n x) ds + 1 t tends to as t, n. By (11.1.2) we ave Ly n,t Lx = T (t)(x n x) (x n x) t Given ε >, fix τ > suc tat 1 τ τ + 1 t T (s)lx ds Lx ε, T (s)x ds x T (s)lx ds Lx. and ten take n N suc tat (Me ωτ + 1) x n x /τ ε. Terefore, Ly n,τ Lx 2ε and te statement follows Generation Teorems In tis section we recall te main generation teorems for C -semigroups. Te most general result is te classical Hille Yosida Teorem, wic gives a complete caracterisation of te generators. For contractive semigroups, i.e., semigroups verifying te estimate T (t) 1 for all t, te caracterisation of te generators provided by te Lumer-Pillips Teorem is often useful. We do not present ere te proofs of tese results, referring e.g. to [EN, II.3].

6 138 Lecture 11 First, we recall te definition of spectrum and resolvent. Te natural setting for spectral teory is tat of complex Banac spaces, ence if X is real we replace it by its complexification X = {x + iy : x, y X} endowed wit te norm x + iy X := sup x cos θ + y sin θ π θ π (notice tat te seemingly more natural Euclidean norm ( x 2 + y 2 ) 1/2 is not a norm in general). Definition Let L : D(L) X X be a linear operator. Te resolvent set ρ(l) and te spectrum σ(l) of L are defined by ρ(l) = {λ C : (λi L) 1 L(X)}, σ(l) = C\ρ(L). (11.2.1) Te complex numbers λ σ(l) suc tat λi L is not injective are te eigenvalues, and te vectors x D(L) suc tat Lx = λx are te eigenvectors (or eigenfunctions, wen X is a function space). Te set σ p (L) wose elements are all te eigenvalues of L is te point spectrum. For λ ρ(l), we set R(λ, L) := (λi L) 1. (11.2.2) Te operator R(λ, L) is te resolvent operator or briefly resolvent. We ask to ceck (Exercise 11.3) tat if te resolvent set ρ(l) is not empty, ten L is a closed operator. We also ask to ceck (Exercise 11.4) te following equality, known as te resolvent identity R(λ, L) R(µ, L) = (µ λ)r(λ, L)R(µ, L), λ, µ ρ(l). (11.2.3) Teorem (Hille Yosida). Te linear operator L : D(L) X X is te generator of a C -semigroup verifying estimate (11.1.1) iff te following conditions old: (i) D(L) is dense in X, (ii) (iii) ρ(l) {λ R : λ > ω}, (R(λ, L)) k L(X) M k N, λ > ω. (λ ω) k Before stating te Lumer Pillips Teorem, we define te dissipative operators. Definition A linear operator (L, D(L)) is called dissipative if for all λ >, x D(L). (λi L)x λ x (11.2.4) Teorem (Lumer Pillips). A densely defined and dissipative operator L on X is closable and its closure is dissipative. Moreover, te following statements are equivalent. (i) Te closure of L generates a contraction C -semigroup. (ii) Te range of λi L is dense in X for some (ence all) λ >.

7 Semigroups of Operators Invariant measures In our lectures we sall encounter semigroups defined in L p spaces, i.e., X = L p () were (, F, µ) is a measure space, wit µ() <. A property tat will play an important role is te conservation of te mean value, namely T (t)f dµ = f dµ t >, f L p (). In tis case µ is called invariant for T (t). Te following proposition gives an equivalent condition for invariance, in terms of te generator of te semigroup rater tan te semigroup itself. Proposition Let {T (t) : t } be a strongly continuous semigroup wit generator L in L p (, µ), were (, µ) is a measure space, p [1, + ), and µ() <. Ten T (t)f dµ = f dµ t >, f L p (, µ) Lf dµ = f D(L). Proof. Let f D(L). Ten lim t (T (t)f f)/t = Lf in L p (, µ) and consequently in L 1 (, µ). Integrating we obtain 1 Lf dµ = lim (T (t)f f)dµ =. t t Let f D(L). Ten te function t T (t)f belongs to C 1 ([, + ); L p (, µ)) and d/dt T (t)f = LT (t)f, so tat for every t, d T (t)f dµ = LT (t)f dµ =. dt X Terefore te function t X T (t)f dµ is constant, and equal to X f dµ. Te operator L p (, µ) R, f (T (t)f f)dµ, is bounded and vanises on te dense subset D(L); ence it vanises in te wole L p (, µ) Analytic semigroups We recall now an important class of semigroups, te analytic semigroups generated by sectorial operators. For te definition of sectorial operators we need tat X is a complex Banac space. Definition A linear operator L : D(L) X X is called sectorial if tere are ω R, θ (π/2, π), M > suc tat (i) ρ(l) S θ,ω := {λ C : λ ω, arg(λ ω) < θ}, (ii) R(λ, L) L(X) M λ ω λ S θ,ω. (11.4.1)

8 14 Lecture 11 Sectorial operators wit dense domains are infinitesimal generators of semigroups wit noteworty smooting properties. Te proof of te following teorem may be found in [EN, Capter 2], [L, Capter 2]. Teorem Let L be a sectorial operator wit dense domain. Ten it is te infinitesimal generator of a semigroup {T (t) : t } tat enjoys te following properties. (i) T (t)x D(L k ) for every t >, x X, k N. (ii) Tere are M, M 1, M 2,..., suc tat (a) T (t) L(X) M e ωt, t >, were ω is te constant in (11.4.1). (b) t k (L ωi) k T (t) L(X) M k e ωt, t >, (11.4.2) (iii) Te function t T (t) belongs to C ((, + ); L(X)), and te equality olds. d k dt k T (t) = Lk T (t), t >, (11.4.3) (iv) Te function t T (t) as a L(X)-valued olomorpic extension in a sector S β, wit β >. Te name analytic semigroup comes from property (iv). If O is an open set in C, and Y is a complex Banac space, a function f : O Y is called olomorpic if it is differentiable at every z O in te usual complex sense, i.e. tere exists te limit f(z) f(z ) lim =: f (z ). z z z z As in te scalar case, suc functions are infinitely many times differentiable at every z O, and te Taylor series k= f (k) (z )(z z ) k /k! converges to f(z) for every z in a neigborood of z. We do not present te proof of tis teorem, because in te case of Ornstein-Ulenbeck semigroup tat will be discussed in te next lectures we sall provide direct proofs of te relevant properties witout relying on te above general results. A more general teory of analytic semigroups, not necessarily strongly continuous at t =, is available, see [L] Self-adjoint operators in Hilbert spaces If X is a Hilbert space (inner product,, norm ) ten we can say more on semigroups and generators in connection to self-adjointness. Notice also tat te dissipativity condition can be reprased in te Hilbert space as follows. An operator L : D(L) X is dissipative iff (see Exercise 11.5) Re Lx, x, x D(L). (11.4.4) Let us prove tat any self-adjoint dissipative operator is sectorial.

9 Semigroups of Operators 141 Proposition Let L : D(L) X X be a self-adjoint dissipative operator. Ten L is sectorial wit θ < π arbitrary and ω =. Proof. Let us first sow tat te spectrum of L is real. x D(L) we ave If λ = a + ib C, for every (λi L)x 2 = (a 2 + b 2 ) x 2 2a x, Lx + Lx 2 b 2 x 2, (11.4.5) ence if b ten λi L is injective. Let us ceck tat in tis case it is also surjective, sowing tat its range is closed and dense in X. Let (x n ) D(L) be a sequence suc tat te sequence (λx n Lx n ) is convergent. From te inequality (λi L)(x n x m ) 2 b 2 x n x m 2, n, m N, it follows tat te sequence (x n ) is a Caucy sequence, ence (Lx n ) as well. Terefore, tere are x, y X suc tat x n x and Lx n y. Since L is closed, x D(L) and Lx = y, ence λx n Lx n converges to λx Lx rg (λi L) and te range of λi L is closed. Let now y be ortogonal to te range of (λi L). Ten, for every x D(L) we ave y, λx Lx =, wence y D(L ) = D(L) and λy L y = λy Ly =. As λi L injective, y = follows. Terefore te range of (λi L) is dense in X. From te dissipativity of L it follows tat te spectrum of L is contained in (, ]. Indeed, if λ > ten for every x D(L) we ave, instead of (11.4.5), (λi L)x 2 = λ 2 x 2 2λ x, Lx + Lx 2 λ 2 x 2, (11.4.6) and arguing as above we deeduce λ ρ(l). Let us now estimate R(λ, L), for λ = ρe iθ, wit ρ >, π < θ < π. For x X, set u = R(λ, L)x. Multiplying te equality λu Lu = x by e iθ/2 and ten taking te inner product wit u, we get wence, taking te real part, ρe iθ/2 u 2 e iθ/2 Lu, u = e iθ/2 x, u, ρ cos(θ/2) u 2 cos(θ/2) Lu, u = Re(e iθ/2 x, u ) x u and ten, as cos(θ/2) >, also wit θ = arg λ. u x λ cos(θ/2), Proposition Let {T (t) : t } be a C -semigroup. Te family of operators {T (t) : t } is a C -semigroup wose generator is L.

10 142 Lecture 11 Proof. Te semigroup law is immediately cecked. Let us prove te strong continuity. Possibly considering te rescaled semigroup e ωt T (t) wit M, ω as in (11.1.1), see Remark , we may assume tat T (t) L(X) M for every t, witout loss of generality, T (t) = T (t) 1 (see Exercise 11.6). For x X we ave wence T (t) x x 2 = T (t) x x, T (t) x x = T (t) x 2 + x 2 x, T (t) x T (t) x, x 2 x 2 ( x, T (t) x + T (t) x, x ) = 2 x 2 ( T (t)x, x + x, T (t)x ) lim sup T (t) x x = t by te strong continuity of T (t), and ten T ( ) x is continuous at. By Lemma , t T (t) x is continuous on [, ) and {T (t) : t } is a C -semigroup. Denoting by A its generator, for x D(L) and y D(A) we ave Lx, y = lim t t 1 (T (t) I)x, y = lim t x, t 1 (T (t) I)y = x, Ay, so tat A L. Conversely, for y D(L ), x D(L) we ave We deduce x, T (t) y y = T (t)x x, y = = T (s)x, L y ds = T (t) y y = LT (s)x, y ds T (s) L y ds, x, T (s) L y ds. wence, dividing by t and letting t we get Ay = L y for every y D(L ) and consequently L A. Te following result is an immediate consequence of Proposition Corollary Te generator L is self-adjoint if and only if T (t) is self-adjoint for every t > Exercises Exercise Let R be endowed wit te Lebesgue measure λ 1, and let f : [a, b] X be a continuous function. Prove tat it is Bocner integrable, tat b a f(t) dt = lim n n i=1 f(τ i ) b a n

11 Semigroups of Operators 143 [ for any coice of τ i ], a + (b a)i n, i = 1,..., n (te sums in tis approxi- a + (b a)(i 1) n mation are te usual Riemann sums in te real-valued case) and tat, setting F (t) = te function F is continuously differentiable, wit a f(s)ds, a t b, F (t) = f(t), a t b. Exercise Prove tat if u C([, + ); D(L)) C 1 ([, + ); X) is a solution of problem (11.1.3), ten for t > te function v(s) = T (t s)u(s) is continuously differentiable in [, t] and it verifies v (s) = T (t s)lu(s) + T (t s)u (s) = for s t. Exercise Let L : D(L) X X be a linear operator. Prove tat if ρ(l) ten L is closed. Exercise Prove te resolvent identity (11.2.3). Exercise Prove tat in Hilbert spaces te dissipativity condition in Definition is equivalent to (11.4.4). Exercise Let {T (t) : t } be a bounded strongly continuous semigroup. Prove tat te norm x := sup T (t)x t is equivalent to and tat T (t) is contractive on (X, ). Bibliograpy [Br] H. Brezis: Functional Analysis, Sobolev spaces and partial differential equations, Springer, 211. [DS1] N. Dunford, J. T. Scwartz: Linear operators I, Wiley, [EN] K. Engel, R. Nagel: One-parameter semigroups for linear evolution equations, Graduate Texts in Matematics, 194, Springer, [L] A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems, Birkäuser, Second edition, Modern Birkäuser Classics, 213. [Y] K. Yosida: Functional Analysis, 6 t ed., Springer, 198.