Strongly continuous semigroups and their generators

Transcription

1 LECTURE 2 Strongly continuous semigroups nd their genertors In the first prt of the Internet Seminr we investigte the liner evolution eqution (or more precisely, the initil vlue or Cuchy problem) u (t) = Au(t), t 0, u(0) = u 0, (2.) on Bnch spce X. Here A is given liner opertor with domin D(A) nd the initil vlue u 0 X (often u 0 D(A)). We wnt to develop systemtic theory for (2.) which will lso be the bsis for our study of semiliner problems. As explined in Lecture, the solution of (2.) will be given by u(t) = T (t)u 0 for n opertor semigroup (T (t)) t 0 = T ( ) on X. Our nlysis strts with the study of such semigroups. For ech semigroup we introduce its genertor A s the derivtive of t T (t) t t = 0, roughly speking. We next investigte some of their bsic properties, solve problem (2.) for the genertor A, nd study simple but instructive clss of exmples, the trnsltion semigroups. In the following lectures, we will tckle the more difficult problem to chrcterize those (given) opertors A which re genertors of strongly continuous semigroup. A few words bout our nottion: Throughout we ssume tht (X, X ) is complex Bnch spce with X 0}, where we mostly write insted of X, if no confusion is to be expected. By B(X, Y ) we denote the spce of ll bounded liner opertors from X into nother Bnch spce Y, setting B(X) = B(X, X). Mostly, the opertor norm is lso designted by. Further, X is the dul spce of X nd I is the identity mp on X. We write x, x := x (x) for x X nd x X. The sclr product on Hilbert spce H is denoted by (x y) H, or simply (x y), for x, y H. By M we designte the chrcteristic function of set M. We put R + = [0, ) nd R = (, 0]. For M R d we write C(M, X) for the vector spce of continuous functions f : M X. We use the subspces C b (M, X) = f C(M, X) f is bounded }, C 0 (M, X) = f C(M, X) f(s) 0 s s nd s s M \ M }, C c (M, X) = f C(M, X) supp f M is compct }, where C b (M) := C b (M, C) etc. Here supp f is the closure in R d of the set s M f(s) 0 }. Below we will repet these definitions in specific cses. The spces C b (M, X) nd C 0 (M, X) re lwys equipped the sup norm f := sup s M f(s) nd then become Bnch spces. We employ nlogous nottions for spces of differentible functions. For instnce, C0 ([0, ]) = f C ([0, ]) f, f C 0 ([0, ]) }. 9

2 If M is Borel subset of R d nd p [, ], we write L p (M) for the usul Lebesgue spce of complex vlued functions with respect to the Lebesgue mesure on M nd endow it with the p norm, given by f p p := M f p dx if p < nd by the essentil supremum for p =. If M = (, b) R, we write C 0 (, b), L p (, b) nd so on. Definition 2.. A mp T ( ) : R + B(X) is clled strongly continuous opertor semigroup or just C 0 semigroup if the following conditions re fulfilled: () T (0) = I nd we hve T (t + s) = T (t)t (s) for ll t, s 0. (b) For ech x X the orbit, defined s the mp T ( )x : R + X, is continuous. The genertor A of T ( ) is given by setting nd defining D(A) := x X the limit Ax := lim t 0 + t t T (t)x, lim t 0 + t (T (t)x x) exists in X} (T (t)x x) for x D(A). We lso sy tht A genertes T ( ). If one replces throughout in this definition R + by R nd t 0 + by t 0, one obtins the concept of C 0 group with genertor A. Property () in Definition 2. is clled the semigroup lw nd (b) is the strong continuity. We point out tht the semigoup does not need to be continuous with respect to the opertor norm, cf. Exmple 2.6. Observe tht the bove definition directly implies tht D(A) is liner subspce of X nd A is liner mp in X which is uniquely determined by the semigroup. Let T ( ) be C 0 semigroup. The semigroup opertors then commute since T (t)t (s) = T (t + s) = T (s + t) = T (s)t (t) (2.2) holds for ll t, s 0. By induction, we obtin ( n ) n T (nt) = T t = T (t) = T (t) n (2.3) for ll n N nd t 0. If T ( ) is even C 0 group, it stisfies j= j= T (t)t ( t) = T (0) = I = T ( t)t (t) (2.4) for ll t R. Hence, T (t) is invertible with inverse T ( t) for every t R. We first look t the finite dimensionl sitution, which is rther simple since here one cn construct the semigroup from the given opertor A by power series. This does not work for unbounded A. Exmple 2.2. Let X = C d, A B(X) = C d d nd set T (t) := e ta t n := n! An for t R. It is known from nlysis courses tht the series converges in B(X), T ( ) stisfies () in Definition 2. nd T ( ) is continuously differentible even in 0 n=0

3 B(X) with d dt eta = Ae ta for ll t R. In prticulr, T ( ) is C 0 group with genertor A. Moreover, for ny given u 0 X the function u : R + X defined by u(t) = e ta u 0 solves the liner ordinry differentil eqution u (t) = Au(t), t R, u(0) = u 0. The sme results hold for ny bounded liner opertor A on Bnch spce X, see Exercise 2.. The simple Definition 2. hs mny stonishing consequences. We first observe tht every C 0 semigroup is exponentilly bounded. This fct then leds to the subsequent bsic definition. Lemm 2.3. Let T ( ) be C 0 semigroup. Then there re M nd ω R such tht T (t) Me ωt holds for ll t 0. Proof. By strong continuity, ech orbit T ( )x is bounded on [0, ]. Hence, there is constnt c > 0 with T (τ) c for ll τ [0, ] due to the principle of uniform boundedness. Let t 0. Tke n N 0 nd τ [0, ) with t = n + τ. Setting ω := log c, we deduce from Definition 2. () nd (2.3) tht T (t) = T (τ)t (n) = T (τ)t () n c c n (n+) log c = e = e tω e ( τ) log c e mxlog c,0} e tω. Definition 2.4. Let T ( ) be C 0 semigroup with genertor A. Then ω 0 (T ) := ω 0 (A) := inf ω R } M ω : T (t) M ω e ωt for ll t 0 is clled the growth bound of T ( ). Lemm 2.3 sys tht ω 0 (T ) <. It my hppen tht ω 0 (T ) =, see Exmple 2.6. The nottion ω 0 (A) will be justified in the next lecture where we show tht n opertor A cn generte t most one C 0 semigroup. We point out tht in generl the infimum in Definition 2.4 is not minimum even in the mtrix cse. For instnce, for X = C 2 (endowed with the -norm) nd A = ( ) 0 0 0, we hve T (t) = e ta = ( ) t 0, so tht T (t) = t + for t 0, while ω 0 (T ) = 0. The next lemm often helps to verify the strong continuity of n opertor semigroup. Lemm 2.5. Let T ( ) : R + B(X) be mp stisfying condition () in Definition 2.. Then the following ssertions re equivlent. () T ( ) is strongly continuous (nd thus C 0 semigroup). (b) It holds lim t 0 + T (t)x = x for ll x X. (c) There re number t 0 > 0 nd dense subspce D X such tht sup 0 t t0 T (t 0 ) < nd lim t 0 + T (t)x = x for ll x D. The nlogous equivlences hold in the group cse. Proof. The impliction () (c) is n immedite consequence of Lemm 2.3. Assertion (b) follows from (c) by the Bnch-Steinhus theorem.

4 So it only remins to conclude () from (b). Fix x X nd t > 0. For h > 0 the semigroup property implies T (t + h)x T (t)x = T (t)(t (h)x x) T (t) T (h)x x, where the right hnd side of this inequlity converges to 0 s h tends to 0. In the cse tht h ( t, 0], we note tht Lemm 2.3 yields T (t + h) Me ω(t+h) Me ω t for some constnts M nd ω R. As result, T (t + h)x T (t)x = T (t + h)(x T ( h)x) Me ω t x T ( h)x 0 s h 0, nd () holds. The ssertions bout groups re shown similrly. In the bove lemm the impliction (c) () cn fil if one omits the boundedness ssumption (cf. Exercise I.5.9(4) in [EN99]). We now exmine bsic clss of exmples for C 0 semigroups, the trnsltion semigroups. They re given by n explicit formul (which is rre exception) nd re thus very convenient to illustrte vrious spects of the theory. Their genertors will be determined in the next lecture. Exmple 2.6 (Left trnsltion semigroups on R nd [0, )). () Let X = C 0 (R) = f C(R) f(s) 0 s s } nd T ( ) be given by (T (t)f)(s) := f(s + t) for t R, f X, s R. We clim tht T ( ) is C 0 group on X. Clerly, T (0) = I nd T (t) is liner isometry on X so tht T (t) =. We further obtin T (t)t (r)f = (T (r)f)( + t) = f( + t + r) = T (t + r)f for ll f X nd r, t R. Hence, T (t)t (r) = T (t + r). We employ Lemm 2.5 to verify the strong continuity. For f C c (R) the function T (t)f converges uniformly to f s t tends to 0 since f is uniformly continuous. It thus remins to check C c (R) = C 0 (R). For ech n N tke cut off function ϕ n C(R) stisfying ϕ n = on [ n, n], 0 ϕ n nd supp ϕ n ( n, n + ). For f C 0 (R), we then hve ϕ n f C c (R) nd f ϕ n f = sup ( ϕ n (s))f(s) sup f(s) 0 s n s n s n. We now conclude tht T ( ) is C 0 group by mens of Lemm 2.5. The sme ssertions hold for X = L p (R) with p < by similr rguments, see Exercise 2.2. In contrst to these results, T ( ) is not strongly continuous on X = L (R). Indeed, consider f = [0,] nd observe tht }, s + t [0, ] T (t)f(s) = [0,] (s + t) = = [ t, t] (s) 0, s + t / [0, ] for s, t R. Thus, T (t)f f = for every t 0. In ddition, T ( ) is not continuous s B(X)-vlued function for X being L p (R) (see Exercise 2.2) or C 0 (R). In fct, for X = C 0 (R) consider for ech n 2

5 N functions f n C c (R) with 0 f n, f n (n) = nd supp f n (n n, n+ n ). We then hve supp T ( 2 n )f n (n 3 n, n n ) for n N, which implies T ( 2 n ) I T ( 2 n )f n f n = for ll n N. (b) Let X = C 0 ([0, )) = f C([0, )) lim s f(s) = 0 }. For t 0 nd s [0, ), we define f(s + t), s + t <, (T (t)f)(s) := 0, s + t. We show tht T ( ) is C 0 semigroup on X. Since f(s + t) 0 s s + t, we hve T (t)f X. Clerly, T (t) is liner nd T (t). Note tht T (t) = 0 whenever t. In this cse, one sys tht T ( ) is nilpotent. As consequence, ω 0 (T ) =. Let t, r 0 nd s [0, ). We then obtin (T (r)f)(s + t), if s + t <, (T (t)t (r)f)(s) = 0, else, f(s + t + r), if s + t <, s + t + r <, = 0, else, = (T (t + r)f)(s). Hence, T ( ) is semigroup (which cnnot be extended to group since e.g. T () = 0 is not bijective). As in () one sees tht C c ([0, )) = f C([0, )) b f (0, ) : supp f [0, b f ] } is dense subspce of X. For f C c ([0, )) nd t (0, b f ), we compute f(s + t) f(s), s [0, t), T (t)f(s) f(s) = 0, s [ t, ) [b f, ], nd deduce lim t 0 T (t)f f = 0 using the uniform continuity of f. According to Lemm 2.5, T ( ) is C 0 semigroup on X. We next stte the solution concept for eqution (2.). Definition 2.7. Let A be liner opertor on X with domin D(A) nd let x D(A). We sy function u : R + X solves the Cuchy problem u (t) = Au(t), t 0, u(0) = x, (2.5) if u C (R +, X) stisfies u(t) D(A) for ll t 0 nd fulfills (2.5). We next show tht if A genertes C 0 semigroup, then the semigroup gives the unique solution of (2.5). Moreover, T (t) nd A commute on D(A). Proposition 2.8. Let A generte the C 0 semigroup T ( ) nd x D(A). Then T (t)x D(A), AT (t)x = T (t)ax for ll t 0 nd the function is the unique solution of (2.5). u : R + X, t T (t)x, 3

6 Proof. ) Let t 0, h > 0 nd x D(A). We obtin h (T (t + h)x T (t)x) = h (T (h) I)T (t)x = T (t) h (T (h)x x) T (t)ax s h 0. The very definition of A yields T (t)x D(A) nd AT (t)x = T (t)ax. Moreover, T ( )x is differentibility from the right. Let 0 < h < t. It further holds h (T (t h)x T (t)x) = T (t h) h (T (h)x x) T (t)ax s h 0, where we hve used Lemm 2.9 below (with S(t, h) = T (t h)). Since T ( )Ax is continuous, we hve shown tht T ( )x C (R +, X) with derivtive d dt T ( )x = AT ( )x; i.e., u solves (2.5). 2) Let v be nother solution of (2.5) nd t > 0. Set w(s) := T (t s)v(s) for s [0, t]. Lemm 2.9 (with S(t, s) = T (t s) nd Y = D(A)) nd the first step now imply tht d ds w(s) = T (t s)v (s) T (t s)av(s) = 0, where the lst equlity follows from the ssumption tht v solves (2.5). So for every x X the sclr function w( ), x is differentible with vnishing derivtive nd thus constnt, which leds to T (t)x, x = w(0), x = w(t), x = v(t), x for ll t 0 nd x X. The Hhn-Bnch theorem now yields T ( )x = v. In generl one relly needs the extr condition tht x D(A) to obtin solution of (2.5). For instnce, if f C 0 (R) \ C (R), then the orbit T ( )f of the trnsltion semigroup on C 0 (R) is not differentible, cf. Exmple 2.6. We continue with the lemm used in the previous proof. Lemm 2.9. Let b > be rel numbers, M = (t, s) [, b] 2 t s }, S : M B(X) be strongly continuous nd f C([, b], X). Then the function g : M X, (t, s) S(t, s)f(s), is lso continuous. Further, let Y X be subspce nd the mp [, t] X, s S(t, s)y, hs the derivtive s S(t, s)y for ech t (, b] nd y Y. Let f C ([, b], X) tke vlues in Y. Then the mp [, t] s g(t, s) is differentible in X with s g(t, s) = S(t, s)f (s) + s S(t, s)f(s). Proof. Observe tht sup (t,s) M S(t, s)x < for every x X by continuity. The number c := sup (t,s) M S(t, s) is finite by the uniform boundedness principle. For (t, s), (t, s ) M we thus obtin S(t, s )f(s ) S(t, s)f(s) c f(s ) f(s) + (S(t, s ) S(t, s))f(s), where the right hnd side of this inequlity tends to 0 s (t, s ) (t, s). To show the second ssertion, fix t (, b] nd tke s, s + h [, t] for h R \ 0}. We compute h (S(t, s + h)f(s + h) S(t, s)f(s)) = S(t, s + h) h (f(s + h) f(s)) + h (S(t, s + h) S(t, s))f(s). 4

7 As h 0, the second clim follows from the first prt nd our ssumptions. In the next lecture we wnt to study further properties of genertors. To this im, we will need severl concepts which we now explin. Here we only recll the bsic definitions, results nd exmples; most proofs nd more detils cn be found in Appendices A nd B. Intermezzo : Closed opertors nd their spectr Let D(A) X be liner subspce nd A : D(A) X be liner. opertor A is clled closed if it holds: If (x n ) n is ny sequence in D(A) such tht the limits lim n x n = x nd lim n Ax n = y exist in X, then x D(A) nd Ax = y. Note tht ny opertor A B(X) is closed with D(A) = X. In the next exmple we introduce the prototype of n unbounded closed opertor. Exmple 2.0. Let X = C([0, ]) nd Af := f with D(A) = C ([0, ]). Tke sequence (f n ) n in D(A) such tht (f n ) n, respectively (f n) n, converge to f, respectively g, in X. It is well known fct tht then f belongs to C ([0, ]) nd f = g (see lso Remrk 2. (f) below), which mens tht A is closed. Second, consider A 0 f := f with D(A 0 ) := f C ([0, ]) f (0) = 0 }. If (f n ) n is sequence in D(A) such tht f n f nd f n g in X s n, then we obtin f C ([0, ]) with f = g s bove. Furthermore, g(0) = f (0) = lim n f n(0) = 0. Consequently g D(A 0 ) nd A 0 f = g, i.e., A 0 is closed. Next, we define the Riemnn integrl for vector vlued functions. Let < b be rel numbers. A (tgged) prtition Z of the intervl [, b] is given by finite sequences (t k ) m k=0 nd (τ k) m k= in [, b] stisfying t k < t k nd τ k [t k, t k ] for ll k,..., m}, where t 0 = nd t m = b. We set δ(z) := mx k=,...,m (t k t k ). For function g C([, b], X) we define the Riemnn sum S(g, Z) (of g with respect to Z) by m S(g, Z) := g(τ k )(t k t k ) X. k= As for continuous rel vlued functions, it cn be shown tht for every sequence (Z n ) n of (tgged) prtitions with lim n δ(z n ) = 0 the sequence (S(g, Z n )) n converges in X nd tht the limit J does not depend on the choice of such (Z n ) n. In this sense we sy tht S(g, Z) converges in X to J s δ(z) 0. The Riemnn integrl g(t) dt is now defined s this limit, i.e., g(t) dt := lim S(g, Z). δ(z) 0 The integrl hs the usul properties known from the rel vlued cse (with similr proofs) like linerity, dditivity nd vlidity of the stndrd estimte g(t) dt (b ) g. The sme definition nd results work for piecewise continuous functions. 5 The

8 In the following remrk we collect the properties of closed opertors nd the Riemnn integrl we need lter. We especilly emphsize tht the fundmentl theorem of clculus is vlid lso in the vector vlued cse (see prt (e)) so tht the substitution rule extends to this setting. The simple property (g) is used very often in these lectures. Remrk 2.. Let A be liner opertor on X. Then the following ssertions hold. () The opertor A is closed if nd only if the grph of A, i.e., the set gr(a) := (x, Ax) x D(A) }, is closed in X X (endowed with the norm given by (x, y) = x + y ) if nd only if D(A) is Bnch spce with respect to the grph norm x A := x + Ax. We write [D(A)] for (D(A), A ). (b) If A is closed with D(A) = X, then A is even continuous ( closed grph theorem ). (c) Let A be injective nd set D(A ) = R(A) := Ax x D(A) }. Then A is closed if nd only if A is closed. (d) Let A be closed nd f C([, b], X) with f(t) D(A) for ech t [, b] such tht Af C([, b], X), where (Af)(t) := Af(t). We then hve f(t) dt D(A) nd A f(t) dt = An nlogous result holds for piecewise continuous functions. (e) For f C([, b], X) the function [, b] X, t t f(τ) dτ, Af(t) dt. is differentible with d t f(τ) dτ = f(t) for ll t [, b]. (2.6) dt For g C ([, b], X) nd t [, b] we hve t g (τ) dτ = g(t) g(). (2.7) (f) Let (f n ) n be sequence in C (J, X), J, nd f, g C(J, X) for n intervl J such tht f n f nd f n g uniformly on J s n. Then f C (J, X) nd f = g. (g) Let f C([, b], X) nd t [, b). Then t+h h t f(s) ds f(t) s h 0 +. Proof. Prts () nd (c) re proved in Lemm A.6 of the ppendix. Prt (b) cn be found in Theorem A.7. (d) Let f be s in the sttement. Clerly, S(f, Z) D(A) for ny prtition Z of [, b] nd m AS(f, Z) = (Af)(τ k )(t k t k ) = S(Af, Z) Af(t) dt k= s δ(z) 0 becuse Af is continuous. The ssertion now follows from the closedness of A. 6

9 (e) Let t [, b] nd h 0 such tht t + h [, b]. We estimte ( t+h t ) f(τ) dτ f(τ) dτ f(t) h = t+h (f(τ) f(t)) dτ h h h t sup f(τ) f(t) 0 τ t h s h 0. So we hve shown (2.6). Putting = t, we lso derive (g). To show (2.7), set ϕ(t) = t g (τ) dτ for t [, b]. Eqution (2.6) implies tht ϕ C ([, b], X) with ϕ = g. Therefore, ϕ g belongs to C ([, b], X) with vnishing derivtive. In the proof of Proposition 2.8 we hve seen tht thus ϕ g is constnt, nd hence (2.7) is true. (f) Formul (2.7) gives t f n (t) = f n () + f n(τ) dτ for ll t J. Letting n 0, we deduce tht f(t) = f() + t g(τ) dτ for ll t J. Hence, f C (J, X) nd f = g due to (2.6). It is delicte mter to dd or multiply closed opertors. The sitution is simpler if one opertor is bounded, see Proposition A.9 in Appendix A. Remrk 2.2. Let A be closed nd T B(X). Then the opertors B = A+T with D(B) = D(A) nd C = AT with D(C) = x X T x D(A) } re closed. This pplies in prticulr to the opertor λi A for λ C. For closed opertor A, we define the resolvent set ρ(a) := λ C λi A : D(A) X is bijective }. We write R(λ, A) for (λi A) if λ ρ(a). This opertor is clled resolvent. The spectrum of A is given by σ(a) := C\ρ(A). Since λi A is closed, R(λ, A) is closed with domin X nd thus bounded thnks to the closed grph theorem (see Remrk 2.(b)). It is known tht ρ(a) is open in C (nd so σ(a) is closed). More precisely, for λ ρ(a) we hve B(λ, R(λ, A) ) ρ(a), (2.8) s one cn see by Neumnn series. Moreover, if T B(X), then σ(t ) is even compct nd lwys non-empty, nd the spectrl rdius of T is given by r(t ) := mx λ λ σ(t ) } = inf n N T n n = lim n T n n. There re closed opertors A with σ(a) = C or σ(a) = (see Exmple B.3 (b) nd (c)). We hve the resolvent eqution R(µ, A) R(λ, A) = (λ µ)r(λ A)R(µ, A) = (λ µ)r(µ, A)R(λ, A) 7

10 for ll λ, µ ρ(a). Furthermore, the mp ρ(a) B(X), λ R(λ, A), is infinitely often differentible (even nlytic) with ( d ) nr(λ, A) = ( ) n n! R(λ, A) n+ (2.9) dλ for ll λ ρ(a) nd n N 0. These results re shown in Theorems B.4 nd B.6 of the ppendix. Exercises Exercise 2.. Let A B(X) nd t R. Show tht the series e ta t k := k! Ak k=0 converges bsolutely in B(X) uniformly for t [ r, r], for ny r > 0. Further show tht ( d ne dt) ta = A n e ta = e ta A n for ll t R nd n N. Let A, B B(X) with AB = BA. Show tht e A+B = e A e B = e B e A. In prticulr e (t+s)a = e ta e sa for t, s R nd e λi+a = e λ e A for λ C. Exercise 2.2. Let p [, ) nd X = L p (R). Set T (t)f = f( + t) for t R nd f L p (R). Show tht (T (t)) t R is C 0 group of isometries on X nd tht the mp R B(X), t T (t), is not continuous. Exercise 2.3. Let p [, ) nd X = L p (0, ). For t 0, s (0, ) nd f L p (0, ) set f(s + t), s + t <, (T (t)f)(s) := 0, s + t. Show tht (T (t)) t 0 is C 0 semigroup on X. Exercise 2.4. Let Ω R d be open, X = C 0 (Ω) nd m C(Ω) such tht sup s Ω Re(m(s)) <. Define T (t)f = e tm f for t 0 nd f X. Show tht T ( ) is C 0 semigroup on X generted by the opertor Af = mf with D(A) = f X mf X }. Exercise 2.5. Let A be closed opertor, C \ 0} nd b C. Define B = A + b with D(B) = D(A). Show tht σ(b) = σ(a) + b nd R(µ, A) =, A) for µ ρ(b). R( µ b 8

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