Semigroups of Linear Operators

Transcription

1 Semigroups of Linear Operaors Sheree L. LeVarge December 4, 23 Absrac This paper will serve as a basic inroducion o semigroups of linear operaors. I will define a semigroup in he conex of a physical problem which will serve o moivae furher (elemenary) heoreical developmen of linear semigroups including he Hille- Yosida Theorem. Applicaions and examples will also be discussed. Inroducion Before defining wha a semigroup is, one needs o recognize heir global imporance. Of course heir imporance canno be fully realized unil we have a clear definiion and developed heory. However, in general, semigroups can be used o solve a large class of problems commonly known as evoluion equaions. These ypes of equaions appear in many disciplines including physics, chemisry, biology, engineering, and economics. They are usually described by an iniial value problem (IVP) for a differenial equaion which can be ordinary or parial. When we view he evoluion of a sysem in he conex of semigroups we break i down ino ransiional seps (i.e. he sysem evolves from sae A o sae B, and hen from sae B o sae C). When we recognize ha we have a semigroup, insead of sudying he IVP direcly, we can sudy i via he semigroup and is applicable heory. The heory of linear semigroups is very well developed []. For example, linear semigroup heory acually provides necessary and sufficien condiions o deermine he well-posedness of a problem [3]. There is also heory for nonlinear semigroups which his paper will no address. This paper will focus on a special class of linear semigroups called C semigroups which are semigroups of srongly coninuous bounded linear operaors. The heory of hese semigroups will be presened along wih some examples which end o arise in many areas of applicaion. 2 Wha is a Semigroup? Le s begin wih he mos basic noion of a semigroup. 2. Definiion (Semigroup) - A semigroup is a se S coupled wih a binary operaion ( : S S S) which is associaive. Tha is, x, y, z S, (x y) z = x (y z). Associaiviy can also be realized as F (F (x, y), z) = F (x, F (y, z)) where F (x, y) serves as he mapping from S S o S [4].

2 A semigroup, unlike a group, need no have an ideniy elemen e such ha x e = x, x S. Furher, a semigroup need no have an inverse. Therefore, many problems which can be solved wih semigroups can only be solved in he forward direcion (e.g. forward in ime). 2.2 Simple Examples Some of he simples examples of semigroups are: 2S = R S = M 2 2 (R) = addiion where M 2 2 (R) = he se of 2 2 marices wih real enries [4]. = marix muliplicaion While we have inroduced he mos general definiion of a semigroup, his paper will focus on semigroups of linear operaors. In paricular, i will provide definiions, heory, examples, and applicaions of semigroups of linear operaors (linear semigroups). 2.3 A More Concree Example To moivae he resuls abou linear semigroups, consider he physical sae of a sysem which is evolving wih ime (according o some physical law) as given by he following IVP (or absrac Cauchy problem): d u() d = A[u()] ( ) () u() = f where u() describes he sae a ime which changes in ime a a rae given by he funcion A. The soluion of () is given by: u() = e A f. (2) A naural firs quesion o ask is, Is () well posed? A well posed problem is one whose soluion exiss and is unique. Semigroup heory can deermine when a problem is well posed and in order o use he heory, we need o know ha we have a semigroup. So o coninue wih (2), le T operae on u as follows: T () : u(s) u( + s). (3) If we assume ha A does no depend on ime, hen T () is independen of s [3]. The soluion, u( + s) a ime + s, can be compued as T ( + s) acing on f. Likewise, if we painsakingly break down he process ino wo seps we have: s Sep: T (s)(f) = u(s) 2 nd Sep: T ()(u(s)) = T ()(T (s)(f)) = u( + s) = T ( + s)(f). 2.4 The Semigroup Propery By ransiionally breaking down he process of evoluion, i is eviden ha we can reach he sae of he sysem a ime + s by eiher going direcly from he iniial condiion o he sae 2

3 a ime + s or by allowing he sae o evolve over s ime unis (aking a snapsho), and hen allowing i o evolve more ime unis. Here he T ( ) is acing like a ransiion operaor []. The uniqueness of he soluion gives reveals he semigroup propery which is given by: T ( + s) = T ()T (s) (, s > ). (4) The semigroup propery (4) of he family of funcions, {T (); }, is a composiion (no a muliplicaion). Noice ha T () is he ideniy operaor (I) (i.e. here is no ransiion a ime zero and he iniial daa exiss) [3]. 2.5 More Properies Now ha we have seen he fundamenal semigroup propery, we wan o undersand how A (which governs he evoluion of he sysem) and T relae o one anoher. We will firs examine he scalar case. Two observaions which may be preliminary indicaors of he relaionship are given as follows: T ()(f) = T ()(u()) = u() = e A f (5) Noice ha u() = T ()(f) solves () and suggess ha: d T ()(f) = A(T ()(f)). (6) d T ()(f) = e A f (7) where A is he derivaive of T (). In addiion, each T () : f e A f is a coninuous operaor on R, (or in an infinie dimensional seing, a Banach space X), which indicaes he coninuous dependence of u() on f [3]. The iniial daa f should belong o he domain of A. Upon inspecion of (7), we have he following resuls: i. T () exhibis he semigroup propery as in (4), ii. T () is a coninuous funcion, iii. T ()f = f, iv. T () : R R is linear provided A is linear. Again, since we are ineresed in linear semigroups, we will assume ha A is linear. These observaions bring forh he noion of C semigroups. 3 C Semigroups Now ha semigroups have been inroduced in he framework of a physical problem, we should formally define a C semigroup; a erm which was inroduced by Hille [3]. Generally we say ha a C semigroup is a srongly coninuous one parameer semigroup of a bounded linear operaor on a Banach Space X. 3

4 3. Definiion (C Semigroup) - A C semigroup (or srongly coninuous semigroup) is a family, T = {T () R + }, of bounded linear operaors from X o X saisfying: i. T ( + s) = T ()T (s), s R +, ii. T () = I, he ideniy operaor on X, and iii. lim + T ()f f for each f X wih respec o he norm on X []. The coninuiy condiion given by (iii) arises naurally as we do no wan our physical sysem in () o breakdown in ime due o small measuremen errors in he iniial sae (for example). From now on hroughou his paper, he word semigroup will mean C semigroup. A more careful inspecion of he definiion of semigroup may provoke he following quesion, Can one replace (iii) by he condiion: lim T () I = (8) where denoes he norm on X? The answer is, NO! Semigroups ha saisfy he propery given in (8) are called uniformly coninuous semigroups of bounded linear operaors. The condiion given in (8) is oo srong for srongly coninuous semigroups []. Uniformly coninuous semigroups are hus a subse of srongly coninuous semigroups. This paper will focus on he larger se of semigroups bu will (as necessary) commen on he smaller se. 3.2 Some Quesions Now ha we have an official definiion of a semigroup, we wan o answer he following hree quesions which aemp o unveil he relaionship beween T () and A: Q. Given he semigroup T (), how can we find he operaor A in e A f? Q2. Which operaors A give rise o which semigroups? Q3. Given A, how can we consruc he corresponding semigroup T ()? The nex secion will address he answers o hese quesions. 4 Semigroup Generaion In his secion we will explore he generaion of semigroups. This will reveal he connecion beween A in () and T () in (3). We will firs examine he answer o (Q) as in he previous secion. To moivae his, (7) suggess ha T () = e A. Noice ha: T () = Ae A = AT () and T () = A. So perhaps we can obain A by A = d d T () =. This leads o he following definiion. 4

5 4. Definiion (Generaor) - Le T be a semigroup. equaion: The (infiniesimal) generaor of T, denoed by A, is given by he Af = lim + A f = lim + T ()f f where he limi is evaluaed in erms of he norm on X and f is in he domain of A iff his limi exiss []. (9) So, according o (9), he generaor A is obained by differeniaing he semigroup T. From his we see ha u( ) = T ( )f solves (). This answers (Q). 4.2 On he Naure of A Thus far, we have seen wo ypes of linear semigroups, uniformly and srongly coninuous semigroups. So in regard o (Q2), we pose he quesion, Which operaors A give rise o hese wo differen ypes of semigroups? Ineresingly, we have ye o discuss wha ype of operaor A is. In paricular, is A a bounded (nice!) or unbounded (no so nice!) operaor? Of course, as i almos always urns ou, ineresing problems are more difficul o work wih. So in general, for mos applicaions, A will be an unbounded operaor. In fac, he difference beween uniformly coninuous and srongly coninuous semigroups is jus he naure of A. Precisely, A is he generaor of a uniformly coninuous semigroup T iff A is a bounded operaor. So, if T is srongly coninuous and fails o be uniformly coninuous, hen T will have an unbounded generaor A [], [2]. The fac ha A can be unbounded in a seing such as ha in () should d no be sarling since we may iniially associae A wih he operaor dx which we know is unbounded. This does no do (Q2) jusice, and we will reurn o his quesion as we develop furher. 4.3 On he Naure of T () To examine (Q3), recall (7). I (i.e. (7)) should no be a clear fac a his ime as i has no been rigorously proved. However, wha should be clear is ha he exponenial in (7) may play a role in uncovering T () from A. Therefore, we need o undersand he exponenial funcion in (perhaps) one of he following conexs: i. e A = (A) n n= n! ( ) ii. lim n A n n ( ) iii. e A = L for λ > Re(A) λ A where L denoes he inverse Laplace ransform [3]. Furhermore, now ha we know ha he boundedness of A deermines he ype (coninuiy) of he semigroup generaed, we should answer his quesion in he conex of bounded/unbounded operaors. Le s look a he easier (bounded) case firs. This leads us o he following heorem where we view he exponenial as a power series as in (i). 5

6 4.4 Theorem - Le A be a bounded operaor from X o X. Then, { T = T () = e A (A) n = n! is a uniformly coninuous semigroup. Proof - n= : R + } () Since A is bounded we know A < and hus (A) n n= n! converges for each o he bounded linear operaor T (). We know he semigroup propery (4) holds since ( i= ) () i i! j= (s) j = j! ( + s) k. k! Clearly T () = I. Finally, o disinguish T () as uniformly coninuous semigroup: k= (A) n T () I = n! n A n n! n= n= = e A and e A + as and he proof is complee [2]. I should be noed, and i is no hard o show, ha A is in fac he gereraor of T () [5]. So we have answered (Q3) in he conex of a bounded generaor A in which case given A, we consruc he semigroup T () as T () = e A. Therefore, for a bounded generaor A, he suggesion in (7) is rue. Furhermore, he map T () = e A is differeniable. Bu how can we consruc he semigroup when A is unbounded? Furhermore, geing back o (Q2), wha properies does A possess o make i a generaor of srongly coninuous semigroups? The answer o his quesion lies deeply wihin he moher heorem of linear semigroups called he Hille-Yosida Theorem. We will devoe he enire nex secion o building up o and presening his heorem. 5 Hille-Yosida Theorem The previous secion showed us he basic resuls for bounded generaors and heir uniformly coninuous semigroups. In his secion, we are forced o look a unbounded generaors o invesigae many ineresing problems. I would be convenien o use he approach we ook for bounded generaors which was using he power series for e A. However, convergence of his 6

7 series is no likely when A is unbounded. Recall ha i remains o be shown wha (exacly) makes A a generaor of a semigroup and, once ha is shown, how do we recover he semigroup T () from he generaor? Le s firs address he quesion of wha A is, exacly. 5. Noes on Resolvens Ulimaely we are looking for relaionships beween A and T (). In doing so, one may sumble upon a connecion beween T () and he resolven operaor of A. Recall he resolven se of A is given by ρ(a) and is he se of complex numbers λ for which λi A is inverible. The resolven of A is a family of bounded linear operaors which is denoed by R(λ, A) and is given by R(λ, A) = (λi A) where λ ρ(a). To see is connecion o T (), consider he following: λ A = e λ e A d where A R and λ C wih Re(λ) > A [3]. This gives rise o he operaor version: R(λ, A)f = e λ T ()fd which is valid provided λ > [3]. So he resolven operaor can be hough of as he Laplace ( ) ransform of he semigroup. Furhermore, in ligh of viewing e A as lim n A n, n we can re-wrie his expression as: [ ( e A = lim A ) ] n [ n ( n ) ] n = lim n n n I A. () So imbedded wihin his formula for e A is he resolven operaor (λi A) where λ = n. Before we proceed furher wih resolven ses (which we will reurn o laer), we will inroduce corollaries and heorems ha will lead up o he Hille-Yosida Theorem. 5.2 Theorem - Le T () be a semigroup. There exis consans ω R and M such ha he following holds: T () Me ω for <. (2) Proof - Choose a consan M such ha T () M for all. Le ω = logm. Then for each > and if n is he leas ineger hen: ( n ) T () = T n = n ( ( )) ( ( )) T n = n T M n M + = Me ω n k= compleing he proof. k= 7

8 5.3 Corollary - If T () is a semigroup hen for each f X, T ()f is a coninuous funcion from R + o X. Proof - Le, h and f X. Then we have: T ( + h)f T ()f = T ()T (h)f T ()f T () T (h)f f Me ω T (h)f f and for h we have: T ( h)f T ()f = T ( h)f T ( h + h)f = T ( h)f T ( h)t (h)f T ( h) f T (h)f Me ω f T (h)f by (2). Thus we have coninuiy [5]. 5.4 Theorem - Le T () be a semigroup generaed by A. Then he following hold: i) For each f D(A), T ()f D(A) (domain of A) and AT ()f = T ()Af (3) ii) For each f D(A) and T ()f D(A), d T ()f = AT ()f = T ()Af. (4) d Proof - i) Le f D(A) and fix. Then, for s >, A s as in (Def. 4.) and using T (s)t () = T ()T (s) = T (s + ): A s T ()f = (T (s)t ()f T ()f) s = (T ()T (s)f T ()f) s = T () (T (s)f f). (5) s As s +, he righ hand side of (5) converges o T ()(Af) (Def. 4.) since f D(A) and T () is coninuous on X. Therefore, lim A st ()f = T ()Af s + which gives T ()f D(A) and AT ()f = T ()Af as desired []. 8

9 ii) Le f D(A) and h >. Consider he righ-hand limi, (T ( + h)f T ()f) lim h + h T ()T (h)f T ()f) = lim h + h ( ) (T (h) I) = lim T ()f = AT ()f = T ()Af h + h since T ()f D(A) by (i) []. Consideraion of he appropriae lef-hand limi can be found in [5]. 5.5 Theorem - Le T () be a semigroup generaed by A. Then he following hold: i) For each f X, ii) For each f X, iii) For each f D(A), lim h h +h T (s)fds = T ()f. (6) ( ) T (s)fds D(A) and A T (s)fds = T ()f f. (7) T ()f T (s)f = Proof - T (τ)afdτ = S s AT (τ)fdτ. (8) i) The proof of (6) follows from he coninuiy of T () given in Corollary 5.3 [5]. ii) Le f X and h >. Then, A h ( ) T (s)fds = T (h) I h Then, by he semigroup propery, T (s)fds = h (T (h) I)T (s)fds. h (T (h) I)T (s)fds = h (T (s + h)f T (s)f) ds 9

10 which gives: h +h h T (s )fds h T (s)fds = h +h T (s)fds h h T (s)fds. (9) Leing h + and applying he Fundamenal Theorem of Calculus o (9) yields T ()f T ()f = T ()f f which proves (ii) []. iii) The proof of (8) follows from inegraing (4) from s o [5]. Recall ha our goal is o describe he A s ha generae he T () s. The following heorem is a precursor o he Hille-Yosida Theorem and provides some addiional informaion perinen o our goal. 5.6 Theorem - If A is he generaor of a semigroup T (), hen D(A) is dense in X and A is a closed operaor. Proof - Firs, o show ha D(A) is dense in X, we mus show ha D(A) = X. Take f an arbirary elemen of X and le f be given by: f = T (s)fds By par (ii) of Theorem 5.5, f D(A) and furhermore, par (i) of he same heorem gives: f = T (s)fds T ()f = f as + Thus, f = lim n f D(A) where f was chosen arbirarily so ha D(A) = X[]. n I remains o be shown ha A is a closed operaor. Recall ha A closed means if f n D(A), f n f, and Af n g, hen f D(A) and Af = g. So le {f n } n= D(A) wih f n f and Af n g as n. By par (iii) of Theorem 5.5, we have: T ()f n f n = T (s)af n ds (2) for each n and >. By Theorem 5.2, we have T (s) Ce ω where C is a consan. Therefore,

11 T (s)af n ds So, leing n in (2) yields T (s)gds T (s) Af n g ds C T ()f f = T ()f f = T ()f f lim + = lim A f = g. T (s)gds T (s)gds + Af n g ds C Af n g as n T (s)gds Therefore, by Def. 4., we conclude ha f D(A) and Af = g. Thus A is closed [3]. 5.7 Theorem - A semigroup is uniquely deermined by is generaor. Proof - Le T and S be wo semigroups having he same generaor A. Le f D(A) and le >. Define u : [, ] X by u(s) = T (s)s( s)f. Then, du(s) ds = T (s)( A)S( s)f + T (s)as( s)f = giving u = consan on [, ]. Therfore, since u is consan, Thus, T = S [3]. T ()f = u() = u() = S()f. Now ha we have inroduced some elemenary heory, we will inroduce a definion which we will use in he Hille-Yosida Theorem since he deails become much easier o work wih. 5.8 Definiion (Semigroup of Conracions) - A semigroup T () is a semigroup of conracions when M = and ω = in (2) [5]. Tha is, T (). As saed in [3], Roughly speaking, for mos purposes i is enough o consider only conracion semigroups. For a full explanaion see [3]. So, wihou furher anicipaion, we will presen he Hille-Yosida Theorem.

12 5.9 Theorem (Hille-Yosida Theorem) - A linear unbounded operaor A is he generaor of a (C ) semigroup iff: i. A is a closed operaor, ii. iii. A has dense domain (D(A)), for each λ >, λ ρ(a), and iv. R(λ, A) λ. The Hille-Yosida Theorem is very powerful as i gives us boh necessary and sufficien condiions. While he proof of his heorem is quie difficul and lenghy, he heorem iself provides a much more jusified answer o (Q2) as i describes A s characer in furher deail. We will prove he necessiy o provide some insigh ino he heorem. The he proof of sufficiency will no be provided bu can be found in [2], [3], and [5]. To achieve sufficiency, a more exensive background (in he form of lemmas) is required. In addiion, he insigh gained from proving sufficiency will no be pu o full use as his paper will no rigorously discuss exacly how we view T () as an exponenial. Proof (Hille-Yosida Theorem) Necessiy. The proofs of (i) and (ii) are given by Theorem 5.6. To prove (iii), noice ha for each λ >, {e λ T () : R} is a semigroup of conracions whose generaor is compued by, e λ T ()f f lim + = lim + λe λ T ()f + e λ Af = λf + Af (2) using L Hôpial s Rule which gives he generaor as A λi wih domain D(A). Applying (7) o his semigroup gives: e λ T ()f + f = (λi A) e λ T ()f + f = e λs T (s)fds e λs T (s)(λi A)fds f X f D(A) Leing A we know e λs T (s)fds D(A) since A is closed. This resul, ogeher wih he dominaed convergence heorem give: f = (λi A) f = (λi A) f = e λs T (s)fds e λs T (s)(λi A)fds f X e λs T (s)fds f X, λ >. f D(A) So we conclude ha (λi A) : D(A) X is (i.e. bijecive) and (λi A) is a bounded linear operaor on X and we have λ ρ(a). This proves (iii). 2

13 To prove (iv), R(λ, A) = (λi A) f Thus, he proof of necessiy is complee [3]. e λs T (s) f ds f λ f X, λ >. One should noice ha he resolven of A is he Laplace Transform of he semigroup.?coincidenly? we should expec o obain he semigroup by invering Laplace ransform (which is one way o view e A ). Probably no coincidence. So while he Hille-Yosida heorem fills in some of he gaps, one perplexing one remains. 5. The Missing Link One missing link remains. We have seen o how o ge from T () o A via semigroup differeniaion: T ()f f Af = lim. + We have also seen how o manuever from he A o is resolven (which can be reversed) as: and also from T () o he resolven of A as: R(λ, A) = (λi A) R(λ, A) = A = λ R(λ, A) e λ T ()d. (22) More, imporanly, we would like o ouch on he answer o (Q3) in which we are supposed o reconsruc T () from A. In his secion, we will provide resuls in he absence of rigor o give he reader an idea of how his can be accomplished. Recall ha when A is bounded, we had T () = e A = (A) n n= n!. To handle he case of A unbounded, we can ry o approximae A by a sequence {A n } n N of bounded operaors and hope ha: e A = lim n ean. One such approximaion of A, ermed he Yosida approximaion, is given by: A λ = λar(λ, A) = λ 2 R(λ, A) λi. Ineresingly, T ()f = lim λ e A λ f for f X [5]. Therefore, in some sense, a srongly coninuous semigroup wih generaor A is obained as he limi of a sequence of uniformly coninuous semigroups generaed by he bounded linear operaors given by he A λ s. As previously suggesed in (), one may also accomplish his ask of going from A o T () by: ( T ()f = lim I ) n [ n ( n )] n n n A f = lim n R, A f for f X as given in [5]. This mehod uses form (ii) of he exponenial formula given in Sec. 4.3 coupled wih he resolven se. Therefore, despie he lack of rigor, his secion gives he reader an idea of how we view T () as an exponenial. 3

14 6 Applicaions and Examples As was saed a he beginning of he paper, recognizing problems o which semigroup heory can be applied o is imporan as he heory accompanying semigroups can be a powerful ool. In fac semigroup heory can deermine if a problem is well-posed. This gives rise o he following Theorem. 6. Theorem (Well Posed Theorem) - The IVP given by () (wih A being linear) is well posed iff A is he generaor of a semigroup T. In his case he unique soluion of () is given by u() = T ()(f) for f in he domain of A [3]. This urns ou o be quie imporan as i provides boh necessary and sufficien condiions o deermine if a problem is well-posed. We will now inroduce some examples of semigroups. Many examples fall ino he caegories of: ranslaions, fracional inegraion, harmonic funcions, sochasic processes, difussion equaions and ergodic heory [4]. We will look a hree examples. 6.2 The Hea Equaion We are ineresed in using our knowledge of semigroups in a slighly more concree example. In paricular, we will look a he soluion of he hea equaion and show i is given by a semigroup. In his seing, le X = L p (R), p <. Recall ha he hea equaion as given by: u = u xx x (23) u(x, ) = f. Using Fourier Transform mehods, he soluion o (23) can be wrien as u(x, ) = 4π e (x y)2 4 f(y)dy. (24) The hea kernel is given by K (s) = 4π e (s)2 4 equaion as a convoluion: and we can wrie he soluion o he hea So he soluion of (23) is a semigroup on X wrien as u(x, ) = K f. (25) T ()f(s) = e (s r)2 4 f(r)dr >, x R, and f X (26) 4π and we se T () = I. This ((26)) is called he Gauss-Weiersrass semigroup. To show i saisfies he semigroup propery we mus show: T (a + b)f(s) = T (a)t (b)f(s). (27) 4

15 Symbolically, T (a + b)f(s) is given by: T (a + b)f(s) = K a+b f(s). (28) Likewise, symbolically wriing T (a)t (b)f(s) we have: T (a)t (b)f(s) = T (a)[k b f(s)] = K a [K b f(s)] = [K a K b ] f(s) (29) since he operaion of convoluion is associaive. Therefore, o show ha (27) holds, i suffices o show ha K a+b (x) = K a K b (x) (3) which is equivalen o showing: 4π(a + b) e x2 4(a+b) = 4πa 4πb e x2 4a e (x y) 2 4b dy. (3) Alhough his requires many manipulaions, we will go hrough he argumen o prove ha we do in fac have a semigroup. Working wih he righ-hand side of (3) (wihou he consan) gives: R e [y2 (a+b) 2axy+x 2 a] 4ab dy = e x2 a 4ab = e x2 x 4b e 2 a 4b(a+b) where u = R e (a+b) 4ab [y xa a+b ]2 dy = e x2 4(a+b) R e (a+b) 4ab [y 2 2xa a+b y] dy = e x2 4b R e (a+b) 4ab u 2 du = e x2 4(a+b) e (a+b) 4ab [y xa a+b ]2 + x2 a 4b(a+b) dy R ( ) y xa a+b a+b. Now making he change of variables = 4ab u we have: e x2 4(a+b) ab 2 e 2 d = e x2 4(a+b) abπ 2 a + b R a + b R e ( a+b 4ab u ) 2 du by evaluaing he Gaussian inegral. Now in order o show (3) we need o verify ha: abπ = 2. 4π(a + b) a + b 4πa 4πb By simple calulaion of he above, we see ha (3) holds and he semigroup propery is verified for he hea equaion. 6.3 Poisson Semigroup We inroduce he Poisson semigroup wihin he space X = L p (R), p <. For >, define T () on X by: T ()f(x) = π 2 f(y)dy x R and f X. + (x y) 2 5

16 We have T ()f = P f where he kernel is given as: P (x) = π 2 + x 2. We can evaluae he Fourier ransform of P, denoed by F, as: (FP )(x) = π 2 + u 2 eixu du. This can be evaluaed using sandard conour inegraion mehods and he Residue Theorem. Using he Residue Theorem: π lim e iux u i u + i = e x = π 2i 2πi e x which gives (FP )(x) = e x for x. To accoun for x R, (FP )(x) = e x. So for f X, we have: (F(T ()f))(x) = e x (Ff)(x) x R. Since e x s e x = e x (s+), he semigroup propery is saisfied. The Poisson semigroup arises ( in many insances ) since he kernel, P x, is a fundamenal soluion o Laplace s equaion 2 u x + 2 u 2 = in he region {(x, ) x R, > } Translaion Semigroups Now we will inroduce a class of semigroups called ranslaion semigroups. In his seing, le X = C ([, )) be he Banach space of funcions f which are coninuous on [, ) (coninuous on he righ a ) and for which f(x) M < as x wih respec o he sup norm. For, define T () on X by: (T ()f)(x) = f(x + ). Here he operaor T () ranslaes he funcion f C o he lef by unis and forms a semigroup. The semigroup propery is saisfied as ranslaion by + s unis is he same as a ranslaion by unis followed by a ranslaion by s unis. Also, T () = I is saisfied. Furhermore, lim + T ()f = f since: lim sup f(x + ) f(x) =. + x In addiion, our ranslaion semigroup forms a conracion semigroup since T ()f = f which gives T () =. 6.5 Wha Can Semigroups Do? This secion is inended o provide a shor lis of where (specifically) semigroups arise. This lis should serve as evidence of how semigroups make heir mark in many disciplines. 6

17 . Feller Markov Processes 2. Conrol Theory 3. Populaion Growh Models 4. Linear Transpor (Bolzmann) Equaions 5. Delay Differenial Equaions 6. Inegro-Differenial Equaions These are jus o name a few. To conclude, evoluion equaions arise in many disciplines of science. An absrac way o sudy and dissec hese equaions is hrough semigroups. Using semigroups is advanageous as he associaed heory is quie rich. Sudying semigroups, as I have for his paper, heighen your awareness of heir prevalence hroughou applied mahemaics. 7 References [] Belleni-Morane, A. and A. McBride. Applied Nonlinear Semigroups, Mahemaical Mehods in Pracice, John Wiley & Sons, Chicheser, 998. [2] Engel, K. and R. Nagel. One-Parameer Semigroups for Linear Evoluion Equaions, Graduae Texs in Mahemaics, Springer, New York, 995. [3] Goldsein, J. Semigroups of Linear Operaors and Applicaions, Oxford U. Press, New York, 985. [4] Hille, E. Wha is a Semi-group?, in Sudies in Real and Complex Analysis (pp 55-66) by Hirschman, Vol 3, Mahemaical Associaion of America, Prenice-Hall, Englewood Cliffs, 965. [5] Pazy, A. Semigroups of Linear Operaors and Applicaions o Parial Differenial Equaions, Applied Mahemaical Sciences, Springer-Verlag, New York,