Yosida 1 Functional_Analysis

7/26/2019 Yosida 1 Functional_Analysis

1/468


2/468


3/468


4/468


5/468


6/468


7/468


8/468


9/468


10/468


11/468


12/468


13/468


14/468


15/468


16/468


17/468


18/468


19/468


20/468


21/468


22/468


23/468


24/468


25/468


26/468


27/468


28/468


29/468


30/468


31/468


32/468


33/468


34/468


35/468


36/468


37/468


38/468


39/468


40/468


41/468


42/468


43/468


44/468


45/468


46/468


47/468


48/468


49/468


50/468


51/468


52/468


53/468


54/468


55/468


56/468


57/468


58/468


59/468


60/468


61/468


62/468


63/468


64/468


65/468


66/468


67/468


68/468


69/468


70/468


71/468


72/468


73/468


74/468


75/468


76/468


77/468


78/468


79/468


80/468


81/468


82/468


83/468


84/468


85/468


86/468


87/468


88/468


89/468


90/468


91/468


92/468


93/468


94/468


95/468


96/468


97/468


98/468


99/468


100/468


101/468


102/468


103/468


104/468


105/468


106/468


107/468


108/468


109/468


110/468


111/468


112/468


113/468


114/468


115/468


116/468


117/468


118/468


119/468


120/468


121/468


122/468


123/468


124/468


125/468


126/468


127/468


128/468


129/468


130/468


131/468


132/468


133/468


134/468


135/468


136/468


137/468


138/468


139/468


140/468


141/468


142/468


143/468


144/468


145/468


146/468


147/468


148/468


149/468


150/468


151/468


152/468


153/468


154/468


155/468


156/468


157/468


158/468


159/468


160/468


161/468


162/468


163/468


164/468


165/468


166/468


167/468


168/468


169/468


170/468


171/468


172/468


173/468


174/468


175/468


176/468


177/468


178/468


179/468

168 VI. Fourier Transform and Differential Equations

We have

2T ( I )

/ en(2 T- t ) /

l(u) l ( t -1 ~ )

du dt = 0,

and, by the change of variables u = T - v , t = 2 T - v - w, we obtainJ J en(v+w) I ( T- v) I ( T- w) dv dw = 0..1

where .1 is the triangle v + w > 0, v < T, w < T i n the v - w plane.Let .1' be the triangle v + w ~ 0, v > - T, w > - T. Then the join.1 + .1' is the square -T < v, w < T. The above equality shows thatthe integral

of en(v+w) I ( T- v) I ( T- w) over .1 + .1' is equal tothe inte

gral over .1'. The integral over L1+ .1' is the product of two single integrals, and in the integral over .1' we have en(v+w) < 1. Thus

I nu I ( T- u) dul 2= I J J en(v+w) I ( T- v) I ( T- w) dv dwI1 - T d +LI '

< J f l i ( T- v ) I ( T- w ) j dvdw :S: 2T 2 -A 2 ,Ll'

where A is the maximum of ll(t) Ifor 0 < t < 2T, and 2T 2 is the areaof .1'. We thus have

j_fenu i (T-u)du l< V 2 T A ,and, moreover, j_fenu I ( T- u) dul< T A. Therefore

Ifnu I ( T- u ) dul = Lf-_/I


180/468

6. M :ikus i.D.ski' s Opera tional Calcu lu s 169

T hu s , b y (f *g ) (t) = 0, w e h a ve [( / * g1 ) * (f * g1 ) ] (t) = 0 , and so,by the speci a l cas e p r o v e d ab o ve, (f * g1 ) (t) = 0, th a t is,

I J ( t - u ) ug( u ) du = 0 (0 < t < o o ).0

Fro m th i s we o btain , s imi la r ly t o abov e ,I

J ( t - u ) u 2 g(u) du = 0 (0 ~ t < oo).0

R e p e a t ing the a r g u m e n t , w e f ind th at, J ( t - u ) u "g(u ) du = 0 (O < t< o o; n = 1, 2, . . . .

0

H e n ce, b y L e r c h 's th e o r e m p r ov e d ab o ve, w e ob t ain

f ( t - u )g ( u )= O fo r o : : : ; ;u < t< o o .

I f a u 0 ex is ts for w h ic h g (u 0 ) =I=0, then f ( t - u 0 ) = 0 fo r a l l t > u 0 ,tha t is, f(v) = 0 for a l l v > 0. T here fore, w e h a ve e i th e r l( v) = 0 fora ll v > 0 o r g (v) = 0 f o r all v > 0.

6 . M ik u sm s ki s O p erat io nal C alcu lu s

In h is " E le c t r o m a g n e t ic Theo ry", L ondo n (18 99), the p h ys ic is t0. HEAV ISIDE in augu rated an o p e ra t io n a l ca lc ulus w hich he su c cessf u llya ppl ie d to o rdin a ry l ine a r dif f e ren t ia l e q u a t ion s co n n ec ted with e lec tr o

te c hn ica l p roble m s. I n his c alcu lu s o cc ured c e r t a i n o p e ra to r s w h o sem e a n i n g i s not a t a ll o bvio u s. T he int erpret ation o f su c h o p e r a to r s asgiv e n b y HEAVISIDE him se lf is diff ic u lt to jus t i f y . T he i nt erp re t atio ngi v en b y his su cces sors i s unc l e a r w ith r e g a r d t o its r a n ge o f v a l i d i ty ,sinc e i t is b a se d upon the theor y of L ap lace t r ans f o r ms . The theor y ofco n v olu t io n q u o t ien ts d u e to J. MIKUSINSKI prov id e s a c lear and s im pleb as is fo r a n ope r a t i ona l ca lcu lus ap p licab le to ordi na ry d if fe ren t ia l equ at ion s w ith co n stant c oeffi c i ents, a s w e ll as to c e r t a in pa rtia l diffe rent i a l e qu a t ion s wi th con st ant co e fficie n ts, d if feren c e equ a t ion s and in t e g r a l

e qu a t i ons .

T h e co nv olut io n q uo tien ts . We d e no t e by C th e to talit y o f c o mple x

v a l u e d fu nct io n s I (t) def in e d for 0 < t < oo . In th i s sec ti on, w e w r i t e such fu n c t ions by {f(t)} o r s imp ly by I; w h e reas, f(t) w ill m ean th e va lu e att of t h e fun c t ion l(t) . W e w ri t e {l(t )} {g(t)} or s im p ly I g for th e con v o-

lu t ion fun c t ion { / l( t - s) g( s) ds}. i .e.,{f(t)} {g(t)} = {( /* g ) (t)} = { / (t -s) g (s ) ds}. (1)


181/468


We have, as proved in Chapter VI, 3,

f g = g f (commutativity),f (g h) = (/ g) h (associativity).

(2)

(3)

We can define, beside the convolution product f g, the sum through{f(t)} + {g(t)} = {f(t) + g(t)}, (4)

and we have the distributive law

h . (/ + g) = h 1+ h g. (5)Hence C is a ring with respect to the addition f + g and the multiplication f g. The zero of this ring is represented by the function which isidentically zero; we shall thus denote this function by 0. This ring C iswithout zero factor, that is, in C, f g = 0 implies either f = 0 or g = 0.This is a consequence of Titchmarsh's theorem. Hence, by introducing

the convolution quotient ffg = j_ of two functions f, g E C with g:j= 0,g

we obtain a commutative field Q:

afb = cjd is equivalent to a d = be, and, in particular,ajb = cis equivalent to a = be,

a c a c7 ) - ; [ = l ) . d ,

a c a d + h cb + d = - -b - d - ~ '

ajb = a cfb c (c=i= 0).

(6)

(7)

(8)

(9)

The operator. We shall call the quotient ajb an "operator". Anya E C gives an example of operators, since, by (9), we can identify a witha bfb (b =i=0).

The unit- or ()-operator. The operator cjc (c =i=0) is the unit of the

multiplication in the field Q. For, we have, by (7) and (9),(10)

We have, by (6), cjc = bfb. We call cjc the unit- or ()-operator, and we shallhenceforth denote this operator by 1:

a a a1 b = b 1 = b (11)

The operator 1 = cjc does not belong to C, for, if we take for c the func-

tion {1}, then {1}/{1} = {f(t)} E C implies that {1} {f(t)} = { / 1 f(s) ds}

= { / (s) d,.,}= {1}, which is surely a contradiction.


182/468

6. M i kusin s ki's O p erati o nal C alc ulu s 17 1

T he ope ra tor o f in t eg rati on . W e sh all den o te by h the o p e r a to r d efi ned by the fu nc tion {1 }:

h = {1}, (12)

an d ca ll h the opera tor of integ r ation, sin ce, fo r a ny f E C, we h av e, a sab ove ,

h {f(t)} = {1} {f(t)} = { / (s) d s}. (13)

R em a rk . In his b ook , r efer re d to a bov e, MIK U SINS K I us es th e sy m b o l

l for {1}. W e h er e us e the sy m b ol h in h o n o u r o f H e av i s id e an d fo r ty p og rap h ic al c on ve ntio n . A lo c ally i n teg ra b le fu n c t io n {f(t)}, t > 0, m a y b e

id ent i fie d w i th the o p er a t o r { j (s) ds} fh. T h u s t h e co nvo lu t ion q u o t i en tis an ot h er k i n d o f "ge ne raliz ed f u n c t ion " .

T h e scal ar ope ra tor. L et X b e a n y c om ple x n u m b e r, a nd {X}t h e fu n ct io n w hi ch i s iden tic all y e q u a l to X, T he n t h e op era to r

(14)

is c alle d a scal a r ope rator, b eca us e w e h av e

[1X] + [{J] = [1X + {J], [1X ]. [fJ] = [1X{J], [1X ]. {f( t)} = {1X /(t)}. (15) Pro of .

[1X] + [{J] = { ~} + { }= {1X t {3} = [1X + {J],[ J [{J] - {1X}. {{3 }- {1X{3 t}- h . {1X{3} - {1X { 3}- [ {J]X - h2 - h2 - h2 - h - X ,

{ / iX f(s) ds} [1 X ]{f ( t )}={ 1 X}y ( t )}= 0 h =h {~ f (t )} = {1 X/ ( t) } .

R em a rk . A s a co roll ar y , w e ob tain

{ a (t)} {1Xa (t)}[XJ {b (t)} = {b (t)f, (16)

so t hat th e ef fect o f th e o p er a t o r [X] is ex ac t l y the X- tim es m u lti p l ic a

tio n . W e c an thu s id en t i f y t he s ca lar op erat o r {1} /{1 }w ith the u n it op era

to r 1, an d t he o p era t o r [X] w it h th e n u m b e r X,

T h e o per at or o f diffe re ntia ti on. W e sh all d e no t e th e o p e ra to r 1 / h b y

t he sy m b ol s:s = 1/h = 1/{1}. (17)

s is c all ed the ope rator o f diffe renti a tion, sinc e, if f = {I (t)} E C h as a c o n t in u ou s d eriv a t ive f' = {/ ' (t)}, th e n

s f = f' + /(0), w he re /( 0) ='-' {f(O)} fh. (18)


183/468

172 V I. Fourier T ransform an d Different ia l Equa tion s

Proof. M ul tip ly both sid es of the e q ua t ion

{f( t )} = {/ {0)} + { / ' (s) ds} = h /{0) + h {/' {t)}by s and use the fa c t s h = h s = 1.

Corollary 1. I f f = { / (t)} has a cont inuou s n- th der i va t ive jfnl = {/(n)(t)}, t hen/(n) = Sn . f - Sn-1 f (0) - sn - 2 . f' {0) - . . . - s / ( n -2 ) {0) - / (n -1 ) {0)'

where jUl (0) is th e o p e ra tor of the jUl (0)-times m ultiplicati on , (19)

tha t is, jUl (0) = {jU l (O)}jh.

Proof. F o r n = 2, we h av e

s 2 f = s (s f) = s (/' + f(O)) = s -/ ' + s /{0 ) = f" + f' (0) + s /{0 ) .T h e general ca se m ay b e p ro v ed by in duct ion.

C orollary 2. W e have{20)

an d m o re general ly ,f tfl-1 l

1 / ( s -


184/468

7. Sobo lev's Lemma

Example 2. L et .?.b e a cons tan t =F 0. Solve t h e equa t ion

x" (t) + .?.2 x(t) = 0, x(O) =ex , x ' (0) = {3.So lution. The correspond ing operato r equation is

s2 x-cxs -{3 + ?.2 x = 0, i.e., (s2 + ?.2 ) x = cxs + {3.Henc e, expandin g in to p ar t i al f raction, we obta in

cxs+P y


185/468

174 V I. Fourier T ransform a nd Different ial E quation s

we h av e th e follo wing re su l t which is o f fu nd a mental imp or tance mm o d e rn tr eatmen t of p artial diff erential equ ations.

T h eorem (Sob olev 's Lemm a). Let G be a b ound ed o p en do main of Rn . Let a fu n c t ion u (x) b elong to Wk (G) f o r k > 2- 1 n + a, where a is anin teg e r ~ 0. W e thus assu m e t hat th e d is t r ibu t io n a l der iv a t iv e s o fu (x) of o rd e r u p to a nd in c lud ing k a ll belong to L2 (G). T h en , for an y opensu b se t G1 of G su ch th at the clo sure G ~is a c o m p a c t su bse t o f G, thereexists a fu nc t ion u 1 (x) E ca (G1) su ch that u (x) = u 1 (x) a. e. in G1

Proof. Le t ex (x) be a f unc t ion E CZO(Rn) s u ch thatG1 C

< ( f /(1 + I Y/ 2) - k l 2 y i ' Y ~ ' . . . ~ l 2 d y j lv (y) (1 + IY/2)k/2l2dy,112. jyj>C jyj>C j


186/468

8. Ga rding's In equality 175

T h e second f a c tor on the right of the la s t inequ a lity is < o o by (1). Th e fir:;t fa ctor is f init e by

dy = dYI dy 2 dy, = r"-1

dr d Q , , 1w here dQ, is th e hype r sur face e le ment o f the surfacer 'of tlre unit sphe r e of R" with the origin 0 as its c entr e,

p rov ide d

2 l q l - - 2 k + n - 1 < - 1 , that is , if k>; + lql.Now, by P la nc he r e l ' s theore m , we have

v(x) = l . i .m. (2:nr" 12 f v(y) e xp ( i (y, x )) dy h->00 [y[ :;;,h

(3)

and so, as in the proo f of th e co m pleteness o f L 2 (R "), w e can choo se asubseque n ce {h'} of p osi t ive inte g ers h suc h t hat

v(x) = F m (2 n )-nfZ J v y) e xp(i (y, x) ) dy for a.e . x E R".h - + 0 0 [y[ :;i;h '

Bu t, sin ce v y) is i n tegrab le o ve r R" as p r ove d a bov e , the r i gh t side ise qu a l to

v1 (x) = (2 n)-" 12 J v y) e x p (i (x, y)) d y ,R n

t hat is, v (x) is equa l to v1 (x) for a.e . x E R". By (2), t he d if f e ren t ia t ion o fv1 (x) under the i n t e g r a l sign is jus t i f ied up to t he order a ; and the r e s u l tof the d ifferent ia t i o n is conti n uous in x. By taking u 1 (x) = v1 (x) forx E G 1 , we have pro ved the T h eorem.

R em ark. For the or ig inal proof, see S. L. So BOLEV [1] , [2] andL. KANTOROV ITCH-G. AK I LOV [1].

8. Garding s Inequal i t y

Cons ider a quadr at ic integr a l form de f in ed for C 0 0 func t ion u (x) =u (x1 , x2 , X 11) w ith co mpact supp ort in a bou nded d o m a in G of R ":

B [u, u] = ~ (c51 D 5 u, D 1u) 0 , (1)[s[,fir:;;,m

where the c omplex-va lued coeff ic ients c51 ar e c on t inu ous on th e clo sureGa of G, a nd (u, v)0 d enotes the sca la r produ ct in L2(G ).

T hen we haveTheorem (L. GA RDIN G [1]). A s u ff ic ient co n dit ion for t h e ex is tenc e of

p o s i t ive co nsta nts c, C so that the in e quality

llu I ~ < c R e B [u, u] + C llu I ~ (2)

holds for all u E ego (G), is th a t, for some posi t ive co n stant Co,Re ~ . C51 ~ s~ ~> c0 I ~ 1m for al l x E G and al l rea l v e ctors

[s[,ltf= m (3) ~ = ( ~ v~ 2 ~ m ).


187/468

176 V I. Fourier T ransform an d Differenti al Equa tion s

R e mark . The inequa lity (2) is c al led C ardin g 's ine qttal ity. I f co nd ition (3) is satisfie d , t h en the diff erent ial ope rator

L = 2: ( -1 )1 11D 1c51D 5 (4) l s l = l l l ~ m

i s said to be strongly ellip tic in G, ass uming that Cst is em o n ca.Proof. We f i rs t pro v e that, for every e > 0, there is a c onstant e (e ) > O

su ch that for ev e ry ego (G) funct i o n u,

(5)

To this pu r pos e, we co ns id er u to b elo ng to ego(Rn), def in i ng its va l ue as0 out side G. B y the Fou rier tr ansformat io n we have

/'-.... I n 12I D 5 u I I ~ =II(D 5 u) I I ~ =[ ;lJ yjiu (y) dy ,R

in vir tue of P lanch ere l ' s theore m . Th u s (5) is a conseq uen ce f rom the factt h at

( 1 ~ -1 j ft yj'i)I ( + ltrfmjfty;e;)(ls I = j ~ s;, It I = j ~ t;)tend s to zer o u n iformly wi th respec t to y = (YI, y2 , Yn) as e too.

S u p p ose that the coefficie n ts c51 are con stant and vanish u n l essIs I = It I = m. By th e F ourie r trans format ion u (x) --+ u ~ ) and Pla nc her el ' s theorem , we h ave, by (3),

R e B [u, u] = Re J 2 : ~ s ~ tl u ( ~ )1 d ~s ,t

;;;:: J Co I ~1m u ~ )1 d ~> C1( Iu 11;.- IIu 11;.-1)'w h ere c1 > 0 i s a c onstant w h i ch is indepen d ent of u . H ence, by (5) , we seeth at (2) is true for our spe cial cas e.

W e next c ons ider th e cas e of variable coeffic ient s c51 S up po se, f i rs t ,

th a t the s upp ort of u is sufficient ly s mal l and co ntained, sa y , in a smal ls ph e re about th e origin. By the prec edi ng case, we have, with a constantc ~> 0 which i s in d epe ndent of u,

c ~llu II;.< R e B [u, u] + R e lsl= if=m J cse(O) - Cse(x)) D 5 u. D 1 u d x- R e 2: J Cse(x) D 5 u . D 1udx + e (e) l l u l l ~-

lsi+Tir


188/468

T h u s , b y

9. Fr ie dric h 's T he o rem

2 j,xI IPI< e Jcx /2 + e- 1 IPj2 w hi c h is v a l id f o r eve r ye > 0,

177

(6)

we obta in lull < co ns tan t R eB [u, u ] + con stant llull - 1 + C (e) l l ~t l l ~ .a nd so , by (5), we obtai n (2).

Next w e co n sider th e g e nera l c ase. C on s t ru c t a parti t ion of unity i n G :

N

1 = j ~ wj, W j E CC0 (G) and W j (x) > 0 i n G,

suc h t ha t the supp o rt of e ach W j ma y b e take n a s sm a ll a s w e p le ase .T h e n , b y L ei b niz ' ru le o f diff e ren t i a t ion o f the prod uct o f func t ions ,

S ch w arz ' in e q u a l i ty a nd the e s t im a te o f th e c as e ob ta ined a bov e, w e h ave

R e B [u, u] = R e J ,1 D 'uD 1 u dx = R e ~ ~ J wjc , 1 D 'u D 1 udxs t s t 1

= Re ~ c,1 D ' (wju) D 1 (wju) dx + O(l lu lim jJullm-1)J s ,t

~ c o n st a n t ( l lw jul l - llwiu[J -1) + O(llul[m llullm-1)J

>co n stan t lull + O (llullm llullm-1) T hu s w e o b tain (2) by (5). W e re mark that the c o n s t a n t s c, C in (2)

d e p e n d u po n c0 , c81 and the d o ma in G.

L et9 Fr ie d ric hs Th eo rem

L = ~ D 'c ' 1 (x) D 1ll,ltf: Om

(1)

b e s t ron g ly e l l ip tic w ith re a l C 0 0 c o effic ients c81 (x) in a bou n d e d o pen d o ma in o f R " . For a g iven loca lly squ a re in t egra b le fu n ct ion l(x) i n G, a loca ll ysq u a r e i n tegra b le fu n ct io n u (x) in G is ca l le d a w ea k s o lu tio n of

L u = f , (2)i f w e ha v e

(u, L*cp)0 = (/, cp)0 , L* = I I ~ (-1)11+1t1 D 1 c81 (x) n , (3) s , lf:;>m

for e v ery cp E ego (G). H ere (/ , g)0 d e n otes the s c a la r p roduc t of th e H il b e r tsp a c e L 2 (G). Th us a w e a k so lu t ion u of (2) is a s olut i on in t he sen se o f th ed i s t r ibu t i on . C once rn ing the d i f f e ren ti ab i l i ty o f the w e a k so lu t ion u, we h av e the fo llow in g fu ndam ental r e su l t :

Th eo rem (K. FRIEDRICHS [1]). An y w e a k s o lut io n u of (2) h a s sq u a re

in teg r a b l e ( dis tr ib ut io n al) d e r iva t ive s o f o rd e r up to (2 m + p) in thed o m a i n G1 ~ G w h e re I has sq u a r e i n tegra b le (d istr ibu t iona l} de r iv a t iv e s o f o r de r up to p . In ot he r word s , an y w e a k so lu t i on u of (2) belo ng s to WP+ 2m(G1) w he n e v e r I belo n gs to JVi'(Gl).12 Y osida, Func t io n a l Ana lysis


189/468

178 V I. F ourier Tran sform a nd D ifferent ial E quation s

Coro llar y. I f p = oo, th e n, b y S obolev 's lem m a , th ere e xists a function u 0 (x )E C"'(G 1 ) s u ch that u ( x ) = u 0 (x) for a .e. x E G1 T h u s , af ter a correc t ion on a se t of m easu r e zer o, any w eak solut io n u (x) of (2) is CC"'in the d o m ain ~ G where I (x) is C 0 0 ; the cor rec t ed s olut ion is h ence agenu i ne solut ion of the differ entia l eq ua t ion (2) in th e d o m ain w here l(x)is C 0 0

Rem ar k. When L = Ll, th e L ap lacian, the above C oro llary is W e y l 'sL emma (see C h a p t e r II, 7). T here is ex tens i v e liter ature conce rnin gthe exte nsions of Wey l ' s L emma to g enera l el l ip t ic ope ra tor L;s u ch ex tens ions a r e s o m e t im e s c alled t he W eyl-Sc h w a rtz theo rem .A m o n g an abun dant literatu re, we refer to the p ap e r s by P. LAx [2],

L. N IRENBE RG [1] an d L. N IREN BERG [2]. T h e proof belo w is d ueto t he p resen t a ut h or (unpu blish ed) . A s im i lar proo f was g iven b y L. BE RS[1]. I t is to be noted tha t a non-diff eren tiable, l ocal ly in teg rab l e func t io nI (x) is a d is t r i b u t i on so lu t ion of the h yperbolic eq uation

o f_ = 0oxoy '

as may be se en f rom

0 = _ [ { _ [ l(x) 8~ t~ : )dy}dx( 0), whe n ev er


190/468

9. F ri edric h 's The orem 179

The c ond i ti on v E W q (G1) th us m ea n s t hat th e F o u r ie r co e ffici e nts v,.o f v (x) def in e d b y th e F o ur i e r e xpa n s ion

v (x),....., . I v,. exp ( ik x)"

(k = (k v . . . kn), x = (xv . . . xn), an d k x = / i.kjxj) (6)sa t i s f y

(7)

F or, by pa rtial i n t e g r a t i on , the F o u r i e r c oeffi c ients o f the dis t r ib u t io n a lde r i va t iv e nv sa t i s f y

(D 5 v (x), e x p (ik . x)) 0 = (-1 )1 1 (v x), n e x p (ik x)) 0

- ( - ")11IIn kY - ( )- Z . J v,., S - S v S2, , Sn ,J = l

a nd so , by P a r se va l ' s r e la t ion f o r t he F our i e r coe fficien ts of D qv E L 2 (G1 ) , we obtai n (7).

I t is con v enie n t to in t rod u c e th e spac e Wq (G1 ) w ith in te ge r q 0, by s ay ing that a s eque n ce {w,.; k = (k v k 2 , , kn)} of co m plex n um b ers w,.w i th w,.=w_,. b e long s to Wq (G1 ) i f (7) ho lds. T h is sp ac e Wq (G1 ) is no r m e d by

{w,.} llq = ( f Iw,. 1 (1 + Ik I2)T'2 B y v i r t u e of the P a r se v a l r e la t ionwi th r e sp e c t to th e c o mp le t e o r t h o n o r m a l sy s t e m { (2n)- n f2 e x p (i k x)} of

L 2(G 1 ) , we se e t hat , w he nq > 0, th e n o rm lv l lq = ( ~ J D5 v(x) l2 dx)1 ' 2l s f ~ qG1

is e q u iva l en t t o th e norm ll{v,.} llq w h e re v(x),....., . I v,. e xp(i k x)."he abo v e pr oo f of (7) sho w s th at, if I E WP (Gl). t he n n IE wP-11 (G l)'

an d q;f E WP (G1 ) for q; E coo (G1 ) . H ence i f f E WP (G1 ) , t h e n , for a ny d if f e ren ti a l op e r a to r N of

o r d e r q wi t h coo(G l) coef f icien ts , N I E wP -Iq l (G l).(8)

To p r ove t he T h e o r e m for o u r per i o dic c a se, w e first s how tha t w e may a s sum e that the w e a k s olu t i o n uE L 2 (G1 ) = WO(G1 ) of (2) bel o ngs to Wm(G 1 ) . T his i s jus t i f ied a s follo w s. L et

u( x) ,.....,. I u,. ex p (ik x ), v(x ) ,.....,. I uk ( l + lk 12 )-m exp(i k x) .k k

T h e n it is ea sy to se e th at v (x) E W 2 m (G1 ) and t hat v is a w e ak s o lu t io n n

of ( I - - L1 m v = u, w here L1de n otes the L a p lac i an . I 8 2 J8 xj. H en c e vi sJ = l

a w e a k so l u t ion o f t he s t ron g ly e l li pt ic e qua t i o n of o r de r 4m :

L ( I -L1)m v = f . (2 ')I f w e c a n show that t his w e a k so lut ion v E W 2m(G1 ) a c tu a l ly b e long s toW 4 m +p (G1 ) , th e n, b y (8), u == ( I - L1)m v be longs to W 4 m+P - 2 m (G1 ) ==

12*


191/468

18 0 VI. Fo ur ier Tra n sform a nd Dif fe renti al Equa tions

W P+2" '( G 1 ) . T here fo re, w itho u t lo s in g th e gen er ality , we m ay as sum etha t t he w eak solu tion u of (2) belo ng s to W "'( G 1 ) , w h ere m is h al f of

th e or der 2m of L .N ext , by G ar ding 's i ne q u ali t y ( 4) for La nd th at for ( I - A)m, we ma y

ap p ly th e L ax- M ilgr am th eo rem (Cha pt er II I , 7) to th e fo llow ing e ff e ct.T h e bil in ear fo rms o n coo (G1) :

(rp,1p)' = (rp, L*1p)0 a nd (rp,1p)" = (rp, (I -L 1)"'1 p) 0 (9)

ca n bo th b e ex tend e d to b e c on tinu ou s b ili near form s on W "' (G1 ) such th at th e re e x i st o ne -o ne b ic onti nu ous l inea r m app ings T , T " o n W "'(G 1 ) onto w m ( G1) sa ti sfyin g t he c ond iti ons

(T' rp, 1p)' = (rp, 1p),., ( T " rp, 1p)" = (rp, 1p),. for rp, 1p E W "' (G1 ) . Th er efor e, the re exis ts a o ne -one b icon tinuo us line a r ma p T,. = T" (T')- 1o n W "'(G 1 ) on to wm (G 1 ) such t hat

(rp, 1p)' = ( T .rp, 1p)" w h enev e r rp, 1pE w m (G1 ) .

W e c an s ho w t ha t

fo r an y i > 1 , T,. m aps W 'H (G1 ) o n to wm+i (G1 ) in a o ne -one and bi con tin uou s wa y .

In fac t, we h ave

(10)

(11)

(rp, L*( I -L1) 1 "P)o = (T,. r p, (I -L1) "'H "P) o for rp, 1pE C 00 (G1 ) .

O n th e o th e r h an d , ther e exis ts, by th e La x -Mil gr am th eore m as a p plied to th e stro ng ly e ll iptic o per at ors (I - A ) i L a n d ( I - A)"' + i, a o ne-o n eb icont in uou s l inea r ma p T m+i of w m H (G1 ) on to wm H (G1 ) such th at

(rp, L * ( I - L1)1"P)o = (T m+i rp, ( I - A)m+i "P)o fo r rp, 1p E C 0 0 (G1 ) .

T h erefo re, th e fun c tion w = (T m H - T ,.) rp is , for any rp E C 0 0 (G1 ) , a

w e a k so lu tion o f (I -A) m +i w = 0. B ut, su c h a w(x) is i den tic ally z ero. F o r , th e Fou ri e r c oe ffici en t w,. of w (x) sat is fies

0 = ( ( I -L 1 ) " '+ i w (x), e xp(ik x)) 0 = (w (x), ( I -A)" ' + i ex p(ik x)) 0

= (1 + Jk J2 )" '+ i (w(x) , ex p ( ik x) ) 0 = (1 + Jk J2 )m+j w,.,an d so w,. = 0 fo r a l l k. T h u s (T m +i - T .) is 0 o n C00 (G1 ) . T h e s pa ceC00 ( G1 ) is den se in W " '+ i (G1 ) ~ wm ( G1 ) , si nc e tri g onom e tric p olyn o mial s

~ w,. ex p (ik x) are de nse in th e spac e wm + i (G1 ) . H en ce T m+i = T, .ikJ


192/468

9. Frie drich's The orem 181

H ence, forT m U ' ~ ck exp (ik x), tp (x) ,_, ~ 'IJ'k e x p (i k x),

k k

we o b ta in , b y P a rseva l ' s re l a tion,

( T mu, (I -L l)m 'IJ')o= ~ ck(1 + lk l2 )m ~ k = ~ fk'iiikk k

By th e arbi t rar i n ess of tp E C"" (G1 ) , we h ave ck (1 + I l2 )m = fk, and so,by fE W P(G 1 ) , we must have Tmu EWP+ Zm(G 1 ) . Hence , by (11),uE WP+Zm (G1 ) .It is to b e no ted that the above co nclusion uE WP+Zm (G1) is true ev en in the case 0 > p > (1 - m ) , i.e., th e case {ik}E WP (G)wi t h 0 > p > (1 - m ) . F or, by p + 2 m > m + 1, we can a p ply (11).

W e finally shal l prove, for t he general n on-periodi c case, our d ifferentia b il i ty theore m . The foll owing argum ent is due t o P. LAx [ 2].W e wa nt t o prove, fo r the gener a l non-perio d ic case, the differentia

bility theo rem in a vicinity of a poin t x 0 of G. Let {3(x) E ~ (G) b e ident ically one in a v ic in i ty of x 0 . D e note {3u by u'. u' is a w eak solu t io n of

Lu' = {31 + Nu , (12)where N is a dif ferential op e rator of or d er a t m o s t equa l to ( 2 m - 1)whose co efficients ar e , together w ith {3(x), zero outs ide some vic in it y Vof x 0 , and the opera to r N is to be app lied in t he se nse of the d istr ibution . We d eno te the d is t r ibut ion {3 f + N u by f'.

Le t th e periodic p arallelogra m G 1 con ta i n V, and im agine th e coefficients o f L so alter e d inside G1 b ut outside V that they become pe riodicwitho ut losing th eir differe n tiabil i ty an d ell iptici ty propertie s . Denotethe so altered L by L'. Th us u' is a w e ak solutio n in G of

L' u' = f', where f' = {3/ + Nu. (13)

W e can thus ap p ly the res u lt obtain ed above for th e periodic c ase too ur w eak solutio n u ' . We ma y assume th at the w ea k solution u ' belongs

to wm (G1 ) . Thus , s i nce N is of o rder < (2 m - 1 ) and with coeffi c ientsvan i sh i n g onts ide V, f' ={ 31 + Nu mu st satisfy, by (8),

f'E WP' (G1 ) wi t h p' = min ( p, m (2 m -1) ) = m i n(p, 1-m);: ;:::: 1 - m .There fore, th e w e a k solution u ' of (13) m ust satisfy

u ' E WP" (G1 ) with p " = m i n (p + 2m , 1 - m + 2m)

= min(p + 2 m , m + 1 ) .

Hence , in a cer ta in v ic in i ty o f x 0 , u has s q uare in teg rable d is t r ib ut ional

der iv a tives up to order p" which is ? (m + 1 ).Thus f' = {31 + N uha s , in a vici n ity of x 0 , square in teg rable d is t r i b ut ional de r ivatives of o rder up to

p"' = min(p, p " - ( 2 m - 1)) 2:":m i n (p, 2 - m ) .


193/468

1 8 2 VI. Fourie r Transfo rm a nd Differ ent ial Equation s

Thus aga in app ly ing the r e sul t a l r eady ob ta i n e d , w e see tha t u ' has , in ace r ta i n v ic i n i ty o f x 0 , s qua r e in teg rab le d is t r i bu t ion a l de r iva t iv e s u p to

o r de r

p(4 ) = mi n(p +2m , 2 - m + 2 m ) = min(p + 2 m , m-+ 2) . R e p ea t in g t he proce ss, we see that u ha s , in a vic in ity of x 0 , sq ua r ein tegr a b l e d is t r ib u t ion a l de r i va t ive s up to o rd e r p + 2m .

1 0 . T he Mal gran ge-Ehr enpr eis The orem

T h e r e is a s t r ik ing diffe rence be tw e en ordina ry differ ent ia l equat ionsand p artial diffe rent ia l equ a t ions . A c lassica l r e su l t of PEANO states th at the ordi nary d iffere ntial e qua t ion dyfdx = f(x, y) ha s a so lut ion under a s ingle c ondi t ion of th e c ontinuit y of the funct io n f. T his re su l thas also be en extend ed to e q ua t io ns of h ighe r o r de r or to sy s t e ms ofe quatio ns. H oweve r, for partial d if ferent i a l eq ua t ion s , th e s i tua t i on i sen t i r e ly d i fferen t. H. LEWY [1] c o ns t r u c ted in 1957 the e qua t i o n

. a u a u 2 ( . ) au I ( )- z - + - - x 1 + 1x 2 - = x3 ,ax l ax2 axaw hich h a s no solu tion at all i f f is n ot an alyt ic , even i f f is C 0 0 L e w y 'se xa m ple led L.

H6RMANDER [3] tod

evelo p a sy s tem at ic metho d ofc o ns t r uc t ing l inea r pa rt ial dif fe ren t ia l equ a t ion s w i th ou t so lution s. T h usi t is impo rtan t to single o ut cla sses o f l inea r part i al diff e ren t ia l equ a t ion swi th solu tions .

Le t P (.;) be a polyno mia l in .;I>.;2 , . ; . and le t P (D) be th e l ine a rd if fe ren t ia l op e r a to r obtain ed b y repla c ing .;j by Dj = - 1 8/oxj. P ( D) mayb e w ritten as

P (D) = I c XD,., w here, f or


194/468

10 . The M algr a nge-E hrenp r eis Th eo re m 183

E xa m pl e. L e t P (D ) b e the Lapla cian L1= 'i; a: in R " wi th n > 3.j= 1 ax,

T he n the dist r ib u t ion 1

E = Tg, where g (x) = ( 2 _ n) S,. I 12 - a nd S,. = the a rea of th e surfa ce of the u nit sp here o f R",

is a fun d a m e n tal so lution of Ll.

P roof . W e ha ve, in the p olar c o ordin ates, d x = lxl"-1 d lxldS,., a nd so th e fun ction g(x) is local ly inte grabl e in R ". H en ce

L1 r , ~, - n(I]J) = lim J I X 12 - n . LlqJ dx' IP E ;1) (R") .e. j.O

1 ~ 1 ; : ; : ;

L et us t ake two po sitive num bers e an d R (> e) suc h that the s u p p (I]J)is cont a ined in th e in ter i or of t he sph ere I I < R . C on sider t he dom ain G :e < JxI< R of R" a nd ap p ly G reen's in teg ral t he o r e ~, obta in ing

J lx 12-n. LlqJ- L1lx12-nqJ) dx = J lx 12-n. ~ ~- a l~ , _, . qJ) dSG ~

where S = 8G is the b o u n d a ry su rfaces given by xl = e and lxl = R ,an d v de notes th e o u tward s norm al to S. Sin ce qJ v anishe s a rou n d I I= R, we have, remem berin g tha t L1lx 1- = 0 fo r x I= 0 and t h a t - :v= al: Iat the poin t s of th e in n e r bo u n d a r y surf a ce [xI = e,

f Jxl2 - LlqJ d x = - ( e 2 - n a ~ dS + ( ( 2 - n) e1 - n 1J dS .Rn l ~i =e I I I ~J =

nW h en e t 0, t h e exp ressio n 81]J/8IxI = . I (xjflx I) 81] /8xj is bou nded

J= 1

a nd the a rea of th e b o u n d a r y surfa ce I I = e is S ,.e "- 1 . C onse quent l ythe first t er m o n th e r igh t t ends to ze ro as e t 0. B y th e c o n t i nu i t y of IPan d a si milar a rgum ent to abov e, the secon d te rm on t he r ig h t t en ds, as e t 0, t o ( 2 - n) S,. IP(0). T hu s Tg is a fu ndam enta l solut i on of Ll.

T h e e xisten ce of a fund a m e n tal so lution for ev ery l i near p artia l diffe

re nt ial equat i o n w i th con s t an t coeffi cients was p roved indep enden t ly by B . MALGRANGE [1 ] an dL . EHRENPREIS [1]aro u n d 1 954--5 5. The expo sitionof th e resu lt giv en bel ow is due to L. HoRMANDER [4].

D ef in ition 2. Set

(1)


195/468

184 VI . F ourier Trans form an d Di fferentia l Eq ua tions

W e sa y that a diff uential op e ra tor Q( D) with c o n s t a n t c o efficie nts isweake r th an P (D) if

Q( ~ )< CP ( ~ ) ,~ E R n, C b eing a pos itive const ant. (2)Th e or em 1. I f Q is a bou n d e d d o m a in of Rn and IE L2 ( J ), then t here

exists a solu t ion u of P (D) u = I in J suc h t ha t Q (D) u E L2 ( J) for all Qw eaker than P . H e r e the differe ntial ope r a t o r s P (D) and Q D) a re to be a p p l ied in th e se nse of the t heory of di s t r ibu t ions .

T he p ro of is b a se d o nTh eo rem 2. For every s > 0, t h e r e is a f u n d a m e n t a l so l ut ion E of

P(D ) su c h that, w ith a const ant C inde p e n d ent of u ,

I (E * u) (0) I < C su p Ju ( ~

+ i1))1/ P ( ~ ) d~ ,

u E C (f (R n). (3)1'11;:;; Rn Here u s th e F o u rier-L aplace t r a n s f o r m of u:

u C) = (2n) -nf 2 J e-i(x ,o u (x) dx, C = ~ + i TJ,R n

and the f i nitene ss of the r ig ht sid e of (3) is a ssured by th e Pa le y - Wi e ne rthe o r e m in C ha p te r VI, 4.

Deductio n o f Theor em 1 from Th eorem 2. Replac e u in (3) by Q D) tf, * v, w h e r e u an d v a re in C( f (Rn). T he n , by (10) i n C h a p t e r V I , 3,

I Q(D) E

*u

*V)

(0) I= I E * Q (D ) u * v) (0)J

< CN (Q(D )tJ

* v),where N (u) = sup J i t ( ~ + i 1))J/P ~) ~ .

1'11;:;; R n

The Four ie r-Lap lace t r a n s f o rm of Q D) u * v i s , by (17) in C ha p ter V I , 2and (15) in C h a p te r V I , 3, e q ua l to (2n t ' 2 Q C) u C) v C). Sin ce, byT aylor's form ula,

1 Q ( ~+ i1)) = ~ (< X ) (-n)"' ' Q ( ~) . w he re (-1 ))"'= I.I (- 1) "';), (4)

"' Jw e h a v e , by (2),

/ Q ( ~ + i1 ) ) // P ( ~ )< C 'for /T JJ


196/468

1 0. The J. \llal gra nge -E h ren pre is T he or e m 185

T here fo re w e hav e p r o v e d

J (Q D) E

* u ) (x) v ( - x ) dxll ~ (e e") IIu e lx ll l l lve l xl l l (u , vE e:;o (R ") ).

R n (fi)

W e sha l l w r i t e L; (Rn) for the H ilbert s pace o f fun c t ion s w (x) norm e d by

(i lw(x) 12 elxl d x t 2 = llw (x )ee lxlllSinc e ego(R") is d ense in L; (Rn) a nd, a s may be p ro v e d e asily , L':.. (Rn) is th e c on j u ga te s pace o f L; (R ,.), w e ob t a in, b y div i d ing (5) by v (x) e lxfllan d tak i ng th e supr e mum over v E ego (Rn),

II( Q (D) E * u) (x) e -efx lll < (ee") ll u (x) e *'JI, u E ego (Rn).H en ce the m a p p i n g

u ~ Q( D ) E * u (6)

ca n b e e xten de d by conti n uity f ro m ego (Rn) to L; (Rn), so tha t i t be co m es a c o n t inu o us l i n ea r m appi n g on L ;(R n) in t o L':.. (R ,.). T h u s , to p ro ve

T h e o rem 1 , we have to tak e 11 = I n . Q and 11 = 0 i n Rn -Q and d e fineu a s e q u a l to u = E * 11 .

Fo r the p roof o f T he o re m 2, we p r e pa r e t h r e e L e m m as .

L e m m a 1 (M A L G R A N G E ) . I f I (z) is a ho lom o rp hic fu nc t i o n of a c o m p lexv a r ia b l e z for I I< 1 a nd p (z) is a p o lynom ial in w h ic h the c oeff ic ientof th e high e s t o r d e r te rm is A, th e n ,.

A 1(0) I< ( 2n)- 1 J l(e6 ) p(e6 ) I dO. (7)_,.Pr oof. L e t z /s be th e zero s of p (z) in the u n it cir c le z I< 1 an d p ut

p( z) = q(z) J -z:zz , l J

Then q(z) is re gu lari n t he u n it circl e and P(z) I= q(z) I or lz I = 1. H e n cewe h a ve

,. ,.( 2 n)- 1 J ll( e 16) p(e 6 ) Id O= ( 2 n)- 1 J l(e6 ) q(e' 6 ) IdO

- n - n

> (2n ) -I [_ l l(e' 6 ) q(e6 ) dOI= 1(0) q(O) IThus Lemm a 1 i s pro ved sin ce Jq(O)/A I is e q ua l t o the prod u ct of thea bso lu te v a lues o f zer o s of p (z) o u ts ide the u n it cir c le.

L e m m a 2. W ith t he no tati on s in L emma 1, we ha ve , if th e deg re e of

p(z) is < m,

1(0 ) p


197/468

186 V I . Fou rier T ra nsform an d Di fferenti al Equations

P roof. We m ay assume that t he de gr ee of p z) ism a nd th atm

p (z) = .II ( z - zj).J = lk

Apply ing th e precedi ng L e m m a 1 to th e po ly nomial . I/ (z-z i) and theJ = l

m

holo m orphic fun c t ion I (z) . I I ( z - zj), we o b t a i n J = k + l

11(0) i = R / i ~ ~(2n)-1 _[ ll(eiB) p(eiB) Id().

A simila r inequa l i ty w ill ho ld fo r any p ro d u c t of ( m - k) of th e n u m b erszio n the left ha nd side. Sin ce p(k) (0) is the sum of m f(m -k) s u ch t e rms ,mul t ip li ed by -1 ) m-k, we ha v e p roved t he i nequa l i t y (8).

L e mm a 3 . L et F (C) = F { ~1 , ~ 2 , . . ~ n ) be ho lo morphic fo r IC =CliI j 12r2 < C2, . . . Cn) a polyn o mial of de g ree< m . Le t f/J (C) = f/J ( ~ vC2 , Cn) b e a non-neg at ive in tegr ab le funct io nw i t h co m p act suppo rt dependin g on ly on 1~1 1 , IC21, . . . , ICnl Th en weh av e

IF(O) D" P(O) I f I C I (' ) f / > ( C )d ~


198/468

10. Th e Ma l grang e-Ehre npreis Th eo re m 187

H en ce, by F o u r ier 's i n teg ra l th eo rem ,

lu(O) 1< ( 2n }-"12

Ju( ~)

1~

< e;_ J (Jl v (~

+ C)1 / P ( ~) d ~ )d ~

ICI;>;

s e ~ f ( f lti ~ + ~ + i 17') 11.P ~ ) d ~ 'd 17' ) ~ . ~ ' +' l ' ' : ; ; e

On t h e o th e r h a nd, w e h a v e

be cau se

D"' ~ + ) = f \ ~ ~~D"'+P ( ~) .

so th at J D"' P ( ~ + ~ )I/P ~) is bo und e d wh en If S e. T here fo re we hav e

lu (O )I als o tend s to ze ro i n th e to polo g y of i) (R "), uni fo rm l y wi th resp ec t to 17fo r 117I s e. H ence, a s in C hap te r V I , 1, we see ea sily tha tv1 ( ~ + i 17), as func tion of ~, ten ds to ze ro in the to po log y of 6 (R") ,unif or m ly i n 17fo r 117 I < e. T he ref ore, b y (12), L defin es a d is tr ib u ti onTE i ) (R ") ' . T hus , by (5) in C h apte r VI , 3.

L (v) = (T * v) (0) = ( f * v) (0 ). (13)W e h a ve th u s pro ve d Th e orem 2 b y ta k in g E = f. (3) is cl ea r fr om (11 ).


199/468


11. Differential Operators with Uniform Strength

The existence theorem of the preceding section may be extended to alinear differential operator

P(x, D ) = ~a.,(x) D" (1)IX

whose coefficients a., (x) are continuous in an open bounded domain Q ofRn.

Definition. P (x, D) is said to be of uniform strength in Q, if

(2)

where P(x, $)is defined by (f IP'"')(x, $) I2Y'2 considering x as parameters.Examples. The differential operator P(x, D ) = ~ Dsas 1 (x) D1

J s J , J t J ~ m 'with real, bounded C"" coefficients as,t (x) = a1,s(x) in Q is strongly ellipticin Q (see Chapter VI, 9) if there exists a positive constant


200/468

12. T he Hypoellipt ic ity (Horma nder's Theo r em) 189

B y the resul t of the prec e ding sectio n , there exi s ts a bounde d l inearopera to r T o n L2 {Q1 ) into L2 (Q 1 ) such th a t

P 0 (D) T f = I for all / E L2 { Q 1 ) , (6)and th at t he o perators Pi (D) T are a ll bounded a s opera to r s on L2 (Q1 )in to L2 Q 1 ) . Here J 1 is a n y open su b domain ofQ . W e hav e on ly to tak eT f a s the restr ic t ion to Q 1 of E * /1 w here / 1 = I in Q 1 an d / 1 = 0 inR n - Q 1 .

The eq u at ion P(x, D ) u = f is e quivalent t o

(7)

W e sha ll seek a solution o f the form u = Tv. Subs tituting th i s in (7) , we ob ta in , by (6),N

v + . ~ bi(x) Pj(D) T v = f.J = l

(8)

Let the s um of the n orms of th e b o u n d e d l inea r oper a tors P j (D) T onL2 (Q 1 ) i nto L2 {Q1 ) be d e n o ted by C. Since bj (x) is contin u ous an dbj(x 0 ) = 0, we may choose Q 1 3 x 0 so small that

C jbj(x) I< 1/N w h enever x E J 1 ( j = 1, 2, . . . n).W e may assume tha t the abov e inequali t ie s ho ld wh e never x be longsto th e c o m p a c t closure o f Q 1 . Thus the n o r m of t he operator

N . ~ bj (x) Pj (D ) T is less t han 1, and so the equ a t ion {8) m ay be solv ed

J = l

by N e u m a n n 's series (Th eorem 2 in C hap te r I I, 1):

v = (1 +.I bjP j(D) T)- 1 = Af,;= 1

where A is a boun e d l inear op er ator o n L2 (Q 1 ) in to L 2 (Q 1 ) . H enceu = TAf is t he required solution of P(x, D) u = f.

12 . T h e Hypo elli pticity (H o rmande r s Theore m )

W e h ave define d in Chap te r II , 7 the no t ion o f hypoel l ip t ic i ty ofP(D) an d prove d H ormande r' s theorem to the ef fe ct that, if P (D) ishypo e lliptic, then there ex ists, for an y large po s it ive const a nt C1 , ap ositive c on stant c2 su c h that, fo r all solution s c= ~ + rJ of the alg ebrai c equatio n P (C)= 0,

(1)

To pro ve convers e ly that (1) implies th e hypoel l i p ticity of P (D),

we pre pare the Lemma (H6RMANDER [1]), (1) implie s th at

J 0 j P < ' "> ( ~ ) I2 / , . ~0 P ' ' " > ( ~) I2 - +0 as ~ E R n , ~ j - +o o . (2)


201/468

190 V I. Fourier Transform a nd Different ia l Equa tion s

Proof . W e first s how t hat, fo r an y r ea l v ecto r g E R", we have

P (; + 0){ P (; ) ~ 1 as ; E R", j;j oo. (3)We maya ssume thatt hecoo rdina tesare socho sen that e = (1, 0, 0, . . . , 0).We h a ve , b y (1 ),

P(; + 'Y}) :f= 0 w he n I YJI < c1 a nd [;I > c2.T hen th e ineq uality j ; - C' I > c1 ho lds i" j;j > c1 + c2 an d p (C') = 0.F o r , setti ng C' = ; + 'Yj', w e ha ve e i t he r I'YJ'I:;::: c1 o r e lse WI < c2 sotha t j ; - ; 'j > C1 . G iv ing fi xed va lues to, ; 2 , ; 3 , . . . ;n we c a n wr i te

mP (;) = C I I ( ~1 tk), C :f= 0,

k=1

w here (tk, ; 2 , . . . ;n) is a ze ro of P. H ence w e ha v e j t k - ; 1 j > C1 if;j > C 1 + C 2 T h u s

~ ( ~ + 6>) = r r~ 1+ .1 - t k = rr(l+ _1 )P ( ~ ) k= 1 ~ 1 -tk k=1 ~ 1 -tk

sati sfies

I ~ ~ )O) -11 < m C 1 1(1 + C11 m- 1 if [;I> c1 + C2.As w e ma y take C 1 a rbitrarily la rge by ta ki n g C 2 s uff ic ient ly large , w eh ave p r o v e d (3).

W e h a ve , by T a y lor ' s formu la,(cx)

P ( ~+ 'YJ)= ~ ( < X ~ p( ~ )'YJ"', p ( ~ )= ( i )


202/468

1 2. T h e H yp o ell ip ti c i ty (H o rm a nd er 's T he o re m ) 19 1

Pro of. B y ap p ly in g L e ibn iz ' f o rm ula of d iffe re n t ia t i o n o f pro du c t o f

fun ctio ns, we see tha t p


203/468

192 VI. Fourier T ransform a nd Different ia l Equa tion s

we se e b y p (D) Uo E Hfoc (Q ) tha t, w h e n e v er cpl E ego (.Ql),

P (D ) cp1 u 0 E W k,, 2 (Rn) w ith k1 = mi n(s, k).

Thus the Fo urier tran sfor m u l ( ~ ) of ul( x) = cpl(x) Uo(x) satisfies

f I P ( ~ )ul ~ )12 (1 + 1 l 2)k, a ~ < ooRR

and hen ce, by Lemm a 2,

J P(a:)( ~ ) ul ~ )12 (1 + I ~l2)k,+ p d ~ < oo, tha t is,Rn

p(a: ) (D) u 1 E Wk,+ p,Z (Rn) for every IX# 0.

(9)

(10)

(11)

Let .Q2 be any o p en s u bdomain o f .Q1 su c h th a t its c l o su r e ~is com pacta nd co ntain ed in.Q 1 Th en , for a nyc p 2 E ego(.Q2 ) , we prov e, b y (8) a nd (11)as ab ov e , that

P (D) cp2 u 1 E W k 2 (Rn) with k 2 = m i n (s, k1 + p) a nd henc eP("')(D) cp2u 1 E W k,+p,Z (Rn) fo r every IX # 0 .

R ep eating the argum ent a fini te num b er of t im es , we see tha t, for anyopen s ubdom ain .Q' of .Q such that it s closu re i s comp act a n d containe d i n .Q,

p( a:) (D) cpu E ws,Z (Rn) for a l l IX # 0 w h e never p E ego (Q') .

T hus, p (a:) ( ~ ) =co nst ant =l=0 gives cpu E ws,z (Rn).Proof of L emma 1. T he Fo urier tr a nsform o f "PI is

(2n)- n f 2 J J ('Y/) 1 ~ - YJ)d YJ(s ee Theo re m 6 in Chap te r VI, 3)RR

and th us we hav e to sh o w t hat, for s 0 ,

f (1 + ~ 12f I f .;p(nll ~ - n) dnjz d < oo .R n Rn

By Sc h wa r z' inequ ality , th i s can b e estim ated f rom abov e by

R[(1 + 1 ~ \2)8 [J \.fP(n) Idn R[l.fP(n) 1 1 / ( ~ -n) l2 dn] a ~ = i l.fP(n) Idn[ i i (1 + 1 ~ 12)8 l.fP(n) I 11( ~ - n) 1 a~ dn}W e then mak e use of th e ineq uality

(1 + ~ ~12)s < 4lsl (1 + IYJ2)lsl (1 + ~ ~_ YJ12)'whic h m ay be prov ed by

1 + I; a 1 + I ; - n a1 + l$-n la < 4(1 + ln l2), 4(1 + 1 ~ 1 2) > 1 + lnla .

By (13), t he r ight s id e of (12) is esti mate d by J ~ ('Y/)Idn-t imesRn

(12)

(13)


204/468

1. D ual Operator s 193

T h i s i n t e g r a l converge s since /E W 8 2 (Rn) and .fP('YJ)E 6(Rn ).W e ha ve th us p r ov e d ou r Theorem.

Furthe r R esearche s

1. A l ine a r pa rtial d i fferent ia l o perator P(x , D) with C 0 0 (Q) coeff icients is s a id to be for mally hyp oe lliptic in Q ~ Rn if th e following tw ocondit i o ns are sa t i s f ied: i) P(x 0 , D ) is hyp o elliptic for eve ry f ixed x 0 EQand ii) P ( x0 , ~ ) = O ( P (x ' , ~ )) a s ~ E R n ,1 ~ 1- + = for eve r y fixed x0 a ndx ' E Q . L. HoRMANDER [5] an d B. MALGRANGE [2] h a v e p r ove d that,for su c h an op erat or P (x, D), any dis t r ib u t ion solut i o n u E 'Il (Q) ' of thee qu a t ion P(x, D) u = I is coo af ter co r rec t ion on a se t o f me a sure zeroin t he open su b doma in ~ Q where I is C 0 0 T h e p roof above f o r the constant c oefficients case may b e modif ied so as to app ly for the formallyhy p oell ipt ic c ase, see. e.g ., J. PEETRE [1].

2. I t was prov ed essentia l ly by I. G. PETROWSKY [1] that a l l dis tr ibutio n solut ion s u E 'Il (Rn) of P ( D ) u = 0 are a na lytic func t io ns in Rniff t he homoge n eous part p m ( ~ )of p m f the h ig h est degree m does notvanish for ~ E Rn. I f th is condit io n is satisfie d th e n P (D) is said to be(an alytically) e lliptic. I t is proved tha t in such a case the d egree m is even a nd P (D) is hypoell ipt i c . I t is to b e noted that the hypo e l l ip t ic i ty

of an a n a l y t ica l ly e l l ip tic operato r P (D) c a n a lso be prov e d by Fr iedr i c hs 'Theorem in Cha p te r V I, 9. For, b y the non-v anishing of P m ( ~ ) , we e asilysee, b y the F o u r i e r t r ansfor m a t ion , tha t P (D ) or -P D) is s t ronglyelli p tic. For th e proof of P e t r ow sky ' s T heorem, se e , e.g., L. HoRMANDER[6], F. TRiNES [1] and C. B. MORREY-L. NIRENBERG [1].

VII . Dual Op erators

1 . D ua l O perators

T he n o tio n of the transposed m atrix may b e extende d to the not io n ofd ua l operato r t hrough

T heorem 1. L et X, Y be locally con v ex l inear t o pological s p aces andx;, Y ; their s t r o n g du a l sp a ces, respec t ive ly. Let T be a l inea r operator on D (T) ~ Xin to Y . Consider th e poin ts {x', y } of the pr oduct spac e ~ X Y ~sa t isfyin g the condi t i o n

< T x, y') =


205/468

194 VII. Dual Opera tors

T h us the " if" part is clea r f rom t he c ontin uity o f the l inea r func t i ona lx'. A s sume that D ( T t =F X . T h e n there ex i sts, by the Hahn-Ba na chth e orem , a n x ~=F 0 suc h that


206/468

2. Ad joint Operat or s 195

Proposit ion 1. T h e m a pping T -+ T' of L (X , Y) in to L (Y;, x;) isno t , i n g eneral, co n t inuous in th e simple co nvergenc e t opology of o pera

tors, th at is, lim T nx = T x for all xE X does not nec essarily im p lyl im T ~ y = T ' y ' fo r a l ly ' E Y', in the s tr ong du a l to pology o f x;.

n - > 0 0

Theo rem 2'. Let T be a bou nde d l inear o p era tor on a no rmed line arspace X in to a n o rm ed l ine ar s pace Y . T h en th e d u a l opera to r T' is abo unded l i nea r opera t o r on y; in to x; such that

II T I /=IIT ' l l Proof. From the defin ing rela tion


207/468

196 V II . Dual Oper ators

D efin ition I Le t X, Y be H i lbert spac es, a nd T a lin ea r o p eratordef in ed on D (T) ~ X into Y. Let D (T t = X and let T' be the d ualoperator o f T . Thus ( Tx , y ' ) = ( x, T'y ') fo r xED(T) , y 'ED(T') . I fw e deno te b y ] th e on e-to-on e no rm-pres erv i n g co n ju g a t e l inear corr espon dence x ; 3 I++ y1 E X (defined in C orollary 1 in Chapte r II I, 6), th en(Tx, y') = y' (Tx) = (Tx, ] yy ') , (x, T 'y' ) = (T 'y ' ) (x) = (x ,JxT 'y ' ) .

We hav e thus

(T x, J yy') = (x ,] x T ' y') , t hat is, (Tx, y) = (x , ] x T ' T;}y).In the special c ase wh en Y = X , we w rite

T* = fxT 'Tx ?and call T * the adjoin t operator of T.

Rema rk. I f X is th e c om p lex Hilb e rt space (l2 ) , we see, as in th e E xample in the pre ced i ng sect i on, th at th e matrix co r resp onding to T* is th e trans posed con jugatP- matrix of th e matr ix co rres ponding to T.

As in the case of dual o p e ra to rs , w e c an p ro v eT heorem I T * e xists iff D ( T t = X . I t is def i ned as follo ws: y E X

is in the d o m ain of D (T*) if f there exist s a y* E X s u ch t hat

(T x, y) = (x, y*) ho lds for all x E D ( T ) , (1)

a nd we defin e T* y = y* .W e ca n re wr i te th e ab ove the orem in term s of the gr ap h G (A) of th e

l in ear op e ra tor A (the graph was introdu ced in C h ap te r I I, 6 ):The orem 2 . We i n t r o d u c e a con t inu ous l in ear op e rator V on X X X

into X X X byV{ x, y} = {- y , x}. (2)

Then (V G (T ))j_ is the g r aph of a li near operato r if f D (T t = ~ X, an d , in f ac t , we have

G ( T*) = (V G (T) ) j_ . (3)

Pro of. The condi t i o n { - T x , x} _L {y, y*} is eq u iv a len t to (T x,y) =(x, y*). Thus T h e o rem 2 i s pro ved b y T h eo rem 1.

C oroll ary. T* is a clos ed l inea r o p era to r, s ince the o rt h o g o nal com plem ent of a lin ear subspac e is a closed l in ear subs pace .

T heor em 3. Let T b e a l inear operator on D (T) ~ X i nto X su ch thatD ( T t = X . Then T admi ts a cl osed lin ear ex tensi o n i ff T** = (T*)*ex i s ts , i.e., iff D (T*t = X .

Pro of. Su fficiency. W e have T** ~ T b y de fini t ion, and T** = (T * )*

is clos ed by t he a b o v e Co roll ary.N ecessity. Let 5 be a clo sed ext ens i on o f T . Then G (S ) c o n ta in s

G ( T t as a clos ed l inea r su bspace, and so G (T t. is the g ra ph of a lin ea ro p e ra to r . But G (T t = G(T)j_ j_ = (G(T)j _)j_ by the co ntinuity of t he


208/468

3. Symmetr ic Operators an d Self -adjoi nt Operators 197

scala r p roduc t , and , mo reover, by VG (T*) = G(T)_j_, we obtain(VG(T*)) _ _ = G(T)_j__j_. Therefo re, (VG(T*) )_ _ is t he g raph of a l inear

opera to r. Thus by Theorem 2 D ( T * t = X and T * * exists.Corollary . Under the condition th at D ( T t = X , T is closed line a r iff

T = T * * .Proof. T h e sufficien cy is clear. N ecessity is p roved by o bserving th e

fo rm ula G ( T t = G (T**) ob ta ined a b ove. For, G (T) = G ( T t implies t hat T = T* *.

Theorem 4. An eve rywhere def ined closed l inear opera to r T i s a c ont inuous l inear oper a tor.

Proof. C lear from th e closed gra p h theorem .

T h eorem 5. I f Tis a bou n ded l inear o perator, th e n T* is also b o u n d e dl inea r and

IITII=liT* II ( 4)Proof. Sim ilar to the case of d ua l operators .

3 . S y m m et r i c Oper a tors and S e l f -ad jo in t Operators

A H e r m i ti a n m atrix is a ma trix which is e qua l to it s t r anspose d con ju gate matrix . I t is kn o w n that su ch a matrix can be t ra n s fo rmedinto a diagon a l matri x by a suitabl e (complex) ro tation of the vec to r sp ace on whi ch the matrix operates a s a l inear op e rator. The n otion of the Hermi t i a n matr ices i s ex tended t o the not i o n of self-a d joint oper a torsin a Hi lbert space.

Definitio n 1. Let X be a Hi lbe rt space. A l in ea r opera t o r on D (T) ~ X

in to X is called s y m m e tr ic if T* ~ T, i.e., if T* is an exte n sion ofT. N oteth a t the con d ition of the existence of T* impli es that D ( T t = X .

Pr o position 1 I f T is sym metr ic , then T * * is als o symmetr i c .P roof. Sinc e Tis s y m m e tric, we ha v e D (T*) ~ D (T) and D ( T t = X .

Hence D (T * t = X a nd s o T * * = (T*)* ex ists. T * * is sure ly an e x ten

sion o f T and so D (T**) ~ D (T). Thus , again by D ( T t = X , we haveD (T * * t = X a nd so T * * * = (T**) * exists. W e have, fro m T* ~ T,T ** ~ T * and hen c e T*** ;{ T * * which p roves that T** is sym m etr ic .

Corollar y . A s y m m e tr ic operat o r T has a c losed sym m e tr ic extens ionT * * = (T*)*.

Definition 2. A l inea r opera to r T is called se lf-adjoint if T = T*.Proposit ion 2. A se lf -adjoint op e rator is clo sed. An ever y where defi n ed

s y m m e tr ic operato r is bou nde d and self-a d joint.Pro of. Being t he ad jo in t of itself, a self-adjoin t opera to r i s closed.

T he las t a ssertion is p r o v e d by th e fact th at a n everywhe re defined c losedo p erato r is b o und ed (clo sed graph th eorem).

Exampl e 1 (integ ra l operato r of the H ilber t -Schm idt type). Le t - o o < a < b < oo and consider L2 (a , b). L et K(s, t) b e a complex -


209/468

198 VII. D ual Opera tors

val ued measur able functio n for a < s, t < b such th a tb b

J J jK(s, t) j2 ds dt < CXJ. a aFor a ny x(t) E P( a, b), we de fine th e o p erato r K b y

b

(K x) (s) = J K(s , t) x(t) dt.a

W e h ave, by Sch warz' i nequ ali ty an d th e Fubin i-To nelli th eore m ,b b b b

jj(K x)( s)j2ds -: : ; ; .J J jK (s , t ) j 2 dtd sJ jx (t) \ 2 dt.a a a a

Hence K is a boun ded linear o pera tor on L2 (a, b) into L 2(a , b) su ch that

( bb )1/2IIK II < j j jK (s, t) j2 ds dt . I t is easy to see that t h e o p erato r K*b

is de fined b y (K*y) (t) = J K(s, t) y(s) ds. H ence K is s elf-a djoint i ffa

K (s, t) = K (t, s) for a. e. s, t.Exa mple 2 (the coo rdinate o p era to r in qu a ntum m e chanics ). L e t

X = L2 ( - CXJ,CXJ). Le t D = {x(t); x (t) and t x(t) both E P - CXJ,CXJ)}.

Th en th e op erator T de fined b y T x(t) = t x(t) on D is se lf-adjoi nt.Pr oof. I t is clear that Da = X , since th e l inear c om b ination s ofdefinin g fu nctions of fi nite in terva ls are s tron gly den se in P(-CX J,CXJ).L et y E D (T*) and set T*y = y* . T hen, fo r all x E D = D (T),

0 0 0 0

J tx(t) y(t) dt = J x(t) y* (t) dt.- 00 - 00

I f we ta k e f or x(t) th e d efining fun ction o f the in te rv a l [eX, t0 ] , we hav et 0 t 0

J t y (t) dt = J y* (t) dt, a nd henc e, by differe ntia tion, t0 y (t0 ) = y* {t0 )"' "'for a .e. t0 T hus y ED a nd T*y (t) = t y (t). C onve rsely, i t is c lear th aty E D im plies t hat y ED (T*) and T* y (t) = t y (t).

Ex ample 3 (the m o m en t u m opera to r in q uantu m m echani cs). L etX = L 2 -CXJ, CXJ).L et D b e th e total ity of x(t) E P(-C X J, CXJ) such thatx (t) is abs olute ly cont inuo us on e very finite inter va l wi th th e der ivat ivex ' (t) E L 2 ( - CXJ, CXJ). T h en the ope rator T def ined b y T x(t) = i - 1 x ' (t) on D is s elf-adj oint.

Proo f. Let a con tinuo us fun ction x,. (t) b e de fined b y

x,. (t) = 1 fortE [eX, t0 ] ,

x,. (t) = 0 f or t -::;;.eX - n - 1 a nd for t > t0 + n - 1 ,x,.( t) is a linear func tion on [ iX - n - 1 , eX] and on [t0 , t0 + n - 1 ] .


210/468

3. Symmetric O perators an d Self-ad jo i nt O perators 199

T h en the lin ea r combin atio ns of fun cti ons of th e fo rm xn (t) w i th differe n tvalue s o f c:x,t0 an d n are de nse in L 2 ( - oo , oo). Th u s D is den se in X.

Let y E D (T*) an d T*y = y* . The n for any x E D,0 0 0 0

J i - 1 x' (t) y (t) d t = J x (t) y* (t) dt.-oo -oo

I f we ta ke Xn (t) fo r x (t), we ob ta i n"' t ,+n-t oo

n J i - 1 y ( t ) d t - n J i - 1 y ( t ) d t = J Xn (t)y*(t) dt,.: - n- 1 t0 o o

'and so , b y l e t t ing n - + o o, we o b ta in i - 1 (Y (c:x)- y (t0 ) ) = J y* (t) d t for"'a.e. c:xand t0 I t is cle ar, by Schw arz ' inequal ity , that y* (t) is integr ab l e

over a ny fi nite inter va l . Thus y (t0 ) is abso lu t ely cont i nuo us in t 0 ov erany finit e in te rva l, a nd so w e h ave i - 1y ' (t0 ) = y* (t0 ) for a.e. t0 . H enceyE D a nd T*y( t) = i - 1 y'(t ). Let, con ver sely, yE D . Then , b y partialintegrat ion ,

b b

J - 1 x ' (t) y(t) dt = i- 1 [x(t) y ( t) ] ~ + J x( t ) ( i- 1y' (t)) d t.a a

B y t he in tegra bi l i ty of x(t)y(t) ov er ( -oo ,o o) , we se e that0 0 0 0

l im I x(t) y ( t ) ] ~I= 0, and so J i - 1 x' (t) y(t ) d t = J x( t ) ( i- 1y' (t)) dt.aj , -o o ,b too -oo -ooThus y E D (T*) an d T*y (t) = i-1y' (t).

Th eorem 1. I f a s elf-adjoi n t o pera tor T a dmi t s th e inverse T - 1 , thenT - 1 is also selfadjoint.

P r oof. T = T * is equiv ale n t to (V G(T)).l. = G( T ). We h av e alsoG(T- 1) = V G ( - T ) . H ence, by ( - T ) * = - T * = - T , (V G(-T)) ..L =G ( -T ) and so

( V G (T - 1 ))1. = G(-T) . .L = (VG(-T) ).l...L = V G ( - T ) = G (T -1

) ,

that is , (T-1)* = y-1.

We have used, in the abov e p roof, the fac t that ( V G ( - T ) / = V G ( - T )in v irtue of t he closedne ss of ( -T) .

Corollary. A s ym metr ic o pe r a tor T in a Hi lbe r t sp ace X is se lf-adjoint if D (T) = X o r if R(T) = X .

Proo f. T he case D (T) = X wa s proved alr eady. W e s hall prov e thecas e R(T) = X . T x = 0 imp lies 0 = (T x, y) = (x, Ty) for all

y ED (T ), and so, by R (T) = X, we m ust have x = 0. Ther efo re thein ve rse T - 1 ex ist s which i s s u rely sym m e tr ic w i th T. D ( T - 1 ) = R (T) = X ,and so the eve rywhere def ined sym m e tr ic oper a to r T - 1 mu st be self-a djoin t. Hence T = (T -1) - 1 is s elf-adjoi n t by Theore m 1.


211/468

200 V II . Dual Op er ators

W e ca n co n s t ru c t self -adjoint o p e ra to rs f r om a closed l ine a r o p e r a to r. More p reci sely, we h ave

T heorem 2 (J. VON N E UM ANN [5]) . For any c lose d l inear operator T in a Hilbe rt sp ace X s u c h th at D ( T t = X , th e o p era to rs T*T and I T *are self-adj oint , and ( I + T* T) a nd ( I + TT *) both a dm it bou nd edlinear in vers es.

P roo f. W e kn o w that, in the product spac e X x X , G(T) and VG (T*)ar e cl osed l in ear subspac es o r thogon a l to each o ther and s pannin g th e wh ole produ ct spac e X x X . H ence, fo r any hE X, we have th e u n iq u e lyd e te r m i n e d dec omposi t ion

{h, 0} = {x, Tx} + { - T* y , y} wi th x E D (T), y ED (T*) . (1)T hu s h = x - T*y, 0 = Tx + y. Th eref ore

x E D (T* T) a nd x + T* T x = h . (2)B ecau se of the uniquen ess of deco m po sit ion (1), x is un iquely determ i n e d by h, and so the eve ry wh ere def ined inverse (I + T* T) -1 e xists .

For any h, k E X, let

X = ( I + T *T)- 1h, y = ( I + T* T )- 1k.

T h en x and y E D (T* T) and , b y the clo sedness of T, (T*)* = T . H e n c e

(h, ( I + T*T)- 1k) = ( ( I + T *T) x, y) = (x, y) + (T *Tx, y)= (x, y) + (Tx, Ty) = (x, y) + (x, T*Ty )= (x, ( I + T*T ) y) = ( ( I + T* T)-1 h, k),

w hich pr o v es that th e operato r ( I + T * T)- 1 is s elf-adjo int. As an e veryw he re def in ed s elf -adjo int o p e ra to r , (I + T * T )- 1 is a bou nd ed o p e ra t o r.By T he o re m 1, its inve rse (I + T* T) an d hen ce T* T are self-adj oint .

Since Ti s closed , we have (T*)* = T , a nd so, by w ha t was pr ovedabo ve, TT * = (T*)* T* is self -adjoin t and I + TT *) has a boundedl inea r invers e.

We next give a n ex am p le of a no n-se lf-adjoin t, sy m m e t r i c operat or : Examp le 4 . Let X = P(O , 1). L et D be the t otalit y of abso lu te ly

con t inu o u s functio n s x(t) E P (O, 1) su c h that x(O) = x(1 ) = 0 an dx'(t)EL2 (0 , 1). Then th e opera to r T 1 defin ed by T 1x( t) = i - 1x'(t ) on D = D (T 1 ) is sym m etric but not self-ad joint .

Pr oof. We sha l l p ro v e that T i = T 2 , where T 2 is def ined by:T 2 x ( t ) = i - 1x '( t) on D(T 2 )={x( t)E L2(0,1 );x (t) isa b so lu te ly c on t in u o u s su ch that x' (t) E L2 (0, 1)}.

Since D = D (T 1 ) is dense in L2 (0, 1), the operato r T i is defi ned. Le t y ED (T i) a nd se t T fy = y*. T h e n , for a n y x E D = D (T 1),

1 1

J i - 1 x ' (t) y (t) dt = J x (t) y* (t) dt.0 0


212/468

3. S ymme tric O perato rs a nd Se l f- adjo in t Oper a tors 201

By pa rt ial in t e gra t i o n, we ob ta in , rem em b er ing x(O) = x(1) = 0,

1 1 t

J x (t) y* (t) dt = - J x ' (t) Y* (t) dt , w he re Y* (t) = J y* (s) ds. 0 0 01

H ence , by x (1) = J x'( t) dt = 0, w e ha v e , for any c onsta nt c, 0

1

J x ' (t) ( Y* (t) - i - 1 y (t) - C) dt = 0 for a l l x E D (T J .0

I

O n the ot he r ha nd, fo r any z (t) E L2 (0, 1), the func t i on Z (t) = J (t) d t -0

1

tJ

z( t) dt s u r e l y belo n gs to D{T 1) .

H e nce, t a k in g Z(t) for the a b o ve

0

x (t), we o btain

j {z ( t ) - j z(t ) dt} (Y*(t ) - i - 1 y ( t ) - C) dt = 0.1

I f we ta ke t he co n stant c i n su c h a w ay th at J Y * ( t ) - i - 1 y( t) - c ) dt = 0,0

th en1

f z (t) ( Y* (t) -i - 1 y

(t) -C ) dt = 0,

0

an d so, b y t he a r b i t r a r ine s s of z E L 2 (0, 1) , we m us t have Y*(t) =I

J y* (t) d t= i - 1 y (t) + c. H ence y E D (T 2) and T 2 y = y*. T h is p r ov e s 0

th at T'f ~ T 2 I t is a ls o c le a r, by pa rt i al in t e g ra t i o n , th at T 2 ~ Tt andso T 2 = Tt.

The o rem 3 . I f H is a b o u n d e d se lf - ad jo i n t ope r a to r, th en

IIH JI = s u p I (H x, x)l. (3)11..11:;3;1

Proo f . S e t sup I( Hx, x) l = y . T h e n , b y I( H x,x) l < J IH xJI Ji xJI ,11..11:;3;1

y < JIHII. F o r an y r e a l nu m b er A., w e hav e

I (H (y lcz), y lcz) I = I(H y, y) 2/c R e(H y , z) + c2 (H z, z) I

< y IY 1czll2H e n c e

I4A R e(H y, z) I:::;; y(II Y + 1czll2 + ly -/cz ll 2) = 2y(IIY II2 + )c211zJI2).By t aking A = IIYII/II zll, w e obta in I R e(H y ,z)l < y IIY IIIIzl l. H en c e ,b y su b stitut i ag zei9 for z, we o btain I (Hy, z) I < y IIY I l l i z i I, and so

(Hy, H y) = IH y JI 2 < y IY III H y ll, i .e ., [[ H [[ < Y


213/468

202 VII. Dual O perators

4 . U nita ry Op erato rs. The Ca yley Tr ansf orm

A sy m met r i c op er ator is no t necessari ly a b o u n d e d o perator. V a riousinve st igat ions o f a s y mme tr i c o p e ra t o r H may b e m a d e t h r o u g h t hec o n t inuous o p era to r ( H - i l ) (H + i i ) - 1 calle d the Ca yley transfo rm of H . We shal l be gin with t he no t ion o f isometric op erators.

D e finition 1. A b o u n d ed l i ne ar opera t o r T on a H i l ber t space X intoX is cal le d (bounded ) isometri c if T leaves the scalar pr o d u c t invar iant :

(Tx , T y) = ( x , y ) forall x , y E X . (1)

If, i n pa r t icular, R (T) = X , th en a (bou n ded) isom et r i c opera t o r Tis called a u n i ta ry oper a t or.

Pro position 1. For a bou n d ed l inear o pera to r T , condi t ion (1) ise qu iva len t to th e cond i t ion of the i somet ry

IIT x II= IIx II for al l x E X .Pr oof. I t is cle a r that (1) im plies (2). W e have , b y (2),

4 Re(T x, Ty) = II T (x + y) 112 -IIT(x-y ) 112= IIx + y 112 - IIx - y 112 = 4 Re (x, y) .

(2)

B y t ak i n g iy i n place of y, we also obtain 4Im (Tx, Ty) = 4 Im(x,y) ,

and so (2) implie s (1).Pr opositio n 2. A b o u n d ed li near opera to r on a Hi l b e r t space X int o Xis unitar y iff T * = T-1.

Pr oof. I f T is u n i t a r y, t h en T - 1 surely exi sts in v i r tue of condi t ion ( 2),a nd D(T-1) = R(T ) = X . Moreover , by (1), T* T = I and s o T * = T -1.Con v ersely, the c ondi t ion T * = T - 1 im p lies T* T = I which is the cond ition of t he invarian c e of the sc alar prod u c t. Moreov e r, T * = T - 1im p lies that R (T) = D (T- 1 ) = D ( T*) = X an d hence we see that T mu st be un i t a ry .

Exa m pl e 1. L et X = L 2 ( - = , =) .Then, for any real nu m ber a, t heo p e ra to r T d ef ined by T x(t) = x( t + a ) on P -= , = ) i s un itary.

Exampl e 2. T h e Fo u r i e r t r an s fo rm on L 2 (R") o nt o L 2 (Rn) is u nitary,sin ce i t leaves the scalar p ro duct (/ ,g )= J (x) g (x) dx inva ri a nt.

Rn

D efinitio n 2. Let X b e a H i lber t sp ac e. A l inea r o p e ra t o r T definedo n D (T) ~ X i n t o X suc h that D ( T t = X is c al led norm a l if

TT* = T* T . (3)Se lf -adjoint o p e rators an d un itary o p e rators are n ormal .

T he Cay ley T ransfor m

Th eorem 1 J. voN NEUMANN [1]). Let H be a closed sym m etr ico p e ra t o r in a Hi lber t space X . T h en t he co n t inuous (b u t no t neces sa rily


214/468

4. Unitary Operators. T he Cayley T ransform 203

everywhere defined) in v erse (H + i i ) - 1 exists , a nd the o p era to rU J l = (H - i i ) (H + i i ) - 1 w ith th e d o main D (U JI) = D((H + il )-1) (4)is cl osed iso metri c (IIUI x II= IIx II),an d ( I - U JI)-1 exists.W e have , mo re over,

H = i ( I + UJI) ( I - UJI)-1.Thu s , in par t ic u lar, D (H) = R ( I - U JI) is dense in X .

D efinition 3. U JI is calle d the C ayley transf orm of H.

P roof of The orem 1 We have , for a n y x E D(H ),

(5)

((H i f ) x, (H i f ) x) = (Hx , Hx) (H x , ix) (ix, H x ) + (x, x).The sy m metry c o ndition fo r H impl ie s (H x, ix ) = - i (H x, x) =- i ( x , H x ) = - ( i x , H x ) and so

II (H i f ) xll2= 11Hxll2 + llxll2 (6)H ence (H + i f ) x = 0 implies x = 0 and so the inver se (H + iJ) -1exists. S ince II(H + i i ) x II> IIx II, the inv erse (H + i J )- 1 is cont inuous .By (6) , it is clear t hat IIU HYII = IIy II,i. e ., U JI is iso m etric.

U JI is c losed. For , let (H + i f ) Xn = Yn--')- y and ( H - i i ) Xn = Zn--')- zas n- ' ) -oo . Then we have, by (6), 11 Yn-Ymll 2 = IIH(xn-Xm )l l 2 +

jIX n - Xm 2 , and so ( x n - xm) --')- 0, H (x n - Xm)--')- 0 a s n, m--')- oo. SinceH is closed, w e must h av e x = s-lim Xn E D (H) and s-lim H Xn = H x.1>---700 1>-- -700

Th us (H + i i ) Xn ->- y = (H + i i ) x, ( l - i f ) Xn--')- z = ( H - i i ) X andso U JIY = z. T h is proves t hat U I is c losed.

From y = (H + i i ) x and U HY = ( H - i i ) x, w e obtain 2- 1 I - Uu) y = ix an d 2- 1 (I + Uu) y = H x. Thus ( I - Uu) y = 0im plies x = 0 and so ( I + UJI) y = 2H x = 0 which i m plies y =2- 1 ( ( 1 - UJI) y + ( I + UJI) y) = 0. T he refore t he inverse ( I - Uu )- 1e xists. By th e same ca l c ula t ion as a bove, we ob tain

Hx = 2- 1 (1 + UJI) y = i ( I + UJI) ( I - Uu)- 1 x, tha t is,H = i ( I + Uu) ( I - Uu )- 1 .

The orem 2 (J. VON N E U M A N N [1]). Let U b e a closed is ometr ic op e rator such that R ( I - U)a = X . T hen the re e xis ts a un iq ue ly de te r m inedc losed sym m etr ic opera to r H whos e Cayley tr a nsform is U.

Pro of. W e f ir s t show that the invers e ( I - U}- 1 exists. Sup p ose tha t( I - U) y = 0. Fo r a ny z =( I - U) wE R ( I - U), we have, by theisometric pr o perty of U, (y , w) = (U y, Uw) as in Section 1. Hence

(y, z) = (y, w ) - (y, Uw) = (U y , U w ) - (y, Uw) = ( U y - y, Uw) = 0.

H ence, by th e conditio n R ( I - U t = X , y mu st b e = 0. T hus ( I - U )- 1

exi s ts. Pu t H = i (I + U) ( I - U}- 1 . T h e n D (H) = D ( ( I - U)- 1 ) =


215/468

204 VII. D ual Ope ra tors

R ( I - U) is dense in X . W e first p rov e tha t H is sym met r i c . Le tx.,yE D(H) = R I U) an d put x = I - U) u , y = I - U) w. Th en ( V u, U w ) = (u, w) impl ies tha t

(H x, y) = ( i (I + U) u, ( 1 - U) w) = i(( Uu , w ) - (u, U w) )= I - U) u , i ( I + U) w) = (x, H y).

Th e p roof of U H = U is ob t a i n ed as fol lows. F or x = I - U) u, we hav eH x = i ( I + U) u a nd so (H + i i ) x = 2i u, (H - i i ) x = 2 iU u . Thu s D ( U H ) = {2 iu; u E D( U )} = D(U) , a nd UH ( 2 iu) = 2 i U u = U(2i u ).H e nce UH = U .

T o com plete th e p roof o f The orem 2, we show tha t H is a clo sed

o perato r. In f act , H is th e op er ator w hich m a ps I - U) u o n to i ( I + U) u .I f I - U) Un and i (I + U) Un both conv e rge a s n ~ oo, t he n U n and U un b oth c onver ge as n ~ o o. H e nce b y the closur e pr o perty o f U , w em us t h ave

U n ~U , I - U) U n ~ I - U) U , i ( I + U) U n ~ i ( I + U) U .This p rove s that H is a clos ed op e rator .

F o r the struc ture o f th e ad j o i n t o p e ra to r of a s y m m etr ic o p e ra t o r,w e ha v e

T h eore m 3(J.

voN NEUMANN [1]). L et H b e a clos ed sy m m e t r i coper at or in a H i lber t spac e X. For the C ayley t ran s fo rm U H =

( H - i i ) (H + i i ) - 1 of H , we s e t(7)

T h e n we h av e

X fi = {x E X; H *x = i x}, XJi = {xE X; H *x = - i x } , (8)a nd th e elem ent x of D (H*) is uniq ue ly e xpres sed as

x = x 0 + x 1 + x 2 , w here x 0 E D (H), x 1 E Xfi, x 2 E XJi so th a tH *x = H x 0 + i x 1 + ( - i x 2 ) . (9) Pro of . xE D(UH )j_ = D((H + i i ) - 1)j_ impli e s (x , (H + i i ) y ) = 0

fo r all y E D( H). H ence (x, Hy ) = (x, i y) = (i x, y) and so xE D( H* ),H* x = i x . T he l a s t con dit ion impli e s ( x, (H + i I) y) = 0 fo r ally E D ( H),i.e. x E D ( (H + i J ) - 1)j_ = D( U H)j_ . Thi s prov es th e firs t ha l f of (8 );th e latt er hal f may b e p r o ved s imila r ly.

S ince U H is a c losed i some tric o perato r, we see t ha t D (U H) and R (U H) are c losed l inear subsp aces of X. Henc e any elem ent x E X i s un ique l y deco mpose d as t he sum of a n elem ent of D(UH ) and a n e l emen t of D ( U H) j_. I f we a pp ly th i s o r thogo n al de comp o sition to t he e le men t( H * + i i ) x, we o b tain

(H * + i i ) x = (H + i i ) x 0 + x' w here x0 E D (H) , x' E D (UH) J .


216/468

4. Unitary Operators. The Cayley Transform 205

But we have (H + i l ) x0 = (H* + i l ) x0 by x0 E D(H) and H ~ H*.We have also H* x' = ix ' by x' ED (U H)l. and (8). Thus

x' = (H* + iJ) x1 , x1 = (2i)- 1 x' E D(UH)l. ,and so

(H* + i l ) x = (H* + i l ) x0 + (H* + i l ) x1 wherex0 ED(H), x1 ED(UH)l.

Therefore ( x - x0 - x1 ) E R (U H)l. by H* ( x - x0 - x1) = i x - x0 - x1 )and (8). This proves (9). The uniqueness of the representation (9) isproved as follows. Let 0 = x0 + x1 + x2 with x0 ED (H), x 1 ED (UH) 1,x2 E R(UH)l.. Then, by H* x 0 = H x 0 , H*x 1 = ix 1 , H* x 2 = i x 2 ,

0 = (H* + i l ) 0 = (H* + i l ) (x0 + x1 + x2) = (H + i l ) x0 + 2ix 1 .But by the uniqueness of the orthogonal decomposition of X as the sumof D (U H) and D (UH)l., we obtain (H + i l ) x0 = 0, 2ix 1 = 0. Since theinverse (H + i J) - 1 exists, we must have x0 = 0 and so x2 = 0 - x0 -x1 = 0 - 0 - 0 = 0.

Corollary. A closed symmetric operator H in a Hilber t space X is selfadjoint iff its Cayley transform UHis unitary.

Proof. The condition D (H) = D (H*) is equivalent to the conditionD (U H)l. = R (U H)l. = {0}. The last condition in tum is equivalent to thecondition that U H is unitary, i.e. the condition that U H maps X ontoX one-one and isometrically.

5. The Closed Range Theorem

The closed range theorem of S. BANACH [1] reads as follows.Tl.eorem. Let X and Y be B-spaces, and T a closed linear operator

defined in X into Y such that D ( T t = X. Then the following propositions are all equivalent:

R (T) is closed in Y, (1)R (T') is closed in X' , (2)

R(T) = N(T') l . = {yE Y; (y, y*) = 0 for all y* E N{T')}, (3)

R (T') = N (T)l. = {x* EX'; (x, x*) = 0 for all x E N (T)}. (4)Proof. The proof of this theorem requires five steps.The first step. The proof of the equivalence (1) ++ {2) is reduced to

the equivalence (1) ++ (2) for the special case when T is a continuouslinear operator such that D (T) = X .

The graph G = G (T) of Tis a closed linear subspace of X X Y, and soG is a B-space by the norm ll{x,y}II= [[xII+ flyIIof X X Y. Consider acontinuous linear operator S on G into Y:

S{x, Tx} = T x .


217/468

206 VII. Dual Operators

Then the dual operator S' of S is a continuous linear operator on Y' intoG', and we have


218/468

5. The Closed Range Theorem 207

and so, since R (T1} = R (T) is dense in Y 1 = R (Tt , Yi must be 0. Therefore, the condition that R (T') = R ( T ~ )(proved above) is closed, implies

that T ~ is a continuous linear operator on the B-space (R (Tt) ' = Y ~onto the B-space R ( T ~ )in a one-one way. Hence, by the open mappingtheorem, ( T ~ ) -1 is continuous.

We then prove that R(T) is closed. To this purpose, it suffices toderive a contradiction from the condition

here exists a positive constant e such that the image{T 1 x; llx II< e}is not dense in all the spheres IIy llOO

Since {T 1 x; IIx II< e}a is a closed convex, balanced set of the B-space Y 1there exists, by Mazur's theorem in Chapter IV, 6, a continuous linearfunctional In on the B-space Y 1 such that

fn(Yn) > sup lfn(Tlx) I (n = 1, 2, . . . .llxll:::;;e

Hence IIT ~ f nII< e-1 IIIn 1111YnII,and so, by s-lim Yn = 0, T ~does not have1 r-->OO

a continuous inverse. This is a contradiction, and so R (T) must be closed.The fourth step. We prove (1} ---* {3). First, it is clear, from

( T x , y * ) = l x , T ' y * ) , xED(T), y*ED(T') ,

tha tR(T) c;;,N(T')j_. We show that (1) impliesN(T')j_ c;;,R(T). Assumethat there exists ay 0 E N(T')j_ withy 0 E R(T). Then, by the Hahn-Banachtheorem, there exists a YriE Y' such that (y 0 , Yri># 0 and (T x, Yri)= 0for all x E D (T). The latter condition implies (x, T' Yri>= 0, x E D (T),and hence T' Yri= 0, i.e., y0 E N (T') j_. This is a contradiction and so wemust have N(T')j_ s;;R(T).

The implication (3} _,.. (1) is clear, since N(T')l_ is closed by virtueof the continuity in y of (y, y*).

The fifth step. We prove (2)--* (4). The inclusion R(T') ~ N(T)l_ isclear as in the case of (3). We show that (2) implies that N(T)l_ ~ R(T') .To this purpose, let x* E N (T) 1_, and define, for y = T x, the functional/ 1 (y) of y through / 1 (y) = (x, x*). I t is a one-valued function of y, since

Tx = Tx' implies ( x - x ' ) EN(T) and so, by x*EN (T)l_, ( (x-x ' ) ,x*) = 0.Thus / 1 (y) is a linear functional of y. (2) implies (1), and so, by the openmapping theorem applied to the operator 5 in the first step, we maychoose the solution x of the equation y = T x in such a way that s-lim


219/468

208 VII. Dual Operators

y = 0 implies s-lim x = 0. Hence / 1 (Y) = Re(Tu, u) > c llull 2 for all uE D(T).Hence IIT u II > c IIu II, u E D (T), and so T admits a continuous inverse.Thus, by the preceding Corollary, R (T') = X . Hence R (T*) = R (T') = X .

Remark. A linear operator T on D (T) ~ X into X is called accretive(the terminology is due to K. FRIEDRICHSand T. KATo) if

Re(Tu,u) > 0 for all u ED( T) . (8)T is called dissipative (the terminology is due to R. S. Phillips) if - Tis accretive.

References for Chapter VIIFor a general account concerning Hilbert spaces, see M. H. STONE

[1], N. I. ACHIESER-1.M. GLASMAN [1] and N. DUNFORD-J. SCHWARTZ[5]. The closed range theorem is proved essentially inS. BANACH [1].


220/468

1. T he Reso lvent and S pectru m 209

V III. R eso lvent a nd Spectr um

Let T be a l inea r o pera tor whose d oma in D (T ) and r ange R (T) bothlie in th e sa me c om p lex l inear t o pologica l s p ace X. W e c onsider the l inea ro pe r ator

T;. = U -T

Documents

Yosida 1 Functional_Analysis