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The Spectrum of a Type of Integral Operator Tao, Andrew Yau-Shun 1972

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THE SPECTRUM OF A TYPE OF INTEGRAL OPERATOR by ANDREW YAU-SHUN TAO A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1972 In present ing th is thes is in pa r t i a l f u l f i lmen t o f the requirements fo r an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on of th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of M&~& 1 C* The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date September 27, 1972 11 Supervisor: Dr. D. W. Boyd ABSTRACT The purpose of this thesis i s to determine the spectrum of a type of integral operator called "Convolution Operator" acting on L^(0, °°) . In the special case p = 2 , we have a tool called the Mellin trans-form, this enables us to analyze the spectrum i n more detail. A number of papers have dealt with special cases of our result, the reader i s referred to the bibliography of a paper by D. W. Boyd; (to appear). Spectra of Convolution Operators. i i i TABLE OF CONTENTS PAGE CHAPTER I Introduction 1 .1.1 Notation, Convolution Operators 1 .1.2 Fourier Transforms on 5 .1.3 Convolutions 7 .1.4 Inversion Formula 8 2 CHAPTER II Spectrum of the Operator Acting on L (0, ») 11 2 .2.1 Fourier Transforms on L (-<», °°) 11 2 2.2 Mellin Transforms on L (0, °°) and their relation to 17 t Fourier ^ Transforms on L*" (-», -•<*>) 2.3 Hilbert Space Techniques 20 2.4 Spectrum of T 23 CHAPTER III Wiener's Theorem 26 3.1 Commutative Banach Algebras with unit 26 3.2 Maximal ideals and the Canonical Homomorphism 28 ,3.3 V as a commutative Banach Algebra with unit 31 3.4 Wiener's theorem 32 CHAPTER IV Spectrum of T 37 4.1 Essentially Convolutions 37 4.2 Spectrum of T . 39 CHAPTER I INTRODUCTION 1.1 Notation-convolution operators As indicated i n the t i t l e , we are going to determine the spectrum of a type of i n t e g r a l operator: we state our conclusion i n 4.2.4. The i n t e g r a l operator we are interested i n i s of the following form. Let k be a complex valued measurable function defined on (0, °°) such that -1 |k(s) | s Pds < ~ . For any f e L P ( 0 , °°) (1 <. p £ °°) , define Tf e L P(o CO to be Tf(t) = /gk(s)f(st)ds ; 'we c a l l such an operator T a convolution operator, since i t w i l l be seen i n chapter 4 section 1 that this operator i s closely related to convolution . The impatient reader may read 4.1 ri g h t away to see hew this i s done. In t h i s f i r s t chapter we investigate some properties of T i n general and also Fourier transforms on L^ (-<», °°) as w e l l . The l a t t e r w i l l be of great use. In chapter 2, we study s p e c i f i c a l l y the case p = 2 because i t i s simpler, but i n order to determine the spectrum i n general ( i . e . 1 <^  p _< 0°) which i s the aim of chapter 4, we need a deep theorem due i n substance to Wiener, the proof of which takes the whole of chapter 3. ) 1.1.1 Notation LP(-°°, ») (1 <_ p <_ ») . In the case p ^  » , this i s the space of a l l complex valued measurable functions f on (-<», ») such that / |f | P < I Denote the norm by = if ^ J f | ^ • In the case p = °° , L (-«>, ») denotes the space of a l l complex valued measurable functions f such that f i s es s e n t i a l l y bounded; the norm of f i s denoted by IlfIL = ess sup |f(x)| . L^(P> ro) This is the same as the above, except that our functions are now defined on (0, °°) . The norm of f e L P(0, °°) is again denoted by 1 ||f.|| = [ / " | f | P ] P i n case 1 <_ p < °° , and jjf|| = ess sup |f (x) j . p (J 0 0 R(T) The range of a function T . N(T) null space of the linear transformation T . A*" The collection of a l l elements perpendicular to A .(i.e. perpendicular to every element i n A) where A is a subset of a Hilbert space. E closure of E XT, characteristic function of E C The collection of a l l complex numbers . p(X, T) = {X e C: T - X is one-one, (T-X) - 1: R(T-X) -> X is bounded, R(T-X) = X} where X is a normed linear space and T an operator on i t . The set p(X, T) is called the resolvent of T . a(X, T) = C - p(X, T) . a(X, T) i s called the spectrum of T . Co(X, T) = {A eC: (T-X) one-one (T-X)" 1: R(T-X) -> X i s not bounded R(T-X) = X}. Ca(X, T) is called the continuous spectrum of T . Ra(X, T) = {X eC: (T-X) is one-one R(T-X) ± X}. Ra(X, T) is called the i residual spectrum of T Po(X, T) = {A e C: T-X is not one-one. Pa (X, T) i s called the point spectrum of T . Clearly the three sets Ca(X, T) , Ra(X, T) , Pa(X, T) are disjoint and their union is a(X, T) . It is time to go back and see whether our convolution operator T really makes good sense. 1.1.2. Proposition (1) Tf e L P(0, <*>) T: L P(0, ~) L F(0, ~) -JL (2) f|T|| <_ /™|k(s)|' s pds . k is called the kernel of T . Proof Case 1. 1 < p < 0 0 . I I Let q be the conjugate of p (i.e. q + P =1) then for any g e L^(0, °°) , k(s)f(st)g(t) i s absolutely integrable on R 2 . In fact /^|k(s)f(st)g(t)|dsdt = /Q /Q |k(s) f(st) g(t)| dt ds = /Q|k(s)| [/JJ|f(st)||g(t)|dt]ds r 1 < /~jk(s) i [/~|f (st) | Pdt] P||g(t)j| pds [Holder's inequality] . -1 -1 >./~|k(s)| S Pi|f|gig||qds = ||f||p||g||q /"|k(8)|8 P d S < - ... (1) -Thus, by Fubini's theorem, letting g(t) = Xr n n+i] ' t* i e f u n c t i ° n OO CO /pk(s)f(st)g(t)ds = Xr n n +2]^Qk(s)f(st)ds must be defined for almost a l l t; OO moreover, i t i s measurable. Letting n=0,l,2, ... we see that /gk(s)f(st)ds i s well "defined and measurable. Now, for any g e L^(0, 00) /^|Tf (t) |g(t) < /o|g(t)| | /Qk(s)f(st)ds | dt < /~ /~|k(s) f(st) g(t)|ds dt -1 ± H f H P Nl q ~0\k^\s P d s hy <!> Thus by the "converse of Holder's inequality" (see [1] theorem (9)) -1 |Tf| e L P(0, o o ) and fjTf || p < |jf||p /~|k(s) |s pds . But this means Tf e L P(0, oo) and since |||Tf ||| = ||Tf]| , [|T|| < /o|k'(s)|s Pds . Case 2. p = 1 Now the function k(s)f(st) i s absolutely integrable because /Q /JJ|k(s) | |f (st) |ds dt = /JJ /Q|k(s) | |f (st) |dt ds = /o|k(s)| [/~|f(st)|dt]ds = / J l k C s ) ^ " 1 ! ^ ! ^ ds = f l f j ^ / p l k C s ) ^ " ^ < « . . . . (2) CO So / k(s)f(st)ds i s defined and integrable o flTff^ = J /~k(s)f(st)ds | dt < /" /~|k(s)| |f(st)|ds dt • < Ufj^ /~|k(s)|s - 1ds by (2) |(T||< / ^ ( s ^ d s Case 3. p = 0 0 . _^ The condition /Q|k(s)|s pds < oo should mean J"Q|k(s)|ds < oo . |Tf(t)| = /~k(s)f(st)ds <. /™|k(s)| ess sup |f|ds = [ / ; |k (8 ) |ds ] f m . ess sup (Tf| < [/™|k(s)|ds] f w . " =>• IlTfll^ < [/Q|k(s)|ds] Hfll^  => ||T|| < /~|k(s)|ds 1.1.3 Lemma Let be defined as f t(x) = f(x-t) where f e L (-<», °°) ; then || f t - f [| -> 0 as t -> 0 Proof Given e > 0 , choose g e L^C-oo, co) such that g is compactly supported and continuous and such that [|f-g|]^ < e then l l f t ~ 8 tlli = /l|f(x-t)-g(x-t)|dx = £ j f < u > "8( u)|du = l l f - g ^ < £ . We choose [a, b] so that for |t| <1 , g (x) - g(x) = g(x-t) -g(x) vanishes outside i t . Obviously g(x-t) -g(x) i s continuous and therefore uniformly con-tinouus, i f , at the same time we require that t i s small enough so that |g(x-t)-g(x)| < _e_ Vx b-a then |)g -gf^  = /^|g(x-t) -g(x)|dx <_ e : i i f H ^ i M J i i + , M i i ; + i i 8 - f i i i e • 2 1.1.4 Lemma f £ L (-00, °°) , again define f to be f f c(x) = f(x-t) then ' . | | f t - f | l 2 0 as t -v 0 Proof Same proof, using'the fact that continuous compactly supported functions are dense. 1.2 Fourier Transforms on 1?" The material i n sections 1.2, 1.3, 1.4 can be found i n standard text-books. For example, [2], [3], [6], [9]. 1.2.1 Definition For f e L"^-00, °>), the Fourier transformof f i s de-fined by f ( t ) = / f(x)e dx . Since e is bounded f is defined for 6 a l l r e a l numbers t 1.2.2 Theorem ( i ) f + g = f + .g ( i i ) of = of ( i i i ) f(x) = f(-x) . Proof easy. 1.2.3 Theorem f i s uniformly continuous. Proof | f (x+t) - f ( x ) | < /"Jf(0 e 1 ( x + t ) C - f (D e i x ? | d£ - /l|f(OMe 1 5 t-l||e l x 5|d5 - / : j f ( 0 | | e i ? t - l | d ? This l a s t expression does not depend on x ; i f we show that i t goes to •zero -fhen-we'-are "done. -I'e1 "^—:l'| •<-2 --The-"integrand-is -bounded by 2|f (?) | which i s integrable. By the Lebesgue dominated convergence theorem li m /-Jf<0||e l 5 t-l|d5- /"„ lim |f (O | |e 1 5 t-l|d5 = 0 . t->0 t->0 1.2.4 Theorem ( i ) | f ( x ) | < ||f|j \/x . ( i i ) f ->• f i n L 1 norm n => Vs , f n(s) + f(s) Proof easy. 1.2.5 Theorem I f F e L 1 ^ , =°) , F(x) e L 1 (-<*>, °°) , then, F i s d i f f e r -entiable and F = iCF Proof F(x+t) -F(x) = / ^ F ( k ) e 1 X g ( e l € t - l ) d? t ~°° t Now the integrand i s bounded by an integrable function because .|F(Oe l x ere 1 S t-l] < |F(5)||5| F'(x) = / " j L i m F ( 5 ) e i x C ( e 1 ^ - ! ) dcj ~°°t->0 t = r F(5)i Se^dC - I£F(X) —OO 1.2.6 Theorem (Riemann-Lebesgue lemma) f (x) -»• 0 as |x| 0 0 . Proof -f(x) = /" f ( C ) ( - e i x C ) d c : = /" f ( ? ) e i x ( C + 2>d5 • -1 1 1 1 1 —00 ~oo = / _ w f ( u - x ) e du - £ (x) - (-f (x) ) < l1m f (u) - f (u- e ± X U du By lemma 1.1.3 21 f (x) | -> 0 as |x| « . 1.3 Convolutions .1.3.1 D e f i n i t i o n . For f , g e L"^ -00, ») the convolution of f with denoted by f * g i s defined f * g(t) = /"^f(t-u)g(u)du for r e a l 1.3.2 Theorem ( i ) f * g(t) i s w e l l defined for almost a l l t , F * G E LV-, ») W i t h ||f * g\\± < ||f||1|| gp i ( i i ) f * g = g * f ( i i i ) f *(g+h) = f * g + f * h (iv) a(f*g) = (af)*g a a complex number (v) f"*"£ = f g (vi) ( f * g ) * h = f * (g*h) 8 Proof ( i ) C l e a r l y f ( t - u ) g ( u ) i s a b s o l u t e l y i n t e g r a b l e so f * g i s de-f i n e d a.e. and i n t e g r a b l e , w i t h |[f * g|| <_ / " w / " ^ l f (t-u) | |g(u)|du dt = C / l | f ( t - u ) | | g ( u ) | d t du = [ 1 ^ I J g j ^ ( i i ) ( i i i ) and ( i v ) easy. CO CO 1 V"t" CO CO -] V"f" (v) f * g ( t ) = / ^ f (x-y)g(y)e <iy dx = f_m f_J(x-y)g(y)e1XCdx dy = l O ^ ) e i y t d y ] [ 0 ( u ^ e l u t d u ^ = f ( t ) • g ( t ) ( v i ) f * ( g * h ) ( t ) = /°° f (t-x)g(x-u)h(u)du dx = /" h(u) /°° f (t-x)g(x-u)dx du —CO —CO « o o — o o The change of order of i n t e g r a t i o n can be j u s t i f i e d by showing t h a t the i n t e g r a n d i s a b s o l u t e l y i n t e g r a b l e to | f | * (|g| * |h|) ( t ) < » f o r almost a l l t . 1.3.3 Remarks Theorem 1.3.2 i s "important" i n that i t shows L^ " i s a commutative Banach Algebra (see chapter 3 ) . 1.4 Inverse formula 1.4.1 D e f i n i t i o n A summability k e r n e l on the r e a l l i n e i s a c o l l e c t i o n of continuous f u n c t i o n s {k } on R . (The parameter X may be d i s c r e t e or continuous) s a t i s f y i n g ( i ) /" k. (x)dx = 1 — o o \ ( i i ) / "Ik (x)|dx i s bounded as X -*• °° . — CO y\ ( i i i ) \/6 > 0 l i m /,-. |k (x)|dx = 0 . X-x» ' ' 1.4.2 D e f i n i t i o n The F e j e r ' s k e r n e l {K } i s d e f i n e d by ' A K x ( x ) - XK(Xx) (X>0) x 2 where KM = |; ^ ( l - l d , ^ = ^ One can check that f° K(x)dx = 1 -co / ! j K x ( x ) | d K = / : j K ( u ) | d u = 1 ' i x ^ l V * 0 l d x = f | x | > 6 K ( u ) d u + 0 a S • X + m ' Therefore the F e j e r ' s k e r n e l i s a summability k e r n e l . 1.4.3 Theorem For t h i s F e j e r ' s k e r n e l K and f o r f e L^"(-«>, ») A || K x * f - f || 0 as X » . Proof P i c k e > 0 then 6 > 0 such that | t | < 6 - > | | f ( t - x ) -f(T) | | 2 < e • Choose s u f f i c i e n t l y l a r g e X so that /| t j > ( j . K x ( t ) d t < e t h e n , f o r such a X . f * Kx- f ||x = / _ / " o o f ( x - t ) K A ( t ) d t - f ( x ) | d x 1 CJf(*-t'> - f ( x ) | K x ( t ) d t dx = r j ( f t - f ) ( x ) |K x(t)dx dt = ' l y ^ l l v f H i d t + f \ t \ > ^ ¥ c flfd t < .e + e 2||f|| 1e(l+2||ff^) 1.4.4 Theorem Let {K^} be the F e j e r ' s k e r n e l , then V F ( T ) = h f-X(1~ - ^ ) f A ( O e i t C d 5 f o r f e LV", -) Proof K * f ( t ) = / X K ( X ( x - t ) ) f ( x ) d x . A - c o ( n o t i c e that K(y) = K(-y)) - '1 / ^ ( l - l E l ^ ^ f M d E dx " '1 A ^ J I L ) = 1 U < X " t ) f Wdu d X - £ Jiut<i- ¥» o'-w* *• _ -L- f A /-I I U 1 N "rt \ ~ i u t j 1.4.5 Conclusion ( i ) l i m \- f X_x(l- ^ ~ ) f (Oe~ i x 5d 5 = f (x) L 1 X-x» Proof Immediate consequence of theorem 1.4.3 and 1.4.4 ( i i ) I f f e I/Vco, ») , then T f (£)e~ixCd5 = f (x) for almost a l l x . ( i i i ) I f f = 0 then f = 0 a.e. 11 CHAPTER II SPECTRUM OF THE OPERATOR ACTING ON L 2(0, ~) The principal result of the f i r s t section i s the fact that the Fourier 2 transform is a 1-1 correspondence of L (-«>, <») to i t s e l f . Moreover, i t i s within a constant multiple of being an isometry. 2 In the next section, we shal l define the Mellin transform on L (0, °°) and show that i t i s closely related to the Fourier transform. Finally we use this together with some Hilbert space techniques to determine the 2 spectrum of our integral operator T acting on L (0, °°) . 2 2.1 Fourier transforms on L (-», QQ) Readers may consult [3]. 2.1.1 Theorem If f e !?'(-<*>, °°) i s continuous at x , then 2l /-X ( 1" ^^Oe'^dx f (x) as X + - . Proof For X > 0 , Let I (x) = -^ A (1- -ifL)f (C)i i x 5d5 A Zir —A A .i - ; "f( t )A(i- J f)? ! ( M , d{ dt 2TT —°° —X X Since A ( l - l f L ) i i ^ - t ) d c . 2[l-cosX(x-t) ] T / "V 1 ^ ,^1-COSX (x~t) ,. 1 ° ° , , ^ l-COS>t ,^ I (x) = - / f (t) - ' dt = - / f (x-t) 5 dt X(x-t) Xt But I.(x) = - /" f ( x + t ) 1 " C O S - X t dt as well. A Tf - 0 0 ^ 2 A t 12 Thus 21 fx) = - ! m (f(x+t) + f ( x - t ) ) 1 " 0 0 3 X t dt A TT -oo 2 At I X(x) = 1 /™(f (x+t) + f C x - t ) ) 1 " 0 0 3 ^ dt TT 'Xt oc>2.—COS At TT Integrating by parts, / ^ — d t = X t Therefore I Ax) - f(x) = - A f ( x + t ) + f(x-t) -2f (x) ] 1 " C 0 S 0 X t c 17 0 xt 2 1 5 CO = — 7Q + = + . 6 > 0 is to be determined Let <j)(t) = |f(x+t) + f(x-t) -2f(x)| and *(t) = <Kf )dt» Then, for -j- < 6 . | l j 1 | < ; ; * ( t , ^ d t - i J + / J - J 3 + J 4 X Since given any e > 0 , <_ e for every sufficiently small So l e t us choose 6 mentioned above to be such that 0 < t < 6 2 Since 1-cos 8 < —- and > Q 2 Xt1 < ' 2Xt 2 2 X Now 1-cos 6 < 2 1 j < / 2(j»(t) d t = fS 2di|>(t) = 2it>(g) 2iKX) + 4 /* dt 4 ~ i X t 2 i Xt 2 X6 2 i X X t 13 X6 7 A t J < 2 e + 4 /,6 —j dt < 2e + 4e = 6e Thus '|TTJ j <_ 6jz At < ^ [ " ^ / ^ | f (x+t) [dt + f&\£(x-t) |dt + 2 f ( x ) as A -> co . |lx(x) - f(x)| < | j j + 1J2I <|f e + | I 2 | which i s very small provided X large . 2.1.2 Theorem For f e L/n L2(-«, «) , | | f j j 2 = j[f|| 2 Proof Define f(x) = f(-x) then | f | 2 = f f = f * f by 1.2.2 ( i i i ) and 1.3.2(v) f * f i s continuous at 0 because |f * f(h) - f * f ( 0 ) | <. f j f (h-u)f (u) - f (-u)f (u) |du ±Hf_h ~ f l l 2 |ifjl2 - 0 as h -> 0 . by Holder's inequality and 1.1.4. Therefore by the previous theorem 14 lim ~ / A ( l - -i|J-)f * f (Od? = f * f(0) = ; ^ f ( - u ) f ( u ) d u = \\f\\22 .... (1) CO I A I 2 OO ,A|2 Next we c l a i m f_m\£ \ < 0 0 . Otherwise / |f | = » ; then, ' V 1 _ * f (5)-d5 = / X < 1 - -^-) |f | 2dC > / ^ a ^ j f a f d s >|/2 A |f| 2 2 2 ->- oo C o n t r a d i c t i n g ( 1 ) Hence l i . ±j / <1- " f ) |f | 2 d 5 - £ / l | f | 2 X-x» Th i s together w i t h (1) gives /"^| f | 2 = ||fj| 2 , which i s the d e s i r e d r e s u l t . 2.1.3 P r o p o s i t i o n We can extend the F o u r i e r transform f -> f to the whole 2 of L (-», oo) w i t h ( i ) f + g = f + g ( i i ) of = af , a a complex number ( i i i ) ||f|| 2 = ^ ||f ||2 1 2 2 2 Proof Since L n L (-oo, oo) i s dense i n L (-oo, oo) g i v e n any f e L (-oo, oo) X 2 choose f e L n L (-oo, oo) s o t h a t ||f - f|L -»• 0 n n 2 2 Thus f i s Cauchy, now, f are a l l i n L (-», °°) by the previous theorem, moreover f i s Cauchy as w e l l because the F o u r i e r transform i n 15 this case i s norm preserving except for a constant. 2 f converges to some g e l (0, ») i.e. (|fn - g|i 2 - 0 this g w i l l be called the Fourier transform of f , denoted by f . It can be checked easily that g i s not dependent upon f , thus f i s well defined. The proof of (i) ( i i ) and ( i i i ) are l e f t to the reader as easy exer-cises. 2 2.1.4 Theorem For f , g e L (-», ») . _CO _ CO _ Proof Consider f„ = Y, „ „,f % = X[-M,M]8 then fjj » % e L*("*°°> °°) > which enables us to switch the following integral OO / \ C O OO T v f " - C 0 N ( x ) e i x t g M ( t ) d t dx - ON ^ . Now, apply Holder's inequality and l e t N ->- » 16 Apply Holder's inequality and l e t M -»• » ; this time CO " CO ~ '_cof 8 M * '-» f 8 This completes the proof, ~ 2 2.1.5 theorem The mapping f -»- f from L (-<», °°) into i t s e l f i s actually onto. A 2 — Proof Let f e L (-«>, <») and l e t h = f then h e L2(-«>, «,) and ..J| h[|.2 = /2ir]|h|j 2 = j | f | l 2 f - h|i2 = /(f-h)(f-h) = |(f||2 - /fh - /fh + ||h||2 - 2[|f|i2 - i f h - /fh .... (1) Bit /fh = /fh by the previous theorem 2ir A 2 A putting this back into (1) we notice that ||f - h||2 = 0 f = h . We summarize 2.16 theorem (PI a n r W p l ) 2 The mapping f + f .of L (-«>, ~) into i t s e l f i s l i n e a r and norm pre-serving except for a constant; therefore i t i s 1-1 . I t i s also onto. 17 Lastly, i t is just the ordinary Fourier transform on L i n case 1 2 f e L r\ L (-<», «>) . 2 2.2 Mellin transforms on L (0, °°) and their Relation to Fourier Trans-2 forms on L (-°°, ro) ~" 2 2.2.1 Definition Let <j> e L (0, ») . The Mellin transform M(j> of ij> is 2 defined as an element i n L (-», oo) by: To show that this makes sense, we relate i t back to the Fourier trans-2 form on L (-oo, co) we developed i n the last section. F i r s t of a l l , let 2 2 us set up a 1-1 correspondence between L ( 0 , oo) and L ( - ° ° , oo) x_ 2 x 2 For <j) e L ( 0 , oo) consider A <fi(x) = <Ji(e )e . x a real number clearly A i s linear and measurable M<J>(x) = l i m dy 0 A (H 2 = Cl*(eX>|2eXdx = /ol<l»(")|2du A «J> e L 2 ( - o o , «,) and ||A<{>||2 = ||<j>||2 Therefore A is 1-1 Lastly we claim that A i s onto, the reason being that A -1 i s given by A - 1 f (x) = f(log x)^r- . Thus A is an isometry between L ( 0 , m) and L (-«>, oo) 2 2 Now for $ e L ( 0 , » ) , A$ e L (-*>,°o) 18 1 2 Consider (A<fi) = X r -i i i A<f>e L p, L (-oo, «o) n [-log n, log n] \ > / Then A<b ->„ A* as n -> <» . L But A<J> = / l o g nA<Kt)e i t Xdt -/ ]°& nUet)e1+ ^dt r n -log n r -log n y x / n ~2 + ± X = / <{>(u)u du n Hence M<{) = A<j>. Since A and -A are both isometries, aside from the fact that s\ i n -volves a constant /2TT , M i s an isometry up to a constant. We have |[M<f>||2 = /2TT || ( f i j ^ • diagrammatically: L 2 ( 0 , ~) M—> L2(-«°, oo) , 2 T 2 , S A = Fourier transform on L L (-oo, oo) 2.2.2 Definitions and Propositions -1 1 00 . | 2 oo — j [ r — 2 Suppose /Q|k(s)js ds < oo and denote o)(x) = /gk(s)s ds for a l l r e a l numbers T . -1_ co 2 (i ) |o)(x)| <_ /Q|k(s)|s ds which i s clear enough x x 2 1 ( i i ) Consider K(x) = k(e )e . Then K i s measurable and K e L (-°o, oo) x -1_ ,oo i x. i 2 , oo i , , . i 2 lKHl = / _ J k ( e x ) | e % I x = / Q|k(u)|u zdu 19 1XT K(x) = / K ( x ) e dx = / k ( e A ) x . x (2 + i t ) dx 0 0 2 = / Qk(u)u du = w(—r) By the Riemann Lebesgue lemma CO(T) -»- 0 as | T | -> <» . By 1 . 2 . 3 , ID i s u n i f o r m l y c o n t i n u o u s . 2 2 . 2 . 3 Theorem For g e L (0 , «>) , MTg = o> -Mg , where T i s our con-v o l u t i o n o p e r a t o r . P r o o f L e t s show t h a t the theorem i s t r u e f o r 1 2 8a = X [ l , a ] 8 e L L ^ a > 1 ) a „ ~ f + i x r B -. / 0 y J ^ - C t ^ C y t H t J d y = /Q /Q y k ( t ) g a ( y t ) d y dt . The i n t e r c h a n g e of order of i n t e g r a t i o n can be j u s t i f i e d because the . i n t e g r a n d i s a b s o l u t e l y i n t e g r a b l e t o - 1 1 u z | g a ( u ) | d u f / " | k ( t ) | t 2 d t < oo C o n t i n u i n g on we get !0 ;0  7 k ( t ) g a ( y t ) d y dt C O I CO = 'o k ( t )l/o : y g a ( y t ) d y d t 20 / Qk(t) .""/Xv 2 + i T / Ndx W g a ( x ) ~ dt -1 7 - IT / Q k ( t ) t dt 3 f~ 7 + IT J x 2 0 g (x)dx 3. /• \ ra 2 + 1 T g (x)dx = w(x)J^ x & a v -I a 2 Consider the last expression / x 1 a + i x g (x)dx is the pointwise 9. N ~2 + i T 2 limit of /. x 8 (x)dx . So i t must also be the L l i m i t , N which i s "Kg . . a ~— + -t N 2 Tg (y)dy The very f i r s t expression i s the pointwise limit of y a N 2 Thus i t is equal to the L limit which is MTg proving MTg = co * Mg 2 As a •> » , g + g in the L norm. Since M and T are both con-tinuous and to is bounded coMg coMg MTg -> MTg . _ • Si This completes the proof. As a reference for this section, see [4], [6]. 2.3 HUbert space Techniques 2 Clearly L (0, °°) i s a Hilbert space with inner product defined by CO <x, y> =/Qxy . A reference for this section is [10]. 21 2.3.1 Lemma Let A be a s e l f - a d j o i n t operator on a H i l b e r t space H ( i i ) Im <Ax, y> •= Re(-i<Ax, y>) .2.3.2 D e f i n i t i o n An operator T on a H i l b e r t space i s sa i d to be normal ft ft ft i f i t commutes w i t h i t s a d j o i n t T . i . e . TT = T T 2.3.3 Lemma T i s normal <=> ^ x || Tx (| = (JT x|j. 2 Proof I f T i s normal, then Tx|| = <Tx, Tx> = <T Tx, x> = <TT x, x> = <T x, T x> = T x * Conversely, i f Tx = T x then ft ft ft ft ft ft <(TT -T T)x, x> = <TT x, x> - <T Tx, x> = <T x, T x> - <Tx, Tx> i s s e l f - a d j o i n t . 2.3.4 Theorem I f T i s a normal operator on a H i l b e r t space H , then i t s r e s i d u a l spectrum i s empty. * * — Proof Since (T-X) = T - X , t h e r e f o r e one can e a s i l y check that (T-X) i s ft normal as w e l l . Now x e N(T-X) <=> Tx - Xx = 0 <=> T x - Xx = 0 * — (by the previous lemma) <=> x e N(T - X) . of <Ax, x> = 0 Vx, then <Ax, y> = 0 \/x, y . Proof This f o l l o w s from the f o l l o w i n g ( i ) Re <Ax, y> = <A(x+y), x+y> — <A(x-y), x-y> 4 ft ft * by the previous lemma TT = T T s i n c e TT - T H(T-X) = N(T - X) (1) 22 * _ i Next we c l a i m that N(T - X) = R(T - X) (2) * — F i r s t , take x e N(T - X) , then f o r y e R(T- X) choose (T - X ) X r y * — <x, y> = l i m <x, (T-X)x n> = l i m <(T - X ) x , X R > = 0 x i s perpendicular to any y i n R(T - X) Conversely, i f x e R(T - X) then <(T*- X)x, (T*- X)x> = <x,(T - X)(T*- X)x> = 0 Comparing (1) and (2) , we see that N(T - X) = R(T - X)" Suppose the r e s i d u a l spectrum i s nonempty say, X i s i n i t then R(T - X) £ H R(T - X) >^ {0} => N(T - X) ^ {0} => X i s i n the p o i n t spectrum. This i s a c o n t r a d i c t i o n . o 2.3.5 P r o p o s i t i o n Let T be the c o n v o l u t i o n operator a c t i n g on L (0, °°) . 00 & T x ( t ) = / n k ( s ) x ( s t ) d x . Then T i s given by T*x(t) = f°Q ^  k ( i ) x ( u t ) d u * Proof T i s a l s o a c o n v o l u t i o n operator because -1 . " -1 k ( i ) | u 2 d u = /Q|k(s)| s 2 d s < » . * To show T i s r e a l l y the a d j o i n t , c onsider <Tx, y> = fQ / 0 k ( s ) x ( s t ) y ( t ) ds dt 23 = /" /"k(^(u) M t T f d t U U t t = /Qx(u)/~k(^)7(0 ~ du = /Qx(u)/pk(^)y"(uIT^ du = /QX(U)[/~ J k(i) .y.(.us)ds]du = <x, T y> 2.3.6 Theorem the convolution operator T i s normal. ft i i Proof TT x(t) = J-pkCs)/^ ± k(^) x(ust)du ds = ~ k(i)[/Qk(s)x(ust)dsjdu = T*Tx(t) The interchange of order of integration can be j u s t i f i e d for almost a l l ft t because the integrand i s absolutely integrable to |T||T ||x|(t) where * 1 1 |T| and |T | are operators with kernels |k| and |— k(—) | respectively. 2.4 Spectrum of T In this last section, we determine the spectrum of T and the com-ponents of the spectrum as well. 2.4.1 Theorem a ( L 2 ( 0 , °°), T) = '{U>(T): T real}u{0} = R(a))U{0} . Proof Since U(T) i s continuous and tends to zero as |T| -> 0 0 . 0 i s the only possible limit point of R(io) . (i) Suppose X £(R(u))C{0}) then 3e > 0 such that |x - W(T)| >. e for a l l x . 24 For any g e L (0, ») , apply theorem 2.2.3 and the fact that M is al-most an isometry to obtain N(T - A ) g i l 2 = fe"M(T - X ) g i ?2 = fe l i a ,- M g - ™&h 1 /" |ai(x)-X| Mg(x) dx ^ 1 2 > fe e||Mg||2 - e||g||2 Hence T - X is 1-1 and (T - X) i s bounded, since the residual spec-2 trum i s empty ,by theorems 2.3.4 and 2.3.6 , X ep(L (0, °°) , T) . 2 ( i i ) To prove the other containment, we need only show that R(co) C a(L (0, °°)» T) because 0 i s a limit point of R(co) and the spectrum i s closed, given W(TQ) , given e > 0 . Choose 6 > 0 so that |T-T0| < 6 => |w(x) - OJ(T0) I < e . Consider J F , , , .,E L (-00, °°) -17' XtTft-«, t n+ 6] This function i s equal to Mg for some g e L (0, ») . Ii[T-a,(x^]g|l2 =7 2 7llM(T-a)(x 0))g|| 2 r T +6 1 -- fe w^o» M4 = k L / T ^ ( t t ) - " ( T o ) i 2 - ^ 2 d u J 2 * U 2o ~* but l[gj|2 = ||Mgl|2 = 1 /2TT Thus (T-CO(XQ)) 1 even i f i t exists cannot possibly be continuous /, u)(T 0)e a(L 2(0, °°) , T) . 25 2.4.2 Theorem ( i ) Ra(L 2(0, °°) , T) = 0 . 2 ( i i ) Pa(L (0, «) , T) = {X: {x: CO(T) = X} has pos i t i v e measure) ( i i i ) Ca(L 2(0, ») , T) = {X: X e R(xo)V{0}, {x: w(x) = X} has measure Proof ( i ) Immediate consequence of theorems 2.3.4 and 2.3.6 ( i i ) Suppose {x: co(x) = X} has measure 0 then [T-X]g = -0 => ||M(T-X)g|j2 = 0 => [u)-X]Mg = 0 - ; => [u>(x)-X]g(x) = 0 a.e. Since io(x)-X / 0 a.e. Mg(x) = 0 a.e. => g = 0 /.A i s not i n the point spectrum Conversely, i f , { . . T: w(x) .= .X}-4ias - positive measure-, l e t «us 'choose-a >subset --E of •fr : O J ( T ) = X} such that E has f i n i t e p o s i t i v e measure. Choose 2 1 1 g e M O , °°) so that Mg = xE then ]|g||2 Mg 2 = -^(measure E) g i 0 .hit ||(T-X)gi|2 = ^||M(T-X)g|| 2 = ^ j|coMg-XMg||2 Since [u(x) -X]Mg(x) =0 i f x e {x: u)(x) = X and [w(x) -X]Mg(x) =0 i f x f {x: u)(x) = i } - E [w-X] Mg = 0 => (T-X)g =0 Therefore X i s i n the point spectrum ( i i i ) This follows from ( i ) and ( i i ) . 26 CHAPTER I I I WIENER'S THEOREM The reason we discuss Banach algebras i n th i s chapter i s because L 1 i t s e l f i s a Banach algebra, and i n order to prove Wiener's theorem, we need to know some properties of L 1 which are closely related to i t s Banach algebra structure. The p r i n c i p a l r e s u l t of the theory of commutative Banach algebra with unit i s at the end of section 2. We apply t h i s to a p a r t i c u l a r Commutative Banach algebra with unit to prove Wiener's theorem, which w i l l be needed i n chapter 4. The proof employed i n this chapter i s based on [5] 3.1 Commutative Banach Algebras with unit 3.1.1 Definitions ( i ) A Banach Algebra i s a Banach space, say B , together with a binary operation * , call e d m u l t i p l i c a t i o n defined on B such that a, b, c B and X any scalar (1) a*(b+c) = a*b + a*c (2) (b+c)*a = b*a+c*a (3) A(a*b) = (Aa)*b (4) a*(b*c) = (a*b)*c (5) ||a*b|| < ||aftUb|| ( i i ) A commutative Banach algebra i s a Banach algebra such that m u l t i p l i c a t i o n i s commutative. ( i i i ) I f there i s a m u l t i p l i c a t i v e unit i n B then i t i s called a Banach algebra with unit of this i s the case, we require the norm of 27 the u n i t t o be 1 . ( i v ) An i d e a l of a Banach A l g e b r a B i s a subspace I of B such t h a t xy e I and yx e I whenever x e I , y e B . (v) A maximal i d e a l i s a proper i d e a l . 3 . 1 . 2 Proposition ( i ) M u l t i p l i c a t i o n i s cont inuous . T h i s can be seen b y : I f a a and b -> b then n n II b n - a b j | < | | a n - a|j|jb || + ||aj||b n - b|)+ 0 ( i i ) l l x n | | <^  ||x|\n where n i s a p o s i t i v e i n t e g e r . T h i s can be proved e a s i l y by i n d u c t i o n . ( i i i ) L e t e be the u n i t of a commutative Banach A l g e b r a B of e - x < 1 then x 1 , the m u l t i p l i c a t i v e i n v e r s e of x , e x i s t s . To see t h i s , c o n s i d e r x^ d e f i n e d by e + (e-x) + . . . + ( e - x ) n then i s Cauchy because i f m > n , then || x - x II < ||e-x|\n + 1 -r—7J rr-* 0 as m, n-> «> . " m n l l _ 1 11 l-He-x|| Thus x^ converges to y say x * x = (e +(e-x)+ . . . + ( e - x ) n ) ( e - ( e - x ) ) n e - ( e - x ) -> e but x ^ * x -> y * x as w e l l / , y * x = x * y = e . 28 (iv) Let I be a proper closed ideal of a commutative Banach Algebra B with unit e , then the quotient algebra B/I = {x + I: x e B} is again a commutative Banach Algebra with unit. To see this, f i r s t of a l l , i t i s well known that B/I i s a Banach space and also a commutative ring with unit. We need only to show that || XY || <. ||X||/|Y|| X, Y £ B/I . And that the unit e + I has norm 1 . ||XY|t = inf ||xy(| < inf ||x|//Jy|| xeX xeX yeY yeY < inf |ix|| inf ||y|| = ||X|| ||Y|| xeX yeY To show |je + I || = 1 f i r s t of a l l ||e + l | | <_ 1 because e e l . Conversely i f e + x e e + I where x z I , then jje + x jj < 1 => x ^ = -(-x) exists because (-x) ^ by the previous proposition, exists. I, being a proper ideal, cannot possibly contain an invertible element. .". j|e + x|J >^  1 . 3.2 Maximal ideals and the Canonical Homomorphism 3.2.1 Proposition Every proper ideal I of a commutative Banach algebra B with unit e i s contained i n some maximal ideal of B . This can be proved using Zorn's lemma. Let P = {A C B: A i s a proper ideal that contains I}, Then P is par t i a l l y ordered by set inclusion. P f <f>' . One easily checks that every chain i s bounded above, because i f C i s a chain then \JC = L/C is an upper bound. Thus P CeC contains a maximal element which must be a maximal ideal. 29 3.2.2 Theorem A maximal i d e a l of a commutative Banach algebra B with unit e i s closed. Proof Let M denote the maximal i d e a l , then M i s also an i d e a l because (i ) x, y e M and A scalar => ^xn-^ » e ^ x x , y -> y Ax + y -> Ax + y n n n Jn J => Ax + y e M ( i i ) x e B , y e M => {y^} e M y^ ->• y xy e M xy -> xy .".xy e M . n n Just suppose M i s not closed then M ^ M . This forces M = B , so e e M => j x e M such that lie — x ||< 1 => x exists contradicting M being proper. 3.2.3 Theorem An element x. of a commutative Banach algebra B with unit e i s i n v e r t i b l e <=> i t i s i n no maximal i d e a l of B . Proof Suppose x 1 e x i s t s , then i t cannot possibly be i n any maximal i d e a l , this i s w e l l known. Conversely, i f x 1 does not e x i s t , then consider I = {zx: z e B} . I i s c l e a r l y a subspace and zx e I , y e B => y(zx) = (zx)y = (yz)x e I . /. I i s an i d e a l . I i s proper because e e l . Thus by 3.2.1 I i s contained i n some maximal i d e a l , therefore, x i s i n that maximal i d e a l as w e l l because x = ex e I . 30 3.2.4 Theorem I f a commutative Banach Algebra B w i t h u n i t e has the property that every non zero element i s i n v e r t i b l e , then B i s isomorphic to the complex numbers. Proof Let x e B then the spectrum of x i s non-empty ( i . e . there e x i s t s some complex number A such t h a t (x-Ae) 1 does not e x i s t ) . Since B i s a f i e l d (x -Ae) ^ does not e x i s t => x - Ae = 0 => x = Ae . Therefore every element can be w r i t t e n i n the form Ae , c l e a r l y t h i s A i s unique. The correspondence Ae •+ X i s then an isomorphism. 3.2.5 P r o p o s i t i o n s Let M be a maximal i d e a l of a commutative Banach Algebra B w i t h u n i t e , then ( i ) there i s a homorphism form „B to ,C... .,We,,denot:e t h i s by M and w r i t e M.(x) f o r x e B . We a l s o use the n o t a t i o n x(M) = M(x) ( i i ) M(e) = e(M) = 1 ( i i i ) M(x) = x(M) = 0 <=> x e M ( i v ) M i s continuous, HM||_< 1 Proof ( i ) By 3.2.2 M i s c l o s e d ; thus B/M i s a commutative Banach Algebra w i t h u n i t . B/M i s a l s o a f i e l d because M i s maximal. By 3.2.4 there i s an isomorphism <j> from B/M to C . Le t p be the c a n o n i c a l p r o j e c t i o n of B onto B/M then c e r t a i n l y <{>p i s a homomorphism from B to C t h i s i s c a l l e d the c a n o n i c a l hom- omorphism . ( i i ) M(e) = $p(e) = <j>(e + M) = 1 31 ( i i i ) M(x) = 0 <=> <J>p(x.) =0 <=> (f>(x + M) =0 <=> x + M = M <=> x e M ( i v ) M i s c l e a r l y l i n e a r s i n c e Hp"|l £ 1 114^11= 1 • i l M | l ^ 1 and M i s c o n t i n u o u s . 3 .3 V as. a Commutative Banach A l g e b r a w i t h u n i t 3 . 3 . 1 Theorem L e t • y e L^(-°° , °°) and x a bounded f u n t i o n i n L"*"(-°o, » ) then x * y i s c o n t i n u o u s . Proof L e t C be such t h a t |x(t) | _< C then | y * x ( t + A t ) - y * x ( t ) | <, / " J y ( t + A t - s ) x ( s ) - y ( t - s ) x ( s ) |ds <. C / ^ | y ( t + A t - s ) - y ( t - s ) |ds = C/™ | y ( u - A t ) - y ( u ) |du =-'ej|yAt-yJ|1 ->0-as At + 0 L^(-°° , <») i s a commutative Banach A l g e b r a where the m u l t i p l i c a t i o n i s j u s t the o r d i n a r y c o n v o l u t i o n but L ^ does not have a u n i t . 3 . 3 . 2 Theorem L^(-°° , <=°) does not have a u n i t . Proof Suppose n o t , l e t e be the u n i t c o n s i d e r Xrn, i ] e ^ • Then e * X [ 0 1 ] = X [ 0 1] "*"S c o n t - ^ n u o u s by the p r e v i o u s theorem, t h i s i s not t r u e because every cont inuous f u n c t i o n d i f f e r s from v , , , on a s e t of A [ o , l ] p o s i t i v e measure. 3 . 3 . 3 P r o p o s i t i o n Even though L^(-<», <*>)• d o e s n ' t have a u n i t , we may a d j o i n one i n t o i t f o r m a l l y , s a y , l e t e be the u n i t and c a l l the r e -s u l t a n t s e t of t h i s f o r m a l a d j u n c t i o n V . 32 Therefore V = {Ae + f: A a complex number f e L 1} with operations and norm defined as. follows ( A ^ + f 1 ) + (A2e + f 2 ) = (.X±+ A 2)e + ( f ^ y A x(Ae + f) = x^xe + Ajf ( A 1 6 + f l ) A ( A 2 e + f2) = X l X 2 e + ( A l f 2 + X 2 f l + f l * V I Ae + f I = A + " f i t One easily verifies that V is a commutative Banach Algebra with unit and L 1 is a subalgebra. V i s the algebra we are interested i n . After this formal adjunc-tion 'of a unit, we may apply the results of the previous sections to this V , and prove Wiener's theorem. 3.4 Wiener's Theorem 3.4.1 Lemma Let A be a real number and C < A then , v , . . xrc,A+h] " x r c a i r,. Xr K l ( t - A ) = lim x r, m * *Z (t) [a,b] ^ [a,b] h Proof We need only consider sufficiently small h say 0 < h < b-a . By considering various cases we see that the function i n the limit sign looks l i k e 33 <lX[a,b] x E C X + h t X h x r C t X l - X r a - b 1 ( t - X ) H h = h -> 0 The lemma can e a s i l y be extended to step f u n c t i o n s i n p l a c e of Y , , , A [ a , b ] I t can be f u r t h e r extended. 3.4.2 Lemma Let Z e L*(-», «>) then lemma 3.4.1 i s s t i l l t r u e i f we r e -P l a ° e X [ a , b ] b y 2 ' Proof Approximate Z by step f u n c t i o n s x . Since x (t-X)-> Z(t-X) n n i n L"'" norm (or V norm) i l i m x « " x j c , x + h ] - x [ c , x ] _ z * x[c,x+h] " X f c . x l , 'b* n h ro "h 1 < x - Z because the norm of Y R R , , ^, . - Y R „ , , i s 1 . X [ C , x t h ] " x f C , X l h The lemma f o l l o w s .3.4.3 Lemma I f I i s a closed i d e a l of V and i f Z e I , then f o r any r e a l number X , Z (t) = Z(t-X) e l . A Proof I f Z e I then Z * x[C,X+h1" xrc.x1 e l h Since I i s c l o s e d , by the previous lemma Z e I . A 3.4.4 N o t a t i o n Let s be a r e a l number. Me -'• {Xe + f e V: X +f(x) = 0} . s 3.4.5 Propositions (i) The M 's are a l l distinct. 1 <> s ( i i ) None of the M g is equal to . i s ^ i s 2 t Proof (i) Suppose s^ £ 31 such that e £ e Let f be the characteristic function of a small interval around t then f ( X ; L ) = f ( s 2 ) . - f ( s , ) e + f i s i n M: but not i n M 1 Bx S 2 ( i i ) For given real numbers s^  j f £ L such that f(s) f 0 - f(s)e + f £ M but not i n L1 . s 3.4.6 Theorem t \ M are maximal ideals of V and these are a l l the s maximal ideals. Proof L\ M are a l l proper subspaces of V . It is easy to verify s that they are ideals. To show that Mg i s maximal. Suppose M i s an ideal such that M s ^  M then z\ \e + £ e M such that A + f(s) = 0 /,[A + f (s)]e e M =>M = V . To show L^ " i s maximal. Suppose M is an ideal such that L 1 ^ M then Ae + f e M such that A £ 0 . Ae e M => M = V Now, let's show these are a l l the maximal ideals. 35 Let M be a maximal i d e a l of V other than L 1 then 3 Z e L 1 ^ Z e M /. M(Z) f 0 Define \(X) = ^ f x ? We w i l l show xM  = e^ S^  f o r some r e a l number s M(Z) To begin w i t h , x i s continuous s i n c e X •*• Z i s continuous by lemma 1.1.3. A l s o , s i n c e 2 * Z u = /" o oZ(t-y-X)Z(y-u)dy = F_Z (t-u-v-X)Z (v)dv = Z A + u * Z . We have xMxM = X ( A + U ) F i n a l l y , by lemma 3.4.2. Apply M on both s i d e s M(Z ) = M(Z)lim M X[C,X+h] ~ X[C,X] h->0 h So |M(Z A)[ < |M(Z)NlM||j| XrCX+h^| " X r C X l ||= |M(Z)| hence |X(A)|<.1 but X(A)X(-A) = x(0) = 1 t h e r e f o r e | X ( X ) | = 1 A continuous f u n c t i o n s a t i s f y i n g t h i s c o n d i t i o n together w i t h i s X x(A)x(u) = x( A+u) must be of the f o r the form xM = e f ° r some r e a l number s . From (1) , we a l s o o b t a i n dX M ^ X [ C X]^ = X ^ ' I n t e S r a t i n 8 b o t h s i d e s M ( X [ C , b ] } ~ M ( x [ C , a ] ) = C e±S x [ a , b ] d X T h u S M ( X [ a , b ] ) = X f a , b ] ( s ) 36 The above i s not only t r u e f o r x a^ but a l s o t r u e f o r any f e lA . L a s t l y f o r Ae + f e V . M(Ae + f ) = A + f ( s) Xe + f e M <=> X + f ( s) = 0 M = M . s 1 3.4.1 (Wiener) Let X f 0 and f c L be such that f + X i s never 0 then 3 g e L 1 such t h a t Proof Xe + f i s not i n any maximal i d e a l of V . So i t s i n v e r s e A^e + g e x i s t s where g e 1?~ . (Xe + f ) * ( X i e + g) = e . => XX = 1 and . A j f + Ag + f * g » 0 A f + Xg + f g = 0 . 37 CHAPTER IV SPECTRUM OF T In this last chapter., we determine o(L P(0, oo) , T) i n general (i.e. 1 < p < °°) . Recall the definition of T our convolution operator from section 1.1. 4.1 Essentially Convolutions Let us set up a 1-1 correspondence between L P(0, oo) and L P ( - c o , o o ) , then manufacture an operator on L p ( - o o , oo) that corresponds to T on L P(0, oo) . It should be understood that — = 0 i n case p = oo . Define P A: L P(0, C O ) - » L P ( - ° ° , oo) via x_ •Af (x) <= .f<(e*>eP . -x Clearly A i s an isometry. Define K(x) = k(e X ) e ^  then K e L^ " -1 with ||Kl^ = /~|k(u)|u p du . Define A^ .: L P ( - o o , oo) -> L P ( - o o , co) by A^x) = K * x . Then for f e L P(0, »)j A ^ f ( t ) = /^K(t-y)Af(y)dy -_t = r k(e y" t)e ye q f(e y)dy CO -t 0 V Jt = e P /™k(V)f(V efc)dV = e P Tf(e t) = ATf(t) Therefore i s the r i g h t operator^ we have A^ = ATA -1 A^ A = AT . 4.1.2 Observations ( i ) Diagrammatically T L P ( 0 , co) ^ L P ( 0 , ») A A LP( - c o , oo) V LP( - o o , oo) ( i i ) For any K e «) and 1 <. p <. 0 0 . -1 Let k(x) = K ( - l o g x) then /"|k(u)|u p d u < » . The c o n v o l u t i o n operator w i t h k e r n e l k then corresponds to ( i i i ) From ( i i ) and s e c t i o n 4.1, we o b t a i n the Young's i n e q u a l i t y For F E L 1 3 G E L P . 1 n y i < iiKid ( i v ) For F, G e L and H e L (F*G) H = F *(F * H) (v) For G, F e L and H e L P ( G + F ) * H = G * H + F * H 39 (vi) a(L P (-<*>, »), Aj,) « a(L P(0, «) , T) So we can determine the spectrum of and infer the spectrum of from the relation of K and k . 4.2 Spectrum of T P 4.2.1 Theorem a(LF(-c°, «,) , A^) C. R(K) Proof We use Wiener's theorem i n this proof. Take X £ R(K) . Since 0 is the only possible limit point of the range of K , this means -X j 0 and K(£) -A =/ 0 tyreal numbers £ -By Wiener's theorem, 3G e L^(-°°, °°) such that -—^ = - 7" + G(£) K(0~X -1 P —X I + A i s an operator on L (-00, °°) . We have (A^-XI) (-X-1I + Ag) = - x " 1 ^ + VV" a A g + 1 (-A"1! + A G) (Aj^-Al) = -A'\+A GYAA G •+ I By the associativity and commutativity of convolutions, both of the sabove expressions reduce to I . (Aj^-XI)"1: LP(-<», ») L P ( - ~ , «>) exists By the opening mapping theorem (A^-XI) i s continuous X e p(LP(-«>, «°) , A^) . 40 4.2.2 Lemma Let 1 < p < °° . Let K e L 1 ( - o o , « ) Suppose |xK(x) |dx = M < °° . Then, for each real number £ and 6 > 0 there exist functions u . £ L P ( - o oJ <») with ||ul| = 1 such that 6 6 'p ^ i s bounded as 6 0 Proof 6>0. £ is a real number Let V,(x) - /f V^dn = l e ' ^ s i n 6 x In case 6 = 1 , V.e L P . Therefore V. e L P with 1 o V. = 6 1 _-V. .... (1) 6 p j> x _p By changing the order of integration K * V.(x) = / ^ e i r i X K ( n ) d n o £—o Let E fx) = K( 5)V fx) - K * V fx) = /f^ e i n x ( K ( 5 ) - K ( n ) ) d r , . 0 0 0 t,-o Now K is differentiable and |K*( n)| <_ M . /, |E (x)| <. M62 . Also, upon integration by parts, |E^(x)| _< 46M|x| 1 Therefore /*jE f i(x) | pdx < < 4(M6 2) Pdx + / j x j > 4 (46M|x|~1)Pdx i I = 6 2 p- 1[M P(6+-^ T)] p-1 V Let u = _6 41 Recalling (1) We obtain p^ — ^ "^l'^p ' p-"*" /which i s bounded 4.2.3 Theorem c ( L P ( - c o , 0 0 ) , ^ ) = R(K) Proof By 4.2.1 and the fact that the spectrum is closed we need only to show R(K) C a ( L P ( — , «) , A^) . Consider the case 1 < p < » . Let K be v, , K , then * n *[-n,n] ^^l-xK (x)| < 00. For a given real number E, make use of the previous lemma; there is a u e L P which has norm such that n || K * u - K (5)u II < - . 11 n n n x^' n"p n Secondly ||.A^  - A^ |/ <_ ||K - 0 as a + » . n Lastly |Kn(c) - UO | < ||K.^  - /.[|K * un - K( 5)u n i | p < ||K * u - K * u |L + ||K * U - K (OuJ| +/|K (?)u - K(g)u || -»• 0 — " n n n"p n n .n * ri'p n ^ n ^ n Mp A Therefore K(0 must be i n the spectrum. iEx * In case p = °° , f(x) = e i s an eigenvector with eigenvalue K(^) .»*• K(0 i s the spectrum . 1 C O Lastly i f p = 1 , then the dual space of L i s L . Therefore 1 CO 1 * <r(L , Ay) = o(L , Ay) . but i s just ^ ( _ x ) i-e. If we define K to be K*(0 = K(-0 then A£ = A ^ . 42 1 A A * Thus a(L , = {0}\J R(K ) = {0} U R(K) . This completes the proof. 4.2.4 Conclusion -x Since K(0 = /" k(e~ X)e q e i x ? d x . -15 = /gk(u)u p du -1 IT CO p Therefore i f we define W(T) = /gk(u) F du Then, a(L P(0, «) , T) = COJ(T) : T a r e a l number} U {0} . = R(w)L/{0} . One sees that theorem 2.4.1 i s a special case. BIBLIOGRAPHY G. H. Hardy, J . E. Littlewood, G. Polya, Inequalities, Cambridge 1934. Y. Katznelson, Introduction to Harmonic Analysis, Wiley, 1968. S. Goldberg, Fourier transforms, Cambridge Tracts No. 52, 1962. H. Kober, On a theorem of Schur and on Fractional Integrals of Purely  Imaginary Order, Trans. Amer. Math. Soc. 50(1941) 160-174. I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative normed  rings, Chelsea, 1964. E. G. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford, 1937. D. W. Boyd, The Spectrum of the Cesaro Operator, Acta S c i . Math. (Szeged)29, (1968), 31-34. D. W. Boyd, Spectra of Convolution Operators, (to appear). K. Ross, Fourier Series and Integral, Department of Math., Yale University, 1965. K. Yosida, Functional Analysis. Springer-Verlag, New York Inc. 1968. 

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