Spectral properties of a polynomial operator

Numer. Math. 13, 247--259 (1969)

Spectral Properties of a Polynomial Operator* M. V, PATTABHIRAMAN and P. LANCASTER

Received June 28, 1968

1. Introduction

The purpose of this paper is to obtain a spectral decomposition for the inverse of a polynomial operator D (~), given by

D(2) == 2 A o + ; t l - X A l + ... + A t (i)

where the Ai 's (i = 0, t, 2 . . . . . l) are bounded linear operators from a complex Banach space B into itself, A o has a bounded inverse and 2 belongs to C, the set of complex numbers. The case when the A { s are square matrices and D(~) is a simple larnbda matr ix has been considered b y LANCASTER [4]. KUMMER [3] has considered the case when the Ai 's are closed operators on a Hilbert space H.

If the complex number 2 and the non-zero element q of B have the property tha t D (2) q = 0, we shall call 2 and q a characteristic value and characteristic vector

respectively. We define the resolvent set and spectrum of D (2) in the following way. The resolvent set consists of those 2 e ~ for which D -1 (~) exists as a bounded operator with domain dense in B. The spectrum of D (2) is then the complement of the resolvent set in cal. Clearly the characteristic values of D (2) are in the spectrum of D (2). I f A is a bounded linear operator on B, the resolvent set of A is defined as above except tha t D -1 (2) is replaced by (12 - - A ) -1. The operator ( 1 2 - - A ) -1 is called the resolvent of A and will be written R(2; A). We denote by ~ " the nullspace of ( I 2 - - A ) ~ and it is obvious that ~0_(~/~1 ( . . . . I f there is a least positive integer v such tha t , f ~ = _ ~ , + l then v is called the index of 2.

We shall use the operational calculus for bounded linear operators developed by DUNFORD and SCHWARTZ [{], for example. If / is a function analytic in an open set U containing the spectrum of A, and if the boundary c# of U is a rectifi- able curve, t h e n / ( A ) is defined by

/ ( A ) = ~ / . I(2) R(2; A) d2. (2)

2. An Equivalent Operator We form the product space B----B 1 × B~× ... × B t, (B i = Bi) where any ele-

ment ~ B is of the form

= (x (~), x (~) . . . . , x(tl), x (i) ~ B .

* This paper forms part of the doctoral dissertation submitted to the University of Calgary by the first author under the supervision of the second. The research has been supported by the University of Calgary and The National Research Council of Canada.

18 Numer. Math., Bd. 13

248 M . V . PATTABHIRAMAN and P. LANCASTI~R:

We make B a normed linear space by defining

0~X t (1) A_ y(1), y(1)), fl~C,

II~l]~= fI.">H% .. . + ll.c*>l]~

We introduce the following operators

d : B -+ /~ , where d ~ = ~ is given by

y(kt = A o xq-k+~) +Alx(Z-k+~l + ... +Ak_Ix(1) ( k = t, 2 . . . . . 1). (3)

qY: B - + B, where ~ ~ = fi is given by

yfk)=--Aox(~-k~--AlX(l-k+l) . . . . . Ak_lx (~-1) ( k = l , 2 . . . . . ( l - - 1)), (4)

y(0 = A~ x ~.

ocPk: B--> B, ..~k ~" = x(k) (k : : f, 2 . . . . . l), (5) and

Ji: B - ~ B , ~ x = (0,0 . . . . . x, 0 . . . . . 0) (f = t , 2 . . . . . l), (6)

with x in the ]th position. The operators ~ and g have the following mat r ix representat ions:

/ O 0 . . . O A o \

d [ ° ° AoA 1

/ 0 0 . . . 0 - - A o 0 \ (7)

~ ( 0 0 ... --Ao --AI 0 ~

\ - - A 0 --A 1 . . . . . . - -A,_2 0 / \ 0 0 . . . . . . 0 A l /

We call ~ and ~ the selection and insertion operators respectively. I t can easily be verified tha t ~ and ~ are bounded linear operators with norm one and tha t 5 P k J / = O a ] I where I is the ident i ty opera tor in B. From ~ / x = 0 it follows tha t ~, = 0 and this implies tha t ~/-x exists.

(a) ae-~ 4 = ,61Ao ~, Lemma 2.1. (8)

(b) ~ / - ~ _ ~ = - - ~ A o ~ & A o ~ + 4 A o ~.

The proof is obta ined af ter simple manipulat ions with the definitions of the operators and is left to the reader.

L e m m a 2.2. The set o/ characteristic values o/ D(2) coincides with the set o/ characteristic values o~ the pencil (~¢). +ca).

Proo]. I t can easily be seen t h a t D ( a ) x = 0 implies (ag2+~)()]-lx,)]-2x, . . . . x) = 0 and conversely t ha t (ag2 +¢g) (x (1), x (2) . . . . . x I0) = 0 implies D (2) x (° = 0. Hence if ~t is a characterist ic value of D (~) with associated characterist ic vector x, then ~t is a characterist ic value of (a~c~t +ca) with (21-1x, 21-2x . . . . . 2x, x) as an

Spectral Properties of a Polynomial Operator 249

associa ted character is t ic vector. Conversely if 2 is a character is t ic value of ( d ) . -[-(~) wi th ~ as an associated character is t ic vector , then 2 is a character is t ic value of D (2) wi th the last component x (0 of 2 as an associated character is t ic vector. (Note t ha t x~ = 0 would imply ~ = 0.)

Lemma 2.3. (~ ' ) . +c~) ~ = ~ is solvable/or arbitrary ~ E B i]] D (~) / ---- g is solvable /or arbitrary g~ B.

Pro@ Let ( d 2 +cg) ~ =2Y be solvable for a rb i t r a ry ~c B. This implies t ha t g i v e n ~ = (y(1), y(2) . . . . . y(Z)) we can f ind an unique ~ B such t h a t ( d 2 +~f) ~ = 2¢. Let g~.B be a rb i t r a r i ly given. Choose~ ~ (0, 0 . . . . . 0, g). Let ~ = (x (1) . . . . . x/~)) be the solut ion corresponding to this ~. Using the s t ruc ture of ~¢ and cg, i t is found t ha t x(1)=2Z-~x Iz), . . . , x IZ- l l=2x q) and tha t ( 2 ~ A 0 + - . - + A ~ ) x ( Z l = g . Thus we see t ha t the last component x Iz) of the solut ion of (~¢2 + c ~ ) ~ = ~ is the solut ion of D (2) / = g. The converse proposi t ion is proved along similar lines.

Combining Lemmas 2.2 and 2-3 we have

Theorem 2.1. The resolvent sets and hence the spectra o/ the operators D (2) in B and ( ~ 2 +cg) in B coincide.

Theorem 2.2. With operators d , ~ , 5~1. and ~ as de/ined in 3, 4, 5 and 6, we have, /or any 2 in the resolvent set o~ D (2)

~. (~¢ ~ + ~)-~ 4 = 2 ~-j D-~ (2).

Pro@ ~ j ( d 2 + ~ ) - ~ 4 x ==~9~-(d2 +~) -1 (0 , 0 . . . . . 0, x). F r o m the s t ructure of ~ and ~ , i t can be proved t h a t

(~ /2 +c#)-~ (0, 0 . . . . . 0, x) = (X~-~y, 2 - 2 y . . . . . 2y, y)

where D (2) y = x. Now cjj (2l-1 y . . . . . y) = 2 t - i y = },~-i D -t (2) x which gives

G ( ~ 2 +~)-~S,=Z'-~D-~(2) . (9) ~,1

Pu t t i ng ~' = l in Theorem 2.2, we get

D -~ (2) =- 2~ (~/2 + ~6') -~ ~ = ~ (I 2 + d -1 ~) -~ ~ , a ~ . (10)

We wri te ~ = - - ~ - ~ f . Rela t ion (10) implies t ha t D-~(2) is analyt ic in 2 if R(2 ; !Lf) is and fur thermore t ha t if 2 i is a pole of R(2; 5(,) of order r, then 2/ is a pole of D -~ (2) of order r. Bu t we prove ra ther more:

Theorem 2.3. The complex number h i is a pole o/ D -~ (2) o/order r i / / 2 i is a pole o / R (,~; oW) o /o rde r r.

Pro@ In view of (10) we have only one s ta tement to prove, namely tha t if 2 i is a pole of D-~ (2) of order r, then 2 i is a pole of R(2; 2~) of order r. Consider the equat ion (._~/2 + c ~ ) ~ = ~ . I t can easily be proved tha t

x") = D-~(2) {yt0 +2y(~-~) + ... +2-~y(~)},

x (~- ~) - - 2 x (~) - - A o 1 y(~), (t 1)

x(X) __ 2z-~ x~) _ 2 t - ~ A o 1 ym . . . . . A o ~ y(~-a).

18"

250 M . V . PATTABHIRAMAN and P. LANCASTER:

Note tha t the vector ~ = (x m, x(*),..., x (o) can be wri t ten in the form

~=~xCl~ +4x<~ + . . . +4xl*l

and tha t the ].th component y(il of:~ can be wri t ten as S~/:~. We thus have

= [4D-1(4 ) (2 ' -* 5~1 + , t t - ~ 5~ + . . . + a t i l t _ ~ - -5~)

+ J~_t{D-~ (4) (at 5~ + . . - + 4 5~tt ) - - A o ~ oc~} (12)

+ ~ {D -1 (,~) (,~='-= 5~ 1 + . . . + 4 '-~ ~ ) - - 2~-=Ao a 5f~ . . . . . A o a 5~}]).

Examin ing the nature of the operators mul t ip lying D -~ (4) on the left and on the right it is clear tha t if D -~ (,t) has a pole of order r at ~ti, then the same is t rue of ( a /2 +*g)-* and hence of R (4; 05f).

If R(4; .~') has a pole of order r at 4/, it is known (Theorem VII .3A8 of ~l]) t h a t 2i is a characterist ic value of c f of index r. Henceforward, 4~ will always denote an isolated characterist ic value.

3. The Pseudo-Projections We define

~ I f 2~i D-l(a) da (13) o

where ~. is a sufficiently small circle wi th centre 4 i containing no other points of the spec t rum of D(X). Relat ion (10) implies t h a t D-I(~) is analyt ic in ~ on ~. and hence the integral in (13) is well defined. We now prove a l emma establishing the relation between P~. and the operator J j defined by

, f J/= 2~i R(,~; ~) d2. (t4) O

Lemma 3.4. Let D -1 (,~) have a simple pole at ~i and let the operators d , 5°k, ~ , P] and J~ be as defined in (3), (5), (6), (13) and (14) respectively. Then we have

(a) 5pk j id - l~= l-k ~j Pj , (k = t , 2 . . . . . l) (b) N j j - 1 6 = Zj,-,~

Proo/. (a) For any x~ B

5<, 1 5 ¢ k J j d - ~ x = k { ~ f ( I~ - - .W) - lda}d - l~x . 05) o

Since (12-- .2°) -1 is continuous in the uniform opera tor topology and ~ is a bounded opera tor defined on all of B we m a y rewrite (t 5) in the form

O


Using Theorem 2.2, we get

~ ~(~'4+~')-1~xd'1= 2 ~ 2-kD-l(4)dX x. rj rj

According to definition (13), if D -I (4) is expanded in a Laurent series around 4i, then P/ is the coefficient of (4- 4,) -I. Since the pole of D -I (4) at 4~ is simple, it follows that in the expansion of 4~-kD -I (4) around 7j. the coefficient of (4 -- 9'i)-I is 4}-kPj. Thus using Cauchy's theorem

1 f4Z-~D- l (~) d,~ z-k =~j P,. (k=~ 2, ~l) 2zti . 1 ' ' " "

rt

and the first par t of the lemma is proved.

(b) For any xEB, ~9~zfjs/-l~x =~9~Jj~¢-1(0, 0 . . . . . x, 0 . . . . . 0) with x in the k th position. Thus

, f ~ I J j ~ - I ~ x = ~ - i ~ / ( ~ 4 -1-(~)-1 (0, 0 . . . . . X, 0 . . . . . 0) d ~ . rj

Let (~44 +(d)-l(0, 0 . . . . . x, 0 . . . . . 0) = (y(1), y(2) . . . . . y(O). Using the structure of -~4 and ~ and after some simplifications, we get y(O = 4~-kD-1 (~)x. Thus

5~z (~44 + ~ ) 4 (0, 0 . . . . . x, 0 . . . . . 0) = 5~z(y (1) . . . . . y(0) = y(O = ).~-~D-I(~).

Hence

2d-i . +~) -1 (0, 0 . . . . . x, 0 . . . . . O) d 4 = 2~-i-. )'~-kD-~(2) xd) . = ).}-kP i x rj rj

giving

Our next two results bring out a generalization and an extension of familiar properties in the linear case. In particular note that, if D ( R ) = I 4 - - A in The- orem 3.4, then P~ = Pj.

Theorem 3.4. I / ;~i is a simple pole o /D- I (~) and the operator Pj is as de/ined in (t3), then

P,m(z ; )P j = p, where

D1(4) = l ~Z-lA o + (l - - 1) Z*-2A~ + ... +A~_I .

Pro@ Par t I. I t is easily seen that

D -1 (~) -- D -x (/2) = - - (4 - -~) D -1 (4) Q (4,/x) D -1 (~) (t 7) where

Q ( 4 , / ~ ) = ( ) ~ - 1 + 4 ~ - 2 / ~ + . . . + / ~ - I ) A o_t_(;t ~ - 2 + ' ' ' + / A ~ - 2 ) A I + ' ' ' - + - A t - 1 . (18)

Put t ing 4 = / * in (18), we get

Q (4, ~) = 14 l-1 + ( 1 - 1 ) 2 -~A 1 + . . . + A,_~ = D ~(~). (t9)

252 M . V . PATTABHIRAMAN a n d P. LANCASTER:

and hence Q (h, ~) = D~(~) + (h - -~) (7 (~., ~)

for some polynomial ~). Since D-1(2) has a simple pole a t h i it follows tha t D~(hi) + 0 and then t h a t Q(h, hi) does not have 2 - - h i as a factor. Thus,

' fD-*(h) O(h, h,) dh=-pjDl(hi) (20) 2:~4 0

Similarly we have , f 2=i , Q(h,#) D4(#) d#=Q(h, hj) P i. (2t) i)

Par t I I . Let _P~ be a closed curve enclosing h i and containing no other member of the spec t rum of D (h). Let -P2 be another closed curve around h i contained in the interior of ~ .

,Z

Fig. 1

We then have

pj = ~i~l f -z-~--i-~ f D-l(/t) dl*" D-1 (2) d.~ = I" 1 I'=

Hence, using (20) and (21) we have

F1

& r.

l ffD-l(a)Q(a,t~)D-l(p~)d.~.dl~ (22) F~ F~

, f f D-~(,)--D-~O,) 4 ~ 2 i 2 ,~--I.~ dhdt* /'1P.*

j f,% ' f.-.(.)..f" _ 1 "D-I(I) d 2 + 4 = ' i ' r t T, F~ F~

We have . a--/~ - - 0 as h is outside -P= (see figure) and . a - / , - - 2ozi. Hence Fi F1

(22) gives

' f ' tl P j D ~ ( h / ) P / = 0 + ~ T - / ¢ • 2 ~ i D-l(#)dt* -- 2~i F~ F s


T h e o r e m 3.5. Let P~. be as defined in (t3). I/`1 i is a simple pole o~ D-t(,~), then the range o/P~ is the null@ace o[ D (,~j).

Pro@ As `1j is a simple pole of D -1 (`1) we have

Pj D-~ (4) = 2 - - ~ - / + s;(4)

where S i ()0 is analyt ic at `1~-. Premul t ip lying by D (~t), we have

( `1- 4;) I - . D (`1) {P,. + (4 --`1;) Sj(`1)}.

Put t ing `1 = `1i, we have 0 = D (~j) P,.

whence x = Pj y + n (`1j) x --- 0, giving ~(Pj) _Ji+ (D (,lj)) where ~(Pj) and +4Z(D (),j)) s tand for the range of Pj and the nullspace of D (,~]) respectively.

Conversely, we have f rom

(`1 - - Xj) Z - - {g. + (;, - - `1j) Sj (`1)} D (~),

expanding D (`1) in powers of (`1 --`1i) and equat ing coefficients of the first power,

I = Pj D ~ (`1j) + Sj (`1i) D (`11)"

Hence n (`1i) x = 0 implies x --. Pjy where y = D 1 (`1i) x. Thus ~ ( D (`1i)) -(~ (P])" II

4. Spectral Reso lut ion of a Part icular Funct ion of a Bounded Linear Operator

Let A be a bounded linear operator from a complex Banach space B into itself. Let the characteristic values `1, of A be denumerable and be enumerated so tha t I,tll => 1`121 => . . . . Let {/~} be a subsequence of {1`1~1} with/z 1 > / ~ > . . . and such that , for each 1"= 1, 2 . . . . . there is a ' k ' (depending on 1') for which I ,t, [ =/zk. Let there be m k members of {`in} with modulus /z , , k = t, 2 . . . . . We make the following assumptions regarding the spectrum of A.

(i) The spect rum consists of only characteristic values with the possible exception of zero.

(ii) The characterist ic values `1~ are isolated, enumerabte in number (23) and have zero as their only limit point.

(iii) The resolvent R (4; A) has simple poles a t `11, ̀ 12 . . . . .

Let 4 d , = # , , --#~+1 and let A~ be the circle in the complex plane with centre a t the origin and radius

r~ -- 2 -- #,~÷1 + 2d , .

For k = t , 2 . . . . let ~ be the circle with centre `1k and radius ek=min(d~ , d~_l) where l`1kl = /~ , and we define d o = d x. The radius m a y have to be smaller than ~ if we want to ensure tha t there is just one characteristic value in each ~ . This can obviously be done as tile characteristic values are isolated. Let U~ be the interior of ~ , ~ . . . . . /',~+,~,+ ... +,,~ and the circle with 0 as centre and r ad ius /~+1 + d, .

254 M, V. PATTABHIRAMAN and P. LANCASTER:

\zt , , , \ \ " , '%~ X L I ; T J; -j j

Fig. 2

In the sketch we have # . - 1 = I~Jl, m = I~,1, and/~.+1----12,1 Then if zEA,~ and eEU'~, we have tz-~.l >d~. By LemmaVII .6Af of [~3

there is a constant k. such that for zcA,, fiR(z; A)II <=k,dd,,. We therefore have

[' f ] 2xr"'maxl[(z)l'maxtlR(z;A)lt ~ ( [(z) R(z;A) dz < ~ " 2,71; zE'dn zEZln 'ln

kn __<r~.~. max l/(z) 1

=k, , O*.+S.+~) 4 . maxl / (z)[ .

We will be particularly interested in functions of the type

~f L ( z ) - a - , ' z4:2, r > o .

r~ In this case, if zEz]. and 2¢z]~, we have ]/,(z)] ~ tl21_r.i

2-~ f D(z) R(z; A) dz <2(1- ')k~ (/2n-}-/Zn+l)r+l ~ = ( ~ - ~ + ~ ) 1~1

and hence

(24)

We want the bound on the right of (24) to approach zero as n -+ oo. This will happen when ~ ~ 0 iff

l i m {k. (""+~'+l)'+~t~ j = O .

Sufficient conditions for this to be the case are provided by hypothesis (ii) of (23) together with {kn} bounded and

lim (~"+t*"+*)'+l - - O. .-.oo (~.--~.+l)


However , i t should be no ted t ha t even if {k,} is no t bounded we m a y have

[ ' f ~ dz -+O, -2-~T , (z) R (z; A) as n ~ oo A~

for suff icient ly large r.

Choosing the contour C to be the bounda ry of U" and wri t ing

(25)

we have

N = m l + m ~ + ... +m~,

, f /,(A) -~ -2-~-i. /,(z) R(z; A) dz C

N

-- 2~-i /,(z) n(z; A) dz + ~ . /,(z) R(z; A) dz. j =1 O ~-

If the norm of the las t t e rm tends to zero as n - ~ oG, and if condi t ion (iii) of (23) is satisfied, then using Theorem 5.9.3 of [2] and Cauchy 's residue theorem, we obta in

oo •

/ ,(A)--~A'(I ~ - - A)- = ~ ~ _ ~ ]i j = l 1

where *

L - - ~=i . rj

and the convergence is in the uniform opera tor topology. We thus have the following theorem.

Theorem 4.6. Let A be a bounded linear operator satis/ying conditions ( i-- i i i ) Z r

0/(23). Let/,(z) denote the/unction/~(z) = i-- z ' z 4= ), where r is a positive integer

and let condition (25) be satisfied/or this value o/r. Then

(3O y

/ r (A)=A~( I2_A) - I=i~=I ~_i f j

the convergence being in the uni/orm operator topology.

I t should be po in ted out t ha t the summat ion in the above series is in order of decreasing 12jl 's and t h a t ~ / s hav ing the same modulus should be b racke ted together .

5. Spectral Resolution of D -1 (~)

We can now ob ta in expressions for D -1 (),) in t e rms of t h e opera tors Pj defined

in (t3).

* Since the results of this section will be applied to the operator -~ as defined in §2, we trust tha t no confusion will arise on using the same symbol here as in (t4).


Theorem 5.7. Let the spectrum o/ D(4) satis/y hypotheses (i--iii) o/(23) and let Theorem. 4.6 (applied to the operator ~ ) be valid/or r = 1. We then have

2-k+lD-~(a)= ~ i_; . P,, 2~k_<l , (26)

2tD_l(4) = ~, a} p .+Ao l ' l ~ l (27)

the convergence being in the uni/orm topology o/ operators and the terms in the summation being grouped according to the magnitude o~ ]4il.

Pro@ From the identity ( [2 - -~ f ) (14--oW)-l= I, we have

(I.~ -- ~ ) -i = I -j- .~(I 4 -- ~o) -1 (28)

Since we suppose Theorem 4.6 valid for r = t,

where ]: is defined in (t4), and (28) can be written

oo a 4 (i + i + F x2: : J; (291

operating on (29) with S# k on the left and d - l ~ on the right (we may take them

under the summation sign as 2 ~ - ~ ] Ji is convergent in the uniform operator l

topology) and using Theorem 2.2, Lemmas t and 4, we obtain

OO ,,~,I --,/~ + 1 ~ l - - k + l n - i (,~) = ~ ~ Ao 1 +j~4_i~2_~;_ p/.

If k > t , ~ = 0 and we obtain (26). The case k = t yields (27) and the proof is complete.

Theorem 5.8. Suppose that the spectrum o /D (4) satisfies the hypotheses el (23) and that Theorem 4.6 (applied to the operator .Lf ) is valid/or r = 2. We then have

4'-~+*D-~ (~) = 2 ,t_ ,~-7 Pi (3 =<k_--<l), (30) j=l

;t} D-~ ().) = ~, ?-"~-~7 P / + Aol (l=>2), (3t) /=1

).}+~ 4Z+~D-l(,~):j2j'l-~-~y-_j~jPi+,~Aoi--AolA1A~i (l>__ 1). (32)

Pro@ The identity

(14 - - .oc#)-1 ~. @22 (I 4 -]- ,,.~) "~ ~ ,,.o ~O2 (I ~ - - .~)-1 (33)


is easily verified. From Theorem 4.6, we have

~ 2 (I ?~ -- ~LP)-l -~i~=l-~T, i Ji (34)

and following the same steps used in proving Theorem 5.7, we obtain

Using the structure of ~ and ~, we can prove that

( ~ - - W ) ~ - 1 ~ / = ~ / - -1+ a ,~/ (36)

and then using Theorem 2.2, Lemmas I and 4, we have oo

~'-k+~ l)-l (]L) = ~9°k (-- ~ Aol AI Ao14:- d~2Aol + )"f11A'ol) + 2 )~}-~+~ Pi" i=x A--Ai

For k ~3 , we have ~ = 3 f k J2 = 0 and we get (30). The cases k = 2 and k = I give the results (31) and (32) respectively.

6. Application to an Integral Equation

VVe now give an example which illustrates the use of one of the expressions obtained in the preceding theorems.

Consider the integral equation

1 1 $ (x) -- v f A (x, y) 4) (Y) dy -- v 2 f B (x, y) 4) (Y) dy = /(x) (37)

0 0

where A (x, y) and B (x, y) are symmetric L 2 kernels and in addition B is positive semidefinite. We let A and B denote the operators on the separable Hilbert space R, given by

1 1 A / = = f A ( x , y ) / ( y ) d y and B / = f e ( x , y ) / ( y ) d y . (38)

0 0

We assume that A and Bt are self adioint operators of finite double norm (ref. p. 208 of [8]). With the above notation, we can rewrite (37) as

(I - - v A - -v a B) ~b = / . (39)

Letting 2 = t/v (39) becomes

D (),) d) = (I~" --)~A -- B) 4) = 23/•

MOLtEn [5] has linearized the above problem by introducing an equivalent self- adjoint operator L in the product space R = R × R, given by

(A L = B J " (40)


He proves tha t the characteristic values of D (~) coincide with those of L, tha t L

has a finite double norm and tha t ~. ~ < oo where the 2~'s are the eigenvalues

of L, all of which are real. Moreover L is compact and self-adjoint and hence every non-zero characteristic value is real and has index one. Thus conditions (i--ill) of (23) are satisfied. From Theorem 6.4--C of [6] we know tha t there exists a bound on the norm of the resolvent of L, given by

I IR(~,L)U< = @~1 { t + s u p I~jt_~ (4t) j 1~-~,:1 J" In order to apply Theorem 5.8 we need condition (25) which is satisfied if (in

{ (~.+~.÷1)'+1[ the nota t ion of §4) ~ - ~ _ ~ j - + 0 and {k,} is bounded. I t can be proved

using (4t) tha t these sufficient conditions are satisfied for r => 2 if we can make

the fur ther hypothesis tha t w -- O (n) as n - + oo. Applying Eq. (3 t) we Pn -- ~n+l have in this case,

={j=~ ~ ' J ~ / } + 1 , / . (42) ¢ = h - l (~) (~2 /) = ~2 D- l (~) [

By Theorem 3.5 the range of P / i s the nullspace of D (Xi) and hence

mt

P/[ = ~ a/k q~:k (43) k=l

where the q~/k's are a set of characteristic vectors corresponding to ;l k Thus we have from (42)

co 2 ~=X ~ (ajlq}il + .. . +aj,~d#j,,~) + / . (44)

Enumera t ing the $:k's as q9 i , 4~2 . . . . and similarly the a/k's we can rewrite (44) as

* = 2 ¢ ~ h ~h*h + / (45) h=l

(and it is still being understood that certain terms are grouped under the summation).

In the case l = t (i.e. the case B = 0) and under the assumption that The- orem 4.6 is valid for r = 1 we can apply Theorem 5.7 to obtain

oo

q~ : h ~ _ i ~ ah qgh + / . (46)

Remembering tha t ~ = 1/v and )t h = 1 [v h the above becomes

co gh

which is Eq. (t0) on p. t t 5 of TRICOMI [7], derived for the same example by the Hilbert-Schmidt theory.

Spec t ra l P roper t i e s of a P o l y n o m i a l Ope ra to r 259

Acknowledgement. The au t ho r s are gra tefu l to H. D. URSELL for m a n y cons t ruc t ive cr i t ic isms.

References

t . DUNrORD, N., and J . T . SCHWARTZ: L inea r opera tors , vol. I. New York : In t e r - science 1966.

2. HILLE, E., and R . S . PmLIPS: Func t iona l ana lys i s and semigroups. Amer ican M a t h e m a t i c a l Society Col loquium Publ ica t ions , 1965.

3. KUMMER, H. : Zur p r a k t i s c h e n B e h a n d l n n g n i ch t l i nea re r E i g e n w e r t a u f g a b e n abge- schlossener l inearer Opera toren . Mi t t e i lungen aus dem Mathem. Seminar Giel3en t 964.

4. LANCASTER, P. : Inve r s ion of l a m b d a mat r ices a n d appl ica t ions to the t h e o r y of non- l inea r v ib ra t ions . Arch ive for Ra t iona l Mechanics a n d Analys is 6, 1 0 5 - 1 t 4 (1960).

5. MOLLER, P. H. : E i g e n w e r t a b s c h ~ t z u n g e n ffir Gle ichungen v o m Typ (A~I - AA -- B) x = 0 . Arch ly der M a t h e m a t i k 12, 307- -310 (196t) .

6. TAYLOR, A. E. : I n t r o d u c t i o n to func t iona l analysis . New York : J o h n Wi ley & Sons 1957.

7. TRICOMI, F. G.: I n t e g r a l equat ions . New York : In te r sc ience Pub l i shers 1957. 8. ZAANE~, A. C. : L inea r analysis . A m s t e r d a m : N o r t h Ho l l and Pub l i sh ing Co. 1960.

Dr. M. V. PATTABHIRAMAN 6-3-1191/6 B e g a m p e t H y d e r a b a d 16 I n d i a

Dr. P. LANCASTER D e p a r t m e n t of M a t h e m a t i c s The U n i v e r s i t y of Calgary Calgary, Alber ta , C a n a d a

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