O P E R A T I O N A L C A L C U L U S O:F L I N E A R U N B O U N D E D

O P E R A T O R S A N D S E M I G R O U P S

G . I . L a p t e v

The theory of continuous semigroups of ope ra to r s (both in Banach and a lso in local ly convex spaces) is suff icient ly wel l detai led (see [1]-[5]). In [1] an example was given of an a b s t r a c t Cauchy p rob lem with an ope ra to r which genera tes an unbounded semigroup , and the d e s i r e t o cons t ruc t a genera l theory of such p rob l ems was exp re s sed . In the p r e s e n t pape r is p roposed a scheme for the invest igat ion of p rob l ems of such type which is based on the const ruct ion of a v a r i a n t of operat ional calculus (a m o r e genera l approach has been proposed by the author in [6]).

A closed ope ra to r a is cons idered in a locally convex space P for which (k-a) "1 is defined in some sec to r of the complex plane. Then a s e t is chosen on which an opera t ional calculus can be cons t ruc ted for a definite c lass o f functions (§ 1). A m o r e r e s t r i c t ed set p o s s e s s e s the s a m e p rope r ty that the r e s t r i c t i on of the opera to r a to this se t is the genera t ing ope ra to r of an analyt ic semigroup of continuous ope ra to r s in the cor responding topology (§2). In §3 this fact is applied to the quest ion of the solvabi l i ty of the c o r r e s - ponding a b s t r a c t Cauchy p rob l em. In §4 the Cauchy p r o b l e m connected with the so -ca l l ed dis t r ibut ion s e m i - groups of exponential type is c o n s i d e r e d [7-11]. In addition, s o m e r e s u l t s of [8]-[11] a r e sharpened and ampl i f ied . As an example , the Cauchy p rob l em for c o r r e c t (in the sense of Pet rovski i ) s y s t e m s of par t ia l di f ferent ia l equations with constant coefficients is s tudied in §5.

:§1 . O p e r a t i o n a l C a l c u l u s

L e t P be a local ly convex countably complete l inear topological space and let a be a c losed ope ra to r ac t ing in i t with d o m a i n D(~).

We let Z denote the open s ec to r s of the complex plane which contain the negative rea l axis and a r e s y m m e t r i c with r e spec t to it; Z = {~: larg (k -c r ) l> 0, a > 0 , 0 < 0 <Tr}. ~, will a lways denote the c losed complement sec to r : ~-' = {k : l a r g (k-a)~ ~ 0}.

Now, in the sec to r 2~ 0 (with p a r a m e t e r s a 0 and 00), let an ope ra to r function J k be defined which s a t i s - fies the following condit ions.

1% For each k ~ r 0 t h e ope ra to r JTt ac ts in the space P and is c losed.

2°°

3 ° .

If an e lement Jkx has been de te rmined , then it belongs to D(t~) and

(~ - - a) Jxx = x. (1.1)

If the function gxx has been defined at the points X, v fi ~0, then it s a t i s f i e s the Hi lber t identity

J s ~ - - Jvx =- (v - - ~)JvY~x (~, v ~ ~o)- (1.2)

T h e se t of s ec to r s ~ c ~0 for which cr< ~r 0 and 0 > 00 will be denoted by ~/0.

Let H0 be the s e t of al l e lements x E P for which the function Jkx is defined and analy t ic in the sec to r ~0 both for each ~ ~ 1I n and continuous s e m i n o r m p{x),

sup(l-t71 ~, I) P(Jxx) -~ Mp (x) < ~ . : (1.3)

Computing Center , Latvian State Univers i ty . Trans la ted f r o m + Funktsional 'nyi Analiz i Ego P r i l o - zheniya, Vol. 4, No. 4, pp. 3 ! -40 , Oc tobe r -December , 1970. o r ig ina l a r t i c l e submit ted F e b r u a r y 17, 1969.

© 1971 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 ~]est 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission of the publisher. A copy o[ this article is available from the publisher for $15.00.

290

We will cons ider the family Y of numer ica l valued functions (g(X) of the complex va r i ab le X, each of which is analyt ic and sa t i s f i es the following inequali ty in some sec to r l , (22 ~ ~I0):

I~ (X) l<c l~ l -~ (Xe X'). (1.4)

Definition. fo rmula

Let the function ~(D ~ ~'. Then the ope ra to r ~0(a) is defined on e lements of H0 by the

~p(a)x = -~ t / I ~ (~,) J~xd~. ,

F

where the contour F of the s ec t o r Z, is t r a v e r s e d so that Z ' r ema ins on the teft.

By the inequality,

p(~p Ca) x) ~ c ~] L [-x (1 -t- I kl)-lMp (x) l d~l F

the in tegra l (1.57 converges absolu te ly and is independent of the choice of Z 6 ~ro.

We note that the o p e r a t o r s ~p(a) cannot be continuous in P in the g e n e r a l ca se .

THEOREM 1.1. Let an e l emen t x E H0, let the functions ~P, @ E ~ ' , and let or,/3 be a r b i t r a r y c o m - plex numbers . Then

a) q~(a) x (:Ho, @(a) x (:Ho;

b) ~ + ~Vp ~ ,9" and (aq~ + ~BC#) (a) x := atp (a) x + [~Kb (a) x;

c) (p*(..~ and (~,)(~)x =~(a) , (a )x .

vary . is chosen so that i ts ve r t ex a and genera t ing angle 0 sa t i s fy the inequali t ies a t < a < a 0 and 00 < 0< 01. Then

lv-~l-f-<c(z+[~l)-* (v~.,~,~r).

Here c is independent of u and X for fixed r. and F .

Let us cor~ tder the function

r

I t c a n be r ep re sen ted in the fo rm:

X(a)x:=_~t/ i ~ ( k ) , . t,i~x--.&,xl dk (v E ~), P

since the added t e r m is analyt ic in the sec to r ~-' and its in tegra l vanishes . By the Hi lber t identity,

(.)x = jJ:d X F

By assumpt ion the ope ra to r Jv is c losed. The re fo re it can be taken out f r o m under the in tegral s ign. This gives X(a)x = Jv cp(g)x.

F r o m the integral r ep re sen ta t i on of the function X(a) i ts analyt ic i ty in each s ec to r Z follows and thus in x 0. The inequali ty

p (x (,,)x) ~< ' ! - [ sup I 'v - - ~ ' l - ' ~ - x t xer ) I [ ~ (~')1 p (3~'x) I d~'l" I "

is obtained f rom this s a m e represen ta t ion .

By (1.7), sup (l + ] v l) p (x (a) x) < ~o, i .e . , the e lement (P(a)x belongs to the se t H0. This p roves pa r t (a) of the t heo rem, z P a r t (b) is obvious; (c) is proved in the usual manner by using the Hilber t identity (1.2).

(i .5)

(1.67

Proof . We will show that ~(a) x E H0. Let us choose a s ec to r Z l E~10 in wh ich the p a r a m e t e r u will Le t the boundary of this s e c t o r be defined by the equation a r g (} , -a ) =~ 8. The contour F in (1.57

(1.7)

291

~§2. T h e S p a c e o f A n a l y t i c i t y o f t h e R e s o l v e n t

Le t H denote the se t Qf al l e l emen t s x E P for which

1) the funct ion Jxx is def ined and ana ly t ic in the s e c t o r Z0;

2) in e a c h s e c t o r Y. E~I 0,

lira 7~J~x = x (XE~). (2.1)

I t follows f r o m (1) and (2) tha t for any cont inuous s e m i n o r m p(x) and s e c t o r ~- EU0, the fol lowing quant i ty is f inite:

h (x) = sup (1 q- [ ~. 1) p (Y~x) = sup [p (J~x) d- p (~J~x)]. (2.2)

We note that since J~x is analytic in Z 0, the function p(J~x) + p(7~JXx) is subharmonic, and therefore

, the seminorm h(x) can be computed by the formula

h(x) sup(l+l~,[)p(J~x) , (2.3) ~,EI'

w h e r e F is the con tour bounding the s e c t o r r..

I t fol lows f r o m the condi t ion (2.1) and the def ini t ion (2.2) tha t

p(x) <~ h(x ) . (2.4)

THEOREM 2.1. The se t H with the s y s t e m of s e m i n o r m s h(x) of the f o r m (2.2) is a countably c o m - p le te , loca l ly convex space .

P roo f . It is obvious that H is a s e p a r a b l e loca l ly convex space , and t h e r e f o r e it is suf f ic ient to p rove its c o m p l e t e n e s s .

Le t Xn be a Cauchy sequence in the space H, i .e . ,

sup (1 + I ~ J) P (J~x, . - - & x , . ) = h (xn - - x , . ) - - . O.

for n, m --~ ~o for a fixed s e c t o r Z and s e m i n o r m p.

I t f o l l ows . f rom (2.4) that the sequence Xn is a l so a Cauchy sequence in the space P . By the count - able c o m p l e t e n e s s of this space , an e l emen t x = l im Xn ex i s t s . We will show that x is the l imi t of the s e - quence Xn in the topology of the space H and that it be longs to this space .

I t follows f r o m (2.2) that fo r each ~ E ~',

p (J~xn - - J~,xm) "~ h (xn --Xrn) "--* O, p (~,J~,xn - - #~J~.Xm) ,~ h ( X n - Xm) "--'>0. (2,5)

By the f i r s t of these re la t ions the sequence J~xn is Cauchy in the space P for each ~ E ~'. Since the o p e r a t o r J~ is c losed , yx = l im J?~xn =J~x.

Since the e s t i m a t e s (2,5) a r e un i fo rm wi th r e s p e c t to ~ E Z, it fol lows that fo r a g iven s e m i n o r m p, a pos i t ive in teger N ex is t s such that for a)l n > N and for ~ E Z,

p (J~xn - - J~.x) ~ 1 and p (~.J~.xn - - ;kl~.x) < 1.

Adding these two inequal i t ies and taking the m a x i m u m with r e s p e c t to X, we obtain h ( x n - x ) -<2, i .e . , the sequence Xn c o n v e r g e s to the e l emen t x in the topology gene ra t ed by the s e m i n o r m s (2.2).

The l imi t s J x x and ~Jxx of the un i fo rmly conve rgen t sequences will be ana ly t i c funct ions in each s e c t o r Z E ~I0 and thus a l so in al l of E 0.

F ina l ly , i f [XJ is taken suf f ic ien t ly l a rge , we have pC~J~nx-xn) < 1 f r o m (2.1) for each e l e men t o f the sequence Xn. If we r e g a r d n > N as b e f o r e , then f r o m the p r eced ing inequal i t ies we obtain

p (~J~x - - x) ~ p (~Jxx - - ~.Jxx.) + p (~.J~x. - - x,,) + p (x,, - - x) , ( 4.

But this means that the e l emen t x, in fact , s a t i s f i e s (2.1). T h u s x E H and the t h e o r e m is p roved .

292

THEOREM 2.2. For each v E Z0the o p e r a t o r Jv maps the space H into i t se l f . F o r each2; E ~I~ and for each s e m i n o r m h which is cont inuous in H,

l imh (vJvx - - x) = 0 (v ~ Z) (2.6) M~o

and a s e m i n o r m h'(x) ex i s t s which is cont inuous in H such that

h ( J , ~ x ) < (1 + Iv D-~h' (x) (v ~ Z). (2.7)

P roo f . Le t x 6 H. Since the funct ion Jkx is ana ly t i c , and f r o m the Hi lbe r t ident i ty (1.2), it follows that the funct ion Jk(Jvx) is ana ly t i c in ~. Fo r the e l emen t Jhx, (2.1) wil l follow at once f r o m (2.6), by (2.4). T h e r e f o r e we tu rn to the p r o o f of (2.6) and (2.7). I f the s e c t o r Z 2 is conta ined in the s e c t o r Z 1, then a c - co rd ing to (2.2), the inequal i ty hz2(x) - h ~ l ( x ) wil l hold for the c o r r e s p o n d i n g s e m i n o r m s . Thus , fo r the p r o o f of {2.6) and (2.7), it is pos s ib l e to r e s t r i c t o u r s e l v e s only to s e m i n o r m s h computed a c c o r d i n g to (2.3) with the c o n t o u r s F suf f ic ien t ly c lo se to the bounda ry of the s e c t o r Z 0" A s e c t o r Z 1 E ~I 0 is chosen such that it conta ins the s e c t o r ~-, and such that i ts boundary F 1 is loca ted at a pos i t ive angle to the bounda ry of the s e c t o r Z, and at a pos i t ive d i s tance f r o m it. Then, f o r the points u E Z and k E F l,

[ ~ _ v [ - x < { : (1+ l ~t)- ' , (2.8)

(~+Ivl) -~. This a l lows us to use the Hi lber t ident i ty in the fo rm:

We have

J~Jvx : : i Jxx + t _ ~ ~ - ~ J~x.

Using (2.3) and (2.8) we obtain at once that the f i r s t t e r m does not exceed chz(x) and that the second t e r m does not exceed chz l (x ) . Denot ing the s u m of these two s e m i n o r m s by h ' (x) , we obtain (2.7).

For the p r o o f of the l imi t r e l a t ion (2.6) we use the ident i ty

to obtain

h (vJ~x - - x) - sup ~ p (vJvX - - ~.J~x). i ~ - - v !

Le t us a s s u m e that the r i g h t - h a n d s ide does not vanish when v E ~ and I v ] ~ . Then, sequences Vn and Xn (n = 1, 2 . . . . . ) ex i s t fo r which t Unt - ~ and

l + l ~ , l [~n--Vnl p ( v , J v x - - ~ J ~ , x ) > 6 ~ O ( n - 1 , 2 . . . . ). (2.9)

Since x E H, p(VJux) - cons t and p(kJkx) -<const . T h e r e f o r e , the f a c t o r (1 ÷ tknl) lkn-Vnl - I in (2.9) is bounded below for al l n = 1, 2 . . . . . whence [Xnl -~oo for n ~ o o . But then fo r the second f ac to r , by (2.1) we have for n --..co,

p ( v ~ J ~ x - - ~.nJ~,x) ~ p ( V J ~ X - - X) + p (~nJ~,x - - x) --> O.

In addit ion, (1 + [ XnI) [kn'Vn1-1 r e m a i ns u n i f o r m l y bounded above by (2.8). By the s a m e token (2.9) c a n - not be sa t i s f ied and this con t r ad ic t ion p r o v e s the t h e o r e m .

The t h e o r e m jus t p roved m e a n s the fol lowing (see [4], p. 301): the r e s t r i c t i o n of the o r ig ina l o p e r a - t o r ~ to the space H has a dense domain of def ini t ion and cont inuous r e s o l v e n t in the s e c t o r Z0 for which the fami ly of o p e r a t o r s kJk is equicont inuous in each c losed s u b s e c t o r ~ ( ~ E ~I0).

§ 3 . T h e C a u c h y P r o b l e m

We will explain the connec t ion of the in t roduced se t H with the Cauchy p r o b l e m

d--Lx -- - - a x (t), x (0) = xo. (3.1) dt

293

L e t us r e c a l l t ha t a w e a k s o l u t i o n of the Cauchy p r o b l e m on [0, ~o) is a func t ion x(t) wi th v a l u e s in P, con t inuous on [0, o0), and hav ing a s t r o n g d e r i v a t i v e w h i c h s a t i s f i e s (3.1) on (0, oo) ( see [1]).

We a s s u m e tha t the c l o s e d o p e r a t o r a d o e s not h a v e e i g e n v a l u e s in the s e c t o r ~.0 wi th p a r a m e t e r s a 0 and 00 < 7r/2. Then the o p e r a t o r ( X - a ) -1 is de f ined in th i s s e c t o r and i t m i g h t not b e a con t inuous o p e r a - t o r . L e t JA ( X - a ) -1. Th i s o p e r a t o r i s c l o s e d a s the i n v e r s e of a c l o s e d o p e r a t o r , and i t s a t i s f i e s t he r e l a t i o n ( A - a ) J X x = x by de f in i t i on . If we de f ine Jkx and Jr (x) fo r s o m e e l e m e n t x, then in the i den t i t y x = (X-v ) JAx ÷ ( v - a ) J x x , the o p e r a t o r J v can be a p p l i e d to the l a s t t e r m s , and then i t can be a p p l i e d to the whole r e l a t i o n , which g i v e s the H i l b e r t i den t i t y . By the s a m e t o k e n the cond i t i ons 1°-3 ° § 1 a r e s a t i s f i e d f o r the o p e r a t o r s JA, and the s p a c e H can b e c o n s t r u c t e d in t e r m s of t h e m .

L e t ~-+ deno t e the s e c t o r {z: ~ a r g z l < ( T / 2 ) - 00}.

T H E O R E M 3.1. L e t the o p e r a t o r a have no e i g e n v a l u e s in the s e c t o r x 0. Then the w e a k Cauchy p r o b l e m (3.1) has a unique s o l u t i o n on [0, ~o) which a d m i t s an a n a l y t i c con t i nua t i on in the s e c t o r x + wi th the e s t i m a t e

p ( x ( z ) ) ~ c e -~°ReZ, c : c ( x o , argz , p) , (3.2)

i f and on ly i f x 0 EH.

P r o o f . L e t x(t) b e a s o l u t i o n wi th the p r o p e r t i e s s t a t e d in the cond i t i ons of the t h e o r e m . L e t

o o

d~x° : - - S e~tx (t) d t (Re ~. < %). (3.3) o

Since the so lu t i on is unique fo r a g iven x0, the o p e r a t o r Jxx0 is de f ined un ique ly . We i n t e g r a t e b y p a r t s (us ing (3.1)) to ob t a in

oo

• ~.x~l 1 Jxx o := - - -~ ~., e~tax (t) dt.

o

Since the o p e r a t o r a i s c l o s e d , i t fo l lows tha t jAx 0 E D(a) and ( X - a ) J A x ~ = x 0.

Thus Jxx0 c o i n c i d e s wi th the r e s o l v e u t of the o p e r a t o r a . I t fo l lows f r o m the e s t i m a t e (3.2) tha t t h i s funct ion can be a n a l y t i c a l l y con t inued in the s e c t o r ~0 i f in (3.3) we i n t e g r a t e o v e r the r a y s a r g z = ~ , ~0 [< ( t r /2)- 0. By the s a m e token , x 0 s a t i s f i e s the f i r s t cond i t i on in the de f i n i t i on of the s e t H. We v e r i f y (2.1) for i t . L e t X = - r , w h e r e r > 0 . Then

oo

~,J~x o : r J" e-~tx (t) dt. o

Since x(t) is bounded and con t inuous a t z e r o ,

lira L/xx = A- lira x (t) = x (0).

Now, (2.1) fo l lows for a n y s e c t o r ~ f r o m known t h e o r e m s of P h r a g m e n - L i n d e l S f t ype .

By the s a m e token the n e c e s s i t y o f the cond i t ions of the t h e o r e m is p r o v e d .

Now we p r o v e the s u f f i c i e n c y . L e t x 0 E H. I t fo l lows f r o m a s s u m p t i o n s m a d e in the p r e c e d i n g s e c - t ion tha t the r e s t r i c t i o n of the o p e r a t o r a to the s e t H g e n e r a t e s a s u b g r o u p e - z a in th i s s p a c e which i s a n a l y t i c in the s e c t o r 2] + , and is s t r o n g l y cont inuous a t z e r o , wi th the e s t i m a t e

h (e-Zaxo) ~< cz -*° Re ~ h' (xo), c = c (arg z).

By (2.4), the funct ion x(z) = e-Zax0 wi l l be a n a l y t i c in ~.+ and s t r o n g l y cont inuous a t z e r o , i f we c o n - s i d e r i t to be a funct ion wi th v a l u e s in the o r i g i n a l s p a c e P . In p a r t i c u l a r , th i s funct ion g i v e s the s o l u t i o n o f the Cauchy p r o b l e m (3.1). I t s u n i q u e n e s s fo l lows f r o m the fac t t ha t t he i n t e g r a l s (3.3) v a n i s h i d e n t i c a l l y fo r a s o l u t i o n with a z e r o i n i t i a l cond i t ion . Hence x(t) ==-0.

Th i s p r o v e s the t h e o r e m .

294

§ 4 . A n a l y t i c D i s t r i b u t i o n S e m i g r o u p s

Let P be a countably complete , local ly convex space and l e t a be an opera tor acting in it which sa t i s - fies the following conditions

1) The opera tor a is closed.

2) In some sec tor Z 0 of the complex plane which contains the negative real axis, there a re no eigen- values of the opera tor .

3) For all k E ~0the operator Jk = (X-a) -1 is weaker than the opera tor an for some integer n >- 0, which by definition means that: D(J~) ~ D(an).

4) For each sec to r Z E ~I0 and seminorm p, a s eminorm p ' exists such that for all x E D(an),

p (J~x) ~ (1 + l ~ [)-i ~ p, ~akx) (~. ~ ) (4.1) k = 0

The se t D{an) can be considered to be a l inear topological space with the sy s t em of seminorms

qp(x) : ~.~ p(akx) (x6D(a")). (4.2) k ~ O

As has been established in the preceding section, the operator Jk has the proper t ies 10-3 ° of § 1. It follows f rom (4.1) that D(an) c H0 and that

Mp (x) .~< qp, (x). (4.3)

Thus, it is possible to const ruct functions of the opera tor a on the set D(an) and, in par t icular , f r ac - tional powers and semigroups generated by them. By (1.6) and (4.3), these functions will be continuous opera tors f rom D(a n) into the original space P.

The following theorem gives the connection (introduced in the second section) of the space H with known sets .

THEOREM 4.1. The space H contains the region D(am) beginning with m = n + 1 as a dense subject of it.

Proof . Let x E D(an+l). In view of the identity k J k x - x = Jk (ax) for each closed sec tor Z E~I 0,

n + l

p ( a t ~ - x) .< (1 + I ~. I) -1 ~ , p' (a'x) ~ o. k = l

for Ixl - ® .

This means that x E H, i .e. , that D(a n+l) c H. Fur the rmore , for any x E H, according to Theorem 2.2, h (h lkx-x ) --* 0 for I kl -"* ,o.

Since the element Jhx E D(a), the set D(a) N H is dense in the space H in the topology of the latter, whence by induction the whole space D(a m) N H(m = 1, 2 . . . . ) is dense. According to previous r emarks , these sets coincide with D(am) beginning with m = n + 1, which completes the proof of the theorem.

In conjunction with Theorem 3.1, this gives the following theorem.

THEOREM 4.2. Let the operator a sat isfy the conditions (1)-(4), where the genera tor angle 00 < 0r/2) for the sec to r Z 0. Then the Cauchy problem (3.1) has one and only one solution for every element x0 E D(an+l). This solution admits an analytic continuation into some sec tor Z+ which contains the positive rea l axis, and continuously depends on the initial data in the following sense: if xg k) E D(a n+i) and if x~ ~ 0 in the topology of the space D(an), defined by the seminorms (4.2), then the corresponding solutions xk(t) --* 0 in the topology of the original space P.

Here the continuous dependence on the initial data (in the stated sense) follows f rom the imbedding inequality

n

h (x) = sup (I + I ~. I) p (,I~x) < y, p' (a~x) = q (x).

295

Now we recal l the necessa ry and sufficient condition [8]-[10] that the operator a generate an analy- tic distr ibution semigroup of exponential type in the Banach space E: D(a) is dense in E and in each closed subsector ~- of some open sec tor Z0 bounded f rom the right by the contour a rg (X-~ 0) = ~v 0 < (7r/2), and the inequality ]l(k-a)-lU-< c ( l + IA}) n-I (n > I; X q x) is sat isf ied.

, For conve~aence we assume that ~0 <0 . For x ~ De.n), the inequality ~ IX-.)-txl l ~ - ~ , ~ llak.ll

I " I k__ 0

(x E D(an)) follows f rom the identity

(~,--a) -1 x = i x + t---ax + . , . + t.-~--a"-Ix + l.-~-(Z,--a)-ta"x X X 2 ~n £n

which permi ts us to apply the resul ts obtained above (see [8]-[11]).

§ 5 . C o r r e c t ( i n t h e P e t r o v s k i i s e n s e ) S y s t e m ( a n a l y t i c c a s e )

Following [1] (Ch. 1, §8), here we will consider ~ - theory and assume that a Fourier t r ans fo rm with respect to the space variable is a l ready applied to the sys tem. Here the matr ix of differentiation of the sys tem turns into the opera tor of multiplication on an m-dimensional m a t r i x - a ( p ) , whose coefficients depend polynomially on n pa rame te r s p = (Pl . . . . . Pn). The Cauchy problem takes the form

d--L = - -a (p )x ( t , p), x(O, p) = xo(P). (5.1) dt

We introduce the notation

A (x , p ) = I x - - a {P) l = x ~ - - s l (o) : - ' - . . . - - s~ ( p ) ( 5 . 2 )

for the charac te r i s t i c polynomial of the mat r ix a(p).

Here the numerica l coefficients sk(p) equal the sum of the principal minors of k-th o rde r of the mat r ixa (p ) . For the augmented matr ix , which sat isf ies the relation (X-a(p))b(k, p) = A (X, p), the follow- ing representa t ion is well known (see [12]) :

b(~., p) = x m-1 + ~:- 'b~ (p) + . . . + b.,-~ (p), (5.3)

where

bk (p) : : a k (p) - - s. (p) a k - l (p) - - . . . " s k (p). (5.4)

Now let us introduce a fundamental res t r ic t ion. We will assume that the eigenvalues Xl(p) . . . . . Xm(p) of the matr ix tz (p) lie in the sec tor t a rg (X-~0) I < 0o < (7r/2) (~0 >0). The complementary sec tor to this open sector will be denoted by x 0. If we choose a sec tor £ E~I 0, then for X E Z" each factor in the representat ion

A (~., p) = (L - - ~1 (p)) . . . (j, - - L,n (p)) (5.5)

will sa t i s fy the two inequalities

IX -X~ (p ) l>c ( !+ l~ l } I x - - xk(p)l> clX~(p)l. (5.6)

B y V i ~ t a ' s T h e o r e m , Sk(P) = ( - -1 ) k ~, X,, (p) . . . £;k(P)- i,<...<i/~

For fixed p we choose the maximal in modulus t e r m in this sum. For definiteness let this be Am-k+l(P) • • • Xm(P). Then

Isk(p)l ~<cl ~Lk+~(p)l . . . IZ~ (P)I-

If we write the product A ' l (k , P)Sk(p) in the form

( x - - x, (p))-' .. ( x - x~_k (?))- ' • 'k ( ' ) 0 , - x=_~+~ ( p ) ) . (~ - ~= (p))

and apply the f i rs t of the inequalities (5.6) to the f i rs t group of factors and the second of the inequalities (5.6) to the second group of factors , then we obtain the est imate

I A- ' (~, p) sk 09) 1 "<~ c (1 + } ~. D~=-k):

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Now it follows f r o m (5.4) that

< tlx @)ll.,- i ta-~(x, p)b~,(p)x(p)ll. < c I la '@)~(s') l l , , , + -~ l i a r - ' @)~(P)II,,, + " " + <~+1~.1),,,_ k ( l + I t~ 1)" ( l + I ~, I) ' ' - ~

The index by the sign of the n o r m shows that the n o r m under cons idera t ion is in an m-d imens iona l space .

Since (~-a(p)) -1 = A-t(2t, p)b(X, p), it follows f r o m (5.3) that

C C C I[(~.-a(p))-~x(p)ll,~<~ltx(p)ll~ + ~ [ ; a ( p ) x ( p ) ; l m + . . . 4- - - - ila~-'(p)x(p)l~. (5,7) (1-t- I ~. I) m

I f x E D(am-1), then in the space ~ (Rm) we finally obtain ttv following inequality f r o m (5.7):

C C C

II ( ~ - a) -~ xll < , + I ~1 II x II + ( , + I ~ 1) ' It a x II + - . . 4 - - Ila~-~x/\- (5 .8) ( t - l - I ~' t)"

Thus the conditions (1)-(4) of the p reced ing sec t ion hold for the case under considera t ion. We can apply T h e o r e m s 3.1 and 4.1 for the semigroup e - t a . In p a r t i c u l a r , we obtain

THEOREM 5.1. Let the e igenvalues of the m a t r i x a ( p ) lie in the sec to r l a r g (X'-cr 0) I - 0o < (rr/2). Then the Cauchy p r o b l e m (5.1) will have an analyt ic solution for al l initial values x 0 ~ D(am), where m is the o rde r of the s y s t e m .

We note that the inequality (5.8) is s t r onge r than the inequali ty (4.1), which pe rmi t s us to obtain addi- t ional informat ion (see [6]). Thus, the Cauchy p r o b l e m (5.1) will have a solution for a l l x E D(am-t) , and for sma l l , pos i t ive t ,

II e -'ax II "-<< c (11 x II + t II a%il + - - . + tm-" II a m-~x li).

In conclusion we cons ider such an example . Let the ma t r i x a(p) have the fo rm,

Obvious ly the ope ra t o r and its r e so lven t define unbounded ope ra to r s in the space L2(-**, **) X L~(-*% **): Since a 2 = 0, the genera t ing subgroup has the f o r m e - ta = l - t a , whence it is seen that the ope ra to r e - t a is defined for a l l t > 0 on D(/) and only the re .

The author thanks S. G. Kre in for d i scuss ion of the resu l t s and constant in te res t in this work.

L I T E R A T U R E C I T E D

1. S . G . Krein , L inea r Different ia l Equations in Banach Space [in Russian] , Nauka, Moscow (1967). 2. E. Hille and P. Phi l l ips , Functional Analysis and Semigroups [Russian t rans la t ion] , IL , Moscow

(1962) . 3. M . A . K r a s n o s e l ' s k i i , P. P . Zabre iko , E. I . Pus ty l 'n ik , and P. E. Sobolevskii , In tegra l Opera to r s in

Spaces of Summable FUnctions [in Russian], Nauka, Moscow (1966). 4. K. Yosida, FUnctional Analys is [Russian t rans la t ion] , Mir , Moscow (1967). 5. N. Dunford and J . Schwartz , L inea r Opera to r s . Genera l Theory [Russian t rans la t ion] , IL, Moscow

(1962). 6. G . I . Laptev , "On the theory of opera t ional calculus of l inear unbounded o p e r a t o r s , " Dokl. Akad.

Nauk SSSR, 8_~5, No. 4, 760-763 (1969). 7. J . L i o n s , " L e s semigroupes d is t r ibut ions ," Por tyga l . Math. , 1._99, 141-164 (1960). 8. D. Fuj iwara , "A cha rac te r i za t ion of exponential dis t r ibut ion s emig roups , " J. Math. Soc. Japan, 18,

No. 3, 267-274 (1966). 9. G. DaPra to and U. Mosco, "Sernigruppi dis t r ibuzioni anal i t ic i , " Ann. Scuola Norm. Super . P i sa , 19,

367"396 (1965). 10. G. DaPra to and U. Mosco, "Regolar izzazione dei semigruppi dis t r ibuzioni anal i t ic i , " Ann. Scuola

Norm. Supe r .P i s a , 1_.99, 563-576 (1965).

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11.

12.

G. DaPrato, "R-semigruppi analitici ed equazioni di evoluzione in LP," Ricerche Mat., 16, No. 2, 233- 249 (1967). F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).

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