On the spectrum of cosine operator functions

0378-620X/89/050713-1251.50+0.20/0

Integral Equations and Operator Theory Vol. 12 (1989)

(c) 1989 Birkh~user Verlag, Basel

ON THE SPECTRUM OF COSINE OPERATOR FUNCTIONS Carlos

Lizama

In t h i s paper we c h a r a c t e r i z e the spectrum of strongly continuous cosine functions, d e f i n e d in a H i l b e r t s p a c e , in terms of p r o p e r t i e s of the infinitesimal generator. INTRODUCTION Throughout cosine

function,

generator

A.

this

defined

p(A)

For f u n d a m e n t a l

is

facts

work C ( t ) ,

teR

in a H i ] b e r t

the resolvent on s t r o n g l y

is

a strongly

space H, w i t h set

and

o(A)

continuous

continuous

infinitesimal

the

cosine

spectrum.

functions

see

[2,8,11,12] The p r o b l e m o f

determining

some k n o w l e d g e of t h e o p e r a t o r (see E I , 3 , 7 , 9

] ).

known L9] to

sion

fails

the

[2,9]

).

second o r d e r

In p a r t i c u l a r , (*)

1~ p ( C ( 1 ) )

several



of

this

equation

was p r o v e d

that

{ -4~'n'}

in

for

the class general

of

result

a certain uniformly is

true:

kind

Hilbert

2 p ( A ) and

class

of

in c o n e c t i o n

u":Au+f

neZ and t h a t ,

by many a u t h o r s

are cases where t h e r e v e r s e

results

differential

it

given

o(C(t))

but t h e r e

(see

o(C(t)),

A, has been s t u d i e d

hold,

In ~ I ]

spectrum

The i n c l u s i o n :

ch{ tVo--~CE}-- } c is

the

incluwith

are p r e s e n t e d .

spaces we have:

Supln(4~'n'

+ A)-I~O 6 ~ c h

that

~6 + {e 2w6 + T } ~ / z } ,

6>0

chAe p ( C ( 6 ) ) . PROOF. Let s>O and ~eC f i x e d . We define: =

ews

+

4e2ws

+1

,

~

=

-

ews

+

~e2ws

+1

for

aach

, we h a v e

Lizama

715

Observe t h a t

leRep and t h i s

~-~

~B= I and

e-Re~ I > 2e ws

_

yields:

(3)

Ich~l.

e ws
I and w>O such

i ! M e ws

and t h a t f o r a r b i t r a r y r e a l numbers t o , t I . . . . . . t n and S l , S 2 , . . , s n with O!to~t1~ ..... ~tn:t, O~sj~tj-tj_ I ( j = 1 , 2 . . . . . n ; t n = t ) we have: IC(Sn)C(Sn. I) .... C(Sl)C(to)

~ ~ M e wt

(see [ 2 ]

t h e n , d e n o t i n g by r t h e s p e c t r a l r a d i u s of C ( s ) , and s j = s ( k = 0 , 1 . . . . . n ; j = 1 , 2 . . . . . n) t h a t : (5)

r(C(s))

Combining

(3)

p.60)

we have f o r

tk:kS

< e ws.

and (5) t h e

lemma i s p r o v e d .

The f o l l o w i n g lemma shows t h a t i t i s a l w a y s p o s s i b l e t o translate the conditions in (2) on t h e r e s o l v e n t of A, from t h e strip S t = ( ~ e C / IReul ! i n ( e W t + ~ e2Wt + I )} t o i t s complement. LEMMA 2 .

(6)

Let

[+-~ + 2 ~ i n / t }

(7)

Supl(•

~ G C and

f

n~Z

+ 2~inlt)[

c

-

p(A)

t>O f i x e d

and

6uppo6e

and

[+-~ + ~ i n l t )

z - A

]-II

nG Z

then,

there

exist6

reC

and

Tn eL(H),

n e Z

6ueh

that:

< +|

that:

716

Lizama

(8)

chTt

(9)

(•

~ p(C(~)) 2~in/t)'

(I0)(•

c p(A)

neZ

-

211in/t)[(•

+

+ 2~in/~)'-A

for as163ne Z. Moreover, PROOF.

]-I = Tn ( • 2 4 7 2 4 7

-I

Sup ITn i 0 such t h a t

E > (I/t)In(e

wt +#f e 2wt + I )

and d e f i n e : (11)

9 = ~ + i

then by lemma 1, Now, by (1)

(8)

holds.

we have: r

(12)

{(•

+ 2~in/t)2-A} = (•

holds.

•

(13)

Pn :•

sh{(•

2~in/t)s}

C(t-s)x

ds =

0

+ 2~in/t){

+ 2~in/t

Finally,

t

| J

Note t h a t

Im(u)

# 0 for

ch~tx - C(t)x} all

n~Z, then

, neZ , xeH. by (8)

we have t h a t

(9)

we d e f i n e : + 2~in/t

;

~n =+z-

+ 2~tn/t

, nGZ.

We have: (14)

pn(p~ - A ) - I : ( p n / ~ n ) [ ( ~ - P ~ ) ( P ~ - A ) - I + I ] = n ( ~ - A )

-I

Define: (15)

Tn

then using (11) i t are bounded. Then, and (15)

imply

(10)

:

(~n-

Pn' / ~ n )pn(P~ - A) -1 + Pn/~n

i s easy to see t h a t { p n / ~ n } n e Z a n d { ~ n - p ~ / ~ n } n e z (7) i m p l i e s t h a t S u p I T n l < +| . T h e r e f o r e (14) and the p r o o f

isn~mplete.

Lizama

717

Let peC and t>O f i x e d

(16)

F(p,s)x

= Vt/2

and d e f i n e

e ps C ( t - l s l ) x

the function: ; IslO, x~H,~GC. =+" -|

PROOF. D e f i n e Xn=-:~+ 2 ~ i n / t and ~n=~ + 2 ~ i n l t f o r a l l i s known t h a t f o r f u n c t i o n s f , g G L 2 ( ] O , t [ ; H ) t h e Parseval's h o l d s in the Form

neZ. I t formula

Ft

+|

Jo I f ( s ) x l '

(17)(I/t)

ds : -~X

rt

( 1 8 ) ( I i t ) ] 0 Ig(s)xl' then,

(19)

with

f(s)x

ds

:

+~ -| Z

rt

=_|

hand,

ds~'

sin(2~ns/t)g(s)x

dsl ~ ,

0

and g ( s ) x

: ch(ps)C(t-s)x

tt

ds +

t JO ~ c h ( ~ s ) C ( t - s ) x

the following

sh(knS)

+ sh(PnS)

sh(XnS)

-

Then u s i n g

(19)

sh(PnS)

I'

we have

ds :

=

cos(2~ns/t)

2 sh(Ps)

[ {I -|

law,

," neZ , seR.

we have

t

0

(1/2)

hold

sin(2~ns/t)

and the p a r a l l e l o g r a m

I Ish(.s)C(t-s)xl'

t

identities

= 2i c h ( ~ s )

t

=

cos(2~ns/t)f(s)x 0

os(2~ns/t)sh(~s)C(t-s)xds~' +_.Z ~ 0sin(211ns/t)ch(ps)C(t-s)x ds~ '

On t h e o t h e r

(20)

t

?

I(1/t)

= sn(ps)C(t-s)x

Jo l s h ( ~ s ) C ( t - s ) x i '

t

i

~(I/t)

ds + t I Ich(ps)C(t-s)xl'

s h ( ~ n S ) C ( t _ s)x d s l ' 0

0

ds :

+ I rt s h ( P n S ) C ( t - s ) x d s l ' } 0

718

Lizama

Now, left

of

(20),

and s h ( ~ s )

(21) -|Z ~ and t h e

using

the

and a p p l y i n g

- ch(~s)

that

the

: e -~s

i>oo

result

fact

-an

:

identities

~-n

for

sh(~s)

all

neZ i n

+ ch(~s)

:

the e ~s

we o b t a i n :

s

ds~ 2 = (t/2){

~e~Sc(t-s)x~'ds + 0 ~e- ~Sc(t-s)x~'ds}

follows.

MAIN RESULT.

THEOREM 4. Le~ H be a H i s of a 6 ~ r o n g s fos (22)

eontinuou~

6tatement6

co6ine

6 p a c e and A t h e g e n e r a t o r

function

O(t),

t~R.

Then t h e

arc equivas

{ e p(C(t))

(23) { 4 ' /

eh~t

= ~ } e p(A) -

and

Sup Ix(x ' - A)-II ~hxt=~

::MO and observe t h a t the s o l u t i o n s of the equation ch~t={ are of the form: (24) Xk =-# + 2 ~ i k / t By lemma I ,

, ~k =

# + 2~ik/t

we can assume t h a t

We d e f i n e F(#,s)

as in (16),

; keZ , DeC.

IReptl < In(e wt + ~e2wt + I ).

then, there e x i s t s constants e>O and

0>0 such t h a t : (25)

(I/~2)IC(t)xl ' ~

?

~F(~,s)x~' ds s 621ch~tx - C ( t ) x l 2, xeH.

t

In f a c t ,

in order to prove the i n e q u a l i t y on the l e f t

in (25),

observe t h a t D'A1embert's f u n c t i o n a l equation gives: C ( t ) x = 2 C(t) C ( t - s ) x - C ( t - 2 s ) x By i n t e g r a t i o n over [ O , t ]

,

s e R , xeH.

and using the f a c t t h a t C ( t )

is even,

we

Lizama

719

we o b t a i n : [

t C(t)x : 2 I

J

t

c(t)c(t-s)x 0

ds-

rt I C(s)x ds J0

, x~H

then, using (4) and HOlder's i n e q u a l i t y , we have t h a t there e x i s t s a constant KI>O such t h a t : t (26)

IC(t)xi'


0

(25).

the

following

two c a s e s :

# 0

We h a v e : (34)

Ichptx~ 2 - ~C(t)x~' = 2 Re< chptx, chptx - C ( t ) x > - ~ch~tx - C(t)x~ 2 _< 2 Ichptx~ ~chptx - C(t)x~ - Wchptx - C(t)x~'

Hence a c o m b i n a t i o n

(35) Ichptxi 2 < ~' -

< ~'

B'IchlJtx

of

i

t

(25)

and

IF(u,s)xI'

(34)

yields:

ds +

Ichptxi'

Ic(t)xl'

t

- C(t)x I'

+ 21chptx ~ Ichptx-C(t)x~-~chptx-C(t)x~

~

Lizama

721

then (35) O~ (~2B'

implies

that:

- 1)~chptx-C(t)x~'

+ 2 i c h p t x ~ ~ c h p t x - C ( t ) x ~ - ~ c h ~ t x l 2.

Hence: (36)

Jchptx-C(t)x~

~ (I + e B ) - 1 1 c h ~ t l

~x~

Now, we d e f i n e C*(s) := ( C ( s ) ) * . Then, i t f o l l o w s [ 9 ] t h a t C*(s) is a s t r o n g l y c o n t i n u o u s cosine f u n c t i o n w i t h g e n e r a t o r A*. Hence a s i m i l a r argument i m p l i e s : (37)

~ ( c h ~ t - C ( t ) ) * x i ~ (I + ~B) -11ch~t I ~x~

I t f o l l o w s from (36) and (37) (see [ 6 ] p.92) and the p r o o f is complete in t h i s case.

Note t h a t (38)

CASE 2. chpt : O. D'Alembert's functional

that

{= chpt~ p ( C ( t ) )

equation gives:

0 ~ p ( C ( t ) ) •

e

p(C(t/2)).

On the Other hand, we obtain from the i d e n t i t y : (I/2)chpt = (ch(pt/2) -VZ/2)(ch(pt/2) + ;~/2) (39){p'/chpt=O}

= {p'/ch(pt/2)=#E/2} U {p2/ch(pt/2)=-#~/2}

Now, by h y p o t h e s i s : {p'/ Then (39)

chpt:O}

c p(A)

and

Suplp(p' chpt=O

A)-II