DSpace at VNU: Linear equations with polyinvolutions

VNU JOURNAL OF SCIENCE. Nat, Sci,. t XV.

- 1999

L IN E A R E Q U A T IO N S W IT H

P O L Y IN V O L U T IO N S

T ia u T h i Tao
FHcuity o f MHthciiiHtics
Hanoi ưnivcTsity o f Scỉviìce- V N Ư
he (I l i n e a r s pace o v e r c . D e n o t e hy L q ( X )

A bstract.

I.et X

operators A

G L (X

—* .Y) Wifh d o m A =

X

m i d by X

L e t S i , . . . . 5,„ be i n v o l u i i o n s o f o r d e r s 7Í1 , . . . ,

t h e sef o f (ill !nie(ir

s u b a ỉ g eh r a o f L()( A) .

respectively, s a tisfy in g S , . S j ~

S j . S i for any } . j = 1 , 2 ......... ??/. ('onsider the equaiwn
. . . 5 ; ; r . r = ,v.

A{S).r=

(0)

k - 1,IM

e

w h er e

i/ ^

XỤị, —

I n t his p a p e r we p r e s e n t a m e t h o d to r ed uc e t h e e q u a t i o n ( 0 ) to t h e s y s t e m o f
equations without a n y involuHon.

T h e n w e are able to g i v e all soỉutiOTìs o f EquatìOìì

(0) in a close.d f o r w .

1. S o m e fu n d a m en tal p rop erties o f p o ly in v o lu t io n o p e r a to r s
a. An op erato r s G Lq { X) is said to bo an involution of onloi Ì) if .S'” “ / ami .S’^

I
= 1, / ; - 1.
Suppose th a t s is an involution of order 1 Ì then
P. — _ y
It

27T/

^

f = erp —

n

k= \

are callocl projections aysoc’iat(‘cl with

s.

The projections P i , . . . , p,, satisfy th e following proportios:
1. P j P j =

where

is the Kron ock or s y m b o l .

2 . ± r , = /,

.7=1

3. SPr
Hence, it implies th at
S = Y,^'Pr.

5"

ĩ=\

X = 0

Xj,

1=1

w h eio

A', = P j X

7=1

42

(j = Ty>).


L i n e a r E q u a t i o n s w i th P o l y i n v o l u t i o n s

43

b . Lot Si , . . . , Syji be co m m u tativ e involution operators of orders 77.1, . . . , respectively
and Pkj^Uk = 1,7/Ả:) be projections associated with Sk{k = l , m ) . We denote

k = l,m } ,

r ^ {(?■) = (il,
cim
oc(») _ CM

,

— Plji } •'-I Pmj„^ 1 1 ^

s '^'ki

A = { ( jz ,- ..,J r n )|P o ) 7 ^ 0 } ,
27T7
e

P ro p o sitio n .
a) ^

— eJ

VVe have the following relations
p „ , = I.

U)eA

^>) P{ i ) P{ 3) — ị

-f c \ _

I

0 A'(J)

cj ^ =

/

F(J)

{{i)\ =' U ) i k = jk V Ả := l,m )

i f (?:) =

0 );

where X(j) =

0)eA

Tlfc
Proof: a) From ^

= / VẲ: = 1, m an d the com m utativity of Pf^j^ we get
m

ntf.

k = i Jk = ĩ

j f c = T ^ (j)eA
fc= 1. rn

b) It n a t u r a l l y h o l d s b y the c o m m u t a t i v i t y a n d th.e a s s o c i a t i v i t y o f p r o j ec t io n s

c) It is im plied from a) and b)

□

2. T he equation (1) can be re w ritte n in th e from

=

A{S)x=

(1)

(•)€/
where A(.) =
We consider th e eq u atio n (1) under th e assumptions; V P(j), P(^),
G

A, ( 0 €

r, 3

where (j), (Ả:)

€ X satisfying
= ^{r)U)ik)P{k) (o rP (j)^ (,) =

A^)U)wPik))
(fc)€A

. 'f^0)^(«)0)(fc)'f’(í) = 0 V(/)

(k)

Acting on b o th sides of equ ation (1) by P(j), we obtain the system

(2)


TYan Thi Tao

44

Pu) E
{^)e^

= P(j)y

vơ) G A

Y1

^ P{])y V(j) e A

(t)er(Ẳ:)€A

E

E

V0 ' ) € A

=

(i)er{fc)€A

=> E

E

■4(.)o k * ) '''‘" ' ’’ (‘ ) = n , ) ! /

V(j) e A,

(3)

(r)€r(fc)6A
w h ere

3-(fc) = P(k)X € X(fc).

L enim a 1. ỉ f the condition (2) is satisfied, then the eqìiãtỉon (1) has a solution T e X if
and only i f sys tem (3) has a soiỉiủioíi (^(A:))(A:)eA ^

of (1) then (P{k)'^ĩ'){k)eA

Ỵ
(k)eA

^ioreover, if

T

is a solution

^ soiiitioii of (3) Hiid conversely, i f (j'(k)}(k)€A ^

^(A) is a
(*)€A

solution o f (3) then

X

= ^
j:(A:) is a sohition o f (1).
{k)eA

Proof: It is obvious t h a t if the equation (1) has a solution
is a solution of (3).

X

then from (3), {P{k)-^){k)eA

Con ver sely, s u p p o s e th a t t h e s y s t e m (3) has a s o lu ti o n (j^(A:))(ye)6A ^

We

(t)6A
prove that

X

=

^

is a solution of the equation (1).

{k)eA

Iiidet'd, since •'(it) t ^{k)

) t A (Iiicl ^ ( k ) ~
(fc)€A

= -Tịk)- FurtherrnoiP, i ^( k) ) { k) e^

V(J) € A,

5;

^ s o lu ti o n o f (3), which implies th at

= Pu)y

5;

{t)er{k)eA

(r)er(fc)€A

^

J2

(*)er(fc)€A

Y 1 ^ i ’){j)wP{k)S^' ^T = F(j)ị/

(i)e r(/c )6 A

^

= -^0 )^

(t)e r

^0) Z !
(j)eA

=> Y 1

(i)er

^ 0 )2^
(j)eA

Thus,
(fc)GA

is a solution of (1). Lem m a 1 is proved

Y.
(Oer

□

= y-


L i n e a r E q u a ti o n s w ith P o l y i n v o l u t i o n s

45

L e m m a 2. Suppose thnt the condi t i on (2) is satisfied. I f the system (3) has solutions
ãỉKÌ ( /■(ả-))(a-)€A ^
one its solution in the space
soliJtion of (3) in the space

then (Pạ-)-'ĩ'ik))(k)eA

^

(k)eA
Proof : L(*t (-T{k)){k-)eA ^

^ solution of system (3), i. e., v ( j ) 6 A
^(00)(A-)''**'^^’^-'>"(A-) = P u y y

(Oer (Ẳ-)€A
^

^

X]

(i)er(A :)eA

(0er(.)6A

= Puyy

{ i ) eA

(0.A^
jiV

(t)er(A-)eA

Y1 12
( kAc= ,\

Y1

(Ắ'}

( t ) €A

(Oer(A)eA

From Condition (2), it follows th at the left side of the last equality belongs to ker Pịj)
and Its light side belongs to

Note th a t Kor Pị^j) =

Xị,)

^

=> kerf(^) n.Y(^) ^

ij)
{ 0 } we get

^\ỉ)y ~ ^
X!
(0er(A-)6A

^ u

(0€r(Ẳ)€A

i.(\. (^^(A-Ị-í^íAOÌíẢìeA ^ solution of systoni (3) □
Combining two rosults just obtained yields ih(* following
T h e o r e m . Siipposc tiiHt Coiiditioli (2) is satisfied. The eqiiatioii (1) ÌÌH.S sohitioiis if and
only if the svstciii (3) Ììrìs soììitioĩi. Moreover, if .1' is H sobition o f (Ĩ) tiicn {P{k-)-^')(k ) e \
a solution o f (3) and coiiverscly, //"

^ soìĩìtion o f (3) then .1' = ^
{k-)eA

ri sollỉĩioỉi o f ( 1 ).
R e m a r k . I f *4(,) ((;) 6 r) are com m utative with the operators Sk- (A' — 1, 777) then the
system (3) becomes the iiKlepondont system
E
(Oer

= Puyy

VU) G A,

3. E x a m p l e s .
E x a m p l e 1 . Consider the Volterra - C arlem an integral equation of the form


IVtzn Thi Tao

16

ự:>(.r,y,t) ~ y

y

/

A',,(.7:./y. ^ r ) ^ [ f i , ( . r ) , / i , ( y ) ,

r]r/r

= !/(.r.Ịj,t).

(4)

\vh(‘i
1) g{.r. Ị/ , t ). K, i ( . v ,

Ịj,t,T)

aif' c o n t i n u o u s f u n c t i o n s V/ = 1, n,

2) a ( .r ) , f3{y) ai(' Ca r lp m a n t i a n s f o n n a t i o i i s o f or doi
a(A. + i)(.r) = a[QA.(.T)],QA-{.r) /
â ( k + i ) { y ) = d{ í h- { y) ] . f l k- { y) /
3)

y,t,T)

,T
ỉ/

if
if

V

J

= l, ? n.

a n d ni l e s p p c t i v r l y on

1 < Ả-< n,

Q„(:r) = ,r.

1 < A' < i n,

0 „ , { y ) = y.

R, i.e.,

a r e in v a r i a n t u n d e r t h o t r a n s f o r m a t i o n s a { . r ) , f3{y).

W e define the o p erators V , W , A j j E Loi-r) as follows:
{Vip){.r,yJ) = ự>a{.r),y,t

( U » ( . r , i/,0 = ự > \ x j 3 { y ) j ] ,
A,,^){x,y,f) =

/

K^j{x,yJ,T)ip{x,y,T)dT.

./0
Then the equation (4) can be rew ritten in the foini
n

v ,w

We note th a t

771

are com m utative involution o p e ra to rs of order

V and 7Ì1 respec­

tively and the o perators A,J also com m ute w ith V, \V {i — 1, n , j — 1, m).
Denote by

—

l , n ) and

Qị^{ỊJ-

—

1, 77?) th e pro jection s associated with the V,

w

respectively.
From the above obtained results, in order to s tu d y of E q u a tio n (4), we can s tu d y
the system of the independent equations

-

/

M^f,{T,y,f,T)ip^,,{T,y,T)dT = g^f,{x,y,t)

V ;/=

1, n , / i = 1 ,7 » ,

If)
n

wheiT

rfi

1= 1 7=1
27T?
f i = e x p ------,

n

27T/

f2 = exp

rn

E x a m p l e 2; Consider th e Fredholm - Carleinari integral e q u atio n of form
n

2

ip{x

K^j { x, t , T) t f [ a^{. T) ,

Pj{T)]dr

= g{xj),

t = l j = l ‘' - i

where
1) g { x , y , t ) and K i j { x , t , r ) are continuous functions X e R ; t , r e [ - 1 , 1
2)

q (t )

is Carlem an transform ation of order n.

3 )/? (V --^

02Ìt) = m t ) )

= t^

We define th e operators V, w , Ai j as follows:

(5


L inear Equations w ith P o lyin vo lu tio n s

47

{Vip){x,t) =
{ Wự>) { x , f ) =

-t),

Ị

{A, j i p ) { x , i ) =

V?: = T7n; j =

K,j{x,f,T)^{x,r)dT,

1, 2.

T h en the e q u atio n (5) can be re w ritte n in the form:

We get
V/" = / ,

^

yịy ^

Denote by p ^ ( u = 1, 7?) the projections associated with V and

Qi = ị n - W ) ,
e = exp

Q^ = ụ i + W),

2ni
—

n

.

We prove th at the condition (2) is satisfied. Indeed, p uttin g
] ^

^

” k = l 1=1
then

f P l' Q s-^7 j p u Q r — ^ijiysf^rf^^Qr‘>
\ Pi/Qs ^ijusfir PoQi)
v.4,;(/ =

^ 1, 2)

and

V P^,P^(/v,/i

T h ( ' 1 ont IB t o n l i o w t h a t

1
^

"

—0

if

^

or

r; /

r

1,7?,) Q s . Q r { s . r = 1,2).

a iv tlir iiitcgiul o p c ia to iy . w\- have

“

rl

^

/

K , , [ n f , . { : T ) j 3i { t ) , T ] < f ị . r , 0 2 - l Ì T ) ] d T

” k = \ 1=1
Putting

Ơ = /32_ ;( r )

r = Pi{ơ),fÍT =

the light side OI
of rn
thee lasT
last eequality
q u a l ity ccan
a n aalso
lso be w rritten
i t t e n as
2n

'

P u tting

/
A-=l /-1

K,Aa,{x),m ,Pi{cTM x,a)da.


Tran Jhi Tao

48
We get

Thus, instead
id of studying the equation (5) we can stu d y the following onv

- Ẻ Ề Ê Ẻ i

i 1 1 (-

2= 1 J = 1 ^ = 1 r = l

A-=l i = l

X

(6)

X K i j { a k { x ) , l3 i ( f ) , P i { T ) ] i p ^ , r { : r , T ) d T

- P^Qsgi^J)

V i/= l , n ; s = 1, 2

Equation (5) has solutions if and only if the system (6 ) has solutions. Moreover, if
s=\

2

solutions of (6 ), then
TỈ

2

l/=l S=1
is a solution of (5).

REFERENCE
1] Nguven Van Mail. Generalized algebraic elernents and Linear singular infecral equa­
tions with t r a n s f o r m e d arguments. Warsaw 1989.
2] D. P. Rolewicz. Equations in linear spaces. A m sterdam - Warsaw 1968.
3] D. P. Rolewicz. Algebraic Analysis. A m sterdam - Warsaw 1987.
T A P C H Í K H O A H O C D H Q G H N , K H T N , t.xv, n ^l - 1999

P H Ư Ơ N G T R ÌN H T U Y Ế N TÍN H V Ớ I CÁC T O Á N T Ử ĐA P H Ố l HỢP

Trần T h ị T ạo
Khoa Toán - Cơ - Tin bọc
Dại học Khoã học Tựnbièn - Đ H Q G HàNội
X là m ộ t khòrig g i a n t u y ế n t ín h trên t r ư ờ n g c . L o { X) là t ậ p tất cả rác t o á n tư

tuvến tính A trẻn X với dom i4 = X . X Ik đại số con của Lq{ X) . Giả sử 5 i , . . . , Sn, là
các toán tử đối hợp cấp r ?i , . . .

tư ơ n g ứng đòi một giao hoán với nhau.

Xét phương trình

t ^ —1, »1

fc=(l ,m)

trong đó
e X{ i k =

l,Tik,k

= l , m)

và x, y € X .

Nội dung của bài báo này là đ ư a phương trìn h (*) về hệ p h ư an g trình không còn
toán tử đối h ạ p m à tính giải đ ư ợ c của nó khả thi h ơ n nhiều, đồng thời cho mối liên hệ
giữa cấu trúc nghiệm của p h ư a n g trìn h (*) vái cấu trú c nghiệm của hệ đó.