Integr. equ. oper. theory 36 (2000) 201-211 0378-620X//020201-11 $1.50+0.20/0 9 Birkhfiuser Verlag, Basel, 2000
Integral Equations and Operator Theory
CONVOLUTION EQUATIONS ON FINITE INTERVALS AND FACTORIZATION OF MATRIX FUNCTIONS 1 I. Feldman, I. Gohberg, N. K r u p n i k
Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.
w Introduction. Let us introduce the following notations:
L n x m is the space of n x m matrix-valued functions with entries from L 1 ( - c r oc); ] is the Fourier transform of f E L~xm : ](A) =
F
f(t)ei)'tdt;
O0
W '~x'~ is the Wiener algebra of continuous n x n matrix-valued functions of the form c + ](A), in which c is a constant n x n matrix and f E Lnx~; W~_ xn (W__ nxn) is the subalgebra of W '~x~ consisting of those functions of the form c + ](A) that f ( t ) = 0 for t<0(fort>0). If A is any algebra then G.A will denote the group of invertible elements in' A. The factorization of matrix functions A E G W nxn is its representation in the form
A = A_DA+ where A - E GW~_ xn, and D is a diagonal matrix function
D(A) = diag
~
,...,
~-~
1This research was partially supported by the Israel Science Foundation funded by the Israel Academy of Sciences and Humanities.
202
Feldman, Gohberg and Krupnik
with k; > ... _> kn integers called the partial indices of A. In the case kl . . . . .
kn = 0
so that A = A_ A+, then A is said to admit a canonical factorization. In this paper, the convolution equation on a finite interval
~o(t) +
//
ko(t - s)~o(s)ds = f(t)
is studied. Here f(t) E Ln• a) is a given function, ~(t) E Lnxl(O, a) is the solution and ko(t) E L ~ x n ( - a , a). The following matrix function from G W 2n• (1)
A(A) = [ -e-i~/~(A) t I + k(A)
- I +/r -eia;'/r
] J
where k(,~) denotes the Fourier transform of k and k E L,~xn is any extension of k0 to the real line, plays the main role in the study of the operator (2)
(B(~)(t) = ~(t) +
/o~ko(t - s)~(s)ds.
Since det A(A) = 1, then E kj = - Z kj k~>o kj
scheme
for factorization of a certain class of 2 • 2
matrices. The last section deals with the discrete version.
w
F o r m u l a t i o n of t h e m a i n t h e o r e m s .
T h e o r e m 1. The sum of positive partial indices of A()~) defined by (1) does not depend
on the extension of ko(t) to the real line and therefore the matrix function A(A) admits ~or does not admit) a canonical factorization independently from how the matrix function ko(t) (ltI < a) was extended to the real line. T h e o r e m 2. Let ko($) E Lnx~(-a,a). The operator =
+
~0a k o ( t -
Fcldman, Gohberg and Krupnik is invertible in L~•
203
a) ~ and only ~ the partial indices of the matrix function -I+k(A) -e~~
A(A) = [ - e - +i ~k(A) ^( A ) I equal zero. Let this condition be fulfilled, and the equality
(3)
1 [_r+A+I
A(A)=A_(A)A+(A) = [ I+A~IAg 1
I + A~-2 j
A~k(oc ) = 0
(j,k =
A22
j
1,2)
gives the canonical factorization of A( )~). Denote (4)
~1(~) = &+2(~) + P~
~2(~) = A=+~(~)+ P~
Then the inverse operator B -1 is defined by the equality ( B - l ~ ) ( t ) = ~(t) +
/o ~
7(t, s)~(s)ds,
where ~(t, s) = ~ 2 ( t - s - a) + p 2 ( t - ~ + a)
min(t,s)
(s)
[w2(t - u - a)p2(u - s + a) + w l ( t
-- U -- a)pl(u
--
8
+ a)]du.
max(t,s)-a
This formula is similar to the Gohberg-Heinig formula from [GH2]. T h e o r e m 3. If n = 1, then the kernel of the inverse operator B -1 is defined by the formula ~(t, s) = ~ l ( t - ~) + ~ a ( t - ~ - a)
(6)
min(t,s)
+
/
[w2(t - u - a)wl (u - s) - wl (t - u - a)w2 (u - s)]du
max(t,s)--a
and v(t,O)=~(t--a),
"y(0, s) = ~ox(-s).
Formula (6) actually coincides with the Gohberg-Sementsul formula (see [GF], p. 100).
204
w
Feldman, Gohberg and Kmpnik
P r o o f s of t h e m a i n t h e o r e m s . We need the following statement which is proved in [FGK2] (see also [KF]).
T h e o r e m 4. Let V, V -1, C be some linear bounded operators acting in the Banach space
E, V - 1 V = I and Q, P denote the following projections: Q = V V -1, P = I - Q. Then (i) the operator T = P C P + Q is invertible in E if and only if the operator
M=
C
is invertible in E x E. (ii) If M is invertible and M-1
2 "~- [Mjk]j,k=l
,
then T -1 = PMI~P + Q. Proof of Theorem 1. Let ko(t) E L n x n ( - a , a) and the operator B is defined by equality (2). Denote by k(t) E Lnxn any extension of k0 to the real line and consider in the space Lnx 1(0, o~) the following operators (Rqa)(t) =
; 0,
fO~ k(t -
s)qo(s)ds,
0< t< a
(v.v)(t)
( u _ ~ ) ( t ) = ~(t + a), ~(t-a),
t>a,
Q=UaU-a,
P=I-Q.
The operator B defined by (2) actually coincides with the operator P ( I + R)P. By Theorem 4 the operator T = P ( I + R ) P + Q is invertible if and only if the operator (7)
M = [ U-aI+R
u~O]
is invertible. In addition, it is easy to see that in general (8)
d i m k e r M = dimkerB,
dim cokerM = dim cokerB.
Feldman, Gohberg and Krupnik
205
Since (9)
M = M1MoM2
where MI =
'~ ,
Mo=
I +R
-RU~ J '
M2=
0
I
and the operators M1 and M2 are evidently invertible we get (10)
dimkerM0 = dimkerB,
dim cokerM0 = dim cokerB.
The symbol of the operator Mo is the matrix function A(A) defined by (1) and therefore ill)
kj = dimkerB.
E k~ >0
Since the operator B does not depend on the extension of k0(t), Theorem 1 is proved. Proof of Theorem 2. The first statement of the theorem follows from equalities (9) and
(n), Let A(A) admit the factorization (3) and we now find the operator M -1 where M is defined by (7). The symbol of the operator M is the matrix function (12)
G(A)
o]
[ e-ia;~i
t I + ~(A)
e'~
According to equalities (3), (4) and (9) G(A) = G _ (A)G+(A) =
[G;k]21[G+k]2
where
(13) G_(A): [I+Sh(A) A21(A)
e-"~'I+~2(A)] I+A~_2(A) j,
[ G+(A)=
A+~(A) I+A+I(A)
-I+/51(A) ] e~a~i+~2(A ) .
It is well known (see, for example, [GF], p.197) that the operator M -1 can be found by the formula M -1 = W < , W < l
where WE is the Wiener-Hopf operator with symbol F. Denote =
= [H~j,.
206
Feldman, Gohberg and Krupnik
Then the block/I//12 of the operator M -~ is the following operator MI2 = WH+ WH5 + WH+~WH~~9
(14)
The equalities
[Cj~]l
+ 2
=
G(A)[H~]~
imply that (15)
H+
=
e~aAG~l ,
H~
-_ e - ~ G + 12,
H + = ei~X G -12,
H~
= e - ~ G + 22,
and according to (13),
H~ = e ~ ( I + &~), H~ = e-~"x(-I + f,~),
Hence by (14)
M n = (Ua + We~a~al)(-U-a + W e - , ~ ) + (I + W~.~c~2)(I + W e - , ~ 2 ). Taking into account the equalities PUa = U-aP -- 0, we get
Note that from equalities (15) it follows that
suppwj(t) C [-a,O],supppj(t) C [0, a]
(j = 1,2).
By Theorem 4 the operators B -1 and PM12P coincide and therefore the formula (5) is proved.
Proof of Theorem 3. Since det G_(A) = det G+(A) = 1, then
G$'
L_O2 '
O,,
'
From the equality G+ = G-_IG, it follows that
(16) and from (13) and (16), it follows that /)1 (A) =
--eiaA~b2(A),
[-O~
O~, ]
Feldman, Gohberg and Krupnik
So pl(t) = - w 2 ( t - a),
207
p2(t) = w l ( t - a). If we substitute the functions - w ~ ( t - a)
and w l ( t - a) in formula (5) instead o f p l ( t ) and p2(t), we get formula (6). Directly from (6) it follows that ~'(t, O) = w2 (t - a) +
w2(t - u - a)wl (u)du -
wl (t - u - a)w2 (u)du.
a
After a change of variable u = t - r - a in the second integral we obtain that both integrals are equal. Hence v(t, 0) = w2(t - a). Analogously, we get ~/(O,s)=wl(-s),
7(t,a)=wl(t-a),
~/(a,s)=w2(-s).
From these equalities, in particular, it follows that ~'(t, 0) = ~/(a, a -
t),
~/(0, s) =
~ ( a - s, a )
On the other hand, the functions ~(t, 0) and 7(0, s) evidently satisfy the following equations "y(t, 0) +
/o ~ko(t -
//
"~(0, s) +
r)7(r, O)dr = - k o ( t )
ko(r - s)7(O, r)dr = - k 0 ( - s ) ;
and so (6) actually coincides with the Gohberg-Sementsul formula from [GF]. w
A computational
scheme.
If n = 1 and the function It(A) is meromorphic in the upper (lower) half plane, the factorization of the matrix function A(A) defined by (1) can be obtained in an explicit form. This can be done using the results from [FGK1]. We present in a short form the scheme for the matrix (17)
A(~)=
whereb, c, d c W ,
b=p/q,
q(A)=l-[\
c(k)
d(k)
p(A) E W +
A+i
]
,
Imaj>0,
Emj=m.
j=l
1
We assume that the matrix A(A) admits a canonical factorization. Let us introduce the following notations: P+
c+
(t)ei~tdt
= c+
//
f(t)ei~tdt
P_ = I -
P+
208
Feldman, Gohberg and Krupnik
m
m
ca,xj (18)
/-(~)
=
0
k
E ca,xj ,
h+(,X) =
0
(:, + i)m
1-I(~ - ~j)mj 1
with cj E C, A(A) = det A(A), A(A) = .&_ (A).5+(A) is the factorization of A(A),
/'~ _--1(~) C(/~))~J (j = 0 , 1 , . . . , m ) , E(:' - ~,)m.
1
m
(19)
m
--
m
0
C
+
0
~(~) = h + ( ~ ) - ; ( ~ ) g + ( ~ ) . The function u(A) depends on the constants cj (j = 0, 1 , . . . , m + 1) which have to satisfy the following conditions
(20)
u"(aj)=O
(r=O,l,...,mj-1;
j=l,2,...,k).
The equalities (20) can be considered as a linear system of m equations with m + 2 unknowns cj (j = 0, 1,... , m + 1). The rank of the matrix of this system is equal to m, and therefore m of the unknowns can be expressed via the remaining two. For example, let c 2 , . . . , c2m+l be expressed via Co and cl, then the functions f - ( A ) and g-(A) defined by (18) and (19) can be represented in the form f-(~)
: r
+ cla12(~)
g - (,X) = ~0 a2-~ (~) + ~ 1 % (~)
with some functions ayk(A ) C W_ (j, k = 1, 2). In this case, the matrix (17) admits a canonical factorization A(A) = A_ (A)A+(A) where
A_(A) = [ %- ( ~ ) ] j , k2 = l .
Feldman, Gohberg and Krupnik
w
209
I n v e r s i o n o f finite b l o c k T o e p l i t z m a t r i c e s . In this section we consider the block Toeplitz matrix of the form
(21)
A = [aj_k]j~,k=l
where a s (j = - n +
1,... , n -
1) are m x m matrices with complex elements. We find the
inverse matrix A-1 in terms of factorization of the following 2m x 2m matrix function
A(t) = [ t-'~I
(22)
0
[ a(t)
tnI
]
where n-1
(23) j=-n+l
T h e o r e m 5. The block Toeplitz matrix a
n
A = [ j-k]j,k=l is invertible if and only if the matrix function
L ~(t)
t
admits a canonical factorization relative to the unit circle. Let this condition be fulfilled, and let the canonical factorization be 2 + (t)lj,k=l 2 A(t) = A _ (t)A+ (t) = [A -jk(t)]j,k=l [Ajk
(24) where (25) j=0
j=0
j=0
j=l
Then the inverse matrix A -1 is defined by the equality
A-l=
r................... xn0 0]rznz , Xn-1
Xn
9 9 9
0
0
Zn
zl1
...
Z2
(26) [ Yn +
y~_~
y~
...
...
0] o
Iv~
v~-~
"'"
v~l
Yl
Y2
...
Yn
LO
0
...
vn J
[
0
210
Feldman, Gohberg and Krupnik
Proof. Let f(t) be an arbitrary m • m matrix function continuous on the unit circle and fj (j = 0, : k l , . . . ) its Fourier coefficients. Denote by e2m the space of column vectors of length rn with components from the Hilbert space g2 of all square summable complexvalued sequences and denote by Tf acting in g~ the block Toeplitz operator generated by f (its symbol):
7"f = [fj-k]j,~=l. Consider in the space g2m the following operators: T~ (the matrix function a(t) is defined by (23)), V = Tt,1, V -1 = T w , i , Q = V V -1, P = I - Q. The matrix A defined by (21) actually coincides with PTaP. By Theorem 4, the operator T = P T ~ P + Q is invertible if and only if the operator
is invertible. Since the symbol of the operator M is the matrix function (22), the first statement of the theorem is proved. As is well known, the operator M -1 can be found by the formula
M-~ = TA;,TA:, where A• are factors from (24). Denote A_-1 = [H~]~, A+ 1 = [Hjk]l. + 2 Then the block M12 of M -1 is the following operator (27)
M12 =
THS TH5 + TH+ TH;2.
Just as in the proof of Theorem 2, we get the equalities
H+ = t'A11,
H~2 = t-'~A+,2,
H + = tnA[2,
H ~ = t-~A+2;
and according to equalities (25) and (27)
M12 = TS, TS~ + TI3TI, where
j=0
5=0
j=O
j=O
By Theorem 4 the matrices A -1 and PM12P actually coincide and so equality (26) is proved. Formula (26) is similar to the Gohberg-Heinig formula from [GHI]. If n = 1, formula (26) can be simplified. Such a simplified formula was obtained in [FGK2].
Feldman, Gohberg and Krupnik
211
REFERENCES
[FGK1] Feldman, I. Gohberg, I., Krupnik, N., [FGK2]
A method of explicit factorization of matrix functions and its applications, Integr. Eq. Oper. Theory 18 (1994), 277-302. Feldman, I. Gohberg, I., Krupnik, N., On explicit ]actorization and applications, Integr. Eq. Oper.
Theory 21 (1995), 430-459. Gohberg, I., Feldman, I., Convolution equations and projection methods for their solutions, Transl. Math. Monographs, vol. 41, Amer. Math. Soc, Providence, RI, 1974. [GH1] Gohberg, I., Heinig G., Inversion of finite Toeplitz matrices composed from elements of noncommutative algebra, Rev. Roum. Math. Pures Appl. 19 (1974), 623-663. [GH2] Gohberg, I., Heinig G., On matrix integral operators on finite intervals with kernels depending on the difference of arguments, Rev. Roum. Math. Pures Appl. 20 (1975), 55-73. [KF] Krupnik, N., Feldman, I., On the relation between factorizations and inversion of finite Toeplitz matrices, Izv. Akad. Nauk Mold. SSR, Fiz-Tekh Mat. 3 (1985), 20-25.
[CF]
I. Feldman and N. K r u p n i k D e p a r t m e n t of M a t h e m a t i c s and Computer Science Bar-Ilan University 52900 R a m a t - G a n Israel I. Gohberg School of M a t h e m a t i c a l Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University 69978 Tel-Aviv Israel
MSC 47A79
Submitted: March 10, 1999