Integr Equat Oper Th Vol. 25 (1996)

0378-620X/96/010073-21 $1.50+0.20/0 (c) Birkhauser Verlag, Basel

ON CANONICAL FACTORIZATION RATIONAL MATRIX FUNCTIONS

OF

I. G o h b e r g a n d Y. Zucker

This paper concerns the problem of canonical factorization of a rational matrix function W(A) which is analytic but may be not invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.

0. I n t r o d u c t i o n The problem of canonical factorization for a rational matrix function W(•) with respect to a bounded contour when W(oo) is invertible was solved in the book [BGK] (see also [GGK]), where the necessary and sufficient condition for the existence of canonical factorization and explicit formulas for the factors were given. The case when the matrix W(c~) is not invertible is more involved, it was treated in [C]. The purpose of the present paper is to obtain a more complete analog of the theorem of canonical factorization for the case of general matrix W(oo) which contains explicit formulas for factors in terms of the given data. This problem is solved in Section 1. In Section 3 the problem of symmetric canonical factorization for rational matrix functions selfadjoint on the unit circle is considered. Section 2 concerns rational matrix functions selfadjoint on the unit circle and has a preliminary character. In Section 4 the canonical factorization of rational matrix functions positive on the unit circle and analytic at infinity is considered. It is our pleasure to thank M.A. Kaashoek for usefull comments and discussions.

1. C a n o n i c a l F a c t o r i z a t i o n Let F be a bounded Canchy contour with A+ and A_ as inner and outer domain. Assume that 0 E A+ and ~ E A_.

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Gohberg and Zucker

Consider a rational m x m matrix function W(A) without poles on r which is analytic (but not necessarily invertible) at oo. As is well known (see [BGK]) such a function can be represented in the following form (1.1)

W ( A ) = D + C(AIn - A ) - I B ,

A q[ a ( A )

where 0 = (A, B, C, D; C n, C m) is a finite dimensional node and A has no spectrum on P. Moreover, the node 0 can always be chosen to be minimal and in that case the eigenwalues of A coincide with the poles of W. The representation (1.1) is called the realization of W(A). In this paper we look for explicit Wiener-Hopf canonical factorization of W(A) with respect to F in terms of the representation (1.1). The following theorem is the main result of this section. THEOREM 1.1. Let W ( A ) = D + C( AIn - A ) - I B be a rationaJ m x m matrix function, where A has no spectrum on P and det W(A) has no zeroes on P. Let P = ~

(A - A ) - l d A

be the spectral projection of A corresponding to A +. Then the operator (1.2)

px = p_

~

(A- A)-'BW(A)-~C(&-

A)-ldA

is a projection and W ( A) admits a right canonical factorization relative to F if and only if

(1.3)

C~ = ImP+KerP x .

//'(1.3) is satisfied and H is the projection ofC n Mong I m P onto Ker p x , then a right canonical factorization of W relative to r is given by W(A) = W_(A)W+(A), where

(1.4) (1.5)

W+(A) = D + CII(A - A ) - I B , W_(,~) = I + C(A - A ) - I ( I - H)(B + R ( D - I))

and

(1.6)

R = -2zr---~

(A - A ) - I B W ( A ) - l d A .

In the proof of Theorem 1.1 it will be shown that the formulas (1.2) and (1.6) can also be rewritten in the following form (1.7)

P• = 2--~ i f r [I, 01 [ A - M Cn

(1.8)

n = ~

[z.

o]

B]-I[/~]

D

Im

dA .

Gohbcrg and Zucker

75

One can also give an alternative formula for W_(A) which uses the projection H only, namely -1

B

CH

"

The proof of this formula will be presented at the end of the proof of Theorem 1.1. In the proof of this theorem we use the result of [GK] which deals with the realization of W(,~) in another form, namely

W(A) = I + C ( A G - A ) - I B , where the pencil )~G - A is invertible on F (F-regular). The result is as follows. THEOREM 1.2 [GK]. Let 9 be a rational m x m matrix function without poles on the contour F, and let ~ be given in realized form:

~(~)=I+C(~G-A)-aB,

(1.10)

~9

Put A x = A - BC. Then ~ admits a right canonical factorization relative to I' if and only if the following two conditions hold true: (i) the pencil (G - A x is F-regular, (ii) C" = / m H 1 4 K e r I I ~ and C" = ImH~4KerII~. Here n is the order of the matrices G and A, and H1 = ~

( ( G - A)-~Gd~,

II2

H i = 2~r---i (~G - A•

=

r

.

relative to F is obtained

= r

O_(~)=I+C(~G-A)-'(I-p)B,

~6F, ~6F,

O_(~) -1 = I - C ( I - r ) ( ~ G - A • 0+(~)-1 = I - C ( ~ G - A •

A)-ld( ,

II~ = 27ril f r G(~G - A•

In that case a right canonical factorization r by taking

(1.11) (1.12) (1.13) (1.14)

1 frG((G-

~eF, (9

Here r is the projection of C n a l o n g / m i l l onto KerH~ and p is the projection along ImH2 onto KerH~. Furthermore, the two equalities in (ii) are not independent, in fact, the t]rst equality in (ii) implies the second and conversely.

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Gohberg and Zucker

PROOF OF THEOREM 1.1. The first step in the proof is to rewrite W(A) in the following form

and to put

'

0

Xm

'

Im-D

"

Then we obtain a representation (1.16)

w ( ~ ) = xm + 6 ( ~

- ~)-1~,

~ e r.

Since a(AG - ,4) = a ( A ) , we have A

a(lG- A ) :

(1.i7)

r = 0.

Now Theorem 1.2 gives necessary and sufficient conditions for the existence of canonical factorization for W(A). Note first that condition (i) of Theorem 1.2 for the pencil (1.16)

~(~

.4•

-

n r = 0

is equivalent to the invertibility of W(A) on F ([GK], Theorem 4.2). Since we assume that det W(A) has no zeroes on F, this condition automatically holds. Next using (1.15) we compute that

0

I

(I - O)C

Since AG - ,~x is invertible on F, we conclude that [ A -CAI

BD] is also invertible on

F. Next consider condition (ii) of Theorem 1.2. We obtain

1

[0

Gohberg and Zucker

77

where P = ~x f r ( AIn - A ) - l d A is a spectral projection of A with respect to A+. Using (1.19) we find

II~ = ~ / f r ( ~ G - ]~x)-lGd~ =

(1.20)

=2,,,rill[-/ -I0] [A-cAI B]-l[/ ~]dA=

i ~ [I 0] [A-hi

= 2--~

C

B] -1

C

D

[~ 00] [~• 0] 0

ds =

where P x and Q are some operators. Since II~( is a

projection

F•

'

we have

--

I Thus

(1.21)

(px)2=px,

Qp•

=Q

.

From the first equality follows that P x is a projection. Recall that condition (ii) of Theorem 1.2 in the considered case has the form (1.22)

ImII15cKerH~( = Cn+ 'n .

One easily sees that

Taking into account (1.21) we obtain

KerII~=Ker[~ • 00]=Ker[QPp•215 00]= __

Therefore condition (1.22) can be rewritten in the form (1.23)

[t~P]-~ ([Ke0PX] + [C0m])--C

n+m

and hence it is equivalent to the condition ImP-]-KerP x

=

C" .

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Gohberg and Zucker

The projection P• can easily be computed from (1.20). Indeed p•

0].2~fr[/

~] [A c A I

,00]~ [0,]

DB]-I [0

=

~ / , , 0,[~, ~] '[~]~ This gives the formula (1.7). To obtain formula (1.2) for p x note that

[A?,. ~]= [C,A" ~,.) 1 o] [Ao~,~ O+C/~,n0 A) 1~][ ~"0

(A - a•

]

Im

j,Acr

and hence

[A c M "

(1.24)

[~

Im

0

D -1 =

C(AI, - A) -~

W(A) -1

Im

'

"

It follows that

0 i

[~A0

~,~) 1] [~,~'A, 1

i

-----~ ~ ( A -

AI)-' + ~ ~ ( A - A)-IBW(A)-Ic(A - A)-ldA =

= P - ~

(A - A)-IBW(A)-IC(A - A)-IdA

and formula (1.2) is obtained. Let r be the projection of C "+m along IrnII1 onto KerH~. Denote by H the projection of C ~ along ImP onto KerP x. Then from (1.23) we see that (1.25)

r =

[~ 0] Im

"

Consider now the dual condition in (ii) of Theorem 1.2, namely (1.26)

ImII2+KerII~ = C n+m ,

Gohberg and Zucker

79

where

I15 = ~ 1

fr ~ ( ~

- ~)-ld~,

II~' = ~

1 fr ~(~ _ ~•

Using (1.15) and (1.19) it is easy to see that in this case H2 = Ha and

[; 00][

II~ = ~ f 1r

--C

-0I

][

A -C AI

dA =

(1.27) / fr[0/ = 2-~

01 0 [A-A C/

B ] -a dX = [PX D

0R] ,

where P• is the same operator as in (1.7) and R is an operator acting from C 'n to C n. From (1.27) it is easy to compute that

This is the formula (1.8). compute that

Now using the decomposition (1.24) we can immediately

R -

2,~i

(~ - A)-IBW(a)-ld~

which gives the formula (1.6) for R. Since (H~)2 = II~ we obtain from (1.27) that p x R = R. Next we find KerH~ = I m ( I - II~') = I m (1.28) -

I m [In

--

P•

[

In - P• 0

-R] Im

=

-R

--/{

0

Hence the condition (1.26) is equivalent to the condition (1.29) which is the same as (1.30) Since the subspace

consists of the zero vector only the second sum in (1.30) is always a direct sum. Therefore the condition (1.30) is equivalent to the condition (1.3). Assume now that this condition

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Gohberg and Zucker

is fulfilled and let p be the projection of C "+m along I m I I 2 onto Ker II~. Then from (1.29) we see that for any x E C n and any y E C rn there exist x l E I m P and x2 E K e r P x such that

(:)

and hence Xl + x 2 = x + R y . So z2 = II(x + R V ) where II is a projection of Cn along I m P onto K e r P x . It follows that X

and therefore p has the form

(1.31)

P=

n 0

(n-

I.)R 1 I,. J

and I - p has the form (1.32)

[ I , I- I 0

I-p=

(I, OII)R]

Next we deduce formulas for the factors from formulas (1.11) and (1.12) of Theorem 1.2. Using (1.12), (1.15) and (1.25) we find

w + ( ~ ) = zm + ~ r ( ~

- ~)-l~

= I m + [ - C , I,,1 = I,n + C I I ( A - A ) - I B

=

I,,

0

-Im

Im

- D

+ D - I,n = D + CII(A - A ) - l B .

Similarly, using (1.11), (1.15) and (1.32) we compute that W _ ( A ) = I,n + C ( A G - A - ) - * ( I - p ) B =

-- r,~ + c ( a - A)-I(X, - II)(B + R(D - I , . ) ) .

=

Gohberg and Zucker

81

As was mentioned above one can also give an alternative formula (1.9) for W_(A) which uses the projection II only.

To obtain this formula we first compute

W_(A) -1 from the formula (1.13) of Theorem 1.2. Taking into account (1.19) we find

W-()~) -1 = I m

-

C( I - r)( AG

-

./~X)-l~ =

--,-+ tc~.-~),o1[~-~ ~]_1[o.-Zm]

"

From the last formula using the general formula for the inverse of the function of form (1.10) ([GK], Theorem 4.2) we obtain

w_(~) = I n +

01))-1[o

~ ([~

+ [c(I. - n),o] (~ [ I~ Since

[A

B

B = [ CII + D C ( I .

- H)

=

0

we have

w_(~,) = i , . +

+ [c(z. - n),0] [ - C ( I . z. - H) = Im-

[ C ( I , - II), 0] [A - A/~

[ vii

o](~[~ 00][:~ g]) 1[o~ ]: g] '[o~,.]

This proves the formula (1.9). The theorem is proved. Condition (1.3) and formula (1.7) for projection P• were earlier obtained in Theorem 8 of [C] using a different approach. Consider now the ease when the matrix D is invertible. Let us show that in this case the formulas of Theorem 1.1 coincide with the formulas of Theorem XIII.6.1 in [GGK]. As was mentioned in [C], if D is invertible the following identity holds

[a~ ~1: [~ o][~0 ~ ol][5 o o]

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Gohberg and Zucker

where A • = B D - 1 C .

Substituting this identity into the definition (1.7) of p x it is easy

to see that px=

2 lr i ~r (AIn - AX)-ldA

is just the spectral projection of A x with respect to A+, i.e. this is the same p x as in Theorem XIII.6.1 of [GGK]. Since formula (1.4) for W+(A) is exactly the same formula as in [GGK], we have only to analyze formula (1.5) for W_(A). Let

2--~

(1.33)

=

'

where PX(A) and R(A) are some matrix functions. From the definitions (1.7) and (1.8) of P x and R it is clear that

From (1.33) it follows that + R(

)D =

O .

Integrating this identity over F we find pXB+RD=O.

Hence, if D is invertible we obtain (1.34)

R = -pXBD-1

.

Taking into account that I - H = ( I - H ) P • we see that in this case W_(A) = I m + C(A - A ) - l ( I n - I I ) ( B + R ( D - I n ) ) = = I+ C(A- A)-I(I-

II)P•

pXBD-I(D-

= I + C()~ - A ) - I ( I - I I ) ( p x B - p x s

= I + C(A- A)-I(I-

I)) =

+ pXBD-1)

=

H ) p X B D -1 = I + C ( A - A ) - x ( I -

This formula for W _ ( A ) coincides with the formula for

W_()~) of

I I ) B D -1

Theorem XIII.6.1 of

lOCK]. The formulas (1.4) and (1.5) can also be obtained by the following limit procedure which was suggested by M.A. Kaashoek. Let W(A) = D + C(AI,, - A ) - I B

be

Gohberg and Zucker

83

a rational m x rn matrix function, where A has no spectrum on the bounded Cauchy contour F and det W(A) has no zeros on F. Consider the function F(s

-A)

-~ -(AI,-A)-~BW(A)-IC(AIn

-A)

-~ .

If D is invertible, then F(A) = ( h i , - A• -~, where A • = A - B D - 1 C

(see L e m m a

XIII.5.3 in [GGK]). Hence in that case (1.35) P •

1/

2~ri

(A-AX)-ldA = P-

~

F and (1.36) P •

-1 -

1/

()~I,~-A)-~BW()~)-IC(M,-A)-ld)~

F

1/

2~i

(()~ - A ) - I B D - 1

- ()~ - A ) - I B W ( ) O - 1 C ( ) ~

- A)-IBD-1)d)~

=

r = 27r---~

()~ -- A ) - I B D - I

- ()~ - A ) - I B W ( ) O - 1

(W()Q - D)D -1

d)~ =

1/ r

=

2,%5

-

r

We will use (1.35) and (1.36) to give an alternative way to derive the formulas (1.4) and

(1.5). Let E ~ 0 be sufficiently small such that D + eI,., is invertible. Set W,(A) = D + el,,, + C ( A I n - A ) - ~ B , A•

= A-

B(D

+ eIm)-lC

.

Without loss of generality we m a y assume that det We(A) # 0 for A E F. This allows us to write (hi,, - AX(e)) -1 = ()~In - A ) - 1 - ( A I n - A ) - I B W ~ ( ) ~ ) - I C ( A I ,

- A) -1

and hence P•

1/

= P - ~i

(hi,, - A)-IBW,(~)-IC(AIn

- A)-ld~,

F px(E)B(D+Eim)_

1 = 27r--~ 1 / ()~ - A ) - I B W * ( $ ) - I d $ F

"

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Gohberg and Zucker

Let p x be defined by formula (1.2) and R by formula (1.6), then it is clear that lim P •

~---*0

= p•

and lim P x ( e ) B ( D + sire) -1 = - R . e~0

Now assume that (1.3) holds. Then C n = ImP3rKerP•

for e suffciently small, and

hence We(A) admits a canonical factorization for e sufficiently small, =

where + CH,(A - A ) - I B ,

W+,e(A) = (D + r

W-,e(~) -- I + C(.~ - A ) - I ( / -

H,)B(D +

Here He is a projection of C n along I m P onto KerP•

r

-1

.

Taking limits when r I 0 we

get W(A) = W_(A)W+(A) for A E F, where (1.37)

W+(A) = D + CI'I(A - A ) - I B ,

(1.38)

W_(A) = !im (Ira + C(A - A ) - I ( I - I I , ) B ( D + d i n ) -1) = = limn (Ira + C(A - A ) - I ( I - II,)P•162 0

+r

-1) =

= I m - C(A - A ) - ' ( I - I I ) R and H is a projection of C n along I m P onto K e r P •

From (1.37) it is easy to see that

W+,~(A) converges to W+(A) uniformly on F and hence because of m a x i m u m module principle W+,,(A) converges to W+(A) uniformly on F + . Therefore W+(A) is analytic in F+ and continuous up to the boundary, i.e. has no poles in F + . From W = W_W+ we see that det W+(A) r 0 for A e F. Since [argdet W+,,(A)]r = 0 and converges is uniform, we conclude that [argdet W+(A)]r = 0. Hence det W+(A) has no zeroes in F+. So, W+(A) has no poles and zeroes in F + . Similarly, W_(A) has no poles and zeroes on F _ and hence formulas (1.37) and (1.38) give a canonical factorization of W on F. It remains to remark that as was proved above

P•

=0 ,

and hence the formula (1.38) coincides with the formula (1.5). Indeed I+ C(A- A)-I(I-

H ) ( B + R(D - I)) =

= I+ C(A- A)-'(I= I-

C(A- A)-I(I-

H)(P•215 H)R.

+ RD) - P•

=

Gohberg and Zucker

85

We end this section with the following corollary of Theorem 1.1. C O R O L L A R Y 1.3. /s both the operators I + P - p x and I - P + P• are invertible then W(A) admits a right canonical factorization with the factors given by (1.4) and (1.5). P R O O F . Indeed, from the invertibility of I + P - P • it follows that ImP+ KerP • = Cn and from the invertibility of I - P + P x we see that I m P N K e r P • = (0) . Hence I m P J c K e r P x = Cn and the statement of the corollary follows from Theorem 1.1. 2. R a t i o n a l M a t r i x Functions Selfadjoint o n t h e U n i t Circle The rational matrix function is called selfadjoint with respect to the unit circle if

Consider a rational m x m matrix function W(A) which is analytic at oo. Let W(A)=D+C(AI,-A)-IB be a minimal realization of W(A). In Section 3 we will use the following proposition. P R O P O S I T I O N 2.1. Let W(A) = D + C(A - A ) - I B be a minimal realization of a rational m • m matrix function. Then W(A) is selfadjoint with respect to the unit circ/e if and only if there exists a (unique) invertible selfadjoint matrix H such that (2.2)

A'HA=H,

C=iB*HA,

D=D*+iB*HB.

For the case when D is invertible this statement can be found in [LRR], Section 3, p.787 and for the case D = I in [GLR1]. P R O O F . Suppose that (2.1) holds. Then

({)"

x-01imW(A) = 1 ~ W i.e. A = 0 is not a pole of W(A).

= D*

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Gohberg and Zucker

Since the realization is minimal, A is invertible. It follows that W(A) = W

= D* + B*

- A*

C* =

= D* + A B * ( I - A A * ) - I C * = = D* + B * A * - ~ ( A A * - I + I ) ( I -

AA*)-Ic

* =

= (D* - B * A * - ~ C *) - B*A*-~(A - A * - ~ ) - I A * - t C *

is another minimal realization of W(A). Hence it is similar to the realization W(A) = D + C ( A - A ) - i B and the similarity is unique. Therefore there exists (a unique) invertible matrix H1 such that D=D*

-B

.

A .-i C , ,

C = -B*A*-IH1,

A=H~IA*-IHi

B = H~IA*-I C * .

This is equivalent to: D = D* + C A - 1 B and (2.3)

A * H 1 A = H1,

C =-B*H1A,

C = B*H~A .

Note, that (2.3) can be rewritten in the following form (2.4)

A*(-H;)A=-H;,

C=B*(-H;)*A,

C=-B*(-H;)A.

Comparing (2.4) with (2.3) we find that (-H~*) is also a solution of (2.3).

Since the

similarity is unique, we have H1 = - H ;

.

Denote H = ill1. Then H* = H and A'HA=H,

C = iB*HA,

D = D* + i B * H B .

The converse statement is easily checked if we take into account that the first equality in (2.2) implies that A is invertible. Note, that the first identity in (2.2) means that A is H-unitary, i.e. A is unitary in the indefinite inner product [x, y] = < H x , y > on C n given by H. Here < -,- > denote the usual inner product on C". For the theory of indefinite inner products and H-unitary and H-selfadjoint operators and matrices we refer to [GLR2]. The last identity in (2.2) can be rewritten as

Gohberg and Zucker

87

It means that the imaginary part of matrix D is equal to 89

and the real part can

be chosen arbitrarily.

3. Symmetric Faetorizatlon In what follows we denote by F a unit circle. Let W(A) be a rn x rn selfadjoint on the unit circle rational matrix function which does not have poles and zeroes on the unit circle and is analytic at infinity (but not necessarily invertible). Such a function can be represented in the form (3.1)

W(A) = D + C(A - A ) - ~ B ,

)~ q~ a(A) ,

where A has no eigenvalues on the unit circle and there exists an invertible selfadjoint matrix H such that (3.2)

A'HA=H,

C = iB*HA,

D = D* + i B * H B .

Note, that for the minimal realization these properties are always fulfilled. Here we do not require the minimality. Given such a realization, we look for a symmetric canonical Wiener-Hopf factorization (3.3)

W(A) = W+

XW+(A),

A9 F

where W+(A) is a rational matrix function, which is analytic and invertible on the closed unit disc and X is a constant selfadjoint invertible matrix. L E M M A 3.1. Let P be a spectraJ projection of A with respect to the open unit disc and let p x be a projection defined by (1.2). The following identities hold (3.4)

H P = ( I - P)*H

(3.5)

HP • = (I- P• P R O O F . From (3.2) it follows that A is invertible and hence H A -1 = A*H

or

(A-A*)-IH=H(A-A-1) -1, AEF. Integrating over the unit circle we find

2,

i

-

H

=

H.

-

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Gohberg and Zucker

or

P*H = H. ~

=g

(AA-

I-~i

I+ I)(AA-

(l~-d)-ld#

I)-ldA =

=g(I-P)

and hence HP = (I-

P)*H.

To prove (3.5) we use the equality (1.2) for the projection P• (3.6)

It follows that

p• = p + ~"

where I" denotes the following integral i f (3.7) I = ~ ]r(A - A ) - I B W - 1 C ( A

- A)-ldA

I

From (3.6) and (3.4) we obtain (3.8)

H P • = H ( P + "I) = H P + H ' I = ( I - P ) * H + H ' I .

Next, using (3.2) we find that H~=

H. ~

(~ - A ) - ~ B W ( a ) - ~ C ( a

= H. ~ = ~

- A)-~da =

(~ - H - I A * - ~ H ) - ~ B W ( a ) - ~ C ( ~ ,

- H-~A*-~H)-laa

(aA* - r)-~A*HBW-I(~,)CH-IA*(),A

9 - r)-~a~. H.

Since (see (3.2)) A * H B = iC*

and

C I I - 1 A * = iB*

we have H'[:

2~r i

--1

= -2--~

-

(I - AA*)-IC*W(A)-IB*(I-

(# -

C*W

2~rfr (~-A)-IB W

) , A * ) - l d A 9H = --1

B * ( # - g * ) - l d # 9g =

C(A-A) - 1

d,k.H--

=

Gohberg and Zucker

89

Taking into account that W

H~= -

~

= W(A) we finally obtain

(~ - A ) - I B W ( ~ ) - l C ( ~

- A)-lda

]

.H = -PH.

So, we proved that (3.9)

HI'= -FH.

Now from (3.8), (3.9) and (3.6) we find H P • = ( I - P ) * H - "I*H = ( I - P - "D*H = ( I - P X ) * H

and the lemma is proved. T H E O R E M 3.2 T h e rational m a t r i x f u n c t i o n W(A) a d m i t s a s y m m e t r i c canonical W i e n e r - H o p f factorization i f and o n l y i f

(3.10)

I m P N K e r P x = (0).

I f this condition hotds t h e n C n = I m P S r K e r P x and such a factorization is given b y -

(3.11)

W(A) =

D + CH

- A

B

].

(O-

C I I A - I B ) -~

[O+ ClI(~- A)-~B],

where II is the p r o j e c t i o n of C" along I m P o n t o K e r P x . This theorem is the anMog for the case of general matrix D of Theorem 0.1 in [KR] which concerns the case when D is invertible. P R O O F . For a linear operator T acting on C" denote by T[*] the adjoint of T with respect to the indefinite inner product Ix, y] = < H x , y > given by H. It is easy to see that the statement of L e m m a 3.1 can be rewritten as P[*] = I - P , (P•

=I-P•

.

But then ( I M P ) [• = KerP[*] = K e r ( I - P ) = I m P

(KerP) [• = IMP[*] = I m ( I - P ) = K e r P and the same is true for P • also.

,

90

Gohberg and Zucker

Thus ImP

n K e r P • = ( I M P ) [• n ( K e r P •177 = ( I m P + K e r P x)[•

and hence ImP

n K e r P x = (0) r

ImP + KerP • = C ~ .

Therefore I m P - i - K e r P x = C"

if and only if

n K e r P x = (0).

ImP

This proves the first part of the theorem. To obtain the formula for symmetric factorization we use the result of Theorem 1.1 for the general case which states that

w(x)=w_(~)w+(A), where (3.12)

W+(A) = D + C H ( A - A ) - I B ,

(3.13)

W_(A) = I+ C(A - A)-I(I

- H)(B + R(D - I))

and R is given by (1.6). From (3.13) it follows that w_(oo) = I.

On the other hand, we know that there exists a symmetric canonical factorization of the form (3.14)

W(X) = W+

XW+(A)

with the same W+(A) as in (3.12) and some selfadjoint invertible matrix X. Hence w_(~) = w+

x

and for A ~ oo we obtain

z = (w+(0))*x. Since A is invertible, from (3.12) follows W+(0) = D - C H A - 1 B

.

Gohberg and Zucker

91

Therefore X = ((D - C I I A - 1 B ) - I ) * .

Since X is selfadjoint the s a m e is true for (D - C H A - 1 B ) -1 a n d hence (3.15)

X = ( D - C I I A - 1 B ) -1 .

F o r m u l a (3.11) now follows from (3.14), (3.12) and (3.15). T h e t h e o r e m is proved. T h e following corollary is the analog of Corollary 1.3 for the selfadjoint case. C O R O L L A R Y 3.3. I f at least one of the operators I + P - P x and I - P + P x is invertible then W(A) admits a symmetric canonical factorization which is given by

(3.11). P R O O F . As was proved above the condition I m P N K e r P • = (0)

holds if a n d only if the condition I m P + K e r P • = C"

holds. If at least one of the operators I + P - P • a n d I - P + P • is invertible then at least one of these conditions is fulfilled. Hence the other is also fulfilled and s t a t e m e n t of the Corollary follows from T h e o r e m 3.2.

4. R a t i o n a l M a t r i x Functions Positive o n the U n i t Circle T h e r a t i o n a l m x m m a t r i x function W(A) is called positive on the unit circle if (4.1)

>O

( y E C n,

yr

AEF).

Such a function has no poles and zeroes on the unit circle a n d is a u t o m a t i c a l l y selfadjoint on the unit circle. T h e following theorem deals with the rational m a t r i x function positive on the unit circle a n d analytic at oo. T H E O R E M 4.1. Let W(A) = D + C(A - A ) - I B be a minimal realization of a positive on the unit circle rational matrix function. Then W(A) admits a symmetric canonical Wiener-Hopf factorization and this factorization is given by (3.11).

P R O O F . Since W(A) has no poles and zeroes on the unit circle a n d the realization is m i n i m a l we conclude t h a t A has no s p e c t r u m on the unit circle. Since W(A) is

92

Gohberg and Zucker

selfadjoint on the unit circle, we know from Proposition 2.1 t h a t there exists a (unique) invertible selfadjoint m a t r i x H such t h a t

A'HA=H,

(4.2)

C = iB*HA,

D = D* + i B * H B

.

A p p l y i n g T h e o r e m 3.2 we see t h a t it is enough to prove t h a t

I m P 0 K e r P x = (0) . Take x E ( I m P f') K e r R x ). T h e n from (3.6) we have

[p x x, x] = [Px, x] + ['ix, x] , where I ' i s defined by (3.7). Obviously, [P•

x] = O. Since ( I M P ) [a-] = I m P we also o b t a i n [Px, x] = [x, x] = 0

and therefore [ i x , x] = o .

Using the definition (3.7) of I" we find (4.3)

< (H.

i

~--~ ~ F ( A - A ) - I B W ( A ) - I c ( A -

A)-ldA)

x,x >=O .

F r o m (4.2) it easy to c o m p u t e t h a t for A E F the following is true (H(A - A ) - I B ) * = B*(~ -

=B*(I_HA-1H-1)

A*)-1H= -1 H = ) ~ B * H A ( A

-/~)-1

=

---- i,kC(,k - A) -1 . Using the last identity and (4.3) we o b t a i n

r ~ < (C(),- A)-I)*W(A)-~C(,k

- A ) - a x , z > dA = 0

or <( W ( / ~ ) - I

(C(,~ -

a)-lx),C()~

- a)-lx

> dA = O .

Let ,k = e '~, 0 _< ~ < 2rr. T h e n we get

fO rt < w(ei~) -1 (C(e icP- A ) - l x ) ,

C(e i~ -- A ) - l x

> d~ = 0 .

Gohberg and Zucker

93

Since W(A) is positive on the unit circle the function under the integral is non-negative. Since the integral is equal to zero, the function is identically equal to zero. Hence C(A-A)-lx=o,

A6F.

But C(A - A ) - l x is a rational matrix function. It follows that C(A-A)-lx=0

for all

A•a(A).

Taking the Laurent expansion of both sides in a neighborhood of infinity we obtain CAkx=O,

k=O, 1,2,... .

But then because the realization is minimal we have OO

x 6 N KerCAk = (0). k=0

So, x = 0 and the theorem is proved. REFERENCES

[BCK] [C] [GGK] [GK]

Bart, H., Gohberg, I. and Kaashoek, M.A., Minimal factorization of matrix and operator functions. OT1, Birkhs Basel, 1979. Cohen, N., On minimal factorizations of rational matrix functions. Integral Equations and Operator Theory, Vol.6, 647-671 (1983). Gohberg, I., Goldberg, S., Kaashoek, M.A., Classes of linear operators. OT49, Birkh/iuser, Basel, 1990. Gohberg, I., Kaashoek, M.A., Block Toeplitz operators with rational symbols. Operator Theory: Advances and Applications 35, Birkh/iuser, Basel, 385-440

(1988).

[GLR1]

[GLR2] [KR]

[LRR]

Gohberg, I., Lancaster, P.,Rodman, L., A sign characteristic for selfadjoint rational matrix functions. Lecture Notes in Control and Information Sciences, Mathematical Theory of Networks and Systems, Springer-Verlag, 58, 263-269 (1984). Gohberg, I., Lancaster, P.,Rodman, L., Matrices and indefinite scalar products. OT8, Birkhs Basel, 1983. Kaashoek, M.A., Ran, A.C.M., Symmetric Wiener-Hopf factorization of selfadjoint rational matrix functions and realization. Operator Theory: Advances and Applications 21, Birkhs Basel, 373-409 (1986). Lancaster, P., Raa, A.C.M., Rodman, L., Hermitian solutions of the discrete algebraic Riccati equation. Int. J. Control, Vol.44, 777-802 (1986). School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, 69978, Israel.

AMS classification:

47A68,

Submitted:

1995

June

12,

47A56,

93B28,

15A99