GENERALIZED CANONICAL FACTORIZATION OF MATRIX AND ...

Integr Equat Oper Th Vol. 15 (1992)

0378-620X/92/040673-2451.50+0.20/0 (c) 1992 Birkh~user Verlag, Basel

G E N E R A L I Z E D C A N O N I C A L F A C T O R I Z A T I O N OF M A T R I X AND O P E R A T O R F U N C T I O N S W I T H D E F I N I T E H E R M I T I A N PART. Andrd C.M. R a n and Leiba R o d m a n *

It is proved that rational matrix functions with definite hermitian part on the real line admit a generalized canonical factorization. The functions are allowed to have poles on the real line. A generalization of this result to a class of operator functions is obtained as well. 0.INTRODUCTION Let W(A) be a n x n rational matrix function (with complex coefficients). A factorization

w(~) = w+(~)w_(~)

(o.1)

is called a left canonical pseudo-spectral factorization of W(A) (with respect to the real line) if the following conditions are satisfied: (i) W+(,X) (resp. W_(A)) is a rational matrix function without poles and zeros in the open upper (resp. open lower) balfplane; (ii) at every point Ao on the real line and at infinity, the faetorization (0.1) is minimal. Recall (see [BGK] Sections 4.1 and 4.3) that the factorization (0.1) is called m i n i m a l at Ao if -1 rank

rank

wr o .. wL1 w: ... .

.

".

o) o .

l~p .

0 .

W;W$_ 1

+ rank

:

w;* Partially supported by an N S F grant

=

0 .... W;- ... :

"..

w;-..,

0 o :

w~

(o.2)

674

Ran

and

Rodman

where the matrices W~, W +, Wt- are taken from the Laurent series P - -

-

q

w+(A) =

- a 0 ) - k w +, ]r

=

oc

(A

-

For further properties and many applications of factorizations that are minimal at a given point see the books [BGK, GLR1, BGR]. A right canonical pseudo-spectral factorization is given by W(A) = W_(A)W+ (A), with the same properties (i) and (ii). If W(A) has no poles and zeros on the real line, then left or right canonical pseudo-spectral factorizations becomes left or right canonical Wiener-Hopf factorization, see [CG], also [BGK]. The concept of canonical pseudo-spectral factorization was introduced and studied in [Rol, Ro2] in connection with Wiener-Hopf integral equations. Note also that a rational matrix function W(A) which admits a (left or right) canonical pseudo-spectral factorization is necessarily regular (i.e. det W(A) is not identically zero), and any canonical pseudospectral factorization is minimal on the extended complex plane. We now state one of the main results of this paper. THEOREM 1.1. Let W(A) be an n • n rational matrix functzon. Assume that the hermitian part w . ( A ) :=

+ w(x))*)

has no zeros on the real line and at znfinity, and that WH( A) is pos,twe definite for all real A (that are not poles of W u ( A)). Then W( A) admzts left, and mght canonical pseudo-spectral factorizations. We emphasize that the case where WH(A) has poles on the real line or at infinity is not excluded in Theorem 1.1. In case W(A) does not have poles and zeros on the real line this theorem is proved for a larger class of functions in IS]. For a more general reference see [GK], see also [CG], [LS]. In this case the factorizations are canonical. The proof of this theorem is based on two important ingredients: the general factorization result of [BGK] and the theory of invariant subspaces for matrices in an indefinite inner product space (see [IKL]). In case W(A) is analytic at infinity, and the hermitian part WH(A) is a constant matrix we shall provide formulas in realization form for the factors (see Theorem 1.3 below). The paper contains also a generalization to operator functions under strong hypotheses (see Section 4). The paper is organized as follows. Section 1 is devoted to the proof of Theorem 1.1. The next two sections provide further information concerning canonical pseudo-spectral

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factorizations of rational matrix functions with positive definite hermitian part. In Section 2 we give formulas for factors and consider the factorizations of matrix polynomials, while in Section 3 problems of uniqueness and stability of factorizations are studied. Finally, in Sections 4 and 5 we extend several of these results to a certain class of operator valued functions. 1. MAIN RESULT: MATRIX FUNCTIONS In this section Theorem 1.1 will be proved, as well as an important special case. It will be convenient to prove first a lemma. LEMMA 1.2. Let U(A) be an n • n rational m a t r i x f u n c t i o n which is strictly proper, z.e., V ( cc ) = O, and such that U ( A ) = - U ( A )*. Let D be an n x n m a t r i x such that D + D* zs positive definite (notation: D + D* > 0). Then the f u n c t i o n Y()~) = D + U(A) has no zeros on the real line and at infinity. P r o o f . We will use a minimal realization of V(A): Y()~) = D + C ( A I - A ) - I B ,

(1.1)

where C, A and B are of size n x m, m • n and m x m, respectively. We refer the reader to Chapter 7 in [GLR1] for the basic properties of minimal realizations. Here we only remark that minimality means that the size of A is the smallest among all representations of V(A) in the form (1.1); also, the minimality of (1.3) is equivalent to the requirement that the matrices

\

cCA

/

and(B

,AB,

...,

AP-IB)

(1.2)

!

CA p-1 / have full column rank and full row rank, respectively, for sufficiently large integer p. Using the property U(A) = -(U(A))* we obtain from (1.1) V(A) = D - B * ( A I - A * ) - I C *,

(1.3)

which is again a minimal realization. By the state space isomorphism theorem, there is a unique invertible H such that HA = A'H,

H B = C*,

C H -1 = - B * .

(1.4)

Taking adjoints in (1.4) we have H*A = A'H*,

C H *-1 = B*,

C* = - H * B .

(1.5)

Hence, by the uniqueness of H we have H = - H * . Thus, i H is hermitian. The first equality in (1.4) shows that A is i H - s e l f a d j o m t , i.e., A is selfadjoint in the indefinite scalar product [., .] in C m induced by i l l : [x, y] = ( i H x , y),

676

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and

Rodman

where (., .) is the standard scalar product in C "~. Arguing by contradiction assume that D + U(A) has a real zero Ao. Then Ao is an eigenvalue of the matrix A • = A - BD-1C

(see, e.g., Theorem 7.2.3 in [GLR1]); the invertibility of D is a consequence of D having positive definite hermitian part. So, (A - B D - 1 C ) x = A0x,

x # 0.

Use (1.1) to rewrite this in the form (A - H - 1 C * D - 1 C ) x -- ),oX.

Premultiply by ill; after some simple algebra one sees that [Ax,x] = A0[x,x] + i ( C * D - 1 C x , x ) .

(1.6)

The left hand side is real by iH-selfadjointness of A. The matrix D has positive definite hermitian part; hence the same is true of D -1, and thcrefore the right-hand side of (1.6) has a non-zero imaginary part if C x ~ O. We conclude therefore that C x = O. But then A x = Aox, and the one-dimensional subspace span {x} is an A-invariant subspace contained in Ker C. This contradicts the minimality of realization (1.1). We first prove now the following special case of Theorem 1.1, which is of interest in its own right, in particular because in this case we have formulas for the factors in a canonical pseudo-spectral factorization. We will need some terminology concerning indefinite scalar products. Let G be an invertible m x m hermitian matrix, inducing the indefinite scalar product [x, y]G = ( c x , y), x, y e c m A subspace M C C m is called G-nonnegative (resp., G-nonpositive) if [x,x]a _> 0

for all x 6 M

(resp., Ix, x]c _< 0 for all x 6 M). A subspace M _C C ~ is called maximal G-nonnegative if it is G-nonnegative and is not contained in any bigger G-nonnegative subspace. The definition of a maximal G-nonpositive subspace is analogous. T H E O R E M 1.3 Let U(s = C(AIm - A ) - I B be a minimal realizatwn of an n x n rational matrix function such that U(A) = - U ( A ) * . Let D be an n x n matmx such that D + D* > O. Let H be a skew-hermitian inve~ible matrix such that H A = A ' H , H B --C*. Then there exists an A-invamant maximal iH-nonnegatwe subspace M such that a(AIM ) C {A 6 C I Im A < 0}, and there exists an A • = A - B D - 1 C - m v a r i a n t maximal iH-nonpositive subspace M x such that a(A•215 C {A 6 C [ Im A > 0}, and moreover any two such subspaces M and M • are direct complements to each other: M + M x = C m.

(1.7)

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677

L e t r be the p r o j e c t i o n along M

onto M •

and put

W_(A) = D + C~r(AI - 7rATr)-I~rBD

(1.8)

14%(A) = I + C ( I - 7r)(AI - A I M ) - I ( I - 7r)B.

(1.9)

Then, wzth W(A) = D + U(A), we have that w(:~) = w + ( ~ ) w _ ( ~ )

(1.1o)

is a left canonzcal p s e u d o - s p e c t r a l f a c t o m z a t i o n o f W(A).

P r o o f . The existence and uniqueness of H as desired has already been observed (see (1.4), (1.5)). To continue, note that A is iH-selfadjoint. By a well-known result in the theory of selfadjoint operators in an indefinite scalar product space (see, e.g., Section 1.3.12 in [GLR2], or Section IX.7 in [B]), there exists an A-invariant subspace M with the following properties (i) all the eigenvalues of AIM lie in the closed lower halfplane; (ii) M contains all the spectral A-invariant subspaces corresponding to the eigenvalues s with Im A0 < 0; (iii) M is maximal iH-nonnegative. It is well-known (see, e.g., Section 1.1.3 in [GLR2]) that (iii) is equivalent to (1.11) and (1.12) below: Ix, x] > 0 for all x E M (1.11) and dim M = k+,

(1.12)

thc number of positivc eigcnvahles of i H (counted with multiplicities). Consider now A x = A - B D - 1 C . We have HA x - A•

= H(A - BD-1C)

- (A* - C * D - I * B * ) H

= HA - HBD-1C

- A*H + C*D-I*B*H

= _C*(D -1 + D - I * ) c , where (1.4) and (1.5) was used to obtain the latter equality. Since D has positive definite hermitian part, so does D -1, and therefore H A x - A X * H is negative semidefinite. This means that Im [ A X x , x ] 0 and U(A) is a skew-hermitian rational matrix function with value 0 at infinity. Easy examples show that left canonical pseudo-spectral factorizations

= w+

(3.2)

(:9

with W+ (oc) = I are generally non-unique (of course, analogous examples can be arranged for right canonical pseudo-spectral factorizations): E x a m p l e 3.1 Let --A-1 / w(A)--

1

"

682

Ran

Here A=O,

B=[,

H=

( 0 1 01 ) '

and

Rodman

ol),

c=(O1

and W(,X) has the minimal realization W(..k) = I + C ( M - A ) - I B .

Further,

All A-invariant maximal iH-nonnegative subspaces M such that o'(AtM ) C {A 6 e l

Im A < O}

are given by the formula

where the imaginary part of x 6 C is nonpositive. The subspaee M • is unique (spanned by the eigenvector of A x corresponding to the eigenvalue i), and is given by

As expected (by Theorem 1.3), M ~ - M • = C2, and we have a continuum of left canonical pseudo-spectral faetorizations of W(A) given by (1.8) - (1.10), for the continuum of the choices of x, resulting in:

w _ ( ~ ) = i + ~. x _ i W+(A) = I +

(x -1),

, _x -_il ( : ) (1i X"

forthecaseM= span{(lz)},while

w_(~) = I + y w+(~)--=I+~ incascM=

span{(O) 1 }

0 o

-1)

Ran and Rodman

683

Given W(A) as in (3.1) let U()~) = C ( M - A ) - I B

be a minimal realization of U(A). Then, as in Section 1, there exists unique invertible hermitian matrix i H such that HA = A'H,

H B = C*,

C H -1 = - B * .

Theorem 1.3 shows that every pair of subspaces (M, M x), where M is A-invariant maximal iH-nonnegative, M x is AX-invariant maximal iH-nonpositive and a(AIM) C {A e (3 [ I m A < 0}, a(A x l M •

ImA>0},

{)~6CI

gives rise to a left canonical pseudo-spectral factorization of W(A) by the formulas (1.8) - (1.10). Left canonical pseudo-spectral factorizations that arise in this way will be called special. (Special factorizations of selfadjoint rational matrix functions and operator polynomials have been studied extensively; see, for example, books [GLR2,R,M] and papers [KL,RR1,MMR]). Note that not necessarily all left canonical pseudo-spectral factorizations are obtained as special factorizations. Indeed, a left canonical pseudospectral factorization is obtained from any choice of A-invariant subspace M such that a(AIM ) C {A 6 C I I m A _< 0} and M 4 - M x = C m where M x is the unique A• subspace with a(A x IMX) C {A e C I Im A > 0}. The special factorization W(A) = W+(A)W_(A),

W+(cx)) = I

(3.3)

will 1,e called stable if the following holds: Given minimal realizations W(A) = D + C(AI - A ) - I B ,

(a.4)

W+(A) = I + C + ( M - A + ) - I B + , W_($) = D + C _ ( M - A _ ) - I B _ ,

(3.6)

for every e > 0 there exists ~ > 0 such that every rational matrix function W(A) having minimal realization of the form

with/)+D*

> 0, liD - DI[ + I]C - e l i +

lIB -

Bit + IIA - All
0 for every x E Y, and maximal J-nonnegatwe if Y is J-nonnegative and is not contained in any strictly larger J-nonnegative subspace. P r o o f . Let P+,Po and P_ be the spectral projections of A corresponding to the spectrum of A in the open upper halfplane, the real line and the open lower halfplane, respectively. Then the A-invariant subspaces Im P+ and hn P_ are J-isotropic, i.e., [x, X] = 0 for all x 6 Im P+ and for all x E I m P_ (see, e.g., Corallary II.3.11 in [AI]), and X is a J-orthogonal direct sum of Im P + + Im P_ and Im P0. In other words, X = ( Im P+ 4 Im P _ ) 4 ImPo

(4.4)

and [x, y] = 0 whenever x E Im P+-~ Im P_, P+ + P_ is J-selfadjoint. Now

y 6 I m Po. Indeed, by Corollary II.3.12 in [AI] the sum

[PoY,(P+ + P_)x] = [(P+ + P-)Poy, x] = 0 for any x, y 6 X. Also, Po = I - (P+ + P_) is J-selfadjoint as well, and therefore the subspace Im Po is J-nondegenerate, i.e., if xo E Im Po is such that [Xo, y] = 0 for all y E I m Po then xo - 0. Now consider Ao:=AiImpo: ImPo~ ImPo J0:--=/}()JlIr~Po : I m P o ~

ImPo.

688

R a n and R o d m a n

Clearly, Ao is Jo-selfadjoint and (by the hypothesis on the real spectrum of A) the subspace Im Po is finite dimensional. As it is also J0~nondegenerate, we can apply Pontryagin's theorem (see, e.g., Theorem 1.3.20 in [GLR2]): there is an Ao-invariant subspace M0 which is maximal Jo-nonnegative. Put L=Mo+

Im P_

The subspace L is A-invariant, and as the sum is a J-orthogonal direct sum we have that L is J-nonnegative. It remains to prove that L is maximal J-nonnegative. Arguing by contradiction, let L' be a strictly bigger J-nonnegative subspace. Take x 6 L ' \ L ; without loss of generality we can assume x 6 Im Po + Im P+. Write x = Xo + x+ where Xo 6 Im Po, x+ 6 Im P+. We can further assume that x+ ~ 0. (Indeed, if L' C Im P_ + I m Po, we obtain easily a contradiction with the maxim ality of Mo as a Jo-nonnegative subspace). By Corallary II.3.12 in [AI] the sum Im P+~- Im P_ is J-nondegenerate; so there exists y E Im P+ ~- Im P_ such that y] > o.

As hn P§ is J-isotropic, we may assume y E I m P_. Now x - a y E U for all a 6 C, and by the J-nonnegativeness of L ~ we have 0 0 such that /

NC(AI - A ) - l x l l 2 dk > cIIxiI2 oo

(4.5)

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for all x 6 X. P r o o f . Arguing by contradiction, assume there is a sequence {X n}n=l in X such that Ilxnll ---- 1 for all n and

/_

~ I[C(AX - A ) - ~ x n l l ~ dA

--*

as n ---* ~ .

0

(4.6)

It is easy to see (for example, arguing by contradiction), that lim

]lC(AoI

-

n----) o o

A)-~x~][ - -

0

(4.7)

for every real A0. We use now the Banach generalized limits construction (see [Be], also Appendix to Chapter 3 in [R]), which has been frequently used in the studies of operator valued functions (see, e.g., [Ra,KMR]). Thus consider the Banach space B x of all bounded sequences (an),,~__l of elements in X, with the supremum norm. Let N be the subspace of all oc a = ( Ot n)n=l in B such that lim,~_~ an = 0. Define )( to be the factor space B x / N , which is a Banach space with the quotient norm. Define G in the same way starting with the Hilbert Nspace G. The operators C and A induce naturally (bounded) operators C : )~ --* G a n d A : X ---* )~ by C~ = the equivalence class of (Can)~=l .4~ = the equivalence class of (Ao'n)~=l, where ~ is the equivalence class of ( n)n=~" It is easy to see that the passage from an operator D to its Banach ~eneralized limit extension/9 preserves addition and multiplication of operators, and that I = I. Thus, the operator

/ )

: .X ---* GP

(4.8)

oo is left invertible. Now, let ~" be the equivalence class of the sequence {X n}n=1 introduced at the beginning of this proof. By (4.7) we have

C ( A I - .~)-1~ = 0

(4.9)

for all real )~. Developing (4.9) in the Taylor series centered at infinity, we obtain

C A ~ = 0,

j = 0,1,...,

which in view of (4.8) implies 5 = 0. But this contradicts the equalities llXnH = 1 for all n.

9

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The proof of Lemma 4.3 shows that it is valid for Banach space operators as well. Moreover, one can replace the left hand side of (4.5) by

f

~ I[C(AI - A)-Ixll 2 dA, o

where Ao is a fixed real number, and remove the hypothesis that a ( A ) N R = ~. We have presented, however, Lcmma 4.3 in the form it will be used in this paper. P r o o f o f T h e o r e m 4.1. As in the proof of Theorem 1.3, the operator A is i H selfadjoint and - A • is/H-dissipative. Using condition (iii) and appealing to Lemma 4.2 we find an A-invariant maximal iH-nonnegative subspace M such that

r

) C {A e C I ImA_O.

692

Ran

Indeed, suppose x,~ + y , --+ 0 where max(llx~ll,

Ily,~H) =

and

Rodman

1. Then

o o

for n large enough by (4.11) unless yn ~ 0. But then also x,~ ~ 0. So we have a contradiction. By a theorem in [GM], the sum M + M • is a closed subspace. It is not difficult to see (Theorem V.4.4 in [B]) that (ill) -a M • is an A-invariant maximal iH-nonpositive subspace, and ( i H ) - I ( M • is an A• maximal (iH)-nonnegative subspace, and therefore ( i g ) - l M • N ( i H ) - I ( M • • = (0), which can be proved analogously to (4.10). But

(iH)-I M • A ( i g ) - l ( M X ) • = ( i H ) - I ( M + MX) • and therefore M + M • is dense in X. Combining with the previously proven closedness of M + M • we obtain (4o14). Once (4.14) is proved, the result of the factorization of W(A) given by formulas (1.8) (1.10) follows from Theorem 1.5 in [BGK]. That this factorization is left canonical pseudospectral follows in turn using the formulas (1.8) - (1.10), as in the finite dimensional case.

Note that the hypotheses of Theorem 4.1 are more restrictive than those of Theorem 1.1 in the sense that WH()Q is constant in Theorem 4.1, in contrast with Theorem 1.1. We believe however that a full generalization of Theorem 1.1 to the infinite dimensional case is possible, and the following conjecture is valid: C O N J E C T U R E 4.4. Let W( A) be a regular fimtely meromorphic operator function in a neighborhood of R U { ~ } . Then W(A) admits left and right canonical pseudo-spectral factorizations provided its hermitian part WH(A) satz,sfies the following properties:

fl) WH( A) is a regular finitely meromorphie function in a neighborhood of R U { ~ } Oi) WH(A) has no zeros on the real line and at infinity (iii) (WH()Qx,x) > 0 for all non-zero x 6 G and all )~ 6 R U {oc} whzch are not poles of

w.(A). Observe that under the hypotheses of Conjecture 4.4 the function W(A) has no zeros on the real line and at infinity (just as in the finite dimensional case, L e m m a 1.2). Indeed, if A0 C R U {~c} were a zero of W(A), then, by using the local factorization of regular finitely meromorphic functions [GS], we would obtain a G-valued analytic function x(A) in a neighborhood of A0 such that IIx(A0)[I = 1 and W(A)x(A) ~ 0 as A ---* Ao. The equality

(w(~)x(:~),~(:~)) : (w.(A).(;~), x(A)) + (w~(A)z(~), ~(~)),

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and Rodman

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where W s ( A ) = g(14 1 , ( A ) - (W(A))*), implies that (WH(A)x(A),x(A)) ~ 0 as A ~ )%. Therefore, by the local factorization result for WH(A) [GS]. it follows that A0 is a zero of WH(/~), a contradiction with (ii). 5. UNIQUENESS AND STABILITY: THE OPERATOR FUNCTIONS In this section we extend the uniqueness and stability results of Section 3 to canonical pseudo-spectral factorizations of the class of operator valued functions introduced in the previous section. Thus, we consider operator valued functions of the form W(A) = D - B * H ( A I - A ) - I B ,

(5.1)

where A, H and D satisfy the properties (i) - (iv) of Section 4. As the proof of Theorem 4.1 shows, every pair of subspaces (M, M • ), where M is A-invariant maximal iH-nonnegative with a(AIM) C {A e C I ImA_ 0 there exists ~ > 0 such that every operator function W(A) of the form W(A) = / ) - B * H ( A I - A ) - ~ B with ArA = A*[t,

H = -At*

(5.6)

and lID - D[I + [IA - All + l[/} - BII + I[~ r - HII < d

(5.7)

admits a special factorization w(~) = w+(~)w_(~), where W+(A) have minimal realizations of the form

W-t- (,'~) = /~4- q- Ci(/~I -- fl~--b)-lB:t:,

L)+ = I

with

llc~ - c , II + I1-~, - & l l + IIB~ - B,,]I < E for v = + , - . Note that if (5 > 0 is sufficiently small the conditions (5.6) and (5.7) guarantee that W(A) satisfies the conditions (i) - (iv) of Section 4. Observe also that in view of Proposition 5.1 the definition of a stable special faetorization is independent of the choice of the minimal realizations (5.1), (5.4) arid (5.5). THEOREM 5.2. Let (5.3) be a special factorizatzon of W(A). The following statements are equivalent:

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(1) The special factorizatwn of W()~) is unique. (:i) The special facto ation of W( ,s stable. (iii) Let Mo be the (finite dimensional) sum of the spectral A-invariant subspaces corresponding to the real points in a(A), and let Po be the corresponding spectral projection in Mo. Then the pair (AIMo,Po(iM)IMo) satisfies the sign condition. The proof is again essentially the same as the proof of Theorem 2.2 in [RR1], using the easily verificd facts that (under the conditions (i) - (iv) of Section 4) the spectral Ainvariant subspace corresponding to the part of a(A) in the open upper and open lower halfplane are well-behaved under small perturbations of A, and the spectral A• subspace corresponding to the part of a(A • in the open upper halfplane is well-behaved under small perturbations of A, B and H. The counterpart of Theorem 5.2 concerning stability of right canonical pseudo-spectral factorizations is valid as well. We leave to the interested reader the statement of this counterpart. AKNOWLEDGEMENT. We are grateful to Prof. C.V.M. van der Mee whose question prompted us to investigate the problem treated in this paper. REFERENCES

[hi]

T.Ya. Azizov, I.S. Iovidov. Linear Operators in Spaces with an Indefinite Metric. J. Wiley and Sons, New-York, 1989. (Translation from Russian). [BGR] J.A. Ball, I. Gohberg, L. Rodman. Interpolation of Rational Matrix Functions. OT 45, Birkhs Verlag, Basel, 1990. [BGK] H. Bart, I. Gohberg, M.A. Kaashoek. Minimal Factorization of Matrix and Operator Functions. OT 1, Birkh~user Verlag, Basel, 1979. [Be] S.K. Berberian. Approximate proper vectors. Proc. AMS 13 (1962), 111-114. [S] J. Bognar. Indefinite Inner Product Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78, Springer Verlag, 1974. [CG] K. Clancey, I. Gohbcrg. Factorization of Matrix Functions and Singular Integral Operators. OT 3, Birkh/iuser Verlag, Basel, 1981. [CK] I. Gohberg, M.G. Krcin. Systems of integral equations on a half line with kernels depending on the difference of arguments. Uspahi Mat. Nauk 13 (1958), no. 2(80). 3-72~ English transl. Amer. Math. Soc. transl. (2) (1960), 217-287. [CLR1] I. Gohberg, P. Lancaster, L. Rodman. Invariant Subspaces of Matrices with Applications. J. Wiley and Sons, New-York. 1986. [ LR2] I. Gohberg, P. Lancaster, L. Rodman. Matrices and Indefinite Scalar Products. OT 8, Birkhs Verlag, Basel, 1983. [GM] I. Gohberg, A.S. Markus. Two theorems on the gap between subspaces of a Banach space. Uspehi Mat. Nauk 14 (1959), 135-140 (Russian). [GS] I. Gohberg, E.I. Sigal. An operator generalization of the logarithmic residue theorem and thc theorem of Rouch4. Math. USSR Sbornik 13 (1971), 603625(Russian).

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Ran and Rodman

[HI J.W.

Helton. Systems with infinite-dimensional state space: the Hilbert space approach. Proc. IEEE, 64 (1976), 145-160. [IKL] I.S. Iohvidov, M.G. Krein, H. Langer. Introduction to Spectral Theory of Operators in Spaces with Indefinite Metric. Reihe "Math. Forschung', Akademie Verlag, Berlin, 1981. [KMR] M.A. Kaashoek, C.V.M. van der Mee, L. Rodman. Analytic operator functions with compact spectrum, I. Spectral nodes, linearization and equivalence. Integral Equations and Operator Theory 4 (1981), 504-547. [KL] M.G. Krein, H. Langer. On some mathematical principles in the linear theory of damped oscillations of continua I, II. Integral Equations and Operator Theory 1 (1978), 364-399; 539-566 (translation from Russian; original 1965). Ins] G.S. Litvinchuk, I.M. Spitkovskii. Factorizations of Measurable Matrix Functions. OT 25, Birkhs Verlag, Basel, 1987. [M] A.S. Markus. Introduction to Spectral Theory of Polynomial Operator Pencils. AMS Transl. of Math. Monographs, Vol. 71, 1988 (Translation from Russian) [MMR] A.S. Markus, V.I. Matsaev, G.I. Russu. On some generalizations of the theory of strongly damped bundles to the case of bundles of arbitrary order. Acta Sci. Math. (Szeged) 34 (1973), 245-271 (Russian). IRa] A.C.M. Ran. Minimal factorization of selfadjoint rational matrix functions. Integral Equations and Operator Theory 5 (1982), 850-869. [RR1] A.C.M. Ran, L. Rodman. Stability of invariant maximal semidefinite subspaces. II. Applications: selfadjoint rational matrix functions~ algebraic Riccati equations. Linear Algebra and its Applications 63 (1984), 133-173. [RR2] A.C.M. Ran, L. Rodman. Stability of invariant maximal scmidefinite subspaces. I. Linear Algebra and its Applications 62 (1984), 51-86. [a] L. Rodman. An Introduction to Operator Polynomials. OT 38, Birkhs Verlag, Basel, 1989. [aol] I Roozemond. Systems of Non Normal and First Kind Wiener-Hopf Equations. PH.D. Thesis, Vrije Universiteit, Amsterdam, 1987. [Ro2] L. Roozemond. Canonical pseudo-spectral factorization and Wiener-Hopf integral equations. Operator Theory: Advances and Applications 21 (1986), 127-156. Yu. L. Smul'yan. Riemann's problem for Hermitian matrices. Uspehi Mat. Nauk [S] 9 (1954), 243-248 (Russian). [T] R.C. Thompson. Pencils of complex and real symmetric and skew-matrices. Linear Algebra and its Applications 147 (1991), 323-371. Andr4 C.M. Ran Vrije Universiteit Faculteit Wiskunde en Informatica De Boelelaan 1081a 1081 HV Amsterdam The Netherlands

Leiba Rodman Dept. of Mathematics College of William and Mary Williamsburg, VA23187-8795 U.S.A.

MSC: 47 A 68, 47 B 50 Submitted: July 17, 1991. Revised: July 24, 1991