0378-620X/82/060850-20501.50+0.20/0 © 1982 Birkhauser Verlag, Basel

Integral Equations and Operator Theory Vol.5 (1982)

MINIMAL

FACTORIZATION

OF S E L F A D J O I N T

RATIONAL

MATRIX

FUNCTIONS

A. C.M.

Ran

INTRODUCTION In this their for

paper

minimal

such

function

selfadjoint

factorizations

functions W(h),

(cf.

analytic

are

[i]). at

For

factorizations

of the

are

described.

description

spaces

which

ciate main heavily inner

are

invariant

operator

on the

product

type

space

for W(h)

[3,6,7,121). of t h e o r e m s

earlier

for

matrix

last

section

generalizations case

are

W(~)

(cf.

Some

which

with

of our m a i n

matrix

L(~)

= I

of c e r t a i n

El])

or the

and

sub-

asso-

leans

in an i n d e f i n i t e

of our r e s u l t s have

polynomials

model

= I, all m i n i -

operator

matrices

and

rational

in t e r m s

the m a i n

generalizations

dimensional

the n o d e

= L,(h)L(h)

is g i v e n

natural

selfadjoint

using

and w i t h

of s e l f a d j o i n t

(cf.

functions

a nonnegative

W(~)

under

of a n o d e

theory

matrix

studied

infinity

mal

The

rational

been

(see

[6,7]).

results

are

proved

to the

In the infinite

given.

i. P R E L I M I N A R I E S A system and A linear

9 = (A,B,C,D;X,Y),

: X ÷ X, B operators,

dimensional,

then

: Y ÷ X, C is c a l l e d

where

: X ÷ Y and a node.

~ is c a l l e d

X and D

Y are B a n a c h

spaces,

: Y ÷ Y are b o u n d e d

If X and

Y are

finite dimensional.

finite The

transfer

Ran

851

function W0(1)

(i.1) Here

of ~ is given

by

we(1) = D+C(II-A)-~B ~(A)

denotes

resolvent

the

spectrum

(l ~ ~(A)

of A. The

set of A, will be denoted

0 i = (Ai,Bi,Ci,Di;Xi, Y~ and there

exists

(i =1,2)

a bounded,

set ~ \ u(A

by p(A).

are

similar i f D 1 = D 2

called

invertible

, the

Two nodes

operator

S

: X + X such

that (1.2) The

A I : S-IA2 S,

operator

B I : S-IB2,

C I_ : C~S.

a node similarity. The node

S is called

8 is called

minimal if 'i)

n

(i.3)

Ker CA j : (0)

j:0 ii)

span

U

Im AJB

: X.

j:0 In case

the

first

identity

holds

observable. If the operator

is again

a node. W8(1)

We write

It is clear

that

BD -I,

-D-IC,

if and only

is called

then

D-i;

A x for the operator

is invertible

and in that case W8(l)-I

then the node

D is invertible,

8 x : ( A - BD-IC,

show that

only,

X,Y)

A - BD-IC.

if I { 0(A)

One

can

n o(AX),

= W0x(1).

the transfer

function

of a finite

dimensio-

nal, m i n i m a l node is a r a t i o n a l matrix function which is analytic at infinity. The converse is also true: for every r a t i o n a l matrix function Wwhich

is analytic at infinity,

sional,

minimal

node

8 with W = W 0 (see

Let H be a bounded, X. Then a bounded HA : A*H or, inner

The in

[i].

[3] and

[x,y]

theory For the [6,7].

selfadjoint

operator

equivalently,

product

one can construct a finite d i m e n -

of nodes

Section

and invertible

A on X is called

on X given

and transfer

of H - s e l f a d j o i n t

2.1). operator

on

H-selfadjoint if

if A is selfadjoint

:

theory

[i],

in the indefinite

by H. functions operators

can be found we refer

to

Ran

852

We i n t r o d u c e

the

following

B 2. : Xj. + Y is a b o u n d e d matrix

[B 1 .

where

Cj

matrix

is a l i n e a r

is a b o u n d e d

[C 1 ..... Cn]T

operator

and

the r e s t r i c t i o n

Throughout function

W(h)

which

operator,

operator

col

(Cj)j=l,n

denotes

... • X n.

SELFADJOINT we

is a n a l y t i c

where

0 = ( A , B , C , D ; g n , g m)

section

we

= D+C(~I

always

proposition

(see

assume [4],

the

If A

of X, t h e n

: X ÷ Y

A M will

RATIONAL

consider at ~.

MATRIX

a rational

Such

FUNCTIONS

m × m matrix

a function

can be

that

W(h)

right

hand

the node

that

dimensional

0 to be m i n i m a l .

Theorem

the

node.

6.1)

function

gives

W(h)

Our

In this first

conditions

on

is s e l f a d j o i n t .

selfadjoint if

= w(Y) ~

side

PROPOSITION

is a finite

is c a l l e d

w(~)

n-A)-IB,

cf. [5],

C and D in order

(2.~)

The

the

where

as w(~)

Recall

denotes

• X n to Y, and linear

: X ÷ X1 •

section

(2.1)

A, B,

operator,

M is a s u b s p a c e

ABOUT

this

n (Bj)j=I,

row

of A to M.

2. P R E L I M I N A R I E S

written

linear

B. n]. m a. p p.i n g. X. 1 •

: X + Xj

operator

denote

.

notations:

of

2.1.

(2.2)

Let

will

W(~)

be d e n o t e d

by W

= D + C(hl n - A)-IB,

(~). where

( A , B , C , D ; ~ n , ~ m) is a finite dimensional minimal node.

e =

Then W(~)

is selfadjoint if and only if there exists a (unique) invertible selfadjoint operator H : ~n + ~n such that (2.3)

HA PROOF.

= A'H,

Suppose

HB = C*~

W(~)

is s e l f a d j o i n t .

0" = (A * , C ~ ,B * ,D;~n,~ m)

is also

node

~* are

for W.

H be the

Hence

0 and

similarity

the r e l a t i o n s

in

between

(2.3)

C = B'H,

hold.

One

a finite similar

e and Using

easily

dimensional

([1]

0 *. T h e n these

D = D ~.

Theorem

checks

3.1)

H is i n v e r t i b l e

relations

that

minimal Let and

it is e a s i l y

Ran

853

seen that

also

H* is a node

of the m i n i m a l i t y unique

(see

easily

checked.

Let

of e and

[1], T h e o r e m

the r a t i o n a l

let H be the unique identity

in

(2.3)

.

If,

allows (see

is also

us to apply

[3,6,7])

to the

of A - s i g n

definition

of the

with

9 is m i n i m a l

of W ([1],

section

the e i g e n v e c t o r s

~J

function

~(~)

~ ¢m is called

at h

the eigenvalues

operators

For the

[6,7]. of A coincide

In the r e m a i n i n g

detail

the relations

eigenvectors

on the other

part between

of A on one hand,

hand.

Recal!

pole function of W if W(~)~(~)

that

a

co@fficients

is analytic

# 0.

~

O

be a pole

of the

function

(2.1).

In a n e i g h b o u r h o o d

we can write W(h)

where

to

This

in Section

be used.

we refer

3.3).

=

~ho Let

O

of H - s e l f a d j o i n t

= (~-~o)~1 + (~-~o)2~2 + ... with

and lim W(h)~(h) o

of h

a

of W(I)

HA × : (AX)*H.

of H will

in more

and g e n e r a l i z e d

and the p o l e f u n c t i o n s vector

Theorem

we describe

means

[x,y]

then the o p e r a t o r

i.e.,

characteristic

and

first

which

product

A and A x . Especially,

characteristic

sign

inner

theory

6" is is

be selfadjoin~,

A is H - s e l f a d j o i n t ,

spectral

9 and

converse

in (2.~j.~ The

D is invertible,

operators

the node

the poles

of this

us that

9". Because

between

(2.1)

appearing

H-selfadjoint,

the

the notion

Since

function

in the indefinite

in addition,

9 and

and so H : H*. The

similarity

tells

between

9* the similarity

3.1),

matrix

that A is selfadjoint

A × : A - BD-1C

similarity

q is some

:

Z j=-q

positive

(~-~o)JWj integer.

As W(h)

: D + C(Iln-A)-IB,

we have (2.4)

W_j

where

P

i.e.,

for

O

is the

= C ( A - ho)J-lPoB ,

spectral

projection

of A c o r r e s p o n d i n g

to (~ ), O

r sufficiently

s°

: ~1

small

- o J:r

4

Ran

854

Take a polefunction

~(h)

= (h-~o)k~ k + (k-ho)k+l~k+l

+

and put (2.5)

x. = ~ P (A-ko)V-a-lB~ v J v:k o '

Then Xo,...,Xk_ i is a Jordan (A-ho)VXk_ 1 = Xk_l_ v and i.e.,

j = 0,

such that

(A-ho)X ° : 0. If this chain is maximal,

find a pole function

(2.5) holds.

Indeed,

there exist ~k,~k+l,... non zero),

:

~ j=k

= ~j=k

: lim

(1.3)

ii),

of ~j are

(h-~o

and Cxj•

(2.6)

(2.5) and

holds:

we get Cx o ~ 0 from minimality.

is a pole

function.

In particular,

Any pole function

(2.5) will be called

to the J o r d a n

by (2.4)

of formula

identity

Define

W(l)~¢h).

satisfying

corresponding

(2.5) holds

h+ho

CAx ° = hoCXo,

we see that ~(h)

~(~)

=

a pole f u n c t i o n

Xo,...,Xk_ 1. Note that in this

chain

(2.5), =

(W~)j denotes

of W(X)~(~)

it is clear that

(k-ho)J~j ' then because

o

where

: (h-ho)k~ k + ..~

(A-~)J-kB~j o "

(A~h ° )v Xk_ 1 = Xk_l_v,

Cx

case,

~(k)

of relation

(in fact only a finite number

(A-ho)X ° = 0 the following

~j=k

because

chain for A in h o.

such that Xk-1

Since

"

Xk_ 1 @ Im (A-ho) , then ~k ~ 0.

Then one can always

~(~)

k-i

chain for A in ho' i.e.,

Conversely , let Xo,...,Xk_ 1 be a Jordan

Since

.... '

x v=k

w

J-v~v

=

(w~)

J . .j .= .0 ,

the j-th co@fficient

'k - ~ ,

in the Taylor

expansion

at O

THEOREM

2.2. Let W(h)

= D + C(~I

(A,B,C,D;¢n,~ m) is a m i n i m a l selfadjoint

operator

given

chain for A in h 0 E ~, ponding

pole

function

and

node.

by

for W(h)

- A)-IB,

where

8 =

Let H be the unique

(2.3).

let ~(h)

n

Let

invertible,

Xo,...,Xk_ i be a Jordan

= ~j=k

at h o. Then

(k-~o)J~j

be a corres-



has

Ran

855

a zero the

of o r d e r

following

at

least

k at

A

and

0

its

Taylor

=

= (A-Io )k + . . . + < H X k _ l , X k _ l >

of order

at

has

a zero

The

fact

So,

by

Since

W(A)~(~)is

least

k at

of o r d e r

that

h

at

O'

(2.3)

also

least

is real

O

formulas

h

analytic

:

(2.6)

we know

k at

and

On the

other

hand

(2.8)

that

by

desired

In this (3.1)

us c o m p u t e Further,

O

< H x i , X k _ l >. PoXk_l =

<~j,C(A-lo)J-i-lXk_l>

C X k _ ~ + i : (W~)k_~+i,

<~j'

(W~

:

Xkl I "

=

so

)k-j+i>"

Z

( Z

Z:k

\v=k

(2.7)

<(W~)£_v,~

>](l-ho "

)£.

Xk_2,Xk_l > =

: < H X k _ l , X i >,

formulas

section =

we

assume

Im

+ C(ll n

8 : ( A , B , C , I m ; ¢ n , ¢ m)

factors

Let

= P*H.

~(K)>

and

so < H x i , X k _ l > is real (2.8),

one

gets

the

FACTORIZATIONS

w(~)

We are

O


a zero

result.

3. M I N I M A L

where

function

has

, we have

=
comparing

and ~({)

O

+ ...

we have:

~ Then,

A

(A-ho)2k-i

<~j'CXk-j+i>"

that

at

ho" HP

(2.5

k+i = Z j:k

(2.7)

Note

Z

j=k

the

implies

= ~ k+i j=k

From

has

form:



PROOF.

expansion

interested which

and W 2 are

are

assumed

that -

A)-IB,

is a m i n i m a l ,

in f a c t o r i z a t i o n s of the

same

type

to be r a t i o n a l

W(A) as W,

finite

dimensional

: WI(A)W2(h) i.e.,

m × m matrix

the

node.

into

factors

functions

W1

that

Ran

856

are analytic tion

at ~ and have

is called minimal

more

degree

[1],

Section

4.3).

factors

we have

the

All minimal (3.1)

of the minimal 1.1)

projection

node

that

(3.2) Every

supporting

factorization

same d e f i n i t i o n

to

degree

of

of W in three

or

of minimality. matrix

of the supporting

0 : (A,B,C,I

~cn,gm).

Recall

~ of cn is called

function

projections (see

[1],

a supporting

0 if

7] a Ker ~,

projection

of W(1)

of W is equal

of the rational

a projection

A[Ker

degree

For a f a c t o r i z a t i o n

in terms

for the node

at ~. Such a factoriza-

m

of W I and the M c M i l l a n

factorizations

can be d e s c r i b e d

Section

I

if the M c M i l l a n

the sum of the M c M i l l a n W 2 (cf.

the value

A×[Im

7] c Im 7.

~ of 0 gives

in the f o l l o w i n g

rise

to a minimal

way:

w(~) : [Im+C(~-A)-I(I-~)B][I m+C~(I-A)-IB]. Conversely, this way Theorem

every

minimal

to a unique

supporting

function

W(h)

similarity

this

is selfadjoint,

between

THEOREM

section

3.1.

8 and

node

minimal

factorization

supporting

for

W,.

projection

corresponds 8 (see

in

[I],

W,

formula

= K(h)L(h)

H denotes

projection

W : K.L,

for the

Then T ~s a

to the f a c t o r i z a t i o n

node whose

projection

is minimal.

transfer

to check

for 8" c o r r e s p o n d i n g

Now use that W is selfadjoint

between

is a s u p p o r t i n g

8 and

8"

projection

It follows for

function

for 8 c o r r e s p o n d i n g

it is s t r a i g h t f o r w a r d

projection

the

be the c o r r e s p o n d i n g

this f a c t o r i z a t i o n

8" is a minimal

matrix

(2.3)).

for 8 c o r r e s p o n d i n g

= L,K,.

similarity

H-I(I-w*)H

symbol

the rational

of W. Put T = H-I(I-w*)H.

is a supporting

factorization H is the

and the

z is a supporting

to the f a c t o r i z a t i o n l-w*

that

In particular,

The node Since

we assume

8* (cf.

8, and let W(h)

= L,(h)K,(1). PROOF.

that

projection

Let ~ be a supporting

minimal

equals

of W(1)

4.8).

Throughout

W(1)

factorization

that

~ and that

to the

and that • =

W : L,K,

Ran

857

is the

corresponding

THEOREM

factorization.

Let

3.2.

~ be a supporting p r o j e c t i o n for the minimal

node @, and let W(1) = K ( 1 ) L ( h ) be the c o r r e s p o n d i n g factorization. Then there exists a r a t i o n a Z m (3.3)

W(1)

is a minimal [Ker ~]±

Let

w].

zation

W(A)

forward

In that

to c h e c k

for

if and

H-l[Im

7] ± and

H[Ker

of the

theorem

except

N : L ~ I W L -I, To function

Ker

T]

= [Ker

N : N,.

7]

and

a [Im ~]±

3.1 we k n o w

@ corresponding

N(A)

T = H-I(T ~-K * )H.

that

that

N : N..

details) only

It

This

the

factori-

It is s t r a i g h t -

that

K.(k)

= N(A)L(A)

that

Im T =

T.

follows

K] ! .

that

if

~ n Ker

But

to the

= K . ( A ) L ( A ) -i.

(see [i!] for more

Im ~ c Im ~, we u s e

i~ H [ I m

)H. F r o m T h e o r e m

Put

factorization

(3.4)

case

projection

= L.(A)K.(A).

is a m i n i m a l

Next

if and only

T = H "(i-K

< is a s u p p o r t i n g

such

= L,(1)N(h)L(~)

factorization

a HIKer

PROOF.

× m matrix function N(A)

proves

latter

all

fact

statements

is

clear

from

and W : W,.

find m i n i m a l (3.1)

factorizations

we have

the m i n i m a l

node

nonnegative

and m a x i m a l

to c o n s t r u c t

@. Here

and A ×, r e s p e c t i v e l y

of the

we

shall

(cf.

[6],

supporting

use

nonpositive

rational

the

projections

existence

subspaces

Section

matrix

V.2,

of m a x i m a l

invariant

and

for

[3],

for A

Theorem

IX.7.2).

THEOREM

3.3.

subspace,

and

subspace.

Then {n

So,

Let M 1 be an A-invariant,

let M 2 be an A×-invariant,

nonnegative

nonpositive

: M1 • M 2 .

if ~ is the p r o j e c t i o n

a supporting

maximal

maximal

projection,

of ~n along

and hence

M 1 onto

W admits

M2,

then

a minimal

~ is

factori-

Ran

858

zation

W(/)

maximal

sets

contain

pairs

in ~ ( A )

that

PROOF. M i is

On

M I the

First

x c M 1 and

respectively, numbers,

the set of non-real

we

show

nonnegative Schwarz

if ~ and a × are a r b i t r a r y

conjugate

the set of n o n - r e a l

As

Since

and d ( A × ) ,

of c o m p l e x

M 1 and M 2 such o, and

Moreover,

: WI(1)W2(1).

and

zeros that

M 2 is

inequality Ax

~ Mi,

poles

M1 n M2 :

(0).

nonpositive, for

the

for

shows

all

x

that

c H 1 n M 2 we

for

all

~×.

Choose

we

x

have

c M 1 n M 2.



H-inner

: 0.

product.

get:

II 2 ~ . So

do not

of W 2 is the set

of W 2 is the set

holds

we

which

then one can choose

have



x c M 1 n M 2 we

: 0.

: 0.

have

In

the



same

: 0.

way

It

one

follows

that:

o : <~{x,x> and x

hence

H I n M 2 c Ker

{ M I n M2,

and

all

x

and

so

C.

: <~BCx,x>

But

M l n M 2 is

{ M l n M 2 we

have

then

A×x

= = Ax-BCx

A-invariant.

cAnx

: 0.

= llCxll 2,

-- A x

Hence

for

for all

all n

-> 0

So

oo

M1 n M2 c

N

Ker

CA J

:

(0).

j=°0 Now maximal from

from

Chapter

from

this

Next that

M 1 and

o(A × M2)\R of

non-real

real

let

zeros

follows.

we

V,

get

Cn

the

4.4,

it

1.2,

we

same

dimension

fellows

know

that

that

every

as

MI,

and

H-1M2

is

be

of

H-IM

as

chosen of

in

the

such

= n-

theorem.

that

minimality

W 2 equals

W 2 equals

= dim

[6]

we

know

: ~

and

8,

the

im

= M 1 • M 2.

o × be

= 0 × . Because

of

has

Theorem

M i = dim

M 2 can

Lemma

Hence:

o and

poles

IX,

subspace

nonnegative.

dim and

Chapter

nonnegative

[3],

maximal

[3],

~(A of

a ( A I M i) \ ~

o ( A × M2 ] k W .

From

From M1)\~

the

node

and

the

set

this

the

theorem

of

set non-

Ran

859

A result

for m a t r i x

proved

in

finite

dimensional

W(X)

[6],

Chapter

V.

See also

version

RATIONAL

In this

we consider

with

section

the p r o p e r t y

selfadjoint.

[10],

to T h e o r e m

Theorem

MATRIX

3.3 is

3 for an in-

case.

FUNCTIONS

rational

m× m matrix

functions

that

0 ~

Such a function

similar

in the p o l y n o m i a l

4. N O N - N E G A T I V E

(4.1)

the

polynomials

~ +~

is called

(y ~ ¢m, ~ ~ ~ ) .

non-negative

As in section

and is a u t o m a t i c a l l y

3 we assume

that

W(A)

is given

in

form w(~)

where

=

Im

+ C(hl n

8 : (A,B,C,Im;{n,@ m)

the unique

similarity

A)-IB,

is a m i n i m a l

between

the nodes

node.

Further,

H will

8 and

8" (cf.

formula

be

(2.3)). THEOREM mal node mal

4.1.

8, and

Let

let W(h)

factorization.

Then

Ker ~ are maximal (4.2)

(4.2)

I ~n, and hence

if

imply

4.2.

(4.2)

G and

that W(h) ~*. Then

eigenvalues teristic

H[Ker

holds. both

Let W(~)

( A , B , C , I m ; ~ n , ~ m) is a finite assume

the minimini-

if Im ~ and

~]

Since

of T h e o r e m

3.2 that

H is invertible

Im w and Ker ~ have

= I m + C(hl n -A)-IB,

dimensional Let

minimal

H be

multiplicities

of A and A × are even,

of H and the A×-sign

= [Ker ~]±

the

dimension

neutral.

is non-negative.

the partial

if and only

from the proof

both are maximal

PROPOSITION

for

i.e.,

follows

that

projection

be the c o r r e s p o n d i n g

= L,(h)

7] = [Im ~]±,

It easily

K = L, if and only relations

= K(h)L(h) K(h)

H-neutral,

H[Im PROOF.

~ be a supporting

and both

where node,

0 = and

the similarity corresponding the A-sign

characteristic

between

to real

charac-

of H consist

of the

Ran

860

integers+2

only.

PROOF.

It is sufficient

For A x it follows negative. basis

to prove the p r o p o s i t i o n

by considering

Let io be a real eigenvalue

such that with respect

form (see

[7], Section

Xo,...,Xk_ i is a maximal of this basis,

Jordan

obtained

a Jordan

(A,H) has canonical

In particular,

this means

that if

chain for A in ~o' which is part

then is either

of +i's and -~'s, teristic

of A, and choose

to this basis

1.2).

for A.

W(1) -I, which is also non-

in this way,

+i or -i. The sequence is the A-sign charac-

of H.

Let Xo,...,Xk_ i be as in the previous paragraph, and let k+l . .. ~(I) = (I- lo)XO k + (h- l o) ~k+l + "'" o e a c o r r e s p o n d i n g pole function for W(1) at I (see Section 2). From T h e o r e m 2.2 we know O

that in a n e i g h b o u r h o o d

of I

O



we have: : (h-ho)kh(1),

where the scalar function h is analytic .

Consider

= +2. This proves

corresponding

THEOREM

4.3. Let W(1)

minimal node.

values of A (resp. (resp.

function

M~).

of +2's only.

factorizations

is analytic

: Im + C ( h l n - A ) - I B

where

of the type

at ~ and L(~)

= I

m"

be a non-negative

9 = (A,B,C,Im;~n,~ m) is a finite

Let M+ (resp.

of A (resp.

M~) be the image of the

A x) corresponding

to the eigen-

A x) in the open upper half plane.

N x) be an arbitrary

M+ (resp.

and that the

rational m x m matrix function W(h) the

such that L(h)

spectral projection

of A are even,

all minimal

rational matrix function, dimensional,

Using

that the partial m u l t i p l i c i t i e s

of H consist

For a n o n - n e g a t i v e next theorem describes = L,(h)L(1)

and h(l o) =

one sees that k is even and

to real eigenvalues

A-sign c h a r a c t e r i s t i c

W(h)

O

the real i in this neighbourhood.

the fact that W is n o n - n e g a t i v e

in I

A (resp.

AX)-invariant

Let N

subspace

of

Then there exists a unique rational m x m matrix

L(I) such that W(h)

lytic at infinity and L(~)

: L,(h)L(h)~

the function L is ana-

= Im, the factorization

W(h)

=

Ran

861

L,(I)L(1) for

is m i n i m a l

8 satisfies

(4.3)

the c o r r e s p o n d i n g

the f o l l o w i n g

Zm ~

Conversely,

and

M +× : N ×

N

all minimal

supporting

projection

identities: Kern

n M+

: N.

factorizations

of this

type

: L,(1)L(/)

is a m i n i m a l

are o b t a i n e d

in this way. PROOF. with

Suppose

L analytic

ponding such

W(/)

at i n f i n i t y

supporting

that

of the

the

N and

to

show

that

there

that

AM I c ML; A×M2

M 2 n M ~ = N × . Note and

(AX,H)

Further, number

again

According supporting Theorem

to W(1) (4.3)

uniqueness the

This

proves

the

last

part

theorem. unique

the

conditions

~ in

projection the The

[~2]

M~

{n

onto

that

from

the

construction

follows

from

the

uniqueness

M 2 such

n M+

: N,

i in

[L2].

case

the

~n.

So,

to

: Mi • M2 " Let

M2,

then

: Im,

for

w be

it follows

factorization

L(~)

follow

i in

the pairs

in this

is equal

corresponding fact

M~

4.2

of T h e o r e m 4.2,

we have

along

Theorem

M I and

and

of P r o p o s i t i o n

of P r o p o s i t i o n

in T h e o r e m

that

subspaces

HM i c M'±i (i : L,2)

HM. : M. ±. 1 l to T h e o r e m 3.3,

4.L

We apply

is equal

and

the

fact

of L(h),

and

the

of M i and M 2. This

that

proves

theorem. COROLLARY

rational mal node.

4.4.

matrix Let

Let W(/)

function,

: im + C ( l l n - A ) - I B

where

~ and o x be maximal which

do not contain

numbers.

there

exists

Then

with: ~) W(/) 2)

: L,(1)L(1)

sets

a unique

zeros

of L is o×,

in a ( A ) \ ~

pairs

and o(A x) \ ~ ,

of c o m p l e x

rational

is a minimal

the set of non real poles non real

be a n o n - n e g a t i v e

@ : (A,B,C ,Im; ~ n ,~ m ) is a mini-

respectively,

L(/)

: I .Let w be the c o r r e s m @. T h e n we can take N and N ×

hold.

because

satisfy

: L,(k)L(1).

holds

c M2,

because

we have

exist

that

v appearing

i : 1,2,

from

(4.3)

N × be as in the

[12]

the

for

theorem.

Let

(A,H)

and L(~)

projection

identities

factorization

conjugate

matrix fuction

factorization,

of L is o, and

the set of

Ran

862

3)

L(~)

PROOF.

= I

Take

m M x to be the

image

of the

spectral

projection

X

of A x c o r r e s p o n d i n g projection Apply

Theorem

supporting

4.3 with

projection

in the upper the

to o , and M to be the image

of A c o r r e s p o n d i n g this

choice

obtained

half plane,

lower half plane,

to 5.

then

then

the structure Im P(~o,A)

of such

P(~o,A)M

~)XR

Let ~ ~ ~. If ~ is o o c N c Ker 7. If ~o is in

= (0) and hence

that

subspaces

P(~o,A)Ker

H-neutral (cf.

= a. Indeed,

since

~ = (0).

Since

we can use

to conclude

In the

derive

P(~o,A)N = (0).

subspace,

[12])

M c Ker 7.

that M x c Im ~. From this we will and 0(AIKer

let ~ be the

in this way.

maximal

c Ker ~. Hence

of N and N x, and

Im P(~o,A)

From Ker ~ n M+ = N it follows Ker ~ is an A-invariant,

of the spectral

Put N = M n M+ and N x = M x n M+.X

same way

that

that

one gets

~(AX llm ~) \ ~

M x c !m ~ it is clear

= ox that

×

c o(AXllm there

7)k~.

Suppose

a pair

of eigenvalues

exists

and ~o

~ o(AXllm

7 ) k~.

this

Then

there exist

dicts

way one sees that set of non real

(4.3) • Hence o(AiKer

zeros

of non real poles the

from

[3],

x and y in Im ~ with

the r e l a t i o n

~)\~

of L(~)

of L(~)

inclusion.

~o' ~o of A x such

Chapter



~o ~ x

Since

!!, T h e o r e m

= I. This

= o. Because ~(AXllm

o(AiKer

3.5,

contra-

o × = o( A x Ilm ~ ) \ ~ " In the

equals

equals

Then

that

Im P(~o,A x) n Im ~ ~ (0).

Im P(~o,A x) c Im ~, it follows that

is a strict

same

of minimality,

the

7)\~,

set

7)k~.

and the

This

proves

corollary. Using

deduce

a standard

from Corollary

rational

matrix

Corollary

technique

functions

4.4 is somewhat

version

of

[13],

similar

to C o r o l l a r y

(see

4.4 a similar

Theorem

[l],

with n o n - v a n i s h i n g stronger 3.4.

than

A result

4,4 is proved

in

[13],

Theorem

3.3 for an infinite

From

[14],

Theorem

I, one can deduce

4.4 above.

the

1.5),

one

can

for n o n - n e g a t i v e determinant.

finite

for matrix [6], Chapter

with

than Corollary

section

statement

dimensional

a somewhat

dimensional polynomials VIII

(compare

version).

weaker

statement

Ran

863

5. THE

INFINITE

In this

DIMENSIONAL

section

a number

sections

will

Consider

the o p e r a t o r

be extended

(5.1)

W(1)

where

A

: X ÷ X,

operators we assume operator

that H

: Y ÷ X and C

between

there

this

Hilbert

exists

implies

The existence tor H

to the infinite

the

following

: X ÷ Y are bounded spaces.

Throughout

an invertible,

HB : C ,

case.

bounded

linear

this

section

selfadjoint

that W(1)

function

strong

(5.3)

C = B*H. = W(~)*

of an invertible,

: X ÷ X satisfying

versely,

dimensional

function

HA : A'H, that

from the previous

: X ÷ X such that

(5.2) Note

of results

= Iy + C(~I x - A)-IB, B

acting

CASE

(5.2)

W(1)

version

there

bounded

follows

exists

automatically

is selfadjoint

of m i n i m a l i t y a Jo such

selfadjoint

and,

if,

operacon-

in addition,

the

is fullfilled:

that

col

(CAJ)j_ J?0 and

•

col To see this 2.1,

using

one

(B*A*$)j

can apply

[9], T h e o r e m

As in Sections K(1)L(I) jection

there

of W, with (5.4) Suppose spaces

same

exists K(~)

K(~)

: L(~)

W(1)

as in P r o p o s i t i o n

study

: Iy.

factorizations

W(1)

If ~ is a supporting

A x

a corresponding

[Im ~]

c

Im ~,

factorization

W(1)

= K(1)L(1)

: Iy, namely

= [Iy + C ( I l x - A ) - I ( I - ~ ) B ] [ I Y + C~(IIx-A)-IB].

~ is a supporting

projection

Im ~ and Ker z satisfy

the

for

e, and

identities

let the

=

pro-

i.e.,

~] c Ker ~,

= L(~)

invertible

arguments

4 we shall

for 0 : (A,B,C,Iy;X,Y), A[Ker

then

the

left

3.b.1.

3 and

of W, with

are

sub-

Ran

864

(5.5)

H[Im

Then

the

W(h)

: L,(h)L(h).

to ~ is g i v e n

compute

L,(I

from

(5.5)

H[Ker

factorization

Indeed,

first

of all

by

(5.4).

Put:

(5.2).

One

that

= [Ker

the

w]±

is of the

form

factorization

L(1)

corres-

: Iy + C~(hl X - A)-IB,

finds

= Iy + C H - I ( h l x -

it follows

w]

of W(I)

, using

L,(~) Now

: [Im ~]~ ,

corresponding

ponding and

w]

HAH-I)-I~*HB.

~ : H-I(I-z*)H.

So

L.(~) : Iy+C<~ZX-A)-~
Hence

= L,(A)L(A).

Before

we

come

introduce

extensions

Let

denote

g lim

defined

to the m a i n ~ and

a fixed

for b o u n d e d

vector

space

Clearly,

this

on B. Let and

define

in the denote

operator (nA ~~)

then

Its

the in X.

class

continuous

way

completion

operator

extension

if M is a c l o s e d M may

Recall ~

to X

functional

of B/N.

x : (Xn) n of B,

of x in B/N.

a bounded

that

= 0, Define

let ~ = (Xn) n that

a bounded

on B/N by A(Xn) n : (see

[2])

is also

linear

subspace

of X,

with

a closed

linear

be i d e n t i f i e d

of X.

LEMMA operator

Consider

bilinear

x { B such

space

an e l e m e n t

Further,

in a n a t u r a l

subspace

For

A on X induces

by A.

[2]).

[2]),

(Xn) n of e l e m e n t s

symmetric

of all

Hilbert

equivalence

on B/N.

denoted

(see

(cf.

c B define

negative

subspace

way.

limit

we

: g l i m < X n , Y n >.

X to be the

same

the

section

numbers.

sequences

y : (Yn)n

is a non

N be the

generalized

of c o m p l e x

B of all b o u n d e d



of this

~ of X and Y, r e s p e c t i v e l y

Banach

sequences

If x : (Xn) n ~ B and

theorems

5.1.

on X,

an A - i n v a r i a n t ,

Let and

H be a s e l f a d j o i n t , i n v e r t i b l e a n d b o u n d e d let A be H - s e l f a d j o i n t

maximal

H-nonnegative

and b o u n d e d .

subspace.

Let

Consider

M be

M as a

Ran

865

subspace of X. Then M is an A-invariant, subspace. PROOF. bounded,

note

Section

1.11).

X_. Then

and there

exists

Im ( I ) ( s e e Take

Let

[3],

P+M = X+

it follows

P+M = X+.

l~deed, I inverse (K)"

inverse

decomposition

P+ be the f u n d a m e n t a l

Section

(see

[3],

operator

Chapter K

of X (see

projector

onto

V, T h e o r e m

4.2),

: X+ ÷ X_ such that

M =

11.11).

(K)

=

= g lim < H X n , X n >

_> 0.

that M is ~ - n o n n e g a t i y e .

the o p e r a t o r Then

I

P+ M

P + M = P+ M

"~-'(K)"Hence

P+M

= X+.

One easily

decomposition that

that

one gets

M is ~ - i n v a r i a n t

LEMMA

5.2.

that

: ~ + X+ is invertible,

Theorem

V. 4.2.

We claim

: M ÷ X+ is invertible,

~+ • X_ is a f u n d a m e n t a l fact

and

x = (Xn) n in M. Then

From this ~

invertible

and H - s e ~ ± a d j o l n t .

a bounded



with

H is selfadjoint,

X = X+ • X_ be a f u n d a m e n t a l

X+ along

~

that

and ~ is b o u n d e d

Let [3],

First

maximal H-nonnegative

of X. Again

~_, is" m a x i m a l

is easily

checks

with

that

using

~ =

[3],

H-nonnegative.

The

seen.

Let M 1 and M 2 be closed subspaces

of X. Suppose

M1 n M2 = (0). Then M I n M 2 = (0) and M I + M 2 is closed. PROOF.

Let

~ ( M 1 , M 2) = inf In order necessary Lemma

to show that and

(IIx+yll i x ~ Ml,Y ~ M2, max

(llxll,llyll) = i).

M i n M 2 = (0) and M 1 + M 2 is closed

sufficient

to show that

~ ( M I , M 2) > 0, see

it is [8],

1. Suppose

such that max

~ ( M 1 , M 2) = 0. Let

(Xln)n

c M 1 and

(X2n) n a M 2 be

(llxlnlI,llX2nlI) = 1 and X l n + X2n ÷ 0. Put ~ 1 =

(Xln) n and x 2 = (X2n) n. Then

x i # 0 and ~2 # 0, ~1 ~ MI'

~2 ~ M2 and ~1 + ~x 2 = (Xln + X2n) = 0. Hence is a c o n t r a d i c t i o n . Hence D ( M 1 , M 2) > 0. Now we come to the main

theorems

~1 ~ Mi n M2' which

of this

section.

Ran

866

THEOREM

5.3.

let P (resp.

Assume ~(A)

n ~

corresponding to the part of a(A) half plane.

: ~ and o(A x)

pX) be the spectral projections

: ~, and

n ~

of A (resp. A x)

(resp. o(AX))

in the upper

Then X = Im pX • Ker P, and W factorizes W(1)

as follows:

: L.(1)L(1),

where L is an operator function which is analytic and invertible in the closed lower half plane, an

including ~. Moreover,

if G is

operator function with the same properties as L, then there

exists a unitary operator U such that G(h) PROOF.

As b o t h

= UL(h)

A and A x are H - s e l f a d j o i n t

for all h.

(see

(5.2)),

we

have:

(5.6) First

we

show

the S c h w a r z

H[Im

pX] : [Im pX]±

that

N = Im pX

inequality

HIKer

n Ker

P = (0).

on Ker P one

in the

same way



Further in the

:

Ker

lower

open

upper

P n Im pX

Using

half

:

and the

consider

the

Also

and A X - i n v a r i a n t .

~(AIN)

c ~ ( A I K e r p), w h i c h

~ ( A x N ) c ~ ( A X l l m pX),

p l a n e " As A x N = A IN' this

(5.6)

P ¢Im

one

pX)±

invertibility

Ker P • Im px Consider

: O,

means

is

which that

(o).

the r e l a t i o n s H-l(Ker

F r o m this

A-invariant

plane.

half



: [Icxll2.

N is b o t h

A : A x on N. Now open

is in the N

C. H e n c e

x c N. U s i n g

= 0. H e n c e

o : So N c Ker

Take

sees

II 2 s < H A x , A x > . and

P] : [Ker P]±

node

sees : Ker

that P n Im pX

of H it

follows

= (0).

that

= X.

~ : (A,B,C,I;X,Y),

where

X and ~ are

the

Ran

867

extensions

of X and Y, r e s p e c t i v e l y ,

Then,

the o p e r a t o r

with

are ~(A)

satisfied. n ~

H induced

Furthermore,

: ~ and ~(~x)

n ~

which

were

introduced

by H, the relations

since

~(~)

= ~(A),

before.

(5.2)

for

we have

: ~. Then ~ and ~x can be obtained ×

in the same way as P and P , r e s p e c t i v e l y , projections

corresponding

to A and ix,

i.e.,

as spectral

respectively.

Then,

in

×

the

same way as Ker P n Im P

Using

Lemma

5.2 we get

that

(0),

:

one

sees

Ker ~ n Im ~x

Ker P • Im pX is closed.

Hence

=

(0).

X =

Ker P • Im pX. Let w be the p r o j e c t i o n a supporting beginning

projection

of this

factorization outside

is analytic including

form W(1)

p) and invertible in the

Im P×.

Then ~ is

from the d i s c u s s i o n that

the

at the

corresponding

= L.(X)L(1).

Now L is analytic

outside

Im pX).

closed

~(AX

Hence

G is an o p e r a t o r for all

lower half plane,

function

with

I in a n e i g h b o u r h o o d

the

same properties

of the real

axis,

in-

~, we have

(5.7) left hand

in the closed and invertible is an entire, extended

G.(X)-IL,(I)

: G(h)L(X) -1 .

side

equality

of this

upper

half plane,

in the

closed

everywhere

complex

plane,

G(1)L(1) -i is unitary. Next,

invertible

we generalize

hand

operator

is analytic

and from This

is analytic

the right

lower half plane.

which

G(1)L(1) -i is constant,

(5.7)

proves

Theorem

and invertible side

is analytic

So G(h)L(h) -I

function

on the

at infinity.

we conclude

Hence

that

the theorem.

3.3 to the infinite

dimensional

case. THEOREM

function,

5.4.

Let W(h)

: Iy + C ( I I x - A ) - I B

where 8 : (A,B,C,I;X,Y)

strong minimality maximal

L

infinity.

as L, then

The

Ker P onto

8, and

it follows

and invertible

Assume

cluding

section

is of the

~(AIKer

along

for

condition

nonnegative

(5.3).

subspace,

be an operator

is a node satisfying

the

If M& is an A-invariant,

and M 2 is an AX-invariant,

Ran

868

maximal

nonpositive

projection for

along

e, and hence

K(~) = L(~) PROOF.

subspace,

then X = M 1 • M 2. Let

W admits

W(h)

a factorization

= Iy, c o r r e s p o n d i n g

H-1M~

= K(k)L(h),

with

to ~.

As in the first part of the proof of Theorem

one shows that M 1 n M 2 = (0). Since H - 1 M ~ maximal

~ be the

M I onto M 2. Then ~ is a s u p p o r t i n g p r o j e c t i o n

3.3

is A-invariant

and

nonpositive (see [3], Chapter V, Theorem 4.4), and A x is -invariant and maximal nonnegative we also have

n H invertible

<01 this implies

Consider observable.

Now H IM

nHIM

I(MIoM2

that X = M 1 • M 2.

the node ~ = (A,B,C,I;X,Y).

Indeed,

col

Since H is

We claim that ~ is

(CA j)jj°=0 is injective

and has closed

range. Hence 0 is not in the approximate point spectrum of col (CA j )j=0" Jo From Theorem 1 in [2] it follows that 0 is not in the approximate col

point

spectrum

(~AJ)j=0J° is injective, From Lemma

nonnegative,

and

of col

which

( ~ J )j=0j jo Hence

implies

that

5.1 we know that M1 is A-invariant M2 is ~×-invariant

Using the observability

and maximal

of ~ one can prove

using Lemma

and maximal

nonpositive.

in the same way

as in the first part of the proof of Theorem (0). Finally,

¢ is observable

3.3 that M1 n M2 =

5.2, we get that M 1 • M 2 is closed.

Hence X = M 1 • M 2. This proves

the theorem.

Acknowledgement I am greatly indebted to I. @ohberg, M.A. Kaashoek and L. Rodman for many valuable suggestions, and many discussions on the contents of this paper.

REFERENCES H. Bart, I. Gohberg, M.A. Kaashoek: Minimal factorization of matrix and operator functions; Operator Theory: Advances and Applications, Vol. 1; Birkhauser Verlag 1979. S.K. Berberian: Approximate 13 (1962), 111-114.

proper vectors;

Proc.

A.M.S.,

Ran

869

J. Bogn~r: Indefi~ite inner product spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78; Springer Verlag 1974. .

R.W. Brockett, R.A. Skoog: A new perturbation theory for the synthesis of nonlinear networks; Mathematical aspects of Electrical Network Analysis, IAM-A.M.S. Proceedings V. 3, 17-33.

.

P.A. Fuhrmann: On symmetric rational transfer functions; Preprint, Dep. of Math. Ben Gurion University of the Negev, 1981.

.

I. Gohberg, P. Lancaster, L. Rodman: Spectral analysis of selfadjoint matrix polynomials; Research paper no. 419, Dep. of Math. and Stat., University of Calgary, 1979.

.

I. Gohberg, P. Lancaster, L. Rodman: Spectral analysis of selfadjoint matrix polynomials; Annals of Math.; 112 (1980), 33-71.

.

I.C. Gohberg, A.S. Markus: Two theorems on the gap between subspaces of a Banach space; Uspehi Mat. Nauk; 14 (1959), 135-140,[Russian].

,

J.W. Helton: Discrete time systems, operator models and scattering theory; J. of Funct. Anal.; 16 (1974), 15-38.

10.

H. Langer: Factorization of operator pencils; Acta Sci. Math.; 38 (1976), 83-96.

11.

A.C.M. Ran: Factorization of selfadjoint rational matrix and operator functions; Rapport hr. 173, Wiskundig Seminarium Vrije Universiteit Amsterdam, 1981.

12.

L. Rodman: Maximal invariant neutral subspaces and an application to the algebraic Riccati equation.

13.

M. Rosenblum, J. Rovnyak: The factorization problem for nonnegative operator valued functions; Bull. of the A.M.S.; 77 (1971), 287-318.

14.

L.A. Sahnovic: On the relation between the theory of differential equations and factorization problems: Siberian Math. J., Vol. 21 nr. 5 (1980) 169-173 [Russian].

A.C.M.

Ran

Wiskundig Seminarium Vrije Universiteit Amsterdam,

Submitted:

The Netherlands

January 31, 1982