Operators and Function Theory (NATO Science Series C)

-f

k(t-s)<j>(s)ds = f (t) ,

0

~

t <

00

0

As before, in (1.1) the function k(t) is an mXm matrix function of which the entries are in Ll (~,oo). We assume that the symbol weAl is rational. We shall combine the canonical factorization theorem of Section 1.7 with the general factorization method outlined in Section 1.1 to solve equation (1.1). The first step is to make a minimal realization of the symbol: 0.2)

Since weAl is continuous on the real line, the minimality of the realization (1.2) implies that A has no ~igenvalues on the real line. x

Next, assume that also A = A-BC has no eigenvalues on the real line (which is equivalent to the requirement that det WeAl ~ 0, ~ < A < 00) • x x Let M (resp., M ) be the spectral subspace of A (resp., A ) x corresponding to the eigenvalues of A (resp., A ) in the upper (resp., x lower) half plane. Assume X = M $ M , where X is the state space of the realization (1.2). Put I + CIT(A-A)-l B,

~

<

A< ~

<

00,

A<

00,

where IT is the projection of X along M onto MX. Then weAl = W_(A)W+(A) is a canonical factorization relative to the real line of the symbol of (1.1) (see Theorem 1.7.1). It follows that (1.1) is uniquely

57

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

solvable in Lm[O,OO)

(see Section 1.1). To find the resolvent kernel we 1 -1 have to compute the inverse Fourier transforms of W (A)- and W+(A) . To do this one first checks that p

-<XJ

< A<

00,

x Next, using the special properties of the spaces M and M , one proves that o x W (A)-l I + f ieiAtce -iA SrBdt, 00

W+(A)-l

=

I -

f

x ieiAtC(I_TI)e-iA t dt .

o Finally, one applies formula (1.1.7) the resolvent kernel.

~o

obtain from these identities

Let WeAl = I + C(A-A)-l B be a minimal realization of the symbol of equation (1.1), let x be its state space, and let AX = A-BC.

THEOREM 1.1.

In order that for each f E Lm[O,OO) equation (1.1) has a unique solution p ~ E Lm[D,oo) it is necessary and sufficient that p

(i)

AX has no real eigenvalues,

(ii)

x

x

= M $ M ,

where M (resp., MX) is the spectral subspace corresponding to the .

x

eigenvalues of A (resp., A ) in the upper (resp., lower) half plane. In that case for a given f E Lm[O,OO), the solution of (1.1) is given by 00

~(t)

= f(t)

where (1.2)

+

f

p

y(t,s)f(s)ds,

o :,;

t <

00,

o

, _itAx iSA x 1Ce TIe B, 0:,; s < t < 00, { x x _iCe-itA (I_TI)e isA B, o :,; t < s <

yet,s)

00,

x

with TI being the projection of x along M onto M • Let us illustrate the above theorem for the following simple equation: (1.3)

~(t)

-

~

00

f

e-lt-sl~{S)dS

= f (t) ,

o :,;

t <

00.

0

Here c is a positive real constant and the given function f, the unknown

58

M.A.KAASHOEK

~

and the kernel k are scalar functions. To solve (1.3) is to construct a realization of its symbol:

the first step

Put

[~

A

Then wO.)

C

[-c

oJ.

1 + C(A-A)-l B is a minimal realization and

A

x

A -

BC

x

It follows that A has no real eigenvalues if and only if 0 < c < 1. Therefore, assume 0 < c < 1. Let M and MX be as in Theorem 1.1. One computes that

It follows that ~2 = M $ MX, and thus equation (1.3) is uniquely solvable in L [0,00). The projection IT of ~2 along M onto MX is given by p

IT

1

a+l

1

[

-ia

i] a'

where a = 11-c. Using this in (1.2) one finds that for 0 < c < 1 the resolvent kernel of (1.3) is equal to yet,s) =

2~

{(1_a2)e-a/t-sl - (1_a)2 e -a(t+s)},

where, as before, a = 11-c . Condition (i) in Theorem 1.1 is equivalent to the condition that the symbol weAl admits a (possibly non-canonical) Wiener-Hopf factorization, which, in turn, is equivalent to the requirement that the integral operator I-K corresponding to the left hand side of (1.1) is a Fredholm operator on Lm[O,OO). Furthermore, in that case, using the p

formulas for the factorization indices given at the end of Section 1.7 one deduces: x x dim --x- , dim M n M , codim Im(I-K) dim Ker (I-K) M+M and thus

59

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

index (I-K) where P and P

x

x

rank P - rank P ,

are the Riesz projections corresponding to the eigenx

values of A and A 11.2.

=

in the upper half plane (see formula (1.7.4».

A half space problem in transport theory

Wiener-Hopf integral equations appear in a natural way in various half space transport problems. In fact, the simple equation in the example of the previous section is used to solve the problem ~f radiation diffusion in a one dimensional medium (see [36J, p. 310). The Milne problem, which is one of the classical transport problems in Astrophysics, was the starting point for the original Wiener-Hopf paper [39J. However, in general, the symbols of the Wiener-Hopf equations that appear in transport problems are non-rational. Nevertheless, with some modification, the general approach of the previous section also works in this context. To illustrate this consider the following plane symmetric version of the one-dimensional linear transport equation:

a~

]lax (x,]l) + ~(x,]l) =

(2.1)

f1

g(]l,]lI)~(X,]lI)diJ',

-1 ~]l ~ 1, x ~ O.

-1

One thinks of (2.1) as a balance equation which describes the timeindependent transport of particles through a homogeneous, semi-infinite medium. The unknown function ~ represents an averaged density of the particles. The integral in the right hand side of (2.1) describes .the scattering in the medium and the corresponding scattering kernel g is real symmetric. The variable x is the optical depth, which is a position coordinate inside the medium. Further, ]l = cos ~, where ~ describes the direction of the velocity vector. The problem is to find the unknown function ~ under certain boundary conditions. Here we shall consider the so-called half space problem in which the unknown function ~ is required to satisfy (2.2)

with

~+

given, and a certain growth condition for x

+

00

which we shall

specify later. Condition (2.2) tells us that the density of the incoming particles is known. By writing ~(x) (]l) = ~(x,]l), we may consider the unknown function ~ as a vector function on 0 ~ x ~ 00 with values in the Hilbert space L2 [-1,lJ. This allows one to rewrite equation (2.1) as an operator differential equation: (2.3)

T

~(X) + ~(x)

=

BO~(X) ,

o :;

x <

00,

Where the derivative is taken with respect to the norm in L2 [-1,lJ. The

60

M.A.KAASHOEK

operators T and BO' which act on the space L 2 [-1,lJ, are defined by 1

(Tf) (].l)

].If(].l),

(BOf)

(].l)

f

=

g(].l,].l')f(].l)d].l.

-1

Note that because of the symmetry conditions on the scattering kernel g the operator BO is selfadjoint. In what follows we shall assume that I-B O is strictly positive, which in the transport theory means that we are dealing with an absorbing medium. The equation (2.3) has to be considered together with the boundary (2.4)

Here P+ is the orthogonal projection of L2 [-1,lJ onto the space consisting of all functions that vanish a.e. on [-l,OJ and ~+ is a given function in L2 [-1,lJ which is zero on [-l,OJ. If one assumes, as a boundary condition at infinity, that the unknown function Wis (Bochner-) integrable on [0,00), then (2.3) together with the boundary condition (2.4) is equivalent to a WienerHopf integral equation: 00 (2.5)

w(x) -

f

o

k(x-y)w(y)dy

with an operator-valued (2.6)

W(A)

=I

L 1 -kernel,

- (I-iAT)

-1

o

= f(x),

$

x < 00,

of which the symbol is equal to ~

BO'

< A < 00.

The right hand side of equation (2.5) is given by

f (x) (].l)

{ ~ + (].l) exp( - fr) , o

,

-1 $ ].l < O.

To solve (2.5) we want to construct, if possible, a right canonical factorization of the symbol W(A). To do this we employ the method of Theorem 1.7.1 although here W(A) is not rational. First of all we rewrite the symbol W(A) as a transfer function by setting: 'V

W(A) = W(-

1 x-)

(I-B O)

-1

= I + C (A-A)

-1

B,

where (2.7)

A =

-iT,

C

=

A.

Note that the invertibility of I - BO is guaranteed by our hypothesis that I - BO is strictly positive. The operators A, Band C are operators

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

61

acting on the infinite dimensional Hilbert space L2 [-1,1] and the associate operator

x A

(2.8)

=A

- Be

=

(I-B O)

-1

A.

x Next, we want to find the spectral subspaces M and M . The main x

difficulty here is that the spectra of A and A both cut through the real line at zero. However, since A = -iT it is natural to take for M the spectral subspace of T corresponding to the spectrum of T on the negative part of the real line. In other words, we take M = Ker P+. Since I - BO is strictly positive, the operator iA

x

is selfadjoint with

respect to the equivalent inner part 1

[h 1 ,h 2 ] =

f

-1

((I-B O)h 1 ) {t)h 2 {t)dt.

x So there is also a natural choice for the space M , namely the spectral x subspace of the selfadjoint operator iA corresponding to the positive x part of the real"line. Let M and M be chosen in this way. Then the next important step is to prove the decomposition: (2.9)

Assume (2.9) holds true. Then W{A) admits a canonical factorization relative to the real line and the factors can be constructed in precisely the same way as in Theorem 1.7.1. It follows that W{A) admits a canonical factorization. In fact, let IT be the projection of L2 [-1,1] x along M onto M and put -1

I - iATIT{I-iAT)

W+{A)

I - BO - iAT{I-iAT)-l{I-IT)B O•

-

Then W{A)

-1

W (A)

= W_{A)W+{A)

(I-B O)

BO'

is a right canonical factorization of the symbol

W{A). On the basis of this factorization, using the classical method described in Section 1.1, it is not difficult to prove that equation (2.5) has a unique (Bochner-) integrable solution ~ on 0 $ x < 00, which is given by -1 -xT (I-B O) ~(x) e IT+. The matching of the subspaces in (2.9), which is the main problem here, can be proved for a large class of scattering kernels g. For example, if n g

(ll, II ' )

~

j=O

a j P j (ll)Pj (ll') ,

62

M. A.

where p.

J

(~)

KAASHOEK

is the j-th normalized Legendre polynomial and a. < 1 J

(j = O,l, ••. ,n), then (2.9) holds true (see, e.g., [26J). In [33J the decomposition (2.9) is proved for all scattering kernels g such that I - BO is strictly positive and BO

=

ITlaD for some 0 < a < 1 and some

bounded operator D on L2 [-1,lJ. 11.3.

Wiener-Hopf equations with analytic symbols

In this section the symbol weAl of the equation (1.1) is assumed to be analytic in a strip around the real line, i.e., for some positive number h the function weAl is analytic on 11m AI < h. The Wiener-Hopf equation considered in the previous section is an example of an equation with such a symbol. In fact, the symbol weAl in (2.6) is analytic on 11m A I < 1. To extend the theory of rational symbols discussed in Section 11.1 to analytic symbols, the first problem is to realize weAl as a transfer function, i.e., to construct a representation (3.1)

weAl = I + C(A-A)

-1

B,

Since weAl is non-rational one cannot expect a realization (3.1) with a finite dimensional state space. To solve the realization problem one has to distinguish between two cases. First, assume that the symbol weAl is analytic on the real line and at infinity. So there exists an open set ~ on the Riemann sphere such that. the real line and the point infinity are in ~. Let X be the linear space of all ~-valued analytic functions on ~ such that II fez) II Illflll

sup ZE~

< co, max(1,llw(z)ll)

and consider the following operators: A: X + X, (Af) (z) = zf(z) - (z-a)f(co); B: a:m + X, (By) (z) = (z-a)[W(z)y - yJ; C: X + a:m, Cf = f(co). Here a is an a~bitrary complex number in a: \ ~. Then the space X endowed with the norm 111.111 is a Banach space, the operators A, Band Care bounded linear operators, ~ \ {co} belongs to the resolvent set peA) of A and (3.2)

weAl = I + C(A-A)

-1

B,

A€

~ \

{co}.

In particular, the operator A in (3.2) has no spectrum on the real line. We conclude that a symbol weAl which is analytic on the real line and at infinity admits a representation of the form (3.1) with A, Band

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

63

e bounded linear operators, where the operator A, which act~ on an infinite dimensional Banach space X, has no spectrum on the real line. As soon as one has such a representation for the symbol one can analyse equation (1.1) in essentially the same way as in the rational case. For M one takes 1m P, where P is the Riesz projection corresponding to the part of the spectrum of A in the upper half plane, and one proves that equation (1.1) is uniquely solvable in Lm[O,OO) if and only if AX = A-Be p x x has no spectrum on the real line and X = M eM, where M = Ker P and x x P is the Riesz projection of A corresponding to the part of the spectrum in the upper half plane. The formula for the resolvent kernel is then the same as in the rational case. Next, we drop the condition of analyticity at infinity. Then, in general, the symbol weAl cannot admit a representation (3.1) with A, B and e bounded. Indeed, if (3.1) holds true with A, Band e bounded linear operators, then the Neumann expansion of the resolvent of a bounded linear operator implies that weAl is analytic at infinity. So, in this case the best one can hope for is a realization with unbounded operators. For the symbol of the Wiener-Hopf equation in the transport problem of Section 11.2, a realization with unbounded operators comes automatically from the data of the problem. In fact, (2.6) may be rewritten in the form: WeAl = I - iT

-1

(A+iT

-1 -1

)

BO'

where T- 1 is the unbounded inverse of T. In the general case the realization has to be constructed. Here we shall present the solution of the realization problem for a symbol that is analytic in a strip around the real line under some additional conditions on the kernel k(t). The conditions we shall work with are sufficient but not necessary for all considerations that follow. We shall require that on E \ {a} the kernel k has a continuous second derivative which is exponentially decaying and has at most a jump discontinuity at zero. These conditions imply that there exists W > 0 such that (3.3)

j

0,1,2.

In particular, the s¥IDh0l weAl of the corresponding Wiener-Hopf equation is analytic in 11m AI < w. Now, let X be the linear space of all ~-valued functions f on

E \ {a} such that eWitif(t) is bounded and uniformly continuous on E \ {a}. The space X endowed with the norm

III f III

sup e wt IIf(t) II t#O

is a complex Banach space. Let A be the unbounded operator 1d~ on X.

64

M. A. KAASHOEK

Thus the domain V(A) of A consists of all f € X such that f is differentiable and f' € X, and for such a function Af = if'. Further, define B:

,fI1

+ X,

(By) (t) = k' (t)y, 00

C: X + a;m,

Cf = i f f (t) dt. _00

The operator A is a densely defined closed linear operator such that the strip 11m AI < W is in the resolvent set of A, the operators Band Care bounded and by a direct computation one checks that (3.4)

I + C(A-A)-1 B ,

W(A)

11m

AI < w.

Note that the operators CA and AB are also bounded. The main operator A appearing in (3.4) has several interesting properties. To describe these let P be the projection of X defined by (Pf) (t)

=

f (t)

(t < 0),

(Pf) (t) = 0

(t

> 0).

The projection P commutes with A, and hence relative to the decomposition X = 1m P $ Ker P the operator A can be written as an 2x2 diagonal operator: (3.5)

A

0 1 (1m P A+

[A_

o

$

It turns out that -iA+ and iA

Ker P + 1m P

$

Ker P).

are generators of exponentially decaying

Co-semigroups. In fact: (e -i~f) (s)

f (t+s)

(t,s > 0),

(e -itA_f) (s)

f (t+s)

(t,s < 0).

Any closed densely defined operator A on X of the form (3.5) with -iA and iA generators of exponentially decaying Co-semigroups will + be called a bigenerator. If A is a bigenerator, then the projection P of X that induces the partioning (3.5) is uniquely determined by A and is called the separating projection. A bounded linear operator A on X is a bigenerator if and only if A has no spectrum on the real line, and in that case the correspon~ing separating projection is precisely the spectral projection associated with the part of the spectrum of A in the upper half plane, that is, P = __1__ . 2m.

f

r

(A-A)-1 dA ,

where r is a suitable contour in the upper half plane around the spectrum of A in the upper half plane.

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

65

In order that the symbol wO.) admits a realization

THEOREM 3.1. (3.6)

-CO

< A<

00,

where A is a bigenerator and B and c are bounded linear operators such that AB and CA are bounded it is necessary and sUfficient that on ~ \ {a} the kernel k has a continuous second derivative which is exponentially decaying and has at most a jump discontinuity at zero. In that case =

{

k (t)

(3.7)

iCe

-itA . (I-P)B,

_ice-l.tApB,

where

P

t

> 0,

t

< 0,

is the separating projection of

In (3.7) the expression e

-itA

A.

(I-P) has to be understood as the semiqroup

(I-p)e-i~(I-P). Similarly, e-itAp

= pe-itA-p. Theorem 3.1 solves the realization problem. The next step is to use the realization (3.6) to analyse the equation (1.1).

THEOREM 3.2. Let W(A) = I + C(A-A)-l B be a realization of the symbol of equation (1.1), where A is a bigenerator on X and B and C are bounded linear operators such that AB and CA are bounded. Put AX = A - BC. In order that for each f E Lm[O,OO) the equation (1.1) has a unique solution p , E Lm[O,OO) it is necessary and sUfficient that p X

is a bigenerator,

(i)

A

(11)

x =M

X

eM,

with M = 1m P and MX = Ker pX, where P (resp., pX) is the separating projection associated with the bigenerator A (resp., AX). In that case for a given f E Lm[O,OO) the solution of (1.1) is equal to p 00

,(t) = f(t) +

where

J y(t,s)f(s)ds, o

o

$

t

<

00,

X

ice-i(t-S)A (I_px)B + h(t,s), yet,s) • {

with h(t,s) X

along

M

X

_ice-i(t-S)A pXB + h(t,s),

= _iCe-itAx(I_Px ) (I_n)eiSAxpxB x

onto

o

$

s < t <

00,

o

$

t <

00,

S <

and n is the projection of

M •

x In the previous theorem the main difficulty is to prove that A is a biqenerator if (1.1) is uniquely solvable and to construct the

66

M. A. KAASHOEK

corresponding separating projection pX. It turns out that the property "A is a bigenerator" is stable under perturbation of a certain type. THEOREM 3.3.

Let

A

be a bigenerator on x, and let

operator on X such that

~

the strip let

11m AI

(3.8)

sUI? IAI2 11m AI~h

h

V«Ax)2) = V(A 2 ).

be a closed linear

AX

Assume that for some

is contained in the resolvent sets of II (A_Ax )-l -

(A-Al- 1 11

<

A

h > 0

and

AX,

and

00

Then AX is a bigenerator and the corresponding separating projection is given by X

P x

Now let W(A) be as in Theorem 3.2, and assume that equation (1.1) is uniquely solvable in Lm[O,OO). Then there exists a strip around the . p real line such that for A in this strip W(A) is invertible and (3.9)

X -1

(A-A)

-

(A-A)

-1

(A-A)

-1

-1

BW(A)

C(A-A)

-1

•

By using the boundedness of AB and CA, one can deduce from (3.9) that X

the evaluation (3.8) holds true, which yields that A

is a bigenerator.

11.4. Wiener-Hopf equations and time-invariant linear systems So far the results discussed were based essentially on factorization. In the present section we come to a method in which factorization does not appear at all. As before we deal with the Wiener-Hopf integral equation (1.1). In Section 1.1.1 the Fourier transform has been used to reduce equation (1.1) to the vector-function equation: (4.1)

...00

<

A<

00.

Here W(A) is the symbol of equation (1.1). The functions F+(A) and ~+(A) are the Fourier transforms of the given function f and the unknown ~, respectively, where both f and ~ are assumed to be zero on the negative part of the real line. The function F (A) is the Fourier transform of a function fO with support on t ~ O. Now, let us assume that W(A) is rational. Then W(A) admits a realization: (4.2)

W(A)

I + C(A-A)

-1

B,

...00

<

A<

00,

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

67

with a finite dimensional state space ~n and A has no real eigenvalues. Introduce the auxiliarly function

Then (4.1) can be rewritten in the form \X(A) = AX(A) - B~ (A), {

= -CX(A)

F+(A)

+ :+(A) - F (A).

By applying the inverse Fourier transform one obtains the linear system: x(t)

-iAx(t) + iB¢(t),

f(t)

-Cx(t) + ¢(t) - fO(t),

{

which has to be considered on the real line. Recall that we are interested only in functions on the positive half line. By restricting to t ~ 0 one can drop the unknown function fO which is zero on t ~ O. So we are led to the system {

(4.3)

x(t)

-iAx(t) + iB¢ (t),

t

~

0,

f(t)

-Cx(t) + ¢(t),

t

~

0,

A further analysis shows that (4.3) has to be considered with the following. boundary condition at zero: (4.4)

(I-P)x(O) = 0,

where P is the spectral projection of A corresponding to the eigenvalues in the upper half plane. The system (4.3) together with the boundary condition (4.4) is equivalent to equation (1.1) in the following sense: If for a given f

E

Lm[O,OO) the function ¢ p

(1.1), then there exists a unique x

E

E

Lm[O,OO) is a solution of p

Ln[O,OO) satisfying (4.4) such that p

(4.3) with input ¢ has output f, and, conversely, if there exists :XE

Ln[O,OO) satisfying (4.4) such that (4.3) with input ¢ E Lm[O,OO) has p

p

output f, then ¢ is a solution of (1.1). By using this equivalence the first step to solve equation (1.1) is to interchange in (4.3) the role of the input and the output. This leads to the following system: (4.5)

{

Ht)

-i(A-BC)x(t) + iBf(t),

t

~

0,

¢(t)

Cx(t) + f (t),

t

~

0,

which has to be considered together with the boundary condition (4.4) •

68

M.A.KAASHOEK

Next, one uses the variation of constants formula to find the solutions of the first equation in (4.5) that are in Ln[O,OO) and satisfy the p

boundary condition (4.4). In this way one obtains conditions for the solvability of (1.1), formulas for its solutions and various other characteristics of equation (1.1) in terms of the operators A, Band C. As an example we mention the following theorem.

Let weAl I + C(A-A)-l B be a reaLization of the symboL of equation (1.1) with state space ~n~ and assume that A and AX have no x reaL eigenvaLues. Let p (resp.~ p ) be the spectraL subspace of A x (resp.~ A ) corresponding to the eigenvaLues in the upper haLjpLane. Then~ given f € Lm[O,OO)~ equation (1.1) is soLvabLe in Lm[O,OO) if and onLy if p p THEOREM 4.1.

00

J

o

e

isAx

Bf(s)ds

x

€

1m P + Ker P ,

and in that case the generaL soLution of (1.1) is equaL to .

x

t

I

~(t) = f(t) + ce-~tA x + i

x

.

ce-~(t-S)A Bf(s)ds,

o where x is any v2ctor in 1m P such that x + i

co

I

o

e

iSAX

Bf(s)ds

is in

Ker pX. In particuLar~ if T is the integraL operator on Lm[O,OO) defined by p 00

~(t) -

(T~) (t)

I

o

o

k(t-s)~(s)ds,

~

t

<

00,

then Ker T

= {~ I ~(t) = ce- itA 00

1m T

= {f I I

o

e

i~x

x

Bf(s)ds

x, €

x

1m P n Ker P }, x 1m P + Ker P }, X €

and T is a FredhoLm operator with index x index T = rank P - rank P •

One of the main features of the method described here is that it can also be applied in cases when the factorization method cannot be used. For example, the reduction of a Wiener-Hopf integral equation with rational symbol to a linear system can be repeated for convolution equations on a finite interval, T

(4.6)

~(t) -

f

o

k(t-s)~(s)ds

= f(t),

and leads to a system of the form:

0 s t ~ T,

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

0 5 t 5 T,

{ ,'t) = -iAx,t) + i",t), (4.7)

f(t)

=

-Cx(t) +

=

(I-P)x(O)

0,

~(t),

PX(T)

69

0 5 t 5 T,

=

O.

The system (4.7) and the integral equation (4.6) are equivalent whenever the kernel k(t) of (4.6) admits the following representation: (4.8)

k(t)

{

iCe

-itA

. (I-P)B,

_ice-l.tApB,

t > 0, t < 0,

where A, Band C are matrices of sizes nXn, nXm and mXn, respectively, and P is a projection on ~n which commutes with A. If k(t) can be extended to an integrable function on the real line of which the Fourier transform K(A) is rational, then a realization of the symbol I-K(A) yields a representation (4.8) in which A has no real eigenvalues and P is the Riesz projection corresponding to the eigenvalues of A in the upper half plane. By using the equivalence between (4.6) and (4.7) the invertibility properties of (4.6) can be derived from a study of the inverse system:

{

x(t)

= -i(A-BC)x(t)

~(t)

= Cx(t) - f(t),

(I-P)x(O),

PX(T) =

+ iBf(t),

o o

5 t 5 T, 5 t

5 T,

o.

As an example we mention the following theorem.

Asswne that the ke1'nel k(t) of the equation (4.6) admits the representation (4.8), where P is a projeation aommuting with A. . . PUt Ax = A-BC. Then for eaah f E Lm[ O,T J equat~on (4.6) ha s a un~que p solution ~ E Lm[O,TJ if and only if the operator p

THEOREM 4.2.

1m P

-)0

1m P

1m P

is invertible. In that aase the solution of (4.6) is given by t"

~(t)

f(t) +

I

y(t,s)f(s)ds,

o

where y't,s)

={

x x ice-itA TIe iSA B, s < t, x _itA iSA x -iCe (I-TI)e B, s > t,

and TI is the projeation of ~n onto 1m P defined by

M.A.KAASHOEK

70

TIx

=x

- S

-1

Pe

iTA _iTA X e x,

The interplay between systems and integral operators will be the main topic of the third lecture. 11.5.

Further remarks and references

The main reference for Section 11.1 is §4.5 in [3]. The final formula for the resolvent is taken from [4]. Transport theory is a branch of mathematical physics which originated from and has many points of contact with astrophysics and nuclear physics (see, e.g., the recent book [30] and the references given there). For Section 11.2 the main source is [3], Chapter 6, but see also [33], which has, among many other things, the final formulas of the solution and of the resolvent kernel. The importance of the matching of the subspaces in (2.9) was known in transport theory and was proved, for example, in [26] (see also [25]) in connection with another method to solve half space problems. The solution of the realization problem for symbols that are analytic on the real line and at infinity, which is taken from [4], is closely related to the idea of linearization (see [20]); in fact the formulas for A, Band C in (3.2) come from the linearization theorem in [11] (see also [3], Sections 2.3 and 2.4). Symbols that are not analytic at infinity form the main topic of [9]; Theorems 3.1 - 3.3 are taken from [9]. The connections between convolution equations and time invariant linear systems, discussed in Section 11.4, have been developed in [4, 5]. The discrete versions of the results of Sections 11.1 and 11.4 can be found in [3, 4, 5].

III.

SYSTEMS WITH BOUNDARY CONDITIONS AND INTEGRAL OPERATORS

In this third lecture integral operators are viewed as transfer operators of time-varying linear systems with boundary conditions. For such systems the concepts of realization, similarity, and cascade connection will be discussed. Invertibility of integral operators is studied in terms of inversion of systems and factorization in terms of decoupling. On a finite interval the integral operators involved are compact and have a semi-separable kernel. The half line case concerns a large class of non-compact integral operators which includes WienerHopf integral operators with rational symbols. 111.1.

Integral operators as transfer operators

On a finite interval a time varying linear system with boundary conditions has the following state space representation:

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

X(t) = A(t)x(t) + B(t)u(t), { e yet) = e(t)x(t) + D(t)u(t), N1x(a) + N2 x(b)

71

a

:s;

t

:s;

b,

a

:s;

t

:s;

b,

O.

Here A(t) is an nXn matrix function of which the entries are integrable on a :s; t :s; b. The input coefficient B(t) and the output coefficient e(t) are matrices of sizes nXm and mxn, respectively, and the entries of B(t) and e(t) are square integrable on a :s; t :s; b. (For simplicity, the number of input variables is taken to be equal to the number of output variables.) The extePnaZ coefficient ott) is assumed to be measurable and essentially bounded on a :s; t :s; b. We shall refer to A(t) as the main coefficient. The boundary value matrices are constant and have size nXn. In what follows we shall represent the system e by its four coefficients and two boundary value matrices and write e = (A(t) ,B(t) ,e(t) ,O(t) iNl ,N 2 ). To ensure that in e the state x(t) is uniquely determined by the input u(t), the boundary conditions of e are assumed to be weZZ-posed which means that the homogeneous equation itt)

= A(t)x(t),

a

:s;

t

:s;

b, N1x(a) + N2X(b)

= 0,

has no non-zero solution. The well-posedness of the boundary conditions implies that the output y is a linear function of the input u, i.e., the input/output behaviour of the system is described by a linear operator Te' y = Teu, which we sha~l call the transfer operator of e. To determine the transfer operator of e one uses the fundamentaZ matrix of the system e, which, by definition, is the unique absolutely continuous solution of the matrix differential equation: (1.1)

U(t) = A(t)U(t), a

:s;

t

:s;

b, Uta) = I,

where I is the nXn identity matrix. In terms of the fundamental matrix well-posedness is equivalent to 0.2)

By using the variation of constants formula one determines the state x as a function of the input u. By inserting the formula for the state in the second equation of the system one computes that the transfer operator is an integral operator b a :s; t :s; b, Y (t) = (TeU) (t) = O(t)u(t) + f k(t,s)u(s)ds, a with kernel (1. 3)

k (t,s)

{"(t'U(t' ('-P'U(5,-'.(5', -C(t)U(t)PU(s)-l B(s),

aSs < t S b, a S t < s S b,

72

M. A. KAASHOEK

The operator P, which plays an important role in what follows, is well-defined because of (1.2). Using P the boundary conditions of 8 can be rewritten in the form: (I-P)x(a) + PU(b)-l x (b)

0,

and for that reason P is called the canonical boundary value of 8. If 8 is a causal system (i.e., Nl = I, N2 = 0), then P for an

anticausal system (i.e., Nl

= I)

0, N2

operator

the operator P

0, and

=

I. In

general, P is not a projection; in fact (1.4)

From the integrability conditions on the input and output coefficients it follows that the kernel (1.3) of the transfer operator is a Hilbert-Schmidt kernel, and hence we shall consider T8 as an operator on L~[a,b]. The time invariant case (when the coefficients do not depend on t) is of special interest. For example, consider ( x(t) = Ax(t) + Bu(t), (1.5)

,j

l

yet) = Cx(t) + Du(t), N1 X(0) + N2 X(T)

O~t~T,

o

~

t

~ T,

= o.

The fundamental matrix of (1.5) is U(t) e the transfer operator of (1.5) is given by

tA

and hence the kernel of

O~s
(1.6) O~t<S~T,

h were Po

=

( N1+ N 2 e TA) -I N2 e TA

' , h A, If , 1n a dd"1t1on, Po commutes w1t

then kO(t,s) depends on the difference of the arguments only and the corresponding transfer operator is a convolution operator. In fact, if POA = APO' then kO(t,s) = h(t-s), where

{

Ce

h(t) "

tA

(I-PO)B,

-ce~oB,

t > 0, t < O.

Conversely, if the transfer operator of (1.5) is a convolution operator and the triple (A,B,C) is minimal in the sense that

73

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

n-l

n

n-l Ker CA j = (0),

V

1m AjB =

a: n ,

j=O

j=O

then Po commutes with A. If Po is a projection commuting with A, then the boundary conditions of (1.5) can be rewritten as (I-P)x(O) = 0,

PX(T)

=

0,

and we come to systems that were considered in Section 11.4. To see which integral operators on L~[a,b] can be realized as transfer operators, note that the kernel k in (1.2) is semi-separable, i.e., it admits a representation of the form:

(1. 7)

k(t,s) = {

Fl (t)G 1 (s),

a :-:; s < t :-:; b,

-F 2 (t)G 2 (s),

a :-:; t <

S

:-:;

b,

where for V = 1,2 the matrix functions Fv (') and Gv (.) are square integrable on [a,b] and have sizes mxnv and nvxm, respectively. For example, to write (1.2) in the form (1.7) one may take C(t)u(t) (I-P),

u (t) -l B (t)

C(t)U(t)P,

U(t) -l B(t).

Conversely, if T on

,

m

L2 [a,b] is an integral operator, b

(T<j» (t) = O(t)<j>(t) +

f

k(t,s)<j>(s)ds,

a :-:; t :-:; b,

a

such that k is semi-separable and OCt) is measurable and essentially bounded, then T is the transfer operator of a system e, in which case e is called a realization of T. In fact, if k is given by (1.7), then T = Te with (1.8)

e

(A(t) ,B(t) ,C(t) ,O(t); I-Q, QU(b) -1),

where A(t) is an arbitrary integrable nXn matrix function, U(t) is determined by A(t) via (1.1) and B(t)

U(t)[G 1 (t)j, G2 (t)

C(t) = [F 1 (t)

Q = [00

F 2 (t)] U(t)-l

Note that the canonical boundary value operator P of (1.8) is equal to Q, and hence P is a projection. In (1.8) we may take A(t) 0, which

=

M.A.KAASHOEK

74

=

implies that U(t) I. So the transfer operator always has a realization 8 of which the main coefficient A(t) is identically zero and the canonical boundary value operator is a projection. The different choices of A(t) in (1.8) correspond to system similarities. Two time varying systems 8 1 and 8 2 ,

v

1,2,

=

are called similar (notation: 8 1 ~ 8 2 ) if D1 (t) = D2 (t) a.e. on a ~ t ~ b and there exist an invertible nXn matrix E and an absolutely continuous nXn matrix function S(t), a ~ t ~ b, such that det S(t) F 0, a ~ t ~ b, and S(t)A 1 (t)S(t)-1 + S(t)S(t)-1, S(t)B 1 (t),

C1 (t)S(t)-1,

EN;ll S (a) -1,

EN~1)S(b)-1,

almost everywhere on a ~ t ~ b. This notion of similarity appears in a natural way when in 8 the state x(t) is replaced by z(t) = S(t)x(t). Well-posedness of the boundary conditions is preserved under similarity and similar systems with well-posed boundary conditions have similar canonical boundary value operators and the same transfer operator. 111.2.

An application: inversion

Consider the integral equation: b

(2.1)


J

k(t,s)
f(t),

=

a

~

t

~

b.

a

We assume that in (2.1) the kernel is semi-separable; so, in other words, the kernel k is the kernel of the transfer operator of a system 8 with well-posed boundary conditions: x(t) { 8 f(t)

=

A(t)x(t) + B(t)
a

=

C(t)x(t) +
a :-; t

~

t :-; b, ~

b,

o.

N1x(a) + N2x(b)

The problem is to solve (2.1) in L~[a,b]. To do this, we use the equivalence between 8 and (2.1). First we interchange in 8 the role of the input and the output, which leads to the inverse system: x(t)

(A(t) - B(t)C(t»x(t) + B(t)f(t),

a


-C(t)x(t) + f(t),

a :-; t :-; b,

N1x(a) + N2x(b)

=

o.

~

t :-; b,

75

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

Note that the coefficients of e

x

satisfy the desired integrability

conditions. It turns out that the boundary conditions of eX are wellposed if and only if Te is invertible, and in that case (Te)

-1

= Tex.

X

X X Let u (t) denote the fundamental matrix of e ; we shall refer to U (t) as the assoeiate fundamental matrix of e.

Let k be the kernel of the tr>ansfer> oper>ator> of the system e = (A(t) ,B(t) ,C(t) ,I;N1,N2~ and let UX(t) be the fundamental matr>ix of e. Then, given f E L~[a,b~ equation (2.1) has a unique solution ~ E

THEOREM 2.1.

L~[a,bJ if and only if

~ 0, and in that ease

det(N 1 + N2 Ux (b» b

~(t)

= f(t)

J y(t,s)f(s)ds,

-

a ~ t ~ b,

a

with r>esolvent ker>nel =

{

y(t,s)

XX x -1 C(t)U (t) (I-P )U (s) B(s),

s < t,

x x x -1 -C(t)U (t)P u (s) B(s),

s > t,

Note that in the above theorem the invertibility of an integral operator acting on an infinite dimensional linear space is reduced to the x invertibility of a finite matrix. The matrix N1 + N2 U (b) can be used to derive other invertibility properties of equation (2.1). For example, for f E L~[a,bJ equation (2.1) is solvable in L~[a,bJ if and only if x b x -1 x N2 U (b) U (s) B(s)f(s)ds E Im(N 1 + N2 U (b». a x The connections between N1 + N2 U (b) and equation (2.1) are quite

J

deep. To illustrate this further, let us put a spectral parameter (2.1) :

~

in

b

(2.2)

~(t) + ~

f

k(t,s)~(s)ds = f(t),

a

x

a

~

t

~

b.

x

To deal with (2.2) one has to replace U (t) by U (t;~) which, by definition, is the unique absolutely continuous solution of

d~ UX(t;~)

=

(A(t) -

~B(t)C(t»Ux(t;~),

{ U

Put

S(~)

(a;~)

= I.

x N1 + N2 U (b;~). The function

S(~)

is an nXn matrix function

76

M. A. KAASHOEK

of which the entries are analytic on the full complex plane. One can prove that the integral operator I + ~K defined by the left hand side of (2.2) and the analytic matrix function S(~) are related in the following way:

(2.3)

[ I+~K o

1=

0

E(~)

s (~) [ 0

~

E

a:.

Ia:n

Here E(~) and F(~) are invertible operators which depend analytically on the parameter ~ and which can be described explicitly in terms of the coefficients A(t), B(t), C(t) of the realization 8. The analytic equivalence (2.3) allows one to derive the spectral data of the integral operator K (i.e., its eigenvalues, eigenvectors, generalized eigenvectors, Jordan chains, etc.) from the corresponding data of the matrix function S(~). For this reason S(~) is called the ind~cator of the integral operator K. 111.3.

Integral equations with a semi-separable kernel on the half line

In this section we consider the case of a semi-infinite time interval. We start with an integral equation 00

(3.1 )

~(t) +

f

k(t,s)~(s)ds = f(t),

O!>t
o

of which the mXm matrix kernel admits the following semi-separable representation:

(3.2)

k(t,s) = {

c (t) u (t) (I-P) u (s) -1 B (s)

,

-1

-c (t) u (t) PU (s)

B (s) ,

o

!> s < t <

00,

o

!> t < S <

00

Here U(t) is determined by the differential equation: (3.3)

U(t)

= A(t)U(t),

0 !> t <

U(O)

00,

=

I,

and for simplicity we shall assume that in (3.2) the operator P, which acts on

a: n ,

is a projection. with the representation (3.2) of the kernel

x

we associate an nXn matrix function U (t) defined by: (3.4)

·X

u

(t)

(A(t) - B(t)C(t»Ux(t), 0 !> t <

00,

Ux(O)

1.

x

We shall refer to U (t) as the associate fundamentaZ matrix. Assume that in (3.3) the entries of A(t) are integrable on o !> t < 00 and that the entries of B(t) and C(t) are in L 2 [0,OOJ. Then equation (3.1) can be analysed in precisely the same way as equation (2.1) in the previous section. In fact, under the stated integrability

x

conditions, the functions U(t) and U (t) are well-defined and the limits

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

lim t-+oo

u(oo)

x

u (00)

U (t),

lim

U

x

77

(t)

t-+<x>

exist and are invertible matrices, which allows one to repeat the arguments of the previous section. one only has to replace a by 0, b by 00, Nl by I-P and N2 by PU(OO)-l, and one sees that the invertibility properties of (3.1) are determined by the matrix I-P + PU(OO)-l UX (oo). Note that under the above integrability conditions the integral operator K corresponding to the integral in the left hand side of (3.1) has a Hilbert-Schmidt kernel and hence K cannot be a Wiener-Hopf integral operator. To get a class of integral equations of the form (3.1) with non-compact kernels including, for example, some Wiener-Hopf equations, it is necessary to look for other conditions on A(t), B(t) and C(t). In what follows we shall assume that B(t) and C(t) are essentially bounded, that A(t) is locally integrable and that the projection I-P is an exponential dichotomy for the differential equation x(t) = A(t)x(t) , o ~ t < 00. The latter condition means th~t there exist positive constants L, M, a and S such that (3.Sa) (3.Sb)

II II

u (t) (I-P) u (s) -111 u (t) PU (s) -111

~ Le -a (t-s) ,

~ Me -S (s-t) ,

o

~

s < t < 00,

o

~

t < S < 00

Since A(t) is locally integrable, the function U(t) is well-defined by (3.3). If (3.Sa) and (3.Sb) hold true we shall also say that I-P is. an exponential diohotomy for u(t). The inequalities (3.Sa) and (3.Sb) do not determine P uniquely, but only its kernel. In fact, if I-P is an exponential dichotomy for U(t), then Ker P

{x E ~n

=

I

U(t)x E L7[0,00)},

and any projection IT of ~n such that Ker IT = Ker P is again an exponential dichotomy for U(t). Note that our conditions allow that A(t), B(t) and C(t) are -itA independent of t. If A(t) = -iA does not depend on t, then U(t) = e -itA The function U(t) = e has an exponential dichotomy if and only if A has no eigenvalues on the real line, and in that case I-P is an exponential dichotomy for U(t) = e- itA if and only if Ker P is the space spanned by the eigenvectors and generalized eigenvectors of A corresponding to eigenvalues in the lower half plane. It follows that Our class of kernels includes convolution kernels of the form k(t,s) = h(t-s), where h (t)

{

ice-itA(I_P)B,

t > 0,

-iCe -itA --1>B,

t < 0,

M.A.KAASHOEK

78

such that A has no real eigenvalues and P is the spectral projection corresponding to the eigenvalues of A in the upper half plane. The integral equations corresponding to the kernels k(t,s) = h(t-s) are precisely the Wiener-Hopf integral equations with rational symbols. To study the integral equation (3.1) the following equivalence with time varying systems is most useful. THEOREM 3.1. Let the kePnel k(t,s) be of the form (3.2), where B(t) and C(t) are essentially bounded and l-P is an exponential dichotomy

for u(t)

with

U(t)

as

in

(3.3). Then, given f

€

Lm[O,OO), the p

function ~

€

Lm[O,oo) is a solution of (3.1) if

a unique x

€

Ln[O,oo) such that the time varying system

~

p

and only if there exists

p

os:t
itt) = A(t)x(t) + B(t)u(t), { yet) = C(t)x(t) + u(t), (l-P)x(O)

with input u =

~

os:t
= 0,

responses with output y = f.

From Theorem 3.1 and using the idea of cascade connection, it can be proved that the class of all integral operators T on Lm[O,OO) , p

00

(T~) (t) = a~(t) +

f

o

k(t,s)~(s)ds,

os:t
where a € ~ and k(t,s) is any kernel of the type considered in the previous theorem, forms an algebra. To analyse equation (3.1) one interchanges in ~ the role of the input and the output which leads to x

x

the study of the associate fundamental matrix U (t). Note that U (t), which is defined by (3.4), exists, because under our conditions on A(t) , B(t) and C(t) the matrix function A(t) - B(t)C(t) is locally integrable. The following theorems are samples of the results one obtains by analysing (3.1) in terms of ~.

Let the kePnel k(t,s) be of the form (3.2), where B(t) and are essentially bounded and l-P is an exponentially dichotomy for U(t). In order that for any f € L~[O,OO) the equation (3.1) has a unique solution ~ € L~[O,OO) it is necessary and SUfficient that the associate fundamental matrix UX(t) has an exponential dichotomy l_p x such that

THEOREM 3.2. c(t)

~n

=

lm P e Ker pX

In that case the solution

~

is given by

00

~(t) = f(t)

-

f

o

y(t,s)f(s)ds,

o

s: t <

00,

79

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

with resolvent kernel _

XX -1 C(t)U (t) (l-Il)U (s) B(s), x x-1 -C(t)U (t)IIU (s) B(s),

{

y(t,s) -

o

~

s < t

<

00,

o

~

t < s <

00,

where II is the projection of ~n along Ker pX onto 1m P. THEOREM 3.3. Let the kernel k(t,s) be of the form (3.2), where B(t) and C(t) are essentially bounded and I-P is an exponential dichotomy

for U(t). In addition, assume that UX(t) has an exponential dichotomy x m I-P. Then the integral operator l+K on L 1 CO,(0), 00

f

(Kq,) (t) =

o

k(t,s)q,(s)ds,

~

t <

00,

o

is a Fredholm operator, Ker (I+K) = {q,

I

L7CO,(0)

E

x

C(t)U (t)x,

q,(t)

x

X E 1m P n Ker P },

1m (I+K)

{f

00

m

E

If

L 1 CO,(0)

x x

P u (s)

-1

B(s)f(s)ds

o

E X

E

1m P + Ker P },

and its Fredholm index is equal to index (l+K)

rank P

Furthermore, if for f

E

x

rank P.

-

L~CO,oo) the equation (l+K)¢

f

is solvable in

L7CO,00), then one of its solutions is given by 00

=

q,(t)

f(t) -

f

where

x

{

y(t,s)

Here II x

P

x

~

-

x

_C(t)U (t) (l-II)u (s)

-1

~

t < 00,

B(s),

_C(t)U x (t)TIu x (S)-l B (s),

+ (l-P

S = P P: 1m P

o

y(t,s)q,(s)ds,

o

x

)8

+ x

P , with

8

+

o

~

s < t <

o

~

t < s < 00

00,

a generalized inverse of the operator

x

1m P •

Theorem 3.2 and Theorem 3.3 hold true also for L:CO,oo) and with some modifications for LmCO,OO), 1 < P < P

00.

80

M. A. KAASHOEK

III.4.

Cascade connection and factorization of operators

We return to time varying systems with well-posed boundary conditions on a finite interval a ~ t ~ b. For ~ = 1,2 let 8~ = (A~(t) ,B~(t) ,C~(t), O~(t)i

(~)

N1

(~)

,N 2

) be such a system. By definition (cf., Section I.S),

the cascade connection (or product) of 8 1 and 82 is the system 8 (A(t) ,B(t) ,C(t) ,OCt) iN1 ,N 2 ) with

=

B1 (t)C 2 (t)j, A(t)

B(t) A 2 (t)

C(t)

0 1 (t)C 2 (t)],

OCt)

o 1,2.

j

N. J

(2) N. J

We shall write 8

=

8 1 8 2 , It is not difficult to see that 8

=

8 1 8 2 is

the system with well-posed boundary conditions which one gets if the output of 8 2 is equal to the input of 8 1 , It follows that (4.1)

We shall use cascade decompositions to construct factorizations of the operator I+K, when K is an integral operator with a semi-separable kernel. Let 8 be a realization of I+K, i.e., the operator I+K is the transfer operator of 8. Let 8 1 and 8 2 be realizations of the operators I+K 1 and I+K 2 , respectively. Since similar systems have the same transfer operator, it is clear from (4.1) that 8

~

8 1 82 implies that

(4.2) So we are interested in factorizations of 8 modulo similarity. Let 8 = (A(t) ,B(t),C(t) ,IiN 1 ,N 2 ) be a system with well-posed

=

boundary conditions of which the external coefficient OCt) I. Let U(t) be the fundamental matrix of 8, and let P be its canonical boundary value operator. A projection IT of onto

~n

~n1 e ~n2 along ~n1

~n2 is called a supporting projection of 8 if P P

) 12j: ~nl

P 22

e ~n2

+

~nl e ~n2

81

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

and the Riccati matrix differential equation R(t)

=

[1 1

R(t) ]U(t) -l B (t)C(t)U(t)

(-~~t)],

{

(4.3)

R(a) = 0, has a solution R(t): ~n2 + ~nl on a $ t $ b. with the supporting projection IT we associate two systems, namely:

Here R(t) is the solution of (4.3) on a $ t $ b. Let us assume that in what follows all systems have external coefficient equal to I.

Let 8 be a system with well-posed boundary conditions. If IT is a supporting projection of 8, then 8 ~ In(8)r n (8). Conversely, if 8 ~ 8 1 8 2 , then there exists a supporting projection IT such that

THEOREM 4.1.

Let us specify Theorem 4.1 for causal/anticausal factorization. So, consider 8 ~ 8_8+, where 8 is causal and 8+ is anticausal, i.e., the systems 8

and 8+ are of the form

,c 1 (t)

8

(A 1 (t) ,B 1 (t)

8+

(A 2 (t) ,B 2 (t) ,C 2 (t) ,1;0,1 2 ),

,1;1 1 ,0),

Let IT be a supporting projection, and assume that l r n (8)

~

(8) ~ 8 and n 8+. Since similar systems have similar canonical boundary

value operators, one sees that Pll

= 0

and P22

= 12 ,

and hence P12

= 0,

because of the boundary condition at b in the Riccati equation (4.3). It follows that the canonical boundary value operator P is equal to the supporting projection IT. Next, one uses the conditions of solvability of an initial value Ricatti equation to find out what it means that P is a supporting projection. In this way one comes to the following theorem.

Let 8 be a system with well-posed boundary conditions, let U(t) be the fundamental matrix and UX(t) the associate fundamental matrix of 8. Then 8 ~ 8_8+, where 8_ is causal and 8+ is anticausal, if and only if the canonical boundary value operator P of 8 is a projection THEOREM 4.2.

M. A. KAASHOEK

82

and (4.4)

det (I-P+U(t)

In that case 8 8

~

8 8

- +

(0,

=

-1 x

U (t)P)

~

a ::; t ::; b.

0,

with

(I-P+R(t))U(tJ- 1 B(t), C(t)U(t)

I

'

I; I Ker P' 0),

IKer P (0, PU(t)-l B (t), C(t)U(t) (P-R(t))i

8+

I; 0, 11m p), 1m P

where (4.5)

R(t) = (I-P) (I-P+U(t)

-1 x

U (t)P)

-1

a ::; t ::; b.

P,

Let us apply Theorem 4.2 to integral operators. Let K be an integral operator with a semi-separable kernel. Choose (which is always possible) a realization 8 of I+K of which the canonical boundary value operator P is a projection. Let U(t) be the fundamental matrix of 8 and UX(t) its associate fundamental matrix. Assume (4.4) holds true. Then formula (4.1) and Theorem 4.2 imply that (4.6)

I+K

=

(I+K ) (I+K ),

-

+

where

t

(K _ CP) (t)

1 C(t)U(t) (I-P+R(s))U(s)- B(s)<j>(s)ds,

f a

(K+<j» (t) = -

a

f

C(t)U(t) (P-R(t) )U(s)

-1

B(s)<j>(s)ds,

t with R(t) given by (4.5). Formula (4.6) is a so-called LU-factorization of I+K. It can be proved that, conversely, if I+K admits an LUfactorization, then necessarily (4.4) holds true. The class of time invariant systems is not closed under the factorizations considered in Theorem 4.2, that is, if 8 ~ 8 8 and 8

- +

is time invariant, then it does not follow that 8_ and 8+ are also time invariant. Consider, for example, on system:

8

( [ -1

°

1

0] [-1 l ' 1 j'

[1

° :;

1J, 1;

t

::; 1 the time invariant

[°1

0] [0 0'

°

0] ) 1 .

The canonical boundary value operator P of 8, its fundamental matrix x U(t) and its associate fundamental matrix U (t) are given by:

83

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

P

=

[

°0

OJ l ' U(t)

=

It follows that det (I-P+U(t) Hence 8 ~ 8 8 with - +

[e-°

U

but there is no factorization e

U (t)P)

e

~

x

COS

t

[

(t)

-i sin t

-1 x

8

111.5.

t

-t

e

-t

cos t

#

-i sinttJ. cos

° for °

~ t

~ 1.

1; 1,0),

8- e + with e

and e+ time independent.

Remarks and references

probably the first systematic paper on time varying systems with wellposed boundary conditions is [31J, where one also will find a number of examples of such systems. Note, however, that Sturm-Liouville two point boundary value problems can be put into this context, in which case the Green's function becomes the kernel of a transfer operator. In the analysis we have given the variable t does not have to be a time variable, which allows one to include many non-trivial examples, e.g., transport problems in non-homogeneous media. Sections 111.1 and 111.2 are fully covered by [18J, which also treats discrete time systems with well-posed boundary conditions. For time varying systems the problems of minimality and minimal realization, which are not discussed here, have several different and interesting aspects. One aspect, connected with dilation, reduction and similarity, is completely clarified in [19J. The results of Section 111.3 are taken from [21J. The notion of exponentially dichotomy, which plays an essential role in Section 111.3 (and in [21J) is due to J.L. Massera and J.J. Schaffer [32J (see, also [15, 16J). The discrete analogues of the results of Section 111.3, which require the analysis of the notion of dichotomy for difference equations, also appear in [21J. The coupling method from [7J, which allows one to study various classes of integral operators in terms of simpler operators and often just in terms of matrices, plays an important role in the proofs of the theorems in Sections 111.2 and 111.3. Section 111.4 concerns joint work with I. Gohberg which is still in progress and aims at a complete theory of minimal factorization of integral operators, of which LU factorization is a special case. LIST OF REFERENCES 1.

2.

Adams, G.T.: 'A nonlinear characterization of stable invariant subspaces', IntegraL Equations and Opepatop Theopy ~ (1983), pp. 473-487. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Operator polynomials as inverses of characteristic functions', IntegraL Equations and

84

M A.KAASHOEK

OperatDr Theory 1 (1978), pp. 1-18. 3.

4.

5.

6.

7.

8.

Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal factorization of matrix and operator functions. Operator Theory: Advances and Applications, Vol. 1, Birkhauser Verlag, Basel, 1979. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Wiener-Hopf integral equations, Toeplitz matrices and linear systems.' In: Toeplitz Centennial (Ed. I. Gohberg), Operator Theory: Advances and Applications, Vol. 4, Birkhauser Verlag, Basel, 1982; pp. 85-135. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Convolution equations and linear systems', Integral Equations and Operator Theory ~ (1982), pp. 283-340. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Wiener-Hopf factorization of analytic operator functions and realization', Wiskundig Seminarium der Vrije Universiteit, Rapport nr. 231, Amsterdam, 1983. Bart, H., Gohberg, I., Kaashoek, M.A.: 'The coupling method for solving integral equations'. In: Topics in Operator Theory, Systems and Networks, The Rehovot Workshop (Ed. H. Dym, I. Gohberg). Operator Theory: Advances and Applications, Vol. 12, Birkhauser Verlag, Basel, 1984, pp. 39-73. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Wiener-Hopf factorization and realization'. In: Mathematical Theory of Networks and Systems,

Proceedings of the MTNS-83 International Symposium, Beer Sheva, Israel ted. P. Fuhrmann), Lecture Notes in Control and Information 9.

10.

11.

12.

13.

14. 15. 16.

17. 18.

Sciences, nr. 58. Springer Verlag, Berlin, 1984; pp. 42-62. Bart, H., Gohberg, I., Kaashoek, M.A.: 'Spectral analysis of systems of Wiener-Hopf integral equations with symbols that are analytic in a strip around the real line', in preparation. Bart, H., Gohberg, I., Kaashoek, M.A., Van Dooren, P.: 'Factorizations of transfer functions', Siam J. Control Optimization l~ (1980), pp. 675-696. -Boer, H. den: 'Linearization of operator functions on arbitrary open sets', Integral Equations and Operator Theory l (1978), pp. 19-27. Brodskii, M.S.: Triangular and Jordan representations of linear operators. Transl. Math. Monographs 32, Amer. Math. Society, Providence R.I., 1971. Brodskii, V.M., Gohberg, I.C., Krein, M.G.: 'On characteristic functions of an invertible operator', Acta Sci. Math. (Szeged) 32 (1971), pp. 141-164. Cohen, N.: 'On minimal factorizations of rational matrix functions', Integral Equations and Operator Theory § (1983), pp. 647-671. Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin, 1978. Daleckii, Ju. L., Krein, M.G.: Stability of solutions of differential equ.ations in Banach space. Amer. Math. Soc. Transl. 43, American Mathematical Society, Providence R.I., 1974. Foia~, C., Salinas, N.: 'Stably invariant subspaces of Hilbert space operators'. Kansas University, preprint, 1980. Gohberg, I., Kaashoek, M.A.: 'Time varying linear systems with boundary conditions and integral operators, I. The transfer operator and its properties', Integral Equations and Operator

MINIMAL FACTORIZATION, LINEAR SYSTEMS AND INTEGRAL OPERATORS

Theory

19.

20.

21.

22.

23.

24. 25.

26.

27. 28. 29. 30.

31. 32. 33.

34. 35. 36.

Z

85

(1984), pp. 325-391. Gohberg, I., Kaashoek, M.A.: 'Time varying linear systems with boundary conditions and integral operators, II. Similarity and reduction', Wiskundig Seminarium der Vrije Universiteit, Rapport nr. 261, Amsterdam, 1984. Gohberg, I., Kaashoek, M.A., Lay, D.C.: 'Equivalence, linearization and decompositions of holomorphic operator functions', J. Funct. Anal. ~§ (1978), pp. 102-144. Gohberg~ I., Kaashoek, M.A., Schagen, F. van: 'Non-compact integral operators with semi-separable kernels and their discrete analogues: inversion and Fredholm properties', Integral Equations and Operator Theory (to appear). Gohberg, I.C., Krein, M.G.: 'Systems of integral equations on a half line with kernels depending on the difference of arguments', Uspehi Mat. Nauk l~ (1958) no. 2 (80), pp. 3-72 (Russian) = Amer. Math. Soc. Transl. (2) 1~ (1960), pp. 217-287. Gohberg, I.C., Krein, M.G.: Tneory and applications of Volterra operators in Hilbert space. Transl. Math. Monographs 24, Amer. Math. Soc., Providence R.I., 1970. Gohberg, I., Lancaster, P., Rodman, L.: Matrix polynomials, Academic Press, New York N.Y., 1982. Hangelbroek, R.J.: 'A functional analytic approach to the linear transport equation'. Ph.D. Thesis, University of Groningen, The Netherlands, 1973 = Transport Theory and Statist. Phys. 5 (1976), pp. 1-85. Hangelbroek, R.J., Lekkerkerker, C.G.: 'Decompositions of a Hilbert space and factorization of a W-A determinant', Siam J. Math. Analysis § (1977), pp. 459-472. Kailath, T.: Linear Systems, Prentice-Hall, Inc., Englewood Cliffs N.J., 1980. Kalman, R.E.: 'Mathematical description of linear dynamical systems', Siam J. Control 1 (2) (1963), pp. 152-192. Kalman, R.E., Falb, P.L., Arbib, M.A.: Topics in mathematical system theory, McGraw-Hill, New York, 1969. Kaper, H.G., Lekkerkerker, C.G., Hejtmanek, J.: Spectral methods in linear transport theory. Operator Theory: Advances and Applications, Vol. 5, Birkhauser Verlag, Basel, 1982. Krener, A.J.: 'Boundary value linear systems', Asterisque Z~/Z~ (1980), 149-165. Massera, J.L., Schaffer, J.J.: Linear differential equations and function spaces, Academic Press, New York, 1966. Mee, C.V.M. van der: Semigroup and factorization methods in transport theory. ph.D. Thesis Vrije Universiteit, Amsterdam, The Netherlands, 1981 = Math. Centre Tracts 146, Mathematisch Centrum, Amsterdam, 1981. Ran, A.C.M.: semidefinite invariant subspaces; stability and applications. Ph.D. Thesis Vrije Universiteit, Amsterdam, 1984. Rosenbrock, H.H.: State-space and multivariable theory, Nelson, London, 1970. Sobolev, V.V.: A treatise on radiative transfer. Van Nostrand, Princeton N.J., 1963.

86

37. 38. 39.

M.A.KAASHOEK

Sz.-Nagy, B., Foia~, C.: Harmonic analysis of operators on Hilbert space. North-Holland Publ. Co., Amsterdam, 1970. Van Dooren, P.: 'Factorization of a rational matrix: the singular case', Integral Equations and Operator Theory (to appear). Wiener, N., Hopf, E.: 'Ueber eine Klasse singularer Integralgleichungen', Sitze Ber. Preuss. Akad. Wiss., Phys.-Math. Klasse 31 (1931), pp. 696-706.

HA-PLITZ OPERATORS: A SURVEY OF SOME RECENT RESULTS

N.K.Nikol'skii Leningrad Branch, Steklov Math Institute Fontanka 27, Leningrad, 191011, USSR

O. Introduction, and formulae linking Hankel and Toeplitz operators 1. Singular numbers 2. Inverse spectral problems 3. Toeplitz operators and exponential bases 4. Spectral multiplicity problem

O.

INTRODUCTION

Originally, Hankel and Toeplitz operators are defined as operators acting on t~ and having matrices with entries depending only on the sum or, respectively, difference of indices: r={ tn+Kl)t. K~O' T = {t n - K} 11.,K~O • Many close relation~ of such operators (matrices) to various problems of algebra, analysiS, differential equations were discovered as early as in the last century. But the spectral theories of Hankel and Toeplitz operators start their deVelopment only in the late 50th. Now they are joined 87

88

N. K. NIKOL'SKII

within the spectral theory of ~nkel and Toeplitz operators. Por most problems of the theory is is more 9; II~ convenient to take the Pourier transfonD. t - + H ,

)( ={xnJ~~o t ~ X= and T

~ :x:,~v D ~ n~o

on the Hardy space

1.l1tt

and to consider

,

2

H=

{

f

r

~

e

L ('If)

:

A

0, tv -< 0 } ; here 1r = { ~ e; {, : I ~ I = 1 J is the unit circle, ~(n) stands for the n-th Fourier coefficient. Moreover, instead of r we consider the operator H from H? to H: = Lt.C]I') e H~ which coincides with r up to an isometric factor and has the same matr1x(rKTK )J but with respect to the f

(11-) =

pair of bases {zl-t. ll-t. ~ 0

and {l Kf"1 JI(~O, Z

being the

identi ty mapping of 'If, X(~) ~ ~, ; e 71' • In other words, H=1r where}f = Z f (~), ie L?-('il') • It is clear that '} is an involution on L({,('!r> inII~ 112terplacing M and N_ • The following two theorems fonD. a background of the theory. There are many sources where the interested reader can find the proofs as well as numerous comments and subsequent developments concerning these theorems, [9], [10J , [17] , [18J , [22J , [24J , [25J • See also [16] for further information. In the statements written below all operators are supposed to be defined only on the set ~A of all polynomials in % • Theorem (Nehari, 1957). A Hankel operator ~ Ii : ~~- W~ is continuous iff there exists a function