Integral Equations and Operator Theory Vol. 13 (1990)

0378-620X/90/060836-1351.50+0.20/0 (c) 1990 Birkhauser Verlag~ Basel

L E F T A N D R I G H T B L A S C H K E - P O T A P O V P R O D U C T S

A N D A R O V - S I N G U L A R M A T R I X - V A L U E D F U N C T I O N S

V i c t o r K a t s n e l s o n ~

If J is an indefinite s ignature matr ix, then there exists a J contractive holomorphic matr ix valued funct ion W(z) in the open uni t disc which can be expressed as a left Blaschke-Potapov product: W(z) = B(l)(z) , bu t not as a right Blaschke-Potapov product: W(z) = E(z)B(r) (z), where B(r)(z) is a right Btaschke-Potapov product and E(z)

is a so called Arov singular matr ix function. In fact B(*)(z) may be chosen to obta in any Arov singular matr ix funct ion E(z) in the second representat ion. This phenomenon and mult ipl icat ive representat ions of Arov singular functions are discussed.

1 ~ In* " pape r we a ~hls present n u m b e r of results on left and right Blaschke-

Potapov products without proof. The proofs will be presented elsewhere. We begin

with some notat ions. Let ID denote the open uni t disc in the complex plane r let

"IF -- 0ID denote its boundary: 9F = {t E (~ : It[ = 1} and let rn(dt) denote normalized

Lebesgue measure on 9r. Let .A4n denote the set of all n x n matrices and let I E/~An

denote the ident i ty matr ix.

Let J E J~n be a matrix, which satisfies the two condit ions

j = j . , j 2 = ! . (1.1)

Such a matr ix is called a signature matrix.

A matr ix W C M n , is called J-contractive, if

J - W*JW > 0 .

* This paper was written while the author was a guest of the Department of Theoretical Mathematics of The Weizmann Institute of Science, Rehovot. The author would like to thank the Department for its support and hospitality.

Katsnelson 837

and J -un i ta ry if

J - W * J W = 0 .

In this p a p e r we shall deal wi th meromorph ic J - con t r ac t i ve ma t r ix -va lued

funct ions W ( z ) , t ha t is wi th funct ions W : ID --+ .hdn, which are meromorph ic in ID and

sat isfy the condi t ions

J - W * ( z ) J W ( ~ ) >__ o ( z ~ m) . ( i .2)

We suppose also tha t det W ( z ) ~ 0 so tha t the inverse m a t r i x W ( z ) -1 is

meromorph ic .

We shall s tudy the fac tor iza t ion proper t ies of J - con t r ac t i ve mat r ix -va lued

ana ly t ic funct ions.

2 ~ The BIaschke factor is the complex-vMued funct ion of the complex

r I~[ (if r163 b(~;O- f - - ~ (

var iable z:

b(z; O) = z .

(zi)

Let {Zk} be a sequence of poin ts in ID which sat isfy the Blaschke condi t ion

E (1 --Iz~l) < ~ - (2,2) k

Then the p roduc t

b(~) = I X b(z; zk) (2.3) k

converges to a non-zero analy t ic function.

{zk}) . The p roduc t b(z) is called the Bla~chke product (which is based on the sequence

The Blaschke p roduc t is a cont rac t ive funct ion in ID.

ho lomorph ic funct ion in D then the rad ia l l imit

s(t) d2--- f lira s(rt) (t E ~ ) rT1

An ana ly t ic funct ion s(z) : ID --~ r is called contractive if

14z)l < 1 (Izl < i ) . (2.4)

If s(z) is a bounded

(2.5)

838 Katsnelson

exists for almost every t E ~]r. This function (which is defined for almost every t in the

sense of Lebesgne measure):is called the boundary value of the function s(z).

If b(z) is a Blaschke product then the modulus of its boundary value is equal

to unity:

]b(t)[ = 1 ( fo r a.e. t e ~]F) . (2.6)

Let s(z) ~ 0 be a holomorphic contractive complex-valued function in Is I t

is well known that for the set {zk} of its roots the Blaschke condition (2.2) is satisfied. So

we can build the Blaschke product b(z) with this set of roots. The fact0rization theorem

of Riesz and Nevanlinna asserts that every analytic-contractive function s(z) in ID admits

a factorization of the form

s ( 4 = b(z), e(z) (2.7)

where b(z) is the Blaschke product (which is based on the set {%} of the roots of the

function s(z)) and e(z) is a contractive holomorphic function in ID with no roots in ID:

Thus the inverse function e(z) -1 is also holomorphic in ID.

I t may happen that in (2.7) the factor b(z) is absent (if the function s(z) has

no roots i n W ) or that the factor r is absent (i.e., (~(z) = eo,~sta~t).

DEFINIT ION. The representation (2.7) for ho!omorphic contractive complex-

valued functions s(z) in ID is called a multiplicative representation.

The multiplicative representation of every such analytic cont rac t ive function

s(z) is unique.

DEFINIT ION. A contractive holomorphic matrix-function in ID is called a

Blasehke product if the factor e(z) in its multiplicative representat ion is a constant:

e(z)=_u, lul=l

THE C R I T E R I O N OF A BLASCHKE P R O D U C T (F. Riesz, !1]).

A holomorphic contractive function s(z) in ID is a Blasehke product if and only if

~r~-o /~r Inls(rt)lm(dt) = 0 . (2.8)

3 ~ For analytic J -contract ive matrix-valued functions in ]]3 there exists a

rich factorization theory. The foundations of this theory was built by V.P. Potapov, [2].

Katsnelson 839

The difficulties in the creation of such a theory are caused first of all by the

noncommutativity of matrix multiplication. The indefiniteness of the signature matrix

Y is a cause of supplementary difficulties.

In the matrix ease the role of the Blaschke factor in the scalar case is now

played by the so-called Blaschke-Potapov factor.

In the scalar case the Blaschke factor b(z, ~) is defined only by its root - the

point 4. In the matrix ease the Blaschke-Potapov factor is defined not only by some

point r but also by some matrix P. Moreover, in the matrix case there exist three kinds

of Blaschke-Potapov factors.

The Blaschke-Potapov factor of the first kind has the form

B(z; P) = z + (b(z, 0 - 1)P (3.1)

where b(z, ~) is a scalar Blaschke factor, !~1 < 1, and a P is a matrix which satisfies the

conditions

JP>O, p2=p . (3.2)

The inverse function

B(z; ~, p ) - I = I Jr (b(z, r _ 1)P (3.3)

So the Blaschke-Potapov factor of the first kind is a holomorphic function in

the unit disk, and its inverse B(z; {, p ) - I has a simple pole at the point ~. It follows

easily that

J - B*(z; 4, P ) J B ( z ; (, P) = (1 - [b(z, ( ) [ 2 ) j p (3.4)

and hence that

J - B * ( z ; ~ , P ) J B ( z ; ( , P ) ~ >-0' if I z l < l (3.5) = 0 , if I z l = l [

The Blaschke-Potapov factor of the second kind is of the form

B(z; ~, P) = I -- (b(z, ~)-~ - 1)P (3.6)

where b(z, ~) is a scalar Blaschke factor, and the matrix P satisfies the conditions

J P > O , p2 = _ p . (3.7)

840 Katsnelson

The inverse function has the form

B(z; r P ) - I = r - (b(z, r - 1)P (a.s)

Finally, the Blaschke-Potapov factor of the third kind has the form

B(z; r P) : I - r + z p (3.9)

where s is a point on the unit circle [~1 = 1, and the matrix P satisfies the conditions

J P >_ O , p2 = o . (3.10)

In this case, r + z p (a.1!)

B ( z ; r p ) - i = I + ( _ z

In the positive definite case ( J = I ) only Blaschke-Potapov factors of the first

kind exist. In the negative definite ease ( J = - I ) only the Blaschke-Potapov factors of

{he second kind exist. In the case J r + I Blaschke-Potapov factors of all three kinds

exist.

4 ~ Let {Zk} be a sequence of points in ID (not necessarily distinct), let {Pk}

be a sequence of matrices which satisfy the conditions

JPk >_ o , Pg = e ~ . (4.1)

and assume that

E (1 -[ZkI)[[PkH < oc (4.2) l<k<oo

Let B(z; zk, Pk) denote the corresponding Blaschke-Potapov factors: of the

first kind. We want to form the product of such factors. Since the multiplication of

matrices is noneommutative, the order of these factors is essential: the product depends

upon the order of multiplication.

Let us define a left Blaschke-Potapov product B (g) (z) and a right Blaschke-

Potapov product B(r) (z):

B(e)(*) = l ] B(~; ~k, Pk) = B(z; zl , P,)B(~; ~2, e2) . �9 k>l

(4.3)

Katsnelson 841

+ . -

B (r) (z) = H B(z; zk, Pk) . . . . B(z; z2, P2)B(z; z l , P1) �9 (4.4) k>_l

If the number of the factors is infinite, then the infinite product is defined as

a limit of "partial products":

= lira / r t - - r

where

= lira (4 .5)

- . - + +.__

1-I II B(z;zk,P ). (4.6) t<k<m l<k<m

The condition (4.2) guarantees that both Blaschke-Potapov products converge

absolutely and uniformly on each compact set inside ]I).

Since a Blaschke-Po~apovfactor of the first kind is u holomorphic J-contractive

matrix-valued function in 113, B(D (z) and B(~) (z) are holomorphic J-contractive matrix-

valued functions in 113, and (B(g)(z)) -1 , (B(r)(z) -1 are meromorphic in I13. They are

holomorphic everywhere in 111) except at the points Zl, z2<-- where they have poles.

Since it is J-contractive every Blaschke-Potapov product B(z) (both left and right) has

a boundary value

B( 0 dej ~ m B( r0 , (t e ~r) (4.7) r 1-0

where the limit exist for almost every point t 6 ~J]? in the sense of Lebesgue measure.

(Here B(t) stands for either B(l)(t) or B(~)(t)). Thus the function B(t) is defined almost

everywhere on ql'. Arov proved [5], that the values of the function B(t) are J-unitary:

J - B*( t )JB( t ) = 0 (for almost all t e ~lr)

or equivalently

J - B( t )JB*( t ) = 0 (for almost all t E ~ ) .

5 ~ A key role in the factorization theory of analytic J-contractive matrix-

valued functions is played by the splitting off of elementary factors from the matrix-valued

function.

Let W(z) : ID ~ 3,4n be a J-contractive analytic matrix-valued function. We

suppose for simplicity that the function W(z) is holomorphic in ID, and that the poles

Zl ,Z2, . . . of W(z) -1 are simple.

842 Katsnelson

It is POsSible to split-off a Blaschke-Potapov factor B(z; Zl, P~g)) of th e first

kind from the left hand side of W to obtain

W(z) = a(z; ~, Pl(e)). w(*)(z) , (5.1)

where W (g) (z) is a matrix-valued function which is J-contractive and holomorphic in

113 and W(g)(z) -~ has poles only at the points z2,za,... , but not at the point z 1.

Analogously it is possible to split off a Slaschke-Potapov s B(z; zl, p~r) os thefirs t

kind from the right hand side of W(z) to obtain

W(z) = }9"1 (r) (z)B(z; Zl, pl(r)) , (5.2)

where the matrix-function W (r) (z) is J-contractive and holomorphie in 1I), and W (r) (z ) - !

has poles only at the points z2, za, �9 �9 -, -- �9 , but not the point z 1.

At the next step it is possible to split-off a Blaschke-Potapov factor 2~(z; z2, P(Q

from the left of W (*) (z), and to split-off a Blaschke-Potapov factor B(z; z2, ~ ) ) from

the right of W(r)(z). After n such steps we obtain two representations for W(z):

where .._+ + - -

~ 7 ) _ - ~ ,(z;~,eff)), B(:)(z)= 1-[ B(z;z~,Pk +)) (5.4) l<k<_n l<k<n

are Blaschke-Potapov products of the first kind, and W(O(z), W(r)(Z) are J-contractive

holomorphic matrix-value d functions in ID and (W(e)(z))-! , (W(r)(z)) "1 are h010mo>

phic everywhere in ID except at the points Znq-l~ Zn+2~.... These points are the poles

of the functions (W(g)(z)) -1, (w(r ) (z l ) "1.

products

B(~)(z) = nm B~)(~) , n - - - + ~

converge.

Potapov proved that the infinite Blaschke

B(r)(z) = lira B(r)(z)

THEOREM. (On the existence of multiplicative representations. V. Potapov

[2]). Every m~tri~ ~I~ed f~nction W(,) which is J-,o,tractive ~ d holomorp,~i, in •

can be represented in each of the following two forms:

w ( ~ ) = B (~)(~)E (~)(~) (5.5)

Katsnelson 843

and w ( ~ ) = E(r) ( z ) B + ) ( z ) , (5.6)

where B(*)(z) [B(r)(z)] is a left [right] Bla, chke-Potapov product of BIasehke-Potapov

factors of the first kind, E(O(z) [E(r)(z)] is J-contractive and holomorphic in ID and

E(l)(z) -1 [E(r)(z) -1] is holomorphic in 119.

We shall refer to (5.5) [(5.6)] as a left [right] multiplicative representation of

W(z) , when the corresponding factors have the properties described in the statement of

the last theorem.

In the construction of the left [right] multiplicative representation it is possible

to change the order of enumerating the poles {zk}. But the result does not depend on

the order apart from a unitary constant factor.

THEOREM. (Uniqueness of multiplicative representation. V. Potapov, [2],

4, ~2, Theorem 3). I f W ( z ) = B~Z)(z)WJ~)(z) (j = 1,2) are two left multiplicative Ch.

representations of the same holomorphic J-contractive matrix-valued function W(z ) in

TD (corresponding to distinct enumerations of the set of poles of W ( z ) - l ) , then

. ~ ) ( z ) = B~)(z)u(~) , w~(~)(~) = ( u ( ~ ) ) - ~ w / ) ( ~ ) ,

where U (z) is a J-unitary constant matrix.

y w ( z ) = w (~) z~B (~) z 2) are j ( , j ( ) (j = 1, two right muItiplicative representa-

tions of the same holomorphic J-contractive matrix-valued function W(z ) in ID (corre-

sponding to distinct enumerations of the set of poles of W ( z ) - l ) , then

B~r)(z) = u ( r ) B ~ ) ( z ) , E~)(~) = E~) ( z ) (U(~) ) - I ,

where U (r) is a Y-unitary constant matrix.

This theorem makes the following definition meaningful. A matrix-valued

function W ( z ) which is J-contractive and holomorphic in ID is called a left Blaschke-

Potapov product if the factor E(z) (z) in its left multiplicative expansion (5.5) is a J-unitary

constant. The matrix-valued function W(z ) which is J-contractive and holomorphic in

ID is called a right Blaschke-Potapov product if the factor E (r) in its right multiplicative

expansion is a J-unitary constant.

PROBLEM. Is it possible for a holomorphic J-contractive matrix-valued

function W ( z ) to be a left BIaschke-Potapov product and not a right Blaschke-Potapov product?

844 Katsnelson

In other words, i8 the equality

= E ( z ) B ( ) , (5 .7)

possible, when B(D(z) is a left Blaschke-Potapov product, B(r)(z) i8 a right Bla~chl~e-

Potapov product~ and E(z) is a nonconstant J-contractive matrix-valued function such that both E(z) and E(z) -1 are holomorphic in ID?

In the sc~ar case (5.7) is obviously impossible. It is also impossible in the

matr ix case when J is definite (that is J = I or J = - I ) . This is because the J

contractive matrix-valued function W(z) is contractive when J -- I and hence W(z) is a left (right) Blaschke-Potapov product if and only if det W(z) is a Blaschke-Potapov

product.

It turns out tha t in the indefinite ease ( J ~ • (5.7) is possible with noncon-

s tant E(z). It is also possible to describe in function-theoretic terms which matrix-valued

functions E(z) ma y occur in (5.7).

D.Z. Arov, [4], gave a criterion for a holomorphic J -contrac t ive matr ix function

to be a Blaschke-Potapov product of the first kind. But this criterion is formulated in

such a way that we are not able to apply it for our purposes.

6 ~ DEFINITION. (See for example, [3], Ch. II, w Le~ f(z) be a

holomorphic complex-vMued function (scalar) in ID. The function f(z) belongs to the

class N + of V.I. Smirnov if the family In + [f(rt)[, 0 < r < 1, is uniformlyintegrabte on ~r.

This means that for any ~ > 0 there exists a 5 = 5(~) : (e C ~]r, e is measurable, re(e) <

e) - - > ,

f ln+[f(rt)[m(dt) < e (0 <_ r < 1) .

The etass N+ of V,I. Smirnov is a subclass of the well known ctass N of R.

Nevanlinna (see [3], Ch. II, w

Each of the functions f(z) ~ 0 of Smirnov class admits a multiplicative rep-

resentation of the form

where #s(dt) > 0 is singular (with respect to Lebesgue measure) and b(z) is a Blaschke

product . Conversely, every function f(z) of the form (6.1), where b(z) is a Blaschke

product and #s(dt) is a positive singular measure, is a function of the class N + of V.L

Katsne!son 845

Smirnov. Thus the possibility of obtaining a representation of the form (6.1) is a criterion

for belonging to the class N+.

DEFINITION. Let F(z) : 119 --~ M n be a holomorphic matrix-valued func-

tion. The function F(z) belongs to the class N+(n) of V.I. Smirnov if the family

ln+t[F(rt)[[, 0 ~ r < 1, is uniformly integrable on T'. That is if for every r > 0

there exists a 5 = 5(e): (e C T', e measurable, re(e) < ~) ==~

fein+IIF(rt)llrn(dt) < ~ V r e [0,1) .

f From this definition it follows that the matrix-valued function F(z) = [fpq~z)]l<_p,q<_n

belongs to the matrix class N+(n) if and only if all its entries fp,q(Z), I <_ p, q < n belong

to the scalar class N+.

If Fl(z) , F2(z) e N+(n), then Fl(Z) + F~(z) ~ N+ a~d E l ( z ) . F2(z) ~ N+.

7 ~ DEFINITION. (See [4]). Let J E A4n be a fixed signature matrix.

The matrix-valued function E(z) : 11) --~ M n is called Arov-singular if

(i) E(z) is J-contractive for z ~ n) .

(ii) The boundary values E(t) are J-unitary for almost all t E ~r .

(iii) Both the functions E(z), E(z) -1 are holomorphic inside ID.

(iv) Both the ~netio~s E(~), E(z) -~ belong to the das~ lV+(~) of V. Smirnov.

In the definite case (J = I or J = - I ) there are no non-constant Arov-singular

matrix functions. On the contrary~ in the indefinite case (Y ~ =t=!) there exist infinitely

many nonconstant Arov-singular matrix-functions.

THEOREM 1. Let

w ( z ) = B(~)(z) ,

be a left Blaschlce-Potapov product of the first kind with right multipIicatiw representation

Then E(z ) is Arov-singular.

THEOREM II.

W ( z ) = E ( z ) B (r)(z) .

Let E(z) : ]D --+ Jt4n be any Arov-singuIar matrix-valued

function. Then there exists a right Blaschke-Potapov product B(r)(z) of the first kind

~uch that the matrix-function

W(z) d~j E( z ) . B(~)(z)

846 gatsnelson

is a left Blaschke-Potapov product of the first kind

w ( z ) = B(~) (z) .

In other words,

sentable in the form

every Arov-singular matrix-valued function E(z) is repre-

E(z ) = B(~)(z) �9 B(r)(z) -1 ,

where B(O(z),

spectively.

7 ~ T H E O R E M III. Let F l ( z ) ,F2(z ) , . . . ,Fn(z) , ' . "

matrix-functions such that

II s " F~(0)i I < c o . k

B (r)(z) are left and right Blaschke-Potapov products of the first kind re-

be Arov-gngular

Then the products F(Z)(z) = ~I Fk(z) and F(r)(z) = [I Fk(z) converge uniformly on

each compact subset of ID, and F(O(z), F(r)(z) are both Arov'singular matrix,fanctions.

It is obvious tha t a Blaschke-Potapov factor of the third kind is an Arov-

singular matrix-function.

COROLLARY. Every convergent (left or right) Blaschke-Potapov product

which consists only of factors of the third kind is an Arov-singular function.

8 ~ The class of Arov-singular matrix-functions is not exhausted by Blaschke-

Po tapov products of the third kind. It also contains all the functions which are repre-

sented by multiplicative integrals of the form

g

/0- { E(z) = ~xp o 0 - ) - z

or the form l

E(z) = e~p [ 0 ( ~ ) - -

when H ( r ) and O(r ) are subject to the following special restrictions:

The function H(~-) in (8.1) or (8.2) is a matrix-valued function, 2ft = : [0,g] -+

3dn which satisfies the conditions of

J - positivity : H(T)J > 0 (0 < ~" < e) , (8.3)

Katsnelson 847

Nilpotency : H(v ) 2 = 0 (0 < r < t ) , (8.4)

Normal izat ion: sp H ( v ) J = 1 (0 <_ 7 <_ s (8.5)

The function @(T) = e iO(r) where 0 : [0,~] --* [0, 2~r] is a nondecreasing func-

t ion of T which is described below.

Let r : [0, 2~ l -* [0, e I be a nondeereasing function, which is normalized

b y t h e cond i t i on r - 0) = r 0 < t _< e. W e shal l define 0 ( r ) : [0, e] --+ [0, 2=]

as the generalized inverse of r This generalized inverse function O(t) will be defined

everywhere on [0, el even if the function r has jumps and intervals of constancy.

For ~- 6 [0, 2~r] put

0(r) a J inf{t : r _> ~} (8.6)

This generalized inverse 0(v) is nondecreasing on [0,g], and satisfies the normalization

condition

0 ( 7 - 0 ) = 0 ( ~ ) 0 < ~ _ < e .

The nondecreasing function ~b(t) : [0, 2~r] -* [0, t] is called singular if its deriva-

tive r vanishes at almost every (with respect to Lebesgue measure) value of its argu-

ment t.

C O N J E C T U R E . Let the matrix-valued function H ( r ) : [0, el- [0, 2=] satisfy

the conditions (S.S), (8.4) and (8.5), and let the funetio e ( , ) = exp{iO(,)},

0(7) is the generalized inverse of a singular nondecreasing function r : r =

[0, g], r 0) = r Suppose further that the matrix-valued function E ( z ) : / D --+ A/In,

is represented by either a left mult~)licative integral (8.1) or a right multiplicative integral

(8.2). Then the matrix-function E(z) is an Arov-singular matrix-function.

We are not able to prove this conjecture in complete generality. But we are

able to prove an impor tan t part icular case.

Let {Ok} be a sequence of distinct points in the segment [0, 2~r], le~ {~k} be a

sequence of positive numbers which satisfy the condition

k

and put

k:Ok<t

848 Katsnelson

This function r is cMled a jump function. Such a jump function r is nondecreasing,

and is normalized by the condition r - 0) = r and r 2~v]) = [0,~].

THEOREM IV. Let r be a jump function, ~b([0,2~r]) = [0,g], 8(7) be the

generalized inverse of r @(r) = exp{iS(t)}, and H(T) : [0, g] --+ dvln be a measurable

matrix-function which ~atisfies the conditions (8.3), (8.,~) and (8.5).

Then the matrix-function E(z) of the form {8.1j or (3.2) i8 an drov-singular

matrix-function.

The last result shows that the class of Arov-singular matrix-valued functions

is rich.

R E F E R E N C E S

1. Riesz F., 0ber die Randwerte eines analytischen Function, Mathematische Zeitschrift. 18, (1923), 87-95.

2. Potapov V.P., The muttipIicative structure of J-contractive matrix functions. Trudy Moskov. Math. ObgS. 4, (1955), 125-236.

3. Garnett J., Bounded Analytic Functions. Academic Press, New York, 1981.

�9 4. Arov D.Z., Three problems about J-inner matrix functions. Lecture Notes in Mathematics. 1043, Springer (1984), 164-168.

5. Arov D.Z., On boundary values of convergent sequences of meromorphic matrix functions. Mathem. Zametky. 25 (1979), 335-339,

Faculty of Mathematics Kharkov State University Kharkov, U.S.S.R.

Submitted: May 24, 1990