on the bitangential interpolation problem for contractive valued ...

c Copyright by Theta, 2000

J. OPERATOR THEORY 44(2000), 277–301

ON THE BITANGENTIAL INTERPOLATION PROBLEM FOR CONTRACTIVE VALUED FUNCTIONS IN THE POLYDISK D. ALPAY, J.A. BALL and V. BOLOTNIKOV

Communicated by Nikolai K. Nikolskii Abstract. We solve the bitangential interpolation problem with a finite number of interpolating points for the multivariable analogue of the Schur class introduced by J. Agler. The description of all solutions is parametrized by a Redheffer transform whose entries are given explicitly in terms of the interpolation data. Keywords: Multidimensional unitary system, transfer or characteristic function, contour integral interpolation conditions, linear fractional parametrization. MSC (2000): Primary 47A57; Secondary 32B99.

1. INTRODUCTION

In this paper we study the bitangential interpolation problem for the class of contractive valued functions in the polydisk introduced by J. Agler in [2]. This class, which we denote by Sdp×q and call the Schur class of the polydisk, consists of all Cp×q -valued functions S analytic on the d-fold polydisk Dd :  Dd = z = (z1 , . . . , zd ) ∈ Cd : |zk | < 1 and such that sup kS(rT1 , . . . , rTd )k 6 1 r 0

(k = 1, . . . , d)

and satisfy the generalized Stein identity (2.8)

d X k=1

(Mk Λk Mk∗ − Nk Λk Nk∗ ) = XX ∗ − Y Y ∗

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On the bitangential interpolation problem

where  (2.9)

Mk =

InL 0

  Ak 0 , Nk = 0 Bk

0 InR



 , X=

   XL YL , Y = . XR YR

Proof. Here we check the necessity of conditions (2.7), (2.8). The proof of the sufficiency part is postponed up to Section 4 where it will be obtained as a consequence of stronger results. Let S be a solution of Problem 1.2, that is let relations (1.1), (1.15)-(1.17) be in force. The expression in the left hand side of (1.17) is necessarily nonnegative which gives (2.7). Substituting the partitionings (1.13) and (2.9) into (2.8) we conclude that (2.8) is equivalent to the following three equalities (2.10)

d X

L ∗ ∗ ∗ (ΛL k − Ak Λk Ak ) = XL XL − YL YL

k=1

(2.11)

d X

∗ LR ∗ ∗ (ΛLR k Bk − Ak Λk ) = XL XR − YL YR

k=1

(2.12)

d X

R ∗ ∗ ∗ (ΛR k − Bk Λk Bk ) = YR YR − XR XR .

k=1

Applying (2.6) to F (z) = G(z) = XL Hk1 (z) and using (1.18) we get ΓA,A {zk ω ¯ k XL Hk1 (z)Hk1 (ω)∗ XL∗ } = Ak ΓA,A {XL Hk1 (z)Hk1 (ω)∗ XL∗ }A∗k ∗ = Ak ΛL k Ak .

(2.13)

Summing up equalities (1.18) and (2.13) for all k and subtracting the second sum from the first one, we obtain   X d d X 1 1 ∗ ∗ L ∗ (2.14) ΓA,A XL (1 − zk ω ¯ k )Hk (z)Hk (ω) XL = (ΛL k − Ak Λk Ak ). k=1

k=1

On the other hand the interpolation condition (1.15) implies  ΓA,A XL (Ip − S(z)S(ω)∗ )XL∗  (2.15) = ΓA,A XL XL∗ − RA (XL S(z))(RA (XL S(ω)))∗ = XL XL∗ − YL YL∗ . By (1.1), the expressions in the left hand sides of (2.14) and (2.15) are equal. The equality of the right hand side expressions leads to (2.10). In much the same way, applying (2.6) to F (z) = G(z) = YR Hk2 (z) and using (1.20) and (1.16) we get (2.12). Finally, in view of (1.19), ∗ ΓA,B {(zk − ω ¯ k )XL Hk1 (z)Hk2 (ω)∗ XR } ∗ ∗ = Ak ΓA,B {XL Hk1 (z)Hk2 (ω)∗ XR } − ΓA,B {XL Hk1 (z)Hk2 (ω)∗ XR }Bk∗ LR ∗ = Ak ΛLR k − Λk Bk

which being summed up over all k = 1, . . . , d, gives   X d d X ∗ LR ∗ (2.16) ΓA,B XL (zk − ω ¯ k )Hk1 (z)Hk2 (ω)∗ XR = (Ak ΛLR k − Λk Bk ). k=1

k=1

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On the other hand the interpolation conditions (1.15) and (1.16) imply (2.17)

ΓA,B {XL (S(z) − S(¯ ω ))YR∗ } = RA (XL S(z))YR∗ − XL RB (YR S(¯ ω )∗ )∗ ∗ = YL YR∗ − XL XR .

By (1.1), the expressions in the left hand sides of (2.16) and (2.17) are equal. The equality of the right hand side expressions leads to (2.11). Thus, the equalities (2.10)–(2.12) hold and therefore (2.8) is in force. Note that conditions (2.7), (2.8) for the Nevanlinna-Pick interpolation problem were first obtained in [1]. 3. THE UNIVERSAL UNITARY COLLIGATION ASSOCIATED WITH THE INTERPOLATION PROBLEM

We recall that a d-variable colligation is defined as a quadruple d n o M Ω= H= Hk , F, G, U k=1

consisting of three Hilbert spaces H (the state space) which is specified to have a fixed d-fold orthogonal decomposition, F (the input space) and G (the output space), together with a connecting operator       A B H H (3.1) U= : → . C D F G The colligation is said to be unitary if the connecting operator U is unitary. A colligation d o n M e e= ek , F, G, U e= H H Ω k=1

is said to be unitarily equivalent to the colligation Ω if there is a unitary operator e such that α:H→H     α 0 e α 0 αPk = Pek α (k = 1, . . . , d) and U=U 0 IG 0 IF e onto where Pk and Pek are orthogonal projections from H onto Hk and from H ek , respectively. The characteristic function of the colligation Ω is defined as H (3.2)

SΩ (z) = D + CZ(z)(IH − AZ(z))−1 B,

where Z is defined as in (3.26). Thus, Theorem 1.1 claims that a Cp×q -valued function S analytic in Dd belongs to the class Sdp×q if and only if it is the characteristic function of some d-variable unitary colligation    d M A B (3.3) Ω= H= H k , Cq , Cp , . C D k=1

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On the bitangential interpolation problem

Remark 3.1. Unitary equivalent colligations have the same characteristic function. In this section we associate certain finite dimensional (i.e. with finite dimensional state space, input space and output space) unitary colligation to Problem 1.2. It turns out that the characteristic function of this colligation (which is rational, according to (3.2)) is the transfer function of the Redheffer transform describing all solutions of Problem 1.2. We assume that necessary conditions (2.7) and (2.8) for Problem 1.2 to have a solution, are in force. Let Mk , Nk , X and Y be the matrices defined by (2.9) and let (3.4)

1/2

1/2

1/2

W1 = (M1 Λ1 , . . . , Md Λd )

1/2

and W2 = (N1 Λ1 , . . . , Nd Λd ).

The identity (2.8) provides the linear map  ∗  ∗ W1 W2 (3.5) V: f −→ f Y∗ X∗ to be an isometry from  ∗ W1 ⊂ Cnd+q DV = Ran Y∗

 onto

Ran

W2∗ X∗



⊂ Cnd+p

(here and in what follows we set n := nL +nR ). The verification is straightforward: for every choice of f, g ∈ Cn ,  ∗   ∗    ∗   ∗   W1 W1 W2 W2 f, g − f, g X∗ Y∗ Y∗ X∗ =g

∗

d X

Mk Λk Mk∗

+YY

∗



f −g

k=1

∗

d X

 Nk Λk Nk∗ + XX ∗ f = 0.

k=1

Let ∆(z) be the Cn×n -valued function defined by (3.6)

∆(z) =

d X

Mk Λk (Mk∗ − zk Nk∗ ) + Y Y ∗ .

k=1

Another representation of ∆, (3.7)

∆(z) =

d X

(Nk − zk Mk )Λk Nk∗ + XX ∗ ,

k=1

follows from (2.8) and (3.6). Note that ∆ takes the nonnegative value at the origin, (3.8)

∆(0) =

d X

Mk Λk Mk∗ + Y Y ∗ =

k=1

d X

Nk Λk Nk∗ + XX ∗ ,

k=1

which on account of (3.4) can be written as (3.9)

∆(0) = W1 W1∗ + Y Y ∗ = W2 W2∗ + XX ∗ .

Let rank ∆(0) = r 6 n and let Q ∈ Cn×r be a matrix such that (3.10)

rank Q∗ ∆(0)Q = rank ∆(0) = r.

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The latter relation implies in particular, that Q∗ ∆(0)Q > 0, which allows to define the pseudoinverse matrix ∆(0)[−1] as ∆(0)[−1] = Q(Q∗ ∆(0)Q)−1 Q∗ .

(3.11)

Note that (3.11) determines the Moore-Penrose pseudoinverse ([11]) if the columns of Q span the range Ran ∆(0) of ∆(0). Since V is an isometry, it follows from (3.9) that (3.12)

dim DV = dim RV = rank(W1 , Y ) = rank(W2 , Y ) = r.

Introducing the orthogonal complements ⊥ DV = Cnd+q DV

nd+p and R⊥ RV V =C

we obtain as a corollary of (3.12) that (3.13)

⊥ q 0 := dim DV = nd + q − r

and p0 := dim R⊥ V = nd + p − r.

⊥ is given by Remark 3.2. The orthogonal projection PD⊥ of Cnd+q onto DV V the formula  ∗ W1 (3.14) PD⊥ = Ind+q − ∆(0)[−1] (W1 , Y ) V Y∗

whereas the orthogonal projection PR⊥ of Cnd+p onto R⊥ V is given by the formula V  ∗ W2 (3.15) PR⊥ = Ind+p − ∆(0)[−1] (W2 , X). V X∗ 0

0

We let T1 ∈ C(nd+q)×q and T2 ∈ C(nd+p)×p be isometric matrices whose ⊥ columns span DV and R⊥ ⊥ and PR⊥ can V respectively. Then the projections PDV V be represented as PD⊥ = T1 T1∗ ,

(3.16)

PR⊥ = T2 T2∗ .

V

V

On the other hand, the following equalities (3.17)

T1∗ T1 = Iq0 ,

T2∗ T2 = Ip0 ,

(W1 , Y )T1 = 0,

(W2 , X)T2 = 0

hold by construction. Lemma 3.3. The operator (3.18)

U0 =

U11 U21 U31

U12 U22 U32

U13 U23 0

! :

Cnd Cq 0 Cp

! −→

Cnd Cp0 Cq

with entries specified by the rules    ∗ U11 U12 W2 (3.19) = ∆(0)[−1] (W1 , Y ) U21 U22 X∗   U13 ∗ (U31 , U32 ) = T1 , (3.20) = T2 U23 is a unitary extension of the isometry V defined by (3.5).

!

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On the bitangential interpolation problem

Proof. It follows from (3.9),   U11 U12 U11 U21 U22 U21  ∗  U11 U12 U11 U21 U22 U21

(3.19) that ∗  ∗  U12 W2 ∆(0)[−1] (W2 , X) = X∗ U22   ∗ U12 W1 ∆(0)[−1] (W1 , Y ) = Y∗ U22

which together with (3.14)–(3.16) and (3.20) imply   ∗   U11 U12 U11 U12 U13 ∗ ∗ + (U13 , U23 ) = Ind+q U21 U22 U21 U22 U23  ∗    ∗  U11 U12 U11 U12 U31 (U31 , U32 ) = Ind+p . + ∗ U32 U21 U22 U21 U22 It follows immediately from the two latter equalities and (3.17) that U0 is unitary. To show that U0 is an extension of V , it suffices to check that   ∗  ∗ U11 U12 W1 W2 (3.21) x = x (∀x ∈ Cn ). U21 U22 Y∗ X∗ Let Q ∈ Cn×r be a matrix satisfying (3.10). Then every vector x ∈ Cn can be represented as (3.22)

x = Qf + g

for

f ∈ Cr

and g ∈ Ker ∆(0).

By (3.9), (3.23)

W1∗ g = 0,

W2∗ g = 0

and X ∗ g = 0

(∀g ∈ Ker ∆(0)).

Substituting (3.19) and (3.22) into the right hand side of (3.21) and taking into account (3.11) and (3.23) we get   ∗  ∗ U11 U12 W1 W2 x= ∆(0)[−1] (W1 W1∗ + Y Y ∗ )x U21 U22 Y∗ X∗  ∗ W2 = Q(Q∗ ∆(0)Q)−1 Q∗ ∆(0)(Qf + g) X∗  ∗  ∗  ∗ W2 W2 W2 = Qf = (Qf + g) = x X∗ X∗ X∗ and the lemma follows. The unitary operator U0 specified by (3.18)–(3.20) is the connecting operator of the finite dimensional unitary colligation  q  p   d M C0 C0 nd n C , , , U0 . (3.24) Ω0 = C = Cp Cq k=1

According to (3.2), the characteristic function of this colligation is given by   Σ11 (z) Σ12 (z) Σ(z) = Σ21 (z) Σ22 (z)     (3.25) U22 U23 U21 = + (I − Z(z)U11 )−1 Z(z)(U12 , U13 ) U32 0 U31

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where  (3.26)

z1 In

 ..

Z(z) = 

.

. zd In

The formula (3.25) presents a unitary realization of the function (nd+q+p−r)×(nd+q+p−r) Σ ∈ Sd . By Theorem 1.1,   U21 ∗ ∗ ∗ I −Σ(z)Σ(ω)∗ = (I −Z(z)U11 )−1(I − Z(z)Z(ω)∗ )(I −U11 Z(ω)∗ )−1(U21 , U31 ) U31   U21 Σ(z)−Σ(¯ ω) = (I −Z(z)U11 )−1(Z(z)−Z(ω)∗ )(I − U11 Z(ω)∗ )−1 (U12 , U13 ) U31  ∗  U12 ∗ −1 ∗ (I −Z(z)U11 ) (I −Z(z)Z(ω)∗ )(I − U11 Z(ω)∗ )−1(U12 , U13 ). I −Σ(¯ z ) Σ(¯ ω) = ∗ U13 Using Schur blockwise matrix multiplication, the latter three relations can be written as   Σ(z) − Σ(¯ ω) Ip − Σ(z)Σ(ω)∗ Σ(¯ z )∗ − Σ(ω)∗ Iq − Σ(¯ z )∗ Σ(¯ ω)    1  d X Fk (z) 1 − zk ω ¯ k zk − ω ¯k = ◦ (3.27) (Fk1 (ω)∗ , Fk2 (ω)∗ ) zk − ω ¯ k 1 − zk ω ¯k Fk2 (z) k=1

where (3.28)

Fk1 (z)



U21 U31





∗ U12 ∗ U13



=

(I − Z(z)U11 )−1 Pk

and (3.29)

Fk2 (z)

=

∗ −1 (I − Z(z)U11 ) Pk .

4. EXPLICIT FORMULAS FOR THE CHARACTERISTIC FUNCTION OF THE UNIVERSAL UNITARY COLLIGATION

In this section we give explicit formulas for the block entries Σjk of the characteristic function Σ defined via (3.25). Let ∆(z) be the function given by (3.6). Its value at zero ∆(0) has been already used for the construction of the colligation (3.24). It follows immediately from (3.8), that Ker ∆(0) = {f ∈ Cn : Λk Mk∗ f = 0 (∀k), Y ∗ f = 0} (4.1)

= {f ∈ Cn : Λk Nk∗ f = 0 (∀k), X ∗ f = 0} = {f ∈ Cn : Λk Mk∗ f = 0, Λk Nk∗ f = 0 (∀k), Y ∗ f = 0, X ∗ f = 0}. Lemma 4.1. For every z ∈ Dd it holds that

(4.2)

Ker ∆(z) = Ker ∆(z)∗ = Ker ∆(0).

289

On the bitangential interpolation problem

Proof. Let f ∈ Ker ∆(0). Then (4.3)

Λk Mk∗ f = 0,

Λk Nk∗ f = 0 (k = 1, . . . , d),

Y ∗ f = 0,

X ∗f = 0

and by (3.6), ∆(z)f = ∆(z)∗ f = 0 at every point z. Therefore, Ker ∆(0) ⊆ Ker ∆(z) ∩ Ker ∆(z)∗ .

(4.4)

Now suppose that ∆(z)f = 0 for some choice of f ∈ Cn and z ∈ Dd . Then using representations (3.7) and (3.6) for ∆(z) and ∆(z)∗ respectively, we get 0 = f ∗ (∆(z) + ∆(z)∗ )f = f ∗

d X

(Nk − zk Mk )Λk Nk∗ + XX ∗

k=1

+

d X

 (Mk − z¯k Nk )Λk Mk∗ + Y Y ∗ f

k=1

=f

∗

d X

2

(1 − |zk |

)Mk Λk Mk∗

+

k=1

d X

(zk Mk − Nk )Λk (¯ zk Mk∗ − Nk∗ )

k=1 ∗

+ XX + Y Y

∗



f.

Therefore, relations (4.3) hold and thus, ∆(0)f = 0. Using the same chain of equalities we conclude that ∆(z)∗ f = 0 also implies ∆(0)f = 0. Thus, Ker ∆(z) ∪ Ker ∆(z)∗ ⊆ Ker ∆(0) which together with (4.4) implies (4.2). Thus, we have shown that ∆(z) has a nonnegative real part in Dd . Let Q ∈ C be the matrix from the representation (3.11) of the pseudoinverse ∆[−1] (0). According to (3.11) we introduce the function n×r

(4.5)

∆(z)[−1] = Q(Q∗ ∆(z)Q)−1 Q∗ ,

which is analytic in Dd on account of (4.2). Remark 4.2. Let PKer ∆(0) denote the orthogonal projection from Cn onto Ker ∆(0). Then
(4.6) I − ∆(z)∆(z)[−1] = I − ∆(z)∆(z)[−1] PKer ∆(0) . Proof. Take the representation (3.22) of any vector x ∈ Cn . On account of (4.2), ∆(z)g = 0 and using (4.5) we get

I − ∆(z)∆(z)[−1] ∆(z)x = I − ∆(z)Q(Q∗ ∆(z)Q)−1 Q∗ ∆(z)(Qf + g) = 0. Since x is an arbitrary vector, the latter equality means that for every z ∈ Dd ,
(4.7) I − ∆(z)∆(z)[−1] PRan ∆(z) = 0, where PRan ∆(z) denote the orthogonal projection from Cn onto Ran ∆(z). In view of (4.2), PRan ∆(z) = I − PKer ∆(z)∗ = I − PKer ∆(0) which being substituted into (4.7), leads to (4.6).

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Lemma 4.3. The following resolvent-like identity ∆(z)[−1] − ∆(0)[−1] = ∆(0)[−1] W1 Z(z)W2∗ ∆(z)[−1]

(4.8)

holds for every point z = (z1 , . . . , zd ) ∈ Dd . Proof. We begin with the equality ∆(0) − ∆(z) =

(4.9)

d X

zk Mk Λk Nk∗ = W1 Z(z)W2∗

k=1

which follows directly from (3.6), (3.8) and (3.4). Using (3.11) and (4.5) we get ∆(z)[−1] − ∆(0)[−1] = Q(Q∗ ∆(0)Q)−1 Q∗ (∆(0) − ∆(z))Q(Q∗ ∆(z)Q)−1 Q∗ , which together with (4.9) implies (4.8). To establish the explicit formula for Σ(z) in terms of the interpolation data we apply the matrix equality (I + BA−1 C)−1 = I − B(A + CB)−1 C

(4.10) to the matrices

A = Q∗ ∆(0)Q,

B = −Z(z)W2∗ Q,

C = Q∗ W1 .

Taking into account (4.9) and (4.5) we get (I − Z(z)U11 )−1 = I + Z(z)W2∗ Q(Q∗ (∆(0) − W1 Z(z)W2∗ )Q)−1 Q∗ W1 = I + Z(z)W2∗ Q(Q∗ ∆(z)Q)−1 Q∗ W1

(4.11)

= I + Z(z)W2∗ ∆(z)[−1] W1 . Furthermore, in view of (3.19), (4.8) and (4.11), it follows that

(4.12)


U21 (I − Z(z)U11 )−1 = X ∗ ∆(0)[−1] W1 I + Z(z)W2∗ ∆(z)[−1] W1  = X ∗ ∆(0)[−1] + ∆(0)[−1] W1 Z(z)W2∗ ∆(z)[−1] W1 = X ∗ ∆(z)[−1] W1 .

Similarly,  (I − U11 Z(z))−1 U12 = I + W2∗ ∆(z)[−1] W1 Z(z) W2∗ ∆(0)[−1] Y (4.13)


= W2∗ ∆(0)[−1] + ∆(z)[−1] W1 Z(z)W2∗ ∆(0)[−1] Y = W2∗ ∆(z)[−1] Y

and therefore, (4.14) (I − Z(z)U11 )−1 Z(z)U12 = Z(z)(I − U11 Z(z))−1 U12 = Z(z)W2∗ ∆(z)[−1] Y. Taking adjoints in (4.13) and replacing z by z¯ we obtain (4.15)

∗ ∗ −1 U12 (I − Z(z)U11 ) = Y ∗ (∆(¯ z )[−1] )∗ W2 .

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On the bitangential interpolation problem

Now we give explicit formulas for block entries Σjk of the function Σ: using (3.20), (3.25), (4.11), (4.12) and (4.14) we get Σ11 (z) = U22 + U21 (I − Z(z)U11 )−1 Z(z)U12 (4.16)

= X ∗ ∆(0)[−1] Y + X ∗ ∆(z)[−1] W1 Z(z)W2∗ ∆(0)[−1] Y
= X ∗ ∆(0)[−1] + ∆(z)[−1] W1 Z(z)W2∗ ∆(0)[−1] Y = X ∗ ∆(z)[−1] Y

(4.17)

Σ12 (z) = U23 + U21 (I − Z(z)U11 )−1 Z(z)U13  
U13 = U21 (I − Z(z)U11 )−1 Z(z), Ip U23
= X ∗ ∆(z)[−1] W1 Z(z), Ip T2

(4.18)

Σ21 (z) = U32 + U31 (I − Z(z)U11 )−1 Z(z)U12   (I − Z(z)U11 )−1 Z(z)U12 = (U31 , U32 ) Iq   ∗ [−1] Z(z)W ∆(z) Y ∗ 2 = T1 Iq

(4.19)

Σ22 (z) = U31 (I − Z(z)U11 )−1 Z(z)U13  
Ind ∗ = T1 I + Z(z)W2∗ ∆(z)[−1] W1 (Ind , 0)Z(z)T2 . 0

5. DESCRIPTION OF THE SET OF ALL SOLUTIONS

In this section we describe the set of all solutions of Problem 1.2. The main result is: Theorem 5.1. Let Σ be the function decomposed as in (3.25) into four blocks Σjk specified by (4.16)–(4.19). Then S is a solution of Problem 1.2 if and only if S has a representation of the form (5.1)

S(z) = Σ11 (z) + Σ12 (z)E(z)(Iq0 − Σ22 (z)E(z))−1 Σ21 (z) 0

0

where the free parameter E sweeps through the set Sdp ×q . The transformation (5.1) acting on the free parameter E is sometimes called the Redheffer transformation with the transfer function Σ. The proof of Theorem 5.1 is divided in a number of steps and relies on the following auxiliary results.

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D. Alpay, J.A. Ball and V. Bolotnikov 0

0

S be of the form (5.1). Then S ∈ Sdp×q



Σ11 Σ12 Σ21 Σ22 and moreover,

Lemma 5.2. Let E ∈ Sdp ×q , let Σ =



(p+q 0 )×(q+p0 )

∈ Sd

and let

I − S(z)S(ω)∗ = Ψ(z)(I − E(z)E(ω)∗ )Ψ(ω)∗ (5.2)

∗



+ (I, Ψ(z)E(z))(I − Σ(z)Σ(ω) )

I E(ω)∗ Ψ(ω)∗



S(z) − S(¯ ω ) = Ψ(z)(E(z) − E(¯ ω ))Φ(¯ ω) (5.3)

 + (I, Ψ(z)E(z))(Σ(z) − Σ(¯ ω ))

I E(¯ ω )Φ(¯ ω)



I − S(¯ z )∗ S(¯ ω ) = Φ(¯ z )∗ (I − E(¯ z )∗ E(¯ ω ))Ψ(¯ ω) (5.4)

+ (I, Φ(¯ z )∗ E(¯ z )∗ )(I − Σ(¯ z )∗ Σ(¯ ω ))

where (5.5) Ψ(z) = Σ12 (z)(I −E(z)Σ22 (z))−1

and



I E(¯ ω )Φ(¯ ω)

 ,

Φ(z) = (I −Σ22 (z)E(z))−1 Σ21 (z).

Proof. Using the functions Ψ and Φ from (5.5) and taking into account the identity E(z)(Iq0 − Σ22 (z)E(z))−1 = (Ip0 − E(z)Σ22 (z))−1 E(z), one can represent the function S of the form (5.1) as (5.6) S(z) = Σ11 (z) + Ψ(z)E(z)Σ21 (z) = Σ11 (z) + Σ12 (z)E(z)Φ(z) The following identities (5.7) Σ12 (z) + Ψ(z)E(z)Σ22 (z) = Ψ(z), Σ21 (z) + Σ22 (z)E(z)Φ(z) = Φ(z) follow immediately from (5.5) and imply together with (5.6), that   Σ11 (z) Σ12 (z) (5.8) (I, Ψ(z)E(z))Σ(z) = (I, Ψ(z)E(z)) = (S(z), Ψ(z)) Σ21 (z) Σ22 (z) and  (5.9)

Σ(z)

I E(z)Φ(z)



 =

Σ11 (z) Σ12 (z) Σ21 (z) Σ22 (z)



I E(z)Φ(z)



 =

S(z) Φ(z)

Using (5.8) we get (I, Ψ(z)E(z))(I − Σ(z)Σ(ω)∗ )



I E(ω)∗ Ψ(ω)∗



= I + Ψ(z)E(z)E(ω)∗ Ψ(ω)∗ − S(z)S(ω)∗ − Ψ(z)Ψ(ω)∗ which is equivalent to (5.2). Next, on account of (5.8) and (5.9),   I (I, Ψ(z)E(z))(Σ(z) − Σ(¯ ω )) E(¯ ω )Φ(¯ ω)     I S(¯ ω) = (S(z), Ψ(z)) − (I, Ψ(z)E(z)) E(¯ ω )Φ(¯ ω) Φ(¯ ω) = S(z) − S(¯ ω ) − Ψ(z)(E(z) − E(¯ ω ))Φ(¯ ω)

 .

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On the bitangential interpolation problem

which is equivalent to (5.3). In much the same way one can check (5.4) with help 0 0 of (5.9). Let E ∈ Sdp ×q . Then   Ip0 − E(z)E(ω)∗ E(z) − E(¯ ω) E(¯ z )∗ − E(ω)∗ Iq0 − E(¯ z )∗ E(¯ ω) (5.10)   1  d  X Gk (z) 1 − zk ω ¯ k zk − ω ¯k = ◦ (G1k (ω)∗ , G2k (ω)∗ ) zk − ω ¯ k 1 − zk ω ¯k G2k (z) k=1

for some operator-valued functions G1k (z) and G2k (z) analytic on Dd . Substituting the latter representation together with (3.27) into (5.2)–(5.4) we get   Ip − S(z)S(ω)∗ S(z) − S(¯ ω) S(¯ z )∗ − S(ω)∗ Iq − S(¯ z )∗ S(¯ ω) (5.11)    1  d X 1−z ω Hk (z) zk − ω ¯k k ¯k = ◦ (Hk1 (ω)∗ , Hk2 (ω)∗ ) zk − ω ¯ k 1 − zk ω ¯k Hk2 (z) k=1

where Hk1 (z) = (Ψ(z)G1k (z), (I, Ψ(z)E(z))Fk1 (z))

(5.12) and

Hk2 (z) = (Φ(¯ z )∗ G2k (z), (I, Φ(¯ z )∗ E(¯ z )∗ )Fk2 (z)).

(5.13)

Therefore, S ∈ Sdp×q . Lemma 5.3. Let Σ11 be the function given by (4.16). Then (5.14)

RA (XL Σ11 (z)) = YL

RB (YR Σ11 (¯ z )∗ ) = XR .

and

Proof. First we note that in view of (4.1), Ker ∆(0) ⊆ Ker Y ∗ ∩ Ker X ∗ , and therefore, (5.15)

PKer ∆(0) Y = 0,

X ∗ PKer ∆(0) = 0

where PKer ∆(0) denotes the orthogonal projection from Cn onto Ker ∆(0). Using (3.6) and taking into account (4.6) we get XX ∗ ∆(z)[−1] = ∆(z)∆(z)[−1] −

d X

(zk Mk − Nk )Λk Nk∗ ∆(z)[−1]

k=1

(5.16) =I−

d X


(zk Mk − Nk )Λk Nk∗ ∆(z)[−1] − I − ∆(z)∆(z)[−1] PKer∆(0) .

k=1

In much the same way, it follows from (3.7) that

(5.17)

∆(z)[−1] Y Y ∗ = I − ∆(z)[−1]

d X

Mk Λk (Mk∗ − zk Nk∗ )

k=1


− PKer∆(0) I − ∆(z)[−1] ∆(z) .

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Using (4.16), (5.16) and the first relation in (5.15), we get XΣ11 (z) = XX ∗ ∆(z)[−1] Y = Y −

d X

(zk Mk − Nk )Λk Nk∗ ∆(z)[−1] Y.

k=1

Substituting (1.13) and (2.9) into the latter equality we obtain XL Σ11 (z) = YL −

d X

LR ∗ [−1] (zk InL − Ak )(ΛL Y. k , Λk )Nk ∆(z)

k=1

Applying the operator RA to both parts of the latter equality, taking into account that ∆(z)[−1] is analytic in Dd and using (2.1), (2.2), we get RA (XL Σ11 (z)) = RA (YL ) −

d X


LR ∗ [−1] RA (zk InL − Ak )(ΛL Y = YL k , Λk )Nk ∆(z)

k=1

which proves the first relation in (5.14). The proof of the second assertion of lemma is quite similar: using (4.16), (5.17) and the second relation in (5.15) we get Σ11 (z)Y ∗ = X ∗ ∆(z)[−1] Y Y ∗ = X ∗ − X ∗ ∆(z)[−1]

d X

Mk Λk (Mk∗ − zk Nk∗ ).

k=1

Substituting partitions (1.13) and (2.9) into the latter equality and comparing the ”right” blocks we obtain  LR  d X Λk ∗ Σ11 (z)YR∗ = XR − X ∗ ∆(z)[−1] Mk (Bk∗ − zk InR ). ΛR k k=1

Therefore, YR Σ11 (¯ z )∗ = XR +

d X


∗
∗ ∗ R (zk InR − Bk ) (ΛLR z )[−1] X k ) , Λk Mk ∆(¯

k=1

and applying the operator RB to both parts of this latter equality we come to RB (YR Σ11 (¯ z )∗ ) = XR which ends the proof of lemma. Lemma 5.4. Let Σ12 and Σ21 be the functions defined by (4.17) and (4.18), respectively. Then (5.18)

RA (XL Σ12 (z)) = 0

and

RB (YR Σ21 (¯ z )∗ ) = 0.

Proof. We begin with equalities (5.19)

(W2 , X)T2 = 0 and

(W1 , Y )T1 = 0

which hold by definition of T1 , T2 . Note also that in view of (4.1), Ker ∆(0) ⊆ Ker W1∗ ∩ Ker W2∗ , and therefore, (5.20)

PKer ∆(0) W2 = 0,

W1∗ PKer ∆(0) = 0.

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On the bitangential interpolation problem

Using (4.17), (5.16) and taking into account the first relations from (5.19), (5.20) we get XΣ12 (z) = X(X ∗ ∆(z)[−1] W1 Z(z), Ip )T2 d    X = I− (zk Mk − Nk )Λk Nk∗ ∆(z)[−1] W1 Z(z), X T2 k=1



= W1 Z(z) − W2 −

d X

 (zk Mk − Nk )Λk Nk∗ ∆(z)[−1] W1 Z(z), 0 T2 .

k=1

Using the explicit formulas (3.4) for W1 , W2 one can rewrite the latter equality as 1/2 1/2
XΣ12 (z) = (z1 M1 − N1 )Λ1 , . . . , (zd Md − Nd )Λd , 0 T2 d X  − (zk Mk − Nk )Λk Nk∗ ∆(z)[−1] W1 Z(z), 0 T2 . k=1

Substituting partitions (1.13), (2.9) into the latter equality and comparing the upper blocks we obtain 1/2 1/2
XL Σ12 (z) = (z1 InL − A1 , 0)Λ1 , . . . , (zd InL − Ad , 0)Λd , 0 T2 d X  LR ∗ [−1] (zk InL − Ak )(ΛL − W1 Z(z), 0 T2 . k , Λk )Nk ∆(z) k=1

Applying the operator RA to both parts of the latter equality, taking into account that ∆(z)[−1] is analytic in Dd and using (2.2), we get the first relation from (5.18). The second one is obtained in much the same way. Lemma 5.5. Fk1 (z) and Fk2 (z) are the functions given by (3.28) and (3.29) and let Γ be the operator defined via (1.12). Then   ∗  XL 1 1 ∗ (5.21) ΓA,A (XL , 0)Fk (z)Fk (ω) = ΛL k 0   ∗ YR ΓA,B (XL , 0)Fk1 (z)Fk2 (ω)∗ (5.22) = ΛLR k 0   ∗ YR (5.23) ΓB,B (YR , 0)Fk2 (z)Fk2 (ω)∗ = ΛR k. 0 Proof. Substituting (4.12) and (4.15) into (3.28) and (3.29), respectively, we obtain (5.24)

(Ip , 0)Fk1 (z) = X ∗ ∆(z)[−1] W1 Pk ,

(Iq , 0)Fk2 (z) = Y ∗ (∆(¯ z )[−1] )∗ W2 Pk .

Multiplying both sides of the first equality in (5.24) by X from the left an using (5.16) and the second relation in (5.20) we receive d   X (X, 0)Fk1 (z) = XX ∗ ∆(z)[−1] W1 Pk = I − (zk Mk − Nk )Λk Nk∗ ∆(z)[−1] W1 Pk k=1

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and the comparison of the upper blocks leads to d   X LR ∗ [−1] (XL , 0)Fk1 (z) = (InL , 0) − (zk InL − Ak )(ΛL W1 Pk . , Λ )N ∆(z) k k k k=1

Applying the operator RA to both parts of the latter equality, taking into account that ∆(z)[−1] is analytic in Dd and using (2.1), (2.2), we get (5.25)

RA ((XL , 0)Fk1 (z)) = (InL , 0)W1 Pk .

Similarly, using the second equality from (5.24) together with (5.17) and the first relation in (5.20) we get (Y, 0)Fk2 (z) = Y Y ∗ (∆(¯ z )[−1] )∗ W2 Pk d  X
∗  = I− W2 Pk (Mk − zk Nk )Λk Mk∗ ∆(¯ z )[−1] k=1

and the comparison of the lower blocks leads to d  X
∗ R
∗
 [−1] ∗ (zk InR − Bk ) ΛLR M ∆(¯ z ) (YR , 0)Fk2 (z) = (0, InR ) + , Λ W2 Pk . k k k k=1

Applying the operator RB to both parts of the latter equality, we get (5.26)

RB ((YR , 0)Fk2 (z)) = (0, InR )W2 Pk .

By definition (1.12) of Γ and in view of (5.25), (3.4) and (2.9),   ∗    XL InL 1 1 ∗ ∗ ΓA,A (XL , 0)Fk (z)Fk (ω) = (InL , 0)W1 Pk W1 0 0   InL ∗ = (InL , 0)Mk Λk Mk = ΛL k, 0      0 0 ΓA,B (XL , 0)Fk1 (z)Fk2 (ω)∗ = (InL , 0)W1 Pk W2∗ YR∗ InR   0 ∗ = (InL , 0)Mk Λk Nk = ΛLR k , InR which prove (5.21) and (5.22). The equality (5.23) is verified in much the same way. Now we can prove the necessity part of Theorem 5.1. Lemma 5.6. Every function S of the form (5.1) satisfies the interpolation conditions (1.15), (1.16) and (1.17). Proof. Let Ψ and Φ be the functions defined in (5.5). Since the functions Ψ(z)E(z) and E(z)Φ(z) are analytic in Dd it follows from (5.18) and by Lemma 2.1 that RA (XL Σ12 (z)E(z)Φ(z)) = 0 and

RB (YR Σ21 (¯ z )∗ E(¯ z )∗ Ψ(¯ z )∗ ) = 0.

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On the bitangential interpolation problem

Now it follows immediately from (5.14) and representations (5.6) that S satisfies interpolation conditions (1.15) and (1.16). It remains to show that the functions Hk1 (z) and Hk2 (z) from the representation (5.11) and defined by (5.12), (5.13), respectively, satisfy the interpolation conditions (1.18)–(1.20). It follows from (5.5) and (5.18) that (5.27)

and RA (XR Φ(¯ z )∗ ) = 0.

RA (XL Ψ(z)) = 0

By Lemma 2.1, we get from (5.12), (5.13) and (5.27) (5.28)

RA (XL Hk1 (z)) = RA (0, (XL , 0)Fk1 (z))

(5.29)

RA (YR Hk2 (z)) = RA (0, (YR , 0)Fk2 (z))

where Fk1 (z) and Fk2 (z) are the functions from the representation (3.27) given by formulas (3.28) and (3.29), respectively. By definition (1.12) of Γ and in view of (5.28), (5.29),   ∗  XL 1 1 ∗ ∗ 1 1 ∗ ΓA,A {XL Hk (z)Hk (ω) XL } = ΓA,A (XL , 0)Fk (z)Fk (ω) 0   ∗ YR ΓA,B {XL Hk1 (z)Hk2 (ω)∗ YR∗ } = ΓA,B (XL , 0)Fk1 (z)Fk2 (ω)∗ 0   ∗ YR ΓB,B {YR Hk2 (z)Hk2 (ω)∗ YR∗ } = ΓB,B (YR , 0)Fk2 (z)Fk2 (ω)∗ 0 which together with (5.21)–(5.23) imply (1.18)–(1.20). Lemma 5.7. Let S be a solution of Problem 1.2. Then it can be represented 0 0 in the form (5.1) for some choice of the parameter E ∈ Sdp ×q . Proof. By the construction, the coefficient matrix Σ of the transformation (5.1) is the characteristic function of the unitary colligation  ∗Ω 0 defined by (3.24). W1 It is easily seen from (3.4) that the domain DV = Ran of the isometry V Y∗ L  d 1/2 defined by (3.25), is the subspace of Ran Λk ⊕ Ran Y ∗ . k=1

It was shown in [7] that a function S is of the form (5.1) for some choice of 0 0 the parameter E ∈ Sdp ×q if and only if S is the characteristic function of a unitary colligation    d M e B e A q p e e e Hk , C , C , e e (5.30) Ω= H= C D k=1

with the state space (5.31)

e= H

d M

ek H

ek = Ran Λ1/2 ⊕ Nk , H k

of the form

k=1

and such that  (5.32)

e B e A e e C D



 Ran

W1∗ Y∗

 = V.

298

D. Alpay, J.A. Ball and V. Bolotnikov

e is the coupling of the colligation Ω0 and some In other words, the colligation Ω unitary colligation d n o M 0 0 Ω0 = N = N k , Cp , Cq , U 0 k=1 0

0

where the dimensions p and q of the input and the output spaces are given by (3.13). At that, the parameter E in (5.1) is the characteristic function of Ω0 . The mentioned result is a multivariable analogue of the corresponding 1variable result of D. Arov and L. Grossman. We refer also to [9] for application of these ideas to the one-variable abstract interpolation problem. By Theorem 1.1, S is the characteristic function of some unitary colligation Ω of the form (3.3). In other words, S admits a unitary realization (1.3) with the state space H decomposed into a d-fold orthogonal sum (1.2), and the equality (1.1) holds for functions Hk1 and Hk2 defined via (1.8). The functions Hk1 and Hk2 are analytic and take respectively L(Hk ; Cp ) and L(Hk ; Cq ) values in Dd . Then the function H1 and H2 defined as (5.33)

H 1 (z) = H11 (z)P1 + · · · + Hd1 (z)Pd = C(IH − Z(z)A)−1 ,

(5.34)

H 2 (z) = H12 (z)P1 + · · · + Hd2 (z)Pd = B ∗ (IH − Z(z)A∗ )−1 ,

are analytic and respectively L(H; Cp )- and L(H; Cq )-valued in Dd . With respect to the decomposition (1.2), H j has the following block-operator form H j (z) = (H1j (z), . . . , Hdj (z))

(5.35)

(j = 1, 2).

The interpolation conditions (1.15), (1.16) and (1.17) which are assumed to be   A B satisfied by S, force certain restrictions on the connecting operator U = . C D Substituting (1.3) into (1.15) and (1.16) we get RA (XL (D + C(IH − Z(z)A)−1 Z(z)B)) = YL and RB (YR (D∗ + B ∗ Z(z)(IH − A∗ Z(z))−1 C ∗ )) = XR which are equivalent on account of (5.33) and (5.34) to XL D + RA (XL H 1 (z)Z(z))B = YL

(5.36) and

YR D∗ + RB (YR H 2 (z)Z(z))C ∗ = XR ,

(5.37)

respectively. It also follows from (5.33) and (5.34) that C + H 1 (z)Z(z)A = H 1 (z),

B ∗ + H 2 (z)Z(z)A∗ = H 2 (z)

and therefore, that (5.38)

XL C + RA (XL H 1 (z)Z(z))A = RA (XL H 1 (z))

and (5.39)

YR B ∗ + RB (YR H 2 (z)Z(z))A∗ = RB (YR H 2 (z)).

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On the bitangential interpolation problem

The equalities (5.36) and (5.38) can be written in matrix form as   A B = (RA (XL H 1 (z)), YL ), (5.40) (RA (XL H 1 (z)Z(z)), XL ) C D whereas the equalities (5.37) and (5.39) are equivalent to  ∗  A C∗ 2 (5.41) (RB (YR H (z)Z(z)), YR ) = (RB (YR H 2 (z)), XR ). B ∗ D∗ Since the functions Hkj are analytic in Dd , the following operators Tk1 := RA (XL Hk1 (z)),

(5.42)

Tk2 := RB (YR Hk2 (z))

are bounded and act from Hk into CnL and into CnR , respectively. It follows from (5.35) and (5.42) that (5.43)

RA (XL H 1 (z)) = (T11 , . . . , Td1 )

and RB (YR H 2 (z)) = (T12 , . . . , Td2 ).

Using (5.35) and (1.4) we get H j (z)Z(z) = (z1 H1j (z), . . . , zd Hdj (z)) (j = 1, 2) and therefore, (5.44)

RA (XL H 1 (z)Z(z)) = (A1 T11 , . . . , Ad Td1 )

(5.45)

RB (YR H 2 (z)Z(z)) = (B1 T12 , . . . , Bd Td2 ).

Substituting (5.43)–(5.45) into (5.40) and (5.41) we obtain   A B 1 1 (A1 T1 , . . . , Ad Td , XL ) = (T11 , . . . , Td1 , YL ) C D and (B1 T12 , . . . , Bd Td2 , YR )



A∗ B∗

C∗ D∗



= (T12 , . . . , Td2 , XR ).



 A B is unitary, we conclude from these two relations that C D for every choice of f ∈ Cn ,    1∗ ∗  1∗ T1 A1 T12∗ T1 T12∗ B1∗   .    . A B   f.  f =  ..  .. (5.46)  1∗ ∗ C D  T 1∗ T 2∗ B ∗  2∗  T A T 1 d d d d d ∗ XR XL∗ YL∗ YR∗ Since the operator

Now we use the interpolation conditions (1.17): substituting (5.42) into (1.17) we obtain the following factorizations Lk L∗k = Λk 

(k = 1, . . . , d),

 1

Tk and where Tk1 and Tk2 are defined in (5.42). Therefore, the Tk2 linear transformations Uk defined by the rule

where Lk =

(5.47)

1/2

Uk : L∗k f → Λk f

(f ∈ Cn )

300

D. Alpay, J.A. Ball and V. Bolotnikov 1/2

is the unitary map from Ran L∗k onto Ran Λk . Setting Nk := Hk Ran L∗k

ek := Ran Λ1/2 ⊕ (Hk Ran L∗k ) = Ran Λ1/2 ⊕ Nk , and H k k

ek : Hk → H ek by the rule let us define the unitary map U n ∗ ek g = Uk g for g ∈ Ran Lk , (5.48) U g for g ∈ Nk . The operator (5.49)

e := U

d M

ek : H → H e := U

k=1

d M

ek H

k=1

is unitary and satisfies e Pk = Pek U e U

(k = 1, . . . , d)

e onto where Pk and Pek are orthogonal projections from H onto Hk and from H ek , respectively. Introducing the operators H (5.50)

e=U e AU e ∗, A

e=U e B, B

e = CU e ∗, C

e =D D

e via (5.31). By definition, Ω e is unitarily equivalent we construct the colligation Ω e has the same to the initial colligation Ω defined in (3.3). By Remark 3.1, Ω characteristic function as Ω, that is, S(z). To apply the general result from [7], it suffices to check (5.32). But this equality easily follows from (5.47)–(5.50). Theorem 5.1 is a consequence of Lemmas 5.6 and 5.7. REFERENCES 1. J. Agler, Some interpolation theorems of Nevanlinna-Pick type, preprint. 2. J. Agler, On the representation of certain holomorphic functions defined on a polydisk, in Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, (Ed. L. de Branges, I. Gohberg and J. Rovnyak), vol. 48, Oper. Theory Adv. Appl., Birkh¨ auser Verlag, Basel 1990, pp. 47–66. 3. D. Alpay, V. Bolotnikov, Interpolation in the Hardy space of the bidisk, Proc. Amer. Math. Soc. 127(1999), 1789–1799. 4. J. Ball, I. Gohberg, L. Rodman, Interpolation of Rational Matrix Functions, Birkh¨ auser Verlag, Basel 1990. 5. J. Ball, I. Gohberg, L. Rodman, Tangential interpolation problems for rational matrix functions, Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence 1990, pp. 59–86. 6. J. Ball, I. Gohberg, L. Rodman, Two-sided Lagrange-Sylvester interpolation problems for rational matrix functions, Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence 1990, pp. 17–83. 7. J. Ball, T. Trent, Unitary colligations, reproducing kernel Hilbert spaces and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157(1998), 1–61. 8. V. Bolotnikov, H. Dym, On degenerate interpolation, entropy and extremal problems for matrix Schur functions, Integral Equations Operator Theory 32(1998), 367–435.

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9. V. Katsnelson, A. Kheifets, P. Yuditskii, An abstract interpolation problem and the extension theory of isometric operators, in Topics in Interpolation Theory, (Ed. H. Dym, B. Fritzsche, V. Katsnelson, and B. Kirstein), Oper. Theory Adv. Appl., vol. 95, Birkh¨ auser Verlag, Basel 1997, pp. 283–297. Translated from: Operators in function spaces and problems in function theory, pp. 83–96 (Naukova-Dumka, Kiev, 1987. Edited by V.A. Marchenko). 10. A. Nudelman, On a generalization of classical interpolation problems, Sov. Math. Dokl. 23(1997), 125–128. 11. R.A. Penrose, A generalized inverse for matrices. Proceedings Cambridge Philos. Soc. 51(1955), 406–413. DANIEL ALPAY Department of Mathematics Ben–Gurion University of the Negev Beer-Sheva 84105 ISRAEL

JOSEPH A. BALL Department of Mathematics Virginia Polytechnic Institute Blacksburg, VA 24061–0123 USA

VLADIMIR BOLOTNIKOV Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot 76100 ISRAEL Received June 15, 1998; revised November 25, 1998.