A commutant lifting theorem on the polydisc

A COMMUTANT LIFTING THEOREM ON THE POLYDISC

J.A. Ball, W.S. Li, D. Timotin and T.T. Trent

0. Introduction Interpolation problems for bounded analytic functions inthe unit disk have been studied for at least one century. The simplest ones are theNevanlinna-Pick case, in which the constraints on the functions are the values in afinite number of points, and the Caratheodory-Fejer, where the first finite numberof Taylor coefficients of the development of a function are prescribed. In all thesecases, one imposes a size constraint on the function: say, its supremum norm shouldbe smaller than 1.

Starting with the paper of Sarason ([S]), it has been realized that there exists anatural operatorial frame which unifies all function theoretic problems. The settingis the following: the algebra of bounded analytic functions is identified with thealgebra of analytic multiplication (Toeplitz) operators acting on the Hardy Hilbertspace H2(D), while the interpolation conditions are translated in the existenceof a subspace of H2, semiinvariant with respect to Toeplitz operators, and thecompression of the multiplication operator to this subspace. The most generalresult in this direction is the intertwining lifting theorem of Sz-Nagy and Foias([SNF]), which has found subsequently many applications, including applied areaslike system theory.

The generalization of these interpolation theorems to several variables (that is,to bounded analytic functions on the polydisc) is a relatively new subject. To para-phrase the one-dimensional result, it would imply the consideration of a subspaceM ⊂ H2(Dd), semiinvariant to the d operators of multiplication by the variables,and of a contraction X on M commuting with the compression of these multipli-cations. The problem would be to lift X to a contractive Toeplitz operator on thewhole H2(Dd).

The most simple case, the Nevanlinna-Pick problem, has been solved only re-cently ([Ag], [AgMC], [BT]). It also points out that a direct analogue of the one-dimensional problem is not possible, and that we have either to restrict the hypoth-esis or to relax the conclusion.

The second alternative has been achieved, most notably by M. Cotlar and C. Sa-dosky in [CS1], where they obtain two-variable commutant lifting theorems in themore general context of abstract scattering systems. When specialized to the inter-twining lifting problem, their result produces two “partial” interpolating Toeplitz

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1

2 J.A. BALL, W.S. LI, D. TIMOTIN AND T.T. TRENT

operators. Also, partial results on 2-dimensional commutant lifting (in a purelyoperator-theoretic setting) have been given by Arocena [Ar].

This paper adopts the other point of view; inspired by the quoted results on theNevanlinna-Pick problem, we will impose supplementary conditions on the contrac-tion X. These turn out to be necessary and sufficient for a conclusion similar tothat of the one-dimensional intertwining liftings.

However, there is still a subtility, which already plays an important role in theseminal paper [Ag]. It turns out that there is an essential difference between dimen-sion 2 and larger dimensions. This is related to the validity of the (generalized) VonNeumann inequality in the case of two variables (a consequence of Ando’s theorem[An]). The inequality is not true in general in higher dimensions ([V]), where onehas to restrict oneself to a special subclass of bounded analytic functions.

Thus, our main result (Theorem 5.1 below) is a complete analogue of the one-dimensional result only in the case of d = 2. However, the introduction of therestricted interpolation class Sd(E , E∗) (see Section 3) seems to be relevant fromthe point of view of interpolation. Also, there is an interesting relation to multi-dimensional linear systems,to which we will allude in Section 4; for more completeresults, see [BT].

Finally, let us note that the basic ideas in the development of this paper appearin [Ag], which proves a representation formula for functions in the restricted inter-polation class (formula (3.2) below). This formula is implicitely reproved on theway towards Theorem 5.1.

1. Preliminaries Let E and E∗ be separable Hilbert spaces. We will beconcerned with analytic functions defined on the polydisc Dd = z = (z1, . . . , zd) ∈Cd : |zj | < 1. The main notations are the following: H2(Dd, E) is the Hilbertspace of E-valued analytic functions f : Dd → E with power series expansion f(z) =∑

n1,...nd≥0 zn11 · · · znd

d an, an = an1,... ,nd∈ E , and ‖f‖2 =

∑n1,... ,nd≥0 ‖an‖2 < ∞;

H∞(Dd,L(E , E∗)) is the Banach algebra of all bounded analytic functions φ : Dd →L(E , E∗) with the norm ‖φ‖∞ = sup‖φ(z)‖ : z ∈ Dd and H∞

1 (Dd,L(E , E∗)) isthe unit ball of H∞(Dd,L(E , E∗)). For φ analytic (vector or operator-valued) andr < 1, we denote, as usual, φr(z) = φ(rz1, . . . , rzd).

If φ ∈ H∞(Dd,L(E , E∗)), the Toeplitz operator Tφ : H2(Dd, E) → H2(Dd, E∗) isdefined by

(Tφf)(z) = φ(z)f(z), z ∈ Dd, f ∈ H2(Dd, E).

In particular, the multiplications by the coordinate functions zi on H2(Dd, E) willbe denoted by Si: (Sif)(z) = zif(z) for i = 1, . . . , d. The multiplications bythe coordinate function zi on H2(Dd, E∗) will be denoted by S∗i. It is well knownthat the Toeplitz operators Tφ, φ ∈ H∞(Dd,L(E , E∗)) are characterized by thecommutation relations S∗iTφ = TφSi, i = 1, . . . , d.

A COMMUTANT LIFTING THEOREM ON THE POLYDISC 3

Lemma 1.1. If φ ∈ H∞(Dd,L(E , E∗)), then TφrT∗φr

→ TφT ∗φ in the ultraweak

topology.

Proof. All operators in question are uniformly bounded, so it is enough to checkweak convergence on a dense set. Define kw;ξ(z) = 1

1−w1z1· · · 1

1−wdzdξ; then kw;ξ ∈

H2(Dd, E), and their linear span (for w ∈ Dd, ξ ∈ E) is dense. The reproducingkernel property of these functions implies that

T ∗φkw;ξ = kw;φ(w)∗ξ. (1.1)

It follows that the relation

〈TφrT ∗φr

f, f ′〉 → 〈TφT ∗φf, f ′〉

becomes, for f = f ′ = kw;ξ,

‖kw;φ(rw)∗ξ‖2 → ‖kw;φ(w)∗ξ‖2,

and is consequently immediately verified. The Lemma is proved. Q.E.D.

Basic data for interpolation will be subspaces M ⊂ H2(Dd, E) and M∗ ⊂H2(Dd, E∗) such that for any i = 1, . . . , d, S∗i M ⊂ M and S∗∗iM∗ ⊂ M∗. LetP (resp. P∗) be the orthogonal projection of H2(Dd, E) onto M (resp. H2(Dd, E∗)onto M∗). For i = 1, . . . , d, denote also Ti = PSi|M, T∗i = P∗S∗i|M∗; obviouslyT ∗i = S∗i |M, T ∗∗i = S∗∗i|M∗.

Specializing to E = E∗ = C we obtain scalar functions; since the considerationof the vector-valued case asks only to be careful with notations, we have found itworth to present the more general case.

2. Conjugacy operators If R is an operator on a Hilbert space H, thecorresponding conjugacy operator on L(H) is MR(X) = RXR∗, X ∈ L(H). It isimmediate that MR is monotone and multiplicative, and ‖MR‖ = ‖R‖.

In connection with positivity conditions, we will be often interested by operatorsof the type I −MR. If the spectral radius of R is strictly smaller than 1, then thesame is true about MR; consequently, I −MR is invertible, with inverse given bythe familiar series ΣR =

∑k≥0(MR)k. In this case, if ∆ ∈ L(H) is positive, then

ΣR(∆) is a norm convergent, increasing sum of positive linear operators.We will use the same notation ΣR(∆) in case the sum is only weakly (equivalently,

strongly) convergent. Also, if R1, . . . , Rs are commuting operators, we will say thatΣR1 · · ·ΣRs

(∆) is convergent if the increasing multiple series of positive operators∑k1,... ,ks

Rk1 · · ·Rks∆R∗k1 · · ·R∗ks


is strongly convergent. The partial sums of ΣR(∆) are

Σ(N)R (∆) =

N∑k≥0

(MR)k(∆)

Suppose now that we have, as in the preceding section, invariant subspacesM⊂ H2(Dd, E) and M∗ ⊂ H2(Dd, E∗).

Lemma 2.1. Let X ∈ L(M), TiX = XTi for i = 1, . . . , d. For each h, k ∈M,

〈(X∗h)(0), (X∗k)(0)〉 = 〈(I −MT1) · · · (I −MTd)(XX∗)h, k〉.

Proof. It is enough to compute, taking into account the fact that

〈(I −MT1) · · · (I −MTd)(XX∗)h, k〉 = 〈(I −MS1) · · · (I −MSd

)(XX∗)h, k〉.

Q.E.D.

To state more conveniently the next result, we will introduce some supplementarynotations. Denote, for ∆ ≥ 0,

Π(∆) = ΣT1 · · ·ΣTd(∆)

andΠ′

i(∆) = ΣT1 · · ·ΣTi−1ΣTi+1 · · ·ΣTd(∆)

Also, we will have notations for “partial sums”:

Π(N)(∆) = Σ(N)T1

· · ·Σ(N)Td

(∆)

andΠ′(N)

i (∆) = Σ(N)T1

· · ·Σ(N)Ti−1

Σ(N)Ti+1

· · ·Σ(N)Td

(∆).

Lemma 2.2. Let Γi ∈ L(M) (i = 1, . . . , d), Γi ≥ 0; suppose Π′i(Γi) is convergent

for each i. If

Γ1 − T1Γ1T∗1 + · · ·+ Γd − TdΓdT

∗d ≤ 0. (2.1)

then Γ1 = · · · = Γd = 0.

Proof. Denote Gi = Π′i(Γi). Equation (2.1) may be written as

d∑i=1

(I −MTi)(Γi) ≤ 0.


Fix N ∈ N, and apply to this inequality the composition of the d monotone opera-tors Σ(N)

Ti. We obtain

d∑i=1

Π′(N)i (I − (MTi

)N+1)(Γi) ≤ 0.

Fixing h ∈M, then

0 ≥d∑

i=1

〈Π′(N)i (Γi)h, h〉 −

d∑i=1

〈Π′(N)i (Γi)T ∗N+1

i h, T ∗N+1i h〉

≥d∑

i=1

〈Π′(N)i (Γi)h, h〉 −

d∑i=1

〈GiT∗N+1i h, T ∗N+1

i h〉.

When N → ∞, the last term tends to∑d

i=1〈Gih, h〉, since Ti ∈ C.0; thus∑di=1 Gi ≤ 0. But Gi ≥ 0 by its definition, so we must have Gi = 0, and conse-

quently Γi = 0 for all i = 1, . . . , d. Q.E.D.

3. Fractional transforms A basic role in this context is played by Agler’srepresentation theorem, which allows one to write certain functions in H∞

1 (Dd,L(E , E∗))as certain type of fractional transforms (or, in system theory language, a realiza-tion as the transfer function of a d-dimensional system). Precisely, this meansthe existence of a Hilbert space H = H1 ⊕ · · · ⊕ Hd and a unitary operatorU ∈ L(H⊕ E ,H⊕ E∗), which with respect to this decomposition is written

U =[

A BC D

]; (3.1)

if we define Z = Z(z) ∈ L(H) = L(⊕d

i=1Hi) by Z(⊕di=1hi) = ⊕d

i=1zihi, then therepresentation formula is

φ(z) = D + C(I − ZA)−1ZB. (3.2)

For further use, note that corresponding to the direct sum decomposition of Hwe have decompositions A = [Aij ]i,j=1,...,d, B = [Bi]i=1,... ,d (column matrix) andC = [Cj ]j=1,... ,d (row matrix).

A basic fact about fractional transformations is that, if X ∈ L(H) is a strictcontraction, then Y = D + C(I − XA)−1XB is also a strict contraction; thisfollows from a straightforward computation which yields

I − Y ∗Y = B∗(I −X∗A∗)−1(I −X∗X)(I −AX)−1B.


It follows then from formula (3.2) that φ ∈ H∞1 (Dd,L(E , E∗)). But there is more

that can be said: the representation (3.2) implies a von Neumann type inequalityfor the function φ.

For any φ ∈ H∞1 (Dd,L(E , E∗)) and (R1, . . . , Rd) a commuting collection of strict

contractions on a Hilbert space K we may define φ(R1, . . . , Rd) as an operator inL(E , E∗)⊗ L(K) ≡ L(E ⊗ K, E∗ ⊗K), either by the Cauchy formula

φ(R1, . . . , Rd) =1

(2πi)d

∫Td

φ(z)⊗ (z1 −R1)−1 · · · (zd −Rd)−1 dz1 · · · dzd,

or by direct replacement in the power series; that is, if φ(z) =∑

zk11 · · · zkd

d φk1,...,kd,

then φ(R1, . . . , Rd) =∑

φk1,...,kd⊗Rk1

1 · · ·Rkd

d .Let us define, as in [BT], Sd(E , E∗) ⊂ H∞(Dd,L(E , E∗)) as the class of func-

tions φ ∈ H∞(Dd,L(E , E∗)) with the property that ‖φ(R1, . . . , Rd)‖ ≤ 1 for any(R1, . . . , Rd) a commuting collection of strict contractions on a Hilbert space K.This class has been introduced by Agler ([Ag]). By taking Rj = zjI, it is immediatethat Sd(E , E∗) ⊂ H∞

1 (Dd,L(E , E∗)).

Lemma 3.1. If φ ∈ H∞(Dd,L(E , E∗)) is given by formula (3.2), then φ ∈ Sd(E , E∗).

Proof. The representation (3.2) implies that

φ(R1, . . . , Rd) = D ⊗ IK + (C ⊗ IK)(I − R(A⊗ IK))−1R(B ⊗ IK), (3.3)

where R ∈ L(H⊗K) is defined, via the identification

H⊗K = (d⊕

j=1

Hj)⊗K ≡d⊕

j=1

Hj ⊗K,

by the formula R =⊕d

j=1 IHj⊗Rj . This is also a fractional transform, correspond-

ing to the unitary operator

U ⊗ IK =[

A⊗ IK B ⊗ IKC ⊗ IK D ⊗ IK

]∈ L(E ,H⊗K).

applied to the strict contraction X = R ∈ L(H⊗K). According to the remark at thebeginning of the section, the result Y = φ(R1, . . . , Rd) is also a strict contraction.Q.E.D.

The main part of Agler’s theorem is the reverse of Lemma 3.1. Its proof will beincluded in Theorem 5.1. Meanwhile, keeping the same notations, we will obtainin this section other consequences of representation (3.2).


Lemma 3.2. Define Ω = [Ω1 · · · Ωd] : H1 ⊕ · · · ⊕ Hd → H2(Dd, E∗) by Ω(ξ)(z) =C(I − ZA)−1ξ, z = (z1, . . . , zd) ∈ Dd. Then Ω is contractive and the relation

[S∗1Ω1 · · · S∗dΩd](ξ)(z) = C(I − ZA)−1Zξ is satisfied.

Proof. Consider the analytic function F (z) defined on the polydisc by the formula

F (z) = 〈(I + ZA)(I − ZA)−1ξ, ξ〉.

For any r < 1 we have, by Cauchy’s formula,

1(2π)d

∫ 2π

0

· · ·∫ 2π

0

F (reit1 , . . . reitd) dt1 · · · dtd = F (0) = ‖ξ‖2,

and thus

1(2π)d

∫ 2π

0

· · ·∫ 2π

0

<F (reit1 , . . . reitd) dt1 · · · dtd = F (0) = ‖ξ‖2.

But, for z = (reit1 , . . . reitd),

<F (z) =12〈[(I + ZA)(I − ZA)−1 + (I −A∗Z∗)−1(I + A∗Z∗)]ξ, ξ〉

= 〈(I −A∗Z∗)−1(I − r2A∗A)(I − ZA)−1ξ, ξ〉= 〈(1− r2)(I − ZA)−1ξ, (I − ZA)−1ξ〉

+ 〈r2C∗C(I − ZA)−1ξ, (I − ZA)−1ξ〉≥ ‖rC(I − ZA)−1ξ‖2.

It follows then that, again for z = (reit1 , . . . reitd),

1(2π)d

∫ 2π

0

· · ·∫ 2π

0

‖C(I − ZA)−1ξ‖2 dt1 · · · dtd ≤1r2‖ξ‖2.

Making r → 1, we obtain that Ω is a contraction.Consider then ξ = ξ1 ⊕ · · · ⊕ ξd ∈ H1 ⊕ · · · ⊕ Hd. If z = (z1, . . . , zd), we have

[S∗1Ω1 · · · S∗dΩd](ξ)(z) = [S∗1Ω1 · · · S∗dΩd](ξ1 ⊕ · · · ⊕ ξd)(z)

= [S∗1Ω1 · · · S∗dΩd](ξ1 ⊕ 0 · · · ⊕ 0)(z)

+ · · ·+ [S∗1Ω1 · · · S∗dΩd](0⊕ · · · 0⊕ ξd)(z)

= z1Ω(ξ1 ⊕ 0 · · · ⊕ 0)(z) + · · ·+ zdΩ(0⊕ · · · 0⊕ ξd)(z)

= z1C(I − ZA)−1(ξ1 ⊕ 0 · · · ⊕ 0)

+ · · ·+ zdC(I − ZA)−1(0⊕ · · · 0⊕ ξd)

= C(I − ZA)−1Z(ξ1 ⊕ 0 · · · ⊕ 0)

+ · · ·+ zdC(I − ZA)−1Z(0⊕ · · · 0⊕ ξd)

= C(I − ZA)−1Zξ.

Q.E.D.


Lemma 3.3. Suppose φ ∈ H∞1 (Dd,L(E , E∗)) is given by (3.2). With the notations

of Lemma 3.2, denote the adjoint of the contractive map Ω : H → H2(Dd, E∗) to be

Φ = ⊕iΦi : H2(Dd, E∗) → ⊕iHi. If Γi = Φ∗i Φi, then the limits Gi = Π′i(Γi) exist in

the strong operator topology, and I − TφT ∗φ = G1 + · · ·Gd.

Proof. Let h ∈ H2(Dd, E∗) and x ∈ E . One can also view the vector x as a constantfunction in H2(Dd, E); then, using Lemma 2.1,

〈(T ∗φh)(0), x〉E = 〈(T ∗φh)(0), x〉H2(Dd,E) = 〈h, Tφx〉H2(Dd,E∗)

= 〈h, (D + C(I − ZA)−1ZB)x〉= 〈D∗h, x〉E∗ + 〈B∗

1Φ1S∗∗1h, x〉+ · · ·+ 〈B∗

dΦdS∗∗dh, x〉

= 〈D∗(h(0)), x〉E + 〈(B∗1Φ1S

∗∗1 + · · ·+ B∗

dΦdS∗∗d)h, x〉E ,

and hence we have

(T ∗φh)(0) = D∗(h(0)) + (B∗1Φ1S

∗∗1 + · · ·+ B∗

dΦdS∗∗d)h. (3.4)

Also using Lemma 3.2,

Ω(ξ)(z) =C(I − ZA)−1ξ

=C(I − ZA)−1(I − ZA + ZA)ξ

=Cξ + [S∗1Ω1 · · · S∗dΩd]Aξ

and thus

〈h, Ω(ξ)(z)〉H2(Dd,E∗) =〈h, Cξ〉H2(Dd,E∗) + 〈h, [S∗1Ω1 · · · S∗dΩd]Aξ〉H2(Dd,E∗)

=〈h(0), Cξ〉E∗ + 〈Φ1S∗∗1h⊕ · · · ⊕ ΦdS

∗∗dh, Aξ〉H1⊕···⊕Hd

=〈C∗h(0), ξ〉H + 〈A∗(Φ1S∗∗1 ⊕ · · · ⊕ ΦdS

∗∗d)h, ξ〉H.

Therefore

Φ1h⊕ · · · ⊕ Φdh = C∗h(0) + A∗(Φ1S∗∗1h⊕ · · · ⊕ ΦdS

∗∗dh). (3.5)

Together with (3.4), we see that U∗ maps Φ1S

∗∗1h...

ΦdS∗∗dh

h(0)

into

Φ1h...

Φdh

(T ∗φh)(0)

.


Since U is unitary, it follows that

‖h(0)‖2 + ‖Φ1S∗∗1h‖2 + · · ·+ ‖ΦdS

∗∗dh‖2 = ‖(T ∗φh)(0)‖2 + ‖Φ1h‖2 + · · ·+ ‖Φdh‖2.

(3.6)Set Γi = Φ∗i Φi and ∆1 = Φ∗2Φ2. Together with Lemma 1.2, (3.6) is equivalent to

(I −MS∗1) · · · (I −MS∗d)(I −MTφ

)(I) = (I −MS∗1)(Γ1) + · · · (I −MS∗d)(Γd).

Applying Π(N) to both sides, we obtain, for any f ∈ H2(Dd, E∗),

〈(I −MTφ)(I −MSN+1

∗1) · · · (I −MSN+1

∗d)(I)f, f〉 =

d∑i=1

〈Π′i(N)(I −MSN+1

∗i)(Γi)f, f〉.

Suppose now that f ∈⋂d

i=1 ker SN+1∗i . The above relation becomes then

〈(I − TφT ∗φ )f, f〉 =d∑

i=1

Π′i(N)(Γi)f, f〉.

Since such vectors are dense in H2(Dd, E∗), it follows that there exist boundedpositive operators Gi, i = 1, . . . , d, such that

∑di=1 Π′

i(N)(Γi) → Gi, and I−TφT ∗φ =

G1 + · · ·+ Gd. The lemma is proved. Q.E.D.

4. Intermezzo: linear systems There is a nice system theory interpretationof formula (3.2) that is worth mentioning because the supplementary light it throwson the results already obtained. More details can be found in [BT].

The basic notion is that of d-variable unitary colligation; this is a quadruple(H = ⊕d

j=1Hj , E , E∗,U) as in the preceding section. ThenHj is the j-th partial statespace, E is the input space, E∗ is the output space, A = [Aij ]i,j=1,...,d is the mainoperator of the colligation, B = [Bi]i=1,... ,d (column matrix) is the input operator,C = [Cj ]j=1,... ,d (row matrix) is the output operator and D is the feedthroughoperator.

One associates to the colligation a d-dimensional discrete time linear system,for which the time variable n is a d-tuple of integers n = (n1, . . . , nd). The inputsignal u(n) has values in E , the output signal y(n) has values in E∗, while thestate vector x(n) ∈ H has components xj(n) ∈ Hj . As in [BT], denoting σj(n) =(n1, . . . , nj−1, nj + 1, nj−1, . . . , nd), the equations of the system are

x1(σ1(n)) = A11x1(n) + · · ·+ A1dxd(n) +B1u(n)...

......

...xd(σd(n)) = Ad1x1(n) + · · ·+ Addxd(n) +Bdu(n)

y(n) = C1x1(n) + · · ·+ Cdxd(n) +Du(n)

(4.1)


Then the function φ defined by (3.2) is the transfer function of this system; itgives the relation between the input and the output of the system in the case of zeroinitial conditions. Specifically, suppose for every j = 1, . . . , d, xj(n) = 0 whenevernj = 0. Then relations (4.1) specify y(n) for n ∈ Nd as functions of u(n), n ∈ Nd.If u(z) =

∑n∈Nd znu(n), and y(z) =

∑n∈Nd zny(n), then

y(z) = φ(z)u(z).

Thus Lemma 3.1 states, in a different language, that the transfer function ofa unitary colligation is in Sd(E , E∗). Conversely, given φ ∈ H∞

1 (Dd,L(E , E∗)), acolligation that has φ as a transfer function is called a realization of φ; thus, Agler’sresult stated above is that any function in Sd(E , E∗) has a unitary realization.

There is also an interpretation of Lemma 3.2 in this context. Suppose all inputsu(n) are 0, x(0, . . . , 0) = ξ, while xj(n) = 0 if nj = 0 and n 6= (0, . . . , 0). Weobtain then a sequence of outputs y(n) depending on ξ; the map that associates thissequence to a given ξ is called the observability operator. Its “frequency domain”transform ξ 7→ y(z) is exactly the operator Ω of Lemma 2.1; the lemma states thenthat the observability operator of a unitary colligation is contractive.

5. Main theorem This section contains the main result of the paper. Itrepresents a commutant lifting theorem for d-variables on H2(Dd, E). Applied toa particular case, it reproves Agler’s theorem in a slightly different language. Allnotations remains the same as in the preceding sections.

Theorem 5.1. Let X ∈ L(M,M∗), ‖X‖ ≤ 1, and T∗iX = XTi for i = 1, . . . , d.

Then the following are equivalent:

(i) there exists φ ∈ Sd(E , E∗) such that XP = P∗T∗φ ;

(ii) there exist a Hilbert space H = H1 ⊕ . . . ⊕ Hd and a unitary operator U ∈

L(H⊕ E ,H⊕ E∗), U =[

A BC D

], such that, if

φ(z) = D + C(I − ZA)−1ZB, (70)

then XP = P∗T∗φ ;

(iii) there exist Gi ∈ L(M∗), i = 1, . . . , d, such that Gi ≥ 0,∏

j 6=i(I−MT∗j)(Gi) ≥ 0

for all i, and I −XX∗ = G1 + . . . + Gd.

Proof. (i) ⇒ (iii). Define a cone C ⊂ L(H2(Dd, E∗)) by

C = R = G1 + · · ·+ Gd : Gi ≥ 0,∏j 6=i

(I −MS∗j)(Gi) ≥ 0 for all i = 1, . . . , d.

C is a convex set; in order to show that it is weak-∗ closed, it is enough (by Eberlein-Shmulyan theorem) to show that the intersection of C with each ball is weak-∗


closed. But this is easy: if the net G(α)1 + . . . + G

(α)d is uniformly bounded and

tends ultraweakly to R, then the bounded nets G(α)i (i = 1, . . . , d) contain subnets

that are convergent to Gi respectively. Then the decomposition R = G1 + . . . + Gd

shows that R ∈ C.Let T0 = R ∈ L(H2(Dd, E∗)) : RSi = SiRforalli (i.e., T0 is the subalgebra of

L(H2(Dd, E∗)) consisting of the bounded Toeplitz operators). Now, the two mainobservations about C are the following:

(a) RR∗ ∈ C for each R ∈ T0; indeed∏j 6=i

(I −MS∗j )(RR∗) = R( ∏

j 6=i

(I −MS∗j )(I))R∗ ≥ 0,

since∏

j 6=i(I − MS∗j)(I) is the projection onto

⋂j 6=i ker S∗j . Therefore RR∗ =∑

i Gi with G1 = RR∗, Gi = 0 for i > 1 is a valid decomposition.(b) RR∗ − S∗iRR∗S∗∗i ∈ C for each R ∈ T0: Indeed

∏j 6=i

(I −MS∗j)(RR∗ − S∗iRR∗S∗∗i) =

d∏j=1

(I −MS∗j)(RR∗)

= Rd∏

j=1

(I −MS∗j)(I)R∗ ≥ 0,

since∏d

j=1(I − MS∗j)(I) is the projection onto the constants. Therefore RR∗ −

S∗iRR∗S∗∗i =∑

i Gi with Gi = RR∗ − S∗iRR∗S∗∗i, Gj = 0 for j 6= i is a validdecomposition.

If Y = Tφ, we intend to show that I − Y Y ∗ ∈ C. We will do this by a sepa-ration argument; take therefore a trace-class operator Ω on H2(Dd, E∗) for which<[Tr(ΩK)] ≥ 0 for all K ∈ C; we have to check that <[Tr(Ω(I −Y Y ∗)] ≥ 0. Definea Hermitian form on T0 via

[R,R′] =12[Tr(ΩRR′∗) + Tr(ΩR′∗R)]

(so [R,R] = <[Tr(ΩRR∗)]). The Hermitian form [·, ·] is positive semidefinite onT0. The usual completion procedure yields a Hilbert space T with T0 embeddedpartially isometrically as a dense subspace.

Let us then define, for R ∈ T0, Ci(R) = S∗iR. Then (a) and (b) imply thatCi maps contractively T0 into itself (in the [·, ·]-norm), and hence extend uniquelyby continuity to define contraction operators (also denoted by Ci) on T . It isalso immediate that Ci and Cj commute. Therefore, φ ∈ Sd(E , E∗) implies that‖φ(rC1, . . . , rCd)‖ ≤ 1 for all r < 1.

Now, the Taylor series development of φ shows that

φ(rC1, . . . , rCd)(I) = φr(S1, . . . , Sd) = Yr.


Therefore, applying Lemma 1.1,

Tr(Ω(I − Y Y ∗)) = limr↑1

Tr(Ω(I − YrY∗r ))

= limr↑1[I, I]H − [φ(rC1, . . . , rCd)(I), φ(rC1, . . . , rCd)(I)]H ≥ 0.

The separation argument implies then that I − Y Y ∗ ∈ C as desired.Consequently, there exist Gi ∈ L(H2(Dd, E∗)) satisfying Gi ≥ 0,

∏j 6=i(I −

MS∗j)(Gi) ≥ 0, such that I − Y Y ∗ = G1 + . . . + Gd. If P∗ denotes the orthogonal

projection onto M∗, then I −XX∗ = P∗(I − TφT ∗φ )P∗ = P∗G1P∗ + . . . + P∗GdP∗.Denoting Gi = P∗GiP∗, then∏

j 6=i

(I −MT∗j )(Gi) = P∗( ∏

j 6=i

(I −MS∗j )(Gi))P∗ ≥ 0

where we have used the relations P∗S∗i(I−P∗) = 0 for i = 1, . . . , d. This concludesthe proof that (i) implies (iii).

(iii) ⇒ (ii). We begin by applying∏

j(I −MT∗j) to both sides of the equality

I −XX∗ = G1 + . . . + Gd. Defining Γi =∏

j 6=i(I −MT∗j )(Gi), it follows that

∏j

(I −MT∗j )(I −XX∗) =∑

i

(I −MT∗i)(Γi). (5.1)

For further use, note that

Π′i(N)(Γi) =

∏j 6=i

((I −MT

(N+1)∗j

)(Gi) ≤ Gi, (5.2)

and thus the series Π′i(Γi) is convergent for all i = 1, . . . , d.

Applying Lemma 2.1 to (5.1) yields, for h ∈M∗,

‖h(0)‖2 − ‖(X∗h)(0)‖2 = ‖Γ1/21 h‖2 − ‖Γ1/2

1 T ∗∗1h‖2 + . . . + ‖Γ1/2d h‖2 − ‖Γ1/2

d T ∗∗dh‖2.

By rearranging the terms, we immediately see that the map Γ1/2

1 T ∗∗1h...

Γ1/2d T ∗∗dh

h(0)

7→

Γ1/2

1 h...

Γ1/2d h

(X∗h)(0)

.

defines an isometry from Γ1/21 T ∗∗1h⊕ · · · ⊕ Γ1/2

d T ∗∗dh⊕ h(0) : h ∈ M∗ into M∗ ⊕· · ·⊕M∗⊕E . Extend the isometry to a unitary operator U fromH1⊕· · ·⊕Hd⊕E∗ toH1⊕ · · · ⊕Hd⊕E where Hi are some Hilbert spaces containing M∗; by composing


Γ1/2i with the embeddings of M∗ into Hi, we obtain mappings Φi : M∗ → Hi

(i = 1, . . . , d), such that

U∗

Φ1T

∗∗1h...

ΦdT∗∗dh

h(0)

7→

Φ1h

...Φdh

(X∗h)(0)

.

Formulas (5.2) imply that the series

Π′i(Φ

∗i Φi) < ∞ (5.3)

for all i = 1, . . . , d.

Consider then the familiar decomposition U =[

A BC D

], and define the function

φ ∈ H∞1 (Dd,L(E , E∗)) by φ(z) = D + C(I − ZA)−1ZB. We have

(X∗h)(0) = D∗(h(0)) + (B∗1Φ1T

∗∗1 + · · ·+ B∗

dΦdT∗∗d)h, (5.4)

and Φ1h...

Φdh

= C∗h(0) + A∗

Φ1T∗∗1h...

ΦdT∗∗dh

. (5.5)

If the maps Φ′i ∈ L(H2(Dd, E∗),Hi) are associated with φ as in Lemma 3.3,formulas (3.4) and (3.5) become

(T ∗φh)(0) = D∗(h(0)) + (B∗1Φ′1T

∗∗1 + · · ·+ B∗

dΦ′dT∗∗d)h, (5.6)

and Φ′1h...

Φ′dh

= C∗h(0) + A∗

Φ′1T∗∗1h...

Φ′dT∗∗dh

. (5.7)

Also, Lemma 3.3 implies that

Π′i(Φ

′i∗Φ′i) < ∞ (5.8)

Define Ψi = Φi−Φ′i|M∗ For any i = 1, . . . , d, Ψ∗i Ψi ≤ (Φi−Φ′i)

∗(Φi−Φ′i)|M∗+(Φi + Φ′i)

∗(Φi + Φ′i)|M∗, so we have Ψ∗i Ψi ≤ 2(Φ∗i Φi + Φ′i

∗Φ′i)|M∗. From formulas(5.3) and (5.8) it follows that the sums Π′

i(Ψ∗i Ψi) are strongly convergent.

On the other hand, equations (5.5) and (5.7) imply Ψ1h...

Ψdh

= A∗

Ψ1T∗∗1h...

ΨdT∗∗dh


Therefore, since A∗ is a contraction,

Ψ∗1Ψ1 + · · ·+ Ψ∗

dΨd ≤ T∗1Ψ∗1Ψ1T

∗∗1 + · · ·+ T∗dΨ∗

dΨdT∗∗d,

or(I −MT∗1)(Ψ

∗1Ψ1) + · · ·+ (I −MT∗d

)(Ψ∗dΨd) ≤ 0.

Apply Lemma 2.2 to Ψ∗i Ψi, it follows that Ψi = 0, consequently, Φi = Φ′i|M∗.

Now, (5.4) and (5.6) imply that (X∗h)(0) = (T ∗φh)(0) for every h ∈M∗. Thus forevery x ∈ E , considered as a constant function in H2(Dd, E), we have 〈T ∗φh, x〉 =〈X∗h, x〉, and so 〈T ∗φS∗n1

∗1 · · ·S∗nd

∗d h, x〉 = 〈X∗T ∗n1∗1 · · ·T ∗nd

∗d h, x〉 for all nonnega-tive integers n1, . . . , nd. Since T∗iX = XTi, this implies 〈T ∗φh, Sn1

1 · · ·Snd

d x〉 =〈X∗h, Sn1

1 · · ·Snd

d x〉 and so T ∗φh = X∗h. Therefore XP = P∗Tφ.

Finally, (ii) ⇒ (i) is a consequence of Lemma 2.0. The theorem is completelyproved. Q.E.D.

Note that, for M = H2(Dd, E) and M∗ = H2(Dd, E∗), the equivalence of (i)and (ii) is Agler’s theorem. The class Sd(E , E∗) is therefore characterized by thefractional representation formula (3.2).

6. Particular cases

6.1 The first most noteworthy remark concerns the case d = 2. Indeed, inthis case Ando’s Theorem on the existence of a joint unitary dilation for a pairof commuting contractions implies the validity of the von Neumann inequality.Consequently, Sd(E , E∗) = H∞(Dd,L(E , E∗)), and all results are valid for generalbounded analytic functions; no exotic subclass is involved. Naturally, many of thepreceding formulas also simplify in this particular case: for instance, the productΠ′

i reduces to a single term, a.s.o.

6.2 An important particular case happens if the space M∗ is finite dimensional.Note that in this case the fact that T∗1, . . . , T∗d ∈ C.0 implies that the spectral radiiof these two operators are strictly smaller than 1. Thus, as remarked in Section 2,I − MT∗i

is invertible as an operator on L(M∗) and its inverse is given by ΣT∗i.

We may then obtain the following corollary of Theorem 5.1.

Proposition 6.2.1. If M∗ is finite dimensional, then any of the three conditions

in the statement of theorem 5.1 is equivalent to the existence of Γi ∈ L(M∗),Γi ≥ 0, such that

(I−MT∗1) · · · (I−MT∗d)(I−XX∗) = (I−MT∗1)(Γ1)+ · · ·+(I−MT∗d

)(Γd). (6.1)

Proof. It is easy to show the equivalence of the new hypothesis with condition (iii)in Theorem 5.1. Indeed, in order to obtain (6.1) from (iii) of Theorem 5.1, we apply


(I − MT∗1) · · · (I − MT∗d) to the relation I − XX∗ = G1 + · · · + Gd, and defined

Γi =∏

j 6=i(I−MT∗j)(Gi). Conversely, we may apply (I−MT∗1)

−1 · · · (I−MT∗d)−1

to (6.1), and define Gi = Π′i(Γi) =

∏j 6=i(I − MT∗j

)−1(Γi), this yields condition(iii). Q.E.D.

Formula (6.1) may be used to obtain tangential Nevanlinna-Pick interpolationresults, as in [BT]. Suppose we are given, for any p = 1, . . . , n, distinct pointsw(p) = (w(p)

1 , . . . , w(p)d ) ∈ Dd and vectors ξ(p) ∈ E , ξ

(p)∗ ∈ E∗. The tangential

Nevanlinna-Pick problem asks for the existence of a function φ ∈ H∞1 (Dd,L(E , E∗)),

such thatφ(w(p))∗ξ(p)

∗ = ξ(p). (6.2)

Proposition 6.2.2. The tangential Nevanlinna-Pick problem is solvable if and

only if there exist, for i = 1, . . . , d, positive matrices (Γipq)

np,q=1, such that for any

p, q = 1, . . . , n

1− 〈ξ(p), ξ(q)〉 = (1− w(p)1 w

(q)1 )Γ1

pq + · · ·+ (1− w(p)2 w

(q)2 )Γd

pq. (6.3)

Proof. The proof follows the argument that is well known in the one-dimensionalcase. Define M ⊂ H2(Dd, E) and M∗ ⊂ H2(Dd, E∗) to be the (finite dimensional)subspaces spanned by the functions kw(p);ξ(p) and k

w(p);ξ(p)∗

(p = 1, . . . , n) respec-tively. According to formula (1.1), a function φ ∈ H∞(Dd,L(E , E∗)) satisfies theinterpolation property (6.2) if and only if T ∗φk

w(p);ξ(p)∗

= kw(p);ξ(p) .Define then the operator X : M→M∗ by means of its adjoint according to the

formulaX∗k

w(p);ξ(p)∗

= kw(p);ξ(p) .

It is easy to check that X∗T ∗i = T ∗∗iX∗, and consequently we may apply Proposition

6.2.1, which yields Γi ∈ L(M∗), Γi ≥ 0 that satisfy (6.1). If we denote

Γipq = 〈Γikw(p);ξ

(p)∗

, kw(q);ξ

(q)∗〉,

then (6.1) is equivalent to (6.3). Q.E.D.

Another case in which M∗ is finite dimensional is the multi-dimensional Cara-theodory-Fejer problem. For simplicity, we will consider only the scalar-valued case;this consists in finding a function φ ∈ Sd = Sd(C, C) for which a finite number ofFourier coefficients are given: if φ(z) =

∑zk11 · · · zkd

d φk1,...,kd, then φk1,...,kd

aregiven for k = (k1, . . . , kd) ∈ κ ⊂ Zd. The finite set κ is not arbitrary; it hasto satisfy the condition “k ∈ κ and k′i ≤ ki for any i = 1, . . . d implies k′ ∈ κ”.In particular, acceptable sets are the rectangle 0 ≤ ki ≤ Ki and the simplexk1 + · · · kd ≤ K.


The solution of the Caratheodory-Fejer problem is obtained by considering inTheorem 5.1 the subspaceM = M∗ to be equal to the polynomials with coefficientsin κ; then the condition on κ implies the invariance of this space to S∗i . Theoperator X ∈ L(M) has, with respect to the natural basis formed by the monomialsek(z) = zk1

1 · · · zkd

d (k ∈ κ), the matrix coeficients Xk,k′ = φk−k′ if k ≥ k′ and 0otherwise. Condition (iii) in Theorem 5.1 becomes a condition on matrices withcardκ rows and columns. In this case there is no simpler alternate equivalentcondition similar to (6.3); the reason is that the reproducing kernels do not form agenerating set. One can write formulas in terms of generalized reproducing kernels,but they become significantly more complicated.

6.3 The space M∗ in the Nevannlinna-Pick problem is the linear span of a finitecollection of reproducing kernels kw,ξ. It is interesting to note that the proof ofTheorem 5.1 simplifies in case the reproducing kernels form a generating set inM∗ (which is the case, for instance, also when M∗ = H2(Dd, E∗): the situation ofAgler’s theorem).

The simplification appears in the proof of the implication (iii) ⇒ (ii). In formula(5.5) we may choose h = kw,ξ, and use the relations T∗ikw,ξ = wikw,ξ to obtain

Φ(kw,ξ) = C∗kw,ξ(0) + A∗Z(w)∗Φ(kw,ξ),

whenceΦ(kw,ξ) = (I −A∗Z(w)∗)−1C∗kw,ξ(0).

The same formula may be obtained for Φ′ instead of Φ, starting with (5.7). There-fore Φ = Φ′ on all reproducing kernels; if these generate M∗, we obtain Φ = Φ′|M∗

without using Lemma 2.2.

6.4 The Nevanlinna-Pick interpolation problem can be generalized to the casewhen the values of the function are prescribed on an arbitrary set W ⊂ Dd. Againwe will restrict ourselves, for simplicity, to the scalar-valued case. The scalar re-producing kernels are then

kw(z) = kw;1(z)

and we defineM = M∗ =

∨w∈W

kw. (6.4)

Formula (1.1) says that the functions kw are eigenvectors for any T ∗φ ∈ L(H2),φ ∈ H∞, with eigenvalues φ(w). Thus M is invariant for all adjoints of Toeplitzoperators on H2.

Suppose now the “interpolation data” are given in the form of a function ξ :W → C. We may define a linear operator Yξ on a dense set in M, extending bylinearity the formula

Yξ(kw) = ξ(w)kw. (6.5)

On this dense set, Yξ commutes with all Toeplitz operators.


Proposition 6.4.1. With the above notations, the following are equivalent:

(i) there exists φ ∈ Sd, φ(w) = ξ(w) for every w ∈ W ;

(ii) Yξ defined above extends to a bounded operator on M (denoted with the

same letter), and there exist Gi ∈ L(M∗), i = 1, . . . , d, such that Gi ≥ 0,∏

j 6=i(I−MT∗j )(Gi) ≥ 0 for all i, and I − Y ∗

ξ Yξ = G1 + . . . + Gd;

(iii) there exist functions γi(v, w), i = 1, . . . , d, v, w ∈ W , such that all finite

principal submatrices of γi are positive definite, and

1− ξ(v)ξ(w) =d∑

i=1

(1− viwi)γi(v, w).

Proof. The equivalence of (i) and (ii) is a reformulation, in this context, of Theorem5.1. That (ii)⇒(iii) is proved as in 6.2. The only implication that deserves adiscussion is (iii)⇒(ii). We can apply Proposition 6.2.1 to each finite subset F ofW , and obtain functions φF that satisfy φF (z) = ξ(z) for z ∈ F . Since all φF areuniformly bounded by 1, a Montel-type argument yields a single function φ whichis the limit in Dd of a subnet of φF . Then φ(z) = ξ(z) for all z ∈ W . Q.E.D.

Proposition 6.4.1 is a general result, that applies to any set W and function ξ(w).Certain more concrete conditions are important in applications. First, suppose thatthere exists a function F ∈ H∞, such that F (w) = ξ(w) for every w ∈ W . Then Y

is the restriction of T ∗F to a dense set, and is consequently automatically bounded;condition (ii) in Proposition 6.4.1 can be rephrased accordingly. Condition (i) statesthen the existence of a function φ ∈ Sd such that φ(w) = F (w) for every w ∈ W .

Secondly, we may assume that W is actually the zero set Z(b) of a given analyticfunction b ∈ H∞; a convenient supplementary assumption in this context is that W

has no accumulation points on Td. This is the case studied, for instance, in [CS2].It is interesting to point out the case when condition (i) in Proposition 6.4.1 can bereformulated to the existence of φ ∈ Sd such that φ(z)−F (z) = b(z)K(z) for someK ∈ H∞. In this case, we need a supplementary assumption on b, which shouldimply that any analytic function in Dd which is 0 on Z(b) factors through b. Sucha condition can be given locally: for any w ∈ Z(b) the decomposition of the germbw into irreducible elements should be square free. For all these facts, see [GR, ch.II].

In particular, the assumption about b is verified if the germ bw is irreducible ineach w ∈ Z(b). A sufficient condition for this is that all zeroes of b are simple (thismeans that the Taylor series development around any point of Z(b) has its firstnonzero term of degree 1). Another example is given by b(z1, z2) = z2

1 − 2z32 .

The case when some germs bw have decompositions consisting of more than onefactor is also interesting for applications. The main example when this condition issatisfied consists in the multidimensional Blaschke products. These are functions

b(z1, . . . , zd) = b1(z1) · · · bd(zd), (6.6)


where each bi is a (one-dimensional) Blaschke product with simple zeros (see [CS1],[CS2]); we will use them in the next subsection in order to obtain boundednessresults about big Hankel operators.

An open problem is to obtain a solution criterion in terms of finitely many finitematrices in case F and b are rational (and hence are given in terms of finitely manyfinite matrices (AF , BF , CF , DF ) and (Ab, Bb, Cb, Db) according to the formulasF (z) = DF + CF (I − Z(z)AF )−1Z(z)BF and b(z) = Db + Cb(I − Z(z)Ab)−1Bb).

6.5 The last result can be applied to big Hankel operators (this is actually re-versing the path in [CS2]). Considering the embedding of H2(Dd) in L2(Td), letus denote by P+ the orthogonal projection onto H2(Dd), and by S′i ∈ L(H2(Dd)⊥)the compressions of multiplications with the variables to H2(Dd)⊥. A big Hankeloperator H : H2(Dd) → H2(Dd)⊥ is characterized by the commutation relationHSi = S′iH, for any i = 1, . . . , d.

In particular, a class of Hankel operators can be obtained by defining, for anyφ ∈ L∞(Td), Hφ(f) = (I − P+)φf . Then φ is called a symbol of Hφ; it is uniquelydefined modulo H∞. It is known ([CS1]) that not all big Hankel operators havebounded symbols.

Suppose now b is an analytic function in Dd, with the property that, if f isanalytic in Dd, and f(w) = 0 for w ∈ Z(b), then f = bf for some analytic f . (Wehave discussed in the preceding subsection this condition.) Suppose also that b isinvertible in a neighborhood of Td; it follows then immediately that, if f ∈ H2,then f ∈ H2. Also, if M is defined by (6.4), then f ∈ H2 M is equivalent tof = 0 on Z(b); therefore,

M = H2 bH2 (6.7)

(bH2 is a closed subspace of H2).Let us suppose now b is given by formula (6.6); then b satisfies all the above

assumptions and, moreover, it is inner. Let ξ(w) be defined, as above, on Z(b),and Yξ given by formula (6.5). Multiplication by b is an isometry on L2(Td); wemay consider the linear operator K defined on a dense set in H2⊥ by the formulaKf = YξP+bf ; it takes values in H2. If K extends to a bounded operator, thenits adjoint H is a Hankel operator. We may then give the following corollary ofProposition 6.4.1.

Corollary 6.5.1. With the above notations, the following are equivalent:

(i) H has a bounded symbol of the form bφ, with φ ∈ Sd.

(ii) Yξ defined above extends to a bounded operator on M (denoted with the

same letter), and there exist Gi ∈ L(M∗), i = 1, . . . , d, such that Gi ≥ 0,∏

j 6=i(I−MT∗j

)(Gi) ≥ 0 for all i, and I − Y ∗ξ Yξ = G1 + . . . + Gd;

(iii) there exist functions γi(v, w), i = 1, . . . , d, v, w ∈ W , such that all finite


principal submatrices of γi are positive definite, and

1− ξ(v)ξ(w) =d∑

i=1

(1− viwi)γi(v, w).

It is easy to see that any symbol of H, if it exists, must be of the form bF , whereF is actually analytic in Dd and F (w) = ξ(w) for any w ∈ Z(b). Variations on thisproblem are given by Cotlar and Sadosky in [CS2].

6.6 A corona-type operatorial result can be obtained along the lines of The-orem 5.1. For this, and supplementary results regarding corona-type operatorialproblems, see [T].

REFERENCES

[Ag] J. Agler, On the representation of certain holomorphic functions defined ona polydisc, in Topics in Operator Theory: Ernst D. Hellinger Memorial Volume,Operator Theory: Advances and Applications Vol. 48, Birkhauser-Verlag, Basel,1990, pp. 47-66.

[AgMc] J. Agler and J.E. McCarthy, Nevanlinna-Pick interpolation on the bidisk,preprint, 1997.

[An] T. Ando, On a pair of commutative contractions, Acta Sci. Math. 24 (1963),88-90.

[Ar] R. Arocena, On the Nagy-Foias-Parrott commutant lifting theorem, in Har-monic Analysis and Operator Theory, Contemporary Mathematics, Vol. 189, 1995,pp. 55-64.

[BT] J.A. Ball and T.T. Trent, Unitary colligations, reproducing kernel Hilbertspaces, and Nevanlinna-Pick interpolation in several variables, J. Functional Anal-ysis, to appear.

[CS1] M. Cotlar and C. Sadosky, Transference of metrics induced by unitary cou-plings, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy-Foias theo-rem for two pairs of dilation, J. Functional Analysis 111 (1993), 473-488.

[CS2] M. Cotlar and C. Sadosky, Nehari and Nevanlinna-Pick problems and holo-morphic extensions in the polydisc in terms of restricted BMO, J. Functional Anal-ysis 124 (1994), 205-210.

[GR] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables,Prentice-Hall, Englewood Cliffs, N.J., 1965.

https://www.researchgate.net/publication/238860021_Transference_of_Metrics_Induced_by_Unitary_Couplings_a_Sarason_Theorem_for_the_Bidimensional_Torus_and_a_Sz-Nagy-Foias_Theorem_for_Two_Pairs_of_Dilations?el=1_x_8&enrichId=rgreq-3e14d13f560a7d31ac8a291cfafb0126-XXX&enrichSource=Y292ZXJQYWdlOzI0NDk1MzUxMztBUzoxMDY4NDY5Mzc0MTk3NzZAMTQwMjQ4NTcwMzE0Ng==



https://www.researchgate.net/publication/223076637_Nehari_and_Nevanlinna-Pick_Problems_and_Holomorphic_Extensions_in_the_Polydisk_in_Terms_of_Restricted_BMO?el=1_x_8&enrichId=rgreq-3e14d13f560a7d31ac8a291cfafb0126-XXX&enrichSource=Y292ZXJQYWdlOzI0NDk1MzUxMztBUzoxMDY4NDY5Mzc0MTk3NzZAMTQwMjQ4NTcwMzE0Ng==




[R] W. Rudin, Function Theory in Polydiscs, W.A. Benjamin, New York–Amsterdam1969.

[S] D. Sarason, Generalized interpolation in H∞, Trans. Am. Math. Soc. 127(1967), 2, 179-203.

[SNF] B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space,North-Holland, Amsterdam, 1970.

[V] N. Varopoulos, Sur une inegalite de Von Neumann, C.R. Acad. Sc. Paris 277(1973), Ser. A, 19-22.

[T] T.T. Trent, Function Theory Problems and Operator Theory, preprint.

https://www.researchgate.net/publication/260401617_Function_Theory_Problems_and_Operator_Theory?el=1_x_8&enrichId=rgreq-3e14d13f560a7d31ac8a291cfafb0126-XXX&enrichSource=Y292ZXJQYWdlOzI0NDk1MzUxMztBUzoxMDY4NDY5Mzc0MTk3NzZAMTQwMjQ4NTcwMzE0Ng==

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