Nevanlinna-Pick Interpolation on the Bidisk

Jim Agler ^{∗} John E. McCarthy ^{†} University of California at San Diego, La Jolla, California 92093 Washington University, St. Louis, Missouri 63130

Abstract We solve the matrix-valued Nevanlinna-Pick problem in the space of bounded analytic functions on the bidisk, and give a description of all the interpolating functions. We also prove the Toeplitz-corona theorem on the bidisk.

1 Nevanlinna-Pick on the Disk

The classical Nevanlinna-Pick problem on the disk 𝔻 is: given n distinct points λ_{1}, …, λ_{n} in 𝔻 and n complex numbers z_{1}, …, z_{n}, find the analytic function ϕ of smallest norm (in the space H^{∞}(𝔻) of bounded analytic functions on 𝔻 equipped with the supremum norm) that maps λ_{i} to z_{i}. The problem was first studied by G. Pick [12] who proved that there is a function of norm 1 interpolating the λ_{i}’s to the z_{i}’s if and only if the Pick matrix is positive definite, i.e.

(\frac{1 − \overline{z_{i}}z_{j}}{1 − \overline{λ_{i}}λ_{j}} )_{i,j=1}^{n} ≥ 0.  (1.1)

The problem has been extensively studied since then, both by function theoretic and operator theoretic techniques. The optimal solution has certain qualitative properties that we shall consider (recall that a function in H^{∞}(𝔻) is called inner if its boundary values have modulus one almost everywhere on the circle):

Theorem 1.2 Suppose that inf{∥ϕ∥ : ϕ(λ_{i}) = z_{i}, i = 1, …, n} = 1. Then

(i) There is a unique ϕ in H^{∞}(𝔻) of norm one that interpolates the data. (ii) The function ϕ is rational and inner.

∗Partially supported by the National Science Foundation.

†Partially supported by National Science Foundation grant DMS 9531967.

See e.g. [14], [11] or [10] for proofs of this theorem. Condition (ii) can be phrased as saying that ϕ is a finite Blaschke product, but we write it this way to facilitate extension to higher dimensions.

On the bidisk, the analogue of Pick’s theorem was proved by the first author [1]:

Theorem 1.3 Let z_{1}, …, z_{n} be complex numbers. A necessary and sufficient condition for there to exist a function ϕ in H^{∞}(𝔻^{2}) with ∥ϕ∥ ≤ 1 and ϕ(α_{i}, β_{i}) = z_{i}, i = 1, …, n, is that there exist positive n-by-n matrices Γ and Δ such that

(1 − \overline{z_{i}}z_{j}) = (1 − \overline{α_{i}}α_{j})Γ_{ij} + (1 − \overline{β_{i}}β_{j})Δ_{ij}

(For another treatment of the proof of this theorem and interpolation on the bidisk, see [9]). A problem with this theorem, however, is that even if Γ and Δ are known, one does not know what ϕ is, and cannot describe qualitative properties in the manner of Theorem 1.2.

Using operator theoretic ideas, we are able to describe all the solutions to the interpolation problem on the bidisk (Theorem 2.10). Our approach works equally well for matrixvalued interpolation, and gives a new proof of Theorem 1.3 even in the scalar case. Thus we have the first matrix-valued Nevanlinna-Pick theorem on the bidisk (Theorem 2.1).

On the bidisk, the solution of smallest norm to a Nevanlinna-Pick problem may or may not be unique, as easy examples show; but from our description of all solutions we can show that some optimal interpolant will be rational and inner (Corollary 2.13).

The corona problem on the disk is to decide, given two functions ϕ_{1} and ϕ_{2} in H^{∞}(𝔻), whether there exist functions ψ_{1} and ψ_{2} in H^{∞}(𝔻) so that ψ_{1}ϕ_{1} + ψ_{2}ϕ_{2} = 1. Carleson’s corona theorem [8] asserts that the obvious necessary condition |ϕ_{1}| + |ϕ_{2}| ≥ ε > 0 is sufficient; but the smallest possible norm of the pair ψ does not depend just on ε. The Toeplitz-corona theorem on the disk, proved in different ways by several authors [6], [17], [15], [13], does give optimal bounds for solving the corona problem, but at the expense of replacing a functional inequality by an operator inequality.

Let σ denote Lebesgue measure on the circle, and define the Hardy space H^{2}(σ) to be the closure in L^{2}(σ) of the analytic polynomials. For f ∈ L^{∞}(σ), the Toeplitz operator T_{f} is the operator on H^{2}(σ) that sends a function g to P (fg), where P is the orthogonal projection from L^{2}(σ) onto H^{2}(σ). The Toeplitz-corona theorem is this:

Theorem 1.4 Let ϕ_{1}, ϕ_{2} be in H^{∞}(𝔻), and δ > 0. Then the following are equivalent:

(i) For any points λ_{1}, …, λ_{n} in the disk,

(\frac{\overline{ϕ_{1}(λ_{i})}ϕ_{1}(λ_{j}) + \overline{ϕ_{2}(λ_{i})}ϕ_{2}(λ_{j}) − δ}{1 − \overline{λ_{i}}λ_{j}} )_{i,j=1}^{n} ≥ 0

(ii) The Toeplitz operators satisfy the inequality

T_{ϕ_{1}}T_{ϕ_{1}}^{∗} + T_{ϕ_{2}}T_{ϕ_{2}}^{∗} ≥ δI.

(iii) There exist functions ψ_{1}, ψ_{2} in H^{∞}(𝔻) such that

ψ_{1}ϕ_{1} + ψ_{2}ϕ_{2} = 1

and

sup_{z∈𝔻}[|ψ_{1}(z)|^{2} + |ψ_{2}(z)|^{2}] ≤ \frac{1}{δ}.

We prove a bidisk version of this theorem (Theorem 3.1). Both the Toeplitz-corona theorem and the description of all solutions of the Nevanlinna-Pick problem rely heavily on a representation theorem of the first author describing bounded analytic matrix-valued functions on the bidisk [2]. As this theorem is essential for this paper, we include a proof (Theorem 4.1).

Remark: For notational convenience, we restrict ourselves to the bidisk. All the theorems and proofs in this paper go through, however, to the polydisk 𝔻^{d}, provided the algebra H^{∞}(𝔻^{2}) is replaced by the algebra H^{∞}(F_{d}), introduced in [1]. Here F_{d} is the collection of all d-tuples of commuting contractions, and H^{∞}(F_{d}) is the set of all holomorphic functions on the polydisk whose norm, defined as

∥ϕ∥_{d,∞} = sup {∥ϕ(T)∥ | T ∈ F_{d}},

is finite. By von Neumann’s inequality [18] and Andô’s theorem [3], when d = 1 or 2 the algebras H^{∞}(𝔻^{d}) and H^{∞}(F_{d}) are isometrically isomorphic; but this is no longer true for d ≥ 3. As our methods are heavily operator theoretic, they extend naturally to H^{∞}(F_{d}), but not to H^{∞}(𝔻^{d}).

Note: After writing this paper, we learned that most of the results have also been found independently by J. Ball and T. Trent [7].

Acknowledgement: We would like to thank the referee for many valuable comments, including suggesting the statement of Theorem 3.12.

2 Solutions on the Bidisk

We shall first prove an extension of Theorem 1.3 to the matrix case. Let points λ_{i} on 𝔻^{2} be written as (α_{i}, β_{i}). Let I_{k} denote the k-by-k identity matrix, and let H^{∞}(𝔻^{2}, M_{k}) denote the bounded analytic functions on the bidisk taking values in the space of k-by-k matrices.

Theorem 2.1 Let z_{1}, …, z_{n} be k-by-k matrices. A necessary and sufficient condition for there to exist a function ϕ in H^{∞}(𝔻^{2}, M_{k}) with ∥ϕ∥ ≤ 1 and ϕ(α_{i}, β_{i}) = z_{i}, i = 1, …, n, is that there exist positive nk-by-nk matrices Γ and Δ such that, as i, j range from 1 to n and l, m range from 1 to k,

(I_{k} − z_{i}^{∗}z_{j})_{l;m} = (1 − \overline{α_{i}}α_{j})Γ_{i,l;j,m} + (1 − \overline{β_{i}}β_{j})Δ_{i,l;j,m}.  (2.2)

Proof: (⇐): Choose nk vectors {u_{i,l}}_{i=1,...,n;l=1,...,k} in a Hilbert space H_{1} so that

Γ_{i,l;j,m} = ⟨u_{j,m}, u_{i,l}⟩.

Define a_{i} to be the matrix whose columns are u_{i,1} through u_{i,k}; so a_{i} is a linear operator from ℂ^{k} to H_{1} and

a_{i}^{∗}a_{j} = [⟨u_{j,m}, u_{i,l}⟩]_{l,m=1}^{k}

is the k-by-k submatrix of Γ corresponding to fixing i and j and letting l and m range from 1 to k.

Similarly, choose vectors v_{i,l} in H_{2} so that

Δ_{i,l;j,m} = ⟨v_{j,m}, v_{i,l}⟩,

and let

b_{i} = [v_{i,1} … v_{i,k}].

Then (2.2) can be rewritten so that, for each i, j, the following k-by-k matrices are equal

[I_{k} + \overline{α_{i}}α_{j}a_{i}^{∗}a_{j} + \overline{β_{i}}β_{j}b_{i}^{∗}b_{j}] = [z_{i}^{∗}z_{j} + a_{i}^{∗}a_{j} + b_{i}^{∗}b_{j}].  (2.3)

Let {c_{i}}_{i=1}^{n} be arbitrary vectors in ℂ^{k}, and define a linear transformation L on a subspace of ℂ^{k} ⊕ H_{1} ⊕ H_{2} by

L : sp \{( \begin{matrix}I_{k} \\ α_{i}a_{i} \\ β_{i}b_{i} \end{matrix} ) c_{i} : 1 ≤ i ≤ n, c_{i} ∈ ℂ^{k}\} → sp \{( \begin{matrix}z_{i} \\ a_{i} \\ b_{i} \end{matrix} ) c_{i} : 1 ≤ i ≤ n, c_{i} ∈ ℂ^{k}\}

∑_{i=1}^{n} ( \begin{matrix}I_{k} \\ α_{i}a_{i} \\ β_{i}b_{i} \end{matrix} ) c_{i} ↦ ∑_{i=1}^{n} ( \begin{matrix}z_{i} \\ a_{i} \\ b_{i} \end{matrix} ) c_{i}  (2.3)

Then (2.3) says that L is an isometry. Extend L to a unitary U from all of ℂ^{k} ⊕ H_{1} ⊕ H_{2} to ℂ^{k} ⊕ H_{1} ⊕ H_{2}. Let M = H_{1} ⊕ H_{2}, and with respect to the decomposition of the space ℂ^{k} ⊕ M, write U as

U = \begin{aligned}& ℂ^{k} M \\ M^{ℂ^{k}} & ( \begin{matrix}A & B \\ C & D \end{matrix} ) . \end{aligned}  (2.5)

For λ = (α, β) in 𝔻^{2}, let E_{λ} be the operator on M that is multiplication by α on H_{1} and multiplication by β on H_{2}. Define

ϕ(λ) = A + BE_{λ}(1 − DE_{λ})^{−1}C.  (2.6)

Note first that as λ ↦ E_{λ} is analytic and contraction valued, (1 − DE_{λ}) is invertible for λ in 𝔻^{2}, so ϕ is an analytic M_{k}-valued map. Moreover, using the fact that U is unitary, so U^{∗}U = I, a calculation gives

1 − ϕ(λ)^{∗}ϕ(λ) = ((1 − DE_{λ})^{−1}C)^{∗}(1 − E_{λ}^{∗}E_{λ})((1 − DE_{λ})^{−1}C).  (2.7)

This is positive for λ in the bidisk, so ϕ is in H^{∞}(𝔻^{2}, M_{k}) and ∥ϕ∥ ≤ 1.

Let u_{i} be a_{i} ⊕ b_{i}, a linear map from ℂ^{k} to M. In terms of A, B, C, D, (2.3) becomes

(A + BE_{λ_{i}}u_{i})c_{i} = z_{i}c_{i}  (2.8)
(C + DE_{λ_{i}}u_{i})c_{i} = u_{i}c_{i}  (2.9)

Solving, (2.9) gives

u_{i} = (1 − DE_{λ_{i}})^{−1}C,

and substituting this equation into (2.8) gives

z_{i} = A + BE_{λ_{i}}(1 − DE_{λ_{i}})^{−1}C = ϕ(λ_{i}).

So ϕ interpolates, as desired.

(⇒): If ϕ is any function in the ball of H^{∞}(𝔻^{2}, M_{k}), then there are Hilbert spaces H_{1} and H_{2} and a unitary operator U on ℂ^{k} ⊕ (H_{1} ⊕ H_{2}) such that when U is written as in (2.5), then ϕ is given by (2.6). This was proved by the first author in [2]; as the result is central to this paper, we include a proof at the end of the paper (Theorem 4.1).

So suppose there exists a ϕ that interpolates the data, and write it in the above way. Let u_{i} = (1 − DE_{λ_{i}})^{−1}C, a linear operator from ℂ^{k} to H_{1} ⊕ H_{2}, and decompose u_{i} as a_{i} ⊕ b_{i}, where a_{i} maps ℂ^{k} to H_{1} and b_{i} maps ℂ^{k} to H_{2}. Define the nk-by-nk positive matrix Γ by letting its (i, j)^{th} k-by-k block be a_{i}^{∗}a_{j}, and similarly define Δ by letting its (i, j)^{th} block be b_{i}^{∗}b_{j}.

Then the fact that ϕ interpolates means that (2.3) holds with U in place of L; and because U is unitary, this means (2.3) holds. So (2.2) holds as desired. □

We can now describe all the interpolating functions, in both the scalar and matrix-valued case. Let L(H_{1}, H_{2}) denote the bounded linear operators from H_{1} to H_{2}.

Theorem 2.10 Suppose there is a function in H^{∞}(𝔻^{2}, M_{k}) of norm at most one that maps the points (α_{i}, β_{i}) to z_{i}, for i = 1, …, n; so (2.2) holds. Choose operators a_{i} in L(ℂ^{k}, H_{1}) and b_{i} in L(ℂ^{k}, H_{2}), for some auxiliary Hilbert spaces H_{1} and H_{2}, so that a_{i}^{∗}a_{j} = Γ_{ij} and b_{i}^{∗}b_{j} = Δ_{ij}. Let L be the isometry on the subspace of ℂ^{k} ⊕ H_{1} ⊕ H_{2} given by

L : sp \{( \begin{matrix}I_{k} \\ α_{i}a_{i} \\ β_{i}b_{i} \end{matrix} ) c_{i} : 1 ≤ i ≤ n, c_{i} ∈ ℂ^{k}\} → sp \{( \begin{matrix}z_{i} \\ a_{i} \\ b_{i} \end{matrix} ) c_{i} : 1 ≤ i ≤ n, c_{i} ∈ ℂ^{k}\}

( \begin{matrix}I_{k} \\ α_{i}a_{i} \\ β_{i}b_{i} \end{matrix} ) c_{i} ↦ ( \begin{matrix}z_{i} \\ a_{i} \\ b_{i} \end{matrix} ) c_{i}

Let U be any unitary that extends L to all of ℂ^{k} ⊕M, where M = H_{1}⊕H_{2}. For λ = (α, β) a point of 𝔻^{2}, let E_{λ} be the operator on M that is multiplication by α on H_{1} and multiplication by β on H_{2}. Write U, with respect to the decomposition ℂ^{k} ⊕ M, as

U = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .  (2.11)

Then

ϕ(λ) = A + BE_{λ}(1 − DE_{λ})^{−1}C  (2.12)

is a function in the unit ball of H^{∞}(𝔻^{2}) that interpolates the data. Moreover, all solutions of the interpolation problem arise in this way, for some choice of Γ, Δ, {a_{i}}, {b_{i}} and some extension U.

Proof: That one can choose a_{i} and b_{i} follows as in the proof of Theorem 2.1. The fact that L is an isometry is precisely the content of (2.2). Repeating the proof of Theorem 2.1 yields that ϕ defined by (2.12) is of norm at most one and interpolates.

Conversely, suppose ϕ is any function of norm at most one that interpolates. Using the representation theorem 4.1 again, just as in the proof of necessity of Theorem 2.1, one gets U, a_{i} and b_{i} (and hence also Γ and Δ) so that the theorem holds. □

This description of the solutions allows us to show that there is always a rational inner solution. (An inner function in H^{∞}(𝔻^{2}) is one whose boundary values on 𝕋^{2}, the distinguished boundary of the bidisk, have modulus one almost everywhere. A function in H^{∞}(𝔻^{2}, M_{k}) is called inner if its boundary values on 𝕋^{2} are unitary almost everywhere).

Corollary 2.13 Suppose there exists some function in H^{∞}(𝔻^{2}, M_{k}) of norm less than or equal to 1 that interpolates the points λ_{i} to z_{i}. Then there is a rational inner function that interpolates.

Proof: Choose H_{1} and H_{2} to be finite dimensional (they can clearly always be chosen to have dimension nk at most), and then ϕ in (2.12) is rational.

As λ tends to a point on the distinguished boundary of the bidisk, E_{λ}^{∗}E_{λ} tends to the identity, so equation (2.7) shows that, as long as (1 − DE_{λ}) remains invertible, |ϕ(λ)| tends to 1. Since the determinant of (1−DE_{λ}) is a polynomial, its zero set is of Lebesgue measure zero in the distinguished boundary, so |ϕ(λ)| tends to 1 almost everywhere. □

3 The Toeplitz-corona theorem on the bidisk

Let μ be any finite measure on 𝕋^{2}. Let H^{2}(μ) denote the closure of the analytic polynomials in L^{2}(μ), and for any function f in L^{∞}(μ) define the Toeplitz operator T_{f}^{μ} on H^{2}(μ) by

T_{f}^{μ}g = P_{H^{2}(μ)}fg.

In the statement of the folowing theorem, we shall let λ = (α, β) and z = (a, b) be points on the bidisk. Saying Γ(z, λ) is a positive semi-definite kernel means that for any points λ_{1}, …, λ_{n}, and any constants c_{1}, …, c_{n}, the sum

∑_{i,j=1}^{n} c_{i}\overline{c_{j}}Γ(λ_{i}, λ_{j}) ≥ 0.

Theorem 3.1 Let ϕ_{1} and ϕ_{2} be in H^{∞}(𝔻^{2}), and δ > 0. The following are equivalent:

(i) There exist positive semi-definite kernels Γ(z, λ) and Δ(z, λ) such that

\overline{ϕ_{1}(λ)}ϕ_{1}(z) + \overline{ϕ_{2}(λ)}ϕ_{2}(z) − δ = (1 − \overline{α}a)Γ(z, λ) + (1 − \overline{β}b)Δ(z, λ).  (3.2)

(ii) There exist functions ψ_{1}, ψ_{2} in H^{∞}(𝔻^{2}) such that

ψ_{1}ϕ_{1} + ψ_{2}ϕ_{2} = 1  (3.3)

and

sup_{z∈𝔻^{2}}[|ψ_{1}(z)|^{2} + |ψ_{2}(z)|^{2}] ≤ \frac{1}{δ}.  (3.4)

(iii) For all measures μ on 𝕋^{2},

T_{ϕ_{1}}^{μ}T_{ϕ_{1}}^{μ∗} + T_{ϕ_{2}}^{μ}T_{ϕ_{2}}^{μ∗} ≥ δI.  (3.5)

Proof: We shall prove (i) ⇔ (ii) and (ii) ⇒ (iii) ⇒ (i). Although logically redundant, we include the proof (ii) ⇒ (i) because we think it will help the reader to understand the interconnections; see also Remark 1 after the proof.

(i) ⇒ (ii): Choose vector-valued functions f_{1} and f_{2} so that

⟨f_{1}(z), f_{1}(λ)⟩ = Γ(z, λ)

and

⟨f_{2}(z), f_{2}(λ)⟩ = Δ(z, λ)

(one can always do this: see e.g. [5]). Let

f(λ) = ( \begin{matrix}f_{1}(λ) \\ f_{2}(λ) \end{matrix} ) ,
E_{λ}f(λ) = ( \begin{matrix}αf_{1}(λ) \\ βf_{2}(λ) \end{matrix} ) ,

and

ϕ(λ) = ( \begin{matrix}ϕ_{1}(λ) \\ ϕ_{2}(λ) \end{matrix} ) .

Then (3.2) says that there is an isometry

( \begin{matrix}A & B \\ C & D \end{matrix} ) : ( \begin{matrix}ϕ(λ) \\ λf(λ) \end{matrix} ) ↦ ( \begin{matrix}δ^{\frac{1}{2}} \\ f(λ) \end{matrix} ) .

Solving, we get

δ^{\frac{1}{2}} = [A + BE_{λ}(1 − DE_{λ})^{−1}C]ϕ(λ)

Because ( \begin{matrix}A & B \\ C & D \end{matrix} ) is an isometry, the 1-by-2 vector [A + BE_{λ}(1 − DE_{λ})^{−1}C] is of norm less than or equal to one. Let

ψ(λ) = (ψ_{1}(λ) ψ_{2}(λ)) = δ^{− \frac{1}{2}} [A + BE_{λ}(1 − DE_{λ})^{−1}C].

Then (3.3) and (3.4) are satisfied.

(ii) ⇒ (i): By hypothesis, δ^{1/2}ψ is in the ball of H^{∞}(𝔻^{2}, L(ℂ^{2}, ℂ^{1})), the bounded analytic functions mapping the bidisk into the linear maps from ℂ^{2} to ℂ^{1}. Therefore by 4.1 there is a Hilbert space M = H_{1} ⊕ H_{2} and a unitary

( \begin{matrix}A & B \\ C & D \end{matrix} ) : ℂ^{2} ⊕ M → ℂ^{2} ⊕ M,

such that

δ^{\frac{1}{2}} ( \begin{matrix}ψ_{1} & ψ_{2} \\ 0 & 0 \end{matrix} ) = A + BE_{λ}(1 − DE_{λ})^{−1}C.

Define

f = (1 − DE_{λ})^{−1}C ( \begin{matrix}ϕ_{1}(λ) \\ ϕ_{2}(λ) \end{matrix} ) ,

and decompose f as f_{1} ⊕ f_{2} with respect to H_{1} ⊕ H_{2}. Let Γ(z, λ) = ⟨f_{1}(z), f_{1}(λ)⟩, and Δ(z, λ) = ⟨f_{2}(z), f_{2}(λ)⟩. Then equation (3.2), which we wish to establish, is the same as

δ + f^{∗}f = ϕ^{∗}ϕ + f^{∗}E_{λ}^{∗}E_{λ}f.  (3.6)

Using the fact that U^{∗}U = 1 as in (2.7) and the formula for f we get

f^{∗}(1 − E_{λ}^{∗}E_{λ})f = ϕ^{∗}((1 − DE_{λ})^{−1}C)^{∗}(1 − E_{λ}^{∗}E_{λ})((1 − DE_{λ})^{−1}C)ϕ
= ϕ^{∗} (1 − δ ( \begin{matrix}\overline{ψ_{1}} & 0 \\ \overline{ψ_{2}} & 0 \end{matrix} ) ( \begin{matrix}ψ_{1} & ψ_{2} \\ 0 & 0 \end{matrix} )) ϕ
= ϕ^{∗}(1 − δψ^{∗}ψ)ϕ
= ϕ^{∗}ϕ − δ

so (3.6) does hold, as desired.

(ii) ⇒ (iii): Equation (3.4) gives that

(T_{ψ_{1}}^{μ∗} T_{ψ_{2}}^{μ∗}) ( \begin{matrix}T_{ψ_{1}}^{μ} \\ T_{ψ_{2}}^{μ} \end{matrix} ) ≤ \frac{1}{δ}

for any measure μ supported on \overline{𝔻^{2}}. Therefore

( \begin{matrix}T_{ψ_{1}}^{μ} \\ T_{ψ_{2}}^{μ} \end{matrix} ) (T_{ψ_{1}}^{μ∗} T_{ψ_{2}}^{μ∗}) ≤ \frac{1}{δ} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,

and so

(T_{ϕ_{1}}^{μ} T_{ϕ_{2}}^{μ}) ( \begin{matrix}T_{ψ_{1}}^{μ} \\ T_{ψ_{2}}^{μ} \end{matrix} ) (T_{ψ_{1}}^{μ∗} T_{ψ_{2}}^{μ∗}) ( \begin{matrix}T_{ϕ_{1}}^{μ∗} \\ T_{ϕ_{2}}^{μ∗} \end{matrix} ) ≤ \frac{1}{δ}(T_{ϕ_{1}}^{μ}T_{ϕ_{1}}^{μ∗} + T_{ϕ_{2}}^{μ}T_{ϕ_{2}}^{μ∗}).

But the left-hand side of this inequality is the identity by (3.3), so (3.5) is established. (iii) ⇒ (i): Let K(z, λ) be any kernel such that

K(z, λ)(1 − \overline{α}a) ≥ 0  (3.7)

and

K(z, λ)(1 − \overline{β}b) ≥ 0  (3.8)

Choose a vector valued function f : 𝔻^{2} → H such that

K(z, λ) = ⟨f(z), f(λ)⟩.

Let T_{1} and T_{2} be operators on H defined by

T_{1} : f(z) ↦ af(z)
T_{2} : f(z) ↦ bf(z)

and extended to be linear on the span of the range of f (which we shall take to be all of H). Then T_{1} and T_{2} commute, and by (3.7) and (3.8) they are contractions. Therefore by the commutant lifting theorem [16] they have extensions to commuting co-isometries S_{1}^{∗} and S_{2}^{∗}. Now (3.5) says that for all cyclic pairs of commuting isometries, and therefore for all commuting isometries S_{1} and S_{2}, we have

ϕ_{1}(S_{1}, S_{2})ϕ_{1}(S_{1}, S_{2})^{∗} + ϕ_{2}(S_{1}, S_{2})ϕ_{2}(S_{1}, S_{2})^{∗} ≥ δI.  (3.9)

We actually need a slight variant of (3.9). For ϕ in H^{∞}(𝔻^{2}), define ϕ̆(z) := \overline{ϕ(\overline{z})}. We introduce this notation because ϕ̆(S_{1}, S_{2})^{∗} = ϕ(S_{1}^{∗}, S_{2}^{∗}) is an extension of ϕ(T_{1}, T_{2}).

As (3.5) holds for all μ, it also holds if ϕ_{1} and ϕ_{2} are replaced by ϕ̆_{1} and ϕ̆_{2}, because if μ̆ is the measure defined by μ̆(E) = μ(\overline{E}), with \overline{E} here the complex conjugate of E, then

⟨T μ̆ ; ϕ̆T μ̆∗ ; ϕ̆p̆, p̆⟩_{L^{2}(μ̆)} = ⟨T_{ϕ}^{μ}T_{ϕ}^{μ∗}p, p⟩_{L^{2}(μ)}.

Therefore we also have

ϕ̆_{1}(S_{1}, S_{2})ϕ̆_{1}(S_{1}, S_{2})^{∗} + ϕ̆_{2}(S_{1}, S_{2})ϕ̆_{2}(S_{1}, S_{2})^{∗} ≥ δI.  (3.10)

Now

⟨f(z), [ϕ_{1}(T_{1}, T_{2})^{∗}ϕ_{1}(T_{1}, T_{2}) + ϕ_{2}(T_{1}, T_{2})^{∗}ϕ_{2}(T_{1}, T_{2}) − δ]f(λ)⟩
=  [ϕ_{1}(z)\overline{ϕ_{1}(λ)} + ϕ_{2}(z)\overline{ϕ_{2}(λ)} − δ]K(z, λ).

So from (3.10), we have that

[ϕ_{1}(z)\overline{ϕ_{1}(λ)} + ϕ_{2}(z)\overline{ϕ_{2}(λ)} − δ]K(z, λ) ≥ 0  (3.11)

whenever K is a kernel such that (3.7) and (3.8) hold.

(Notice that if K is only defined on a subset S × S of 𝔻^{2} × 𝔻^{2}, and if (3.7) and (3.8) hold on S × S, then the preceding argument shows that (3.10) implies that (3.11) holds on S × S).

We claim first that (3.2) holds on any finite set. For, fix a finite set F = {λ_{1}, …, λ_{n}} of distinct points in 𝔻^{2}, and write each λ_{i} as (α_{i}, β_{i}). Let Φ be the n-by-n self-adjoint matrix with (i, j) entry

Φ_{ij} = ϕ_{1}(λ_{j})\overline{ϕ_{1}(λ_{i})} + ϕ_{2}(λ_{j})\overline{ϕ_{2}(λ_{i})} − δ.

Let T be the set of all n-by-n self-adjoint matrices T representable in the form

T_{ij} = (1 − α_{i}\overline{α_{j}})Γ_{ij} + (1 − β_{i}\overline{β_{j}})Δ_{ij},

where Γ and Δ are positive. As T is a closed wedge, if Φ were not in T , there would be a real linear functional on the space of all n-by-n self-adjoint matrices that was positive on T and strictly negative on Φ.

Any such linear functional must be of the form T ↦ tr(W T) for some self-adjoint matrix W. As T contains the set of all positive matrices (let Δ = 0 and Γ be the Schur product of the positive matrix 1/(1 − \overline{α_{i}}α_{j}) with an arbitrary positive matrix), W must be positive. Let K be the transpose of W. Then

∑_{i,j=1}^{n} c_{i}\overline{c_{j}}K_{ij}(1 − α_{i}\overline{α_{j}}) = tr(W T),

where

T_{ij} = (1 − α_{i}\overline{α_{j}})c_{i}\overline{c_{j}}

is in T . So K satisfies (3.7), and similarly also (3.8). Therefore, by the remark after (3.11), Φ · K is positive, so

tr(W Φ) = ∑_{i,j=1}^{n} K(λ_{i}, λ_{j})Φ_{ij} ≥ 0.

Therefore Φ is in T , proving the claim.

Given that (3.2) holds on all finite sets F, we wish to find one Γ and Δ that work for the whole bidisk. This follows from general topology. For each finite set F, let S_{F} denote the set of pairs of positive semi-definite kernels Γ and Δ on F that satisfy (3.2). Then each S_{F} is a non-empty compact set, and for any superset G of F there is a restriction map from S_{G} to S_{F}. Therefore by Kurosh’s theorem [4, p.75], the projective limit of this system is non-empty, which means precisely that there are kernels Γ and Δ defined on the whole disk for which (3.2) holds. □

Remark 1: In the proof that (ii) ⇒ (i) the construction produces an analytic function f, so it follows that if (3.2) holds for any Γ and Δ, then Γ(z, λ) and Δ(z, λ) can be chosen to be analytic in z and conjugate-analytic in λ.

Remark 2: It is obvious that the theorem extends from two to n functions for any n. It is also true that the functions can be taken to be in H^{∞}(𝔻^{2}, M_{k}), with the above proof working mutatis mutandis (note that f_{1} and f_{2} will then take values in the space of linear operators from ℂ^{k} to an auxiliary Hilbert space).

Remark 3: One could combine the preceding theorems into one general result:

Theorem 3.12 Let S ⊆ 𝔻^{2} and let ϕ_{1}, …, ϕ_{p}: S → M_{n×k} and ψ_{1}, …, ψ_{q}: S → M_{m×k} be functions. Then the following are equivalent:

(i) There exist positive kernels Γ, Δ: S × S → M_{k×k} such that

∑_{i=1}^{q} ψ_{i}(λ)^{∗}ψ_{i}(z) − ∑_{j=1}^{p} ϕ_{j}(λ)^{∗}ϕ_{j}(z) = (1 − \overline{α}a)Γ(z, λ) + (1 − \overline{β}b)Δ(z, λ)

for all z = (a, b), λ = (α, β) in S.

(ii) There exists a holomorphic function h: 𝔻^{2} → Ball(M_{np×mq}) such that

h(z) ( ψ_{1}(z) ; ⋮ ; ψ_{q}(z) ) = ( ϕ_{1}(z) ; ⋮ ; ϕ_{p}(z) )

for all z ∈ S.

The proof of Theorem 3.12, and the representation of h as

h(λ) = A + BE_{λ}(1 − DE_{λ})^{−1}C

for an appropriate unitary ( \begin{matrix}A & B \\ C & D \end{matrix} ), require no new ideas.

4 A proof of the representation theorem for H^{∞}(𝔻^{2}, M_{k})

For completeness, we include a proof of the representation theorem from [2].

Theorem 4.1 [2] Let ϕ be in H^{∞}(𝔻^{2}, M_{k}) of norm less than or equal to 1. Then there exist auxiliary Hilbert spaces H_{1} and H_{2} and a unitary operator U on ℂ^{k} ⊕ H_{1} ⊕ H_{2} such that, if U is written as

U =  ( \begin{matrix}& ℂ^{k} & H_{1} ⊕ H_{2} \\ ℂ^{k} & A & B \\ H_{1} ⊕ H_{2} & C & D \end{matrix} )  (4.2)

then

ϕ(λ) = A + BE_{λ}(1 − DE_{λ})^{−1}C.  (4.3)

Here, for λ = (α, β) in 𝔻^{2}, E_{λ} is the operator of multiplication by α on H_{1} and multiplication by β on H_{2}.

Proof: First, we shall give a proof in the scalar case k = 1.

By Andô’s theorem [3], if T_{1} and T_{2} are any commuting contractions, then ∥ϕ(T_{1}, T_{2})∥ ≤ 1. Let K(z, λ) be any positive definite kernel on 𝔻^{2}. Choose a vector valued function f so that K(z, λ) = ⟨f(z), f(λ)⟩. Throughout the proof we shall write z = (a, b) and λ = (α, β). Let T_{1} : f(λ) ↦ αf(λ) and T_{2} : f(λ) ↦ βf(λ).

Then T_{1} and T_{2} are contractions if and only if

K(z, λ)(1 − \overline{α}a) ≥ 0  (4.4)

and

K(z, λ)(1 − \overline{β}b) ≥ 0  (4.5)

The operator ϕ(T_{1}, T_{2}) is a contraction if and only if

K(z, λ)(1 − \overline{ϕ(λ)}ϕ(z)) ≥ 0.  (4.6)

So by Andô’s theorem, the positivity of (4.4) and (4.5) implies the positivity of (4.6). Just as in the proof of Theorem 3.1 (iii) ⇒ (i), this means that if F is any finite set in 𝔻^{2}, there exist positive kernels Γ and Δ on F such that

1 − \overline{ϕ(λ)}ϕ(z) = (1 − \overline{α}a)Γ(z, λ) + (1 − \overline{β}b)Δ(z, λ).  (4.7)

Again, a projective limit argument implies that there is a choice of Γ and Δ such that (4.7) holds on the whole bidisk. Factor Γ and Δ in the now familiar way to get the desired representation of ϕ.

(Matrix Case). The above proof works in essentially the same way. Let K(z, λ) be an M_{k}-valued positive definite kernel, and as in the proof of Theorem 2.1, factor K as f(λ)^{∗}f(z) where f(λ) ∈ L(ℂ^{k}, H). Define T_{1} and T_{2} on H by, for c ∈ ℂ^{k}, T_{1} : f(λ)c ↦ αf(λ)c and T_{2} : f(λ)c ↦ βf(λ)c. Then repeating the proof of the scalar case, one gets

I_{k} − ϕ(z)\overline{ϕ(λ)} = (1 − \overline{α}a)Γ(z, λ) + (1 − \overline{β}b)Δ(z, λ),

and from this, as in the proof of Theorem 2.1, one gets the desired representation of ϕ. □

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