AN INTRINSIC DIFFICULTY WITH INTERPOLATION ON THE BIDISK

JAMES P. SOLAZZO

Abstract. The set of possible values (w_{1}, ..., w_{k}) = (f(x_{1}), ..., f(x_{k})) arising from restricting contractive elements f from some uniform algebra A to a finite set {x_{1}, ..., x_{k}} in the domain is called an interpolation body. When the uniform algebra is the bidisk algebra, Cole and Wermer show that the associated interpolation body is a semi-algebraic set and it is in this sense that the interpolation body is “computable”. Motivated by the work of Cole and Wermer, Paulsen introduced the notion of the Schur ideal which acts a natural “dual” object for these interpolation bodies. From this “duality” a stronger notion of “computability” follows which will allow us to discuss the intrinsic differences between interpolation on the bidisk and interpolation on the disk.

1. Introduction

Let X be compact Hausdorff space and let C(X) denote the continuous complex-valued functions on X. We call A ⊆ C(X) a uniform algebra provided that A is uniformly closed, contains the identity, and separates points in X. For fixed points x_{1}, ..., x_{k} in X, B. Cole, K. Lewis, and J. Wermer in [5] define the interpolation body associated with A and x_{1}, ..., x_{k}, denoted D(A; x_{1}, ..., x_{k}), in the following way: a point w⃗ = (w_{1}, ..., w_{k}) in ℂ^{k} belongs to D(A; x_{1}, ..., x_{k}) if for each ϵ > 0, there exists f in A with ∥f∥_{∞} ≤ 1 + ϵ such that f(x_{i}) = w_{i} for i = 1, ..., k. Equivalently, a point (w_{1}, ..., w_{k}) in ℂ^{k} belongs to D(A; x_{1}, ..., x_{k}) if ∥w_{1}[f_{1}] + … + w_{k}[f_{k}]∥ ≤ 1, where [f_{i}] is in A/I_{x}, f_{i}(x_{j}) = δ_{ij}, and I_{x} is the ideal of functions in A vanishing at the points x_{1}, ..., x_{k}. With this point of view, one can see that an interpolation body D(A; x_{1}, ..., x_{k}) is a natural coordinization of the closed unit ball of the quotient algebra (k-idempotent operator algebra) A/I_{x}.

G. Pick solved the classic interpolation problem from 𝔻 to 𝔻 (open unit disk) in 1916. In [11], G. Pick showed that (w_{1}, ..., w_{k}) in ℂ^{k} is in D(A(𝔻); α_{1}, ..., α_{k}) if and only if the matrix (\frac{1−w̄_{i}w_{j}}{1−ᾱ_{i}α_{j}} ) is positive semidefinite, where A(𝔻) denotes the disk algebra. Note, that G. Pick’s theorem yields the same set of finite conditions for determining if an arbitrary k-tuple (w_{1}, ..., w_{k}) in ℂ^{k} belongs to D(A(𝔻); α_{1}, ..., α_{k}). The author would like to accentuate that the goal of this paper resides in finding a finite (or possibly infinite) number of conditions which will allow us to determine the ball D(A; x_{1}, ..., x_{k}) and more specifically the ball D(A(𝔻^{2}); z_{1}, ..., z_{k}), where A(𝔻^{2}) denotes the bidisk algebra. It is known for a fixed pair of k-tuples, say z_{1}, ..., z_{k} in 𝔻^{2} and w_{1}, ..., w_{k} in ℂ, that there exist finite methods for determining whether one can interpolate the given pair of k-tuples (i.e, ∀ ϵ > 0 ∃ f ∈ A(𝔻^{2}) with ∥f∥_{∞} ≤ 1 + ϵ such that f(z_{i}) = w_{i}). For such constructions, see the work of J. Ball and T. Trent [3] and/or the work of J. Agler and J. McCarthy [2]. However, these finite methods yield unique sets of finite conditions for distinct k-tuples, say (w_{1}, ..., w_{k}) and (w̃_{1}, ..., w̃_{k}) in ℂ^{k}, the target space. This is precisely the intrinsic difficulty we are referring to with interpolation on the bidisk.

In [7], B. Cole and J. Wermer show that D(A(𝔻^{2}); z_{1}, ..., z_{k}) is a semi−algebraic set (i.e, a set determined by a finite collection of polynomial inequalities). Thus, one can determine if an arbitrary k-tuple (w_{1}, ..., w_{k}) in ℂ^{k} is in D(A(𝔻^{2}); z_{1}, ..., z_{k}) by checking a finite number of polynomial inequalities (the same set of polynomial inequalities work for each k-tuple). However, in proving that D(A(𝔻^{2}); z_{1}, ..., z_{k}) is a semi-algebraic set B. Cole and J. Wermer appeal to an existential theorem of A. Tarskie and A. Seidenberg and by doing so avoid the construction of the polynomial inequalities which determine D(A(𝔻^{2}); z_{1}, ..., z_{k}).

In this paper we will show that for three particular points z_{1}, z_{2}, z_{3} in 𝔻^{2} that D(A(𝔻^{2}); z_{1}, z_{2}, z_{3}) fails to be “computable” in a stricter sense, namely, that the largest (affiliated) Schur ideal is not finitely generated. In section 2 we will describe tersely the motivation for studying Schur ideals and make the terminology precise. In section 3 we will bring forward the notion of an interpolation problem being “strongly computable”, as well as, compare and contrast this new notion of computability with the work of B. Cole and J. Wermer in [7]. In section 4 we will present an example which is not strongly computable by using some results from V. Paulsen’s paper [9]. Last, in section 5 we will give an alternate proof of J. Agler’s bidisk interpolation formula, [1], due to V. Paulsen.

2. Preliminaries

Throughout the paper M_{k} will denote the k × k matrices with complex entries and M_{k}^{+} will denote the closed cone of positive semi-definite matrices in M_{k}. If P is in M_{k}, then it will be understood that P is the matrix (p_{ij})_{i,j=1}^{k} simply written as (p_{ij}). If P, Q are in M_{k}, then the Schur product of the matrices P and Q, denoted as P ∗ Q, is defined to be (p_{ij}) ∗ (q_{ij}) = (p_{ij}q_{ij}). The matrix P being positive semi-definite will be denoted as P ≥ 0.

The following theorem is due to B. Cole and J. Wermer [6], in which they give conditions on a point (w_{1}, ..., w_{k}) in ℂ^{k} in order that (w_{1}, ..., w_{k}) belong to the interpolation body D = D(A; x_{1}, ..., x_{k}) where A ⊆ C(X) is an arbitrary uniform algebra and x_{1}, ..., x_{k} in X.

Theorem 1. (Cole-Wermer) If A ⊆ C(X) is a uniform algebra and x_{1}, ..., x_{k} ∈ X, then there exists a set S ⊆ M_{k}^{+} such that

(w_{1}, ..., w_{k}) ∈ D if and only if ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 for all (p_{ij}) ∈ S.

In [9], V. Paulsen observes that Theorem 1 suggests the following “dualities” between subsets of the closed k-polydisk and subsets of M_{k}^{+}. Given a non-empty set S ⊆ M_{k}^{+} define

S^{⊥} = {(w_{1}, ..., w_{k}) ∈ ℂ^{k} : ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 ∀ (p_{ij}) ∈ S}.

Similarly, given a subset D of the closed k-polydisk with 0 ∈ D define

D^{⊥} = {(p_{ij}) : ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 ∀ (w_{1}, ..., w_{k}) ∈ D}.

Since we insist that 0 ∈ D then D^{⊥} is always a set of positive semidefinite matrices. Observing further that the set D^{⊥} has certain properties, V. Paulsen [9] introduces the concept of a Schur ideal, defined below. By studying this duality between hyperconvex sets and Schur ideals and using some results from the theory of abstract operator algebras, V. Paulsen is able to generalize J. Agler’s scalar-valued interpolation results [1] for the bidisk to more general product domains [9]. For another approach to such problems see the paper of A. Tomerlin [14].

Definition 1. Let I ⊆ M_{k}^{+} be a non-empty set. Then I will be called a Schur ideal provided that:

(1) A, B ∈ I ⇒ A + B ∈ I.

(2) A ∈ I, P ∈ M_{k}^{+} ⇒ A ∗ P ∈ I.

If D is a subset of the closed k-polydisk with 0 ∈ D, then D^{⊥} is a Schur ideal. A consequence of Theorem 1 is that if D = D(A; x_{1}, ..., x_{k}), then D = D^{⊥⊥}.

Definition 2. A Schur ideal I ⊆ M_{k}^{+} is said to be finitely generated provided that there exist P_{1}, ..., P_{m} ∈ M_{k}^{+} such that

I = ⟨P_{1}, ..., P_{m}⟩ = {∑_{i=1}^{m} P_{i} ∗ R_{i} : R_{i} ∈ M_{k}^{+}}.

Definition 3. Let A ⊆ C(X), x_{1}, ..., x_{k} ∈ X, and I ⊆ M_{k}^{+} be a Schur ideal. Then the Schur ideal I is said to be affiliated with the interpolation body D(A; x_{1}, ..., x_{k}) associated with A and x_{1}, ..., x_{k} provided that I^{⊥} = D(A; x_{1}, ..., x_{k}).

Another consequence of Theorem 1 is that each interpolation body D(A; x_{1}, ..., x_{k}) has an affiliated Schur ideal, namely D(A; x_{1}, ..., x_{k})^{⊥}. Note, we have that ⟨(\frac{1}{1−ᾱ_{i}α_{j}})⟩ is the unique affiliated Schur ideal for the interpolation body D(A(𝔻); α_{1}, ..., α_{k}), [13]. In Definition 3 we could have easily defined the notion of an affiliated set instead of an affiliated Schur ideal. However, the interpolation body D(A(𝔻); α_{1}, ..., α_{k}) has infinitely many affiliated sets.

3. Semi-algebraic vs. Strongly Computable

In this section we will introduce the notion of an interpolation problem being “strongly computable” as well as compare and contrast this new notion with the work of B. Cole and J. Wermer in [7].

Throughout this section we will assume that D = D(A; x_{1}, ..., x_{k}) for some uniform algebra A and for fixed points x_{1}, ..., x_{k} in X. This assumption will allow us to make use of the fact that D^{⊥⊥} = D.

Definition 4. A Schur ideal I is said to be affiliated with the set D provided that

I^{⊥} = D.

One can verify that if I is any other affiliated Schur ideal with respect to the set D, then I ⊆ D^{⊥}. In this sense, D^{⊥} is the largest affiliated Schur ideal with respect to the set D. This motivated the following definition.

Definition 5. The set D is strongly computable provided that D^{⊥} is a finitely generated Schur ideal.

Remark: If D is stongly computable, then ∃ P_{1}, ..., P_{m} ∈ M_{k}^{+} such that D^{⊥} = ⟨P_{1}, ..., P_{m}⟩. Let (w_{1}, ..., w_{k}) ∈ ℂ^{k} and consider the following implications:

(1 − w̄_{i}w_{j}) ∗ P_{l} ≥ 0 ⟹ ∀ Q ∈ M_{k}^{+}, (1 − w̄_{i}w_{j}) ∗ Q ∗ P_{l} ≥ 0
for l = 1, ..., m  for l = 1, ..., m
⟹ ∀ Q_{1}, ..., Q_{m} ∈ M_{k}^{+},

(1 − w̄_{i}w_{j}) ∗ ∑_{l=1}^{m} Q_{l} ∗ P_{l} ≥ 0

⟹ ∀ P ∈ D^{⊥}, (1 − w̄_{i}w_{j}) ∗ P ≥ 0.

Since D^{⊥⊥} = D we have that (w_{1}, ..., w_{k}) ∈ D. Thus, if D^{⊥} is finitely generated, then it is sufficient to check finitely many positivity conditions to determine if (w_{1}, ..., w_{k}) ∈ D.

It turns out that, if D = D(A(𝔻); α_{1}, ..., α_{k}), then D^{⊥} = ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩, see Proposition 4 in section 4. Hence, D = D(A(𝔻); α_{1}, ..., α_{k}) is strongly computable. Moreover, if I is any other Schur ideal affiliated with D = D(A(𝔻); α_{1}, ..., α_{k}), then I is the Schur ideal ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩ , see [13]. However, for a general uniform algebra A, it is still not known whether I^{⊥} = D(A; x_{1}, ..., x_{k}) implies that I = D(A; x_{1}, ..., x_{k})^{⊥}; we leave this issue for future work. In section 4 we will show that for three particular points z_{1}, z_{2}, z_{3} in 𝔻^{2} that the ball D(A(𝔻^{2}); z_{1}, z_{2}, z_{3}) is not strongly computable.

B. Cole and J. Wermer prove the following theorem in [7].

Theorem 2. (Cole-Wermer) If z_{1} = (α_{1}, β_{1}), ..., z_{k} = (α_{k}, β_{k}) ∈ 𝔻^{2}, then D(A(𝔻^{2}); z_{1}, ..., z_{k}) is a semi-algebraic set in ℂ^{k}.

Loosely speaking, a set X ⊆ ℝ^{k} is semi-algebraic provided that X is determined by a finite collection of polynomial inequalities (for a brief exposition of the theory of semi-algebraic sets see [7]). Thus, despite our example where D(A(𝔻^{2}); z_{1}, z_{2}, z_{3}) is not strongly computable, it seems plausible to assume that there may exists a finitely generated Schur ideal, say I, such that I ⊆ D^{⊥} and I^{⊥} = D(A(𝔻^{2}); z_{1}, z_{2}, z_{3}). This was the motivation behind the next definition.

Definition 6. The set D is said to be computable provided there exists a finitely generated Schur ideal I_{0} with I_{0}^{⊥} = D.

Clearly, if a set D is strongly computable, then the set D is computable. In the next proposition we will show that if a set D is computable, then the set D is semi-algebraic.

Proposition 3. If D is computable, then D is a semi-algebraic set.

Proof: We note that a matrix P ∈ M_{k} is positive semi-definite [8] if and only if P = P^{∗} and the principal minors of P are non-negative, where the principal minors of P are the scalars Δ_{l}(P) defined by

Δ_{l}(P) = det [ \begin{matrix}p_{11} & … & p_{1l} \\ ⋮ & & ⋮ \\ p_{l1} & … & p_{ll} \end{matrix} ] , 1 ≤ l ≤ k.

The set D being computable implies that there exists a finitely generated Schur ideal I_{0} with I_{0}^{⊥} = D. Thus, we can choose P_{1}, ..., P_{m} ∈ M_{k}^{+} such I_{0} = ⟨P_{1}, ..., P_{m}⟩. But,

(w_{1}, ..., w_{k}) ∈ D ⟺ ((1 − w̄_{i}w_{j})p_{ij}^{l}) ≥ 0, for l = 1, ..., m,
⟺ Δ_{1}((1 − w̄_{i}w_{j})p_{ij}^{l}) ≥ 0, for l = 1, ..., m
Δ_{2}((1 − w̄_{i}w_{j})p_{ij}^{l}) ≥ 0, for l = 1, ..., m
⋮

Δ_{k}((1 − w̄_{i}w_{j})p_{ij}^{l}) ≥ 0, for l = 1, ..., m.

Thus, the real polynomials Δ_{n}^{l}(w_{1}, ..., w_{k}, w̄_{1}, ..., w̄_{k}) for n = 1, ..., k and l = 1, ..., m determine the set D. Hence the set D is semi-algebraic in ℂ^{k}. □

We mentioned previously that D^{⊥} is the largest affiliated Schur ideal with respect to the set D. Thus, if D^{⊥} is finitely generated, then the set D is semi-algebraic. Hence, a set being strongly computable (and/or computable) is a stricter notion of computability than a set being semialgebraic.

4. Example

In this section we will show that for three particular points z_{1}, z_{2}, z_{3} in 𝔻^{2} that the interpolation body D(A(𝔻^{2}); z_{1}, z_{2}, z_{3}) is not strongly computable. Proving that D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} is infinitely generated does not rely on any particularly deep theorems in analysis but is rather a straight forward argument. The subtlety lies in determining what form the Schur ideal D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} takes. Theorem 5 below, does exactly that for any fixed set of points z_{1}, ..., z_{k} in 𝔻^{2}, where k is arbitrary. Theorem 5 is a special case of a more general theorem, due to V. Paulsen in [9]. Theorem 5 does even more than tell us what form D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} takes! By using some elementary facts about dual cones and a factorization theorem for polynomials in two complex-variables we will see in section 5 that J. Agler’s bidisk interpolation formula [1] follows as a corollary to Theorem 5 quite naturally. We will begin this section with a proposition about P ick bodies (i.e, D(A(𝔻); α_{1}, ..., α_{k})).

Proposition 4. Let α_{1}, ..., α_{k} ∈ 𝔻. Then

⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ = D(A(𝔻); α_{1}, ..., α_{k})^{⊥} = \{(\frac{1}{1 − ᾱ_{i}α_{j}} )\}^{⊥⊥}.

The proof of Proposition 4 will be omitted. However, the key ingredient in proving Proposition 4 is Pick’s theorem. Note, throughout the remainder of the paper we will often interchange ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩ with D(A(𝔻); α_{1}, ..., α_{k})^{⊥} and vice versa when convenient.

Theorem 5. (Paulsen) Let z_{1} = (α_{1}, β_{1}), ..., z_{k} = (α_{k}, β_{k}) ∈ 𝔻^{2}. Then

D(A(𝔻^{2}); z_{1}, ..., z_{k}) = (⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩)^{⊥} .

A proof of Theorem 5 will be provided in section 5.

Theorem 6. Let z_{1} = (0, 0), z_{2} = (\frac{1}{\sqrt{2}}, 0), z_{3} = (0, \frac{1}{\sqrt{2}}). Then the Schur ideal

D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} = ⟨( \begin{matrix}1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{matrix} )⟩ ⋂ ⟨( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 2 \end{matrix} )⟩

is infinitely generated.

Corollary 7. The set D(A(𝔻^{2}); (0, 0), (\frac{1}{\sqrt{2}}, 0), (0, \frac{1}{\sqrt{2}})) is not strongly computable.

This example exhibits the intrinsic difference between interpolation on 𝔻^{2} and interpolation on 𝔻 (i.e, D(A(𝔻); α_{1}, ..., α_{k})^{⊥} is necessarily finitely generated and D(A(𝔻^{2}); z_{1}, ..., z_{k})^{⊥} is not necessarily finitely generated). However, D(A(𝔻^{2}); (0, 0), (\frac{1}{\sqrt{2}}, 0), (0, \frac{1}{\sqrt{2}})) may be computable since D(A(𝔻^{2}); z_{1}, ..., z_{k}) is necessarily a semi-algebraic set [7]; we leave this issue for future work.

We will need three lemmas in order to prove Theorem 6. The proofs of the first two lemmas will be omitted since they are straight ahead calculations. Throughout the remainder of this section z_{1} = (0, 0), z_{2} = (\frac{1}{\sqrt{2}}, 0), and z_{3} = (0, \frac{1}{\sqrt{2}}).

Lemma 1. If θ ∈ [0, 2π], then S_{θ} ∈ D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} where

S_{θ} = ( \begin{matrix}1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & 1 & \frac{1+e^{iθ}}{3} \\ \frac{1}{\sqrt{3}} & \frac{1+e^{−iθ}}{3} & 1 \end{matrix} ) .

Lemma 2. Let P = (p_{ij})_{i,j=1}^{3} ≥ 0. Let

v_{1} = ( \frac{1−2e^{iθ}}{\sqrt{3}} ; 2e^{iθ} ; −1 ) and v_{2} = ( \frac{2−e^{iθ}}{\sqrt{3}} ; e^{iθ} ; −2 ) , for a fixed θ.

8 JAMES P. SOLAZZO

If ( \begin{matrix}1 & 1 & 1 \\ 1 & 1/2 & 1 \\ 1 & 1 & 1 \end{matrix} ) ∗ P v_{1} = 0 and ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1/2 \end{matrix} ) ∗ P v_{2} = 0, then

P = tS_{θ} where t ≥ 0 and S_{θ} = ( \begin{matrix}1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & 1 & \frac{1+e^{iθ}}{3} \\ \frac{1}{\sqrt{3}} & \frac{1+e^{−iθ}}{3} & 1 \end{matrix} ) .

Lemma 3. Let Q_{1}, ..., Q_{m} ∈ D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} and

S_{θ} = ∑_{i=1}^{m} P_{i} ∗ Q_{i}

where P_{1}, ..., P_{m} ≥ 0. Then P_{i}∗Q_{i} = t_{i}S_{θ} where t_{i} ≥ 0 for i = 1, ..., m.

Proof:

S_{θ} = ∑_{i=1}^{m} P_{i} ∗ Q_{i} ⟹ ( \begin{matrix}1 & 1 & 1 \\ 1 & 1/2 & 1 \\ 1 & 1 & 1 \end{matrix} ) ∗ S_{θ} = ∑_{i=1}^{m} ( \begin{matrix}1 & 1 & 1 \\ 1 & 1/2 & 1 \\ 1 & 1 & 1 \end{matrix} ) ∗ P_{i} ∗ Q_{i}
and ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1/2 \end{matrix} ) ∗ S_{θ} = ∑_{i=1}^{m} ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1/2 \end{matrix} ) ∗ P_{i} ∗ Q_{i}.

Now for i = 1, ..., m we will let

R_{i} = ( \begin{matrix}1 & 1 & 1 \\ 1 & 1/2 & 1 \\ 1 & 1 & 1 \end{matrix} ) ∗ P_{i} ∗ Q_{i} and T_{i} = ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1/2 \end{matrix} ) ∗ P_{i} ∗ Q_{i}.

For i = 1, ..., m each Q_{i} ∈ D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} ⟹ ∃ U_{i} ≥ 0 such that

Q_{i} = U_{i} ∗ ( \begin{matrix}1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{matrix} ) .

Hence, R_{i} = P_{i} ∗ U_{i} ≥ 0, for i = 1, ..., m. Similarly, T_{i} ≥ 0, for i = 1, ...m. Let v_{1}, v_{2} be as in Lemma 2. Then

( \begin{matrix}1 & 1 & 1 \\ 1 & 1/2 & 1 \\ 1 & 1 & 1 \end{matrix} ) ∗ S_{θ}v_{1} = 0 and ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1/2 \end{matrix} ) ∗ S_{θ}v_{2} = 0.

But this implies

(1)  ⟨∑_{i=1}^{m} R_{i}v_{1}, v_{1}⟩ = 0 ⟹ R_{i}v_{1} = 0 for i = 1, ..., m

and

(2)  ⟨∑_{i=1}^{m} T_{i}v_{2}, v_{2}⟩ = 0 ⟹ T_{i}v_{2} = 0 for i = 1, ..., m.

Thus, by Lemma 2 and (1), (2) above we have that P_{i} ∗ Q_{i} = t_{i}S_{θ} and t_{i} ≥ 0 for i = 1, ..., k. Note, that ∑_{i=1}^{m} t_{i} = 1. □

Proof: (Theorem 6) Suppose that D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} is finitely generated. Then there exists Q_{1}, ..., Q_{m} ∈ M_{3}^{+}, such that D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} = ⟨Q_{1}, ..., Q_{m}⟩ . For each θ ∈ [0, 2π] we can choose P_{1}, ..., P_{m} ≥ 0 such that

S_{θ} = ∑_{i=1}^{m} P_{i} ∗ Q_{i}, ⟹ P_{i} ∗ Q_{i} = t_{i}S_{θ}, t_{i} ≥ 0 and ∑_{i=1}^{m} t_{i} = 1,
for i = 1, ..., m (Lemma 3),
⟹ ∃ k ∈ {1, ..., m} such that t_{k} ≠ 0,
⟹ S_{θ} = t_{k}^{−1}P_{k} ∗ Q_{k}.

Therefore, for all θ ∈ [0, 2π], there exists k_{θ} ∈ {1, ..., m} and R_{θ} ≥ 0 such that

S_{θ} = R_{θ} ∗ Q_{k_{θ}}.

Note, if θ ≠ π, then Q_{k_{θ}} has all non-zero entries. Now, define Γ_{k} = {θ : k_{θ} = k} for k = 1, ..., m. Since the Γ_{k′s} cover the unit circle, we can choose k_{o} ∈ {1, ..., m} and {θ_{n}} ⊆ Γ_{k_{o}} such that θ_{n} ⟶ π as n ⟶ ∞ and for each n ∈ ℕ, θ_{n} ≠ π. Thus, for each n ∈ ℕ we can choose R_{n} ≥ 0 such that

S_{θ_{n}} = R_{n} ∗ Q_{k_{o}}.

The fact that there exists an n ∈ ℕ such that θ_{n} ≠ π implies that Q_{k_{o}} = (q_{ij}^{k_{o}}) has all non-zero entries. Thus, we can define

Q̂_{k_{o}} = (\frac{1}{q_{ij}^{k_{o}}} ) .

But for each n ∈ ℕ we have that S_{θ_{n}} = R_{n} ∗ Q_{k_{o}}, this implies for each n ∈ ℕ that R_{n} = Q̂_{k_{o}} ∗ S_{θ_{n}} and

S_{θ_{n}} ⟶ S_{π} as n ⟶ ∞ ⟹ R_{n} ⟶ R_{π} as n ⟶ ∞,
⟹ S_{π} = R_{π} ∗ Q_{k_{o}}.

Since Q_{k_{o}} has all non-zero entries we may assume that

Q_{k_{o}} = ( \begin{matrix}1 & a & b \\ ā & 1 & z \\ b̄ & z̄ & 1 \end{matrix} ) , where a, b, z ∈ ℂ.

But Q_{k_{o}} ∈ D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} implies that the following two matrices

( \begin{matrix}1 & a & b \\ ā & \frac{1}{2} & z \\ b̄ & z̄ & 1 \end{matrix} ) and ( \begin{matrix}1 & a & b \\ ā & 1 & z \\ b̄ & z̄ & \frac{1}{2} \end{matrix} )

are positive semi-definite. Thus, |a|^{2} ≤ \frac{1}{2} and |b|^{2} ≤ \frac{1}{2}. We also have that

S_{π} = ( \begin{matrix}1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & 1 & 0 \\ \frac{1}{\sqrt{3}} & 0 & 1 \end{matrix} ) ⟹ R_{π} = ( \begin{matrix}1 & \frac{1}{a\sqrt{3}} & \frac{1}{b\sqrt{3}} \\ \frac{1}{ā\sqrt{3}} & 1 & 0 \\ \frac{1}{b̄\sqrt{3}} & 0 & 1 \end{matrix} ) .

Thus, we have the following two implications,

(3)  R_{π} ≥ 0 ⟹ 1 − \frac{1}{3|a|^{2}} − \frac{1}{3|b|^{2}} ≥ 0 and
(4)  |a|^{2} ≤ \frac{1}{2} ⟹ \frac{1}{3|a|^{2}} ≥ \frac{2}{3}.

But (3) and (4) imply that |b|^{2} ≥ 1, which contradicts the fact that |b|^{2} ≤ \frac{1}{2}. Hence, D(A(𝔻^{2}); z_{1}, z_{2}, z_{3})^{⊥} is not finitely generated. □

5. Agler’s Bidisk Interpolation Formula

In this section we will first state a factorization theorem for polynomials in two complex variables with norm strictly less than 1, due to D. Blecher and V. Paulsen [4]. This theorem together with the matrixvalued version of Pick’s theorem ,[12], and the fact that D^{⊥⊥} = D for an arbitrary interpolation body will enable us to prove Theorem 5. Again, Theorem 5 is a special case of a more general theorem in [9]. The proof of Theorem 5 is much simpler than the proof of the general theorem and is due to V. Paulsen.

Following the proof of Theorem 5 are two corollaries (9 and 10) which follow from Theorem 5 (not Theorem 8). We will omit the proof of Corollary 9 (V. Paulsen’s bidisk interpolation formula) and before proving Corollary 10 (J. Agler’s bidisk interpolation formula) we will discuss how the two corollaries tell us when we can or can’t interpolate two sets of k-tuples. We will then recall some elementary facts about dual cones and end the paper with an alternate proof of J. Agler’s bidisk interpolation formula, due to V. Paulsen.

Theorem 8. (Blecher-Paulsen) Let p(α, β) be a polynomial in two complex-variables with ∥p∥ = sup{|p(α, β)| : |α|, |β| ≤ 1} < 1. Then p factors as follows:

p(α, β) = F_{1}(α)G_{1}(β)F_{2}(α)G_{2}(β) … F_{m}(α)G_{m}(β)
1 × l_{1} l_{1} × l_{2} l_{2} × l_{3} l_{3} × l_{4} … l_{2m−1} × 1

where ∥F_{i}∥, ∥G_{i}∥ < 1 ∀i and entries of F_{i} and G_{i} are polynomials.

Proof: (Theorem 5) Let D = D(A(𝔻^{2}); z_{1}, ..., z_{k}), where z_{i} = (α_{i}, β_{i}). Looking at functions constant in β implies that D(A(𝔻); α_{1}, ..., α_{k}) ⊆ D which implies that D^{⊥} ⊆ D(A(𝔻); α_{1}, ..., α_{k})^{⊥}. Similarly, D^{⊥} ⊆ D(A(𝔻); β_{1}, ..., β_{k})^{⊥}. Therefore,

D^{⊥} ⊆ D(A(𝔻); α_{1}, ..., α_{k})^{⊥} ∩ D(A(𝔻); β_{1}, ..., β_{k})^{⊥}

which implies that

(⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩)^{⊥} ⊆ D^{⊥⊥} = D.

To show the other containment let Q = (q_{ij}) ∈ D(A(𝔻); α_{1}, ..., α_{k})^{⊥}∩ D(A(𝔻); β_{1}, ..., β_{k})^{⊥} and let p(α, β) be a polynomial such that ∥p∥ < 1. Let w_{i} = p(α_{i}, β_{i}) and suppose ((1 − w̄_{i}w_{j})q_{ij}) ≥ 0. Then we have the following:

((1 − w̄_{i}w_{j})q_{ij}) ≥ 0 ⇒ (q_{ij}) ∈ D^{⊥}
⇒ D(A(𝔻); α_{1}, ..., α_{k})^{⊥} ∩ D(A(𝔻); β_{1}, ..., β_{k})^{⊥}
⊆ D^{⊥}
⇒ D^{⊥⊥} = D
⊆ (D(A(𝔻); α_{1}, ..., α_{k})^{⊥} ∩ D(A(𝔻); β_{1}, ..., β_{k})^{⊥})^{⊥}.

Thus, we need only prove that ((1−w̄_{i}w_{j})q_{ij}) ≥ 0. Assume p(α, β) = F (α)G(β).

1 − w̄_{i}w_{j} = 1 − \overline{p(α_{i}, β_{i})}p(α_{j}, β_{j})
= 1 − G^{∗}(β_{i})F^{∗}(α_{i})F (α_{j})G(β_{j})
= 1 − G^{∗}(β_{i})G(β_{j}) + G^{∗}(β_{i})[I − F^{∗}(α_{i})F (α_{j})]G(β_{j})

Now Schur product (1 − w̄_{i}w_{j}) with (q_{ij}) where (q_{ij}) = (\frac{p_{ij}^{1}}{1−ᾱ_{i}α_{j}} ) or

(q_{ij}) = (\frac{p_{ij}^{2}}{1−β̄_{i}β_{j}} ) where (p_{ij}^{1}), (p_{ij}^{2}) ≥ 0, to get

((1 − w̄_{i}w_{j})q_{ij}) = ((1 − G^{∗}(β_{i})G(β_{j}))q_{ij})

+(G^{∗}(β_{i})[(I − F^{∗}(α_{i})F (α_{j}))q_{ij}]G(β_{j}))

= (\frac{1 − G^{∗}(β_{i})G(β_{j})}{1 − β̄_{i}β_{j}}p_{ij}^{2})

+ (G^{∗}(β_{i}) [\frac{I − F^{∗}(α_{i})F (α_{j})}{1 − ᾱ_{i}α_{j}}p_{ij}^{1}] G(β_{j})) ≥ 0.

To prove the above inequality we are using three separate facts. First, we are using the matrix-valued version of Pick’s theorem [12]. Next, we are using two facts from matrix analysis. First, the Schur product of two positive semi-definite matrices is positive semi-definite and if P is positive semi-definite, then X^{∗}P X is positive semi-definite for X ∈ M_{k,m}, m arbitrary [8]. Now, by induction on p(α_{i}, β_{i}) = F (α_{i})H(α_{i}, β_{i}), where H(α_{i}, β_{i}) has n − 1 factors, we are done. □

Now that we have proven Theorem 5 the following corollaries follow easily.

Corollary 9. (Paulsen) Let z_{1} = (α_{1}, β_{1}), ..., z_{k} = (α_{k}, β_{k}) ∈ 𝔻^{2} and w_{1}, ..., w_{k} ∈ 𝔻. Then (w_{1}, ..., w_{k}) ∈ D(A(𝔻^{2}); z_{1}, ..., z_{k}) if and only if (1 − w̄_{i}w_{j}) ∗ Q ≥ 0 ∀ Q satisfying

(1) (1 − ᾱ_{i}α_{j}) ∗ Q ≥ 0,

(2) (1 − β̄_{i}β_{j}) ∗ Q ≥ 0.

Corollary 10. (Agler) Let z_{1} = (α_{1}, β_{1}), ..., z_{k} = (α_{k}, β_{k}) ∈ 𝔻^{2} and w_{1}, ..., w_{k} ∈ 𝔻. Then (w_{1}, ..., w_{k}) ∈ D(A(𝔻^{2}); z_{1}, ..., z_{k}) if and only if there exists positive semi-definite matrices P and Q such that

(1 − w̄_{i}w_{j}) = ((1 − ᾱ_{i}α_{j})p_{ij}) + ((1 − β̄_{i}β_{j})q_{ij}).

The above two corollaries tell us the following. If one chooses the correct P, Q ≥ 0 satisfying

(1 − w̄_{i}w_{j}) = ((1 − ᾱ_{i}α_{j})p_{ij}) + ((1 − β̄_{i}β_{j})q_{ij}),

then J. Agler’s formula tells us we can interpolate (z_{1}, ..., z_{k}) and (w_{1}, ..., w_{k}). However, one would have to exhaust an infinite number of possible P, Q ≥ 0 to show one can’t interpolate (w_{1}, ..., w_{k}) using Corollary 10.

Conversely, suppose Q satisfies

(1 − ᾱ_{i}α_{j}) ∗ Q ≥ 0 and (1 − β̄_{i}β_{j}) ∗ Q ≥ 0.

If (1 − w̄_{i}w_{j})∗Q is not positive semi-definite, then V. Paulsen’s formula tells us we can’t interpolate (w_{1}, ..., w_{k}). On the other hand, to show that we can interpolate (w_{1}, ..., w_{k}) we would need to verify (1−w̄_{i}w_{j})∗ Q ≥ 0 for all Q satisfying conditions (1) and (2) in Corollary 9. But by Theorem 6 we have that for three particular points in the bidisk that there are infinitely many Q′s satisfying conditions (1) and (2) in Corollary 9.

Before proving Agler’s bidisk interpolation formula (Corollary 10) we need to recall some facts about dual cones.

Given a set S let S^{+} = {y : 0 ≤ y · x ∀ x ∈ S}.

i) S^{+} is a cone ii) S ⊆ S^{++} iii) S^{++} = cone(S) - smallest closed cone containing S (Krein-

Milman).

Theorem 11. Let C_{1} and C_{2} be closed cones. Then

(C_{1} ∩ C_{2})^{+} = (C_{1}^{+} + C_{2}^{+})^{−}.

Consider the space of k × k Hermitian matrices M_{k}^{h} as a real Hilbert space with pairing A ⊙ B = ∑_{i,j=1}^{k} a_{ij}b_{ij} = ⟨A ∗ Be, e⟩ where e is the vector of all ones and ⟨ , ⟩ denotes the usual inner product on ℂ^{k}.

Proposition 12. Let I ⊆ M_{k}^{+} be a Schur ideal. Then I is a cone and

I^{+} = {H ∈ M_{k}^{h} : H ⊙ P ≥ 0 ∀ P ∈ I}
= {H ∈ M_{k}^{h} : H ∗ P ≥ 0 ∀ P ∈ I}.

Moreover, (w_{1}, ..., w_{k}) ∈ I^{⊥} if and only if (1 − w̄_{i}w_{j}) ∈ I^{+}.

We are now able to prove Agler’s bidisk interpolation formula.

Proof: (Corollary 10) Let D = D(A(𝔻^{2}); z_{1}, ..., z_{k}). By Theorem 5

D = (⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩)^{⊥} .

But by Proposition 12

(w_{1}, ..., w_{k}) ∈ D ⇔ (1−w̄_{i}w_{j}) ∈ (⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩)^{+} .

But by Theorem 11

(⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩)^{+} = ⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩^{+}+⟨(\frac{1}{1 − β̄_{i}β_{j}} )⟩^{+} .

Next observe that

⟨(\frac{1}{1 − ᾱ_{i}α_{j}} )⟩^{+} = \{H ∈ M_{k}^{h} : H ∗ (\frac{1}{1 − ᾱ_{i}α_{j}} ) ≥ 0\}
= {(1 − ᾱ_{i}α_{j}) ∗ P : P ≥ 0}.

This completes the proof of Corollary 10.□

References

[1] J. Agler, Interpolation, preprint.

[2] J. Agler and J.E. McCarthy, Nevanlinna-Pick Interpolation on the bidisk., preprint.

[3] J. Ball and T. Trent, Unitary Colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Functional Anal. 157 (1998), 1-61.

[4] D. Blecher and V. Paulsen, Explicit Construction of Universal Operator Algebras and an Application to Polynomial Factorization, Proc. Amer. Math. Soc. 112 (1991), 839-850.

[5] B. Cole, K. Lewis and J. Wermer, Pick Conditions on a Uniform Algebra and von Neumann Inequalities, J. Functional Anal. 107 (1992), 235-254.

[6] B. Cole and J. Wermer, Pick Int, von Neumann Inequalities, and Hyperconvex sets, Complex Potential Theory, P.M Gauthier (ed.), NATO Adv. Sci. Inst. Ser. C 439, Kluwer, Dordrecht, (1994), 89-129.

[7] B. Cole and J. Wermer, Boundaries of Interpolation bodies, Houston J. Math., 23 (1997), 499-527.

[8] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1999.

[9] V.I. Paulsen, Matrix-valued interpolation and hyperconvex sets, Int. Eqn. and Op. Thy., to appear.

[10] V.I. Paulsen, Operator Algebras of Idempotents, J. Functional Anal., to appear.

[11] G. Pick, Ueber die Beschraenkungen analytisher Funktionen, welche durch vorgegebene Funktionswete bewirkt werden, Math.Ann., 77 (1916), 7-23.

[12] D. Sarason, Generalized interpolation in H^{∞}, Trans. Amer. Math. Soc., 127 (1967), 179-203.

[13] J.P. Solazzo, A Uniqueness Result Concerning Schur Ideals, Proc. Amer. Math. Soc. (to appear).

[14] A.T. Tomerlin, Products of Nevanlinna-Pick Kernels and Operator Colligations, Int. Eq. and Op. Thy. 38, 2000, to appear.

Department of Mathematics, University of Georgia, Athens, GA 30602 E-mail address: solazzo@math.uga.edu