Pick interpolation in several variables

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb{C}^d$.


Introduction
Suppose z 1 , . . . , z n are distinct points in the complex open disk D and w 1 , . . . , w n are complex numbers. The classic Pick interpolation theorem for D states the following: there is a holomorphic function f on D satisfying f (z i ) = w i for i = 1, . . . , n and sup{|f (z)| : z ∈ D} ≤ 1 if and only if the matrix (1 − w i w j )(z i z j ) −1 is positive semidefinite.
It is well known that, in general, the analogue of Pick's theorem does not hold for domains other than the disk. The single matrix present in Pick's theorem is typically replaced with an infinite family of matrices. Abrahamse's interpolation theorem for multiply connected regions [1] was the first appearance of this phenomenon, where the family of matrices are naturally parametrized by a polytorus. Cole, Lewis and Wermer [7] approached the Pick problem in substantial generality by considering the problem for any uniform algebra. They showed that a solution exists when the Pick matrices associated to a large class of measures are all positive semidefinite. In [2], Agler and McCarthy carried out a deep analysis of the Pick problem for the bidisk. Though an infinite family of Pick matrices once again appeared, their simultaneous positivity was reforumulated as a factorization problem for H ∞ (D 2 ). These results also hold for the polydisk, but the associated Pick theorem is not given in terms of the H ∞ (D d ) norm. See [3] and the references therein for a detailed treatment of all of these results.
In this paper, we present two Pick-type theorems in a multivariable setting. For d ≥ 2, the open unit ball and polydisk in C d will be denoted B d and D d , respectively. For a bounded domain Ω ⊂ C d and Lebesgue measure µ on Ω, the Bergman space L 2 a (Ω) is defined as those functions which are analytic on Ω and contained in L 2 (Ω, µ). If Ω is either B d by D d , the Hardy space H 2 (Ω) is defined as the closure of the multivariable analytic polynomials in L 2 (∂Ω, θ), where θ is Lebesgue measure on ∂Ω. The algebra of bounded analytic functions on Ω will be denoted H ∞ (Ω). In Theorem 1.1 below, a Pick theorem for the polydisk and unit ball is obtained. Theorem 1.1 improves upon existing interpolation theorems for these domains by explicitly describing the associated family of Pick matrices in terms of a certain class of absolutely continuous measures. Moreover, no distinction is made between the d = 2 and d > 2 case. In Theorem 1.2 a Pick result for any bounded domain in C d is established using Bergman spaces. As is the case with Theorem 1.1, the associated Pick matrices arise from absolutely continuous measures on Ω. Both theorems are also valid for any weak * -closed subalgebra of H ∞ (Ω), a feature which is typically absent in Pick interpolation results. In both theorems, the symbol k ν refers to a reproducing kernel function described in Section 2.
Suppose Ω is either D d or B d and that z 1 , . . . , z n ∈ Ω and w 1 , . . . , w n ∈ C. Let A be any weak * -closed subalgebra of H ∞ (Ω). There is a function ϕ ∈ A with sup z∈Ω |ϕ(z)| ≤ 1 and ϕ(z i ) = w i for i = 1, . . . , n if and only if the matrix n i,j=1 ≥ 0 is positive semidefinite for every measure of the form ν = |f | 2 θ, where θ is Lebesgue measure on ∂Ω and f ∈ H 2 (Ω).

Theorem 1.2.
Suppose Ω is a bounded domain in C d and that z 1 , . . . , z n ∈ Ω and w 1 , . . . , w n ∈ C. Let A be any weak * -closed subalgebra of H ∞ (Ω). There is a function ϕ ∈ A with sup z∈Ω |ϕ(z)| ≤ 1 and ϕ(z i ) = w i for i = 1, . . . , n if and only if the matrix n i,j=1 ≥ 0 is positive semidefinite for every measure of the form ν = |f | 2 µ, where µ is Lebesgue measure on Ω and f ∈ L 2 a (Ω). Both Theorem 1.1 and Theorem 1.2 provide significant simplifications of the Cole-Lewis-Wermer approach for the algebra H ∞ (Ω) and its subalgebras. Even though an infinite family of kernel functions is still required, they are given a concrete description. These theorems will be established using some recent results of Davidson and the author [8], where a general framework for Pick-type theorems was developed using the theory of dual operator algebras and their preduals. Dual algebra techniques in Pick interpolation can also be seen in the paper of McCullough [10].
In Section 2, we will briefly discuss reproducing kernel Hilbert spaces and their multipliers. A natural notion of equivalence between reproducing kernel Hilbert spaces is introduced and applied to cyclic subspaces of H 2 (Ω) and L 2 a (Ω) in Theorem 2.6. In order to prove the desired results, some dual algebra results of Bercovici-Westood [6] and Prunaru [11] are invoked which, when combined with the other results in Section 2, establish the desired results.

Reproducing kernel Hilbert spaces and their multipliers
We say that a Hilbert space H of C-valued functions on X is a reproducing kernel Hilbert space if point evaluations are continuous. For every x ∈ X, the reproducing kernel at x is the function k x ∈ H which satisfies f (x) = f, k x for f ∈ H. The associated positive definite kernel on X × X is given by k(x, y) := k y , k x . A multiplier ϕ of H is a function on X such that ϕf ∈ H for every f ∈ H. Each multiplier ϕ induces a bounded multiplication operator M ϕ on H. The multiplier algebra of H, denoted M (H), is the operator algebra consisting of all such M ϕ . The multiplier algebra is unital, maximal abelian and closed in the weak operator topology. The adjoints of multiplication operators are characterized by the fundamental identity M * ϕ k x = ϕ(x)k x .
respectively. When Ω is either D d or B d , the multiplier algebra M (H 2 (Ω)) is equal to H ∞ (Ω), and this identification is isometric: If Ω is any bounded domain in C d , the kernel function for the Bergman spaces L 2 a (Ω) is generally difficult to compute. Some familiar examples are given by It is easy to verify that M (L 2 a (Ω) = H ∞ (Ω) and that M ϕ = ϕ ∞ for any ϕ ∈ H ∞ (Ω). See the book of Krantz [9] for detailed treatment of these spaces. This note will concern itself only with reproducing kernel Hilbert spaces of analytic functions. We further assume that H is endowed an L 2 norm, i.e. there is some set ∆ along with a σ-algebra of subsets and measure µ such that H is a closed subspace of L 2 (∆, µ). The set ∆ may play different roles depending on the context. For the Hardy space of the polydisk or unit ball, ∆ is taken to be either ∂D 2 or ∂B d , respectively and µ is the corresponding Lebesgue measure on these sets. For the Bergman space L 2 a (Ω), we simply take ∆ = Ω and µ to be Lebesgue measure on Ω. We also assume that H contains the constant function 1, so that every multiplier of H is contained in H.
Given an algebra of multipliers A on H and a measure ν on ∆, let A 2 (ν) denote the closure of A in L 2 (∆, ν). The measure ν is said to be dominating for X (with respect to A) if A 2 (ν) is a reproducing kernel Hilbert space on X. We will write k A 2 (ν) x for the reproducing kernel on this space, or more simply as k ν x when the context is clear. The associated positive definite kernel function on X × X will be denoted k ν (x, y) := k ν y , k ν x A 2 (ν) . For any such ν, A is obviously an algebra of multipliers on A 2 (ν).
Remark 2.2. Our notion of a dominating measure differs slightly from Cole-Lewis-Wermer [7]. In their setting, A is a uniform algebra and ∆ is the maximal idea space of A. A measure µ on ∆ is said to be dominating for a subset Λ of ∆ if there is a constant C such that |ϕ(λ)| ≤ C ϕ L 2 (µ) for every λ ∈ Λ. Their theorem solves the so-called weak Pick problem for A: Given ǫ > 0, w 1 , . . . , w n ∈ C and λ 1 , . . . , λ n ∈ ∆, there is a function ϕ ∈ A with ϕ(λ i ) = w i for i = 1, . . . , n and ϕ < 1 + ǫ if and only if for every measure µ which is dominating for {λ 1 , . . . , λ n }.
A unital, weak- * closed subalgebra of B(H) will be called a dual algebra. The predual A * of a dual algebra may be identified canonically with a quotient of the trace class operators on H. A dual algebra A is said to have property A 1 (1) if every π ∈ A * with π < 1 may be written as π(A) = Ax, y H for some x, y ∈ H which satisfy x y < 1. See the manuscript of Bercovici, Foiaş and Pearcy [5] for a detailed treatment of dual algebras and the structure of their preduals.
If H is a reproducing kernel Hilbert space and A ⊂ M (H) is a dual algebra, we call A a dual algebra of multipliers. If L is an invariant subspace of A, then L is also a reproducing kernel Hilbert space with the kernel function at x given by P L k x . Clearly A is also a dual algebra of multipliers on L, and we denote the multiplication operator associated to ϕ as M L f . It follows that (M L ϕ ) * P L k x = ϕ(x)P L k x for x ∈ X. Note that if f (x) = 0 for every f ∈ L, then P L k x = 0. Since we are principally concerned with evaluation of multipliers, it is useful to extend the kernel function for L to points which are annihilated by every function in L. If f ∈ H, we denote the closed cyclic subspace generated by A and f as A[f ]. The following lemma appears as Lemma 2.1 in [8].

Lemma 2.3. Suppose A is a dual algebra of multipliers on a reproducing kernel Hilbert space H and let I x denote the ideal of functions in
If L = A[f ] for f ∈ H, we will use the notation k f x and M f ϕ for k L x and M L ϕ , respectively. The following result of Davidson and the author ( [8], Theorem 3.4) gives a Pick theorem for dual algebras of multipliers which have property A 1 (1). A 1 (1). Then the following statement holds: given x 1 , . . . , x n ∈ X and w 1 , . . . , w n ∈ C, there is a multiplier ϕ ∈ A such that ϕ(x i ) = w i and M ϕ ≤ 1 if and only if the matrix

Theorem 2.4. Suppose that H is both a reproducing kernel Hilbert space over a set X and that A is a dual algebra of multipliers on H which has property
When H is contained in an ambient L 2 space, we seek a nicer description of the cyclic subspaces A[f ]. Suppose H and K are reproducing kernel Hilbert spaces on X with kernels k and j, respectively, and that U : H → K is a unitary map. We say that U is a reproducing kernel Hilbert space isomorphism if for every x ∈ X there is a non-zero scalar c x such that U k x = c x j x . We require the following easy proposition, the details of which may be found in ( [3], Section 2.6).
Proposition 2.5. Suppose H and K are reproducing kernel Hilbert spaces on a set X and that U : H → K is a reproducing kernel Hilbert space isomorphism. As sets, the multiplier algebras of H and K are equal. Moreover, the map U induces a unitary equivalence between M (H) and M (K).
The following result shows that cyclic subspaces may naturally be identified with H under a different norm. Recall that if H ⊂ L 2 (∆, µ), A ⊂ M (H) and ν is some measure on ∆ , then A 2 (ν) is the closure of A in L 2 (∆, ν). Theorem 2.6. Suppose H is a reproducing kernel Hilbert space of analytic functions on some bounded domain Ω. Suppose further that there is a measure space (∆, µ) such that H is a closed subspace of L 2 (∆, µ). If A is any dual algebra of multipliers on H, then A 2 (|f | 2 µ) is a reproducing kernel Hilbert space on the set Ω f := {z ∈ Ω : f, k f z H = 0}. The reproducing kernel for A 2 (|f | 2 µ) is given by Moreover, there is a reproducing kernel Hilbert space isomorphism In particular, any measure of the form |f | 2 µ is dominating for Ω f with respect to A.
Proof. Define a linear map V : It is clear by the definition of the norms involved that V extends to a unitary We claim that U := V * is the required isomorphism. For notational convenience, let which proves the theorem.
Corollary 2.7. Suppose A and H satisfy the hypotheses of Theorem 2.6. Then the matrix is positive semidefinite if and only if is positive semidefinite.
Proof. By Theorem 2.6, the matrix [(1 − w i w j )j f (z i , z j )] is the Schur product of , the latter of which is manifestly positive semidefinite.
We can now summarize the results of this section so far.
Theorem 2.8. Let (∆, B, µ) be a σ-finite measure space such that H is both a reproducing kernel Hilbert space over a set X and a closed subspace of L 2 (∆, µ).
Suppose that A is a dual algebra of multipliers on H which has property A 1 (1). Then the following statement holds: given x 1 , . . . , x n ∈ X and w 1 , . . . , w n ∈ C, there is a multiplier ϕ ∈ A such that ϕ(x i ) = w i and M ϕ ≤ 1 if and only if the matrix is positive semidefinite for every measure of the form ν = |f | 2 µ where f ∈ H.
Proof. If such a ϕ exists, then M f ϕ is a contraction for every f ∈ H. It follows that the operator I − M f ϕ (M f ϕ ) * is positive semidefinite. Taking an inner product against finite spans of the functions k f z implies that the matrices are positive semidefinite. Now apply Corollary 2.7 and note that k |f | 2 µ = j f . The nontrivial direction is obtained by combining Theorem 2.4 and Corollary 2.7.
We will use the following important factorization result of Bercovici-Westwood ( [6], Theorem 1). Recall that θ is Lebesgue measure on ∂Ω.
Property A 1 (1) is hereditary for weak- * closed subspaces ( [5], Proposition 2.04), and so we may combine the above result with Theorem 2.8, which proves Theorem 1.1. In order to prove Theorem 1.2, we employ a versatile factorization theorem of Punraru ( [11], Theorem 4.1). This result applies to any instance where H is a reproducing kernel Hilbert space on a measure space (X, B, µ) and H is a closed subspace of L 2 (X, µ). In particular, it applies to any Bergman space, but not to Hardy space. Any reproducing kernel Hilbert space which satisfies these hypothesis always has the property that M ϕ = sup x∈X |ϕ(x)| for any multiplier ϕ. Theorem 2.10 (Prunaru). Let (X, B, µ) be a σ-finite measure space such that H is a reproducing kernel Hilbert space over X and a closed subspace of L 2 (X, µ) Then the multiplier algebra M(H) has property A 1 (1).
In fact, Prunaru shows that M (H) satisfies a much stronger predual factorization property known as X(0, 1). In the particular case where H is a Bergman space, Bercovici proves a result similar to Theorem 2.10 in [4]. An application of Theorem 2.8 gives us what we need.
Corollary 2.11. Let (X, B, µ) be a σ-finite measure space such that H is both a reproducing kernel Hilbert space over X and a closed subspace of L 2 (X, µ). Suppose A is a dual algebra of multipliers on H. If x 1 , . . . , x n ∈ X and w 1 , . . . , w n ∈ C, then there is a function ϕ ∈ A with ϕ ∞ ≤ 1 and ϕ(x i ) = w i for i = 1, . . . , n if and only if [(1 − w i w j )k ν (x i , x j )] n i,j=1 ≥ 0 for every measure of the form ν = |f | 2 µ, for f ∈ H. Theorem 1.2 now follows from Corollary 2.11 by taking X to be a bounded domain Ω and setting H = L 2 a (Ω).