Complete Nevanlinna-Pick Kernels

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

Abstract We give a new treatment of Quiggin’s and McCullough’s characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.

0 Introduction

Let X be an infinite set, and k a positive definite kernel function on X, i.e. for any finite collection x_{1}, …, x_{n} of distinct points in X, and any complex numbers {a_{i}}_{i=1}^{n}, the sum

∑_{i,j=1}^{n} a_{i}ā_{j}k(x_{i}, x_{j}) ≥ 0,  (0.1)

with strict inequality unless all the a_{i}’s are 0. For each element x of X, define the function k_{x} on X by k_{x}(y) := k(x, y). Define an inner product on the span of these functions by

⟨∑ a_{i}k_{x_{i}} , ∑ b_{j}k_{y_{j}}⟩ = ∑ a_{i}b̄_{j}k(x_{i}, y_{j}),

and let H = H_{k} be the Hilbert space obtained by completing the space of finite linear combinations of k_{x_{i}}’s with respect to this inner product. The elements of H can be thought of as functions on X, with the value of f at x given by ⟨f, k_{x}⟩.

A multiplier of H is a function ϕ on X with the property that if f is in H, so is ϕf. The Nevanlinna-Pick problem is to determine, given a finite set x_{1}, …, x_{n} in X, and numbers λ_{1}, …, λ_{n}, whether there exists a multiplier ϕ of norm at most one that interpolates the data, i.e. satisfies ϕ(x_{i}) = λ_{i} for i = 1, …, n.

∗Partially supported by the National Science Foundation

†Partially supported by National Science Foundation grant DMS 9531967.

If ϕ is a multiplier of H, we shall let T_{ϕ} denote the operator of multiplication by ϕ. Note that the adjoint of T_{ϕ} satisfies T_{ϕ}^{∗}k_{x} = \overline{ϕ(x)}k_{x}. So if ϕ interpolates the data (x_{i}, λ_{i}), then the n-dimensional space spanned by {k_{x_{i}} : 1 ≤ i ≤ n} is left invariant by T_{ϕ}^{∗}, and on this subspace the operator T_{ϕ}^{∗} is the diagonal

( \overline{λ_{1}} ; ... ; \overline{λ_{n}} )  (0.2)

with respect to the (not necessarily orthonormal) basis {k_{x_{i}}}. For a given set of n data points (x_{1}, λ_{1}), …, (x_{n}, λ_{n}), let R_{x,λ} be the operator in (0.2), i.e. the operator that sends k_{x_{i}} to \overline{λ_{i}}k_{x_{i}}. A necessary condition to solve the Nevanlinna-Pick problem is that the norm of R_{x,λ} be at most 1; the kernel k is called a Nevanlinna-Pick kernel if this necessary condition is also always sufficient.

Notice that R_{x,λ} is a contraction on sp{k_{x_{i}} : 1 ≤ i ≤ n} if and only if (1 − R_{x,λ}^{∗}R_{x,λ}) is positive on that space. As

⟨(1 − R_{x,λ}^{∗}R_{x,λ}) ∑_{i=1}^{n} a_{i}k_{x_{i}}, ∑_{j=1}^{n} a_{j}k_{x_{j}}⟩ = ∑_{i,j=1}^{n} a_{i}\overline{a_{j}}(1 − λ_{j}\overline{λ_{i}})⟨k_{x_{i}}, k_{x_{j}}⟩,

it follows that the contractivity of R_{x,λ} on sp{k_{x_{i}} : 1 ≤ i ≤ n} is equivalent to the positivity of the n-by-n matrix

((1 − λ_{j}\overline{λ_{i}})⟨k_{x_{i}}, k_{x_{j}}⟩)_{i,j=1}^{n} .  (0.3)

The classical Nevanlinna-Pick theorem asserts that the Szegö kernel

k(x, y) = \frac{1}{1 − x̄y}

on the unit disk is a Nevanlinna-Pick kernel. The condition is normally stated in terms of the positivity of (0.3), but as we see that is equivalent to the contractivity of (0.2).

The matrix-valued Nevanlinna-Pick problem is as follows. Fix some auxiliary Hilbert space, which for notational convenience we shall assume to be the finite-dimensional space ℂ^{ν}. The tensor product H ⊗ ℂ^{ν} can be thought of as a space of vector valued functions on X. A multiplier of H⊗ ℂ^{ν} is now a ν-by-ν matrix valued function Φ on X with the property that whenever

( f_{1} ; ⋮ ; f_{ν} ) ∈ H ⊗ ℂ^{ν}

then

Φ ( f_{1} ; ⋮ ; f_{ν} ) ∈ H ⊗ ℂ^{ν}.

The matrix Nevanlinna-Pick problem is to determine, given points x_{1}, …, x_{n} and matrices Λ_{1}, …, Λ_{n}, whether there is a multiplier Φ of norm at most one that interpolates: Φ(x_{i}) = Λ_{i}.

Fix a (not necessarily orthonormal) basis {e^{α}}_{α=1}^{ν} for ℂ^{ν}. As before,

T_{Φ}^{∗}k_{x} ⊗ v = k_{x} ⊗ Φ(x)^{∗}v,

so if M is the span of {k_{x_{i}} ⊗ e^{α} : 1 ≤ i ≤ n, 1 ≤ α ≤ ν}, a necessary condition for the Nevanlinna-Pick problem to have a solution is that the nν-by-nν matrix

R_{x,Λ} : k_{x_{i}} ⊗ e^{α} ↦ k_{x_{i}} ⊗ Λ_{i}^{∗}e^{α}  (0.4)

be a contraction. We shall call the kernel k a complete Nevanlinna-Pick kernel if, for all finite ν and all positive n, the contractivity of R_{x,Λ} is also a sufficient condition to extend Φ to a multiplier of all of H × ℂ^{ν} of norm at most one.

In Section 1 we give a classification of all complete Nevanlinna-Pick kernels. This was originally done by S. McCullough in [7] in the context of the Carathéodory interpolation problem. The Nevanlinna-Pick problem was studied by P. Quiggin, who in [9] established the sufficiency of the condition in Theorem 1.2. Necessity was proved independently by Quiggin in [10] and by McCullough in [8].

In Section 2 we show that if a kernel has the Nevanlinna-Pick property for row vectors of length ν, then it has the Nevanlinna-Pick property for μ-by-ν matrices for all μ. In particular, having the vector-valued Nevanlinna-Pick property is equivalent to having the complete Nevanlinna-Pick property.

In Section 3, we show that all complete Nevanlinna-Pick kernels have the form

k(x, y) = \frac{\overline{δ(x)}δ(y)}{1 − F (x, y)}

where δ is a nowhere vanishing function and F : X × X → 𝔻 is a positive semi-definite function.

In Section 4 we introduce the universal complete Nevanlinna-Pick kernels a_{m} defined on the unit ball 𝔹_{m} of an m-dimensional Hilbert space (m may be infinite) by

a_{m}(x, y) = \frac{1}{1 − ⟨x, y⟩}.

These kernels are universal in the sense that, up to renormalization, every complete Nevanlinna-Pick kernel is just the restriction of an a_{m} to a subset of 𝔹_{m}.

1 Characterization of Complete Nevanlinna-Pick kernels

To simplify notation, we shall let k_{i} denote k_{x_{i}}, and k_{ij} denote ⟨k_{i}, k_{j}⟩ = k(x_{i}, x_{j}). First we want a lemma that says that we can break H up into summands on each of which k is irreducible, i.e. k_{ij} is never 0. For convenience, we shall defer the proof of the lemma until after the proof of the theorem.

Lemma 1.1 Suppose k is a Nevanlinna-Pick kernel on the set X. Then X can be partitioned into disjoint subsets X_{i} such that if two points x and y are in the same set X_{i}, then k(x, y) ≠ 0; and if x and y are in different sets of the partition, then k(x, y) = 0.

A reducible kernel will have the (complete) Nevanlinna-Pick property if and only if each irreducible piece does, so we shall assume k is irreducible.

Theorem 1.2 A necessary and sufficient condition for an irreducible kernel k to be a complete Nevanlinna-Pick kernel is that, for any finite set {x_{1}, …, x_{n}} of n distinct elements of X, the (n − 1)-by-(n − 1) matrix

F_{n} = (1 − \frac{k_{in}k_{nj}}{k_{ij}k_{nn}} )_{i,j=1}^{n−1}  (1.3)

is positive semi-definite.

Proof: Let x_{1}, …, x_{n−1} and Λ_{1}, …, Λ_{n−1} be chosen, let M be the span of {k_{i} ⊗ e^{α} : 1 ≤ i ≤ n − 1, 1 ≤ α ≤ ν}, and define R_{x,Λ} on M by (0.4). The operator R_{x,Λ} is a contraction if and only if I − R_{x,Λ}^{∗}R_{x,Λ} ≥ 0. Calculate

⟨(I − R_{x,Λ}^{∗}R_{x,Λ}) ∑_{i,α} a_{i}^{α}k_{i} ⊗ e^{α} , ∑_{j,β} a_{j}^{β}k_{j} ⊗ e^{β}⟩ = ∑_{i,α,j,β} a_{i}^{α}ā_{j}^{β}k_{ij}(⟨e^{α}, e^{β}⟩ − ⟨Λ_{j}Λ_{i}^{∗}e^{α}, e^{β}⟩) (1.4)

A necessary and sufficient condition to be able to find a matrix Λ_{n} so that the extension R_{x,Λ}^{∼} of R_{x,Λ} that sends k_{x_{n}} ⊗ e^{α} to k_{x_{n}} ⊗ Λ_{n}^{∗}e^{α} for each α has the same norm as R_{x,Λ} is: whenever Λ_{1}, …, Λ_{n−1} are chosen so that

I − R_{x,Λ}^{∗}R_{x,Λ} ≥ 0  (1.5)

on ∨{k_{i} ⊗ e^{α} : 1 ≤ i ≤ n − 1, 1 ≤ α ≤ ν}, then

P − (P R_{x,Λ}^{∼}P)^{∗}(P R_{x,Λ}^{∼}P) ≥ 0,  (1.6)

where P is the orthogonal projection from ∨{k_{i} ⊗ e^{α} : 1 ≤ i ≤ n, 1 ≤ α ≤ ν} onto the orthogonal complement of ∨{k_{n} ⊗e^{α} : 1 ≤ α ≤ ν}. (This was first proved in [1] in the scalar case, and a proof of the matrix case is given in [3]. Notice that (1.6) does not depend on the choice of Λ_{n}. We use ∨ to denote the closed linear span of a set of vectors.)

That (1.5) always implies (1.6) for any choice of x and Λ is not only necessary, but also sufficient for k to be a complete Nevanlinna-Pick kernel. Sufficiency is proved by an inductive argument that if one can always extend a multiplier defined on a finite set to any other point without increasing the norm, then one can extend the multiplier to all of X. In the absence of any a priori simplifying assumptions about the multiplier algebra of H being large, the proof of this inductive argument is subtle, and is originally due to Quiggin [9, Lemma 4.3].

Using the fact that

P (k_{i} ⊗ e^{α}) = (k_{i} − \frac{k_{in}}{k_{nn}} k_{n}) ⊗ e^{α},

we can calculate that

⟨(P − (P R_{x,Λ}^{∼}P)^{∗}(P R_{x,Λ}^{∼}P) ∑_{i,α} a_{i}^{α}k_{i} ⊗ e^{α} , ∑_{j,β} a_{j}^{β}k_{j} ⊗ e^{β}⟩

equals

∑_{i,α,j,β} a_{i}^{α}ā_{j}^{β}k_{ij} (1 − \frac{k_{in}k_{nj}}{k_{ij}k_{nn}} ) [⟨e^{α}, e^{β}⟩ − ⟨Λ_{j}Λ_{i}^{∗}e^{α}, e^{β}⟩]  (1.7)

Comparing (1.4) and (1.7), we see that we want that whenever the matrix whose (i, α)^{th} column and (j, β)^{th} row is given by

k_{ij}(⟨e^{α}, e^{β}⟩ − ⟨Λ_{j}Λ_{i}^{∗}e^{α}, e^{β}⟩)  (1.8)

is positive, then the Schur product of this matrix with F_{n} ⊗ J is positive, where J is the ν-by-ν matrix all of whose entries are 1. As the Schur product of two positive matrices is positive, the positivity of (1.3) is immediately seen to be a sufficient condition for k to be a complete Nevanlinna-Pick kernel.

We shall prove necessity by induction on n. The case n = 2 holds by the Cauchy-Schwarz inequality. So assume that F_{n−1} is positive, and we shall prove that F_{n} is positive.

Note first the sort of matrices that can occur in (1.8). For each i and α, one can choose the vector Λ_{i}^{∗}e^{α} arbitrarily. In particular, let G be any positive (n−1)-by-(n−1) matrix, let ε > 0, and choose {e^{α}} so that ⟨e^{α}, e^{β}⟩ = εδ_{α,β} + 1. Choose vectors v_{i} so that ⟨v_{i}, v_{j}⟩ = G_{ij}. Let ν = n − 1, and choose Λ_{i}^{∗} to be the rank one matrix that sends each e^{α} to v_{i}. Then (1.8) becomes

k_{ij}(εδ_{α,β} + 1 − G_{ij}).  (1.9)

We know that F_{n} has the property that if G is a positive matrix and the (n−1)ν-by-(n−1)ν matrix (1.9) is positive, then the Schur product of F_{n} ⊗J with (1.9) is also positive. Denote by K the (n − 1)-by-(n − 1) matrix whose (i, j) entry is k_{ij}, and let · denote Schur product. By letting ε tend to zero, we get that whenever G ≥ 0 and

[K · (J − G)] ⊗ J ≥ 0,

then

[F_{n} ⊗ J] · ([K · (J − G)] ⊗ J) ≥ 0,

which is the same as saying

K · (J − G) ≥ 0 ⇒ F_{n} · K · (J − G) ≥ 0.  (1.10)

Let L be the rank one positive (n − 1)-by-(n − 1) matrix given by

L_{ij} = \frac{k_{i(n−1)}k_{(n−1)j}}{k_{(n−1)(n−1)}} ,

and let G be the matrix given by

G_{ij} = 1 − \frac{L_{ij}}{k_{ij}} .

Then G is the matrix that agrees with F_{n−1} in the first (n − 2) rows and columns, and all the entries in the (n−1)^{st} row and column are zero. Therefore G is positive by the inductive hypothesis. Moreover, K · (J − G) = L and so is positive. Therefore F_{n} · L is positive. But L is rank one, so 1/L (the matrix of reciprocals) is also positive, and therefore

F_{n} · L · 1/L = F_{n} ≥ 0,

as desired. □

Proof of Lemma 1.1: Let X_{x} = {y : k(x, y) ≠ 0}. We need to show that for any two points x and y, the sets X_{x} and X_{y} are either equal or disjoint. This is equivalent to proving that if k(x, z) ≠ 0 and k(y, z) ≠ 0, then k(x, y) ≠ 0.

Assume this fails. Consider the 2-by-2 matrix T^{∗} defined on the linear span of k_{x} and k_{y} by

T^{∗}k_{x} = k_{x}
T^{∗}k_{y} = −k_{y}

This has norm one, because k(x, y) = 0. By the hypothesis that k is a Nevanlinna-Pick kernel, T^{∗} can be extended to the space spanned by k_{x}, k_{y} and k_{z} so that the new operator has the same norm and has k_{z} as an eigenvector. But for this to hold, from equation (1.7) we would need

( \begin{matrix}0 & 2 \\ 2 & 0 \end{matrix} ) · ( \begin{matrix}k_{xx} − \frac{|k_{xz}|^{2}}{k_{zz}} & k_{xy} − \frac{k_{xz}k_{zy}}{k_{zz}} \\ k_{yx} − \frac{k_{yz}k_{zx}}{k_{zz}} & k_{yy} − \frac{|k_{yz}|^{2}}{k_{zz}} \end{matrix} ) ≥ 0.  (1.11)

But the Schur product of the two matrices in (1.11) is zero on the diagonal, non-zero off the diagonal, and therefore cannot be positive. □

By the same argument as in the theorem, an irreducible kernel will have the (scalar) Nevanlinna-Pick property if and only if whenever G is positive and rank one, (1.10) holds. We do not know how to classify such kernels in the sense of Theorem 1.2.

The positivity of F_{n} can be expressed in other ways. The proof that F_{n} being positive is equivalent to 1/K having only one positive eigenvalue below is due to Quiggin [9].

Corollary 1.12 A necessary and sufficient condition for the irreducible kernel k to have the complete Nevanlinna-Pick property is that for any finite set x_{1}, …, x_{n}, the matrix

H_{n} := (\frac{1}{k_{ij}} )_{i,j=1}^{n}

has exactly one positive eigenvalue (counting multiplicity).

Proof: As all the diagonal entries of H_{n} are positive, H_{n} must have at least one positive eigenvalue.

The condition that F_{n+1} be positive is equivalent to saying

M_{n} := (\frac{k_{n+1,n+1}}{k_{i,n+1}k_{n+1,j}} − \frac{1}{k_{ij}} )_{i,j=1}^{n} ≥ 0,  (1.13)

because k_{i,n+1}k_{n+1,j} is rank one so its reciprocal is positive. But (1.13) says that H_{n} is less than or equal to a rank one positive operator, so has at most one positive eigenvalue.

Conversely, any symmetric matrix

( \begin{matrix}A & B \\ B^{∗} & C \end{matrix} )

with C invertible is congruent to

( \begin{matrix}A − BC^{−1}B^{∗} & 0 \\ 0 & C \end{matrix} ) .

(The top left entry is called the Schur complement of C.) Applying this to H_{n} with C the (n, n) entry, we get that H_{n} is congruent to

( \begin{matrix}−M_{n−1} & 0 \\ 0 & \frac{1}{k_{nn}} \end{matrix} ) .

So if H_{n} has only one positive eigenvalue, −M_{n−1} must be negative semi-definite, and therefore F_{n} must be positive semi-definite. □

As an application of the Corollary, consider the Dirichlet space of holomorphic functions on the unit disk with reproducing kernel k(w, z) = \frac{1}{w̄z} log \frac{1}{1−w̄z}. It is shown in [1] that this is a Nevanlinna-Pick kernel, and in the course of the proof it is established that 1 − 1/k is positive semi-definite (because all the coefficients in the power series are positive). It then follows at once from Corollary 1.12 that the Dirichlet kernel is actually a complete Nevanlinna-Pick kernel.

2 Vector-valued Nevanlinna-Pick kernels

Let M_{μ,ν} denote the μ-by-ν matrices. Let us say that a kernel k has the n-point M_{μ,ν} Nevanlinna-Pick property if, for any points x_{1}, …, x_{n+1}, and any matrices Λ_{1}, …, Λ_{n} in M_{μ,ν}, the operator

T^{∗} : sp{k_{x_{i}} : 1 ≤ i ≤ n} ⊗ ℂ^{μ} → sp{k_{x_{i}} : 1 ≤ i ≤ n} ⊗ ℂ^{ν}
k_{x_{i}} ⊗ v ↦ k_{x_{i}} ⊗ Λ_{i}^{∗}v ∀ v ∈ ℂ^{μ}

has an extension T̃^{∗} from sp{k_{x_{i}} : 1 ≤ i ≤ n + 1} ⊗ ℂ^{μ} to sp{k_{x_{i}} : 1 ≤ i ≤ n + 1} ⊗ ℂ^{ν} that sends k_{x_{n+1}} ⊗ v to k_{x_{n+1}} ⊗ Λ_{n+1}^{∗}v for some Λ_{n+1}, and such that T̃^{∗} has the same norm as T^{∗}. We shall say that k is a vector-valued Nevanlinna-Pick kernel if k has the n point M_{1,ν} Nevanlinna-Pick property for all n and ν.

Theorem 2.1 Let ν ≥ n − 1. Then k has the n-point M_{μ,ν} Nevanlinna-Pick property for some positive integer μ if and only if it has the property for all positive integers μ.

Proof: It is clear that the n-point M_{μ,ν} Nevanlinna-Pick property implies the n-point M_{π,ν} Nevanlinna-Pick property for all π smaller than μ. So it is sufficient to prove that the n-point M_{1,ν} Nevanlinna-Pick property implies the n-point M_{μ,ν} Nevanlinna-Pick property for all μ.

As in the proof of Theorem 1.2, the kernel k has the n-point M_{μ,ν} Nevanlinna-Pick property if and only if the positivity of the matrix

[k_{ij}(⟨e^{α}, e^{β}⟩_{ℂ^{μ}} − ⟨Λ_{j}Λ_{i}^{∗}e^{α}, e^{β}⟩_{ℂ^{ν}})]_{i,j=1;α,β=1}^{i,j=n;α,β=μ}  (2.2)

implies the positivity of the Schur product of (2.2) with F_{n+1} ⊗J_{μ}. Again, as in the proof of Theorem 1.2, this implies that whenever K ·(J_{n}−G) is positive, then so is F_{n+1}·K ·(J_{n}−G), for G any positive n-by-n matrix of rank less than or equal to max(ν, n).

So, if k has the n-point M_{1,ν} Nevanlinna-Pick property, then we can choose G to be the rank (n − 1) matrix used in the proof of Theorem 1.2, and conclude that F_{n+1} has to be positive. But the positivity of F_{n+1} clearly implies that k has the n-point M_{μ,ν} Nevanlinna-Pick property for all values of μ and ν. □

Corollary 2.3 The kernel k is a complete Nevanlinna-Pick kernel if and only if it is a vector-valued Nevanlinna-Pick kernel.

See [3] for another approach to describing M_{ν,ν} Nevanlinna-Pick kernels when there is a distinguished operator (or tuple of operators) acting on H for which all the k_{x}’s are eigenvectors.

3 Representation of Complete Nevanlinna-Pick kernels

It is a consequence of Theorem 1.2 that all complete Nevanlinna-Pick kernels have a very specific form.

Theorem 3.1 The irreducible kernel k on X is a complete Nevanlinna-Pick kernel if and only if there is a positive semi-definite function F : X × X → 𝔻 and a nowhere vanishing function δ on X so that

k(x, y) = \frac{\overline{δ(x)}δ(y)}{1 − F (x, y)}.  (3.2)

Proof: (Sufficiency): If k has the form of (3.2), then 1/k is a rank-one operator minus a positive operator, so has exactly one positive eigenvalue, and the result follows from Corollary 1.12.

(Necessity): Suppose k is a complete Nevanlinna-Pick kernel. Fix any point x_{0} in X. Then the kernel

F (x, y) = 1 − \frac{k(x, x_{0})k(x_{0}, y)}{k(x, y)k(x_{0}, x_{0})}  (3.3)

is positive semi-definite by Theorem 1.2. Let

δ(x) = \frac{k(x_{0}, x)}{\sqrt{k(x_{0}, x_{0})}}.

It is immediate that equation (3.2) is satisfied. As k(x, x) is positive and finite for all x, F (x, x) must always lie in [0, 1); as F (x, y) is a positive semi-definite kernel, it follows that |F (x, y)| < 1 for all x, y. □

Any positive definite kernel k(x, y) can be rescaled by multiplying by a nowhere-vanishing rank-one kernel \overline{δ(x)}δ(y). Let j(x, y) = \overline{δ(x)}δ(y)k(x, y). Then the Hilbert space H_{j} is just a rescaled copy of H_{k}: a function f is in H_{k} if and only if δf is in H_{j}, so H_{j} = δH_{k}. The multipliers of H_{k} and H_{j} are the same, and one space has the complete Nevanlinna-Pick property if and only if the other one does (the matrices F_{n} are identical, as the scaling factors cancel). We shall say that the kernel k is normalized at x_{0} if k(x_{0}, x) = 1 for all

x; this is equivalent to scaling the kernel by \frac{\sqrt{k(x_{0}, x_{0})}}{k(x_{0}, x)}, and means that in (4.1) δ can be

chosen to be one, and F (x, y) becomes 1 − \frac{1}{k(x, y)}.

4 The Universal Complete Nevanlinna-Pick Kernels

It follows from Theorem 3.1 that there is a universal complete Nevanlinna-Pick kernel (actually a family of them, indexed by the cardinal numbers). Let l_{m}^{2} be m-dimensional Hilbert space, where m is any cardinal bigger than or equal to 1. Let 𝔹_{m} be the unit ball in l_{m}^{2}, and define a kernel a_{m} on 𝔹_{m} by

a_{m}(x, y) = \frac{1}{1 − ⟨x, y⟩}  (4.1)

Let H_{m}^{2} be the completion of the linear span of the functions {a_{m}(·, y) : y ∈ 𝔹_{m}}, with inner product defined by ⟨a_{m}(·, y), a_{m}(·, x)⟩ = a_{m}(x, y). We shall show that the spaces H_{m}^{2} are universal complete Nevanlinna-Pick spaces.

Theorem 4.2 Let k be an irreducible kernel on X. Let m be the rank of the Hermitian form F defined by (3.3). Then k is a complete Nevanlinna-Pick kernel if and only if there is an injective function f : X → 𝔹_{m} and a nowhere vanishing function δ on X such that

k(x, y) = \overline{δ(x)}δ(y) a_{m}(f(x), f(y)).  (4.3)

Moreover if this happens, then the map k_{x} ↦ \overline{δ(x)}(a_{m})_{f(x)} extends to an isometric linear embedding of H_{k} into δH_{m}^{2}.

If in addition there is a topology on X so that k is continuous on X × X, then the map f will be a continuous embedding of X into 𝔹_{m}.

Proof: (Sufficiency): Any kernel of the form (4.3) is of the form (3.2).

(Necessity): Suppose k is a complete Nevanlinna-Pick kernel. As F is positive semidefinite, there exists a Hilbert space of dimension m (which we shall take to be l_{m}^{2}) and a map f : X → l_{m}^{2} so that F (x, y) = ⟨f(x), f(y)⟩. Moreover, as F takes value in 𝔻, f actuallly maps into 𝔹_{m}. It now follows from Theorem 3.1 that k has the form (4.3).

The linear map that sends k_{x} to the function \frac{\overline{δ(x)}}{1 − ⟨f(x), ·⟩} is an isometry on ∨\{k_{x} : x ∈ X} by (4.3) and gives the desired embedding.

If f(x) = f(y) then k_{x} = k_{y}; as k is positive definite, this implies x = y.

Finally, f can be realised as the composition of the four maps

x ↦ k_{x}
k_{x} ↦ \overline{δ(x)}a_{m}(f(x), ·)
\overline{δ(y)}a_{m}(y, ·) ↦ a_{m}(y, ·)
a_{m}(y, ·) ↦ y

The fourth map is continuous by direct calculation, the second is an isometry by the theorem, and the first and third maps are continuous if k is continuous. □

Note that if one first normalizes k at some point, δ can be taken to be 1 in Theorem 4.2. For m = 1, the space H_{m}^{2} is the regular Hardy space on the unit disk. For larger m, it is

a Hilbert space of analytic functions on the ball 𝔹_{m}. Thus every reproducing kernel Hilbert space with the complete Nevanlinna-Pick property is a restriction of a space of analytic functions.

It was shown in [2] that the Sobolov space W^{1,2}[0, 1], the functions g on the unit interval for which ∫_{0}^{1} |g|^{2} + |g′|^{2}dx is finite, has the Nevanlinna-Pick property. It follows from [9, Cor. 6.5] that the condition of Corollary 1.12 is satisfied, so the Sobolov space has the complete Nevanlinna-Pick property. We can normalize W^{1,2}[0, 1] at 1 say, by calculating that k_{1}(t) = cosinh(1) cosh(t), and hence δ(t) = \sqrt{sinh(1) cosh(1)}cosech(t). Therefore there is a continuous embedding f : [0, 1] → 𝔹_{ℵ_{0}} so that if g is any function in W^{1,2}[0, 1], then (δ.g) ∘ f^{−1} extends off the curve f([0, 1]) to be analytic on all of 𝔹_{ℵ_{0}} - even though δ.g need not be analytic in any neighborhood of the unit interval on which it is originally defined.

After normalization, every separable reproducing kernel Hilbert space with the complete Nevanlinna-Pick property is the restriction of the space H_{ℵ_{0}}^{2} to a subspace spanned by a set of kernel functions, which is why we call this space universal. The kernel k is just the restriction of a_{ℵ_{0}} to a subset of 𝔹_{ℵ_{0}}.

Let A be a normed algebra of functions on a set X with the complete Nevanlinna-Pick property, i.e. there exists a positive definite function k on X × X such that there is a function f in A ⊗ M_{k} of norm at most one and with f(x_{i}) = Λ_{i} if and only if the nk-by-nk matrix

k(x_{i}, x_{j}) ⊗ [I_{k} − Λ_{i}^{∗}Λ_{j}]

is positive. It is then immediate that A is the multiplier algebra of H_{k}, and k is a complete Nevanlinna-Pick kernel. If H_{k} is separable, k is therefore the restriction of a_{ℵ_{0}} to some subset of 𝔹_{ℵ_{0}}. By the Pick property, every function in A extends to an element of the multiplier algebra of H_{ℵ_{0}}^{2} without increasing the norm. So every separably acting algebra with the complete Nevanlinna-Pick property embeds isometrically in the multiplier algebra of H_{ℵ_{0}}^{2}.

It is probably the universality of the kernel a_{m} which is responsible for the recent surge of interest in it - see e.g. [3, 4, 5, 6].

References

[1] J. Agler. Some interpolation theorems of Nevanlinna-Pick type. Preprint, 1988.

[2] J. Agler. Nevanlinna-Pick interpolation on Sobolov space. Proc. Amer. Math. Soc., 108:341–351, 1990.

[3] J. Agler and J.E. McCarthy. Nevanlinna-Pick kernels and localization. To appear.

[4] A. Arias and G. Popescu. Noncommutative interpolation and Poisson transforms. To appear.

[5] W.B. Arveson. Subalgebras of C*-algebras III: Multivariable operator theory. To appear.

[6] K.R. Davidson and D.R. Pitts. Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. To appear.

[7] S. McCullough. Carathéodory interpolation kernels. Integral Equations and Operator Theory, 15(1):43–71, 1992.

[8] S. McCullough. The local de Branges-Rovnyak construction and complete Nevanlinna-Pick kernels. In Algebraic methods in Operator theory, pages 15–24. Birkhäuser, 1994.

[9] P. Quiggin. For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integral Equations and Operator Theory, 16(2):244–266, 1993.

[10] P. Quiggin. Generalisations of Pick’s Theorem to Reproducing Kernel Hilbert Spaces. PhD thesis, Lancaster University, 1994.