Commutant lifting and Nevanlinna-Pick interpolation in several variables

TL;DR

This paper extends commutant lifting and Nevanlinna-Pick interpolation theorems to multipliers between vector-valued Drury-Arveson space and various reproducing kernel Hilbert spaces over the unit ball, including Hardy and Bergman spaces.

Contribution

It establishes a unified framework for commutant lifting and interpolation in several variables for a broad class of Hilbert spaces.

Findings

Generalized commutant lifting theorem for vector-valued spaces

Nevenlinna-Pick interpolation results in several variables

Applicable to Hardy, Bergman, and weighted Bergman spaces

Abstract

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

Equations

$$\phi (z_i)=w_i,$$

$${\left[\begin{array}{c}\frac{1-w_i{\bar{w}}_j}{1-z_i{\bar{z}}_j}\end{array}\right]}_{i,j=1}^n,$$

$$k_m(\mathbf{z},\mathbf{w})=(1-\sum _{i=1}^n{z}_i{\bar{w}}_i{)}^{-m}(\mathbf{z},\mathbf{w}\in {\mathbb{B}}^n),$$

$$X(P_{{\mathcal{Q}}_1}{M}_{z_i}{|}_{{\mathcal{Q}}_1})=(P_{{\mathcal{Q}}_2}{M}_{z_i}{|}_{{\mathcal{Q}}_2})X,$$

$$X=P_{{\mathcal{Q}}_2}{M}_{\phi }{|}_{{\mathcal{Q}}_1}.$$

$$H_n^2\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\phi ({\mathbf{z}}_i)=w_i,$$

$${\left[\begin{array}{c}\frac{1}{(1-⟨{\mathbf{z}}_i,{\mathbf{z}}_j⟩{)}^m}-\frac{w_i{\bar{w}}_j}{1-⟨{\mathbf{z}}_i,{\mathbf{z}}_j⟩}\end{array}\right]}_{i,j=1}^n,$$

$$k_1(\mathbf{z},\mathbf{w})=(1-\sum _{i=1}^n{z}_i{\bar{w}}_i{)}^{-1}(\mathbf{z},\mathbf{w}\in {\mathbb{B}}^n).$$

$$k(\mathbf{z},\mathbf{w})=k_1(\mathbf{z},\mathbf{w})\tilde{k}(\mathbf{z},\mathbf{w})(\mathbf{z},\mathbf{w}\in {\mathbb{B}}^n).$$

$$(M_{z_i}f)(\mathbf{w})=w_{i}f(\mathbf{w}),$$

$$\sum _{i=1}^n{M}_{z_i}{M}_{z_i}^{\ast }\le I_{{\mathcal{H}}_k}.$$

$${\mathbb{B}}^n\times {\mathbb{B}}^n\ni (\mathbf{z},\mathbf{w})\mapsto k(\mathbf{z},\mathbf{w})I_{\mathcal{E}}.$$

$${\mathbb{B}}^n\times {\mathbb{B}}^n\ni (\mathbf{z},\mathbf{w})\mapsto k_1(\mathbf{z},\mathbf{w})I_{\mathcal{E}}.$$

$$k_{\lambda }(\mathbf{z},\mathbf{w})=(1-\sum _{i=1}^n{z}_i{\bar{w}}_i{)}^{-\lambda }(\mathbf{z},\mathbf{w}\in {\mathbb{B}}^n).$$

$$\Pi T_i^{\ast }=(M_{z_i}\otimes I_{{\mathcal{E}}_{\ast }}{)}^{\ast }\Pi ,$$

$${\Pi }_k:{\mathcal{H}}_k\otimes \mathcal{E}\to H_n^2\otimes {\mathcal{E}}_{\ast },$$

$${\Pi }_k(M_{z_i}\otimes I_{\mathcal{E}}{)}^{\ast }=(M_{z_i}\otimes I_{{\mathcal{E}}_{\ast }}{)}^{\ast }{\Pi }_k,$$

$$i_{\mathcal{Q}}:\mathcal{Q}\hookrightarrow \mathcal{H}.$$

$$i_{\mathcal{Q}}{i}_{\mathcal{Q}}^{\ast }=P_{\mathcal{Q}}.$$

$$(M_{z_i}\otimes I_{\mathcal{E}}{)}^{\ast }\mathcal{Q}\subseteq \mathcal{Q}(i=1,\dots ,n).$$

$$X(P_{{\mathcal{Q}}_1}{M}_{z_i}{|}_{{\mathcal{Q}}_1})=(P_{{\mathcal{Q}}_2}{M}_{z_i}{|}_{{\mathcal{Q}}_2})X,$$

$$\Phi ({\mathcal{H}}_{k_1}\otimes {\mathcal{E}}_1)\subseteq {\mathcal{H}}_{k_2}\otimes {\mathcal{E}}_2.$$

$${\mathcal{M}}_1(H_n^2\otimes {\mathcal{E}}_1,{\mathcal{H}}_k\otimes {\mathcal{E}}_2)=\{\Phi \in \mathcal{M}(H_n^2\otimes {\mathcal{E}}_1,{\mathcal{H}}_k\otimes {\mathcal{E}}_2):\Vert M_{\Phi }\Vert \le 1\}.$$

$$X(M_{z_i}\otimes I_{{\mathcal{E}}_1})=(M_{z_i}\otimes I_{{\mathcal{E}}_2})X,$$

$$XT=\hat{T}X.$$

$$\mathcal{Q}=\Pi \mathcal{H}\text{mbox}and\hat{\mathcal{Q}}=\hat{\Pi }\hat{\mathcal{H}},$$

$$\tilde{X}=\hat{\Pi }X{\Pi }^{\ast }{|}_{\mathcal{Q}},$$

$$\tilde{X}(P_{\mathcal{Q}}V{|}_{\mathcal{Q}})=(P_{\hat{\mathcal{Q}}}\hat{V}{|}_{\hat{\mathcal{Q}}})\tilde{X}.$$

$$\tilde{X}=(\hat{\Pi }{\hat{\Pi }}^{\ast })\hat{\Pi }X{\Pi }^{\ast }{|}_{\mathcal{Q}}=P_{\hat{\mathcal{Q}}}(\hat{\Pi }X{\Pi }^{\ast }){|}_{\mathcal{Q}},$$

Full text

1. Introduction

Let $\mathbb{D}=\{z\in\mathbb{C}:\displaystyle|z|<1\}$ be the open unit disc in the complex plane $\mathbb{C}$. The classical Nevanlinna–Pick interpolation theorem [13, 15] states: Given distinct $n$ points $\{z_{i}\}_{i=1}^{n}\subseteq\mathbb{D}$ (initial data) and $n$ points $\{w_{i}\}_{i=1}^{n}\subseteq\mathbb{D}$ (target data), there exists a $\varphi\in H^{\infty}(\mathbb{D})$ such that $\|\varphi\|_{\infty}\leq 1$ and such that

[TABLE]

for all $i=1,\ldots,n$, if and only if the Pick matrix

[TABLE]

is positive semi-definite. Here we denote by $H^{\infty}(\mathbb{D})$ the Banach algebra of all bounded analytic functions on $\mathbb{D}$ equipped with the norm $\|\varphi\|=sup\{|\varphi(z)|:z\in\mathbb{D}\}$, $\varphi\in H^{\infty}(\mathbb{D})$. In his seminal paper [16], Sarason proved the commutant lifting theorem for compressions of the shift operator to shift co-invariant subspaces of the Hardy space which gives a simpler and elegant proof of the Nevanlinna–Pick interpolation theorem.

Sarason’s approach to the commutant lifting theorem, along with its direct application to Nevanlinna–Pick interpolation theorem, is deeply connected with a number of classical problems in function theory and operator theory and have been studied extensively in the past few decades (cf. [9]). There also has been a great deal of interest in analyzing the possibilities of commutant lifting theorem and interpolation (and other related problems) in the setting of general reporducing kernel Hilbert spaces over domains in $\mathbb{C}^{n}$, $n\geq 1$ (for instance, see [4], [8], [2], [5] and [7]).

In this paper we make a contribution to a commutant lifting theorem and a version of Nevanlinna-Pick interpolation interpolation in several variables. To be more precise, let $m\geq 1$ and let $\mathcal{H}_{m}$ denotes the reproducing kernel Hilbert space corresponding to the kernel $k_{m}$ on $\mathbb{B}^{n}$, where

[TABLE]

and $\mathbb{B}^{n}=\{\bm{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}:\displaystyle\sum_{i=1}^{n}|z_{i}|^{2}<1\}$. Recall that $\mathcal{H}_{m}$ is the Drury-Arveson space (popularly denoted by $H^{2}_{n}$), the Hardy space, the Bergman space and the weighted Bergman space over $\mathbb{B}^{n}$ for $m=1$, $m=n$, $m=n+1$ and $m>n+1$, respectively.

Our main results, restricted to $\mathcal{H}_{m}$, $m>1$, can now be formulated as follows:

Commutant lifting theorem (Theorem 3.4): Suppose $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are joint $(M_{z_{1}},\ldots,M_{z_{n}})$ co-invariant subspaces of $H^{2}_{n}(=\mathcal{H}_{1})$ and $\mathcal{H}_{m}$, respectively. Let $X\in\mathcal{B}(\mathcal{Q}_{1},\mathcal{Q}_{2})$ and $\|X\|\leq 1$. If

[TABLE]

for all $i=1,\ldots,n$, then there exists a holomorphic function $\varphi:\mathbb{B}^{n}\rightarrow\mathbb{C}$ such that the multiplication operator $M_{\varphi}\in\mathcal{B}(H^{2}_{n},\mathcal{H}_{m})$, $\|M_{\varphi}\|\leq 1$ (that is, $\varphi$ is a contractive multiplier), and

[TABLE]

Thus, we have the following commutative diagram:

[TABLE]

Given a closed subspace $\mathcal{S}$ of a Hilbert space $\mathcal{H}$ we denote by $P_{\mathcal{S}}$ the orthogonal projection of $\mathcal{S}$ on $\mathcal{H}$.

Nevanlinna–Pick interpolation theorem (Theorem 5.1): Given distinct $n$ points $\{\bm{z}_{i}\}_{i=1}^{n}\subseteq\mathbb{B}^{n}$ and $n$ points $\{w_{i}\}_{i=1}^{n}\subseteq\mathbb{D}$, there exists a contractive multiplier $\varphi$ such that

[TABLE]

for all $i=1,\ldots,n$ if and only if the matrix

[TABLE]

is positive semi-definite. Here $\langle\bm{z},\bm{w}\rangle=\displaystyle\sum_{i=1}^{n}z_{i}\bar{w}_{i}$ for all $\bm{z},\bm{w}\in\mathbb{C}^{n}$.

We make strong use of the commutant lifting theorem in the setting of Drury-Arveson space (see Theorem 2.2) and a refined factorization result (see Theorem 4.2) concerning multipliers between Drury-Arveson space and a large class of analytic reproducing kernel Hilbert space over $\mathbb{B}^{n}$.

The remainder of the paper is organized as follows. Section 2 discusses some useful and known facts about reproducing kernel Hilbert spaces. Section 3 presents the commutant lifting theorem. Section 4 is devoted to factorizations of multipliers. The factorization results obtained here may be of independent interest. Section 5 provides the interpolation theorem.

2. Preliminaries

The Drury-Arveson space over the unit ball $\mathbb{B}^{n}$ in $\mathbb{C}^{n}$ will be denoted by $H^{2}_{n}$. Recall that $H^{2}_{n}$ is a reproducing kernel Hilbert space corresponding to the kernel function

[TABLE]

Let $k:\mathbb{B}^{n}\times\mathbb{B}^{n}\rightarrow\mathbb{C}$ be a kernel such that $k$ is analytic in the first variables $\{z_{1},\ldots,z_{n}\}$. We say that $k$ is regular if there exists a kernel $\tilde{k}:\mathbb{B}^{n}\times\mathbb{B}^{n}\rightarrow\mathbb{C}$, analytic in $\{z_{1},\ldots,z_{n}\}$, such that

[TABLE]

If $k$ is a regular kernel, then $\mathcal{H}_{k}$, the reproducing kernel Hilbert space corresponding to the kernel $k$, will be referred as a regular reproducing kernel Hilbert space.

In the case of a regular reproducing kernel Hilbert space $\mathcal{H}_{k}$, it follows [11] that $M_{z_{i}}$, the multiplication operator by the coordinate function $z_{i}$, is bounded. Note that

[TABLE]

for all $f\in\mathcal{H}_{k}$, $\bm{w}\in\mathbb{B}^{n}$ and $i=1,\ldots,n$. Moreover, it also follows that the commuting tuple $(M_{z_{1}},\ldots,M_{z_{n}})$ on $\mathcal{H}_{k}$ is a row contraction, that is

[TABLE]

If $\mathcal{E}$ is a Hilbert space, then we also say that $\mathcal{H}_{k}\otimes\mathcal{E}$ is a regular reproducing kernel Hilbert space. Note that the kernel function of $\mathcal{H}_{k}\otimes\mathcal{E}$ is given by

[TABLE]

The $\mathcal{E}$-valued Drury-Arveson space, denoted by $H^{2}_{n}(\mathcal{E})$, is the reproducing kernel Hilbert space corresponding to the $\mathcal{B}(\mathcal{E})$-valued kernel function

[TABLE]

To simplify the notation, we often identify $H^{2}_{n}(\mathcal{E})$ with $H^{2}_{n}\otimes\mathcal{E}$ via the unitary map defined by $z^{\bm{k}}\eta\mapsto z^{\bm{k}}\otimes\eta$ for all $\bm{k}\in\mathbb{Z}_{+}^{n}$ and $\eta\in\mathcal{E}$. This also enable us to identify $(M_{z_{1}},\ldots,M_{z_{n}})$ on $H^{2}_{n}(\mathcal{E})$ with $(M_{z_{1}}\otimes I_{\mathcal{E}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}})$ on $H^{2}_{n}\otimes\mathcal{E}$.

Typical examples of regular reproducing kernel Hilbert spaces arise from weighted Bergman spaces over $\mathbb{B}^{n}$. More specifically, let $\lambda>1$, and let

[TABLE]

Then $\mathcal{H}_{k_{\lambda}}$ is a regular reproducing kernel Hilbert space. Note that $\mathcal{H}_{k_{\lambda}}$ is the Hardy space, Bergman space and weighted Bergman space for $\lambda=n$, $n+1$ and $n+1+\alpha$ for any $\alpha>0$, respectively.

Suppose $\mathcal{H}$ and $\mathcal{E}_{*}$ are Hilbert spaces and $(T_{1},\ldots,T_{n})$ is a commuting tuple of bounded linear operators on $\mathcal{H}$. We say that $(T_{1},\ldots,T_{n})$ on $\mathcal{H}$ dilates to $(M_{z_{1}}\otimes I_{\mathcal{E}_{*}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}_{*}})$ on $H^{2}_{n}\otimes\mathcal{E}_{*}$ if there exists an isometry $\Pi:\mathcal{H}\rightarrow H^{2}_{n}\otimes\mathcal{E}_{*}$ such that

[TABLE]

for all $i=1,\ldots,n$ (cf. [17]). We often say that $\Pi:\mathcal{H}\rightarrow H^{2}_{n}\otimes\mathcal{E}_{*}$ is a dilation of $(T_{1},\ldots,T_{n})$.

If $\mathcal{H}=\mathcal{H}_{k}$ is a regular reproducing kernel Hilbert space, then by [Theorem 6.1, [11]], it follows that $(M_{z_{1}}\otimes I_{\mathcal{E}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}})$ on $\mathcal{H}_{k}\otimes\mathcal{E}$ dilates to $(M_{z_{1}}\otimes I_{\mathcal{E}_{*}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}_{*}})$ on $H^{2}_{n}\otimes\mathcal{E}_{*}$ for some Hilbert space $\mathcal{E}_{*}$. More specifically:

Theorem 2.1**.**

Let $\mathcal{E}$ be a Hilbert space. If $\mathcal{H}_{k}$ is a regular reproducing kernel Hilbert space, then there exist a Hilbert space $\mathcal{E}_{*}$ and an isometry

[TABLE]

such that

[TABLE]

for all $i=1,\ldots,n$.

Since $(M_{z_{1}}\otimes I_{\mathcal{E}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}})$ on $\mathcal{H}_{k}\otimes\mathcal{E}$ is a pure row contraction [11], the above result also directly follows from Muller-Vasilescu [12] and Arveson [3].

In what follows, given a Hilbert space $\mathcal{H}$ and a closed subspace $\mathcal{Q}$ of $\mathcal{H}$, we will denote by $i_{\mathcal{Q}}$ the inclusion map

[TABLE]

Note that $i_{\mathcal{Q}}$ is an isometry and

[TABLE]

We now recall the commutant lifting theorem in the setting of the Drury-Arveson space (see [2] or Theorem 5.1, page 118, [5]). A closed subspace $\mathcal{Q}$ of a regular reproducing kernel Hilbert space $\mathcal{H}_{k}\otimes\mathcal{E}$ is said to be shift co-invariant if

[TABLE]

Theorem 2.2**.**

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be Hilbert spaces. Suppose $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are shift co-invariant subspaces of $H^{2}_{n}(\mathcal{E}_{1})$ and $H^{2}_{n}(\mathcal{E}_{2})$, respectively, $X\in\mathcal{B}(\mathcal{Q}_{1},\mathcal{Q}_{2})$ and let $\|X\|\leq 1$. If

[TABLE]

for all $i=1,\ldots,n$, then there exists a multiplier $\Phi\in\mathcal{M}(H^{2}_{n}(\mathcal{E}_{1}),H^{2}_{n}(\mathcal{E}_{2}))$ such that $\|M_{\Phi}\|\leq 1$ and $P_{\mathcal{Q}_{2}}M_{\Phi}|_{\mathcal{Q}_{1}}=X$.

Recall also that, given regular reproducing kernel Hilbert spaces $\mathcal{H}_{k_{1}}\otimes\mathcal{E}_{1}$ and $\mathcal{H}_{k_{2}}\otimes\mathcal{E}_{2}$, a function $\Phi:\mathbb{B}^{n}\rightarrow\mathcal{B}(\mathcal{E}_{1},\mathcal{E}_{2})$ is called a multiplier from $\mathcal{H}_{k_{1}}\otimes\mathcal{E}_{1}$ to $\mathcal{H}_{k_{2}}\otimes\mathcal{E}_{2}$ if

[TABLE]

The multiplier space $\mathcal{M}(\mathcal{H}_{k_{1}}\otimes\mathcal{E}_{1},\mathcal{H}_{k_{2}}\otimes\mathcal{E}_{2})$ is the set of all multipliers from $\mathcal{H}_{k_{1}}\otimes\mathcal{E}_{1}$ to $\mathcal{H}_{k_{2}}\otimes\mathcal{E}_{2}$. In what follows, $\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ will denote the closed ball of radius one:

[TABLE]

We have the following useful characterization of multipliers (cf. Proposition 4.2, [17]): Let $\mathcal{H}_{k}$ be a regular reproducing kernel Hilbert space, and let $X\in\mathcal{B}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$. Then

[TABLE]

if and only if $X=M_{\Phi}$ for some $\Phi\in\mathcal{M}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$.

3. Commutant lifting theorem

We begin with a general result concerning intertwiner of bounded linear operators.

Lemma 3.1**.**

Suppose $\Pi:\mathcal{H}\rightarrow\mathcal{K}$ and $\hat{\Pi}:\hat{\mathcal{H}}\rightarrow\hat{\mathcal{K}}$ are isometries, $V\ \in\mathcal{B}(\mathcal{K})$, $\hat{V}\in\mathcal{B}(\hat{\mathcal{K}})$, $T=\Pi^{*}V\Pi$ and $\hat{T}=\hat{\Pi}^{*}\hat{V}\hat{\Pi}$. Moreover, let $X\in\mathcal{B}(\mathcal{H},\hat{\mathcal{H}})$ satisfies

[TABLE]

If we define

[TABLE]

and

[TABLE]

then $\tilde{X}\in\mathcal{B}(\mathcal{Q},\hat{\mathcal{Q}})$ and

[TABLE]

Proof. Notice that $P_{\mathcal{Q}}=\Pi\Pi^{*}$ and $P_{\hat{\mathcal{Q}}}=\hat{\Pi}\hat{\Pi}^{*}$. Hence

[TABLE]

and in particular

[TABLE]

which shows that $\tilde{X}\in\mathcal{B}(\mathcal{Q},\hat{\mathcal{Q}})$. Moreover

[TABLE]

which completes the proof. ∎

Now we are ready to prove a variation, in terms of dilations, of Theorem 2.2.

Theorem 3.2**.**

Let $\mathcal{H}$ and $\hat{\mathcal{H}}$ be Hilbert spaces. Suppose $T=(T_{1},\ldots,T_{n})$ and $\hat{T}=(\hat{T}_{1},\ldots,\hat{T}_{n})$ are commuting tuples on $\mathcal{H}$ and $\hat{\mathcal{H}}$, respectively, $X\in\mathcal{B}(\mathcal{H},\hat{\mathcal{H}})$, $\|X\|\leq 1$, and

[TABLE]

for all $i=1,\ldots,n$. If $\Pi:\mathcal{H}\rightarrow H^{2}_{n}\otimes\mathcal{E}$ and $\hat{\Pi}:\hat{\mathcal{H}}\rightarrow H^{2}_{n}\otimes\hat{\mathcal{E}}$ are dilations of $T$ and $\hat{T}$, respectively, then there exists a multiplier $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E},H^{2}_{n}\otimes\hat{\mathcal{E}})$ such that

[TABLE]

Proof.

Let

[TABLE]

If

[TABLE]

then by Lemma 3.1, it follows that $\tilde{X}\in\mathcal{B}(\mathcal{Q},\hat{\mathcal{Q}})$ and

[TABLE]

for all $i=1,\ldots,n$. It then follows from the commutant lifting theorem, Theorem 2.2, that

[TABLE]

for some $\Phi\in\mathcal{M}(H^{2}_{n}\otimes\mathcal{E},H^{2}_{n}\otimes\hat{\mathcal{E}})$ and $\|M_{\Phi}\|\leq 1$. Then

[TABLE]

It then follows from

[TABLE]

that

[TABLE]

Thus

[TABLE]

and hence ${X}=\hat{\Pi}^{*}M_{\Phi}\Pi$. ∎

Now let $\mathcal{Q}$ be a shift co-invariant subspace of $\mathcal{H}_{k}\otimes\mathcal{E}$. An isometry $\Pi:\mathcal{Q}\rightarrow H^{2}_{n}\otimes\mathcal{E}_{*}$ is said to be a dilation of $\mathcal{Q}$ if

[TABLE]

for all $i=1,\ldots,n$, that is $(P_{\mathcal{Q}}M_{z_{1}}|_{\mathcal{Q}},\ldots,P_{\mathcal{Q}}M_{z_{n}}|_{\mathcal{Q}})$ on $\mathcal{Q}$ dilates to $(M_{z_{1}}\otimes I_{\mathcal{E}_{*}},\ldots,M_{z_{n}}\otimes I_{\mathcal{E}_{*}})$ on $H^{2}_{n}\otimes\mathcal{E}_{*}$ via the isometry $\Pi$.

Lemma 3.3**.**

Let $\mathcal{H}_{k}$ be a regular reproducing kernel Hilbert space, and let $\mathcal{E}$ and $\mathcal{E}_{*}$ be a Hilbert spaces. Suppose $\mathcal{Q}$ is a shift co-invariant subspace of $\mathcal{H}_{k}\otimes\mathcal{E}$. If $\Pi:\mathcal{H}_{k}\otimes\mathcal{E}\rightarrow H^{2}_{n}\otimes\mathcal{E}_{*}$ is a dilation of $\mathcal{H}_{k}\otimes\mathcal{E}$, then $\Pi_{\mathcal{Q}}:\mathcal{Q}\rightarrow H^{2}_{n}\otimes\mathcal{E}_{*}$, defined by

[TABLE]

is a dilation $\mathcal{Q}$.

Proof.

We first observe that

[TABLE]

Now we compute

[TABLE]

Now

[TABLE]

and so

[TABLE]

for all $i=1,\ldots,n$. This completes the proof of the lemma. ∎

We are now ready to present and prove the commutant lifting theorem.

Theorem 3.4**.**

Let $\mathcal{H}_{k}$ be a regular reproducing kernel Hilbert space, $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be Hilbert spaces, and let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be shift co-invariant subspaces of $H^{2}_{n}\otimes\mathcal{E}_{1}$ and $\mathcal{H}_{k}\otimes\mathcal{E}_{2}$, respectively. Let $X\in\mathcal{B}(\mathcal{Q}_{1},\mathcal{Q}_{2})$, and suppose that $\|X\|\leq 1$ and

[TABLE]

for all $i=1,\ldots,n$. Then there exists a multiplier $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ such that

[TABLE]

Proof.

Observe that the inclusion map $i_{\mathcal{Q}_{1}}:\mathcal{Q}_{1}\hookrightarrow H^{2}_{n}\otimes\mathcal{E}_{1}$ is a dilation of $\mathcal{Q}_{1}$. Let $\Pi_{k}:\mathcal{H}_{k}\otimes\mathcal{E}_{2}\rightarrow H^{2}_{n}\otimes\hat{\mathcal{E}}$ be a dilation of $\mathcal{H}_{k}$ (see Theorem 2.1), that is, $\Pi_{k}$ is an isometry and

[TABLE]

for all $i=1,\ldots,n$ and some Hilbert space $\hat{\mathcal{E}}$. Set

[TABLE]

By Lemma 3.3, it follows that $\Pi_{\mathcal{Q}_{2}}:\mathcal{Q}_{2}\rightarrow H^{2}_{n}\otimes\hat{\mathcal{E}}$ is a dilation of $\mathcal{Q}_{2}$. Then Theorem 3.2 yields

[TABLE]

for some multiplier $\Phi_{1}\in\mathcal{M}(H^{2}_{n}\otimes\mathcal{E}_{1},H^{2}_{n}\otimes\hat{\mathcal{E}})$. Hence

[TABLE]

Since

[TABLE]

we have, using also the adjoint of (3.1),

[TABLE]

for all $i=1,\ldots,n$, that is, $\Pi_{k}^{*}M_{\Phi_{1}}:H^{2}_{n}\otimes\mathcal{E}_{1}\rightarrow\mathcal{H}_{k}\otimes\mathcal{E}_{2}$ intertwines the shifts. Consequently

[TABLE]

for some multiplier $\Phi\in\mathcal{M}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$. Hence

[TABLE]

and thus

[TABLE]

Hence, we have

[TABLE]

Finally

[TABLE]

completes the proof of the theorem. ∎

A simpler way of presenting the above theorem, from Hilbert module point of view, is to say that the following diagram commutes:

[TABLE]

4. Factorizations

Let $k$ be a regular reproducing kernel on $\mathbb{B}^{n}$. Then there exists a positive definite kernel $\tilde{k}:\mathbb{B}^{n}\times\mathbb{B}^{n}\rightarrow\mathbb{C}$ such that

[TABLE]

Let $\mathcal{H}_{\tilde{k}}$ be the reproducing kernel Hilbert space corresponding to the kernel $\tilde{k}$. Suppose $\bm{w}\in\mathbb{B}^{n}$ and $ev(\bm{w}):\mathcal{H}_{\tilde{k}}\rightarrow\mathbb{C}$ is the evaluation map, that is

[TABLE]

Then

[TABLE]

and so

[TABLE]

From Corollary 4.2 in [11] it follows that the map

[TABLE]

defines a coisometry from $H^{2}_{n}\otimes\mathcal{H}_{\tilde{k}}=\mathcal{H}_{k_{1}}\otimes\mathcal{H}_{\tilde{k}}$ to $\mathcal{H}_{k}=\mathcal{H}_{k_{1}\tilde{k}}$. If we view $H^{2}_{n}\otimes\mathcal{H}_{\tilde{k}}$ as a reproducing kernel Hilbert space of functions with values in $\mathcal{H}_{\tilde{k}}$, then the map $\pi$ is actually the multiplier $M_{ev}$; indeed, if we compute the action on reproducing kernels, we have

[TABLE]

This formula may be extended by tensorizing with $I_{\mathcal{E}}$, where $\mathcal{E}$ is a Hilbert space. If we define $\Psi_{k}:\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}\to\mathcal{E}$ by $\Psi_{k}:=ev\otimes I_{\mathcal{E}}$, then $\Psi_{k}$ is obviously also a coisometric multiplier. Taking into account (4.1), we obtain the following theorem (see also [10, Theorem 4.1] and [11, Theorem 6.2]):

Theorem 4.1**.**

Let $k:\mathbb{B}^{n}\times\mathbb{B}^{n}\rightarrow\mathbb{C}$ be a regular kernel, and let

[TABLE]

for some kernel $\tilde{k}$ on $\mathbb{B}^{n}$. Suppose $\mathcal{H}_{\tilde{k}}$ is the reproducing kernel Hilbert space corresponding to the kernel $\tilde{k}$. If $\mathcal{E}$ is a Hilbert space, then there exists a co-isometric multiplier $\Psi_{k}\in\mathcal{M}(H^{2}_{n}\otimes(\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}),\mathcal{H}_{k}\otimes\mathcal{E})$ such that

[TABLE]

In particular, it is instructive to consider the familiar case: weighted Bergman spaces over $\mathbb{B}^{n}$. Let $m>1$ be an integer and let

[TABLE]

Then

[TABLE]

and hence $\Psi_{k_{m}}(\bm{w})^{*}:\mathcal{E}\rightarrow\mathcal{H}_{k_{m-1}}\otimes\mathcal{E}$ is given by

[TABLE]

for all $\bm{z},\bm{w}\in\mathbb{B}^{n}$ and $\eta\in\mathcal{E}$. Note also that

[TABLE]

for all $f\in\mathcal{H}_{k_{m-1}}$, $\eta,\zeta\in\mathcal{E}$ and $\bm{w}\in\mathbb{B}^{n}$.

For this particular case, the representation of $\Psi_{k_{m}}$ has been computed explicitly in Section 4 of [6] and [4].

Now suppose $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are Hilbert spaces, and $k$ is a regular kernel on $\mathbb{B}^{n}$. Let $\Theta:\mathbb{B}^{n}\rightarrow\mathcal{B}(\mathcal{E}_{1},\mathcal{E}_{2})$ be an analytic function. From [14, Theorem 6.28] it follows that $\Theta\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ if and only if

[TABLE]

is a positive definite kernel. By virtue of Theorem 4.1, this is equivalent to positive definiteness of the kernel

[TABLE]

We may then apply [1, Theorem 8.57($(i)\Rightarrow(ii)$)] to obtain the following theorem.

Theorem 4.2**.**

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be Hilbert spaces, and let $\Theta:\mathbb{B}^{n}\rightarrow\mathcal{B}(\mathcal{E}_{1},\mathcal{E}_{2})$ be an analytic function. In the setting of Theorem 4.1, the following conditions are equivalent:

(i) $\Theta\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$,

(ii) there exists $\tilde{\Theta}\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},H^{2}_{n}\otimes(\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}_{2}))$ such that

[TABLE]

More specifically, the multiplier $\Psi_{k}$ makes the following diagram commutative:

H^{2}_{n}\otimes\mathcal{E}_{1}$$M_{\Theta}$$\mathcal{H}_{k}\otimes\mathcal{E}_{2}$$H^{2}_{n}\otimes(\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}_{2})$$M_{\tilde{\Theta}}$$M_{\Psi_{k}}

One should compare Theorems 4.1 and 4.2 with Lemma 4.1 and Theorem 4.2 in [6] and Theorem 2.1 in [4].

5. Nevanlinna-Pick interpolation

We now turn to the interpolation problem. Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be Hilbert spaces. We denote by $\mathcal{B}_{1}(\mathcal{E}_{1},\mathcal{E}_{2})$ the open unit ball of $\mathcal{B}(\mathcal{E}_{1},\mathcal{E}_{2})$, that is

[TABLE]

We aim to solve the following version of Pick-type interpolation problem: Suppose $\{\bm{z}_{i}\}_{i=1}^{m}\subseteq\mathbb{B}^{n}$, $\{W_{i}\}_{i=1}^{m}\subseteq\mathcal{B}_{1}(\mathcal{E}_{1},\mathcal{E}_{2})$ and $m\geq 1$. Find necessary and sufficient conditions (on $\{\bm{z}_{i}\}_{i=1}^{m}$ and $\{W_{i}\}_{i=1}^{m}$) for the existence of a multiplier $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ such that

[TABLE]

for all $i=1,\ldots,m$.

Given such data $\{\bm{z}_{i}\}_{i=1}^{m}\subseteq\mathbb{B}^{n}$, $\{W_{i}\}_{i=1}^{m}\subseteq\mathcal{B}_{1}(\mathcal{E}_{1},\mathcal{E}_{2})$, set

[TABLE]

and

[TABLE]

Obviously $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are shift co-invariant subspaces of $H^{2}_{n}\otimes\mathcal{E}_{1}$ and $\mathcal{H}_{k}\otimes\mathcal{E}_{2}$, respectively. Define $X:\mathcal{Q}_{2}\rightarrow\mathcal{Q}_{1}$ by

[TABLE]

for all $i=1,\ldots,m$ and $\eta\in\mathcal{E}_{2}$. Then

[TABLE]

for all $i=1,\ldots,n$. Then, by Theorem 3.4, $X$ is a contraction if and only if there exists $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ such that

[TABLE]

In particular

[TABLE]

for all $\eta\in\mathcal{E}_{2}$ and $i=1,\ldots,m$, and so $\Phi$ satisfies (5.1). Conversely, if $\Phi$ satisfies (5.1), then it is easy to see that $X$ defines a contractions from $\mathcal{Q}_{2}$ to $\mathcal{Q}_{1}$.

Now $X$ is a contraction if and only if

[TABLE]

for all $\eta_{1},\ldots,\eta_{m}\in\mathcal{E}_{2}$, where the last equality follows from Theorem 4.1.

On the other hand, Theorem 4.2 says that $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ if and only if there exists $\tilde{\Phi}\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},H^{2}_{n}\otimes(\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}_{2}))$ such that

[TABLE]

for all $\bm{z}\in\mathbb{B}^{n}$. Summarizing, we have established the following interpolation theorem:

Theorem 5.1**.**

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be Hilbert spaces, $k$ be a regular kernel on $\mathbb{B}^{n}$, and let

[TABLE]

for some kernel $\tilde{k}$ on $\mathbb{B}^{n}$. Suppose $\{\bm{z}_{i}\}_{i=1}^{m}\subseteq\mathbb{B}^{n}$ and $\{W_{i}\}_{i=1}^{m}\subseteq\mathcal{B}_{1}(\mathcal{E}_{1},\mathcal{E}_{2})$. Then the following conditions are equivalent:

(i) There exists a multiplier $\Phi\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},\mathcal{H}_{k}\otimes\mathcal{E}_{2})$ such that $\Phi(\bm{z}_{i})=W_{i}$ for all $i=1,\ldots,m$.

(ii) \displaystyle\sum_{1\leq i,j\leq m}\langle\Big{(}k(\bm{z}_{i},\bm{z}_{j})I_{\mathcal{E}_{2}}-\frac{W_{i}W^{*}_{j}}{1-\langle\bm{z}_{i},\bm{z}_{j}\rangle}\Big{)}\eta_{j},\eta_{i}\rangle for all $\eta_{1},\ldots,\eta_{m}\in\mathcal{E}_{2}$.

(iii) There exists a multiplier $\tilde{\Phi}\in\mathcal{M}_{1}(H^{2}_{n}\otimes\mathcal{E}_{1},H^{2}_{n}\otimes(\mathcal{H}_{\tilde{k}}\otimes\mathcal{E}_{2}))$ such that

[TABLE]

If $n=1$ and $\tilde{k}(z,w)=(1-z\bar{w})^{-m}$, $m\in\mathbb{N}$ (that is, weighted Bergman space over $\mathbb{D}$ with an integer weight), then a part of Theorem 5.1 was proved by Ball and Bolotnikov [4].

Note that, the positivity condition in part (ii) of Theorem 5.1 does not hold in general:

Example: Consider the regular kernel $k$ as the Bergman kernel on $\mathbb{D}$, that is

[TABLE]

Here

[TABLE]

Then, for a given pair of points $\{w_{1},w_{2}\}\subseteq\mathbb{D}$, condition (ii) in Theorem 5.1 holds for some pair $\{z_{1},z_{2}\}\subseteq\mathbb{D}$ if and only if

[TABLE]

where $`\diamond$’ denotes the Schur product of matrices. However, if $z_{1}=w_{2}=0$ and $z_{2}\neq 0$, then it is easy to see that the positivity condition fails to hold for any $w_{1}\in\mathbb{D}$ such that

[TABLE]

Acknowledgement: The research of the second named author is supported by NBHM (National Board of Higher Mathematics, India) post-doctoral fellowship no: 0204/27-

/2019/R&D-II/12966. The research of the third named author is supported in part by NBHM grant NBHM/R.P.64/2014, and the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2017/000522 and Core Research Grant, File No: CRG/2019/000908, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. The third author also would like to thanks the Institute of Mathematics of the Romanian Academy, Bucharest, Romania, for its warm hospitality during a visit in May 2019.