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1 Introduction↩︎

Interpolation by bounded analytic functions on the unit disc \(\mathbb{D} \subseteq \mathbb{C}\) has long been a central topic in complex analysis and operator theory. Given distinct points \(z_1,\ldots, z_p\in \mathbb{D}\) and scalars \(w_1,\ldots, w_p \in \mathbb{D},\) the classical Nevanlinna-Pick interpolation problem asks for necessary and sufficient condition for the existence of a holomorphic function \(\varphi: \mathbb{D} \to \mathbb{C}\) satisfying \[\sup_{z\in \mathbb{D}} |\varphi(z)|\leq 1, \;\;\text{and}\;\;\varphi(z_i)=w_i\;\; \text{for all}\;\; i=1,\ldots, p.\] The celebrated theorem of Nevanlinna and independently Pick asserts that such a function exists if and only if the associated \(p\times p\) matrix, called the Pick matrix, given by \[\begin{align} \label{pick32matrix} \left(\frac{1-w_i \bar{w}_j}{1-z_i \bar{z}_j} \right)_{i,j=1}^p, \end{align}\tag{1}\] is positive semi-definite.

After nearly four decades, Donald Sarason introduced an operator-theoretic approach to the interpolation problem in [1]. His result, now known as the commutant lifting theorem, may be formulated as follows. Let \(H^2(\mathbb{D})\) be the Hardy space on \(\mathbb{D}\), and let \(T_{z}\) denote the multiplication operator on \(H^2(\mathbb{D})\) by the coordinate function \(z\). A closed subspace \(\mathcal{Q}\) of \(H^2(\mathbb{D})\) is called a quotient module if \(T_{z}^* \mathcal{Q}\subseteq \mathcal{Q}.\) Let \(P_{\mathcal{Q}}\) be the orthogonal projection of \(H^2(\mathbb{D})\) onto \(\mathcal{Q}.\) Sarason’s theorem states that if \(X\) is a bounded linear operator on \(H^2(\mathbb{D})\) satisfying \(X P_{\mathcal{Q}} T_{z}|_{\mathcal{Q}}= P_{\mathcal{Q}} T_z |_{\mathcal{Q}}X\), then there exists a bounded holomorphic function \(\varphi\) on \(\mathbb{D}\) such that \[X=P_{\mathcal{Q}} T_{\varphi}|_{\mathcal{Q}} \;\; \text{and}\;\; \|X\|=\|\varphi\|_{\infty},\] where \(T_{\varphi}\) denotes the multiplication operator by \(\varphi\) on \(H^2(\mathbb{D}).\) As an application, Sarason recovered the Nevanlinna–Pick interpolation theorem.

The multivariable counterparts of both the interpolation and commutant lifting have since attracted considerable attention. In contrast to the disc, interpolation on higher dimensional domains is generally not governed by a single Pick-type matrix. The interpolation problem on the bidisc was solved by Agler and McCarthy in [2]. The same problem on the symmetrized bidisc \(\mathbb{G}_2\) has been resolved recently in [3], [4]. There have also been several efforts to address interpolation and commutant lifting problems in general multivariable settings; see, for instance, [5]–[7]. However, many of these results apply only to restricted classes of functions or operators. More recently, a comprehensive characterization of both interpolation and commutant lifting for the polydisc \(\mathbb{D}^d\) and the Euclidean unit ball \(\mathbb{B}_d\) have been obtained in [8], [9].

The purpose of this paper is to develop commutant lifting and interpolation results on a broad class of domains that arise as the image of a bounded symmetric domain under a proper holomorphic map factored by a finite complex reflection group.

Recall that a bounded domain \(\Omega\subseteq \mathbb{C}^d\) is symmetric if, for every pair of points \(\boldsymbol{z}, \boldsymbol{z}'\in \Omega,\) there exists an involution in the group of biholomorphic automorphisms \(\mathrm{Aut}(\Omega)\) interchanging them. In their standard realization, bounded symmetric domains are circled, convex, and contain the origin. The open unit disc \(\mathbb{D}\), the unit polydisc \(\mathbb{D}^d\), and the Euclidean ball \(\mathbb{B}_d\) in \(\mathbb{C}^d\) are some well studied examples of bounded symmetric domains. For a masterful exposition on bounded symmetric domains, the reader is referred to [10], [11].

Given bounded domains \(\Omega_1,\Omega_2\subseteq \mathbb{C}^d\), a holomorphic map \(\pi: \Omega_1\to \Omega_2\) is said to be proper if the preimage of every compact subset of \(\Omega_2\) is compact in \(\Omega_1.\) Such a map is necessarily surjective, and it follows from [12] that there exists a positive integer \(n\) (referred to as multiplicity of \(\pi\)) such that \[\text{cardinality of } \pi^{-1}(\boldsymbol{w}) \begin{cases} =n, \qquad \boldsymbol{w}\in \Omega_2\setminus\pi(N(J_\pi))\; \text{and}\\ < n, \qquad \boldsymbol{w}\in \pi(N(J_\pi)), \end{cases}\] where \(J_\pi\) is the complex Jacobian determinant of \(\pi\), and \(N(J_\pi)\) denotes its zero set. We say \(\pi\) is factored by automorphisms if there exists a finite subgroup \(G\subseteq \mathrm{Aut}(\Omega_1)\) such that \(\pi^{-1}(\pi(\boldsymbol{z}))=\{\sigma(\boldsymbol{z}): \sigma\in G\}\) for all \(\boldsymbol{z}\in \Omega_1.\) It is known that such a group \(G\) is either a group generated by complex reflections or conjugate to a complex reflection group, where a complex reflection is a finite order linear map \(\sigma\) on \(\mathbb{C}^d\) such that the rank of \(I_d-\sigma\) is \(1\). A group \(G\) generated by finite number of complex reflections is referred as finite complex reflection group.

A bounded domain \(\Omega\) in \(\mathbb{C}^d\) is said to be \(G\)-invariant if \(\sigma\cdot\boldsymbol{z}:=\sigma^{-1}(\boldsymbol{z})\in \Omega\) for all \(\sigma\in G.\) This induces an action on function \(f:\Omega \to \mathbb{C}\) defined by \((\sigma\cdot f)(\boldsymbol{z}):=f(\sigma^{-1}\cdot \boldsymbol{z})\), and we say \(f\) is \(G\)-invariant if \(\sigma \cdot f=f\) for all \(\sigma\in G.\) A fundamental characterization of finite complex reflection groups is that the ring of \(G\)-invariant polynomials is generated by a set of \(d\) algebraically independent homogeneous polynomials (known as the basic polynomials) \(\{\theta_1,\ldots, \theta_d\}\) associated to \(G\) [13]. The mapping \(\boldsymbol{\theta}=(\theta_1,\ldots, \theta_d): \Omega \to \boldsymbol{\theta}(\Omega)\) is proper holomorphic and factored by \(G\) [12], [14]. Any other proper holomorphic map factored by \(G\) factors through \(\boldsymbol{\theta}\) via a biholomorphism. Consequently, we consider \(\boldsymbol{\theta}(\Omega)\) as the image of \(\Omega\) under a proper holomorphic map factored by \(G.\) Although the map \(\boldsymbol{\theta}\) is not unique, the degrees of \(\theta_j\)’s are unique for \(G\) up to order. Throughout this paper, \(\Omega\) denotes a bounded symmetric domain, \(G\) a finite complex reflection group acting on \(\mathbb{C}^d\) under which \(\Omega\) is invariant, and \(\boldsymbol{\theta}(\Omega)\) the associated quotient domain, with typical elements denoted by \(\boldsymbol{\zeta}=(\zeta_1,\ldots, \zeta_d)=\boldsymbol{\theta}(\boldsymbol{z}).\)

Quotient domains have become an active subject of research in function theory and operator theory; see, for instance, [14]–[21], as well as [22] and the references therein. A suitable framework for Hardy spaces on quotient domains, together with the associated theory of Toeplitz operators, was developed in [22] (see also [16]). For our convenience, we recall in Section 2 those aspects of the Hardy space construction that are relevant to the present work.

Let \(H^2(\boldsymbol{\theta}(\Omega))\) denote the Hardy space on the quotient domain \(\boldsymbol{\theta}(\Omega),\) and let \(\boldsymbol{T}_{\boldsymbol{\zeta}}=(T_{\zeta_1}, \ldots, T_{\zeta_d})\) be the commuting tuple of multiplication operators on \(H^2(\boldsymbol{\theta}(\Omega))\) induced by the coordinate functions. We denote by \(H^\infty(\boldsymbol{\theta}(\Omega))\) the Banach algebra of bounded holomorphic functions on \(\boldsymbol{\theta}(\Omega),\) endowed with the supremum norm \(\|\cdot\|_{\infty, \boldsymbol{\theta}(\Omega)}.\) The elements of the closed unit ball \(\mathcal{S}(\boldsymbol{\theta}(\Omega))\) of \(H^\infty(\boldsymbol{\theta}(\Omega))\) are referred to as Schur functions. For any \(\varphi \in H^\infty(\boldsymbol{\theta}(\Omega))\), the analytic Toeplitz operator \(T_{\varphi}\) on \(H^2(\boldsymbol{\theta}(\Omega))\) is the multiplication operator \[T_{\varphi}:H^2(\boldsymbol{\theta}(\Omega)) \to H^2(\boldsymbol{\theta}(\Omega)), \;\; T_\varphi f=\varphi f.\] Moreover, \(\|T_{\varphi}\|=\|\varphi\|_{\infty, \boldsymbol{\theta}(\Omega)}:=\sup\{|\varphi(\boldsymbol{\zeta})|: \boldsymbol{\zeta}\in \boldsymbol{\theta}(\Omega)\}.\) Let \(\mathcal{Q}\) be a quotient module of \(H^2(\boldsymbol{\theta}(\Omega)),\) that is, a closed subspace invariant under \(T_{\zeta_j}^*\) for each \(j=1,\ldots,d.\) Let \(P_{\mathcal{Q}}\) denote the orthogonal projection of \(H^2(\boldsymbol{\theta}(\Omega))\) onto \(\mathcal{Q}.\) For any \(\varphi\in H^\infty(\boldsymbol{\theta}(\Omega))\), the compression \(S_{\varphi}:=P_{\mathcal{Q}} T_{\varphi}|_{\mathcal{Q}}\) defines a bounded operator on \(\mathcal{Q}.\) In particular, \(S_{\zeta_j}=P_{\mathcal{Q}} T_{\zeta_j}|_{\mathcal{Q}}\in \mathcal{B}(\mathcal{Q})\) for all \(j=1,\ldots,d.\) It readily follows that \(S_{\varphi }\) commutes with \(S_{\zeta_j}\) for all \(j,\) and that \(\|S_{\varphi}\| \leq 1\) whenever \(\varphi \in \mathcal{S}(\boldsymbol{\theta}(\Omega)).\) Motivated by this observation, we define a module map as any operator \(X\in \mathcal{B}(\mathcal{Q})\) satisfying \(XS_{\zeta_j}=S_{\zeta_j}X\) for all \(j=1,\ldots,d.\) Thus every Schur function \(\varphi\in \mathcal{S}(\boldsymbol{\theta}(\Omega))\) induces a contractive module map \(S_{\varphi}\) on \(\mathcal{Q}\). Let \(\mathcal{B}_1(\mathcal{Q})=\{X \in \mathcal{B}(\mathcal{Q}): \|X\|\leq 1\}.\) We say that a contractive module map \(X\in \mathcal{B}_1(\mathcal{Q})\) is liftable or admits a lift if there exists a function \(\varphi \in \mathcal{S}(\boldsymbol{\theta}(\Omega))\) such that \(X=S_\varphi.\)

The main objective of this paper is twofold. First, we establish several equivalent characterizations for the existence of Schur class lifts of contractive module maps on quotient modules of \(H^2(\boldsymbol{\theta}(\Omega))\). These results are developed in Section 3. In the special case of the quotient domain \(\boldsymbol{\theta}(\mathbb{D}^d)\), arising from the polydisc \(\mathbb{D}^d,\) we obtain a commutant lifting criterion expressed in terms of inner functions on \(\boldsymbol{\theta}(\mathbb{D}^d)\) (see Theorem 8).

Secondly, in Section 4, we apply these commutant lifting characterizations to interpolation problems. More precisely, for prescribed distinct points in \(\boldsymbol{\theta}(\Omega)\) and prescribed target values in the unit disc, we derive criteria for the existence of a Schur class interpolant in \(\mathcal{S}(\boldsymbol{\theta}(\Omega)).\) The symmetrized bidisc \(\mathbb{G}_2\) and the tetrablock \(\mathbb{E}\), which arise in connection with \(\mu\)-synthesis, are notable examples of quotient domains that have attracted considerable interest in geometric function theory and operator theory. The general results obtained in this paper apply in particular to these domains and therefore yield interpolation criteria on \(\mathbb{G}_2\) and \(\mathbb{E}.\)

2 Preliminaries↩︎

In this section, we recall the Hardy space on the quotient domain \(\boldsymbol{\theta}(\Omega)\) and describe its boundary realization. Recall that \(\boldsymbol{\theta}=(\theta_1,\ldots, \theta_d):\Omega \to \boldsymbol{\theta}(\Omega)\) is the proper holomorphic map factored by the finite complex reflection group \(G.\) We write \(\partial\Omega\) and \(\partial\boldsymbol{\theta}(\Omega)\) for the Shilov boundaries of \(\Omega\) and \(\boldsymbol{\theta}(\Omega)\), respectively. Since the quotient map \(\boldsymbol{\theta}\) extends as a proper holomorphic map of the same multiplicity to suitable neighborhoods of the corresponding closures, it follows from [23] that \(\partial\boldsymbol{\theta}(\Omega)=\boldsymbol{\theta}(\partial\Omega).\)

Let \(\mathbb{K}=\{\phi\in \mathrm{Aut}(\Omega): \phi(0)=0\}\) be the isotropy subgroup at the origin, and let \(\mu\) denote the normalized \(\mathbb{K}\)-invariant Lebesgue measure on \(\partial\Omega.\) Then the Hardy space \(H^2(\Omega)\) consists of all holomorphic functions \(f:\Omega\to \mathbb{C}\) such that \[\|f\|_{H^2(\Omega)}^2=\sup_{0<r<1} \int_{\partial\Omega} |f(r\boldsymbol{z})|^2d\mu(z) <\infty.\] Let \(L^2(\partial\Omega)\) denote the space of square-integrable functions with respect to \(\mu.\) By [24], every \(f\in H^2(\Omega)\) has a boundary value function \(f^* \in L^2(\partial\Omega)\) satisfying \[\lim_{r \to 1^-} \int_{\partial\Omega}|f_r-f^*|^2 d\mu =0,\quad \text{and} \quad \|f\|_{H^2(\Omega)}=\|f^*\|_{L^2(\partial\Omega)},\] where the radial slice \(f_r:\partial\Omega \to \mathbb{C}\) is given by \(f_r(\boldsymbol{z})=f(r\boldsymbol{z})\) for \(0<r<1\) (see also [25]). Thus, the boundary value map \(i: H^2(\Omega) \to L^2(\partial\Omega)\), defined by \(i(f)=f^*\), embeds \(H^2(\Omega)\) isometrically into \(L^2(\partial\Omega)\). We denote its range by \(H^2(\partial\Omega)\).

The space \(H^2(\Omega)\) is a reproducing kernel Hilbert space. Its reproducing kernel, denoted by \(\mathbb{S}_\Omega\), is the Szegö kernel of \(\Omega.\) The corresponding Poisson kernel \(P:\Omega \times \partial\Omega \to \mathbb{C}\) is given by \[P( \boldsymbol{z}, \boldsymbol{\xi})=\frac{|\mathbb{S}_\Omega(\boldsymbol{z}, \boldsymbol{\xi})|^2}{\mathbb{S}_\Omega(\boldsymbol{z},\boldsymbol{z})}, \qquad \boldsymbol{z}\in \Omega,\; \boldsymbol{\xi} \in \partial\Omega.\] By [26] (see also [27]), any \(f\in H^2(\Omega)\) can be recovered from its boundary value \(f^* \in L^2(\partial\Omega)\) via the Poisson integral formula: \[f(\boldsymbol{z})=\int_{\partial\Omega} P(\boldsymbol{z}, \boldsymbol{\xi}) f^*(\boldsymbol{\xi})d\mu(\boldsymbol{\xi}), \qquad \boldsymbol{z}\in \Omega.\] Moreover, \(f_r\) converges to \(f^*\) almost everywhere on \(\partial\Omega\) (see [28]), that is, \[f^*(\boldsymbol{z})=\lim_{r \to 1^-} f(r\boldsymbol{z})\qquad \text{for a.e}\; \boldsymbol{z}\in \partial \Omega.\]

We now recall the Hardy space on the quotient domain. Using the change of variables formula, define a measure \(\mu_{\boldsymbol{\theta}}\) supported on \(\partial \boldsymbol{\theta}(\Omega)\) by \[\int_{\partial\boldsymbol{\theta} (\Omega)} f d\mu_{\boldsymbol{\theta}}= \int_{\partial\Omega} (f\circ \boldsymbol{\theta}) |J_{\boldsymbol{\theta}}|^2 d\mu,\] where \(J_{\boldsymbol{\theta}}\) denotes the complex Jacobian determinant of the proper holomorphic map \(\boldsymbol{\theta}: \Omega \to \boldsymbol{\theta} (\Omega).\) The Hardy space \(H^2(\boldsymbol{\theta}(\Omega))\) is then defined as the Hilbert space of holomorphic functions \(f\) on \(\boldsymbol{\theta}(\Omega)\) for which \(J_{\boldsymbol{\theta}} (f\circ \boldsymbol{\theta})\in H^2(\Omega)\). Furthermore, the norm of a function \(f\in H^2(\boldsymbol{\theta}(\Omega))\) is given by \[\begin{align} \label{eq324} \|f\|_{H^2(\boldsymbol{\theta}(\Omega))}=\frac{1}{c_{\boldsymbol{\theta}}} \left\{\sup_{0<r<1} \int_{\partial\Omega}|f\circ \boldsymbol{\theta}(r\boldsymbol{z})|^2 |J_{\boldsymbol{\theta}}(r\boldsymbol{z})|^2 d\mu(\boldsymbol{z}) \right\}^{1/2}, \end{align}\tag{2}\] where the normalization by \(c_{\boldsymbol{\theta}}=\|J_{\boldsymbol{\theta}}\|_{H^2(\Omega)}\) ensures \(\|1\|_{H^2(\boldsymbol{\theta}(\Omega))}=1.\) It follows from [22] that polynomials are dense in \(H^2(\boldsymbol{\theta}(\Omega))\) and that it is a reproducing kernel Hilbert space with the reproducing kernel \(\mathbb{S}_{\boldsymbol{\theta}(\Omega)}:\boldsymbol{\theta}(\Omega) \times \boldsymbol{\theta}(\Omega) \to \mathbb{C}\) given by \[\begin{align} \label{Hardy32kernel} \mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\boldsymbol{\theta}(\boldsymbol{z}), \boldsymbol{\theta}(\boldsymbol{w}))=\frac{c_{\boldsymbol{\theta}}^2}{|G|}\frac{1}{J_{\boldsymbol{\theta}}(\boldsymbol{z})\overline{J_{\boldsymbol{\theta}}(\boldsymbol{w})}} \sum_{\sigma\in G} {\rm sgn\,}(\sigma^{-1}) \mathbb{S}_{\Omega}(\sigma^{-1}\cdot \boldsymbol{z}, \boldsymbol{w}),\qquad \boldsymbol{z}, \boldsymbol{w}\in \Omega. \end{align}\tag{3}\]

Let \(L^2(\partial\boldsymbol{\theta}(\Omega))\) denote the space of square-integrable functions with respect to \(\mu_{\boldsymbol{\theta}}.\) For \(f\in L^2(\partial\boldsymbol{\theta}(\Omega)),\) the norm is given by \[\begin{align} \label{eq325} \|f\|_{L^2(\partial\boldsymbol{\theta}(\Omega))}= \frac{1}{c_{\boldsymbol{\theta}}} \|J_{\boldsymbol{\theta}} (f\circ \boldsymbol{\theta})\|_{L^2(\partial \Omega)}= \frac{1}{c_{\boldsymbol{\theta}}}\left( \int_{\partial\Omega} |f(\boldsymbol{\theta}(\boldsymbol{z}))|^2 |J_{\boldsymbol{\theta}}(\boldsymbol{z})|^2 d\mu(\boldsymbol{z})\right)^{1/2}. \end{align}\tag{4}\]

Recall that the proper map \(\boldsymbol{\theta}=(\theta_1,\ldots, \theta_d) :\Omega \to \boldsymbol{\theta}(\Omega)\) consists of homogeneous polynomials; let \(k_j\) denote the degree of \(\theta_j\) for each \(1\leq j\leq d.\) If \(f\in H^2(\boldsymbol{\theta}(\Omega))\), then \(J_{\boldsymbol{\theta}}(f \circ \boldsymbol{\theta}) \in H^2(\Omega)\), and its boundary value \((J_{\boldsymbol{\theta}}(f \circ \boldsymbol{\theta}) )^*\) belongs to \(H^2(\partial\Omega).\) Since \(J_{\boldsymbol{\theta}}\) is a homogeneous polynomial, the following limit exists for almost every \(\boldsymbol{z}\in \partial\Omega\): \[\begin{align} \label{eq3216}(J_{\boldsymbol{\theta}}(f \circ \boldsymbol{\theta}) )^*(z)=\lim_{r \to 1^-} J_{\boldsymbol{\theta}}(r\boldsymbol{z})(f \circ \boldsymbol{\theta})(r\boldsymbol{z})=J_{\boldsymbol{\theta}}(\boldsymbol{z}) \lim_{r \to 1^-}f(r^{k_1} \theta_1(\boldsymbol{z}), \ldots, r^{k_d}\theta_d(\boldsymbol{z})). \end{align}\tag{5}\] Let \(V\subseteq \partial\Omega\) be the set of points where the limit in 5 does not exist. Then the set \(V\) is \(G\)-invariant and \(\mu(V)=0\). Since the zero set \(N(J_{\boldsymbol{\theta}})\) of \(J_{\boldsymbol{\theta}}\) is \(G\)-invariant with \(\mu(N(J_{\boldsymbol{\theta}}))=0\), we define a function \(\tilde{f}^*\) on \(\partial\boldsymbol{\theta}(\Omega)\) by \[\tilde{f}^* (\boldsymbol{\zeta})=\begin{cases} \frac{(J_{\boldsymbol{\theta}}(f \circ \boldsymbol{\theta}) )^*(z)}{J_{\boldsymbol{\theta}}(\boldsymbol{z})}, \; \text{ if } \boldsymbol{\zeta}=\boldsymbol{\theta}(\boldsymbol{z})\in \partial\boldsymbol{\theta}(\Omega) \text{ with } \boldsymbol{z}\notin V\cup N(J_{\boldsymbol{\theta}})\\ 0, \qquad\qquad\;\; \text{otherwise} \end{cases}.\] This gives a well defined element of \(L^2(\partial\boldsymbol{\theta}(\Omega)).\) Given a function \(f:\boldsymbol{\theta}(\Omega) \to \mathbb{C}\) and \(0<r<1\), we define the slice \(\tilde{f}_r:\partial\boldsymbol{\theta}(\Omega)\to \mathbb{C}\) by \[\tilde{f}_r(\boldsymbol{\zeta})=f(r^{k_1}\zeta_1, r^{k_2} \zeta_2,\ldots, r^{k_d} \zeta_d),\qquad \boldsymbol{\zeta}=(\zeta_1,\ldots,\zeta_d)\in \partial\boldsymbol{\theta}(\Omega).\] If \(f\in H^2(\boldsymbol{\theta}(\Omega)),\) a routine verification shows that \(\tilde{f}_r \in L^2(\partial\boldsymbol{\theta}(\Omega))\) for all \(0<r<1\), and that \(\tilde{f}^*\) is measurable and square integrable with respect to \(\mu_{\boldsymbol{\theta}}\), satisfying \[\lim_{r \to 1^-} \int_{\partial\boldsymbol{\theta}(\Omega)} |\tilde{f}_r-\tilde{f}^*|^2d\mu_{\boldsymbol{\theta}}=0.\] Moreover, for all \(\boldsymbol{\zeta}=\boldsymbol{\theta}(\boldsymbol{z})\) with \(\boldsymbol{z}\notin V\cup N(J_{\boldsymbol{\theta}})\), we have \(\tilde{ f}^*(\boldsymbol{\zeta})=\lim_{r\to 1^-} \tilde{f}_r(\boldsymbol{\zeta})\), and \(J_{\boldsymbol{\theta}} (\tilde{f}^* \circ \boldsymbol{\theta}) \in H^2(\partial\Omega).\)

Define \(H^2(\partial\boldsymbol{\theta}(\Omega)):=\{f\in L^2(\partial\boldsymbol{\theta}(\Omega)): J_{\boldsymbol{\theta}}(f \circ \boldsymbol{\theta}) \in H^2(\partial\Omega)\}.\) The following result establishes an isometric embedding of \(H^2(\boldsymbol{\theta}(\Omega))\) onto the closed subspace \(H^2(\partial\boldsymbol{\theta}(\Omega))\) of \(L^2(\partial\boldsymbol{\theta}(\Omega)).\)

Let \(L^\infty(\partial\boldsymbol{\theta}(\Omega))\) denote the algebra of essentially bounded \(\mu_{\boldsymbol{\theta}}\)-measurable functions on \(\partial\boldsymbol{\theta}(\Omega)\), and let \(L^\infty(\partial\Omega)^{G}=\{ f\in L^\infty (\partial\Omega): \sigma(f)=f\; \mu\text{-a.e. for all } \sigma\in G\}.\) There exists an isometric \(*\)-isomorphism from \(L^\infty(\partial\boldsymbol{\theta}(\Omega))\) onto \(L^\infty(\partial\Omega)^{G}\) (see [22]). We define \[H^\infty(\partial \boldsymbol{\theta}(\Omega)):=H^2(\partial\boldsymbol{\theta}(\Omega)) \cap L^\infty(\partial\boldsymbol{\theta}(\Omega)).\] Note that \(H^\infty(\boldsymbol{\theta}(\Omega)) \subseteq H^2(\boldsymbol{\theta}(\Omega)).\) The following result shows that the boundary value map \(i'\) restricts to an isometric isomorphism between \(H^\infty(\boldsymbol{\theta}(\Omega))\) and \(H^\infty(\partial\boldsymbol{\theta}(\Omega)).\)

3 Commutant lifting↩︎

In this section, we characterize contractive module maps on quotient modules of \(H^2(\boldsymbol{\theta}(\Omega))\) that admit Schur-class lifts. More precisely, given a quotient module \(\mathcal{Q}\) of \(H^2(\boldsymbol{\theta}(\Omega))\), we classify all contractive module maps on \(\mathcal{Q}\) that arise as compressions \(S_{\varphi}=P_{\mathcal{Q}}T_{\varphi}|_{\mathcal{Q}}\) for some \(\varphi\in \mathcal{S}(\boldsymbol{\theta}(\Omega))\).

By Proposition 1, the Hardy space \(H^2(\boldsymbol{\theta}(\Omega))\) is canonically identified with its boundary realization \(H^2(\partial\boldsymbol{\theta}(\Omega)).\) Accordingly, we make no distinction between a function in \(H^2(\boldsymbol{\theta}(\Omega))\) and its boundary realization unless necessary. Thus, every quotient module of \(H^2(\boldsymbol{\theta}(\Omega))\) may be regarded as a closed subspace of \(H^2(\partial\boldsymbol{\theta}(\Omega))\). Henceforth, \(\mathcal{Q}\) denotes a nonzero quotient module of \(H^2(\partial\boldsymbol{\theta}(\Omega))\). Since \(\mathcal{Q}^\perp\) is a submodule, the nontriviality of \(\mathcal{Q}\) implies that \(P_{\mathcal{Q}}1\neq 0.\) Consequently, \[\begin{align} \label{Q32description}\mathcal{Q} = \overline{\operatorname{span}}\left\{P_{\mathcal{Q}}\boldsymbol{\zeta}^\alpha P_{\mathcal{Q}}1: \alpha\in \mathbb{Z}_+^d\right\}. \end{align}\tag{6}\] For a closed subspace \(\mathcal{E}\subseteq L^2(\partial\boldsymbol{\theta}(\Omega))\), set \(\mathcal{E}^{conj}=\{\bar{f}:\, f\in \mathcal{E}\}.\) Further, let \[H_0^2(\partial\boldsymbol{\theta}(\Omega))=\left\{f\in H^2(\partial\boldsymbol{\theta}(\Omega)):\langle f,1\rangle_{H^2(\partial\boldsymbol{\theta}(\Omega))}=0\right\}.\] Under the boundary identification, this corresponds to the subspace of functions in \(H^2(\boldsymbol{\theta}(\Omega))\) vanishing at the origin. The following orthogonality relation will be used repeatedly.

It follows from Lemma 1 that \[H^2(\partial\boldsymbol{\theta}(\Omega)) + H^2(\partial\boldsymbol{\theta}(\Omega))^{conj} = H_0^2(\partial\boldsymbol{\theta}(\Omega)) \oplus H^2(\partial\boldsymbol{\theta}(\Omega))^{conj}\] is a closed subspace of \(L^2(\partial\boldsymbol{\theta}(\Omega))\). Define \[\mathcal{M}_{\boldsymbol{\theta}(\Omega)} = L^2(\partial\boldsymbol{\theta}(\Omega)) \ominus \big( H^2(\partial\boldsymbol{\theta}(\Omega)) + H^2(\partial\boldsymbol{\theta}(\Omega))^{conj} \big).\] By construction, the spaces \(H^2_0(\partial\boldsymbol{\theta}(\Omega)),\;\mathcal{M}_{\boldsymbol{\theta}(\Omega)},\) and \(\mathcal{Q}^{conj}\) are mutually orthogonal in \(L^2(\partial\boldsymbol{\theta}(\Omega)).\) Accordingly, we define their algebraic direct sum in \(L^1(\partial\boldsymbol{\theta}(\Omega))\): \[\begin{align} \label{eq3214} \mathcal{M}_{\mathcal{Q}} =\mathcal{Q}^{conj} \dotplus \mathcal{M}_{\boldsymbol{\theta}(\Omega)} \dotplus H_0^2(\partial\boldsymbol{\theta}(\Omega)). \end{align}\tag{8}\] We now establish our first characterization of commutant lifting on quotient modules of \(H^2(\boldsymbol{\theta}(\Omega))\) (c.f [8]).

Building on Theorem 3, we next reformulate the contractivity of the functional \(X_{\mathcal{Q}}\) as a distance condition in \(L^1(\partial\boldsymbol{\theta}(\Omega)).\) Recall that \(X_{\mathcal{Q}}\) is completely determined by the vector \(\psi=XP_{\mathcal{Q}}1 \in \mathcal{Q}.\) It follows from the proof of Theorem 3 that \(X_{\mathcal{Q}}\) vanishes on \(\mathcal{M}_{\boldsymbol{\theta}(\Omega)}\dotplus H^2_0(\partial\boldsymbol{\theta}(\Omega))\) and on the orthogonal complement of \(\overline{\psi}\) in \(\mathcal{Q}^{conj}.\) Hence, the kernel of \(X_{\mathcal{Q}}\) is given by \[\widetilde{\mathcal{M}}_{\mathcal{Q}, X}:=\mathrm{ker}X_{\mathcal{Q}}=(\mathcal{Q}^{conj}\ominus \mathrm{span}\{\overline{\psi}\}) \dotplus \mathcal{M}_{\boldsymbol{\theta}(\Omega)}\dotplus H^2_{0}(\partial\boldsymbol{\theta}(\Omega)).\] This leads to the following geometric characterization of commutant lifting.

3.1 Polydisc↩︎

It was shown by Rudin [29] that Schur functions on the Euclidean unit ball \(\mathbb{B}_d\) can be approximated, in the weak-\(*\) topology, by inner functions. More recently, this approximation phenomenon was used to obtain a characterization of commutant lifting on \(\mathbb{B}_d\) in terms of inner functions (see [9]). Motivated by these developments, we establish an analogous characterization of Schur-class liftability for quotient domains arising from the polydisc. We begin by fixing the relevant class of quotient domains and by formulating the corresponding notion of inner function.

Throughout this subsection, we take \(\Omega=\mathbb{D}^d\), and let \(G=G(m,n,d)\) be one of the imprimitive finite complex reflection groups in the Shephard-Todd classification [13], where \(m\) and \(n\) are positive integers such that \(n\) divides \(m\). Let \(\alpha\) be a primitive \(m\)-th root of unity. The group \(G(m,n,d)\) consists of transformations of the form \[\mathbb{C}^d \ni \boldsymbol{z}=(z_1,\ldots,z_d)\longmapsto (\alpha^{k_1} z_{\tau(1)}, \ldots, \alpha^{k_d} z_{\tau(d)}),\] where \(\tau\in \Sigma_d\) and \(k_1,\ldots, k_d\) are integers such that \(k_1+\cdots+k_d\) is divisible by \(n.\)

The associated basic polynomial map \(\boldsymbol{\theta}=(\theta_1,\ldots,\theta_d):\mathbb{D}^d\to \boldsymbol{\theta}(\mathbb{D}^d)\) can be given explicitly: for each \(1\leq j\leq d-1\), \(\theta_j\) is the \(j\)-th elementary symmetric polynomial in \(z_1^m,\ldots,z_d^m\), and \(\theta_d(\boldsymbol{z})=(z_1\cdots z_d)^{m/n}.\) The map \(\boldsymbol{\theta}:\mathbb{D}^d\longrightarrow \boldsymbol{\theta}(\mathbb{D}^d)\) is proper and is factored by \(G(m,n,d)\). In the special case \(m=n=1\), the quotient domain \(\boldsymbol{\theta}(\mathbb{D}^d)\) is the symmetrized polydisc. For notational simplicity, we shall write \(G\) for \(G(m,n,d)\) throughout the rest of this subsection.

Recall that the Shilov boundary of the quotient domain \(\boldsymbol{\theta}(\mathbb{D}^d)\) is \(\partial\boldsymbol{\theta}(\mathbb{D}^d)=\boldsymbol{\theta}(\mathbb{T}^d).\) In analogy with the polydisc, we say that a function \(f\in H^\infty(\boldsymbol{\theta}(\mathbb{D}^d))\) is inner if the corresponding boundary function \(\tilde{f}^*\in H^\infty(\boldsymbol{\theta}(\mathbb{T}^d))\) is unimodular \(\mu_{\boldsymbol{\theta}}\)-a.e. on \(\boldsymbol{\theta}(\mathbb{T}^d).\) Let \[\mathcal{I}(\boldsymbol{\theta}(\mathbb{D}^d))=\{f\in H^\infty(\boldsymbol{\theta}(\mathbb{D}^d)): f\circ\boldsymbol{\theta} \text{ is an inner function on } \mathbb{D}^d\}.\] By Proposition 2, for every \(f\in H^\infty(\boldsymbol{\theta}(\mathbb{D}^d))\), \(\tilde{f}^* \circ \boldsymbol{\theta}=(f \circ \boldsymbol{\theta})^*\) almost everywhere on \(\mathbb{T}^d\) and therefore the class of inner functions on \(\boldsymbol{\theta}(\mathbb{D}^d)\) coincides with \(\mathcal{I}(\boldsymbol{\theta}(\mathbb{D}^d)).\)

We first prove that every Schur function on \(\boldsymbol{\theta}(\mathbb{D}^d)\) is a weak-\(*\) limit, in \(L^\infty(\boldsymbol{\theta}(\mathbb{T}^d)),\) of inner functions on \(\boldsymbol{\theta}(\mathbb{D}^d).\) To achieve this, we establish a preliminary result analogous to [30]. The proof follows essentially the same line of argument as in Rudin’s theorem, with suitable modifications to incorporate the \(G\)-invariance condition. Recall that a function \(f\) is \(G\)-invariant if \(\sigma(f)=f\) for all \(\sigma \in G.\)

We now show that inner functions are weak-\(*\) dense in the Schur class on \(\boldsymbol{\theta}(\mathbb{D}^d).\)

The weak-\(*\) density of inner functions established above yields the following approximation characterization for commutant lifting. For a subset \(S\subseteq L^1(\boldsymbol{\theta}(\mathbb{T}^d))\) and a function \(\psi \in H^2(\boldsymbol{\theta}(\mathbb{D}^d))\), we say a sequence \(\{f_n\}\subseteq \mathcal{S}(\boldsymbol{\theta}(\mathbb{D}^d))\) converges to \(\psi\) in the weak-\(*\) topology determined by \(S\) if \[\lim_{n\to\infty}\int_{\boldsymbol{\theta}(\mathbb{T}^d)} f_n g\, d\mu_{\boldsymbol{\theta}}=\int_{\boldsymbol{\theta}(\mathbb{T}^d)} \psi g\, d\mu_{\boldsymbol{\theta}}\qquad \text{for all } g\in S.\]

The weak-\(*\) convergence condition in Theorem 8 can be further simplified. Recall from the proof of Theorem 3 that the associated functional \(X_{\mathcal{Q}}\) vanishes on \(\mathcal{M}_{\boldsymbol{\theta}(\mathbb{D}^d)}\dotplus H^2_0(\boldsymbol{\theta}(\mathbb{T}^d)).\) Consequently, convergence against \(\mathcal{M}_{\mathcal{Q}}\) reduces exclusively to convergence determined by \(\mathcal{Q}^{conj}.\) This observation leads to the following immediate corollary.

4 Interpolation↩︎

In this section we apply the commutant lifting results of Section 3 to interpolation problems on quotient domains. This approach is motivated by Sarason’s operator-theoretic proof of the classical Nevanlinna–Pick interpolation theorem [1], and by subsequent multivariable extensions for the polydisc and the Euclidean unit ball; see, for instance, [8], [9]. Interpolation on the symmetrized bidisc, which is a proper image of the bidisc, was previously studied in [3], [4] using realization formulas for Schur functions. In contrast, our framework relies on commutant lifting and applies uniformly to quotient domains, including the symmetrized bidisc and the tetrablock.

The primary objective of this section is to characterize the existence of a function \(\varphi\in \mathcal{S}(\boldsymbol{\theta}(\Omega))\) satisfying \[\varphi(\boldsymbol{\zeta}_i)=w_i, \qquad i=1,\ldots,p,\] for a prescribed set of distinct points \(\mathcal{Z}=\{\boldsymbol{\zeta}_1,\ldots,\boldsymbol{\zeta}_p\}\subseteq \boldsymbol{\theta}(\Omega)\) and target scalars \(w_1,\ldots,w_p\in\mathbb{D}.\)

We begin by recalling that \(H^2(\boldsymbol{\theta}(\Omega))\) is a reproducing kernel Hilbert space with reproducing kernel \(\mathbb{S}_{\boldsymbol{\theta}(\Omega)}\) is as given in 3 . For each \(\zeta \in \boldsymbol{\theta}(\Omega)\), the kernel function \(\mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot,\boldsymbol{\zeta})\) satisfies the fundamental reproducing property \[\langle{f},\,{\mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot,\boldsymbol{\zeta})} \rangle_{H^2(\boldsymbol{\theta}(\Omega))}= f(\boldsymbol{\zeta})\qquad \text{for all}\; f\in H^2(\boldsymbol{\theta}(\Omega)).\] In particular, \[\begin{align} \label{eq3210} T_{\zeta_i}^* \mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot, \boldsymbol{\eta})=\bar{\eta}_i \mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot, \boldsymbol{\eta}), \qquad \boldsymbol{\eta}=(\eta_1,\ldots, \eta_d)\in \boldsymbol{\theta}(\Omega), \quad i=1,\ldots,d. \end{align}\tag{15}\] Given the set of \(p\) distinct points \(\mathcal{Z}\), we define the subspace \[\mathcal{Q}_{\mathcal{Z}}=\mathrm{span}\{\mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot, \boldsymbol{\zeta}_i): i=1,\ldots, p\}.\] It follows from 15 that \(\mathcal{Q}_{\mathcal{Z}}\) is a finite-dimensional quotient module of \(H^2(\boldsymbol{\theta}(\Omega)).\) Its orthogonal complement is the space of functions vanishing on the interpolation nodes: \[\mathcal{Q}_{\mathcal{Z}}^\perp =\left\{ g\in H^2(\boldsymbol{\theta}(\Omega)): g(\boldsymbol{\zeta}_i)=0 \text{ for all } i=1,\ldots,p\right\}.\] We refer to quotient modules of this form as finite-dimensional zero-based quotient modules. Since each kernel function \(\mathbb{S}_{\boldsymbol{\theta}(\Omega)}(\cdot,\boldsymbol{\zeta}_j)\) is bounded, we have \(\mathcal{Q}_{\mathcal{Z}} \subseteq H^\infty(\boldsymbol{\theta}(\Omega)).\)

Let \(X\in \mathcal{B}(\mathcal{Q}_{\mathcal{Z}})\) be a bounded module map, and set \(\psi =XP_{\mathcal{Q}_{\mathcal{Z}}} 1\). Then \(\psi \in \mathcal{Q}_{\mathcal{Z}}\subseteq H^\infty (\boldsymbol{\theta}(\Omega)).\) As \(X\) is a module map, for any \(\alpha \in \mathbb{Z}_+^d\) we obtain \[\begin{align} X(P_{\mathcal{Q}_{\mathcal{Z}}} \boldsymbol{\zeta}^\alpha P_{\mathcal{Q}_{\mathcal{Z}}} 1) =X (S_{\boldsymbol{\zeta}}^\alpha P_{\mathcal{Q}_{\mathcal{Z}}} 1)= S_{\boldsymbol{\zeta}}^\alpha X P_{\mathcal{Q}_{\mathcal{Z}}} 1= S_{\boldsymbol{\zeta}}^\alpha \psi=S_{\psi} P_{\mathcal{Q}_{\mathcal{Z}}} {\boldsymbol{\zeta}}^\alpha P_{\mathcal{Q}_{\mathcal{Z}}}1. \end{align}\] Since \(\mathcal{Q}_{\mathcal{Z}} =\overline{\mathrm{span}} \{P_{\mathcal{Q}_{\mathcal{Z}}} \boldsymbol{\zeta}^\alpha P_{\mathcal{Q}_{\mathcal{Z}}} 1: \alpha \in \mathbb{Z}_+^d \}\), it follows that \(X=S_{\psi}\) on \(\mathcal{Q}_{\mathcal{Z}}.\) Thus, every bounded module map on \(\mathcal{Q}_{\mathcal{Z}}\) admits an \(H^\infty(\boldsymbol{\theta}(\Omega))\)-lift. However, such a lift need not preserve the Schur-class norm structure on \(\boldsymbol{\theta}(\Omega).\) Thus the interpolation problem reduces to deciding when the \(H^\infty(\boldsymbol{\theta}(\Omega))\)-lift can be chosen from the Schur class.

Using the commutant lifting framework developed in Section 3, we now establish the following interpolation theorem.

We next derive a geometric formulation of the interpolation problem using the distance characterization of commutant lifting. As shown in the proof of Theorem 9, the existence of a function \(\varphi \in \mathcal{S}(\boldsymbol{\theta}(\Omega))\) satisfying \(\varphi(\boldsymbol{\zeta}_i)=w_i\) for all \(i=1,\ldots, p\) is equivalent to the fact that the operator \(X_{\mathcal{Z,W}}\) defined by 16 admits a lift to \(\varphi.\) In particular, such a lift necessarily implies that \(\|X_{\mathcal{Z,W}}\|\leq 1.\) Applying Theorem 5 to \(X_{\mathcal{Z,W}}\) therefore yields the following interpolation criterion.

We now pass to quotient domains of the polydisc associated with the finite complex reflection groups \(G(m,n,d)\). For these domains, Theorem 8 gives a commutant lifting criterion in terms of inner functions. Applying that criterion to the finite-dimensional zero-based quotient modules determined by the interpolation nodes yields the following inner-function characterization of Schur-class interpolation.

Acknowledgment↩︎

The author expresses his sincere gratitude to his PhD supervisor, Professor Surjit Kumar, for many valuable suggestions and comments during the preparation of this draft.

References↩︎

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1. Support for the work of the author was provided in the form of a Prime Minister’s Research Fellowship (PMRF / 2502827).↩︎