Pick Bodies in Carathéodory Hyperbolic Domains

• Pick bodies are sets representing boundary data for bounded holomorphic functions meeting precise interpolation constraints on Carathéodory hyperbolic domains.
• Their geometric structure as convex, compact Reinhardt sets enables the translation of complex interpolation problems into operator-theoretic frameworks.
• The analysis employs Schur–Agler theory and kernel positivity to bridge classical function theory with modern operator algebra, clarifying conditions for classical and multivariate interpolation.

Pick bodies on Carathéodory hyperbolic domains form a central object at the intersection of complex analysis, operator theory, and interpolation geometry. For a domain $\Omega \subset \mathbb{C}^m$, these sets encapsulate the locus of data admitting bounded holomorphic interpolants subject to precise extremal and geometric constraints. The characterization and realization of Pick bodies, as well as their operator-theoretic correspondences, yield deep insights into the structure and function theory of several complex variables, especially when $\Omega$ is Carathéodory-hyperbolic (Biswas, 31 Jan 2026).

1. Carathéodory-Hyperbolic Domains and Carathéodory Pseudodistance

A domain $\Omega \subset \mathbb{C}^m$ is Carathéodory-hyperbolic when its Carathéodory pseudodistance $c_\Omega^*(z, w)$ becomes a true metric, i.e., $c_\Omega^*(z, w) > 0$ for $z \neq w$. This pseudodistance is defined as

$$c_\Omega^*(z, w) = \sup \{ m(f(z), f(w)): f \in \mathcal{O}(\Omega, \mathbb{D})\},$$

where $\mathbb{D} = \{\zeta \in \mathbb{C}: |\zeta| < 1\}$ and $$m(\zeta_1, \zeta_2) = |(\zeta_1 - \zeta_2)/(1 - \overline{\zeta}_1 \zeta_2)|$$ is the classical Poincaré distance on $\mathbb{D}$.

The generalized Carathéodory function is also significant:

$$c_\Omega^*(z_1; z_2, \ldots, z_n) = \sup \{ |f(z_1)| : f \in \mathcal{O}(\Omega, \mathbb{D}),\ f(z_j) = 0\ \text{for}\ j = 2,\dots,n \}.$$

These metrics provide the foundational geometric context for the interpolation problems under consideration.

2. Definition and Geometric Structure of Pick Bodies

Given distinct points $z_1, \ldots, z_n \in \Omega$, the Pick body or interpolation body is defined as

$$D_\Omega(z_1, \ldots, z_n) = \{ (f(z_1), \ldots, f(z_n)) : f \in \mathcal{O}(\Omega, \overline{\mathbb{D}}) \} \subset \mathbb{C}^n.$$

Geometrically, $D_\Omega(z_1, \ldots, z_n)$ is a compact, convex, complete Reinhardt subset of $\mathbb{C}^n$. Functionally, it represents all data $w = (w_1, \ldots, w_n)$ such that the interpolation task $f(z_j) = w_j$ admits a holomorphic solution $f:\Omega \to \overline{\mathbb{D}}$.

3. Operator-Theoretic Realization and Kernel Intersections

For Carathéodory-hyperbolic $\Omega$, there exists a family $\mathcal{K}_\Omega$ of positive-definite $n \times n$ kernels $K = (K_{i, j})$ such that

$$D_\Omega(z_1, \ldots, z_n) = \bigcap_{K \in \mathcal{K}_\Omega} D_K,$$

where each kernel–Pick body is

$$D_K = \{ w \in \mathbb{C}^n : [(1 - w_i \overline{w}_j) K_{i, j}]_{i, j=1}^n \geq 0 \}.$$

Each $D_K$ corresponds to the closed unit ball of a contraction $T_w$ on an $n$-dimensional Hilbert space $H_K$ (with reproducing kernel $K$), where $T_w k(z_j) = \overline{w}_j k(z_j)$. The interpolation body thus encodes a universal operator-theoretic test for contractivity, linking classical analytic function theory to modern operator algebra techniques (Biswas, 31 Jan 2026).

The construction leverages Schur–Agler theory: $\mathcal{O}(\Omega,\overline{\mathbb{D}})$ is treated as a Schur–Agler class generated by test functions vanishing at a fixed basepoint. Admissible kernels satisfy kernel positivity with respect to all these test functions, and, after restriction and suitable perturbation for definiteness, $\mathcal{K}_\Omega$ provides the desired decomposition.

4. Realization Conditions for Classical Pick Bodies

A sufficient condition exists for realizing the Pick body on $\Omega$ as the classical Pick body on the unit disc. Let $K$ be an extremal kernel on $\{z_1, \ldots, z_n\}$ such that $K(z_i, z_i) = 1$, and suppose $D_\Omega(z) = D_K$. If there exists $\alpha = (\alpha_1, \ldots, \alpha_n) \in D_K$ fulfilling

$$m(\alpha_1, \alpha_j) = c^*_\Omega(z_1, z_j),\quad j = 2, \ldots, n,$$

then

$$D_\Omega(z_1, \ldots, z_n) = D_{\mathbb{D}}(\alpha_1, \ldots, \alpha_n),$$

where

$$D_{\mathbb{D}}(\alpha_1, \ldots, \alpha_n) = \left\{ w \in \mathbb{C}^n:\ \left[ \frac{1 - w_i \overline{w}_j}{1 - \alpha_i \overline{\alpha}_j} \right] \geq 0 \right\}.$$

There is then a holomorphic map $\varphi:\mathbb{D} \to \Omega$ sending $\alpha_j \to z_j$ and producing a biholomorphism between the interpolation problems. Proof proceeds by induction on $n$, utilizing classical two-point Pick theory and Szegő-like factorization for extremal kernels, with verification through principal minor positivity (Biswas, 31 Jan 2026).

5. Concrete Examples and Special Cases

The structure of Pick bodies is particularly transparent in low-dimensional and special configurations:

• $n = 2$: $$D_\Omega(z_1, z_2) = \{ (w_1, w_2) \in \overline{\mathbb{D}}^2 : m(w_1, w_2) \leq c_\Omega^*(z_1, z_2) \} \cup \{ (e^{i\theta}, e^{i\theta}) : \theta \in \mathbb{R} \}$$, reproducing the classical lens region.
• $n = 3$: If an extremal kernel $K$ and $\alpha \in \partial D_\Omega(z) \cap \mathbb{D}^3$ exist such that at least two coordinate pairs attain the two-point boundary, all three points are collinear on a disc geodesic, and $$D_\Omega(z_1, z_2, z_3) = D_\mathbb{D}(\alpha_1, \alpha_2, \alpha_3)$$.
• Polydisc type: If $K$ is diagonal and positive-definite, $D_K$ is the closed unit polydisc, characterizing extremal behavior in each coordinate. Points lying on coordinate projections of a Carathéodory geodesic may thus be lifted to $D^2$ or higher polydiscs.

Case	Pick Body Description	Notable Structure
$n=2$	$$\{(w_1,w_2): m(w_1, w_2) \leq c_\Omega^*(z_1, z_2)\}$$	Lens with diagonal circle
$n=3$ with extremals	$D_\mathbb{D}(\alpha_1,\alpha_2,\alpha_3)$	Disc geodesic realization
Diagonal kernel/polydisc	Unit polydisc $D^n$	Extremal in each coordinate

6. Open Problems and Research Directions

Several fundamental questions remain open:

• For a general Carathéodory-hyperbolic domain, the full structure of admissible kernels $\mathcal{K}_\Omega$ is not characterized. A geometric classification in terms of Carathéodory extremal mappings is a notable open problem.
• For the bidisc $D^2$, the Agler decomposition yields orthogonal test-function families. It is unresolved whether general $\Omega$ admit finite Agler decompositions, and how such decompositions constrain $D_\Omega(z)$.
• The equivalence between the existence of Carathéodory geodesics through $z_1, ..., z_n$ and realizability of the Pick body by a single kernel prompts the search for geometric necessary and sufficient conditions on $\Omega$ for the Pick body to coincide with a classical Pick body in $\mathbb{D}$.
• In higher dimensions, the structure and operator-theoretic model of extremal Sarason interpolation functions are not fully classified, nor is their interaction with Pick bodies at arbitrary $n$-tuples.
• Extensions to infinite-dimensional settings such as matrix balls or noncommutative domains are open, motivating potential generalizations of these results.

This suggests a rich interplay between several complex variables, operator models, and complex geometry, with much remaining to be classified regarding extremal functions, kernel structures, and geometric realization conditions (Biswas, 31 Jan 2026).