---
title: Multivariate Carleman Condition
url: https://www.emergentmind.com/topics/multivariate-carleman-condition
type: topic
---

# Multivariate Carleman Condition

The multivariate Carleman condition constitutes a criterion for the solvability and determinacy of multidimensional moment problems, extending the foundational role of Carleman's classical condition from the univariate setting. In the two-dimensional moment problem, it involves specific slicewise growth constraints on a positive semidefinite double sequence of moments $(s_{m,n})_{m,n\ge0}$, determining whether a positive Borel measure $\mu$ on $\mathbb{R}^2$ with prescribed moments exists and is unique. When these conditions are met, all solutions are analytically parameterized via contractive operator-valued functions, with generalized resolvent theory providing the bridge between moment data and operator extensions [2508.10823].

## 1. Formal Statement of the Multivariate Carleman Condition

Let $(s_{m,n})_{m,n\ge0}$ be a positive semidefinite double sequence governing the two-dimensional moment problem. The multivariate Carleman-type condition is imposed as a countable family over $m$:
\[
\sum_{k=1}^\infty \frac{1}{\sqrt[2k]{\,s_{2m,2k}+s_{2m+2,2k}\,}} = \infty,\quad m=0,1,2,\dots
\]
A slightly stronger slicewise analogue, sometimes employed for analysis of the second coordinate operator, is:
\[
\sum_{k=0}^\infty \frac{1}{\sqrt[2k]{\,s_{2m,2k}\,}} = \infty,\quad m=0,1,2,\dots
\]
The stipulated condition requires divergence of the reciprocal root sum for every fixed $m$, thus regulating the growth rate of the moment sequence along the $n$-axis for each $m$ slice.

## 2. Generalization of the Classical One-Dimensional Criterion

In one dimension, Carleman's criterion for determinacy of the Hausdorff moment problem reads:
\[
\sum_{n=1}^\infty s_{2n}^{-1/(2n)} = \infty
\]
The multivariate analog generalizes this principle by imposing a corresponding slicewise condition. For each fixed $m$, the sequence $\{s_{2m,2k}+s_{2m+2,2k}\}_k$ is treated as an effective one-dimensional moment sequence, and Carleman's condition is required independently for each such slice. This prevents unbounded measure mass from escaping in any direction. Operator-theoretically, it enforces essential self-adjointness of the relevant coordinate multiplication operators, securing the existence of a unique joint spectral measure.

## 3. Determinacy and Operator-Theoretic Characterization

With positivity and Carleman-type conditions satisfied, the main determinacy theorem establishes equivalence among:
1. Uniqueness (determinacy) of the representing measure for the two-dimensional moment problem.
2. Existence of a canonical solution, i.e., a unique pair of commuting self-adjoint extensions acting in the original Hilbert space.
3. Self-adjointness of the first coordinate operator $A_1$.

The proof framework exploits Chernoff's quasi-analytic-vector theorem to obtain essential self-adjointness of $A_2$, followed by operator conjugation and commutative factorization to construct self-adjoint extensions. The freedom in extending $A_1$ correlates directly with non-uniqueness; if $A_1$ is not self-adjoint, there exists an analytic family of distinct commuting self-adjoint extensions, yielding a continuum of representing measures.

## 4. Analytic Parameterization in the Indeterminate Case

When $A_2$ achieves self-adjointness, one passes to the Cayley transforms:
\[
V_1=(A_1+iI)(A_1-iI)^{-1},\quad V_2=(A_2+iI)(A_2-iI)^{-1}
\]
with $V_1$ and $V_2$ being an isometry and a unitary, respectively. The space of all solutions is then in bijection with analytic, contractive operator-valued maps
\[
\Phi_z \in \mathcal S_{V_1,V_2}(\mathbb{D};N_0(V_1),N_\infty(V_1))
\]
where $\Phi$ acts from the defect space $N_0(V_1)$ to $N_\infty(V_1)$ and commutes appropriately with $V_2$. Each $\Phi$ produces a generalized resolvent and integral kernel reconstructing a representing measure. Thus, indeterminacy is fully captured by analytic Schur-class functions of a single complex variable.

## 5. Generalized Resolvents for Commuting Symmetric Operators

Given commuting closed symmetric operators $A_1,A_2$ with Cayley transforms $V_1,V_2$, any commuting self-adjoint extensions $B_1,B_2$ in an enlarged space yield a generalized resolvent via
\[
\mathbf R_{s;\lambda_1,\lambda_2} = P_H\,(I+\lambda_1B_1)(B_1-\lambda_1I)^{-1}\,(I+\lambda_2B_2)(B_2-\lambda_2I)^{-1}|_H
\]
A key result establishes correspondence—through a two-variable Shtraus–Chumakin analogue—between such resolvents and those of $(V_1,V_2)$. The explicit analytic parametrization leverages contractive operator-valued Schur functions, positioning the generalized resolvent as the central mediator between moment constraints and spectral data.

## 6. Canonical Solutions, Sharpness, and Necessity

Canonical solutions described in Theorem 6.2 are constructed via arbitrary unitary multipliers in the direct-integral fibers for $V_2$. These exemplify the continuum of solutions available when $A_1$ is not self-adjoint. The necessity of the multivariate Carleman condition is underscored by its role in guaranteeing essential self-adjointness of $A_2$; relaxation of the condition for any $m$ yields possible loss of self-adjointness and breakdown of solution existence or uniqueness. The operator-theoretic parametrization and analogues to Nevanlinna theory in multiple variables rely fundamentally on these slicewise Carleman-type constraints [2508.10823].

Source: https://www.emergentmind.com/topics/multivariate-carleman-condition