Multivariate Carleman Condition

• Multivariate Carleman condition is a criterion for the determinacy of moment problems, imposing slicewise growth constraints on a positive semidefinite double sequence.
• It generalizes Carleman’s one-dimensional condition by applying similar restrictions to each fixed slice, ensuring the uniqueness of the representing measure.
• Satisfying the condition guarantees essential self-adjointness of relevant operators and enables analytic parametrization via contractive operator-valued functions.

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 (Zagorodnyuk, 14 Aug 2025).

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 (Zagorodnyuk, 14 Aug 2025).