[Paper Review] The truncated matrix trigonometric moment problem: the operator approach

Sergey M‎. ‎Zagorodnyuk|arXiv (Cornell University)|Dec 7, 2010

Spectral Theory in Mathematical Physics1 references3 citations

TL;DR

This paper presents a bijective parameterization of all solutions to the truncated matrix trigonometric moment problem using an operator-theoretic approach based on generalized resolvents of isometric operators. By constructing a closed isometric operator from the moment data and representing solutions via analytic operator-valued functions in the unit disk, the authors establish a one-to-one correspondence between contractive analytic functions and solutions, resolving the long-standing question of bijectivity in parameterization.

ABSTRACT

In this paper we study the truncated matrix trigonometric moment problem. We obtained a bijective parameterization of all solutions of this moment problem (both in nondegenerate and degenerate cases) via an operator approach. We use important results of M.E.~Chumakin on generalized resolvents of isometric operators.

Motivation & Objective

• To provide a complete and bijective parameterization of all solutions to the truncated matrix trigonometric moment problem.
• To unify the treatment of both degenerate and non-degenerate cases under a single operator-theoretic framework.
• To resolve the open question of whether existing parameterizations are bijective by constructing a new, explicitly bijective correspondence.
• To extend Chumakin’s results on generalized resolvents to the matrix trigonometric moment problem context.
• To offer a transparent, unified view of the solution space through abstract operator theory, avoiding step-by-step algorithmic constructions.

Proposed method

• Construct a closed isometric operator $ A $ from the given moment sequence $ \{S_n\}_{n=0}^d $, using a Hilbert space of vector-valued trigonometric polynomials.
• Define the generalized resolvent $ \mathbf{R}_\zeta = [E - \zeta(U \oplus \Phi_\zeta)]^{-1} $ for $ \zeta \in \mathbb{D} $, where $ \Phi_\zeta $ is an analytic operator-valued function with values in contractions from $ H \ominus D(A) $ to $ H \ominus R(A) $.
• Reconstruct the matrix spectral measure $ M(t) $ via the Stieltjes inversion formula: $ \int_0^{2\pi} \frac{1}{1 - \zeta e^{it}} dm_{k,j}(t) = (\mathbf{R}_\zeta x_k, x_j)_H $ for $ \zeta \in \mathbb{C} \setminus \partial\mathbb{D} $.
• Use the duality relation $ \mathbf{R}_{1/\bar{\zeta}} = E_H - \mathbf{R}_\zeta^* $ to ensure consistency across the unit circle.
• Establish that different analytic functions $ \Phi_\zeta $ yield different solutions, proving bijectivity of the parameterization.
• Leverage the uniqueness of generalized resolvents for isometric operators to ensure that every solution arises from a unique $ \Phi_\zeta $.

Experimental results

Research questions

• RQ1 Is there a bijective parameterization of all solutions to the truncated matrix trigonometric moment problem when $ T_d \geq 0 $?
• RQ2 Can the solution space be uniformly described for both degenerate and non-degenerate cases using a single operator-theoretic framework?
• RQ3 Does the generalized resolvent approach of Chumakin extend to matrix-valued moment problems with full generality?
• RQ4 Can the correspondence between solutions and analytic operator-valued functions be shown to be one-to-one?
• RQ5 Can the operator-theoretic viewpoint provide a transparent, unified picture of the solution set without algorithmic step-by-step construction?

Key findings

• The paper establishes a bijective correspondence between all solutions of the truncated matrix trigonometric moment problem and analytic operator-valued functions $ \Phi_\zeta $ that are contractive and analytic in the unit disk $ \mathbb{D} $.
• Each solution $ M(t) $ is uniquely reconstructed from the generalized resolvent $ \mathbf{R}_\zeta = [E - \zeta(U \oplus \Phi_\zeta)]^{-1} $ via the integral formula $ \int_0^{2\pi} \frac{1}{1 - \zeta e^{it}} dm_{k,j}(t) = (\mathbf{R}_\zeta x_k, x_j)_H $.
• The parameterization is valid for all $ T_d \geq 0 $, covering both degenerate and non-degenerate cases uniformly.
• The generalized resolvent representation ensures that distinct $ \Phi_\zeta $ functions yield distinct solutions, proving the bijectivity of the parameterization.
• The method provides a transparent, global view of the solution space by embedding the moment problem into the theory of isometric operators and their generalized resolvents.
• The construction avoids algorithmic or iterative procedures, offering instead a direct, structural characterization of all solutions through analytic operator functions.