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Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$

A. Arziev, K. Kudaybergenov. P. Orinbaev

TL;DR

The paper generalizes Mercer’s theorem to vector-valued RKHSs defined over Kaplansky–Hilbert modules on $L_\infty(\Omega)$, driven by a measurable kernel $K(\omega,t,s)$ and a partial integral operator. Using an isometric isomorphism between the module and measurable sections of a Hilbert bundle, it reduces the problem to fiberwise classical Mercer theory and introduces a vector-valued lifting to preserve inner-product structure. The main result characterizes the vector-valued RKHS by the equivalence of three properties: basis completeness of the eigenfunctions, injectivity of the adjoint embedding, and a pointwise Mercer-type decomposition of the kernel, with measurability and essential boundedness conditions on the eigenvalues. This framework provides a constructive spectral description for multivariate, parameter-dependent kernels and paves the way for applications in functional regression and spectral analysis of operator families in random environments.

Abstract

The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable functions. The analysis focuses on a partial integral operator with a positive definite kernel depending on a measurable parameter, and establishes the equivalence of three fundamental properties: the completeness of the system of eigenfunctions in the corresponding vector-valued space, the injectivity of the adjoint embedding operator, and the existence of a pointwise spectral decomposition of the kernel in terms of the eigenvalues and eigenfunctions of a parameterized family of operators. The proof relies on constructing an isometric isomorphism between the Kaplansky-Hilbert module and the space of measurable sections of a Hilbert bundle, thereby reducing the problem to the application of the classical Mercer theorem on each fiber of the bundle. Furthermore, a formalism of vector-valued lifting is developed to guarantee the coherence of inner product structures between the original module and its bundle representation.

Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$

TL;DR

The paper generalizes Mercer’s theorem to vector-valued RKHSs defined over Kaplansky–Hilbert modules on , driven by a measurable kernel and a partial integral operator. Using an isometric isomorphism between the module and measurable sections of a Hilbert bundle, it reduces the problem to fiberwise classical Mercer theory and introduces a vector-valued lifting to preserve inner-product structure. The main result characterizes the vector-valued RKHS by the equivalence of three properties: basis completeness of the eigenfunctions, injectivity of the adjoint embedding, and a pointwise Mercer-type decomposition of the kernel, with measurability and essential boundedness conditions on the eigenvalues. This framework provides a constructive spectral description for multivariate, parameter-dependent kernels and paves the way for applications in functional regression and spectral analysis of operator families in random environments.

Abstract

The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable functions. The analysis focuses on a partial integral operator with a positive definite kernel depending on a measurable parameter, and establishes the equivalence of three fundamental properties: the completeness of the system of eigenfunctions in the corresponding vector-valued space, the injectivity of the adjoint embedding operator, and the existence of a pointwise spectral decomposition of the kernel in terms of the eigenvalues and eigenfunctions of a parameterized family of operators. The proof relies on constructing an isometric isomorphism between the Kaplansky-Hilbert module and the space of measurable sections of a Hilbert bundle, thereby reducing the problem to the application of the classical Mercer theorem on each fiber of the bundle. Furthermore, a formalism of vector-valued lifting is developed to guarantee the coherence of inner product structures between the original module and its bundle representation.

Paper Structure

This paper contains 8 sections, 6 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial Integral Operators
  4. Vector-Valued Reproducing Kernel Hilbert Spaces
  5. Definition and Fundamental Properties
  6. Vector-Valued Mercer Theorem
  7. Proof of Theorem \ref{['MT']}
  8. Conclusion

Key Result

Theorem 1

Let $T : L_{2,\infty}(\Omega \times S) \to L_{2,\infty}(\Omega \times S)$ be a partial integral operator of the form where the kernel $K : \Omega \times S^{2} \to \mathbb{C}$ satisfies the assumptions of Lemma 1 and generates a vector-valued RKHS $\mathbf{R} \subset L_{2,\infty}(\Omega \times S)$. Let $(\lambda_n)_{n \in \mathbb{N}} \subset L_\infty(\Omega)$ and $(x_n)_{n \in \mathbb{N}} \subset

Theorems & Definitions (15)

  • Theorem 1
  • Example 1
  • Remark 1
  • Definition 1
  • Proposition 1
  • Lemma 1
  • proof
  • Definition 2
  • Remark 2
  • Proposition 2
  • ...and 5 more