Mercer Operator-Valued Kernels

• General Mercer operator-valued kernels are extensions of classical Mercer kernels mapping into operator algebras, enabling spectral decompositions and convergence analyses.
• They underpin the construction of reproducing kernel Hilbert and Banach spaces, supporting structured learning, probabilistic modeling, and infinite-dimensional analysis.
• These kernels facilitate advanced applications in quantum information, operator learning for PDEs, and non-commutative analysis through factorization and Radon–Nikodym frameworks.

General Mercer operator-valued kernels are a rigorous and versatile generalization of the classical Mercer theorem to the setting where the kernel maps a product space into operator algebras, such as the bounded or trace-class operators on a separable Hilbert space. These kernels are foundational in modern analysis, probability, machine learning, operator theory, and quantum information, as they enable spectral expansions, structured learning, and probabilistic modeling in vector-valued and infinite-dimensional contexts.

1. Mathematical Structure of Mercer Operator-Valued Kernels

Let $K : X \times X \to \mathcal{L}(H)$ be a kernel mapping from a (possibly non-compact or infinite-dimensional) input space $X$ to the bounded linear operators on a separable Hilbert space $H$. Such a kernel is called (Hermitian) symmetric if $K(x, y) = K(y, x)^*$ and positive definite if for any choice of points $x_1,\dots,x_n \in X$ and vectors $a_1,\dots, a_n \in H$, the block matrix $[K(x_i, x_j)]$ satisfies

$$\sum_{i,j=1}^n \langle a_i, K(x_i, x_j) a_j \rangle_H \geq 0.$$

The associated integral operator $\mathcal{A}_K$ acts on $L^2(X; H)$ as

$$(\mathcal{A}_K f)(x) = \int_X K(x, y) f(y) \, d\mu(y)$$

with integration in the Bochner sense.

Under continuity and positivity assumptions, and compactness of the associated integral operator (which can be achieved via trace-class or Hilbert–Schmidt regularity of $K$), a Mercer-type expansion applies: $$K(x, y) = \sum_{j=1}^\infty \lambda_j \Phi_j(x) \otimes_H \Phi_j(y)$$ where the $\lambda_j > 0$ are the eigenvalues of $\mathcal{A}_K$ (countable and tending to zero), and the $\Phi_j \in L^2(X; H)$ are orthonormal eigenfunctions in $L^2(X; H)$. The convergence of the series is absolute and uniform under sufficient regularity, generalizing the scalar case to the operator-valued context (Santoro et al., 2023, Zweck et al., 9 Aug 2024).

In the matrix-valued case ($K : X \times X \to M_n(\mathbb{C})$), similar expansions hold component-wise, with convergence guaranteed under continuity and positivity (Vito et al., 2011, Neuman et al., 27 Mar 2024). In the $C^*$-algebraic setting, operator-valued kernels may map to $L(\mathfrak{A}, L(H))$ for a unital $C^*$-algebra $\mathfrak{A}$, with positivity expressed via sums involving $K(s_i, s_j)(a_i^* a_j)$ (Jorgensen et al., 27 May 2025).

2. Reproducing Kernel Hilbert and Banach Spaces

An operator-valued positive definite kernel $K$ characterizes a unique reproducing kernel Hilbert space (RKHS) $H_K$ of functions $f: X \to H$ with the reproducing property

$$f(x) = \langle f, K_x \rangle_{H_K}, \quad K_x(y) := K(y, x),$$

with the inner product structure determined by the kernel (Vito et al., 2011, Jorgensen et al., 23 Apr 2024, Jorgensen et al., 15 May 2024). For diagonal kernels $K(x, x') = k(x, x') \cdot I$ with $k$ a scalar Mercer kernel, $H_K$ is isometric to the Hilbert–Schmidt operators from $H_k$ (scalar RKHS) to $H$ (Yang et al., 25 Apr 2025, Yang et al., 14 Sep 2025).

Beyond Hilbert spaces, generalized Mercer kernels enable the construction of reproducing kernel Banach spaces (RKBSs), where the kernel may be expressed as

$K(x, y) = \sum_n \phi_n(x) \psi_n(y)$

(not necessarily symmetric or orthogonal) and the Banach space norm may be $\ell^p$-based, enabling sparsity and more general geometries (Xu et al., 2014).

3. Spectral Decompositions and Factorizations

The cornerstone of Mercer theory is spectral decomposition. For operator-valued kernels with compact, self-adjoint, positive integral operators, the Mercer expansion holds with absolute and uniform convergence (Santoro et al., 2023, Zweck et al., 9 Aug 2024). In non-Hermitian or more general settings, one can obtain absolutely convergent bilinear series expansions under additional smoothness (such as the $K^\infty$-Mercer type) (Novitskii, 2012).

A powerful structural insight is the factorization of operator-valued kernels via an explicit RKHS and a family of evaluation operators $V_x : H \to \mathcal{H}_{\tilde K}$ such that

$K(x, y) = V_x^* V_y.$

This factorization is fundamental in analysis and quantum information, closely paralleling Stinespring dilations and GNS constructions (Jorgensen et al., 23 Apr 2024, Jorgensen et al., 5 May 2024, Jorgensen et al., 15 May 2024, Jorgensen et al., 27 May 2025). In the $C^*$-algebraic case, the factorization takes the form $K(s, t)(a) = V(s)^* \pi(a) V(t)$ with a *-representation $\pi$.

When one operator-valued kernel $K$ is dominated by another $L$ (in the sense $K \leq L$), the Radon–Nikodym theory applies: there exists a unique positive operator $A$ on the larger RKHS such that $K(x, y) = V_L(x)^* A V_L(y)$ (Jorgensen et al., 15 May 2024, Jorgensen et al., 27 May 2025). This result generalizes the classical Radon–Nikodym theorem to the non-commutative setting.

4. Operator-Valued Mercer Kernels in Stochastic Processes and Functional Analysis

Mercer operator-valued kernels are the basis for constructing Hilbert space-valued Gaussian processes $W_x$ with covariance

$$\mathbb{E}[\langle a, W_x \rangle_H \langle W_y, b \rangle_H] = \langle a, K(x, y) b \rangle_H,$$

with explicit series representations via the orthonormal basis of the associated RKHS and independent standard normals (Jorgensen et al., 23 Apr 2024, Jorgensen et al., 5 May 2024, Jorgensen et al., 15 May 2024). The generalized Karhunen–Loève theorem for random flows in Hilbert spaces yields the expansion

$$\chi(t) = \sum_{j=1}^\infty \langle \chi, \Phi_j \rangle_{L^2(\mathcal{T}, H)} \Phi_j(t)$$

with uncorrelated coefficients and uniform convergence over time (Santoro et al., 2023).

These structural properties guarantee optimal finite-dimensional approximations and underpin functional data analysis for infinite-dimensional observations (e.g., in stochastic partial differential equations, time-evolving fields in physics, and quantum systems).

In operator theory, Mercer-type factorizations enable explicit dilation constructions: for contractions or completely positive maps, the construction of minimal unitary dilations or Stinespring representations is driven by such kernel decompositions (Jorgensen et al., 23 Apr 2024, Jorgensen et al., 27 May 2025).

5. Learning and Approximation with Operator-Valued Kernels

Vector-valued and operator-valued kernels underpin structured prediction and operator learning frameworks. In functional regression and operator learning, one seeks to estimate nonlinear operators $h^\dagger : X \to Y$ between infinite-dimensional Banach or Hilbert spaces. If $h^\dagger$ lies in the RKHS induced by a Mercer OVK, iterative algorithms such as regularized stochastic gradient descent exhibit near-optimal, dimension-free polynomial convergence rates, with explicit high-probability bounds on prediction and estimation error (Yang et al., 25 Apr 2025, Yang et al., 14 Sep 2025). The framework supports misspecification quantification via vector-valued interpolation spaces and is applicable to integral operator learning, encoder–decoder architectures, and regression problems for complex outputs.

In machine learning, operator-valued RKHSs enable kernel-based approaches for multi-task, multi-output, and function-valued prediction, including structured output embedding and support vector machines (Xu et al., 2014, Kadri et al., 2015). Notably, random Fourier feature constructions have been extended to the operator-valued setting, enabling kernel approximation and scalable algorithms while retaining convergence guarantees (Brault et al., 2016, Minh, 2016).

Advances such as entangled kernels extend beyond separable or "diagonal" forms, enabling input-dependent output correlation structures crucial for realistic multi-output learning (Huusari et al., 2021).

6. Generalizations, Regularity, and Universality Conditions

Several fundamental generalizations of Mercer’s theorem have been established for operator-valued kernels:

• Matrix-valued and $C^*$-algebraic kernels: The Mercer–Young theorem has been extended to matrix-valued kernels on separable metric spaces, equating pointwise and integral positive definiteness (Neuman et al., 27 Mar 2024).
• Operator-valued kernels on infinite-dimensional spaces: Absolute and uniform convergence of Mercer expansions requires the kernel be continuous and positive definite on $X \times X$ (compact), with suitable regularity (such as Hölder continuity with exponent $> \frac12$ and boundedness in trace class norm) to ensure the integral operator is trace class (Zweck et al., 9 Aug 2024).
• Universality and differentiability: Necessary and sufficient conditions for universality and $C^q$-universality of operator-valued kernels hinge on energy integrals involving Radon measures, with strict positive definiteness in each direction implying universality (Guella, 2020).
• Banach space settings: The notion of generalized Mercer kernels and the construction of RKBSs with $L^p$-type geometry extend the spectral representation paradigms and enable the design of sparse and regularized learning algorithms with Banach-valued outputs (Xu et al., 2014).

7. Advanced Topics and Applications

Mercer operator-valued kernels play foundational roles in the following advanced areas:

• Operator learning for PDEs and dynamical systems: The kernel-based stochastic approximation paradigm enables the learning of nonlinear operators such as solution maps to PDEs (e.g., Navier–Stokes) with rigorous finite-sample guarantees and empirical validation (Yang et al., 14 Sep 2025).
• Quantum measurement and inverse problems: The explicit factorization of operator-valued kernels supports kernel-based approaches to quantum state tomography, optimization of quantum gates, and analysis of positive operator-valued measures (POVMs), bridging RKHS theory and quantum information (Jorgensen et al., 5 May 2024, Jorgensen et al., 15 May 2024).
• Non-commutative Radon–Nikodym theory: The domination and factorization results yield Radon–Nikodym theorems for completely positive maps between operator algebras, linking kernel theory to non-commutative probability and operator algebra (Jorgensen et al., 27 May 2025).

A plausible implication is that these structural advances contribute not only to theory but also provide practical methodologies for high-dimensional, structured, and quantum learning problems.

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Summary Table: Key Structural Elements

Element	Mathematical Object	Reference(s)
Mercer Expansion	$$K(x,y) = \sum_j \lambda_j \Phi_j(x) \otimes \Phi_j(y)$$	(Vito et al., 2011, Santoro et al., 2023, Zweck et al., 9 Aug 2024)
RKHS Construction	$H_K$ of $H$-valued functions, reproducing via $K$	(Vito et al., 2011, Jorgensen et al., 23 Apr 2024, Jorgensen et al., 5 May 2024)
Factorization	$K(x,y) = V_x^* V_y$	(Jorgensen et al., 23 Apr 2024, Jorgensen et al., 15 May 2024)
Gaussian Process Covariance	$$\mathbb{E}[\langle a, W_x \rangle_H \langle W_y, b \rangle_H] = \langle a, K(x,y) b \rangle_H$$	(Jorgensen et al., 23 Apr 2024, Jorgensen et al., 5 May 2024)
Radon–Nikodym Domination	$$K \leq L \iff \exists A \geq 0, K(x,y) = V_L(x)^* A V_L(y)$$	(Jorgensen et al., 27 May 2025)
Trace Class/Regularity	$K$ continuous, Hermitian, Hölder $>\frac12$ for trace class	(Zweck et al., 9 Aug 2024)

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General Mercer operator-valued kernels form the analytic, probabilistic, and algorithmic backbone of modern operator theory, machine learning, and quantum information, providing spectral, probabilistic, and algebraic tools for infinite-dimensional and highly structured problems.