Hermitized Resolvent Representation

• Hermitized resolvent representation is a method that transforms non-selfadjoint or matrix-valued resolvents into symmetric forms using operator pencils and Laplace-type integral representations.
• It facilitates spectral analysis and functional calculus in complex settings such as Clifford and quaternionic modules, as well as the explicit construction of self-adjoint extensions in QFT-like Hamiltonians.
• This framework also underpins the synthesis of multidimensional positive-real functions and provides explicit orthogonal representations in the matrix moment problem via Hermitized continued-fraction expansions.

A Hermitized resolvent representation is a class of operator-theoretic and function-theoretic constructions in which a given resolvent or transfer function—often arising in non-selfadjoint, noncommutative, or matrix-valued analysis—is expressed as or factorized through a "hermitized" (i.e., symmetric or self-adjoint) operator pencil, quadratic operator, or continued-fraction scheme. These representations are fundamental in several areas: semigroup theory on Clifford or quaternionic modules, truncation of matrix moment problems, analysis of QFT-like Hamiltonians, and the structural theory of rational positive-real functions in several variables. The Hermitized resolvent formalism encodes positivity, spectral, and analytic information in a manner that is directly adapted to operator and matrix calculus, particularly when underlying non-commutativity, multivariable, or indefinite signature is present.

1. Hermitized Resolvent in the Context of Clifford and Quaternionic Functional Calculus

For a closed, right-linear operator $A$ generating a strongly continuous semigroup $T(t)=e^{tA}$ on a Banach two-sided $C\ell(0,n)$-module $X$, the quadratic (hermitized) resolvent is defined as

$$Q_q(A) = \left(A^2 - 2\,\mathrm{Re}(q)A + |q|^2I\right)^{-1},$$

with $q$ residing in the quadratic cone $QA \subset C\ell(0,n)$, $a = \mathrm{Re}(q)$, and $b = |\mathrm{Im}(q)|$ (Ghiloni et al., 2021).

The key feature is the Laplace-type integral representation valid for $a > \omega$ (where $\omega$ is the semigroup growth bound),

$Q_q(A)\,x = \int_{0}^{\infty} T(t)\,g_q(t)\,x\,dt,$

where $g_q(t)=t\,e^{-a t}\,\sinc(b t)$ is the unique solution of $g_q''(t) + 2a\,g_q'(t) + (a^2 + b^2)g_q(t) = 0$, $g_q(0)=0$, $g_q'(0)=1$. This realizes the quadratic resolvent directly in terms of $T(t)$ and the kernel $g_q$. The classical S-resolvent $C_q(A)$ is related by

$C_q(A) = Q_q(A) q^c - A Q_q(A)$

and itself possesses a Laplace representation

$$C_q(A) = \int_{0}^{\infty} T(t) e^{-tq} dt \quad (\mathrm{Re}(q) > \omega).$$

Hermitized resolvents underlie the spectral mapping theorem for the S-spectrum $\sigma_S(A)$, yield norm bounds $\|Q_q(A)\| \le M/(a-\omega)^2$, and establish the foundations for fractional powers and subordination semigroups in the noncommutative setting (Ghiloni et al., 2021).

2. Hermitized Kreĭn-Type Resolvent Formula for QFT-like Hamiltonians

For operators of the form $H + A^* + A$ with $H$ self-adjoint and $A$ an $H$-small operator between scales $S_{1/2}$ and $\mathcal{H}$, the self-adjoint extension $H_s$ admits an explicit domain description: $$\mathrm{dom}(H_s) = \{ \psi \in \mathcal{H} : \psi = \phi + G\xi,\, \phi \in \mathrm{dom}(H),\, \xi \in D_s \},$$ with $G$ the boundary map $(A R_\lambda)^*$, and $D_s$ the defect subspace where $(1-G)\xi \in \mathrm{dom}(H)$.

The core Hermitized resolvent formula is

$$(H_s-z)^{-1} = R_z + G_z\, [O_s + M_z]^{-1}\, G_{\bar z}^*$$

for $z \in \rho(H_s)\cap\rho(H)$, with $R_z = (H-z)^{-1}$, $G_z = (A R_z)^*$, and block-operators $M_z$ and $O_s$ encoding the boundary coupling and symmetric extension parameters (Posilicano, 2023).

This Hermitized representation is essential for analytic control: it reveals norm-resolvent convergence of ultraviolet cutoff Hamiltonians $H_n = H + A_n^* + A_n - E_n$ to $H_s$, provided $A_n\to A$ and $E_n$ are appropriate counterterms. The formulation generalizes to renormalization schemes in quantum field theory Hamiltonians, such as the Nelson model, where all limiting and regularization aspects are encoded by the Hermitized (boundary-augmented) resolvent structure (Posilicano, 2023).

3. Long (Hermitized) Resolvent Representation of Matrix-Valued Positive-Real Functions

Given a rational, homogeneous positive-real matrix-valued function $f(x)$ of several variables, the Hermitized (long-resolvent) representation is

$f(x) = D - B^* [L(x)]^{-1} B,$

where $L(x) = A_0 + \sum_{k=1}^{d}x_k A_k$ with each $A_k = A_k^*\succeq 0$ (Hermitian positive semi-definite) (Bessmertnyi, 2021). Alternatively, when partitioned as

$$L(x) = \begin{pmatrix}A_{11}(x) & A_{12}(x) \ A_{21}(x) & A_{22}(x)\end{pmatrix},$$

the Schur complement formula gives

$$f(x) = A_{11}(x) - A_{12}(x)[A_{22}(x)]^{-1}A_{21}(x).$$

The positivity of all $A_k$ is essential for representing homogeneity, symmetry, and positive-realness of $f$ in the right poly-halfplane. The connection is rigorously established using sum-of-squares (SOS) decompositions for the Wronskians $$W_k[q,P](z) = q(z)\partial_k P(z) - P(z)\partial_k q(z)$$, which are proven to be SOS forms if $f$ is rational, homogeneous, positive-real, and $d\geq 3$ (Bessmertnyi, 2021).

This representation provides canonical parameterizations for multidimensional circuit synthesis, matrix interpolation, and operator model theory.

4. Hermitized Factorization in the Matrix Moment Problem

In the truncated matrix Stieltjes moment problem, the resolvent matrix $U_m(z)$, constructed via specific orthogonal matrix polynomials $\phi_n$, $\psi_n$, $\chi_n$, and $\theta_n$ derived from a Stieltjes-definite sequence $(s_0,\dots,s_m)$, admits a Hermitized continued-fraction factorization (Choque-Rivero et al., 2016): $$U_m(z) Q_m(z) = \prod_{\ell=0}^{m} \left\{ \begin{pmatrix}I&0\ -z DSp^{-(\ell)}_0 & I\end{pmatrix}\begin{pmatrix}0 & \psi_0^{(\ell)}\ -z [\psi^{(\ell)}_0]^{-1*} & 0\end{pmatrix} \right\},$$ where $DSp^{\pm}$ are Dyukarev–Stieltjes parameters and each factor is Hermitian (semi-)definite for $z \in \mathbb{C} \setminus [0,\infty)$. The conjugating matrices $P_n$, $Q_n(z)$ ensure Hermiticity at each continued-fraction step.

Hermitization elucidates the analytic and spectral properties of the solution set: all solutions are generated through resolvent matrices corresponding to nonnegative Hermitian measures; moreover, the second-kind polynomials $Q_n$ become orthogonal with respect to an associated Hermitian measure (Choque-Rivero et al., 2016).

5. Structural and Analytic Implications

The Hermitized resolvent approach systematically "symmetrizes" analytic and algebraic structures—typically non-selfadjoint or indefinite—making spectral analysis, functional calculus, and positivity properties transparent. Concrete implications include:

• Direct Laplace-type integral representations for quadratic and polynomial resolvents, simplifying domain characterization and norm estimation (Ghiloni et al., 2021).
• Parameterization of all self-adjoint extensions of non-standard Hamiltonians, including ultraviolet-renormalized objects in QFT, by explicit boundary or defect operators (Posilicano, 2023).
• Synthesis of multidimensional positive-real functions and realization theory via Schur complements and matrix pencils (Bessmertnyi, 2021).
• Matrix moment theory reformulated entirely in terms of nonnegative Hermitian factors, offering spectral-theoretic representability and explicit orthogonality relations (Choque-Rivero et al., 2016).

A plausible implication is that these structures provide a canonical bridge between analytic positivity and algebraic (operator-theoretic) positivity across several inter-related domains.

6. Connections, Generalizations, and Applications

Hermitized resolvent representations underpin:

• Noncommutative functional calculi: The S-resolvent and its quadratic predecessor $Q_q(A)$ serve as the basic analytic tool for slice-regular functional calculi on Clifford and quaternionic modules (Ghiloni et al., 2021).
• Operator extension theory: Kreĭn-type Hermitized formulas, allowing for the explicit construction and approximation of self-adjoint extensions and their resolvents, play a central role in quantum physics and abstract boundary analysis (Posilicano, 2023).
• Systems and circuit theory: The long-resolvent realization enables explicit synthesis of multidimensional positive-real transfer functions, essential in network realization and passive system theory (Bessmertnyi, 2021).
• Moment problems: The Hermitized continued-fraction expansion attests to the deep links between positivity, moment determinacy, and spectral representation in the matrix-valued context (Choque-Rivero et al., 2016).

These methodologies are now standard in rigorous operator theory, multidimensional system theory, and mathematical physics, particularly wherever positivity in noncommutative or non-selfadjoint settings must be explicitly represented in analytic or algebraic form.