Resolvent-Generated Generalized Spectral Operators for Nonlinear Dynamical Systems via Koopman Semigroups

Abstract

Spectral methods form a cornerstone of linear dynamics, where evolution is resolved into harmonic modes governed by eigenvalues and spectral measures of normal operators. For nonlinear dynamical systems, however, the harmonic eigenfunction paradigm typically breaks down: Koopman operators are often non-normal, may possess a continuous spectrum, and rarely admit complete eigenbases on natural observable spaces. This work develops a resolvent-centered operator-theoretic framework for generalized spectral representations of nonlinear flows through their associated Koopman $C_0$ semigroups. Rather than relying on diagonalization, we construct resolvent-generated generalized spectral operators that yield weak integral representations of the semigroup valid in non-normal and continuous-spectrum regimes. We show that, under mild polynomial resolvent growth bounds along vertical lines, these spectral distributions become finite complex Radon measures on bounded spectral regions, thereby recovering a measure-theoretic interpretation analogous to classical spectral integrals. In the normal case, the framework reduces to the standard spectral theorem. The resulting resolvent-based perspective naturally incorporates pseudospectral amplification and transient growth, providing a unified description of both asymptotic and non-modal dynamics.

1. Introduction

Spectral representations constitute one of the most powerful analytical paradigms in the study of dynamical systems. In linear settings, classical harmonic analysis and eigenfunction expansions yield complete and physically interpretable descriptions of evolution [1,2], underpinning modal decomposition, stability analysis, and a broad range of applications. In that regime, the state can often be expanded in eigenmodes of a finite-dimensional matrix or a normal operator on a Hilbert space [3,4], and long-time behavior is read directly from spectral data.

For nonlinear dynamical systems, the direct extension of this harmonic paradigm encounters structural obstacles. First, the superposition principle fails at the level of the state. Second, even when one studies linearizations along trajectories, the resulting time-dependent operators need not admit a global diagonalization [5]. Third, when nonlinear dynamics are lifted to infinite-dimensional linear operators acting on observables, most prominently through Koopman’s approach [6,7], the induced operators are typically non-normal and may possess continuous spectra, lack a complete eigenbasis, and exhibit strong transient amplification phenomena more faithfully captured by resolvent norms and pseudospectral structure than by eigenvalues alone [8]. These issues become acute in chaotic systems, where mixing, stretching and folding, and the coexistence of multiple time scales frustrate purely eigenfunction-based analyses. For example, for mixing measure-preserving systems, the Koopman operator on $L^2$ often exhibits purely continuous spectra (see, e.g., [9,10]).

Koopman’s seminal insight [6] is that while the nonlinear state evolution is not linear, the evolution of observables composed with the flow is linear. This yields a linear (often infinite-dimensional) representation of nonlinear dynamics. Mezić and collaborators further developed the modern spectral viewpoint, connecting Koopman operators with ergodic theory, model reduction, and data-driven decomposition [7]. In its simplest form, the Koopman program seeks eigenfunctions whose level sets are invariant or coherent and whose eigenvalues encode growth and oscillation rates. Yet in many realistic systems, eigenfunctions may be scarce or fail to span natural function spaces, and the operator may be far from diagonalizable. In such cases, a more general representation theory is needed, one that retains the operator-theoretic advantages of the Koopman lift while replacing strict harmonic decompositions by generalized spectral structures compatible with non-normality, the continuous spectrum, and weak notions of mode expansion.

The purpose of this paper is to develop a rigorous and detailed operator-theoretic framework for generalized spectral operators associated with nonlinear dynamical systems through their Koopman semigroups. The term generalized is used here in a precise functional-analytic sense: rather than requiring diagonalization in an orthonormal eigenbasis, we build representation tools based on semigroup theory, resolvent analysis, weak integral decompositions, and generalized eigenstructures (including Jordan-type behavior in appropriate settings), formulated in Banach and Hilbert space frameworks adapted to dynamics. This approach is designed to (i) keep the nonlinear system arbitrary throughout the theoretical development, (ii) separate analytic hypotheses (regularity of the flow, choice of observable space, invariance, and continuity properties) from spectral conclusions, and (iii) naturally accommodate non-harmonic and pseudospectral phenomena without forcing assumptions that are rarely satisfied in nonlinear, chaotic regimes.

The present work is not intended as a replacement of classical semigroup or spectral theory, but rather as a systematic extension of Koopman operator analysis beyond the harmonic eigenfunction paradigm that dominates much of the existing literature. Classical Koopman theory, beginning with [6,11] and developed extensively in [7,12,13,14], emphasizes eigenfunctions and modal decompositions whenever available. However, for generic nonlinear and chaotic systems, the Koopman generator is non-normal, may possess a predominantly continuous spectrum, and typically lacks a complete eigenbasis. In such regimes, eigenfunction expansions provide, at best, partial information.

While the functional-analytic foundations of $C_0$ semigroups and resolvent theory are classical (see, e.g., [3,15,16,17]), pseudospectral analysis of non-normal operators is well developed in linear settings [8,18], and there are solid background developments (e.g., [19,20]) partially supporting the further developments that we seek to contribute; to the best of our knowledge, a unified framework that places resolvent-based functional calculus, weak spectral representations, and pseudospectral structure at the center of Koopman analysis for nonlinear flows has not yet been systematically developed.

The goal of this paper is to address these gaps by developing a resolvent-centered framework for Koopman spectral analysis that remains meaningful in the presence of non-normal generators and a continuous spectrum. Several ingredients used in our construction build from classical grounds, including the semigroup-generator correspondence for $C_0$ semigroups [15,21], Bromwich-type inversion formulas for semigroup representations [15], and the Dunford–Taylor functional calculus for operators [3]. However, their systematic organization in a Koopman setting leads to an innovatively useful structural viewpoint.

The contribution of this paper is, therefore, conceptual and structural: we develop a coherent operator-theoretic framework in which (i) nonlinear dynamics are treated through their associated Koopman semigroups on general Banach or Hilbert spaces of observables, (ii) the generator resolvent is elevated to a primary spectral object, enabling generalized functional calculi and weak integral representations without diagonalizability assumptions, and (iii) non-normal and pseudospectral effects are incorporated intrinsically into the spectral description rather than regarded as perturbative anomalies. In contrast to approaches focused primarily on discrete eigenvalues or finite-dimensional truncations, the framework presented here is designed to remain valid in settings with a continuous spectrum, Jordan-type behavior, or strong transient amplification. In this sense, the work provides a systematic extension of Koopman spectral analysis that retains the structural advantages of classical operator theory while removing assumptions that are rarely satisfied in nonlinear and chaotic regimes.

Formally, the novelty lies in defining and developing resolvent-generated generalized spectral operators for Koopman generators and in showing how weak semigroup representations, functional calculi, and pseudospectral structure can be treated in a single, generator–resolvent framework without diagonalizability assumptions.

More precisely, we show that weak spectral objects associated with the Koopman generator can be constructed directly from the resolvent. The central analytical result (Theorem 1) establishes that under polynomial resolvent growth along vertical lines, these objects admit a representation by finite complex Radon measures on bounded spectral windows. This provides a bridge between resolvent-based representations and measure-valued spectral descriptions that are compatible with non-normal dynamics.

The results presented here are intended primarily as structural tools. They provide a framework within which both classical spectral expansions and more general continuous-spectrum phenomena can be analyzed in a unified manner, particularly in situations where eigenfunction-based approaches are insufficient due to non-normality or strong transient growth. In this respect the approach complements existing Koopman operator methods developed in the dynamical systems literature [7,13,19].

To illustrate the framework concretely, this paper also includes a finite-dimensional non-normal example explicitly showing how a resolvent structure captures generalized spectral behavior beyond eigenvalue analysis.

The manuscript is structured as follows. Section 2 frames the present contribution relative to previous works, namely, related lines of research. Then, building from established functional-analytical, operator-theoretic, semigroup-theoretic, nonlinear dynamical, and mathematical physical developments [1,3,4,15,16,21,22], Section 3, Section 4 and Section 5 provide the theoretical and methodological background from which the main advances of the present contribution later emerge in Section 6, Section 7 and Section 8. In particular, Section 3 overviews the formulation of nonlinear flows and their induced Koopman evolution on observables. Section 4 establishes the functional-analytic setting, discussing choices of observable spaces, continuity properties, and dual pairings needed for weak representations. Section 5 contextualizes the Koopman semigroup, its infinitesimal generator and semigroup-generator relations. Section 6 develops generalized spectral operator constructions, including resolvent-based representations and pseudospectral diagnostics. Section 7 presents generalized invariant structures and resolvent-based spectral selection, including an explicit treatment of a finite-dimensional non-normal example. Section 8 presents principled computational strategies for approximating generalized spectral quantities in a manner consistent with the underlying operator-theoretic framework, and Section 9 wraps this contribution with concluding remarks and perspectives.

2. Framing Relative to Previous Works

While succinctly addressed in the introduction, it is important to reiterate and further elaborate on the distinctions between the framework proposed in the present contribution and two related lines of research, namely transfer operators and resonance theory on the one hand, and finite-dimensional or data-driven Koopman approximations on the other.

In the ergodic and hyperbolic setting, spectral analyses of dynamical systems are often conducted via Perron–Frobenius or transfer operators acting on anisotropic Banach spaces, leading to the theory of Ruelle–Pollicott resonances and decay of correlations (see, e.g., [23,24,25,26,27,28,29]). These approaches typically rely on delicate functional-analytic constructions tailored to hyperbolic structure and aim at identifying discrete resonances that govern statistical properties such as mixing rates.

However, the perspective adopted in the present contribution is different in scope and emphasis. Rather than constructing specialized anisotropic spaces to recover discrete resonance spectra, we formulate a semigroup-based Koopman framework on general Banach or Hilbert spaces of observables and develop generalized spectral operators directly from the resolvent of the Koopman generator. The resulting weak spectral representations are not restricted to hyperbolic systems, do not presuppose discrete resonance expansions, and remain meaningful in the presence of a continuous spectrum or strong non-normality. In this sense, the present work complements resonance theory by discussing a resolvent-centered formulation that remains meaningful beyond settings where discrete Ruelle–Pollicott spectra can be isolated.

A second major line of research concerns finite-dimensional and data-driven approximations of Koopman operators, including dynamic mode decomposition (DMD) [30], extended DMD, and related modal extraction techniques [7,12,13,14]. More recently, nonlinear and machine-learning extensions have been developed, including kernel-based and neural approaches that learn finite-dimensional Koopman representations directly from data [31,32,33,34]. These methods approximate the action of the Koopman operator or its generator on a chosen observable subspace and typically interpret the resulting matrix eigenvalues as approximations of Koopman spectra. While considerably successful in applications, such approaches inherently produce non-normal finite-dimensional operators whose spectra may be highly sensitive to truncation, discretization, and noise.

The framework developed in the present study is conceptually distinct: rather than prioritizing discrete eigenvalues extracted from finite-dimensional projections, we place the resolvent and pseudospectrum of the (possibly infinite-dimensional) Koopman generator at the center of analysis. This shift emphasizes quantities that are stable under weak limits and more directly connected to transient growth and non-modal dynamics, thereby providing a theoretical foundation for interpreting and, where necessary, reinterpreting modal information obtained from finite-dimensional or data-driven approximations. In particular, convergence of resolvents and pseudospectra, rather than pointwise convergence of eigenvalues, emerges as the structurally appropriate notion of spectral approximation in non-normal Koopman settings [8,18].

In settings where transfer-operator methods yield discrete Ruelle–Pollicott resonances, one may view such resonances as appearing through the singularity structure of the relevant resolvents; the present paper does not assume hyperbolicity or discrete resonance expansions, but the resolvent-generated viewpoint is compatible with those regimes when additional dynamical hypotheses are imposed.

The central technical contribution of this work is the formulation of resolvent-generated generalized spectral operators for Koopman generators associated with nonlinear flows. Rather than postulating eigenfunction expansions, the construction proceeds directly from the analytic structure of the generator resolvent. This yields weak spectral representations of the Koopman semigroup that remain valid in the presence of non-normality, a continuous spectrum, or Jordan-type behavior.

More precisely, we show that for any $C_0$ Koopman semigroup, the Bromwich inversion formula induces a generalized spectral operator acting on observables, with semigroup evolution represented as a weak superposition of exponential contributions over a complex spectral parameter space. Furthermore, under mild polynomial resolvent growth bounds along vertical lines, these generalized spectral distributions become finite complex Radon measures on bounded spectral regions, thereby recovering a measure-theoretic interpretation analogous to classical spectral integrals. In the normal-operator case, the framework collapses to the usual projection-valued spectral theorem, while in strongly non-normal regimes, it naturally incorporates pseudospectral amplification through the resolvent.

In this way, the theory developed here interpolates rigorously between classical harmonic spectral analysis and the resolvent-dominated behavior typical of nonlinear and chaotic dynamics.

3. Nonlinear Dynamical Systems and Induced Linear Operators

3.1. Flows, Cocycles, and Evolution Maps

Let $\mathcal{X}$ be a smooth manifold or an open subset of ${\mathbb{R}}^d$ equipped with its Borel $\sigma$-algebra. Consider an autonomous nonlinear dynamical system

$$\dot{x}\left(t\right)=f\left(x\right(t\left)\right),$$

(1)

where $f:\mathcal{X}\to {\mathbb{R}}^d$ is at least locally Lipschitz, so that for each initial condition $x_0\in \mathcal{X}$, there exists a unique maximal solution $t\mapsto {\phi }^t\left(x_0\right)$ defined on a (possibly finite) interval containing $t=0$. When solutions exist globally for all $t\ge 0$ (or for all $t\in \mathbb{R}$), the maps ${\phi }^t:\mathcal{X}\to \mathcal{X}$ form a semiflow (or flow) satisfying

$${\phi }^0=\mathrm{id},\hspace{1em}\hspace{1em}{\phi }^{t+s}={\phi }^t\circ {\phi }^s,$$

(2)

for all times that both sides are defined. In what follows, we focus primarily on forward-time evolution (a semiflow), which is the natural setting for strongly continuous semigroups on observable spaces. When a global two-sided flow exists, analogous constructions hold on $t\in \mathbb{R}$ with corresponding group properties.

We emphasize that the Koopman framework is not tied to differentiability of f beyond what is needed to define the flow. Higher regularity becomes relevant later when identifying a generator in terms of Lie derivatives and when selecting function spaces on which that generator is densely defined and closed.

3.2. Observable Spaces and the Koopman Evolution

An observable is a function $g:\mathcal{X}\to \mathbb{C}$ representing a measurable quantity of interest. The nonlinear evolution of the state induces a linear evolution on observables by composition with the flow:

$$\left(U^{t}g\right)\left(x\right) := g\left({\phi }^t\right(x\left)\right).$$

(3)

Whenever $U^t$ maps a chosen function space $\mathcal{F}$ into itself, the family ${\left\{U^t\right\}}_{t\ge 0}$ defines a linear operator semigroup on $\mathcal{F}$. Linearity is immediate from the definition. The semigroup property follows from (2):

$$U^{t+s}g=U^t\left(U^{s}g\right),\hspace{1em}\hspace{1em}{U}^{0}g=g.$$

(4)

The analytical content is in the choice of $\mathcal{F}$ and the topology in which $t\mapsto U^t$ is continuous. For example, if $\mathcal{F}=C_b\left(\mathcal{X}\right)$ is the space of bounded continuous observables on a locally compact $\mathcal{X}$, then $U^t$ preserves continuity when ${\phi }^t$ is continuous in x. If $\mathcal{F}=L^p(\mathcal{X},\mu )$ for an invariant (or quasi-invariant) measure $\mu$, then $U^t$ can often be defined as a bounded operator (or at least an isometry) depending on the invariance and nonsingularity properties of ${\phi }^t$ with respect to $\mu$.

A central observation motivating the generalized spectral framework developed below is that the Koopman operator family $\left\{U^t\right\}$ is typically non-normal on natural spaces of observables. In dissipative or chaotic systems, this non-normality may produce strong transient amplification and spectral sensitivity that cannot be inferred from eigenvalues alone. These phenomena are more accurately captured by resolvent behavior and the pseudospectral structure.

4. Function Spaces, Continuity, and Dual Pairings

4.1. Banach and Hilbert Settings for Observables

To place (3) in a rigorous operator-theoretic framework, we fix a Banach space $(\mathcal{F},{∥·∥}_{\mathcal{F}})$ of observables, such that for each $t\ge 0$, $U^t:\mathcal{F}\to \mathcal{F}$ is a bounded linear operator. The specific choice is problem-dependent, but several recurring settings are particularly useful.

If $\mathcal{X}$ is locally compact and ${\phi }^t$ is continuous, one may take $\mathcal{F}=C_0\left(\mathcal{X}\right)$, the continuous functions vanishing at infinity, or $C_b\left(\mathcal{X}\right)$, the bounded continuous functions. In dissipative systems possessing a compact global attractor, it is also natural to restrict observables to the attractor and use $C\left(\mathcal{A}\right)$ with the supremum norm.

If $(\mathcal{X},\mathcal{B},\mu )$ is a measure space and the flow is $\mu$-nonsingular (i.e., $\mu \left(A\right)=0\Rightarrow \mu \left({\phi }^{-t}\right(A\left)\right)=0$ for appropriate inverses), one may define $U^t$ on $L^p\left(\mu \right)$ by composition almost everywhere. When $\mu$ is invariant and $p=2$, the Koopman operators are isometries on $L^2\left(\mu \right)$ but not necessarily normal unless an additional structure holds. For mixing systems, the spectrum on $L^2\left(\mu \right)$ can be purely continuous, which is precisely the regime where generalized representation theory is needed.

In what follows, we do not commit to a single $\mathcal{F}$ globally. Instead, we make the analytic hypothesis of each theorem explicit in terms of boundedness, strong continuity, domain density, and dual pairing properties.

4.2. Strong Continuity and Basic Semigroup Hypotheses

The foundational analytical requirement for a semigroup approach is strong continuity: we say that ${\left\{U^t\right\}}_{t\ge 0}\subset \mathcal{L}\left(\mathcal{F}\right)$ is a strongly continuous (or $C_0$) semigroup if

$$\underset{t\downarrow 0}{lim}{∥U^{t}g-g∥}_{\mathcal{F}}=0,\hspace{1em}\hspace{1em}\forall g\in \mathcal{F}.$$

(5)

In applications, strong continuity typically follows from continuity of the flow combined with density and approximation properties of the chosen observable space. When $\mathcal{F}=C_0\left(\mathcal{X}\right)$ and ${\phi }^t$ is continuous, standard arguments yield strong continuity under mild compactness or uniformity conditions on ${\phi }^t$ over bounded time intervals. In $L^p\left(\mu \right)$ settings, strong continuity can be more delicate and may require additional assumptions, such as continuity of $t\mapsto g\circ {\phi }^t$ in $L^p$ for g in a dense subspace.

Once strong continuity holds, semigroup theory furnishes an infinitesimal generator A capturing the local-in-time behavior of $U^t$, as well as resolvent operators ${(\lambda I-A)}^{-1}$ and a functional calculus that can be used to define generalized spectral operators. These constructions, along with their non-normal and weak integral extensions, are developed in subsequent sections.

4.3. Dual Pairings and Weak Notions of Representation

Many generalized spectral decompositions are most naturally expressed not as norm-convergent expansions in $\mathcal{F}$ but as weak representations against test functionals. For this reason, we will systematically employ dual pairings between $\mathcal{F}$ and a chosen dual space ${\mathcal{F}}^{\ast }$ (or, more generally, between two spaces in duality).

Given a Banach space $\mathcal{F}$, its continuous dual ${\mathcal{F}}^{\ast }$ consists of all continuous linear functionals $\mathcal{l}:\mathcal{F}\to \mathbb{C}$. We write $⟨\mathcal{l},g⟩:=\mathcal{l}\left(g\right)$. The adjoint (or dual) operators $U^{t\ast }:{\mathcal{F}}^{\ast }\to {\mathcal{F}}^{\ast }$ are defined by

$$⟨U^{t\ast }\mathcal{l},g⟩:=⟨\mathcal{l},U^{t}g⟩,\hspace{1em}\hspace{1em}\mathcal{l}\in {\mathcal{F}}^{\ast }, g\in \mathcal{F}.$$

(6)

Weak representations of $U^t$ are then statements of the form

$$⟨\mathcal{l},U^{t}g⟩={\int }_{\Sigma }\Phi (t,\sigma ) d{\mu }_{\mathcal{l},g}(\sigma ),$$

(7)

where $\Sigma$ is a spectral parameter set and ${\mu }_{\mathcal{l},g}$ is a complex measure depending on the dual pair $(\mathcal{l},g)$. Such representations generalize classical spectral integrals (which arise for normal operators via projection-valued measures) and remain meaningful in non-normal and continuous-spectrum regimes. Establishing conditions under which a relation of the form (7) is valid, and clarifying how it interfaces with resolvents and pseudospectra, is a principal goal of the generalized spectral theory developed later.

In the next section, we proceed from these foundational settings to the Koopman semigroup-generator correspondence, define the infinitesimal generator rigorously (including Lie-derivative expressions on appropriate cores), and develop the resolvent framework that supports generalized spectral operator constructions beyond the harmonic/eigenfunction paradigm.

5. Koopman Semigroups and Infinitesimal Generators

5.1. Strongly Continuous Koopman Semigroups

Let $(\mathcal{F},\parallel ·{\parallel }_{\mathcal{F}})$ be a Banach space of observables, such that for each $t\ge 0$, the Koopman operator

$$\left(U^{t}g\right)\left(x\right)=g\left({\phi }^t\left(x\right)\right)$$

(8)

defines a bounded linear operator on $\mathcal{F}$. Assume throughout this section that ${\left\{U^t\right\}}_{t\ge 0}$ forms a strongly continuous semigroup on $\mathcal{F}$, i.e.,

$$U^0=I,\hspace{1em}\hspace{1em}{U}^{t+s}=U^t{U}^s,\hspace{1em}\hspace{1em}\underset{t\downarrow 0}{lim}{\parallel U^{t}g-g\parallel }_{\mathcal{F}}=0 \forall g\in \mathcal{F}.$$

(9)

Strong continuity is the minimal assumption allowing a differential-in-time description of the Koopman evolution. In particular, it ensures that the semigroup admits an infinitesimal generator in the sense of semigroup theory. Importantly, this requirement is compatible with a wide range of observable spaces used in nonlinear dynamics, including $C_0\left(\mathcal{X}\right)$ and $C\left(\mathcal{A}\right)$ on compact invariant sets, and suitable $L^p$ spaces when the flow is nonsingular with respect to a reference measure.

While each $U^t$ is induced by a nonlinear flow, the family $\left\{U^t\right\}$ is linear and autonomous. This observation permits the full machinery of linear operator theory to be applied, provided one works in function spaces adapted to the dynamics rather than attempting to linearize the state evolution itself.

5.2. Definition of the Infinitesimal Generator

The infinitesimal generator A of the Koopman semigroup ${\left\{U^t\right\}}_{t\ge 0}$ is defined in the standard way for $C_0$ semigroups.

Definition 1.

The infinitesimal generator $A:\mathrm{dom}\left(A\right)\subset \mathcal{F}\to \mathcal{F}$ is defined by

$$Ag:=\underset{t\downarrow 0}{lim}{\displaystyle \frac{U^{t}g-g}{t}},$$

(10)

with domain

$$\mathrm{dom}(A):=\left\{g\in \mathcal{F}:\underset{t\downarrow 0}{lim}{\displaystyle \frac{U^{t}g-g}{t}} \mathit{exists} \mathit{in} \mathcal{F}\right\}.$$

(11)

The operator A is linear, densely defined, and closed. The density of $\mathrm{dom}\left(A\right)$ follows from general semigroup theory and does not require additional structure beyond strong continuity. Closedness means that if $g_n\to g$ in $\mathcal{F}$ and $A{g}_n\to h$ in $\mathcal{F}$, then $g\in \mathrm{dom}\left(A\right)$ and $Ag=h$.

The Koopman semigroup may be formally written as

$$U^t=e^{tA},$$

(12)

but this exponential notation must be interpreted in the semigroup sense rather than via a convergent power series, since A is generally unbounded.

5.3. Generator as a Lie Derivative

When the underlying flow is sufficiently smooth and the observable space is chosen accordingly, the generator A admits a concrete representation as a Lie derivative along the vector field f defining the dynamics.

Assume that $\mathcal{X}\subset {\mathbb{R}}^d$ is open, that $f\in C^1(\mathcal{X};{\mathbb{R}}^d)$, and that $\mathcal{F}$ contains a dense subspace of continuously differentiable observables with appropriate decay or boundedness properties. For $g\in C^1\left(\mathcal{X}\right)$, differentiation along trajectories yields

$${\displaystyle \frac{d}{dt}}g({\phi }^t(x)){|}_{t=0}=\nabla g(x)·f(x).$$

(13)

Consequently, for such observables,

$$Ag=f·\nabla g$$

(14)

(see, e.g., [15,25]). Equation (14) should be interpreted with care. While it provides an explicit expression for A on smooth observables, the generator is generally defined on a larger domain obtained by closure in $\mathcal{F}$. The Lie derivative form thus serves as a core representation rather than a complete characterization of $\mathrm{dom}\left(A\right)$. Establishing that a given subspace is a core for A is a nontrivial analytical step and depends on both the dynamics and the choice of $\mathcal{F}$.

5.4. Growth Bounds and Spectral Bounds

The norm growth of the Koopman semigroup is controlled by the generator through classical results of semigroup theory. Define the growth bound

$${\omega }_0:=inf\left\{\omega \in \mathbb{R}:\exists M\ge 1 \mathrm{such} \mathrm{that} \parallel U^t\parallel \le M{e}^{\omega t} \forall t\ge 0\right\}.$$

(15)

This quantity measures the maximal exponential rate at which observables may grow under composition with the flow.

The spectrum $\sigma \left(A\right)$ of the generator is a closed subset of $\mathbb{C}$, and its spectral bound is defined by

$$s\left(A\right):=sup\{\Re \lambda :\lambda \in \sigma (A\left)\right\}.$$

(16)

For $C_0$ semigroups, one always has $s\left(A\right)\le {\omega }_0$. Equality need not hold, particularly for non-normal generators, and the gap between ${\omega }_0$ and $s\left(A\right)$ is one manifestation of non-normal transient growth. This distinction is crucial in nonlinear dynamics, where Koopman operators are rarely normal and where transient amplification of observables may dominate long-time asymptotics.

5.5. Resolvent of the Koopman Generator

For $\lambda \in \mathbb{C}$ with $\Re \lambda$ sufficiently large, the resolvent of the generator is defined by

$$R(\lambda ,A):={(\lambda I-A)}^{-1}.$$

(17)

The resolvent set $\rho \left(A\right)$ consists of those $\lambda$ for which this inverse exists as a bounded operator on $\mathcal{F}$.

For $C_0$ semigroups, the resolvent admits the integral representation

$$R(\lambda ,A)g={\int }_0^{\infty }{e}^{-\lambda t}{U}^{t}g dt,\hspace{1em}\hspace{1em}\Re \lambda >{\omega }_0,$$

(18)

where the integral converges in the Bochner sense. Formula (18) provides a direct link between the Koopman evolution and generalized spectral analysis: it expresses the resolvent as a Laplace transform of the semigroup and serves as the primary analytic object for defining spectral measures, functional calculi, and weak representations in non-normal settings.

In particular, even when A has no eigenvalues or only a small point spectrum, the resolvent remains well defined and informative. Its norm and analytic structure encode both spectral and pseudospectral properties, which are essential for understanding transient behavior in chaotic and strongly nonlinear systems.

5.6. Eigenfunctions and Limitations of Harmonic Representations

A function $g\in \mathrm{dom}\left(A\right)$ is called a Koopman eigenfunction with eigenvalue $\lambda \in \mathbb{C}$ if

$$Ag=\lambda g,$$

(19)

or equivalently,

$$U^{t}g=e^{\lambda t}g \forall t\ge 0.$$

(20)

Eigenfunctions provide invariant coordinates and play a central role in classical Koopman spectral theory. However, for generic nonlinear systems, the set of eigenfunctions is typically small and may fail to span a dense subspace of $\mathcal{F}$. Moreover, even when eigenfunctions exist, the generator may not be diagonalizable, and Jordan-type behavior or continuous spectral components may dominate.

These limitations motivate the development of generalized spectral operators that do not rely on a complete set of eigenfunctions. Instead, the resolvent-based approach developed here allows one to represent the Koopman evolution through integral transforms and weak decompositions that remain valid in the absence of harmonic completeness.

5.7. Adjoint Semigroups and Dual Generators

Let ${\mathcal{F}}^{\ast }$ denote the continuous duality of $\mathcal{F}$. The adjoint operators $U^{t\ast }:{\mathcal{F}}^{\ast }\to {\mathcal{F}}^{\ast }$ form a weak-∗ continuous semigroup, given in the same form as (6), by

$$⟨U^{t\ast }\mathcal{l},g⟩=⟨\mathcal{l},U^{t}g⟩,\hspace{1em}\hspace{1em}\mathcal{l}\in {\mathcal{F}}^{\ast }, g\in \mathcal{F}.$$

(21)

Under appropriate conditions, this adjoint semigroup admits a generator $A^{\ast }$ that is the Banach-space adjoint of A. Dual generators play a crucial role in weak spectral representations, where one pairs observables with test functionals or distributions rather than seeking norm-convergent expansions.

In the next section, we build on the generator and resolvent framework developed here to introduce generalized spectral operators: weak integral representations of the Koopman semigroup, resolvent-based functional calculi, and the systematic incorporation of non-normal and pseudospectral effects. This will establish the abstract spectral machinery required before any concrete dynamical application is considered.

6. Generalized Spectral Operators for Koopman Generators

6.1. Motivation and Scope

Classical spectral theory for linear operators is largely built around normal or diagonalizable operators acting on Hilbert spaces. In this setting, the spectral theorem provides a projection-valued measure, allowing operators and semigroups to be represented as integrals over the spectrum, yielding harmonic decompositions with strong convergence properties. For Koopman generators associated with nonlinear dynamical systems, such assumptions are rarely satisfied. The operators are typically non-normal, may have a continuous or residual spectrum, and often lack a complete set of eigenfunctions.

The aim of this section is to develop a generalized notion of spectral representation adapted to this setting. Rather than seeking norm-convergent expansions in terms of eigenfunctions, we work with resolvent-based constructions, weak integral representations, and functional calculi defined on appropriate spaces of test functionals. This approach preserves much of the interpretive power of spectral analysis while remaining applicable to non-normal operators with a complicated spectral structure.

6.2. Resolvent as a Primary Spectral Object

Let A denote the infinitesimal generator of a strongly continuous Koopman semigroup ${\left\{U^t\right\}}_{t\ge 0}$ on a Banach space $\mathcal{F}$. As shown above, the resolvent (17) is well defined and bounded for all $\lambda$ in the resolvent set $\rho \left(A\right)$, which contains the half-plane $\{\Re \lambda >{\omega }_0\}$, where ${\omega }_0$ is the growth bound of the semigroup.

In non-normal settings, the resolvent plays a more fundamental role than the spectrum alone. The spectrum identifies values of $\lambda$ where the resolvent fails to exist, but the magnitude and analytic behavior of $R(\lambda ,A)$ away from the spectrum encode information about transient growth, sensitivity to perturbations, and non-modal dynamics. These features are invisible to purely eigenvalue-based analyses.

The resolvent identity

$$R(\lambda ,A)-R(\mu ,A)=(\mu -\lambda )R(\lambda ,A)R(\mu ,A),$$

(22)

together with analyticity of $\lambda \mapsto R(\lambda ,A)$ on $\rho \left(A\right)$ forms the backbone of the generalized functional calculus developed below.

6.3. Generalized Functional Calculus

Given a closed operator A with a nonempty resolvent set, one may define operator-valued functions of A via contour integrals involving the resolvent. Let $\Omega \subset \mathbb{C}$ be an open set containing $\sigma \left(A\right)$, and let f be a function holomorphic on $\Omega$. The Dunford–Taylor functional calculus defines

$$f(A):={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }f(\lambda ) R(\lambda ,A) d\lambda ,$$

(23)

where $\Gamma$ is a positively oriented contour enclosing $\sigma \left(A\right)$ and lying within $\Omega$.

For Koopman generators, this calculus allows one to define operators such as $e^{tA}$ (recovering the semigroup), fractional powers, and resolvent-weighted transforms without requiring diagonalizability. Although the resulting operators may not admit a simple modal interpretation, they provide a consistent and rigorous framework for analyzing the evolution of observables under nonlinear dynamics.

In practice, the contour $\Gamma$ may be chosen to lie in regions where the resolvent is well controlled, making (23) meaningful even when the spectrum itself is highly nontrivial.

6.4. Weak Spectral Representations

To obtain representations analogous to spectral integrals for normal operators, we work in a weak sense using dual pairings. Let ${\mathcal{F}}^{\ast }$ be the continuous dual of $\mathcal{F}$, and let $⟨·,·⟩$ denote the duality pairing.

A weak spectral representation of the Koopman semigroup is an identity of the form

$$⟨\mathcal{l},U^{t}g⟩={\int }_{\Sigma }\Psi (t,\sigma ) d{\mu }_{\mathcal{l},g}(\sigma ),$$

(24)

valid for all $t\ge 0$, where

• $\Sigma$ is a spectral parameter set (typically a subset of $\mathbb{C}$),
• $\Psi (t,\sigma )$ is a kernel encoding temporal evolution (often $e^{\sigma t}$ or a generalized exponential), and
• ${\mu }_{\mathcal{l},g}$ is a complex measure depending bilinearly on $(\mathcal{l},g)$.

Unlike the projection-valued measures arising in the spectral theorem for normal operators, the measures ${\mu }_{\mathcal{l},g}$ need not be positive, orthogonal, or uniquely determined. Their role is to encode how observables and test functionals probe the spectral content of the generator through the resolvent.

Such representations are naturally derived from inverse Laplace transforms of the resolvent:

$$⟨\mathcal{l},U^{t}g⟩={\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma -i\infty }^{\gamma +i\infty }{e}^{\lambda t} ⟨\mathcal{l},R(\lambda ,A)g⟩ d\lambda ,$$

(25)

(see, e.g., [15,16] where $\gamma >{\omega }_0$. Formula (25) is a Bromwich-type inversion integral and provides a canonical weak representation of the Koopman semigroup in terms of the generator resolvent.

6.5. Resolvent-Generated Generalized Spectral Operators

The weak inversion Formula (25) suggests that the natural spectral object associated with a non-normal Koopman generator is not a projection-valued measure in the classical sense, but rather a resolvent-generated distribution on the complex plane. We formalize this idea as follows.

Definition 2 (Generalized spectral operator).

Let A be the infinitesimal generator of a $C_0$ Koopman semigroup ${\left\{U^t\right\}}_{t\ge 0}$ on a Banach space $\mathcal{F}$, with growth bound ${\omega }_0$. A generalized spectral operator associated with A is a linear mapping

$$\mathcal{S}:\mathcal{F}⟶{\mathcal{D}}^{\prime }(\Sigma )$$

(26)

into the space of distributions (or complex measures) on a spectral parameter set $\Sigma \subset \mathbb{C}$, satisfying the following properties:

(i) 

(Resolvent generation) For every $g\in \mathcal{F}$ and $\mathcal{l}\in {\mathcal{F}}^{\ast }$,

$$⟨\mathcal{l},R(\lambda ,A)g⟩={\int }_{\Sigma }{\displaystyle \frac{1}{\lambda -\sigma }} d{\left(\mathcal{S}g\right)}_{\mathcal{l}}(\sigma ),$$

(27)

for all λ in the resolvent set with $\Re \lambda >{\omega }_0$, where ${\left(\mathcal{S}g\right)}_{\mathcal{l}}$ denotes the scalar distribution obtained by pairing $\mathcal{S}g$ with ℓ.

(ii) 

(Semigroup reconstruction) For all $t\ge 0$ and all $\mathcal{l}\in {\mathcal{F}}^{\ast }$,

$$⟨\mathcal{l},U^{t}g⟩={\int }_{\Sigma }{e}^{\sigma t} d{\left(\mathcal{S}g\right)}_{\mathcal{l}}(\sigma ),$$

(28)

with convergence understood in the weak sense.

Condition (i) states that $\mathcal{S}$ is determined by the analytic structure of the resolvent, rather than by diagonalization of A. Condition (ii) expresses the Koopman evolution as a generalized superposition of spectral contributions, without requiring orthogonality, completeness, or norm convergence.

Proposition 1 (Weak spectral representation via resolvent inversion).

Let A generate a $C_0$ semigroup ${\left\{U^t\right\}}_{t\ge 0}$ on a Banach space $\mathcal{F}$. Then, for each $g\in \mathcal{F}$ and $\mathcal{l}\in {\mathcal{F}}^{\ast }$, the weak Bromwich representation (25) induces at least one scalar distribution ${\left(\mathcal{S}g\right)}_{\mathcal{l}}\in {\mathcal{D}}^{\prime }(\Sigma )$, with $\Sigma =\mathbb{C}$, such that the semigroup reconstruction Formula (28) holds in the weak sense.

In this way, the semigroup evolution admits a resolvent-generated generalized spectral representation consistent with Definition 2.

The role of Proposition 1 is not to establish this inversion result, but to interpret the corresponding weak resolvent representation as a generalized spectral object acting on observables. This provides the entry point for the measure-theoretic refinement developed in Theorem 1.

The generalized spectral operator is not, in general, unique as a distribution on $\Sigma$; however, its action on resolvent kernels and semigroup orbits is uniquely determined by the generator A.

Remark 1.

If A is normal on a Hilbert space, then $\mathcal{S}$ reduces to the classical projection-valued spectral measure provided by the spectral theorem [3]. In that case, the above definition recovers the standard spectral representation

$$U^t={\int }_{\sigma \left(A\right)}{e}^{\sigma t} dE(\sigma ),$$

(29)

and ${\left(\mathcal{S}g\right)}_{\mathcal{l}}$ is induced by the orthogonal spectral measure associated with A. Thus, the present notion extends classical spectral operators strictly beyond the normal and diagonalizable setting.

Lemma 1 (Scalar resolvent growth).

Let A be the generator of a $C_0$ semigroup on a Banach space $\mathcal{F}$, and assume that there exist constants $C>0$ and $m\ge 0$, such that

$$\parallel R(\gamma +i\eta ,A)\parallel \le C{(1+|\eta \left|\right)}^m$$

(30)

for all $\eta \in \mathbb{R}$ and some fixed γ larger than the spectral bound of A. Then for every $g\in \mathcal{F}$ and every $\mathcal{l}\in {\mathcal{F}}^{\ast }$, the scalar function

$$\eta \mapsto ⟨\mathcal{l},R(\gamma +i\eta ,A)g⟩$$

(31)

satisfies the polynomial bound

$$\left|⟨\mathcal{l},R(\gamma +i\eta ,A)g⟩\right|\le C\parallel \mathcal{l}\parallel \parallel g\parallel (1+{\left|\eta \right|)}^m.$$

(32)

In particular, this scalar resolvent function grows at most polynomially in $\left|\eta \right|$.

Proof. 

The result follows immediately from the definition of the operator norm:

$$|⟨\mathcal{l},R(\gamma +i\eta ,A)g⟩|\le \parallel \mathcal{l}\parallel \parallel R(\gamma +i\eta ,A)g\parallel \le \parallel \mathcal{l}\parallel \parallel R(\gamma +i\eta ,A)\parallel \parallel g\parallel .$$

(33)

The assumed resolvent growth bound therefore implies the stated estimate.    □

Theorem 1 (Radon-measure regularity under polynomial resolvent growth).

Let A be the generator of a $C_0$ semigroup on a Banach space $\mathcal{F}$, and suppose that the resolvent satisfies the polynomial growth bound

$$\parallel R(\gamma +i\eta ,A)\parallel \le C{(1+|\eta \left|\right)}^m$$

(34)

for $\eta \in \mathbb{R}$ and some γ larger than the spectral bound of A.

Then for every $g\in \mathcal{F}$ and $\mathcal{l}\in {\mathcal{F}}^{\ast }$, the scalar resolvent function

$$F_{g,\mathcal{l}}\left(\eta \right)=⟨\mathcal{l},R(\gamma +i\eta ,A)g⟩$$

(35)

defines a tempered distribution in the variable η.

Moreover, when restricted to any bounded spectral window $\Delta \subset \mathbb{C}$, the corresponding generalized spectral object admits a representation by a finite complex Radon measure on Δ.

This result provides the key analytical step that upgrades the weak, distributional spectral representation in Proposition 1 to a measure-theoretic one on bounded spectral regions.

Proof. 

By the previous lemma, the scalar function (35) satisfies the polynomial bound

$$|F_{g,\mathcal{l}}{\left(\eta \right)|\le C \parallel \mathcal{l}\parallel \parallel g\parallel (1+\left|\eta \right|)}^m.$$

(36)

Hence, $F_{g,\mathcal{l}}$ grows at most polynomially in $\left|\eta \right|$. It therefore defines a tempered distribution in the spectral variable $\eta$.

The semigroup representation obtained via Bromwich inversion expresses the pairing $⟨\mathcal{l},T\left(t\right)g⟩$ in terms of the inverse Fourier transform of the function $F_{g,\mathcal{l}}\left(\eta \right)$. In distributional form, one may write

$$⟨\mathcal{l},T\left(t\right)g⟩={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{e}^{(\gamma +i\eta )t}{F}_{g,\mathcal{l}}(\eta ) d\eta ,$$

(37)

where the integral is understood in the sense of tempered distributions.

Consider now a bounded spectral window $\Delta \subset \mathbb{C}$, corresponding to a bounded interval in $\eta$ along the vertical line $\lambda =\gamma +i\eta$. Multiplying $F_{g,\mathcal{l}}$ by a smooth cutoff function supported in this interval yields a compactly supported distribution in $\eta$. Standard results from distribution theory imply that such distributions admit representations in terms of finite Radon measures and their derivatives. In the present scalar setting, this yields a finite complex Radon measure representation on $\Delta$.

Consequently, the restriction of the generalized spectral object to any bounded spectral window admits a representation by a finite complex Radon measure.    □

As a consequence of the previous theorem, under polynomial resolvent growth along vertical lines, the weak spectral representation

$$⟨\mathcal{l},U^{t}g⟩={\int }_{\Sigma }{e}^{\sigma t} d{\left(\mathcal{S}g\right)}_{\mathcal{l}}(\sigma )$$

(38)

may be interpreted locally as a genuine measure-theoretic superposition rather than merely a distributional identity. In particular, on bounded spectral windows, the generalized spectral objects ${\left(\mathcal{S}g\right)}_{\mathcal{l}}$ admit representations by finite complex Radon measures.

Equation (38) requires that the resolvent of the generator satisfy a polynomial bound along a vertical line in the resolvent set,

$$\parallel R(\gamma +i\eta ,A)\parallel \le C{(1+|\eta \left|\right)}^m,\hspace{1em}\hspace{1em}\eta \in \mathbb{R},$$

(39)

for some $\gamma$ greater than the spectral bound of A. Estimates of this type are standard in semigroup theory and arise naturally in a number of settings.

First, if A generates an analytic semigroup on a Banach space, then resolvent estimates of the form $\parallel R(\lambda ,A)\parallel \le C/\left|\lambda \right|$ hold uniformly outside sectors containing the spectrum; see, for example, [15,21]. More generally, polynomial resolvent bounds occur for generators whose high-frequency behavior is controlled by differential operators of finite order.

Second, when the Koopman generator arises from a smooth flow on a compact manifold and acts on sufficiently regular observable spaces (such as $C^k$ or Sobolev spaces), the generator may be identified with a first-order differential operator corresponding to the Lie derivative along the underlying vector field. In such cases, resolvent growth along vertical lines is governed by the differential order of the operator, naturally leading to polynomial bounds in $\left|\eta \right|$; see the general discussion in [15,21].

Third, in Hilbert-space settings where the Koopman operator is unitary (for example, when the dynamics preserves an invariant probability measure), the generator is skew-adjoint and the resolvent satisfies the uniform estimate

$$\parallel R(\gamma +i\eta ,A)\parallel \le {(\gamma -\omega )}^{-1},$$

(40)

for $\gamma$ greater than the spectral bound $\omega$. In this case, the polynomial bound holds with $m=0$.

These examples indicate that assumption (38) is compatible with several standard classes of Koopman generators. In the present work, the assumption serves to identify a broad regime in which the weak resolvent-based spectral objects constructed earlier admit the measure-theoretic interpretation established in Theorem 1.

6.6. Non-Normality and Pseudospectra

For non-normal operators, the spectrum alone provides an incomplete description of dynamical behavior. The $\epsilon$ pseudospectrum of A is defined by

$${\sigma }_{\epsilon }(A):={\{\lambda \in \mathbb{C}:\parallel (\lambda I-A)}^{-1}\parallel >{\epsilon }^{-1}\} \cup \sigma (A).$$

(41)

Regions of large resolvent norm correspond to spectral instability and transient amplification, even when $\Re \lambda$ is negative or bounded away from the spectrum.

In the Koopman context, pseudospectra quantify how observables may experience transient growth under nonlinear dynamics before asymptotic decay or mixing dominates. This is particularly relevant for chaotic systems, where stretching and folding mechanisms produce strong non-normal effects.

Generalized spectral operators naturally incorporate pseudospectral information through the resolvent. Weak representations such as (25) depend not only on the location of singularities of $R(\lambda ,A)$ but also on its magnitude along integration contours. As a result, transient dynamics are encoded directly in the generalized spectral representation, without requiring explicit eigenmodes.

6.7. Comparison with Classical Spectral Theory

In the special case where A is normal on a Hilbert space, the generalized framework developed here collapses to the classical spectral theorem. The measures ${\mu }_{\mathcal{l},g}$ become induced by a projection-valued spectral measure, the kernel $\Psi (t,\sigma )=e^{\sigma t}$, and weak representations coincide with norm-convergent spectral integrals.

For Koopman generators arising from generic nonlinear systems, however, the generalized framework is strictly richer. It accommodates a continuous spectrum, Jordan-type behavior, and resolvent growth, all of which are essential for a faithful description of nonlinear and chaotic dynamics.

In the next section, we specialize the generalized spectral operator framework to concrete classes of observable spaces and generators and discuss invariant subspaces and spectral projections in the generalized sense.

7. Generalized Invariant Structures and Resolvent-Based Spectral Selection

The measure-theoretic regularity established in Theorem 1 provides a natural setting in which contour constructions based on the resolvent can be interpreted through scalar spectral measures on bounded spectral windows. In this section we explore how such constructions lead to generalized invariant structures in observable space and how they may be used to separate dynamical contributions associated with different regions of the spectrum.

7.1. Invariant Subspaces for Koopman Semigroups

Invariant subspaces play a central role in both classical and generalized spectral theories. For a Koopman semigroup ${\left\{U^t\right\}}_{t\ge 0}$ acting on a Banach space of observables $\mathcal{F}$, a closed subspace $\mathcal{M}\subset \mathcal{F}$ is said to be invariant if

$$U^t\mathcal{M}\subset \mathcal{M} \mathrm{for} \mathrm{all} t\ge 0.$$

(42)

Equivalently, $\mathcal{M}$ is invariant under the generator A in the sense that $\mathrm{dom}\left(A\right)\cap \mathcal{M}$ is dense in $\mathcal{M}$ and $A\left(\mathrm{dom}\right(A)\cap \mathcal{M})\subset \mathcal{M}$.

In classical Koopman theory, invariant subspaces are often generated by eigenfunctions and their products, leading to finite- or countable-dimensional decompositions. In the generalized setting considered here, invariant subspaces may be infinite-dimensional, non-orthogonal, and defined only in a weak or resolvent-based sense. Nevertheless, they provide a natural organizational principle for decomposing the space of observables into dynamically meaningful components.

7.2. Spectral Subspaces via Resolvent Integrals

Let A be the generator of a strongly continuous Koopman semigroup on $\mathcal{F}$, and let $\Delta \subset \mathbb{C}$ be a bounded open set whose boundary $\partial \Delta$ lies entirely in the resolvent set $\rho \left(A\right)$. One may formally associate with $\Delta$ the contour operator

$$P_{\Delta }:={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial \Delta }R(\lambda ,A) d\lambda .$$

(43)

When the contour integral (43) converges in the operator norm, the resulting operator coincides with the classical Dunford–Riesz spectral projection associated with $\Delta$ in the sense of the Dunford–Taylor functional calculus [3]. In that case, one has $P_{\Delta }^2=P_{\Delta }$, and $P_{\Delta }$ commutes with A on $\mathrm{dom}\left(A\right)$.

In the Koopman settings considered here, however, operator-norm convergence of (43) cannot generally be guaranteed. The resolvent may exhibit substantial growth near spectral boundaries, and pseudospectral effects may strongly influence the behavior of the integral. Accordingly, the contour construction should primarily be viewed as a formal spectral selector whose rigorous interpretation is given through weak pairings with observables and dual elements.

In the normal operator case, $P_{\Delta }$ coincides with the spectral projection associated with $\Delta$ in the sense of the spectral theorem. For non-normal Koopman generators, the contour construction retains many of the same formal features, but its interpretation is more subtle. Rather than isolating eigenspaces alone, it may capture generalized modes, Jordan chains, or continuous spectral contributions.

7.3. Weak Spectral Projections

In many Koopman settings, the contour integral (43) cannot be shown to converge in operator norm on the full space $\mathcal{F}$. This may occur because the resolvent grows rapidly near parts of the spectrum or because $\partial \Delta$ passes through regions of strong pseudospectral amplification.

In such situations, one may nevertheless define the contour construction in a weak sense through scalar pairings. Given $\mathcal{l}\in {\mathcal{F}}^{\ast }$ and $g\in \mathcal{F}$, define

$$⟨\mathcal{l},P_{\Delta }g⟩:={\displaystyle \frac{1}{2\pi i}}{\int }_{\partial \Delta }⟨\mathcal{l},R(\lambda ,A)g⟩ d\lambda ,$$

(44)

whenever the scalar contour integral converges.

This construction defines a bilinear form that can often be interpreted as an element of the bidual ${\mathcal{F}}^{\ast \ast }$ or of a suitable distributional completion of $\mathcal{F}$. In this setting, the operators $P_{\Delta }$ should be viewed as weak spectral selectors rather than genuine projection operators. They isolate contributions associated with regions of the complex plane without requiring a full operator-level spectral decomposition.

Weak spectral selectors are particularly useful in chaotic or strongly non-normal systems, where spectral boundaries may be blurred by pseudospectral effects and classical spectral projections need not exist on the full observable space.

7.4. Decomposition of Observables

Let ${\left\{{\Delta }_j\right\}}_{j\in J}$ be a collection of disjoint spectral regions whose union covers a portion of the complex plane relevant to the dynamics; for instance, regions associated with slow decay, oscillatory behavior, or rapid transients.

Using the weak contour constructions introduced above, one may formally separate contributions associated with these spectral regions. At the level of scalar observables, this leads to a weak decomposition of the form

$$⟨\mathcal{l},g⟩=\sum _{j\in J}⟨\mathcal{l},P_{{\Delta }_j}g⟩+⟨\mathcal{l},r⟩,$$

(45)

where the remainder r accounts for spectral contributions outside the selected regions.

Equation (45) should be interpreted as a weak spectral decomposition rather than a strict operator-level identity. In particular, the sum need not converge in norm, and the contour operators $P_{{\Delta }_j}$ need not satisfy projection properties unless additional conditions ensure convergence of the Dunford–Riesz integral in the operator norm.

7.5. Invariant Manifolds in Observable Space

Generalized spectral subspaces provide a bridge between operator theory and geometric notions of invariance in state space. While Koopman operators act on observables rather than states, invariant subspaces of observables often correspond to invariant sets, foliations, or coherent structures in the underlying dynamics.

For example, if $\mathcal{M}\subset \mathcal{F}$ is an invariant subspace generated by observables that are constant along a family of invariant manifolds in state space, then $U^t$ acts trivially on $\mathcal{M}$ up to exponential scaling. In the generalized setting, such structures may be detected through contour-based spectral selectors associated with eigenvalues near the imaginary axis or through resolvent growth indicating slow-manifold dynamics.

Importantly, the generalized spectral framework does not require these invariant structures to be finite-dimensional or smoothly parameterized. This flexibility is essential for treating chaotic attractors, where invariant sets often possess fractal geometry and support continuous spectra.

7.6. Time-Asymptotic Versus Transient Behavior

A key advantage of the generalized spectral approach is its ability to distinguish between asymptotic and transient dynamics. Spectral components associated with $\Re \lambda <0$ govern decay rates, while those near $\Re \lambda =0$ control long-time persistence and recurrence. Pseudospectral regions with a large resolvent norm but negative real part may nevertheless generate substantial transient growth before eventual decay.

Through representations such as

$$⟨\mathcal{l},U^{t}g⟩={\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma -i\infty }^{\gamma +i\infty }{e}^{\lambda t}⟨\mathcal{l},R(\lambda ,A)g⟩ d\lambda ,$$

(46)

one explicitly sees how both spectral location and resolvent magnitude shape the observed dynamics. This perspective is particularly relevant in nonlinear systems where transient amplification plays a dominant role in transport, predictability, and regime transitions.

The constructions in this section are formulated at the level of scalar observables and therefore apply to any Koopman semigroup satisfying the standing assumptions of strong continuity together with appropriate resolvent bounds. In concrete applications, the main challenge lies in identifying observable spaces and test functionals for which the resulting generalized spectral objects, or their associated weak spectral selectors, can be rigorously computed, approximated, or meaningfully interpreted.

7.7. A Finite-Dimensional Non-Normal Example

Finite-dimensional Jordan blocks provide canonical models of non-normal generators and serve as prototypes for the transient behavior frequently observed in numerical approximations of Koopman operators associated with nonlinear flows.

We illustrate the preceding constructions with a simple non-normal generator given by the Jordan block

$$A=\left(\begin{array}{cc}\alpha & 1\\ 0& \alpha \end{array}\right),\hspace{1em}\hspace{1em}\alpha \in \mathbb{C}.$$

(47)

Although finite-dimensional, this example already displays the key features that motivate the generalized framework: the operator is not normal, it possesses only one eigenvalue, and yet its semigroup exhibits behavior that cannot be described solely in terms of a basis of eigenvectors.

Let $\mathcal{F}={\mathbb{C}}^2$ and define the Koopman-type semigroup by

$$U^t=e^{tA},\hspace{1em}\hspace{1em}t\ge 0.$$

(48)

Since $A=\alpha I+N$, where

$$N=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\hspace{1em}\hspace{1em}{N}^2=0,$$

(49)

one obtains

$$U^t=e^{\alpha t}{e}^{tN}=e^{\alpha t}\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right).$$

(50)

Thus, even though the spectrum of A consists only of the single point $\left\{\alpha \right\}$, the semigroup contains the polynomial prefactor t, which produces transient growth when $\Re \alpha <0$ is small in magnitude and reveals the non-semisimple structure of the generator.

The resolvent can also be computed explicitly. For $\lambda \in \rho \left(A\right)=\mathbb{C}\setminus \left\{\alpha \right\}$,

$$R(\lambda ,A)={(\lambda I-A)}^{-1}=\left(\begin{array}{cc}{(\lambda -\alpha )}^{-1}& {(\lambda -\alpha )}^{-2}\\ 0& {(\lambda -\alpha )}^{-1}\end{array}\right).$$

(51)

This formula immediately shows that the resolvent contains a second-order pole at $\lambda =\alpha$. The diagonal terms correspond to the usual simple-pole behavior associated with an eigenvalue, whereas the off-diagonal term ${(\lambda -\alpha )}^{-2}$ records the Jordan-chain structure. In particular, the resolvent captures more information than the point spectrum alone.

To connect this example with the weak spectral framework developed above, let $g={(g_1,g_2)}^{\top }\in {\mathbb{C}}^2$ and let $\mathcal{l}=({\mathcal{l}}_1,{\mathcal{l}}_2)\in {\left({\mathbb{C}}^2\right)}^{\ast }$ be identified with a row vector. Then

$$⟨\mathcal{l},R(\lambda ,A)g⟩={\displaystyle \frac{{\mathcal{l}}_1{g}_1+{\mathcal{l}}_2{g}_2}{\lambda -\alpha }}+{\displaystyle \frac{{\mathcal{l}}_1{g}_2}{{(\lambda -\alpha )}^2}}.$$

(52)

This scalar resolvent pairing exhibits both a simple-pole contribution and a second-order contribution. The first term corresponds to the spectral value $\alpha$ itself, while the second term reflects the generalized eigenvector structure responsible for the polynomial factor in (50).

Let $\Delta \subset \mathbb{C}$ be a small bounded open set containing $\alpha$ and no other spectral values, and let $\Gamma =\partial \Delta$ be positively oriented. Since the present example is finite-dimensional, the Dunford–Riesz integral converges in the operator norm, so the contour operator

$$P_{\Delta }={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }R(\lambda ,A) d\lambda$$

(53)

is a genuine spectral projection. Using (51) and the residue theorem, one finds

$$P_{\Delta }=I,$$

(54)

because the entire spectrum of A is contained in $\Delta$. If instead, $\Delta$ does not contain $\alpha$, then $P_{\Delta }=0$. Thus, in this finite-dimensional case, the contour construction reproduces the classical spectral projection.

However, Formula (52) shows that the weak resolvent-based perspective contains additional structural information: the contour integral detects not only the location of the eigenvalue but also the order of the pole and hence the nontrivial Jordan structure. Indeed, by the inverse Laplace formula,

$${\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{\lambda t}{(\lambda -\alpha )}^{-1} d\lambda =e^{\alpha t},\hspace{1em}\hspace{1em}{\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{\lambda t}{(\lambda -\alpha )}^{-2} d\lambda =t{e}^{\alpha t},$$

(55)

so that the second-order pole produces exactly the polynomial prefactor appearing in (50).

This example illustrates in concrete form the main theme of the paper. A purely eigenvalue-based description would identify only the spectral point $\alpha$, whereas the resolvent-centered framework distinguishes between semisimple and non-semisimple behavior through the analytic structure of the resolvent. In more general Koopman settings, especially for non-normal or infinite-dimensional generators, one should not expect such contour constructions to produce operator-valued projections on the full observable space. Nevertheless, the scalar resolvent pairings used throughout this paper continue to provide meaningful spectral selectors and a natural description of transient amplification and a generalized spectral structure.

This finite-dimensional example serves only as an illustration, but it explicitly shows how the analytic structure of the resolvent encodes dynamical features that extend beyond a purely eigenfunction-based description.

In the present finite-dimensional setting, the generalized spectral objects reduce to atomic measures concentrated at the spectral value $\alpha$, with the higher-order pole structure reflected in the associated generalized-mode contributions.

Although the present example is finite-dimensional, similar phenomena shall arise in numerical approximations of infinite-dimensional Koopman generators. In particular, discretizations shall yield non-normal matrices whose resolvent structure reflects the generalized spectral behavior discussed above.

8. Numerical and Computational Aspects of Generalized Spectral Operators

8.1. Motivation and Computational Challenges

The preceding sections developed an abstract framework for generalized spectral analysis based on resolvents, weak spectral representations, and contour constructions. The finite-dimensional example of the previous section illustrates how the analytic structure of the resolvent encodes dynamical information such as transient growth and Jordan-chain behavior. In practical applications, however, Koopman semigroups and their generators are typically infinite-dimensional and must be approximated numerically from either explicit models or data.

Unlike classical spectral problems for normal matrices, Koopman generators are frequently unbounded and strongly non-normal. Numerical analysis must therefore contend simultaneously with the discretization error, truncation effects, and strong sensitivity associated with pseudospectral behavior.

The goal of this section is to outline principled computational strategies for approximating generalized spectral quantities in a manner consistent with the underlying operator-theoretic framework. Rather than proposing a single algorithm, we emphasize structural considerations that ensure that numerical procedures reflect genuine properties of the Koopman generator rather than artifacts of discretization.

8.2. Finite-Dimensional Approximations of Koopman Operators

A common numerical strategy is to approximate the Koopman semigroup $\left\{U^t\right\}$ by finite-dimensional operators acting on a chosen subspace ${\mathcal{F}}_N\subset \mathcal{F}$. Let $\{{\psi }_1,\dots ,{\psi }_N\}$ be a collection of observables spanning ${\mathcal{F}}_N$, and define a projection ${\Pi }_N:\mathcal{F}\to {\mathcal{F}}_N$. One then approximates $U^t$ by

$$U_N^t:={\Pi }_N{U}^t{|}_{{\mathcal{F}}_N},$$

(56)

which may be represented as an $N\times N$ matrix once a basis is fixed.

Such approximations underlie a broad class of methods, including Galerkin schemes, extended dynamic mode decomposition (EDMD), and related data-driven approaches. From the perspective of generalized spectral theory, the crucial point is that $U_N^t$ approximates the action of $U^t$ only on a restricted observable subspace, and its spectrum need not converge pointwise to the spectrum of $U^t$ as $N\to \infty$. In non-normal settings, convergence of resolvents and pseudospectra is often a more robust diagnostic.

A fundamental difficulty in finite-dimensional Koopman approximations is spectral pollution [35,36]: the spectrum of ${\left(U^t\right)}^N$ or $A_N$ may contain spurious eigenvalues that do not converge to points in the spectrum of the infinite-dimensional operator as $N\to \infty$. This phenomenon is particularly acute for non-normal operators, where small perturbations may produce large spectral shifts [8,17].

For this reason, pointwise convergence of discrete eigenvalues is generally neither necessary nor sufficient for the convergence of Koopman spectral information. A more structurally appropriate requirement is the convergence of resolvents on compact subsets of the resolvent set, namely

$$\parallel {(\lambda I-A_N)}^{-1}-{(\lambda I-A)}^{-1}\parallel \to 0$$

(57)

uniformly for $\lambda$ in compact subsets of $\rho \left(A\right)$. Such resolvent convergence implies stability of weak spectral representations and ensures convergence of contour-based functional calculi [3,17].

In non-normal settings, the convergence of pseudospectra, rather than the convergence of eigenvalues, provides a more robust and dynamically meaningful criterion. Indeed, the stability of $\epsilon$-pseudospectral sets as N increases reflects the stability of resolvent norms and therefore of generalized spectral operators.

8.3. Approximation of the Generator

Instead of approximating $U^t$ directly, one may seek a finite-dimensional approximation $A_N$ of the Koopman generator A. Formally, this may be achieved by

$$A_N:={\Pi }_{N}A{|}_{{\mathcal{F}}_N},$$

(58)

whenever the action of A on the basis functions is accessible; for example, through analytic differentiation or numerical estimation of Lie derivatives.

Generator-based approximations offer conceptual advantages in the generalized spectral framework. First, resolvents ${(\lambda I-A_N)}^{-1}$ can be computed explicitly for each N, enabling direct approximation of generalized spectra and pseudospectra. Second, time evolution may be reconstructed via matrix exponentials $e^{t{A}_N}$, allowing consistent comparison between semigroup-based and generator-based computations.

Care must be taken, however, since $A_N$ is typically a highly non-normal matrix even when A itself is defined on a Hilbert space. Consequently, eigenvalues of $A_N$ may converge slowly or erratically, while pseudospectral features often stabilize more reliably as N increases.

When approximating the Koopman generator A by finite-dimensional operators $A_N$, the central analytical object is again the resolvent. If $A_N$ converges to A in the strong resolvent sense,

$${(\lambda I-A_N)}^{-1}g\to {(\lambda I-A)}^{-1}g \mathrm{for} \mathrm{all} g\in F,$$

(59)

for each $\lambda \in \rho \left(A\right)$, then weak generalized spectral representations induced by Bromwich inversion converge correspondingly. Strong resolvent convergence is here understood in the standard sense for closed operators on Banach spaces.

Stronger conclusions follow if resolvent convergence holds in the operator norm on compact subsets of $\rho \left(A\right)$. In that case, Dunford–Taylor contour integrals defining functional calculi converge, and generalized contour-based spectral components associated with bounded spectral regions stabilize as $N\to \infty$ [3,17]. This provides a rigorous justification for computing contour-based spectral components numerically, even in the absence of reliable eigenvalue convergence.

Conversely, failure of resolvent stability manifests as large pseudospectral regions of $A_N$, indicating strong transient amplification and sensitivity. Within the present framework, such behavior should not be regarded as numerical pathology, but as a faithful reflection of intrinsic non-normal dynamics encoded in the Koopman generator.

8.4. Numerical Resolvent Estimation

Central to the generalized spectral framework is the resolvent

$$R(\lambda ,A)={(\lambda I-A)}^{-1}.$$

(60)

Numerically, one approximates $R(\lambda ,A)$ by computing ${(\lambda I-A_N)}^{-1}$ for a sequence of increasing dimensions N. This enables the estimation of resolvent norms, resolvent growth regions, and approximate pseudospectra.

In practice, resolvent norms may be estimated using induced matrix norms or singular value decompositions:

$$\parallel {(\lambda I-A_N)}^{-1}\parallel \approx {\sigma }_{min}{(\lambda I-A_N)}^{-1},$$

(61)

where ${\sigma }_{min}$ denotes the smallest singular value. Regions of small ${\sigma }_{min}$ correspond to a large resolvent norm and thus to pseudospectral amplification.

From the Koopman perspective, such computations provide insight into transient dynamics that are invisible to eigenvalue plots alone. In particular, they reveal directions in observable space that experience strong short-time growth even when asymptotic decay dominates.

8.5. Approximation of Weak Spectral Representations

Weak spectral representations of the form

$$⟨\mathcal{l},U^{t}g⟩={\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma -i\infty }^{\gamma +i\infty }{e}^{\lambda t}⟨\mathcal{l},R(\lambda ,A)g⟩ d\lambda$$

(62)

can be approximated numerically by replacing A with $A_N$ and truncating the Bromwich contour. The resulting approximation

$$⟨\mathcal{l},U_N^{t}g⟩\approx {\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma -i\Omega }^{\gamma +i\Omega }{e}^{\lambda t}⟨\mathcal{l},R(\lambda ,A_N)g⟩ d\lambda$$

(63)

captures the dominant generalized spectral contributions over finite time horizons.

This approach emphasizes the resolvent as the fundamental computational object. Rather than attempting to extract individual eigenmodes, one computes integrals of resolvent-weighted contributions, which are often more stable under discretization and better aligned with the weak nature of the theoretical framework.

8.6. Pseudospectral Diagnostics and Robustness

Pseudospectra provide a natural diagnostic tool for assessing the robustness of numerical approximations. If a finite-dimensional approximation $A_N$ exhibits large regions of pseudospectral amplification far from its eigenvalues, then individual eigenmodes should not be interpreted as physically meaningful. Instead, generalized spectral measures based on resolvent norms or contour integrals offer a more faithful representation of the dynamics.

From a computational standpoint, the convergence of pseudospectral contours as N increases is often a stronger and more informative criterion than the convergence of eigenvalues. This aligns closely with the generalized spectral philosophy developed in earlier parts of the paper.

8.7. Data-Driven Considerations

In many applications, the flow ${\phi }^t$ is not known explicitly, and only time-series data are available. In such cases, empirical approximations of Koopman operators and generators may be constructed from sampled trajectories. While the mathematical setting differs from model-based discretizations, the generalized spectral framework remains applicable.

Data-driven approximations typically yield finite-dimensional operators that are inherently non-normal and noisy. The emphasis on resolvents, weak representations, and pseudospectra is therefore particularly well suited to such contexts. Rather than interpreting extracted eigenvalues literally, one analyzes resolvent growth and generalized spectral projections to identify dynamically relevant structures.

8.8. Summary and Connection to Theory

The numerical strategies outlined in this section are guided by the operator-theoretic principles developed earlier in the paper. By focusing on resolvents, weak representations, and pseudospectra rather than on strict eigenfunction expansions, computational methods can remain faithful to the true structure of Koopman generators associated with nonlinear and chaotic dynamics.

This articulation between theory and computation is essential for ensuring that numerical results reflect intrinsic dynamical features rather than artifacts of discretization or basis selection.

9. Conclusions and Perspectives

This work develops a resolvent-centered framework for generalized spectral operators associated with nonlinear dynamical systems through their Koopman semigroups. Departing from the classical harmonic paradigm of spectral analysis, which relies on normality, diagonalizability, and complete eigenfunction expansions, we formulate an operator-theoretic approach that remains valid in the presence of non-normal generators, a continuous spectrum, and strong transient dynamics. The central analytical result shows that polynomial resolvent growth along vertical lines upgrades weak spectral representations to finite Radon measures on bounded spectral windows, thereby connecting distributional spectral objects with measure-valued spectral descriptions.

At the core of the framework lies the interpretation of nonlinear dynamics as a strongly continuous linear semigroup acting on appropriately chosen spaces of observables. By focusing on the infinitesimal generator of this semigroup and, crucially, on its resolvent, we have shown how many of the conceptual advantages of spectral analysis can be retained without imposing restrictive assumptions that are rarely satisfied by nonlinear or chaotic systems. Weak spectral representations, generalized functional calculi, and resolvent-based projections provide a consistent and flexible language for describing the evolution of observables in settings where classical Koopman eigenfunctions are sparse or altogether absent.

A central theme throughout this paper has been the role of non-normality. For Koopman generators associated with dissipative and chaotic flows, spectral information alone is insufficient to characterize dynamics. Pseudospectra and resolvent growth capture transient amplification and sensitivity phenomena that dominate observable behavior over finite time horizons. By incorporating these effects naturally into generalized spectral operators, the framework presented here aligns more closely with the qualitative and quantitative features observed in nonlinear dynamics.

The finite-dimensional Jordan-block example presented in this paper illustrates this mechanism explicitly, showing how the analytic structure of the resolvent captures generalized spectral behavior that extends beyond purely eigenvalue-based descriptions.

On the computational side, we emphasized numerical strategies that are faithful to the generalized spectral viewpoint. Finite-dimensional approximations, whether model-based or data-driven, inevitably introduce non-normality and truncation effects. By prioritizing resolvent estimation, pseudospectral diagnostics, and weak integral representations over direct eigenmode extraction, numerical methods can better reflect intrinsic dynamical structure and avoid misleading modal interpretations.

Overall, the generalized spectral operator framework presented here offers a simple bridge between rigorous functional analysis and the practical needs of nonlinear dynamics. By relinquishing the requirement of harmonic completeness while retaining the structural insights of spectral theory, it provides a coherent and adaptable foundation for the analysis of complex dynamical systems across theory, computation, and applications.

As part of a broader formal development in progress, the present communication is intentionally structural rather than exhaustive. In this effort, we do not yet attempt to provide a final construct of optimal anisotropic function spaces for specific classes of hyperbolic systems, nor do we yet provide sharp resolvent growth estimates for particular nonlinear flows. Likewise, the uniqueness of the generalized spectral distributions beyond their action through resolvent and semigroup pairings is not yet pursued here. At this stage, the framework is designed to identify minimal operator-theoretic structures; namely, semigroup generation, resolvent analyticity, and weak inversion, which remain valid across a broad class of nonlinear dynamical systems. Refinements tailored to specific dynamical regimes (e.g., uniformly hyperbolic systems, stochastic perturbations, or infinite-dimensional PDEs) require additional structural assumptions and are under development.

Several further directions for future research emerge naturally from this work. One avenue is the systematic development of numerical algorithms for approximating resolvents and generalized spectral measures directly from data, with theoretical guarantees on convergence in weak or pseudospectral senses. Another is the extension of the framework to infinite-dimensional systems such as partial differential equations, where Koopman generators act on function spaces of observables over infinite-dimensional state spaces. Further connections with ergodic theory, including further elaborated relationships between generalized spectral operators and mixing rates or decay of codependencies, are also under development, along with broader extensions beyond ergodic theory, involving more general algebraic constructs and recent methodological developments in categorical and information physical intelligence [37,38].

As a final remark, we note that the perspective discussed in this paper is not a replacement for classical spectral theory, but rather a contribution to a formal extension of it. In regimes where classical harmonic Koopman analysis applies, the generalized framework collapses to familiar results. In the more common regimes where it does not, generalized spectral operators such as those contributed in this work can offer a principled and formal alternative.