Koopman spectral analysis of skew-product dynamics on Hilbert $C^{*}$-modules

Operator-theoretic methods have been used extensively in analysis and computational techniques for dynamical systems. Let $f:\mathcal{X}\to\mathcal{X}$ be a dynamical system on a state space $\mathcal{X}$. Then, the Koopman operator $U_{f}$ associated with $f$ is defined as a composition operator on an $f$-invariant function space $\mathcal{F}$ on $\mathcal{X}$,

for $v\in\mathcal{F}$ [1, 2]. In many cases, $\mathcal{F}$ is chosen as a Banach space or Hilbert space, such as the Lebesgue spaces $L^{p}(\mathcal{X})$ for a measure space $\mathcal{X}$ and the Hardy space $H^{p}(\mathbb{D})$ on the unit disk $\mathbb{D}$, where $p\in[1,\infty]$. Meanwhile, the Perron–Frobenius, or transfer, operator associated with $f$ is defined as the adjoint $P_{f}$ of the Koopman operator acting on the continuous dual $\mathcal{F}^{\prime}$ of $\mathcal{F}$, i.e., $P_{f}\nu=\nu\circ U_{f}$ for $\nu\in\mathcal{F}^{\prime}$ [3]. In a number of important cases (e.g., $\mathcal{F}=L^{p}(\mathcal{X})$ with $p\in[1,\infty)$ or $\mathcal{F}=C(\mathcal{X})$ for a compact Hausdorff space $\mathcal{X}$), $\mathcal{F}^{\prime}$ can be identified with a space of measures on $\mathcal{X}$; the transfer operator is then identified with the pushforward map on measures, $P_{f}\nu=\nu\circ f^{-1}$. When $\mathcal{F}$ has a predual, $\mathcal{F}_{*}\subseteq\mathcal{F}^{\prime}$ it is common to define $P_{f}$ as the predual of the Koopman operator, i.e., $(U_{f}v)\nu=v(P_{f}\nu)$; an important such example is $\mathcal{F}=L^{\infty}(\mathcal{X})$ with $\mathcal{F}_{*}=L^{1}(\mathcal{X})$. A central tenet of modern ergodic theory is to leverage the duality relationships between $f:\mathcal{X}\to\mathcal{X}$, $U_{f}:\mathcal{F}\to\mathcal{F}$, and $P_{f}:\mathcal{F}^{\prime}\to\mathcal{F}^{\prime}$ to characterize properties of nonlinear dynamics such ergodicity, mixing, and existence of factor maps, using linear operator-theoretic techniques [4].

Starting from work in the late 1990s and early 2000s [5, 6, 7, 8], operator-theoretic techniques have also proven highly successful in data-driven applications [9, 10, 11, 12]. A primary such application is the modal decomposition (e.g., [13, 14, 15, 16, 17, 18]). This approach applies eigenvalue decomposition to the Koopman operator to identify the long-term behavior of the dynamical system. Assume $\mathcal{F}$ is a Hilbert space equipped with an inner product $\left\langle\cdot,\cdot\right\rangle$, and $U_{f}$ is normal, bounded, and diagonalizable, with eigenvalues $\lambda_{1},\lambda_{2},\ldots\in\mathbb{C}$ and corresponding basis of orthonormal eigenvectors $v_{1},v_{2},\ldots\in\mathcal{F}$. Then, for $u\in\mathcal{F}$ and a.e. $x\in\mathcal{X}$ we have $u(f^{i}(x))=U_{f}^{i}u(x)=\sum_{j=1}^{\infty}\lambda_{j}^{i}v_{j}(x)\left\langle v_{j},u\right\rangle$, where $i\in\mathbb{N}$ represents discrete time. Therefore, the time evolution of observables is described by the Koopman eigenvalues and corresponding eigenvectors. By computing the eigenvalues of the Koopman operator, we obtain oscillating elements and decaying elements in the dynamical system.

Several attempts have been made to generalize the above decomposition to the case where the Koopman operator has continuous or residual spectrum. Korda et al. [19] approximate the spectral measure of the Koopman operator on $L^{2}(\mathcal{X})$ for measure-preserving dynamics using Christoffel–Darboux kernels in spectral space. Slipantschuk et al. [20] consider a riddged Hilbert space and extend the Koopman operator to a space of distributions so that it becomes compact. Colbrook and Townsend [21] employ a residual-based approach that consistently approximates the spectral measure by removing spurious eigenvalues from DMD-type spectral computations. Spectrally approximating the Koopman operator in measure-preserving, ergodic flows by compact operators on reproducing kernel Hilbert spaces (RKHSs) has also been investigated [22]. However, dealing with continuous and residual Koopman spectra is still a challenging problem.

On the transfer operator side, popular approximation techniques are based on the Ulam method [23]. The Ulam method has been shown to yield spectrally consistent approximations for particular classes of systems such as expanding maps and Anosov diffeomorphisms on compact manifolds [5]. In some cases, spectral computations from the Ulam method has been shown to recover eigenvalues of transfer operators on anisotropic Banach spaces adapted to the expanding/contracting subspaces of such systems [24]; however, these results depend on carefully chosen state space partitions that may be hard to construct in high dimensions and/or under unknown dynamics. Various modifications of the basic Ulam method have been proposed that are appropriate for high-dimensional applications; e.g., sparse grid techniques [25].

We focus on measure-preserving skew-product systems in discrete time, $T(y,z)=(h(y),g(y,z))$, or continuous-time, $\Phi_{t}(y,z)=(h_{t}(y),g_{t}(y,z))$, on a product space $\mathcal{X}=\mathcal{Y}\times\mathcal{Z}$. Here, $\mathcal{Y}$ and $\mathcal{Z}$ are measure spaces, oftentimes referred to as the “base” and “fiber”, respectively. In such systems, the driving dynamics on $\mathcal{Y}$ is autonomous, but the dynamics on $\mathcal{Z}$ depends on the configuration $y\in\mathcal{Y}$. In many cases, one is interested in the time-dependent fiber dynamics, rather than the autonomous dynamics on the base. A typical example of skew-product dynamics is Lagrangian tracer advection under a time-dependent fluid flow [26, 27, 28], where $\mathcal{Y}$ is the state space of the fluid dynamical equations of motion and $\mathcal{Z}$ is the spatial domain where tracer advection takes place.

A well-studied approach for analysis of skew-product systems involves replacing the spectral decomposition of Koopman/transfer operators acting on functions on $\mathcal{X}$ by decomposition of associated operator cocycles acting on functions on $\mathcal{Y}$ using multiplicative ergodic theorems. In a standard formulation of the multiplicative ergodic theorem, first proved by Oseledets [29], one considers an invertible measure-preserving map $h:\mathcal{Y}\to\mathcal{Y}$ and the cocycle generated by a matrix-valued map $A$ on $\mathcal{Y}$. The multiplicative ergodic theorem then shows the existence of subspaces $\mathcal{V}_{1}(y),\ldots,\mathcal{V}_{k}(y)$ such that $A(y)\mathcal{V}_{j}(y)=\mathcal{V}_{j}(h(y))$. The subspace $\mathcal{V}_{j}(y)$ is called an Oseledets subspace (or equivariant subspace). Each Oseledets subspace has an associated Lyapunov exponent and associated covariant vectors, which are the analogs of the eigenvalues and eigenvectors of Koopman/transfer operators, respectively, in the setting of cocycles. Since its inception, the multiplicative ergodic theorem has been extended in many ways to infinite-dimensional operator cocycles [30, 31, 32, 33, 34]. Under appropriate quasi-compactness assumptions, it has been shown that the Lyapunov exponent spectrum is at most countably infinite and the associated Oseledets subspaces are finite-dimensional, e.g., [34, Theorem A].

A primary application of Oseledets decompositions is the detection of coherent sets and coherent structures in natural and engineered systems [35]. A family of sets $\{\mathcal{S}(y)\}_{y\in\mathcal{Y}}$ is called coherent if $\nu(\mathcal{S}(y)\bigcap g(h(y),\mathcal{S}(y)))/\nu(\mathcal{S}(y))$ is large for a reference measure $\nu$ on $\mathcal{Z}$. If an Oseledets subspace $\mathcal{V}(y)$ with respect to the transfer operator cocycle $U_{g(y,\cdot)}^{-1}$ is represented as $\mathcal{V}(y)=\operatorname{Span}\{v_{y}\}$ for a covariant vector $v_{y}\in L^{2}(\mathcal{Z})$ satisfying $U_{g(y,\cdot)}v_{h(y)}=v_{y}$, then setting $\mathcal{S}(y)$ to a level set of $v_{y}$ leads to a family of coherent sets. Finite-time coherent sets and Lagrangian coherent structures as the boundaries of the finite-time coherent sets have also been studied [26, 36, 37].

In this paper, we investigate a different approach to deal with continuous and residual spectra of Koopman operators on $L^{2}$ associated with skew-product dynamical systems. We propose a new decomposition, called eigenoperator decomposition, which reconstructs the Koopman operator from multiplication operators acting on certain subspaces, referred to here as generalized Oseledets spaces. These multiplication operators are obtained by solving an eigenvalue-type equation, but they can individually have continuous spectrum. Intuitively, this decomposition provides a factorization of the (potentially continuous) spectrum of the underlying Koopman operator into the spectra of eigenoperator families.

Our approach is based on the theory of Hilbert $C^{*}$-modules [38], which generalizes Hilbert space theory by replacing the complex-valued inner product by a product that takes values in a $C^{*}$-algebra. In this work, we employ the $C^{*}$-algebra of bounded linear operators on $L^{2}(\mathcal{Z})$, denoted by $\mathcal{B}(L^{2}(\mathcal{Z}))$. A standard operator-theoretic approach for skew-product dynamics is to define the Koopman or transfer operator on the product Hilbert space $\mathcal{H}=L^{2}(\mathcal{Y})\otimes L^{2}(\mathcal{Z})$ [28, 39]. In contrast, here we consider the Hilbert $C^{*}$-module $\mathcal{M}=L^{2}(\mathcal{Y})\otimes\mathcal{B}(L^{2}(\mathcal{Z}))$ over $\mathcal{B}(L^{2}(\mathcal{Z}))$. By considering $\mathcal{B}(L^{2}(\mathcal{Z}))$ instead of $L^{2}(\mathcal{Z})$, we aim to push information about the continuous spectrum of the Koopman operator onto the $C^{*}$-algebra $\mathcal{B}(L^{2}(\mathcal{Z}))$.

In more detail, starting from discrete-time systems, we define a $\mathcal{B}(L^{2}(\mathcal{Z}))$-linear operator $K_{T}$ on $\mathcal{M}$, which can be thought of as a lift of the standard Koopman operator on $\mathcal{H}$ to the Hilbert $C^{*}$-module setting. In addition, $K_{T}$ can be used to reconstruct the Koopman operator of the full skew-product system on $\mathcal{X}$. We show that $K_{T}$ admits a decomposition

where $\hat{M}_{i,j}$ is a $\mathcal{B}(L^{2}(\mathcal{Z}))$-linear multiplication operator (which we call eigenoperator), and $\hat{w}_{i,j}\in\mathcal{M}$ are eigenvectors associated with the operator cocycle on $L^{2}(\mathcal{Z})$ induced by the skew-product dynamics. We also derive an analogous version of (1) for continuous-time systems, formulated in terms of the generator of the Koopman group $\{K_{\Phi_{t}}\}_{t\in\mathbb{R}}$ acting on $\mathcal{M}$. A schematic overview of our approach for the continuous-time case is displayed in Fig. 1.

The eigenoperator decomposition (1) and its continuous-time variant have associated equivariant subspaces of $L^{2}(\mathcal{Z})$ as in the multiplicative ergodic theorem. In particular, to each eigenoperator $\hat{M}_{i,j}$ there is an associated family $\{\mathcal{V}_{j}(y)\}_{y\in\mathcal{Y}}$ of closed subspaces $\mathcal{V}_{j}(y)\subseteq L^{2}(\mathcal{Z})$ such that $U_{g(y,\cdot)}$ maps vectors in $\mathcal{V}_{j}(h(y))$ to vectors in $\mathcal{V}_{j}(y)$. Since we consider cocycles generated by unitary Koopman/transfer operators, the equivariant subspaces $\mathcal{V}_{j}(y)$ can be infinite-dimensional. Therefore, we call them generalized Oseledets subspaces. Spectral analysis of $\hat{M}_{i,j}$ then reveals coherent structures under the skew-product dynamics.

The rest of this paper is organized as follows. In Section 2, we derive our eigenoperator decomposition for discrete-time systems, and establish the correspondence between $K_{T}$ and the Koopman operator. We illustrate the decomposition in Section 3 by means of analytical examples with fiber dynamics on abelian and non-abelian groups. In these examples, the generalized Oseledets subspaces can be constructed explicitly, which provides intuition about the behavior of eigenoperator decomposition. In Section 4, we describe the construction of the infinitesimal generator and the associated eigenoperator decomposition for continuous-time systems. Section 5 contains numerical applications of the decomposition for continuous-time systems to simple time-dependent flows in two-dimensional domains. Section 6 contains a conclusory discussion. The paper includes an Appendix collecting auxiliary results.

Let $\mathcal{Y}$ and $\mathcal{Z}$ be separable measure spaces equipped with measures $\mu$ and $\nu$, respectively and let $\mathcal{X}=\mathcal{Y}\times\mathcal{Z}$, the direct product measure space of $\mathcal{Y}$ and $\mathcal{Z}$. Let $h:\mathcal{Y}\to\mathcal{Y}$ be a measure preserving and invertible map and let $g:\mathcal{X}\to\mathcal{Z}$ be a measurable map such that $g(y,\cdot)$ is measure preserving and invertible for any $y\in\mathcal{Y}$. Consider the following skew product transformation $T$ on $\mathcal{X}$:

We consider the Koopman operator $U_{T}$ on $L^{2}(\mathcal{X})$. Note that since $L^{2}(\mathcal{Y})$ and $L^{2}(\mathcal{Z})$ are separable, their tensor product $L^{2}(\mathcal{Y})\otimes L^{2}(\mathcal{Z})$ satisfies

Since $T$ is measure preserving, the Koopman operator $U_{T}$ is an unitary operator, but $U_{T}$ does not always have an eigenvalue decomposition since it has continuous spectrum in general.

We extend the Koopman operator $U_{T}$ to an operator on a Hilbert $C^{*}$-module. We first introduce Hilbert $C^{*}$-module [38, 40].

Let $\mathcal{A}$ be the $C^{*}$-algebra $\mathcal{B}(L^{2}(\mathcal{Z}))$. Let

i.e., the (right) Hilbert $\mathcal{A}$-module defined by the tensor product of the Hilbert $\mathbb{C}$-module $L^{2}(\mathcal{Y})$ and (right) Hilbert $\mathcal{A}$-module $\mathcal{A}$ [38]. We now define an operator on a Hilbert $C^{*}$-module.

The well-definedness of $K_{T}$ is not trivial. The following proposition shows the well-definedness of $K_{T}$ as an operator from $\mathcal{M}$ to $\mathcal{M}$.

The next proposition shows the relationship between $U_{T}$ and $K_{T}$, which enables us to connect existing studies of Koopman operators with our framework.

We derive a decomposition of $K_{T}$ called the eigenoperator decomposition. We first derive a fundamental decomposition using a cocycle on $\mathcal{Y}$. Then, we refine the decomposition using generalized Oseledets subspaces.