Avoiding spectral pollution for transfer operators using residuals
This provides robust spectral estimation tools for applications in dynamical systems, though it is incremental as it extends existing methods to transfer operators.
The paper tackles the problem of spectral pollution in finite-dimensional approximations of transfer operators, which introduces spurious eigenvalues and compromises spectral computations in nonlinear dynamical systems. It presents algorithms that avoid spectral pollution, demonstrating accuracy in case studies such as Blaschke maps and a protein folding model.
Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.