Residual SCI Upper Bounds And Lower Witnesses For Koopman Approximate Point Spectra In $L^p$ For $1<p<\infty$: Extended Version
For researchers in dynamical systems and numerical analysis, this work extends SCI theory from Hilbert to Banach spaces, providing rigorous computational guarantees for spectral approximation of Koopman operators.
This paper provides upper bounds on the Solvability Complexity Index (SCI) for computing approximate point spectra of Koopman operators in L^p spaces (1<p<∞), distinguishing two types of pseudospectra. The results show that these spectral sets can be computed with finite-dimensional approximations, with explicit error bounds depending on the regularity of the underlying map.
We study residual computation of approximate point spectral sets of bounded Koopman operators $\mathcal K_F$ on $L^p(\mathcal X,ω)$, $1<p<\infty$, where $\mathcal X$ is a compact metric space and $ω$ is a finite Borel measure. The input is the underlying map $F : \mathcal X \to \mathcal X$, accessed through point evaluations, and the output metric is the Hausdorff metric on non-empty compact subsets of $\mathbb C$. For a bounded operator $T$, we distinguish the regularized approximate point $\varepsilon$-pseudospectrum $R_{\mathrm{ap},\varepsilon}(T)$ from the closed approximate point $\varepsilon$-pseudospectrum $C_{\mathrm{ap},\varepsilon}(T)$. The latter is the direct closed lower-norm analogue of the approximate point $\varepsilon$-pseudospectrum used in the $L^2$ Koopman SCI theory. Using continuous finite-dimensional dictionaries and tagged quadrature residuals, we prove SCI upper bounds for $R_{\mathrm{ap},\varepsilon}(T)$, $C_{\mathrm{ap},\varepsilon}(T)$, and $σ_{\mathrm{ap}}$ on four natural classes of maps: continuous nonsingular maps, maps with a prescribed modulus of continuity, measure-preserving maps, and maps satisfying both measure preservation and a prescribed modulus.