Residual SCI Upper Bounds And Lower Witnesses For Koopman Approximate Point Spectra In $L^p$ For $1<p<\infty$: Extended Version

Abstract

We study residual computation of approximate point spectral sets of bounded Koopman operators $\mathcal K_F$ on $L^p(\mathcal X,\omega)$, $1<p<\infty$, where $\mathcal X$ is a compact metric space and $\omega$ 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 $\sigma_{\mathrm{ap}}$ 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.