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 $\epsilon$-pseudospectrum $R_{\mathrm{ap},\epsilon }(T)$ from the closed approximate point $\epsilon$-pseudospectrum $C_{\mathrm{ap},\epsilon }(T)$. The latter is the direct closed lower-norm analogue of the approximate point $\epsilon$-pseudospectrum used in the $L^2$ Koopman SCI theory. Using continuous finite-dimensional dictionaries and tagged quadrature residuals, we prove SCI upper bounds for $R_{\mathrm{ap},\epsilon }(T)$, $C_{\mathrm{ap},\epsilon }(T)$, 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.

Extended and revised version: Distinguishes regularized and closed approximate point epsilon-pseudospectra, replaces the earlier Haar/conditional-expectation approach by continuous-dictionary quadrature residuals, revises lower-bound claims, leaves the no-modulus exact-spectrum sharpness problem open; Extended version also adds Appendix A on natural failed approaches