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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.16016 |
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| _version_ | 1866909036828426240 |
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| author | Sorg, Christopher |
| author_facet | Sorg, Christopher |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residual SCI Upper Bounds And Lower Witnesses For Koopman Approximate Point Spectra In $L^p$ For $1<p<\infty$: Extended Version Sorg, Christopher Spectral Theory Computational Complexity Numerical Analysis Dynamical Systems 47B33, 47A10, 03D78, 37M25, 46E30, 65F15 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. |
| title | Residual SCI Upper Bounds And Lower Witnesses For Koopman Approximate Point Spectra In $L^p$ For $1<p<\infty$: Extended Version |
| topic | Spectral Theory Computational Complexity Numerical Analysis Dynamical Systems 47B33, 47A10, 03D78, 37M25, 46E30, 65F15 |
| url | https://arxiv.org/abs/2509.16016 |