Residual SCI Upper Bounds And Lower Witnesses For Koopman Approximate Point Spectra In L^p For 1<p<infty: Extended Version
Pith reviewed 2026-05-18 15:51 UTC · model grok-4.3
The pith
Continuous finite-dimensional dictionaries and tagged quadrature residuals yield upper bounds on approximate point spectra of Koopman operators in L^p spaces.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Using continuous finite-dimensional dictionaries and tagged quadrature residuals, we prove SCI upper bounds for R_ap,ε(T), C_ap,ε(T), and σ_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.
What carries the argument
Continuous finite-dimensional dictionaries paired with tagged quadrature residuals, which produce explicit upper bounds on the Hausdorff distance to the target spectral sets.
If this is right
- Upper bounds hold for both the regularized and closed approximate point ε-pseudospectra on continuous nonsingular maps.
- The same residual method supplies upper bounds when the map satisfies a prescribed modulus of continuity.
- Measure-preserving maps admit computable upper bounds on the same spectral sets.
- Maps that are both measure-preserving and modulus-controlled receive the tightest residual upper bounds among the four classes.
- Lower witnesses accompany the upper bounds to certify the quality of the approximation.
Where Pith is reading between the lines
- The residual technique may adapt to other classes of operators that admit finite-dimensional dictionary approximations on L^p spaces.
- Numerical experiments on standard dynamical systems such as circle rotations could test how quickly the Hausdorff error decreases with dictionary dimension.
- The distinction between regularized and closed pseudospectra suggests different stability properties under perturbation of the map.
Load-bearing premise
The map is available only through point evaluations and the finite Borel measure makes the Koopman operator bounded on L^p for 1 < p < ∞.
What would settle it
For a concrete measure-preserving map whose approximate point spectrum is known exactly, compute the dictionary-based residual upper bound and check whether the true spectrum lies inside the claimed Hausdorff neighborhood.
read the original 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.
Editorial analysis