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.

翻译：我们研究有界Koopman算子$\mathcal K_F$在$L^p(\mathcal X,ω)$空间（$1<p<\infty$）中近似点谱集的残差计算，其中$\mathcal X$是紧度量空间，$ω$是有限Borel测度。输入是通过点评估访问的底层映射$F : \mathcal X \to \mathcal X$，输出度量是$\mathbb C$上非空紧子集的Hausdorff度量。对于有界算子$T$，我们将正则化近似点$\varepsilon$-伪谱$R_{\mathrm{ap},\varepsilon}(T)$与闭近似点$\varepsilon$-伪谱$C_{\mathrm{ap},\varepsilon}(T)$加以区分。后者是$L^2$ Koopman SCI理论中使用的近似点$\varepsilon$-伪谱的直接闭下范数模拟。利用连续有限维字典和标记正交残差，我们证明了$R_{\mathrm{ap},\varepsilon}(T)$、$C_{\mathrm{ap},\varepsilon}(T)$和$σ_{\mathrm{ap}}$在四类自然映射类上的SCI上界：连续非奇异映射、具有指定模连续性的映射、保测映射以及同时满足保测性和指定模连续性的映射。