We study the computation of the approximate point spectrum and the approximate point $\varepsilon$-pseudospectrum of bounded Koopman operators acting on $L^p(\mathcal{X},\omega)$ for $1<p<\infty$ and a compact metric space $(\mathcal{X}, d_{\mathcal{X}})$ with finite Borel measure $\omega$. Building on finite sections in a computable unconditional Schauder basis of $L^p(\mathcal{X},\omega)$, we design residual tests that use only finitely many evaluations of the underlying map and produce compact sets on a planar grid, that converge in the Hausdorff metric to the target spectral sets, without spectral pollution. From these constructions we obtain a complete classification, in the sense of the Solvability Complexity Index, of how many limiting procedures are inherently necessary. Also we analyze the sufficiency and existence of a Wold-von Neumann decomposition analog, that was used in the special $L^2$-case. The main difficulty in extending from the already analyzed Hilbert setting $(p=2)$ to general $L^p$ is the loss of orthogonality and Hilbertian structure: there is no orthonormal basis with orthogonal coordinate projections in general, the canonical truncations $E_n$ in a computable Schauder dictionary need not be contractive (and may oscillate) and the Wold-von Neumann reduction has no directly computable analog in $L^p$. We overcome these obstacles by working with computable unconditional dictionaries adapted to dyadic/Lipschitz filtrations and proving stability of residual tests under non-orthogonal truncations.

翻译：我们研究了作用于 $L^p(\mathcal{X},\omega)$ ($1<p<\infty$) 的有界 Koopman 算子的近似点谱与近似点 $\varepsilon$-伪谱的计算问题，其中 $(\mathcal{X}, d_{\mathcal{X}})$ 为具有有限 Borel 测度 $\omega$ 的紧度量空间。基于 $L^p(\mathcal{X},\omega)$ 的一个可计算无条件 Schauder 基中的有限截断，我们设计了残差检验方法，该方法仅使用底层映射的有限次求值，并在平面网格上生成紧集，这些紧集在 Hausdorff 度量下收敛到目标谱集，且无谱污染。从这些构造出发，我们得到了关于需要多少极限过程在本质上必要的完整分类，该分类依据可解性复杂度指标的意义。同时，我们分析了在特殊 $L^2$ 情形中使用的 Wold-von Neumann 分解类似物的充分性与存在性。将已分析的 Hilbert 空间情形 ($p=2$) 推广到一般 $L^p$ 空间的主要困难在于正交性与 Hilbert 结构的缺失：一般而言不存在具有正交坐标投影的正交基，可计算 Schauder 字典中的标准截断 $E_n$ 未必是压缩的（且可能振荡），并且 Wold-von Neumann 约化在 $L^p$ 中没有直接可计算的类似物。我们通过使用适应于二进/Lipschitz 滤的可计算无条件字典，并证明非正交截断下残差检验的稳定性，从而克服了这些障碍。