Solvability Complexity Index Classification For Koopman Operator Spectra In $L^p$ For $1

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.