We study the computation of the approximate point spectrum and the approximate point -pseudospectrum of bounded Koopman operators acting on Lᵖ (X, ω) for 1<p< and a compact metric space (X, dₗ) with finite Borel measure ω. Building on finite sections in a computable unconditional Schauder basis of Lᵖ (X, ω), 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 L²-case. The main difficulty in extending from the already analyzed Hilbert setting (p=2) to general Lᵖ is the loss of orthogonality and Hilbertian structure: there is no orthonormal basis with orthogonal coordinate projections in general, the canonical truncations Eₙ in a computable Schauder dictionary need not be contractive (and may oscillate) and the Wold-von Neumann reduction has no directly computable analog in Lᵖ. 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.
C Sorg (Fri,) studied this question.