Koopman Spectral Computation Beyond The Reflexive Regime: Endpoint Solvability Complexity Index And Type-2 Links

Abstract: We study the Solvability Complexity Index (SCI) of Koopman operator spectral computation in the information-based framework of towers of algorithms. Given a compact metric space $(\mathcal{X},d)$ with a finite Borel measure $ω$ on $\mathcal{X}$ and a continuous nonsingular map $F:\mathcal{X}\to \mathcal{X}$, our focus is the Koopman operator $\mathcal{K}F$ acting on $L^p(\mathcal{X},ω)$ for $p\in{1,\infty}$ for the computational problem [ Ξ{σ{\mathrm{ap}}}(F) :=σ{\mathrm{ap}}!\bigl(\mathcal{K}F\bigr), ] with input access given by point evaluations of $F\mapsto F(x)$ (and fixed quadrature access to $ω$). We clarify how the $L^1$ case can be brought into the same oracle model as the reflexive regime $1<p<\infty$ by proving a uniform finite-dimensional quadrature compatibility, while highlighting the fundamentally different role played by non-separability at $p=\infty$. Beyond Koopman operators, we also construct a prototype family of decision problems $(Ξ_m){m\in\mathbb N}$ realizing prescribed finite tower heights, providing a reusable reduction source for future SCI lower bounds. Finally, we place these results deeper in the broader computational landscape of Type-2/Weihrauch theory.