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

TL;DR

This paper investigates the computational complexity of spectral analysis of Koopman operators on different function spaces, extending existing methods beyond reflexive regimes and establishing new theoretical frameworks for understanding endpoint solvability.

Contribution

It introduces a unified approach to Koopman spectral computation across $L^p$ spaces, constructs prototype decision problems for complexity analysis, and connects these findings to Type-2/Weihrauch theory.

Findings

Unified framework for $L^1$ and $L^p$ spectral computation

Prototype decision problems for complexity bounds

Deeper connection to Type-2/Weihrauch theory

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 $\omega$ 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 $L^p(\mathcal{X},\omega)$ for $p\in\{1,\infty\}$ for the computational problem \[ \Xi_{\sigma_{\mathrm{ap}}}(F) :=\sigma_{\mathrm{ap}}\!\bigl(\mathcal{K}_F\bigr), \] with input access given by point evaluations of $F\mapsto F(x)$ (and fixed quadrature access to $\omega$). 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 $(\Xi_m)_{m\in\mathbb N}$ 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.