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

Christopher Sorg

TL;DR

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 $p=\infty$.

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.

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

TL;DR

The L^1 case can be brought into the same oracle model as the reflexive regime by proving a uniform finite-dimensional quadrature compatibility, while highlighting the fundamentally different role played by non-separability at .

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 with a finite Borel measure on and a continuous nonsingular map , our focus is the Koopman operator acting on for for the computational problem with input access given by point evaluations of (and fixed quadrature access to ). We clarify how the case can be brought into the same oracle model as the reflexive regime by proving a uniform finite-dimensional quadrature compatibility, while highlighting the fundamentally different role played by non-separability at . Beyond Koopman operators, we also construct a prototype family of decision problems 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.
Paper Structure (29 sections, 42 theorems, 247 equations, 1 table)

This paper contains 29 sections, 42 theorems, 247 equations, 1 table.

Table of Contents

  1. Introduction
  2. Spectral Computation And The Solvability Complexity Index
  3. Koopman Operators, Endpoint Spaces, And The Evaluation Device
  4. Why We Work On The Cantor Space
  5. What This Document Contributes
  6. Message For Dynamical Systems And Data-Driven Koopman Methods
  7. SCI And Type-2 / Weihrauch: Scope And What Is Genuinely New
  8. Koopman Framework On L1(X,w) And On Loo(X,w)
  9. Koopman Operator On L1(X,w)
  10. Koopman Operator On Loo(X,w)
  11. Upper Bound Meta-Theorem
  12. The Problematic F=Loo(X,w) Case
  13. The Arithmetical Case
  14. The General Case
  15. Reduction To Koopman Approximate Point Spectrum In The m-class
  16. ...and 14 more sections

Key Result

Proposition 2.1

Let $F$ be nonsingular and $\rho_{F}\in L^{\infty}(\omega)$. Then $\mathcal{K}_{F}:L^{1}(\omega)\to L^{1}(\omega)$ is bounded and In particular, if $F$ is measure–preserving ($\rho_{F}=1$ a.e.), then $\mathcal{K}_{F}$ is an isometry on $L^{1}(\omega)$.

Theorems & Definitions (98)

  • Proposition 2.1: Boundedness on $L^{1}$
  • Corollary 2.3: Composition stability in $L^{1}$
  • Proposition 2.4: Well-definedness and boundedness on $L^{\infty}$
  • Lemma 2.6: Composition stability in $L^{\infty}$
  • Remark 2.7: When is $\mathcal{K}_F$ an isometry on $L^\infty$?
  • Theorem 3.2: Meta theorem for Koopman approximate point spectra on Banach function spaces
  • proof
  • Remark 3.3: Examples and non-examples for \ref{['F1']}-\ref{['F4']}
  • Lemma 3.4: Uniform Riemann-sum approximation of the $L^1$-norm on finite $W\subset C(\mathcal{X})$
  • proof
  • ...and 88 more