SCI lower bounds for Koopman spectral computations

Establish lower bounds on the minimal number of successive limits (Solvability Complexity Index classification) required to compute spectral properties of Koopman operators from data-driven approximations, thereby determining the intrinsic algorithmic difficulty of these spectral computations within the SCI framework.

Background

The paper discusses that reliable computation of spectral quantities for Koopman operators in infinite-dimensional settings often requires taking multiple limits, such as a large-data limit followed by a large-dictionary limit, and that these limits generally do not commute. This motivates the use of the Solvability Complexity Index (SCI) to classify problems by the minimal number of successive limits required.

Within this framework, some spectral quantities of Koopman operators are known to need two successive limits, while others appear to belong to higher SCI levels. However, a rigorous classification demands establishing lower bounds on the required number of limits, which remains unresolved in general and is identified as an active research direction. Clarifying these lower bounds would formalize which Koopman spectral computations are fundamentally harder and guide algorithm design.