and uniqueness corollary for Floquet normal forms.

Our main results also imply several existence and uniqueness corollaries relevant to the “applied Koopmanism” literature, which has experienced a surge of interest initiated by [@dellnitz1999approximation; @mezic2004comparison; @mezic2005spectral]—motivated largely by data-driven applications— more than 70 years after Koopman’s seminal work [@koopman1931hamiltonian].[^1] Existence and uniqueness results are desirable since, in analyzing the theoretical properties of any algorithm for computing some quantity, it is desirable to know whether the computation is well-posed [@hadamard1902problemes], and in particular whether the quantity in question *exists* and is *uniquely determined*. Our results yield precise conditions under which various quantities in this literature—including targets of numerical algorithms—exist and are unique, and are especially relevant to work on [*principal eigenfunctions*]{} and [*isostables*]{} for point attractors [@mohr2016koopman; @mauroy2013isostables] and to work on [*isostable coordinates*]{} for periodic orbit attractors in [@wilson2016isostable; @shirasaka2017phase; @wilson2018greater; @monga2019phase]. Isostables and isostable coordinates are useful tools for nonlinear model reduction, and it has been proposed that these objects could prove useful in real-world applications such as treatment design for Parkinson’s disease, migraines, cardiac arrhythmias [@wilson2016isostable-pde], and jet lag [@wilson2014energy].

We give an intrinsic definition of principal eigenfunctions for nonlinear dynamical systems which generalizes the definition for linear systems 