In Koopman analysis, any non-linear dynamical system can be exactly represented as an infinte dimensional linear operator acting on a Hilbert space of all functions. I am looking to prove that the same property holds (or doesn't hold) when you look at only invertible functions.
I think this comes down to the question of showing equality between the span of invertible functions and the span of all functions - but I am not really familiar with how to do this, or if this is even a well posed question as I have presented it