Several recent papers have been published in computer graphics based on the idea of a functional map, which is usually defined as follows. Let $M$ and $N$ be manifolds. For a bijective continuous map $T:M \to N$, the functional map corresponding to $T$ is defined by $$T_F : L^2(M,\mathbb{R}) \to L^2(N,\mathbb{R})~~~~T_F(f)=f \circ T^{-1}$$ This definition has led to many significant computational results in the past two years (i.e. see http://www.lix.polytechnique.fr/~maks/papers/obsbg_fmaps.pdf), however not much is yet known about functional maps theoretically, i.e. how best to formulate them rigoroulsy, what properties they satisfy, etc. Functional maps are formally similar to the transpose of a linear map (or rather its inverse), however the spaces $M$ and $N$ of most interest are compact manifolds rather than vector spaces.

I would be very interested to learn of any related concepts or mathematics literature that would help to better understand functional maps. Thank you.