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.

Edit: This question has been significantly revised.

Some recent developments in computational geometry (for example see http://geometry.stanford.edu//papers/fmfrmbs-obsbg-12/fmfrmbs-obsbg-12.pdf) are based on the idea of considering the pullback of a map between two manifolds. As the pullback is a linear map (insofar that it is well-defined), it is much more friendly to work with than the original map, and many tools from Hilbert spaces can now be exploited. However, the "price paid" is that now you are perhaps working in a higher dimensional space, or some information about the original map is lost.

I am very interested to learn of any other mathematics literature that may be related to these ideas or this approach. Thank you very much.

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.

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.