Spectral theorem for linear maps (not operators)

Question

Supppose $\{U_n\}_{n=1}^d$ are finite dimensjonal Hilbert spaces with dimensions $\{k_n\}_{n=1}^d$. $T_n: U_{n+1} \to U_{n}$ are a set of linear transformations. I am looking for something like the spectral theorem, to prove that a point is an attractor. Does a suitable result exist for linear tranformations of the type $T_n$, even though it is not a linear operator?