Transfer operator of a probability-preserving map

The transfer operator $P_T$ of a probability-preserving transformation is the Hilbert-space adjoint $P_T=U_T^*$ on $L^2$ . With inner products linear in the first argument, it is characterized by

\int (P_T f)\overline h\,d\mu =\int f\,\overline{h\circ T}\,d\mu.

It is a positive contraction and $P_T1=1$ . If $T$ is invertible, $P_Tf=f\circ T^{-1}$ .

Doubling-map example

For $T(x)=2x\bmod1$ ,

P_T f(x)=\tfrac12\bigl(f(x/2)+f((x+1)/2)\bigr).

The operator averages over the two preimages. It is not composition with a nonexistent inverse map. The same duality extends to a positive $L^1$ operator transporting densities relative to the invariant probability measure.