A Noether theorem for stochastic operators on Schatten classes

We show that a stochastic (Markov) operator S acting on a Schatten class C1 satisfies the Noether condition (i.e. S′(A)=A and S′(A2)=A2, where A∈C∞ is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H,〈⋅,⋅〉)), if and only if S(EA(G)XEA(G))=EA(G)S(X)EA(G) for any state X∈C1 and all Borel sets G⊆R, where EA(G) denotes the orthogonal projection coming from the spectral resolution A=∫σ(A)zEA(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.