  We study pseudodifferential operators on a hyperbolic surface using `Zelditch
quantization'. We motivate and study the trace of $A_2^* A_1(t)$, where $A_2$
is a fixed operator and the Zelditch symbol of $A_1(t)$ evolves by geodesic
flow. We find conditions under which the trace decays exponentially as $t \to
\pm \infty$.
