Geodesic flow and decay of traces on hyperbolic surfaces

The paper’s Theorem 6 asserts exactly the two claims at issue—trace class and exponential decay of Trace((A2^Γ)^*A1^Γ(t))—under the same symbol, analyticity, and mean-zero hypotheses. Its proof proceeds by (i) Zelditch quantization and a correlation formula (Lemma 7, Corollary 8), (ii) anisotropic Sobolev bounds for the distributions T_{j,a2}, and (iii) exponential decay of correlations for the geodesic flow; the summation over j then converges by Weyl asymptotics. The candidate solution follows the same architecture: the same quantization and decomposition, the same regularity inputs, and the same exponential mixing step (via standard contact Anosov flow theory). Minor stylistic differences aside (the candidate cites Liverani; the paper cites Faure–Sjostrand/Nonnenmacher–Zworski and notes some details briefly), the arguments are substantively aligned and correct.