Consider a closed Riemannian manifold $(M,g)$ and the heat equation $\partial_tu = \Delta_g u$ on it. Let $P_t$ be the heat semigroup generated by the equation. Of course the laplacian is symmetric with respect to the $L^2(dvol_g)$ inner product. Is there an easy way to see that also $P_t$ is symmetric? On what space of functions is it self-adjoint?
2026-03-26 08:14:26.1774512866
Heat semigroup is self-adjoint
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