To do the next few exercises, you will have to know some functional analysis, specifically, the spectral theorem for self-adjoint operators. See Lax XXXXXXXXXXLet E be a Dirichlet form with domain...

To do the next few exercises, you will have to know some functional analysis, specifically, the spectral theorem for self-adjoint operators. See Lax (2002). Let E be a Dirichlet form with domain D(E) and let L be the infinitesimal generator of the semigroup Pt that corresponds to L. Let E(dλ) be a spectral resolution of the identity for −L. (The operator L is a negative operator, so −L is a positive one.) Then a consequence of the spectral theorem is that.