Construction of a unique self-adjoint generator for a symmetric local semigroup

AbstractA definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ⩽ t ⩽ T for some T > 0) on a Hilbert space H, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on H such that P(t) is a restriction of e−tH. As an application it is proven that H0 + V is essentially self-adjoint, where e−tH0 is an Lp-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L2 + ε and e−δV ∈ L1 for some ε, δ > 0