Essential self-adjointness of operators in ordered hilbert space

LetH ₀0 be a self-adjoint operator acting in a spaceL ²(M, ). It is assumed thatH ₀ e=0, wheree is strictly positive, and that exp(–tH ₀) is positivity preserving fort0. LetV be a real function onM such that its positive part is inL ²(M,e ²) and its negative part is relatively small with respect toH ₀. ThenH=H ₀+V is essentially self-adjoint on the intersection of the domains ofH ₀ andV. This result is applied to Schrödinger operators and to quantum field Hamiltonians.