= 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), namely so that the surjectivity condition is satisfied. Hence, by the reflexive version of the Lumer–Phillips theorem A generates a contraction semigroup. There are many more examples where a direct application of the Lumer–Phillips theorem gives the desired result. In conjunction with translation, scaling and perturbation theory the Lumer–Phillips theorem is the main tool for showing that certain operators generate strongly continuous semigroups. The following is an example in point. A normal operator (an operator that commutes with its adjoint) on a Hilbert space generates a strongly continuous semigroup if and only if its spectrum is bounded from above. Notes References Semigroup theory Theorems in functional analysis