Generalization on LP - Contractivity of Semigroups Commutators and C 0-Semigroups of Resolvent Estimates

Abstract

We derive a pointwise estimate on the absolute difference between two corresponding diffusion kernels of two diffusion semigroups , as well as an L” —operator norm bound. We show that linear partial differential operators of order higher than two can not generate contraction semigroups on the Lebesgue space except for some fourth order operators in a restricted compact interval . We consider a comparison between two semigroups , a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions . We give a sufficient condition for the criterion in the setting of square field operator. We also consider the essential self a djointness of a perturbed semigroup . We discuss the existence and the continuity of the boundary values problem on the Lebesgue space of the resolvent of a self — adjoint operator of the conjugate operator method . we allow the conjugate operator to be the generator of a Co —semigroup and that first commutator is not comparable to the self —adjoint operator . Strong application include the spectral theory of zero mass quantum field models are considered .