LpContraction Semigroups for Vector Valued Functions

We discuss semigroups acting on vector valued functions. We consider a comparison theorem between two semigroups: a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions. The criterion is given by the abstract Kato theorem. By using this, we give a sufficient condition for the criterion in the setting of square field operator. We also consider the essential self-adjointness of a perturbed semigroup.