Chapter Two CONTRACTIVITY PROPERTIES Let H be a Hilbert space over K = R or C. Denote by a a sesquilinear form on H. We assume that a is densely defined, accretive, continuous, and closed (see (1.2)-(1.5)). Denote by A its associated operator. We have seen in the previous chapter that -A generates a strongly continuous semigroup (e -tA ) t0 on H. Assume now that H = L 2 (X, µ, C), where (X, µ) is a -finite measure space. Several properties of the semigroup like positivity, L p -contractivity, domination, and so on can be characterized in terms of the operator A. However, in most applications, one does not precisely know the operator A. Typical situations where this occurs are when A is an elliptic operator with measurable coefficients and acts on L 2 (), where is any open subset of R n (see Chapter 4). Thus, characterizations in terms of the generator cannot be applied in several situations. On the contrary, in most situations one knows the form a. 1 Therefore, criteria for properties of the semigroup (e -tA ) t0 would be more useful and powerful if they are given in terms of the form. In the present chapter, we give criteria in terms of the form a for posi- tivity, irreducibility, and L -contractivity of the semigroup (e -tA ) t0 . We also study the domination property of semigroups by using the associated forms. The results are in the spirit of the famous Beurling-Deny criteria. The latter characterize the sub-Markovian property of semigroups associ- ated with symmetric forms. We will consider forms that are not necessarily symmetric and recover the Beurling-Deny criteria. The method used here works also for semigroups acting on vector-valued functions. The approach is based on criteria for invariance of closed convex sets of H under the action of the semigroup. These criteria hold in a general setting. The previously mentioned properties of the semigroup are obtained as particular cases, by choosing the appropriate convex set. It should be emphasized that all the results in the present chapter hold in both complex or real spaces. We do not distinguish the two cases unless we mention this explicitly. As in the previous chapter, we write real and imaginary parts of elements of K without assuming K = C. In the case 1 Usually, one starts by defining the form, hence its expression and domain are known. The associated operator A is given by Definition 1.21. In several situations, one cannot describe A precisely.