Domination of semigroups generated by regular forms

Abstract

We give a representation for regular forms associated with dominated C 0 -semigroups which, in turn, characterises domination of C 0 -semigroups associated with regular forms. In addition, we prove a relationship between the positivity of (dominated) C 0 -semigroups and the locality of the associated forms.A characterisation of domination. Throughout, we let (Ω, A, µ) be a topological measure space. By this we mean that Ω is a topological space, A is the Borel σ-algebra, and µ is a (positive) Borel measure on (Ω, A). Without loss of generality, we assume that µ has full support in the sense that there exists no nonempty open subset U ⊆ Ω which has zero measure. Otherwise, we replace Ω with Ω \ U, where U is the largest open subset which has measure zero. All vector spaces in this note are complex vector spaces.We now state our main result: Theorem 1.1. Let a : D(a) × D(a) → C and a : D( a) × D( a) → C be two sesquilinear, Hermitean, closed, accretive, and densely defined forms on L 2 µ (Ω) such that the associated self-adjoint C 0 -semigroups (denoted by T and T respectively) are real. Assume that the semigroup T is positive, and that the form a is A-regular Date: 1st December 2021. 2020 Mathematics Subject Classification. 47D03, 31C15.

Evolution equations with eventually positive solutions

“…and is given by Av = −v (4) for each v ∈ dom(A). The C 0 -semigroup (e tA ) t ∈ [0, ∞) is not positive; this can for instance be seen by associating a sesquilinear form to −A and using the so-called Beurling-Deny criterion [30,Corollary 2.18].…”

“…Compare also [4,Section 4.2] for a related discussion. An example of eventual positivity for different non-local boundary conditions which lead to a non-self-adjoint realisation of the Laplace operator can be found in [14,Theorem 4.3].…”