Essentially Coercive Forms and asympotically compact semigroups

Analysis Seminar
Wednesday, March 11, 2020 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 005
Isabelle Chalendar – Université Paris-Est - Marne-la-Vallée
Shahaf Nitzan

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that $\dist(S(t),{\mathcal K}(H))\to 0$ as $t\to\infty$, where ${\mathcal K}(H)$ denotes the space of all compact operators on the underlying Hilbert space.