Seminar 4. Fundamental Properties of $C_0$-Semigroups, Part 2

This session is dedicated to a further investigation of the properties of $C_0$-semigroups. The main focus is a detailed analysis of the generator: its graph, the graph norm, and the concept of a core. The relationship between the convergence of generators, semigroups, and resolvents is explored. The theory of invariant subspaces is discussed separately, including the construction of the subspace of strong continuity for an arbitrary semigroup. In conclusion, fundamental integral formulas linking the semigroup, its generator, and the resolvent are proved, including the Taylor formula.