Seminar 3. Fundamental Properties of $C_0$-Semigroups, Part 1

This session will systematise the fundamental principles of the theory of strongly continuous ($C_0$) semigroups of linear operators. The primary focus is on the interrelations between the three key objects: the semigroup itself, its generator, and the resolvent. We will cover the properties of the generator (dense definition, closedness) and its resolvent, including the latter's integral representation via the semigroup. For a rigorous exposition, the concepts of the Bochner integral for vector-valued functions and spaces of strongly continuous operator-valued mappings will be employed. The session aims to build a comprehensive understanding of the basic framework of semigroup theory.