# Lecture 2

Dear Participants,

this week we move to a more abstract setting and introduce the concept of semigroups.

Why this rather drastic change? It may seem unclear why we decide to leave the parabolic differential operator which we met in the first lecture to introduce something abstract.

The reason is that semigroups of bounded operators are everywhere and in particular they appear naturally in the study of elliptic and parabolic PDEs.

This first lectures on the semigroups deals with the so-called strongly continuous (or -) semigroups.

We begin by proving their main basic properties and then we show that to any strongly continuous semigroup one can associate uniquely a closed operator (the infinitesimal generator). A question arises naturally. Can we characterize all the operators which are infinitesimal generators of strongly continuous semigroups? The answer is positive and it is the well celebrated Hille-Yosida theorem.

Together with this lecture

we upload on wiki an appendix where you may find all the results on calculus and integration for functions with values in Banach space, which we use in this lecture and more generally in this ISem.

We kindly ask the team of Tübingen to provide us the official solutions to the exercises in this lecture.

We would also like to thank you (Jürgen, Sascha, Sebastian, Gabel and Jamil) for using the Discussion Board, where we received very useful remarks, corrections and suggestions, which will be incorporated in corrected versions of the manuscript.