# Lecture 3

Dear Participants,

as announced, this week we keep on the study of semigroups introducing the so called sectorial operators and the associated analytic semigroups. Looking at the resolvent set and the resolvent operator of infinitesimal generators of $C_0$-semigroups and of sectorial operators a basic difference immediately appears: the spectrum of the infinitesimal generator $A$ of a $C_0$-semigroup contains a halfplane and on this halfplane the norm of the resolvent operator $R(\lambda,A)$ is controlled from above by the inverse of the REAL PART of $\lambda$. On the contrary, the resolvent set of a sectorial operator $A$ contains a sector of the complex space and on this sector the norm of $R(\lambda,A)$ is controlled from above by the inverse of the MODULUS of $\lambda$. This difference is crucial and leads to many important differences between strongly continuous and analytic semigroups. Typically, if $\{T(t)\}$ is a $C_0$-semigroup and $x\in X$, then the function $t\mapsto T(t)x$ is just continuous $[0,+\infty)$ and $T(t)x$ does not belong $D(A)$ unless $x$ itself belongs to $D(A)$. On the other hand if $\{T(t)\}$ is an analytic semigroup, then the function $t\mapsto T(t)$ is analytic in a sector, which properly contains the halfline $(0,+\infty)$, with values in $L(X)$ (this is the reason why these semigroups are called analytic) and for each $x\in X$ and $t>0$, $T(t)x$ belongs to the intersection of all the powers of its generator $A$.

We anticipate that typically solutions of parabolic equations with smooth enough coefficients are associated with analytic semigroups. This is the reason why in this lecture we introduce this class of semigroups.

Now, it is time to stop writing and let you enter in the "world" of analytic semigroups. Please download here below the file and... enjoy the reading!

This time we kindly ask the team from Ulm to prepare and post on internet the official solutions of this lecture.