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# 25th Internet Seminar: "Spectral theory for Operators and Semigroups"

This year's topic is “Spectral theory for operators and semigroups” and is suited for master students, Ph.D. students and young post-docs with a basic knowledge on functional analysis and complex analysis of one variable.

As usual (and as many of you may know), the ISem has three phases:

- Phase 1 (October 2021 to February 2022): A weekly lecture will be provided via the ISem website. This material forms a proper self-contained course with study material and exercises. There will also be a forum to discuss the material.

- Phase 2 (March 2022 to June 2022): You will have the chance to work on projects in small international groups. Of course, there will be a supervisor for each project.

- Phase 3 (June 20 to June 24, 2022): There will be a final one-week workshop in Salerno (Italy), where the project teams of Phase 2 will present their projects and additional lectures will be delivered by leading experts.

Registration will open by early September and will be possible until the mid/end of October.

### ISem Team 2021/2022

#### Virtual Lecturers

• Angela Albanese
• Luca Lorenzi
• Elisabetta Mangino
• Abdelaziz Rhandi

#### Organizer

• Loredana Caso
• Federica Gregorio
• Cristian Tacelli

### Description of the course

Starting from basic concepts of operator theory, the course concentrates on spectral theory for linear operators and strongly continuous semigroups on Hilbert and in general on Banach spaces.

The main task of the spectral theory of linear operators is the following:

Given a linear operator $A$ with domain $D(A)$ on a Banach space $X$, determine the set, called the resolvent set of $A$, $$\rho(A):=\{\lambda \in \mathbb C : (\lambda I-A) \text{ has a bounded inverse in }X\},$$ its complement $\sigma(A)$ in $\mathbb C$ called the spectrum of $A$ and, consequently, investigate the relationship between the operator $A$ and the operators $(\lambda I-A)^{-1}$, if they exist.

An important application of the spectral theory is the study of the long time behaviour of the solution $u(t)=e^{tA}u_0$ to the initial value problem $$\left\{\begin{array}{ll} u'(t)=Au(t),\quad t\ge 0,\\ u(0)=u_0\in D(A), \end{array} \right.$$ where the operators $e^{tA}$ form a so-called strongly continuous semigroup generated by $A$.

In this course we shall develop the basic tools from spectral theory for linear operators and apply them to strongly continuous semigroups.

Topics to be covered include:

• Compact operators and Riesz-Schauder theory;
• Spectral representation theorem for bounded and unbounded operators;
• Strongly continuous and analytic semigroups;
• Spectral mapping theorems for semigroups;
• Asymptotic behaviour of strongly continuous and analytic semigroups;
• Long time behaviour of solutions to nonhomogeneous abstract Cauchy problems.

Some of these topics will be elaborated on in Phase 2, where the students will have the possibility to work on projects which are related to active research. 