Operator Semigroups and Applications

7th international Internet Seminar 2003-2004
Operator Semigroups and Applications
Strongly continuous semigroups of linear operators on Banach spaces are a powerful tool to
treat linear initial value (Cauchy) problems and to investigate the qualitative properties of
their solutions.
Indeed, many initial value problems, originally given as a partial differential equation, an integrodifferential
equation or a functional differential equation, can be written as an Abstract
Cauchy Problem
(ACP)
{ ˙u(t) = Au(t), t ≥ 0,
u(0) = x,
for some linear operator A on a Banach space X and a function u : R + → X. The starting
point of the theory is the fact that (ACP) is wellposed if and only if the operator A is the generator
of a strongly continuous one-parameter semigroup (T (t)) t≥0 of bounded linear operators
on X. The Hille-Yosida Theorem characterizes these generators and Liapunov-type theorems
give conditions on the generator yielding, e.g., stability and other qualitative properties of the
semigroup.
The aim of Phase 1 of this Internet Seminar is to give a streamlined introduction into this
theory concentrating on the major results such as
◦ the Hille-Yosida and Lumer-Phillips theorems,
◦ analytic semigroups,
◦ perturbation theory,
◦ the Trotter-Kato approximation theorem,
◦ stability theory.
A basic course on functional analysis and operator theory should suffice as a prerequisite. As
our main reference we will use
K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations, Springer-
Verlag, New York 2000.
In Phase 2 we will offer study projects on more advanced and specialized topics of semigroup
theory. These projects will be proposed and coached by the local coordinators and will emphasize
the applications of the theory to evolution equations.
For Phase 3 (June 13-19, 2004) we invite the participants to the Heinrich-Fabri Institut at
Blaubeuren, Germany, where they will meet their (up to then mostly ”virtual”) teammates
and the other participants, and present their projects. Some leading researchers will expose
further perspectives on semigroup methods for evolution equations.

Structure of the Internet Seminar
Phase 1: October–February
The organizers will assist the participants to become acquainted with the BSCW program. A
detailed manual for BSCW will be provided. We will then send study material (as dvi-, ps- or
pdf-files) on a weekly basis to the participants and local coordinators. This material includes the
basic text, additional exercises, bibliographic references and open problems. The participating
students, supervised by their local coordinators, will study and discuss this material. Each
participant is strongly encouraged to communicate via BSCW with the other participants on
questions, solutions, and new problems.
Phase 2: February–June
The local coordinators will propose study projects to the students. Guided by the coordinators,
the teams prepare seminar talks and inform the other participants about their work via written
status reports.
Phase 3: June
This is when the ’virtual’ seminar becomes ’real’ and all participants are invited to a joint
workshop. The students meet their teammates, present their seminar talks, and attend survey
talks on recent research by some of the coordinators. The workshop takes place June 13-19,
2004 at the Heinrich-Fabri-Institut Blaubeuren, Germany. The living expenses are covered by
the organizer; travel expenses are the responsibility of each individual participant.
Potential Participants
The Internet Seminar is aimed at graduate and Ph.D. students. The participants are expected
to have some basic knowledge from functional analysis and operator theory. They should have
a local coordinator responsible for scientific, technical, and bureaucratic matters. The Internet
Seminar uses the BSCW (‘Basic Support for Cooperative Work’) software to facilitate the
electronic communication between participants. To use this tool, the student needs access to
a recent version of an internet browser. Mathematical texts should be edited in Tex or in Latex.
Each local coordinator should
Local Coordinators
◦ provide access to the internet to his students
◦ help his students with mathematical and technical problems
◦ ensure that his students receive full credit for the seminar.
Scientific Committee
Wolfgang Arendt (Ulm)
András Bátkai (Budapest)
Klaus-J. Engel (L’Aquila)
Alessandra Lunardi (Parma)
Giorgio Metafune (Lecce)
Rainer Nagel (Tübingen)
Abdelaziz Rhandi (Marrakesh)
Lutz Weis (Karlsruhe)
http://www.fa.uni-tuebingen.de/tulka/isem