ISem23

Dear Participants of ISem 23,

We have just uploaded the third lecture, which you can find at the following link

https://www.mat.tuhh.de/veranstaltungen/isem23/_media/lecture_03.pdf

or in the forum.

This lecture is focused on the construction of the time derivative as a normal, unbounded, continuously invertible operator in some weighted, vector-valued Hilbert space.

Our strategy to construct the time derivative operator is to define the `anti-derivative' operator first. For this, we need some preliminaries on Banach space-valued integration. Thus the first part of Lecture 3 will be devoted to the so-called Bochner integral and the definition of Bochner–Lebesgue spaces.

The most striking property of the derivative operator introduced will be its continuous invertibility. More so, it will turn out that the Hilbert space, the derivative operator is defined in allows for some flexibility in the weighting function, which in turn results in the anti-derivative operator to have small norm. This is of profound usefulness in the next lecture, where we discuss ordinary differential equations with possibly infinite dimensional state space.

We encourage the participants to contribute to the forum.

We welcome contributions to the exercises by any team, which will be checked by us and feedback will be provided. We particularly ask the team from Chemnitz to take note of the exercises and to provide `official' solutions for us to upload.

With the best regards,

the virtual lecturers