Title: Spectral characterisations of stable operator semigroups

Abstract:

Many time-evolution phenomena can be modeled by the abstract Cauchy problem, whose well-posedness is directly linked to the existence of a suitable operator semigroup, and the classical solutions are then given by its orbits. Since the seminal work of Hille and Yoshida, operator semigroups have been extensively studied through the spectral properties of their infinitesimal generators.

A central question in this theory concerns the long-term behaviour of solutions. In this context, an operator semigroup is called strongly stable if each of its orbits converges to zero as time tends to infinity. In the 1980s, Arendt and Batty, and independently Lyubich and Vu, established a celebrated sufficient spectral condition for strong stability. This result kicked off a rich line of research leading to numerous further criteria, yet a complete spectral characterisation of strong stability had not been found.

In this talk, we introduce the "local pseudo-function spectrum" of the infinitesimal generator and prove that an operator semigroup is strongly stable if and only if this spectrum is empty. We then continue to demonstrate how this characterisation allows us to recover and improve many known sufficient conditions for strong stability, including the Arendt-Batty-Lyubich-Vu theorem. Among others, our techniques use vector-valued optimal forms of the Ingham-Karamata Tauberian theorem.

This work is based on collaborative work with Morgan Callewaert and Jasson Vindas.

2. abstract of the talk by David Berger:

Title: Weighted psi-Bessel-potential spaces, Lévy processes and nonlocal gradients

Abstract: 
We define weighted psi-Bessel-potential spaces on domains and discuss embedding properties. Furthermore, we have a short look on connections to possible extensions of nonlocal gradients and prove an existence result for solutions of a nonlocal variant of a weighted p-Laplace equation.