# Dortmund-Hagen-Wuppertal Analysis Meeting am 24. Januar 2019

Gemeinsames Treffen der Arbeitsgruppe Funktionalanalysis an der Bergischen Universität Wuppertal, der Lehrgebiete Analysis, Angewandte Stochastik und Stochastik der Fernuniversität Hagen, sowie des Lehrstuhls für Analysis, Mathematische Physik & Dynamische Systeme der Technischen Universität Dortmund. Ausgerichtet diesmal an der Fakultät für Mathematik der TU Dortmund.

### Veranstaltungsort

TU Dortmund
Campus Nord
Hörsaal 1 im Chemiegebäude

### Zeitplan der Vorträge

• 14.30 Michael Hartz
• 15.30 Thomas Kalmes
• 16.30 Kaffee und Tee
• 17.00 Robert Nabiullin
• anschließend, gegen 19.00, Abendessen in einem Restaurant

Ansprechpartner: Birgit Jacob (Wuppertal), Delio Mugnolo (Hagen), Ivan Veselić (Dortmund)

### Vorträge und Abstracts

• Michael Hartz (Hagen): Interpolierende Folgen und das Kadison-Singer-Problem

Eine Folge $$(z_n)$$ in der Einheitskreisscheibe ist eine interpolierende Folge für $$H^\infty$$, falls es zu jeder beschränkten Folge $$(w_n)$$ eine beschränkte holomorphe Funktion $$f$$ auf der Einheitskreisscheibe mit $$f(z_n) = w_n$$ für alle $$n$$ gibt. Diese Folgen wurden von Lennart Carleson charakterisiert. Ich werde über eine Verallgemeinerung von Carlesons Satz auf andere Klassen von Funktionen reden. Diese Verallgemeinerung nutzt die Lösung des Kadison-Singer-Problems von Marcus, Spielman und Srivastava. Der Vortrag beruht auf einer gemeinsamen Arbeit mit Alexandru Aleman, John McCarthy und Stefan Richter.

• Thomas Kalmes (Chemnitz): On surjectivity of partialdifferential operators with a single characteristic direction and on Rungepairs for such operators

We report on recent results concerning partial differential operators with constant coefficients $$P(\partial)$$ for which the characteristic set $$\{\xi\in\mathbb{R}^d;\,P_m(\xi)=0\}$$ of its symbol $$P\in\mathbb{C}[X_1,\ldots,X_d]$$ is a one-dimensional subspace of $$\mathbb{R}^d$$. Here $$P_m$$ denotes the principal part of $$P$$, i.e. $$P_m(\xi):=\sum_{|\alpha|=m}a_\alpha \xi^\alpha$$ for $$P(\xi)=\sum_{|\alpha|\leq m}a_\alpha\xi^\alpha$$ with minimal $$m\in\mathbb{N}_0$$. Among others, this class of partial differential operators contains the time-dependent free Schrödinger operator as well as non-degenerate parabolic operators like the heat operator. We characterize those open subsets $$X$$ of $$\mathbb{R}^d$$ for which $$P(\partial)$$ is surjective on $$C^\infty(X)$$, or equivalently on $$\mathscr{D}_F'(X)$$, the space of distributions of finite order on $$X$$. Moreover, we give a sufficient geometrical/topological condition for pairs of open subsets $$X_1\subseteq X_2$$ of $$\mathbb{R}^d$$ to be $$P$$-Runge pairs, which means that every smooth solution, resp. distributional solution, of the equation $$P(\partial)u=0$$ in $$X_1$$ can be approximated by smooth solutions, resp. distributional solutions, of the same equation in $$X_2$$. This condition is in the spirit of Runge's Approximation Theorem from complex analysis which deals with the case when $$P(\partial)$$ is the Cauchy-Riemann operator. Finally, we show that under the additional assumption of semi-ellipticity for such a differential operator surjectivity on $$C^\infty(X)$$ implies that its kernel $$C_P^\infty(X)=\{f\in C^\infty(X);\,P(\partial)f=0\}$$ has the linear topological invariant $$(\Omega)$$ of Vogt and Wagner which plays a prominent role when dealing with the question of surjectivity of $$P(\partial)$$ on spaces of vector valued smooth functions. Via Grothendieck-Köthe duality this can be interpreted as an abstract version of another classical theorem from complex analysis, namely Hadamard's Three Circles Theorem. See here for a PDF version of this abstract.

• Robert Nabiullin (Wuppertal): Input-to-state stability for infinite-dimensional linear systems

In this talk we study the notion of input-to-state stability for linear systems on Banach spaces with a possibly unbounded control operator. This class of systems includes for instance boundary control problems, which are described by evolution partial differential equations. Our main interest lies in the connection between input-to-state stability and integral input-to-state stability for bounded inputs. We show that the latter is equivalent to input-to-state stability with respect to some Orlicz space. For the strong versions of those stability notions this equivalence in general does not hold. Assuming that the semigroup associated with the system is strongly stable, we show that the infinite-time admissibility with respect to an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable. The converse fails in general. This talk is based on joint work with Birgit Jacob, Jonathan R. Partington, and Felix L. Schwenninger.

### Kontakt

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers .

Sekretariat: Janine Textor
Raum M 620

E-mail:
janine.textor@tu-dortmund.de

Tel. (0231) 755-3063
Fax (0231) 755-5219