# Johannes Kepler Symposium für Mathematik

Im Rahmen des Johannes-Kepler-Symposiums für Mathematik wird Dr. Richard Lechner, Institute of Analysis, am Mon, Sept. 23, 2019 um 17:15 Uhr im S2 053 einen öffentlichen Vortrag (mit anschließender Diskussion) zum Thema "Factorization of the identity operator" halten, zu dem die Veranstalter des Symposiums,

O.Univ.-Prof. Dr. Ulrich Langer,
Univ.-Prof. Dr. Gerhard Larcher
A.Univ.-Prof. Dr. Jürgen Maaß, und
die ÖMG (Österreichische Mathematische Gesellschaft)

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Factorization of the identity operator

Our starting point is the 1960 paper Projections in certain Banach spaces''~\cite{pelczynski:1960}
by Pe{\l}czy{\'n}ski, where he proved that the spaces $\ell^p$, $1\leq p < \infty$ are prime. Since
then, great advances in numerous papers were made: the most significant, influential and closely
related to our work are

\begin{enumerate}
[(1)]
\item $\ell^\infty$ is prime: Lindenstrauss 1967, On complemented subspaces of
$m$''~\cite{lindenstrauss:1967};

\item $L^p$, $1\leq p <\infty$ is primary: Enflo (published by Maurey in 1975), Sous-espaces
compl\'ement\'es de $L^p$, d'apr\es P.~Enflo''~\cite{maurey:1975:2};

\item the space of bounded analytic functions on the poly-disc is primary: Bourgain 1983, `On the
primarity of $H^{\infty }$-spaces''~\cite{bourgain:1983}.
\end{enumerate}

The common thread running through this literature are factorization problems, which usually are the
main challenge to overcome. This habilitation thesis addresses two basic types of factorization
problems: infinite dimensional factorization problems and finite dimensional, quantitative
factorization problems in several classical Banach spaces; among the results are the following:

\begin{enumerate}
[(1)]
\item biparameter $\bmo$ is primary~\cite{lechner:mueller:2015};

\item the identity operator in mixed norm Hardy and $\bmo$ spaces factors through operators with
large diagonal~\cite{laustsen:lechner:mueller:2015,lechner:2018:factor-mixed};

\item $SL^\infty$ is primary~\cite{lechner:2018:SL-infty};

\item $\ell^p(X)$, $1 \leq p \leq \infty$ is primary, whenever $X$ is a dual Banach space which has
a non-$\ell^1$-splicing subsymmetric weak$^*$ Schauder basis~\cite{lechner:2018:subsymm};

\item a two person game approach to factorization
problems~\cite{lechner:motakis:mueller:schlumprecht:2018};

\item improving super-exponential quantitative estimates for factorization problems in one- and
twoparameter Hardy spaces and $SL^\infty$ to polynomial
estimates~\cite{lechner:2018:1-d,lechner:2018:2-d}.
\end{enumerate}