Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics Dr. Richard Lechner, Institute of Analysis, will give a public talk (followed by a discussion) on Mon, Sept. 23, 2019 at 15:15 o'clock at S2 053 on the topic of "Factorization of the identity operator" . The organziers of the symposium,

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

hereby cordially invite you.

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“ [pelczynski:1960] by Pelczynski, 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

  1. $\ell^\infty$ is prime: Lindenstrauss 1967, „On complemented subspaces of $m$“ [lindenstrauss:1967];
  2. $L^p$, $1\leq p < \infty$ is primary: Enflo (published by Maurey in 1975), „Sous-espaces complémentés de $L^p$, d'après P. Enflo“ [maurey:1975:2];
  3. the space of bounded analytic functions on the poly-disc is primary: Bourgain 1983, „On the primarity of $H^{\infty }$-spaces“ [bourgain:1983].

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:

  1. biparameter BMO is primary [lechner:mueller:2015];
  2. the identity operator in mixed norm Hardy and BMO spaces factors through operators with large diagonal [laustsen:lechner:mueller:2015,lechner:2018:factor-mixed];
  3. $SL^\infty$ is primary [lechner:2018:SL-infty];
  4. $\ell^p(X)$, $1 \leq p \leq \infty$ is primary, whenever $X$ is a dual Banach space which has a non-$\ell^1$-splicing subsymmetric weak$^*$ Schauder basis [lechner:2018:subsymm];
  5. a two person game approach to factorization problems [lechner:motakis:mueller:schlumprecht:2018];

  6. improving super-exponential quantitative estimates for factorization problems in one- and twoparameter Hardy spaces and $SL^\infty$ to polynomial estimates [lechner:2018:1-d,lechner:2018:2-d].