## Workshop in Analysis and Probability Seminar

**Date:** August 22, 2018

**Time:** 3:00PM - 3:50PM

**Location:** BLOC 220

**Speaker:** Richard Lechner, Johannes Kepler Universität Linz

**Title:** *Dimension dependence of factorization problems*

**Abstract:** Abstract: For each $n\in\mathbb{N}$, let $(e_j)_{j=1}^n$ denote a normalized $1$-unconditional basis for the $n$-dimensional Banach space $X_n$. We consider the following question: What is the smallest possible dimension $N=N(n)$ such that the identity operator on $X_n$ factors through any operator having large diagonal on $X_N$ ? For one- and two-parameter dyadic Hardy spaces and $SL^\infty$, we improve the best previously known \emph{super-exponential} estimates for $N=N(n)$ to \emph{polynomial} estimates. References: R. Lechner. Dimension dependence of factorization problems: Hardy spaces and $SL_n^\infty$. ArXiv e-prints https://arxiv.org/abs/1802.02857, Feb. 2018. R. Lechner. Dimension dependence of factorization problems: bi-parameter Hardy spaces. ArXiv e-prints https://arxiv.org/abs/1802.05994, Feb. 2018.