Hello,
The next Banach spaces webinar is on Friday January 15 at 9AM Central time. Please join us at
https://unt.zoom.us/j/83807914306
Speaker: Richard Lechner (Johannes Kepler Universität Linz) Title: Restriced invertibility, subsymmetric bases and factorization
Abstract: Given an unconditional normalized basis $(e_j)_{j=1}^n$ of a Banach space $X_n$, we consider conditions under which an operator $T\colon X_n\to X_n$ with ``large diagonal'' can be inverted when restricted to $X_\sigma = [e_j : j\in\sigma]$ for a ``large'' set $\sigma\subset {1,\ldots,n}$ (restricted invertibility). We then discuss restricted invertibility and its close connection to finite dimensional quantitative factorization.
In the second part of the talk, we show that subsymmetric Schauder bases $(e_j)$ of an infinite dimensional Banach space $X$ have the factorization property, i.e.@ the identity $I_X$ on $X$ factors through every bounded operator $T\colon X\to X$ with large diagonal. In Banach spaces with a Schauder basis, this type of result can often be proved using gliding-hump techniques, but in non-separable Banach spaces gliding-hump techniques seem unfeasible. However, if $(e_j^*)$ is a non-$\ell^1$-splicing (there is no disjointly supported $\ell^1$-sequence in $X$) subsymmetric weak$^*$ Schauder basis for the dual $X^*$ of $X$, $(e_j^*)$ also has the factorization property.
For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach
Thank you, and best regards,
Bunyamin Sari