This is an announcement for the paper “A Toolkit for Constructing Dilations on Banach Spaces” by Stephan Facklerhttps://arxiv.org/find/math/1/au:+Fackler_S/0/1/0/all/0/1, Jochen Glückhttps://arxiv.org/find/math/1/au:+Gluck_J/0/1/0/all/0/1.

Abstract: We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if $X$ is a super-reflexive Banach space and $T$ is contained in the weakly closed convex hull of all invertible isometries on $X$, then $T$ admits a dilation to an invertible isometry on a Banach space $Y$ with the same regularity as $X$. The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.08547