A Toolkit for Constructing Dilations on Banach Spaces

TL;DR

This paper introduces a new structure theoretic approach to dilation theory on Banach spaces, establishing conditions under which operators can be dilated to invertible isometries, thus unifying and extending classical results.

Contribution

It develops a novel framework for dilation theory on Banach spaces, generalizing classical theorems to super-reflexive spaces with a new structural perspective.

Findings

Operators in the weakly closed convex hull of invertible isometries admit dilations.

Classical dilation theorems are special cases of the new theory.

The approach applies to Banach spaces with the same regularity as the original space.

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.