A Category of Banach Space Functors

TL;DR

This paper introduces a sheaf theoretic framework for Banach space functors, enabling new functional calculus tools and exponential maps for operators, with applications in dynamical systems and KAM theory.

Contribution

It develops a novel algebraic approach to Banach space functors, extending functional calculus and exponential maps to infinite-dimensional Lie group actions.

Findings

Established a sheaf theoretic perspective for functional analysis.

Proved the existence of exponential maps for generalized operators.

Applied framework to dynamical systems and KAM theory.

Abstract

We introduce a sheaf theoretic viewpoint on functional analysis designed for infinite dimensional Lie group actions. We develop functional calculus for Banach valued functors and, in particular, prove the existence of an exponential map for a certain class of operators that generalise first order partial differential operators. This algebraic framework can be used in dynamical systems and KAM theory to provide normal forms and versal deformations. it is used in our papers on the subject such as the Herman conjecture paper, the versal deformation theorem for vector fields etc.