Linear connections for reproducing kernels on vector bundles

TL;DR

This paper establishes a canonical correspondence between reproducing kernels on infinite-dimensional Hermitian vector bundles and linear connections, providing a functorial relationship and examples including classical and dilation theory kernels.

Contribution

It introduces a novel functorial framework linking reproducing kernels to linear connections on vector bundles, expanding the understanding of their geometric and analytical interplay.

Findings

Constructs a canonical functor between kernels and connections.

Provides examples including classical, homogeneous, and dilation theory kernels.

Analyzes covariant derivatives associated with these connections.

Abstract

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.