Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

TL;DR

This paper explores the geometry of subspaces in reproducing kernel Hilbert spaces, establishing conditions for minimal geodesics between subspaces defined by vanishing functions, with applications to Hardy spaces and eigenvalue estimates.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of geodesics in the Grassmann manifold of RKHS, and analyzes their properties in specific examples like Hardy spaces.

Findings

Conditions for existence and uniqueness of geodesics

Relations between geodesic distance and other metrics

Eigenvalue estimates for operators defining geodesics

Abstract

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.