Manifolds of Hilbert space projections

TL;DR

This paper studies manifolds of orthogonal projections in Hilbert spaces generated by unimodular function groups, explicitly characterizing their closures and symmetries, with applications to classical spheres in functional analysis.

Contribution

It provides explicit descriptions of the closures of projection manifolds and classifies certain cases up to unitary equivalence, revealing their geometric and symmetry properties.

Findings

Closures of projection manifolds are n-balls and n-spheres.

The Fourier-Plancherel 2-sphere and hyperbolic 3-sphere are special cases with automorphism groups.

Some manifolds are classified up to unitary equivalence.

Abstract

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(\lambda)H^2(R) of L^2(R). For classes of admissible functions \psi_i the strong operator topology closures of M and M \cup M^\perp are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular the Fourier-Plancherel 2-sphere and the hyperbolic 3-sphere of Katavolos and Power appear as distinguished special cases admitting nontrivial unitary automorphisms groups which are explicitly described.