Manifolds of Hilbert space projections

Abstract

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_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.