Manifolds of Hilbert space projections
Abstract
The Hardy space H2(R) for the upper half plane together with a unimodular function group representation u(λ) = (i(λ11 + ... + λnn)) for λ in Rn, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H2(R) of L2(R). For classes of admissible functions i the strong operator topology closures of M and M M 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.