Manifolds of Hilbert Space Projections.

Abstract

The Hardy space H2 () for the upper half-plane together with a multiplicative group of unimodular functions u() = exp(i(11 + ... +nn)), with n, gives rise to a manifold of orthogonal projections for the subspaces u() H2 () of L2 (). For classes of admissible functions i the strong operator topology closures of and are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals (E. M. STEIN, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43 (Princeton University Press, Princeton, NJ, 1993)) 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 (A. KATAVOLOS and S. C. POWER, Translation and dilation invariant subspaces of L2(), J. reine angew. Math. 552 (2002) 101–129) appear as distinguished special cases admitting non-trivial unitary automorphism groups, which are explicitly described