Levene, R. H. and Power, Stephen C. (2010) Manifolds of Hilbert Space Projections. Proceedings of the London Mathematical Society, 100 (2). pp. 485509. ISSN 1460244X
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Abstract
The Hardy space H2 () for the upper halfplane 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 nballs and nspheres. The arguments used are direct and rely on the analysis of oscillatory integrals (E. M. STEIN, Harmonic analysis: realvariable 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 2sphere and the hyperbolic 3sphere 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 nontrivial unitary automorphism groups, which are explicitly described.
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