Proceedings of the London Mathematical Society Advance Access published online on August 24, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp035
© 2009 London Mathematical Society
Manifolds of Hilbert space projections
School of Mathematics
Trinity College Dublin
Dublin
Ireland
levene@maths.tcd.ie
Department of Mathematics and Statistics
Lancaster University
Lancaster
LA1 4YF
United Kingdom
Received 21 May 2008. Revision received 19 May 2009.
The Hardy space H2 (
) for the upper half-plane together with a multiplicative group of unimodular functions u(
) = exp(i(1
1 + ... +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.
2000 Mathematics Subject Classification 47B38, 46E20, 58D15.
The first named author was supported by Engineering and Physical Sciences Research Council under grant EP/D050677/1.