Geometric Structures in Functional Analysis - Quantization of Infinite-Dimensional Manifolds

CNCSIS grant PNII - Programme `Idei' (code 1194)

Brief description of the research project

The usefulness of the methods and ideas with a geometric flavor is well known to all of the mathematicians interested in areas like complex analysis, functional analysis, or mathematical physics. Examples in this connection are manifold: just recall the notions of convexity in the several variable complex analysis, the Krein-Milman theorem in functional analysis, or the properties of the moment map in hamiltonian mechanics and in the method of geometric quantization. The significance of geometric structures in many areas employing functional analytic methods is however much deeper than just convexity properties of subsets of various topological vector spaces. Geometric features, and particularly the symmetry properties of the geometric objects that underlie the problems we wish to address, are often reflected in properties of certain function spaces or even in the spectral theory of operators on these function spaces. This is a central theme in several areas of contemporary research in mathematics and physics, including the representation theory of finite- or infinite-dimensional Lie groups, the theory of spaces of analytic functions, harmonic analysis on groups and on bounded symmetric domains, spectral geometry of Riemannian manifolds etc. Our aim in the present research project is to point out new geometric features of certain functional analytic objects such as

reproducing kernels,

spectral functions of operators,

embeddings of Hilbert and Krein spaces,

generalized Toeplitz operators,

invariant subspaces of Hilbert space operators.

Each of these objects plays an important role in the method of geometric quantization of the symplectic manifolds in Hamiltonian mechanics, and the results obtained in the first stages of the present project should lay the foundations for methods of geometric quantization applying to wide classes of infinite-dimensional symplectic manifolds.

The main lines of research in our project are the following ones:

1. Reproducing kernels and geometric quantization in infinite dimensions.

2. The Horn conjecture in the framework of von Neumann algebras.

3. Closely contained Hilbert and Krein spaces.

4. Spectral theory of generalized Toeplitz operatori.

5. Invariant subspace problems for operators on Hilbert space.

Project manager

Dr. Daniel Beltita, Scientific Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Research team

Ph.D. student

Mihaela Ivan

Research financially supported in 2009 - 1st phase of the project

I. Beltita, D.Beltita, Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups. Annals of Global Analysis and Geometry 36 (2009), no. 3, 293-322.

I. Beltita, D. Beltita, Uncertainty principles for magnetic structures on certain coadjoint orbits. Journal of Geometry and Physics 60 (2010), no. 1, 81-95.

I. Beltita, D. Beltita, A survey on Weyl calculus for representations of nilpotent Lie groups. In: P. Kielanowski, S.T.Ali, A. Odzijewicz, M. Schlichenmaier, Th. Voronov (eds.), ``XXVIII Workshop on Geometrical Methods in Physics'', AIP Conf. Proc., Amer. Inst. Phys., 1191, Melville, NY, 2009, pp. 7-20.

I. Beltita, D. Beltita, Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups. Analele Universitatii din Bucuresti. Seria Matematica 1 (LIX) (2010), no. 1, 17-46.

P. Cojuhari, A. Gheondea, Embeddings, operator ranges, and Dirac operators. Complex Analysis and Operator Theory (to appear).

Research financially supported in 2010 - 2nd phase of the project

I. Beltita, D. Beltita, Modulation spaces of symbols for representations of nilpotent Lie groups. Journal of Fourier Analysis and Applications 7 (2011), no. 2, 290-319.

I. Beltita, D. Beltita, On Weyl calculus in infinitely many variables. In: P. Kielanowski, V. Buchstaber, A. Odzijewicz, M. Schlichenmaier, Th. Voronov (eds.), ``XXIX Workshop on Geometrical Methods in Physics'', AIP Conf. Proc., Amer. Inst. Phys., 1307, Melville, NY, 2010, pp. 19-26.

D. Beltita, J.E. Galé, Universal objects in categories of reproducing kernels. Revista Matemática Iberoamericana 27 (2011), no. 1, 123-179.

D. Beltita, K.-H. Neeb, Geometric characterization of hermitian algebras with continuous inversion. Bulletin of the Australian Mathematical Society 81 (2010), no. 1, 96-113.

H. Bercovici, B. Collins, K. Dykema, W.S. Li, D. Timotin, Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor. Journal of Functional Analysis 258 (2010), no. 5, 1579-1627.

B. Prunaru, Toeplitz and Hankel operators associated with subdiagonal algebras. Proceedings of the American Mathematical Society 139 (2011), 1387-1396.

B. Prunaru, Polynomial approximation and generalized Toeplitz operators. Annals of the University of Bucharest (mathematical series) 1 (LIX) (2010), no. 1, 135-144.

Research financially supported in 2011 - 3rd phase of the project

I. Beltita, D. Beltita, Continuity of magnetic Weyl calculus. Journal of Functional Analysis 260 (2011), no. 7, 1944-1968.

I. Beltita, D. Beltita, On differentiability of vectors in Lie group representations. Journal of Lie Theory 21 (2011), no. 4, 771-785.

I. Beltita, D. Beltita, Algebras of symbols associated with the Weyl calculus for Lie group representations. Monatshefte für Mathematik (to appear).

B. Prunaru, A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces. Canadian Mathematical Bulletin (to appear).