2007 Fiscal Year Final Research Report Summary
Study of Analysis on Manifolds
| Project/Area Number |
17540202
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
FURUTANI Kenro Tokyo University of Science, Departmen of Mathenratics, Professor (70112901)
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| Co-Investigator(Kenkyū-buntansha) |
IWASAKI Chisato University of Hyogo, Dept. Math., Professor (30028261)
MORIMOTO Tohru Nara Women's Unirersity, Depatment of Mathematics, Professor (80025460)
BAUER Wdfram Tokyo Univ. Science, Dept. Math., Researcher (80453819)
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| Project Period (FY) |
2005 – 2007
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| Keywords | spectral flow / elliptic openrator / Afrya-Patodi-singer bourdary conditior / Seta-wgataviged Deteiminant / Apkevical Apace form / Hilbert-Schmidt operator / Toplity openata / Beverin trans form |
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| Research Abstract |
(1) We proved a variant of the spectral flow formula for one-parameter family of selfadjoint elliptic operators defined on a closed manifold. When we decompose the manifold into two connected components, the spectral flow of this family of the operators is the sum of two spectral flows of the restricted families onto each component with suitable elliptic boundary conditions on the common boundary and a correction term. This result made clear that a relation between Cauchy data spaces and global elliptic boundary conditions and their geometric role, and that the correction term is expressed as the H〓rmnader index in the infinite dimension. (2) We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms MG. Extending results in the paper by C. NASH AND D. O'CONNOR; Determinants of Laplacians on lens spaces, (J. Math. Phys. 36 (1995) ) , the zeta-regularized determinant of the Laplacian on MG is obtained explicitly from these formulas. In particular, our method applies the 〓 where G is the dihedral group. As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in A. IKEDA; (Osaka J. Math. 17 (1983) ) . (3) Let H be a reproducing kernel Hilbert space contained in a wider space L^2 (X, μ) . We study the Hilbert-Schmidt property of Hankel operators H_g on H with bounded symbol g by analyzing the behavior of. the iterated Berezin transform. We determine symbol classes S such that for g ∈ S the Hilbert-Schmidt property of H_g implies that H_g is a Hilbert-Schmidt operator as well. We apply this general result to the cases of Bergman spaces over strictly pseudo convex domains in C^n, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric.
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