Science China Mathematics

, Volume 61, Issue 4, pp 745–768 | Cite as

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

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Abstract

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i MM (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.

Keywords

submodules Kreĭn spaces reproducing kernels defect operators Lorentz group little Lorentz group 

MSC(2010)

47A15 

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Notes

Acknowledgements

This work was supported by Grant-in-Aid for Young Scientists (B) (Grant No. 23740106). The first author and the third author thank Yixin Yang for valuable discussions and in particular ideas leading to Lemma 7.4.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of InsuranceCentral University of Finance and EconomicsBeijingChina
  2. 2.National Defense AcademyYokosukaJapan
  3. 3.Department of MathematicsState University of New York at AlbanyAlbanyUSA

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