On the dependence of the orthogonal projector on deformations of the scalar product

Zbigniew Pasternak-Winiarski

Studia Mathematica (1998)

  • Volume: 128, Issue: 1, page 1-17
  • ISSN: 0039-3223

Abstract

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We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.

How to cite

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Pasternak-Winiarski, Zbigniew. "On the dependence of the orthogonal projector on deformations of the scalar product." Studia Mathematica 128.1 (1998): 1-17. <http://eudml.org/doc/216474>.

@article{Pasternak1998,
abstract = {We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.},
author = {Pasternak-Winiarski, Zbigniew},
journal = {Studia Mathematica},
keywords = {scalar product; orthogonal projector; dependence of projectors on scalar products; bounded positive and invertible operators; orthogonal projectors; Taylor expansion},
language = {eng},
number = {1},
pages = {1-17},
title = {On the dependence of the orthogonal projector on deformations of the scalar product},
url = {http://eudml.org/doc/216474},
volume = {128},
year = {1998},
}

TY - JOUR
AU - Pasternak-Winiarski, Zbigniew
TI - On the dependence of the orthogonal projector on deformations of the scalar product
JO - Studia Mathematica
PY - 1998
VL - 128
IS - 1
SP - 1
EP - 17
AB - We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.
LA - eng
KW - scalar product; orthogonal projector; dependence of projectors on scalar products; bounded positive and invertible operators; orthogonal projectors; Taylor expansion
UR - http://eudml.org/doc/216474
ER -

References

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  1. [1] S. Bergman, The Kernel Function and Conformal Mapping, Math. Surveys Monographs 5, Amer. Math. Soc., 2nd rev. ed., 1970. Zbl0208.34302
  2. [2] S. G. Krantz, Function Theory of Several Complex Variables, Interscience-Wiley, New York, 1982. Zbl0471.32008
  3. [3] J.-P. Labrousse, The general local form of an analytic mapping into the set of idempotent elements of a Banach algebra, Proc. Amer. Math. Soc. 123 (1995), 3467-3471. Zbl0855.47009
  4. [4] A. Odzijewicz, On reproducing kernels and quantization of states, Comm. Math. Phys. 114 (1988), 577-597. Zbl0645.53044
  5. [5] A. Odzijewicz, Coherent states and geometric quantization, ibid. 150 (1992), 385-413. 
  6. [6] K. Maurin, Analysis, Part I, Elements, PWN-Reidel, Warszawa-Dordrecht, 1976. 
  7. [7] Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), 110-134. Zbl0739.46010
  8. [8] Z. Pasternak-Winiarski, On reproducing kernels for holomorphic vector bundles, in: Quantization and Infinite-Dimensional Systems, J.-P. Antoine, S. Twareque Ali, W. Lisiecki, I. M. Mladenov and A. Odzijewicz (eds.), Plenum Press, New York, 1994, 109-112. Zbl0980.32501
  9. [9] Z. Pasternak-Winiarski, Bergman spaces and kernels for holomorphic vector bundles, Demonstratio Math. 30 (1997), 199-214. Zbl0877.32003
  10. [10] J. H. Rawnsley, Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. 28 (1077), 403-415. Zbl0387.58002
  11. [11] K. Yosida, Functional Analysis, Springer, Berlin, 1965. 

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