# The geometry of Kato Grassmannians

Bogdan Bojarski; Giorgi Khimshiashvili

Open Mathematics (2005)

- Volume: 3, Issue: 4, page 705-717
- ISSN: 2391-5455

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topBogdan Bojarski, and Giorgi Khimshiashvili. "The geometry of Kato Grassmannians." Open Mathematics 3.4 (2005): 705-717. <http://eudml.org/doc/268855>.

@article{BogdanBojarski2005,

abstract = {We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.},

author = {Bogdan Bojarski, Giorgi Khimshiashvili},

journal = {Open Mathematics},

keywords = {35E15; 58D15},

language = {eng},

number = {4},

pages = {705-717},

title = {The geometry of Kato Grassmannians},

url = {http://eudml.org/doc/268855},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Bogdan Bojarski

AU - Giorgi Khimshiashvili

TI - The geometry of Kato Grassmannians

JO - Open Mathematics

PY - 2005

VL - 3

IS - 4

SP - 705

EP - 717

AB - We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.

LA - eng

KW - 35E15; 58D15

UR - http://eudml.org/doc/268855

ER -

## References

top- [1] P. Abbondandolo and P. Majer: “Morse homology on Hilbert spaces”, Comm. Pure Applied Math., Vol. 54, (2001), pp. 689–760. http://dx.doi.org/10.1002/cpa.1012 Zbl1023.58003
- [2] J. Avron, R. Seiler and B. Simon: “The index of a pair of projections”, J. Func. Anal., Vol. 120, (1994), pp. 220–237. http://dx.doi.org/10.1006/jfan.1994.1031 Zbl0822.47033
- [3] B. Bojarski: “Abstract linear conjugation problems and Fredholm pairs of subspaces” (Russian), In: Differential and Integral equations, Boundary value problems. Collection of papers dedicated to the memory of Academician I, Vekua, Tbilisi University Press, Tbilisi, 1979, pp. 45–60.
- [4] B. Bojarski: “Some analytical and geometrical aspects of the Riemann-Hilbert transmission problem”, In: Complex analysis. Methods, trends, applications, Akad. Verlag, Berlin, 1983, pp. 97–110.
- [5] B. Bojarski: “The geometry of the Riemann-Hilbert problem”, Contemp. Math., Vol. 242, (1999), pp. 25–33. Zbl0948.58021
- [6] B. Bojarski: “The geometry of Riemann-Hilbert problem II”, In: Boundary value problems and integral equations. World Scientific, Singapore, 2000, pp. 41–48.
- [7] B. Bojarski and G. Khimshiashvili: “Global geometric aspects of Riemann-Hilbert problems”, Georgian Math. J., Vol. 8 (2001), pp. 799–812. Zbl0998.58001
- [8] B. Bojarski and A. Weber: “Generalized Riemann-Hilbert transmission and boundary value problems. Fredholm pairs and bordisms”, Bull. Polish Acad. Sci., Vol. 50, (2002), pp. 479–496. Zbl1019.58008
- [9] B. Booss and K. Wojciechowsky: Elliptic boundary value problems for Dirac operators, Birkhäuser, Boston, 1993.
- [10] M. Dupré and J. Glazebrook: “The Stiefel bundle of a Banach algebra”, Integr. Eq. Oper. Theory, Vol. 41, (2000), pp. 264–287. http://dx.doi.org/10.1007/BF01203172 Zbl0997.46047
- [11] J. Eells: “Fredholm structures” Proc. Symp. Pure Math., Vol. 18, (1970), pp. 62–85.
- [12] J. Elworthy and A. Tromba: “Differential structures and Fredholm maps on Banach manifolds”, Proc. Symp. Pure Math., Vol. 15, (1970), pp. 45–94. Zbl0206.52504
- [13] D. Freed: “The geometry of loop goups”, J. Diff Geom., Vol. 28, (1988), pp. 223–276.
- [14] D. Freed: “An index theorem for families of Fredholm operators parametrized by a group”, Topology, Vol. 27, (1988), pp. 279–300. http://dx.doi.org/10.1016/0040-9383(88)90010-9
- [15] T. Kato: Perturbation theory for linear operators, Springer, Berlin 1980. Zbl0435.47001
- [16] G. Khimshiashvili: On the topology of invertible linear singular integral operators, Springer Lecture Notes Math., Vol. 1214, (1986), pp. 211–230. Zbl0608.47052
- [17] G. Khimshiashvili: On Fredholmian aspects of linear conjugation problems, Springer Lect. Notes Math., Vol. 1520, (1992), pp. 193–216. http://dx.doi.org/10.1007/BFb0084722
- [18] G. Khimshiashvili: “Homotopy classes of elliptic transmision problems over C *-algebras”, Georgian Math. J., Vol. 5, (1998), pp. 453–468. http://dx.doi.org/10.1023/B:GEOR.0000008116.53411.d0 Zbl0923.46048
- [19] G. Khimshiashvili: “Geometric aspects of Riemann-Hilbert problems”, Mem. Diff. Eq. Math. Phys., Vol. 27, (2002), pp. 1–114. Zbl1091.30040
- [20] G. Khimshiashvili: “Global geometric aspects of linear conjugation problems”, J. Math. Sci., Vol. 118, (2003), pp. 5400–5466. http://dx.doi.org/10.1023/A:1025884428386 Zbl1058.58001
- [21] A. Pressley and G. Segal: Loop groups, Clarendon Press, Oxford, 1986. Zbl0618.22011
- [22] K. Wojciechowski: “Spectral flow and the general linear conjugation problem”, Simon Stevin Univ. J., Vol. 59, (1985), pp. 59–91.

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