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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Abbondandolo and P. Majer: “Morse homology on Hilbert spaces”, Comm. Pure Applied Math., Vol. 54, (2001), pp. 689–760.
J. Avron, R. Seiler and B. Simon: “The index of a pair of projections”, J. Func. Anal., Vol. 120, (1994), pp. 220–237.
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.
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.
B. Bojarski: “The geometry of the Riemann-Hilbert problem”, Contemp. Math., Vol. 242, (1999), pp. 25–33.
B. Bojarski: “The geometry of Riemann-Hilbert problem II”, In: Boundary value problems and integral equations. World Scientific, Singapore, 2000, pp. 41–48.
B. Bojarski and G. Khimshiashvili: “Global geometric aspects of Riemann-Hilbert problems”, Georgian Math. J., Vol. 8 (2001), pp. 799–812.
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.
B. Booss and K. Wojciechowsky: Elliptic boundary value problems for Dirac operators, Birkhäuser, Boston, 1993.
M. Dupré and J. Glazebrook: “The Stiefel bundle of a Banach algebra”, Integr. Eq. Oper. Theory, Vol. 41, (2000), pp. 264–287.
J. Eells: “Fredholm structures” Proc. Symp. Pure Math., Vol. 18, (1970), pp. 62–85.
J. Elworthy and A. Tromba: “Differential structures and Fredholm maps on Banach manifolds”, Proc. Symp. Pure Math., Vol. 15, (1970), pp. 45–94.
D. Freed: “The geometry of loop goups”, J. Diff Geom., Vol. 28, (1988), pp. 223–276.
D. Freed: “An index theorem for families of Fredholm operators parametrized by a group”, Topology, Vol. 27, (1988), pp. 279–300.
T. Kato: Perturbation theory for linear operators, Springer, Berlin 1980.
G. Khimshiashvili: On the topology of invertible linear singular integral operators, Springer Lecture Notes Math., Vol. 1214, (1986), pp. 211–230.
G. Khimshiashvili: On Fredholmian aspects of linear conjugation problems, Springer Lect. Notes Math., Vol. 1520, (1992), pp. 193–216.
G. Khimshiashvili: “Homotopy classes of elliptic transmision problems over C *-algebras”, Georgian Math. J., Vol. 5, (1998), pp. 453–468.
G. Khimshiashvili: “Geometric aspects of Riemann-Hilbert problems”, Mem. Diff. Eq. Math. Phys., Vol. 27, (2002), pp. 1–114.
G. Khimshiashvili: “Global geometric aspects of linear conjugation problems”, J. Math. Sci., Vol. 118, (2003), pp. 5400–5466.
A. Pressley and G. Segal: Loop groups, Clarendon Press, Oxford, 1986.
K. Wojciechowski: “Spectral flow and the general linear conjugation problem”, Simon Stevin Univ. J., Vol. 59, (1985), pp. 59–91.
Author information
Authors and Affiliations
About this article
Cite this article
Bojarski, B., Khimshiashvili, G. The geometry of Kato Grassmannians. centr.eur.j.math. 3, 705–717 (2005). https://doi.org/10.2478/BF02475627
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.2478/BF02475627
Keywords
- Fredholm pair of subspaces
- Fredholm operator
- index
- compact operator
- restricted Grassmannian
- Hilbert manifold
- homotopy groups
- Fredholm structure