REFERENCES
J. Adams, Lectures on Lie Groups, Benjamin, New York–Amsterdam (1969).
J. Adams, P. Lax, and R. Phillips, “On matrices whose real linear combinations are nonsingular, ”Proc. Amer. Math. Soc., 16, 318–322(1965).
W. Adams, C. Berenstein, P. Loustaunau, I. Sabadini, and D. Struppa, “Regular functions of several quaternionic variables and the Cauchy–Fueter complex, ”J. Geom. Anal., 9, 1–15(1999).
W. Adams, P. Loustanau, V. Palamodov, and D. Struppa, “Hartogs' phenomenon for polyregular functions and projective dimension of related modules ofer a polynomial ring, ”Ann. Inst. Fourier, 47, 623–640(1997).
V. Adukov, “On the factorization of analytic matrix-valued functions, ”Teor. Math. Fiz., 118, 324–336(1999).
H. Alexander, “Gromov 's method and Bennequin 's problem, ”Invent. Math., 125, 135–148(1996).
D. Anosov and A. Bolibruch, The Riemann–Hilbert Problem, Vieweg, Braunschweig (1994).
M. Atiyah and R. Bott, “Yang–Mills equations on Riemann surfaces, ”Phil. Trans. Roy. Soc. London, A308, 523–615(1982).
J. Avron, R. Seiler, and B. Simon, “The index of a pair of projections, ”J. Func. Anal., 120, 220–237(1994).
V. E. Balabaev, “Canonical rst-order elliptic systems, ”Differ. Uravn., 27, 634–648(1991).
P. Baum, R. Douglas, and M. Taylor, “Cycles and relative cycles in K-homology, ”J. Differ. Geom., 30, 761–804(1989).
H. Begehr, “Boundary-value problems for analytic and generalized analytic functions, ”In:Complex Analysis. Methods, Trends, and Applications, Akad. Verlag, Berlin (1983), pp. 150–165.
H. Begehr and G. C. Wen, Nonlinear Elliptic Boundary-Value Problems and Their Applications, Longman, New York (1996).
H. Begehr and Z. Xu, “Nonlinear half-Dirichlet problems for rst-order elliptic equations in the unit ball of ℛm (m ≥3), ”Appl. Anal., 45, 3–18(1992).
L. Bers, Theory of Pseudo-Analytic Functions, New York (1953).
M. Bershadsky, V. Sadov, and C. Vafa, “D-branes and topological eld theory, ”Nucl. Phys. B, 463, 420–434(1996).
E. Bishop, “Differentiable manifolds in complex Euclidean space, ”Duke Math. J., 32, 1–21(1965).
A. V. Bitsadze, Introduction to the Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1969).
B. Blackadar, K-Theory for Operator Algebras, Springer-Verlag (1986).
B. Bojarski, “On the stability of Hilbert problem for holomorphic vectors, ”Soobshch. Akad. Nauk Gruz. SSR, 21, 391–398(1958).
B. Bojarski, “Abstract linear conjugation problems and Fredholm pairs of subspaces, ”In:Differential and Integral Equations. Boundary-Value Problems. Coll. Pap. dedicated to the memory of Academician I. Vekua [in Russian], 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, and Applications, Akad. Verlag, Berlin (1983), pp. 97–110.
B. Bojarski, “The geometry of the Riemann–Hilbert problem, ”Contemp. Math., 242, 25–33(1999).
B. Bojarski, “The geometry of the 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., 8, 799–812(2001).
A. Bolibruch, “The Riemann–Hilbert problem, ”Russ. Math. Surv., 45, 1–47(1990).
A. Bolibruch, “Construction of a Fuchsian equation from a monodromy representation, ”Math. Notes Acad. Sci. USSR, 48, 1090–1099(1990).
B. Booss and K. Wojciechowsky, Elliptic Boundary-Value Problems for Dirac Operators, Birkh ¨auser, Boston (1993).
R. Bott, “The space of loops on a Lie group, ”Mich. Math. J., 5, 31–64(1958).
A. B ¨ottcher and B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin (1990).
A. B ¨ottcher, S. Grudsky, and I. Spitkovsky, “Matrix-valued functions with arbitrarily prescribed left and right partial indices, ”Integr. Eqs. Oper. Theory, 36, 71–91(2000).
F. Bracks, R. Delange, and F. Sommen, Clifford Analysis, Pitman (1982).
K. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkh ¨auser, Basel (1981).
A. Connes, Noncommutative Differential Geometry, Academic Press, London (1994).
G. Corach, H. Porta, and L. Recht, “Di. erential geometry of systems of projections in Banach algebras, ”Pac. J. Math., 143, 209–228(1990).
P. Deift, “Orthogonal polynomials and random matrices:A Riemann–Hilbert approach, ”Lect. Notes Math., 3 (1999).
P. Deift, L. Li, and C. Tomei, “Matrix factorizations and integrable systems, ”Commun. Pure Appl. Math., 42, 443–521(1989).
P. Deligne, Equations differentielles a points singuliers reguliers, Springer Lect. Notes Math.(1970).
A. Dzhuraev, Methods of Singular Integral Equations [in Russian], Nauka, Moscow (1987).
S. Disney, “The exponents of loops on a general complex linear group, ”Topology, 12, 297–315(1973).
H. Doi, “Nonlinear equations on a Lie group, ”Hiroshima Math. J., 17, 535–560(1987).
S. Donaldson, “Boundary-value problems for Yang–Mills elds, ”J. Geom. Phys., 8, 89–122(1992).
A. Douady, “Le probleme des modules pour les sous-espaces analytique compacts, ”Ann. Inst. Fourier, 1, 1–95(1966).
R. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators, Am. Math. Soc., Providence, Rhode Island (1973).
M. Dupre and J. Glazebrook, “The Stiefel bundle of a Banach algebra, ”Integr. Eqs. Oper. Theory, 41, 264–287(2000).
A. S. Dynin, “On the theory of pseudo-differential operators on a manifold with boundary, ”Dokl. Akad. Nauk SSSR, 186, 251–253(1969).
J. Eells, “A setting for global analysis, ”Bull. Am. Math. Soc., 72, 751–802(1966).
J. Eells, “Fredholm structures, ”Proc. Symp. Pure Math., 18, 62–85(1970).
J. Elworthy and A. Tromba, “Differential structures and Fredholm maps on Banach manifolds, ”Proc. Symp. Pure Math., 15, 45–94(1970).
T. Ehkhardt and I. Spitkovsky, “Factorization of piecewise constant matrix-valued functions and systems of linear differential equations, ”Algebra Analiz, 13, 56–123(2001).
B. V. Fedosov, “Analytic formulas for the index of elliptic operators, ”Trans. Moscow Math. Soc., 30, 159–240(1974).
I. Feldman and A. Markus, “On some properties of factorization indices, ”Integr. Eqs. Oper. Theory, 30, 326–337(1998).
F. Forstneri ¡c, “Analytic discs with boundary in a maximal real submanifold of ℂ 2, ”Ann. Inst. Fourier, 37, 1–44(1987).
F. Forstneri ¡c, “Polynomial hulls of sets bered over the circle, ”Indiana Univ. Math. J., 37, No. 4, 869–889(1988).
F. Forstneri ¡c, “Complex tangents of real surfaces in complex surfaces, ”Duke Math. J., 67, 353–376(1992).
M. Fr ´echet, “La paraanalitikaj funkcioj en n dimensioj, ”J. Reine Angew. Math., 193, 122–141(1956).
D. Freed, “The geometry of loop groups, ”J. Di. er. Geom., 28, 223–276(1988).
D. Freed, “An index theorem for families of Fredholm operators parametrized by a group, ”Topology, 27, 279–300(1988).
D. Freed and E. Witten, “Anomalies in string theory with D-branes, ”Asian J. Math., 3, 819–852(1999).
D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).
J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Univ. Press, Cambridge (1990).
G. Giorgadze, “On the monodromy of principal bundles of Riemann surfaces, ”Bul l. Georgian Acad. Sci., 141, 473–476(1991).
G. Giorgadze, “On some geometrical properties of generalized analytic functions, ”Sem. Repts. I. Vekua Inst. Appl. Math. Tbilisi State Univ., 13, 6–10(1998).
J. Globevnik, “Perturbation by analytic discs along maximal real submanifolds of ℂ N, ”Math. Z., 217, 287–316(1994).
J. Globevnik, “Perturbing analytic discs attached to maximal real submanifolds of ℂ N, ”Indag. Math., 7, 37–46(1996).
I. Gokhberg and M. Krein, “Systems of integral equations on a half-line with kernels depending on the difference of the arguments, ”Amer. Math. Soc. Transl., 14, 217–287 (1960).
M. Golubitsky and V. Guillemin, Stable Maps and Their Singularities, Springer-Verlag, Berlin (1973).
G. Gong, “Relative K-cycles and elliptic boundary conditions, ”Bull. Am. Math. Soc., 28, 104–108(1993).
M. Gromov, “Pseudo-holomorphic curves in symplectic manifolds, ”Invent. Math., 82, 307–347(1985).
M. Gromov, Partial Differential Relations, Springer-Verlag, Berlin (1987).
F. Harvey and H. Lawson, “Calibrated geometries, ”Acta Math., 148, 47–157(1982).
Yu. Ilyashenko, “The nonlinear Riemann–Hilbert problem, ”Tr. Mat. Inst. Ross. Akad. Nauk, 213, 10–34(1999).
G. Ishikawa and T. Ohmoto, “Local invariants of singular surfaces in an almost complex four-manifold, ”Ann. Global Anal. Geom., 11, No. 2, 125–133(1993).
G. Kasparov, “Hilbert C*-modules:Theorems of Stinespring and Voiculescu, ”J. Operator Theory, 4, 133–150(1980).
T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin (1966).
G. Khimshiashvili, “On the homotopy structure of invertible singular integral operators, ”Bull. Acad. Sci. Georgian SSR, 108, 273–276(1982).
G. Khimshiashvili, “Polysingular operators and topology of invertible singular operators, ”Z. Anal. Anwendungen, 5, No. 1, 139–145(1986).
G. Khimshiashvili, “On a Riemann boundary-value problem with coefficients in a compact Lie group, ”Tr. Inst. Prikl. Mat. Tbilis. Gos. Univ., 3, 31–34(1988).
G. Khimshiashvili, “On the homotopy structure of invertible singular operators, ”In:Complex Analysis and Applications, So a (1989), pp. 230–234.
G. Khimshiashvili, “On the topology of the generalized Riemann–Hilbert problem, ”Soobshch. Akad. Nauk Gruz. SSR, 135, 241–244(1989).
G. Khimshiashvili, “On the Riemann–Hilbert problem for a compact Lie group, ”Dokl. Akad. Nauk SSSR, 310, 1055–1058(1990).
G. Khimshiashvili, “Linear conjugation problems and Fredholm structures, ”Tr. Inst. Prikl. Mat. Tbilis. Gos. Univ., 5, 189–192(1990).
G. Khimshiashvili, “Lie groups and transmission problems on Riemann surfaces, ”Contemp. Math., 131, 164–178(1992).
G. Khimshiashvili, “Remarks on EPI-algebras and category-like invariants, ”Proc. A. Razmadze Math. Inst., 112, 91–102(1995).
G. Khimshiashvili, “Elliptic boundary-value problems for Cauchy–Riemann systems, ”In:Proc. Symp. Anal. Num. Meth. Quatern. Clifford Anal., Seiffen, Germany, 1996, Freiberg (1996), pp. 77–83.
G. Khimshiashvili, “Elliptic transmission problems over C *-algebras, ”Georgian Math. J., 5, 511–521(1998).
G. Khimshiashvili, “Hyperholomorphic cells and Riemann–Hilbert problems, ”In:Proc. Conf. Clifford Anal. Appl., Kluwer Academic (2001), pp. 110–124.
G. Khimshiashvili, “Hyperholomorphic cells and Fredholm operators, ”Georgian Math. J., 8, No. 2, 223–236(2001).
G. Khimshiashvili, “Signature formula for topological invariants, ”Proc. A. Razmadze Math. Inst., 125, 1–121(2001).
G. Khimshiashvili and E. Wegert, “Analytic discs and totally real surfaces, ”Bul l. Georgian Acad. Sci., 162, 41–44(2000).
G. Khimshiashvili and E. Wegert, “Complex points of planar endomorphisms, ”Bul l. Georgian Acad. Sci., 165, 26–29(2002).
G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, “Boundary-value problems for analytic and harmonic functons in domains with nonsmooth boundaries. Applications to conformal mappings, ”Mem. Differ. Eq. Math. Phys., 14, 1–195(1998).
N. Kuiper, “The homotopy type of the unitary group of Hilbert space, ”Topol ogy, 3, 19–30(1965).
S. Lang, Introduction to Differentiable Manifolds, Interscience, New York (1966).
A. Lecki and Z. Szafraniec, “An algebraic method for calculating the topological degree, ”Banach Center Publ., 35, 73–83(1996).
L. Lempert, “Loop spaces as complex manifolds, ”J. Di. er. Geom., 38, 519–543(1993).
L. Lempert, “The Virasoro group as a complex manifold, ”Math. Res. Lett., 2, 479–495(1995).
L. Lempert, “The problem of complexifying a Lie group, ”Contemp. Math., 205, 169–176(1997).
G. S. Litvinchuk, Boundary-Value Problems and Singular Integral Equations with Shift [in Russian], Nauka, Moscow (1977).
G. S. Litvinchuk and I. Spitkovsky, Factorization of Measurable Matrix Functions, Birkh ¨auser, Basel (1987).
J. Lu and A. Weinstein, “Poisson-Lie groups, dressing transformations, and Bruhat decompositions, ”J. Differ. Geom., 31, 501–526(1990).
V. Manuilov and E. Troitsky, Hilbert Modules over C*-algebras, Mir, Moscow (2001).
M. Martin and N. Salinas, “Di. erential geometry of generalized Grassmann manifolds in C*-algebras, ”In:Operator Theory:Advances and Applications, 80, Birkh ¨auser, Basel (1995), pp. 206–243.
J. Mickelsson, Current Algebras and Groups, Plenum Press, New York (1989).
J. Mickelsson and S. Rajeev, “Current algebras in (d +1)-dimensions and determinant bundles over infinite dimensional Grassmanians, ”Commun. Math. Phys., 116, 365–400(1988).
J. Mingo, “K-theory and multipliers of stable C *-algebras, ”Trans. Amer. Math. Soc., 299, 397–411(1987).
A. Mishchenko, “Banach algebras, pseudodiffrential operators and their applications in K-theory, ”Usp. Mat. Nauk, 34, 67–79(1979).
A. Mishchenko and A. Fomenko, “The index of elliptic operators over C *-algebras, ”Izv. Akad. Nauk SSSR, Ser. Mat., 43, 831–859(1979).
G. Misiolek, “The exponential map on a free loop space is Fredholm, ”Geom. Func. Anal., 7, 954–969(1997).
G. Misiolek, “Exponential maps of Sobolev metrics on loop groups, ”Proc. Am. Math. Soc., 127, 2475–2482(1999).
G. Misiolek, “Shallow water equation as geodesicflow on the Bott–Virasoro group, ”J. Geom. Phys., 24, 203–208(1998).
N. I. Muskhelishvili, Singular Integral Equations [in Russian], Noordho., Groningen (1946).
R. Narasimhan and T. Seshadri, “Stable and unitary vector bundles on a compact Riemann surface, ”Ann. Math., 82, 540–564(1965).
E. I. Obolashvili, “The solution of some boundary problems for generalized analytic vectors, ”Differ. Equations, 10, 80–83(1974–75).
R. Palais, “On the homotopy type of certain groups of operators, ”Topol ogy, 3, 271–279(1965).
R. Palais and Ch.-L. Terng, “Critical point theory and submanifold geometry, ”Lect. Notes Math., 1353 (1988).
V. Palamodov, “Deformation theory, ”In:Itogi Mauki i Tekhn. Fundam. Napravl., 10 (1986).
V. Palamodov, “Holomorphic synthesis of monogenic functions of several quaternionic variables, ”J. Anal. Math., 78, 177–204(1999).
S. Pr ¨ossdorf, Einige Klassen singul ¨arer Gleichungen, Akad. Verlag, Berlin (1974).
A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford (1986).
M. Rieffel, “Dimension and stable rank in the K-theory of C *-algebras, ”Proc. Lond. Math. Soc., 46, 301–333(1983).
Yu. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Reidel, Dordrecht (1988).
H. R ¨ohrl, “Holomorphic vector bundles over Riemann surfaces, ”Bull. Am. Math. Soc., 68, 125–160(1962).
I. Sabadini, M. Shapiro, and D. Struppa, “Algebraic analysis of the Moisil–Theodorescu system, ”Complex Variables, 40, 333–357(2000).
S. Shatz, “On subbundles of vector bundles over ℂℙ1, ”J. Pure Appl. Algebra, 10, 315–322(1977).
M. Shapiro and N. Vasilevski, “Quaternionic hyperholomorphic functions, singular integral operators, and boundary-value problems, I, II, ”Complex Variables, 27, 17–44, 67–96 (1995).
A. I. Shnirelman, “The degree of a quasi-ruled mapping and the nonlinear Hilbert problem, ”Mat. Sb., 89(131), 366–389(1972).
B. Simon, Trace Ideals and Their Applications, Cambridge Univ. Press, Cambridge (1979).
Yu. Solovjev and E. Troitsky, C*-algebras and Index Theory [in Russian], Nauka, Moscow (1997).
M. Spera and T. Wurzbacher, “The differential geometry of Grassmanian embeddings of based loop groups, ”Differ. Geom. Appl., 13, 43–75(2000).
I. Spitkovsky and A. Tashbaev, “Factorization of certain piecewise constant matrix-valued functions and its applications, ”Math. Nachr., 151, 241–261(1991).
E. Stein and G. Weiss, “Generalization of Cauchy–Riemann equations and representations of the rotation group, ”Am. J. Math., 90, 163–196(1968).
I. Stern, “Boundary value and initial value problems in complex analysis, ”Pitman Res. Notes Math., 256, 159–183(1991).
I. Stern, “On the existence of Fredholm boundary value problems for generalized Cauchy–Riemann systems, ”Complex Variables, 21, 19–38(1993).
N. N. Tarkhanov, “A remark on the Moisil–Theodorescu system, ”Sib. Mat. Zh., 28, No. 3, 208–213(1987).
Ch.-L. Terng, “Proper Fredholm submanifolds of Hilbert space, ”J. Differ. Geom., 29, 9–47 (1989).
K. Thomsen, “Non-stable K-theory for operator algebras, ”K –Theory, 3, 1–25 (1990).
E. Troitsky, “On the contractibility of general linear groups of a Hilbert C*-module, ”Funkts. Anal. Prilozh., 20, 58–64(1986).
I. N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford (1962).
N. P. Vekua, Systems of Singular Integral Equations, Noordho., Groningen (1967).
E. Wegert, Nonlinear Boundary-Value Problems for Holomorphic Functions and Singular Integral Equations, Akad. Verlag, Berlin (1992).
E. Wegert, G. Khimshiashvili, and I. Spitkovsky, “On noncompact totally real manifolds globally bered by analytic discs, ”In:Proc. Conf. Int. Soc. Anal. Alg. Appl., Kluwer (1999).
E. Wloka, Boundary-Value Problems for Partial Differential Equations, Vieweg, Braunschweig (1992).
K. Wojciechowsky, “Spectral flow and the general linear conjugation problem, ”Simon Stevin Univ. J., 59, 59–91(1985).
K. Wojciechowsky, “A note on the space of pseudodifferential projections with the same principal symbol, ”J. Operator Theory, 15, 201–216(1985).
L. von Wolfersdorf, “A class of nonlinear Riemann–Hilbert problems for holomorphic functions, ”Math. Nachr., 116, 89–107(1984).
L. von Wolfersdorf, “On the theory of the nonlinear Hilbert problem for holomorphic functions, ”In: H. Begehr and A. Jeffrey, Partial Differential Equations with Complex Analysis, Longman, London (1992), pp. 134–149.
D. Wilkins, “The Grassman manifold of a C*-algebra, ”Proc. Roy. Irish Acad., 90, 99–116(1990).
T. Wurzbacher, “Symplectic geometry of the loop space of a Riemannian manifold, ”J. Geom. Phys., 16, 345–384(1995).
R. Yang, “BCL index and Fredholm tuples, ”Proc. Am. Math. Soc., 127, 2385–2393(1999).
I. Zakharevich, “Kronecker webs, bihamiltonian structures, and the method of argument translation, ”Transform. Groups, 6, 267–300(2001).
S. Zhang, “Factorizations of invertible operators and K-theory of C*-algebras, ”Bull. Am. Math. Soc., 28, 75–83(1993).
Rights and permissions
About this article
Cite this article
Khimshiashvili, G. Global Geometric Aspects of Linear Conjugation Problems. Journal of Mathematical Sciences 118, 5400–5466 (2003). https://doi.org/10.1023/A:1025884428386
Issue Date:
DOI: https://doi.org/10.1023/A:1025884428386