Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-30T10:15:48.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Graph Subspaces and the Spectral Shift Function

Published online by Cambridge University Press:  20 November 2018

Sergio Albeverio ,
Konstantin A. Makarov  and
Alexander K. Motovilov
Sergio Albeverio
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany, website:, http://wiener.iam.uni-bonn.de/albeverio/albeverio.html e-mail: albeverio@uni-bonn.de
Konstantin A. Makarov
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA, website:, http://www.math.missouri.edu/people/kmakarov.html e-mail: makarov@math.missouri.edu
Alexander K. Motovilov
Affiliation:
Bogoliubov Laboratory of Theoretical Physics, JINR, Joliot-Curie str. 6, 141980 Dubna, Russia, website:, http://thsun1.jinr.ru/˜motovilv e-mail: motovilv@thsun1.jinr.ru
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

Keywords

47B4447A1047A2047A40

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 449 - 503
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Adamjan, V. and Langer, H. The spectral shift function for certain operator matrices. Math. Nachr. 211(2000), 5–24.Google Scholar
[2] Adamjan, V. and Langer, H. Spectral properties of a class of operator-valued functions. J. Operator Theory 33(1995), 259–277.Google Scholar
[3] Adamyan, V. M., Langer, H., Mennicken, R. and Saurer, J. Spectral components of selfadjoint block operator matrices with unbounded entries. Math. Nachr. 178(1996), 43–80.Google Scholar
[4] Adamyan, V., Langer, H. and Tretter, C. Existence and uniqueness of contractive solutions of some Riccati equations. J. Funct. Anal. 179(2001), 448–473.Google Scholar
[5] Adamyan, V. M., Mennicken, R. and Saurer, J. On the discrete spectrum of some selfadjoint operator matrices. J. Operator Theory 39(1998), 3–41.Google Scholar
[6] Adams, T. A nonlinear characterization of stable invariant subspaces. Integral Equations Operator Theory 6(1983), 473–487.Google Scholar
[7] Akhiezer, N. I. and Glazman, I. M., Theory of linear operators in Hilbert space. Dover Publications Inc., New York, 1993.Google Scholar
[8] Atkinson, F. V., Langer, H., Mennicken, R. and Shkalikov, A. A. The essential spectrum of some matrix operators. Math. Nachr. 167(1994), 5–20.Google Scholar
[9] Bhatia, R., Davis, C. and McIntosh, A. Perturbation of spectral subspaces and solution of linear operator equations. Linear Algebra Appl. 52/53(1983), 45–67.Google Scholar
[10] Bhatia, R. and Rosenthal, P. How and why to solve the operator equation AX − XB = Y. Bull. London Math. Soc. 29(1997), 1–21.Google Scholar
[11] Birman, M. and Solomjak, M., Stieltjes double-operator integrals. Topics in Mathematical Physics 1, Consultants Bureau, New York, 1967, 25–54.Google Scholar
[12] Birman, M. S. and Pushnitski, A. B. Spectral shift function, amazing and multifaceted. Integral Equations Operator Theory 30(1998), 191–199.Google Scholar
[13] Birman, M. S. and Yafaev, D. R. Spectral properties of the scattering matrix. Algebra i Analiz (6) 4(1992), 1-27 (Russian); English transl., St. Petersburg Math. J. 4(1993), 1055–1079.Google Scholar
[14] Birman, M. S. and Yafaev, D. R. The spectral shift function. The work of M. G. Krein and its further development. Algebra i Analiz (5) 4(1992), 1-44 (Russian); English transl., St. Petersburg Math. J. 4(1993), 833–870.Google Scholar
[15] Callier, F. M., Dumortier, L. and Winkin, J. On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite-dimensional systems. Integral Equations Operator Theory 22(1995), 162–195.Google Scholar
[16] Carey, R. W. and Pincus, J. D. Unitary equivalence modulo the trace class for self-adjoint operators. Amer. J. Math. 98(1976), 481–514.Google Scholar
[17] Curtain, R. F. Old and new perspectives on the Positive-real Lemma in systems and control theory. Z. Angew. Math. Mech. 79(1999), 579–590.Google Scholar
[18] Daleckii, Y. On the asymptotic solution of a vector differential equation. Dokl. Akad. Nauk SSSR 92(1953), 881–884.Google Scholar
[19] Davis, C. and Kahan, W. M. Some new bounds on perturbation of subspaces. Bull. Amer. Math. Soc. 75(1969), 863–868.Google Scholar
[20] Davis, C. and Kahan, W. M. The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7(1970), 1–46.Google Scholar
[21] Davis, C. and Rosenthal, P. Solving linear operator equations. Canad. J. Math. (6) XXVI(1974), 1384–1389.Google Scholar
[22] Friedrichs, K. O. On the Perturbation of Continuous Spectra. Comm. Pure Appl. Math. 1(1948), 361–406.Google Scholar
[23] Gesztesy, F. and Makarov, K. A. Some applications of the spectral shift operator. Operator theory and its applications, Fields Inst. Commun. 25(2000), 267–292.Google Scholar
[24] Gesztesy, F. and Makarov, K. A. The Ξ operator and its relation to Krein's spectral shift function. J. Anal. Math. 81(2000), 139–183.Google Scholar
[25] Gesztesy, F., Makarov, K. A. and Motovilov, A. K. Monotonicity and concavity properties of the spectral shift function. Canad. Math. Soc. Conference Proceedings Series, Providence, RI, 29(2000), 207–222.Google Scholar
[26] Gesztesy, F., Makarov, K. A. and Naboko, S. N. The spectral shift operator. Operator Theory: Advances and Applications, Birkhäuser, Basel, 108(1999), 59–90.Google Scholar
[27] Gesztesy, F. and Simon, B. The xi function. Acta Math. 176(1996), 40–71.Google Scholar
[28] Goedbloed, J. P., Lecture notes on ideal magnetohydrodynamics. Rijnhiuzen Report, Form Instutuut voor Plasmafysica, Niewwegein, 1983, 83–145.Google Scholar
[29] Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear non-selfadjoint operators. Trans. Math. Monographs 18, Amer. Math. Soc., Providence, 1969.Google Scholar
[30] Heinz, E. Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann. 123(1951), 415–438.Google Scholar
[31] Helton, J. and Howe, R. Traces of commutators of integral operators. Acta Math. 135(1975), 271–305.Google Scholar
[32] Ionesco, V., Oară, C. and Weiss, M., Generalized Riccati Theory and Robust Control. A Popov Function Approach. John Wiley & Sons, Chichester, 1999.Google Scholar
[33] Kantorovich, L. V. and Akilov, G. P., Functional Analysis, Third Edition. Nauka, Moscow, 1984, Russian.Google Scholar
[34] Kato, T., Perturbation theory for linear operators. Springer-Verlag, New York, 1966.Google Scholar
[35] Kostrykin, V. Concavity of eigenvalue sums and the spectral shift function. J. Funct. Anal. 176(2000), 100–114.Google Scholar
[36] Krein, M. G., On certain new studies in the perturbation theory for self-adjoint operators. In: M. G. Krein, Topics in Differential and Integral Equations and Operator Theory, (ed., I. Gohberg), Birkhäuser, Basel, 1983, 107–172.Google Scholar
[37] Krein, M. G. On perturbation determinants and a trace formula for certain classes of pairs of operators. Amer.Math. Soc. Transl. Ser. 2 145(1989), 39–84.Google Scholar
[38] Krein, M. G. On perturbation determinants and a trace formula for unitary and self-adjoint operators. Dokl. Akad. Nauk SSSR 144(1962), 268–271.Google Scholar
[39] Krein, M. G., On some new investigations in perturbation theory. First Math. Summer School, Kiev, 1963, 104–183, Russian.Google Scholar
[40] Krein, M. G. On the trace formula in perturbation theory. Mat. Sb. 75 33(1953), 597–626.Google Scholar
[41] Lancaster, P. and Rodman, L., Algebraic Riccati equations. Clarendon Press, Oxford and Oxford University Press, New York, 1995.Google Scholar
[42] Lasiecka, I., Mathematical Control Theory of Coupled PDEs. CBMS-NSF Regional Conference Series in Applied Math. 75, SIAM, Philadelphia, 2002.Google Scholar
[43] Lauric, V. and Pearcy, C. M. Trace-class commutators with trace zero. Acta Sci. Math. (Szeged) 66(2000), 341–349.Google Scholar
[44] Lifschitz, A. E., Magnetohydrodynamics and spectral theory. Kluwer Academic Publishers, Dordrecht, 1989.Google Scholar
[45] Lifshits, I. M. On a problem of perturbation theory. Uspekhi Mat. Nauk (1) 7(1952), 171–180.Google Scholar
[46] Lifshits, I. M. Some problems of the dynamic theory of nonideal crystal lattices. Nuovo Cimento Suppl. Ser. X 3(1956), 716–734.Google Scholar
[47] Lumer, G. and Rosenblum, M. Linear operator equations. Proc. Amer. Math. Soc 10(1959), 32–41.Google Scholar
[48] Malyshev, V. A. and Minlos, R. A. Invariant subspaces of clustering operators. I. J. Stat. Phys. 21(1979), 231-242; Invariant subspaces of clustering operators. II. Comm. Math. Phys. 82(1981), 211–226.Google Scholar
[49] Markus, A. S. and Matsaev, V. I. On the basis property for a certain part of the eigenvectors and associated vectors of a selfadjoint operator pencil. Math. USSR Sb. 61(1988), 289–307.Google Scholar
[50] Markus, A. S. and Matsaev, V. I. On the spectral theory of holomorphic operator-valued functions in Hilbert space. Funct. Anal. Appl. (1) 9(1975), 73–74.Google Scholar
[51] McEachin, R. Closing the gap in a subspace perturbation bound. Linear Algebra Appl. 180(1993), 7–15.Google Scholar
[52] Mennicken, R. and Motovilov, A. K. Operator interpretation of resonances arising in spectral problems for 2 X 2 operator matrices. Math. Nachr. 201(1999), 117–181.Google Scholar
[53] Mennicken, R. and Motovilov, A. K. Operator interpretation of resonances generated by 2 X 2 matrix Hamiltonians. Theoret. and Math. Phys. 116(1998), 867–880.Google Scholar
[54] Mennicken, R. and Shkalikov, A. A. Spectral decomposition of symmetric operator matrices. Math. Nachr. 179(1996), 259–273.Google Scholar
[55] Motovilov, A. K., Potentials appearing after removal of the energy-dependence and scattering by them. In: Proc. of the Intern. Workshop. Mathematical aspects of the scattering theory and applications., St. Petersburg University, St. Petersburg, 1991, 101–108.Google Scholar
[56] Motovilov, A. K. Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian. J. Math. Phys. 36(1995), 6647–6664; Elimination of energy from interactions depending on it as a resolvent. Theoret. and Math. Phys. 104(1995), 989-1007.Google Scholar
[57] Phóng, V. Q. The operator equation AX − XB = C with unbounded operators A and B and related abstract Cauchy problems. Math. Z. 208(1991), 567–588.Google Scholar
[58] Pushnitski, A. B. A representation for the spectral shift function in the case of perturbations of fixed sign. St. PetersburgMath. J. 9(1998), 1181–1194.Google Scholar
[59] Pushnitski, A. B. Estimates for the spectral shift function of the polyharmonic operator. J. Math. Phys. 40(1999), 5578–5592.Google Scholar
[60] Pushnitski, A. B. Integral estimates for the spectral shift function. St. PetersburgMath. J. 10(1999), 1047–1070.Google Scholar
[61] Pushnitski, A. B. Spectral shift function of the Schrödinger operator in the large coupling constant limit. Comm. Partial Differential Equations 25(2000), 703–736.Google Scholar
[62] Pushnitski, A. B. The spectral shift function and the invariance principle. J. Funct. Anal. 183(2001), 269–320.Google Scholar
[63] Rosenblum, M. On the operator equation BX − XA = Q. Duke Math. J. 23(1956), 263–269.Google Scholar
[64] Simon, B. Spectral averaging and the Krein spectral shift. Proc. Amer.Math. Soc. 126(1998), 1409–1413.Google Scholar
[65] Staffans, O. J. Quadratic optimal control of well-posed linear systems. SIAM J. Control Optim. 37(1998), 131–164.Google Scholar
[66] Sz.-Nagy, B. Über die Ungleichung von H. Bohr. Math. Nachr. 9(1953), 255–259.Google Scholar
[67] Virozub, A. I. and Matsaev, V. I. The spectral properties of a certain class of selfadjoint operator functions. Funct. Anal. Appl. 8(1974), 1–9.Google Scholar
[68] Wiess, G. The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I. Trans. Amer.Math. Soc. 246(1978), 193–209.Google Scholar
[69] Yafaev, D. R., Mathematical Scattering Theory. Amer.Math. Soc., Providence, RI, 1992.Google Scholar