zbMATH — the first resource for mathematics

Spectral properties of operators with polynomial invariants in real finite-dimensional spaces. (English. Russian original) Zbl 1209.47005
Proc. Steklov Inst. Math. 268, 148-160 (2010); translation from Trudy Mat. Inst. Steklova 268, 155-167 (2010).
The author considers linear operators lying in the orthogonal group of a quadratic form and studies those spectral properties of such operators that can be expressed in terms of the signature of this form. He shows that, in the typical case, these transformations are symplectic. Some of the results can be extended to the general case when the operator admits a homogeneous form of degree \(\geq 3\).

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A10 Spectrum, resolvent
34C14 Symmetries, invariants of ordinary differential equations
47A75 Eigenvalue problems for linear operators
Full Text: DOI
[1] V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Nauka, Moscow, 1972; Wiley, New York, 1975).
[2] J. Williamson, ”On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems,” Am. J. Math. 58(1), 141–163 (1936). · JFM 63.1290.01
[3] V. V. Kozlov, ”Linear Systems with a Quadratic Integral,” Prikl. Mat. Mekh. 56(6), 900–906 (1992) [Appl. Math. Mech. 56, 803–809 (1992)].
[4] V. N. Rubanovskii, ”On Bifurcation and Stability of Stationary Motions in Certain Problems of Dynamics of a Solid Body,” Prikl. Mat. Mekh. 38(4), 616–627 (1974) [Appl. Math. Mech. 38, 573–584 (1974)]. · Zbl 0312.70007
[5] G. Frobenius, ”Über lineare Substitutionen und bilineare Formen,” J. Math. 84, 1–63 (1878). · JFM 09.0085.02
[6] L. S. Pontryagin, ”Hermitian Operators in Spaces with Indefinite Metric,” Izv. Akad. Nauk SSSR, Ser. Mat. 8(6), 243–280 (1944). · Zbl 0061.26004
[7] D. Carlson and H. Schneider, ”Inertia Theorems for Matrices: The Semidefinite Case,” J. Math. Anal. Appl. 6, 430–446 (1963). · Zbl 0192.13402
[8] H. K. Wimmer, ”Inertia Theorems for Matrices, Controllability, and Linear Vibrations,” Linear Algebra Appl. 8, 337–343 (1974). · Zbl 0288.15015
[9] V. I. Arnol’d, ”Conditions for Nonlinear Stability of Stationary Plane Curvilinear Flows of an Ideal Fluid,” Dokl. Akad. Nauk SSSR 162(5), 975–978 (1965) [Sov. Math., Dokl. 6, 773–777 (1965)].
[10] O. Taussky, ”A Generalization of a Theorem of Lyapunov,” J. Soc. Ind. Appl. Math. 9, 640–643 (1961). · Zbl 0108.01202
[11] A. Ostrowski and H. Schneider, ”Some Theorems on the Inertia of General Matrices,” J. Math. Anal. Appl. 4, 72–84 (1962). · Zbl 0112.01401
[12] H. Weyl, The Classical Groups: Their Invariants and Representations (Princeton Univ. Press, Princeton, NJ, 1939; Inostrannaya Literatura, Moscow, 1947). · Zbl 0020.20601
[13] V. V. Kozlov and A. A. Karapetyan, ”On the Stability Degree,” Diff. Uravn. 41(2), 186–192 (2005) [Diff. Eqns. 41, 195–201 (2005)]. · Zbl 1090.34564
[14] F. R. Gantmacher, The Theory of Matrices (Nauka, Moscow, 1988; AMS Chelsea Publ., Providence, RI, 1998). · Zbl 0085.01001
[15] G. W. Hill, ”On the Part of the Motion of the Lunar Perigee Which Is a Function of the Mean Motions of the Sun and Moon,” Acta Math. 8(1), 1–36 (1886). · JFM 18.1106.01
[16] V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Mosk. Gos. Univ., Moscow, 1991; Am. Math. Soc., Providence, RI, 1991). · Zbl 0751.70009
[17] M. G. Krein, ”On an Application of the Fixed-Point Principle in the Theory of Linear Transformations of Spaces with an Indefinite Metric,” Usp. Mat. Nauk 5(2), 180–190 (1950) [Am. Math. Soc. Transl., Ser. 2, 1, 27–35 (1955)].
[18] S. V. Bolotin and V. V. Kozlov, ”Asymptotic Solutions of Equations of Dynamics,” Vestn. Mosk. Univ., Ser 1: Mat., Mekh., No. 4, 84–89 (1980) [Mosc. Univ. Mech. Bull. 35 (3–4), 82–88 (1980)]. · Zbl 0457.70022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.