Abstract
An important task of the spectral analysis is to construct the model representations for linear operators realizing them via the operator of multiplication by an independent variable in a particular function space. Unlike those spectral decompositions for self-adjoint (unitary) operators constructed by von Neumann, similar representations for non-self-adjoint (non-unitary) operators are rather arduous to obtain. In the fifties Livšic [20] undertook investigations in this direction and devised a theory of characteristic functions and a theory of triangular models of linear operators. Later, in mid-sixties, a theory of dilations for semigroups of contractions was created by Sz.-Nagy and Foias [1]. Simultaneously, Lax and Phillips [2] had shaped a geometric theory of scattering of acoustic waves at bounded obstacles. Further development in these three areas supplied a basis for creating a method of studying non-self-adjoint operators and building correspondent models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B.Sz. -Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publ. Co., Amsterdam, and Akademiai Kiado, Budapest, 1970.
P.D. Lax and R.S. Phillips, Scattering theory, Academic Press, NY, 1967.
P.D. Lax and R.S. Phillips, Scattering theory for automorphic functions, Annals of Math. Studies, no. 87, Princeton University Press, Princeton, NJ, 1976.
B.S. Pavlov, Dilation theory and spectral analysis of non-self-adjoint differential operators, Theory of operators in linear spaces (Proc. Seventh Winter school, Drogobych, 1974), Tsentral. Ekonom.-Mat. Inst. Akad. Nauk SSSR, Moscow 1976, pp.3–69; English transl. in Amer. Math. Soc. Transl. (2) 115 (1980).
B.S. Pavlov and L.D. Faddeev, Scattering theory and automorphic functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27 (1972), 161–193; English transl. in J. Soviet Math. 5:4 (1975).
B.S. Pavlov, Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model, Itogi Nauki i Tekhniki: Ser. Sovr. Probl. Mat.: Fund. Napravleniya vol. 65, VINITI, Moscow 1991, pp. 95–163 (in Russian).
B.S. Pavlov and S.I. Fedorov, The group of shifts and harmonic analysis on a Riemann surface of genus one, Algebra i Analiz 1:2 (1989),132–169; Engish transl. in Leningrad Math J. 1 (1990).
S.I. Fedorov, Harmonic analysis in a multiply connected domains. I, Mat. Sb. 181:6 (1990), 833–864 (in Russian).
S.I. Fedorov, Harmonic analysis in a multiply connected domains. II, Mat. Sb. 181:7 (1990), 867–910 (in Russian).
M.A. Semenov-Tyan-Shanskii, Harmonic analysis on Riemannian symmetric spaces of negative curvature, and scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 562–592; English transl. in Math. USSR-Izv. 10 (1976).
V.A. Zolotarev, Time cones and a functional model on a Riemann surface, Mat. Sb. 181 (1990), 965–995; English transl. in Math. USSR-Sb. 71 (1992).
V.A. Zolotarev, The Lax-Phillips scattering scheme on groups and a functional model of a Lie algebra, Mat. Sb. 183:5 (1992), 115–144; English transl. in Russian Acad. Sci. Sb. Math. 76 (1993).
V.A. Zolotarev, A functional model for the Lie algebra ISO(1,1) of linear non-selfadjoint operators, Mat. Sb. 186:1 (1995), 79–106; English transl. in Russian Acad. Sci. Sb. Math. 186 (1995).
D.P. Zhelobenko and A.I. Shtern, Representations of Lie groups, Nauka, Moscow 1983 (in Russian).
N.Ya. Vilenkin and A.U. Klimuk, Representations of Lie groups and special functions, Vol. I. Simplest Lie groups, special functions, and integral transforms. Itogi Nauki i Tekhniki: Ser. Soar. Probl. Mat.: Fund. Napravleniya, vol.59, VINITI Moscow 1990, pp. 145–264; English transl. in Mathematics and its Applications (Soviet Series), vol. 72, Kluwer, Dordrecht, 1991.
N.Ya. Vilenkin, Special functions and theory of group representations, Nauka, Moscow 1965; English transl. Amer. Math. Soc., Providence, RI, 1968.
B.Ya. Dubrovin, A.T. Fomenko, and S.P. Novikov, Modern geometry. Methods and applications; English transl. Graduate Texts in Math., vol. 93, Springer-Verlag, New York, 1984.
A.T. Fomenko, Symplectic geometry. Methods and applications, Moscow, Izdat. Mosc. Univ., 1988 (in Russian).
I. Springer, Introduction to the theory of Riemann surfaces, Moscow, 1960 (in Russian).
B.A. Dubrovin, Theta-functions and non-linear equations, Uspehi Mat. Nauk 2(218):36 (1981), 11–80 (in Russian).
N.K. Nikol’skii and S.V. Khrushchev, A functional model and some problems of the spectral theory of functions, Trudy Mat. Inst. Steklov 176 (1987), 97–210; English transl. in Proc. Steklov Inst. Math. 1988:3 (176).
V.L. Ostrovskii and Yu.S. Samoilenko, Families of unbounded selfadjoint operators that are connected by non-Lie relations, Funktsional. Anal. i Prilozhen. 23:2 (1989), 67–68; English transl. in Funct. Anal. Appl. 23 (1989).
Sh. Mizohata, The theory of partial differential equations, Cambridge University Press, 1973.
K. Godunov, Equations de la physique mathématique, 2nd ed., Nauka, Moscow 1979; French transl. of 1st ed., Editions Mir, Moscow 1973.
M.S. Livsic and A.A. Yantsevich, Operator colligations in Hilbert spaces, Izdat. Khar’kov. Univ., Kharkov 1971; English transl. Wiley, New York, 1979.
M.S. Livsic, Commuting nonselfadjoint operators and collective motions of systems, Commuting non-self-adjoint operators in a Hilbert space, Lecture Notes in Mathematics, no. 1272, Springer-Verlag, Berlin 1987, pp. 1–38.
V.S. Vladimirov, Equations of mathematical physics, Nauka, Moscow, 1988 (in Russiasn).
M.S. Livsic, N. Kravitsky, A. Markus, V Vinnikov, Theory of commuting nonselfadjoint operators, Kluwer, Dordrecht, 1995.
V. Vinnikov, Commuting nonselfadjoint operators and algebraic curves, Oper. Theory Adv. Appl. (1992), 348–371.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Zolotarev, V.A. (2001). A Functional Model for the Lie Algebra SL(2, ℝ) of Linear Non-self-adjoint Operators. In: Alpay, D., Vinnikov, V. (eds) Operator Theory, System Theory and Related Topics. Operator Theory: Advances and Applications, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8247-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8247-7_25
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9491-3
Online ISBN: 978-3-0348-8247-7
eBook Packages: Springer Book Archive