The functional model for maximal dissipative operators (translation form): An approach in the spirit of operator knots
HTML articles powered by AMS MathViewer
- by Malcolm Brown, Marco Marletta, Serguei Naboko and Ian Wood PDF
- Trans. Amer. Math. Soc. 373 (2020), 4145-4187 Request permission
Abstract:
In this article we develop a functional model for a general maximal dissipative operator. We construct the selfadjoint dilation of such operators. Unlike previous functional models, our model is given explicitly in terms of parameters of the original operator, making it more useful in concrete applications.
For our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti-dissipative adjoint and make use of the Štraus characteristic function in our setting. Explicit formulae are given for the selfadjoint dilation, its resolvent, a core and the completely non-selfadjoint subspace; minimality of the dilation is shown. The abstract theory is illustrated by the example of a Schrödinger operator on a half-line with dissipative potential, and boundary condition and connections to existing theory are discussed.
References
- T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1989. Translated from the Russian by E. R. Dawson; A Wiley-Interscience Publication. MR 1033489
- Jussi Behrndt, Mark M. Malamud, and Hagen Neidhardt, Scattering theory for open quantum systems with finite rank coupling, Math. Phys. Anal. Geom. 10 (2007), no. 4, 313–358. MR 2386256, DOI 10.1007/s11040-008-9035-x
- Jussi Behrndt, Mark M. Malamud, and Hagen Neidhardt, Scattering matrices and Weyl functions, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 568–598. MR 2448240, DOI 10.1112/plms/pdn016
- Jussi Behrndt, Mark M. Malamud, and Hagen Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, J. Funct. Anal. 273 (2017), no. 6, 1970–2025. MR 3669028, DOI 10.1016/j.jfa.2017.06.001
- M. S. Brodskiĭ, Triangular and Jordan representations of linear operators, Translations of Mathematical Monographs, Vol. 32, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin. MR 0322542
- Malcolm Brown, James Hinchcliffe, Marco Marletta, Serguei Naboko, and Ian Wood, The abstract Titchmarsh-Weyl $M$-function for adjoint operator pairs and its relation to the spectrum, Integral Equations Operator Theory 63 (2009), no. 3, 297–320. MR 2491033, DOI 10.1007/s00020-009-1668-z
- Malcolm Brown, Marco Marletta, Serguei Naboko, and Ian Wood, Boundary triplets and $M$-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc. (2) 77 (2008), no. 3, 700–718. MR 2418300, DOI 10.1112/jlms/jdn006
- Kirill D. Cherednichenko, Alexander V. Kiselev, and Luis O. Silva, Functional model for extensions of symmetric operators and applications to scattering theory, Netw. Heterog. Media 13 (2018), no. 2, 191–215. MR 3811560, DOI 10.3934/nhm.2018009
- K. Cherednichenko, A. Kiselev, L. Silva, Functional model for boundary value problems and its application to the spectral analysis of transmission problems, https://arxiv.org/abs/1907.08144
- Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
- V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1–95. MR 1087947, DOI 10.1016/0022-1236(91)90024-Y
- V. A. Derkach and M. M. Malamud, Characteristic functions of almost solvable extensions of Hermitian operators, Ukraïn. Mat. Zh. 44 (1992), no. 4, 435–459 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 4, 379–401. MR 1179404, DOI 10.1007/BF01064871
- V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), no. 2, 141–242. Analysis. 3. MR 1318517, DOI 10.1007/BF02367240
- Bruno Després, Lise-Marie Imbert-Gérard, and Ricardo Weder, Hybrid resonance of Maxwell’s equations in slab geometry, J. Math. Pures Appl. (9) 101 (2014), no. 5, 623–659 (English, with English and French summaries). MR 3192426, DOI 10.1016/j.matpur.2013.10.001
- W. N. Everitt, On a property of the $m$-coefficient of a second-order linear differential equation, J. London Math. Soc. (2) 4 (1971/72), 443–457. MR 298104, DOI 10.1112/jlms/s2-4.3.443
- Marco Falconi, Jérémy Faupin, Jürg Fröhlich, and Baptiste Schubnel, Scattering theory for Lindblad master equations, Comm. Math. Phys. 350 (2017), no. 3, 1185–1218. MR 3607473, DOI 10.1007/s00220-016-2737-1
- Alexander Figotin and Aaron Welters, Dissipative properties of systems composed of high-loss and lossless components, J. Math. Phys. 53 (2012), no. 12, 123508, 40. MR 3405898, DOI 10.1063/1.4761819
- V. I. Gorbachuk and M. L. Gorbachuk, Boundary value problems for operator differential equations, Mathematics and its Applications (Soviet Series), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated and revised from the 1984 Russian original. MR 1154792, DOI 10.1007/978-94-011-3714-0
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- A. N. Kočubeĭ, Extensions of symmetric operators and of symmetric binary relations, Mat. Zametki 17 (1975), 41–48 (Russian). MR 365218
- Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
- Yu. L. Kudryashov, Symmetric and selfadjoint dilations of dissipative operators, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 37 (1982), 51–54 (Russian). MR 701998
- H. Langer, Ein Zerspaltungssatz für Operatoren im Hilbertraum, Acta Math. Acad. Sci. Hungar. 12 (1961), 441–445 (German, with Russian summary). MR 139954, DOI 10.1007/BF02023926
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
- M. S. Livšic, On a certain class of linear operators in Hilbert space, Mat. Sbornik, 19 (1946), no. 2, 239–262.
- M. S. Livšic, On spectral decomposition of linear nonself-adjoint operators, Mat. Sbornik N.S. 34(76) (1954), 145–199 (Russian). MR 0062955
- M. S. Livšic, Operators, oscillations, waves (open systems), Translations of Mathematical Monographs, Vol. 34, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Scripta Technica, Ltd. English translation edited by R. Herden. MR 0347396
- V. È. Lyantse and O. G. Storozh, Metody teorii neogranichennykh operatorov, “Naukova Dumka”, Kiev, 1983 (Russian). MR 757535
- Graeme W. Milton, Nicolae-Alexandru P. Nicorovici, Ross C. McPhedran, and Viktor A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2064, 3999–4034. MR 2186014, DOI 10.1098/rspa.2005.1570
- S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86–114, 203 (Russian). Boundary value problems of mathematical physics, 10. MR 573902
- S. Naboko and R. Romanov, Spectral singularities, Szőkefalvi-Nagy-Foias functional model and the spectral analysis of the Boltzmann operator, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 473–490. MR 1902818
- N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
- B. S. Pavlov, The operator-theoretical significance of the transmission coefficient, Problems in mathematical physics, No. 7 (Russian), Izdat. Leningrad. Univ., Leningrad, 1974, pp. 102–126, 183 (Russian). MR 0512675
- B. S. Pavlov, Conditions for separation of the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 123–148, 240 (Russian). MR 0365199
- B. S. Pavlov, Dilation theory and spectral analysis of nonselfadjoint differential operators, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974) Central. Èkonom. Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 3–69 (Russian). MR 0634807
- B. S. Pavlov, Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in its eigenfunction, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 511–536, 631 (Russian). MR 0510053
- Boris Pavlov, Resonance quantum switch: matching domains, Surveys in analysis and operator theory (Canberra, 2001) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 127–156. MR 1953482
- Vladimir Ryzhov, Functional model of a class of non-selfadjoint extensions of symmetric operators, Operator theory, analysis and mathematical physics, Oper. Theory Adv. Appl., vol. 174, Birkhäuser, Basel, 2007, pp. 117–158. MR 2330831, DOI 10.1007/978-3-7643-8135-6_{9}
- Vladimir Ryzhov, Functional model of a closed non-selfadjoint operator, Integral Equations Operator Theory 60 (2008), no. 4, 539–571. MR 2390443, DOI 10.1007/s00020-008-1574-9
- A. V. Štraus, Characteristic functions of linear operators, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 43–74 (Russian). MR 0140950
- A. V. Štraus, Extensions and characteristic function of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 186–207 (Russian). MR 0225193
- Béla Sz.-Nagy and Ciprian Foiaş, Sur les contractions de l’espace de Hilbert. IV, Acta Sci. Math. (Szeged) 21 (1960), 251–259 (French). MR 126149
- Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy, Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647, DOI 10.1007/978-1-4419-6094-8
- A. S. Tikhonov, An absolutely continuous spectrum and a scattering theory for operators with spectrum on a curve, Algebra i Analiz 7 (1995), no. 1, 200–220 (Russian); English transl., St. Petersburg Math. J. 7 (1996), no. 1, 169–184. MR 1334157
Additional Information
- Malcolm Brown
- Affiliation: School of Computer Science, Cardiff University, Queen’s Buildings, 5 The Parade, Cardiff CF24 3AA, United Kingdom
- MR Author ID: 271038
- Email: Malcolm.Brown@cs.cardiff.ac.uk
- Marco Marletta
- Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, United Kingdom
- Email: MarlettaM@cardiff.ac.uk
- Serguei Naboko
- Affiliation: Department of Math. Physics, Institute of Physics, St. Petersburg State University, 1 Ulianovskaia, St. Petergoff, St. Petersburg, 198504, Russia
- Email: sergey.naboko@gmail.com
- Ian Wood
- Affiliation: School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, CT2 7FS, United Kingdom
- MR Author ID: 739890
- Email: i.wood@kent.ac.uk
- Received by editor(s): November 23, 2018
- Received by editor(s) in revised form: September 23, 2019
- Published electronically: March 3, 2020
- Additional Notes: The second and third authors gratefully acknowledge the support of the Leverhulme Trust, grant RPG167, and of the Wales Institute of Mathematical and Computational Sciences. The third author also gratefully acknowledges support by the RFBR 19-01-00657A grant and the Knut and Alice Wallenberg Foundation.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4145-4187
- MSC (2010): Primary 47A20, 47A48, 47B44
- DOI: https://doi.org/10.1090/tran/8029
- MathSciNet review: 4105520