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7 - Boundary triplets and Weyl functions. Recent developments

Published online by Cambridge University Press:  05 November 2012

By
V.A. Derkach ,
S. Hassi ,
M.M. Malamud  and
H.S.V. de Snoo
Edited by
Seppo Hassi ,
Hendrik S. V. de Snoo  and
Franciszek Hugon Szafraniec
V.A. Derkach
Affiliation:
Donetsk National University
S. Hassi
Affiliation:
University of Vaasa
M.M. Malamud
Affiliation:
Donetsk National University
H.S.V. de Snoo
Affiliation:
University of Groningen
Seppo Hassi
Affiliation:
University of Vaasa, Finland
Hendrik S. V. de Snoo
Affiliation:
Rijksuniversiteit Groningen, The Netherlands
Franciszek Hugon Szafraniec
Affiliation:
Jagiellonian University, Krakow
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Summary

Abstract Selfadjoint extensions of a closed symmetric operator in a Hilbert space with equal deficiency indices are described by means of ordinary boundary triplets. In certain problems the more general notion of a boundary triplet of bounded type is needed. It will be shown that such triplets correspond in a certain way with the, in general infinite dimensional, graph perturbations of selfadjoint operators or relations. However, when considering selfadjoint exit spaces extensions for a symmetric operator by means of the so-called Kreĭn's formula one meets the notion of boundary relation (even when the deficiency indices are finite). Whereas ordinary boundary triplets and boundary triplets of bounded type correspond to bounded unitary or unitary operators in a Kreĭn space, respectively, boundary relations correspond to general unitary relations in a Kreĭn space, which are not necessarily single-valued. It is shown that the study of isometric relations in a Kreĭn space has useful applications. This present overview of recent developments includes illustrations, for instance, by means of elliptic differential operators and Schrödinger operators with local point interactions.

Introduction

The extension theory of densely defined symmetric operators was developed in the 1930s by J. von Neumann. A complete description was given for all selfadjoint extensions in terms of the defect subspaces; see [von Neumann, 1932; Stone, 1932]. Then M.H. Stone suggested to J.W. Calkin to develop another approach based on the notion of “abstract boundary conditions”, which reduces the extension problem to a description of the hyper-maximal symmetric subspaces of some auxiliary Hilbert space.

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Operator Methods for Boundary Value Problems , pp. 161 - 220
Publisher: Cambridge University Press
Print publication year: 2012

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References

Achieser, N.I., and Glasmann, I.M. 1981. Theorie der linearen Operatoren im Hilbertraum, 8th edition, Akademie Verlag, Berlin.
Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., and Holden, H. 2005. Solvable Models in Quantum Mechanics, Second Edition, AMS Chelsea Publ.
Albeverio, S., Kostenko, A.S., and Malamud, M.M. 2010. Spectral theory of semi-bounded Sturm-Liouville operators with local interactions on a discrete set. J. Math. Physics, 51, 1–24.Google Scholar
Albeverio, S., and Kurasov, P. 2000. Singular Perturbations of Differential Operators and Schrödinger Type Operators, Cambridge University Press.
Amrein, W.O., and Pearson, D.B. 2004. M-operator: a generalisation of Weyl-Titchmarsh theory. J. Comp. Appl. Math., 171, 1–26.Google Scholar
Arens, R. 1961. Operational calculus of linear relations. Pacific J. Math., 11, 9–23.Google Scholar
Arlinskii, Y., Belyi, S., and Tsekanovskii, E. 2011. Conservative realizations of Herglotz-Nevanlinna functions, Birkhäuser Verlag.
Ashbaugh, M.S., Gesztesy, F., Mitrea, M., Shterenberg, R., and Teschl, G. 2010. The Krein-von Neumann extension and its connection to an abstract buckling problem. Math. Nachr., 210, 165–179.Google Scholar
Azizov, T. Ya., and Iokhvidov, I.S. 1989. Linear operators in spaces with indefinite metric, John Wiley and Sons, New York.
Bade, W.G., and Freeman, R.S. 1962. Closed extensions of the Laplace operator by a general class of boundary conditions. Pacific J. Math., 12, 395–410.Google Scholar
Beals, R. 1965. Non-local boundary value problems for elliptic operators. Amer. J. Math., 87, 315–362.Google Scholar
Behrndt, J., Derkach, V.A., Hassi, S., and de Snoo, H.S.V. 2011. A realization theorem for generalized Nevanlinna pairs. Operators and Matrices, 5, 679–706Google Scholar
Behrndt, J., Hassi, S., and de Snoo, H.S.V. 2009. Boundary relations, unitary colligations, and functional models. Complex Analysis Operator Theory, 3, 57–98.Google Scholar
Behrndt, J., Hassi, S., de Snoo, H.S.V., and Wietsma, H.L. 2010. Monotone convergence theorems for semibounded operators and forms with applications. Proc. Royal Soc. Edinburgh, 140A, 927–951.Google Scholar
Behrndt, J. and Langer, M. 2007 Boundary value problems for elliptic partial differential operators on bounded domains. J. Functional Analysis, 243, 536–565.Google Scholar
Bennewitz, C. 1972. Symmetric relations on a Hilbert space. Lect. Notes Math., 280, 212–218.Google Scholar
Berezanskii, Yu. M. 1965. Expansions in eigenfunctions of selfadjoint operators, Naukova Dumka, Kiev (Russian). English translation in Amer. Math. Soc.Providence, RI, 1968.
Birman, M.Sh. 1956. On the self-adjoint extensions of positive definite operators. Mat. Sb., 38, 431–450 (Russian).Google Scholar
Bognar, J. 1974. Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 78, Springer-Verlag, New York-Heidelberg.
Brown, B.M., Grubb, G., and Wood, I. 2009. M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr., 282, 314–347.Google Scholar
Brown, M., Hinchcliffe, J., Marletta, M., Naboko, S., and Wood, I. 2009. The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum. Integral Equations Operator Theory, 63, 297–320.Google Scholar
Brown, M., Marletta, M., Naboko, S., and Wood, I. 2008. Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. London Math. Soc., 77, 700–718.Google Scholar
Bruk, V.M. 1976. On a class of problems with the spectral parameter in the boundary conditions. Mat. Sb., 100, 210–216.Google Scholar
Bruning, J., Geyler, V., and Pankrashkin, K. 2007. Cantor and band spectra for periodic quantum graphs with magnetic fields. Commun. Math. Phys., 269, 87–105.Google Scholar
Calkin, J.W. 1939. Abstract symmetric boundary conditions. Trans. Amer. Math. Soc., 45, 369–442Google Scholar
Coddington, E.A. 1973. Extension theory of formally normal and symmetric subspaces. Mem. Amer. Math. Soc., 134, 1–80.Google Scholar
Derkach, V.A. 2009. Abstract interpolation problem in Nevanlinna classes. Oper. Theory Adv. Appi., 190, 197–236.Google Scholar
Derkach, V.A., Hassi, S., Malamud, M.M., and de Snoo, H.S.V. 2000. Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topology, 6, 24–55.Google Scholar
Derkach, V.A., Hassi, S., Malamud, M.M., and de Snoo, H.S.V. 2004. Boundary relations and their Weyl families. Doklady Russian Akad. Nauk, 39, 151–156.Google Scholar
Derkach, V.A., Hassi, S., Malamud, M.M., and de Snoo, H.S.V. 2006. Boundary relations and Weyl families. Trans. Amer. Math. Soc., 358, 5351–5400.Google Scholar
Derkach, V.A., Hassi, S., Malamud, M.M., and de Snoo, H.S.V. 2009. Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys., 16, 17–60.Google Scholar
Derkach, V.A., Hassi, S., Malamud, M.M., and de Snoo, H.S.V. 2012. Graph perturbations of selfadjoint operators and relations, and boundary triplets.
Derkach, V.A., Hassi, S., and de Snoo, H.S.V. 2003. Singular perturbations of self-adjoint operators. Mathematical Physics, Analysis and Geometry, 6, 349–384.Google Scholar
Derkach, V.A., and Malamud, M.M. 1985. Weyl function of Hermitian operator and its connection with characteristic function. Preprint 85-9 (104) Donetsk Fiz-Techn. Institute AN Ukrain. SSR, (Russian).
Derkach, V.A., and Malamud, M.M. 1987. On Weyl function and Hermitian operators with gaps. Doklady Akad. Nauk SSSR, 293, 1041–1046.Google Scholar
Derkach, V.A., and Malamud, M.M. 1991. Generalized resolvents and the boundary value problems for hermitian operators with gaps. J. Funct. Anal., 95, 1–95.Google Scholar
Derkach, V.A., and Malamud, M.M. 1995. The extension theory of hermitian operators and the moment problem. J. Math. Sciences, 73, 141–242.Google Scholar
Dijksma, A., Langer, H., and de Snoo, H.S.V. 1987. Symmetric Sturm-Liouville operator with eigenvalue depending boundary conditions. Canadian Math. Soc. Conference Proceedings, 8, 87–116.Google Scholar
Dijksma, A., Langer, H., and de Snoo, H.S.V. 1988. Hamiltonian systems with eigenvalue depending boundary conditions. Oper. Theory Adv. Appl., 35, 37–83.Google Scholar
Dijksma, A., and de Snoo, H.S.V. 1974. Self-adjoint extensions of symmetric subspaces. Pacific J. Math., 54, 71–100.Google Scholar
Exner, P. 2004. Seize ans après. Appendix K to “Solvable Models in Quantum Mechanics” by Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., and Holden, H., Second Edition, AMS Chelsea Publ.
Gesztesy, F., and Mitrea, M. 2008. Robin-to-Robin maps and Kreïn type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, arXiv.org:0803.3072, 2008, 1–26.
Gesztesy, F., and Mitrea, M. 2011. A description of selfadjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains. J. Math. Anal, Appl., 113, 53–172.Google Scholar
Glazman, I.M. 1950. On the theory of singular differential operators. Uspekhi Matem-aticheskikh Nauk, 5, 102–135.Google Scholar
Gorbachuk, M.L. 1971. Self-adjoint boundary value problems for differential equation of the second order with unbounded operator coefficient. Functional Anal. Appl., 5, (1971), 10–21.Google Scholar
Gorbachuk, V.l., and Gorbachuk, M.L. 1984. Boundary problems for differential operator equations, Naukova Dumka, Kiev (Russian).
Gorbachuk, M.L., Gorbachuk, V.I., and Kochubei, A.N. 1989. The theory of extensions of symmetric operators, and boundary value problems for differential equations. Ukrain. Math. Zh., 41, 1299–1313 (Russian).Google Scholar
Graff, A.A. 1946. To the theory of linear differential systems in one-dimensional domain. Mat. Sb., 18, 305–327Google Scholar
Grubb, G. 1968. A characteriz264 f the non local boundary value problems associated with an elliptic operator. Ann. Scuola Normale Superiore de Pisa, 22, 425–513.Google Scholar
Grubb, G. 2009. Distributions and Operators, Vol 552, Graduate Texts in Mathematics, 552, Springer, New York.
Hassi, S., Malamud, M.M., and Mogilevskiĭ, V. I. 2005. Generalized resolvents and boundary triplets for dual pairs of linear relations. Meth. Funct. Anal. Topology, 11, 170–187.Google Scholar
Hassi, S., Kaltenbäck, M., and de Snoo, H.S.V. 1997. Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions. J. Operator Theory, 37, 155–181.Google Scholar
Hassi, S., Kaltenback, M., and de Snoo, H.S.V. 1998. Generalized Kreĭn-von Neumann extensions and associated operator models. Acta Sci. Math. (Szeged), 64, 627–655.Google Scholar
Hassi, S., Langer, H., and de Snoo, H.S.V. 1995. Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1). 15th Oper. Theory Conf. Proc., 115–145.
Hassi, S., and de Snoo, H.S.V. 1997. One-dimensional graph perturbations of selfad-joint relations. Ann. Acad. Sci. Fenn. Ser. A I Math., 22, 123–164.Google Scholar
Hassi, S., de Snoo, H.S.V., Sterk, A.E., and Winkler, H. 2007. Finite-dimensional graph perturbations of selfadjoint Sturm-Liouville operators, in: Operator Theory, Structured Matrices, and Dilations (Tiberiu Constantinescu Memorial Volume), Theta Series in Advanced Mathematics, 205–228.
Hassi, S., de Snoo, H.S.V., and Szafraniec, F. H. 2012. Infinite-dimensional perturbations, maximally nondensely defined symmetric operators, and some matrix representations.
Ismagilov, R.S., and Kostjuchenko, A.G. 2010. Asymptotics ofthe spectrum of Sturm-Liouville operator with point interaction. Funct. Analysis Appl., 44, 14–20.Google Scholar
Kac, I.S. 1963. Spectral multiplicity of a second-order differential operator and expansion in eigenfunctions. Izv. Akad Nauk. SSSR Ser. Mat, 27, 1081–1112.Google Scholar
Kochubei, A.N. 1975. On extentions of symmetric operators and symmetric binary relations. Matem. Zametki, 17, 41–48.Google Scholar
Kochubei, A.N. 1979. Symmetric operators and nonclassical spectral problems. Math. Notes, 25, 425–434.Google Scholar
Kochubei, A.N. 1989. One-dimensional point interactions, Ukrain. Math.J., 41, 1391–1395.Google Scholar
Kostenko, A.S., and Malamud, M.M. 2010. 1-D Schroädinger operators with local point interactions on a discrete set. J. Differential Equations, 249, 253–304.Google Scholar
Kreĭn, M.G. 1944. On hermitian operators with defect indices (1, 1). Dokl. Akad. Nauk SSSR, 43, 339–342.Google Scholar
Kreĭn, M.G. 1946. On resolvents of Hermitian operator with deficiency index (m, m). Dokl. Akad. Nauk SSSR, 52, 657–660.Google Scholar
Kreĭn, M.G. 1947. Theory of self-adjoint extensions of semibounded hermitian operators and applications, II. Mat. Sb., 21, 365–404.Google Scholar
Kreĭn, M.G., and Langer, H. 1971. On defect subspaces and generalized resolvents of Hermitian operator in Pontryagin space. Funkts. Anal. i Prilozhen, 5, 59–71;Google Scholar
Kreĭn, M.G., and Langer, H. 1971. On defect subspaces and generalized resolvents of Hermitian operator in Pontryagin space. Funkts. Anal. i Prilozhen, 5, 54–69 (Russian).Google Scholar
English translation in Funct. Anal. Appl., 5, (1971), 136–146;
English translation in Funct. Anal. Appl., 5, (1971), 217–228.
Kreĭn, M. G., and Ovcarenko, I. E. 1977. Q-functions and sc-resolvents of non-densely defined Hermitian contractions (Russian). Sibirsk. Mat. Zh., 18, 1032-1056, 1206.Google Scholar
Kreĭn, M. G., and Ovcarenko, I. E. 1978. Inverse problems for Q-functions and resolvent matrices of positive Hermitian operators (Russian). Dokl. Akad. Nauk SSSR, 242, 521–524.Google Scholar
Langer, H., and Textorius, B. 1977. On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math., 72, 135–165.Google Scholar
Lions, J.L., and E., Magenes, E. 1972. Non-homogeneous boundary value problems and applications, Vol. 1, Springer, Berlin, 1972
Lyantse, V.E., and Storozh, O.G. 1983. Methods ofthe Theory of Unbounded Operators, Nauk Dumka, Kiev (Russian).
Malamud, M.M. 1992. On the formula of generalized resolvents of a nondensely defined Hermitian operator. Ukr. Mat. Zh., 44, 1658–1688.Google Scholar
Malamud, M.M. 2010. Spectral theory of elliptic operators in exterior domains. Russ. J. Math. Phys., 17, 96–125Google Scholar
Malamud, M.M., and Mogilevskii, V.I. 2002. Krein type formula for canonical resolvents of dual pairs of linear relations. Meth. Funct. Anal. Topology, 8, 72–100.Google Scholar
Malamud, M.M., and Neidhardt, H. 2011. On the unitary equivalence of absolutely continuous parts of self-adjoint extensions. J. Funct. Anal., 260, 613–638.Google Scholar
Mikhailets, V.A. 1994. One-dimensional Schroädinger operator with point interactions. Dokl. Math., 335, 421–423.Google Scholar
Mirzoev, K. A., and Safonova, T. A. 2011 Singular Sturm-Liouville operators with potential distribution in space ofvector functions, Dokl Russian Academy., 441, 165–168.Google Scholar
Mogilevskiĭ, V.I. 2006. Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers. Meth. Funct. Anal. Topology, 12, 258–280.Google Scholar
Mogilevskiĭ, V.I. 2009. Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices. Meth. Funct. Anal. Topology, 15, 280–300.Google Scholar
Mogilevskiĭ, V.I. 2010. Description of generalized resolvents and characteristic matrices of differential operators by means of a boundary parameter. Mat. Zametki, 90, 558–583.Google Scholar
Naĭmark, M.A. 1943. On spectral functions of a symmetric operator. Izv. Akad. Nauk SSSR, Ser. Matem., 7, 285–296.Google Scholar
Naĭmark, M.A. 1969. Linear Differential Operators, Nauka, Moscow (Russian). English translation by F. Ungar Pub. Co., New York, 1967.
von Neumann, J. 1932. Uäber adjungierte Operatoren. Ann. Math., 33, 294–310.Google Scholar
Phillips, R.S. 1959. Dissipative operators and hyperbolic systems of partial differential equations. Trans. Amer. Math. Soc., 90, 193–254.Google Scholar
Phillips, R.S. 1961. The extension of dual subspaces invariant under an algebra. Proc. Internat. Symp. Linear Spaces, Jerusalem 1960, Academic Press, 363–398.
Posilicano, A. 2001. A Krein-like formula for singular perturbations of selfadjoint operators and applications. J. Funct. Anal., 183, 109–147.Google Scholar
Posilicano, A. 2008. Self-adjoint extensions of restrictions. Oper. Matrices, 2, 483–506.Google Scholar
Rofe-Beketov, F.S. 1969. On selfadjoint extensions of differential operators in a space of vector-functions. Teor. Funkts., Funkts. Anal. i Prilozhen, 8, 3–24.Google Scholar
Ryzhov, V. 2009. Spectral boundary value problems and their linear operators. arXiv:0904.0276v1, (2009), 1–38.
Saakyan, Sh. N. 1965. Theory of resolvents of a symmetric operator with infinite defect numbers. Akad. Nauk Armjan. SSR Dokl., 41, 193–198 (Russian).Google Scholar
Shmuljan, Yu.L. 1976. Theory of linear relations, and spaces with indefinite metric. Funkcional. Anal. i Priložen, 10, 67–72 (Russian).Google Scholar
Sorjonen, P. 1980. Extensions of isometric and symmetric linear relations in a Krein space. Ann. Acad. Sci. Fenn. Ser. A I Math., 5, 355–375.Google Scholar
Stone, M.H. 1932. Linear transformations in Hilbert space and their applications to analysis. Amer. Math. Soc. Colloquium Publ., 15, New York.Google Scholar
Štrauss, A.V. 1954. Generalized resolvents of symmetric operators. Izv. Akad. Nauk. SSSR, Ser. Mat., 18, 51–86 (Russian).Google Scholar
English translation in Math. USSR- Izvestija, 4, (1970), 179–208.
Triebel, H. 1978. Interpolation Theory, Function Spaces, Differential Operators, Berlin.
Višik, M. I. 1952. On general boundary problems for elliptic differential equations. Trudy Moskov. Mat. Obžsc., 1, 187–246 (Russian).Google Scholar

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