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3 - Boundary triplets and maximal accretive extensions of sectorial operators

Published online by Cambridge University Press:  05 November 2012

By
Y. Arlinskiĭ
Edited by
Seppo Hassi ,
Hendrik S. V. de Snoo  and
Franciszek Hugon Szafraniec
Y. Arlinskiĭ
Affiliation:
East Ukrainian National University
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 This chapter is a survey of results related to the problem of a description of all maximal accretive extensions for a densely defined sectorial operator; the problem in more generality was originally posed by R. Phillips. We also treat maximal sectorial extensions. Our approach uses the concepts of boundary pairs and boundary triplets.

Introduction

A linear operator S in a complex Hilbert space h is called accretive [Kato, 1995] if Re (Su, u) ≥ 0 for all u ∈ dom S. An accretive operator S is called maximal accretive (m-accretive) if one of the following equivalent conditions is satisfied [Kato, 1995; Lyantse, 1954; Phillips, 1959a, b, 1969]:

  • the operator S is closed and has no accretive extensions in h;

  • ρ(S) ∩ ≠ 0, where π_ denotes the open left half-plane;

  • the operator S is densely defined and closed, and S* is accretive;

  • the operator −S generates a one-parameter contractive semigroup T(t) = exp(−tS), t ≥ 0.

The resolvent set ρ(S) of an m-accretive operator contains the open left half-plane and

The class of m-accretive operators plays an essential role in differential equations, scattering theory, stochastic processes, passive linear systems and, for instance, hydrodynamics. It should be noted that Phillips calls an operator A dissipative if −A is accretive. Nowadays the term dissipative is used for operators A with Im (Af, f) ≥ 0.

The Phillips problem is to find all m-accretive extensions of a densely defined accretive operator; cf. [Phillips, 1959a, b, 1969].

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Operator Methods for Boundary Value Problems , pp. 35 - 72
Publisher: Cambridge University Press
Print publication year: 2012

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References

Ando, T., and Nishio, K. 1970. Positive selfadjoint extensions of positive symmetric operators. Tohóku Math. J., 22, 65–75.Google Scholar
Arlinskiĭ, Yu. M. 1987. A class of contractions in Hilbert space, Ukrain. Mat. Zh., 39, 691–696 (Russian). English translation in Ukrainian Math. J., 39, 560–564.Google Scholar
Arlinskiĭ, Yu. M. 1988. Positive spaces of boundary values and sectorial extensions of nonnegative operator. Ukrain. Mat. Zh., 40, 8–14 (Russian). English translation in Ukrainian Math. J., 40, 5–10.Google Scholar
Arlinskiĭ, Yu. M. 1991. Characteristic functions of operators of the class C(α). Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 13–21 (Russian).Google Scholar
Arlinskiĭ, Yu. 1992. On class of extensions of a C(α)-suboperators. Dokl. Akad. Nauk Ukr., No. 8, 12–16 (Russian).Google Scholar
Arlinskiĭ, Yu. 1995. On proper accretive extensions of positive linear relations. Ukrain. Mat. Zh., 47, 723–730.Google Scholar
Arlinskiĭ, Yu. 1996. Maximal sectorial extensions and associated with them closed forms, Ukrain. Mat. Zh., 48, 723–738 (Russian). English translation in Ukrainian Math. J., 48, 809-827.Google Scholar
Arlinskiĭ, Yu. 1997. Extremal extensions of sectorial linear relations. Mat. Stud., 7, 81–96.Google Scholar
Arlinskiĭ, Yu. 1998, On m-accretive extensions and restrictions. Methods Funct. Anal. Topology, 4, 1–26.Google Scholar
Arlinskiĭ, Yu. 1999a. On functions connected with sectorial operators and their extensions. Integral Equations Operator Theory, 33, 125–152.Google Scholar
Arlinskiĭ, Yu. 1999b. On a class of nondensely defined contractions on a Hilbert space and their extensions. J. Math. Sci. (N.Y.), 97, 4390–4419. English translation from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 50, Funktsional'nyi Analiz-5, 1997.Google Scholar
Arlinskiĭ, Yu. 1999c. Maximal accretive extensions of sectorial operators. Doctor Sciences Dissertation, Institute of Mathematics Ukraine National Academy of Sciences, 288 p. (Russian).
Arlinskiĭ, Yu. 2000a. Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Math. Nachr., 209, 5–36.Google Scholar
Arlinskiĭ, Yu. 2000b. M-accretive extensions of sectorial operators and Kreĭn spaces. Oper. Theory Adv. Appl., 118, 67–82.Google Scholar
Arlinskiĭ, Yu. 2006. Extremal extensions of a C(α)-suboperator and their representations. Oper. Theory Adv. Appl., 162, 47–69.Google Scholar
Arlinskiĭ, Yu., Hassi, S., Sebestyen, Z., and de Snoo, H.S.V. 2001. On the class of extremal extensions of a nonnegative operators. Oper. Theory Adv. Appl., 127, 41–81.Google Scholar
Arlinskiĭ, Yu., Kovalev, Yu., and Tsekanovskiĭ, E. 2012. Accretive and sectorial extensions of nonnegative symmetric operators, Complex Anal. Oper. Theory, to appear.Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 1984a. On sectorial extensions of positive hermitian operators and their resolvents. Dokl. Nats. Akad. Nauk Armen., 5, 199–202 (Russian)Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 1984b. On resolvents of m-accretive extensions of symmetric differential operator. Math. Phys. Nonlin. Mech., 1, 11–16 (Russian).Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 1988. Quasi-self-adjoint contractive extensions of Hermitian contraction. Theor. Functions, Funk. Anal. i Prilozhen., 50, 9–16 (Russian).Google Scholar
English translation in J. Math. Sci. (N. Y.), 49 (1990), 1241–1247.
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 2002. On the theory of nonnegative extensions of a nonnegative symmetric operator. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, No.11, 30–37.Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 2003. On von Neumann's problem in extension theory of nonnegative operators. Proc. Amer. Math. Soc., 131, 3143–3154.Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 2005. The von Neumann problem for nonnegative symmetric operators. Integral Equations Operator Theory, 51, 319–356.Google Scholar
Arlinskiĭ, Yu., and Tsekanovskiĭ, E. 2009. Krein's research on semi-bounded operators, its contemporary developmentsand applications. Oper. Theory Adv. Appl., 190, 65–112.Google Scholar
Arsene, G., and Gheondea, A. 1982. Completing matrix contractions. J. Operator Theory, 7, 179–189.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
Berezansky, Yu. 1968. Expansions in eigenfunction of selfadjoint operators, Transl. of Math. Monographs, 17, Amer. Math. Soc., Providence.
Birman, M. S. 1956. On the selfadjoint extensions of positive definite operators. Mat. Sb., 38, 431–450 (Russian).Google Scholar
Bruk, V. M. 1976. On a class of boundary value problems with spectral parameters in the boundary conditions. Mat. Sb., 100, 186–192 (Russian).Google Scholar
Calkin, J. W. 1939. Symmetric boundary conditions. Trans. Amer. Math. Soc., 45, 369–442.Google Scholar
Crandall, M. 1969. Norm preserving extensions of linear transformations on Hilbert spaces. Proc. Amer. Math. Soc., 21, 335–340.Google Scholar
Coddington, E. A., and de Snoo, H. S. V. 1978. Positive selfadjoint extensions of positive symmetric subspaces. Math. Z., 159, 203–214.Google Scholar
Davis, C., Kahan, W. M., and Weinberger, H. F. 1982. Norm preserving dilations and their applications to optimal error bounds. SIAM J. Numer. Anal., 19, 445–469.Google Scholar
Derkach, V., and Malamud, M. 1991. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 95, 1–95.Google Scholar
Derkach, V., and Malamud, M. 1995. The extension theory of Hermitian operators and the moment problem. J. Math. Sci. (N. Y.), 73, 141–242.Google Scholar
Derkach, V., Malamud, M., and Tsekanovskiĭ, E. 1988. Sectorial extensions of a positive operator and characteristic function. Soviet Math. Dokl., 37, 106–110.Google Scholar
Derkach, V., Malamud, M., and Tsekanovskiĭ, E. 1989. Sectorial extensions of positive operator. Ukrain. Mat. Zh., 41, 151–158 (Russian). English translation in Ukrainian Math. J., 41, 136-142.Google Scholar
Evans, W. D., and Knowles, I. 1985. On the extension problem for accretive differential operators. J. Funct. Anal., 63, 276–298.Google Scholar
Evans, W. D., and Knowles, I. 1986. On the extension problem for singular accretive differential operators. J. Differential Equations, 63, 264–288.Google Scholar
Gorbachuk, M. L., and Gorbachuk, V. I. 1984. Boundary value problems for differential-operator equations, Naukova Dumka, Kiev (Russian).
Gorbachuk, M. L., Gorbachuk, V. I., and Kochubeĭ, A. N. 1989. Extension theory of symmetric operators and boundary value problems. Ukrain.Mat. Zh., 41, 1298–1313 (Russian).Google Scholar
Grubb, G. 1968. A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup., Pisa (3), 22, 425–513.Google Scholar
Grubb, G. 1970. Les problémes aux limites généraux d'un opérateur elliptique provenant de la théorie variationnelle. Bull Sci. Math., 91, 113–157.Google Scholar
Grubb, G. 1971. On coerciveness and semiboundedness of general boundary problems. Israel J. Math., 10, 32–95.Google Scholar
Grubb, G. 1973. Weakly semibounded boundary problems and sesquilinear forms. Ann. Inst. Fourier (Grenoble), 23, 145–194.Google Scholar
Grubb, G. 1974. Properties of normal boundary problems for elliptic even-order systems. Ann. Scuola Norm. Sup., Pisa (4), 1, 1–61.Google Scholar
Grubb, G. 1983. Spectral asymptotics for the “soft“ self-adjoint extension of a symmetric elliptic differential operator. J. Operator Theory, 10, 2–20.Google Scholar
Grubb, G. 2006. Known and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc. (N.S.), 43, 227–230.Google Scholar
Hassi, S., Malamud, M. M., and de Snoo, H. S. V. 2004. On Kreĭn's extension theory of nonnegative operators. Math. Nachr., 274/275, 40–73.Google Scholar
Hassi, S., Sandovici, A., de Snoo, H. S. V., and Winkler, H. 2007 A general factorization approach to the extension theory of nonnegative operators and relations. J. Operator Theory, 58, 351–386.Google Scholar
Kato, T. 1995. Perturbation theory for linear operators, Springer-Verlag, New York.
Kochubeĭ, A. N. (1975). Extensions of symmetric operators and symmetric binary relations. Mat. Zametki, 17, 41–48 (Russian). English translation in Math. Notes, 17, 25–28.CrossRef
Kochubeĭ, A. N. 1979. On extensions of positive definite symmetric operator. Dokl. Akad. Nauk Ukr. SSR, Ser. A, 169–171 (Russian).
Kolmanovich, V., and Malamud, M., 1985. Extensions of sectorial operators and dual pairs of contractions, Manuscript No. 4428-85, Deposited at VINITI, 1–57 (Russian).
Kreĭn, M. G. 1947a. The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, I. Mat.Sb., 20, 431–495 (Russian).Google Scholar
Kreĭn, M. G. 1947b. The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, II. Mat.Sb., 21, 365–404 (Russian).Google Scholar
Kreĭn, M.G., and Ovčarenko, I. E. 1976. On generalized resolvents and resolvent matrices of positive Hermitian operators. Soviet Math. Dokl., 231, 1063–1066 (Russian).Google Scholar
Kreĭn, M. G., and Ovčarenko, I. E. 1978. Inverse problems for Q-functions and resolvents matrices of positive Hermitian operators. Soviet Math. Dokl., 242, 521–524 (Russian).Google Scholar
Lions, J.-L., and Magenes, E. 1968. Problémes aux limites non homogénes et applications, vol. 1, Editions Dunod, Paris. Russian translation in Mir, Moscow, 1971.
Lyantse, V. E. 1954. On a boundary problem for parabolic systems of differential equations with a strongly elliptic right-hand side. Mat.Sb., 35, 367–368 (Russian).Google Scholar
Lyantse, V. E., and Storozh, O. G. 1983. Methods of the theory of unbounded operators, Naukova Dumka, Kiev (Russian).
Malamud, M. 1989. On extensions of Hermitian and sectorial operators and dual pairs of contractions. Dokl. Akad. Nauk SSSR, 39, 253–254 (Russian).Google Scholar
Malamud, M. 1992. On some classes of Hermitian operators with gaps. Ukrain. Mat. Zh., 44, 215–234 (Russian).Google Scholar
Malamud, M. 2001. On some classes of extensions of sectorial operators and dual pair of contractions. Oper. Theory Adv. Appl., 124, 401–448.Google Scholar
Malamud, M. 2006. Operator holes and extensions of sectorial operators and dual pair of contractions. Math. Nachr., 279, 625–655.Google Scholar
Mikhailets, V.A. 1980. Spectral analysis of diffential operators. Sbornik Nauch. Trud., Kiev, Inst. of Math. of Ukrainian Acad. of Sci., 106–131 (Russian).
Mil'yo, O. Ya., and Storozh, O. G. 1991. On general form of maximal accretive extension of positive definite operator. Dokl. Akad. Nauk Ukrain. SSR, Ser. A, 6, 19–22 (Russian).Google Scholar
Mil'yo, O. Ya., and Storozh, O. G. 1993. Maximal accretive extensions of positive definite operator with finite defect number, Lviv University, 31 pages, Deposited in GNTB of Ukraine 28.10.93, no 2139 Uk93 (Russian).
Mil'yo, O. Ya., and Storozh, O. G. 1999a. Maximal accretiveness and maximal nonnegativeness conditions for a class of finite-dimensional perturbations of a positive definite operator. Mat. Stud., 12, 90–100.Google Scholar
Mil'yo, O. Ya., and Storozh, O. G. 1999b. Sectoriality and solvability conditions for differential-boundary operators of Sturm-Liouville type with multipoint-integral boundary conditions. J. Math. Sci. (N. Y.), 96, 2961–2965.Google Scholar
Mil'yo, O. Ya., and Storozh, O. G. 2002. Sturm-Liouville type differential boundary operator on semiaxis with twopoint integral boundary conditions. Ukrain. Mat. Zh., 54, 1480–1485 Russian). English translation in Ukrainian Math. J., 54, 1793–1801.Google Scholar
Phillips, R. 1959. Dissipative operators and hyperbolic systems of partial differential equations. Trans. Amer. Math. Soc., 90, 192–254.Google Scholar
Phillips, R. 1959. Dissipative operators and parabolic partial diferential equations. Comm. Pure Appl. Math., 12, 249–276.Google Scholar
Phillips, R. 1969. On dissipative operators. Lectures in differential Equations, 3, 65–113.Google Scholar
Pipa, H. M., and Storozh, O. G. 2006. Accretive perturbations of some proper extensions of a positive definite operator. Mat. Stud., 25, 181–190.Google Scholar
Rofe-Beketov, F. S. 1985. Numerical range of a linear relation and maximal relations. Theory of Functions, Funct. Anal. Appl., 44, 103–112 (Russian).Google Scholar
Sandovici, A. 2006. Contributions to the extension theory in Hilbert spaces, PhD Dissertation, Groningen University, 193 p.
Shmul'yan, Yu., and Yanovskaya, R. 1981. On blocks of contractive operator matrix. Izv.Vuzov Math., 72–75 (Russian).
Shtraus, A. V. 1973. On extensions of semibounded operators. Dokl. Akad. Nauk SSSR, 211, 543–546 (Russian).Google Scholar
Storozh, O. G. 1990. Extremal extensions of nonnegative operator and accretive boundary problems,. Ukrain. Mat. Zh., 42, 857–860 (Russian). English translation in Ukrainian Math. J., 42, 758–760.Google Scholar
Sz.-Nagy, B., and Foias, C. 1970. Harmonic analysis of operators on Hilbert space, North-Holland, New York.
Tsekanovskiĭ, E. 1985. Characteristic function and description of accretive and sectorial boundary value problems for ordinary differential operators. Dokl. Akad. Nauk Ukrain. SSR, Ser. A, 6, 21–24.Google Scholar
Vishik, M. I. 1952. On general boundary conditions for elliptic differential operators. Trudy Moskov. Mat. Obsc., 1, 187–246 (Russian).Google Scholar
English translation in Amer. Math. Soc. Transl. 24 (1963), 107–172.
Wei, G., and Xu, Z. 2005. An expanded Phillips theory and its application to differential operators. J. Funct. Anal., 222, 29–60.Google Scholar
Xu, Z., Wei, G., and Fergola, P. 2002. Maximal accretive realizations of regular Sturm-Liouville differential operators. J. London Math. Soc., 66, 175–197.Google Scholar

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