Abstract
We present the solution to the Phillips–Kato restricted extension problem about description and parametrization of the domains of all maximal accretive and sectorial quasi-self-adjoint extensions \({\widetilde S (S\subset \widetilde S\subset S^*)}\) of a closed, densely defined nonnegative operator S in some Hilbert space. This description and parametrization are presented in terms of some sort of an analogy of von Neumann’s formulas for quasi-self-adjoint extensions. We use the approach proposed by Arlinskiĭ and Tsekanovskiĭ (Integr Equ Oper Theory 51:319–356, 2005) and our new formulas match the corresponding ones in the case of nonnegative self-adjoint extensions of S. An application to operators corresponding to finite number δ′-interactions on the real line is given as well as to the parametrization of all resolvents of maximal accretive extensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. In: Texts and Monographs in Physics. Springer, Berlin-New York (1988)
Ando T., Nishio K.: Positive self-adjoint extensions of positive symmetric operators. Tohóku Math. J. 22, 65–75 (1970)
Arlinskiĭ, Yu.M.: A class of contractions in a Hilbert space. Ukrain. Math. J. 39(6), 691–696 (1987, in Russian). English translation in Ukr. Math. J. 39(6), 560–564 (1987)
Arlinskiĭ, Yu.M., Positive spaces of boundary values and sectorial extensions of nonnegative symmetric operators. Ukrain. Math. Zh. 40(1), 8–14 (1988, in Russian). English translation in Ukr. Math. J. 40(1), 5–10 (1988)
Arlinskiĭ Yu.M.: Characteristic functions of operators of the class C(α). Izv. Vyssh. Uchebn. Zaved. Mat. N.2, 13–21 (1991)
Arlinskiĭ Yu.M.: On proper accretive extensions of positive linear relations. Ukr. Math. J. 47(6), 723–730 (1995)
Arlinskiĭ, Yu.: Maximal sectorial extensions and closed forms associated with them. Ukr. Math. J. 48(6), 723–739 (1996, in Russian). English translation in Ukr. Math. J. 48(6), 809–827 (1996)
Arlinskiĭ Yu.: Extremal extensions of sectorial linear relations. Matematychnii Studii 7(1), 81–96 (1997)
Arlinskiĭ Y., Hassi S., Sebestyen Z., de Snoo H.: On the class of extremal extensions of a nonnegative operators. Oper. Theory: Adv. Appl. 127, 41–81 (2001)
Arlinskiĭ, Yu., Kovalev, Y., Tsekanovskiĭ, E.: Quasi-self-adjoint maximal accretive extensions of nonnegative symmetric operators. TEKA Kom. Motor. i Energ. Roln.–OL PAN, XA, 6–14 (2010)
Arlinskiĭ Yu.M., Tsekanovskiĭ E.R.: Non-self-adjoint contractive extensions of Hermitian contractions and M.G. Kreĭn’s theorems. Uspekhi mat. nauk 37(1), 131–132 (1982)
Arlinskiĭ, Yu.M., Tsekanovskiĭ, E.R.: Generalized resolvents of quasi-self-adjoint contracting extensions of a Hermitian contraction. Ukr. Mat. Zh. 35(5) (1983)
Arlinskiĭ Yu., Tsekanovskiĭ E.: On sectorial extensions of positive hermitian operators and their resolvents. Dokl. Akad. Nauk Armenian SSR 5, 199–202 (1984)
Arlinskiĭ, Yu.M., Tsekanovskiĭ, E.R.: Quasi-self-adjoint contractive extensions of Hermitian contractions. Teor. Funkts., Funkts. Anal. Prilozhen 50, 9–16 (1988, in Russian). English translation in J. Math. Sci. 49(6), 1241–1247 (1990)
Arlinskiĭ, Yu., Tsekanovskiĭ, E.: On the theory of non-negative self-adjoint extensions of a non-negative symmetric operator. Report of National Academy of Scinces of Ukraine, vol. 11, pp. 30–37 (2002)
Arlinskiĭ Yu., Tsekanovskiĭ E.: On von Neumann’s problem in extension theory of nonnegative operators. Proc. AMS 131(10), 3143–3154 (2003)
Arlinskiĭ Yu., Tsekanovskiĭ E.: The von Neumann problem for nonnegative symmetric operators. Integr. Equ. Oper. Theory 51, 319–356 (2005)
Arlinskiĭ Yu., Tsekanovskiĭ E.: Krein’s research on semi-bounded operators, its contemporary developments, and applications. Oper. Theory: Adv. Appl. 190, 65–112 (2009)
Arsene Gr., Geondea A.: Completing matrix contractions. J. Oper. Theory 7, 179–189 (1982)
Coddington E.A., de Snoo H.S.V.: Positive self-adjoint extensions of positive symmetric subspaces. Math. Z. 159, 203–214 (1978)
Crandall M.G.: Norm preserving extensions of linear transformations in Hilbert space. Proc. Am. Math. Soc. 21, 335–340 (1969)
Davis Ch., Kahan W.M., Weinberger H.F.: Norm preserving dilations and their applications to optimal error bounds. SIAM J. Numer. Anal. 19(3), 445–469 (1982)
Derkach, V.A., Malamud, M.M.: Weyl function of Hermitian operator andits connection with the characteristic function. Preprint 85-9, Fiz.-Tekhn. Inst. Akad. Nauk Ukraine, p. 50 (1985, in Russian)
Derkach V.A., Malamud M.M.: Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95(1), 1–95 (1991)
Derkach V.A., Malamud M.M.: The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73(2), 141–242 (1995)
Derkach, V.A., Malamud, M.M., Tsekanovskiĭ, E.R.: Sectorial extensions of positive operators and characteristic functions. Ukr. Math. J. 41(2), 151–158 (1989, in Russian). English tranlation in Ukr. Math. J. 41(2), 136–142 (1989)
Douglas R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966)
Evans W.D., Knowles I.I.: On the extensions problem for accretive differential operators. J. Funct. Anal. 63(3), 276–298 (1985)
Evans W.D., Knowles I.: On the extension problem for singular accretive differential operators. J. Differ. Equ. 63, 264–288 (1986)
Fillmore P.A., Williams J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Gesztesy F., Kalton N., Makarov K., Tsekanovskiĭ E.: Some aplications of operator-valued Herglotz functions. Oper. Theory, Adv. Appl. 123, 271–321 (2001)
Gesztesy F., Tsekanovskiĭ E.: On matrix-valued Herglotz functions. Math. Nachr. 218, 61–138 (2000)
Gorbachuk, M.L., Gorbachuk, V.I.: Boundary value problems for differential-operator equations. Naukova Dumka Kiev (1984, in Russian)
Gorbachuk V.I., Gorbachuk M.L., Kochubeĭ A.N.: Extension theory of symmetric operators and boundary value problems. Ukr. Mat. Z. 41(10), 1298–1313 (1989)
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg (1995)
Kochubei, A.N.: Extensions of a positive definite symmetric operator. Dokl. Akad. Nauk Ukrain. SSR, Ser. A, 3, 168–171 (1979, in Russian)
Kreĭn, M.G.: The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications I. Mat. Sbornik 20(3), 431–495 (1947, in Russian)
Kreĭn, M.G.: The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications, II. Mat. Sbornik 21(3), 365–404 (1947, in Russian)
Kreĭn M.G., Langer H.: Über die Q-function eines Π-Hermiteschen operators im raum Πκ. Acta Sci. Math. Szeged 34, 191–230 (1973)
Kreĭn, M.G., Langer, H.: On defect subspaces and generalized resolvents of Hermitian operator in the space Πκ. Fuct. Anal. Appl. 5(3), 54–69 (1971, in Russian)
Kreĭn, M.G., Ovčarenko, I.E.: On Q-functions and sc-resolvents of nondensely defined Hermitian contractions. Siberian Math. J. 18, 728–746 (1977, in Russian)
Malamud, M.M.: On some classes of Hermitian operators with gaps. Ukr. Mat. J. 44(2), 215–234 (1992, in Russian)
Mikhailets, V.A.: Solvable and sectorial boundary value problems for the operator Sturm-Liouville equation. Ukr. Math. Zh. 26, 450–459 (1974, in Russian)
Mil’yo, O.Ya., Storozh, O.G.: On the general form of a maximally accretive extension of a positive-definite operator. Dokl Akad. Nauk Ukraine 6, 19–22 (1991, in Russian)
Phillips R.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc. 90, 192–254 (1959)
Phillips, R.: On dissipative operators. In: Lectures in Differential Equations, vol. II, pp. 65–113. Van Nostrand-Reinhold, New York (1965)
Rofe-Beketov, F.S.: Numerical range of a linear relation and maximal relations. Theory of functions. Funct. Anal. Appl. 44, 103–112 (1985, in Russian)
Sz.-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North-Holland, New York (1970)
Shmul’yan Yu.L., Yanovskaya R.N.: Blocks of a contractive operator matrix. Izv. Vyssh. Uchebn. Zaved. Mat. 7, 72–75 (1981)
Tsekanovskiĭ, E.: Non-self-adjoint accretive extensions of positive operators and theorems of Friedrichs-Kreĭn-Phillips. Funk. Anal. i Prilozhen. 14(2), 87–89 (1980, in Russian)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Guest Editors L. Littlejohn and J. Stochel.
Dedicated to Franek Szafraniec on the occasion of his 70th birthday anniversary.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Arlinskiĭ, Y., Kovalev, Y. & Tsekanovskiĭ, E. Accretive and Sectorial Extensions of Nonnegative Symmetric Operators. Complex Anal. Oper. Theory 6, 677–718 (2012). https://doi.org/10.1007/s11785-011-0169-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-011-0169-7
Keywords
- Symmetric operator
- Quasi-self-adjoint extensions
- Friedrichs extension
- Kreĭn-von Neumann extension
- m-accretive operator
- m-sectorial operator