Abstract
Here the basic properties of two classes of (multi-valued) operators between Kreĭn spaces are presented: the symmetric and isometric relations. Both types of multi-valued operators (relations) naturally appear for instance when studying differential equation; for example Sturm–Liouville equations with an indefinite weight.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arens, R.: Operational calculus of linear relations. Pacific J. Math. 11, 9–23 (1961)
Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Spaces with an Indefinite Metric. Wiley, Chichester (1989)
Bennewitz, C.: Symmetric relations on a Hilbert space. In: Conference on the Theory of Orinary and Partial Differential Equations. Lecture Notes in Mathematics, vol. 280, pp. 212–218. Springer, Berlin (1972)
Calkin, J.W.: Abstract symmetric boundary conditions. Trans. Am. Math. Soc. 45, 369–442 (1939)
Calkin, J.W.: General self-adjoint boundary conditions for certain partial differential operators. Proc. N.A.S. 25, 201–206 (1939)
Ćurgus, B., Langer, H.: A Kreĭn space approach to symmetric ordinary differential operators with an indefinite weight function. J. Diff. Equ. 79, 31–61 (1989)
Daho, K., Langer, H.: Sturm–Liouville operators with an indefinite weight function. Proc. Roy. Soc. Edinburgh Sect. A 78, 161–191 (1977)
Derkach, V.: Boundary triplets, weyl functions, and the Kre\(\breve{\imath }\) n formula. In: Alpay, D. (ed.) Operator Theory, chapter 10, pp. 183–218, Springer, Basel (2015). doi: 10.1007/978-3-0348-0692-3_32
Derkach, V.A., Hassi, S., Malamud, M.M., de Snoo, H.S.V.: Boundary relations and their Weyl families. Trans. Am. Math. Soc. 358, 5351–5400 (2006)
Derkach, V.A., Malamud, M.M.: The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73, 141–242 (1995)
Dijksma, A., Langer, H., de Snoo, H.S.V.: Selfadjoint \(\Pi _{\kappa }\)-extensions of symmetric subspaces: an abstract approach to boundary problems with spectral parameter in the boundary condition. Integr. Equ. Oper. Theory 7, 459–515 (1984)
Dijksma, A., Langer, H., de Snoo, H.S.V.: Unitary colligations in \(\Pi _{\kappa }\)-spaces, characteristic functions and Štraus extensions. Pacific J. Math. 125, 347–362 (1986)
Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Kreĭn spaces I. Oper. Theory Adv. Appl. 24, 145–166 (1987)
Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Kreĭn spaces II. Ann. Acad. Sci. Fenn. Ser. A I 12, 199–216 (1987)
Fillmore, P.A., Williams, J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Gheondea, A.: Canonical forms of unbounded unitary operators in Kreĭn spaces. Publ. Res. Inst. Math. Sci. 24, 205–224 (1988)
Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. 2nd edn. Kluwer Academic Publishers Group, Dordrecht (1991)
Hassi, S., Wietsma, H.L.: On Calkin’s abstract symmetric boundary conditions. Lond. Math. Soc. Lect. Note Ser. 404, 3–34 (2012)
Iokhvidov, I.S.: Unitary operators in a space with an indefinite metric. N.I.I. Mat. i Mekh. Khar’kov Gas. Univ. Mat. Obsch 21, 79–86 (1949, in Russian)
Iokhvidov, I.S.: On the spectra of Hermitian and unitary operators in a space with an indefinite metric. Doklady Akad. Nauk USSR 71, 1950 (1950, in Russian)
Iokhvidov, I.S., Kreĭn, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Akademie, Berlin (1982)
Kreĭn, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Upsekhi Mat. Nauk 3, 3–93 (1948, in Russian). [English translation in Am. Math. Soc. Trans. 26 (1950)]
Langer, H.: Zur Spektraltheorie verallgemeinerter gewöhnlicher Differentialoperatoren zweiter Ordnung mit einer nichtmonotonen Gewichtsfunktion. Ber. Univ. Jyväskylä Math. Inst. Ber. 14, 58 pp. (1972)
Langer, H.: Spectral Functions of Definitizable Operators in Kreĭn Spaces. Lecture Notes in Mathematics, vol. 948. Springer, Berlin (1982)
Nakagami, Y.: Spectral analysis in Kreĭn spaces. Publ. Res. Inst. Math. Sci. 24, 361–378 (1988)
Pontryagin, L.S.: Hermitian operator in spaces with indefinite metric. Izvestiya Akad. Nauk USSR, Ser. Matem 8, 243–280 (1944, in Russian)
Shmul’jan, Yu.L.: Theory of linear relations, and spaces with indefinite metric. Funkcional. Anal. i Priložen 10, 67–72 (1976, in Russian)
Sorjonen, P.: On linear relations in an indefinite inner product space. Ann. Acad. Sci. Fenn. Ser. A I 4, 169–192 (1978/1979)
Sorjonen, P.: Extensions of isometric and symmetric linear relations in a Kreĭn space. Ann. Acad. Sci. Fenn. Ser. A I 5, 355–375 (1980)
Wietsma, H.L.: Representations of unitary relations between Kreĭn spaces. Integr. Equ. Oper. Theory 72, 309–344 (2012)
Trunk, C.: Locally definitizable operators: the local structure of the spectrum. In: Alpay, D. (ed.) Operator Theory, chapter 12, pp. 241–260, Springer, Basel (2015). doi: 10.1007/978-3-0348-0692-3_38
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Basel
About this entry
Cite this entry
Wietsma, H.L. (2015). Symmetric and Isometric Relations. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_42
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0667-1_42
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0666-4
Online ISBN: 978-3-0348-0667-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering