Abstract
Here an introduction to linear spaces with an indefinite inner product, the so-called Kreĭn spaces, and to multi-valued operators between them is given. More precisely, the basic properties of these indefinite inner product spaces, which are generalizations of Hilbert spaces, are described, where the similarities to and differences compared with Hilbert spaces are emphasized. Secondly, the basic properties of multi-valued operators, which are generalizations of (linear) operators, are presented; such operators appear naturally, for instance, in the treatment of differential equations.
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References
Arens, R.: Operational calculus of linear relations. Pac. 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 Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol. 280, pp. 212–218. Springer, Berlin (1972)
Bognár, J.: Indefinite Inner Product Spaces. Springer, Berlin (1974)
Cross, R.: Multi-valued Linear Operators. Marcel Dekker, New York (1998)
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)
Hassi, S., Sebestyén, Z., de Snoo, H.S.V.: Lebesgue type decompositions for nonnegative forms. J. Funct. Anal. 257, 3858–3894 (2006)
Iokhvidov, I.S., Kreĭn, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Akademie, Berlin (1982)
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1966)
Pontryagin, L.S.: Hermitian operator in spaces with indefinite metric. Izvestiya Akad. Nauk USSR Ser. Matem 8, 243–280 (1944) (In Russian)
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Wietsma, H. (2015). Multi-valued Operators/Linear Relations Between Kreĭn Spaces. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_39
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DOI: https://doi.org/10.1007/978-3-0348-0667-1_39
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Print ISBN: 978-3-0348-0666-4
Online ISBN: 978-3-0348-0667-1
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