Abstract
We deal with the Kreîn-Langer problem for\(\mathcal{L}(\mathcal{H})\)-valued functions on the band (−2a, 2a)×Γ, where\(\mathcal{L}(\mathcal{H})\) is the algebra of continuous linear operators on a Hilbert space\(\mathcal{H}\),a a finite positive number and Γ a topological Abelian group. We show that every weakly continuous κ-indefinite function\(f:( - 2a,2a) \times \Gamma \to \mathcal{L}(\mathcal{H})\) admits a strongly continuous κ-indefinite continuation to ℝ × Γ with the same indefiniteness index κ. We give a parametrization of the extensions in terms of operator-valued Schur functions.
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References
D. Alpay, A. Dijksma, J. Rovnyak andH. de Snoo, “Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces,” Operator Theory: Adv. Appl., Vol. 96, Birkhäuser, Basel, 1997.
T. Andô, “Linear Operators on Krein Spaces,” Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979.
T. Ya. Azizov andI.S. Iokhvidov, “Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric,” Nauka, Moscow, 1986; “Linear Operators in Spaces with Indefinite Metric”, Wiley, New York, 1989 [English transl.].
J. Bognár, “Indefinite Inner Product Spaces,” Springer-Verlag, Berlin, 1974.
R. Bruzual andS.A.M. Marcantognini, Local semigroups of isometries in Πκ-spaces and related continuation problems for κ-indefinite Toeplitz kernels,Integral Equations and Operator Theory 15 (1992), 527–550.
A. Dijksma, H. Langer andH.S.V. de Snoo, Generalized coresolvents of standard isometric operators and generalized resolvents of standard symmetric relations in Krein spaces,Operator Theory: Adv. Appl. 48 (1990), Birkhäuser, Basel, 261–274.
M.A. Dritschel andJ. Rovnyak, Extension theorems for contraction operators on Kreîn spaces,Operator Theory: Adv. Appl. 47 (1990) Birkhäuser, Basel, 221–305.
—, “Operators on Indefinite Inner Product Spaces,” Lectures on Operator Theory and its Applications, Fields Institute Monographs, Vol. 3, AMS, Providence, RI, 1996.
J. Friedrich, Nevanlinna parametrization of extensions of positive definite operator-valued functions,Rev. Roumaine Math. Pures Appl. 35, No. 3 (1990). 235–247.
J. Friedrich andL. Klotz, On extensions of positive definite operatorvalued functions,Rep. Math. Phys. 26, No. 1 (1988), 45–65.
M. Grossmann andH. Langer, Über indexerhaltende Erweiterungen eines hermiteschen operators in Pontrjaginraum,Math. Nachrichten 64 (1974) 289–317.
I.S. Iokhvidov, M.G. Kreîn andH. Langer, “Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric,” Akademie-Verlag, Berlin, 1982.
M.G. Kreîn andH. Langer, On some continuation problems which are closely related to the theory of operators in spaces Πκ. IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions,J. Operator Theory 13 (1985), 299–417.
S.A.M. Marcantognini andM.D. Morán, Commuting unitary Hilbert spaces extensions of a pair of Kreîn space isometries and the Kreîn-Langer problem in the band,Journal of Functional Analysis 138 (1996) 379–409.
M.D. Morán, Unitary extensions of a system of commuting isometric operators,Operator Theory: Adv. Appl. 61 (1993), 163–169.
M.A. Naîmark, An analogue of Stone's Theorem in a space with an indefinite metric,Soviet Math. Dokl. 7, No.5 (1966), 1366–1367.
Z. Sasvári, “Positive definite and definitizable functions,” Akademie-Verlag, Berlin, 1994.
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Bruzual, R., Marcantognini, S.A.M. The Kreîn-Langer problem for Hilbert space operator valued functions on the band. Integr equ oper theory 34, 396–413 (1999). https://doi.org/10.1007/BF01272882
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DOI: https://doi.org/10.1007/BF01272882