On the Loewner problem in the class 𝐍_{𝜅}

Alpay, D.; Dijksma, A.; Langer, H.

doi:10.1090/S0002-9939-01-06345-6

On the Loewner problem in the class $\mathbf {N}_{\kappa }$
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Proc. Amer. Math. Soc. 130 (2002), 2057-2066 Request permission

Abstract:

Loewner’s theorem on boundary interpolation of $\mathbf {N}_{\kappa }$ functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function $f$, which is to be extended to an $\mathbf {N}_{\kappa }$ function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which $f$ is defined contains an accumulation point at which $f$ satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of $\mathbf {N}_{\kappa }$ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

References

D. Alpay, V. Bolotnikov, A. Dijksma, and J. Rovnyak, Some extensions of Loewner’s theory of monotone operator functions, to appear in J. Funct. Anal.
Daniel Alpay, Aad Dijksma, and Heinz Langer, Classical Nevanlinna-Pick interpolation with real interpolation points, Operator theory and interpolation (Bloomington, IN, 1996) Oper. Theory Adv. Appl., vol. 115, Birkhäuser, Basel, 2000, pp. 1–50. MR 1766806
Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432, DOI 10.1007/978-3-0348-8908-7
D. Alpay and J. Rovnyak, Loewner’s theorem for kernels having a finite number of negative squares, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1109–1117. MR 1473653, DOI 10.1090/S0002-9939-99-04618-3
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556
Aad Dijksma and Heinz Langer, Notes on a Nevanlinna-Pick interpolation problem for generalized Nevanlinna functions, Topics in interpolation theory (Leipzig, 1994) Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 69–91. MR 1473251, DOI 10.1007/bf03398509
A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, A factorization result for generalized Nevanlinna functions of the class $\scr N_\kappa$, Integral Equations Operator Theory 36 (2000), no. 1, 121–125. MR 1736921, DOI 10.1007/BF01236290
Aad Dijksma, Heinz Langer, and Henk de Snoo, Hamiltonian systems with eigenvalue depending boundary conditions, Contributions to operator theory and its applications (Mesa, AZ, 1987) Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 37–83. MR 1017665
Aad Dijksma, Heinz Langer, and Henk de Snoo, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr. 161 (1993), 107–154. MR 1251013, DOI 10.1002/mana.19931610110
A. Dijksma and H. S. V. de Snoo, Symmetric and selfadjoint relations in Kreĭn spaces. I, Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985) Oper. Theory Adv. Appl., vol. 24, Birkhäuser, Basel, 1987, pp. 145–166. MR 903069
I. S. Iohvidov, M. G. Kreĭn, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Mathematical Research, vol. 9, Akademie-Verlag, Berlin, 1982. MR 691137
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
M. G. Kreĭn and H. Langer, Über die $Q$-Funktion eines $\pi$-hermiteschen Operators im Raume $\Pi _{\kappa }$, Acta Sci. Math. (Szeged) 34 (1973), 191–230 (German). MR 318958
M. G. Kreĭn and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. MR 461188, DOI 10.1002/mana.19770770116
M. G. Kreĭn and Heinz Langer, On some extension problems which are closely connected with the theory of Hermitian operators in a space $\Pi _{\kappa }$. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I, Beiträge Anal. 14 (1979), 25–40 (loose errata). MR 563344
M. G. Kreĭn and H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space $\Pi _{\kappa }$, Acta Sci. Math. (Szeged) 43 (1981), no. 1-2, 181–205. MR 621369
Heinz Langer, A characterization of generalized zeros of negative type of functions of the class $N_\kappa$, Advances in invariant subspaces and other results of operator theory (Timişoara and Herculane, 1984) Oper. Theory Adv. Appl., vol. 17, Birkhäuser, Basel, 1986, pp. 201–212. MR 901070, DOI 10.1007/978-3-0348-7698-8_{1}5
K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.
H. Langer and A. Schneider, On spectral properties of regular quasidefinite pencils $F-\lambda G$, Results Math. 19 (1991), no. 1-2, 89–109. MR 1091959, DOI 10.1007/BF03322419
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Béla Sz.-Nagy and Adam Korányi, Operatortheoretische Behandlung und Verallgemeinerung eines Problemkreises in der komplexen Funktionentheorie, Acta Math. 100 (1958), 171–202 (German). MR 130577, DOI 10.1007/BF02559538