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Loewner's theorem for kernels
having a finite number of negative squares


Authors: D. Alpay and J. Rovnyak
Journal: Proc. Amer. Math. Soc. 127 (1999), 1109-1117
MSC (1991): Primary 30E05, 47A57; Secondary 46C20, 47B50
DOI: https://doi.org/10.1090/S0002-9939-99-04618-3
MathSciNet review: 1473653
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Abstract | References | Similar Articles | Additional Information

Abstract: By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.


References [Enhancements On Off] (What's this?)

  • 1. D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Reproducing kernel Pontryagin spaces, Holomorphic Spaces (S. Axler, J. McCarthy, and D. Sarason, eds.), MSRI Publications, vol. 33, Cambridge University Press, New York, 1998, pp. 425-444.
  • 2. -, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory: Adv. Appl., Vol. 96, Birkhäuser, Basel, 1997. CMP 97:17
  • 3. D. Alpay and H. Dym, On a new class of reproducing kernel spaces and a new generalization of the Iohvidov laws, Linear Algebra Appl. 178 (1993), 109-183. MR 94g:46034
  • 4. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 14:479c
  • 5. L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), Wiley, New York, 1966, pp. 295-392. MR 39:6109
  • 6. L. E. Dickson, On the rank of a symmetrical matrix, Annals of Math. 15 (1913), 27-28.
  • 7. W. F. Donoghue, Jr., Another extension of Loewner's theorem, J. Math. Anal. Appl. 110 (1985), 323-326. MR 87b:30058
  • 8. R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 87e:15001
  • 9. A. Korányi, On a theorem of Löwner and its connections with resolvents of selfadjoint transformations, Acta Sci. Math. Szeged 17 (1956), 63-70. MR 18:588c
  • 10. M. G. Kre[??]in and H. Langer, Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume ${\Pi}_{\kappa }$, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, Amsterdam, 1972, pp. 353-399. Colloq. Math. Soc. János Bolyai, 5. MR 54:11103
  • 11. K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216.
  • 12. M. Rosenblum and J. Rovnyak, Restrictions of analytic functions. I, Proc. Amer. Math. Soc. 48 (1975), 113-119. MR 53:3764a
  • 13. -, Hardy classes and operator theory, Oxford Mathematical Monographs, Oxford University Press, New York, 1985, Dover reprint, 1997. MR 87e:47001; MR 97j:47002
  • 14. L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math. 13 (1964), 115-256. MR 31:3835
  • 15. P. Sorjonen, Pontrjaginräume mit einem reproduzierenden Kern, Ann. Acad. Sci. Fenn. Ser. A I Math. (1975), no. 594, 30 pp. MR 53:8875

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Additional Information

D. Alpay
Affiliation: Department of Mathematics Ben-Gurion University of the Negev P. O. Box 653 84105 Beer-Sheva, Israel
Email: dany@math.bgu.ac.il

J. Rovnyak
Affiliation: Department of Mathematics University of Virginia Charlottesville, Virginia 22903-3199
Email: rovnyak@Virginia.EDU

DOI: https://doi.org/10.1090/S0002-9939-99-04618-3
Keywords: Loewner, L\"owner, Pontryagin space, reproducing kernel, negative squares, Pick, Schur, Nevanlinna.
Received by editor(s): July 25, 1997
Additional Notes: The second author was supported by the National Science Foundation under DMS–9501304.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

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