Abstract
We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.
Similar content being viewed by others
References
[ABDS1]D.Alpay, P.Bruinsma, A.Dijksma, H. de Snoo:Interpolation problems, extensions of symmetric operators and reproducing kernel spaces I, Oper. Theory Adv. Appl. 50 (1991), 35–82, Birkhäuser Verlag, Basel.
[ABDS2]D.Alpay, P.Bruinsma, A.Dijksma, H. de Snoo:Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II, Integral Equations Operator Theory 14 (1991), 465–500.
[ADSR1]D.Alpay, A.Dijksma, H.de Snoo, J. Rovnyak:Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl. 96, Birkhäuser Verlag, Basel 1997.
[ADSR2]D.Alpay, A.Dijksma, H.de Snoo, J.Rovnyak:Reproducing kernel Pontryagin spaces, preprint.
[ADSR3]D.Alpay, A.Dijksma, H.de Snoo, J.Rovnyak:Complementation, realization and factorization in reproducing kernel Pontryagin spaces, preprint.
[Be]Yu.M.Berezansky:Expansions in eigenfunctions of selfadjoint operators, Amer. Math. Soc. Transl. 17, Providence, Rhode Island 1968.
[Bo]R. Boas:Entire functions, Academic Press, New York 1954.
[B]J.Bognar:Indefinite inner product spaces, Springer Verlag, Berlin 1974.
[dB1]L.de Branges:Some mean squares of entire functions, Proc. Amer. Math. Soc. 10 (1959), 833–839.
[dB2]L.de Branges:Some Hilbert spaces of entire functions, Proc. Amer. Math. Soc. 10 (1959), 840–846.
[dB3]L.de Branges:Some Hilbert spaces of entire functions, Trans. Amer. Math. Soc. 96 (1960), 259–295.
[dB4]L.de Branges:Some Hilbert spaces of entire functions II, Trans. Amer. Math. Soc. 99 (1961), 118–152.
[dB5]L.de Branges:Some Hilbert spaces of entire functions III, Trans. Amer. Math. Soc. 100 (1961), 73–115.
[dB6]L.de Branges:Some Hilbert spaces of entire functions IV, Trans. Amer. Math. Soc. 105 (1962), 43–83
[dB7]L.de Branges:Hilbert spaces of entire functions, Prentice-Hall, London 1968.
[dB8]L.de Branges:Complementation in Krein spaces, Trans. Amer. Math. Soc. 305 (1988), 277–291.
[Br]P.Bruinsma:Interpolation problems for Schur and Nevanlinna pairs, Doctoral Dissertation, University of Groningen 1991.
[DS1]A. Dijksma, H.de Snoo:Symmetric and selfadjoint relations in Krein spaces I, Oper. Theory Adv. Appl. 24 (1987), 145–166, Birkhäuser Verlag, Basel.
[DS2]A.Dijksma, H.de Snoo:Symmetric and selfadjoint relations in Krein spaces II, Ann. Acad. Sci. Fenn. Ser. A I 12 (1987), 199–216.
[DLS]A.Dijksma, H.Langer, H.de Snoo:Generalized coresolvents of standard isometric operators and generalized resolvents of standard symmetric relations in Krein spaces, Oper. Theory Adv. Appl. 48 (1990), 261–274, Birkhäuser Verlag, Basel.
[DK]H.Dym, H.McKean:Gaussian processes, function theory, and the inverse spectral problem, Academic Press, New York 1976.
[GG]M.L.Gorbachuk, V.I.Gorbachuk:M.G.Krein's lectures on entire operators, Oper. Theory Adv. Appl. 97, Birkhäuser Verlag, Basel 1997.
[HSW]S.Hassi, H.de Snoo, H.Woracek:Some interpolation problems of Nevanlinna-Pick type. The Krein-Langer method, to appear in Oper. Theory Adv. Appl.
[H]W.K.Hayman:Meromorphic functions, Oxford Mathematical Monographs, 1964.
[IKL]I.S.Iohvidov, M.G.Krein, H.Langer:Introduction to the spectral theory of operators in spaces with an indefinite metric, Akademie Verlag, Berlin 1982.
[KW]M.Kaltenbäck, H. Woracek:Generalized resolvent matrices and spaces of analytic functions, Integral Equations Operator Theory.
[K1]M.G.Krein:On a continuation problem for positive definite functions, Dokl. Akad. Nauk SSSR 26 (1940), 17–21.
[K2]M.G.Krein:On Hermitian operators with defect numbers one, Dokl. Akad. Nauk SSSR 43 (1944), 339–342.
[K3]M.G.Krein:One remarkable class of Hermitian operators, Dokl. Akad. Nauk SSSR 44 (1944), 191–195.
[K4]M.G.Krein:On a continuation problem for spiral arcs in a Hilbert space, Dokl. Akad. Nauk SSSR 45 (1944), 147–150.
[K5]M.G.Krein:A contribution to the theory of entire functions of exponential type, Izv. Akad. Nauk SSSR 11 (1947), 309–326.
[K6]M.G.Krein:The principle aspects of the theory of representations of Hermitian operators whose defect index is (m, m), Ukrain. Mat. Zh. 1 (1949), 3–66.
[KL1]M.G.Krein, H.Langer:Über die Q-Funktion eines Π-hermiteschen Operators im Raume Πκ, Acta Sci. Math. (Szeged) 34 (1973), 191–230.
[KL2]M.G.Krein, H.Langer:Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.
[KL3]M.G.Krein, H.Langer:Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren, J. Funct. Anal. 30 (1978), 390–447.
[KL4]M.G.Krein, H.Langer:On some continuation problems which are closely related to the theory of operators in spaces Π κ.IV., J. Operator Theory 13 (1985), 299–417.
[KL5]M.G.Krein, H.Langer:Continuation of Hermitian positive definite functions and related questions, unpublished manuscript.
[L]H.Langer:Spectral functions of definitizable operators in Krein spaces, Lecture Notes in Math. 948 (1982), 1–46, Springer Verlag, New York.
[LTx]H.Langer, B.Textorius:On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72 (1977), 135–165.
[Le]B.Levin:Nullstellenverteilung ganzer Funktionen, Akademie Verlag, Berlin 1962.
[W]H.Winkler:Zum inversen Spektralproblem für zweidimensionale kanonische Systeme, Doctoral Dissertation, Technical University of Vienna 1993.