Entries of indefinite Nevanlinna matrices
HTML articles powered by AMS MathViewer
- by H. Woracek
- St. Petersburg Math. J. 26 (2015), 757-783
- DOI: https://doi.org/10.1090/spmj/1357
- Published electronically: July 27, 2015
- PDF | Request permission
Abstract:
In the early 1950’s, M. G. Krein characterized the entire functions that are an entry of some Nevanlinna matrix, and the pairs of entire functions that are a row of some Nevanlinna matrix. In connection with Pontryagin space versions of Krein’s theory of entire operators and de Branges’ theory of Hilbert spaces of entire functions, an indefinite analog of the Nevanlinna matrices plays a role. In the paper, the above-mentioned characterizations are extended to the indefinite situation and the geometry of the associated reproducing kernel Pontryagin spaces is investigated.References
- Damir Z. Arov and Harry Dym, On three Krein extension problems and some generalizations, Integral Equations Operator Theory 31 (1998), no. 1, 1–91. MR 1618628, DOI 10.1007/BF01203457
- Damir Z. Arov and Harry Dym, $J$-contractive matrix valued functions and related topics, Encyclopedia of Mathematics and its Applications, vol. 116, Cambridge University Press, Cambridge, 2008. MR 2474532, DOI 10.1017/CBO9780511721427
- D. Alpay, A. Dijksma, and H. Langer, Factorization of $J$-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387 (2004), 313–342. MR 2069282, DOI 10.1016/j.laa.2004.02.037
- 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
- N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
- Andrew G. Bakan, Polynomial approximation in $L_p(\textbf {R},d\mu )$. I. General results and representation theorem, Nats. Akad. Nauk Ukraïn. Īnst. Mat. Preprint 7 (1998), 45. MR 1734341
- Christian Berg, Indeterminate moment problems and the theory of entire functions, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 27–55. MR 1379118, DOI 10.1016/0377-0427(95)00099-2
- Christian Berg and Henrik L. Pedersen, On the order and type of the entire functions associated with an indeterminate Hamburger moment problem, Ark. Mat. 32 (1994), no. 1, 1–11. MR 1277917, DOI 10.1007/BF02559520
- Christian Berg and Henrik L. Pedersen, Nevanlinna matrices of entire functions, Math. Nachr. 171 (1995), 29–52. MR 1316350, DOI 10.1002/mana.19951710104
- Louis de Branges, The Bernstein problem, Proc. Amer. Math. Soc. 10 (1959), 825–832. MR 114080, DOI 10.1090/S0002-9939-1959-0114080-0
- Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
- Vladimir Derkach, Seppo Hassi, and Henk de Snoo, Operator models associated with Kac subclasses of generalized Nevanlinna functions, Methods Funct. Anal. Topology 5 (1999), no. 1, 65–87. MR 1771251
- 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
- M. L. Gorbachuk and V. I. Gorbachuk, M. G. Krein’s lectures on entire operators, Operator Theory: Advances and Applications, vol. 97, Birkhäuser Verlag, Basel, 1997. MR 1466698, DOI 10.1007/978-3-0348-8902-5
- I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
- M. Großmann and H. Langer, Über indexerhaltende Erweiterungen eines hermiteschen Operators im Pontrjaginraum, Math. Nachr. 64 (1974), 289–317 (German). MR 370257, DOI 10.1002/mana.19740640118
- Michael Kaltenbäck, Indefinite analogues of $j_{pq}$-contractive matrix functions and linear fractional transformations, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 349–371. MR 1916585
- F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Izdat. “Mir”, Moscow, 1968 (Russian). Translated from the English by I. S. Iohvidov and G. A. Karal′nik; Edited and supplemented by I. S. Kac and M. G. Kreĭn. MR 0243149
- M. Krein, A contribution to the theory of entire functions of exponential type, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 11 (1947), 309–326 (Russian, with English summary). MR 0022252
- M. G. Kreĭn, On the indeterminate case of the Sturm-Liouville boundary problem in the interval $(0,\infty )$, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 293–324 (Russian). MR 0052004
- 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 H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. II. Verallgemeinerte Resolventen, $u$-Resolventen und ganze Operatoren, J. Functional Analysis 30 (1978), no. 3, 390–447 (German). MR 518342, DOI 10.1016/0022-1236(78)90064-2
- 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 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 II, Beiträge Anal. 15 (1980), 27–45 (1981). MR 614775
- 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
- Mark G. Kreĭn and Heinz Langer, On some continuation problems which are closely related to the theory of operators in spaces $\Pi _\kappa$. 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), no. 2, 299–417. MR 776000
- Mark G. Krein and Heinz Langer, Continuation of Hermitian positive definite functions and related questions, Integral Equations Operator Theory 78 (2014), no. 1, 1–69. MR 3147401, DOI 10.1007/s00020-013-2091-z
- M. Kaltenbäck and H. Woracek, Generalized resolvent matrices and spaces of analytic functions, Integral Equations Operator Theory 32 (1998), no. 3, 282–318. MR 1652697, DOI 10.1007/BF01203772
- M. Kaltenbäck and H. Woracek, On extensions of Hermitian functions with a finite number of negative squares, J. Operator Theory 40 (1998), no. 1, 147–183. MR 1642542
- M. Kaltenbäck and H. Woracek, Pontryagin spaces of entire functions. I, Integral Equations Operator Theory 33 (1999), no. 1, 34–97. MR 1664343, DOI 10.1007/BF01203081
- M. Kaltenbäck and H. Woracek, Pontryagin spaces of entire functions. II, Integral Equations Operator Theory 33 (1999), no. 3, 305–380. MR 1671482, DOI 10.1007/BF01230736
- Michael Kaltenbäck and Harald Woracek, Pontryagin spaces of entire functions. III, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 241–310. MR 1991668
- Michael Kaltenbäck and Harald Woracek, Unique prime factorization in a partial semigroup of matrix-polynomials, Discuss. Math. Gen. Algebra Appl. 26 (2006), no. 1, 21–43. MR 2227628, DOI 10.7151/dmgaa.1103
- Michael Kaltenbäck and Harald Woracek, Pontryagin spaces of entire functions. VI, Acta Sci. Math. (Szeged) 76 (2010), no. 3-4, 511–560. MR 2789685
- Michael Kaltenbäck and Harald Woracek, Pontryagin spaces of entire functions. V, Acta Sci. Math. (Szeged) 77 (2011), no. 1-2, 223–336. MR 2841150
- Michael Kaltenbäck, Henrik Winkler, and Harald Woracek, Singularities of generalized strings, Operator theory and indefinite inner product spaces, Oper. Theory Adv. Appl., vol. 163, Birkhäuser, Basel, 2006, pp. 191–248. MR 2215864, DOI 10.1007/3-7643-7516-7_{9}
- 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
- B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
- M. Langer and H. Woracek, Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints. (to appear)
- Matthias Langer and Harald Woracek, A characterization of intermediate Weyl coefficients, Monatsh. Math. 135 (2002), no. 2, 137–155. MR 1894093, DOI 10.1007/s006050200011
- Matthias Langer and Harald Woracek, The exponential type of the fundamental solution of an indefinite Hamiltonian system, Complex Anal. Oper. Theory 7 (2013), no. 1, 285–312. MR 3010801, DOI 10.1007/s11785-011-0152-3
- Matthias Langer and Harald Woracek, Indefinite Hamiltonian systems whose Titchmarsh-Weyl coefficients have no finite generalized poles of non-positive type, Oper. Matrices 7 (2013), no. 3, 477–555. MR 3052315, DOI 10.7153/oam-07-29
- Henrik L. Pedersen, Logarithmic order and type of indeterminate moment problems. II, J. Comput. Appl. Math. 233 (2009), no. 3, 808–814. MR 2583020, DOI 10.1016/j.cam.2009.02.049
- Reinhold Remmert, Classical topics in complex function theory, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. MR 1483074, DOI 10.1007/978-1-4757-2956-6
- Marvin Rosenblum and James Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1307384, DOI 10.1007/978-3-0348-8520-1
- Harald Woracek, Nevanlinna-Pick interpolation: the degenerated case, Linear Algebra Appl. 252 (1997), 141–158. MR 1428633, DOI 10.1016/0024-3795(95)00700-8
- Harald Woracek, Existence of zerofree functions $N$-associated to a de Branges Pontryagin space, Monatsh. Math. 162 (2011), no. 4, 453–506. MR 2784432, DOI 10.1007/s00605-010-0203-2
- Harald Woracek, An inverse spectral theorem for Kreĭn strings with a negative eigenvalue, Monatsh. Math. 167 (2012), no. 1, 105–149. MR 2942966, DOI 10.1007/s00605-011-0351-z
Bibliographic Information
- H. Woracek
- Affiliation: Institut for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10/101, 1040 Wien, Austria
- Email: harald.woracek@tuwien.ac.at
- Received by editor(s): July 26, 2013
- Published electronically: July 27, 2015
- Additional Notes: The author gratefully acknowledges the support of the Austrian Science Fund (FWF), project I 1536–N25, and the Russian Foundation for Basic Research (RFBR), project 13-01-91002-ANF
- © Copyright 2015 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 757-783
- MSC (2010): Primary 46C20; Secondary 34B20, 30D10, 30D15
- DOI: https://doi.org/10.1090/spmj/1357
- MathSciNet review: 3442847