Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem


Author: S. M. Malamud
Journal: Trans. Amer. Math. Soc. 357 (2005), 4043-4064
MSC (2000): Primary 15A29; Secondary 30C15, 30C10
DOI: https://doi.org/10.1090/S0002-9947-04-03649-9
Published electronically: September 23, 2004
MathSciNet review: 2159699
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.


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

  • 1. N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965. MR 32:1518
  • 2. A. Aziz and N. A. Rather, On an inequality of S. Bernstein and the Gauss-Lucas theorem. Rassias, Themistocles M. (ed.) et al., Analytic and geometric inequalities and applications. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 478, 29-35 (1999). MR 2001e:30006
  • 3. T. Bergkvist and H. Rullgard, On polynomial eigenfunctions for a class of differential operators, Math. Res. Let. v. 9, pp. 153-171, 2002. MR 2003d:34192
  • 4. J. Borcea, Maximal and inextensible polynomials and the geometry of the spectra of normal operators. preprint, math.CV/0309233.
  • 5. J. Borcea and B. Shapiro, Hyperbolic polynomials and spectral order, C. R. Math. Acad. Sci. Paris 337 (2003), no. 11, 693-698.
  • 6. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I. Springer, 1987. MR 88d:46105
  • 7. T. Craven and G. Csordas, The Gauss-Lucas theorem and Jensen polynomials. Trans. Amer. Math. Soc. 278 (1983), no. 1, 415-429. MR 85d:30031
  • 8. N. G. de Bruijn and T. A. Springer, On the zeros of a polynomial and of its derivative II, Indagationes Math. 9, 264-270 (1947). MR 9:30b
  • 9. M. G. de Bruin, K. G. Ivanov and A. Sharma, A conjecture of Schoenberg, J. Inequal. Appl. 4, No. 3 (1999), 183-213. MR 2000k:30007
  • 10. D. Dimitrov, A refinement of the Gauss-Lucas theorem, Proc. Amer. Math. Soc. 126, No.7 (1998), 2065-2070.MR 98h:30005
  • 11. W. F. Donoghue, Monotone Matrix functions and analytic continuation. Springer, 1974. MR 58:6279
  • 12. Ky Fan and G. Pall, Imbedding conditions for Hermitian and normal matrices, Canad. J. Math. 9 (1957), 298-304. MR 19:6e
  • 13. P. Fischer and J. A. R. Holbrook, Balayage defined by the nonnegative convex functions, Proc. AMS 79 (1980), 445-448.MR 81f:46012
  • 14. F. Gesztesy and B. Simon,$m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73, 267-297 (1997).MR 99c:47039
  • 15. G. H. Hardy, J. E. Littlewood and G. Polya, Inequalites. Cambridge, 1988.MR 89d:26016
  • 16. L. Hörmander, Notions of convexity. Birkhäuser, 1994.MR 95k:00002
  • 17. H. Hochstadt, On the construction of a Jacobi matrix from spectral data, Lin. Algebra and Appl., 8 (1974), 435-446.MR 52:3199
  • 18. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge Univ. Press, Cambridge, 1993. MR 91i:15001
  • 19. B. Kostant, On the convexity, the Weyl group and the Iwasawa decomposition, Annales scientificues de l'Ecole Normale Superieure, 6, 413-455 (1973). MR 51:806
  • 20. M. M. Malamud, On the formula of generalized resolvents of a nondensely defined Hermitian operator, Ukr. Matem. Zhurn. vol. 44, N12 (1992), 1658-1688 (in Russian) (translation in Ukr. Math. J. v. 44 (1992), 1522-1547). MR 95i:47046
  • 21. S. M. Malamud, Operator inequalities, converse to the Jensen inequality, Mathematical Notes, v. 69, No. 4 (2001), 633-637. MR 2002g:47031
  • 22. S. M. Malamud, A converse to the Jensen inequality, its matrix extensions and inequalities for minors and eigenvalues, Linear Algebra and Applications, v. 322, (2001), 19-41. MR 2001m:15048
  • 23. S. M. Malamud, An inverse spectral problem for normal matrices and a generalization of the Gauss-Lucas theorem, math.CV/0304158
  • 24. S. M. Malamud, Analog of the Poincare separation theorem for normal matrices and the Gauss-Lucas theorem, Funct. Anal. Appl., v. 37, No 3 (2003), 72-76.
  • 25. A. S. Markus, Eigenvalues and singular values of the sum and product of linear operators, Russian Math. Surveys 19 (1964), 91-120.MR 29:6318
  • 26. M. Marcus and H. Minc, A survey on matrix theory and matrix inequalities. Allyn and Bacon, 1964. MR 29:112
  • 27. A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications. Acad. Press, 1979. MR 81b:00002
  • 28. G. Mason and B. Shapiro, A note on polynomial eigenfunctions of a hypergeometric type operator, Experimental mathematics, 10, 609-618. MR 2003e:33038
  • 29. M. J. Miller, Maximal polynomials and the Ilieff-Sendov conjecture, Trans. Amer. Math. Soc. 321 (1990), 285-303. MR 90m:30007
  • 30. P. Pawlowski, On the zeros of a polynomial and its derivatives, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4461-4472. MR 99a:30008
  • 31. R. Pereira, Differentiators and the geometry of polynomials, J. Math. Anal. Appl. 285 (2003), 336-348.
  • 32. R. R. Phelps, Lectures on Choquet's Theorem. Van Nostrand-Reinhold (1966). MR 33:1690
  • 33. G. Polya and G. Szego, Problems and theorems in analysis, vol. II. Springer, 1976. MR 53:2
  • 34. B. Shapiro and M. Shapiro, This strange and misterious Rolle's Theorem, preprint, math.CA/0302215
  • 35. Harold S. Shapiro, Spectral aspects of a class of differential operators, Operator Theory Adv. Appl., 132, pp. 361-385. Birkhäuser, Basel, 2002.MR 2003i:47035
  • 36. G. Schmeisser, The conjectures of Sendov and Smale, Approx. Theory: A volume dedicated to Blagovest Sendov, DARBA, Sofia, 2002, 353-369.MR 2004c:30007
  • 37. S. Sherman, On a theorem of Hardy, Littlewood, Polya, and Blackwell, Proc. Nat. Acad. Sci. USA 37 (1951), 826-831.MR 13:633g
  • 38. I.J. Schoenberg, A conjectured analogue of Rolle's theorem for polynomials with real or complex coefficients, Amer. Math. Mon. 93, 8-13 (1986). MR 87d:30010
  • 39. D. Tischler, Critical points and values of complex polynomials, J. Complexity 5 (1989), pp. 438-456. MR 91a:30004

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 15A29, 30C15, 30C10

Retrieve articles in all journals with MSC (2000): 15A29, 30C15, 30C10


Additional Information

S. M. Malamud
Affiliation: Departement Mathematik, HG G33.1, ETH-Zentrum, Raemistrasse 101, 8092 Zürich, Switzerland
Email: semka@math.ethz.ch

DOI: https://doi.org/10.1090/S0002-9947-04-03649-9
Keywords: Zeros of polynomials, normal matrices, inverse spectral problem, majorization
Received by editor(s): July 6, 2003
Received by editor(s) in revised form: November 7, 2003
Published electronically: September 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society