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

Malamud, S.

doi:10.1090/S0002-9947-04-03649-9

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem
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Trans. Amer. Math. Soc. 357 (2005), 4043-4064 Request permission

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

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
A. Aziz and N. A. Rather, On an inequality of S. Bernstein and the Gauss-Lucas theorem, Analytic and geometric inequalities and applications, Math. Appl., vol. 478, Kluwer Acad. Publ., Dordrecht, 1999, pp. 29–35. MR 1785860
Tanja Bergkvist and Hans Rullgård, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002), no. 2-3, 153–171. MR 1909635, DOI 10.4310/MRL.2002.v9.n2.a3
J. Borcea, Maximal and inextensible polynomials and the geometry of the spectra of normal operators. preprint, math.CV/0309233.
J. Borcea and B. Shapiro, Hyperbolic polynomials and spectral order, C. R. Math. Acad. Sci. Paris 337 (2003), no. 11, 693–698.
Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics. 1, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987. $C^\ast$- and $W^\ast$-algebras, symmetry groups, decomposition of states. MR 887100, DOI 10.1007/978-3-662-02520-8
Thomas Craven and George Csordas, The Gauss-Lucas theorem and Jensen polynomials, Trans. Amer. Math. Soc. 278 (1983), no. 1, 415–429. MR 697085, DOI 10.1090/S0002-9947-1983-0697085-9
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
M. G. de Bruin, K. G. Ivanov, and A. Sharma, A conjecture of Schoenberg, J. Inequal. Appl. 4 (1999), no. 3, 183–213. MR 1734160, DOI 10.1155/S1025583499000363
Dimitar K. Dimitrov, A refinement of the Gauss-Lucas theorem, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2065–2070. MR 1452801, DOI 10.1090/S0002-9939-98-04381-0
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
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
P. Fischer and J. A. R. Holbrook, Balayage defined by the nonnegative convex functions, Proc. Amer. Math. Soc. 79 (1980), no. 3, 445–448. MR 567989, DOI 10.1090/S0002-9939-1980-0567989-9
Fritz Gesztesy and Barry Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267–297. MR 1616422, DOI 10.1007/BF02788147
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR 944909
Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301332
Harry Hochstadt, On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl. 8 (1974), 435–446. MR 382314, DOI 10.1016/0024-3795(74)90077-9
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR 1084815
Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 364552
M. M. Malamud, On a formula for the generalized resolvents of a non-densely defined Hermitian operator, Ukraïn. Mat. Zh. 44 (1992), no. 12, 1658–1688 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 12, 1522–1547 (1993). MR 1215039, DOI 10.1007/BF01061278
S. M. Malamud, Operator inequalities that are inverse to Jensen’s inequality, Mat. Zametki 69 (2001), no. 4, 633–637 (Russian); English transl., Math. Notes 69 (2001), no. 3-4, 582–586. MR 1846005, DOI 10.1023/A:1010228718572
S. M. Malamud, A converse to the Jensen inequality, its matrix extensions and inequalities for minors and eigenvalues, Linear Algebra Appl. 322 (2001), no. 1-3, 19–41. MR 1804111, DOI 10.1016/S0024-3795(00)00207-X
S. M. Malamud, An inverse spectral problem for normal matrices and a generalization of the Gauss-Lucas theorem, math.CV/0304158
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.
A. S. Markus, Eigenvalues and singular values of the sum and product of linear operators, Uspehi Mat. Nauk 19 (1964), no. 4 (118), 93–123 (Russian). MR 0169063
Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
Gisli Másson and Boris Shapiro, On polynomial eigenfunctions of a hypergeometric-type operator, Experiment. Math. 10 (2001), no. 4, 609–618. MR 1881761
Michael J. Miller, Maximal polynomials and the Ilieff-Sendov conjecture, Trans. Amer. Math. Soc. 321 (1990), no. 1, 285–303. MR 965744, DOI 10.1090/S0002-9947-1990-0965744-X
Piotr Pawlowski, On the zeros of a polynomial and its derivatives, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4461–4472. MR 1473453, DOI 10.1090/S0002-9947-98-02291-0
R. Pereira, Differentiators and the geometry of polynomials, J. Math. Anal. Appl. 285 (2003), 336-348.
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Die Grundlehren der mathematischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0396134
B. Shapiro and M. Shapiro, This strange and misterious Rolle’s Theorem, preprint, math.CA/0302215
Harold S. Shapiro, Spectral aspects of a class of differential operators, Operator methods in ordinary and partial differential equations (Stockholm, 2000) Oper. Theory Adv. Appl., vol. 132, Birkhäuser, Basel, 2002, pp. 361–385. MR 1924990
Gerhard Schmeisser, The conjectures of Sendov and Smale, Approximation theory, DARBA, Sofia, 2002, pp. 353–369. MR 1947846
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
I. J. Schoenberg, A conjectured analogue of Rolle’s theorem for polynomials with real or complex coefficients, Amer. Math. Monthly 93 (1986), no. 1, 8–13. MR 824585, DOI 10.2307/2322536
David Tischler, Critical points and values of complex polynomials, J. Complexity 5 (1989), no. 4, 438–456. MR 1028906, DOI 10.1016/0885-064X(89)90019-8