Arkiv för Matematik

  • Ark. Mat.
  • Volume 44, Number 2 (2006), 211-240.

Spectral order and isotonic differential operators of Laguerre–Pólya type

Julius Borcea

Full-text: Open access


The spectral order on Rn induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$-orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

Article information

Ark. Mat. Volume 44, Number 2 (2006), 211-240.

Received: 18 May 2004
Revised: 5 September 2005
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

2006 © Institut Mittag-Leffler


Borcea, Julius. Spectral order and isotonic differential operators of Laguerre–Pólya type. Ark. Mat. 44 (2006), no. 2, 211--240. doi:10.1007/s11512-006-0017-6.

Export citation


  • Ando, T., Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118 (1989), 163–248.
  • Ando, T., Majorizations and inequalities in matrix theory, Linear Algebra Appl. 199 (1994), 17–67.
  • Arnold, V. I., Hyperbolic polynomials and Vandermonde mappings, Funktsional. Anal. i Prilozhen. 20 (1986), 52–53 (Russian). English transl.: Funct. Anal. Appl. 20 (1986), 125–127.
  • Borcea, J., Convexity properties of twisted root maps, arXiv: math.CA/0312321.
  • Borcea, J., Differential preservers of majorization, hyperbolic polynomial pencils and Lax–Gårding convexity, In preparation.
  • Borcea, J., Brändén, P. and Shapiro, B., Pólya–Schur master theorems for circular domains and their boundaries, arXiv: math.CV/0607416.
  • Borcea, J., Brändén, P. and Shapiro, B., Classification of hyperbolicity preservers I: the Weyl algebra case, arXiv: math.CA/0606360.
  • Borcea, J. and Shapiro, B., Hyperbolic polynomials and spectral order, C. R. Math. Acad. Sci. Paris 337 (2003), 693–698.
  • Carnicer, J. M., Peña, J. M. and Pinkus, A., On some zero-increasing operators, Acta Math. Hungar. 94 (2002), 173–190.
  • Craven, T. and Csordas, G., Differential operators of infinite order and the distribution of zeros of entire functions, J. Math. Anal. Appl. 186 (1994), 799–820.
  • Craven, T. and Csordas, G., Composition theorems, multiplier sequences and complex zero decreasing sequences, in Value Distribution Theory and Its Related Topics, Adv. Complex Anal. Appl., 3, pp. 131–166, Kluwer, Boston, MA, 2004.
  • Dean, A. M. and Verducci, J. S., Linear transformations that preserve majorization, Schur concavity, and exchangeability, Linear Algebra Appl. 127 (1990), 121–138.
  • Dym, H. and Katsnelson, V., Contributions of Issai Schur to analysis, in Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., 210, pp. xci–clxxxviii, Birkhäuser, Boston, MA, 2003.
  • Gårding, L., An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965.
  • Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1988.
  • Jonathan, D. and Plenio, M. B., Entanglement-assisted local manipulation of pure quantum states, Phys. Rev. Lett. 83 (1999), 3566–3569.
  • Korobeinik, Y. F., On the question of the representation of an arbitrary linear operator in the form of a differential operator of infinite order, Mat. Zametki 16 (1974), 277–283 (Russian). English transl.: Math. Notes 16 (1975), 753–756.
  • Kozitsky, Y., Oleszczuk, P. and Wołowski, L., Infinite order differential operators in spaces of entire functions, J. Math. Anal. Appl. 277 (2003), 423–437.
  • Latorre, J. I. and Martín- Delgado, M. A., Majorization arrow in quantum-algorithm design, Phys. Rev. A 66 (2002), 022305, 5 pp.
  • Levin, B. Ya., Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1980.
  • Marshall, A. W. and Olkin, I., Inequalities: Theory of Majorization and Its Applications, Mathematics in Science Engineering, 143, Academic Press, New York–London, 1979.
  • Nielsen, M. A. and Vidal, G., Majorization and the interconversion of bipartite states, Quantum Inf. Comput. 1 (2001), 76–93.
  • Peetre, J., Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211–218; Erratum, ibid. 8 (1960), 116–120.
  • Pólya, G. and Schur, I., Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113.
  • Rahman, Q. I. and Schmeisser, G., Analytic theory of polynomials, London Math. Soc. Monogr., 26, Oxford Univ. Press, Oxford, 2002.