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Transactions of the American Mathematical Society

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



Approximation by polynomials with
nonnegative coefficients and
the spectral theory of positive operators

Authors: Roger D. Nussbaum and Bertram Walsh
Journal: Trans. Amer. Math. Soc. 350 (1998), 2367-2391
MSC (1991): Primary 30C10, 47B15, 47B65
MathSciNet review: 1433126
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Abstract: For $\Sigma $ a compact subset of $\mathbf{C}$ symmetric with respect to conjugation and $f: \Sigma \to \mathbf{C}$ a continuous function, we obtain sharp conditions on $f$ and $\Sigma $ that insure that $f$ can be approximated uniformly on $\Sigma $ by polynomials with nonnegative coefficients. For $X$ a real Banach space, $K \subseteq X$ a closed but not necessarily normal cone with $\overline{K - K} = X$, and $A: X \to X$ a bounded linear operator with $A[K] \subseteq K$, we use these approximation theorems to investigate when the spectral radius $\text{\rm r}(A)$ of $A$ belongs to its spectrum $\sigma (A)$. A special case of our results is that if $X$ is a Hilbert space, $A$ is normal and the 1-dimensional Lebesgue measure of $\sigma (i(A - A^{*}))$ is zero, then $\text{\rm r}(A) \in \sigma (A)$. However, we also give an example of a normal operator $A = - U -\alpha I$ (where $U$ is unitary and $\alpha > 0$) for which $A[K] \subseteq K$ and $\text{\rm r}(A) \notin \sigma (A)$.

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  • 1. R. Askey, Orthogonal Polynomials and Special Functions, Regional Conference Series in Applied Math., SIAM, Philadelphia, 1975. MR 58:11288
  • 2. L. Bieberbach, Lehrbuch der Funktionentheorie Vol. II, Teubner, Leipzig, 1931. MR 40:2823b (later ed.)
  • 3. E. Bishop, A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629-642. MR 22:191
  • 4. F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc. (3) 8 (1958), 53-75. MR 19:1183c
  • 5. L.-K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translation from the Russian, Amer. Math. Soc., Providence, R. I., 1963. MR 30:2162
  • 6. S. Kantorovitz, Classification of operators by means of their operational calculus, Transactions Amer. Math. Soc. 115 (1965), 194-224. MR 33:7855
  • 7. R. D. Nussbaum, Eigenvectors of order-preserving linear operators, [accepted by] J. London Math. Soc..
  • 8. H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann. 138 (1959), 259-286. MR 21:5135
  • 9. H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. MR 33:1689
  • 10. L. Schwartz, Théorie des Distributions, Hermann & Cie.; Tome I, 1957; Tome II, 1959. MR 21:6534; MR 22:8322
  • 11. E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, 1939 (corr. printing 1968).
  • 12. J. F. Toland, Self-adjoint operators and cones, J. London Math. Soc. (2) 53 (1996), 167-183. MR 96k:47040
  • 13. D. V. Widder, The Laplace Transform, Princeton Univ. Press, 1941. MR 3:232d
  • 14. K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin, 1980. MR 82i:46002

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Additional Information

Roger D. Nussbaum
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101

Bertram Walsh
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101

Keywords: Polynomial approximation with nonnegative coefficients, positive linear operators, spectral radius
Received by editor(s): December 26, 1995
Received by editor(s) in revised form: July 1, 1996
Additional Notes: The first author was partially supported by NSF grant DMS 9401823
Dedicated: Dedicated to Helmut H. Schaefer on the 70th anniversary of his birth
Article copyright: © Copyright 1998 American Mathematical Society