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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators
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by Roger D. Nussbaum and Bertram Walsh PDF
Trans. Amer. Math. Soc. 350 (1998), 2367-2391 Request permission

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 $\mathrm {r}(A)$ 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 $\mathrm {r}(A) \in \sigma (A)$. 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 $\mathrm {r}(A) \notin \sigma (A)$.
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Additional Information
  • Roger D. Nussbaum
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
  • MR Author ID: 132680
  • Email: nussbaum@math.rutgers.edu
  • Bertram Walsh
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
  • Email: bwalsh@math.rutgers.edu
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 2367-2391
  • MSC (1991): Primary 30C10, 47B15, 47B65
  • DOI: https://doi.org/10.1090/S0002-9947-98-01998-9
  • MathSciNet review: 1433126