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Transactions of the American Mathematical Society
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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Additional Information

Roger D. Nussbaum
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
Email: nussbaum@math.rutgers.edu

Bertram Walsh
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
Email: bwalsh@math.rutgers.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01998-9
PII: S 0002-9947(98)01998-9
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