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

DOI:
https://doi.org/10.1090/S0002-9947-98-01998-9

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 $\mathrm {r}(A)$ 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

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