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 | References | Similar Articles | Additional Information

Abstract: For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

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

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