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

Roger D. Nussbaum, Bertram Walsh

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

For S a compact subset of C symmetric with respect to conjugation and f ∑ → C a continuous function, we obtain sharp conditions on / and S that insure that / can be approximated uniformly on £by polynomials with nonnegative coefficients. For X a real Banach space, K ⊃ X a closed but not necessarily normal cone with K -K = X, and A : X → X a bounded linear operator with A[K] ⊃ K, we use these approximation theorems to investigate when the spectral radius i(A) of A belongs to its spectrum a(A). A special case of our results is that if X is a Hubert space, A is normal and the 1-dimensional Lebesgue measure of cr(i(A - A)) is zero, then r(A) £ff(A). However, we also give an example of a normal operator A-al (where U is unitary and a 0).for which A[K] ⊃ K and r(/l) a(A).

Original languageEnglish (US)
Pages (from-to)2367-2391
Number of pages25
JournalTransactions of the American Mathematical Society
Volume350
Issue number6
DOIs
StatePublished - 1998

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Polynomial approximation with nonnegative coefficients
  • Positive linear operators
  • Spectral radius

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