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).
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Polynomial approximation with nonnegative coefficients
- Positive linear operators
- Spectral radius