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For a compact subset of C symmetric with respect to conjuga- tion and f : ! Ca continuous function, we obtain sharp conditions on f and that insure that f can be approximated uniformly on by polynomials with nonnegative coecients. For X a real Banach space, K X ac losed but not necessarily normal cone with K K = X ,a nd A : X ! Xa bounded linear operator with A(K) K, we use these approximation theorems to in- vestigate when the spectral radius r(A )o f Abelongs to its spectrum (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 (i(A A)) is zero, then r(A) 2 (A). However, we also give an example of a normal operator A = U I (where U is unitary and >0) for which A(K) K and r(A) = 2 (A).

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