Assume that A is a nonnegative linear relation in a complex Hilbert space \({{\mathfrak {H}}}\) and p(A) is a polynomial in the variable A. The main goal of this note is to identify certain classes of polynomials which preserve the nonnegativity of the variable A, i.e. if A is nonnegative then also p(A) is nonnegative. Three such classes of polynomials are identified. Let \(\tilde{A}\) be a nonnegative selfadjoint extension of A and let p one of the polynomial belonging to one of the aforementioned classes. It is shown that the polynomial \(p(\tilde{A})\) is a nonnegative selfadjoint extension of p(A).
