NULL SPACES AND RANGES OF POLYNOMIALS OF OPERATORS MANUEL GONZÁLEZ We give an elementary proof of the fact that given two polynomials P, Q without common zeros and a linear operator A, the operators P(A) and Q(A) verify some properties equivalent to the pair (P(A), Q(A)) being non-singular in the sense of J .L . Taylor . From these properties we derive expressions for the range and null space of P(A) and spectral mapping theorems for polynomials of continuous (or closed) operators in Banach spaces .