Null spaces and ranges of polynomials of operators (Q1118794)

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.