Suppose A and B are continuous linear operators mapping a complex Banach space X into itself. For any polynomial pC, it is obvious that when A commutes with B, then p(A) commutes with B. To see that the reverse implication is false, let A be nilpotent of order n. Then An commutes with all B but A cannot do so. Sufficient conditions for the implication: p(A) commutes with B implies A commutes with B: were given by Embry [2] for the case p(λ) = λn and Finkelstein and Lebow [3] in the general case. The latter authors proved in fact that if f is a function holomorphic on σ(A) and if f is univalent with non-vanishing derivative on σ(A), then A can be expressed as a function of f(A).