Weyl's theorem holds for algebraically hyponormal operators

In this note it is shown that if T is an "algebraically hyponormal" operator, i.e., p(T) is hyponormal for some nonconstant complex polynomial p, then for every f is an element of H(sigma(T)), Weyl's theorem holds for f(T), where H(sigma(T)) denotes the set of analytic functions on an open neighborhood of sigma(T).