Let A be a finite dimensional Hopf algebra with antipode s over a field k. In (6) the structure of s and s2 is seen to be closely related to the important algebraic features of A and A*. This paper furthers the investigation. More generally "locally finite" bialgebra endomorphisms t:A ->A of an arbitrary Hopf algebra A are considered (this means A is the sum of the finite dimensional t-invariant subspaces). Applications are given to finite dimensional Hopf algebras. We show that the antipode s of an arbitrary Hopf algebra over a field is injective (bijective) if the same is true when it is restricted to the coradical. We prove that s is bijective if it is a locally finite operator, or if the coradical is finite dimensional. Examples of finite dimensional Hopf algebras over Q (which can be defined over Z) are constructed with antipode of order 6 and 8, respectively. These examples show that the relationship between the order of the antipode and the primitive roots of unity in the ground field is not as intricate as indicated by some of the results of (6).