ISOMETRIC SHIFT-OPERATORS ON C(X)

Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtaine d many significiant results concerning shift operators on Banach space s. Using a result of Holsztynski they classify isometric shift operato rs on C(X) for any compact Hausdorff space X into two (not necessarily disjoint) classes. If there exists an isometric shift operator T: C(X ) --> C(X) of type II, they show that X is necessarily separable. In c ase T is of type I, they exhibit a paticular infinite countable set D = {p, psi-1(p), psi-2(p), psi-3(p)....} of isolated points in X. Under the additional assumption that the linear functional GAMMA carrying f is-an-element-of C(X) to Tf(p) is-an-element-of C is identically zero , they show that D is dense in X. They raise the question whether D wi ll still be dense in X even when GAMMA not-equal 0. In this paper we g ive a negative answer to this question. In fact, given any integer l g reater-than-or-equal-to 1, we construct an example of an isometric shi ft operator T: C(X) --> C(X) of type I with X\D having exactly l eleme nts, where DBAR is the closure of D in X.