CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator B-mu : Sigma(X) -> Sigma(X) defined by B-mu(x(0), x(1),...) = (mu(0)x(1), mu(1)x(2),...), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that B-mu(N) is distributionally chaotic and Li-Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.