  We show that if an infinite-dimensional Banach space X has a symmetric basis
then there exists a bounded, linear operator R : X --> X such that the set
  A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X.
Moreover, if x in X\A then some subsequence of (R^n(x)) converges weakly to x.
This answers in the negative a recent conjecture of Prajitura. The result can
be extended to any Banach space containing an infinite-dimensional,
complemented subspace with a symmetric basis; in particular, all 'classical'
Banach spaces admit such an operator.
