Subnormal weighted shifts on directed trees whose nth powers have   trivial domain
  It is shown that for any positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is closed and densely defined while its (n + 1)th power has trivial domain. Similar result for composition operators in L2-spaces is established. 
Piotr Budzynski, Zenon Jan Jablonski, Il Bong Jung, and Jan Stochel
math.FA