m-ISOMETRIC WEIGHTED SHIFTS AND REFLEXIVITY OF SOME OPERATORS

For a positive integer m, a bounded linear operator T on a Hubert space H is called an m-isometry, if $\sum\nolimits_{k = 0}^m {{{\left( { - 1} \right)}^{m - k}}} \left( {\begin{array}{*{20}{c}}m\\k \\\end{array} } \right){T^{*k}}{T^k} = 0$. We characterize all misometric unilateral weighted shift operators that are not m - 1-isometries in terms of their weight sequences. Then we prove the reflexivity of some classes of operators: (1) All nonnegative integer powers of m-isometric unilateral weighted shifts. (2) The contractions whose spectrum are all the closed unit disc. (3) All non-negative integer powers of hyponormal m-isometries.