On reflexivity of hyponormal and weighted shift operators

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r(A) such that its spectrum is the closed disc {z:|z| ≤ r(A)} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.