Generation and commutation properties of the Volterra operator

Let V be the classical Volterra operator on L ₂(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert?CSchmidt operators and as well in the space ${\mathcal{B}(L_2(0,1))}$ equipped with the weak operator topology. Moreover, the algebra generated by V ² and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.