The Euler and Navier-Stokes equations for an incompressible fluid in two dimensions with periodic boundary conditions are considered. Concerning the Euler equation, previous works analyzed the associated (first order) Liouville operator L as a symmetric linear operator in a Hilbert space L 2 ðm g Þ with respect to a natural invariant Gaussian measure m g (given by the enstrophy), with the domain subspace of cylinder smooth bounded functions and have shown that there exist self-adjoint extensions of L. For the Navier-Stokes equation with a suitable white noise forcing term, the associated (second order) Kolmogorov operator K has been considered on the same domain as the sum of the Liouville operator L with the Ornstein-Uhlenbeck operator Q corresponding to the Stokes operator and the forcing term; existence of a C 0 -semigroup of contraction in L 2 ðm g Þ with generator extending the operator K has been proven. In this paper it is proven that both L and K are bounded by naturally associated positive Schro¨dinger-like operators, which are essentially self-adjoint on a dense subspace of cylinder functions. Other uniqueness results concerning L, respectively, K are also given.