  In the framework of loop quantum gravity, we define a new Hilbert space of
states which are solutions of a large number of components of the
diffeomorphism constraint. On this Hilbert space, using the methods of
Thiemann, we obtain a family of gravitational scalar constraints. They preserve
the Hilbert space for every choice of lapse function. Thus adjointness and
commutator properties of the constraint can be investigated in a
straightforward manner. We show how the space of solutions of the symmetrized
constraint can be defined by spectral decomposition, and the Hilbert space of
physical states by subsequently fully implementing the diffeomorphism
constraint. The relationship of the solutions to those resulting from a
proposal for a symmetric constraint operator by Thiemann remains to be
elucidated.
