Operators in divergence form and their Friedrichs and Krein extensions

For a densely defined nonnegative symmetric operator $A = L_2^*L_1 $ in a Hilbert space, constructed from a pair $L_1 \subset L_2$ of closed operators, we give expressions for the Friedrichs and Krein nonnegative selfadjoint extensions. Some conditions for the equality $(L_2^* L_1)^* = L_1^* L_2$ are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given.