This is a question from Altland and Simons book "Condensed Matter Field Theory". In the second exercise on page 64, the book claims that if we define [itex] \hat P_s, \hat P_d [/itex] to be the operators that project onto the singly and doubly occupied subspaces respectively, then at half-filling the following equality holds [itex] \hat P_s \hat H_t \hat P_s = 0 [/itex] and [itex] \hat P_d \hat H_t \hat P_d =0 [/itex], where [itex]\hat H_t[/itex] is the hopping term in the Hubbard model. I am having trouble to see why these two conditions are true. Do I have to write out the projection operator in terms of creation and annihilation operators and directly calculate? Thanks in advance.