Briefly, the single-particle Green function is the ground-state expectation value of two operators, one creation and one annihilation operator. Product of two single-particle Green functions is therefore described by an expression involving four operators. The two expressions cannot a priori describe the same quantity (in fact, they are dimensionally not equivalent; in the energy-momentum representation the single-particle Green function has the dimension of time). I would recommend you to write down the spectral representation of the single-particle Green function and work out how the single-particle density of states is deduced from it (in doing so, making use of the equality 1/(x - i 0+) = P(1/x) + i π δ(x), where 0+ denotes a positive infinitesimal, P the 'principal value' and δ the Dirac delta-function). Following this, you will immediately realise the answer to your question.

Briefly, the single-particle Green function is the ground-state expectation value of two operators, one creation and one annihilation operator. Product of two single-particle Green functions is therefore described by an expression involving four operators. The two expressions cannot a priori describe the same quantity (in fact, they are dimensionally not equivalent; in the energy-momentum representation the single-particle Green function has the dimension of time). I would recommend you to write down the spectral representation of the single-particle Green function and work out how the single-particle density of states is deduced from it (in doing so, making use of the equality 1/(x - i 0+) = P(1/x) + i π δ(x), where 0+ denotes a positive infinitesimal, P the 'principal value' and δ the Dirac delta-function). Following this, you will immediately realise the answer to your question.