As we know, the left pseudo-inverse of matrix $A$ is $A^{\dagger}_l = (A^TA)^{-1}A^T,$ and we have $A^{\dagger}_lA = I$. The right pseudo-inverse of matrix $A$ is $A^{\dagger}_r = A^T(AA^T)^{-1},$ and we have $AA^{\dagger}_r = I$.
Can we assume $A^{\dagger}_l$ and $A^{\dagger}_r$ to be close enough that at some times we can exchange them, such as assuming $A^{\dagger}_rA\approx I$?
Is there any reference that discuss this issue?