For Hermitian positive semidefnite matrices $A$ and $B$. Let $C=A+B$, if $C$ is invertible, then we can easily get $$I=X+Y,$$ where $X=C^{-\frac{1}{2}}AC^{-\frac{1}{2}}$ and $Y=C^{-\frac{1}{2}}BC^{-\frac{1}{2}}$.
But if $C$ is not invertible, I can use the Moore-Penrose generalized inverse $C^+$ to get the required result $$I=X+Y$$ with $X=(C^+)^{\frac{1}{2}}A(C^+)^{\frac{1}{2}}$ and $Y=(C^+)^{\frac{1}{2}}B(C^+)^{\frac{1}{2}}$.
In this example. it is trivial that $range(A)\subseteq range(C)$ and $range(B)\subseteq range(C)$. But if both $A$ and $B$ are not Hermitian positive semidefinite matrices, $range(A)\varsubsetneq range(C)$, $range(B)\varsubsetneq range(C)$, and $C=A+B$ is not invertible, can I use the Moore-Penrose generalized inverse to get $I=X+Y$?