If $A+B\ge C$, can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?

Question

Let $A$ and $B$ be two positive operators on a Hilbert space. $C$ is a positive operator with $A+B\ge C$. Can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?

Answer 1

No, it's not true in general. Take for instance $$ A=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\ B=\begin{bmatrix} 0&0\\0&1\end{bmatrix}, \ C=\begin{bmatrix} 1/2&1/2\\1/2&1/2\end{bmatrix}. $$ Then $C\leq A+B$, and if $0\leq A_1\leq A$ then $A_1=t A$ for some $t\in[0,1]$. Similarly, $0\leq B_2\leq B$ implies $B=s B$ for some $s\in [0,1]$. It is not possible then to have $C=A_1+B_1=tA+sB$.

Answer 2

Set $A,B,C>0$, observe the problem $$ \lambda=\min\left\{\frac{x^tCx}{x^tAx} \colon x^tx=1\right\}.$$ If $\lambda >0$, define $$ A_1= \lambda A, A_2=C-A_1.$$