Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some constant $c \geq 0$, where $\left| \! \left| \cdot \right| \! \right|$ is the operator norm,
\begin{align\*}
 \left| \! \left| \sqrt{A+B} C \right| \! \right| \leq c \left| \! \left| \sqrt{A} C \right| \! \right| + c \left| \! \left| \sqrt{B} C \right| \! \right| ?
 \end{align\*}



A norm inequality for operators
$$
\|\sqrt{A+B}Cx\|^2=(\sqrt{A+B}Cx,\sqrt{A+B}Cx)=\\
((A+B)Cx,Cx)=(ACx,Cx)+(BCx,Cx)=\|\sqrt{A}Cx\|^2+\|\sqrt{B}Cx\|^2,
$$
taking the supremum over unit vectors $x$ we get
$$
\|\sqrt{A+B}C\|^2\leqslant \|\sqrt{A}C\|^2+\|\sqrt{B}C\|^2\leqslant (\|\sqrt{A}C\|+\|\sqrt{B}C\|)^2.
$$


