A norm inequality for operators

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*}

• Should $A$ be $B$ in the last term? – Nate Eldredge Jul 8 at 13:34
• Yes thanks (and sorry) – Glm Jul 8 at 13:36
• Also, do you need some more assumptions to ensure that $A+B$ is self adjoint? – Nate Eldredge Jul 8 at 13:37
• Yes you can assume that – Glm Jul 8 at 13:38

$$\|\sqrt{A+B}Cx\|^2=(\sqrt{A+B}Cx,\sqrt{A+B}Cx)=\\ ((A+B)Cx,Cx)=(ACx,Cx)+(BCx,Cx)=\|\sqrt{A}Cx\|^2+\|\sqrt{B}Cx\|^2,$$ taking the supremum over unit vectors $$x$$ we get $$\|\sqrt{A+B}C\|^2\leqslant \|\sqrt{A}C\|^2+\|\sqrt{B}C\|^2\leqslant (\|\sqrt{A}C\|+\|\sqrt{B}C\|)^2.$$
For any unit vector $$x\in L^2(\mathbb{R}^n)$$, $$\|\sqrt{A+B}\,Cx\|^2=(\sqrt{A+B}\,Cx,\sqrt{A+B}\,Cx) =(ACx,Cx)+(BCx,Cx)=\|\sqrt A\,Cx\|^2+\|\sqrt B\,Cx\|^2 \le(\|\sqrt A\,Cx\|^2+\|\sqrt B\,Cx\|)^2 \le(\|\sqrt A\,C\|+\|\sqrt B\,C\|)^2.$$ So, $$\|\sqrt{A+B}\,C\|\le\|\sqrt A\,C\|+\|\sqrt B\,C\|.$$