Let A, B, and C be positive definite autoadjoint operators on a Hilbert space H, such that $D(A)\subset D(C) \subset D(B)$. We denote $\tilde {A}=C^{-1}A$ y $\tilde {B}=C^{-1}B$ where $C^{-1}$ is bounded, $\tilde {A}$ is a positive definite self-adjoint bijection operator and $\tilde {B}$ it is a positive definite self-adjoint operator. Show that it is true that: $${\|{\tilde {A}}^{1/2}u\|}^2+{\|v\|}^2\leq C({\|{\tilde {A}}^{1/2}v\|}^2+{\|{\tilde {A}}u+{\tilde {B}}v\|}^2)$$ for all $u, v \in H$.
Remember that:
$C^{-1}$ is bounded; if and only if, exists M>0 such that $\|C^{-1}u\|\leq M\|u\|$ $\forall u\in H$.
$A$ is a self-adjoint; if and only if, $(Au,v)=(u,Av)$(consider that $A$ is a self-adjoint, implies that $A^{1/2}$ is self-adjoint).
$A$ is a positive definite operator; if and only if, $ 0\leq (Au,u)$ for all $u\in D(A)$.