Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$

Let $T$ be a densely defined and closed operator from some Hilbert space $H$ into $H^n$. It is a classical theorem by John von Neumann that $T^*T$ is self-adjoint with domain $D(T^*T).$

I wonder whether it is true that for some $C>0$

$$\Vert \langle AT,T \rangle x \Vert \le C (\Vert T^*T x \Vert + \Vert x \Vert) \text{ and all } x \in D(T^*T).$$

Similarly, it seems natural to ask whether we also have that

$$\Vert T^*T x \Vert \le C (\Vert \langle AT,T \rangle x \Vert + \Vert x \Vert) \text{ and all } x \in D(\langle AT,T \rangle)?$$