Let $T\in B(H)$, where $H$ is a (separable) complex Hilbert space. Let $A$ be an unbounded self-adjoint operator acting in $H$. We denote by $D(A)$ its domain. We endow it with the graph norm, e.g., $\|f\|+ \|A f\|$. It is a Hilbert space.
Suppose that $T \in B(D(A))$ and that $T$ bijective from $H$ to $H$.
Since $T$ is a homeomorphism from $H$ to $H$ it is clear that the closure of $TD(A)$ is dense in $H$ for the norm of $H$.
Is it true in general that $TD(A)$ is dense in D(A), where the space is endowed with the graph norm ?