Bounded Linear Operator and the Adjoint

Question

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset \mathcal{D}(T)$, where $T$ is a bounded linear operator.

Given the aforementioned setup is it true that:

1) The domain $\mathcal{D}(T)$ is dense in $\mathcal{H}$.

2) The linear operator $S$ is bounded on $\mathcal{D}(S)$.

The reason that I ask 1) and 2) is that I am trying to deduce information about the adjoint operators $S^*$ and $T^*$ given the data.

Answer 1

If $T$ is bounded then $D(T)=H$ by definition. Therefore, $2$ is false as well.

Answer 2

You might be looking for...

A bounded operator is closable with: $$\|T\|<\infty:\quad\mathcal{D}(\overline{T})=\overline{\mathcal{D}(T)}$$

(Or similarly, a bounded operator is closed iff its domain is so.)