Regular Set of a closed Operator

Question

Let $\mathcal{H}$ be a (separable) Hilbert space and $A : D(A) \to \mathcal{H}$ a closed densly defined operator. Define the regular set by $$\rho(A) := \{z \in \mathbb{C} : z - A \text{ bijection with bounded inverse}\}.$$

Now as I read in my script and a book, by the closed graph theorem, the last condition is redundant.

(Closed Graph Theorem) Let $X,Y$ be Banach spaces and $A : X \to Y$ linear. The following two conditions are equivalent.

• The Graph of $A$ is closed in $X \times Y$.
• $A$ is continuous.

And also the inverse mapping theorem, which states that if $A$ is a continuous linear bijection, then so is $A^{-1}$. Now combining these yields the result. However, I have a little problem: In order to apply these two theorems, we need a Banach space, but $D(A)$ is not necessarily one, isn't it? I mean it is just a subspace of $\mathcal{H}$.

Answer 1

The graph-norm on $D(A)$ is defined by $\Vert \cdot \Vert_{A} := \Vert \cdot \Vert_{\mathcal{H}}+\Vert A \cdot \Vert_{\mathcal{H}}$.

Then $(D(A),\Vert \cdot \Vert_{A}$) is complete if and only if $A$ is a closed operator and the inclusion $(D(A),\Vert \cdot \Vert_{A}) \hookrightarrow \mathcal{H}$ is continuous. With this information you can apply the closed graph theorem.