Adjoint Operators

Definition (Unbounded Operator, Domain)

Let $X$ and $Y$ be normed spaces, and let $D$ be a linear subspace of $X$. An unbounded linear operator from $X$ to $Y$ with domain of definition $D$ is a linear operator $T:D\to Y$. We write $D=\mathcal{D}(T)$ for the domain of $T$.

Definition (Densely Defined Operator)

An unbounded linear operator $T$ from $X$ to $Y$ is said to be densely defined if its domain $\mathcal{D}(T)$ is a dense linear subspace of $X$.

We primarily consider unbounded linear operators on Hilbert space.

Definition (Graph)

Let $X$ and $Y$ be normed spaces. The graph of an unbounded linear operator $T$ from $X$ to $Y$ is the set $\mathcal{G}(T)=\left\{(x,Tx):x\in \mathcal{D}(T)\right\}\subset X\oplus Y$.

We note that the graph of an unbounded linear operator from $X$ to $Y$ is a linear subspace of the normed space $X\oplus Y$. As such, it may or may not be closed.

Definition (Closed Operator)

Let $X$ and $Y$ be normed spaces. An unbounded linear operator $T$ from $X$ to $Y$ is said to be closed if its graph $\mathcal{G}(T)$ is a closed linear subspace of $X\oplus Y$.

If $T$ is an unbounded linear operator from $X$ to $Y$, then the graph closure $\overline{\mathcal{G}(T)}$ is a closed linear subspace of $X\oplus Y$. This subspace may or may not be the graph of an unbounded linear operator from $X$ to $Y$. If it is, that operator is closed.

Definition (Closable Operator)

Let $X$ and $Y$ be normed spaces. An unbounded linear operator $T$ from $X$ to $Y$ is called closable (or pre-closed) if there exists an unbounded linear operator $\overline{T}$ from $X$ to $Y$ such that $\mathcal{G}(\overline{T})=\overline{\mathcal{G}(T)}$.