I am reading Reed & Simon – Methods of Modern Mathematical Physics book and, in Chapter VII, the notion of an unbounded linear operator is introduced. Let $\mathscr{H}$ be a Hilbert space and $T$ be a densely defined linear operator with domain $D(T) \subset \mathscr{H}$. According to the authors, $T$ is called closed if its graph:
$$\Gamma(T) := \{\langle \varphi, T\varphi\rangle: \varphi \in D(T)\}$$
is a closed subset of $\mathscr{H}\times \mathscr{H}$, when the latter is equipped with the product topology.
In addition, if $T_{1}$ and $T$ are operators on $\mathscr{H}$ such that $\Gamma(T) \subset \Gamma(T_{1})$, then $T_{1}$ is said to be an extension of $T$.
So far, so good. Now, consider the following definition.
Definition: An operator $T$ is closable if it has a closed extension. Every closable operator has a smallest closed extension, called its closure, which we denote by $\bar{T}$.
So, my first question is: what is the precise meaning of "smallest"? Does it mean that every other closed extension $T_{1}$ of $T$ is such that $\Gamma(\bar{T})\subset \Gamma(T_{1})$? In other words, every closed extension of $T$ is also a closed extension of $\bar{T}$?
As a second question: how to be sure that a smallest closed extension exists? And is it unique or not necessarily?
Best Answer
Your intuition on what is meant by the smallest closed extension is correct.
We can identify operators with their graphs which are subsets of $\mathscr{H} \times \mathscr{H}$. As such, the set of such graphs (and hence operators) inherits a partial order from the power set of $\mathscr{H}\times\mathscr{H}$ given by inclusion. This means that a smallest closed extension is a minimal element of the set of closed extensions for this partial order i.e. a set $\Gamma(\bar T)$ which is the graph of a closed extension of $T$ and is such that if $T_1$ is a closed extension of $T$ then $\Gamma(T) \subseteq \Gamma(\bar T) \subseteq \Gamma(T_1)$.
It remains to see that whenever $T$ is closable, a smallest closed extension exists. For this, note that a linear subspace $\Gamma \subseteq \mathscr{H} \times \mathscr{H}$ is the graph of an operator if and only if it has the property that whenever $(0, y) \in \Gamma$, one has that $y = 0$.
In particular, if $\Gamma$ is a linear subspace of $\Gamma(T_1)$ for some linear operator $T_1$ then $\Gamma$ is the graph of a linear operator since it is easy to see that the above property is inherited by subspaces.
As a result, one can check that the smallest closed extension of a closable linear operator $T$ is the operator with graph $\overline{\Gamma(T)}$ which is the graph of an operator since $\overline{\Gamma(T)} \subseteq \Gamma(T_1)$ for any closed extension $T_1$ of $T$.