Closure of a closable operator

Let $T:\mathcal D(T)\subseteq X\to Y$ be a closable operator between Banach spaces. Its closure $\overline T$ is the operator whose graph is the closure of the graph of $T$ in $X\times Y$ :

\mathcal G(\overline T)=\overline{\mathcal G(T)}.

Equivalently, $x\in\mathcal D(\overline T)$ precisely when some sequence $x_n\in\mathcal D(T)$ satisfies $x_n\to x$ and $Tx_n\to y$ for some $y\in Y$ ; then $\overline Tx=y$ . Closability is exactly what makes this limit $y$ unique. The operator $\overline T$ is closed, extends $T$ , and is contained in every closed extension of $T$ .

Graph characterization

The closure operation adds exactly the limit pairs forced by the original graph. In particular,

\mathcal D(\overline T) =\left\{x:\exists x_n\in\mathcal D(T),\ x_n\to x,\ (Tx_n)\text{ converges}\right\}.

The value of $\overline T$ is determined by the limit of $Tx_n$ , and does not depend on the chosen approximating sequence. If the closed graph contains a nonzero pair $(0,y)$ , no operator can have that graph and $T$ is not closable.

Relation to adjoints

For a densely defined operator between Hilbert spaces, $T$ is closable exactly when the domain of its adjoint $T^*$ is dense. In that case,

\overline T=T^{**}.

This identity includes the domains: it is not merely equality of the formal actions. The adjoint $T^*$ itself is always closed.

Example and warning

On $L^2(\mathbb R)$ , restrict multiplication by the coordinate $x$ to compactly supported smooth functions. Its closure is the maximal multiplication operator

(\overline Tf)(x)=xf(x),\qquad \mathcal D(\overline T)=\{f\in L^2(\mathbb R):xf\in L^2(\mathbb R)\}.

Other closed extensions can exist, especially for differential operators with boundary conditions. The closure is the smallest closed extension, not an arbitrarily chosen self-adjoint extension.

References