Graph of a linear operator

If $X$ and $Y$ are Banach spaces, $T$ is closed exactly when $\mathcal G(T)$ is closed in $X\times Y$. Equivalently, whenever $x_n\to x$ and $Tx_n\to y$, one has $x\in D(T)$ and $Tx=y$. The operator is closable exactly when the closure of $\mathcal G(T)$ is itself the graph of an operator; that operator is the closure $\overline T$.

When $X$ and $Y$ are normed spaces, the graph norm on $D(T)$ may be defined by

The map $x\mapsto(x,Tx)$ identifies this normed space with the graph equipped with the corresponding product norm. If $X$ and $Y$ are Banach spaces, $T$ is closed precisely when $D(T)$ is complete in the graph norm.

The same formula can define different operators when assigned different domains, and their graphs then differ. For example, differentiation on $L^2$ has distinct closed realizations determined by boundary conditions. Graph language makes this domain dependence explicit and is therefore essential for unbounded operators.

1. Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976. DOI record. Relevant: Chapter III on closed operators and their graphs.
2. Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. DOI record. Relevant: Chapter 1 on domains, graphs, and closures.