Graph of an operator

Remarks

It is the graph of the function $T \colon D(T) \to Y$. Since $T$ is linear, $\Gamma(T)$ is a linear subspace of $X \times Y$.

Example: for $X = Y = \R$ and $Tx = 3x$ with $D(T) = \R$, the graph is the line $\set{(x, 3x) : x \in \R}$.

Non-example: the subspace $\set{(0, y) : y \in \R}$ of $\R \times \R$ is not the graph of any operator, since it contains $(0, 1)$, while every graph contains only $(0, 0)$ above $x = 0$.

A linear subspace $G$ of $X \times Y$ is the graph of an operator if and only if $(0, y) \in G$ implies $y = 0$. Then $D(T)$ is the projection of $G$ onto $X$, and $Tx$ is the unique $y$ with $(x, y) \in G$. In this way $S \subseteq T$ holds exactly when $\Gamma(S) \subseteq \Gamma(T)$.