Graph norm - Imported Knowlpedia Content

Definition

Let $T:D(T)\subseteq X\to Y$ be a linear operator between normed vector spaces. The graph norm of $T$ is the norm on $D(T)$ defined by

\lVert x\rVert_T=\lVert x\rVert_X+\lVert Tx\rVert_Y.

Equivalently, it is the norm transported from the graph of $T$ by the linear bijection $x\mapsto(x,Tx)$ , when $X\times Y$ is given the sum norm. The graph norm is stronger than the norm inherited from $X$ : convergence $x_n\to x$ in graph norm means both $x_n\to x$ in $X$ and $Tx_n\to Tx$ in $Y$ .

Completeness and closedness

If $X$ and $Y$ are Banach spaces, then $T$ is closed exactly when $(D(T),\lVert\cdot\rVert_T)$ is a Banach space. Indeed, the graph map identifies the operator domain isometrically with $\Gamma(T)\subseteq X\times Y$ , and a linear subspace of a Banach space is complete precisely when it is closed.

Equivalent choices

When $X$ and $Y$ are Hilbert spaces, one often uses

\lVert x\rVert_{T,2} =\bigl(\lVert x\rVert_X^2+\lVert Tx\rVert_Y^2\bigr)^{1/2}.

This norm is equivalent to the sum graph norm and comes from the graph inner product

\langle x,z\rangle_T=\langle x,z\rangle_X+\langle Tx,Tz\rangle_Y.

The two formulas define the same topology on $D(T)$ , though only the second is induced by an inner product.

Cores

For a closed operator $T$ , a subspace $D_0\subseteq D(T)$ is a core when it is dense in $D(T)$ for the graph norm. Equivalently, the restriction $T|_{D_0}$ is closable and its closure is $T$ . Ordinary density of $D_0$ in $X$ is not enough, because it does not control convergence of the images $Tx$ .

References

Tosio Kato, Perturbation Theory for Linear Operators, Springer, 1995 reprint of the 2nd ed. Springer DOI record. Relevant: Chapter III.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. Springer DOI record. Relevant: Chapter 1.