Closed quadratic form - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space and let $\mathfrak a$ be a densely defined symmetric sesquilinear form, linear in the second variable, with domain $D(\mathfrak a)\subseteq H$ . Suppose $\mathfrak a[u,u]\geq m\|u\|^2$ for some $m\in\mathbb R$ . The associated quadratic form is $q[u]=\mathfrak a[u,u]$ . It is closed if $D(\mathfrak a)$ is complete for the form norm

\|u\|_{\mathfrak a}= \bigl(\mathfrak a[u,u]+(1-m)\|u\|^2\bigr)^{1/2}.

Replacing $m$ by another lower bound gives an equivalent norm. Unlike a closed operator, a closed form is not defined by closure of the graph of the scalar-valued function $q$ .

Representation by an operator

The first representation theorem associates to every densely defined closed lower-semibounded form $\mathfrak a$ a unique self-adjoint operator $A$ with $A\geq m$ . For any $\lambda<m$ ,

D(\mathfrak a)=D\bigl((A-\lambda)^{1/2}\bigr)

and

\mathfrak a[u,v] =\langle(A-\lambda)^{1/2}u,(A-\lambda)^{1/2}v\rangle +\lambda\langle u,v\rangle.

Thus form domains can be larger than operator domains while still determining the operator uniquely Kato, Chapter VI, §2.

Closability and completion

A lower-semibounded form is closable when it has a closed extension; its smallest closed extension is its closure. Equivalently, if $u_n\to0$ in $H$ and $(u_n)$ is Cauchy in the form norm, then $\mathfrak a[u_n,u_n]\to0$ after shifting to a nonnegative form. This is a form criterion and is distinct from closability of an operator, even though the two theories interact through associated operators.

Examples and conventions

Every bounded symmetric form on all of $H$ is closed. On $L^2(\Omega)$ , the Dirichlet energy $\mathfrak a[u,v]=\int_\Omega\nabla u\cdot\overline{\nabla v}$ with domain $H_0^1(\Omega)$ is closed; its restriction to $C_c^\infty(\Omega)$ is typically closable but not closed. More general literature defines closed sectorial forms using the real part of a shifted form. Without symmetry or a sectoriality hypothesis, “closed quadratic form” requires an explicit convention and cannot be inferred from the norm above.

References

Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., corrected reprint, Springer, 1995. DOI record. Relevant: Chapter VI, §§1–2 on closed sectorial forms and representation theorems.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VIII on closed semibounded quadratic forms and associated self-adjoint operators.