Friedrichs extension - Imported Knowlpedia Content

Definition

Let $A$ be a densely defined symmetric operator on a complex Hilbert space with $\langle Ax,x\rangle\geq m\lVert x\rVert^2$ for some real $m$ . The form

\mathfrak a[x,y]=\langle Ax,y\rangle,\qquad x,y\in\mathcal D(A),

is closable. The Friedrichs extension $A_F$ is the unique self-adjoint operator associated by the representation theorem to the resulting closed quadratic form. It is a self-adjoint extension of $A$ and satisfies $A_F\geq mI$ . Thus it preserves the original lower bound while replacing the initial operator domain by the domain determined by the closed form.

Construction from the form norm

Choose $c>1-m$ and complete $\mathcal D(A)$ in the norm

\lVert x\rVert_{\mathfrak a,c}^2 =\langle Ax,x\rangle+c\lVert x\rVert^2.

The closure of $\mathfrak a$ has this completion as its form domain. The first representation theorem then produces $A_F$ , characterized by

\overline{\mathfrak a}[x,y]=\langle A_Fx,y\rangle

for $x\in\mathcal D(A_F)$ and every $y$ in the form domain. Different admissible values of $c$ give equivalent form norms and the same extension Kato, Chapter VI, §2.

Canonical but not generally unique

The Friedrichs construction depends only on the semibounded symmetric operator and selects an extension with the same lower bound. It does not say that $A$ has only one self-adjoint extension. Other boundary conditions may produce other semibounded extensions. Among nonnegative self-adjoint extensions, the Friedrichs and Kreĭn–von Neumann extensions occupy opposite extremal positions in the standard form or resolvent order; they coincide when the original operator is essentially self-adjoint.

Example

Let $A=-d^2/dx^2$ on $L^2(0,1)$ with initial domain $C_c^\infty(0,1)$ . Its quadratic form is

\mathfrak a[f,g]=\int_0^1 f'(x)\overline{g'(x)}\,dx.

The form closure has domain $H_0^1(0,1)$ , and its associated operator is the Dirichlet Laplacian with domain $H^2(0,1)\cap H_0^1(0,1)$ . This is the Friedrichs extension; Neumann and periodic Laplacians are other self-adjoint realizations but are not selected by closing this initial form.

References

Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995. DOI record. Relevant: Chapter VI, §§1–2 on closed forms and representation theorems.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012. DOI record. Relevant: Chapter 10 on the Friedrichs extension.