Friedrichs extension

In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show. An operator T is non-negative if

$$\begin{gathered} ⟨\xi \mid T\xi ⟩\ge 0\xi \in \mathrm{dom} \\ T \end{gathered}$$

Examples

Example. Multiplication by a non-negative function on an L² space is a non-negative self-adjoint operator. Example. Let U be an open set in Rⁿ. On L²(U) we consider differential operators of the form

$[T\varphi ](x)=-\sum _{i,j}{\partial }_{x_i}\{a_{ij}(x){\partial }_{x_j}\varphi (x)\}x\in U,\varphi \in {\mathrm{C}}_c^{\mathrm{\infty }}(U),$

where the functions a_{i j} are infinitely differentiable real-valued functions on U. We consider T acting on the dense subspace of infinitely differentiable complex-valued functions of compact support, in symbols

${\mathrm{C}}_c^{\mathrm{\infty }}(U)\subseteq L^2(U).$

If for each x ∈ U the n × n matrix

$\left[\begin{array}{cccc}{a}_{11}(x)& a_{12}(x)& \cdots & a_{1n}(x)\\ a_{21}(x)& a_{22}(x)& \cdots & a_{2n}(x)\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}(x)& a_{n2}(x)& \cdots & a_{nn}(x)\end{array}\right]$

is non-negative semi-definite, then T is a non-negative operator. This means (a) that the matrix is hermitian and

$\sum _{i,j}{a}_{ij}(x)c_i\overline{c_j}\ge 0$

for every choice of complex numbers c₁, ..., c_n. This is proved using integration by parts. These operators are elliptic although in general elliptic operators may not be non-negative. They are however bounded from below.

Definition of Friedrichs extension

The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces. If T is non-negative, then

$\mathrm{Q}(\xi ,\eta )=⟨\xi \mid T\eta ⟩+⟨\xi \mid \eta ⟩$

is a sesquilinear form on dom T and

$\mathrm{Q}(\xi ,\xi )=⟨\xi \mid T\xi ⟩+⟨\xi \mid \xi ⟩\ge \Vert \xi {\Vert }^2.$

Thus Q defines an inner product on dom T. Let H₁ be the completion of dom T with respect to Q. H₁ is an abstractly defined space; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom T. It is not obvious that all elements in H₁ can be identified with elements of H. However, the following can be proved: The canonical inclusion

$\mathrm{dom}T\to H$

extends to an injective continuous map H₁ → H. We regard H₁ as a subspace of H. Define an operator A by

$$\begin{gathered} \mathrm{dom} \\ A=\{\xi \in H_1:{\varphi }_{\xi }:\eta \mapsto \mathrm{Q}(\xi ,\eta )\text{ is bounded linear.}\} \end{gathered}$$

In the above formula, bounded is relative to the topology on H₁ inherited from H. By the Riesz representation theorem applied to the linear functional φ_ξ extended to H, there is a unique A ξ ∈ H such that

$\mathrm{Q}(\xi ,\eta )=⟨A\xi \mid \eta ⟩\eta \in H_1$

Theorem. A is a non-negative self-adjoint operator such that T₁=A - I extends T. T₁ is the Friedrichs extension of T. Another way to obtain this extension is as follows. Let :$L:H_1\to H$ be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence $L{L}^{*}:H\to H$ is a bounded injective operator with dense image, where $L^{*}$ is the adjoint of $L$ as an operator between abstract Hilbert spaces. Therefore the operator $A:=(L{L}^{*}{)}^{-1}$ is a non-negative self-adjoint operator whose domain is the image of $L{L}^{*}$. Then $A-I$ extends T.

Krein's theorem on non-negative self-adjoint extensions

M. G. Krein has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator T. If T, S are non-negative self-adjoint operators, write

$T\le S$

if, and only if,

• $\mathrm{dom}(S^{1/2})\subseteq \mathrm{dom}(T^{1/2})$
• $⟨T^{1/2}\xi \mid T^{1/2}\xi ⟩\le ⟨S^{1/2}\xi \mid S^{1/2}\xi ⟩\mathrm{\forall }\xi \in \mathrm{dom}(S^{1/2})$

Theorem. There are unique self-adjoint extensions T_min and T_max of any non-negative symmetric operator T such that

$T_{\mathrm{m}\mathrm{i}\mathrm{n}}\le T_{\mathrm{m}\mathrm{a}\mathrm{x}},$

and every non-negative self-adjoint extension S of T is between T_min and T_max, i.e.

$T_{\mathrm{m}\mathrm{i}\mathrm{n}}\le S\le T_{\mathrm{m}\mathrm{a}\mathrm{x}}.$

See also

• Energetic extension
• Extensions of symmetric operators

References

• N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman, 1981.