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

[math]\displaystyle{ \langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname{dom}\ T }[/math]

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

[math]\displaystyle{ [T \phi](x) = -\sum_{i,j} \partial_{x_i} \{a_{i j}(x) \partial_{x_j} \phi(x)\} \quad x \in U, \phi \in \operatorname{C}_c^\infty(U), }[/math]

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

[math]\displaystyle{ \operatorname{C}_c^\infty(U) \subseteq L^2(U). }[/math]

If for each x ∈ U the n × n matrix

[math]\displaystyle{ \begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \cdots & a_{1 n}(x) \\ a_{2 1}(x) & a_{2 2} (x) & \cdots & a_{2 n}(x) \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1}(x) & a_{n 2}(x) & \cdots & a_{n n}(x) \end{bmatrix} }[/math]

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

[math]\displaystyle{ \sum_{i, j} a_{i j }(x) c_i \overline{c_j} \geq 0 }[/math]

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

[math]\displaystyle{ \operatorname{Q}(\xi, \eta) = \langle \xi \mid T \eta \rangle + \langle \xi \mid \eta \rangle }[/math]

is a sesquilinear form on dom T and

[math]\displaystyle{ \operatorname{Q}(\xi, \xi) = \langle \xi \mid T \xi\rangle + \langle \xi \mid \xi \rangle \geq \|\xi\|^2. }[/math]

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

[math]\displaystyle{ \operatorname{dom} T \rightarrow H }[/math]

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

Define an operator A by

[math]\displaystyle{ \operatorname{dom}\ A = \{\xi \in H_1: \phi_\xi: \eta \mapsto \operatorname{Q}(\xi, \eta) \mbox{ is bounded linear.} \} }[/math]

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

[math]\displaystyle{ \operatorname{Q}(\xi,\eta) = \langle A \xi \mid \eta \rangle \quad \eta \in H_1 }[/math]

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 :[math]\displaystyle{ L:H_1\rightarrow H }[/math] be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence [math]\displaystyle{ LL^*:H\rightarrow H }[/math] is a bounded injective operator with dense image, where [math]\displaystyle{ L^* }[/math] is the adjoint of [math]\displaystyle{ L }[/math] as an operator between abstract Hilbert spaces. Therefore the operator [math]\displaystyle{ A:=(LL^*)^{-1} }[/math] is a non-negative self-adjoint operator whose domain is the image of [math]\displaystyle{ LL^* }[/math]. Then [math]\displaystyle{ A-I }[/math] 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

[math]\displaystyle{ T \leq S }[/math]

if, and only if,

• [math]\displaystyle{ \operatorname{dom}(S^{1/2}) \subseteq \operatorname{dom}(T^{1/2}) }[/math]
• [math]\displaystyle{ \langle T^{1/2} \xi \mid T^{1/2} \xi \rangle \leq \langle S^{1/2} \xi \mid S^{1/2} \xi \rangle \quad \forall \xi \in \operatorname{dom}(S^{1/2}) }[/math]

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

[math]\displaystyle{ T_{\mathrm{min}} \leq T_{\mathrm{max}}, }[/math]

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

[math]\displaystyle{ T_{\mathrm{min}} \leq S \leq T_{\mathrm{max}}. }[/math]

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.