Bivector

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

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

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

$[T\varphi ](x)=-\sum _{i,j}{\partial }_{x_i}\{a_{ij}(x){\partial }_{x_j}\varphi (x)\}x\in U,\varphi \in {\mathrm{C}}_0^{\mathrm{\infty }}(U),$

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

${\mathrm{C}}_0^{\mathrm{\infty }}(U)\subseteq L^2(U).$

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 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

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

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.

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.

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

The Friedrichs extension of T is T_max.

See also

• Energetic extension
• Extensions of symmetric operators

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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