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

:$\langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname\ T$

Examples

Example. Multiplication by a non-negative function on an ''L''² space is a non-negative self-adjoint operator.

Example. Let ''U'' be an open set in R^''n''. On ''L''²(''U'') we consider differential operators of the form

: \phix) = -\sum_ \partial_ \ \quad x \in U, \phi \in \operatorname_c^\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

:$\operatorname_c^\infty(U) \subseteq L^2(U).$

If for each ''x'' ∈ ''U'' the ''n'' × ''n'' matrix

:$\begin a_(x) & a_(x) & \cdots & a_(x) \\ a_(x) & a_ (x) & \cdots & a_(x) \\ \vdots & \vdots & \ddots & \vdots \\ a_(x) & a_(x) & \cdots & a_(x) \end$

is non-negative semi-definite, then ''T'' is a non-negative operator. This means (a) that the matrix is hermitian and

:$\sum_ a_(x) c_i \overline \geq 0$

for every choice of complex numbers ''c''₁, ..., ''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

:$\operatorname(\xi, \eta) = \langle \xi \mid T \eta \rangle + \langle \xi \mid \eta \rangle$

is a sesquilinear form on dom ''T'' and

:$\operatorname(\xi, \xi) = \langle \xi \mid T \xi\rangle + \langle \xi \mid \xi \rangle \geq \, \xi\, ^2.$

Thus Q defines an inner product on dom ''T''. Let ''H''₁ be the completion of dom ''T'' with respect to Q. ''H''₁ 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 ''H''₁ can be identified with elements of ''H''. However, the following can be proved:

The canonical inclusion

:$\operatorname T \rightarrow H$

extends to an ''injective'' continuous map ''H''₁ → ''H''. We regard ''H''₁ as a subspace of ''H''.

Define an operator ''A'' by

: $\operatorname\ A = \$

In the above formula, ''bounded'' is relative to the topology on ''H''₁ 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

:$\operatorname(\xi,\eta) = \langle A \xi \mid \eta \rangle \quad \eta \in H_1$

Theorem. ''A'' is a non-negative self-adjoint operator such that ''T''₁=''A'' - I extends ''T''.

''T''₁ is the Friedrichs extension of ''T''.

Another way to obtain this extension is as follows. Let :$L:H_1\rightarrow H$ be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence $LL^*:H\rightarrow H$ 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 $A:=(LL^*)^$ is a non-negative self-adjoint operator whose domain is the image of $LL^*$. 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

:$T \leq S$

if, and only if,

* $\operatorname(S^) \subseteq \operatorname(T^)$
* $\langle T^ \xi \mid T^ \xi \rangle \leq \langle S^ \xi \mid S^ \xi \rangle \quad \forall \xi \in \operatorname(S^)$

Theorem. There are unique self-adjoint extensions ''T''_min and ''T''_max of any non-negative symmetric operator ''T'' such that

:$T_ \leq T_,$

and every non-negative self-adjoint extension ''S'' of ''T'' is between ''T''_min and ''T''_max, i.e.

:$T_ \leq S \leq T_.$

See also

* Energetic extension
* Extensions of symmetric operators

Notes

{{Reflist

References

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

Operator theory
Linear operators