- Friedrichs extension
In

functional analysis , the**Friedrichs extension**is acanonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematicianKurt Friedrichs . This extension is particularly useful in situations where an operator may fail to beessentially self-adjoint or whose essential self-adjointness is difficult to show.An operator "T" is non-negative if

:$langle\; xi\; mid\; T\; xi\; angle\; geq\; 0\; quad\; xi\; in\; operatorname\{dom\}\; T$

**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 considerdifferential operator s of the form:$[T\; phi]\; (x)\; =\; -sum\_\{i,j\}\; partial\_\{x\_i\}\; \{a\_\{i\; j\}(x)\; partial\_\{x\_j\}\; phi(x)\}\; quad\; x\; in\; U,\; phi\; in\; operatorname\{C\}\_0^infty(U),$

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:$operatorname\{C\}\_0^infty(U)\; subseteq\; L^2(U).$

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

:$egin\{bmatrix\}\; a\_\{1\; 1\}(x)\; a\_\{1\; 2\}(x)\; cdots\; a\_\{1\; n\}(x)\; \backslash \; a\_\{2\; 1\}(x)\; a\_\{2\; 2\}\; (x)\; cdots\; a\_\{2\; n\}(x)\; \backslash \; vdots\; vdots\; ddots\; vdots\; \backslash \; a\_\{n\; 1\}(x)\; a\_\{n\; 2\}(x)\; cdots\; a\_\{n\; n\}(x)\; end\{bmatrix\}$

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

hermitian and:$sum\_\{i,\; j\}\; a\_\{i\; j\; \}(x)\; c\_i\; overline\{c\_j\}\; geq\; 0$

for every choice of complex numbers "c"

_{1}, ..., "c"_{n}. This is proved usingintegration 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

:$operatorname\{Q\}(xi,\; eta)\; =\; langle\; xi\; mid\; T\; eta\; angle\; +\; langle\; xi\; mid\; eta\; angle$

is a sesquilinear form on dom "T" and

:$operatorname\{Q\}(xi,\; xi)\; =\; langle\; xi\; mid\; T\; xi\; angle\; +\; langle\; xi\; mid\; xi\; angle\; geq\; |xi|^2.$

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 asequivalence class es ofCauchy sequence s of elements of dom "T". It is not obvious that all elements in "H"_{1}can identified with elements of "H". However, the following can be proved:The canonical inclusion

:$operatorname\{dom\}\; T\; ightarrow\; H$

extends to an "injective" continuous map "H"

_{1}→ "H". We regard "H"_{1}as a subspace of "H".Define an operator "A" by

: $operatorname\{dom\}\; A\; =\; \{xi\; in\; H\_1:\; phi\_xi:\; eta\; mapsto\; operatorname\{Q\}(xi,\; eta)\; mbox\{\; is\; bounded\; linear.\}\; \}$

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

_{1}inherited from "H". By theRiesz representation theorem applied to the linear functional φ_{ξ}extended to "H", there is a unique "A" ξ ∈ "H" such that:$operatorname\{Q\}(xi,eta)\; =\; langle\; A\; xi\; mid\; eta\; angle\; quad\; eta\; in\; H\_1$

**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".**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 operators, write

:$T\; leq\; S$

if, and only if,

* $operatorname\{dom\}(S)\; subseteq\; operatorname\{dom\}(T)$

* $langle\; xi\; mid\; T\; xi\; angle\; leq\; langle\; xi\; mid\; S\; xi\; angle\; quad\; forall\; xi\; in\; operatorname\{dom\}(S)$

**Theorem**. There are unique self-adjoint extensions "T"_{min}and "T"_{max}of any non-negative symmetric operator "T" such that:$T\_\{mathrm\{min\; leq\; T\_\{mathrm\{max,$

and every non-negative self-adjoint extension "S" of "T" is between "T"

_{min}and "T"_{max}, i.e:$T\_\{mathrm\{min\; leq\; S\; leq\; T\_\{mathrm\{max.$

The Friedrichs extension of "T" is "T"

_{min}.**ee also***

Energetic extension

*Extensions of symmetric operators **References*** N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space", Pitman, 1981.

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