Friedrichs extension

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 angle geq 0 quad xi in operatorname{dom} T


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

: [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) \ 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}

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 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{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 as equivalence classes of Cauchy sequences 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 the Riesz representation theorem applied to the linear functional φξ 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


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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • Extensions of symmetric operators — In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self adjoint extensions. This problem arises, for… …   Wikipedia

  • Self-adjoint operator — In mathematics, on a finite dimensional inner product space, a self adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.… …   Wikipedia

  • List of functional analysis topics — This is a list of functional analysis topics, by Wikipedia page. Contents 1 Hilbert space 2 Functional analysis, classic results 3 Operator theory 4 Banach space examples …   Wikipedia

  • Mollifier — A mollifier (top) in dimension one. At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version. In mathematics, mollifiers (also known as approximations to the identity) are smooth… …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • Germany — • History divided by time periods, beginning with before 1556 Catholic Encyclopedia. Kevin Knight. 2006. Germany     Germany     † …   Catholic encyclopedia

  • Prussia — • The Kingdom of Prussia covers 134,616 square miles and includes about 64.8 per cent of the area of the German Empire. Catholic Encyclopedia. Kevin Knight. 2006. Prussia     Prussia      …   Catholic encyclopedia

  • FORME — L’histoire du concept de forme et des théories de la forme est des plus singulières. Nous vivons dans un monde constitué de formes naturelles. Celles ci sont omniprésentes dans notre environnement et dans les représentations que nous nous en… …   Encyclopédie Universelle

  • Operations Magazine — provides Multi Man Publishing with its own house organ for articles and discussion of its wargaming products. The first issue was produced in the summer of 1991 by The Gamers and was printed regularly until The Gamers were taken over by MMP. The… …   Wikipedia