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

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

: [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

References

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


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