- Friedrichs extension
functional analysis, the Friedrichs extension is a canonicalself-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-adjointor whose essential self-adjointness is difficult to show.
An operator "T" is non-negative if
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
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
If for each "x" ∈ "U" the "n" × "n" matrix
is non-negative semi-definite, then "T" is a non-negative operator. This means (a) that the matrix is
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
is a sesquilinear form on dom "T" and
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
extends to an "injective" continuous map "H"1 → "H". We regard "H"1 as a subspace of "H".
Define an operator "A" by
In the above formula, "bounded" is relative to the topology on "H"1 inherited from "H". By the
Riesz representation theoremapplied to the linear functional φξ extended to "H", there is a unique "A" ξ ∈ "H" such that
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. Kreinhas 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
if, and only if,
Theorem. There are unique self-adjoint extensions "T"min and "T"max of any non-negative symmetric operator "T" such that
and every non-negative self-adjoint extension "S" of "T" is between "T"min and "T"max, i.e
The Friedrichs extension of "T" is "T"min.
Extensions of symmetric operators
* N. I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space", Pitman, 1981.
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