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
Example. Multiplication by a non-negative function on an L<sup>2</sup> space is a non-negative self-adjoint operator.
Example. Let U be an open set in R<sup>n</sup>. On L<sup>2</sup>(U) we consider differential operators of the form
where the functions a<sub>i j</sub> 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 â U the n × n matrix
is non-negative semi-definite, then T is a non-negative operator. This means (a) that the matrix is hermitian and
for every choice of complex numbers c<sub>1</sub>, ..., c<sub>n</sub>. 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.
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<sub>1</sub> be the completion of dom T with respect to Q. H<sub>1</sub> 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<sub>1</sub> can be identified with elements of H. However, the following can be proved:
The canonical inclusion
extends to an injective continuous map H<sub>1</sub> â H. We regard H<sub>1</sub> as a subspace of H.
Define an operator A by
In the above formula, bounded is relative to the topology on H<sub>1</sub> inherited from H. By the Riesz representation theorem applied to the linear functional ÃÂ<sub>þ</sub> extended to H, there is a unique A þ â H such that
Theorem. A is a non-negative self-adjoint operator such that T<sub>1</sub>=A - I extends T.
T<sub>1</sub> is the Friedrichs extension of T.
Another way to obtain this extension is as follows. Let : be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence is a bounded injective operator with dense image, where is the adjoint of as an operator between abstract Hilbert spaces. Therefore, the operator is a non-negative self-adjoint operator whose domain is the image of . Then extends T.
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
if, and only if,
Theorem. There are unique self-adjoint extensions T<sub>min</sub> and T<sub>max</sub> of any non-negative symmetric operator T such that
and every non-negative self-adjoint extension S of T is between T<sub>min</sub> and T<sub>max</sub>, i.e.