In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator.
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 hermitian and for every choice of complex numbers c1, ..., cn.
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 H1 be the completion of dom T with respect to Q. H1 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 H1 can be identified with elements of H. However, the following can be proved: The canonical inclusion extends to an injective continuous map H1 → H. We regard H1 as a subspace of H. Define an operator A by In the above formula, bounded is relative to the topology on H1 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 Theorem.
The inclusion is a bounded injective with dense image.
is a bounded injective operator with dense image, where
is a non-negative self-adjoint operator whose domain is the image of
There are unique self-adjoint extensions Tmin and Tmax of any non-negative symmetric operator T such that and every non-negative self-adjoint extension S of T is between Tmin and Tmax, i.e.