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 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 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. A is a non-negative self-adjoint operator such that T1=A - I extends T.
T1 is the Friedrichs extension of T.
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