Friedrichs Extension - Definition of Friedrichs Extension

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 φ_ξ 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.