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Concept# Friedrichs extension

Summary

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
: \langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname{dom}\ T
Examples
Example. Multiplication by a non-negative function on an L2 space is a non-negative self-adjoint operator.
Example. Let U be an open set in Rn. On L2(U) we consider differential operators of the form
: T \phi = -\sum_{i,j} \partial_{x_i} {a_{i j}(x) \partial_{x_j} \phi(x)} \quad x \in U, \phi \in \operatorname{C}_c^\infty(U),
where the functions ai j are infinitely differentiable real-valued functions on U. We consider T acting on the dense subspace of infinit

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