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
where the functions ai j 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 ∈ 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. 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.
Read more about this topic: Friedrichs Extension
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