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Friedrichs extension ：ウィキペディア英語版

Friedrichs extension
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\ T

==Examples==

Example. Multiplication by a non-negative function on an ''L''² space is a non-negative self-adjoint operator.
Example. Let ''U'' be an open set in R^''n''. On ''L''²(''U'') we consider differential operators of the form
:

)(x) = -\sum_ \partial_ \ \phi(x)\} \quad x \in U, \phi \in \operatorname_0^\infty(U),

where the functions ''a''_{''i 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
:

\operatorname_0^\infty(U) \subseteq L^2(U).

If for each ''x'' ∈ ''U'' the ''n'' × ''n'' matrix
:

\begin a_(x) & a_(x) & \cdots & a_(x) \\ a_(x) & a_ (x) & \cdots & a_(x) \\ \vdots & \vdots & \ddots & \vdots \\ a_(x) & a_(x) & \cdots & a_(x)  \end

is non-negative semi-definite, then ''T'' is a non-negative operator. This means (a) that the matrix is hermitian and
:

\sum_ a_(x) c_i \overline \geq 0

for every choice of complex numbers ''c''₁, ..., ''c''_n. 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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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