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In [[functional analysis]], the '''Friedrichs extension''' is a [[canonical form|canonical]] [[self-adjoint operator|self-adjoint]] extension of a non-negative densely defined [[self-adjoint operator#Symmetric operators|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.
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An operator ''T'' is non-negative if
 
:<math> \langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname{dom}\ T </math>
 
==Examples==
 
'''Example'''.  Multiplication by a non-negative function on an ''L''<sup>2</sup> space is a non-negative self-adjoint operator.
 
'''Example'''.  Let ''U'' be an open set in '''R'''<sup>''n''</sup>.  On ''L''<sup>2</sup>(''U'') we consider [[differential operator]]s of the form
 
:<math> [T \phi](x) = -\sum_{i,j} \partial_{x_i} \{a_{i j}(x) \partial_{x_j} \phi(x)\} \quad x \in U, \phi \in \operatorname{C}_0^\infty(U), </math>
 
where the functions ''a''<sub>''i j''</sub> 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
 
:<math> \operatorname{C}_0^\infty(U) \subseteq L^2(U). </math>
 
If for each ''x'' ∈ ''U'' the ''n'' &times; ''n'' matrix
 
:<math> \begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \cdots & a_{1 n}(x) \\ a_{2 1}(x) & a_{2 2} (x) & \cdots & a_{2 n}(x) \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1}(x) & a_{n 2}(x) & \cdots & a_{n n}(x)  \end{bmatrix} </math>
 
is non-negative semi-definite, then ''T'' is a non-negative operator. This means (a) that the matrix is [[hermitian matrix|hermitian]] and
 
:<math> \sum_{i, j} a_{i j }(x) c_i \overline{c_j} \geq 0 </math>
 
for every choice of complex numbers ''c''<sub>1</sub>, ..., ''c''<sub>n</sub>. This is proved using [[integration by parts]].  
 
These operators are [[elliptic operator|elliptic]] although in general elliptic operators may not be non-negative. They are however bounded from below.
 
== 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
 
:<math> \operatorname{Q}(\xi, \eta) = \langle \xi \mid T \eta \rangle + \langle  \xi \mid \eta \rangle </math>
 
is a sesquilinear form on dom ''T'' and
 
:<math> \operatorname{Q}(\xi, \xi) = \langle \xi \mid T \xi\rangle + \langle  \xi \mid \xi \rangle \geq \|\xi\|^2.</math>
 
Thus Q defines an inner product on dom ''T''. Let ''H''<sub>1</sub> be the [[complete space|completion]] of dom ''T'' with respect to Q. ''H''<sub>1</sub> is an abstractly defined space; for instance its elements can be represented as [[equivalence class]]es of [[Cauchy sequence]]s of elements of dom ''T''.   It is not obvious that all elements in ''H''<sub>1</sub> can identified with elements of ''H''.  However, the following can be proved:
 
The canonical inclusion
 
:<math> \operatorname{dom}\ T \rightarrow H </math>
 
extends to an ''injective'' continuous map ''H''<sub>1</sub> → ''H''.  We regard ''H''<sub>1</sub> as a subspace of ''H''.
 
Define an operator ''A'' by
 
: <math> \operatorname{dom}\ A = \{\xi \in H_1: \phi_\xi: \eta \mapsto \operatorname{Q}(\xi, \eta) \mbox{ is bounded linear.} \} </math>
 
In the above formula, ''bounded'' is relative to the topology on ''H''<sub>1</sub> inherited from ''H''. By the [[Riesz representation theorem]]  applied to the linear functional φ<sub>ξ</sub> extended to ''H'', there is a unique ''A'' ξ ∈ ''H'' such that
 
:<math> \operatorname{Q}(\xi,\eta) = \langle  A \xi \mid  \eta \rangle \quad \eta \in H_1 </math>
 
'''Theorem'''. ''A'' is a non-negative self-adjoint operator such that ''T''<sub>1</sub>=''A'' - I extends ''T''.
 
''T''<sub>1</sub> is the Friedrichs extension of ''T''.
 
== Krein's theorem on non-negative self-adjoint extensions ==
 
[[M. G. Krein]] has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator ''T''.
 
If ''T'', ''S'' are non-negative self-adjoint operators, write
 
:<math> T \leq S </math>
 
if, and only if,
 
* <math> \operatorname{dom}(S^{1/2}) \subseteq \operatorname{dom}(T^{1/2}) </math>
 
* <math> \langle T^{1/2} \xi \mid T^{1/2} \xi \rangle \leq  \langle S^{1/2} \xi \mid S^{1/2} \xi \rangle \quad \forall \xi \in \operatorname{dom}(S^{1/2}) </math>
 
'''Theorem'''. There are unique self-adjoint extensions ''T''<sub>min</sub> and ''T''<sub>max</sub> of any non-negative symmetric operator ''T'' such that
 
:<math> T_{\mathrm{min}} \leq T_{\mathrm{max}}, </math>
 
and every non-negative self-adjoint extension ''S'' of  ''T'' is between ''T''<sub>min</sub> and ''T''<sub>max</sub>, i.e.
 
:<math> T_{\mathrm{min}} \leq S \leq T_{\mathrm{max}}. </math>
 
The Friedrichs extension of ''T'' is ''T''<sub>max</sub>.<ref>A. Alfonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators. J. Operator Theory 4 (1980), 251-270.</ref>
 
==See also==
* [[Energetic extension]]
* [[Extensions of symmetric operators]]
 
== Notes ==
{{Reflist}}
 
==References==
* [[Naum Akhiezer|N. I. Akhiezer]] and I. M. Glazman, ''Theory of Linear Operators in Hilbert Space'', Pitman, 1981.
 
[[Category:Operator theory]]

Latest revision as of 09:42, 2 January 2015

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