Adjacency operator: различия между версиями

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'''Adjacency operator''' --- оператор смежности.  
'''Adjacency operator''' — ''[[оператор смежности]].''


A directed infinite graph <math>G</math> is a pair of the set <math>V</math> of the
A directed [[infinite graph]] <math>\,G</math> is a pair of the set <math>\,V</math> of the
countable vertices and the set <math>E</math> of the arrows (arcs) <math>u \leftarrow
countable [[vertex|vertices]] and the set <math>\,E</math> of the arrows (arcs) <math>u \leftarrow v</math>. Let <math>{\mathcal H}</math> be a Hilbert space <math>\ell^{2}(G)</math> on <math>\,V</math> with a canonical basis <math>\{e_{v}| \; v \in V\}</math>. Since every arrow <math>u \leftarrow v \in E</math> induces a dyad <math>e_{u} \otimes e_{v}</math>, where <math>(x \otimes y)z = \langle z,y\rangle x</math> for <math>x, y, z \in {\mathcal H}</math>, the ''' adjacency operator''' <math>\,A(G)</math> is expressed by
v</math>. Let <math>{\mathcal H}</math> be a Hilbert space <math>\ell^{2}(G)</math> on <math>V</math> with a canonical basis <math>\{e_{v}| \; v \in V\}</math>. Since every arrow <math>u
\leftarrow v \in E</math> induces a dyad <math>e_{u} \otimes e_{v}</math>, where <math>(x
\otimes y)z = \langle z,y\rangle x</math> for <math>x, y, z \in {\mathcal H}</math>, the
''' adjacency operator''' <math>A(G)</math> is expressed by


<math>A(G) = \sum_{u \leftarrow v} e_{u} \otimes e_{v}</math>
<math>A(G) = \sum_{u \leftarrow v} e_{u} \otimes e_{v}</math>


if <math>G</math> has a bounded degree.
if <math>\,G</math> has a bounded degree.


Adjacency operators are classified as follows:
Adjacency operators are classified as follows:


* <math>A</math> is '''self-adjoint''' if <math>A = A^{\ast}</math>.
* <math>\,A</math> is '''self-adjoint''' if <math>A = A^{\ast}</math>.
* <math>A</math> is '''unitary''' if <math>A^{\ast}A = AA^{\ast} = I</math>.
* <math>\,A</math> is '''unitary''' if <math>A^{\ast}A = AA^{\ast} = I</math>.
* <math>A</math> is '''normal''' if <math>A^{\ast}A = AA^{\ast}</math>.
* <math>\,A</math> is '''normal''' if <math>A^{\ast}A = AA^{\ast}</math>.
* <math>A</math> is '''hyponormal''' (resp. '''co-hyponormal''') if <math>A^{\ast}A \geq AA^{\ast}</math> (resp. <math>AA^{\ast} \geq A^{\ast}A</math>).
* <math>\,A</math> is '''hyponormal''' (resp. '''co-hyponormal''') if <math>A^{\ast}A \geq AA^{\ast}</math> (resp. <math>AA^{\ast} \geq A^{\ast}A</math>).
* <math>A</math> is '''projection''' if <math>A = A^{\ast} = A^{2}</math>.
* <math>\,A</math> is '''projection''' if <math>A = A^{\ast} = A^{2}</math>.
* <math>A</math> is '''partial isometry''' if <math>A^{\ast}A</math> and <math>AA^{\ast}</math>.
* <math>\,A</math> is '''partial isometry''' if <math>A^{\ast}A</math> and <math>AA^{\ast}</math>.
* <math>A</math> is '''isometry''' (resp. '''co-isometry''') if <math>A^{\ast}A = I</math> (resp.<math>AA^{\ast} = I</math>).
* <math>\,A</math> is '''isometry''' (resp. '''co-isometry''') if <math>A^{\ast}A = I</math> (resp.<math>AA^{\ast} = I</math>).
* <math>A</math> is '''nilpotent''' if there exists a number <math>n</math> such that <math>A^{n} =
* <math>\,A</math> is '''nilpotent''' if there exists a number <math>\,n</math> such that <math>\,A^{n} = 0</math>.
0</math>.
* <math>\,A</math> is '''idempotent''' if <math>\,A = A^{2}</math>.
* <math>A</math> is '''idempotent''' if <math>A = A^{2}</math>.
* <math>\,A</math> is '''positive''' if <math>(Ax|x) \geq 0</math> for <math>x \in H</math>.
* <math>A</math> is '''positive''' if <math>(Ax|x) \geq 0</math> for <math>x \in H</math>.


Here <math>G^{\ast}</math> is the '' adjoint'' graph for a graph <math>G</math>.
Here <math>G^{\ast}</math> is the '' adjoint'' graph for a graph <math>\,G</math>.

Текущая версия от 12:29, 11 ноября 2011

Adjacency operatorоператор смежности.

A directed infinite graph [math]\displaystyle{ \,G }[/math] is a pair of the set [math]\displaystyle{ \,V }[/math] of the countable vertices and the set [math]\displaystyle{ \,E }[/math] of the arrows (arcs) [math]\displaystyle{ u \leftarrow v }[/math]. Let [math]\displaystyle{ {\mathcal H} }[/math] be a Hilbert space [math]\displaystyle{ \ell^{2}(G) }[/math] on [math]\displaystyle{ \,V }[/math] with a canonical basis [math]\displaystyle{ \{e_{v}| \; v \in V\} }[/math]. Since every arrow [math]\displaystyle{ u \leftarrow v \in E }[/math] induces a dyad [math]\displaystyle{ e_{u} \otimes e_{v} }[/math], where [math]\displaystyle{ (x \otimes y)z = \langle z,y\rangle x }[/math] for [math]\displaystyle{ x, y, z \in {\mathcal H} }[/math], the adjacency operator [math]\displaystyle{ \,A(G) }[/math] is expressed by

[math]\displaystyle{ A(G) = \sum_{u \leftarrow v} e_{u} \otimes e_{v} }[/math]

if [math]\displaystyle{ \,G }[/math] has a bounded degree.

Adjacency operators are classified as follows:

  • [math]\displaystyle{ \,A }[/math] is self-adjoint if [math]\displaystyle{ A = A^{\ast} }[/math].
  • [math]\displaystyle{ \,A }[/math] is unitary if [math]\displaystyle{ A^{\ast}A = AA^{\ast} = I }[/math].
  • [math]\displaystyle{ \,A }[/math] is normal if [math]\displaystyle{ A^{\ast}A = AA^{\ast} }[/math].
  • [math]\displaystyle{ \,A }[/math] is hyponormal (resp. co-hyponormal) if [math]\displaystyle{ A^{\ast}A \geq AA^{\ast} }[/math] (resp. [math]\displaystyle{ AA^{\ast} \geq A^{\ast}A }[/math]).
  • [math]\displaystyle{ \,A }[/math] is projection if [math]\displaystyle{ A = A^{\ast} = A^{2} }[/math].
  • [math]\displaystyle{ \,A }[/math] is partial isometry if [math]\displaystyle{ A^{\ast}A }[/math] and [math]\displaystyle{ AA^{\ast} }[/math].
  • [math]\displaystyle{ \,A }[/math] is isometry (resp. co-isometry) if [math]\displaystyle{ A^{\ast}A = I }[/math] (resp.[math]\displaystyle{ AA^{\ast} = I }[/math]).
  • [math]\displaystyle{ \,A }[/math] is nilpotent if there exists a number [math]\displaystyle{ \,n }[/math] such that [math]\displaystyle{ \,A^{n} = 0 }[/math].
  • [math]\displaystyle{ \,A }[/math] is idempotent if [math]\displaystyle{ \,A = A^{2} }[/math].
  • [math]\displaystyle{ \,A }[/math] is positive if [math]\displaystyle{ (Ax|x) \geq 0 }[/math] for [math]\displaystyle{ x \in H }[/math].

Here [math]\displaystyle{ G^{\ast} }[/math] is the adjoint graph for a graph [math]\displaystyle{ \,G }[/math].