Adjacency operator: различия между версиями

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

A directed [[infinite graph]] <math>\,G</math> is a pair of the set <math>\,V</math> of the

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

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

* <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 '''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 '''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 '''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} = 0</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>.

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