Adjacency operator

Adjacency operator --- оператор 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].