Adjacency operator

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Adjacency operatorоператор смежности.

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

A(G) = \sum_{u \leftarrow v} e_{u} \otimes e_{v}

if \,G has a bounded degree.

Adjacency operators are classified as follows:

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

Here G^{\ast} is the adjoint graph for a graph \,G.