On semigroups of normal matrices (Q1399241)

Semigroups of normal complex square matrices having a finite spectrum are studied. It is shown that every member \(A\) of such a semigroup satisfies the condition \(A^*= A^n\), where \(n\) does not depend on the matrix \(A\). It is shown that such a semigroup is completely reducible and each irreducible contraction consists of a group of unitary matrices and, eventually, the zero matrix. In Section 2 it is concerned with the equivalence between the semigroups having finite spectrum and semigroups of power-Hermitian matrices. In Section 3 it is concerned with the reducibility of semigruops of normal matrices.