  The notion of automatic selfadjointness of all ideals in a multiplicative
semigroup of the bounded linear operators on a separable Hilbert space B(H)
arose in a 2015 discussion with Heydar Radjavi who pointed out that B(H) and
the finite rank operators F(H) possessed this unitary invariant property which
category we named SI semigroups (for automatic selfadjoint ideal semigroups).
Equivalent to the SI property is the solvability, for each A in the semigroup,
of the bilinear operator equation A^* = XAY which we believe is a new
connection relating the semigroup theory with the theory of operator equations.
We found in our earlier works in the subject that even at the basic level of
singly generated semigroups, the investigation of SI semigroups led to
interesting algebraic and analytic phenomena when generated by rank one
operators, normal operators, partial and power partial isometries,
subnormal-hyponormal-essentially normal operators, and weighted shift
operators; and generated by commuting families of normal operators. In this
paper, we focus on a separate M_n(C) treatment for singly generated SI
semigroups that requires studying the solvability of the bilinear matrix
equation A^* = XAY in a multiplicative semigroup of finite matrices. This
separate focus is needed because the techniques employed in our earlier works
we could not adapt to finite matrices. In this paper we find that for certain
classes of generators, being a partial isometry is equivalent to generating an
SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts,
and non-normal Jordan matrices. For the key tools used to establish these
equivalences, we developed a number of necessary conditions for singly
generated semigroups to be SI for the very general classes: nonselfadjoint
matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and
Jordan blocks.
