  This extends our new study of the automatic selfadjoint ideal property for
B(H)-operator semigroups introduced to us by Heydar Radjavi (SI semigroups for
short). Our investigation here of singly generated SI semigroups led to
unexpected algebraic and analytic phenomena on the simplicity of SI semigroups
and on the spectral density of their generators. In particular: the SI property
yields for a hyponormal operator, zero planar area measure of its approximate
point spectrum; the same for the essential spectrum of an essentially normal
operator; and that SI semigroups generated by unilateral weighted shifts with
periodic nonzero weights are simple. We also characterized the simplicity of
the SI semigroups generated by certain commuting classes of normal operators.
