Automatic selfadjoint-ideal semigroups for finite matrices

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