The Dual Inequality for Adjoint Products and the Extremal Equality Cases

Let S⊂Mm×n(C) be a linear subspace, and let L(S)=Span⁡{AB†:A,B∈S) and R(S)=Span{A†B:A,B∈S}. We prove the general inequalitymax⁡{dim⁡L(S),dim⁡R(S)}≥dim⁡S, with an elementary proof based on choosing an element of maximal rank and comparing the losses of the two multiplication maps with the compensating directions on the dual side. The argument yields the quantitative refinement max⁡{dim⁡L,dim⁡R}≥dim⁡S+∣a−b ∣, where a and b are the kernel dimensions of the two multiplication maps, and shows that extremal equality forces a=b. We further study the equality case: we exhibit a counterexample showing that extremal equality does not imply the ternary closure SS†S⊆S, and we classify completely the extremal equality in the diagonal case, obtaining that it holds if and only if the non-zero evaluation vectors are distributed along exactly d distinct projective lines.