A Dual Dimension Inequality for Adjoint Products of Matrix Spaces

We study the two adjoint-product spaces associated with a linearsubspace S of the m x n complex matrices: L(S) = span{ A B^dagger : A, B in S }, R(S) = span{ A^dagger B : A, B in S }. The paper proves the universal dimension inequality max{ dim L(S), dim R(S) } >= dim S, valid for arbitrary m, n, and arbitrary linear matrix spaces S, withno assumption on the maximal rank of S and no use of the classicalclassification of spaces of matrices of bounded rank. The proof restson a direct-sum phenomenon: after choosing an element A of S ofmaximal rank, the kernel of one multiplication map produces, afteradjunction and multiplication by A, a subspace of the oppositeadjoint-product space whose support is disjoint from the image of theother multiplication map. This yields the quantitative estimates dim R(S) >= dim S - b_A + a_A, dim L(S) >= dim S - a_A + b_A, where a_A and b_A are the nullities of X -> X A^dagger andX -> A^dagger X, and consequently max{ dim L(S), dim R(S) } >= dim S + |a_A - b_A|. A first consequence is that the extremal equalitydim L(S) = dim R(S) = dim S forces a_A = b_A for every maximal-rankelement A in S. This condition is necessary but not sufficient:we exhibit explicit counterexamples showing that extremality does notimply the ternary closure S S^dagger S subset of S, and that S neednot be a ternary ring of operators. In the diagonal case, however,the equality admits a complete description: for S contained in thediagonal algebra D_n(C), extremality holds if and only if the non-zeroevaluation vectors r_i in C^d are distributed along exactly d distinctprojective lines in Pᵈ⁻¹. Ternary closure imposes in addition theconstancy of the moduli within each projective direction. The paperthus separates the projective data controlling extremality from themodulus data controlling ternary closure, and leaves open the completecharacterization of extremal matrix spaces in higher dimensions.