DIMENSIONAL ASYMMETRY OF LEFT AND RIGHT ADJOINT PRODUCT SPACES OF ISOMETRIES

For an isometry A in M_{n^2 x n}(C) with block rows A_1,...,A_n, consider the two
Gram spaces L_A = span{A_i A_j^*} and R_A = span{A_i^* A_j}. The paper studies
the dimensional asymmetry between them, with emphasis on the question of how
small dim R_A can be when dim L_A is maximal.

For n = 2 a complete classification is given: the minimum is mu(2) = 4, the
realised pairs (dim R_A, dim L_A) are (1,1), (2,2), (4,1), (4,2) and (4,4), and
dim R_A = 3 is impossible.

For n = 3 it is proved that dim L_A = 9 implies dim R_A >= 4, and an exact
rational isometry with dim L_A = 9 and dim R_A = 5 is exhibited; hence the
minimum lies in {4,5}. The lower bound is obtained through structural rank
bounds, a presentation lemma for three-dimensional self-adjoint subspaces of
M_3(C), a no-go theorem for the resulting rigid family, and a reduction via the
multiplicative domain of the ucp map Phi(C) = A^*(C tensor I_3)A, which shows
that any counterexample to dim R_A >= 5 must satisfy dim R_A = 4 and trivial
multiplicative domain. The witness with dim R_A = 5 lies in an explicit family
parametrised by orthonormal frames in R^5, within which the bound 5 is sharp.

For n >= 4 a Weyl-operator construction yields explicit upper bounds that
improve the trivial bound 1+(n-1)^2 for n >= 5.

The conjecture dim R_A >= 5 remains open, reduced to the single residual case
dim R_A = 4 with trivial multiplicative domain.