Geometric means of HPD GLT matrix-sequences: a maximal result beyond invertibility assumptions on the GLT symbols

In the current work, we consider the study of the spectral distribution of the geometric mean matrix-sequence of two matrix-sequences $\{G(A_n, B_n)\}_n$ formed by Hermitian Positive Definite (HPD) matrices, assuming that the two input matrix-sequences $\{A_n\}_n, \{B_n\}_n$ belong to the same $d$-level $r$-block Generalized Locally Toeplitz (GLT) $\ast$-algebra with $d,r\ge 1$ and with GLT symbols $\kappa, \xi$. Building on recent results in the literature, we examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere (a.e.) is necessary. Since inversion is mainly required due to the non-commutativity of the matrix product, it was conjectured that the hypothesis on the invertibility of the GLT symbols can be removed. In fact, we prove the conjectured statement that is \[ \{G(A_n, B_n)\}_n \sim_{\mathrm{GLT}} (\kappa \xi)^{1/2} \] when the symbols $\kappa, \xi$ commute, which implies the important case where $r=1$ and $d \geq 1 $, while the statement is generally false or even not well posed when the symbols are not invertible a.e. and do not commute. In fact, numerical experiments are conducted in the case where the two symbols do not commute, showing that the main results of the present work are maximal. Further numerical experiments, visualizations, and conclusions end the present contribution.