GLT matrix-sequences and few emblematic applications

This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) ∗-algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT matrixsequences and its applications in mathematical physics. For two HPD sequences {An}n ∼GLT κ and {Bn}n ∼GLT ξ in the same d-level, r-block GLT ∗-algebra, we prove that, when κ and ξ commute, the sequence of geometric means {G(An, Bn)}n is a GLT sequence with symbol (κξ)1/2, without requiring the almost-everywhere invertibility of either symbol, thereby settling [37, Conjecture 10.1] for r = 1, d ≥ 1. In degenerate cases, where symbols vanish on sets of positive measure, we identify conditions ensuring that the geometric mean retains a GLT structure in the commuting setting, so having {G(An, Bn)}n ∼GLT G(κ, ξ). Conversely, for r > 1 with degenerate, non-commuting symbols, we provide numerical evidence that the resulting sequence still admits a spectral symbol, with G(κ, ξ) being not well defined. The latter implies that the result {G(An, Bn)}n ∼GLT G(κ, ξ) in the commuting setting is maximal. Numerical experiments in scalar and block settings, in one and two dimensions, confirm the theoretical predictions and illustrate spectral behaviour. We sketch also the case of k ≥ 2 matrix-sequences, by considering the Karcher mean. Preliminary results and numerical experiments indicate that {G(A(1)n, . . . A(k)n)}n ∼GLT G(κ1, . . . κk), if {A(j)n }n ∼GLT κj, for j = 1, . . . , k. The GLT framework is further applied to mean-field quantum spin systems, with particular attention to the quantum Curie–Weiss model. In this context, we show that the structured matrices arising from the model form GLT sequences, enabling an explicit determination of their spectral distributions in both unrestricted and symmetry-restricted cases. Numerical simulations validate the analysis and reveal additional spectral features such as eigenvalue localization and extremal behaviour. In terms of mathematical tools, we use the axioms characterizing the d-level, r-block GLT ∗-algebra, the notion of approximating classes of sequences and the important two-sided ideal of zero-distributed matrix-sequences.