GLT matrix-sequences and few emblematic applications

Abstract

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 sequences and its applications in mathematical physics. For two HPD sequences $\{A_n\}_n \sim_{\mathrm{GLT}} \kappa$ and $\{B_n\}_n \sim_{\mathrm{GLT}} \xi$ in the same $d$-level, $r$-block GLT $*$-algebra, we prove that when $\kappa$ and $\xi$ commute, the geometric mean sequence $\{G(A_n,B_n)\}_n$ is GLT with symbol $(\kappa\xi)^{1/2}$, without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for $r=1$, $d\ge1$. In degenerate cases, we identify conditions ensuring $\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} G(\kappa,\xi)$. For $r>1$ and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to $k\ge2$ sequences via the Karcher mean, obtaining $\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(\kappa_1,\ldots,\kappa_k)$. Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions.