GLT matrix-sequences and few emblematic applications

TL;DR

This thesis develops the spectral theory of structured matrix-sequences within GLT algebras, proving new results about geometric means of Hermitian positive definite sequences and applying these to quantum spin systems.

Contribution

It proves that the geometric mean of commuting HPD GLT sequences is GLT with a specific symbol, settling a conjecture, and extends the theory to multiple sequences and quantum physics applications.

Findings

Geometric mean of commuting HPD GLT sequences is GLT with symbol $( ext{product})^{1/2}$

Numerical evidence suggests non-commuting sequences also admit spectral symbols

Application to quantum Curie--Weiss model shows matrices form GLT sequences with known spectral distributions

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.