Banach spaces of GLT sequences and function spaces

Abstract

Generalized locally Toeplitz (GLT) sequences of matrices originated from the spectral study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate either partial differential equations or fractional differential equations using any discretization by local methods (finite differences, finite elements, finite volumes, isogeometric analysis, etc.). The study of the asymptotic spectral behavior of GLT sequences is important in analyzing the solution of the corresponding partial differential equations and in finding fast and efficient methods for the corresponding large linear systems. Approximating classes of sequences (a.c.s.) and spectral symbols are important notions connected to GLT sequences. Recently, G. Barbarino obtained some results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudometric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various connections between the subalgebras of matrix-sequence spaces and the subalgebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. We also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. As an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.