Perturbations of Hermitian Matrices and Applications to Spectral Symbols

It is often observed in practice that matrix sequences {An}n generated by discretization methods applied to linear differential equations, possess a Spectral Symbol, that is a measurable function describing the asymptotic distribution of the eigenvalues of An. Sequences composed by Hermitian matrices own real spectral symbols, that can be derived through the axioms of Generalized Locally Toeplitz sequences [1]. The spectral analysis of matrix-sequences which can be written as a non-Hermitian perturbation of a given Hermitian matrix-sequence has been performed in a previous work by Leonid Golinskii and the second author [2]. A result was proven but under the technical restrictive assumption that the involved matrix-sequences are uniformly bounded in spectral norm. Nevertheless that result had a remarkable impact in the analysis of spectral distribution and clustering of matrix-sequences coming from various applications, mainly in the context of the numerical approximation of partial differential equations (PDEs) and related preconditioned matrix-sequences. In this presentation, we propose a new result that does not require the boundedness of the sequences and permits to enlarge substantially the class of problems, such as variable-coefficient PDEs and preconditioned matrix-sequences with unbounded coefficients.