Abstract
Given a sequence of matrices (matrix-sequence) {X n }, with X n Hermitian of size d n tending to infinity, we consider the sequence {X n + Y n }, where {Y n } is an arbitrary (non-Hermitian) perturbation of {X n }. We prove that {X n + Y n } has an asymptotic spectral distribution if: {X n } has an asymptotic spectral distribution, the spectral norms \({\|X_n\|,\|Y_n\|}\) are uniformly bounded with respect to n, and {Y n } satisfies a trace-norm assumption. Furthermore, under the above assumptions, the functional ϕ identifying the asymptotic spectral distribution is the same for {X n + Y n } and {X n }. We mention some examples of applications, including the case of matrix-sequences with asymptotic spectral distributions described by matrix-valued functions and the approximation by \({\mathbb{Q}_k}\) Finite Element methods of convection-diffusion equations.
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This work was supported in part by GNCS-INDAM.
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Garoni, C., Serra-Capizzano, S. & Sesana, D. Tools for Determining the Asymptotic Spectral Distribution of non-Hermitian Perturbations of Hermitian Matrix-Sequences and Applications. Integr. Equ. Oper. Theory 81, 213–225 (2015). https://doi.org/10.1007/s00020-014-2157-6
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DOI: https://doi.org/10.1007/s00020-014-2157-6