Given a sequence of matrices (matrix-sequence) {Xn}, with Xn Hermitian of size dn tending to infinity, we consider the sequence {Xn + Yn}, where {Yn} is an arbitrary (non-Hermitian) perturbation of {Xn}. We prove that {Xn + Yn} has an asymptotic spectral distribution if: {Xn} has an asymptotic spectral distribution, the spectral norms (Formula Presented) are uniformly bounded with respect to n, and {Yn} satisfies a trace-norm assumption. Furthermore, under the above assumptions, the functional ϕ identifying the asymptotic spectral distribution is the same for {Xn + Yn} and {Xn}. 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 (Formula Presented) Finite Element methods of convection-diffusion equations.

### Tools for Determining the Asymptotic Spectral Distribution of non-Hermitian Perturbations of Hermitian Matrix-Sequences and Applications

#### Abstract

Given a sequence of matrices (matrix-sequence) {Xn}, with Xn Hermitian of size dn tending to infinity, we consider the sequence {Xn + Yn}, where {Yn} is an arbitrary (non-Hermitian) perturbation of {Xn}. We prove that {Xn + Yn} has an asymptotic spectral distribution if: {Xn} has an asymptotic spectral distribution, the spectral norms (Formula Presented) are uniformly bounded with respect to n, and {Yn} satisfies a trace-norm assumption. Furthermore, under the above assumptions, the functional ϕ identifying the asymptotic spectral distribution is the same for {Xn + Yn} and {Xn}. 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 (Formula Presented) Finite Element methods of convection-diffusion equations.
##### Scheda breve Scheda completa Scheda completa (DC)
2014
15A18; 15A60; 65N30; Primary 47B35; Secondary 47B38
Garoni, C.; SERRA CAPIZZANO, Stefano; Sesana, D.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/1926725`
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