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Pseudospectrum enclosures by discretization. (English) Zbl 1496.47009

Let \(A\) be a matrix (or operator) of finite or infinite dimension. For \(\lambda \in \mathbb{C}\), the resolvent matrix is \(R_\lambda(A)=(A-\lambda)^{-1}\). To understand more about an object represented by a matrix \(A\), we have to analyze not only the eigenvalues, spectrum and resolvent matrix, but also the pseudospectrum. The concept of pseudospectrum of matrices has a number of applications in different fields: dynamical systems, hydrodynamic stability, Markov chains, and non-Hermitian quantum mechanics. Note that the eigenvalues of the resolvent matrix of \(A\) never coincide with the eigenvalues of \(R_\lambda(A)\).
It is well known that for \(\epsilon>0,\) pseudospectrum of \(A\) is given by
\[\sigma_\epsilon(A)=\{\lambda\in \mathbb{C}:\|(A-\lambda)^{-1}\|< \epsilon^{-1}\}.\]
In this paper, a new method is employed to enclose the pseudospectrum via the numerical range of the inverse of the matrix or linear operator.
The results (Theorem 2.2, Theorem 2.5, Theorem 3.6 and Lemma 4.5) will have a significant impact in the area of studying spectral analysis of matrices or linear operators. Also, there are computations given in Example 7.1 and Example 7.2.
Reviewer: Ali Shukur (Minsk)

MSC:

47A10 Spectrum, resolvent
47A12 Numerical range, numerical radius

Software:

Pseudospectra
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Full Text: DOI arXiv

References:

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