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Properties and preservers of the pseudospectrum. (English) Zbl 1245.15028

Let \(M_n\) denotes the set of \(n\times n\) complex matrices equipped with the operator norm \(\|\cdot\|\) induced by the usual vector norm \(\| x\|=(x^*x)^{1/2}\) on \({\mathbb C}^n\): \[ \|A\|=\max \{\|Ax\|: x\in {\mathbb C}^n, \;0<\|x\|\leq1\}. \]
Let \(\varepsilon>0\). The pseudospectrum of a matrix \(A\in M_n\) is defined by \[ \sigma_{\varepsilon} (A) =\{\mu\in {\mathbb C} : \text{ there are } x\in {\mathbb C}^n , E\in M_n \text{ such that } (A+E)x=\mu x \}. \] In the paper it is shown that the pseudospectrum can be used to study the algebraic and geometric properties of matrices. For example, it is shown that one can characterize Hermitian matrices, positive semidefinite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Moreover, the authors study maps \(\Phi: M_n\to M_n\) such that \(\sigma_{\varepsilon} (A*B)=\sigma_{\varepsilon}(\Phi(A)*\Phi(B))\) for all \(A,B\in M_n\), where \(A*B=A+B,\;A-B\) or \(AB\). It is shown that such maps are always unitary similarity transformations followed by some operations such as adding a constant matrix, taking the matrix transpose, or multiplying by a scalar from \(\{-1;1\}\).

MSC:

15A86 Linear preserver problems
15A18 Eigenvalues, singular values, and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B57 Hermitian, skew-Hermitian, and related matrices
15B48 Positive matrices and their generalizations; cones of matrices

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References:

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