On uniform connectivity of algebraic matrix sets. (English) Zbl 1482.47151

Vides, Fredy

Banach J. Math. Anal. 13, No. 4, 918-943 (2019).

Let the space \(M_n(\mathbb{C})^m\) of \(m\)-tuples of \(n\times n\) pairwise commuting normal contractions \(\mathbf{X}=(X_1,\ldots,X_m)\) be equipped with some metric \(\eth\). Given \(r\) complex polynomials \(p_1,\ldots,p_r\) in \(m\) variables, the set of algebraic normal contractions is \(\mathbb{Z}\mathbb{D}_n^m(p_1,\ldots,p_r)=\{\mathbf{X}\in M_n(\mathbb{C})^m: \|p_j(\mathbf{X})\|=0\text{ for } j=1,\ldots,r\}\). The author proves the connectivity of this set in the following sense. Given two elements \(\mathbf{X},\mathbf{Y}\in \mathbb{Z}\mathbb{D}_n^m(p_1,\ldots,p_r)\) with \(\eth(\mathbf{X},\mathbf{Y})\le\delta\), it is possible to connect them with a smooth path that stays in the neighborhood of \(\mathbf{X}\), actually in the ball \(B_\eth(\mathbf{X},\varepsilon)\) for given \(\varepsilon\ge\delta\). The path is uniform because it does not depend of \(n\). In a second part, he also proves uniform connectivity of nearly algebraic normal contractions, in which case it is assumed that \(\|p_j(\mathbf{X})\|\le\varepsilon\) for \(j=1,\ldots,r\) instead of being zero. This is solved by proving that there are algebraic normal contractions \(\tilde{\mathbf{X}}\) and \(\tilde{\mathbf{Y}}\) close to the given nearly algebraic normal contractions \(\mathbf{X}\) and \(\mathbf{Y}\) that satisfy the previous theorem. The motivation for these problems comes, for example, from the theory of structure preserving perturbation theory from linear algebra and how preconditioning influences the result

Reviewer: Adhemar Bultheel (Leuven)

MSC:

47N40	Applications of operator theory in numerical analysis
47A58	Linear operator approximation theory
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A27	Commutativity of matrices

Keywords:

joint spectrum; matrix function; functional calculus; matrix path; eigenvalue clustering

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