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

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

### 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
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