Spectral sets, extremal functions and exceptional matrices

Thomas Ransford; Nathan Walsh

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arXiv · 2011.02845

Spectral sets, extremal functions and exceptional matrices

Thomas Ransford·Nathan Walsh

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Abstract

Let $A$ be a square matrix and let $\Omega$ be an open set in the plane containing the spectrum of $A$. We consider the problem of maximizing the operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $\Omega$ into the closed unit disk. If $f_0$ is extremal for this problem and if $\|f_0(A)\|>1$, then it turns out that the matrix $f_0(A)$ has special properties, among them the fact that its principal left and right singular vectors are mutually orthogonal. We study this class of exceptional matrices $f_0(A)$. In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.

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BibTeX RIS

Thomas Ransford, Nathan Walsh. 2020-11-04. Spectral sets, extremal functions and exceptional matrices. https://arxiv.org/abs/2011.02845

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