Toeplitz condition numbers as an $H^{\infty}$ interpolation problem
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Document type
Journal articles
Abstract
The condition numbers $CN(T)=\left\Vert T\right\Vert .\left\Vert T^{-1}\right\Vert $ of Toeplitz and analytic $n\times n$ matrices $T$ are studied. It is shown that the supremum of $CN(T)$ over all such matrices with $\left\Vert T\right\Vert \leq1$ and the given minimum of eigenvalues $r=min_{i=1..n}\vert\lambda_{i}\vert>0$ behaves as the corresponding supremum over all $n\times n$ matrices (i.e., as $\frac{1}{r{}^{n}}$ (Kronecker)), and this equivalence is uniform in $n$ and $r$. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem.
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