Toeplitz condition numbers as an $H^{\infty}$ interpolation problem

Details Toeplitz condition numbers as an $H^{\infty}$ interpolation problem Toeplitz condition numbers as an $H^{\infty}$ interpolation problem

2011-03-25

arxiv.org/abs/1103.5016v1

Journal of Mathematical Sciences / Journal of Mathematical Sciences (New York) 156, 5 (2009) 819-823

Mathematics

Functional Analysis

Scientific paper

The condition numbers $CN(T)==||T|| .||T^{-1}|| $ 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 $||T|| \leq1$ and the given minimum of eigenvalues $r=min_{i=1..n}|\lambda_{i}|>0$ 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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