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

Abstract

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 $||T||\le 1$ and the given minimum of eigenvalues $r=mi{n}_{i=\mathrm{1..}n}|{\lambda }_i|>0$ 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.