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

The condition numbers $CN(T)=\Vert T\Vert\cdot\Vert T^{-1}\Vert$ of Toeplitz and analyticToeplitz $n\times n$ matrices $T$ are studied. It is shown that the supremum of $CN(T)$ over all such matrices with $\Vert T\Vert\leq1$ and a given minimum of eigenvalues $r=\min_{i=1,\dots,n}|\lambda_i|>0$ behaves as the corresponding supremum over all $n\times n$ matrices (i.e., as $\frac1{r^n}$; Kronecker), and this equivalence is uniform in $n$ and $r$. The proof is based on the use of the Sarason–Sz.-Nagy–Foiaş commutant lifting theorem. Bibl. – 2 titles.