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

The condition numbers $CN(T)==\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }\mathrm{.}\mathrm{\mid }\mathrm{\mid }{T}^{-1}\mathrm{\mid }\mathrm{\mid }$ 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 $\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }\le 1$ and the given minimum of eigenvalues $r=mi{n}_{i=\mathrm{1..}n}\mathrm{\mid }{\lambda }_i\mathrm{\mid }>0$ 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.