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

TL;DR

This paper investigates the behavior of condition numbers of Toeplitz matrices, linking the problem to an $H^{ olinebreak ext{ extbf{ extit{in}}}}$ interpolation problem and establishing uniform bounds based on eigenvalue constraints.

Contribution

It establishes a connection between Toeplitz condition numbers and $H^{ olinebreak ext{ extbf{ extit{in}}}}$ interpolation, providing uniform bounds and a novel proof approach using the Sarason-Sz.Nagy-Foias theorem.

Findings

Supremum of condition numbers behaves as 1/r^n

Uniform bounds in matrix size and eigenvalue constraints

Use of commutant lifting theorem in proof

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 $||T|| \leq1$ 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.