Sharp estimate on the resolvent of a finite-dimensional contraction

TL;DR

This paper derives an asymptotic estimate for the maximum resolvent norm of finite-dimensional contractions with spectral radius less than one, showing it is achieved by an analytic Toeplitz matrix on the unit circle.

Contribution

It provides a precise asymptotic formula for the resolvent norm supremum of finite-dimensional contractions, identifying the extremal matrix as an analytic Toeplitz matrix.

Findings

Supremum of resolvent norm computed asymptotically

Maximum achieved by an analytic Toeplitz matrix

Results apply to contractions with spectral radius less than one

Abstract

We compute an asymptotic formula for the supremum of the resolvent norm ($\zeta$ -T ) -1 over |$\zeta$| $\ge$ 1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a given r $\in$ (0, 1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.