Sharp estimate on the resolvent of a finite-dimensional contraction

Karine Fouchet

doi:10.1016/j.jmaa.2022.126093

paper

Sharp estimate on the resolvent of a finite-dimensional contraction

arXiv:2505.12901 · doi:10.1016/j.jmaa.2022.126093

Abstract

We compute an asymptotic formula for the supremum of the resolvent norm ( $ζ$ -T ) -1 over | $ζ$ | $\geq$ 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.