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Sharp estimate on the resolvent of a finite-dimensional contraction

Karine Fouchet

TL;DR

The paper studies sharp resolvent‑norm estimates for finite‑dimensional contractions with spectral radius bound $r\in(0,1)$. It proves the supremum of the resolvent norm over $|\zeta|\ge 1$ is attained on the unit circle and is achieved by an explicit analytic Toeplitz matrix, with the maximal value expressed as $\mathcal{R}_{n,r}=\|(1-T^{*})^{-1}\|=\frac{1}{1-r}\|X_{1+r}\|$, where $X_{1+r}$ is a simple lower‑triangular Toeplitz matrix. In the large‑$n$ limit, $\mathcal{R}_{n,r} \sim \frac{2}{\pi}\frac{1+r}{1-r}n$, giving a precise linear growth rate in dimension. The approach combines model spaces, Sz.-Nagy–Foiaş commutant lifting, and $H^{\infty}$‑interpolation, and relates the problem to Pták–Young interpolation via an explicit, solvable Toeplitz maximizer. The results provide a sharp, constructive resolution of a Davies–Simon style question in the finite‑dimensional setting with controlled spectral radius.

Abstract

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

Sharp estimate on the resolvent of a finite-dimensional contraction

TL;DR

The paper studies sharp resolvent‑norm estimates for finite‑dimensional contractions with spectral radius bound . It proves the supremum of the resolvent norm over is attained on the unit circle and is achieved by an explicit analytic Toeplitz matrix, with the maximal value expressed as , where is a simple lower‑triangular Toeplitz matrix. In the large‑ limit, , giving a precise linear growth rate in dimension. The approach combines model spaces, Sz.-Nagy–Foiaş commutant lifting, and ‑interpolation, and relates the problem to Pták–Young interpolation via an explicit, solvable Toeplitz maximizer. The results provide a sharp, constructive resolution of a Davies–Simon style question in the finite‑dimensional setting with controlled spectral radius.

Abstract

We compute an asymptotic formula for the supremum of the resolvent norm ( -T ) -1 over || 1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a given r (0, 1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.
Paper Structure (12 sections, 3 theorems, 70 equations)

This paper contains 12 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notation and main result
  3. Motivation
  4. Statement of our result and main ingredients to its proof
  5. Main ingredients
  6. Model spaces and model operators
  7. Link to interpolation theory
  8. Proofs
  9. Proof of the upper bound in Theorem \ref{['prop']}
  10. Proof of the lower bound in Theorem \ref{['prop']}
  11. Proof of Theorem \ref{['thm:Ptak_type_result']}
  12. Proof of Theorem 2

Key Result

Theorem 1

Given $n\geq1$, $r\in(0,\,1)$$\zeta\in\mathbb{C}\setminus\mathbb{D}$ and $T\in\mathcal{C}_{n}$ such that $\rho(T)\leq r$ we have where $T^{*} \in \mathcal{C}_{n}$ is the analytic $n\times n$ Toeplitz matrix and the analytic $n\times n$ Toeplitz matrix $X_{1+r}$ is entry-wise given by: In particular, we have $\mathcal{R}_{n,\,r} = \left|\!\left|(1-T^{*})^{-1}\right|\!\right|_{}$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • proof
  • proof
  • proof