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A remark on the growth rate of operator semigroups under resolvent bounds

Filippo Dell'Oro

TL;DR

This paper studies growth bounds for $C_0$-semigroups on Hilbert spaces under resolvent bounds of the form $\|R(\lambda,A)\|\le g(1/\mathrm{Re}\lambda)$ with a nondecreasing $g$ and $h(t)=g(t)/t$ nondecreasing. The main result proves that if $\mathbb{C}^{+}\subset \rho(A)$ then $\|T(t)\|=O\big(g(t)\frac{h(t)}{\sqrt{\log t}}\big)$ as $t\to\infty$, sharpening known bounds when $h(t)\to\infty$. The proof combines the Fourier transform of the rescaled semigroup, Plancherel's theorem, and dyadic time-slice estimates to convert resolvent growth into time-domain growth. Examples with $g(t)=t\log^{\beta}(t+1)$ ($0<\beta<1/2$) and $g(t)=t\log\log(t+e)$ illustrate sharper rates and demonstrate practical impact for operator theory.

Abstract

We provide a growth bound for the operator norm of $C_0$-semigroups on Hilbert spaces under a corresponding growth bound on the resolvent of the semigroup generator. For some super-linear resolvent growths, our estimate is sharper than the ones currently available in the literature.

A remark on the growth rate of operator semigroups under resolvent bounds

TL;DR

This paper studies growth bounds for -semigroups on Hilbert spaces under resolvent bounds of the form with a nondecreasing and nondecreasing. The main result proves that if then as , sharpening known bounds when . The proof combines the Fourier transform of the rescaled semigroup, Plancherel's theorem, and dyadic time-slice estimates to convert resolvent growth into time-domain growth. Examples with () and illustrate sharper rates and demonstrate practical impact for operator theory.

Abstract

We provide a growth bound for the operator norm of -semigroups on Hilbert spaces under a corresponding growth bound on the resolvent of the semigroup generator. For some super-linear resolvent growths, our estimate is sharper than the ones currently available in the literature.
Paper Structure (3 sections, 1 theorem, 24 equations)

This paper contains 3 sections, 1 theorem, 24 equations.

Table of Contents

  1. Introduction and Main Result
  2. Proof of Theorem \ref{['KREISSLOG']}
  3. Concluding Remarks

Key Result

Theorem 1.1

Let $A$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t\ge 0}$ on a complex Hilbert space $H$ satisfying $\,\mathbb{C}^{+} \subset \rho(A)$. Assume that there exists a function $g:(0,\infty)\to (0,\infty)$ such that Moreover, assume that the function $h(t)=g(t)/t$ is non-decreasing. Then

Theorems & Definitions (1)

  • Theorem 1.1