An inequality concerning the growth bound of a discrete evolution family on a complex Banach space

TL;DR

This paper establishes a new inequality relating the growth bound of a discrete evolution family on a complex Banach space to the operator norm of a convolution operator, advancing understanding of their spectral properties.

Contribution

It proves a novel inequality connecting the uniform growth bound and convolution operator norm for discrete evolution families on Banach spaces.

Findings

The uniform growth bound satisfies an inequality involving the convolution operator norm.

The inequality provides bounds on the spectral growth of discrete evolution families.

The result links operator norms with growth bounds in the context of Banach space evolution.

Abstract

We prove that the uniform growth bound $\omega_0(\mathcal{U})$ of a discrete evolution family $\mathcal{U}$ of bounded linear operators acting on a complex Banach space $X$ satisfies the inequality $$\omega_0(\mathcal{U})c_{\mathcal{U}}(\mathcal{X})\le -1;$$ here $c_{\mathcal{U}}(\mathcal{X})$ is the operator norm of a convolution operator which acts on a certain Banach space $\mathcal{X}$ of $X$-valued sequences.