Operators $L^1 (\mathbb R_+ )\to X$ and the norm continuity problem for semigroups

TL;DR

This paper introduces a novel method for constructing $C_0$-semigroups with controlled resolvent and norm continuity properties, demonstrating that norm-continuity cannot be characterized solely by resolvent decay on vertical lines.

Contribution

The authors develop a new construction technique for $C_0$-semigroups that simultaneously controls resolvent behavior and norm continuity, providing counterexamples to existing characterizations.

Findings

Existence of a $C_0$-semigroup continuous in operator-norm only at $t=0$ with logarithmic resolvent decay.

Existence of a $C_0$-semigroup with no norm continuity for any $t>0$ but near-logarithmic resolvent decay.

These examples show that resolvent decay alone cannot characterize norm-continuity of semigroups.

Abstract

We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that a) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in [0,1]$ such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in \mathbb R_+$ such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.