$\alpha$-admissibility of the right-shift semigroup on $L^2(\mathbb{R}_+)$

TL;DR

This paper demonstrates that the right shift semigroup on $L^2( plus)$ fails the weighted Weiss conjecture for $$, showing that $$-admissibility cannot always be characterized by simple resolvent conditions, contrasting with the unweighted case.

Contribution

It establishes the failure of the weighted Weiss conjecture for the right shift semigroup on $L^2( plus)$ for $$, linking discrete and continuous $$-admissibility via a counterexample.

Findings

The right shift semigroup does not satisfy the weighted Weiss conjecture for $$.

$$-admissibility cannot be characterized solely by resolvent growth conditions.

A counterexample from the unilateral shift on $H^2(D)$ is translated to continuous systems.

Abstract

It is shown that the right shift semigroup on $L^2(\mathbb{R}_+)$ does not satisfy the weighted Weiss conjecture for $\alpha \in (0,1)$. In other words, $\alpha$-admissibility of scalar valued observation operators cannot always be characterised by a simple resolvent growth condition. This result is in contrast to the unweighted case, where 0-admissibility can be characterised by a simple growth bound. The result is proved by providing a link between discrete and continuous $\alpha$-admissibility and then translating a counterexample for the unilateral shift on $H^2(\mathbb{D})$ to continuous time systems.