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

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