Α-admissibility of the Right-shift Semigroup on L(r+)

It is shown that the right shift semigroup on L(R+) does not satisfy the weighted Weiss conjecture for α ∈ (0, 1). In other words, αadmissibility 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 α-admissibility and then translating a counterexample for the unilateral shift on H(D) to continuous time systems.