αα-admissibility of the right-shift semigroup on L2(R+)L2(R+)

It is shown that the right-shift semigroup on L2(R+)L2(R+) does not satisfy the weighted Weiss conjecture for α∈(0,1)α∈(0,1). In other words, αα-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 00-admissibility can be characterised by a simple growth bound. The result is proved by providing a link between discrete and continuous αα-admissibility and then translating a counterexample for the unilateral shift on H2(D)H2(D) to continuous time systems.