  Counterexamples are presented to weighted forms of the Weiss conjecture in
discrete and continuous time. In particular, for certain ranges of $\alpha$,
operators are constructed that satisfy a given resolvent estimate, but fail to
be $\alpha$-admissible. For $\alpha \in (-1,0)$ the operators constructed are
normal, while for $\alpha \in (0,1)$ the operator is the unilateral shift on
the Hardy space $H^2(\mathbb{D})$.
