  We prove a Weiss conjecture on $\beta$-admissibility of control and
observation operators for discrete and continuous $\gamma$-hypercontractive
semigroups of operators, by representing them in terms of shifts on weighted
Bergman spaces and using a reproducing kernel thesis for Hankel operators.
Particular attention is paid to the case $\gamma=2$, which corresponds to the
unweighted Bergman shift.
