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$β$-admissibility of observation and control operators for hypercontractive semigroups

We prove a Weiss conjecture on $β$-admissibility of control and observation operators for discrete and continuous $γ$-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 $γ=2$, which corresponds to the unweighted Bergman shift.