, Volume 40, Issue 2, pp 231-243

The Weiss conjecture on admissibility of observation operators for contraction semigroups

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We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that\(\parallel C\left( {sI - A} \right)^{ - 1} \parallel \leqslant M\sqrt {\operatorname{Re} s} \) for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.