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The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so‐called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let (Tt)t⩾0 be a bounded analytic semigroup with generator –A on some Banach space X. It is proved that if A1/2 is admissible for A, that is, if there is an estimate ∫0∞‖ A1/2e−tAx ‖2dt⩽M2‖ x ‖2, then any continuous mapping C : D(A) → Y valued in a Banach space Y is admissible for A provided that there is an estimate ‖ (−Re(λ))1/2C(λ−A)−1 ‖⩽K. for λ ∈, Re(λ)<0. This holds in particular if (Tt)t⩾0 is a contractive (analytic) semigroup on Hilbert space. In the converse direction, it is shown that this may happen for a bounded analytic semigroup on Hilbert space that is not similar to a contractive one. Applications in non‐Hilbertian Banach spaces are also given.