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Hypercyclicity of
-semigroups is a very unstable property: We give examples to
show that adding arbitrary small constants or a bounded rank one operator to the generator of a
hypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (even
in operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limit
of nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, the
restriction of...
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