We show that orbits of hyponormal operators display simple growth patterns. We then use our orbital-growth observations to prove that hyponormal operators are never supercyclic, which generalizes a result due to Hilden and Wallen [6, p. 564] and answers a question raised by Kitai [7, p. 4.5]. We also establish that every hyponormal operator is "power regular," which means that if T is a hyponormal operator on the Hilbert space H, then lim_n||T~nh||~(1) exists for every h ∈ H. That every normal operator is power regular follows from results in [2] (see also [5]). Interest in the behavior of orbits of bounded linear operators on Hilbert space derives from the invariant subspace (subset) problem for Hilbert space operators, which is to determine whether every bounded linear operator on a separable, infinite-dimensional Hilbert space H must leave invariant some proper, nonzero, closed subspace (subset) of H. Consider, for example, the following simple proposition, well known to operator theorists.
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