Motivated by a recent investigation of Costakis et al. on the notion recurrence in linear dynamics, we study various stronger forms for operators, particular that frequent recurrence. We study, among other things, relationship between type and corresponding hypercyclicity, influence power boundedness, interplay spectral properties. obtain, particular, Ansari- L\'eon-M\"uller-type theorems $\mat...

We study dynamical notions lying between $$\mathcal{U}$$ -frequent hypercyclicity and reiterative by investigating weighted upper densities the unweighted density Banach density. While chaos implies hypercyclicity, we show that does not imply with respect to any Moreover, if T is -frequently hypercyclic (resp. reiteratively hypercyclic) then n-fold product of still U-frequently this implicatio...

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where F$ is the family subsets natural numbers containing arbitrarily long arithmetic progressions. prove several properties operators, we characterize those which are weakly mixing recurrent, show that there but not mixing.

Let {Tt}t≥0 be a hypercyclic strongly continuous semigroup of operators. Then each Tt (t > 0) is hypercyclic as a single operator, and it shares the set of hypercyclic vectors with the semigroup. This answers in the affirmative a natural question concerning hypercyclic C0-semigroups. The analogous result for frequent hypercyclicity is also obtained.