Abstract
Given A the family of weights a=(an)n decreasing to 0 such that the series ∑n=0∞an diverges, we show that the supremum on A of lower weighted densities coincides with the unweighted upper density and that the infimum on A of upper weighted densities coincides with the unweighted lower density. We then investigate the notions of U-frequent hypercyclicity and frequent hypercyclicity associated to these weighted densities. We show that there exists an operator which is U-frequently hypercyclic for each weight in A but not frequently hypercyclic, although the set of frequently hypercyclic vectors always coincides with the intersection of sets of U-frequently hypercyclic vectors for each weight in A.
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