$$\mathcal{U}$$-frequent hypercyclicity notions and related weighted densities

Linear Dynamical Systems

“…With the general machinery of A-hypercyclicity, the results in [41], [54] and [68] for U-frequent and reiterative hypercyclicity follow as special cases. This approach was also employed by Grosse-Erdmann [97] to prove the following theorem.…”

“…If T is reiteratively hypercyclic then so is its inverse. Theorem 3.11 (Ernst, Esser and Menet [68]). If T ∈ L (X) is U-frequently hypercyclic (resp.…”

“…It was observed in [54] that replacing upper density by upper Banach density in Theorem 3.6 gives a Reiterative Hypercyclicity Criterion. We remark that the studies in [54] and [68] employed the general framework of A-hypercyclicity which was introduced in [41], where A is a Furstenberg family. A nonempty family A of subsets of N 0 is a Furstenberg family if it is hereditary upward, that is if A ∈ A and A ⊂ B, then B ∈ A.…”

“…Finally we mention that weighted upper and lower densities were studied in Ernst and Mouze [69] and Menet [113] to identify a fine-grained picture of the dynamical notions that lie between U-frequent and frequent hypercyclicity. In [68] weighted densities were also studied to illustrate the dynamical behaviours lying between reiterative hypercyclicity and U-frequent hypercyclicity.…”

“…We recall that there exist hypercyclic operators T such that T ⊕ T is not hypercyclic [64], [21]. On the other hand, we recall that it was shown in [68] that if T is U-frequently hypercyclic (resp. reiteratively hypercyclic), then T ⊕ T is U-frequently hypercyclic (resp.…”