We prove the following result which extends in a somewhat "linear" sense a theorem by Kierst and Szpilrajn and which holds on many "natural" spaces of holomorphic functions in the open unit disk 𝔻: There exist a dense linear manifold and a closed infinite-dimensional linear manifold of holomorphic functions in 𝔻 whose domain of holomorphy is 𝔻 except for the null function. The existence of a dense linear manifold of noncontinuable functions is also shown in any domain for its full space of holomorphic...
We show that under no hypotheses on the density of the ranges of the mappings involved, an almost-commuting sequence (Tₙ) of operators on an F-space X satisfies the Hypercyclicity Criterion if and only if it has a hereditarily hypercyclic subsequence , and if and only if the sequence (Tₙ ⊕ Tₙ) is hypercyclic on X × X. This strengthens and extends a recent result due to Bès and Peris. We also find a new characterization of the Hypercyclicity Criterion in terms of a condition introduced by Godefroy...
An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is . The results obtained extend or improve earlier work...
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