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Linear Kierst-Szpilrajn theorems

100%
Bernal-González L.
Studia Mathematica
|
2005
|
tom 166
|
nr 1
55-69
EN
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 functions.
2
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Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

63%
Bernal-González L., Prado-Tendero J.
Annales Polonici Mathematici
|
2001
|
tom 77
|
nr 2
169-187
EN
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 $ℂ^N$ 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 $ℂ^N$. The results obtained extend or improve earlier work of several authors.
3
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The Hypercyclicity Criterion for sequences of operators

63%
Bernal-González L., Grosse-Erdmann K.-G.
Studia Mathematica
|
2003
|
tom 157
|
nr 1
17-32
EN
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 $(T_{n_{k}})$, 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 and Shapiro. Finally, we show that a weakly commuting hypercyclic sequence (Tₙ) satisfies the Hypercyclicity Criterion whenever it has a dense set of points with precompact orbits. We remark that some of our results are new even in the case of iterates (Tⁿ) of a single operator T.
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