Common hypercyclic vectors and universal functions

2015/11/19 by Costakis, George, Tsirivas, Nikos
#30E10 #33E05 #41A17 #41A29 #47A16 #65F10 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.06145

Abstract

Let X,Y be two separable Banach or Frechet spaces , and (Tn) , n=1,2,... be a sequence from linear and continuous operators from X to Y . We say that the sequence (Tn) , n=1,2,... is universal , if there exists some vector v in X such that the sequence Tn(v) , n=1,2,... is dense in Y . If X=Y we say that the sequence (Tn) is hypercyclic .More generally we consider an uncountable subset A from complex numbers and for every fixed a in A we consider a sequence (Ta,n) , n=1,2,... from linear and continuous operators from X to Y .The problem of common universal or hypercyclic vectors is whether the uncountable family of sequences of operators (Ta,n) , n=1,2,... share a common universal vector for all a in A .We examine , in this work ,some specific cases of this problem for translation , differential , and backward shift operators . We study also some approximating problems about universal Taylor series .

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