Common hypercyclic vectors and universal functions

TL;DR

This paper investigates the existence of common universal vectors for families of linear operators, such as translation, differential, and shift operators, and explores related approximation problems involving universal Taylor series.

Contribution

It introduces the concept of common hypercyclic vectors for uncountable families of operators and analyzes specific cases like translation, differential, and shift operators.

Findings

Existence of common hypercyclic vectors for certain operator families.

Conditions under which universal Taylor series can approximate functions.

Insights into the structure of universal vectors across different operator classes.

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 .