A vector x in a linear topological space X is called universal for a linear operator T on X if the orbit \ Tⁿx:n ≥ 0\ is dense in X. Our main result gives conditions on T and X which guarantee that T will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Pólya’s theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.
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Gethner et al. (1987) studied this question.