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Somewhere dense orbits are everywhere dense. (English) Zbl 1049.47002
Let \(X\) be an infinite-dimensional locally convex space. A linear continuous operator \(T\) on \(X\) is said to be hypercyclic if there is an \(x\in X\) whose orbit, i.e., Orb\((T,x)=\{x, Tx, T^2x, \ldots\}\), is dense in \(X\). If Orb\((T,x)\) is dense in \(X\), \(x\) is said to be a hypercyclic vector. A linear continuous operator \(T\) on \(X\) is said to be multihypercyclic if there is a finite subset \(\{x_1,\ldots, x_n\}\) of \(X\) such that \(\bigcup_{k=1}^n \text{Orb} (T,x_k)\) is dense in \(X\).
In [J. Funct. Anal. 128, 374–383 (1995; Zbl 0853.47013)], S. I. Ansari solved a longstanding question, proving that: if \(T\) is hypercyclic, then for every positive integer \(n\), the operator \(T^n\) is also hypercyclic and \(T\) and \(T^n\) share the same collection of hypercyclic vectors. Before Ansari’s theorem appeared, D. Herrero [J. Oper. Theory 28, 93–103 (1992; Zbl 0806.47020)] had posed the following related conjecture: if \(T\) is multihypercyclic, then \(T\) is hypercyclic.
Herrero’s conjecture was established independently by G. Costakis [C. R. Acad. Sci., Paris, Sér. I, Math. 330, 179–182 (2000; Zbl 0953.47004)] and A. Peris [Math. Z. 236, 779–786 (2001; Zbl 0994.47011)], proving that the density in \(X\) of \(\bigcup_{k=1}^n \text{Orb}(T,x_k)\) implies the density in \(X\) of Orb\((T,x_k)\) for some \(k\in\{1,\ldots, n\}\).
Answering questions raised by A. Peris (loc. cit.), the authors of the present paper prove that if an orbit Orb\((T,x)\) is somewhere dense (if the set of scalar multiples of elements in Orb\((T,x)\) is somewhere dense), then it is everywhere dense (then the set of scalar multiples of elements in Orb\((T,x)\) is everywhere dense), providing a new sufficient condition for hypercyclicity and a new proof of Herrero’s conjecture.
We point out that J. Wengenroth [Proc. Am. Math. Soc. 131, 1759–1761 (2003; Zbl 1039.47009)] has shown that the same results are valid in the non-locally convex setting.

MSC:
47A16 Cyclic vectors, hypercyclic and chaotic operators
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