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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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