Hypercyclic bilinear operators on Banach spaces (Q2302924)

In [J. Math. Anal. Appl. 399, No. 2, 701--708 (2013; Zbl 1272.47009)], \textit{K.-G. Grosse-Erdmann} and \textit{S. G. Kim} introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. Among other things, they constructed bihypercyclic bilinear operators in arbitrary Banach spaces, but they left open the question whether these operators can be taken to be symmetric. In [Abstr. Appl. Anal. 2014, Article ID 609873, 11 p. (2014; Zbl 1473.47001)], \textit{J. Bès} and \textit{J. A. Conejero} introduced an alternative notion of orbit for multilinear operators to define and study supercyclic and hypercyclic multilinear operators. Among other things, they showed that every separable infinite-dimensional Fréchet space \(X\) supports, for each \(n\geq2\), an \(n\)-linear operator having a residual set of supercyclic vectors. However, they left open the question whether the set of hypercyclic vectors of a hypercyclic multilinear operator on a Fréchet space is residual. They also wondered if there are hypercyclic multilinear operators on Banach spaces. This well-written paper answers the questions mentioned above. The author considers the notion of transitivity for multilinear operators and analyses examples of multilinear hypercyclic operators over non normable Fréchet spaces with and without a residual set of hypercyclic vectors. He provides an example of a multilinear hypercyclic operator without a residual set of hypercyclic vectors and thus answers the first question of Bès and Conejero [loc.\,cit.]. He also proves the existence of multilinear hypercyclic operators on arbitrary infinite dimensional separable Banach spaces, and thus positively answers the second question of Bès and Conejero [loc.\,cit.]. Furthermore, he also shows that there are symmetric bihypercyclic operators in arbitrary separable and infinite dimensional Banach spaces, thus answering the question by Grosse-Erdmann and Kim [loc.\,cit.].