Hypercyclic bilinear operators on Banach spaces

Abstract

We study the dynamics induced by an m-linear operator. We answer a question of Bès and Conejero showing an example of an m-linear hypercyclic operator acting on a Banach space. Moreover we prove the existence of m-linear hypercyclic operators on arbitrary infinite dimensional separable Fréchet spaces. We also prove an existence result about symmetric bihypercyclic bilinear operators, answering a question by Grosse-Erdman and Kim.Partially supported by ANPCyT PICT 2015-2224, UBACyT 20020130300052BA, PIP 11220130100329CO and CONICET. RODRIGO CARDECCIA spaces (including the finite dimensional case). However it is unknown whether the operator can be taken to be symmetric and the following question was posed (see [10, p. 708]).Question 1.1. Let X be a separable Banach space. Does there exist a symmetric bihypercyclic operator in L( 2 X)?Nevertheless the definition of the orbit induced by a multilinear operator is not canonic and other interpretations are available. Whereas the n-state of the iterate of a linear operator depends only on the immediately preceding step (x n = T (x n−1 )), it would be desirable that the n-state of the iterate of an m-linear operator depends only on the mprevious steps. Bès and Conejero [4] defined the orbit induced by a multilinear operator M with initial conditionsSince the orbit in the sense of Bès and Conejero is contained in the orbit in the sense of Grosse-Erdmann and Kim it follows that a hypercyclic bilinear operator is automatically bihypercyclic. This contention implies also that there is again a sense of limit ball for Banach spaces. Every orbit inside ( 1 M m−1 B X ) m tends to zero and therefore the set of hypercyclic vectors is never residual. In [4] examples of multilinear operators over non normable Fréchet spaces where given, including H(C) and C N . It was also proved that every infinite dimensional and separable Banach space supports a supercyclic multilinear operator (i.e. COrb M (x 1−m,...,x0 ) = X). However no example of a hypercyclic multilinear operator on a Banach space or without a residual set of hypercyclic vectors was given and thus the following questions were posed in [4, Section 5]. Question 1.2. Let X be a Fréchet space and M a hypercyclic multilinear operator. Is necessarily the set of hypercyclic vectors residual? Question 1.3. Are there hypercyclic multilinear operators acting on Banach spaces? Of course a positive answer for Question 1.2 implies a negative answer for Question 1.3.The structure of the paper is the following. In Section 2 we propose a notion of transitivity for multilinear operators and analyze examples of multilinear hypercyclic operators over non normable Fréchet spaces with and without a residual set of hypercyclic vectors. In particular we answer Question 1.2 by showing a multilinear hypercyclic operator without a residual set of hypercyclic vectors. In Section 3 we answer Question 1.3 positively.Moreover we construct bilinear hypercyclic operators in arbitrary separable and infinite dimensional Fréchet spaces.In Section 4 we answer...

Orbits of homogeneous polynomials on Banach spaces

“…More recently, Bernardes and Peris showed in [12] the existence of frequently hypercyclic, chaotic and distributionally chaotic (nonhomogeneous) polynomials for a very wide class of infinite-dimensional separable Fréchet spaces. There were also some attempts to extend the concept of hypercyclicity to multilinear operators; see [13,16,22].…”