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Countably Hypercyclic Operators

To appear in J. Operator Theorydvi file   pdf file

A bounded linear operator T is hypercyclic if there is a vector with dense orbit; so if T acts on a Banach space X, then T is hypercyclic if there is a vector x in X such that Orb(T,x) = {x, Tx, T^2x, ...} is dense in X.  An operator is T is finitely-hypercyclic if there are a finite number of orbits whose union is dense.  That is, if there are vectors {x1, x2,...,xn} such that the union of Orb(T,x1), Orb(T,x2),...,Orb(T,xn) is dense.

D. Herrero conjectured in 1991 that a finitely-hypercyclic operator was actually hypercyclic.  In 1999 Alfredo Peris proved Herrero's conjecture.  It was also proven independently by G. Kostakis and later by Bourdon & Feldman.

Several people asked if there was a countable version of Herrero's conjecture.  In this paper we propose a reasonable definition of a countably hypercyclic operator, and establish a Countable Hypercyclicity Criterion very similar to the Hypercyclicity Criterion.

Definition:  An operator T is countably hypercyclic if there is a bounded separated sequence with dense orbit.

A sequence is separated if the distance between any two of its terms is uniformly bounded away from zero.

Recall that for an operator S the parts of the spectrum of S are all the compact sets of the form $\sigma(S|M)$ where $M$ is an invariant subspace for S.

Theorem 1:  If S is a pure hyponormal operator, then its adjoint $S^*$ is countably hypercyclic if and only if every part of the spectrum of S intersects {z : |z| > 1} and at least one part of the spectrum of S intersects {z : |z| < 1}.

In particular, one may easily use adjoints of multiplication operators on Bergman or Hardy spaces to construct countably hypercyclic operators that are not hypercyclic.

Theorem 2: 
(a) If T is a cohyponormal operator and both T and T^{-1} are countably hypercyclic, then T is hypercyclic.
(b) If T is a backward weighted shift and T is countably hypercyclic, then T is hypercyclic.
(c) A hyponormal operator cannot be countably hypercyclic.

The question for bilateral weighted shifts is still open as well as the general question.

The Countably Hypercyclic Criterion: 
The hypercyclicity criterion basically says if an operator has two dense sets Y and Z (we may assume they are actually linear subspaces) such that the forward orbits go to zero on Y and on Z backward orbits go to zero, then our operator is hypercyclic.

The Countably Hypercyclic Criterion says if there are two subspaces Y and Z such that Y is infinite dimensional and Z is dense and forward orbits go to zero on Y and on Z backward orbits go to zero, then our operator is countably hypercyclic.

That is, the only difference is that the set where the forward orbits go to zero is no longer required to be dense, but only infinite dimensional!