Frame-cyclic operators and their properties

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

On Subspace-recurrent Operators

In this article, subspace-recurrent operators are presented and it is showed that the set of subspace-transitive operators is a strict subset of the set of subspace-recurrent operators. We demonstrate that despite subspace-transitive operators and subspace-hypercyclic operators, subspace-recurrent operators exist on finite dimensional spaces. We establish that operators that have a dense set of periodic points are subspace-recurrent. Especially, if $T$ is an invertible chaotic or an invertible subspace-chaotic operator, then $T^{n}$, $T^{-n}$ and $\lambda T$ are subspace-recurrent for any positive integer $n$ and any scalar $\lambda$ with absolute value $1$. Also, we state a subspace-recurrence criterion.

INVERSE OF FREQUENTLY HYPERCYCLIC OPERATORS

We show that there exists an invertible frequently hypercyclic operator on $\ell ^1(\mathbb {N})$ whose inverse is not frequently hypercyclic.

Chaos and frequent hypercyclicity for weighted shifts

Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.