0000000000174354
AUTHOR
Karl-g. Grosse-erdmann
Algebras of frequently hypercyclic vectors
We show that the multiples of the backward shift operator on the spaces $\ell_{p}$, $1\leq p<\infty$, or $c_{0}$, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of $\mathcal{A}$-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator $T$ on any Banach space, $FHC(T)$ is algebrable for a suitable product, and in some cases it is even strongly algebrable.
Algebrability of the set of hypercyclic vectors for backward shift operators
Abstract We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Frechet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case, we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.