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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 coordinatewisemultiplication, do not possess frequently hypercyclic algebras. More generally,we characterize the existence of algebras of $\mathcal{A}$-hypercyclic vectorsfor these operators. We also show that the differentiation operator on thespace of entire functions, when endowed with the Hadamard product, does notpossess frequently hypercyclic algebras. On the other hand, we show that forany frequently hypercyclic operator $T$ on any Banach space, $FHC(T)$ isalgebrable for a suitable product, and in some cases it is even stronglyalgebrable.

Algebras of frequently hypercyclic vectors