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.

中文翻译：

---

频繁超循环向量的代数

我们表明，空间 $\ell_{p}$、$1\leq p<\infty$ 或 $c_{0}$ 上的后移运算符的倍数，当赋予坐标乘法时，不具有频繁的超循环代数. 更一般地，我们描述了这些算子的 $\mathcal{A}$-超循环向量代数的存在性。我们还表明，整个函数空间上的微分算子，当赋予 Hadamard 乘积时，不具有频繁的超循环代数。另一方面，我们证明了对于任何 Banach 空间上的任何频繁超循环算子 $T$，$FHC(T)$ 对于合适的乘积是可代数的，在某些情况下它甚至是强可代数的。