Let $X$ be a Banach space and $\mathcal{A}$ be an operator ideal. A sequence $(x_{n})_{n=1}^{\infty}$ in $X$ is called $\mathcal{A}$-convergent to zero if there exist an operator $S\in \mathcal{A}(Z,X)$ for some Banach space $Z$ and a sequence $(z_{n})_{n=1}^{\infty}$ in $Z$ converging to zero in norm such that $x_{n}=S(z_{n}),n=1,2,...$ We denote the set of all such sequences by $c^{\mathcal{A}}_{0}(X)$. My questions are the following:

$\bullet$ Is $c^{\mathcal{A}}_{0}(X)$ a Banach space with the sup norm?

$\bullet$ What is the dual space of $c^{\mathcal{A}}_{0}(X)$?