From here, it is known that given any formal power series in the following form ($a_0$ may or may not be zero),
$$ g(z) = \sum_{k=0}^\infty a_k z^k \tag{1} \label{pow-ser} $$
there exists another non-trivial formal power series $y(z)$, such that the Cauchy product between $y(z)$ and $g(z)$
$$ y \times g = \sum_{k=0}^\infty c_k z^{k} $$ satsifies that
$$ \sum_{k=0}^\infty |c_k| < \infty. \tag{2} \label{abs-cvg} $$
A related and probably more complicated problem is as follows:
Given a vector of formal power series in the form of $\eqref{pow-ser}$:
$$P=\begin{bmatrix} g_1(z) & g_2(z) & \cdots & g_n(z) \end{bmatrix}$$
does there always exist a non-trivial formal power series $y(z)$, such that every entry of the Cauchy product between $y$ and the vector $P$
$$y \times P$$
satisfies $\eqref{abs-cvg}$?
The answer is negative already for $\color{blue}{n=2}$, even if we restrict $g_1(z)$ and $g_2(z)$ to represent functions of complex variable (analytic near $z=0$). Specifically, we take $\color{blue}{g_1\equiv 1}$, and let $g_2$ be arbitrary for a while. Then $y\times g_1=y$, so the condition $(2)$ implies that $y(z)$ (represents a function that) is analytic on $|z|<1$, and so is $y\times g_2$; thus, $g_2(z)$ is (a restriction of) a quotient of two functions analytic on $|z|<1$, and any singularity it may have (there) must be a pole.
Hence, to build a counterexample, it suffices to take $g_2(z)$ with an essential singularity in $|z|<1$, say $g_2(z)=e^{z/(1-2z)}$. (Or a branching one, such as $\color{blue}{g_2(z)=\log(1-2z)}$ or $g_2(z)=(1-2z)^{1/2}$.)