I'm stuck in the following two questions (All index sets below are natural numbers).
Assume that $a=(a_n)$ is a sequence of scalars in $\ell^p$ for some $p>1.$ Fix some $q>1$. If $a\ast b\in \ell^q$ for any $b\in \ell^q$, must $a\in \ell^1$?.
Fix $1<p<2$ and assume that $a = (\frac{1}{n})$. Then does there exist $b\in \ell^p$ such that $a\ast b\notin \ell^p$? Or, is $T:\ell^p\rightarrow \ell^p$ defined by $T(b)= a\ast b$ a well-defined linear operator?
Any feedback or reference would be greatly appreciated.