Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X \to Y$. Suppose $\sup_n \sum_k \|A_{n,k}\|<\infty$.
Property: For each sequence $x=(x_1,x_2,\ldots)$ contained in compact of $X$, the image $$ Ax:=\left(\sum_k A_{n,k}x_k:n\ge 1\right) $$ is a well-defined sequence contained in a compact of $Y$. Does such kind of matrices have a name? Or the property is always verified?
(Their property is reminescent of compact operators, i.e., the images of bounded sets are relatively compact.)
The condition $M:=\sup_n \sum_k \|A_{n,k}\|<\infty$ is sufficient for $Ax$ well-defined for all $x \in \ell_\infty(X)$. Then the linear map $x\mapsto Ax$ on $\ell_\infty(X)$ is continuous because $$ \|Ax-Ay\|=\sup_n \|\sum_k A_{n,k}(x_k-y_k)\| \le M\|x-y\|. $$ To conclude, the continuous image of relatively compact sets is relatively compact, see here.