Matrices of bounded linear operators

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.)

• As a side note, such type of matrices appear when checking if a linear map $A:\mathcal A\to\mathcal B$ between $C^*$-algebras $\mathcal A,\mathcal B$ is completely positive although I feel like this is not what you were looking for. – Frederik vom Ende Jan 4 at 9:55
• This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences. – user207096 Jan 4 at 10:31

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.