Let b_{i,j}\in\mathbb{C}, and suppose that for each i we have

\sum_{j=1}^\infty|b_{i,j}|<\infty and \sum_{j=1}^\infty|b_{i,j}|\leq\sum_{j=1}^\infty|b_  {i+1,j}|.

I seek to determine whether a solution X to the equation AX=B exists, where A,B,X are infinite matrices and a_{i,j}=b_{i+1,j}. In other words, I seek to show that there exists some X=(x_{i,j}) with complex entries satisfying the following equation:

\begin{bmatrix}b_{2,1}&b_{2,2}&b_{2,3}&\cdots\\b_{  3,1}&b_{3,2}&b_{3,3}&\cdots\\b_{4,1}&b_{4,2}&b_{4,  3}&\cdots\\\vdots&\vdots&\vdots&\end{bmatrix} \begin{bmatrix}x_{1,1}&x_{1,2}&x_{1,3}&\cdots\\x_{  2,1}&x_{2,2}&x_{2,3}&\cdots\\x_{3,1}&x_{3,2}&x_{3,  3}&\cdots\\\vdots&\vdots&\vdots&\end{bmatrix} =\begin{bmatrix}b_{1,1}&b_{1,2}&b_{1,3}&\cdots\\b_  {2,1}&b_{2,2}&b_{2,3}&\cdots\\b_{3,1}&b_{3,2}&b_{3  ,3}&\cdots\\\vdots&\vdots&\vdots&\end{bmatrix}

What tools do I use to deal with a problem like this? Clearly, if A is invertible then X=A^{-1}B exists. But how do I show that an infinite matrix is invertible? If A is not invertible, do I have other options?

Any help would be much appreciated, thanks!