solution to infinite matrix equation

Let , and suppose that for each we have

and .

I seek to determine whether a solution to the equation exists, where are infinite matrices and . In other words, I seek to show that there exists some with complex entries satisfying the following equation:

What tools do I use to deal with a problem like this? Clearly, if is invertible then exists. But how do I show that an infinite matrix is invertible? If is not invertible, do I have other options?

Any help would be much appreciated, thanks!