Matrix transformations between some classes of sequences

Let A = ( a nk ) be an infinite matrix of complex numbers a nk ( n , k = 1, 2,…) and X , Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y , and we denote it by writing A : X → Y , if for every sequence x = ( x k )∈ X the sequence Ax = ( A n ( x )) is in Y , where A n ( x ) = Σ a nk x k and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A . By ( X , Y ) we denote the class of matrices A such that A : X → Y .