Infinite matrices bounded on weighted ℓ1 spaces

This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ1 spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ1 spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ1 space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ2 (Hilbert) space.