Inequalities Involving Infinite Matrices with Nonnegative Entries

For complete, normally quasinormed subspaces λ, μ of ω, the set of all sequences of scalars, and an infinite matrix A with nonnegative entries, we shall be interested in inequalities of the form (*) ‖ A| x | ‖λ≤K ‖ bx ‖ μ ( x∈ b −1 μ ), $$\left\| {A\left| x \right|} \right\|\lambda \leqslant K{\left\| {bx} \right\|_\mu }\quad \quad \left( {x \in {b^{ - 1}}\mu } \right),$$ , where b ∊ ω, and K is a positive constant. By introducing a method of comparing sequences, we shall obtain results on best possible inequalities of the form (*), best possible not by the smallness of K but by the smallness of the sequence b. Our results have been applied to Hardy’s inequality, and to some of its generalizations. In the process of our investigation, we have also obtained some best possible inequalities of the form (*).