Unconditional Cauchy series and uniform convergence on matrices

The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of P(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem axe proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem axe given.