The role of matrices that are convergent to zero in the study of semilinear operator systems

In this paper we explain the advantage of vector-valued norms and the role of matrices that are convergent to zero in the study of semilinear operator systems by means of some basic methods of nonlinear analysis: the contraction principle, Schauder’s fixed point theorem, the Leray–Schauder continuation principle and Krasnoselskii’s cone fixed point theorem. A vector version of Krasnoselskii’s theorem is also established.