Absolutely simple p-summing operators and applications

In this paper, we are starting to construct a new theory of absolutely simple p-summing operators. We define a significant class of weak operator ideals, namely the class of absolutely simple p-summing operators between arbitrary real Banach spaces and show some basic properties of that class. A key feature of the resulting class is computing simple p-summing norms exactly for any linear operator between finite-dimensional normed spaces, in contrast to the computation of p-summing norms which is in general difficulty or the computation of Lipschitz p-summing norms between particular classes of metric spaces. Building upon S. Kwapień’s result, we figure out the relations between 2-summing norms and simple 2-summing norms and find out the relations between simple p-summing norms and diverse familiar norms of some linear operators. In the end, we present some concluding remarks and introduce some open problems that we think are intriguing.