A remark on p-summing norms of operators

In this paper we improve a result of W. B. Johnson and G. Schechtman by proving that the p-summing norm of any operator with n-dimensional domain can be well-approximated using C(p)n logn(log logn)2 vectors if 1 < p < 2, and using C(p)np/2 logn if 2 < p <∞. 1. p-summing norms Throughout this paper we will follow notations of N. Tomczak-Jaegermann [To]. Definition 1.1. Let X and Y be Banach spaces. An operator u : X −→ Y is called p-summing if there exists a constant c such that for all finite sequences {xj} in X one has ∑ j ||uxj || 1/p ≤ c sup x∗∈X∗ and ||x∗||≤1 ∑ j |(xj , x∗)|p 1/p . The infimum of constants c satisfying this inequality is denoted by πp(u) and is called the p-summing norm of u. Here we would like to discuss the following natural question: Given an operator u with n-dimensional domain, how many vectors do we need to approximate the p-summing norm of u? To present a more precise version of this question we will first give the following definition: Definition 1.2. For positive integer n, let π p (u) is the smallest constant c such that for arbitrary vectors x1, . . . , xn ∈ X one has  n ∑ j=1 ||uxj || 1/p ≤ c sup x∗∈X∗ and ||x∗||≤1  n ∑ j=1 |(xj , x∗)|p 1/p . One can see that ||u|| = π p (u) ≤ π p (u) ≤ · · · ≤ π p (u) ≤ πp(u), 2000 Mathematics Subject Classification. Primary 46E30, 46B07; Secondary 46B09, 60G15.