Products of operator ideals and extensions of Schatten classes

Abstract

We study relations between Schatten classes and product operator ideals, where one of the factors is the Banach ideal ΠE,2 of (E, 2)-summing operators, and where E is a Banach sequence space with l2 ↪ E. We show that for a large class of 2-convex symmetric Banach sequence spaces the product ideal ΠE,2 ○ aq,s is an extension of the Schatten class F with a suitable Lorentz space F. As an application, we obtain that if 2 ≤ p, q < ∞, 1/r = 1/p + 1/q and E is a 2-convex symmetric space with fundamental function λE(n) ≈ n1/p, then ΠE,2 ○ Πq is an extension of the Schatten class r,q (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)