Limit Orders of ϵπ‐Continuous Operators

Abstract

An operator T : X --f Y between BANACH spaces is said to be en-continuous if T 0 T : X @I X --+ Y an Y is continuous, where e and n denote the injective and projective topology, respectively. This concept was motivated by an old question of GROTHEN-DIECK asking for conditions to be imposed on a locally convex space E in order to ensure that E 0. E = E 0% E implies nuclearity of E (see [4] for details). Recently, new interest has been given to this question by PISIER's discovery of infinite dimensional BANACH spaces P such that P @ P = P 8% P (cf. [7]).In this note we compute the limit orders w, p , q) of the class of all en-continuougoperatorg, i.e., for 1 5 p, q I 00 the infimum over all A 1 0 such that each diagonal operator 0,: lp + lg with diagonal u E ia en-continuous. The main result ia:Theorem. Let p , q E [l, 003. Then Noreover, the limit order ie attained.1. Prelimiusries. We shall employ standard notations and notions from BANACH space theory end the theory of operator ideals, as preaented in [6].