ON STRICTLY SINGULAR OPERATORS BETWEEN SEPARABLE BANACH SPACES

摘要

Let X and Y be separable Banach spaces and denote by SS(X, Y) the subset of L(X, Y) consisting of all strictly singular operators. We study various ordinal ranks on the set SS(X, Y). Our main results are summarized as follows. Firstly, we define a new rank rs on SS(X, Y). We show that rs is a co-analytic rank and that it dominates the rank Q introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math. 169 (2009), 221-250]. Secondly, for every 1≤p <+∞, we construct a Banach space Y_p with an unconditional basis such that SS(l_p, Y_p) is a co-analytic non-Borel subset of ,L(l_p, Y_p) yet every strictly singular operator T :l_p→Y_p satisfies Q(T)≤2. This answers a question of Argyros.