SCHREIER SINGULAR OPERATORS

In this paper we further investigate Schreier singular operators introduced in [ADST]. If X and Y are two Banach spaces, a bounded operator T : X -> Y is Schreier singular if for every epsilon > 0 and every basic sequence (x(n)) in X there is a vector of the form x = Sigma(n)(i=1)a(i)x(ki) for some a(1), . . . , a(n) is an element of R and n <= k(1) < ... < k(n) such that parallel to Tx parallel to < epsilon parallel to x parallel to. It was shown in [ADST] that the class of Schreier singular operators on a reflexive space is stable under left and right multiplication by bounded operators. We show that this remains valid for non-reflexive spaces. We also present a characterisation of Schreier singular operators in terms of spreading models. It was shown in [ADST] that if X has few spreading models then the product of any sufficiently many Schreier singular operators is compact. We show that the conclusion remains valid if there are no long chains of spreading models. Finally, we show that this cannot be extended to arbi- trary Banach spaces by presenting an example of a finitely strictly singular operator which is not even polynomially compact.