Composition of operator ideals

Abstract

In [13,14] E. Saksman and H. Tylli and G. Racher in [10] have been studying weak compactness for continuous linear composition operators SRB :l F1;E† !l E1;F† defined by T 7! R T B. Here E, F , E1, F1 are Banach spaces and we suppose that R 2l E;F† and B 2l E1;F1†. If E or F1 has the Dunford-Pettis property, then G. Racher [10] has shown that SRB is weakly compact if and only if R and B are weakly compact operators. In [14] E. Saksman and H. Tylli point out that the exact condition for SRB to be weakly compact is in general quite complicated and to a large extent unknown. In this paper we try to unify these results by considering abstract Banach and operator ideals (see 2.1^2.3 below). Beside the results on weak compactness of the composed operator SRB; we investigate other properties of SRB; namely when the operator is Asplund or Rosenthal. For this purpose we study the so-called fine line betwen Asplund and conditionally weakly compact sets in the space L1 ;F† of measurable functions with yessentially relatively compact range which may be of independent interest and which extends results of H. Collins, W. Ruess and C. Stegall. Section 3 answers the question that the given criteria in part two for the composed operator to be weakly compact, Asplund or Rosenthal are in certain cases necessary. For Banach spaces E, F let f E;F†, k E;F †, w E;F†, r E;F†, d1 E;F† denote the spaces of all finite rank, compact, weakly compact, Rosenthal and Asplund operators from E into F respectively. Let us recall that T is called compact, weakly compact, Rosenthal, Asplund, if it maps the closed unit ball of E onto a relatively compact, a relatively weakly compact, a conditionally weakly compact, an Asplund set in F . A subset K of E is called an Asplund set, if span K0† ; pK0† is separable for all countable K0 K , where pK0 x † :ˆ supx2K0 j j, x 2 span K0† (see [1], pp. MATH. SCAND. 84 (1999), 284^296