Asplund operators and the Szlenk index

Abstract

For α an ordinal, we investigate the class SZ α consisting of all operators whose Szlenk index is an ordinal not exceeding ω α . We show that each class SZ α is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes SZ α and several wellknown closed operator ideals is investigated and quantitative factorization results in terms of the Szlenk index are obtained for the class of Asplund operators.

Direct sums and the Szlenk index

“…Parts (i) of Proposition 1.1 is discussed in [9]. Part (ii) is discussed in [9] in the case of spaces, and the more general case of operators is established in [3,Proposition 2.10]. Part (iii) was proved for K = B E * in [14]; see also p.64 of [10].…”

“…A class of closed operator ideals naturally related to the Szlenk index has been introduced and systematically studied by the present author in [3]. These operator ideals are denoted SZ α , where α is an ordinal, and elements of SZ α are known as α-Szlenk operators.…”

“…These operator ideals are denoted SZ α , where α is an ordinal, and elements of SZ α are known as α-Szlenk operators. The operator ideals SZ α are studied in [3] with regard to their operator ideal properties and their relationship to other closed operator ideals, in particular the class of Asplund operators.…”

“…In [3], results and techniques developed in the current paper are applied to obtain both positive and negative answers to Question 0.1 for the case I = SZ α , with the answer depending upon ordinal properties of α.…”

“…We now outline the structure of the current paper. In Section 1 we detail necessary notation and background results regarding the Szlenk index, including several relevant results from [3]. Our main results are presented in Section 2.…”