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ON STRICTLY SINGULAR OPERATORS BETWEEN SEPARABLE BANACH SPACES

Part of: Set theory Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  22 June 2010

Kevin Beanland  and
Pandelis Dodos
Kevin Beanland
Affiliation:
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, U.S.A. (email: kbeanland@vcu.edu)
Pandelis Dodos
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece (email: pdodos@math.ntua.gr)
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Abstract

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 ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math.169 (2009), 221–250]. Secondly, for every 1≤p<+∞, we construct a Banach space Yp with an unconditional basis such that 𝒮𝒮(ℓp,Yp) is a co-analytic non-Borel subset of ℒ(ℓp,Yp) yet every strictly singular operator T:ℓp→Yp satisfies ϱ(T)≤2. This answers a question of Argyros.

MSC classification

Secondary: 46B28: Spaces of operators; tensor products; approximation properties 03E15: Descriptive set theory
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 285 - 304
Copyright
Copyright © University College London 2010

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