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  • Kevin Beanland (a1) and Pandelis Dodos (a2)


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(p,Yp) is a co-analytic non-Borel subset of ℒ(p,Yp) yet every strictly singular operator T:pYp satisfies ϱ(T)≤2. This answers a question of Argyros.



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  • Kevin Beanland (a1) and Pandelis Dodos (a2)


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