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The ideal of weakly compactly generated operators acting on a Banach space

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  • Tomasz Kania
  • Tomasz Kochanek
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<mark>Journal publication date</mark>1/06/2014
<mark>Journal</mark>Journal of Operator Theory
Issue number2
Volume71
Number of pages23
Pages (from-to)455-477
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We call a bounded linear operator acting between Banach spaces \textit{weakly compactly generated} ($\mathsf{WCG}$ for short) if its range is contained in a~weakly compactly generated subspace of its target space. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a~closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\J_p(\om_1)$, we show how properties of the ideal of $\mathsf{WCG}$ operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the $K_0$-group of $\B(\J_p(\om_1))$ as the additive group of integers and prove automatic continuity of homomorphisms from this Banach algebra.