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Journal of Operator Theory

Volume 71, Issue 2, Spring 2014  pp. 455-477.

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

Authors:  Tomasz Kania (1) and Tomasz Kochanek (2)
Author institution: (1) Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom
(2) Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warszawa, Poland

Summary: 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.

Keywords: operator ideal, weakly compactly generated, WCG space

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