# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2012jun23.1959

**Keywords: **operator ideal, weakly compactly generated, WCG space

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