Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - The ideal of weakly compactly generated operators acting on a Banach space
AU - Kania, Tomasz
AU - Kochanek, Tomasz
PY - 2014/6/1
Y1 - 2014/6/1
N2 - 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.
AB - 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.
KW - operator ideal
KW - weakly compactly generated
KW - WCG space
U2 - 10.7900/jot.2012jun23.1959
DO - 10.7900/jot.2012jun23.1959
M3 - Journal article
VL - 71
SP - 455
EP - 477
JO - Journal of Operator Theory
JF - Journal of Operator Theory
SN - 0379-4024
IS - 2
ER -