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

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The ideal of weakly compactly generated operators acting on a Banach space. / Kania, Tomasz; Kochanek, Tomasz.

In: Journal of Operator Theory, Vol. 71, No. 2, 01.06.2014, p. 455-477.

Research output: Contribution to journalJournal articlepeer-review

### Harvard

Kania, T & Kochanek, T 2014, 'The ideal of weakly compactly generated operators acting on a Banach space', Journal of Operator Theory, vol. 71, no. 2, pp. 455-477. https://doi.org/10.7900/jot.2012jun23.1959

### APA

Kania, T., & Kochanek, T. (2014). The ideal of weakly compactly generated operators acting on a Banach space. Journal of Operator Theory, 71(2), 455-477. https://doi.org/10.7900/jot.2012jun23.1959

### Author

Kania, Tomasz ; Kochanek, Tomasz. / The ideal of weakly compactly generated operators acting on a Banach space. In: Journal of Operator Theory. 2014 ; Vol. 71, No. 2. pp. 455-477.

### Bibtex

@article{5fc4f96fda554a11be86d071dd0f5b41,
title = "The ideal of weakly compactly generated operators acting on a Banach space",
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.",
keywords = "operator ideal, weakly compactly generated, WCG space",
author = "Tomasz Kania and Tomasz Kochanek",
year = "2014",
month = jun,
day = "1",
doi = "10.7900/jot.2012jun23.1959",
language = "English",
volume = "71",
pages = "455--477",
journal = "Journal of Operator Theory",
issn = "0379-4024",
publisher = "Theta Foundation",
number = "2",

}

### RIS

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 -