Journal of Operator Theory

Volume 70, Issue 1, Summer 2013 pp. 291-307.

The Giesy-James theorem for general index $p$, with an application to operator ideals on the $p^{\text{TH}}$ James space

Authors: Alistair Bird (1), Graham Jameson (2), and Niels Jakob Laustsen (3)
Author institution: (1) Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, U.K.
(2) Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, U.K.
(3) Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, U.K.

Summary: A theorem of Giesy and James states that $c_0$ is finitely re\-present\-able in James' quasi-reflexive Banach space~$J_2$. We extend this theorem to the $p^{\text{th}}$ quasi-reflexive James space~$J_p$ for each $p\in(1,\infty)$. As an application, we obtain a new closed ideal of operators on~$J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\ell_\infty^1\oplus\ell_\infty^2\oplus\cdots\oplus \ell_\infty^n\oplus\cdots)_{\ell_p}$ of~$J_p$.

DOI: http://dx.doi.org/10.7900/jot.2011aug11.1936
Keywords: quasi-reflexive Banach space, James space, finite representability of~$c_0$, closed operator ideal

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