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2017 The ideal of unconditionally $p$-compact operators
Ju Myung Kim
Rocky Mountain J. Math. 47(7): 2277-2293 (2017). DOI: 10.1216/RMJ-2017-47-7-2277

Abstract

We investigate the ideal $\mathcal K_{\rm up}$, $1 \leq p \leq \infty $, of unconditionally $p$-compact operators. We obtain the isometric identities $\mathcal K_{\rm up}=\mathcal K_{\rm up}\circ \mathcal K_{\rm up}$, $\mathcal K^{\max }_{\rm up}=\mathcal L^{\rm sur}_{p^*}$, $\mathcal K^{\min }_{\rm up}=\widehat {\otimes }_{/w_{p^*}}$ and $\mathcal K_{\rm up}=\mathcal N_{\rm up}^{\rm Qdual}$ and prove that, if $X^*$ has the approximation property or $Y$ has the $\mathcal K_{\rm up}$-approximation property, then $\mathcal K_{\rm up}(X, Y)$ is isometrically equal to $\mathcal K^{\min }_{\rm up}(X, Y)$, and the dual space $\mathcal K_{\rm up}(X, Y)^*$ is isometric to $(\mathcal L_{p}^{\rm inj})^*(X^*, Y^*)$. As a consequence, for every Banach space $X$, we obtain the isometric identities $\mathcal K_{\rm up}^{\max }(\ell _{1}(\Gamma ), X) =\mathcal L_{p^*}(\ell _{1}(\Gamma ), X)$, $\mathcal K_{\rm up}^{\min }(\ell _{1}(\Gamma ), X) =\ell _{\infty }(\Gamma )\widehat {\otimes }_{w_{p^*}} X$ and $\mathcal K_{\rm up}(\ell _{1}(\Gamma ), X)^* =\mathcal D_{p^*} (\ell _{\infty }(\Gamma ), X^*)$.

Citation

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Ju Myung Kim. "The ideal of unconditionally $p$-compact operators." Rocky Mountain J. Math. 47 (7) 2277 - 2293, 2017. https://doi.org/10.1216/RMJ-2017-47-7-2277

Information

Published: 2017
First available in Project Euclid: 24 December 2017

zbMATH: 06828640
MathSciNet: MR3748231
Digital Object Identifier: 10.1216/RMJ-2017-47-7-2277

Subjects:
Primary: 46B28, 46B45, 46B50, 47L20

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 7 • 2017
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