## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 47, Number 7 (2017), 2277-2293.

### The ideal of unconditionally $p$-compact operators

#### 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 $\mathcal K_{\rm up}(\ell _{1}(\Gamma ), X)^* =\mathcal D_{p^*} (\ell _{\infty }(\Gamma ), X^*)$.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 47, Number 7 (2017), 2277-2293.

**Dates**

First available in Project Euclid: 24 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1514084428

**Digital Object Identifier**

doi:10.1216/RMJ-2017-47-7-2277

**Mathematical Reviews number (MathSciNet)**

MR3748231

**Zentralblatt MATH identifier**

06828640

**Subjects**

Primary: 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] 46B45: Banach sequence spaces [See also 46A45] 46B50: Compactness in Banach (or normed) spaces 47L20: Operator ideals [See also 47B10]

**Keywords**

Unconditionally $p$-summable sequence unconditionally $p$-compact operator Banach operator ideal tensor norm approximation property

#### Citation

Kim, Ju Myung. The ideal of unconditionally $p$-compact operators. Rocky Mountain J. Math. 47 (2017), no. 7, 2277--2293. doi:10.1216/RMJ-2017-47-7-2277. https://projecteuclid.org/euclid.rmjm/1514084428