## Arkiv för Matematik

• Ark. Mat.
• Volume 46, Number 1 (2008), 113-142.

### Geometry of spaces of compact operators

#### Abstract

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces Y. We show that X* has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.

#### Article information

Source
Ark. Mat., Volume 46, Number 1 (2008), 113-142.

Dates
Received: 3 March 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907024

Digital Object Identifier
doi:10.1007/s11512-007-0060-y

Mathematical Reviews number (MathSciNet)
MR2379687

Zentralblatt MATH identifier
1166.46009

Rights
2007 © Institut Mittag-Leffler

#### Citation

Lima, Åsvald; Lima, Vegard. Geometry of spaces of compact operators. Ark. Mat. 46 (2008), no. 1, 113--142. doi:10.1007/s11512-007-0060-y. https://projecteuclid.org/euclid.afm/1485907024

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