Functiones et Approximatio Commentarii Mathematici

A Banach function space $X$ for which all operators from $\ell^p$ to $X$ are compact

Guillermo P. Curbera and Luis Rodríguez-Piazza

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We construct a rearrangement invariant space $X$ on $[0,1]$ with the property that all bounded linear operators from $\ell^p$, $1<p<\infty$, to $X$ are compact, but there exists a non-compact operator from $\ell^\infty$ to $X$. The techniques used allow to give a new proof of the characterization given by Hern\'andez, Raynaud and Semenov of the rearrangement invariant spaces on $[0,1]$ for which the canonical embedding into $L^1([0,1])$ is finitely strictly singular.

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Funct. Approx. Comment. Math., Volume 50, Number 2 (2014), 233-249.

First available in Project Euclid: 26 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 47B07: Operators defined by compactness properties
Secondary: 46B25: Classical Banach spaces in the general theory 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

compact operator strictly singular operator rearrangement invariant space vector measure


Curbera, Guillermo P.; Rodríguez-Piazza, Luis. A Banach function space $X$ for which all operators from $\ell^p$ to $X$ are compact. Funct. Approx. Comment. Math. 50 (2014), no. 2, 233--249. doi:10.7169/facm/2014.50.2.3.

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