Let X and Y be Banach spaces and let 𝓐(X,Y) be a closed subspace of 𝓛(X,Y), the Banach space of bounded linear operators from X to Y, containing the subspace 𝒦(X,Y) of compact operators. We prove that if Y has the metric compact approximation property and a certain geometric property M*(a,B,c), where a,c ≥ 0 and B is a compact set of scalars (Kalton's property (M*) = M*(1, {-1}, 1)), and if 𝓐(X,Y) ≠ 𝒦(X,Y), then there is no projection from 𝓐(X,Y) onto 𝒦(X,Y) with norm less than max|B| + c. Since, for given λ with 1 < λ < 2, every Y with separable dual can be equivalently renormed to satisfy M*(a,B,c) with max|B| + c = λ, this implies and improves a theorem due to Saphar.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-5,
author = {Joosep Lippus and Eve Oja},
title = {On the norm of a projection onto the space of compact operators},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {263-272},
zbl = {1135.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-5}
}
Joosep Lippus; Eve Oja. On the norm of a projection onto the space of compact operators. Studia Mathematica, Tome 178 (2007) pp. 263-272. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-5/