On the norm of a projection onto the space of compact operators

Joosep Lippus, and Eve Oja. "On the norm of a projection onto the space of compact operators." Studia Mathematica 182.3 (2007): 263-272. <http://eudml.org/doc/284969>.

@article{JoosepLippus2007,
abstract = {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.},
author = {Joosep Lippus, Eve Oja},
journal = {Studia Mathematica},
keywords = {compact operators; projections; approximation properties},
language = {eng},
number = {3},
pages = {263-272},
title = {On the norm of a projection onto the space of compact operators},
url = {http://eudml.org/doc/284969},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Joosep Lippus
AU - Eve Oja
TI - On the norm of a projection onto the space of compact operators
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 3
SP - 263
EP - 272
AB - 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.
LA - eng
KW - compact operators; projections; approximation properties
UR - http://eudml.org/doc/284969
ER -