On the norm of a projection onto the space of compact operators
Volume 182 / 2007
Abstract
Let and Y be Banach spaces and let \mathcal{A}(X,Y) be a closed subspace of \mathcal{L}(X,Y), the Banach space of bounded linear operators from X to Y, containing the subspace \mathcal{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 \ge 0 and B is a compact set of scalars (Kalton's property (M^*) = M^*(1, \{-1\}, 1)), and if \mathcal{A}(X,Y) \ne \mathcal{K}(X,Y), then there is no projection from \mathcal{A}(X,Y) onto \mathcal{K}(X,Y) with norm less than \max |B| + c. Since, for given \lambda with 1 < \lambda < 2, every Y with separable dual can be equivalently renormed to satisfy M^*(a,B,c) with \max |B| + c = \lambda, this implies and improves a theorem due to Saphar.
