On the norm of a projection onto the space of compact operators
Volume 182 / 2007
Abstract
Let $X$ 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.
