Duality of measures of non-${\mathcal A}$-compactness
Volume 229 / 2015
Abstract
Let ${\mathcal A}$ be a Banach operator ideal. Based on the notion of ${\mathcal A}$-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-${\mathcal A}$-compactness of an operator. We consider a map $\chi _{\mathcal A}$ (respectively, $n_{\mathcal A}$) acting on the operators of the surjective (respectively, injective) hull of ${\mathcal A}$ such that $\chi _{{\mathcal A}}(T)=0$ (respectively, $n_{\mathcal A}(T)=0$) if and only if the operator $T$ is ${\mathcal A}$-compact (respectively, injectively ${\mathcal A}$-compact). Under certain conditions on the ideal ${\mathcal A}$, we prove an equivalence inequality involving $\chi _{\mathcal A}(T^*)$ and $n_{{\mathcal A}^d}(T)$. This inequality provides an extension of a previous result stating that an operator is quasi $p$-nuclear if and only if its adjoint is $p$-compact in the sense of Sinha and Karn.
