The dual space of (\(\mathcal L(X, Y), \tau _p\)) and the \(p\)-approximation property
According to \textit{D.\,Sinha} and \textit{A.\,K.\thinspace Karn} [``Compact operators whose adjoints factor through subspaces of \(l_p\)'', Stud.\ Math.\ 150, No.\,1, 17--33 (2002; Zbl 1008.46008)], for \(p \geq 1\), a subset \(K\) of a Banach space \(X\) is relatively \(p\)-compact if \(K \subseteq \{ \sum_n a_n x_n : (a_n)_n \in B_{\ell_{p'}} \}\), where \(p' = p/(p-1)\) and \((x_n) \in \ell_p(X)\) (\(c_0(X)\) when \(p = \infty\)). \(\infty\)-compactness is just usual compactness. If the identity operator on \(X\) can be approximated uniformly on every \(p\)-compact set by finite rank operators, then \(X\) is said to have the \(p\)-approximation property (\(p\)-AP). Sinha and Karn proved that every Banach space has the \(p\)-approximation property for every \(1 \leq p \leq 2\). The topology \(\tau_p\) of uniform convergence on \(p\)-compact sets is a locally convex topology on the space of all bounded operators \(L(X,Y)\). The authors give a representation of \((L(X,Y),\tau_p)^*\) in the style of Grothendieck and use it to show that \(X\) has the \(p\)-AP if \(X^*\) has the \(p\)-AP. Using Davie's construction [\textit{A.\,M.\thinspace Davie}, ``The approximation problem for Banach spaces'', Bull.\ Lond.\ Math.\ Soc.\ 5, 261--266 (1973; Zbl 0267.46013)], they show that, given \(2 < p < \infty\), there exists a subspace of \(\ell_{q}\) which fails the \(p\)-AP when \(q > 2p/(p-2)\). It follows that, for every \(2 < p < \infty\), there exists a Banach space \(Y_p\) with the \(p\)-AP such that \(Y^*_p\) fails the \(p\)-AP. They also introduce and study the compact \(p\)-AP and give examples of spaces which fail the compact \(p\)-AP, and of spaces with the compact \(p\)-AP which fail the \(p\)-AP (building on Willis' construction).
scientific article; zbMATH DE number 5801198
scientific article