The \(p\)-approximation property in terms of density of finite rank operators
The paper deals with the concept of \(p\)-approximation property, due to \textit{D.\,P.\thinspace Sinha} and \textit{A.\,K.\thinspace Karn} [Stud.\ Math.\ 150, No.\,1, 17--33 (2002; Zbl 1008.46008)]; this notion is a natural extension of the well-known approximation property in the context of Banach space theory. Let \(1\leq p\leq\infty\) and \(p^{\prime}\) be the conjugate of \(p\). The \(p\)-convex hull of a sequence \((x_{n})\in l_{p}(X)\) is the set \[ p-\text{conv}\{x_{n}\}=\left\{{\displaystyle\sum\limits_{n=1}^{\infty}}a_{n}x_{n};{\displaystyle\sum\limits_{n=1}^{\infty}}\left| a_{n}\right| ^{p^{\prime}}\leq1\right\}. \] If \(p=1\), the second sum is naturally replaced by \(\sup\left| a_{n}\right|\). A set \(K\subset X\) is relatively \(p\)-compact if there exists \((x_{n})\in l_{p}(X)\) (\((x_{n})\in c_{0}(X)\) if \(p=\infty\)) such that \(K\subset p-\text{conv}\{x_{n}\}\). A Banach space \(X\) has the \(p\)-approximation property (\(p\)-\(AP\)) if the identity \(I_{X}\) can be approximated by finite rank operators uniformly on the relatively \(p\)-compact subsets of \(X\). The ideals of bounded, nuclear, compact, \(p\)-summing and finite rank operators are denoted by \(\mathcal{L},\mathcal{N},\mathcal{K},\mathcal{P}_{p}\) and \(\mathcal{F}\), respectively. If \(\mathcal{A}\) is an operator ideal, \(\mathcal{A}^{d}\) denotes its dual operator ideal. An operator \(T\in\mathcal{L}(X,Y)\) is\ \(p\)-compact if \(T(B_{X})\) is relatively \(p\)-compact in \(Y\); the set of all \(p\)-compact operators from \(X\) to \(Y\) is denoted by \(\mathcal{K}_{p}(X,Y).\) The main result of the paper characterizes the \(p\)-\(AP\) in terms of density of finite rank operators in the spaces of \(p\)-compact operators. More precisely, if \(X\) is a Banach space and \(1\leq p\leq\infty\), the authors show that the following statements are equivalent: (a) \(X\) has the \(p\)-\(AP\). (b) \(\mathcal{F}(Y,X)\) is \(\left\|\cdot\right\| \)-dense in \(\mathcal{K}_{p}(Y,X)\) for every Banach space \(Y\). (c) \(\mathcal{F}(Y,X)\) is \(\tau\)-dense in \(\mathcal{K}_{p}(Y,X)\) for every Banach space \(Y\). (d) \(\mathcal{F}(Y,X)\) is \(\tau\)-dense in \(\mathcal{P}_{p}^{d}(Y,X)\) for every Banach space \(Y\). Among other interesting results, the authors also investigate the \(p\)-\(AP\) for duals of Banach spaces. For example, considering the natural embedding \[ \mathcal{N}(X,Y)^{\ast}\subset\mathcal{L}(Y,X^{\ast\ast}), \] it is shown that for Banach spaces \(X\) and \(Y\) and real numbers \(p\geq1,\) the following inclusions hold: (a) If \(X^{\ast\ast}\) has the \(p\)-\(AP\), then \(\mathcal{P}_{p}^{d} (Y,X^{\ast\ast})\subset\mathcal{N}(X,Y)^{\ast}.\) (b) If \(Y^{\ast}\) has the \(p\)-\(AP\), then \(\mathcal{P}_{p}(Y,X^{\ast\ast })\subset\mathcal{N}(X,Y)^{\ast}\).
scientific article; zbMATH DE number 5553661
scientific article