A remark on the approximation of \(p\)-compact operators by finite-rank operators
Let \(X\) and \(Y\) be Banach spaces, \(p \geq 1\), and \(q\) the conjugate index of \(p\), i.e., \(1/p + 1/q = 1\). A linear operator \(T:Y \to X\) is said to be \(p\)-compact if the image \(T(B_Y)\) of the closed unit ball \(B_Y\) is contained in \(\{\sum_n a_nx_n: (a_n) \in B_{\ell_q}\}\) (\((a_n) \in B_{c_0}\) if \(p = 1\)) for some \(X\)-valued absolutely \(p\)-summable sequence \((x_n) \in \ell_p (X)\). The \(\|\cdot\|_{\mathcal K_p}\)-norm of such an operator \(T\) is given by \(\|T\|_{\mathcal K_p} = \inf \|(x_n)\|_p\) where the infimum is taken over all \((x_n) \in \ell_p(X)\). Note that the space \(\mathcal K_p(Y,X)\) of all \(p\)-compact operators contains \(\mathcal F(Y,X)\), the space of finite-rank operators from \(Y\) to \(X\), and is moreover a Banach operator ideal. Recall that \(X\) has the approximation property (\(AP\)) if the identity operator can be approximated by finite-rank operators \(\mathcal F(X,X)\) uniformly on compact subsets of \(X\). The following two successively weaker companions of the \(AP\) have also been studied: (i) \(X\) as the \((p,p)\)-approximation property \(((p,p)\)-\(AP)\) if for every sequence \((x_n) \in \ell_p(X)\) and for every \(\varepsilon >0\) there exists \(S \in \mathcal F(X,X)\) such that \(\|(x_n - Sx_n)\|_p < \varepsilon\). (ii) \(X\) has the \(\mathcal K_p\)-approximation property \((\mathcal K_p\)-\(AP)\) if for every Banach space \(Y\) the space \(\mathcal F(Y,X)\) is \(\|\cdot\|_{\mathcal K_p}\)-dense in \(\mathcal K_p(Y,X)\). The author proves (Proposition \(3\)) that a Banach space has the \((p,p)\)-\(AP\) if and only if it has the \(AP\). In particular, by replacing the \((p,p)\)-\(AP\) with the \(AP\), this improves a result in [\textit{J. M. Delgado}, \textit{C. Piñeiro}, and \textit{E. Serrano}, ``Density of finite rank operators in the Banach space of \(p\)-compact operators'', J. Math. Anal. Appl. 370, No. 2, 498--505 (2010; Zbl 1205.46010)] saying that for a positive measure \(\mu\) closed subspaces of \(L_p(\mu)\) have the \((p,p)\)-\(AP\) exactly when they have the \(\mathcal K_p\)-\(AP\). In the last two sections of the paper some applications and connections to Saphar's approximation properties, \(AP\) of order \(p\), are given, as well as a helpful comparison of \(\mathcal{K}_p\)-operators to the more classical \(p\)-compact operators. Finally the author points out that every Banach space which fails the \(AP\) fails the \((p,p)\)-\(AP\) as well (by Proposition \(3\)), but has the \(\mathcal K_2\)-\(AP\) (since every Banach space has the \(\mathcal K_2\)-\(AP\)). This answers a question in [op. cit.] whether there exists a Banach space with the \(\mathcal K_p\)-\(AP\) which fails the \((p,p)\)-\(AP\).
approximation property defined by a Banach operator ideal
scientific article; zbMATH DE number 5994355
scientific article