The \(E\)-compactness and the \(E\)-approximation property determined by the Banach space \((\sum \ell_p)_q\)
Let \(E\) be a Banach space having a normalised \(1\)-unconditional Schauder basis \((e_n)\), and let \(X\) be a Banach space. The \(X\)-valued sequence \((x_n)\) belongs to \( E(X)\) if the series \(\sum_n \Vert x_n\Vert e_n\) converges in \(E\). The subset \(K \subset X\) is said to be \(E\)-compact if there is \((x_n) \in E(X)\) such that \[ K \subset \{\sum_{n=1}^\infty e^*(e_n)x_n: e^* \in B_{E^*}\}. \] The space \(X\) has the \(E\)-approximation property (abbreviated \(E\)-A.P.), if the identity operator \(I_X\) can be uniformly approximated by finite rank operators on every \(E\)-compact subset of \(X\). The model case \(E = \ell^p\) was considered by \textit{D. P. Sinha} and \textit{A. K. Karn} [Stud. Math. 150, No. 1, 17--33 (2002; Zbl 1008.46008)], and the above generalisations were suggested by \textit{J. M. Kim} and \textit{B. Zheng} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 4, Paper No. 200, 16 p. (2020; Zbl 1468.46031)]. Here the case of the direct sums \(E= \ell^p(\ell^q) := \big(\bigoplus_{\mathbb N} \ell^q\big)_{\ell^p}\) is studied, where \(p = \infty\) or \(q = \infty\) refer to \(c_0\). Main results: (i) Suppose that \(1 \le r \le \infty\) and \(1 \le p < q \le \infty\), and let \(K \subset X\) be a subset. If \(K\) is \(\ell^p(\ell^r)\)-compact, then \(K\) is \(\ell^q(\ell^r)\)-compact, and if \(K\) is \(\ell^r(\ell^p)\)-compact, then \(K\) is \(\ell^r(\ell^q)\)-compact. (ii) Let \(1 < p, q \le \infty\). If the dual space \(X^*\) has the \(\ell^q(\ell^p)\)-A.P., then \(X\) has the \(\ell^q(\ell^p)\)-A.P. as well as the \(\ell^p(\ell^q)\)-A.P.
scientific article; zbMATH DE number 7616148
scientific article