Factorization of operators mapping (F)-spaces into (DF)-spaces (Q793273)

Let \({\mathcal A}\) be an operator ideal in the category of Banach spaces, E, F locally convex spaces, \({\mathcal L}(E,F)\) the space of continuous linear operators from E into F. Then define \({\mathcal A}^ w(E,F)\) to be the set of all \(T\in {\mathcal L}(E,F)\) such that for all Banach spaces \(B_ 1\), \(B_ 2\) and \(R\in {\mathcal L}(B_ 1,E),\quad Q\in {\mathcal L}(F,B_ 2)\) the product QTR belongs to \({\mathcal A}(B_ 1,B_ 2)\); and \({\mathcal A}^ s(E,F)\) to be set of all \(T\in {\mathcal L}(E,F)\) such that there exist Banach spaces \(B_ 1\), \(B_ 2\) and \(R\in {\mathcal L}(E,B_ 1), Q\in {\mathcal L}(B_ 2,F), T_ 0\in {\mathcal A}(B_ 1,B_ 2)\) with \(T=QT_ 0R.\) Obviously \({\mathcal A}^ s(E,F)\subset {\mathcal A}^ w(E,f).\) Main result: If E is a Banach space or a semireflexive (DF)-space, F an (F)-space, and \({\mathcal A}\) a maximal p-normed \((0<p\leq 1)\) ideal, then \({\mathcal A}^ s(E,F)={\mathcal A}^ w(E,F).\) The following corollary is obtained: An (F)-space F has a fundamental system \(\{A_{\alpha}\}\) of the bounded subsets such that the Banach spaces \(F_{A_{\alpha}}\) are Hilbert spaces if and only if the topology of F can be generated by semiscalar products. More on the factorization problem \(''{\mathcal A}^ s(E,F)={\mathcal A}^ w(E,F)''\) is contained in the author's recent book: Locally convex spaces and operator ideals (1983).