The Orlicz-Pettis theorem for multiplier convergent series (Q2832873)

Let \(\lambda\) be a scalar sequence space containing all unit vectors \(e^{j}=\left( \delta_{jk}\right) _{k\in\mathbb{N}}\) \(\left( j\in \mathbb{N}\right) \). Then \(\lambda\) and its \(\beta\)-dual \(\lambda^{\beta }:=\left\{ s=\left( s_{j}\right) :\sum_{j}s_{j}t_{j}\text{ converges for each }t=\left( t_{j}\right) \in\lambda\right\} \) form a dual pair. The space \(\lambda\) is said to have: (a) the signed weak gliding hump property (signed WGHP) if, for every \(t\in\lambda\) and every increasing sequence \(\left( I_{j}\right) \) of intervals, there is a sequence of signs \(\left( s_{j}\right) \) and an index sequence \(\left( n_{j}\right) \) such that the coordinate sum \(\sum_{j}s_{j}\chi_{I_{n_{j}}}t\) is also in \(\lambda\); (b) the infinite gliding hump property (\(\infty\)-GHP) if, for every \(t\in\lambda\) and any increasing sequence \(\left( I_{j}\right) \) of intervals, there exist an index sequence \(\left( n_{j}\right) \) and \(a_{n_{j}}>0\) with \(a_{n_{j} }\rightarrow\infty\) such that every subsequence of \(\left( n_{j}\right) \) has a further subsequence \(\left( p_{j}\right) \) such that the coordinate sum \(\sum_{j}a_{p_{j}}\chi_{I_{p_{j}}}t\) is in \(\lambda\) (here, \(\chi_{I}\) is the characteristic function of \(I\subset\mathbb{N}\)).NEWLINENEWLINELet \(w=w\left( X,X^{\prime}\right) \) be a polar topology defined for all dual pairs \(X,X^{\prime}\). According to \textit{A. Wilansky} [Modern methods in topological vector spaces. Düsseldorf etc.: McGraw-Hill International Book Company (1978; Zbl 0395.46001)], \(w\) is said to be a Hellinger-Toeplitz topology if every weakly (i.e., \(\sigma\left( X,X^{\prime }\right) -\sigma\left( Y,Y^{\prime}\right) \)) continuous linear map \(T:X\rightarrow Y\) is \(w\left( X,X^{\prime}\right) -w\left( Y,Y^{\prime }\right) \) continuous. The following topologies are Hellinger-Toeplitz: \(\beta\left( \cdot,\cdot\right) ,\) the strong topology\(;\) \(\tau\left( \cdot,\cdot\right)\), the Mackey topology; \(\gamma\left( \cdot,\cdot\right) \), the topology of uniform convergence on conditionally weakly\(^{\ast}\) sequentially compact subsets; \(\lambda\left( \cdot,\cdot\right)\), the topology of uniform convergence on weakly\(^{\ast}\) compact subsets.NEWLINENEWLINEA series \(\sum_{j}z_{j}\) in a topological vector space \(Z\) is said to be \(\lambda\) multiplier convergent if the series \(\sum_{j}t_{j}z_{j}\) converges in \(Z\) for each \(t\in\lambda\). For a locally convex space \(Z\), the usual version of the Orlicz-Pettis theorem asserts that a series which is \(\lambda\) multiplier convergent in the weak topology is \(\lambda\) multiplier convergent in the Mackey topology, whenever \(\lambda=m_{0}\), the space of sequences with finite range. By use of the following theorem, the author describes some other topological sequence spaces \(\lambda\) that are relevant in this context.NEWLINENEWLINETheorem 2.1. Let \(w\) be a Hellinger-Toeplitz topology for dual pairs. The following are equivalent: (i) For every dual pair \(X,X^{\prime}\), a series which is \(\lambda\) multiplier convergent for the weak topology \(\sigma\left( X,X^{\prime}\right) \) is \(\lambda\) multiplier convergent with respect to \(w\left( X,X^{\prime}\right)\). (ii) \(\left( \lambda,w\left( \lambda,\lambda^{\beta}\right) \right) \) is an AK-space (i.e., \(\left( e^{j}\right) \) is a Schauder basis).NEWLINENEWLINESo, in order to prove an Orlicz-Pettis theorem for a Hellinger-Toeplitz topology \(w\), it suffices to check the AK-property for \(w\left( \lambda,\lambda^{\beta}\right)\). The author proves the following: (1) if \(\lambda\) has the signed-WGHP, then \(\left( \lambda,\gamma\left( \lambda ,\lambda^{\beta}\right) \right) \) and \(\left( \lambda,\lambda\left( \lambda,\lambda^{\beta}\right) \right) \) (and then also \(\left( \lambda,\tau\left( \lambda,\lambda^{\beta}\right) \right) \)) are AK-spaces (note that \(m_{0}\) has signed-WGHP, \(\left( m_{0}\right) ^{\beta}=\ell\) and \(\left( m_{0},\tau\left( m_{0},\ell\right) \right) \) is an AK-space); (2) if \(\lambda\) has the \(\infty\)-GHP, then \(\left( \lambda,\beta\left( \lambda,\lambda^{\beta}\right) \right) \) is an AK-space; (3) if \(\lambda\) is \(c_{0}\)-factorable (i.e., for every \(t\in\lambda\) there exist \(s\in c_{0}\) and \(u\in\lambda\) with \(t=su\)) and monotone (i.e., \(at\in\lambda\) for all \(t\in\lambda\) and \(a=\left( a_{j}\right) \) with \(a_{j}\in\left\{ 0,1\right\} \)), then \(\left( \lambda,\beta\left( \lambda,\lambda^{\beta }\right) \right) \) is an AK-space.NEWLINENEWLINEFor the entire collection see [Zbl 1353.46002].