A full-invariant theorem and some applications
Let \((X,\tau_{1})\) and \((Y,\tau_{2})\) be two Hausdorff locally convex spaces and denote by \(L(X,Y)\) and \(K(X,Y)\) the spaces of all continuous linear and compact linear operators from \(X\) to \(Y\), respectively. The weak operator topology, strong operator topology and the uniform operator topology on \(K(X,Y)\) are denoted by WOT, SOT and UOT, respectively. Let \(\lambda\) be a space of scalar-valued sequences. The space \(\lambda\) satisfies the signed-weak gliding hump property (s-wghp, in the sequel) if, given any \(t=(t_{i}) \in \lambda\) and any block sequence \((t^{(k)})\) with \(t=\sum_{k=1}^{\infty} t^{(k)}\) (pointwise convergence), each index sequence \(\{m_{k}\}\) has a further subsequence \(\{ n_{k} \}\) and a signed sequence \(\{s_{k}\}\) (\(s_{k}=1\) or \(s_{k}=-1\)) such that \(\sum_{k=1}^{\infty} s_{k} t^{(n_k)}\) is again in \(\lambda\). A series \(\sum_{i} x_{i}\) in \((X,\tau_{1})\) is subseries \(\tau_{1}\)-convergent if for each strictly increasing sequence of positive integers \(\{n_{i}\}\), the series \(\sum_{n_{i}} x_{n_{i}}\) is \(\tau_{1}\)-convergent. A series \(\sum_{i} x_{i}\) in \((X,\tau_{1})\) is said to be \(\lambda\)-multiplier \(\tau_{1}\)-convergent if for each \((t_{i})\in \lambda\), the series \(\sum_{i} t_{i} x_{i}\) is \(\tau_{1}\)-convergent. \(c_{00}\) denotes the space of all scalar sequences that are eventually \(0\). The main result of the paper is the following characterization of a full-invariant property of \(K(X,Y)\). Let \((X,\tau_{1})\) and \((Y,\tau_{2})\) be two Hausdorff locally convex spaces with \(Y\neq \{0\}\). Let \(\lambda\) be a space of scalar-valued sequences satisfying \(c_{00} \subseteq \lambda\) and such that \(\lambda\) satisfies the s-wghp. Then each \(\lambda\)-multiplier WOT-convergent series \(\sum_{i} T_{i}\) in \(K(X,Y)\) is \(\lambda\)-multiplier convergent with respect to all topologies between WOT and UOT if, and only if, each continuous linear operator \(T: (X,\tau_{1}) \to (\lambda^{\beta},\sigma(\lambda^{\beta},\lambda))\) is compact, where \(\lambda^{\beta}\) denotes the \(\beta\)-dual of \(\lambda\). As a consequence, it is obtained that every WOT-subseries convergent series \(\sum_{i} T_{i}\) in \(K(X,Y)\) is UOT-subseries convergent if and only if every continuous linear operator \(T\) from \((X,\tau_{1})\) to \((\ell_{1},\|.\|)\) is compact. Finally, combining their theorems with results from a previous paper, they prove that when \((X,\tau_{1})\) and \((Y,\tau_{2})\) are two Hausdorff locally convex spaces, \((X,\tau_{1})\) is a barrelled space and \(Y\neq \{0\}\), then every WOT-subseries convergent series in \(K(X,Y)\) must be UOT-subseries convergent if and only if \((X^{'},\beta(X^{'},X))\) contains no copy of \((\ell_{\infty},)\). This result shows that the converse of Kalton's Orlicz-Pettis theorem is also valid.
scientific article; zbMATH DE number 1797516