A full-invariant theorem and some applications
type: Item
instance of: scholarly article
title: A full-invariant theorem and some applications
zbMATH Open document ID: 1039.46004
DOI: 10.1016/S0022-247X(02)00077-X
author: Junde Wu
author: Shijie Lu
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-09-08T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 46A03
zbMATH DE Number: 1797516
zbMATH Keywords: \(\lambda\)-multiplier convergent
zbMATH Keywords: subseries convergent
reviewed by: Antonio M. Peralta
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00077-x
OpenAlex ID: W2085385631
description: scientific article; zbMATH DE number 1797516