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Operator valued series and vector valued multiplier spaces. (English) Zbl 1415.46015

Summary: Let \(X\), \(Y\) be normed spaces with \(L(X,Y)\) the space of continuous linear operators from \(X\) into \(Y\). If \(\{ T_j\}\) is a sequence in \(L(X,Y)\), the (bounded) multiplier space for the series \(\sum T_j\) is defined to be
\[ M^\infty \left(\sum T_j\right)=\{\{x\}_j \in l^\infty (X):\sum_{j=1}^\infty T_j x_j\text{ converges} \} \]
and the summing operator \(S:M^\infty (\sum T_j) \rightarrow Y\) associated with the series is defined to be \(S(\{x_j\})=\sum_{j=1}^\infty T_j x_j\). In the scalar case, the summing operator has been used to characterize completeness, weakly unconditionall Cauchy series, subseries and absolutely convergent series. In this paper, some of these results are generalized to the case of operator valued series. The corresponding space of weak multipliers is also considered.

MSC:

46B45 Banach sequence spaces
46A45 Sequence spaces (including Köthe sequence spaces)
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