Multiplier spaces and the summing operator for series

Year, pages: 2016, 687 - 694

If $\{x_j\}$ is a sequence in a normed space $X$, the space of bounded multipliers for the series $\sum_j x_j$ is defined to be $M^{\infty}(\sum x_j)=\{\{t_j\}\in l^{\infty}: \sum_{j=1}^{\infty}t_j x_j converges\}$ and the summing operator $S: M^{\infty}(\sum x_j)\to X$ is defined to be $S(\{t_j\})=\sum_{j=1}^{\infty}t_j x_j$. We show that if $X$ is complete, the series $\sum_j x_j$ is subseries convergent iff the operator $S$ is compact and the series is absolutely convergent iff the operator is absolutely summing. Other related results are established.

Swartz, C. 2016. Multiplier spaces and the summing operator for series. In Mathematica Slovaca, vol. 66, no.3, pp. 687-694. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0170

Swartz, C. (2016). Multiplier spaces and the summing operator for series. Mathematica Slovaca, 66(3), 687-694. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0170