Multiplier spaces and the summing operator for series

Abstract

Abstract If {xj } is a sequence in a normed space X, the space of bounded multipliers for the series ∑ j x j $\sum\limits_j {{x_j}} $ is defined to be M ∞ ( ∑ x j ) = { { t j } ∈ l ∞ : ∑ j = 1 ∞ t j x j converges } ${M^\infty }\big({\sum {x_j}}\big) = \Big\{ {\{ {t_j}\} \in {l^\infty }:\;\sum\limits_{j = 1}^\infty {{t_j}} {x_j}\; \text{converges}} \Big\}$ and the summing operator S : M ∞ ∑ x j → X $S:{M^\infty }\left( {{\mkern 1mu} \sum {{x_j}} } \right) \to X$ is defined to be S ( { t j } ) = ∑ j = 1 ∞ t j x j $S(\{ {t_j}\} ) = \sum\limits_{j = 1}^\infty {{t_j}} {x_j}$ . We show that if X is complete, the series ∑ j x j $\sum\limits_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.