Operator Valued Series and Vector Valued Multiplier Spaces

Document Type: Research articles

Author

Mathematics Department, New Mexico State University Las Cruces‎, ‎NM 88003,USA

Abstract

‎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}(sum T_{j})={{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.‎

Keywords

Article Title [Persian]

سریهای عملگر مقداری

Author [Persian]

• چارلز اسوارتز
Abstract [Persian]

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