This is an announcement for the paper "An operator summability of sequences in Banach spaces" by Anil Kumar Karn and Deba Prasad Sinha.

Abstract: Let $1 \leq p <\infty$. A sequence $\lef x_n \rig$ in a Banach space $X$ is defined to be $p$-operator summable if for each $\lef f_n \rig \in l^{w^*}_p(X^*)$, we have $\lef \lef f_n(x_k)\rig _k \rig _n \in l^s_p(l_p)$. Every norm $p$-summable sequence in a Banach space is operator $p$-summable, while in its turn every operator $p$-summable sequence is weakly $p$-summable. An operator $T \in B(X, Y)$ is said to be $p$-limited if for every $\lef x_n \rig \in l_p^w(X)$, $\lef Tx_n \rig$ is operator $p$-summable. The set of all $p$-limited operators form a normed operator ideal. It is shown that every weakly $p$-summable sequence in $X$ is operator $p$-summable if and only if every operator $T \in B(X, l_p)$ is $p$-absolutely summing. On the other hand every operator $p$-summable sequence in $X$ is norm $p$-summable if and only if every $p$-limited operator in $B(l_{p'}, X)$ is absolutely $p$-summing. Moreover, this is the case if and only if $X$ is a subspace of $L_p(\mu )$ for some Borel measure $\mu$.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46B20, Secondary 46B28, 46B50

Remarks: 16 pages

Submitted from: anilkarn@niser.ac.in

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1207.3620

or

http://arXiv.org/abs/1207.3620