An operator summability of sequences in Banach spaces

TL;DR

This paper introduces the concept of operator $p$-summability for sequences in Banach spaces, explores its properties, and characterizes Banach spaces where weakly $p$-summable sequences are operator $p$-summable.

Contribution

It defines operator $p$-summability, studies its relation to other summability notions, and characterizes Banach spaces where certain summability properties hold.

Findings

Weakly $p$-summable sequences are operator $p$-summable iff all $T o l_p$ are $p$-absolutely summing.

Operator $p$-summable sequences are norm $p$-summable iff all $p$-limited operators are absolutely $p$-summing.

Such spaces are subspaces of $L_p( ext{measure})$ for some Borel measure.

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 $\mu$.