This is an announcement for the paper "On $(p,r)$-null sequences and their relatives" by Kati Ain and Eve Oja.

Abstract: Let $1\leq p < \infty$ and $1\leq r \leq p^\ast$, where $p^\ast$ is the conjugate index of $p$. We prove an omnibus theorem, which provides numerous equivalences for a sequence $(x_n)$ in a Banach space $X$ to be a $(p,r)$-null sequence. One of them is that $(x_n)$ is $(p,r)$-null if and only if $(x_n)$ is null and relatively $(p,r)$-compact. This equivalence is known in the ''limit'' case when $r=p^\ast$, the case of the $p$-null sequence and $p$-compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of $(p,r)$-null sequences.

Archive classification: math.FA

Submitted from: kati.ain@ut.ee

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

http://front.math.ucdavis.edu/1409.6476

or

http://arXiv.org/abs/1409.6476