On $(p,r)$-null sequences and their relatives

Abstract

Let $1\le p<\infty$ and $1\le r\le p^{\ast }$, 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.