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

TL;DR

This paper establishes a comprehensive set of equivalences characterizing $(p,r)$-null sequences in Banach spaces, simplifying previous proofs and extending the understanding of their relation to nullness and relative compactness.

Contribution

It provides a more direct proof of the equivalence between $(p,r)$-null sequences and relative $(p,r)$-compactness, extending known results to broader cases.

Findings

$(p,r)$-null sequences are equivalent to null and relatively $(p,r)$-compact sequences.

The approach simplifies the proof of known equivalences.

Characterizations of unconditional and weak $(p,r)$-null sequences are provided.

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.