On ( p , r ) ‐null sequences and their relatives

Let 1 ≤ p < ∞ and 1 ≤ r ≤ p * , where p * 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 * , 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.