The p-Gelfand Phillips property (1 ≤ p < ∞) is studied in spaces of operators. Dunford -Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T : X → Y is weakly compact, for every Banach space Y .

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The concept of p-L-limited sets and Banach spaces with the p-Llimited property (p ∈ [1, ∞)) are studied. Some characterizations of limited p-convergent operators are obtained. The complementability of some spaces of operators in the space of limited p-convergent operators is also investigated.