Some kind of Bishop–Phelps–Bollobás property

In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces X and Y : property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair ( X , K ) satisfies property 2. Positive results of pairs of Banach spaces ( X , Y ) satisfying property 1 are given and concrete pairs of Banach spaces ( X , Y ) failing both properties are exhibited. A complete characterization of property 1 for the pairs ( ℓ p , ℓ q ) is also provided.