### Denseness of numerical radius attaining holomorphic functions.

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We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and ${\ell}_{\infty}$-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and $X{\oplus}_{\infty}Y$ has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ)...

We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d{*}_{(}w,1)$ into d(w,1), where $d{*}_{(}w,1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.

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