Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP

In this paper, we study the Bishop-Phelps-Bollobás property for operators (BPBp for short). To this end, we investigate the generalized approximate hyperplane series property (generalized AHSP for short) for a pair $(X,Y)$ of Banach spaces, which characterizes when $({\mathrm{\ell }}_1(X),Y)$ has the BPBp. We prove the following results. For a locally compact Hausdorff space $L$, if $(X,{\mathcal{C}}_0(L,Y))$ has the BPBp, then so does $(X,Y)$. Furthermore, if the pair $(X,Y)$ has the generalized AHSP and $\mathcal{L}(X,Z)=\mathcal{K}(X,Z)$, then the pair $(X,Z)$ also has the generalized AHSP, where $Z$ is one of the spaces $\mathcal{C}(K,Y)$, ${\mathcal{C}}_0(L,Y)$, or ${\mathcal{C}}_b(\mathrm{\Omega },Y)$, with $K$ a compact Hausdorff space and $\mathrm{\Omega }$ a completely regular space.