On maps preserving operators of local spectral radius zero

Let L(X)L(X) be the algebra of all bounded linear operators on a complex Banach space X. We describe surjective linear maps ϕ   on L(X)L(X) that satisfyrϕ(T)(x)=0⟹rT(x)=0rϕ(T)(x)=0⟹rT(x)=0 for every x∈Xx∈X and T∈L(X)T∈L(X). We also describe surjective linear maps ϕ   on L(X)L(X) that satisfyrT(x)=0⟹rϕ(T)(x)=0rT(x)=0⟹rϕ(T)(x)=0 for every x∈Xx∈X and T∈L(X)T∈L(X). Furthermore, we characterize maps ϕ   (not necessarily linear nor surjective) on L(X)L(X) which satisfyrϕ(T)−ϕ(S)(x)=0 if and only ifrT−S(x)=0 for every x∈Xx∈X and T,S∈L(X)T,S∈L(X).