Inner local spectral radius preservers.

Let L(X)<inline-graphic></inline-graphic> be the Banach algebra of all bounded linear operators on a complex Banach space X. For an operator T∈L(X)<inline-graphic></inline-graphic>, let ιT(x)<inline-graphic></inline-graphic> denote the inner local spectral radius of T at any vector x in X. We characterize maps ϕ<inline-graphic></inline-graphic> (not necessarily linear nor surjective) on L(X)<inline-graphic></inline-graphic> which satisfy ιT-S(x)=0ifandonlyifιϕ(T)-ϕ(S)(x)=0<graphic></graphic>for every x∈X<inline-graphic></inline-graphic> and T,S∈L(X)<inline-graphic></inline-graphic>. We also describe surjective linear maps ϕ<inline-graphic></inline-graphic> on L(X)<inline-graphic></inline-graphic> for which ϕ(I)<inline-graphic></inline-graphic> is invertible and either ιT(x)=0⟹ιϕ(T)(x)=0<graphic></graphic>for every x∈X<inline-graphic></inline-graphic> and T∈L(X)<inline-graphic></inline-graphic>, or ιϕ(T)(x)=0⟹ιT(x)=0<graphic></graphic>for every x∈X<inline-graphic></inline-graphic> and T∈L(X)<inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]