On maps preserving operators of local spectral radius zero

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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 satisfy

r_ϕ(T)(x)=0⟹r_T(x)=0

r_{ϕ (T)} (x) = 0 ⟹ r_{T} (x) = 0

for every 47a14f0" title="Click to view the MathML source">x∈X

x \in X

and T∈L(X)

T \in L (X)

. We also describe surjective linear maps ϕ on L(X)

L (X)

that satisfy

r_T(x)=0⟹r_ϕ(T)(x)=0

r_{T} (x) = 0 ⟹ r_{ϕ (T)} (x) = 0

for every 47a14f0" title="Click to view the MathML source">x∈X

x \in X

and T∈L(X)

T \in L (X)

. Furthermore, we characterize maps ϕ (not necessarily linear nor surjective) on L(X)

L (X)

which satisfy

r_{ϕ (T) - ϕ (S)} (x) = 0 if and only if r_{T - S} (x) = 0

for every 47a14f0" title="Click to view the MathML source">x∈X

x \in X

and T,S∈L(X)

T, S \in L (X)