Local spectral subspace preservers

Abstract

Let $${\mathcal {B}}(X)$$ be the algebra of all bounded linear operators on Banach space X. For $$T\in {\mathcal {B}}(X)$$ and $$\lambda \in \mathbb {C}$$ , let $$X_{T}(\{ \lambda \})$$ denotes the local spectral subspace of T associated with $$\{\lambda \}$$ . We determine the forms of mappings (not necessarily linear) $$\phi :{\mathcal {B}}(X)\rightarrow {{\mathcal {B}}(X)}$$ that preserve the local spectral subspace of either sum, product of operators or triple product of operators associated with a singleton.