Some spectral domain in approximate point-spectrum-preserving maps on mathcal {B}(mathcal {X})

Abstract

Let mathcal {X} be an infinite-dimensional complex Banach space, mathcal {B}(mathcal {X}) the algebra of all bounded linear operators on mathcal {X}. Denote the spectral domain by sigma _{gamma}(T)={lambda in sigma _{a}(T): T { text{ that is semi-Fredholm and }} asc(T-lambda I)<infty }. In this paper, we characterize the structure of additive surjective maps varphi : mathcal {B}(mathcal {X})rightarrow mathcal {B}(mathcal {X}) with sigma _{gamma}(varphi (T))=sigma _{gamma}(T) for all Tin mathcal{B(X)}.