Linear maps preserving the set of Fredholm operators

Mbekhta, Mostafa

doi:10.1090/S0002-9939-07-08874-0

Linear maps preserving the set of Fredholm operators
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Proc. Amer. Math. Soc. 135 (2007), 3613-3619 Request permission

Abstract:

Let $H$ be an infinite-dimensional separable complex Hilbert space and $\mathcal {B}(H)$ the algebra of all bounded linear operators on $H$. In this paper we characterize surjective linear maps $\phi : F\mathcal {B}(H)\to \mathcal {B}(H)$ preserving the set of Fredholm operators in both directions. As an application we prove that $\phi$ preserves the essential spectrum if and only if the ideal of all compact operators is invariant under $\phi$ and the induced linear map $\varphi$ on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either $ind(\phi (T)) = ind(T)$ or $ind(\phi (T)) = - ind(T)$ for every Fredholm operator $T$.

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