Invertibility preserving linear maps on ℒ(𝒳)

Abstract

For Banach spaces X X and Y Y , we show that every unital bijective invertibility preserving linear map between L ( X ) \mathcal L(X) and L ( Y ) \mathcal L(Y) is a Jordan isomorphism. The same conclusion holds for maps between C I + K ( X ) \mathbb CI+ \mathcal K(X) and C I + K ( Y ) \mathbb CI+\mathcal K(Y) .