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Let X and Y be real or complex Banach spaces. We show that a surjective linear map o: B(X) → B(Y) preserving invertibility in both directions is either of the form o(T) = ATB or the form o(T) = CT'D, where A: X → Y, B: Y → X, C: X' → Y, and D: Y → X' are bounded invertible linear operators. As an application we improve a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.

Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one