摘要
Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B(X) preserves Drazin invertible operators of index non-greater than n in both directions if and only if Ф is either of the form Ф(T) = aATA-1 or of the form Ф(T) = aBT*B-1 where a is a non-zero scalar, A : X → X and B : X* → X are two bounded invertible linear or conjugate linear operators.
Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B(X) preserves Drazin invertible operators of index non-greater than n in both directions if and only if Ф is either of the form Ф(T) = aATA-1 or of the form Ф(T) = aBT*B-1 where a is a non-zero scalar, A : X → X and B : X* → X are two bounded invertible linear or conjugate linear operators.