BANACH SPACES ON WHICH EVERY SURJECTIVE OPERATOR IS INJECTIVE

We say that a Banach space is Hopfian if every surjective bounded linear operator is injective. The first example of an infinite-dimensional Hopfian Banach space was constructed by Gowers and Maurey in 1993 [GM, 11]). This note demonstrates that a Banach space is Hopfian if and only if holds. Additionally, this equivalence is true if holds for every bounded linear operator on . Similarly, holds if and only holds for every bounded linear operator on . Additionally, we show that an infinite-dimensional Banach space is Hopfian if and only if its dual is also Hopfian. In particular, we show that the Banach space is Hopfian if and only if is compact Hausdorff.