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BANACH SPACES ON WHICH EVERY SURJECTIVE OPERATOR IS INJECTIVE
Keywords:
Banach space, bounded linear operator, Drazin inversible operator, descent spectrum equalityDOI:
https://doi.org/10.17654/0972087125018Abstract
We say that a Banach space $X$ is Hopfian if every surjective bounded linear operator $T: X \mapsto X$ 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 $X$ is Hopfian if and only if $\sigma_{R D}(T) \subseteq \sigma_{\text {desc }}(T)$ holds. Additionally, this equivalence is true if $\sigma_{L D}(T)=\sigma_{\text {desc }}(T)$ holds for every bounded linear operator $T$ on $X$. Similarly, $\operatorname{int}(\sigma(T)) \subseteq$ $\sigma_{\text {desc }}(T)$ holds if and only $\operatorname{int}(\sigma(T))=\varnothing$ holds for every bounded linear operator $T$ on $X$.
Additionally, we show that an infinite-dimensional Banach space $X$ is Hopfian if and only if its dual $X^*$ is also Hopfian. In particular, we show that the Banach space $C(K)$ is Hopfian if and only if $K$ is compact Hausdorff.
Received: January 15, 2025
Accepted: April 28, 2025
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