A Positive Answer on Nirenberg’s Problem on Expansive Mappings in Hilbert Spaces

Nirenberg proposed a problem as to whether or not a continuous and expansive operator T : X⟶X (where X is a Hilbert space) is surjective if R(T)∘ ≠∅. I shall give a positive answer for the problem provided that R(T)∘ is unbounded. For contents related to this paper, the reader is referred to the remarks and the study of Asfaw (2021). The present paper gives a complete answer for the problem that has been open for about 47 years.