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

Abstract

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 $R\left(T\right)°\ne \varnothing$. I shall give a positive answer for the problem provided that $R\left(T\right)°$ 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.