Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line segment of H to the weak topology of H (F.E. Browder / G.J. Minty). [Obviously, any linear operator is hemicontinuous.] Intuitively speaking, this is an extremely weak continuity requirement, still very useful, e.g., in the study of nonlinear elliptic boundary value problems.
Now, here is my problem. Let U be an arbitrary (i.e., possibly discontinuous) selfmap of H. Is it true that there exists some hemicontinuous selfmap of H, say V, such that U 2 = V 2 ?