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} ?