Reflexively representable but not Hilbert representable compact flows and semitopological semigroups

We show that for many natural topological groups G (including the group Z of integers) there exist compact metric G-spaces (cascades for G = Z) which are reflexively representable but not Hilbert representable. This answers a question of T. Dow- narowicz. The proof is based on a classical example of W. Rudin and its generalizations. A crucial step in the proof is our recent result which states that every weakly almost periodic function on a compact G-flow X comes from a G-representation of X on reflexive spaces. We also show that there exists a monothetic compact metrizable semitopological semigroup S which does not admit an embedding into the semitopological compact semi- group Θ (H) of all contractive linear operators on a Hilbert space H (though S admits an embedding into the compact semigroup Θ (V) for certain reflexive V).