In the present paper we study the compositions of the piecewise linear
interpolation operator S?n and the Beta-type operator B?n, namely An:= S?n
?B?n and Gn := B?n ? S?n. Voronovskaya type theorems for the operators An and
Gn are proved, substantially improving some corresponding known results. The
rate of convergence for the iterates of the operators Gn and An is
considered. Some estimates of the differences between An, Gn, Bn and S?n,
respectively, are given. Also, we study the behaviour of the operators An
on the subspace of C[0,1] consisting of all polygonal functions with nodes
{0, 1/2,..., n-1/n,1}. Finally, we propose to the readers a
conjecture concerning the eigenvalues of the operators An and Gn. If true,
this conjecture would emphasize a new and strong relationship between Gn and
the classical Bernstein operator Bn.