  The purpose of this article is to develop a technique to estimate certain
bounds for entropy numbers of diagonal operator on spaces of p-summable
sequences for finite p greater than 1. The approximation method we develop in
this direction works for a very general class of operators between Banach
spaces, in particular reflexive spaces. As a consequence of this technique we
also obtain that the entropy number of a bounded linear operator T between two
separable Hilbert spaces is equal to the entropy number of the adjoint of T.
This gives a complete answer to the question posed by B. Carl [4] in the
setting of separable Hilbert spaces.
