Upper and lower bounds are established for the entropy numbers of certain diagonal operators between
Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in
the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question
coincide with those considered by Leopold. The results in the paper improve the previous results in at
least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower
estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the
upper estimates are improvements on the previous ones, the improvement being quite significant in some
cases.