It is well known that an operator u acting between two Banach spaces X, Y is compact if and only if dual operator u· is compact. For any such u: X-t Y and for every e> 0, denote by N (u, e) the minimal cardinality of an e-net, in the metric of Y, of the image u (Bx) of the unit ball Bx of X. Since now the compactness of an operator may be quantified via its metric entropy log N (u, e), one may ask for a quantitative version of the result recalled above, ie, for a comparison of the metric entropies of u and its dual u", It is a conjecture, promoted by B. Carl and A. Pietsch, that the two metric entropies are equivalent in the sense that there exist universal constants a, b> 0 so that a-1 log N (u*, b-1 e)::; log N (u, e)::; alog N (u*, be) holds for any compact operator u and for any e> O. We will refer to it as" the duality conjecture" or the" the duality problem".

Let us observe that for operators acting between Hilbert spaces the metric entropies of u and u· are ezactly the same; this can be seen by considering polar decompositions. Other special cases are settled in [Car],[GKS],[KMT] and [PT]. Also, a form of the duality problem-for operators with fixed rank was considered in [KM](see also [Pi4], Chap. 7). However, in the general setting, the problem of equivalence of the metric entropies is still wide open; even in the form requiring one of the constants a or b (but not both) to be equal to 1. Let us rephrase the problem in terms of the so-called entropy numbers, defined for an operator u by