Ark. Mat., 38 (2000), 37–44 c  2000 by Institut Mittag-Leffler. All rights reserved Polynomials on dual-isomorphic spaces F´ elix Cabello S´ anchez, Jes´ us M. F. Castillo and Ricardo Garc´ ıa( 1 ) In this note we study isomorphisms between spaces of polynomials on Banach spaces. Precisely, we are interested in the following question raised in [5]: If X and Y are Banach spaces such that their topological duals X ′ and Y ′ are isomorphic, does this imply that the corresponding spaces of homogeneous polynomials P ( n X) and P ( n Y ) are isomorphic for every n≥1? D´ ıaz and Dineen gave the following partial positive answer [5, Proposition 4]: Let X and Y be dual-isomorphic spaces; if X ′ has the Schur property and the ap- proximation property, then P ( n X) and P ( n Y ) are isomorphic for every n. Observe that the Schur property of X ′ makes all bounded operators from X to X ′ (and also from Y to Y ′ ) compact. That hypothesis can be considerably relaxed. Follow- ing [6], [7], let us say that X is regular if every bounded operator X →X ′ is weakly compact. We prove the following result. Theorem 1. Let X and Y be dual-isomorphic spaces. If X is regular then P ( n X) and P ( n Y ) are isomorphic for every n≥1. In fact, it is even true that the corresponding spaces of holomorphic maps of bounded type H b (X) and H b (Y ) are isomorphic Fr´ echet algebras. Observe that the approximation property plays no rˆ ole in Theorem 1. This is relevant since, for instance, the space of all bounded operators on a Hilbert space is a regular space (as every C ∗ -algebra [7]) but lacks the approximation property. Our techniques are quite different from those of [5] and depend on certain properties of the extension operators introduced by Nicodemi in [10]. For stable spaces (that is, for spaces isomorphic to its square) one has the following stronger result. Theorem 2. If X and Y are dual-isomorphic stable spaces, then P ( n X) and P ( n Y ) are isomorphic for every n≥1. ( 1 ) Supported in part by DGICYT project PB97-0377.