© Printed in India Nice surjections on spaces of operators

Abstract. A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mas-cioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spaces X, Y. We give necessary and sufficient conditions when nice surjections are given by compo-sition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for L(X, Y) thereby proving an analogue of the result from [9] for Lp (1 < p = 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8]