On an analogue of a theorem by Astala and Tylli

Let ‖ A‖ _e be the essential norm of an operator A and ‖ A‖ _m be the infimum of the norms of restrictions of A to the subspaces of finite codimension. We show that if ‖ A‖ _e< M‖ A‖ _m holds for every bounded noncompact operator A from a Banach space X to every Banach space Y, then the space X has the dual compact approximation property. This is an analogue of a result by Astala and Tylli (J Funct Anal 70(2):388–401, 1987) concerning the Hausdorff measure of noncompactness and the bounded compact approximation property.