On an analogue of a theorem by Astala and Tylli

Let ‖A‖e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert A\Vert _{\mathrm {e}}$$\end{document} be the essential norm of an operator A and ‖A‖m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert A\Vert _m$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert A\Vert _{\mathrm {e}}<M\Vert A\Vert _m$$\end{document} 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.