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We will consider the Banach space valued differential equation y′(t) = Ay(t) , where A is an n× n complex matrix. We give a necessary and sufficient condition in order that the equation have the Hyers-Ulam stability. As a Corollary, we prove that the Banach space valued linear differential equation with constant coefficients y(n)(t) + an−1y(n−1)(t) + · · ·+ a1y′(t) + a0y(t) = 0 has the Hyers-Ulam stability if and only if Reλ = 0 for all the solutions λ of the equation zn +an−1zn−1 + · · ·+a1z+a0 = 0 . Mathematics subject classification (2010): Primary 46B99; Secondary 39B72.

A characterization of the stability of a system of the Banach space valued differential equations