  The purpose of this paper is to present an example of an Ordinary
Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space
$\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fr\'{e}chet sense, such
that the origin is an asymptotically stable equilibrium point but the spectrum
of the linearized operator $DF(0)$ intersects the half-plane $\Re(z)>0$. The
possible existence or not of an example of this kind has been an open question
until now, to our knowledge. An analogous example, but of a non-invertible map
instead of a flow defined by an ODE was recently constructed by the authors in
a recent paper. The two examples use different techniques, but both are based
on a classical example in Operator Theory due to S. Kakutani.
