  A remarkable result of Moln\'ar [Proc. Amer. Math. Soc., 126 (1998), 853-861]
states that automorphisms of the algebra of operators acting on a separable
Hilbert space is stable under "small" perturbations. More precisely, if
$\phi,\psi$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that
$\|\phi(A)-\psi(A)\|<\|A\|$ and $\psi$ is surjective then so is $\phi$. The aim
of this paper is to extend this result to a larger class of Banach spaces
including $\ell_p$ and $L_p$ spaces ($1<p<+\infty$). En route to the proof we
show that for any Banach space $X$ from the above class all faithful, unital,
separable, reflexive representations of $\mathcal B (X)$ which preserve rank
one operators are in fact isomorphisms.
