  We discuss the problem of perturbation of spectral subspaces for linear
self-adjoint operators on a separable Hilbert space. Let $A$ and $V$ be bounded
self-adjoint operators. Assume that the spectrum of $A$ consists of two
disjoint parts $\sigma$ and $\Sigma$ such that $d=\text{dist}(\sigma,
\Sigma)>0$. We show that the norm of the difference of the spectral projections
$\EE_A(\sigma)$ and $\EE_{A+V}\big (\{\lambda | \dist(\lambda, \sigma)$
$<d/2\}\big)$ for $A$ and $A+V$ is less then one whenever either (i)
$\|V\|<\frac{2}{2+\pi}d$ or (ii) $\|V\|<{1/2}d$ and certain assumptions on the
mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.
