On a subspace perturbation problem

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

for $A$ and $A+V$ is less than one whenever either (i) $\Vert V\Vert<\frac{2}{2+\pi}d$ or (ii) $\Vert V\Vert<\frac{1}{2}d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.