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 = dist(sigma, Sigma) > 0. We show that the norm of the difference of the spectral projections E-A(sigma) and E-A+V({lambda\dist(lambda, sigma) < d/2}) for A and A + V is less than one whenever either (i) &PAR;V&PAR; < 2/2+pi d or (ii) parallel toVparallel to < 1/2 d and certain assumptions on the mutual disposition of the sets σ and &USigma; are satisfied.