Alternative proof of the a priori tan I∼ theorem

Let A be a self-adjoint operator in a separable Hilbert space. We assume that the spectrum of A consists of two isolated components sigma(0) and sigma(1) and the set sigma(1) is in a finite gap of the set sigma(1). It is known that if V is a bounded additive self-adjoint perturbation of A that is off-diagonal with respect to the partition spec(A) = sigma(0) a(a) sigma(1), then for , where d = dist(sigma(0), sigma(1)), the spectrum of the perturbed operator L = A+V consists of two isolated parts omega(0) and omega(1), which appear as perturbations of the respective spectral sets s0 and s1. Furthermore, we have the sharp upper bound ||EA(sigma(0)) - EL(omega(0))|| a parts per thousand currency sign sin (arctan(||V||/d)) on the difference of the spectral projections E-A(sigma(0))) and E-L(omega(0))) corresponding to the spectral sets sigma(0) and omega(0) of the operators A and L. We give a new proof of this bound in the case where ||V|| < d.