Notes on the $${\sin 2 \Theta}$$ Theorem.

An analogue of the Davis-Kahan $${\sin 2\Theta}$$ theorem from (SIAM J Numer Anal 7:1-46, ) is proved under a general spectral separation condition. This extends the generic $${\sin 2\theta}$$ estimates recently shown by Albeverio and Motovilov in (Complex Anal Oper Theory 7:1389-1416, ). The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively. [ABSTRACT FROM AUTHOR]