A Generalization of the Tan 2θ Theorem

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂H a closed invariant subspace of A. Assuming that supspec(A0)≤ inf spec(A1), where A0 and A1 are restrictions of A onto the subspaces H0 and H1 = H ⊥ 0 , respectively, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V that are off-diagonal with respect to the decomposition H = H0 ⊕H1. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators A and B = A + V. These results extend the celebrated Davis-Kahan tan2Θ Theorem. On this basis we also prove new existence and uniqueness theorems for contractive solutions to the operator Riccati equation, thus, extending recent results of Adamyan, Langer, and Tretter.