A generalization of the $tan2\mathrm{\Theta }$ Theorem

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and ${\mathfrak{H}}_0\subset \mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup \text{spec}(A_0)\le \inf \text{spec}(A_1)$, where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces ${\mathfrak{H}}_0$ and ${\mathfrak{H}}_1={\mathfrak{H}}_0^{\perp }$, respectively, we study the variation of the invariant subspace ${\mathfrak{H}}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to the decomposition $\mathfrak{H}={\mathfrak{H}}_0\oplus {\mathfrak{H}}_1$. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$ and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated Davis-Kahan $\tan 2\mathrm{\Theta }$ 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.